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Q: How to see changes in flutter app when changing values in Firebase Realtime Database?
I am trying to make these rolling switches to change its value whenever I do any change in Firebase realtime database.
To be more specific, whenever I change the value of Relay1/Data to 0, I want that switch to become inactive.
I've tried and looked everywhere, but I couldn't find any solution.
bool relay1pressed;
final databaseReferenceTest = FirebaseDatabase.instance.reference();
@override
void initState() {
super.initState();
databaseReferenceTest
.child('MedicalCenter')
.once()
.then((DataSnapshot snapshot) {
String value = snapshot.value['Relay1']['Data'];
print('Value is $value');
if (value == '1') {
relay1pressed = true;
} else
relay1pressed = false;
setState(() {
isLoading = true;
});
});
}
//Widget build
StreamBuilder(
stream: databaseReferenceTest
.child('MedicalCenter')
.child('Relay1')
.onValue,
builder: (BuildContext context, AsyncSnapshot<Event> snapshot) {
databaseReferenceTest
.child('MedicalCenter')
.once()
.then((DataSnapshot snapshot) {
String value = snapshot.value['Relay1']['Data'];
print('Value is $value');
if (value == '1') {
relay1pressed = true;
print('relay1 bool $relay2pressed');
} else {
relay1pressed = false;
print('relay1 bool $relay2pressed');
}
});
return LiteRollingSwitch(
value: relay1pressed,
textOn: 'active',
textOff: 'inactive',
colorOn: Colors.deepOrange,
colorOff: Colors.blueGrey,
iconOn: Icons.lightbulb_outline,
iconOff: Icons.power_settings_new,
onChanged: (bool state) {
state
? databaseReferenceTest
.child('MedicalCenter')
.update({'Relay1/Data': '1'})
: databaseReferenceTest
.child('MedicalCenter')
.update({'Relay1/Data': '0'});
A: databaseReferenceTest
.child('MedicalCenter')
.onValue.listen((event) {
var snapshot = event.snapshot;
setState(() {
String value = snapshot.value['Relay1']['Data'];
print('Value is $value');
});
});
this should work...
I just added setState,
A: You're currently using once() to get the value from the database, which means it only reads the current value. If you want to keep monitoring the value, you'll want to use onValue instead.
databaseReferenceTest
.child('MedicalCenter')
.onValue.listen((event) {
var snapshot = event.snapshot
String value = snapshot.value['Relay1']['Data'];
print('Value is $value');
...
});
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 590
|
{-# LANGUAGE BangPatterns #-}
{-# LANGUAGE DeriveDataTypeable #-}
{-# LANGUAGE RankNTypes #-}
module System.IO.Streams.Combinators
( -- * Folds
inputFoldM
, outputFoldM
, fold
, foldM
, fold_
, foldM_
, any
, all
, maximum
, minimum
-- * Unfolds
, unfoldM
-- * Maps
, map
, mapM
, mapM_
, mapMaybe
, contramap
, contramapM
, contramapM_
, contramapMaybe
-- * Filter
, filter
, filterM
, filterOutput
, filterOutputM
-- * Takes and drops
, give
, take
, drop
, ignore
-- * Zip and unzip
, zip
, zipWith
, zipWithM
, unzip
-- * Utility
, intersperse
, skipToEof
, ignoreEof
, atEndOfInput
, atEndOfOutput
) where
------------------------------------------------------------------------------
import Control.Concurrent.MVar (newMVar, withMVar)
import Control.Monad (liftM, void, when)
import Control.Monad.IO.Class (liftIO)
import Data.Int (Int64)
import Data.IORef (IORef, atomicModifyIORef, modifyIORef, newIORef, readIORef, writeIORef)
import Data.Maybe (isJust)
import Prelude hiding (all, any, drop, filter, map, mapM, mapM_, maximum, minimum, read, take, unzip, zip, zipWith)
------------------------------------------------------------------------------
import System.IO.Streams.Internal (InputStream (..), OutputStream (..), fromGenerator, makeInputStream, makeOutputStream, read, unRead, write, yield)
------------------------------------------------------------------------------
-- | A side-effecting fold over an 'OutputStream', as a stream transformer.
--
-- The IO action returned by 'outputFoldM' can be used to fetch and reset the updated
-- seed value. Example:
--
-- @
-- ghci> is <- Streams.'System.IO.Streams.List.fromList' [1, 2, 3::Int]
-- ghci> (os, getList) <- Streams.'System.IO.Streams.List.listOutputStream'
-- ghci> (os', getSeed) \<- Streams.'outputFoldM' (\\x y -> return (x+y)) 0 os
-- ghci> Streams.'System.IO.Streams.connect' is os'
-- ghci> getList
-- [1,2,3]
-- ghci> getSeed
-- 6
-- @
outputFoldM :: (a -> b -> IO a) -- ^ fold function
-> a -- ^ initial seed
-> OutputStream b -- ^ output stream
-> IO (OutputStream b, IO a) -- ^ returns a new stream as well as
-- an IO action to fetch and reset the
-- updated seed value.
outputFoldM f initial stream = do
ref <- newIORef initial
os <- makeOutputStream (wr ref)
return (os, fetch ref)
where
wr _ Nothing = write Nothing stream
wr ref mb@(Just x) = do
!z <- readIORef ref
!z' <- f z x
writeIORef ref z'
write mb stream
fetch ref = atomicModifyIORef ref $ \x -> (initial, x)
------------------------------------------------------------------------------
-- | A side-effecting fold over an 'InputStream', as a stream transformer.
--
-- The IO action returned by 'inputFoldM' can be used to fetch and reset the updated seed
-- value. Example:
--
-- @
-- ghci> is <- Streams.'System.IO.Streams.List.fromList' [1, 2, 3::Int]
-- ghci> (is', getSeed) \<- Streams.'inputFoldM' (\\x y -> return (x+y)) 0 is
-- ghci> Streams.'System.IO.Streams.List.toList' is'
-- [1,2,3]
-- ghci> getSeed
-- 6
-- @
inputFoldM :: (a -> b -> IO a) -- ^ fold function
-> a -- ^ initial seed
-> InputStream b -- ^ input stream
-> IO (InputStream b, IO a) -- ^ returns a new stream as well as an
-- IO action to fetch and reset the
-- updated seed value.
inputFoldM f initial stream = do
ref <- newIORef initial
is <- makeInputStream (rd ref)
return (is, fetch ref)
where
twiddle _ Nothing = return Nothing
twiddle ref mb@(Just x) = do
!z <- readIORef ref
!z' <- f z x
writeIORef ref z'
return mb
rd ref = read stream >>= twiddle ref
fetch ref = atomicModifyIORef ref $ \x -> (initial, x)
------------------------------------------------------------------------------
-- | A left fold over an input stream. The input stream is fully consumed. See
-- 'Prelude.foldl'.
--
-- Example:
--
-- @
-- ghci> Streams.'System.IO.Streams.fromList' [1..10] >>= Streams.'fold' (+) 0
-- 55
-- @
fold :: (s -> a -> s) -- ^ fold function
-> s -- ^ initial seed
-> InputStream a -- ^ input stream
-> IO s
fold f seed stream = go seed
where
go !s = read stream >>= maybe (return s) (go . f s)
------------------------------------------------------------------------------
-- | A side-effecting left fold over an input stream. The input stream is fully
-- consumed. See 'Prelude.foldl'.
--
-- Example:
--
-- @
-- ghci> Streams.'System.IO.Streams.fromList' [1..10] >>= Streams.'foldM' (\x y -> 'return' (x + y)) 0
-- 55
-- @
foldM :: (s -> a -> IO s) -- ^ fold function
-> s -- ^ initial seed
-> InputStream a -- ^ input stream
-> IO s
foldM f seed stream = go seed
where
go !s = read stream >>= maybe (return s) ((go =<<) . f s)
------------------------------------------------------------------------------
-- | A variant of 'System.IO.Streams.fold' suitable for use with composable folds
-- from \'beautiful folding\' libraries like
-- <http://hackage.haskell.org/package/foldl the foldl library>.
-- The input stream is fully consumed.
--
-- Example:
--
-- @
-- ghci> let folds = liftA3 (,,) Foldl.length Foldl.mean Foldl.maximum
-- ghci> Streams.'System.IO.Streams.fromList' [1..10::Double] >>= Foldl.purely Streams.'System.IO.Streams.fold_' folds is
-- ghci> (10,5.5,Just 10.0)
-- @
--
-- /Since 1.3.6.0/
--
fold_ :: (x -> a -> x) -- ^ accumulator update function
-> x -- ^ initial seed
-> (x -> s) -- ^ recover folded value
-> InputStream a -- ^ input stream
-> IO s
fold_ op seed done stream = liftM done (go seed)
where
go !s = read stream >>= maybe (return s) (go . op s)
------------------------------------------------------------------------------
-- | A variant of 'System.IO.Streams.foldM' suitable for use with composable folds
-- from \'beautiful folding\' libraries like
-- <http://hackage.haskell.org/package/foldl the foldl library>.
-- The input stream is fully consumed.
--
-- Example:
--
-- @
-- ghci> let folds = Foldl.mapM_ print *> Foldl.generalize (liftA2 (,) Foldl.sum Foldl.mean)
-- ghci> Streams.'System.IO.Streams.fromList' [1..3::Double] >>= Foldl.impurely Streams.'System.IO.Streams.foldM_' folds
-- 1.0
-- 2.0
-- 3.0
-- (6.0,2.0)
-- @
--
-- /Since 1.3.6.0/
--
foldM_ :: (x -> a -> IO x) -- ^ accumulator update action
-> IO x -- ^ initial seed
-> (x -> IO s) -- ^ recover folded value
-> InputStream a -- ^ input stream
-> IO s
foldM_ f seed done stream = seed >>= go
where
go !x = read stream >>= maybe (done x) ((go =<<) . f x)
------------------------------------------------------------------------------
-- | @any predicate stream@ returns 'True' if any element in @stream@ matches
-- the predicate.
--
-- 'any' consumes as few elements as possible, ending consumption if an element
-- satisfies the predicate.
--
-- @
-- ghci> is <- Streams.'System.IO.Streams.List.fromList' [1, 2, 3]
-- ghci> Streams.'System.IO.Streams.Combinators.any' (> 0) is -- Consumes one element
-- True
-- ghci> Streams.'System.IO.Streams.read' is
-- Just 2
-- ghci> Streams.'System.IO.Streams.Combinators.any' even is -- Only 3 remains
-- False
-- @
any :: (a -> Bool) -> InputStream a -> IO Bool
any predicate stream = go
where
go = do
mElem <- read stream
case mElem of
Nothing -> return False
Just e -> if predicate e then return True else go
------------------------------------------------------------------------------
-- | @all predicate stream@ returns 'True' if every element in @stream@ matches
-- the predicate.
--
-- 'all' consumes as few elements as possible, ending consumption if any element
-- fails the predicate.
--
-- @
-- ghci> is <- Streams.'System.IO.Streams.List.fromList' [1, 2, 3]
-- ghci> Streams.'System.IO.Streams.Combinators.all' (< 0) is -- Consumes one element
-- False
-- ghci> Streams.'System.IO.Streams.read' is
-- Just 2
-- ghci> Streams.'System.IO.Streams.Combinators.all' odd is -- Only 3 remains
-- True
-- @
all :: (a -> Bool) -> InputStream a -> IO Bool
all predicate stream = go
where
go = do
mElem <- read stream
case mElem of
Nothing -> return True
Just e -> if predicate e then go else return False
------------------------------------------------------------------------------
-- | @maximum stream@ returns the greatest element in @stream@ or 'Nothing' if
-- the stream is empty.
--
-- 'maximum' consumes the entire stream.
--
-- @
-- ghci> is <- Streams.'System.IO.Streams.List.fromList' [1, 2, 3]
-- ghci> Streams.'System.IO.Streams.Combinators.maximum' is
-- 3
-- ghci> Streams.'System.IO.Streams.read' is -- The stream is now empty
-- Nothing
-- @
maximum :: (Ord a) => InputStream a -> IO (Maybe a)
maximum stream = do
mElem0 <- read stream
case mElem0 of
Nothing -> return Nothing
Just e -> go e
where
go oldElem = do
mElem <- read stream
case mElem of
Nothing -> return (Just oldElem)
Just newElem -> go (max oldElem newElem)
------------------------------------------------------------------------------
-- | @minimum stream@ returns the greatest element in @stream@
--
-- 'minimum' consumes the entire stream.
--
-- @
-- ghci> is <- Streams.'System.IO.Streams.List.fromList' [1, 2, 3]
-- ghci> Streams.'System.IO.Streams.Combinators.minimum' is
-- 1
-- ghci> Streams.'System.IO.Streams.read' is -- The stream is now empty
-- Nothing
-- @
minimum :: (Ord a) => InputStream a -> IO (Maybe a)
minimum stream = do
mElem0 <- read stream
case mElem0 of
Nothing -> return Nothing
Just e -> go e
where
go oldElem = do
mElem <- read stream
case mElem of
Nothing -> return (Just oldElem)
Just newElem -> go (min oldElem newElem)
------------------------------------------------------------------------------
-- | @unfoldM f seed@ builds an 'InputStream' from successively applying @f@ to
-- the @seed@ value, continuing if @f@ produces 'Just' and halting on
-- 'Nothing'.
--
-- @
-- ghci> is \<- Streams.'System.IO.Streams.Combinators.unfoldM' (\n -> return $ if n < 3 then Just (n, n + 1) else Nothing) 0
-- ghci> Streams.'System.IO.Streams.List.toList' is
-- [0,1,2]
-- @
unfoldM :: (b -> IO (Maybe (a, b))) -> b -> IO (InputStream a)
unfoldM f seed = fromGenerator (go seed)
where
go oldSeed = do
m <- liftIO (f oldSeed)
case m of
Nothing -> return $! ()
Just (a, newSeed) -> do
yield a
go newSeed
------------------------------------------------------------------------------
-- | Maps a pure function over an 'InputStream'.
--
-- @map f s@ passes all output from @s@ through the function @f@.
--
-- Satisfies the following laws:
--
-- @
-- Streams.'map' (g . f) === Streams.'map' f >=> Streams.'map' g
-- Streams.'map' 'id' === Streams.'makeInputStream' . Streams.'read'
-- @
map :: (a -> b) -> InputStream a -> IO (InputStream b)
map f s = makeInputStream g
where
g = read s >>= return . fmap f
------------------------------------------------------------------------------
-- | Maps an impure function over an 'InputStream'.
--
-- @mapM f s@ passes all output from @s@ through the IO action @f@.
--
-- Satisfies the following laws:
--
-- @
-- Streams.'mapM' (f >=> g) === Streams.'mapM' f >=> Streams.'mapM' g
-- Streams.'mapM' 'return' === Streams.'makeInputStream' . Streams.'read'
-- @
--
mapM :: (a -> IO b) -> InputStream a -> IO (InputStream b)
mapM f s = makeInputStream g
where
g = do
mb <- read s >>= maybe (return Nothing)
(\x -> liftM Just $ f x)
return mb
------------------------------------------------------------------------------
-- | Maps a side effect over an 'InputStream'.
--
-- @mapM_ f s@ produces a new input stream that passes all output from @s@
-- through the side-effecting IO action @f@.
--
-- Example:
--
-- @
-- ghci> Streams.'System.IO.Streams.fromList' [1,2,3] >>=
-- Streams.'mapM_' ('putStrLn' . 'show' . (*2)) >>=
-- Streams.'System.IO.Streams.toList'
-- 2
-- 4
-- 6
-- [1,2,3]
-- @
--
mapM_ :: (a -> IO b) -> InputStream a -> IO (InputStream a)
mapM_ f s = makeInputStream $ do
mb <- read s
_ <- maybe (return $! ()) (void . f) mb
return mb
------------------------------------------------------------------------------
-- | A version of map that discards elements
--
-- @mapMaybe f s@ passes all output from @s@ through the function @f@ and
-- discards elements for which @f s@ evaluates to 'Nothing'.
--
-- Example:
--
-- @
-- ghci> Streams.'System.IO.Streams.fromList' [Just 1, None, Just 3] >>=
-- Streams.'mapMaybe' 'id' >>=
-- Streams.'System.IO.Streams.toList'
-- [1,3]
-- @
--
-- /Since: 1.2.1.0/
mapMaybe :: (a -> Maybe b) -> InputStream a -> IO (InputStream b)
mapMaybe f src = makeInputStream g
where
g = do
s <- read src
case s of
Nothing -> return Nothing
Just x ->
case f x of
Nothing -> g
y -> return y
------------------------------------------------------------------------------
-- | Contravariant counterpart to 'map'.
--
-- @contramap f s@ passes all input to @s@ through the function @f@.
--
-- Satisfies the following laws:
--
-- @
-- Streams.'contramap' (g . f) === Streams.'contramap' g >=> Streams.'contramap' f
-- Streams.'contramap' 'id' === 'return'
-- @
contramap :: (a -> b) -> OutputStream b -> IO (OutputStream a)
contramap f s = makeOutputStream $ flip write s . fmap f
------------------------------------------------------------------------------
-- | Contravariant counterpart to 'mapM'.
--
-- @contramapM f s@ passes all input to @s@ through the IO action @f@
--
-- Satisfies the following laws:
--
-- @
-- Streams.'contramapM' (f >=> g) = Streams.'contramapM' g >=> Streams.'contramapM' f
-- Streams.'contramapM' 'return' = 'return'
-- @
contramapM :: (a -> IO b) -> OutputStream b -> IO (OutputStream a)
contramapM f s = makeOutputStream g
where
g Nothing = write Nothing s
g (Just x) = do
!y <- f x
write (Just y) s
------------------------------------------------------------------------------
-- | Equivalent to 'mapM_' for output.
--
-- @contramapM f s@ passes all input to @s@ through the side-effecting IO
-- action @f@.
--
contramapM_ :: (a -> IO b) -> OutputStream a -> IO (OutputStream a)
contramapM_ f s = makeOutputStream $ \mb -> do
_ <- maybe (return $! ()) (void . f) mb
write mb s
------------------------------------------------------------------------------
-- | Contravariant counterpart to 'contramapMaybe'.
--
-- @contramap f s@ passes all input to @s@ through the function @f@.
-- Discards all the elements for which @f@ returns 'Nothing'.
--
-- /Since: 1.2.1.0/
--
contramapMaybe :: (a -> Maybe b) -> OutputStream b -> IO (OutputStream a)
contramapMaybe f s = makeOutputStream $ g
where
g Nothing = write Nothing s
g (Just a) =
case f a of
Nothing -> return ()
x -> write x s
------------------------------------------------------------------------------
-- | Drives an 'InputStream' to end-of-stream, discarding all of the yielded
-- values.
skipToEof :: InputStream a -> IO ()
skipToEof str = go
where
go = read str >>= maybe (return $! ()) (const go)
{-# INLINE skipToEof #-}
------------------------------------------------------------------------------
-- | Drops chunks from an input stream if they fail to match a given filter
-- predicate. See 'Prelude.filter'.
--
-- Items pushed back to the returned stream are propagated back upstream.
--
-- Example:
--
-- @
-- ghci> Streams.'System.IO.Streams.fromList' [\"the\", \"quick\", \"brown\", \"fox\"] >>=
-- Streams.'filterM' ('return' . (/= \"brown\")) >>= Streams.'System.IO.Streams.toList'
-- [\"the\",\"quick\",\"fox\"]
-- @
filterM :: (a -> IO Bool)
-> InputStream a
-> IO (InputStream a)
filterM p src = return $! InputStream prod pb
where
prod = read src >>= maybe eof chunk
chunk s = do
b <- p s
if b then return $! Just s
else prod
eof = return Nothing
pb s = unRead s src
------------------------------------------------------------------------------
-- | Drops chunks from an input stream if they fail to match a given filter
-- predicate. See 'Prelude.filter'.
--
-- Items pushed back to the returned stream are propagated back upstream.
--
-- Example:
--
-- @
-- ghci> Streams.'System.IO.Streams.fromList' [\"the\", \"quick\", \"brown\", \"fox\"] >>=
-- Streams.'filter' (/= \"brown\") >>= Streams.'System.IO.Streams.toList'
-- [\"the\",\"quick\",\"fox\"]
-- @
filter :: (a -> Bool)
-> InputStream a
-> IO (InputStream a)
filter p src = return $! InputStream prod pb
where
prod = read src >>= maybe eof chunk
chunk s = do
let b = p s
if b then return $! Just s
else prod
eof = return Nothing
pb s = unRead s src
------------------------------------------------------------------------------
-- | The function @intersperse v s@ wraps the 'OutputStream' @s@, creating a
-- new output stream that writes its input to @s@ interspersed with the
-- provided value @v@. See 'Data.List.intersperse'.
--
-- Example:
--
-- @
-- ghci> import Control.Monad ((>=>))
-- ghci> is <- Streams.'System.IO.Streams.List.fromList' [\"nom\", \"nom\", \"nom\"::'ByteString']
-- ghci> Streams.'System.IO.Streams.List.outputToList' (Streams.'intersperse' \"burp!\" >=> Streams.'System.IO.Streams.connect' is)
-- [\"nom\",\"burp!\",\"nom\",\"burp!\",\"nom\"]
-- @
intersperse :: a -> OutputStream a -> IO (OutputStream a)
intersperse sep os = newIORef False >>= makeOutputStream . f
where
f _ Nothing = write Nothing os
f sendRef s = do
b <- readIORef sendRef
writeIORef sendRef True
when b $ write (Just sep) os
write s os
------------------------------------------------------------------------------
-- | Combines two input streams. Continues yielding elements from both input
-- streams until one of them finishes.
zip :: InputStream a -> InputStream b -> IO (InputStream (a, b))
zip src1 src2 = makeInputStream src
where
src = read src1 >>= (maybe (return Nothing) $ \a ->
read src2 >>= (maybe (unRead a src1 >> return Nothing) $ \b ->
return $! Just $! (a, b)))
------------------------------------------------------------------------------
-- | Combines two input streams using the supplied function. Continues yielding
-- elements from both input streams until one of them finishes.
zipWith :: (a -> b -> c)
-> InputStream a
-> InputStream b
-> IO (InputStream c)
zipWith f src1 src2 = makeInputStream src
where
src = read src1 >>= (maybe (return Nothing) $ \a ->
read src2 >>= (maybe (unRead a src1 >> return Nothing) $ \b ->
return $! Just $! f a b ) )
------------------------------------------------------------------------------
-- | Combines two input streams using the supplied monadic function. Continues
-- yielding elements from both input streams until one of them finishes.
zipWithM :: (a -> b -> IO c)
-> InputStream a
-> InputStream b
-> IO (InputStream c)
zipWithM f src1 src2 = makeInputStream src
where
src = read src1 >>= (maybe (return Nothing) $ \a ->
read src2 >>= (maybe (unRead a src1 >> return Nothing) $ \b ->
f a b >>= \c -> return $! Just $! c ) )
------------------------------------------------------------------------------
-- | Filters output to be sent to the given 'OutputStream' using a pure
-- function. See 'filter'.
--
-- Example:
--
-- @
-- ghci> import qualified "Data.ByteString.Char8" as S
-- ghci> os1 \<- Streams.'System.IO.Streams.stdout' >>= Streams.'System.IO.Streams.unlines
-- ghci> os2 \<- os1 >>= Streams.'contramap' (S.pack . show) >>= Streams.'filterOutput' even
-- ghci> Streams.'write' (Just 3) os2
-- ghci> Streams.'write' (Just 4) os2
-- 4
-- @
{- Note: The example is a lie, because unlines has weird behavior -}
filterOutput :: (a -> Bool) -> OutputStream a -> IO (OutputStream a)
filterOutput p output = makeOutputStream chunk
where
chunk Nothing = write Nothing output
chunk ch@(Just x) = when (p x) $ write ch output
------------------------------------------------------------------------------
-- | Filters output to be sent to the given 'OutputStream' using a predicate
-- function in IO. See 'filterM'.
--
-- Example:
--
-- @
-- ghci> let check a = putStrLn a ("Allow " ++ show a ++ "?") >> readLn :: IO Bool
-- ghci> import qualified Data.ByteString.Char8 as S
-- ghci> os1 <- Streams.'System.IO.Streams.unlines' Streams.'System.IO.Streams.stdout'
-- ghci> os2 \<- os1 >>= Streams.'contramap' (S.pack . show) >>= Streams.'filterOutputM' check
-- ghci> Streams.'System.IO.Streams.write' (Just 3) os2
-- Allow 3?
-- False\<Enter>
-- ghci> Streams.'System.IO.Streams.write' (Just 4) os2
-- Allow 4?
-- True\<Enter>
-- 4
-- @
filterOutputM :: (a -> IO Bool) -> OutputStream a -> IO (OutputStream a)
filterOutputM p output = makeOutputStream chunk
where
chunk Nothing = write Nothing output
chunk ch@(Just x) = do
b <- p x
if b then write ch output else return $! ()
------------------------------------------------------------------------------
-- | Takes apart a stream of pairs, producing a pair of input streams. Reading
-- from either of the produced streams will cause a pair of values to be pulled
-- from the original stream if necessary. Note that reading @n@ values from one
-- of the returned streams will cause @n@ values to be buffered at the other
-- stream.
--
-- Access to the original stream is thread safe, i.e. guarded by a lock.
unzip :: forall a b . InputStream (a, b) -> IO (InputStream a, InputStream b)
unzip os = do
lock <- newMVar $! ()
buf1 <- newIORef id
buf2 <- newIORef id
is1 <- makeInputStream $ src1 lock buf1 buf2
is2 <- makeInputStream $ src2 lock buf1 buf2
return (is1, is2)
where
twist (a,b) = (b,a)
src1 lock aBuf bBuf = withMVar lock $ const $ do
dl <- readIORef aBuf
case dl [] of
[] -> more os id bBuf
(x:xs) -> writeIORef aBuf (xs++) >> (return $! Just x)
src2 lock aBuf bBuf = withMVar lock $ const $ do
dl <- readIORef bBuf
case dl [] of
[] -> more os twist aBuf
(y:ys) -> writeIORef bBuf (ys++) >> (return $! Just y)
more :: forall a b x y .
InputStream (a,b)
-> ((a,b) -> (x,y))
-> IORef ([y] -> [y])
-> IO (Maybe x)
more origs proj buf = read origs >>=
maybe (return Nothing)
(\x -> do
let (a, b) = proj x
modifyIORef buf (. (b:))
return $! Just a)
------------------------------------------------------------------------------
-- | Wraps an 'InputStream', producing a new 'InputStream' that will produce at
-- most @n@ items, subsequently yielding end-of-stream forever.
--
-- Items pushed back to the returned 'InputStream' will be propagated upstream,
-- modifying the count of taken items accordingly.
--
-- Example:
--
-- @
-- ghci> is <- Streams.'fromList' [1..9::Int]
-- ghci> is' <- Streams.'take' 1 is
-- ghci> Streams.'read' is'
-- Just 1
-- ghci> Streams.'read' is'
-- Nothing
-- ghci> Streams.'System.IO.Streams.peek' is
-- Just 2
-- ghci> Streams.'unRead' 11 is'
-- ghci> Streams.'System.IO.Streams.peek' is
-- Just 11
-- ghci> Streams.'System.IO.Streams.peek' is'
-- Just 11
-- ghci> Streams.'read' is'
-- Just 11
-- ghci> Streams.'read' is'
-- Nothing
-- ghci> Streams.'read' is
-- Just 2
-- ghci> Streams.'toList' is
-- [3,4,5,6,7,8,9]
-- @
--
take :: Int64 -> InputStream a -> IO (InputStream a)
take k0 input = do
kref <- newIORef k0
return $! InputStream (prod kref) (pb kref)
where
prod kref = do
!k <- readIORef kref
if k <= 0
then return Nothing
else do
m <- read input
when (isJust m) $ modifyIORef kref $ \x -> x - 1
return m
pb kref !s = do
unRead s input
modifyIORef kref (+1)
------------------------------------------------------------------------------
-- | Wraps an 'InputStream', producing a new 'InputStream' that will drop the
-- first @n@ items produced by the wrapped stream. See 'Prelude.drop'.
--
-- Items pushed back to the returned 'InputStream' will be propagated upstream,
-- modifying the count of dropped items accordingly.
drop :: Int64 -> InputStream a -> IO (InputStream a)
drop k0 input = do
kref <- newIORef k0
return $! InputStream (prod kref) (pb kref)
where
prod kref = do
!k <- readIORef kref
if k <= 0
then getInput kref
else discard kref
getInput kref = do
read input >>= maybe (return Nothing) (\c -> do
modifyIORef kref (\x -> x - 1)
return $! Just c)
discard kref = getInput kref >>= maybe (return Nothing) (const $ prod kref)
pb kref s = do
unRead s input
modifyIORef kref (+1)
------------------------------------------------------------------------------
-- | Wraps an 'OutputStream', producing a new 'OutputStream' that will pass at
-- most @n@ items on to the wrapped stream, subsequently ignoring the rest of
-- the input.
--
give :: Int64 -> OutputStream a -> IO (OutputStream a)
give k output = newIORef k >>= makeOutputStream . chunk
where
chunk ref = maybe (return $! ()) $ \x -> do
!n <- readIORef ref
if n <= 0
then return $! ()
else do
writeIORef ref $! n - 1
write (Just x) output
------------------------------------------------------------------------------
-- | Wraps an 'OutputStream', producing a new 'OutputStream' that will ignore
-- the first @n@ items received, subsequently passing the rest of the input on
-- to the wrapped stream.
--
ignore :: Int64 -> OutputStream a -> IO (OutputStream a)
ignore k output = newIORef k >>= makeOutputStream . chunk
where
chunk ref = maybe (return $! ()) $ \x -> do
!n <- readIORef ref
if n > 0
then writeIORef ref $! n - 1
else write (Just x) output
------------------------------------------------------------------------------
-- | Wraps an 'OutputStream', ignoring any end-of-stream 'Nothing' values
-- written to the returned stream.
--
-- /Since: 1.0.1.0/
--
ignoreEof :: OutputStream a -> IO (OutputStream a)
ignoreEof s = return $ OutputStream f
where
f Nothing = return $! ()
f x = write x s
------------------------------------------------------------------------------
-- | Wraps an 'InputStream', running the specified action when the stream
-- yields end-of-file.
--
-- /Since: 1.0.2.0/
--
atEndOfInput :: IO b -> InputStream a -> IO (InputStream a)
atEndOfInput m is = return $! InputStream prod pb
where
prod = read is >>= maybe eof (return . Just)
eof = void m >> return Nothing
pb s = unRead s is
------------------------------------------------------------------------------
-- | Wraps an 'OutputStream', running the specified action when the stream
-- receives end-of-file.
--
-- /Since: 1.0.2.0/
--
atEndOfOutput :: IO b -> OutputStream a -> IO (OutputStream a)
atEndOfOutput m os = makeOutputStream f
where
f Nothing = write Nothing os >> void m
f x = write x os
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,902
|
Huck Cycles' electric mopeds go heavy on retro-future style
The American company's handmade bikes offer plenty of customization options, but there's a waiting list.
J. Fergus and Craig Wilson
As people move from filling up their bikes to charging them, the options keep getting cooler. Generally, however, they've sat at two ends of spectrum: the electric bicycle and the electric motorcycle. Huck Cycles, though, is helping swell the ranks of the middle-ground: electric mopeds. Its new line of "bikes for people who don't suck" draws on classic moped design, offers a wide range of price points and power outputs and looks great while doing it.
Eat your heart out, Europe — From Italy and Greece to France and Portugal, when we think of mopeds we think of European vistas and winding, rural roads. That or European teenagers carousing around narrow streets tooting their horns and generally being a nuisance. But there's none of that here. Huck Cycles is an American company selling bikes built by military veterans using mostly American parts and designed for U.S. roads.
Power and pedals — The bikes range from the entry-level 700W Karma with its 20 mph top speed and $3,000 price tag to the popular Rebel's far speedier 50 mph (in 1,000W or 2,000W iterations.) At the top of the pile, there's the blow-your-hair back 3,000W Stinger's 60+ mph top speed and $4,000 starting price. Which you choose will likely depend on the legislation in the state where you plan to ride, how much you're looking to spend, and whether you're after more of an e-bike experience or a moped one. Buyers can also choose from batteries between 52V and 72V.
Huck Cycles
Class 2 compliant — All of Huck's bikes are set to comply with Class 2 regulations (a top speed of 20 mph) at the outset, but the company supplies instructions on how to adjust the limiter so those who are using them in places that allow for higher speeds can do so. Rival rides like the Falcon BLK come with similar regulation-friendly options, but the Falcon suffers next to Huck's offerings due to its starting price of almost $8,000. Gulp.
Huck Cycle's mopeds are handmade to order by the small, in-house team, Electrek reports, which means they can be thoroughly customized. But the combination of labor-intensive work and a lack of desire to become the Ford Motor Company also means Huck Cycle only cranks out about 10 bikes a month. The fledgling company is already at least two months deep in back orders for its various models, so if you want a ride you should act fast, especially if an all-electric two-wheeler is part of your strategy for the end times. And really, along with a way to keep it powered up, it should be.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,942
|
module OpenTelemetry
module Propagator
module B3
VERSION = '0.20.0'
end
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,375
|
from . import sly
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,226
|
\section*{\bibname\markright{\MakeUppercase{\bibname}}}}
\renewcommand\bibname{References}
\defAdv. Sp. Res.{Adv. Sp. Res.}
\defAJ{AJ}
\defApJ{ApJ}
\defAstroph.Sp.Sci.{Astroph.Sp.Sci.}
\defA\&A{A\&A}
\defMNRAS{MNRAS}
\defApJS{ApJS}
\defApJS{ApJS}
\defNature{Nature}
\defMem. SAI{Mem. SAI}
\defIAU Circ.{IAU Circ.}
\defApJLett{ApJLett}
\defA\&AS{A\&AS}
\defPASP{PASP}
\defPASJ{PASJ}
\defPhysRevD{PhysRevD}
\defNewAR{NewAR}
\defARA\&A{ARA\&A}
\defAc. Ast.{Ac. Ast.}
\defA\&A Rev.{A\&A Rev.}
\newcommand{{\textit{Chandra}}}{{\textit{Chandra}}}
\newcommand{{\it RXTE}}{{\it RXTE}}
\newcommand{{\it ASCA}}{{\it ASCA}}
\newcommand{{\it ROSAT}}{{\it ROSAT}}
\newcommand{{\it EINSTEIN}}{{\it EINSTEIN}}
\newcommand{{\it GINGA}}{{\it GINGA}}
\newcommand{{\it BBXRT}}{{\it BBXRT}}
\newcommand{{\it Suzaku}}{{\it Suzaku}}
\newcommand{{\it XMM-Newton}}{{\it XMM-Newton}}
\newcommand{{\it BeppoSAX}}{{\it BeppoSAX}}
\newcommand{{\it Swift}}{{\it Swift}}
\newcommand{\textit{INTEGRAL}}{\textit{INTEGRAL}}
\newcommand{\textit{NuSTAR}}{\textit{NuSTAR}}
\newcommand{\ensuremath{\rm \,M_\odot}}{\ensuremath{\rm \,M_\odot}}
\newcommand{\ensuremath{\rm \,R_\odot}}{\ensuremath{\rm \,R_\odot}}
\newcommand{\ensuremath{\,Z_\odot}}{\ensuremath{\,Z_\odot}}
\newcommand{\ensuremath{\rm \,L_\odot}}{\ensuremath{\rm \,L_\odot}}
\newcommand{\ensuremath{\rm \,M_\odot}}{\ensuremath{\rm \,M_\odot}}
\newcommand{\ensuremath{\,\mathrm{yr}}}{\ensuremath{\,\mathrm{yr}}}
\newcommand{\ensuremath{\,\mathrm{kyr}}}{\ensuremath{\,\mathrm{kyr}}}
\newcommand{\ensuremath{\,\mathrm{Myr}}}{\ensuremath{\,\mathrm{Myr}}}
\newcommand{\ensuremath{\,\mathrm{Gyr}}}{\ensuremath{\,\mathrm{Gyr}}}
\newcommand{\ensuremath{\,\mathrm{km}\,\mathrm{s}^{-1}}}{\ensuremath{\,\mathrm{km}\,\mathrm{s}^{-1}}}
\newcommand{\ensuremath{\,\mathrm{g}\,\mathrm{cm}^{-3}}}{\ensuremath{\,\mathrm{g}\,\mathrm{cm}^{-3}}}
\newcommand{\ensuremath{\,\mathrm{erg}\,\mathrm{s}^{-1}}}{\ensuremath{\,\mathrm{erg}\,\mathrm{s}^{-1}}}
\newcommand{\rm g~s$^{-1}$}{\ensuremath{\,\mathrm{g}\,\mathrm{s}^{-1}}}
\newcommand{\ensuremath{\Msun\mathrm{\; yr}^{-1}}}{\ensuremath{\ensuremath{\rm \,M_\odot}\mathrm{\; yr}^{-1}}}
\newcommand{\ensuremath{\,L_{\rm Edd}}}{\ensuremath{\,L_{\rm Edd}}}
\newcommand{\ensuremath{\,\dot M_{\rm Edd}}}{\ensuremath{\,\dot M_{\rm Edd}}}
\newcommand{\ensuremath{\,T_{\rm eff}}}{\ensuremath{\,T_{\rm eff}}}
\newcommand{\rm M_{\odot}}{\rm M_{\odot}}
\newcommand{$R_{\odot}$}{$R_{\odot}$}
\newcommand{$L_{\odot}$}{$L_{\odot}$}
\newcommand{ergs~s$^{-1}$~cm$^{-2}$}{ergs~s$^{-1}$~cm$^{-2}$}
\newcommand{ergs~s$^{-1}$}{ergs~s$^{-1}$}
\newcommand{$\chi ^{2}$}{$\chi ^{2}$}
\DeclareMathAlphabet\mathbfcal{OMS}{cmsy}{b}{n}
\newcommand{$\mathbfcal{\,S}$}{$\mathbfcal{\,S}$}
\pagenumbering{arabic}
\begin{document}
\begin{center}
{\huge\bf AGN Accretion Discs\footnote{Updated version of Chapter~3 in {\bf Active Galactic Nuclei}; F. Combes ed.; iSTE/Wiley 2022; DOI:10.1002/9781394163724. One figure (Fig. \ref{fig:angmom}) and several references added. {\sl This chapter is a graduate--student level lecture, not a review article.} }}
\vspace{1.0cm}
{\Large Jean-Pierre LASOTA}$^{1,2}$\\
\vspace{1.0cm}
\small{
$^1$ Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences,
Warsaw, Poland\\
\vspace{0.25cm}
$^2$ Institut d'Astrophysique de Paris, CNRS et Sorbonne Universit\'e,
Paris\\
E-mail: lasota@iap.fr\\
}
\end{center}
\begin{figure}[h!]
\centering
\includegraphics[width=4.0cm,height=6.0cm,angle=0]{Active_Galactic_Nuclei.pdf}
\caption*{}
\end{figure}
\section{Introduction}
\label{sec:intro}
{\sl Accretion discs} are ubiquitous in the Universe. The spectacular
ALMA image of the protostellar disc in HL Tau is breathtaking and the M87* black-hole
silhouette observed by the Event Horizon Telescope (EHT) showed us the light emitted by matter as close to the event horizon as possible.
Understanding accretion discs around black holes is interesting in itself because of the fascinating and complex physics
involved but is also fundamental for understanding the coupled evolution of galaxies and their nuclear black holes, i.e.
fundamental for understanding the growth of structures in the Universe. The chance that inflows onto black holes
are strictly radial, as assumed in many models, are slim.
Accretion discs in active galactic nuclei (AGNs) play a central role in the complex engine operating around
the central supermassive black hole. They are the heart of the accretion-ejection processes, playing the role of a rugby-union
fly-half, receiving and passing matter, energy and magnetic fields to various players of the inflow-outflow team.
Accretion disc, in their standard, geometrically-thin version, have the advantage of being the only team-members for
which a well-confirmed, even if not perfect, physical model exists. This chapter concentrates on this model, its properties and observational
consequences, as well as its possible generalisations to situations where it loses its validity.
\section{Black holes}
Black holes which are at the center of all AGN are purely general-relativistic objects and the description of
the properties and behaviour of matter and light in their vicinity requires the use of the formalism of
Einstein's theory of gravitation. Unfortunately, this theory, now used on a daily basis in astronomy
(e.g. observations of the Universe in gravitational waves) and even in everyday life (GPS in our smartphones),
has still not made it to the physics curriculum in most universities. We will therefore present here only
the most important facts about the physics in the vicinity of black holes without derivation and with minimal justification \footnote{An excellent introduction to the general theory of relativity can be found in \citet{Hartle03}}.
\subsection{The horizon}
The Schwarzschild radius (radius of a non-rotating black hole) is
\begin{equation}
R_S=\frac{2GM}{c^2}= 2.95\times 10^{13} \left(\frac{M}{\rm 10^8\ensuremath{\rm \,M_\odot}}\right)\, {\rm cm} = 9.58 \times 10^{-6} \left(\frac{M}{\rm 10^8\ensuremath{\rm \,M_\odot}}\right) \, \rm pc,
\end{equation}
where $G$ is the gravitational constant, $M$ is the mass of the gravitating body and $c$ the speed of light.
Quite often in the literature and also in the present book, one finds the so-called \textsl{ gravitational radius} $R_g=0.5R_S$ used as unit of length.
Here we prefer to use as unit of length the Schwarzschild radius $R_S$.
When the black hole is rotating it is represented by the Kerr solution and the radius of its horizon\footnote{This is the radius of the external horizon. There exists also an inner horizon of no astrophysical interest.} is equal to
\begin{equation}
\label{rh}
R_H = \frac{GM}{c^2}+ \left[\left(\frac{GM}{c^2}\right)^2 -
\left(\frac{J}{Mc}\right)^2 \right]^{1/2}= \frac{GM}{c^2}\left[1 + \left(1 -
a^2 \right)^{1/2}\right],
\end{equation}
where $J$ is the black hole's angular momentum and its dimensionless angular momentum is defined as\footnote{One should be careful not to confuse this
dimensionless ``a'' with the ``a'' often used in the general-relativistic literature defined as $J/Mc$ which has dimension of length.}
\begin{equation}
a:= \frac{Jc}{GM^2}.
\end{equation}
Therefore a horizon, i.e. a black hole, exists only for $0 \leq a \leq 1$. For $a > 1$, the Kerr solution represents a space-time
containing a naked (not covered by a horizon, i.e. observable) singularity and time-like trajectories that violate causality. The
cosmic censorship hypothesis \citep{Penrose0169}, according to which all singularities in the real Universe (except for the Big-Bang) are covered
by an event horizon remains to be proved. However, the third law of black-hole thermodynamics guarantees that no black hole
can be spun-up to $a=1$.
\begin{figure} [h!]
\begin{center}
\includegraphics[angle=0,width=0.9\textwidth]{kerrradiiL.pdf}
\caption{Radii (in units of $R_g$) of the characteristic orbits in the Kerr metric as a function of $a$. The innermost stable circular orbit: ${r}_{\rm ISCO}$, the marginally bound orbit ${r}_{\rm IBCO}$ (marked $\bar{r}_{\rm mb}$), the photon orbit: ${r}_{\rm ph}$ and the black hole horizon: ${r}_{H}$ (marked $\bar{r}_{\rm h}$) The negative
values of $a$ correspond to orbits counterrotating with respect to the black hole.
({\sl Courtesy of \citealt{Sadowski11}}.)}
\label{fig:freq0}
\end{center}
\end{figure}
The numerical value of the maximum angular momentum of a black hole is
\begin{equation}
J_{\rm max}=
\frac{GM^2}{c} = 8.9 \times 10^{48} \left(\frac{M}{\ensuremath{\rm \,M_\odot}}\right)^2
\rm g\ cm^2\ s^{-1}.
\end{equation}
which, for a solar--mass black hole, is comparable to the angular momentum of the Sun: $J_{\odot}= 1.63 \times 10^{48} \rm g\ cm^2\
s^{-1}$, or ${a}_{\odot}= 0.185$.
For a black hole with the mass of the Galaxy ($M_G=1.5 \times 10^{11} \ensuremath{\rm \,M_\odot} $),
the maximum angular momentum would be $2.0 \times 10^{71} \left({M_G}/{\ensuremath{\rm \,M_\odot}}\right)^2
\rm g\ cm^2\ s^{-1}$, so the Galaxy whose angular momentum is $\sim 10^{74}\rm g\ cm^2\ s^{-1}$
would have to lose most of it, if it were to collapse into a black hole. The largest observed mass of a black hole seem to be $6.6 \times 10^{10}\, \ensuremath{\rm \,M_\odot}$ (\citealt{Shemmer1004}).
\subsection{Characteristic orbits}
The space--time around a black hole is curved which implies that particles follow geodesics in a non-euclidean geometry. Far from the
black holes the motion along these geodesics is well approximated by newtonian orbits, but close the horizon the motion of bodies and light
is strongly affected by general-relativistic effects. Whereas in newtonian gravity, the angular momentum on Keplerian orbits decreases monotonically
with the square-root of the distance from the center, in general relativity it has a minimum at the orbit called ISCO, for ``innermost stable circular orbit''.
Closer to the horizon, free circular orbits are unstable, so that a Keplerian accretion disc (see below) has to end at the ISCO. ``Keplerian" means that
matter in the disc moves on approximately free-falling orbit, which is the case for low accretion rates. In the case of high accretion rates, the radial
pressure gradients in the disc are no longer negligible and the inner disc's edge is pushed inwards, but cannot be closer to the horizon than the
innermost bound circular orbit, that we would like to call IBCO, but which is known under the name of ``$R_{\rm mb}$'', ``mb'' corresponding to ``marginally bound".
\subsubsection*{ISCO}
The mathematical expression for the ISCO radius is complex and can be easily found in the literature. For the Schwarzschild solution $R_{\rm ISCO}= 3 R_S$. For
maximally rotating black hole the radius of the counterrotating ISCO is at $R_{\rm ISCO}= 4.5 R_S$, while the radius of a corotating ISCO is formally at
$R_{ISCO}= 0.5R_S=R_H$, but this is the result of a degeneracy for $a=1$ of the coordinate systems used.
For a Schwarzschild black hole the frequency associated with the ISCO at $R_{\rm ISCO}=3R_S$ is
\begin{equation}
\nu_K(R_{\rm ISCO}) = 2.2 \,\left(\frac{M}{10^8 \ensuremath{\rm \,M_\odot}}\right)^{-1}\ \rm \mu Hz.
\end{equation}
A quasi-periodic oscillation (QPO) of $\sim 27 \rm \mu Hz$ that could be associated with the ISCO frequency has been observed
in an active galaxy of type NLS1 (meaning Narrow Line Seyfert 1), whose mass is presumably $1 - 4 \times 10^6\ensuremath{\rm \,M_\odot}$.
\subsubsection*{IBCO}
The IBCO is at radius:
\begin{equation}
R_{\rm IBCO}^{\pm}= \frac{GM}{c^2}\left[2 \mp a + 2\sqrt{1 \mp a}\right],
\end{equation}
which for a Schwarzschild black hole gives $R_{\rm IBCO}=2 R_S$. When $a=1$, for a counterrotating orbit
$R_{\rm IBCO}\approx 2.9 R_S$, in the corotating case, $R_{\rm IBCO}=0.5R_S$ for the reasons mentioned above.
\subsubsection*{Photon orbit}
The photon circular orbit around a black hole is also the Innermost Circular Orbit (ICO),
and its radius is at
\begin{equation}
R_{\mathrm{phot}}^{\pm}= R_S \left(1 + \cos\left[\frac{2}{3}\cos^{-1}\left(\mp {a}\right)\right]\right).
\end{equation}
The ``photon orbit'' is at $1.5 R_S$ for a $a=0$, at $4R_S$ for $a=1$ retrograde orbit and, as all the other characteristic orbits,
at $0.5R_S$ for prograde rotation with $a=1$.
The photon circular orbit is unstable but in a suitable setting an observer could see a ``photon ring" produced by photons
``wrapping around'' $R_{phot}$ before reaching the detector. It has been claimed that such a ring has been observed by the
EHT around the M87 central black hole, but this is far from being certain \citep[see, e.g.][]{Gralla0719}.
\subsection{Binding energy and accretion efficiency}
The binding energy per unit mass of an orbit is (assuming a unit system where c=1),
\begin{equation}
\label{eq:bindingE}
{E}_{\rm bind}=1 - {E}_K,
\end{equation}
where ${E}_K$ is the kinetic energy per unit rest-mass of a Keplerian orbit (in the newtonian case $E_{K}= (GM/2c^2R$).
At the ISCO the binding energy per unit rest-mass is
\begin{itemize}
\item $1 - \sqrt{8/9} \approx 0.06$ for $a=0$
\item $1 - \sqrt{1/3} \approx 0.42$ for $a=1$.
\end{itemize}
Therefore this corresponds to the efficiencies
of accretion in a Keplerian disc around a black hole.
By definition the binding energy at the IBCO is equal to zero. Therefore pressure gradients in an accretion
flow that push the inner disc towards the black hole reduce accretion efficiency.
\subsection{Rotating space}
A purely general relativistic effect, with no newtonian equivalent or analogue, is the rotation of space:
the rotation of a gravitating body ``drags'' with it the surrounding space. The angular velocity of this
dragging is:
\begin{equation}
\label{eq:LTomega}
\Omega_{\mathrm{LT}}\approx \frac{2GJ}{c^2R^3},
\end{equation}
where $J$, as before is the black-hole angular momentum. ``LT'' stands for ``Lense--Thirring'' -- the names
of the two discoverers of this effect in the weak-field approximation of the Einstein's theory of gravitation.
$\Omega_{\mathrm{LT}}$ corresponds to the angular velocity with which the orbits of misaligned
(not in the equatorial plane) test particles precess
around a black hole.
The corresponding precession time is
\begin{equation}
t_{\mathrm{LT}}\approx \frac{1}{\left|\Omega_{\mathrm{LT}}\right|}=\frac{c^{2} R^{3}}{2 G J}=\frac{2}{a}\left(\frac{R}{R_{\mathrm{S}}}\right)^{2} \frac{R}{c},
\end{equation}
which, close to the black hole can be as short as the dynamical time (At the disc inner edge of a $10^8\ensuremath{\rm \,M_\odot}$black hole: $R/c \approx 980$\,s), but since the precession is strongly differential, decreasing with radius,
far from the black hole it is entirely negligible:
\begin{equation}
t_{\mathrm{LT}}\approx 7.1 \times 10^{10} a^{-1}\left(\frac{M}{10^{8} \mathrm{M}_{\odot}}\right)^{-2}\left(\frac{R}{1 \mathrm{pc}}\right)^{3} \text { yrs. }
\end{equation}
However, dissipation in a misaligned disc can lead to the alignment of its innermost regions with the black hole spin, creating a warp, through the so-called
Bardeen-Petterson effect \citep{BP75}, (see below).
\subsection{The ergoregion}
The surface of a black hole rotates with angular velocity
\begin{equation}
\Omega_{\rm H} = \frac{ac}{2R_H},
\end{equation}
where
$\Omega_{\rm H}$ is the angular velocity of the horizon, i.e. the
angular velocity of the horizon-forming light-rays with respect to
infinity. The horizon rotates.
The fact that a rotating black-hole induces rotation of space has two important astrophysical implication. First, it creates a torque
which forces the matter in the tilted disc towards the equatorial plane in the so-called \textsl{Bardeen-Petterson effect}.
This effect, strongest near the rotating body's surface, induces the disc precession.
Second, in the case of a Kerr black hole,
there exists an ergoregion between the horizon and the surface defined by
\begin{equation}
\label{eq:ergoreg}
R_{\rm ergo}(\theta)=\frac{GM}{c^2}\left[1 + \left(1 -
a^2 cos^2\theta\right)^{1/2}\right]
\end{equation}
where being at rest with respect to infinity is impossible: to resist the rotation of space would imply local superluminal
motion.
This implies that the ergoregion contains trajectories with negative mass-energy as seen
from infinity, although the energy
measured in a frame rotating with the space is positive. If e.g., an ingoing particle with energy $E_1 >0$ decays in the ergoregion into two particles, one with energy $E_2>0$, which escapes to infinity,
the other with $\Delta E_H <0$ which falls into the black hole (negative-energy particles cannot leave the
ergoregion because this would require accelerating to speeds larger than the speed of light),
$E_2 = E_1 - \Delta E_H > E_1$, so that the outgoing particle is more energetic than the ingoing one.
This energy has been gained at the expense of the rotational energy of the black hole:
the latter absorbed negative energy and negative angular momentum, since negative-energy particles
counter-rotate with respect to the black-hole spin. These particles carry negative angular momentum.
The electromagnetic version of this \textsl{mechanical Penrose process} allows tapping the black-hole
rotational energy which is supposed to be the source of relativistic
jets observed in some AGN (see Sect. \ref{sec:disc-jets}).
\subsection{Eddington accretion rate}
We will define the Eddington luminosity and accretion rate as
\begin{align}
\dot{M}_{\rm Edd} & =\frac{L_{\rm Edd}}{\eta c^2}=\frac{1}{\eta}\frac{4\pi GM}{c\kappa_{es}} \nonumber \\
& = 1.6\times 10^{26}\eta_{0.1}^{-1} \left(\frac{M}{\rm 10^8 M_{\odot}}\right)\,{\rm g\,s^{-1}} \\
& = 2.5 \, \eta_{0.1}^{-1} \left(\frac{M}{\rm 10^8 M_{\odot}}\right) \rm M_{\odot}\,\rm yr^{-1},
\end{align}
where $\eta=0.1\eta_{0.1}$ is the radiative efficiency of accretion, $\kappa_{es}$ the electron scattering (Thomson) opacity.
We will often use accretion rate measured in units of Eddington accretion rate:
\begin{equation}
\label{eq:mdot}
\dot m=\frac{\dot M}{\dot{M}_{\rm Edd}}\, \ \ \ \ \ \ \ \ \ {\rm and \ use}\ \ \ \ \ \ \ \ m_{8}= \frac{M}{10^8\, \rm M_{\odot}}.
\end{equation}
One can find in the literature definitions of $\dot{M}_{\rm Edd}$ implicitly
using $\eta= 1$, or $\eta=0.2$ so care is recommended
when using published numerical values of the accretion rate in Eddington units.
\section{Disc driving mechanism; viscosity}
\label{sec:viscosity}
Very early in the studies of accretion disc physics, it became clear that, if their driving mechanism
is viscous, an ``anomalous", turbulent viscosity must be at work because astrophysical discs are too large
(and their density too small) for the molecular viscosity to be efficient.
After years of uncertainty, it is now obvious that the
turbulence in ionized Keplerian discs is due to the presence of a Magneto-Rotational Instability
(MRI) also known as the Balbus-Hawley mechanism
\citep{BalbusHawley91}, occurring in \textsl{weakly} magnetised, differentially rotating plasma.
However, despite impressive developments, numerical simulations, even in their global 3D form, still suffer
from weaknesses that, in most cases, make their direct application to real accretion flows almost infeasible.
One of the problems is related to the value of the ratio of the (vertically averaged) total stress to thermal (vertically averaged) pressure
\begin{equation}
\label{eq:numeralpha}
\alpha=\frac{\langle{\tau_{R\varphi}}\rangle_z}{\langle P \rangle_z},
\end{equation}
where the stress $\tau_{R\varphi}$ is the sum of the Maxwell and Reynolds stresses:
\begin{equation}
\tau_{R\varphi}=- \frac{B_{R}B_{\varphi}}{4\pi} + \rho v_R \delta v_{\varphi},
\end{equation}
where indices $r$ and $\varphi$ denote, respectively the radial and azimuthal components
of the magnetic intensity $\mathbf{B}$, $\bf{v}$ and $\mathbf{\delta v}$ vectors, and $\rho$ denotes density.
$\delta v_{\varphi} = v_{\varphi} + 3\Omega_K R/2$ is the difference between the azimuthal velocity and
the mean rotational velocity in the disc, with $\Omega_K$ being the Keplerian orbital angular speed (Eq. \ref{eq: OmegaK}).
$P = P_g + P_{rad}$ is the sum of the thermal pressures; magnetic pressure is not included in this definition.
As we will see in the next section, and in the rest of this Chapter, the $\alpha$ parameter has played a crucial role in the development
of accretion disc astrophysics. Its value ( $0\leq \alpha \leq 1$) can be determined from observations of light variability
of some accreting systems, as will be shown below.
According to most MRI simulation $\alpha \sim 10^{-3}$ whereas observations of ionised discs around white dwarfs in binary
systems called cataclysmic variables, which are the best ``life-size" realisation of MRI discs, unambiguously show that $\alpha\approx 0.1 - 0.2$ \citep{Kotko1209}.
Disc outbursts in X-ray binaries suggest even $\alpha\approx 0.2 - 2.0$\citep{Tetarenko0218}.
The effects of convection at temperatures $\sim 10^4$\,K increase $\alpha$ to values $\sim 0.1$ and a net magnetic field
in the region might bring its value up to the observed one. On the other hand some observations of AGN variability
give values of the viscosity parameter: $0.01 \leq \alpha \leq 0.03$ for $0.01 \leq L/L_{Edd} \leq 1.0$ \citep{Starling0104},
still higher than the
fiducial MRI-simulation value.
Another problem is related to cold discs.
For the standard MRI to work, the degree of ionization in a weakly magnetized,
quasi-Keplerian disc must be sufficiently high to produce the instability that leads
to a breakdown of laminar flow into turbulence;
the latter being the source of viscosity driving accretion onto the central body.
In cold discs the ionized fraction is very small and might be
insufficient for the MRI to operate. In any case in such a disc non-ideal MHD effects are always important. All these problems
still await their solution even if a lot of progress has been made, especially in the context of protostellar accretion discs.
Finally, and very relevant to the subject of this Chapter, there is the question of stability of discs in which the pressure
is due to radiation and opacity to electron scattering. According to theory, such discs should be violently (thermally)
unstable but observations of systems presumed to be in this regime totally infirm this prediction.
\subsection{The $\alpha$--prescription}
The $\alpha$--prescription is a rather simplistic description of the accretion disc
physics but before one is offered better and physically more reliable options its
simplicity makes it the best possible choice and has been the main source of progress
in describing accretion discs in various astrophysical contexts.
One keeps in mind that the accretion--driving viscosity is of magnetic origin, but one
uses an effective hydrodynamical description of the accretion flow.
The hydrodynamical stress tensor is
\begin{equation}
\label{eq:stress}
\tau_{R\varphi}= \rho \nu \frac{\partial v_{\varphi}}{\partial R}=\rho \nu \frac{d\Omega}{d\ln R},
\end{equation}
where $\rho$ is the density, $\nu$ the kinematic viscosity coefficient and $v_\varphi$ the azimuthal velocity ($v_\varphi=R\Omega$).
In 1973 Shakura \& Sunyaev proposed the (now famous) prescription
\begin{equation}
\label{eq:alphaP}
\tau_{R\varphi}=\alpha P,
\end{equation}
where $P$ is the total thermal pressure and $\alpha \leq 1$. This leads to
\begin{equation}
\label{eq:nualphaP}
\nu= \alpha c_s^2 \left[\frac{d\Omega}{d\ln R}\right]^{-1},
\end{equation}
where $c_s = \sqrt{P/\rho}$ is the isothermal sound speed and $\rho$ the density.
For the Keplerian angular velocity
\begin{equation}
\label{eq: OmegaK}
\Omega=\Omega_K=\left(\frac{GM}{R^3}\right)^{1/2}
\end{equation}
this becomes
\begin{equation}
\label{eq:nualphaOmega}
\nu=\frac{2}{3} \alpha c_s^2/\Omega_K.
\end{equation}
Using the approximate hydrostatic equilibrium (Eq. \ref{eq:mec_approx}) one can write this as
\begin{equation}
\label{eq:nu}
\nu \approx \frac{2}{3}\alpha c_s H.
\end{equation}
Multiplying the rhs of Eq. (\ref{eq:stress}) by the ring length ($2\pi R$) and averaging over the (total = $2H$) disc height one obtains the expression for the {\sl total torque}
\begin{equation}
\label{eq:stressH}
{\mathfrak T}= 2\pi R \Sigma \nu R\frac{d\Omega}{d\ln R},
\end{equation}
where
\begin{equation}
\label{eq:Sigma}
\Sigma=\int^{+\infty}_{-\infty}\rho\,dz .
\end{equation}
For a Keplerian disc
\begin{equation}
\label{eq:torqueK}
{\mathfrak T}= 3\pi \Sigma \nu \ell_K,
\end{equation}
($\ell_K=R^2\Omega_K$ is the Keplerian \textsl{specific}(i.e. per unit mass) angular momentum.)
The viscous heating is proportional to $\tau_{r\varphi}(d\Omega/dR)$.
In particular the viscous heating rate per unit volume is
\begin{equation}
\label{eq:visheat1}
q^+= -\tau_{r\varphi}\frac{d\Omega}{d\ln R},
\end{equation}
which for a Keplerian disc, using Eq. (\ref{eq:alphaP}), can be written as
\begin{equation}
\label{eq:qplusalphaP}
q^+=\frac{3}{2}\alpha \Omega_K P,
\end{equation}
and the viscous heating rate per unit surface is therefore
\begin{equation}
\label{eq:vischeat2}
Q^+=\frac{{\mathfrak T\Omega^{\prime}}}{4\pi R}=\frac{9}{8} \Sigma \nu \Omega_K^2.
\end{equation}
(The denominator in the first rhs is $2\times 2\pi R$ taking into account the existence of two disc surfaces.)
\section{Geometrically thin Keplerian discs}
\label{sec:thin}
The 2D structure of geometrically thin, non--self-gravitating, axially symmetric accretion discs
can be split into a 1D+1D structure
corresponding to a hydrostatic vertical configuration and radial quasi-Keplerian viscous flow. These two 1D structures
are coupled through the viscosity mechanism transporting angular momentum and providing the local release
of gravitational energy.
\subsection{Disc vertical structure}
\label{subsec:vstruct.1}
The vertical structure can be treated as a one--dimensional star with two essential differences:
\begin{enumerate}
\item the energy sources are distributed over the whole height of the disc, while in a star they are limited to the nucleus,
\item the gravitational acceleration \textsl{increases} with height because it is given by the tidal gravity of the accretor, while in stars the (self)gravity
decreases as the inverse square of the distance from the center.
\end{enumerate}
Taking these differences into account the standard stellar structure equations adapted to the description of the disc vertical structure are listed
below.
\begin{itemize}
\item Hydrostatic equilibrium
The gravity force is counteracted by the force produced by the pressure gradient:
\begin{equation}
\label{eq:vertichydro}
\frac{dP}{dz}=\rho g_z,
\end{equation}
where $g_z$ is the vertical component (tidal) of the accreting body gravitational acceleration:
\begin{equation}
\label{eq:vertacc}
g_z=\frac{\partial}{\partial z }\left[\frac{GM}{(R^2 + z^2)^{1/2}}\right]\approx \frac{GM}{R^2}\frac{z}{R}.
\end{equation}
The second equality follows from the assumption that $z\ll R$.
Denoting the typical (pressure or density) scale-height by $H$ the condition of geometrical thinness of the disc is $H/R\ll 1$
and writing $dP/dz \sim P/H$, Eq. (\ref{eq:vertichydro})
can be written as
\begin{equation}
\label{eq:mec_approx}
\frac{H}{R}\approx \frac{c_s}{v_K},
\end{equation}
where $v_K=\sqrt{GM/R}$ is the Keplerian velocity and we made use of Eq. (\ref{eq:vertacc}).
From Eq. (\ref{eq:mec_approx}) it follows that
\begin{equation}
\label{eq:dyntime}
\frac{H}{c_s} \approx \frac{1}{\Omega_K}= :t_{\rm dyn},
\end{equation}
where $t_{\rm dyn}$ is the dynamical time.\\
For the parameters of interest
\begin{equation}
\label{eq:dynparam}
t_{\rm dyn}= 1.4 \times 10^3 m_8 r^{3/2} \,\rm s.
\end{equation}
Here, and in what follows $r :=R/R_S$, and $m_8 :=$ M/$10^8\ensuremath{\rm \,M_\odot}$.
Therefore, e.g., for $R\approx 10^{16}$cm (or 330 $R_S$),
and a $10^8\ensuremath{\rm \,M_\odot}$ black hole, the dynamical time is about $3.3$ months.
\item Mass conservation
In 1D hydrostatic equilibrium the mass conservation equation takes the simple form of
\begin{equation}
\label{eq:mass_cons}
\frac{d \varsigma}{dz} = 2 \rho,
\end{equation}
where $\varsigma$ is the surface
density between $-z$ and $+z$.
\medskip
\item Energy transfer - temperature gradient
\begin{equation}
\label{eq:nabla}
\frac{d\ln T}{dz} = \nabla \frac{d \ln P}{dz}.
\end{equation}
For radiative energy transport
\begin{equation}
\label{eq:nablarad}
\nabla_{\rm rad} = \cfrac{}{}{\kappa_{\rm R} P F_{\rm z}}{4 P_r c g_{\rm z}},
\end{equation}
where $P_r$ is the radiation pressure and $\kappa_{\rm R}$ the Rosseland mean opacity.
From Eqs. (\ref{eq:nabla}) and (\ref{eq:nablarad}) one recovers the familiar expression for
the radiative flux
\begin{equation}
\label{eq:radflux}
F_z= - \frac{16}{3}\frac{\sigma T^3}{\kappa_{\rm R}\rho}\frac{\partial T}{\partial z}=- \frac{4\sigma}{3\kappa_{\rm R}\rho}\frac{\partial T^4}{\partial z}
\end{equation}
($F_z$ is positive because the temperature decreases with $z$ so ${\partial T}/{\partial z}<0$).
The photosphere is at optical thickness $\tau\simeq 2/3$ (see Eq. \ref{t2}).
The boundary conditions are: $z = 0$, $F_z = 0$, $T = T_c$, $\varsigma = 0$ at
the disc midplane; at the disc photosphere $\varsigma=\Sigma$ and
$T^4(\tau=2/3) = T^4_{\rm eff}$. For a detailed discussion of radiative transfer, temperature stratification and boundary conditions see Sect. \ref{subsect:rad}.
In the same spirit as Eq. (\ref{eq:mec_approx}) one can write Eq. (\ref{eq:radflux}) as
\begin{equation}
\label{eq:rad_approxx}
F_z\approx \frac{4}{3}\frac{\sigma T_c^4}{\kappa_{\rm R}\rho H}= \frac{8}{3}\frac{\sigma T_c^4}{\kappa_{\rm R}\Sigma},
\end{equation}
where $T_c$ is the mid-plane (``central") disk temperature. Using the optical depth $\tau= \kappa_{\rm R}\rho H= (1/2) \kappa_{\rm R}\Sigma$, this can be written as
\begin{equation}
\label{eq:rad_approx2}
F_z(H) \approx \frac{8}{3}\frac{\sigma T_c^4}{\tau}= Q^-,
\end{equation}
(see Eq. \ref{diff2} for a rigorous derivation of this formula).
In the case of convective energy transport $\nabla=\nabla_{\rm conv}$. Because convection in discs is still not well understood there is no obvious choice for $\nabla_{\rm conv}$ so in practice prescriptions based on
stellar physics, such as the mixing-length approximation, are used, even if it is not obvious that
they apply to accretion discs. Some evidence provided by MRI simulations suggests that they do not.
\item Energy conservation
Vertical energy conservation should have the form
\begin{equation}
\label{eq:strc1}
\frac{dF_{\rm z}}{dz } = q^+ (z),
\end{equation}
where $q^+ (z)$ corresponds to viscous energy dissipation per unit volume.
Note that, in contrast with accretion discs, stellar envelopes are in \textsl{radiative equilibrium} ${dF_{\rm z}}/{dz }=0$
The $\alpha$ prescription does not allow deducing the viscous dissipation stratification ($z$ dependence), it just says that the vertically averaged viscous torque
is proportional to pressure. Most often one assumes therefore that
\begin{equation}
\label{eq:strc2}
q^+ (z) = \frac{3}{2} \alpha \Omega_{\rm K} P (z),
\end{equation}
by analogy with Eq. (\ref{eq:qplusalphaP}) but such an assumption is chosen because of its simplicity and not because of some physical motivation. In fact MRI
numerical simulations suggest that dissipation is not stratified in the same way as pressure.
\item The vertical structure equations have to be completed by the equation of state (EOS):
\begin{equation}
P = P_r + P_g= \frac{4\sigma }{3c}T^4 + \frac{{\cal R}}{\mu} \rho T ,
\label{eq:EOS}
\end{equation}
where ${\cal R}$ is the gas constant and $\mu$ the mean molecular weight,
and an equation describing the mean opacity dependence on density and temperature.
\end{itemize}
\subsection{Disc radial structure}
\label{subsec:radial1}
\begin{itemize}
\item
Continuity (mass conservation) equation has the form
\begin{equation}
\frac{\partial \Sigma}{\partial t} = - \frac{1}{R} \frac{\partial}{\partial
R} (R \Sigma v_{\rm r}) + \frac{S(R,t)}{2 \pi R},
\label{eq:consm}
\end{equation}
where $S(R,t)$ is the matter source (sink) term.
\item Angular momentum conservation
\begin{equation}
\frac{\partial \Sigma \ell}{\partial t} = - \frac{1}{ R} \frac{\partial}{\partial
R} (R \Sigma \ell v_{\rm r}) + \frac{1}{ R} \frac{\partial}{\partial R}
\left(R^3 \Sigma \nu \frac{d\Omega}{dR} \right) + \frac{S_{\ell}(R,t)}{2\pi R}.
\label{eq:consj}
\end{equation}
$\ell$ is the specific (per unit mass) angular momentum. This conservation equation reflects the fact that angular momentum is transported through the disc by a viscous stress $\tau_{r\varphi}=R \Sigma \nu {d\Omega}/{dR}$.
Therefore, if the disc is not considered infinite (recommended in application to real processes and systems) there must be somewhere a sink of this transported angular momentum $S_\ell (R,t)$.
\end{itemize}
For semi-detached binary systems there is both a source (angular momentum brought in by the mass transfer from
the stellar companion) and a sink (tidal interaction taking angular momentum back to the orbit).
In the case of accretion discs in AGN, neither the processes through which matter is fed to the disc,
nor the mechanism removing angular momentum are well established so the form of both $S(R,t)$ and
${S_{\ell}(R,t)}$ are unknown. One can only safely assume that the equations used
in this Section will not apply beyond the radius at which the disc becomes self-gravitating
(see Sect. \ref{subsec:sg}) which defines a natural outer boundary of the disc considered here.
Assuming $\Omega=\Omega_K$, from Eqs. (\ref{eq:consm}) and (\ref{eq:consj}) one can obtain a diffusion equation for the surface density $\Sigma$:
\begin{equation}
\label{eq:eqdiff}
\frac{\partial \Sigma}{\partial t}=\frac{3}{R}\frac{\partial}{\partial R}\left\{ R^{1/2} \frac{\partial}{\partial R}\left[ \nu \Sigma R^{1/2}\right]\right\}.
\end{equation}
Comparing with Eqs. (\ref{eq:consm}) one sees that the radial velocity induced by the viscous torque is
\begin{equation}
v_r= - \frac{3}{\Sigma R^{1/2}}\frac{\partial}{\partial R}\left[ \nu \Sigma R^{1/2}\right],
\label{eq:vr}
\end{equation}
which is an example of the general relation
\begin{equation}
\label{eq:vvisc}
v_{\rm visc}\sim \frac{\nu}{R}.
\end{equation}
Using Eq.(\ref{eq:nu}) one can write
\begin{equation}
\label{eq:vistime1}
t_{\rm vis} := \frac{R}{v_{\rm visc}}\approx \frac{R^2}{\nu}\approx\alpha^{-1}\frac{H}{c_s}\left(\frac{H}{R}\right)^{-2}.
\end{equation}
The relation between the viscous and the dynamical times is
\begin{equation}
t_{\rm vis}\approx \alpha^{-1}\left(\frac{H}{R}\right)^{-2}\, t_{\rm dyn}.
\label{eq:vistime2}
\end{equation}
In thin ($H/R \ll 1$) accretion discs the viscous time is much longer that the dynamical time. In other
words, during viscous processes the vertical disc structure can be considered to be in hydrostatic equilibrium.
\begin{itemize}
\item Energy conservation\\
The general form of energy conservation (thermal) equation can be written as:
\begin{equation}
\label{eq:energy}
\rho T\frac{d s}{dR}:=\rho T\left(\frac{\partial s}{\partial t} +v_r\frac{\partial s}{\partial R}\right)= q^+ - q^- + \widetilde q,
\end{equation}
where $s$ is the entropy density, $q^+$ and $q^-$ are respectively the viscous and radiative energy density, and
$\widetilde q$ is the density of external and/or radially transported energy densities.
The term
\begin{equation}
\label{eq:adv1}
\rho T v_r\frac{\partial s}{\partial R} =: q^{\rm adv}
\end{equation}
describes radial advection of energy.
Using the first law of thermodynamics
$Tds=dU +PdV$
one can write
\begin{equation}
\label{eq:firstlaw}
\rho T\frac{ds}{dt}=\rho\frac{d U}{d t} + P\frac{\partial v_r}{\partial r},
\end{equation}
where $U={{\Re} T_{\rm c} /\mu (\gamma - 1)}$.
Vertically averaging, but taking $T=T_c$, using Eq. (\ref{eq:consm}) and assuming a gas-pressure dominated disc ($P=P_g$),
one obtains
\begin{equation}
\frac{\partial T_{\rm c}}{\partial t} +v_{\rm r} \frac{\partial T_{\rm c}}{\partial R}+
\frac{\Re T_{\rm c}}{\mu c_P} \frac{1}{R} \frac{\partial (R v_{\rm r})}{\partial R}
= 2\frac{Q^+ -Q^-}{c_P \Sigma} + \frac{\widetilde Q}{c_P\Sigma} ,
\label{eq:heat}
\end{equation}
where $Q^+$ and $Q^-$ are respectively the heating and cooling rates per unit surface.
${\widetilde Q}=Q_{\rm out} + J$ with $Q_{\rm out}$ corresponding to energy contributions
by the mass-transfer stream and tidal torques; $J(T,\Sigma)$ represent radial energy fluxes
that are a, more or less, ad hoc addition to the 1D+1D scheme to which they do not belong: indeed
such a scheme assumes that radial gradients ($\partial/\partial R$) of physical quantities can be neglected
when compared with gradients in the vertical direction. That is also the reason that as long as
$H/R \ll 1$, $q^{\rm adv}$ is negligible compared to $q^+$ and $q^{-}$.
\end{itemize}
The viscous heating rate per unit surface can be written as (see Eq. \ref{eq:vischeat2})
\begin{equation}
Q^+=\frac{9}{8} \nu \Sigma \Omega_{\rm K}^2
\label{eq:qplus}
\end{equation}
while the cooling rate over unit surface (the radiative flux) is obviously
\begin{equation}
Q^- =\sigma T_{\rm eff}^4.
\label{eq:qminus}
\end{equation}
In thermal equilibrium one has
\begin{equation}
\label{eq:thermal_equi}
Q^+=Q^-.
\end{equation}
The cooling time can be easily estimated from Eq. (\ref{eq:thermal_equi}).
The energy density to be radiated away is $\rho U \sim \rho c_s^2$,
so the energy per unit surface is $\sim \Sigma c_s^2$ and the cooling (thermal) time is
\begin{equation}
\label{eq:thermal_time}
t_{\rm th}=\frac{\Sigma c_s^2}{Q^-}=\frac{\Sigma c_s^2}{Q^+}\sim \alpha^{-1}\Omega_K^{-1}= \alpha^{-1}t_{\rm dyn}.
\end{equation}
Since $\alpha < 1$, $t_{\rm th}>t_{\rm dyn}$ and during thermal processes the disc can be assumed to be in (vertical)
hydrostatic equilibrium.
For geometrical thin ($H/R\ll 1$) accretion discs one has the following hierarchy of the characteristic times
\begin{equation}
\label{eq:timeshare}
t_{\rm dyn} < t_{\rm th} \ll t_{\rm vis}.
\end{equation}
(This hierarchy is similar to that of characteristic times in stars where the dynamical time
is shorter than the thermal (Kelvin-Helmholtz) time, and the
thermal time is much shorter than the thermonuclear time-scale.)
\subsection{Self-gravity}
\label{subsec:sg}
In this Chapter we are interested in discs that are not self-gravitating, i.e. in discs where the vertical hydrostatic equilibrium is maintained against the pull
of the accreting body's tidal gravity whereas the disc's self-gravity can be neglected. We will see now under what conditions this assumption
is satisfied.
The equation of vertical hydrostatic equilibrium can be written as
\begin{equation}
\frac{1}{\rho}\frac{dP}{dz}= - g=\left(-g_z - g_s \right)= - g_z\left(1+\frac{g_s}{g_z}\right)=:- g_z \left(1 + A\right),
\end{equation}
therefore self-gravity is negligible when $A \ll 1$.
Treating the disc as an infinite uniform plane (i.e. assuming the surface density does not vary too much with radius) one can write its self gravity as
$g_s=2\pi G\Sigma$, whereas the $z$-component of the gravity provided by the central body is $g_z=\Omega_K^2\,z$ (Eq. \ref{eq:vertacc}). Therefore evaluating $A$ at $z=H$
one gets
\begin{equation}
A_H:= \frac{g_s}{g_z}\bigg|_H = \frac{2\pi G \Sigma}{\Omega_K^2 H}.
\end{equation}
$A_H$ is related to the so-called Toomre parameter \citep{Toomre64}
\begin{equation}
\label{eq:toomre}
Q_T:=\frac{c_s\Omega}{\pi G\Sigma},
\end{equation}
widely used in the studies of gravitational stability of rotating systems, through $A_H \approx Q_T^{-1}$.
We will therefore express the condition of negligible self-gravity (gravitational stability) as
\begin{equation}
\label{eq:nosg}
Q_T > 1.
\end{equation}
Notice that the condition of negligible disc self-gravity (Eq. \ref{eq:nosg}) is equivalent to $M_D < (H/R)\,M_{\rm BH}$ and not to just $M_D < M_{\rm BH}$,
as sometimes claimed, where $M_D$ is the disc mass, and $M_{\rm BH}$ is the mass of the accreting body, a black hole in our case.
Using Eqs. (\ref{eq:mec_approx}), (\ref{eq:nu}) and (\ref{eq:amintK}) one can write the Toomre parameter as
\begin{equation}
Q_T=\frac{3\alpha c_s^3}{G\dot M},
\end{equation}
or as function of the mid-plane
temperature $T=10^4\,T_4$K
\begin{equation}
\label{eq:Qdisc}
Q_T\approx 0.5 \,\frac{\alpha\, T^{3/2}_4}{m_8\,\dot m}.
\end{equation}
(where $\dot M= \dot m$ M$_\odot$/yr is the accretion rate, expressed in
solar mass per year, see below).
This shows that hot ionized ($T\gtrsim 10^4$K) discs become self-gravitating for high accretor masses and
high accretion rates. Discs in close binary systems ($m_8 \lesssim 30$) are never self-gravitating for realistic accretion
rates ($\dot m < 1000$, say) and even in intermediate-mass black holes (IMBH) binaries
(if they exist) (hot) discs would also be free of the gravitational instability.
Around a supermassive black hole, however, discs can become self-gravitating quite close to the black hole.
For example when
the black hole mass is $m_8=1$ a hot disc will become self-gravitating at $r \approx 100$,
for $\dot m \sim 10^{-2}$.
In general, geometrically thin, non--self-gravitating accretion discs around supermassive black holes
have a very limited radial extent.
From the frequently used relation (see Sect. \ref{subsec:SS})
\begin{equation}
Q_T \approx 10^4 \alpha^{7/10}m_8^{-13/10}\dot m^{-11/20}r^{-9/8},
\label{eq:selfgagn}
\end{equation}
one obtains for the radius of self-gravitating radius
\begin{equation}
r_{\rm sg} < 4.0 \times 10^3 \alpha^{28/45}m_8^{-52/45}\dot m^{-22/45},
\label{eq:rselfg}
\end{equation}
or, e.g., for $m_8=1$ and $\dot m=0.01$, $R < 0.4$ pc.
The value of the self-gravitating radius plays a fundamental role in determining the maximum mass of a luminous (through accretion of matter) black hole \citep{King0216}.
\subsection{Stationary discs}
\label{subsec:stationary}
In the case of stationary ($\partial/\partial t=0$) discs Eq. (\ref{eq:consm}) can be easily integrated giving
\begin{equation}
\label{eq:accrrate}
\dot M:=2\pi R \Sigma v_{\rm r},
\end{equation}
where the integration constant $\dot M$ (mass/time) is the \textsl{accretion rate}.
Also the angular momentum equation (\ref{eq:consj}) can be integrated to give
\begin{equation}
\label{eq:amint1}
-2\pi R\Sigma v_r\ell + 2\pi R^3\Sigma\nu\frac{d\Omega}{dr}=const.
\end{equation}
Or, using Eq. (\ref{eq:accrrate}),
\begin{equation}
\label{eq:amint2}
-\dot M \ell + {\mathfrak T}={const.},
\end{equation}
where the torque
\begin{equation}
\label{eq:torque}
{\mathfrak T}:= 2\pi R^3\Sigma\nu d\Omega/dr;
\end{equation}
(for a Keplerian disc ${\mathfrak T}= 3\pi R^2\Sigma\nu\Omega_K$).
Assuming that the torque vanishes at the inner disc radius, one gets
$const. =- \dot M \ell_{\rm in}$, where $\ell_{\rm in}$ is the specific angular momentum at the
disc inner edge.
Therefore
\begin{equation}
\label{eq:amintt}
\dot M (\ell -\ell_{\rm in}) ={\mathfrak T}
\end{equation}
which is a simple expression of angular momentum conservation.
For Keplerian discs one obtains an important relation between viscosity and accretion rate
\begin{equation}
\label{eq:amintK}
\nu \Sigma =\frac{\dot M }{3\pi}\left[1 -\left(\frac{R_{\rm in}}{R}\right)^{1/2} \right].
\end{equation}
From Eqs. (\ref{eq:amintK}), (\ref{eq:qplus}), (\ref{eq:qminus}), and the thermal equilibrium equation (\ref{eq:thermal_equi})
it follows that
\begin{equation}
\label{eq:teff}
\sigma T_{\rm eff}^4 =\frac{3}{8\pi}\frac{GM\dot M}{R^3}\left[1 -\left(\frac{R_{\rm in}}{R}\right)^{1/2} \right].
\end{equation}
This assumes only a Keplerian disc in thermal ($Q^+=Q^-$) and viscous ($\dot M=const.$) equilibrium. Since the disc is in thermal equilibrium, the emitted radiation flux cannot contain information about the heating mechanism which explains why the viscosity coefficient is absent from Eq. (\ref{eq:teff}). Steady discs do not provide information about the viscosity operating in discs or the viscosity parameter $\alpha$. To get this information one must consider (and observe) time-dependent states of accretion discs. It is often incorrectly claimed that the $R^{-3/4}$ temperature profile is a signature of the Shakura-Sunyaev solution, but obviously this profile is much more general and independent of the viscosity prescription assumed to obtain this celebrated solution (see below).
Eq. (\ref{eq:teff}) determines indeed a universal radial temperature profile for \textsl{stationary Keplerian accretion discs}
\begin{equation}
\label{eq:temprofile}
T_{\rm eff}\sim R^{-3/4}.
\end{equation}
For an optically thick disc the temperature relations $T\sim T_{\rm eff}$ and $T\sim R^{-3/4}$ should be observed if
stationary, optically thick Keplerian discs do exist in the Universe. And vice versa, if they are observed, this proves that
such discs exist not only on paper. The $R^{-3/4}$ disc--temperature profiles have been clearly observed in bright, eclipsing cataclysmic variables,
it seems, however, that in AGN discs these profiles are a bit steeper, as will be discussed below.
The temperature profile of stationary keplerian accretion disc is given therefore by
\begin{equation}
\label{eq:Teff_value}
T_{\rm eff}=T_{\rm in} \left(\frac{r}{3}\right)^{-3/4},
\end{equation}
where
\begin{equation}
T_{\rm in}=\left(\frac{3GM\dot M}{8\pi \sigma (3R_S)^3}\right)^{1/4} \approx 3.0 \times 10^5\, m_{8}^{-1/2} \dot m^{1/4}\rm K,
\label{eq:Teff_in}
\end{equation}
where we assumed that $R_{\rm in}=3R_S$, i.e. the ISCO for a non-rotating black hole.
Therefore the maximum effective disc temperature $T_{\rm eff}^{\rm {max}}=0.488\, T_{\rm in}$ is located at $r =49/12 \approx 4.1$ and
in AGN corresponds typically to UV radiation.
\subsubsection{The ``no-torque condition"}
There has been a lot of discussion about the inner boundary condition
in an accretion disc around a black hole. The usual reasoning is that for a thin disc the
inner boundary is at ISCO and since circular orbits end there,
the boundary condition should be simply that the ``viscous" torque
vanishes there (there is no orbit below the ISCO to interact with).
In fact, for geometrically thin accretion discs, the no-torque condition is
a simple consequence of conservation of angular momentum \citep{Paczynski0400}.
Numerical simulations of thin accretion discs that do not satisfy this condition
do not conserve angular momentum.
In general, if one does not assume that the torque vanishes at the inner disc edge, Eq. (\ref{eq:amint2})
will take the form
\begin{equation}
\label{eq:amint3}
\dot M (\ell -\ell_{H}) ={\mathfrak T},
\end{equation}
where $l_H$ is the specific angular momentum of the accretion flow at the black-hole surface: torques
must vanish on the horizon, since the horizon is an hypersurface causally detached from the rest of the Universe.
From Eqs. (\ref{eq:amint3}), (\ref{eq:torque}) and the viscosity prescription $\nu \approx \alpha H^2 \Omega$,
one can obtain
\begin{equation}
v_r \approx \alpha ~ H^2 ~ \frac{{\ell}}{{\ell} - {\ell}_H} ~ \frac{d \Omega}{dr}
\approx \alpha ~ H^2 ~ \frac{{\ell}}{{\ell} - {\ell}_H} ~ \frac{ \Omega}{ r }
\approx \alpha ~ v_{\varphi} \left( \frenchspacing{H}{R} \right) ^2 ~ \frac{{\ell}}{{\ell} - {\ell}_H},
\label{vr2}
\end{equation}
where $v_{\varphi}= R \Omega$.
Equation (\ref{vr2}) does not assume that the
radial velocity is small, i.e. this equation holds within the disk as well as
within the stream below the ISCO.
\begin{figure}[h!]
\centering
\includegraphics[width=14cm,height=7cm]{002-figure-angular-momentum.pdf}
\caption{Angular momentum profiles for slim disk solutions with
$\alpha=0.01$ (left panel) and $\alpha=0.1$ (right panel). In both panels,
three curves are presented for sub-Eddingtonian, Eddingtonian, and
super-Eddingtonian accretion rates. Since $H/R \sim \dot M(R)/\dot M_{\rm Edd}$ \citep{FKR}, only the sub-Eddingtonian curve
corresponds to a thin disc, for which clearly the torque vanishes at the inner edge: constant angular momentum below the ISCO. The thin dotted line presents the
Keplerian angular momentum profile. The angular momentum and the radius
are in units of $M$ (i.e. $R_g$, with $c=G=1$).[{\sl From \citep{Abramowicz1010}}]}.
\label{fig:angmom}
\end{figure}
Far out in the disk, where $ {\ell} \gg {\ell}_H$, one
obtains the standard formula (see Eq. \ref{eq:vvisc})
\begin{equation}
v_r \approx \alpha ~ v_{\varphi} \left( \frac{ H}{R } \right) ^2 ,
\hskip 1.0cm R \gg R_{in} .
\end{equation}
The flow crosses the black hole surface at the speed of light and since it is subsonic in the
disc it must somewhere become transonic, i.e. go through a sonic point, which has been shown to be close to
the disc's inner edge.
At the sonic point $ v_r = c_s \approx (H/R)v_{\varphi} $, and
equation (\ref{vr2}) becomes:
\begin{equation}
\frac{v_r }{ c_s} = 1 \approx \alpha ~ \frac{H_{\rm in}}{ R_{\rm in} } ~
\frac{{\ell}_{\rm in}}{{\ell}_{\rm in} - {\ell}_H},
\hskip 1.0cm R = R_{\rm in}.
\label{bep1}
\end{equation}
If the disc is thin, i.e. $ H_{\rm in} / R_{\rm in} \ll 1 $, and the
viscosity is small, i.e. $ \alpha \ll 1 $, then Eq. (\ref{bep1}) implies that
$({\ell}_{\rm in} - {\ell}_H)/{{\ell}_{\rm in}} \ll 1 $,
i.e. the specific angular momentum at the sonic point is almost
equal to the asymptotic angular momentum at the horizon.
In a steady state disc the torque ${\mathfrak T}$ has to satisfy the equation
of angular momentum conservation (\ref{eq:amint2}), which can be written as
\begin{equation}
{\mathfrak T}= \dot M \left( {\ell} - {\ell}_H \right) ,
\hskip 1.0cm
{\mathfrak T}_{in} = \dot M
\left({\ell}_{in} - {\ell}_H \right) .
\label{torquebp}
\end{equation}
Thus it is clear that for a thin, low viscosity disk the `no torque inner
boundary condition' (${\mathfrak T}_{in}\approx 0$) is an excellent approximation
\textit{following from angular momentum conservation}.
However, if the
disk and the stream are thick, i.e. $ H/r \lesssim 1 $, and the
viscosity is high, i.e. $ \alpha \sim 1 $, then the angular momentum can
vary also in the stream in accordance with the simple reasoning presented
above. In such a case the no--stress condition at the disc inner edge might be
unsatisfied.
\subsubsection{Total luminosity and spectrum}
\label{susect:totalL}
The total luminosity of a stationary, geometrically thin accretion disc, i.e. the sum of luminosities of its two surfaces, is
\begin{equation}
\label{eq:lum1}
2\int_{R_{\rm in}}^{R_{\rm out}}\sigma T_{\rm eff}^4\,2\pi RdR= \frac{3GM\dot M}{2}\int_{R_{\rm in}}^{R_{\rm out}}\left[1 -\left(\frac{R_{\rm in}}{R}\right)^{1/2} \right]\frac{dR}{R^2}.
\end{equation}
For $R_{\rm out} \rightarrow \infty$ this become
\begin{equation}
\label{eq:lum2}
L_{\rm disc}=\frac{1}{2}\frac{GM\dot M}{R_{\rm in}}=\frac{1}{2}L_{\rm acc}.
\end{equation}
In the disc the radiating particles move on Keplerian orbits hence they retain half of the potential energy.
When the accreting body is a black hole this leftover energy will be lost. In this case, however, the non-relativistic formula of Eq. (\ref{eq:lum2})
does not apply -- see Eq. (\ref{eq:bindingE}).
For an optically thick disc one can assume that each of its rings radiates as black-body emitter:
\begin{equation}
I_\nu=B_\nu\left[T_{\rm eff}(R)\right]=\frac{2h\nu^3}{c^2(e^{h\nu/kT(R)}-1)}\,\left(\mathrm{erg\,s^{-1}\,cm^{-2}\,Hz^{-1}\,sr^{-1}}\right).
\label{eq:bbdisc}
\end{equation}
This is a rather crude but useful approximation. One should keep in mind, however, that it does not represent real observed disc spectra. Stars are very
optically thick but their spectra do not look like black bodies, because the observed light had to
cross the stellar atmosphere on its way to the detector.
With such an assumption the flux at a given frequency, detected by an observer at a distance $D$ is equal to
\begin{equation}
F_\nu=\frac{2\pi \cos i}{D^2}\int^{R_{\rm out}}_{3R_S}I_\nu R\,dR = \frac{4\pi h \nu^3\,\cos i}{c^2D^2}\int^{R_{\rm out}}_{3R_S}\frac{ R\,dR}{e^{h\nu/kT(R)}-1},
\label{eq:flux_disc}
\end{equation}
where $i$ is the disc inclination.
For frequencies
\begin{equation}
\frac{kT({R_{\rm out}})}{h} \ll \nu \ll \frac{kT_{\rm in}}{h},
\end{equation}
one obtains from Eq. (\ref{eq:flux_disc}) $F_\nu \propto \nu^{1/3}$, known sometimes as the ``disc" spectrum, but such a $\nu^{1/3}$ feature is prominent
only if $T({R_{\rm out}}) \ll T_{\rm in}$ , i.e., if the disc is sufficiently large.
For an AGN disc with $R_{\rm out}\sim 100 R_{\rm in}$ and $\dot m=0.01$ the
optical an UV spectrum will indeed have a 1/3 slope.
\subsection{Size of the disc}
Assuming that thin discs in AGNs emit locally as black-bodies allows one to determine the disc's size and comparing it with observation.
Defining the disc size as corresponding to the radius at which the disc temperature matches the wavelength $R_\lambda$:
\begin{equation}
\label{eq:rlambda}
kT(R_\lambda) = hc/\lambda,
\end{equation}
and using Eq. (\ref{eq:bbdisc}) in the form of
\begin{equation}
I_{\nu}=\frac{2 h_{p} c}{\lambda^{3}}\left[\exp \left(\frac{R}{R_{\lambda_{\text {rest}}}}\right)^{3 / 4}-1\right]^{-1},
\end{equation}
one obtains from Eqs. (\ref{eq:Teff_value}) and (\ref{eq:Teff_in})
\begin{equation}
R_{\lambda}=\left[\frac{45 G \lambda^{4} M \dot{M}}{16 \pi^{6} h c^{2}}\right]^{1 / 3}=2.1 \times 10^{15}\left(\frac{\lambda}{\mu \mathrm{m}}\right)^{4 / 3}m_8^{2 / 3}\left(\frac{L}{\eta L_{E}}\right)^{1 / 3} \mathrm{cm}.
\end{equation}
$\lambda$ is the wavelength in the rest-frame of the AGN. Thus the prediction of the thin-disc model is that the size of the disc $R \sim M^{2/3}$.
Eq. (\ref{eq:rlambda}) assumes that emission at wavelength $\lambda$ originates solely at
radius $R_\lambda$, while in reality this emission comes also from other radii.
Therefore a more appropriated size for comparison with observations would be a flux-weighted mean radius $\langle R_\lambda \rangle = X R_\lambda$.
$X \approx 2 - 3$, but if disc variability is taken into account this factor can be even $\sim 5$. $R_\lambda$ itself depends on the black-hole mass and the accretion rate
(through $\eta$) so that uncertainties in the observed values of this quantities impact the comparison of the model-size with observations. AGN disc sizes are measured
through reverberation mapping and microlensing. Observations are in agreement with the predicted $\sim M^{2/3}$ slope. There is no agreement about the actual size (see, e.g. \citealt{Jha0422} for a recent result). While
microlensing observations and some reverberation estimates claim values 2 -- 3 larger than those predicted by the model, according to other reverberation observations most
measured sizes of AGN discs agree with the thin-disc value. The reasons for the discrepancies (if real) is not clear, but it is possible that the real temperature profile is
shallower than the thin-disc $R^{-3/4}$ which should be the case when the disc is irradiated.
\subsection{Spectral lines from Keplerian accretion discs}
\label{sec:lines}
\begin{figure}
\center
\includegraphics[width=0.35\textwidth]{Disc_line.pdf}
\caption{The loci of constant radial velocity form a dipole field pattern on the surface of a Keplerian disc
(upper figure).
The velocity profile of emission lines from the
disc (lower figure). Emission in the shaded velocity bins arises from the corresponding regions of the disc. This example corresponds to a stellar-mass accretor. [\textsl{Adapted from \cite{HM86}}]}
\label{fig:lineKepl}
\end{figure}
Observations of spectral lines from discs around black holes play an important role as tools allowing to investigate
the disc structure itself but their shapes encode the properties of the accretor, in this case its mass and spin. In practice
this means taking into account the relativistic effects (both special and general) on the propagation of light-rays which
might obfuscate the basic properties of line shapes emitted by an accretion disc. We will therefore begin with the Newtonian case.
Emission lines from a Keplerian accretion discs have a characteristic double-peak structure.
The reason is easy to understand. For a distant observer the frequency of the lines emitted by the disc are shifted
by the Doppler effect, corresponding to velocity
\begin{equation}
\label{eq:vd}
v_{\mathrm{D}}=v_{\mathrm{K}} \sin i \sin \theta,
\end{equation}
where $v_K= R\Omega_K$, $i$ is the disc inclination ($i=90^{\circ}$ for a disc seen edge-on) and $\theta$ is
the azimuth angle relative to the line-of-sight. The total disc emission flux is obtained by integrating the
Doppler-shifted intensities over the whole disc's surface:
\begin{equation}
\label{eq:fnu}
F_{\nu}=\frac{1}{D^{2}}\int {I_{\nu}} R\,{dR}d\theta.
\end{equation}
Here the line specific intensity is $I_\nu= j(R)\phi_\nu$, with
$\int \phi_\nu\,d\nu=1$, $\phi_\nu$ being the emission profile of the line.
$j(R)$ is assumed to be isotropic. Each local line profile $I_\nu$ is shifted from the rest frequency
$\nu^{\prime}$ to the local Doppler frequency
\begin{equation}
\label{eq:vdopp}
\nu_{\mathrm{D}}=\nu^{\prime}\left(1-\frac{v_{\mathrm{D}}}{c}\right).
\end{equation}
As seen on Figure \ref{fig:lineKepl} the loci of constant radial velocity form a ``dipole" pattern on the disc's surface,
The disc's emission-line profile, shown in the lower part of this figure is divided into velocity bins corresponding to the disc regions between consecutive dipole field lines.
The emission in each velocity bin arises from a different region of the disc surface.
Assuming circular orbits, matter crossing the $\theta =0, \pi$ line moves perpendicularly to the line of sight, so
$v_D=0$ and its emission correspond to the line center.
The crescents that are complete near the accretor become truncated at the outer disc radius $R_D$ which produces
cusps in the line profile at $v_D =\pm v_K\left(R_D\right)\,\sin i$. Within the assumptions
of a Keplerian potential, and an almost constant disk surface density, \textsl{double-peaked line profiles thus correspond to the
presence of an outer boundary to the emission-line region}.
The wings of the lines are produced obviously by the fastest moving disc matter, i.e. close to the accretor.
However, in a Keplerian disc, the area of the corresponding crescent $R\Delta R \sim v^{-5}\Delta v$ which explains its rapid decline with increasing velocity.
\subsubsection{General-relativistic spectral line description}
Equations (\ref{eq:vd}), (\ref{eq:fnu}) and (\ref{eq:vdopp}) cannot describe correctly the spectral line close to a black hole
because in their derivation, higher-order terms in $v/c$ and gravitational light-bending have been neglected. The description of
such line profiles requires the use of the full formalism of General Relativity and usually requires numerical calculations.
It what follows we will show how the problem can be simplified even in the case of a Kerr black hole (based on \citealt{HMP94}).
Exceptionally in this subsection we will use units in which c=G=1 in the description of the motion of the emitter.
The specific intensity [erg cm$^{-2}$ s$^{-1}$ Hz$^{-1}$ Sr$^{-1}$], is now written as
\begin{equation}
I_{\nu}=I_{\nu^{\prime}}(1+z)^{-3},
\end{equation}
where $\nu^{\prime}$ is the rest-frame frequency, $\nu$ the frequency
measured at infinity and $1+z = \nu^{\prime}/\nu$ is the redshift factor.
One of the redshift factors in the denominator accounts for the time dilation, while the other two correspond to the angular diameter correction
to the solid angle. Therefore the observed spectral flux is
\begin{equation}
\label{eq:fnugr}
F_{\nu}=\underset{{\text {image }}} {\int} I_{\nu} d \Omega= \underset{{\text {image }}} {\int} I_{\nu^{\prime}}(1+z)^{-3} d \Omega,
\end{equation}
where $d\Omega$ is the solid angle and the integration is over the observed accretion disc image.
Therefore to calculate the observed line profiles one has to calculate the redshift factor $1+z$ which is given by the motion of the emitter in the
gravitational field of the attracting body. In the case of a Keplerian motion around a Kerr black hole, with orbital frequency
\begin{equation}
\label{eq:omegakgr}
\Omega_K=\frac{M^{1 / 2}}{R^{3 / 2}+a M^{3 / 2}},
\end{equation}
the result is
\begin{align}
\label{eq:redshiftgeneral}
& 1+z = \\
& {\frac{1-L_z \Omega_K}{\sqrt{\left(1-2{M}/{R}-R^{2} \Omega_K^{2}\right)+4 a\Omega_K{M^2}/{R} -\left(1+2 {M}/{R}\right) (aM)^{2} \Omega_K^{2}}}},
\end{align}
where $L_z$ is the component of the angular momentum of the emitter with respect to the $z$ (black-hole rotation) axis.
\begin{figure}
\centering
\centering
\includegraphics[width=0.7\textwidth]{LineKerr1.pdf}
\caption{Exact (solid curve) and approximate (dashed line) line profiles emitted by a ring of matter at $r=2.5$ ($R=5GM/c^2$). The disc inclination is $30^{\circ}$. [\textsl{Adapted from \cite{HMP94}}]}
\label{fig:Line1}
\end{figure}
In general, Eq. (\ref{eq:fnugr}) with (\ref{eq:redshiftgeneral}) has to be solved numerically. However, for distances $R > 10$ ($r > 5$)
one can neglect the effects of black-hole rotation (they diminish as $R^{-3}$; see Eq. \ref{eq:LTomega}) and Eqs. (\ref{eq:omegakgr}) and (\ref{eq:redshiftgeneral})
simplify to its Schwarzschild form
\begin{equation}
\Omega_K=\sqrt{\frac{M}{R^{3}}} \quad 1+z=\frac{1-\Phi \Omega_K}{\sqrt{1-3 M / R}}.
\end{equation}
Then, for low inclinations ($i \sim 0$) and large radii, one can easily integrate Eq. (\ref{eq:fnugr}), with $d\Omega \approx \cos i \,rdr\, d\varphi/4\pi D^2$.
Assuming a Dirac-delta intrinsic line profile $I_{\nu^{\prime}}= j(r) \delta(\nu^{\prime} - \nu_0)$ and defining
\begin{equation}
\alpha:=\sqrt{1-3 / \tilde{r}}, \quad \beta:=\sin i / \sqrt{\tilde{r}}, \quad \tilde{r} := R / M,
\end{equation}
one obtains
\begin{equation}
F_{\nu}=\frac{\cos i}{2 \pi D^{2} \nu_{0}} \int_{R_{\mathrm{in}}}^{R_{\mathrm{out}}} \frac{\alpha\left(\nu / \nu_{0}\right)^{3} \phi(R) R\,dR}{\left(\beta^{2}\left(\nu / \nu_{0}\right)^{2}-\left(\alpha-\nu / \nu_{0}\right)^{2}\right)^{1 / 2}},
\end{equation}
which is an asymptotic formula for the spectrum observed at large distances from the black hole.
Close to the black hole, this formula is inappropriate even for low inclinations, since it neglects both
the effects of black-hole rotation and terms of the order of $\left( v/c\right)^3$.
However, it happens to provide
an excellent approximation to the observed line profiles if $\alpha$ and $\beta$ are redefined as
\begin{equation}
\begin{aligned}[b]
\alpha &=\alpha_{0}\\
\beta &=\beta_{0}\left[1+\frac{1.25}{\tilde{r}}+\frac{2.5-3.4 \cos i}{\tilde{r}^{2}}+\frac{3 \sin i}{\tilde{r}^{3}}\right],
\end{aligned}
\end{equation}
for a Schwarzschild metric ($a=0$), and as
\begin{equation}
\begin{aligned}[b]
{\alpha=\alpha_{0}\left[1+\frac{5(\cos i-1)}{\tilde{r}^{5 / 2}}+\frac{6.5-8.5 \cos i}{\tilde{r}^{7 / 2}}\right]} \\
{\beta=\beta_{0}\left[1+\frac{1.25}{\tilde{r}}+\frac{3-3 \cos i}{\tilde{r}^{2}}-\frac{3.5 \sin i}{\tilde{r}^{3}}\right]},
\end{aligned}
\end{equation}
for a Kerr black hole with $a=0.99$. $\alpha_0=(1 - 3v^2)$, $\beta_0=v \sin i$ and $v=R\Omega_K$, with $\Omega_K$
given by Eq. (\ref{eq:omegakgr}).
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{KerrLine2.pdf}
\caption{Same as in Fig. \ref{fig:Line1}, but for $r=12.5$ ($R=25GM/c^2$) and inclination $80^{\circ}$. For such high disc inclinations
the effects of light-bending (producing the two additional inner peaks) cannot be neglected even for relatively large radii. [\textsl{Adapted from \cite{HMP94}}]}
\label{fig:Line2}
\end{figure}
$\alpha$ and $\beta$ are related to the minimum and maximum frequencies of the observed line through
\begin{equation}
\begin{aligned}[b]
\alpha &=\frac{2 \nu_{\min } \nu_{\max }}{\left(\nu_{\min }+\nu_{\max }\right) / \nu_{0}} \\
\beta &=\frac{\nu_{\max }-\nu_{\min }}{\left(\nu_{\min }+\nu_{\max }\right) / \nu_{0}}.
\end{aligned}
\end{equation}
The fit to $\alpha$ and $\beta$ is accurate to better than $1\%$ for $r > 1.5$ in the Schwarzschild metric and
$r > 2.5$ for a Kerr black hole, if $i < 80^{\circ}$.
Finally, the flux as given by Eq. (\ref{eq:fnugr}) has to be renormalised through factors that depend on the radius and inclination.
These factors are
\begin{equation}
\begin{aligned}
C=1 &+\frac{1.25(\cos i-\sin i)+1.6 \tan i}{\tilde{r}^{1-0.19 \sin i}} \\
&+\frac{10-10 \cos i-3.5 \tan i}{\tilde{r}^{2}},
\end{aligned}
\end{equation}
for $a=0$
\begin{equation}
\begin{aligned}
C=1 &+\frac{1.5-2 \sin i+1.2 \tan i}{\tilde{r}^{1-0.25 \sin i}}+\frac{7-7.6 \cos i+2 \sin i}{\tilde{r}^{2}} \\
&-\frac{3.5 \sin i}{\tilde{r}^{3}},
\end{aligned}
\end{equation}
for $a=0.99$. Multiplying by these factors gives the total line fluxes correct within less that $1\%$ for $i < 80^{\circ}$
when $r > 1.5$ in the Schwarzschild metric and and $r > 2.5$ for a Kerr black hole. $\tan i$ in these formulae represents the light
bending effect.
Figures \ref{fig:Line1} and \ref{fig:Line2} show two examples of ``exact" (numerical) and approximated fitting of
the profile of a line emitted by a ring around a $a=0.99$ black hole. For a low inclination ($i = 30^{\circ}$) even
close to the black hole ($r=2.5$) the numerical and approximate profiles are practically indistinguishable. However,
even at relatively large distance from the hole ($r=12.5$) but high inclination ($i = 80^{\circ}$) the imperfections
of the approximate version of the flux are clearly visible.
The double-peaked line shape is, as in the newtonian case, due to the finite extent of the emission region, in this case a ring.
The often used assumption $j(R) \sim R^{-b}$ will not produce double-peak lines if the outer emission region is not spatially
limited.
\subsection{Radiative structure}
\label{subsect:rad}
Here we will show an example of the solution for the vertical thin--disc structure
which exhibits properties impossible to identify when the structure is vertically
averaged. We will also consider here irradiated discs -- such as AGN discs.
We write the energy conservation as :
\begin{equation}
\frac{dF}{ dz} = q^+(R,z),
\label{energy1}
\end{equation}
where $F$ is the vertical (in the $z$ direction) radiative flux
and $q^+(R,z)$ is the viscous heating rate per unit volume.
Eq. (\ref{energy1})
states that an accretion disc is not in radiative equilibrium ($dF/dz\neq 0$), contrary to a
stellar atmosphere. For this equation to be solved, the function
$q^+(R,z)$ must be known. As explained and discussed in Sect. \ref{subsec:vstruct.1} the viscous dissipation is often written as
\begin{equation}
q^+(R,z)= \frac{3}{2} \alpha \Omega_{\rm K} P(z).
\label{voldiss}
\end{equation}
Viscous heating of this form has important implications for the structure of
optically thin layers of accretion discs and may lead to the creation of
coronae and winds. In reality it is an \textsl{ad hoc} formula inspired by
Eq. (\ref{eq:qplusalphaP}). We don't know yet how to describe the viscous
heating stratification in a real \textsl{geometrically thin} accretion disc and Eq. (\ref{voldiss}) just
\textsl{assumes} that it is proportional to pressure. It is simple and convenient
but it is not necessarily true.
When integrated over $z$, the rhs of Eq. (\ref{energy1}) using Eq. (\ref{voldiss})
is equal to viscous dissipation per unit surface:
\begin{equation}
F^+=\frac{3} {2} \alpha \Omega_{\rm K} \int_0^{+\infty} P dz ,
\label{fvis}
\end{equation}
where $F^+=(1/2)Q^+$ because of the integration from $0$ to $+\infty$ while
$Q^+$ contains $\Sigma$ which is integrated from $-\infty$ to $+\infty$ (Eq. \ref{eq:Sigma}).
One can rewrite Eq. (\ref{energy1}) as
\begin{equation}
\frac{dF}{ d\tau} = - f(\tau )\frac{F_{\rm vis} }{ \tau_{\rm tot}},
\label{energy2}
\end{equation}
where we introduced a new variable, the optical depth $d\tau=-\kappa_{\rm R}
\rho dz$, $\kappa_{\rm R}$ being the Rosseland mean opacity and $\tau_{\rm tot}
= \int_0^{+\infty} \kappa_{\rm R} \rho dz$ is the total optical depth.
$f(\tau)$ is of order unity.
At the disc midplane, by symmetry, the flux must vanish: $F(\tau_{\rm tot})=0$,
whereas at the surface, ($\tau=0$)
\begin{equation}
F(0) \equiv \sigma T^4_{\rm eff}= F^+.
\label{fsurface}
\end{equation}
Equation (\ref{fsurface}) states that the total flux at the surface is equal
to the energy dissipated by viscosity (per unit time and unit surface). The
solution of Eq. (\ref{energy2}) is thus
\begin{equation}
F(\tau) = F^+ \left(1 - \frac{\int_0^\tau f(\tau) d\tau}{\tau_{\rm tot}}\right),
\label{flux0}
\end{equation}
where $\int_0^{\tau_{\rm tot}} f(\tau) d\tau = \tau_{\rm tot}$. Since
$f \approx 1$, we have
\begin{equation}
F(\tau) \approx F^+ \left(1 - \frac{\tau}{\tau_{\rm tot}}\right).
\label{fluxd}
\end{equation}
To obtain the temperature stratification one has to solve the transfer
equation. Here we use the diffusion approximation
\begin{equation}
F(\tau) = \frac{4}{3} \frac{\sigma dT^4}{d\tau} ,
\label{diff}
\end{equation}
appropriate for the optically thick discs we are dealing with. The
integration of Eq. (\ref{diff}) is straightforward and gives :
\begin{equation}
T^4(\tau) - T^4(0) = \frac{3}{4} \tau \left(1 - \frac{\tau}{2\tau_{\rm tot}}
\right) T^4_{\rm eff}.
\label{t1}
\end{equation}
The upper (surface) boundary condition is:
\begin{equation}
T^4(0) = \frac{1}{2} T^4_{\rm eff} + T^4_{\rm irr},
\label{bcond2}
\end{equation}
where $T^4_{\rm irr}$ is the irradiation temperature, which depends on $r$,
the albedo, the height at which the energy is deposited and on the shape of
the disc. In Eq. (\ref{bcond2}) $T(0)$ corresponds to
the {\sl emergent} flux and, as mentioned above, $T_{\rm eff}$ corresponds to
the {\sl total} flux ($\sigma T^4_{\rm eff}=Q^+$) which explains the factor 1/2
in Eq (\ref{bcond2}). The temperature stratification
is thus :
\begin{equation}
T^4(\tau) = \frac{3}{4}T^4_{\rm eff}
\left[\tau \left(1 - \frac{\tau}{2\tau_{\rm tot}}\right)
+ \frac{2}{3}\right] + T^4_{\rm irr}.
\label{t2}
\end{equation}
For $\tau_{\rm tot} \gg 1$ the first
term on the rhs has the form familiar from the stellar atmosphere models in the
Eddington approximation.
\textsl{In this case at $\tau=2/3$ one has $T(2/3) = T_{\rm eff}$.}
Also for $\tau_{\rm tot} \gg 1$, the temperature
at the disc midplane is
\begin{equation}
T^4_{\rm c} \equiv T^4(\tau_{\rm tot}) =
\frac{3}{8} \tau_{\rm tot} T_{\rm eff}^4 + T^4_{\rm irr}.
\label{diff2}
\end{equation}
It is clear therefore that for the disc inner structure to be dominated by
irradiation and the disc to be isothermal one must have
\begin{equation}
\frac{F_{\rm irr}}{\tau_{\rm tot}} \equiv \dfrac{}{}{\sigma T^4_{\rm irr}}{\tau_{\rm tot}} \gg F^+
\label{c1}
\end{equation}
and not just $F_{\rm irr} \gg F^+$ as is sometimes assumed. The difference between the two criteria is important in X-ray binary and AGN discs since, for parameters of interest, $\tau_{\rm tot} \gtrsim 10^2 - 10^3$ (see Sect. \ref{subsec:scurve}).
\subsection{Shakura-Sunyaev thin-disc solution}
\label{subsec:SS}
In their seminal and famous 1973 paper, Shakura \& Sunyaev found power-law stationary solutions of the simplified version of the thin--disc equations presented in Sects. \ref{subsec:vstruct.1}, \ref{subsec:radial1} and \ref{subsec:stationary}. The 8 equations for the 8 unknowns $T_c$, $\rho$, $P$, $\Sigma$, $H$, $\nu$, $\tau$ and $c_s$ can be written as
\begin{equation}
\label{}
\Sigma=2H\rho \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \rm (\i) \nonumber
\end{equation}
\begin{equation}
\label{}
H=\frac{c_s R^{3/2}}{(GM)^{1/2}} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \rm (\i\i) \nonumber
\end{equation}
\begin{equation}
\label{}
c_s=\sqrt{\frac{P}{\rho}}\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \rm (\i\i\i) \nonumber
\end{equation}
\begin{equation}
\label{}
P=\frac{{\cal R}\rho T}{\mu} + \frac{4\sigma}{3c}T^4 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \rm(\i v) \nonumber
\end{equation}
\begin{equation}
\label{}
\tau(\rho, \Sigma, T_c)=\kappa_R(\rho,T_c)\Sigma \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \rm(v) \nonumber
\end{equation}
\begin{equation}
\label{}
\nu(\rho,\Sigma,T_c,\alpha)=\frac{2}{3}\alpha c_s H \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \rm(v\i) \nonumber
\end{equation}
\begin{equation}
\label{}
\nu \Sigma =\frac{\dot M }{3\pi}\left[1 -\left(\frac{R_{\mathrm in}}{R}\right)^{1/2} \right] \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \rm(v\i\i) \nonumber
\end{equation}
\begin{equation}
\label{}
\frac{8}{3}\frac{\sigma T_c^4}{\tau}=\frac{3}{8\pi}\frac{GM\dot M}{R^3}\left[1 -\left(\frac{R_{\mathrm in}}{R}\right)^{1/2} \right] . \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \rm(v\i\i\i) \nonumber
\end{equation}
Equations (\i) and (\i\i) correspond to vertical structure equations (\ref{eq:mass_cons}) and (\ref{eq:mec_approx}), Eq. (v\i\i) is the radial Eq. (\ref{eq:amintK}), while Eq. (v\i\i\i) connects vertical to radial equations. Eq. (\i\i\i) defines the sound speed, Eq. (\i v) is the equation of state and (v\i) contains the information about opacities. The viscosity $\alpha$ parametrization introduced in \cite{SS73} provides the closure of the 8 disc equations. Therefore they can be solved for a given set of $\alpha$, $M$, $R$ and $\dot M$.
Power-law solutions of these equations exist in physical regimes where the opacity can be represented in the Kramers form $\kappa=\kappa_0\rho^{n}T^{m}$ and one of the two pressures, gas or radiation, dominates over the other. There are three regimes to be considered:
\begin{center}
$\left. a. \right)$ $P_r \gg P_g$ and $\kappa_{\rm es}\gg \kappa_{\rm ff}$\\
$\left. b. \right)$ $P_g \gg P_r$ and $\kappa_{\rm es}\gg \kappa_{\rm ff}$ \\
$\left. c. \right)$ $P_g \gg P_r$ and $\kappa_{\rm ff}\gg \kappa_{\rm es}$.\\
\end{center}
Regimes $\left. a. \right)$ and $\left. b. \right)$ in which opacity is dominated by electron scattering will be discussed in Sect. \ref{sect:advection}.
Here we will present the solutions of regime $\left. c. \right)$, i.e. we will assume that
\begin{equation}
\label{eq:kramers}
P_r=0 \ \ \ \ {\mathrm{and}} \ \ \ \ \kappa_R=\kappa_{\rm ff}=5\times 10^{24}\rho T_c^{-7/2}\,\rm cm^2g^{-1}.
\end{equation}
The solution for the surface density $\Sigma$, central temperature $T_c$ and the disc relative height (aspect ratio) are respectively
\begin{equation}
\label{eq:SigmaSS}
\Sigma=1.6\times 10^7\,\alpha^{-4/5}m_8^{+1/5}r^{-3/4}{\dot m}^{7/10}f^{7/10}\, \rm g\,cm^{-2},
\end{equation}
\begin{equation}
\label{eq:TcSS}
T_c=8.4\times 10^6\,\alpha^{-1/5}m_8^{-1/5}r^{-3/4}{\dot m}^{3/10}f^{3/10}\,\rm K,
\end{equation}
\begin{equation}
\label{eq:HoverRSS}
\frac{H}{R}= 1.6\,\times 10^{-3}\alpha^{-1/10}m_8^{-1/10}r^{1/8}{\dot m}^{3/20}f^{3/20},
\end{equation}
where $m_8=M/{\rm 10^8 M_{\odot}}$, $r=R /R_S$, and
$f~=~1~ -~ ({R_{\rm in}}/{R})^{1/2}$.
Since
\begin{equation}
\frac{P_g}{P_r}= 1.6 \times 10^{-2} \alpha^{-1/10}m_8^{1/2}r^{3/8}\dot m^{-7/20}f^{-7/20},
\label{eq:pgpr}
\end{equation}
the regime c.) solution exists only for radii
\begin{equation}
r > 6\times 10^4 \alpha^{4/15}m_8^{4/3}\dot m^{14/15}f^{14/15}.
\label{eq: rPr}
\end{equation}
For this solution the viscous time is
\begin{equation}
t_{\rm vis}\approx 26\, \alpha^{-6/5} m_8^{6/5}\dot m^{-3/8}r^{5/4} \rm yr,
\label{eq:viscsol}
\end{equation}
so for the viscous time to be less than the age of the Universe (13.8 Gyr), for an Eddington accretion rate onto a $10^8\ensuremath{\rm \,M_\odot}$ black-hole, the outer disc radius must be
$r_{\rm out}< 10^7 \alpha^{24/25}$, i.e. less than $\approx 100$\,pc, or less than $1\,\rm pc$ if $\alpha$ in AGNs is $\sim 10^{-2}$.
In this solution it was assumed that the opacity is given by the formula
\begin{equation}
\kappa_{\rm R}=5\times 10^{-4}\dot m^{-1/2} r^{3/4} \rm cm^2g^{-1},
\label{eq:kappass}
\end{equation}
so that $\kappa_{\rm R} > \kappa_{\rm es}=0.4\,\rm cm^2g^{-1}$, for
\begin{equation}
r_{\rm Res} > 7400 \dot m^{2/3}.
\label{eq:rkappases}
\end{equation}
Notice that $r_{\rm Res}$ is independent on the accretor's mass.
It is characteristic of the Shakura-Sunyaev solution in this regime that the three $\Sigma$, $T_c$ and $T_{\rm eff}$ radial profiles vary as $ R^{-3/4}$.
(This implies that the optical depth $\tau$ is constant with radius $--$ see Eq. v\i\i\i.)
For high accretion rates and small radii, the
assumption of opacity dominated by free-free and bound-free absorption will brake down, and the solution will cease to be valid.
We will first consider the other disc end: large radii.
There, the temperature will finally drop ($T_c \sim T^{-3/4}$) below $10^4$ K, the disc plasma will recombine leading to a drastic change in opacities which triggers a thermal instability.
\section{Disc instabilities}
\label{sec:DI}
In this section we will present and discuss the disc thermal and the (related) viscous instabilities.
In AGN discs two instabilities are of interest: one is related to the hydrogen ionisation/recombination,
the other occurs in discs where pressure is dominated by radiation. The instability is thermal when it grows
on a thermal timescale, in a geometrically thin disc it means that the timescale is $ \sim (\alpha\,\Omega_K)^{-1}$
(see Eq. \ref{eq:thermal_time}).
\subsection{The thermal instability}
\label{subsec:thermal_instability}
A disc is thermally stable if the radiative cooling varies with
temperature faster than the viscous heating. In other words a disc is thermally stable if
\begin{equation}
\frac{d\ln \sigma T_{\rm eff}^4}{ d\ln T_{\rm c}} > \frac{d\ln Q^+}{d\ln T_{\rm c}}.
\label{stab}
\end{equation}
Using Eq. (\ref{diff2}) one obtains
\begin{equation}
\frac{d\ln T_{\rm eff}^4}{ d\ln T_{\rm c}} =
4\left[1 - \left(\frac{T_{\rm irr}}{T_{\rm c}}\right)^{4}\right]^{-1} -
\frac{d\ln \kappa}{d\ln T_{\rm c}}.
\label{eq:cool}
\end{equation}
One can see from these equations that disc stability is determined by the temperature dependence of
the opacities and that irradiation stabilises accretion discs.
\subsubsection{Thermal instability due to hydrogen ionisation/recombination}
In a gas--pressure dominated disc $Q^+ \sim \rho T\,H \sim \Sigma T \sim T_{\rm c}$ .
At high temperatures
${d\ln \kappa/d\ln T_{\rm c}}\approx - 4$ (see Eq. \ref{eq:kramers}).
A thermal instability arises due to a rapid change of opacities with temperature when hydrogen begins to recombine ($T_c \sim 10^4$K) and
this opacity dependence on temperature is no longer valid. In the instability region, the
temperature exponent becomes large and \textsl{positive}: ${d\ln \kappa/ d\ln T_{\rm
c}} \approx 7 - 10$, and in the end cooling is decreasing with temperature.
Thus this thermal instability should be present in accretion discs with midplane temperature
$T_c$ equal few times $10^4$K and below.
It has been established that it is at the origin of outbursts observed in discs around
stellar-mass black-holes, neutron stars and white dwarfs. Systems containing the first two
classes of objects are known as Soft X-ray transients (SXTs, where ``soft" relates
to their X-ray spectrum), while those containing white-dwarfs are called dwarf-novae
(despite the name that could suggest otherwise, nova and supernova outbursts
have nothing to do with accretion disc outbursts). This range of temperatures is also
present in discs around AGNs and we will present its consequences in Sect. \ref{sec:outbagn}.
\subsubsection{Thermal instability of radiation--pressure dominated discs}
Radiation--pressure dominated ($P=P_{\rm rad}\propto T^4$) accretion discs are thermally unstable when the
opacity is due to electron scattering on electrons. Indeed, in this case
\begin{equation}
\frac{d\ln T_{\rm eff}^4}{ d\ln T_{\rm c}} = 4
\label{eq:cooles}
\end{equation}
because $\kappa_R=\kappa_{\rm es}=const.$, while
in a radiation pressure dominated disc
\begin{equation}
\label{eq:qradp1}
Q^+ \sim HT^4\sim T^8/\Sigma
\end{equation}
so that
\begin{equation}
\frac{d\ln Q^+}{d\ln T_{\rm c}} = 8 \, > \, \frac{d\ln T_{\rm eff}^4}{ d\ln T_{\rm c}}
\label{eq:heates}
\end{equation}
and the disc is thermally unstable. This solution is represented by the middle branch with negative slope on the S-curve in Fig. \ref{fig:adaf1} (see also Eq. \ref{eq:radpdom}) .
The presence of this instability in the model is one of the unsolved problems of the accretion disc theory because it contradicts observations which do not
show any unstable behaviour in the range of luminosities where discs should be in the radiative pressure and electron-scattering opacity domination regime.
This suggests that the disc description in this regime is incomplete.
For example, the radiation-pressure instability can be quenched if the disc vertical support is provided not by radiation but by a magnetic field. This field has to be brought into the disc
from the surrounding medium, because the MRI dynamo is unable to produce a field with the required strength.
The heating rate can be rewritten as
\begin{equation}
Q^{+}= \alpha P_{\mathrm{rad}} H \frac{\mathrm{d} \Omega}{\mathrm{d} r} = \alpha P_{\mathrm{rad}}^{2} \frac{\Sigma}{\Omega_{\mathrm{K}}^{2}} \frac{\mathrm{d} \Omega}{\mathrm{d} r},
\end{equation}
and from
\begin{equation}
\left.\frac{\mathrm{d} \log Q^{+}}{\mathrm{d} \log P_{\mathrm{rad}}}\right|_{\Sigma}=2; \ \ \ \ \ \ \left.\frac{\mathrm{d} \log Q^{-}}{\mathrm{d} \log P_{\mathrm{rad}}}\right|_{\Sigma}=1,
\end{equation}
one deduces, as before, that a radiation-pressure dominated disc with electron-scattering opacity is
thermally unstable.
However, with a magnetic-field pressure $P_{\mathrm{mag}}$, introducing
\begin{equation}
\beta^{\prime}=\frac{P_{\mathrm{mag}}}{P_{\mathrm{rad}} + P_{\mathrm{mag}}}
\end{equation}
one can write the heating rate as
\begin{equation}
Q^{+}=\frac{\alpha P_{\mathrm{rad}}^{2}}{\left(1-\beta^{\prime}\right)^{2}} \frac{\Sigma}{\Omega_{\mathrm{K}}^{2}} \frac{\mathrm{d} \Omega}{\mathrm{d} r},
\end{equation}
from which one obtains
\begin{equation}
\left.\frac{\mathrm{d} \log Q^{+}}{\mathrm{d} \log P_{\mathrm{rad}}}\right|_{\Sigma, P_{\mathrm{mag}}}=2\left(1-\beta^{\prime}\right).
\end{equation}
Radiative cooling is independent of the magnetic pressure, so for $\beta^{\prime} > 0.5$ the heating
rate no longer increases more rapidly with the central radiation
pressure, than the cooling rate and the disc is thermally stable. It is not clear, however, how and
from where a stabilising magnetic field will find itself in the right place and the right moment
to quench the disc's thermal instability (see a related discussion in Sect. \ref{sec:disc-jets}).
\subsection{Thermal equilibria: the $\mathbfcal{\,S}$-curve}
\label{subsec:scurve}
We will consider thermal equilibria of an accretion disc in which
heating is due only to local turbulence, neglecting the
effects of irradiation, because in the case of AGN discs
(contrary to discs in X-ray binaries) irradiation does not make much difference to
the instability's consequences. We put therefore $T_{\rm
irr}=\widetilde Q=0$.
\begin{figure} [!]
\center
\includegraphics[angle=270,width=0.95\textwidth]{fig1b.pdf}
\caption{$\mathbfcal{\,S}$-curves in the $\Sigma - T_{\rm eff}$ plane, for $M = 10^8 \ensuremath{\rm \,M_\odot}$ and $R= 2 \times 10^{16}$ cm.
$\mathbfcal{\,S}$-curves for two values of the viscosity parameter are shown: $\alpha = 0.1$ (left) and $\alpha=0.01$ (right).
The dotted curve corresponds to $Q_T=1$ (see Eq. \ref{eq:toomre}), and the dashed curve to $H/R = 0.1$.
The model applies only to regions below the dashed curved and above the dotted one, i.e., to a geometrically thin ($H/R\ll 1$), non-self-gravitating ($Q_T < 1$) accretion discs. In this case the thin disc approximation
applies to $r \lesssim 600$. [{\sl From \citet{HVL09}}].}
\label{fig:scurve}
\end{figure}
The thermal equilibrium in the disc is defined by the equation $Q^-=Q^+$
(see Eq. \ref{eq:heat}), i.e. by
\begin{equation}
\sigma T_{\rm eff}^4=\frac{9}{8} \nu \Sigma \Omega_{\rm K}^2
\label{termeq}
\end{equation}
(Eq. \ref{eq:vischeat2}).
In general, $\nu$ is a function of density and temperature and in the following we will use
the standard $\alpha$--prescription Eq. (\ref{eq:nualphaP}). The energy
transfer equation provides a relation between the
effective and the disc midplane temperatures so that thermal equilibria
can be represented as a $T_{\rm eff}\left(\Sigma\right)$~ --~relation
(or equivalently a $\dot M(\Sigma)$--relation). In the relevant range of temperatures ($10^3 \lesssim T_{\rm eff}\lesssim 10^5$)
this relation forms an {$\mathbfcal{\,S}$} on the
($\Sigma, T_{\rm eff}$) plane as in Fig. \ref{fig:scurve}. The upper, hot branch corresponds to the Shakura-Sunyaev solution
presented in Section \ref{subsec:SS}. The two other branches correspond to solutions for cold (partially ionised) discs. Near the upper bend of the {$\mathbfcal{\,S}$}
and below it, convection can play an important role in the energy transfer.
Each point on the ($\Sigma, T_{\rm eff}$) $\mathbfcal{\,S}$-curve represents an
accretion disc's thermal equilibrium at a given radius, i.e. a thermal equilibrium of a ring at radius $R$. In other words
each point of the $\mathbfcal{\,S}$-curve is a solution of the $Q^+=Q^-$ equation.
Points not on the $\mathbfcal{\,S}$-curve correspond to solutions of Eq. (\ref{eq:heat}) \textsl{out
of thermal equilibrium}: on the left of the equilibrium curve, cooling dominates over heating, $Q^+ < Q^-$; on the right
heating over cooling $Q^+ > Q^-$. Therefore a positive slope
of the $T_{\rm eff}(\Sigma)$ curve corresponds to \textsl{stable solutions}, since a small increase of temperature of an equilibrium state (an upward perturbation) on the upper branch, say,
will bring the ring to a state where $Q^+ < Q^-$ so it will cool down getting back to equilibrium. In a similar way
a downward perturbation will provoke increased heating bringing back the system to equilibrium.
The opposite is happening along the $\mathbfcal{\,S}$-curve's segment with a negative slope as both a temperature increase and
decrease lead to a runaway. The middle branch of the $\mathbfcal{\,S}$-curve corresponds therefore to \textsl{thermally unstable}
equilibria.
A stable disc equilibrium can be represented only by a point on the lower, cold or
the upper, hot branch of the $\mathbfcal{\,S}$-curve. This means that the surface
density in a stable cold state must be \textsl{lower} than the maximal value on the
cold branch: $\Sigma_{\rm max}$, whereas the surface density in the hot
stable state must be \textsl{larger} than the minimum value on this branch: $\Sigma_{\rm
min}$. Both these critical densities are functions of the viscosity
parameter $\alpha$, the mass of the accreting object, the distance from the
center and depend on the disc's chemical composition. In the case of solar composition the critical surface densities are
\begin{equation}
\label{eq:Sigmacrit}
\Sigma_{\min }=2.90 \times 10^{3} \alpha^{-0.74}{R}_{15}^{1.04} m_{8}^{-0.35} \mathrm{g} \mathrm{cm}^{-2}
\end{equation}
\begin{equation}
\Sigma_{\max }=3.85 \times 10^{3} \alpha^{-0.82} {R}_{15}^{0.99} m_{8}^{-0.33} \mathrm{g} \mathrm{cm}^{-2}
\end{equation}
and the corresponing effective temperatures
\begin{equation}
\label{eq:Tcrit}
T_{\mathrm{eff}}\left(\Sigma_{\min }\right)=4300 R_{15}^{-0.12} \mathrm{K}
\end{equation}
\begin{equation}
\label{eq:Tcritplus}
T_{\mathrm{eff }}\left(\Sigma_{\max }\right)=3300 R_{15}^{-0.12} \mathrm{K},
\end{equation}
where $R=:R_{15}10^{15}\rm cm$.
Using the equation (see Eq. \ref{eq:teff})
\begin{equation}
\dot M= \sigma T_{\rm eff}^4\frac{8\pi R^3}{3GM},
\end{equation}
one obtains for the critical accretion rates
\begin{align}
\dot M^+_{\rm crit} := \dot M({\Sigma_{\rm min}}) = 1.22 \times 10^{22} R_{15}^{2.52} \rm g\,s^{-1} \label{eq:mdotcritplus}\\
\dot M^-_{\rm crit} := \dot M(\Sigma_{\rm max}) = 4.22 \times 10^{21} R_{15}^{2.52} \rm g\,s^{-1} . \label{eq:mdotcritminus}
\end{align}
A stationary accretion disc in which there is a ring with effective temperature contained between the critical values of
Eq. (\ref{eq:Tcrit}) and (\ref{eq:Tcritplus}) cannot be stable. Since in a stationary disc, the effective temperature and the surface density both decrease with radius, the
stability of a disc depends on the accretion rate and the disc size. For a given
accretion rate a stable disc cannot have an outer radius larger than the value corresponding to Eq. (\ref{eq:mdotcritplus}).
A disc is hot and stable if the rate at which mass is brought to its outer edge ($R\sim R_D$) is larger than the critical accretion rate
at this radius $\dot{M}_{\rm crit}^{+}(R_D)$. On the other hand a disc is cold and stable if the mass--arrival rate to the disc is lower than the
critical accretion rate at its inner radius $\dot{M}_{\rm crit}^{-}(R_{\rm in})$.
Taking $R_{\rm in} = 3 R_S$ and $R_D = 2\times 10^{16}\rm cm$, for $10^8\,\ensuremath{\rm \,M_\odot}$ black hole, an accretion disc will be unstable if the mass--arrival rate is contained in the range
\begin{align}
9.6\times 10^{20}\, \rm g\,s^{-1} < \dot M < 2.3 \times 10^{25}\, \rm g\,s^{-1}\, \rm or, \label{eq:stab8}\\
6.8 \times 10^{-7}\, \ensuremath{\rm \,M_\odot}\, \rm yr^{-1} < \dot M < 1.7 \times 10^{-2}\, \ensuremath{\rm \,M_\odot}\, \rm yr ^{-1}
\end{align}
or
\begin{equation}
\label{eq:stab8edd}
6.0 \times 10^{-6} < \dot m < 0.6.
\end{equation}
This means that a typical accretion disc in an AGN \textsl{is thermally unstable}.
\subsection{Variability of unstable AGN discs}
\label{sec:outbagn}
The thermal instability at a certain disc's radius produces a steep temperature gradient. This in turn creates a heating front which
propagates radially bringing the cold disc regions to a hot state in which the heated up
disc's rings are on the upper branch of the $\mathbfcal{\,S}$-curve. The rise of temperature increases the viscosity (Eq. \ref{eq:nualphaOmega}).
In the discs around stellar-mass objects, such as white dwarfs, neutron stars and black holes, this process redistributes the mass of the
disc, creating a quasi-stationary disc configuration with an accretion rate $\dot M \approx \mathrm{const.} \approx \mathrm{few} \times \dot M^+_{\mathrm{crit}}(R_D)$,
the hot critical accretion rate at the outer disc radius. This corresponds to the outburst maximum, from which the accretion rate and luminosity decay
because, by construction, in an unstable disc the mass-feeding rate is less than the hot critical accretion rate $\dot M^+_{\rm crit}(R_D)$ (see Eq. \ref{eq:stab8}).
When the accretion rate drops below this value, a cooling-front forms in the outer disc regions and propagates towards the center, bringing the disc behind it to a cold state, thus switching-off the outburst.
During the following quiescence the disc, emptied during the outburst, fills-up again until it reaches somewhere the critical temperature, leading to the next outburst in the cycle.
A typical dwarf-nova outburst or an X-ray binary transient event is characterised by a fast rise and slow decay which correspond to a viscous decay of a disc with a shrinking outer radius.
\label{sec:variability}
\begin{figure}[!]
\centering
\includegraphics[width=0.7\textwidth]{fig3.pdf}
\caption{Time evolution of an accretion disc around a
$10^8\ensuremath{\rm \,M_\odot}$ black hole. The inner and outer radii are at $10^{14}$ and $10^{16}$ cm,
respectively, and the mean mass feeding rate is $10^{24}$g/s ($\dot m \approx 0.01$). Top panel: visual
magnitude; intermediate panel: accretion rate onto the black hole;
lower panel: radius at which the transition between the hot and cold
regimes takes place corresponding to the heating (cooling) front position. [From \cite{HVL09}]}
\label{fig:sub1}
\end{figure}
As can be seen in Figs. \ref{fig:sub1} and \ref{fig:sub2}, which show the light-curve produced by an unstable disc around a supermassive black hole, the thermal disc instability produces outbursts also in AGNs,
but a closer look shows that they are rather different from the eruptions observed in systems with less massive compact objects.
For relatively high mass-feeding rates ($\dot m \approx 0.01$ -- Fig. \ref{fig:sub1}), the instability results in low-amplitude ($\sim 1$ magnitude) outbursts corresponding to tiny modulations of the accretion rate.
In fact, one sees that the timescale of brightness variations, corresponding to the temperature oscillations, is much shorter than the timescale of the accretion--rate variations.
The basic reason for this is that the
front propagation speed is approximately $\alpha$ times the sound speed
which means that front propagation time-scale is
\begin{equation}
t_{\rm front} \approx \frac{R}{\alpha c_{\rm s}} = \frac{R}{H} t_{\rm th},
\end{equation}
where $t_{\rm th}$ is the thermal time scale. Hence $t_{\rm front}$ is
shorter than the viscous time $t_{\rm visc} = (R/H)^2 t_{\rm th}$ by
a factor $R/H$, i.e. by several orders of magnitude:
\begin{align}
\frac{H}{R} & \approx 3.2 \times 10^{-4}T_4^{1/2} m_8^{-1/2}R_{15}^{1/2}\\
&=5.5 \times 10^{-5} T_4^{1/2}\, r^{1/2}.
\label{eq:153}
\end{align}
The viscous time is therefore
\begin{equation}
t_{\rm vis} \approx 1.5 \times 10^4 \alpha^{-1}\,T^{1/2}_4\,m_8\,r^{1/2} \rm \,yr.
\end{equation}
\begin{figure}[!]
\centering
\includegraphics[width=0.6\textwidth]{fig_std1.pdf}
\caption{Time evolution of an accretion disc with the same parameter
as in Fig. \ref{fig:sub1}, except for the mean mass feeding rate which is here $2\times 10^{22}$g/s ($\dot m \approx 0.0001$). Top panel:
visual magnitude, lower panel: accretion rate onto the black hole.[{\sl From \cite{HVL09}}]}
\label{fig:sub2}
\end{figure}
Hence the front propagates so rapidly that the surface density
does not have time to react to the changes of temperature.
In the case of stellar-mass compact objects, $t_{\rm front}$ is shorter than $t_{\rm visc}$,
but not by such a large amount and strong gradients in the
disk make the effective viscous time comparable to the front
propagation time. The reason is that while in dwarf-nova stars and X-ray binaries the unstable disc
region is at $r \simeq 10^4$, in AGNs it is rather at $r < 300$, much deeper in the gravitational well
of the accretor.
When one lowers the accretion rate to $\dot m \approx 0.0001$, the disc manages to descend to quiescence (Fig. \ref{fig:sub2}), but
it does it through a pure viscous decay, without a cooling front propagating down through the whole disc. The outburst phase lasts $3\times 10^5$ yr,
which is roughly the disc viscous time. During this slow decay from maximum, heating/cooling fronts get reflected 400 times, creating as
before low amplitude optical magnitude oscillations and tiny modulations of the accretion rate.
Neither disc irradiation, nor inner-disc truncation change basically the AGN outburst properties.
This is not surprising since in stellar-mass systems, these effects play an important role
mainly by affecting the cooling from propagation, which in the AGN case are of no importance.
It remains to be seen if in AGNs, thermal-instability outbursts occur in an observable way and even if they are occurring at all.
\section{Beyond thin discs}
\label{sect:advection}
Until now, we have assumed that accretion discs are thin, i.e. that $H/R \ll 1$.
One of the consequences of this assumption was neglecting the advection terms in the energy and momentum equations
for stationary accretion flows.
The vertically averaged advective ``cooling'' term can be written as
\begin{equation}
\label{eq:advH}
Q^{\rm adv}=\frac{\Sigma T v_r}{R}\frac{ds}{d\ln R} = \frac{\dot M}{2 \pi R^2} c_{\rm s}^2 \xi_a,
\end{equation}
where $\xi_a$ is a slowly varying function related to the entropy gradient and is usually $\sim 1$.
One then has
\begin{equation}
\frac{Q^{\rm adv}}{Q^+} \sim \frac{c_s^2}{\Omega_K^2 R^2}\approx \left(\frac{H}{R}\right)^2,
\end{equation}
where we used Eqs. (\ref{eq:qplus}) and (\ref{eq:amintK}). Therefore neglecting the advective term
in the energy equation
\begin{equation}
Q^+= Q^- + Q^{\rm adv}
\end{equation}
is justified if one assumes $H/R \ll 1$.
There are two regimes of parameters where this assumption is
not valid, in both cases for the same reason: low radiative efficiency when the time for radial
motion towards the black hole is shorter than the radiative cooling time. Low density
(low accretion rate), hot, optically thin accretion flows are poor coolers and they are one
of the two configurations were advection instead of radiation is the dominant evacuation-of-energy
(``cooling") mechanism. Such optically thin flows are called ADAFs, for Advection Dominated
Accretion Flows\footnote{Some people prefer to call such flows \textsl{RIAF}s,
from \textsl{Radiatively Inefficient Flows} but I prefer
to stick to the name \textsl{ADAF}, which I had the pleasure to introduce.}.
Also advection dominated are high-luminosity flows, accreting at high rates, but
they are called ``slim discs" to account for their property of not being thin, but still being described
as if this were not of much importance.
We shall start with optically thin flows.
\begin{itemize}
\item ADAFs
Advection Dominated Accretion Flows' (ADAFs) is a term describing accretion
of matter with angular momentum, in which radiation efficiency is very
low. In their applications, ADAFs are supposed to describe inflows onto
compact bodies, such as black holes or neutron stars; but very hot,
optically thin flows are bad radiators in general so that, in
principle, ADAFs are possible also in other contexts. Of course in the
vicinity of black holes or neutron stars, the virial (gravitational)
temperature is $T_{\rm vir}\approx 5 \times 10^{12} (R_S/R)$ K,
so that in optically thin plasmas, at such
temperatures, both the coupling between ions and electrons and the
efficiency of radiation processes are rather feeble. In
such a situation, the thermal energy released in the flow by the
viscosity, which drives accretion by removing angular momentum, is not
going to be radiated away, but will be {\sl advected} towards the
compact body. If this compact body is a black hole, the heat will be
lost forever, so that advection, in this case, acts as sort of a `global'
cooling mechanism. There, advection may act only as
a `local' cooling mechanism. (One should keep in mind that, in
general, advection may also be responsible for heating, depending on
the sign of the temperature gradient $--$ in some conditions, near the black hole,
advection heats up electrons in a two-temperature ADAF).
In general the role of advection in an accretion flow depends on the radiation
efficiency which in turns depends on the microscopic state of matter and
on the absence or presence of a magnetic field. If, for a given
accretion rate, radiative cooling is not efficient, advection is
necessarily dominant, assuming that a stationary solution is possible.\\
\item Slim discs
At high accretion rates, discs around black holes become dominated by radiation
pressure in their inner regions, close to the black hole. At the same time the
opacity is dominated by electron scattering. In such discs $H/R$ is no longer
$\ll 1$. But this means that terms involving the radial velocity are no longer
negligible since $v_r \sim \alpha c_s (H/R)$. In particular, the advective term
in the energy conservation equation $v_r \partial S/\partial R$ (see Eq. \ref{eq:energy})
becomes important and finally, at super-Eddington rates, dominant. When $Q^+=Q^{\rm adv}$
the accretion flow is advection dominated and called a slim disc \citep{Abramowicz0988}.
\item Windy discs
In a super-Eddington accreting disc, the radiative pressure can blow-out matter creating an outflow
that will limit the luminosity to its local Eddington value. In such a case $\dot m \sim r$.
\end{itemize}
\subsection{Advection--dominated--accretion--flow toy models}
One can illustrate fundamental properties of ADAFs and slim discs with a simple toy model.
The advection `cooling' (per unit surface) term in the energy equation
can be written as
\begin{equation}
Q^{\rm adv} = \frac{\dot M}{2 \pi R^2} c_{\rm s}^2 \xi_a
\label{eq:advterm}
\end{equation}
(Eq. \ref{eq:advH}).
Using the (non-relativistic) hydrostatic equilibrium equation
\begin{equation}
\frac{H}{R} \approx \frac{c_{\rm s}}{v_{\rm K}}
\label{eq:hydroeq}
\end{equation}
one can write the advection term as
\begin{equation}
Q^{\rm adv} = \Upsilon \frac{\kappa_{\rm es}c}{2R}\left(\frac{\dot M}{\eta}\right)\xi_a \left(\frac{H}{R}\right)^2
\label{eq:advterm2}
\end{equation}
whereas the viscous heating term can be written as
\begin{equation}
Q^+= \Upsilon\frac{3}{8}\frac{\kappa_{\rm es}c}{R}\left(\frac{\dot M}{\eta}\right) ,
\label{visheat}
\end{equation}
where
\begin{equation}
\label{eq:upsilon}
\Upsilon=\left(\frac{c }{\kappa_{\rm es}r}\right)^2 .
\end{equation}
Since $\xi_a\sim 1$,
\begin{equation}
\label{eq:qadv}
Q^{\rm adv}\approx Q^+\left(\frac{H}{R}\right)^2
\end{equation}
and,
as said before, for geometrically thin discs ($H/R\ll1$) the advective term $Q^{\rm adv}$ is negligible compared
to the heating term $Q^+$ and in thermal equilibrium viscous heating must be compensated by radiative cooling.
Things are different at
very high temperatures, when $(H/R) \sim 1$. Then the advection
term is comparable to the viscous term and cannot be neglected in the equation of thermal equilibrium. In some cases
this term is larger than the radiative cooling term $Q^-$ and (most of) the heat released by viscosity is \textsl{advected}
toward the accreting body instead of being locally radiated away as happens in geometrically thin discs.
From Eq. (\ref{eq:amintK}) one can obtain a useful expression for the square of the relative disc height (or aspect ratio):
\begin{equation}
\left(\frac{H}{R}\right)^2 =
\frac{\sqrt 2}{\kappa_{\rm es}} \left(\frac{\dot m}{\eta}\right)
\left(\alpha \Sigma\right)^{-1}r^{-1/2}.
\label{eq:bizarre}
\end{equation}
Deriving Eq. (\ref{eq:bizarre}) we used the viscosity prescription $\nu=(2/3)\alpha c_{\rm
s}^2/\Omega_K$.
Using this equation one can write for the advective cooling
\begin{equation}
\label{eq:advterm3}
Q^{\rm adv} = \Upsilon\Omega_K \xi_a \left(\alpha \Sigma\right)^{-1}\left(\frac{\dot m}{\eta}\right)^2.
\end{equation}
The thermal equilibrium (energy) equation is
\begin{equation}
\label{eq:energy_eq}
Q^+= Q^{\rm adv}+Q^- .
\end{equation}
The form of the radiative cooling term depends on the state of the accreting matter, i.e. on it temperature, density and chemical composition.
Let us consider two cases of accretion flows:
\begin{description}
\item[$-$] optically thick
\item{} and
\item [$-$] optically thin.
\end{description}
For the optically thick case we will use the diffusion approximation formula
\begin{equation}
\label{eq:rad_approx}
Q^-= \frac{8}{3}\frac{\sigma T_c^4}{\kappa_{\rm R}\Sigma},
\end{equation}
and assume $\kappa_{\rm R}=\kappa_{\rm es}$. With the help of Eq. (\ref{eq:bizarre}) this can be brought to the form
\begin{equation}
\label{eq:slimopthick}
Q^-_{\rm thick}=8 \Upsilon \left(\frac{\kappa_{\rm es} R_S}{c}\right)^{1/2}r^2\Omega_K^{3/2} \left(\alpha \Sigma\right)^{-1/2}\left(\frac{\dot M}{\eta}\right)^{1/2} .
\end{equation}
For the optical thin case of bremsstrahlung radiation we have
\begin{equation}
\label{eq:brems}
Q^-=1.24\times 10^{21}H\rho^2T^{1/2}
\end{equation}
which using Eq. (\ref{eq:153}) can be written as
\begin{equation}
\label{eq:opthin}
Q^-_{\rm thin}=3.4\times 10^{-6}\Upsilon r^2\Omega_K \alpha^{-2}\left(\alpha \Sigma\right)^2.
\end{equation}
\begin{itemize}
\item In the \textsc{optically thick} case we have therefore
\begin{eqnarray}
\label{eq:en_thick}
\xi_a \left(\frac{\dot M}{\eta}\right)^{2} +&& 0.18 r^{1/2} \left(\alpha \Sigma\right) \left(\frac{\dot M}{\eta}\right) +\nonumber \\
&& + 2.3 r^{5/4}\left(\alpha \Sigma\right)^{1/2}\left(\frac{\dot M}{\eta}\right)^{1/2}=0
\end{eqnarray}
\item In the \textsc{optically thin} case the energy equation has the form
\begin{eqnarray}
\label{eq:en_thin}
\xi_a \left(\frac{\dot M}{\eta}\right)^{2} + && 0.18 r^{1/2} \left(\alpha \Sigma\right) \left(\frac{\dot M}{\eta}\right) + \nonumber \\
&&+ 3\times 10^{-6}\alpha^{-2}r^2\left(\alpha \Sigma\right)^3=0
\end{eqnarray}
\end{itemize}
There are two distinct types of advection dominated accretion flows: optically thin and optically thick.
We will first deal with optically thin flows which are the true \textsl{ADAFs}.
\begin{figure} [h!]
\centering
\includegraphics[width=0.9\textwidth]{Slim.pdf}
\caption{Thermal equilibria for optically thick (the right solid S-shaped line) and optically
thin (the left solid line) accretion flows. The upper branches correspond to advection-dominated solution
(ADAF \& Slim). SSD correspond to the Shakura-Sunyaev solution. Flows above the dotted lines $\tau =1 $ are optically thin , $\tau$ being the effective optical
depth calculated for radiation-pressure dominated (upper line) or gas-dominated (lower line) configurations.
This figure corresponds to $M_{\mathrm{BH}} = 10 \ensuremath{\rm \,M_\odot}$, $r=5$, $\alpha=0.1$ and $\xi_a = 1$. {\sl [Adapted from
\citet{Abramowicz0195}].}}
\label{fig:adaf1}
\end{figure}
\subsubsection{Optically thin flows: \textsl{ADAFs}}
\label{subsub:adaf}
For prescribed values $\alpha$ and $\xi_a$, Eq. (\ref{eq:en_thin}) is a quadratic equation in $(\dot m/\eta)$ whose solutions in the form of $\dot m(\Sigma)$ describe thermal equilibria at a given value of $R$. Obviously, for a given $\Sigma$ this equation has at most two solutions.
The solutions form two branches on the $\dot m(\alpha \Sigma)$ -- plane:
\begin{itemize}
\item the ADAF branch
\begin{equation}
\label{eq:adafbranch}
\dot m = 0.53\kappa_{\rm es}\,\eta r^{1/2}\xi_a^{-1} \alpha \Sigma.
\end{equation}
\noindent and\\
\item the radiatively--cooled branch
\begin{equation}
\label{eq:radcoolbranch}
\dot m = 1.9\times 10^{-5}\,\eta r^{3/2}\xi_a^{-1} \alpha^{-2}\left(\alpha\Sigma\right)^2.
\end{equation}
\end{itemize}
From Eqs. (\ref{eq:adafbranch}) and (\ref{eq:radcoolbranch}) it is clear that there exists a maximum accretion rate
for which only one solution of Eq. (\ref{eq:en_thin}) exists. This implies the existence of a maximum accretion rate at
\begin{equation}
\dot m_{\rm max} \approx 1.7 \times 10^3 \eta \,\alpha^2 r^{1/2}.
\label{eq:mdotmax}
\end{equation}
This is where the two branches formed by thermal equilibrium solutions on
the $\dot M (\alpha\Sigma) -$ plane meet as seen on Figure \ref{fig:adaf1}.
The value of $\dot m_{\rm max}$ depends on the cooling mechanism in the accretion flow and the
non-relativistic free-free cooling is not a realistic description of
the emission in the vicinity ($(R/R_S)\lesssim 10^3$) of a black hole. The flow there
will most probably form a two-temperature plasma. In such a case
$\dot m_{\rm max}\approx 10 \alpha^2$, with
almost no dependence on radius. For larger radii $\dot m_{\rm max}$
decreases with radius.
\subsubsection{Optically thick flows: \textsl{slim discs}}
\label{subsub:slim}
Since the first two terms in Eq. (\ref{eq:en_thick}) are the same as in (Eq. \ref{eq:en_thin}), the high $\dot m$, advection dominated solution is
the same as in the optically thin case but now represents the
\begin{itemize}
\item Slim disc branch
\begin{equation}
\tag{\ref{eq:adafbranch}}
\dot m = 0.53\, \kappa_{\rm es}\,\eta r^{1/2}\xi_a^{-1} \alpha \Sigma.
\end{equation}
Now, the full equation (\ref{eq:en_thick}) is a cubic equation in $\dot m^{1/2}$ and on the
$\dot m(\alpha \Sigma)-$ plane its solution forms the
two upper branches of the $\mathbfcal{\,S}$-curve shown in Fig. \ref{fig:adaf1}. The uppermost branch corresponds to slim discs while the branch with negative slope
represents the Shakura-Sunayev solution in the regime \textsl{a.)} (see Sect. \ref{subsec:SS}), i.e. \\
\item a radiatively cooled, radiation-pressure dominated accretion disc
\begin{equation}
\label{eq:radpdom}
\dot m =160\,\kappa^{-1}_{\rm es}\, \eta r^{3/2}\left(\alpha \Sigma\right)^{-1}
\end{equation}
\end{itemize}
\subsubsection{Slim discs and super-Eddington accretion}
From Eqs. (\ref{eq:bizarre}) and (\ref{eq:radpdom}) one obtains for the disc aspect ratio
\begin{equation}
\label{eq:htorradpress}
\frac{H}{R}=0.11\left( \frac{\dot m}{\eta}\right)r^{-1}
\end{equation}
which shows that the height of a radiation dominated disc is constant with radius and proportional to the accretion rate.
But this means that with increasing $\dot m$ advection becomes more and more important (see e.g. Eq. \ref{eq:qadv}) and for
\begin{equation}
\label{eq:adveqplus}
\frac{\dot m}{\eta}\approx 9.2 r
\end{equation}
advection will take over radiation as the dominant cooling mechanism and the solution will represent a slim disc.
Equation (\ref{eq:adveqplus}) can be also interpreted as giving the \textsl{transition radius} between radiatively and
advectively cooled disc for a given accretion rate $\dot m$:
\begin{equation}
\label{eq:advectrans}
{r_{\rm trans}}\approx \frac{0.1}{\eta}{\dot m}
\end{equation}
Another radius of interest is the \textsl{trapping radius} at which the photon diffusion (escape) time $H\tau/c$ is equal to the
viscous infall time $R/v_r$
\begin{equation}
\label{eq:trapp}
R_{\rm trapp}= \frac{H\tau\,v_r}{c}= \frac{H\kappa \Sigma}{c}\frac{\dot M}{2\pi R \Sigma}=\frac{H}{R}\left(\frac{\dot m}{\eta}\right)R_S.
\end{equation}
Notice that both $R_{\rm trans}$ and $R_{\rm trapp}$ are proportional to the accretion rate.
In an advection dominated disc the aspect ratio $H/R$ is independent of the accretion rate:
\begin{equation}
\label{eq:htorslim}
\frac{H}{R}=0.86\, \xi_a\,r^{1/4},
\end{equation}
therefore contrary to radiatively cooled discs, slim discs do not puff up with increasing accretion rate.
Putting (\ref{eq:htorslim}) into Eq. (\ref{eq:trapp}) one obtains
\begin{equation}
\label{eq:trapp2}
{r_{\rm trapp}}= 0.86\,\xi_a^{-1/2}r^{1/4}\left(\frac{\dot m}{\eta}\right).
\end{equation}
Radiation inside the trapping radius is unable to stop accretion and since $R_{\rm trapp}\sim \dot m$ there is no
limit on the accretion rate onto a black hole.
The luminosity of the toy-model slim disc can be calculated from Eqs. (\ref{eq:slimopthick}) and (\ref{eq:adafbranch})
giving
\begin{equation}
\label{eq:Qminusslim}
Q^-=\sigma T^4_{\rm eff}=\frac{0.1}{\xi_a}\frac{L_{\rm Edd}}{R^2},
\end{equation}
which implies $T_{\rm eff}\sim 1/R^{1/2}$.
The luminosity of the slim--disc part of the accretion flow is then
\begin{equation}
\label{eq:luminslim}
L_{\rm slim}=2\int_{R_{\rm in}}^{R_{\rm trans}}\sigma T^4_{\rm eff}2\pi RdR=\frac{0.8}{\xi_a}L_{\rm Edd}\cdot\ln\frac{R_{\rm trans}}{R_{\rm in}}\approx L_{\rm Edd}\ln\dot m,
\end{equation}
where we used Eq. (\ref{eq:advectrans}).
Therefore the total disk luminosity
\begin{eqnarray}
\label{eq:total}
L_{\rm total}&=& L_{\rm thin} + L_{\rm slim}=\\
&& 4\pi\left(\int_{R_{\rm in}}^{R_{\rm trans}}\sigma T^4_{\rm eff} RdR+ \int_{R_{\rm trans}}^{R_{\infty}}\sigma T^4_{\rm eff}RdR\right)\approx L_{\rm Edd}(1 + \ln\dot m), \nonumber
\end{eqnarray}
where $L_{\rm thin}$ is the luminosity of the radiation-cooled disc for which Eq. (\ref{eq:teff}) applies.
Such logarithmic luminosity--accretion-rate dependence is observed in the inner regions of simulated accretion flows.
It is easy to see that the same luminosity formula $L\approx L_{\rm Edd}(1 + \ln\dot m)$ is obtained
when one assumes mass--loss from the disc, resulting in a radially variable accretion rate: $\dot M \sim R$ (see Sect. \ref{sec:disc-jets}).
\section{Disc coronae}
The X-ray emission observed from AGN accretion flows require the presence of emitters other than the optically thick
accretion disc that we have described until now. The highest temperature one can get from the inner regions of such a disc
is $\lesssim 10^6$K (see Eq. \ref{eq:Teff_in}). In analogy to solar and stellar coronae, the optically thin hot structures, much hotter than the underlying
photosphere, the X-ray radiating structure in AGNs and other accreting compact body systems, is called a \textsl{disc corona}.
The main idea is that the hot electrons of such a corona, inverse-Compton upscatter to high energies the soft photons emitted
by the underlying disc.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.6\textwidth]{Corona.pdf}
\caption{Various possible (in principle) disc-coronal structures. [{\sl Courtesy: Chris Done.}]}
\label{fig:corona}
\end{center}
\end{figure}
There is not a single or even leading, or dominating disc corona model. Figure \ref{fig:corona} shows various models or scenarios
that have been proposed in the literature. Various models might correspond to various observed states of the accretion disc. The
bottom configuration corresponds to the oldest proposed model whose idea was that
differential rotation, together with convection and magnetic fields present in the disc,
could produce loop-like structures, forming a magnetically-confined hot corona,
in analogy to what is observed in the Sun. As for the Sun, magnetic-field reconnection
could play a role in accelerating electrons to high energies. More recently, for different physical reasons
(without invoking convection), models of magnetically heated coronae have been constructed.
In this case it is the MRI amplification mechanism that, from the poloidal field, produces large amount of toroidal field in
the upper layers of the disk.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.7\textwidth]{lamp.pdf}
\caption{A scheme representing the lamppost ``coronal" model. The feature called ``corona" is the
lamppost in question. [{\sl From \cite{Caballero2019}}.]}
\label{fig:lamp}
\end{center}
\end{figure}
Other models invoke accretion-disc ``evaporation", mechanism which, for low accretion rates, is supposed to produce ADAFs in the inner part of accretion discs.
In fact an ADAF could play the role of a disc corona, as in the second panel from the top.
However, the most used ``coronal" set-up is the so-called \textsl{lamppost} model, whose principle is shown in Figure \ref{fig:lamp}. As it's name demonstrate,
this model does not have exaggerated ambitions to represent a physical reality, but is extremely useful when considering reverberation mapping. Of course this
lamppost is a corona only in name. Most often one says that it could correspond to the base of a jet.
\section{Discs, winds and jets}
\label{sec:disc-jets}
Accretion in AGN is the source of the inverse phenomenon: outflows (ejections)
that with radiation are the key elements of the feedback cycle
linking the supermassive central black hole to its host galaxy. Radiation and collimated outflows in form of jets
interact with the interstellar medium leading to ejection or heating of the gas,
but accretion disc winds play apparently
the fundamental role in this interaction.
Winds and collimated jets are produced by all systems containing a disc-accreting celestial body, from young stellar objects, through white dwarfs and neutron stars, to black holes of all masses,
so one can expect that a common mechanism operates in accretion discs at all scales. The nature of this mechanism is still subject to controversy.
In the case of winds, three possible mechanisms are invoked. Outer disc regions, most probably eject thermal-driven winds that are accelerated by the thermal gas
pressure. Their velocity is at most $\sim 1000$ km/s.
The radiative pressure due to the intense radiation of most AGN can be a very effective way to drive
an accretion disc wind. For sub-Eddington accretion rates, UV absorption lines are the main source of opacity (as for winds of massive stars),
while near the Eddington luminosity, Compton scattering is the wind blowing driver.
\subsection{The ``forgotten" Shakura-Sunyaev solution}
In their seminal 1973 paper, Shakura and Sunyaev considered also the case of super-Eddington accretion and
found a solution, alternative to the slim-disc solution (which they discarded).
For mass-feeding rates $\dot m > 1$, they identified the \textsl{spherisation} radius where the luminosity is close to the local Eddington value as
\begin{equation}
R_{\mathrm{sph}} \simeq \frac{27}{8} \dot{m} R_{S} \simeq 1 \times 10^{14} \dot{m} m_{8} \,\mathrm{cm}
\end{equation}
Then the solution is obtained by requiring that the local emission within $R_{\mathrm{sph}}$ nowhere exceeds its local
Eddington limit,
This will be true if the outflow is such that the accretion rate through the disc decreases as
\begin{equation}
\dot{m} \simeq \dot{m} \frac{R}{R_{\mathrm{sph}}} .
\end{equation}
Then the total luminosity is
\begin{equation}
L\simeq L_{\rm Edd}(1 + \ln\dot m),
\end{equation}
(see Eq. \ref{eq:total}).
At very high accretion rates ($\dot m \gg 1$) the disc emission will be also strongly beamed by the flow geometry so that an observer situated in the beam of the
emitting system will infer a luminosity
\begin{equation}
\label{eq:beam}
L_{\rm sph}=\frac{1}{b}L_{\rm Edd}(1 + \ln\dot m),
\end{equation}
where $b$ is a beaming factor. Seen from the ``side" such a source with very large apparent luminosities might appear as rather dim.
On the other hand, if e.g., $b\sim 1/\dot m^2$, a stellar mass ultra-luminous X-ray source (by definition $L > 10^{39}$\, erg/s) accreting at
$\dot m \simeq 10^4$, seen along the beam, could have an apparent luminosity $\sim 10^{45}$ erg/s and look like an AGN, but will be positioned
off the galactic center.
\subsection{Relativistic jets}
Magnetic fields may play a fundamental role in the production of disk winds.
In this case the wind is accelerated by the centrifugal force
of the magnetic field lines anchored on the disk and the
magnetic pressure (Blandford-Payne mechanism, cf. \citealp{BP82}).
Magnetic fields
are usually involved at the generation and acceleration phase and are always supposed to play a decisive role in keeping the jet collimated. Quite often in models, the jet is a collimated
part of an outflow from the accretion disc but in other cases winds and jets can be launched by different mechanisms. Such a case is shown schematically in Fig. \ref{fig:jet_wind}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=\textwidth]{Jet_wind.pdf}
\caption{Schematic picture of a disc-wind-jet structure near a rotating black-hole.
[{\sl From \cite{Sadowski1213}}.]}
\label{fig:jet_wind}
\end{center}
\end{figure}
where the relativistic jet is powered by the black-hole rotation while the source of the sub-relativistic wind energy is gravitational.
For launching relativistic jets observed in AGN, the best performing models involve large-scale magnetic fields anchored in the
rapidly rotating matter of the inner parts of accretion discs. In the case of a rotating black hole two sources of jet launching energy are possible: the gravitational energy of accretion and the
black-hole rotational energy that can be tapped through the electromagnetic Penrose process. This last mechanism is possible only when the accretor is a black hole because other rotating compact bodies do not have ergoregions.
The Blandford-Znajek (\citealt{BZ77}, hereafter BZ) mechanism, which is the electromagnetic version of the Penrose process works on the same principle as its mechanical analogue: absorption of negative energy and negative
angular momentum.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=\textwidth]{BZ.pdf}
\caption{Snapshot at $t \approx 7806\, r/c $ from General Relativistic (GR) MHD simulation of the Blandford-Znajek process showing logarithm of rest-mass density in colour (see
the scale on the right-hand side) in both the $z-x$ plane at $y = 0$ (top left-hand panel) and the $y-x$ plane at $z = 0$ (top right-hand panel). The black lines trace
field lines, where the thicker black lines show where field is lightly mass-loaded. In this simulation $a=0.9375$ and accreting matter is in form of an ADAF. [{\sl Adapted from \cite{McKinney12}}]}
\label{fig:BZ}
\end{center}
\end{figure}
A typical jet--launching configuration is shown in Fig. \ref{fig:BZ}, where one sees a poloidal magnetic field penetrating the black hole surface being twisted, forming a toroidal component.
The power extracted by the BZ mechanism is equal to
\begin{equation}
\label{eq:BZpower}
P_{\mathrm{BZ}}=\frac{\kappa}{4 \pi c} \Omega_{\mathrm{H}}^{2} \Phi_{\mathrm{BH}}^{2} f\left(\Omega_{\mathrm{H}}\right),
\end{equation}
where
\begin{equation}
\Phi_{\mathrm{BH}}=(1 / 2) \int_{\theta} \int_{\varphi}\left|B^{r}\right| \mathrm{d} A_{\theta \varphi},
\end{equation}
is the magnetic flux threading one hemisphere of the black-hole horizon. ${d} A_{\theta \varphi}$ is the area element in the $\theta - \varphi$ plane. $\kappa$ is numerical constant depending on the magnetic field geometry ($\kappa = 0.053$ for the so-called split-monopole geometry, such as on Fig. \ref{fig:BZ}.
Quite often Eq. (\ref{eq:BZpower}) is written with $f(\Omega_H) = 1$, which is a good approximation for black-hole spins up to $a \approx 0.95$, but for larger spins, of special interest when considering the BZ mechanism
\begin{equation}
f\left(\Omega_{\mathrm{H}}\right) \approx 1+0.345\left(\Omega_{\mathrm{H}} r_{S} / c\right)^{2}- 0.575\left(\Omega_{\mathrm{H}} r_{S} / c\right)^{4}.
\end{equation}
is a more suitable approximation.
One defines the efficiency of the BZ mechanism as the ratio of the time-averaged electromagnetic flux extracting black-hole rotational energy
to the averaged rate at which the black hole absorbs rest-mass energy:
\begin{align}
\eta_{\mathrm{BZ}} & := \frac{\left\langle P_{\mathrm{BZ}}\right\rangle}{\langle\dot{M}\rangle c^{2}} \times 100 \% \\
&=\frac{\kappa}{16 \pi }\left(\frac{\Omega_{\mathrm{H}} r_{S}}{c}\right)^{2}\left\langle\phi_{\mathrm{BH}}^{2}\right\rangle f\left(\Omega_{\mathrm{H}}\right) \times 100 \%,
\end{align}
where
\begin{equation}
\label{eq:phi}
\phi_{\mathrm{BH}}={\Phi_{\mathrm{BH}}{}\left(\langle\dot{M}\rangle r_{S}^{2} c\right)^{1 / 2}}\approx 10^4 \eta_{0.1}^{1/2} m_8^{1/2} {\dot m}^{1/2}\left(\frac{\Phi_{\mathrm{BH}}}{0.1\,\rm pc^2 G}\right)
\end{equation}
is the dimensionless magnetic flux
threading the black hole.
Therefore the efficiency of the BZ mechanism depends strongly on the black-hole spin and on the magnetic flux threading the black hole surface. In Eq. (\ref{eq:phi}) we have normalised $\Phi_{\mathrm{BH}}$ by its observed interstellar-medium value. In the most efficient configuration the black hole receives as much large-scale
magnetic flux as can be pushed into it by accretion. By
supplying even more flux than this, some of it remains outside the
horizon where it impedes the accreting gas, leading to a ``magnetically arrested
disc" (MAD; \citealt{Tchekhov1111}). It is the MAD configuration that produces the most powerful relativistic jets.
This requires bringing this field to the innermost disc region where it is needed as an extractor of the black hole rotational energy. It's far from obvious how this can be done.
The simplest problem to address in this context is to consider a poloidal field threading a Keplerian accretion disc. The turbulent-viscosity driven accretion tends to drag the field-lines inwards but due to resistivity they diffuse outwards.
Numerical simulation of this problem have shown that the condition for the magnetic field lines to be significantly dragged inwards is
\begin{equation}
\label{eq:drag1}
\mathcal{B}=\frac{3 H\left|v_{R}\right|}{2 \eta}=\left(\frac{H}{R}\right) \left(\frac{\nu}{\eta}\right)= \left(\frac{H}{R}\right)\mathcal{P}_{\rm tm} \gtrsim 1,
\end{equation}
where $\nu$, as before, is the kinematic viscosity coefficient, and
\begin{equation}
\eta=\frac{c^{2}}{ 4 \pi \sigma}
\end{equation}
is the resistivity, with $\sigma$ the electric conductivity. $\mathcal{P}_{\rm tm}$ is known as the magnetic Prandtl number.
The usual definition of the Prandtl number involves microphysical turbulence and viscosity, but here we are interested
in transport coefficients of turbulent origin, hence the letter ``t" in the index of the symbol.
Therefore in thin accretion discs the magnetic field will be dragged inwards only for very large Prandtl numbers:
$\mathcal{P}_{\rm tm} \gg 1$, or $\eta \ll \nu$. In MRI accretion discs $\mathcal{P}_{\rm tm} \gtrsim 1$, so ``thick" discs, such as slim discs or ADAFs are
needed if the magnetic flux is supposed to accumulate at the inner disc, near the black hole surface \citep{Lubow0394}.
Another possibility is that the disc's angular momentum is
not removed by turbulent viscosity, but by e.g., magnetic wind. Then Eq. (\ref{eq:drag1}) does not apply and the condition for field dragging becomes
\begin{equation}
\label{eq:drag2}
\mathcal{B}=\frac{3 H\left|v_{R}\right|}{2 \eta} \gtrsim 1.
\end{equation}
In some models $ \left|v_{R}\right| \sim c_s$ and $\eta \sim Hc_s$, so field dragging is possible.
\bibliographystyle{apalike}
\linespread{0.25}
\renewcommand\bibname{References}
\footnotesize
|
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| 9,030
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{"url":"https:\/\/www.ideals.illinois.edu\/handle\/2142\/19936","text":"## Files in this item\n\nFilesDescriptionFormat\n\napplication\/pdf\n\n9543567.pdf (7MB)\n(no description provided)PDF\n\n## Description\n\n Title: Nuclear magnetic resonance studies of the organic superconductor kappa-(bisethylenedithiotetrathiafulvalene)(2) copper(nitrogen(cyanide)(2))bromine Author(s): DeSoto, Stewart Martin Doctoral Committee Chair(s): Slichter, C.P. Department \/ Program: Physics, Condensed Matter Discipline: Physics, Condensed Matter Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Physics, Condensed Matter Abstract: In this thesis we report $\\sp1$H and $\\sp{13}$C NMR measurements in the normal and superconducting states of the organic superconductor $\\kappa$-(ET)$\\sb2$Cu (N(CN)$\\sb2$) Br (T$\\sb{\\rm c}$ = 11.6 K). These measurements include the temperature and magnetic field dependent lineshapes, Knight shifts, spin-spin relaxation rates and spin-lattice relaxation rates for the $\\sp{-1}$H and $\\sp{13}$C sites located in the organic ET molecules of $\\kappa$-(ET)$\\sb2$CU (N(CN)$\\sb2$) Br for temperatures 2 K 4.3 T). We studied the orientation dependence of this rapid relaxation and found that the extra relaxation disappeared when the magnetic field was oriented very close ($\\le3\\sp\\circ$) to the superconducting layers. This was taken to be strong evidence that fluxoid motion caused the rapid relaxation and that intrinsic pinning of the fluxoids occurred for parallel magnetic fields.We have also studied samples in which the central two carbon sites of the ET molecule were enriched with $\\sp{13}$C (I = 1\/2). Due to the pairing of the ET molecules into dimers, there is a breakdown of the molecular inversion symmetry, and we observe two $\\sp{13}$C resonance lines, labeled Inner and Outer, with different Knight shifts tensors but the same chemical shift tensor.There are several key, transition temperatures in which the $\\sp{13}$C NMR properties show abrupt changes as the sample is cooled. The first occurs at T = 150 K, where the linewidths suddenly broaden, and the relaxation rates $(\\rm 1\/T\\sb1T, 1\/T\\sb2)$ increase. There is another transition at T = 50 K, evident in the relaxation rates and Knight shifts. There is another transition near T = 25K before the superconducting transition at T = 11.6K.In the superconducting state, there is a strong field dependence for the $\\sp{13}$C 1\/T$\\sb1$, with the relaxation rate increasing as the field is increased (opposite to the $\\sp1$H field dependence). The relaxation rate below T$\\sb{\\rm c}$ does not follow the BCS model; there is no increase in the relaxation rate (i.e. coherence peak) just below T$\\sb{\\rm c}$; nor is there an exponential T dependence at low temperature. Finally, the Knight shift below T$\\sb{\\rm c}$ does not follow the BCS weak coupling form, but shows an indication of strong coupling in $\\kappa$-(ET)$\\sb2$Cu (N(CN)$\\sb2$) Br. Issue Date: 1995 Type: Text Language: English URI: http:\/\/hdl.handle.net\/2142\/19936 Rights Information: Copyright 1995 DeSoto, Stewart Martin Date Available in IDEALS: 2011-05-07 Identifier in Online Catalog: AAI9543567 OCLC Identifier: (UMI)AAI9543567\n\ufeff","date":"2016-12-05 00:36:57","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\": 0.4932078421115875, \"perplexity\": 4410.41174956705}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-50\/segments\/1480698541517.94\/warc\/CC-MAIN-20161202170901-00246-ip-10-31-129-80.ec2.internal.warc.gz\"}"}
| null | null |
\section{Introduction}
\label{sec:intro}
Gait is an essential biometric, which has the unique advantage of human identification at a distance without physical contact. Gait empowers many real-world applications such as human retrieval, forensic identification, and serving robots.
Recently, great progress has been made to promote gait recognition from in-the-lab setting~\cite{casiab,oulp,oumvlp} to in-the-wild scenario~\cite{fvg,gait3d,grew,tumgait}. Despite these studies have made significant contributions to recent advances~\cite{gaitset,gaitpart,doumetagait,gaitgl,hopgait}, two inherent problems still remain: (1) \textit{lack of 3D geometry information}, and (2) \textit{poor feasibility in the real-world scenario}.
Existing camera-based methods~\cite{cstl,gaitnet} are counterintuitive to human nature. When recognizing a subject~\cite{gait3d,3dreid}, humans consider not only the 2D appearance characteristics, but also 3D geometry structure information like height, shape, and viewpoints. Differently, camera-based gait recognition methods either capture 2D representations from a single viewpoint as shown in Fig.~\ref{fig:silsmethod}, or exploit 3D representations from estimated 3D pose/mesh models which is usually imprecise in various challenging conditions of low resolution, poor illumination, untrained posture, etc.
Fortunately, 3D sensors provide precise 3D perception like human nature, \eg recognizing a subject from multiple views as illustrated in Fig.~\ref{fig:lidarmethod}.
Existing gait recognition research is mostly based on publicly camera-based datasets, neglecting the fact that walking videos are not guaranteed in unconstrained scenes.
Many factors in the real scenes like poor visibility and complex background~\cite{stcrowd}, cause that gait recognition suffers from accurate human detection and segmentation~\cite{liu2022bevfusion,liang2022gaitedge}.
With visual ambiguity and the abovementioned limitations, we believe that cameras are far behind the requirements of practical gait recognition. Considering the remarkable success of 3D applications~\cite{waymo,kitti,stcrowd}, it is highly desirable to study 3D gait recognition with a new modality, Lidar, to significantly provide 3D structure information and precisely percept humans in the wild. As Lidar sensors have been commonly used in robotics navigation~\cite{jrdbrobot,jrdbrobot2} and autonomous driving~\cite{kitti,waymo}, it also motivates us to endow gait recognition in a future scenario, where facilitating Lidar-based gait recognition on robots.
In this paper, we build the first large-scale Lidar-based dataset to facilitate the research of gait recognition with 3D point clouds, named LIDAR GAIT. Specifically, the dataset is captured in the wild, and collected by a Velodyne VLS128 Lidar sensor and an RGB camera mounted on a mobile robot. Compared to existing datasets listed in Tab.~\ref{tab:dataset}, our LIDAR GAIT dataset has the following distinctive attributes: (1) \textbf{Precision}. The LIDAR GAIT dataset provides 3D point clouds as gait representations with high precision and density, providing precise and robust 3D structure information for recognition. (2) \textbf{Scalability}. The dataset captures 25,279 sequences from 1,050 subjects, which is scalable for statistical evaluation. (3) \textbf{Diversity}. The dataset not only contains realistic challenges including illumination, occlusion, dressing, carrying, and \etal, but it also provides detailed annotations. Therefore, the LIDAR GAIT dataset helps the community to study the impact of diverse challenging factors. (4) \textbf{Multimodality}. Our dataset captured data streams from a Lidar and a camera at the same time, endowing the exploration of sensor fusion for robust gait recognition.
Since point clouds formats differently with pixels in images and point-based gait recognition has been barely investigated, we explore five cutting-edge methods from point-based object classification. Consequently, we observe that all implemented point-based methods perform second-optimal to the methods using silhouettes from the camera, encouraging us to investigate the reason. From analysis, we think the main reason is that those representative point-based methods are specifically designed for object classification. These object classification methods mainly focus on global context modeling and neglects local structure information.
To address this issue, we propose a simple yet effective baseline method, named multi-view projection network (MVPNet). The proposed MVPNet first projects 3D point clouds into depth images from three views. Then it utilizes convolutional networks to efficiently extract spatial features with local connectivity from projection, rather than other point-based methods learning global context from sparse point clouds. Extensive experiments show that (1) 3D structural information significantly contributes to performance improvement, and our proposed MVPNet is effective to maintain 3D structure for gait recognition. (2) Equipped with a Lidar sensor, point-based gait recognition performs stably well on cross-view challenges, demonstrating the effectiveness of Lidar for practical application. (3) There is a lot of room for the exploration of fine-grained feature extraction from sparse point cloud data.
To summarize, our main contributions are as follows:
(1) We carry out one of the first studies of 3D gait recognition with point clouds, bringing precise perception and 3D geometry of humans for better practicality in real-world scenarios.
(2) We build the first large-scale Lidar-based gait recognition benchmark, with various annotations ranging from occlusions, viewpoint, carrying, clothing, distance, and scenes.
(3) We propose a novel point cloud gait recognition framework, named MVPNet, outperforming camera-based methods by a large margin.
\begin{table*}[ht]
\begin{center}
\caption{Comparison of publicly available datasets for gait recognition. }
\label{tab:dataset}
\scalebox{0.83}{
\begin{threeparttable}
\small
\vspace{-5mm}
\begin{tabular}{lcccccccc}
\midrule[1.5pt]
Dataset & Year & Subject \# & Seq \# & View \# & Data Type & 3D structure & Multimodal & In-the-wild \\
\midrule[1.5pt]
CASIA-B~\cite{casiab} & 2006 & 124 & 13,640 & 11 & RGB, Silhouettes & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} \\
CASIA-C~\cite{nightgait} & 2006 & 153 & 1,530 & 1 & Infrared, Silhouettes & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} & \textcolor{green}{\ding{51}} \\
KY4D~\cite{ky4d} & 2010 & 42 & 168 & 16 & Silhouettes, RGB, 3D Volumetrics & \textcolor{green}{\ding{51}} & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} \\
TUM-GAID~\cite{tumgait} & 2012 & 305 & 3,370 & 1 & Audio, Video, Depth & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & \textcolor{red}{\ding{51}} \\
SZTAKI-LGA ~\cite{lidar2} & 2016 & 28 & 11 & 1 & 3D Point Cloud & \textcolor{green}{\ding{51}} & \textcolor{red}{\ding{55}} & \textcolor{green}{\ding{51}} \\
OU-MVLP~\cite{oumvlp} & 2018 & 10,307 & 288,596 & 14 & Silhouettes & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} \\
FVG ~\cite{fvg} & 2019 & 226 & 2856 & 3 & RGB & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} & \textcolor{green}{\ding{51}} \\
PCG ~\cite{lidar1} & 2020 & 30 & 60 & 1 & 3D Point Cloud & \textcolor{green}{\ding{51}} & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} \\
GREW~\cite{grew} & 2021 & 26,345 & 128,671 & 882 & Silhouettes, 2D/3D Skeleton, Flow & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} \\
Gait3D~\cite{gait3d} & 2022 & 4000 & 25309 & 39 & Silhouettes, 2D/3D Skeleton, 3D Mesh & \textcolor{green}{\ding{51}} & \textcolor{red}{\ding{55}} & \textcolor{green}{\ding{51}} \\
OUMVLP-Mesh ~\cite{oumvlpmesh} & 2022 & 10,307 & 288,596 & 14 & 3D Mesh & \textcolor{green}{\ding{51}} & \textcolor{red}{\ding{55}} & \textcolor{red}{\ding{55}} \\ \hline
\textbf{LIDAR GAIT} & - & \textbf{1,050} & \textbf{25,279} & \textbf{12} & \textbf{RGB, Silhouettes, 3D Point Cloud} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}} & \textcolor{green}{\ding{51}}\\
\end{tabular}
\end{threeparttable}
}
\end{center} \vspace{-5mm}
\end{table*}
\section{Related Work}
\label{sec:related}
\noindent\textbf{Gait Recognition.} According to the used representations, gait recognition can be generally divided into 2D and 3D representations-based methods.
The majority of 2D representations-based methods study gait characteristics directly from images, termed appearance-based~\cite{casiaa,gaitset,gaitpart,oumvlp,reversemask,mt3d,survey,metagait} methods, which have made surprising high performance based on silhouettes~\cite{liang2022gaitedge,gei,lixiangpose2} together with other gait templates~\cite{gei,wang2010chrono,MEI}.
The alternative approaches learn human structure~\cite{posegait,gaitgraph,lixiangpose} and dynamics~\cite{gaitgraph} as gait representations, but they are heavily constrained by model-based estimation models.
3D representation methods are generally extracted by sensors~\cite{tumgait,kinect} or estimation models~\cite{posegait,lixiangpose2}. The commonly used 3D sensors such as Kinect, provide 3D structured data, but they only facilitate in an indoor environment~\cite{kinect}. Meanwhile, multi-cameras reconstruction~\cite{tunnel} and 3D estimation models~\cite{gaitgraph,posegait,gaitnet,gait3d,lixiangpose2} provide considerable 3D geometry, but the performance is far behind the requirements of real-world applications as reported in~\cite{grew}.
\noindent\textbf{Gait Recognition Benchmark.} There are three types of publicly available datasets: in-the-lab~\cite{casiab,oumvlp,oulp}, synthetic~\cite{dou2021versatilegait}, and in-the-wild datasets~\cite{grew,gait3d,resgait,tumgait}. The in-the-lab datasets~\cite{casiaa,casiab,oumvlp,oulp}, represented by CASIA series~\cite{casiaa,casiab,nightgait} and OU-ISIR series~\cite{oumvlp,oulp}, advance the investigation of the feasibility of gait recognition.
The recent synthetic datasets~\cite{dou2021versatilegait} are to overcome the difficulty in data acquisition and annotation of gait, providing more synthetic data with a variety of annotations but introducing cross-domain issue~\cite{liang2022gaitedge} at the same time.
The in-the-wild datasets~\cite{resgait,grew,gait3d} are to promote gait recognition research in the unconstrained environment.
The recent works~\cite{lidar1,lidar2,lidarmethod} based on Lidar sensor are closely related to our work, while the main concern is that the existing datasets include at most 30 subjects, which cannot guarantee statistically reliable performance evaluation of Lidar-based gait recognition. Because of insufficient 3D representations for data-driven gait recognition, as shown in Tab.~\ref{tab:dataset}, a dataset with accurate 3D representations is essential.
\noindent\textbf{Point Cloud and 3D Object Classification.} 3D scanners (\eg Lidar and depth camera) project to the targets and then generate point cloud sets. Each point represents a data point in Cartesian coordinates $(X, Y, Z)$. Point cloud data is sparsely distributed, remaining a significant challenge in modeling correlation and geometry. 3D object classification explore projection-based~\cite{mvcnn,simpleview,viewgcn}, point-wise~\cite{pointnet,pointnet++,pointtransformer}, and graph-wise models~\cite{DGCNN,viewgcn} to capture discriminative feature on point cloud data for object classification. In this paper, we select many representative models of 3D object classification and compare them with our proposed method to comprehensively study 3D gait recognition with point clouds.
\section{The LIDAR GAIT Benchmark}
\subsection{Overview of LIDAR GAIT Dataset}
The LIDAR GAIT dataset is constructed by a 128-beam Lidar scanner and a monocular camera mounted at a fixed position on a robot. The LIDAR GAIT dataset includes 1,050 identities, 25,279 sequences, 762,896 point-cloud frames, and 3,044,499 silhouettes in total. Each sequence consists of continuous frames in RGB and points. Moreover, every frame of two modalities is timestamped, serving LIDAR GAIT as a synchronized multimodal dataset. We obtained informed consent from all participants, and we protect personal privacy by blurring faces.
To the end, LIDAR GAIT annotates various variances unincluded in existing in-the-wild datasets~\cite{gait3d,grew,resgait}.
\begin{figure}[t]
\centering
\includegraphics[width=0.95\linewidth]{Figure/setup.pdf}
\caption{Data acquisition setup. Each participant is first instructed to normally walk along four round-trip paths and four one-way paths, then walk again with a random variance along the same paths.}
\label{fig:setup}
\vspace{-5mm}
\end{figure}
\begin{figure*}[t]
\centering
\begin{subfigure}[b]{0.72\textwidth}
\centering
\includegraphics[width=1\textwidth]{Figure/illustration.pdf}
\caption{Samples of 12 views in two conditions for a subject.}
\label{fig:views}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.25\textwidth}
\centering
\includegraphics[width=1\textwidth,height=3.5cm]{Figure/demo.pdf}
\caption{Diverse attributes in LIDAR GAIT.}
\label{fig:demo}
\end{subfigure}
\hfill
\caption{(a) Each participant walks normally (top row), followed by walking with a random variance (bike for this subject) as shown in the bottom row. (b) LIDAR GAIT collects data in both point cloud and RGB modality with diverse realistic variances. \label{fig:dataset_preview}}
\end{figure*}
\subsection{Dataset Construction}
\subsubsection{Data Collection}
The data was collected in July 2022 in three scenes on a university campus using an industrial camera and a Lidar scanner. The camera captured RGB imagery streams at 1,280 x 980 resolution and 30 frames per second (FPS) while the point-cloud streams were recorded at 10 FPS. We synchronized Lidar and the camera to the GPS clock and timestamped each frame, which can help collaborate two modalities for a robust gait recognition system. As shown in Figure~\ref{fig:setup}, each subject first walks in normal condition along the \textit{one-way paths} and the \textit{round-trip paths}, then walks again but with a random attribute, for example, riding a bike as shown in Fig.~\ref{fig:views}. In total, 48 gait sequences
\textit{(= [4 $\times$ 2 (round-trip) + 4 $\times$ 1(one-way)] $\times$ 2 (twice) $\times$ 2 (modality))}
can be captured for each subject. The dataset is captured in the multiple scenes at the campus and includes a variety of in-the-wild variants such as changing lighting, walking in crowds, occlusions, carrying, and complex backgrounds.
\subsubsection{Variances}
As illustrated in Fig.~\ref{fig:dataset_preview}, the proposed dataset concludes the significant challenges: (1) \textbf{View.} The subjects walk along four green round-trip paths, resulting in eight viewpoints ranging from $0^{\circ}$, $45^{\circ}, 90^{\circ}, 135^{\circ}, 180^{\circ}, 235^{\circ}, 270^{\circ}, 315^{\circ}$. The four blue one-way paths generate four extra viewpoints: far-$0^{\circ}$, far-$90^{\circ}$, far-$180^{\circ}$, near-$270^{\circ}$. (2) \textbf{Occlusion.} The portions of the body are obstructed by occlusions as shown in Fig.~\ref{fig:demo}. (3) \textbf{Illumniation.} More than 70 identities are recorded at night. (4) \textbf{Carrying.} The dataset conclude various carrying objects including umbrellas, carton, tote bag, cellphone, clothes, chessboard, kids, balls, etc. (5) \textbf{Clothing and uniform.} Our dataset includes participants wearing different coats but also contains subjects wearing the same uniform. (6) \textbf{Others.} There are other variants in the dataset such as riding a bike, playing ball, walking with kids, etc. More samples of variants can be found in \textbf{the supplementary material}.
\subsubsection{Annotations}
The continuous data streams are first manually segmented into sequences according to the predefined trajectories in Fig.~\ref{fig:setup}. Then, each sequence is labeled by aforementioned variances such as view, occlusion, and illumination. In the end, camera-based sequences and Lidar-based sequences are separately processed to obtain gait representations.
\noindent\textbf{Camera-based Data.} The human boxes and trajectories are automatically generated by YOLOX~\cite{ge2021yolox} and ByteTrack~\cite{zhang2022bytetrack}, respectively. Since the LIDAR GAIT dataset is recorded in a public area, which resulted in multiple people in some frames, and a wrong trajectory when participants overlapped with the passers. We manually select the of-the-interest boxes as the ground truth in such cases. In the end, PaddleSeg~\cite{liu2021paddleseg} is performed to obtain silhouettes in size of $128\times128$. However, detection and segmentation perform well only under good lighting conditions but fail to deliver good performance at night.
\noindent\textbf{Lidar-based Data.} It is relatively easy to obtain human point clouds because there is only one person walking in the experimental area at a time. The point cloud sets of pedestrians are obtained after applying noise removal and ground removal on each frame. To protect the privacy of uninvolved passers, we only release the area range to $[-5, -12m]$ for the X axis, $[-3m, 3m]$ for the Y axis, and $[-2m,3m]$ for the Z axis.
\begin{figure*}[t]
\centering
\begin{subfigure}[b]{0.33\textwidth}
\centering
\includegraphics[width=1.1\textwidth]{Figure/distribution_2.pdf}
\caption{sequence \# over frame \#}
\label{fig:sequenceframe}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.33\textwidth}
\centering
\includegraphics[width=1.1\textwidth]{Figure/distribution_1.pdf}
\caption{frame \# over points/pixels \#}
\label{fig:framepoints}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.33\textwidth}
\centering
\includegraphics[width=1.04\textwidth]{Figure/distribution_3.pdf}
\caption{Attributes distribution}
\label{fig:distribution}
\end{subfigure}
\caption{Statistics about LIDAR GAIT dataset. Lidar modality and RGB modality are represented in blue and yellow, respectively. It shows that LIDAR GAIT dataset is scalable, multimodal, and diverse for the study of 3D gait recognition. Best viewed in color.}
\label{fig:three graphs}
\end{figure*}
\subsection{Dataset Statistics and Evaluation Metrics}
\label{sec:metric}
The statistics about the distributions of sequence, frame, and attribute are detailed in Fig.~\ref{fig:three graphs}. From Fig.~\ref{fig:sequenceframe}, we observe that the sequence scale of the two modalities is equal, but RGB modality has three times of frame number of Lidar modality. From Fig.~\ref{fig:framepoints}, the average pixel number of a silhouette is 3,200, while the average points number of a point cloud frame is 800. The rate of pixels vs points is approximately 1:1 and 4:1 when silhouettes are in the size of $64\times64$ and $128\times128$, respectively. To the end, the distribution of attributes is shown in Fig.~\ref{fig:distribution}, demonstrating the variety of the LIDAR GAIT dataset.
To establish a more challenging and realistic setting, the LIDAR GAIT dataset is evaluated under open-set setting~\cite{nixon2010book,oumvlp}, where train and test set splits are without sample intersection. The LIDAR GAIT dataset is separated into three splits: a train set with 250 identities and 5,988 sequences, a valid set with 250 identities and 5981 sequences, and a test set with 550 identities and 13,310 sequences. During the test, the sequences in normal conditions sets as gallery sets, and the sequences in variant conditions are taken as probe sets. The evaluation protocol follows the cross-view recognition setting as commonly used in CASIA-B~\cite{casiab} and OUMVLP~\cite{oumvlp}, where probe sets of the same view calculate the similarity to gallery sets of each view. To evaluate the impact of attributes, the probe sets are grouped into many subsets according to the attributes, then perform cross-view retrieval task. The prevailing Rank-1 accuracy is adopted as the evaluation metric.
\section{Gait Recognition with Point Clouds}
\subsection{Problem Setting}
In this section, we introduce the problem setting of 3D gait recognition with point clouds. Given a point cloud dataset $\mathcal{P}=\{\mathcal{P}_i^j|i=1,2...,N; j=1,2,...,m_i\}$ with N identities and $m_i$ sequence for each identity $y_i$. Each point cloud sequence $\mathcal{P}_{i}^{j} \in \mathbb{R}^{T \times N \times C} $ is with $T$ frames and $N$ points for each frame, where $C$ is the number of feature channels. Our goal is to learn a network $N_\theta(\cdot)$ that can produce the feature embedding $\mathcal{F}_{i}^{j}$ to represent the associated identity $y_i$.
In this paper, we propose the MVPNet, as shown in Fig.~\ref{fig:method}, to tackle the 3D gait recognition task, formulated as:
\begin{equation}
\mathcal{F}_{i}^{j} = N_\theta(\mathcal{G}(\mathcal{D}(\mathcal{P}_i^j)))
\vspace{-1mm}
\end{equation}
where $\mathcal{D}$ is the essential data augmentation to avoid overfitting and improve the network generalization ability. The function $\mathcal{G}$ operates on point clouds and generates multi-views depth images from multiple virtual cameras.
The feature extractor $N_\theta$ is composed of three components. 1) a feature encoder $\mathcal{E}$ that captures spatially local connectivity from projected multi-view images. 2) a multi-view feature fusion layer $\mathcal{S}$ that combine the learned 2D descriptors from multiple point-views into a compact 3D representation. 3) a temporal aggregation network $\mathcal{T}$ that models dynamical conjunction along sequential input, which can be formulated as:
\begin{equation}
N_\theta(\cdot) = \mathcal{T}(\mathcal{S}(\mathcal{E}(\cdot)))
\vspace{-1mm}
\end{equation}
After point clouds are transformed into depth images, $N_\theta$ is utilized to extract compact embedding as the final representations for recognition. With the spatial representations effectively exploited from multiple viewpoints, we combine the learned 2D descriptors from multiple point-views into a compact spatial-temporal representation, which contains informative 3D geometry. In the subsections below, we will go into further detail about the proposed approach.
\subsection{Multi-view Representations Generation}
\noindent\textbf{Point Cloud.} Point cloud is a set of sparse points in Euclidean space, where each point represents a data point in Cartesian coordinates (X, Y, Z). Lidar will generate points with additional information like intensity, but intensity reflects the fabric of clothing.
To avoid adding gait-irrelated features, we only reserve the 3D coordinates of points as input features. Therefore, the input data for our MVPNet is a sequential point cloud set $\mathcal{P}_{i}^{j} \in \mathbb{R}^{T \times N \times 3} $.
For more details about the impact of the number of T and N, the experiments are presented in \textbf{the supplementary material}.
\begin{figure*}[ht!]
\centering
\includegraphics[width=0.95\linewidth]{Figure/method1.pdf}
\caption{The framework of MVPNet for 3D gait recognition with point clouds. MVPNet receives a sequence of point clouds, then extract and combines representations from range-view, side-view, and bird's-eye-view. }
\label{fig:method}
\end{figure*}
\noindent\textbf{Data Augmentation.} Point cloud augmentation plays a significant role in 3D object classification~\cite{pointnet,pointnet++} from two aspects: 1) data augmentations enlarge the dataset to prevent overfitting from the limited availability of the dataset, 2) and act as the regularizer to learn more representative feature by randomly perturbing points. To fairly evaluate MVPNet with other point-based methods, we adopt data augmentation~\cite{simpleview,pointnet,pointtransformer}, formulated as:
${\mathcal{P}_{i}^j}^{\prime} = \mathcal{D}(\mathcal{P}_{i}^{j})$
where ${\mathcal{P}_{i}^j}^{\prime}$ is the augmented point clouds.
\noindent\textbf{Multi-view Representation.} As point cloud maintains 3D geometric information, Humans also recognize an object in a similar way, which constructs the 3D perception of the target from multiple perspectives. The similarity motivates us to conduct MVPNet for information aggregation from multiple perspectives. Here we acquire multi-view representations from three virtual cameras, \ie range-view, side-view, and bird's-eye-view (BEV) as illustrated in Fig.~\ref{fig:method}. The side-view and BEV cameras are orthogonally placed at a distance of 1.2 units from the pedestrian center~\cite{simpleview}. The range-view depth images are with high density and local connectivity, while the other two orthogonal cameras obtain depth images with sparse connectivity. The depth images can be formulated as:
$\mathcal{G}(\mathcal{P}_{i}^{j}) = \{{I}_{i,r}^{j}, {I}_{i,s}^{j}, {I}_{i,b}^{j}\}$
, where ${I}_{i,r}^{j}, {I}_{i,s}^{j}, {I}_{i,b}^{j}$ represent depth sequences at range-view, side-view, and BEV respectively.
\subsection{Multi-view Representation Learning}
\noindent\textbf{Feature Extractor.} The commonly used backbone~\cite{opengait} is adopted as the feature encoder $\mathcal{E}$. For all three views, feature encoder $\mathcal{E}$ with shared kernels is applied to learn multi-view representations $f_{i,mv}^{j} \in \mathbb{R}^{T \times D \times 3}$, where $D$ is the number of features channel. The multi-view representations $f_{i,mv}^{j}$ consists of three representations $f_{i,r}^{j}, f_{i,s}^{j}, f_{i,b}^{j}$ learned from range-view, side-view, and BEV respectively.
\noindent\textbf{Multi-View Feature Fusion Module} is designed to combine learned multiple representations from different perspectives into a dense representation $F_{i}^{j} \in \mathbb{R}^{T \times D_{fusion}}$, where $D_{fusion}$ refers to the dimension of the fused multi-view features. We study various fusion strategies and present more details and comparisons in \textbf{the supplementary material}.
\noindent\textbf{Temporal Feature Fusion} aggregates sequential dense multi-view representation $F_{i}^{j} \in \mathbb{R}^{T \times D_{fusion}}$ into sequence-level embedding $\mathcal{F}_{i}^{j} \in \mathbb{R}^{D_{fusion}}$, many temporal modeling methods exist in the literature~\cite{gaitset,gaitnet,hopgait}. Set Pooling is adopted for both modalities to fuse dynamic cues to fairly compare with the camera-based method.
\subsection{Traning and Inference}
In this work, the model is trained with a combined loss function, which is formulated as:
\begin{equation}
L = \alpha L_{tri} + \beta L_{ce}
\end{equation}
where $L_{tri}$ is $BA^{+}$ triplet loss~\cite{gaitset}, $L_{ce}$ is the cross-entropy loss~\cite{gaitgl}. $\alpha$ and $\beta$ are the weighted hyperparameters.
During inference, the whole sequence of point clouds is used, then data augmentation $\mathcal{D}$ only applies normalization on the input points. Euclidean distance is used to measure the similarity of each probe-gallery pair and calculate Rank 1 recognition accuracy.
\section{Experiments}
\subsection{Experimental Setup}
The LIDAR GAIT dataset is used to evaluate all the experiments.
We follow the dataset split and evaluation metric as detailed in Sec.~\ref{sec:metric}.
To evaluate the performance of different modalities in the wild, we select the most representative methods for each modality and compare our proposed MVPNet with these methods.
\subsubsection{Comparative Methods}
\textbf{Lidar-based Methods.}
We implement five commonly used approaches in point cloud classification including PointNet~\cite{pointnet}, PointNet++~\cite{pointnet++}, PointTransformer~\cite{pointtransformer}, DGCNN~\cite{DGCNN}, and SimpleView~\cite{simpleview}.
Among them, the methods~\cite{pointnet,pointnet++,pointtransformer} are point-wise models. DGCNN~\cite{DGCNN} is a representative graph-wise model, and SimpleView~\cite{simpleview} is chosen as a representative projection-based method.
\textbf{\textit{Implementation details}.} During training, the frame number, and point number are set to 10 and 512, respectively. All the points and the whole sequence are used for evaluation. For all point-based methods including MVPNet, we apply identical data augmentation including normalization, random scaling, and random shifting during training. Triplet loss with batch size ($p=32, k=8$) is applied for all models, where $p$ and $k$ donate the number of identities and sequences per identity, respectively. SGD~\cite{sgd} with a learning rate of 0.1 optimizes all point-based methods for 30,000 iterations.
\begin{table*}[t]
\centering
\caption{Evaluation of the impact of different attributes. Rank-1 accuracy (\%) on the \textit{valid + test} set is reported.
\label{tab:attributes}}
\scalebox{0.8}{%
\begin{tabular}{ccc|c|cccccc}
\midrule[1pt]
\multicolumn{3}{c|}{Method} & \multicolumn{1}{c|}{\multirow{2}{*}{Mean}} & \multicolumn{6}{c}{Probe Seuqence} \\
\multicolumn{1}{c}{Modality} & Publication & Model & \multicolumn{1}{c|}{} & Bag & Clothing & Umberalla & Uniform & Occlusion & Night \\
\midrule[1pt]
\multirow{4}{*}{Camera} & AAAI2019 & GaitSet~\cite{gaitset} & 64.43 & 68.33 & 35.57 & 64.78 & 63.53 & 65.21 & 23.33 \\
& CVPR2019 & GaitPart~\cite{gaitpart} & 64.18 & 67.65 & 35.91 & 63.08 & 63.62 & 65.28 & 20.18 \\
& GITHUB & OpenGait~\cite{opengait} & 65.85 & 70.01 & 38.04 & 65.96 & 64.96 & 67.05 & 23.82 \\
& ICCV2021 & GaitGL~\cite{gaitgl} & 76.09 & 78.98 & 51.23 & \textbf{76.92} & 76.38 & 77.68 & 21.58 \\ \hline
\multirow{6}{*}{Lidar} & CVPR2017 & PointNet~\cite{pointnet} & 23.63 & 30.00 & 19.57 & 14.52 & 21.40 & 44.96 & 21.17 \\
& NIPS2017 & PointNet++~\cite{pointnet++} & 32.19 & 32.72 & 23.64 & 27.68 & 27.49 & 41.53 & 33.79 \\
& TOG2019 & DGCNN~\cite{DGCNN} & 36.42 & 40.42 & 30.37 & 27.78 & 32.61 & 58.74 & 39.82 \\
& ICML2021 & SimpleView~\cite{simpleview} & 50.75 & 56.98 & 40.99 & 28.54 & 49.62 & 78.99 & 52.77 \\
& ICCV2021 & PointTransformer~\cite{pointtransformer} & 38.24 & 40.93 & 30.79 & 29.16 & 35.67 & 55.10 & 37.99 \\
& - & MVPNet (Range-view) & 79.24 & 80.68 & 56.65 & 60.29 & 73.20 & 92.32 & 83.19 \\
& - & \textbf{MVPNet (Ours)} & \textbf{82.41} & \textbf{83.31} & \textbf{61.91} & 64.11 & \textbf{77.40} & \textbf{94.81} & \textbf{85.53} \\
\midrule[1.5pt]
\end{tabular}
}
\end{table*}
\noindent\textbf{Camera-based Methods.}
To evaluate the performance of the camera-based modality, we implement four cutting-edge methods: GaitSet~\cite{gaitset}, OpenGait~\cite{opengait}, GaitPart~\cite{gaitpart}, and GaitGL~\cite{gaitgl}. \textbf{\textit{Implementation details}.} Since LIDAR GAIT has the equivalent scale of the training set to CASIA-B, the network parametric setting is identical to the configuration for CASIA-B. The frame number in the training is set to 30, and the whole sequence is used in the test evaluation. The silhouettes resolution is in the size of 64$\times$64. The optimizer, weight decay, and initial learning rate (LR) are set to Adam, 0.0005, and 0.0001, respectively. LR is multiplied by 0.1 at the 30,000th and 60,000th iterations, and all camera-based models train with 70,000 iterations in total.
\subsubsection{Implementation Details of MVPNet}
MVPNet is a projection-based method, which transfers point clouds into images and utilizes CNNs to extract appearance features. The commonly used backbone~\cite{gait3d,opengait} is utilized in MVPNet as a feature encoder.
The range-view images are size-normalized by~\cite{oulp}, and other rendered images use point-level normalization~\cite{pointnet}. All the projection-based images from point clouds are with a resolution of $64\times64$. The feature extractor is identical to ~\cite{opengait}. The optimizer and batch size is the same with silhouette-based methods. The weighted parameters for triplet loss and cross-entropy loss are set to 1 and 0.1, respectively. The code for all the experiments is written on codebase~\cite{opengait}.
\noindent\subsection{Comparative Results}
To compare with other point-based methods and understand what factors really impact gait recognition in the wild, we detailed conducted experiments on various realistic factors including illumination, distance, carrying, clothing, and occlusion. We separate all the sequences with the same variance into multiple subsets. For example, to evaluate the impact of illumination, the probe sets only contain sequences with bad illumination conditions, and the gallery sets are all the sequences in the test set with normal conditions. Th cross-view recognition on each subset is evaluated, and the cross-view accuracy matrix is like the results in Fig.~\ref{fig:comparison}. We only report the average of the accuracy matrix in Tab.~\ref{tab:attributes}, and we have the following observations: (1) MVPNet shows its superiority to both other point-based method and camera-based methods, which is mainly beneficial by integration with 3D geometry and multi-view appearance information. (2) MVPNet achieves state-of-the-art results in all conditions except the umbrella subset. It is mainly caused that umbrellas are erased after segmentation on RGB images, while the umbrellas are kept in point sets. (3) The methods using silhouettes make a poor performance at night. Point-based methods provide more promising results, and MVPNet outperforms others by a large margin. (4) All silhouette-learning models~\cite{gaitset,gaitpart,gaitgl,opengait} achieve higher accuracy than point-based models in point cloud classification. This concludes that it is necessary to specifically design point-based models for 3D gait recognition.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{Figure/comparison.pdf}
\caption{Comparison between Lidar and Camera for gait recognition. We report rank-1 accuracy (\%) on cross-view protocol. Best viewed in color and pdf.}
\label{fig:comparison}
\end{figure}
\noindent\textbf{Cross-view Gait Recognition.} We conduct a detailed comparison of two modalities of cross-view gait recognition in Fig.~\ref{fig:comparison}. Note that we utilize identical feature encoders for two modalities to make ablative results, and we can make the following observations: (1) the distance from subjects to sensors indeed impacts the performance for both two modalities. (2) Camera-based method achieves poor performance when query sets are at views of $0^{\circ}, 90^{\circ}, 180^{\circ}$ (see purple pixel in Fig.~\ref{fig:comparison}\textcolor{red}{a}). The same phenomenon can be found on CASIA-B~\cite{gaitset} and OUMVLP~\cite{oumvlp}. However, Lidar-based methods can perform stably in cross-view settings, validating the effectiveness of 3D structure for cross-view gait recognition.
\subsection{Ablation Study}
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{Figure/input.pdf}
\caption{Ablation study on the effectiveness of depth information for performance. Best viewed in color.}
\vspace{-3mm}
\label{fig:input}
\end{figure}
\noindent\textbf{Effectiveness of 3D Geometry Information.}
To evaluate the effectiveness of depth information for gait recognition, we make a comparison among four types of data as input: (1) Camera silhouettes: the camera-based silhouettes are obtained by segmentation results of RGB images. (2) Lidar silhouettes: Lidar silhouettes are obtained by range-view projection of point cloud sets. (3) Lidar depth: the depth information is added. From Fig.~\ref{fig:input}, we can observe that: (1) When depth information is not included, the performance of Lidar silhouettes is much lower than the accuracy of camera silhouettes. This is because the camera has a much higher resolution than Lidar, so the camera can catch the target with more details. (2) Though Lidar generates point clouds in sparse space, the depth of information makes a magnificent improvement to the accuracy. The integration of depth information can improve rank-1 accuracy from 33.55\% to 79.24\%, validating the necessity and effectiveness of 3D information for gait recognition.
\noindent\textbf{Effectiveness of Each View.}
From Fig.~\ref{fig:fourview}, we have the following observations: (1) The comparison between front-view and range-view, indicates that the convolutions can take more advantage from range-view depth maps. It is because the closer view makes the pixels within projected depth images sparser. (2) The side-view depth maps obtained from point cloud projection are reliable and effective, which provide a meaningful cues from another perspective. (3) The accuracy on bird-eye views are relatively low, but BEV images provide interesting evidence that gait recognition is the potential to be achieved at the bird-eye view.
\section{Discussion}
\noindent\textbf{Ethical Discussion.} All the subjects involved in the dataset signed a written consent to agree that their data can be collected, processed, used, and shared for research purposes. The dataset can be distributed only for non-commercial research purposes with the case-by-case dataset access application. The human faces of each RBG image are blurred to protect sensitive privacy. The recorded data can only be used for 20 years. After this date, all data will be deleted and not allowed to be used.
\noindent\textbf{Potential Negative Societal Impacts.} Point cloud gait recognition will endow robotics with the advanced perception of humans, which can benefit healthcare and social security. The main concern is to prevent biometrics security leaks.
\begin{figure}[t]
\centering
\includegraphics[width=0.83\linewidth]{Figure/fourview.pdf}
\caption{Performance comparison between different views. Best viewed in color. }
\vspace{-3mm}
\label{fig:fourview}
\end{figure}
\section{Conclusions}
In this paper, we introduce the Lidar sensor to provide reliable anthropometric parameters for the human body, and to perceive pedestrians in unconstrained scenes. First, we proposed a novel multi-view projection network for point cloud gait recognition, named MVPNet, to exploit 3D human geometry from multi-view representations.
Moreover, we build the first large-scale multimodal 3D point cloud gait recognition dataset, termed LIDAR GAIT, to facilitate the research of gait recognition with point cloud data. LIDAR GAIT contains 25,279 sequences with 1,050 subjects and covers various visibility, views, occlusions, clothing, carry, and scenes.
Lastly, our proposed method achieves remarkable results on the LIDAR GAIT dataset, showing the superior of LIDAR and the effectiveness of MVPNet.
\noindent\textbf{Future Work.} MVPNet has obtained remarkable results in various scenarios, yet it performs not well enough if subjects carry an umbrella. The reason should be that the umbrellas are wrongly included in the point cloud. Better performance can be achieved if the umbrellas are removed from the point clouds. Besides, MVPNet only takes one modality as input currently. LIDAR GAIT dataset is a multimodal dataset with synchronized RGB images and point clouds. Much better results should be achieved if the two modalities are fused.
\setcounter{section}{0}
\renewcommand{\thesection}{\Alph{section}}
\section{Ethical Discussion}
The LIDAR GAIT dataset has been reviewed by Institutional Review Board (IRB), and IRB also approves the use of the dataset. Our data acquisition has been authorized by each subject involved in the dataset. We have applied several processes to protect sensitive privacy, including blurring faces, limiting the duration of data usage, and requiring distribution agreements. Besides, all the data collectors obtained qualification certificates of ethics training.
\section{Ablative Results for Multi-view Fusion}
We study five ways to fuse multiple features $f_{mv} \in \mathbb{R}^{ 3 \times C} $ from three views into a multi-view feature:
\begin{itemize}
\item \textbf{Concatenation.} The given multiple features are naively concatenated together. Then the fused features are in the shape of $ \mathcal{F} \in \mathbb{R}^{ 3 \times C} $.
\item \textbf{Weighted Fusion.} The multiple features from three views are fused by a fully connected network (FCN) along channel dimension. The final features are in the shape of $ \mathcal{F} \in \mathbb{R}^{ C} $.
\item \textbf{Mean Pooling.} Statistically averaging multiple features along view dimension. All the final features of statistical pooling are in the shape of $ \mathcal{F} \in \mathbb{R}^{ C} $.
\item \textbf{Max Pooling.} Statistically selecting maximum features along view dimension.
\item \textbf{Min Pooling.} Statistically selecting minimum features along view dimension.
\end{itemize}
We show the overall rank-1 recognition accuracy evaluated in cross-view protocol over the validation and test set in Tab.~\ref{tab:fusion-table}. It can be seen that: (1) All fusion strategies, excluding min pooling, consistently improve performance, showing the effectiveness of multi-view features for performance improvement. (2) Though the mean pooling and weighted fusion show comparative performance, we choose statistically average pooling as the optimal fusion strategy because channel-wise requires more parameters.
\begin{table}[ht]
\caption{Ablative results of different fusion strategies.\label{tab:fusion-table}}
\scalebox{0.9}{
\begin{tabular}{|l|l|l|l|c|}
\hline
& Fusion & Rank-1 & Rank-5 & Parameters \# \\ \hline
Range-view & - & 79.24 & 92.98 & 3.254M \\
Side-view & - & 29.03 & 54.85 & 3.252M \\
BEV-view & - & 14.33 & 34.26 & 3.252M \\ \hline
MVPNet & Concat & 80.12 & 92.91 & 7.415M \\
MVPNet & Wighted & 82.32 & 94.10 & 5.351M \\
MVPNet & Max & 80.74 & 93.28 & 3.254M \\
MVPNet & Min & 79.25 & 92.58 & 3.254M \\
MVPNet & Mean & \textbf{82.41} & \textbf{94.14} & 3.254M \\ \hline
\end{tabular}
}
\end{table}
\section{Effect of Sequence Length and Point Number per frame}
We study the effect of sequence length and point number per frame during the training process. The sequence length and point number are set to fixed values for the training process. During the inference process, the whole sequence is used, and the point number per frame is the same as the point number in the training process.
It can be seen that: (1) MVPNet performs better when it trains with sequence length less than one gait cycle (10 frames in LIDAR GAIT) as shown in Fig.~\ref{fig:length-effect}. (2) The point number per frame does not impact recognition performance. It is because MVPNet relies on range-view depth images, and the projection-based methods are less sensitive to point numbers than other point-based methods such as DGCNN.
\begin{figure}[ht]
\centering
\resizebox{0.8\columnwidth}{!}{
\begin{tikzpicture}
\scalefont{0.7}
\begin{axis}[
sharp plot,
xmode=normal
xlabel={Number of frames},
ylabel={Rank-1 accuracy(\%)},
width=8cm, height=5cm,
xmin=0,xmax=16,
ymin=80, ymax=85,
xtick={0,3,5,10,15},
ytick={81,82,83,84,85},
xlabel near ticks,
ylabel near ticks,
ymajorgrids=true,
grid style=dashed,
legend style={at={(0.9,1.1)},anchor=south},
]
\addplot+[semithick,mark=x,mark options={scale=0.6}, color=color1] plot coordinates {
(3,82.244)
(5,82.869)
(10,82.41)
(15,80.272)
};
\addlegendentry{MVPNet
\end{axis}
\end{tikzpicture}
}
\caption{The performance of MVPNet training with different sequence length.}
\label{fig:length-effect}
\end{figure}
\begin{figure}[ht]
\centering
\resizebox{0.8\columnwidth}{!}{
\begin{tikzpicture}
\scalefont{0.7}
\begin{axis}[
sharp plot,
xmode=normal
xlabel={Number of points},
ylabel={Rank-1 accuracy(\%)},
width=8cm, height=5cm,
xmin=0,xmax=1050,
ymin=25, ymax=90,
xtick={0,256,512,1024},
ytick={},
xlabel near ticks,
ylabel near ticks,
ymajorgrids=true,
grid style=dashed,
legend style={at={(0.9,1.1)},anchor=south},
]
\addplot+[semithick,mark=x,mark options={scale=0.6}, color=color1] plot coordinates {
(256,81.903)
(512,82.150)
(1024,82.41)
};
\addlegendentry{MVPNet
\addplot+[semithick,mark options={scale=0.3}, color=color2] plot coordinates {
(256,31.292)
(512,36.42)
(1024,31.162)
};
\addlegendentry{DGCNN}
\end{axis}
\end{tikzpicture}
}
\caption{The performance of MVPNet and DGCNN training with different point numbers.}
\label{fig:point-effect}
\end{figure}
We have also studied the impact of sequence length on the inference process. Compared to the camera-based method using silhouettes, we can observe that: (1) Both camera-based and Lidar-based models obtain better performance with the increasing frames of the sequences. (2) When only given one frame for each sequence of probe and gallery, the Lidar-based method can surprisingly achieve $21.44$ \% rank-1 recognition accuracy and $47.31$ \% rank-$5$ recognition accuracy, showing the effectiveness of Lidar-based gait recognition in the few-shot setting.
\begin{figure}
\centering
\resizebox{0.8\columnwidth}{!}{
\begin{tikzpicture}
\scalefont{0.7}
\begin{axis}[
sharp plot,
xmode=normal
xlabel={Number of frames used in inference},
ylabel={Rank-1 accuracy(\%)},
width=8cm, height=5cm,
xmin=0,xmax=30,
ymin=0, ymax=100,
xtick={0,1,3,5,10,15,20,25,30},
ytick={20,40,60,80,100},
xlabel near ticks,
ylabel near ticks,
ymajorgrids=true,
grid style=dashed,
legend style={at={(0.9,1.1)},anchor=south},
]
\addplot+[semithick,mark=x,mark options={scale=0.6}, color=color1] plot coordinates {
(1,21.436)
(3,55.692)
(5,67.742)
(10,77.194)
(15,80.089)
(20,80.940)
(25,81.464)
(30,81.566)
};
\addlegendentry{Lidar-based
\addplot+[dashed,semithick,mark options={scale=0}, color=color1] plot coordinates {
(0,82.41)
(50,82.41)
};
\addlegendentry{All Point Clouds}
\addplot+[semithick,mark options={scale=0.3}, color=color2] plot coordinates {
(1,6.331)
(3,22.920)
(5,36.979)
(10,49.642)
(15,55.812)
(20,58.888)
(25,60.875)
(30,62.079)
};
\addlegendentry{Camera-based}
\addplot+[dashed,semithick,mark options={scale=0}, color=color2] plot coordinates {
(0,65.85)
(50,65.85)
};
\addlegendentry{All Silhouettes}
\end{axis}
\end{tikzpicture}
}
\caption{The performance comparison between Lidar and camera on used frame number in inference.}
\label{fig:label}
\end{figure}
\section{Qualitative results}
To analyze the performance gap between our MVPNet and representative PointNet, We visualize the feature distribution on the LIDAR GAIT dataset. We can observe that MVPNet can capture features with clear discrimination. As shown in Fig.~\ref{fig:mvpnettsne}, MVPNet prominently learns the inter-class margin and makes the intra-class distribution more compact. However, the representative point-wise model, PointNet, can only obtain global features as shown in Fig.~\ref{fig:pointnettsne}. PointNet captures features with less discrimination, and its intra-class features distribute sparsely.
\begin{figure}[ht]
\centering
\begin{subfigure}[b]{0.5
\textwidth}
\centering
\includegraphics[width=1\textwidth]{Figure/pointnet.pdf}
\caption{PointNet}
\label{fig:pointnettsne}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=1\textwidth]{Figure/mvptsne.pdf}
\caption{Our MVPNet}
\label{fig:mvpnettsne}
\end{subfigure}
\caption{Feature distributions are visulized by t-SNE. }
\label{fig:tsne}
\end{figure}
\section{Exemplar Sequences of LIDAR GAIT Dataset}
To demonstrate the necessity of the LIDAR GAIT dataset, We show several exemplar sequences of the LIDAR GAIT dataset under normal, occlusion, and poor illumination conditions in Fig.~\ref{fig:nm-demo}~-~\ref{fig:nt-demo}.
Fig.~\ref{fig:nm-demo} shows that Lidar provides informative geometry as significant cues that extend gait recognition from 2D to 3D space. The most considerable advantage of Lidar for gait recognition is that it allows for perspective from another viewpoint, as shown in the bottom row of Fig.~\ref{fig:nm-demo}.
When the pedestrians are occluded, as shown in Fig.~\ref{fig:oc-demo}, the silhouettes obtained by segmentation methods are typical with lower quality. The conventional segmentation methods are based on 2D cameras, but humans live in 3D space, making it difficult to separate the off-the-interest pedestrian from 2D space. Lidar with precise 3D information can obtain high-quality gait representation under the condition of occlusion.
When the pedestrians are occluded, as shown in Fig.~\ref{fig:oc-demo}, the silhouettes obtained by segmentation methods are typical with lower quality. The conventional segmentation methods are based on 2D cameras, but humans live in 3D space, making it difficult to separate the of-the-interest pedestrian from 2D space. With precise 3D information, Lidar can obtain high-quality gait representation under occlusion.
In Fig.~\ref{fig:demo-others}, we show gait representations in existing in-the-wild datasets, GREW and Gait3D. We can observe that failure gait representations commonly exist because of various factors in the real world. Therefore, it is necessary to investigate a new way to obtain robust gait representation in such complex scenarios.
\begin{figure*}[ht]
\centering
\includegraphics[width=0.95\linewidth]{Figure/demo_nm.pdf}
\caption{Exemplar sequences of LIDAR GAIT dataset under normal conditions. Eight frames in three modalities are visualized. The top three rows show three gait representations in RGB images, silhouettes, and point clouds. The bottom four rows represent gait representations in RGB images, silhouettes, front-view point clouds, and side-view point clouds. It shows Lidar provides informative 3D geometry. (Best viewed in color.)}
\label{fig:nm-demo}
\end{figure*}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.95\linewidth]{Figure/demo_oc.pdf}
\caption{Exemplar sequences of LIDAR GAIT dataset under occlusions. The top three rows show gait representations when another subject overlaps the of-the-interest pedestrian. The bottom three rows show gait representations occluded by a static obstruction. It indicates that Lidar can provide robust gait representations under occlusion conditions. (Best viewed in color.)}
\label{fig:oc-demo}
\end{figure*}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.95\linewidth]{Figure/demo_nt.pdf}
\caption{Exemplar sequences of LIDAR GAIT dataset under poor illumination. When illumination is extremely low, human segmentation is barely performed. In contrast, Lidar provides robust gait representation with point clouds regardless of lighting. (Best viewed in color.)}
\label{fig:nt-demo}
\end{figure*}
\begin{figure*}[ht]
\centering
\begin{subfigure}[b]{1
\textwidth}
\centering
\includegraphics[width=1\textwidth]{Figure/demo_grew.pdf}
\caption{GREW}
\label{fig:demo-grew}
\end{subfigure}
\hfill
\begin{subfigure}[b]{1\textwidth}
\centering
\includegraphics[width=1\textwidth]{Figure/demo_gait3d.pdf}
\caption{Gait3D}
\label{fig:demo-gait3d}
\end{subfigure}
\caption{Failure silhouettes in the existing in-the-wild datasets. The existing in-the-wild datasets face the issues that segmentation methods fail to provide gait representations with high quality by the effect of many real-world factors.}
\label{fig:demo-others}
\end{figure*}
{\small
\bibliographystyle{ieee_fullname}
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{"url":"https:\/\/zbmath.org\/authors\/?q=ai%3Ayang.yong","text":"Yang, Yong\n\nCompute Distance To:\n Author ID: yang.yong Published as: Yang, Yong; Yang, Y.; Yangyong more...less\n Documents Indexed: 264 Publications since 1990 Co-Authors: 217 Co-Authors with 187 Joint Publications 6,084 Co-Co-Authors\nall top 5\n\nCo-Authors\n\n 34 single-authored 11 Qian, Guohua 10 Keller, Thomas Michael 10 Yang, Zailin 8 Jiang, Guanxixi 7 Dong, Jianwei 7 Huang, Shuying 6 Li, Jinbao 6 Li, Lian 6 Lou, Guangpu 6 Peng, Xindong 5 Jin, Ping 5 Lin, Pan 5 Liu, Wende 5 Liu, Yang 4 Guermond, Jean-Luc 4 Liu, Shitian 4 Nguyen Ngoc Hung 4 Popov, Boyan 4 Sun, Baitao 4 Wu, Chufen 3 Anderson, Kevin 3 Guan, Hongbo 3 Jiang, Zhijie 3 Li, Changguo 3 Pei, Yongzhen 3 Rockwell, Donald 3 Song, Yunqiu 3 Zhang, Bingsheng 3 Zhou, Bo 3 Zhu, Huiqing 2 Beck, J\u00f3zsef 2 Betz, Alexander 2 Chao-Haft, Max 2 Chen, William Wai Lim 2 Chen, Xiaoyou 2 Chen, Zengqiang 2 Chung, Jacob N. 2 Crowe, Clayton T. 2 Fang, Yuming 2 Foia\u015f, Ciprian Ilie 2 Gintz, Michael 2 Gong, Ting 2 Gross, Laura K. 2 Guo, Ling 2 Guo, Tianjiao 2 Hao, Wenrui 2 Huang, Suzhou 2 Ji, Shiwei 2 Jolly, Michael S. 2 Kong, Hao 2 Kortje, Matthew 2 Lei, Youming 2 Li, Tianzeng 2 Li, Xinzhu 2 Liang, Chencheng 2 Luo, An 2 Mandal, Satya 2 Pan, Hongfei 2 Phillips, Timothy N. 2 Saavedra, Laura 2 Shi, Changmei 2 Sun, Cheng 2 Tan, Xia 2 Ter-Saakov, Anthony 2 Traverso, L. 2 Troutt, T. R. 2 Tu, Lilan 2 Tu, Wei 2 Wang, Guoyin 2 Wang, Jianzhou 2 Wang, Rong 2 Wang, Yu 2 Wang, Zili 2 Weng, Peixuan 2 Wu, Ying 2 Xia, Tian 2 Yu, Jun 2 Yuan, Zhuzhi 2 Zeng, Ling 2 Zhang, Chongqun 2 Zhu, Junhui 1 Adcock, Thomas A. A. 1 Askari, D. 1 Bai, Yaohui 1 Bessenrodt, Christine 1 Bian, Jinlai 1 Bondeson, Anders 1 Brambleby, R. 1 Cai, Hua 1 Chen, Guangqiu 1 Chen, Jiabo 1 Chen, Kewang 1 Chen, Lansun 1 Chen, Shengli 1 Chen, Yicun 1 Cheng, M. K. 1 Chiclana, Francisco 1 Dai, Jingguo 1 D\u00e9fago, Xavier 1 Dong, Shuqin ...and 161 more Co-Authors\nall top 5\n\nSerials\n\n 15 Journal of Algebra 12 Communications in Algebra 5 Bulletin of the Australian Mathematical Society 5 Journal of Pure and Applied Algebra 5 Monatshefte f\u00fcr Mathematik 5 Acta Mathematica Hungarica 5 Journal of Geodesy 5 Journal of Group Theory 4 Israel Journal of Mathematics 4 Journal of Fluid Mechanics 4 Journal of Lanzhou University. Natural Sciences 4 Mathematics in Practice and Theory 4 Journal of Intelligent and Fuzzy Systems 4 Journal of Algebra and its Applications 3 Wave Motion 3 Chaos, Solitons and Fractals 3 Applied Mathematics and Computation 3 Archiv der Mathematik 3 Journal of Optimization Theory and Applications 3 Journal of Sichuan University. Natural Science Edition 3 Mathematical Problems in Engineering 3 Journal of Northwest Normal University. Natural Science 3 Journal of Jilin University. Science Edition 3 Waves in Random and Complex Media 2 Acta Mechanica 2 International Journal of Multiphase Flow 2 International Journal for Numerical Methods in Fluids 2 International Journal of Solids and Structures 2 Mathematical Methods in the Applied Sciences 2 Physica A 2 Information Sciences 2 Multidimensional Systems and Signal Processing 2 Applied Mathematical Modelling 2 Journal of Guizhou Normal University. Natural Science 2 Pure and Applied Mathematics 2 Fundamenta Informaticae 2 Discrete and Continuous Dynamical Systems. Series B 2 Journal of the Australian Mathematical Society 2 Journal of Shandong University. Natural Science 2 MATCH - Communications in Mathematical and in Computer Chemistry 2 Journal of Control Theory and Applications 2 Communications in Computational Physics 2 International Journal of Biomathematics 2 Communications in Theoretical Physics 2 Open Mathematics 2 AMM. Applied Mathematics and Mechanics. (English Edition) 1 Computers and Fluids 1 Communications in Mathematical Physics 1 IMA Journal of Applied Mathematics 1 International Journal of Systems Science 1 International Journal of Theoretical Physics 1 Journal of Engineering Mathematics 1 Journal of Mathematical Analysis and Applications 1 Journal of the Mechanics and Physics of Solids 1 Nonlinearity 1 Physics Letters. A 1 Rocky Mountain Journal of Mathematics 1 Journal of Geometry and Physics 1 Advances in Mathematics 1 Automatica 1 International Journal for Numerical Methods in Engineering 1 Journal of Applied Probability 1 Journal of Computational and Applied Mathematics 1 Management Science 1 Mathematics and Computers in Simulation 1 Mathematische Zeitschrift 1 Pacific Journal of Mathematics 1 Proceedings of the American Mathematical Society 1 Proceedings of the Edinburgh Mathematical Society. Series II 1 Publicationes Mathematicae Debrecen 1 SIAM Journal on Numerical Analysis 1 Systems & Control Letters 1 Acta Scientiarum Naturalium Universitatis Sunyatseni 1 Applied Mathematics and Mechanics. (English Edition) 1 Acta Automatica Sinica 1 Acta Mathematicae Applicatae Sinica. English Series 1 International Journal of Intelligent Systems 1 Mathematica Applicata 1 Journal of Scientific Computing 1 Economics Letters 1 IEEE Transactions on Signal Processing 1 Applied Intelligence 1 Numerical Algorithms 1 Journal of Physics A: Mathematical and General 1 Pattern Recognition 1 SIAM Journal on Applied Mathematics 1 Analele \u0218tiin\u021bifice ale Universit\u0103\u021bii Al. I. Cuza din Ia\u0219i. Serie Nou\u0103. Matematic\u0103 1 Archive of Applied Mechanics 1 SIAM Journal on Scientific Computing 1 Applied Mathematics. Series A (Chinese Edition) 1 The Journal of Fuzzy Mathematics 1 Journal of Lie Theory 1 Bulletin des Sciences Math\u00e9matiques 1 Engineering Analysis with Boundary Elements 1 Journal of Mathematical Chemistry 1 Journal of Vibration and Control 1 Mathematics and Mechanics of Solids 1 Journal of Shanghai Jiaotong University (English Edition) 1 Soft Computing 1 Algebras and Representation Theory ...and 45 more Serials\nall top 5\n\nFields\n\n 73 Group theory and generalizations\u00a0(20-XX) 30 Computer science\u00a0(68-XX) 27 Partial differential equations\u00a0(35-XX) 21 Fluid mechanics\u00a0(76-XX) 20 Mechanics of deformable solids\u00a0(74-XX) 18 Mathematical logic and foundations\u00a0(03-XX) 18 Systems theory; control\u00a0(93-XX) 17 Biology and other natural sciences\u00a0(92-XX) 14 Numerical analysis\u00a0(65-XX) 12 Game theory, economics, finance, and other social and behavioral sciences\u00a0(91-XX) 11 Dynamical systems and ergodic theory\u00a0(37-XX) 11 Information and communication theory, circuits\u00a0(94-XX) 9 Ordinary differential equations\u00a0(34-XX) 8 Number theory\u00a0(11-XX) 8 Geophysics\u00a0(86-XX) 8 Operations research, mathematical programming\u00a0(90-XX) 6 Combinatorics\u00a0(05-XX) 6 Associative rings and algebras\u00a0(16-XX) 6 Nonassociative rings and algebras\u00a0(17-XX) 5 Commutative algebra\u00a0(13-XX) 5 Global analysis, analysis on manifolds\u00a0(58-XX) 5 Statistics\u00a0(62-XX) 4 Optics, electromagnetic theory\u00a0(78-XX) 4 Quantum theory\u00a0(81-XX) 4 Statistical mechanics, structure of matter\u00a0(82-XX) 3 Linear and multilinear algebra; matrix theory\u00a0(15-XX) 3 Operator theory\u00a0(47-XX) 3 Calculus of variations and optimal control; optimization\u00a0(49-XX) 3 Probability theory and stochastic processes\u00a0(60-XX) 2 Harmonic analysis on Euclidean spaces\u00a0(42-XX) 2 Mechanics of particles and systems\u00a0(70-XX) 2 Classical thermodynamics, heat transfer\u00a0(80-XX) 2 Relativity and gravitational theory\u00a0(83-XX) 1 Order, lattices, ordered algebraic structures\u00a0(06-XX) 1 Algebraic geometry\u00a0(14-XX) 1 Real functions\u00a0(26-XX) 1 Functional analysis\u00a0(46-XX) 1 Geometry\u00a0(51-XX) 1 Differential geometry\u00a0(53-XX)\n\nCitations contained in zbMATH Open\n\n134 Publications have been cited 637 times in 369 Documents Cited by Year\nA second-order maximum principle preserving Lagrange finite element technique for nonlinear scalar conservation equations.\u00a0Zbl\u00a01302.65225\nGuermond, Jean-Luc; Nazarov, Murtazo; Popov, Bojan; Yang, Yong\n2014\nA novel hyperchaos system only with one equilibrium.\u00a0Zbl\u00a01236.37022\nChen, Zengqiang; Yang, Yong; Qi, Guoyuan; Yuan, Zhuzhi\n2006\nOn defining long-range dependence.\u00a0Zbl\u00a00912.60050\nHeyde, C. C.; Yang, Y.\n1997\nA single three-wing or four-wing chaotic attractor generated from a three-dimensional smooth quadratic autonomous system.\u00a0Zbl\u00a01152.37312\nChen, Zengqiang; Yang, Yong; Yuan, Zhuzhi\n2008\nRichards model revisited: validation by and application to infection dynamics.\u00a0Zbl\u00a01337.92219\nWang, Xiang-Sheng; Wu, Jianhong; Yang, Yong\n2012\nOrbits of the actions of finite solvable groups.\u00a0Zbl\u00a01200.20007\nYang, Yong\n2009\nThe multi-fuzzy soft set and its application in decision making.\u00a0Zbl\u00a01426.03035\nYang, Yong; Tan, Xia; Meng, Congcong\n2013\nRegular orbits of finite primitive solvable groups.\u00a0Zbl\u00a01195.20017\nYang, Yong\n2010\nGlobally convergent optimization algorithms on Riemannian manifolds: Uniform framework for unconstrained and constrained optimization.\u00a0Zbl\u00a01153.90017\nYang, Y.\n2007\nRegular orbits of finite primitive solvable groups. II.\u00a0Zbl\u00a01247.20021\nYang, Yong\n2011\nNonsolvable groups with no prime dividing three character degrees.\u00a0Zbl\u00a01325.20005\nQian, Guohua; Yang, Yong\n2015\nForced waves and their asymptotics in a Lotka-Volterra cooperative model under climate change.\u00a0Zbl\u00a01428.35173\nYang, Yong; Wu, Chufen; Li, Zunxian\n2019\nAbelian quotients and orbit sizes of solvable linear groups.\u00a0Zbl\u00a01353.20007\nKeller, Thomas Michael; Yang, Yong\n2016\nLarge character degrees of solvable $$3'$$-groups.\u00a0Zbl\u00a01231.20008\nYang, Yong\n2011\nOn the commuting probability and supersolvability of finite groups.\u00a0Zbl\u00a01300.20034\nLescot, Paul; Nguyen, Hung Ngoc; Yang, Yong\n2014\nRegular orbits of nilpotent subgroups of solvable linear groups.\u00a0Zbl\u00a01222.20010\nYang, Yong\n2011\nGeometrically nonlinear analysis of cylindrical shells using the element-free kp-Ritz method.\u00a0Zbl\u00a01195.74305\nZhao, X.; Yang, Y.; Liew, K. M.\n2007\nBlocks of small defect.\u00a0Zbl\u00a01327.20009\nYang, Yong\n2015\n$$H_{\\infty }$$ controller synthesis for pendulum-like systems.\u00a0Zbl\u00a01157.93375\nYang, Y.; Huang, Lin\n2003\nTwo computationally efficient polynomial-iteration infeasible interior-point algorithms for linear programming.\u00a0Zbl\u00a01406.90075\nYang, Y.\n2018\n3-D container packing heuristics.\u00a0Zbl\u00a01085.90050\nLim, A.; Rodrigues, B.; Yang, Y.\n2005\nAdaptively robust filtering for kinematic geodetic positioning.\u00a0Zbl\u00a01036.86513\nYang, Y.; He, H.; Xu, G.\n2001\nAdjustable soft discernibility matrix based on picture fuzzy soft sets and its applications in decision making.\u00a0Zbl\u00a01361.91034\nYang, Yong; Liang, Chencheng; Ji, Shiwei; Liu, Tingting\n2015\nEpidemic waves of a spatial SIR model in combination with random dispersal and non-local dispersal.\u00a0Zbl\u00a01426.35218\nWu, Chufen; Yang, Yong; Zhao, Qianyi; Tian, Yanling; Xu, Zhiting\n2017\nIntuitionistic fuzzy decision-making with similarity measures and OWA operator.\u00a0Zbl\u00a01323.91020\nSu, Weihua; Yang, Yong; Zhang, Chonghui; Zeng, Shouzhen\n2013\nIntersection theory of algebraic obstructions.\u00a0Zbl\u00a01195.13010\nMandal, Satya; Yang, Yong\n2010\nIntuitionistic fuzzy sets: Spherical representation and distances.\u00a0Zbl\u00a01163.03031\nYang, Y.; Chiclana, F.\n2009\nInterval-valued hesitant fuzzy soft sets and their application in decision making.\u00a0Zbl\u00a01342.68316\nPeng, Xindong; Yang, Yong\n2015\nTime analyticity with higher norm estimates for the 2D Navier-Stokes equations.\u00a0Zbl\u00a01339.35211\nFoias, Ciprian; Jolly, Michael S.; Lan, Ruomeng; Rupam, Rishika; Yang, Yong; Zhang, Bingsheng\n2015\nExperiments on particle dispersion in a plane wake.\u00a0Zbl\u00a01137.76792\nYang, Y.; Crowe, C. T.; Chung, J. N.; Troutt, T. R.\n2000\nThe effect of the consistent mass matrix on the maximum-principle for scalar conservation equations.\u00a0Zbl\u00a01368.65192\nGuermond, Jean-Luc; Popov, Bojan; Yang, Yong\n2017\nA theory of finitely durable goods monopoly with used-goods market and transaction costs.\u00a0Zbl\u00a01232.91451\nHuang, S.; Yang, Y.; Anderson, K.\n2001\nWave-structure interaction: simulation driven by quantitative imaging.\u00a0Zbl\u00a01041.76058\nSirisup, S.; Karniadakis, G. E.; Yang, Y.; Rockwell, D.\n2004\nMinimal energy of bicyclic graphs of a given diameter.\u00a0Zbl\u00a01141.05056\nYang, Yong; Zhou, Bo\n2008\nSynchronization in a fractional-order dynamic network with uncertain parameters using an adaptive control strategy.\u00a0Zbl\u00a01383.34006\nDu, Lin; Yang, Yong; Lei, Youming\n2018\nA decision making approach based on bipolar multi-fuzzy soft set theory.\u00a0Zbl\u00a01305.91125\nYang, Yong; Peng, Xindong; Chen, Hao; Zeng, Ling\n2014\nThe analog of Huppert\u2019s conjecture on character codegrees.\u00a0Zbl\u00a01423.20009\nYang, Yong; Qian, Guohua\n2017\nLarge orbits of subgroups of solvable linear groups.\u00a0Zbl\u00a01302.20006\nYang, Yong\n2014\nAbelian quotients and orbit sizes of linear groups.\u00a0Zbl\u00a01472.20021\nKeller, Thomas Michael; Yang, Yong\n2020\nDynamics of an impulsive control system which prey species share a common predator.\u00a0Zbl\u00a01198.34015\nPei, Yongzhen; Yang, Yong; Li, Changguo\n2009\nRobust estimator for correlated observations based on bifactor equivalent weights.\u00a0Zbl\u00a01158.86340\nYang, Y.; Song, L.; Xu, T.\n2002\nInvariant domains preserving arbitrary Lagrangian Eulerian approximation of hyperbolic systems with continuous finite elements.\u00a0Zbl\u00a01365.65222\nGuermond, Jean-Luc; Popov, Bojan; Saavedra, Laura; Yang, Yong\n2017\nScaling the kernel function based on the separating boundary in input space: a data-dependent way for improving the performance of kernel methods.\u00a0Zbl\u00a01239.68065\nSun, Jiancheng; Li, Xiaohe; Yang, Yong; Luo, Jianguo; Bai, Yaohui\n2012\nA multi-moment transport model on cubed-sphere grid.\u00a0Zbl\u00a01241.76320\nChen, C. G.; Xiao, F.; Li, X. L.; Yang, Y.\n2011\nSolvable permutation groups and orbits on power sets.\u00a0Zbl\u00a01297.20002\nYang, Yong\n2014\nOrbits of finite solvable groups on characters.\u00a0Zbl\u00a01305.20006\nKeller, Thomas Michael; Yang, Yong\n2014\nCohomology of model filiform Lie superalgebras.\u00a0Zbl\u00a01390.17022\nLiu, Wende; Yang, Yong\n2018\nExistence and uniqueness of forced waves in a delayed reaction-diffusion equation in a shifting environment.\u00a0Zbl\u00a01454.35220\nWu, Chufen; Yang, Yong; Wu, Zehao\n2021\nImproving the teleportation scheme of five-qubit state with a seven-qubit quantum channel.\u00a0Zbl\u00a01412.81072\nYang, Yong; Jiang, Min; Zhou, LiuLei\n2018\nWave propagation in two-dimensional anisotropic acoustic metamaterials of K4 topology.\u00a0Zbl\u00a01467.74043\nFallah, A. S.; Yang, Y.; Ward, R.; Tootkaboni, M.; Brambleby, R.; Louhghalam, A.; Louca, L. A.\n2015\nPrescribing zeros and poles on a compact Riemann surface for a gravitationally coupled Abelian gauge field theory.\u00a0Zbl\u00a01077.83061\nYang, Y.\n2004\nShape optimization for radar cross sections by a gradient method.\u00a0Zbl\u00a01074.78006\nBondeson, A.; Yang, Y.; Weinerfelt, P.\n2004\nContinuum theory and phase-field simulation of magnetoelectric effects in multiferroic bismuth ferrite.\u00a0Zbl\u00a01200.74055\nLi, L. J.; Yang, Y.; Shu, Y. C.; Li, J. Y.\n2010\nMultilayer graph cuts based unsupervised color-texture image segmentation using multivariate mixed Student\u2019s $$t$$-distribution and regional credibility merging.\u00a0Zbl\u00a01255.68183\nYang, Yong; Han, Shoudong; Wang, Tianjiang; Tao, Wenbing; Tai, Xue-Cheng\n2013\nExcision in algebraic obstruction theory.\u00a0Zbl\u00a01272.13012\nMandal, Satya; Yang, Yong\n2012\nApplication of a modified Lindstedt-Poincar\u00e9 method in coupled TDOF systems with quadratic nonlinearity and a constant external excitation.\u00a0Zbl\u00a01168.70302\nLim, C. W.; Lai, S. K.; Wu, B. S.; Sun, W. P.; Yang, Y.; Wang, C.\n2009\nGeneralised Clark-Ocone formulae for differential forms.\u00a0Zbl\u00a01331.60106\nYang, Y.\n2012\nOn $$p$$-parts of character degrees and conjugacy class sizes of finite groups.\u00a0Zbl\u00a006850686\nYang, Yong; Qian, Guohua\n2018\nThe largest character degree and the Sylow subgroups of finite groups.\u00a0Zbl\u00a01347.20008\nQian, Guohua; Yang, Yong\n2016\nA microstructure-based description for cyclic plasticity of pearlitic steel with experimental verification.\u00a0Zbl\u00a00988.74504\nPeng, X.; Fan, J.; Yang, Y.\n2002\nWave interaction with a vertical cylinder: Spanwise flow patterns and loading.\u00a0Zbl\u00a00993.76515\nYang, Y.; Rockwell, D.\n2002\nRobust Kalman filter for rank deficient observation models.\u00a0Zbl\u00a00999.86013\nKoch, K. R.; Yang, Y.\n1998\nRobust estimation of geodetic datum transformation.\u00a0Zbl\u00a01001.86507\nYang, Y.\n1999\nFlow past a rotationally oscillating cylinder.\u00a0Zbl\u00a01294.76027\nKumar, S.; Lopez, C.; Probst, O.; Francisco, G.; Askari, D.; Yang, Y.\n2013\nNumerical simulation of water entry with improved SPH method.\u00a0Zbl\u00a01404.76208\nShao, J. R.; Yang, Y.; Gong, H. F.; Liu, M. B.\n2019\nRobust single-image super-resolution based on adaptive edge-preserving smoothing regularization.\u00a0Zbl\u00a01409.94816\nHuang, Shuying; Sun, Jun; Yang, Yong; Fang, Yuming; Lin, Pan; Que, Yue\n2018\nSynchronization of fractional-order hyperchaotic systems via fractional-order controllers.\u00a0Zbl\u00a01419.34164\nLi, Tianzeng; Wang, Yu; Yang, Yong\n2014\nDynamic analysis of a cylindrical cavity in inhomogeneous elastic half-space subjected to SH waves.\u00a0Zbl\u00a01425.74245\nYang, Zailin; Jiang, Guanxixi; Tang, Haiyi; Sun, Baitao; Yang, Yong\n2019\nQuantitative behavior of non-integrable systems. I.\u00a0Zbl\u00a01474.37033\nBeck, J.; Donders, M.; Yang, Y.\n2020\nLarge orbit sizes in finite group actions.\u00a0Zbl\u00a01456.20006\nQian, Guohua; Yang, Yong\n2021\nPermutation characters in finite solvable groups.\u00a0Zbl\u00a01402.20015\nQian, Guohua; Yang, Yong\n2018\nOn $$p$$-parts of conjugacy class sizes of finite groups.\u00a0Zbl\u00a01442.20009\nYang, Yong; Qian, Guohua\n2018\nMixed finite element methods for groundwater flow in heterogeneous aquifers.\u00a0Zbl\u00a01391.76755\nTraverso, L.; Phillips, T. N.; Yang, Y.\n2013\nThe largest character degree, conjugacy class size and subgroups of finite groups.\u00a0Zbl\u00a01402.20016\nQian, Guohua; Yang, Yong\n2018\nBlow-up of spherically symmetric smooth solutions to quantum hydrodynamic models.\u00a0Zbl\u00a01393.35166\nDong, Jianwei; Yang, Yong; Lou, Guangpu\n2018\nGlobal convergence of a class of discrete-time interconnected pendulum-like systems.\u00a0Zbl\u00a01145.93033\nYang, Y.; Duan, Z. S.; Huang, L.\n2007\nComparison study of dynamics in one-sided and two-sided solid-combustion models.\u00a0Zbl\u00a01210.35299\nYang, Y.; Gross, L. K.; Yu, J.\n2010\nTraveling waves in a diffusive predator-prey system of Holling type: point-to-point and point-to-periodic heteroclinic orbits.\u00a0Zbl\u00a01262.92061\nWu, Chufen; Yang, Yong; Weng, Peixuan\n2013\nImplementation of supervisory control using extended finite-state machines.\u00a0Zbl\u00a01156.93302\nYang, Y.; Mannani, A.; Gohari, P.\n2008\nLog-Gabor energy based multimodal medical image fusion in NSCT domain.\u00a0Zbl\u00a01307.92256\nYang, Yong; Tong, Song; Huang, Shuying; Lin, Pan\n2014\nPest management of a prey-predator model with sexual favoritism.\u00a0Zbl\u00a01163.92041\nPei, Yongzhen; Yang, Yong; Li, Changguo; Chen, Lansun\n2009\nAn automatic hybrid method for retinal blood vessel extraction.\u00a0Zbl\u00a01176.92030\nYang, Yong; Huang, Shuying; Rao, Nini\n2008\nA revised TOPSIS method based on interval fuzzy soft set models with incomplete weight information.\u00a0Zbl\u00a01375.68111\nYang, Yong; Peng, Xindong\n2017\nScattering of shear waves by a cylindrical inclusion in an anisotropic half space.\u00a0Zbl\u00a01380.74061\nYang, Zai-lin; Jiang, Guan-xi-xi; Sun, Bai-tao; Yang, Yong\n2017\nComments on: \u201cFuzzy multicriteria decision making method based on the improved accuracy function for interval-valued intuitionistic fuzzy sets\u201d by Ridvan Sahin.\u00a0Zbl\u00a01409.91098\nYang, Yong; Liang, Chencheng; Ji, Shiwei\n2017\nSpectral properties of the temporal evolution of brain network structure.\u00a0Zbl\u00a01374.92026\nWang, Rong; Zhang, Zhen-Zhen; Ma, Jun; Yang, Yong; Lin, Pan; Wu, Ying\n2015\nLarge orbits of odd-order subgroups of solvable linear groups.\u00a0Zbl\u00a01252.20008\nYang, Yong\n2012\nQuantitative behavior of non-integrable systems. II.\u00a0Zbl\u00a01474.37034\nBeck, J.; Donders, M.; Yang, Y.\n2020\nCouple stress-based nonlinear primary resonant dynamics of FGM composite truncated conical microshells integrated with magnetostrictive layers.\u00a0Zbl\u00a01476.83020\nYang, Yong; Sahmani, S.; Safaei, B.\n2021\nScattering of out-plane wave by a circular cavity near the right-angle interface in the exponentially inhomogeneous media.\u00a0Zbl\u00a007211706\nYang, Zailin; Zhang, Chongqun; Yang, Yong; Sun, Baitao\n2017\nPermutation groups and orbits on the power set.\u00a0Zbl\u00a007272801\nYang, Yong\n2020\nOn Huppert\u2019s $$\\rho-\\sigma$$ conjecture.\u00a0Zbl\u00a007493673\nLiu, Yang; Yang, Yong\n2022\nMulti-frame image super-resolution reconstruction based on spatial information weighted fields of experts.\u00a0Zbl\u00a01455.94022\nHuang, Shuying; Wu, Jiajun; Yang, Yong; Lin, Pan\n2020\nOn nilpotent group actions on the characters of solvable groups.\u00a0Zbl\u00a01443.20009\nYang, Yong\n2020\nQuantitative behavior of non-integrable systems. III.\u00a0Zbl\u00a007551258\nBeck, J.; Chen, W. W. L.; Yang, Y.\n2022\nComments on \u201cAn object-parameter approach to predicting unknown data in incomplete fuzzy soft sets\u201d.\u00a0Zbl\u00a01443.03037\nYang, Yong; Song, Juanping; Peng, Xindong\n2015\nNonsolvable groups with five character codegrees.\u00a0Zbl\u00a007327523\nLiu, Yang; Yang, Yong\n2021\nA note on larger orbit size.\u00a0Zbl\u00a01472.20040\nJin, Ping; Yang, Yong\n2020\nOn the number of conjugacy classes of zeros of characters and the length of solvable groups.\u00a0Zbl\u00a01485.20024\nYang, Yong\n2020\nAdaptive feedback synchronization of fractional-order complex dynamic networks.\u00a0Zbl\u00a01387.93100\nLei, Youming; Yang, Yong; Fu, Rui; Wang, Yanyan\n2017\nOn Huppert\u2019s $$\\rho-\\sigma$$ conjecture.\u00a0Zbl\u00a007493673\nLiu, Yang; Yang, Yong\n2022\nQuantitative behavior of non-integrable systems. III.\u00a0Zbl\u00a007551258\nBeck, J.; Chen, W. W. L.; Yang, Y.\n2022\nRegular orbits of finite primitive solvable groups. III.\u00a0Zbl\u00a007420015\nYang, Yong; Vasil’ev, Alexey; Vdovin, Evgeny\n2022\nPermutation groups and set-orbits on the power set.\u00a0Zbl\u00a007453225\nYan, Yanxiong; Yang, Yong\n2022\nExistence and uniqueness of forced waves in a delayed reaction-diffusion equation in a shifting environment.\u00a0Zbl\u00a01454.35220\nWu, Chufen; Yang, Yong; Wu, Zehao\n2021\nLarge orbit sizes in finite group actions.\u00a0Zbl\u00a01456.20006\nQian, Guohua; Yang, Yong\n2021\nCouple stress-based nonlinear primary resonant dynamics of FGM composite truncated conical microshells integrated with magnetostrictive layers.\u00a0Zbl\u00a01476.83020\nYang, Yong; Sahmani, S.; Safaei, B.\n2021\nNonsolvable groups with five character codegrees.\u00a0Zbl\u00a007327523\nLiu, Yang; Yang, Yong\n2021\nOn analogues of Huppert\u2019s conjecture.\u00a0Zbl\u00a007394393\nYang, Yong\n2021\nAbelian quotients and orbit sizes of linear groups.\u00a0Zbl\u00a01472.20021\nKeller, Thomas Michael; Yang, Yong\n2020\nQuantitative behavior of non-integrable systems. I.\u00a0Zbl\u00a01474.37033\nBeck, J.; Donders, M.; Yang, Y.\n2020\nQuantitative behavior of non-integrable systems. II.\u00a0Zbl\u00a01474.37034\nBeck, J.; Donders, M.; Yang, Y.\n2020\nPermutation groups and orbits on the power set.\u00a0Zbl\u00a007272801\nYang, Yong\n2020\nMulti-frame image super-resolution reconstruction based on spatial information weighted fields of experts.\u00a0Zbl\u00a01455.94022\nHuang, Shuying; Wu, Jiajun; Yang, Yong; Lin, Pan\n2020\nOn nilpotent group actions on the characters of solvable groups.\u00a0Zbl\u00a01443.20009\nYang, Yong\n2020\nA note on larger orbit size.\u00a0Zbl\u00a01472.20040\nJin, Ping; Yang, Yong\n2020\nOn the number of conjugacy classes of zeros of characters and the length of solvable groups.\u00a0Zbl\u00a01485.20024\nYang, Yong\n2020\nThe convergence and superconvergence of a MFEM for elliptic optimal control problems.\u00a0Zbl\u00a01488.65614\nGuan, Hongbo; Yang, Yong; Zhu, Huiqing\n2020\nNonlinear derivations of incidence algebras.\u00a0Zbl\u00a01474.16069\nYang, Y.\n2020\nFinite solvable groups with distinct monomial character degrees.\u00a0Zbl\u00a01484.20008\nQian, Guohua; Yang, Yong\n2020\nOn sub-class sizes of finite groups.\u00a0Zbl\u00a01484.20043\nQian, Guohua; Yang, Yong\n2020\nArithmetical conditions of orbit sizes of linear groups of odd order.\u00a0Zbl\u00a01442.20002\nYang, Yong\n2020\nOn the odd order composition factors of finite linear groups.\u00a0Zbl\u00a01479.20017\nBetz, Alexander; Chao-Haft, Max; Gong, Ting; Ter-Saakov, Anthony; Yang, Yong\n2020\nForced waves and their asymptotics in a Lotka-Volterra cooperative model under climate change.\u00a0Zbl\u00a01428.35173\nYang, Yong; Wu, Chufen; Li, Zunxian\n2019\nNumerical simulation of water entry with improved SPH method.\u00a0Zbl\u00a01404.76208\nShao, J. R.; Yang, Y.; Gong, H. F.; Liu, M. B.\n2019\nDynamic analysis of a cylindrical cavity in inhomogeneous elastic half-space subjected to SH waves.\u00a0Zbl\u00a01425.74245\nYang, Zailin; Jiang, Guanxixi; Tang, Haiyi; Sun, Baitao; Yang, Yong\n2019\nConvergence of a homotopy finite element method for computing steady states of Burgers\u2019 equation.\u00a0Zbl\u00a01446.65092\nHao, Wenrui; Yang, Yong\n2019\nTwo computationally efficient polynomial-iteration infeasible interior-point algorithms for linear programming.\u00a0Zbl\u00a01406.90075\nYang, Y.\n2018\nSynchronization in a fractional-order dynamic network with uncertain parameters using an adaptive control strategy.\u00a0Zbl\u00a01383.34006\nDu, Lin; Yang, Yong; Lei, Youming\n2018\nCohomology of model filiform Lie superalgebras.\u00a0Zbl\u00a01390.17022\nLiu, Wende; Yang, Yong\n2018\nImproving the teleportation scheme of five-qubit state with a seven-qubit quantum channel.\u00a0Zbl\u00a01412.81072\nYang, Yong; Jiang, Min; Zhou, LiuLei\n2018\nOn $$p$$-parts of character degrees and conjugacy class sizes of finite groups.\u00a0Zbl\u00a006850686\nYang, Yong; Qian, Guohua\n2018\nRobust single-image super-resolution based on adaptive edge-preserving smoothing regularization.\u00a0Zbl\u00a01409.94816\nHuang, Shuying; Sun, Jun; Yang, Yong; Fang, Yuming; Lin, Pan; Que, Yue\n2018\nPermutation characters in finite solvable groups.\u00a0Zbl\u00a01402.20015\nQian, Guohua; Yang, Yong\n2018\nOn $$p$$-parts of conjugacy class sizes of finite groups.\u00a0Zbl\u00a01442.20009\nYang, Yong; Qian, Guohua\n2018\nThe largest character degree, conjugacy class size and subgroups of finite groups.\u00a0Zbl\u00a01402.20016\nQian, Guohua; Yang, Yong\n2018\nBlow-up of spherically symmetric smooth solutions to quantum hydrodynamic models.\u00a0Zbl\u00a01393.35166\nDong, Jianwei; Yang, Yong; Lou, Guangpu\n2018\nClass 2 quotients of solvable linear groups.\u00a0Zbl\u00a01430.20016\nKeller, Thomas Michael; Yang, Yong\n2018\nNonlinear stability of traveling waves for a multi-type SIS epidemic model.\u00a0Zbl\u00a01382.35316\nLi, Mengqi; Weng, Peixuan; Yang, Yong\n2018\nA variation on a theorem of Gluck.\u00a0Zbl\u00a01427.20014\nYang, Yong\n2018\nEpidemic waves of a spatial SIR model in combination with random dispersal and non-local dispersal.\u00a0Zbl\u00a01426.35218\nWu, Chufen; Yang, Yong; Zhao, Qianyi; Tian, Yanling; Xu, Zhiting\n2017\nThe effect of the consistent mass matrix on the maximum-principle for scalar conservation equations.\u00a0Zbl\u00a01368.65192\nGuermond, Jean-Luc; Popov, Bojan; Yang, Yong\n2017\nThe analog of Huppert\u2019s conjecture on character codegrees.\u00a0Zbl\u00a01423.20009\nYang, Yong; Qian, Guohua\n2017\nInvariant domains preserving arbitrary Lagrangian Eulerian approximation of hyperbolic systems with continuous finite elements.\u00a0Zbl\u00a01365.65222\nGuermond, Jean-Luc; Popov, Bojan; Saavedra, Laura; Yang, Yong\n2017\nA revised TOPSIS method based on interval fuzzy soft set models with incomplete weight information.\u00a0Zbl\u00a01375.68111\nYang, Yong; Peng, Xindong\n2017\nScattering of shear waves by a cylindrical inclusion in an anisotropic half space.\u00a0Zbl\u00a01380.74061\nYang, Zai-lin; Jiang, Guan-xi-xi; Sun, Bai-tao; Yang, Yong\n2017\nComments on: \u201cFuzzy multicriteria decision making method based on the improved accuracy function for interval-valued intuitionistic fuzzy sets\u201d by Ridvan Sahin.\u00a0Zbl\u00a01409.91098\nYang, Yong; Liang, Chencheng; Ji, Shiwei\n2017\nScattering of out-plane wave by a circular cavity near the right-angle interface in the exponentially inhomogeneous media.\u00a0Zbl\u00a007211706\nYang, Zailin; Zhang, Chongqun; Yang, Yong; Sun, Baitao\n2017\nAdaptive feedback synchronization of fractional-order complex dynamic networks.\u00a0Zbl\u00a01387.93100\nLei, Youming; Yang, Yong; Fu, Rui; Wang, Yanyan\n2017\nThe size of the largest conjugacy classes and the Sylow $$p$$-subgroups of finite groups.\u00a0Zbl\u00a01472.20041\nYang, Yong\n2017\nA complex function method of SH wave scattering in inhomogeneous medium.\u00a0Zbl\u00a01384.74021\nYang, Zai-lin; Zhang, Chong-qun; Jiang, Guan-xi-xi; Yan, Pei-lei; Yang, Yong\n2017\nAbelian quotients and orbit sizes of solvable linear groups.\u00a0Zbl\u00a01353.20007\nKeller, Thomas Michael; Yang, Yong\n2016\nThe largest character degree and the Sylow subgroups of finite groups.\u00a0Zbl\u00a01347.20008\nQian, Guohua; Yang, Yong\n2016\nAn iterative learning control design method for nonlinear discrete-time systems with unknown iteration-varying parameters and control direction.\u00a0Zbl\u00a01400.93181\nZhao, Duo; Yang, Yong\n2016\nApplication of complex network method to spatiotemporal patterns in a neuronal network.\u00a0Zbl\u00a01400.92037\nWang, Rong; Li, Jiajia; Wang, Li; Yang, Yong; Lin, Pan; Wu, Ying\n2016\nNonsolvable groups with no prime dividing three character degrees.\u00a0Zbl\u00a01325.20005\nQian, Guohua; Yang, Yong\n2015\nBlocks of small defect.\u00a0Zbl\u00a01327.20009\nYang, Yong\n2015\nAdjustable soft discernibility matrix based on picture fuzzy soft sets and its applications in decision making.\u00a0Zbl\u00a01361.91034\nYang, Yong; Liang, Chencheng; Ji, Shiwei; Liu, Tingting\n2015\nInterval-valued hesitant fuzzy soft sets and their application in decision making.\u00a0Zbl\u00a01342.68316\nPeng, Xindong; Yang, Yong\n2015\nTime analyticity with higher norm estimates for the 2D Navier-Stokes equations.\u00a0Zbl\u00a01339.35211\nFoias, Ciprian; Jolly, Michael S.; Lan, Ruomeng; Rupam, Rishika; Yang, Yong; Zhang, Bingsheng\n2015\nWave propagation in two-dimensional anisotropic acoustic metamaterials of K4 topology.\u00a0Zbl\u00a01467.74043\nFallah, A. S.; Yang, Y.; Ward, R.; Tootkaboni, M.; Brambleby, R.; Louhghalam, A.; Louca, L. A.\n2015\nSpectral properties of the temporal evolution of brain network structure.\u00a0Zbl\u00a01374.92026\nWang, Rong; Zhang, Zhen-Zhen; Ma, Jun; Yang, Yong; Lin, Pan; Wu, Ying\n2015\nComments on \u201cAn object-parameter approach to predicting unknown data in incomplete fuzzy soft sets\u201d.\u00a0Zbl\u00a01443.03037\nYang, Yong; Song, Juanping; Peng, Xindong\n2015\nApproaches to interval-valued intuitionistic hesitant fuzzy soft sets based decision making.\u00a0Zbl\u00a01334.68222\nPeng, Xin-Dong; Yang, Yong\n2015\nOn nilpotent and solvable quotients of primitive groups.\u00a0Zbl\u00a01359.20001\nKeller, Thomas Michael; Yang, Yong\n2015\nA second-order maximum principle preserving Lagrange finite element technique for nonlinear scalar conservation equations.\u00a0Zbl\u00a01302.65225\nGuermond, Jean-Luc; Nazarov, Murtazo; Popov, Bojan; Yang, Yong\n2014\nOn the commuting probability and supersolvability of finite groups.\u00a0Zbl\u00a01300.20034\nLescot, Paul; Nguyen, Hung Ngoc; Yang, Yong\n2014\nA decision making approach based on bipolar multi-fuzzy soft set theory.\u00a0Zbl\u00a01305.91125\nYang, Yong; Peng, Xindong; Chen, Hao; Zeng, Ling\n2014\nLarge orbits of subgroups of solvable linear groups.\u00a0Zbl\u00a01302.20006\nYang, Yong\n2014\nSolvable permutation groups and orbits on power sets.\u00a0Zbl\u00a01297.20002\nYang, Yong\n2014\nOrbits of finite solvable groups on characters.\u00a0Zbl\u00a01305.20006\nKeller, Thomas Michael; Yang, Yong\n2014\nSynchronization of fractional-order hyperchaotic systems via fractional-order controllers.\u00a0Zbl\u00a01419.34164\nLi, Tianzeng; Wang, Yu; Yang, Yong\n2014\nLog-Gabor energy based multimodal medical image fusion in NSCT domain.\u00a0Zbl\u00a01307.92256\nYang, Yong; Tong, Song; Huang, Shuying; Lin, Pan\n2014\nGeneral parameterized time-frequency transform.\u00a0Zbl\u00a01394.94663\nYang, Y.; Peng, Z. K.; Dong, X. J.; Zhang, W. M.; Meng, G.\n2014\nOn whether zero is in the global attractor of the 2D Navier-Stokes equations.\u00a0Zbl\u00a01304.35485\nFoias, Ciprian; Jolly, Michael S.; Yang, Yong; Zhang, Bingsheng\n2014\nRegular and $$p$$-regular orbits of solvable linear groups.\u00a0Zbl\u00a01311.20017\nKeller, Thomas Michael; Yang, Yong\n2014\nThe multi-fuzzy soft set and its application in decision making.\u00a0Zbl\u00a01426.03035\nYang, Yong; Tan, Xia; Meng, Congcong\n2013\nIntuitionistic fuzzy decision-making with similarity measures and OWA operator.\u00a0Zbl\u00a01323.91020\nSu, Weihua; Yang, Yong; Zhang, Chonghui; Zeng, Shouzhen\n2013\nMultilayer graph cuts based unsupervised color-texture image segmentation using multivariate mixed Student\u2019s $$t$$-distribution and regional credibility merging.\u00a0Zbl\u00a01255.68183\nYang, Yong; Han, Shoudong; Wang, Tianjiang; Tao, Wenbing; Tai, Xue-Cheng\n2013\nFlow past a rotationally oscillating cylinder.\u00a0Zbl\u00a01294.76027\nKumar, S.; Lopez, C.; Probst, O.; Francisco, G.; Askari, D.; Yang, Y.\n2013\nMixed finite element methods for groundwater flow in heterogeneous aquifers.\u00a0Zbl\u00a01391.76755\nTraverso, L.; Phillips, T. N.; Yang, Y.\n2013\nTraveling waves in a diffusive predator-prey system of Holling type: point-to-point and point-to-periodic heteroclinic orbits.\u00a0Zbl\u00a01262.92061\nWu, Chufen; Yang, Yong; Weng, Peixuan\n2013\nRichards model revisited: validation by and application to infection dynamics.\u00a0Zbl\u00a01337.92219\nWang, Xiang-Sheng; Wu, Jianhong; Yang, Yong\n2012\nScaling the kernel function based on the separating boundary in input space: a data-dependent way for improving the performance of kernel methods.\u00a0Zbl\u00a01239.68065\nSun, Jiancheng; Li, Xiaohe; Yang, Yong; Luo, Jianguo; Bai, Yaohui\n2012\nExcision in algebraic obstruction theory.\u00a0Zbl\u00a01272.13012\nMandal, Satya; Yang, Yong\n2012\nGeneralised Clark-Ocone formulae for differential forms.\u00a0Zbl\u00a01331.60106\nYang, Y.\n2012\nLarge orbits of odd-order subgroups of solvable linear groups.\u00a0Zbl\u00a01252.20008\nYang, Yong\n2012\nEstimation of response of plate structure subject to low veloctiy impact by a solid object.\u00a0Zbl\u00a01359.74282\nYang, Y.; Lam, N. T. K.; Zhang, L.\n2012\nOrbits of the actions of odd-order solvable groups.\u00a0Zbl\u00a01248.20017\nYang, Yong\n2012\nRegular orbits of finite primitive solvable groups. II.\u00a0Zbl\u00a01247.20021\nYang, Yong\n2011\nLarge character degrees of solvable $$3'$$-groups.\u00a0Zbl\u00a01231.20008\nYang, Yong\n2011\nRegular orbits of nilpotent subgroups of solvable linear groups.\u00a0Zbl\u00a01222.20010\nYang, Yong\n2011\nA multi-moment transport model on cubed-sphere grid.\u00a0Zbl\u00a01241.76320\nChen, C. G.; Xiao, F.; Li, X. L.; Yang, Y.\n2011\nHomology sequence and excision theorem for Euler class group.\u00a0Zbl\u00a01210.13016\nYang, Yong\n2011\nRegular orbits of finite primitive solvable groups.\u00a0Zbl\u00a01195.20017\nYang, Yong\n2010\nIntersection theory of algebraic obstructions.\u00a0Zbl\u00a01195.13010\nMandal, Satya; Yang, Yong\n2010\nContinuum theory and phase-field simulation of magnetoelectric effects in multiferroic bismuth ferrite.\u00a0Zbl\u00a01200.74055\nLi, L. J.; Yang, Y.; Shu, Y. C.; Li, J. Y.\n2010\nComparison study of dynamics in one-sided and two-sided solid-combustion models.\u00a0Zbl\u00a01210.35299\nYang, Y.; Gross, L. K.; Yu, J.\n2010\nOrbits of the actions of finite solvable groups.\u00a0Zbl\u00a01200.20007\nYang, Yong\n2009\nIntuitionistic fuzzy sets: Spherical representation and distances.\u00a0Zbl\u00a01163.03031\nYang, Y.; Chiclana, F.\n2009\n...and 34 more Documents\nall top 5\n\nCited by 664 Authors\n\n 50 Yang, Yong 11 Keller, Thomas Michael 10 Guermond, Jean-Luc 9 Kuzmin, Dmitri 8 Chen, Zengqiang 8 Popov, Boyan 6 Belozyorov, Vasiliy Ye. 6 Qi, Guoyuan 6 Qian, Guohua 5 De Luna, Manuel Quezada 5 Lewis, Mark L. 5 Moret\u00f3, Alexander 5 Nazarov, Murtazo 4 Badia, Santiago 4 Barrenechea, Gabriel R. 4 Dolfi, Silvio 4 Ghaffarzadeh, Mehdi 4 Ghasemi, Mohsen 4 Jiang, Guanxixi 4 Liu, Wende 4 Pacifici, Emanuele 4 Pei, Yongzhen 4 Wang, Jiabing 4 Wang, Zenghui 4 Yang, Zailin 3 Aziziheris, Kamal 3 Bonilla, Jes\u00fas 3 Burman, Erik 3 Cang, Shijian 3 Chinram, Ronnason 3 Chowell, Gerardo 3 Das, Mrinal Kanti 3 Foia\u015f, Ciprian Ilie 3 Li, Jianquan 3 Li, Jinbao 3 Li, Wan-Tong 3 Liu, Yang 3 Ma, Junhai 3 Mahmood, Tahir 3 Mandal, Satya 3 Nguyen Ngoc Hung 3 Sambale, Benjamin 3 Shadid, John N. 3 Song, Yunqiu 3 Titi, Edriss Saleh 3 Tomas, Ignacio 3 Wang, Xiangsheng 3 Wolf, Thomas R. 3 Wu, Chufen 3 Yuan, Rong 2 Akhlaghi, Zeinab 2 Ak\u0131n, Canan 2 Ali, Zeeshan 2 An, Jiayu 2 Ayg\u00fcn, Halis 2 Bai, Jiahui 2 Ballester-Bolinches, Adolfo 2 Biswas, Animikh 2 B\u00fcrger, Raimund 2 \u00c7amliyurt, G\u00fcher 2 Chen, Xiaohong 2 Chen, Xiaoyou 2 Choi, Wonhyung 2 Dadras, Sara 2 Du, Lin 2 Esteban-Romero, Ram\u00f3n 2 Farthing, Matthew W. 2 Frank, Martin 2 Gilboa, Guy 2 Guo, Jong-Shenq 2 Hajduk, Hennes 2 Halasi, Zolt\u00e1n 2 Hao, Wenrui 2 Hsu, Cheng-Hsiung 2 Huo, Bofeng 2 Iampan, Aiyared 2 Ji, Shengjin 2 Jia, Hongyan 2 Jia, Nuo 2 John, Volker 2 Kamac\u0131, H\u00fcseyin 2 Kees, Christopher E. 2 Khan, Muhammad Jabir 2 Knobloch, Petr 2 Kolev, Tzanio V. 2 Kukavica, Igor 2 Kumam, Wiyada 2 Kusch, Jonas 2 L\u0103zureanu, Cristian 2 Li, Changguo 2 Li, Xinzhu 2 Li, Yonghai 2 Liang, Hu 2 Lin, Guo 2 Liu, Chongxin 2 Liu, Ling 2 Lv, Yunfei 2 Mar\u00f3ti, Attila 2 Meng, Hangyang 2 Mishra, Bibekananda ...and 564 more Authors\nall top 5\n\nCited in 151 Serials\n\n 19 Journal of Algebra 18 Nonlinear Dynamics 16 Computer Methods in Applied Mechanics and Engineering 14 Applied Mathematics and Computation 10 Communications in Algebra 9 Journal of Computational Physics 8 Chaos, Solitons and Fractals 8 Soft Computing 7 Journal of Group Theory 6 Mathematical Problems in Engineering 6 Discrete and Continuous Dynamical Systems. Series B 6 Journal of Intelligent and Fuzzy Systems 6 Mathematical Biosciences and Engineering 5 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 5 Nonlinear Analysis. Real World Applications 5 Symmetry 4 Bulletin of the Australian Mathematical Society 4 Israel Journal of Mathematics 4 Archiv der Mathematik 4 Information Sciences 4 Mathematische Zeitschrift 4 Monatshefte f\u00fcr Mathematik 4 Journal of Algebra and its Applications 4 Advances in Difference Equations 4 Waves in Random and Complex Media 3 International Journal of Theoretical Physics 3 Physica A 3 Journal of Computational and Applied Mathematics 3 Journal of Pure and Applied Algebra 3 Quaestiones Mathematicae 3 Applied Mathematical Modelling 3 SIAM Journal on Scientific Computing 3 Computational and Applied Mathematics 3 Chaos 3 Communications in Nonlinear Science and Numerical Simulation 3 International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 3 International Journal of Biomathematics 3 Journal of Theoretical Biology 3 AMM. Applied Mathematics and Mechanics. (English Edition) 2 Computers & Mathematics with Applications 2 Journal of Mathematical Analysis and Applications 2 Mathematical Biosciences 2 ZAMP. Zeitschrift f\u00fcr angewandte Mathematik und Physik 2 Annali di Matematica Pura ed Applicata. Serie Quarta 2 Journal of Differential Equations 2 Mathematics and Computers in Simulation 2 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 2 Proceedings of the American Mathematical Society 2 SIAM Journal on Numerical Analysis 2 Transactions of the American Mathematical Society 2 Physica D 2 Journal of Scientific Computing 2 European Journal of Operational Research 2 Linear Algebra and its Applications 2 Journal of Dynamics and Differential Equations 2 Electronic Journal of Differential Equations (EJDE) 2 Complexity 2 Journal of Mathematical Chemistry 2 Algebras and Representation Theory 2 Discrete Dynamics in Nature and Society 2 Bulletin of the Malaysian Mathematical Sciences Society. Second Series 2 Iranian Journal of Fuzzy Systems 2 AIMS Mathematics 1 International Journal of Modern Physics B 1 Computers and Fluids 1 International Journal of Control 1 Indian Journal of Pure & Applied Mathematics 1 Journal of the Franklin Institute 1 Linear and Multilinear Algebra 1 Rocky Mountain Journal of Mathematics 1 Ukrainian Mathematical Journal 1 Wave Motion 1 Journal of Geometry and Physics 1 Advances in Mathematics 1 Algebra and Logic 1 International Journal of Circuit Theory and Applications 1 Journal of the London Mathematical Society. Second Series 1 Manuscripta Mathematica 1 Numerische Mathematik 1 Proceedings of the Edinburgh Mathematical Society. Series II 1 Results in Mathematics 1 Annales de l\u2019Institut Henri Poincar\u00e9. Analyse Non Lin\u00e9aire 1 Applied Numerical Mathematics 1 Asia-Pacific Journal of Operational Research 1 Applied Mathematics Letters 1 Forum Mathematicum 1 Multidimensional Systems and Signal Processing 1 Neural Computation 1 International Journal of Algebra and Computation 1 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 1 Indagationes Mathematicae. New Series 1 Journal of Mathematical Imaging and Vision 1 Journal of Nonlinear Science 1 Topology Proceedings 1 Journal of Mathematical Sciences (New York) 1 Journal of the Egyptian Mathematical Society 1 Turkish Journal of Mathematics 1 Filomat 1 Journal of Inverse and Ill-Posed Problems 1 Journal of Lie Theory ...and 51 more Serials\nall top 5\n\nCited in 43 Fields\n\n 95 Group theory and generalizations\u00a0(20-XX) 59 Partial differential equations\u00a0(35-XX) 55 Biology and other natural sciences\u00a0(92-XX) 51 Ordinary differential equations\u00a0(34-XX) 48 Dynamical systems and ergodic theory\u00a0(37-XX) 48 Numerical analysis\u00a0(65-XX) 39 Game theory, economics, finance, and other social and behavioral sciences\u00a0(91-XX) 29 Fluid mechanics\u00a0(76-XX) 24 Mathematical logic and foundations\u00a0(03-XX) 22 Computer science\u00a0(68-XX) 16 Operations research, mathematical programming\u00a0(90-XX) 16 Systems theory; control\u00a0(93-XX) 14 Information and communication theory, circuits\u00a0(94-XX) 13 Statistics\u00a0(62-XX) 10 Combinatorics\u00a0(05-XX) 7 Mechanics of deformable solids\u00a0(74-XX) 6 Commutative algebra\u00a0(13-XX) 5 General topology\u00a0(54-XX) 5 Geophysics\u00a0(86-XX) 4 Order, lattices, ordered algebraic structures\u00a0(06-XX) 4 Nonassociative rings and algebras\u00a0(17-XX) 4 $$K$$-theory\u00a0(19-XX) 4 Quantum theory\u00a0(81-XX) 3 General algebraic systems\u00a0(08-XX) 3 Mechanics of particles and systems\u00a0(70-XX) 2 Algebraic geometry\u00a0(14-XX) 2 Associative rings and algebras\u00a0(16-XX) 2 Topological groups, Lie groups\u00a0(22-XX) 2 Difference and functional equations\u00a0(39-XX) 2 Harmonic analysis on Euclidean spaces\u00a0(42-XX) 2 Integral equations\u00a0(45-XX) 2 Probability theory and stochastic processes\u00a0(60-XX) 1 Category theory; homological algebra\u00a0(18-XX) 1 Real functions\u00a0(26-XX) 1 Measure and integration\u00a0(28-XX) 1 Functional analysis\u00a0(46-XX) 1 Operator theory\u00a0(47-XX) 1 Calculus of variations and optimal control; optimization\u00a0(49-XX) 1 Algebraic topology\u00a0(55-XX) 1 Global analysis, analysis on manifolds\u00a0(58-XX) 1 Optics, electromagnetic theory\u00a0(78-XX) 1 Statistical mechanics, structure of matter\u00a0(82-XX) 1 Relativity and gravitational theory\u00a0(83-XX)","date":"2022-09-30 23:05:23","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7023910284042358, \"perplexity\": 11144.551315346136}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-40\/segments\/1664030335504.37\/warc\/CC-MAIN-20220930212504-20221001002504-00444.warc.gz\"}"}
| null | null |
This is about a month late, but I've had a bit of blogger's block this month. I'm not sure why, but my focus has been elsewhere and I've fallen tragically behind.
Before I start feeling sorry for myself or making more excuses, I'm just gonna basically start where I left off, with our trip to Cleveland to help my brother start his new life there!
I told you all about our crazy experience driving to Cleveland - and while that was a bit jarring and definitely stayed with us for quite a while, the visit itself was great.
We went straight to the hotel to settle in and say our happy hellos, and get the weekend started!
Dusty was pretty excited about passing this bridge! He was extremely tempted to take a detour.
It's always exciting to be in a new city. We didn't really know what to expect from Cleveland, but we knew we had a lot to do.
First on our list: eat. We weren't familiar with the best restaurants in town, so we thanked the Lord for the digital age and decided to look a few up on Yelp.
We ended up deciding on a unique spot called "Fat Cats", which was only partially because I was wearing my favorite cat shirt.
It was very funky and artsy, located downtown with a creative menu and fun decor featuring local artists. Hipster heaven!
Love this shot of my beloved bros.
We were all pretty tired, so we enjoyed the sounds of the kitchen and the tasty dishes as we wound down from a long day of travel.
The rest of the weekend was a blur of shopping, napping, eating, laughing, wandering, exploring, moving, decorating, cooking, running, sighing, milk-strawing, chatting, and sleeping.
On Saturday we took the plunge and went to Wal-Mart - the new-apartment-resident's best friend.
We found wild geese congregating in the parking lot, and after shopping for a long time the boys had to get out some pent-up energy by cart-racing in the parking lot. Classy and entertaining.
I've heard a lot of no-good comments about Cleveland from random people, but I honestly enjoyed it!
It was a bit humid but overall the weather was nice, the scenery was lovely and there were a lot of really cool views and buildings, especially around the downtown area.
We had a lot of errands to run so we got to see a lot of the city just by driving to various places.
Dusty, Kevin and I spent a lot of time messing around since mom was the leader and Michael was the new resident, so we were just extras that made things more difficult by napping and goofing around.
We were most concerned with where the nearest Starbucks was at all times, and even made a Starbucks run in the middle of another errand. We were almost overtaken by the extremely confusing twilight zone road that ran in a confusing roundabout circle around the Starbucks, but we finally made the right U-turn and quenched our thirst for iced coffee delights. It's such a joy to be around my little brother. Even if we revert to childhood teasing tendencies, it's kind of the best.
The view from Michael's apartment!
The biggest frustration throughout our time there is that Michael's Uhaul pod never showed up!
All of his furniture and possessions was supposed to be arriving any day, but instead they were giving us the runaround about when it would arrive.
We had to make do with multiple trips to the store, cooking a few meals in his kitchen and setting up the basics in the hopes that it would arrive eventually.
My contribution to his new place - a kitschy gift that he'll love forever.
My favorite moments from the weekend were hanging out in the empty apartment, lounging on the clean carpet, taking naps...(sorry mom!), enjoying the voices of my mom and brothers and smelling her chicken enchiladas cooking in the oven.
My mom also taught my husband how to make perfect sunny-side-up eggs that weekend, which I am very happy about! Yum.
On the ground floor of his apartment building, there was a bar and grill and a little market, where we shopped for a few things like ice and hand soap.
While we were in the dairy section, lamenting over the discontinuation of Nestle's Banana Milk, Kevin discovered a little display with something called "milk straws". He was so adamant to try them, so we finally bought them at one point and spent that afternoon using them to drink milk and glass-bottled Starbucks frapps.
They were these funky little straws with cookies and cream flavored chocolate beads inside, so every time you take a drink the milk would filter through.
Weird and kind of great. Time to blow milk bubbles. Childhood flashback.
Another mystery in the apartment were these odd little bugs stuck to the outside of the windows. There were a lot of them, and they were just kind of chilling there. They were pretty small, but had kind of creepy and colorful little bodies. We were very glad they were on the outside of the windows.
Even so, we kept wondering about them - what were they? Why were there so many?
At one point, us kids decided to wander around the apartment complex to see what else we could find. We were starving, so we a great place to eat was at the top of the list.
We crossed the street to another high structure, and as we stepped onto the new sidewalk, I let out a gasp.
The bugs...those same bugs!...were everywhere.
into the apocalypse of the midges.
We all staggered back, shocked, stumbling into the street, unable to scream, lest they all flow into our mouths.
They covered every inch of the sky-rise building, as far as they eye could see -- they covered every window for 30 stories. The air was thick with them.
We all grabbed the collar of our shirts and frantically covered our mouths, Kevin breathing through his shirt, "Oh god, oh god, they're everywhere...they're everywhere!"
We finally began to walk backwards, turning back as quickly as we could.
They were buzzing around our ears, our eyes, they were all over our clothes.
"I just wanna run. I just want to RUN," Kevin said.
We finally gasped a huge sigh of relief as we tumbled back into the apartment building, deciding to try out the bar and grill on the first floor.
It turned out to be a good choice - it was quiet, cool, and clean. It was late afternoon. We ordered a bunch of appetizers and munched on giant soft pretzels with beer cheese and chicken fingers while we shook from shock and our eyes twitched from haunted buggy thoughts.
Our last full day ended with a fancy steakhouse dinner and a lovely walk around downtown.
Michael had a gift certificate for John Q's, and wanted to use it since apparently it's closing down!
We pulled up and parked across the street, right across the street from this cool place.
Once inside, we settled down for the long haul. Appetizers, entrees, desserts, coffee.
It was the last supper of crazy family time and it was all very bittersweet.
Dusty and I would be heading back home the next morning, but luckily mom and Kevin got to stay for the rest of the week. It's always easier to stagger the goodbyes!
Michael and Kevin reminisced about working at a sorority in Tucson, waiting tables and cooking and cleaning up...crazy stories about the chef and other workers and some of the weird people they encountered.
It was light and fun and nostalgic, and I felt really lucky to be able to have a little glimpse into the life Michael will live for the next five years.
I'm still secretly hoping he realizes how awesome we all are, and will decided to move back to Arizona when all this is over. But in the meantime, we'll just have to send our love across the miles. I definitely have experience with that, but I've never gotten used to it!
After dinner we decided to walk off some of the million calories we ate by walking around and enjoying some evening sights.
This was the courthouse! So cool!
After we walked several blocks and back, we all loaded back into the car and headed back to the hotel.
The next morning dawned drearily, and mom decided she wanted some McDonald's coffee.
Dusty and I jumped at the chance to go, because we had seen the fanciest McDonald's down the road and had to experience it asap.
It had chandeliers and everything! What a place.
Where the poor come to eat poorly and feel rich.
We then all met up at iHop for our last breakfasty meal together.
I was really dreading getting back into the car - I didn't want to say goodbye, and I certainly didn't want to try the 7 hours back home.
Even so, the time had come. We packed up our car and spread the loving goodbyes.
Hopefully it won't be too long until we say hello again!
Road tripping once more, we turned up the tunes, stopped by a Starbucks and just enjoyed the scenery.
It honestly went by in a blur - I hardly remember anything about the drive!
Before we knew it we were edging closer to Lynchburg, and right as we entered Afton, Dusty noticed a sign for Blue Mountain Brewery - an amazing spot we went for his 23rd birthday our first year in Virginia.
All he had to do is look in my direction with a quizzical brow, and we were quickly turning in, right in time for the gorgeous foggy sunset over the mountains.
We weren't terribly hungry, so we snacked one a couple of things (their baked beans are the best ever) and enjoyed the amazing view and weather.
It was the perfect way to end our road trip, savoring the sunset and crossing off another location from our Lynchburg Bucket List.
Although I'm hoping we'll go back at least once before graduation!
The fireflies began popping up, and the entire evening quieted down in a blissful way.
We ended the vacation weekend with happy hearts, happy bellies and happy souls.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,912
|
Larak Rural District () is a rural district (dehestan) in the Shahab District of Qeshm County, Hormozgan Province, Iran. At the 2006 census, its population was 466, in 98 families. The rural district has 1 village.
References
Rural Districts of Hormozgan Province
Qeshm County
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,906
|
\section{Introduction}
\label{intro}
Humans can effortlessly and rapidly recognize surrounding objects~\cite{thorpe1996speed}, despite the tremendous variations in the projection of each object on the retina~\cite{biederman1987recognition} caused by various transformations such as changes in object position, size, pose, illumination condition and background context~\cite{dicarlo2012does}. This invariant recognition is presumably handled through hierarchical processing in the so-called ventral pathway. Such hierarchical processing starts in V1 layers, which extract simple features such as bars and edges in different orientations~\cite{lennie2005coding}, continues in intermediate layers such as V2 and V4, which are responsive to more complex features~\cite{nandy2013fine}, and culminates in the inferior temporal cortex (IT), where the neurons are selective to object parts or whole objects~\cite{tanaka1991coding}. By moving from the lower layers to the higher layers, the feature complexity, receptive field size and transformation invariance increase, in such a way that the IT neurons can invariantly represent the objects in a linearly separable manner~\cite{hung2005fast,rust2010selectivity}.
Another amazing feature of the primates' visual system is its high processing speed. The first wave of image-driven neuronal responses in IT appears around 100 ms after the stimulus onset~\cite{thorpe1996speed,dicarlo2012does}. Recordings from monkey IT cortex have demonstrated that the first spikes (over a short time window of 12.5 ms), about 100 ms after the image presentation, carry accurate information about the nature of the visual stimulus~\cite{hung2005fast}. Hence, ultra-rapid object recognition is presumably performed in a feedforward manner~\cite{dicarlo2012does}. Moreover, although there exist various intra- and inter-area feedback connections in the visual cortex, some neurophysiological~\cite{liu2009timing, freiwald2010functional,dicarlo2012does} and theoretical~\cite{Anselmi2014} studies have also suggested that the feedforward information is usually sufficient for invariant object categorization.
Appealed by the impressive speed and performance of the primates' visual system, computer vision scientists have long tried to ``copy'' it. So far, it is mostly the architecture of the visual system that has been mimicked. For instance, using hierarchical feedforward networks with restricted receptive fields, like in the brain, has been proven useful~\cite{Fukushima1980,LeCun1998,Serre2007.PAMI,Lee2009,Krizhevsky2012,Le2013}. In comparison, the way that biological visual systems learn the appropriate features has attracted much less attention. All the above-mentioned approaches somehow use non biologically plausible learning rules. Yet the ability of the visual cortex to wire itself, mostly in an unsupervised manner, is remarkable~\cite{Ghose2004,Kourtzi2006}.
\headheight=0pt
\lhead{}
\pagestyle{plain}
Here, we propose that adding bio-inspired learning to bio-inspired architectures could improve the models' behavior. To this end,
we focused on a particular form of synaptic plasticity known as spike timing-dependent plasticity (STDP), which has been observed in the mamalian visual cortex~\cite{Meliza2006,Huang2014}. Briefly, STDP reinforces the connections with afferents that significantly contributed to make a neuron fire, while it depresses the others~\cite{Feldman2012}. A recent psychophysical study provided some indirect evidence for this form of plasticity in the human visual cortex~\cite{McMahon2012}.
In an earlier study~\cite{Masquelier2007}, it is shown that a combination of a temporal coding scheme -- where in the entry layer of a spiking neural network the most strongly activated neurons fire first -- with STDP leads to a situation where neurons in higher visual areas will gradually become selective to complex visual features in an unsupervised manner. These features are both salient and consistently present in the inputs. Furthermore, as learning progresses, the neurons' responses rapidly accelerates. These responses can then be fed to a classifier to do a categorization task.
In this study, we show that such an approach strongly outperforms state-of-the-art computer vision algorithms on view-invariant object recognition benchmark tasks including 3D-Object~\cite{4408987,Pepik2014} and ETH-80~\cite{leibe2003analyzing} datasets. These datasets contain natural and unsegmented images, where objects have large variations in scale, viewpoint, and tilt, which makes their recognition hard~\cite{Pinto2008}, and probably out of reach for most of the other bio-inspired models~\cite{pinto2011comparing,ghodrati2014feedforward}. Yet our algorithm generalizes surprisingly well, even when ``simple classifiers'' are used, because STDP naturally extracts features that are class specific. This point was further confirmed using mutual information~\cite{pohjalainen2013feature} and representational dissimilarity matrix (RDM)~\cite{kriegeskorte2008representational}. Moreover, the distribution of objects in the obtained feature space was analyzed using hierarchical clustering~\cite{murtagh2012algorithms}, and objects of the same category tended to cluster together.
\begin{figure*}[!htb]
\centering
\includegraphics[scale=0.45]{Figure-Model2}
\caption{ Overview of our 5 layered feedforward spiking neural
network. The network processes the input image in a multi-scale form, each processing scale is shown with a different color.
Cells are organized in retinotopic maps until the $S_2$ layer (included). $S_1$ cells of each processing scale detect edges from the corresponding scaled image. $C_1$ maps sub-sample the corresponding $S_1$
maps by taking the maximum response over a square neighborhood.
$S_2$ cells are selective to intermediate complexity visual
features, defined as a combination of oriented edges of a same scale(here we
symbolically represented a triangle detector and a square detector).
There is one $S_1$--$C_1$--$S_2$ pathway for each processing scale. Then $C_2$ cells take the maximum
response of $S_2$ cells over all positions and scales and are thus
shift and scale invariant. Finally, a classification is done based
on the $C_2$ cells' responses (here we symbolically represented a
house/non-house classifier). $C_1$ to $S_2$ synaptic connections are learned with
STDP, in an unsupervised manner.
} \label{model_figure}
\end{figure*}
\section{Materials and methods}\label{MaterialsAndMethods}
The algorithm we used here is a scaled-up version of the one presented in~\cite{Masquelier2007}. Essentially, many more C2 features and iterations were used. Our code is available upon request. We used a five-layer hierarchical network $S_1$ $\rightarrow$ $C_1$ $\rightarrow$ $S_2$ $\rightarrow$ $C_2$ $\rightarrow$ $classifier$, largely inspired by the HMAX model~\cite{Serre2007.PAMI} (see Fig.~\ref{model_figure}). Specifically, we alternated simple cells that gain selectivity through a sum operation, and complex cells that gain shift and scale invariance through a max operation. However, our network uses spiking neurons and operates in the temporal domain: when presented with an image, the first layer's $S_1$ cells, detect oriented edges and the more strongly a cell is stimulated the earlier it fires. These $S_1$ spikes are then propagated asynchronously through the feedforward network. We only compute the first spike fired by each neuron (if any), which leads to efficient implementations. The justification for this is that later spikes are probably not used in ultra-rapid visual categorization tasks in primates~\cite{Thorpe1989}. We used restricted receptive fields and a weight sharing mechanism (i.e. convolutional network). In our model, images are presented sequentially and the resulting spike waves are propagated through to the $S_2$ layer, where STDP is used to extract diagnostic features.
More specifically, the first layer's $S_1$ cells detect bars and edges using Gabor filters. Here we used $5\times 5$ convolutional kernels corresponding to Gabor filters with the wavelength of 5 and four different preferred orientations ($\pi/8,\pi/4+\pi/8 , \pi/2+\pi/8, 3\pi/4+\pi/8$). These filters are applied to five scaled versions of the original image: 100\%, 71\%, 50\%, 30\%, and 25\% (each processing scale declared by a different color in Fig.~\ref{model_figure}). Hence, for each scaled version of the input image we have four $S_1$ maps (one for each orientation), and overall, there are 4$\times$5 $=$ 20 maps of $S_{1}$ cells (see the $S_{1}$ maps of Fig.~\ref{model_figure}). Evidently, the $S_1$ cells of larger scales detect edges with higher spatial frequencies while the smaller scales extract edges with lower spatial frequencies. Indeed, instead of changing the size and spatial frequency of Gabor filters, we are changing the size of input image. This is a way to implement scale invariance at a low computational cost.
Each $S_{1}$ cell emits a spike with a latency that is inversely proportional to the absolute value of the convolution. Thus, the more strongly a cell is stimulated the earlier it fires (intensity-to-latency conversion, as observed experimentally~\cite{Celebrini1993,Albrecht2002,Shriki2012}). To increase the sparsity at a given scale and location (corresponding to one cortical column), only the spike corresponding to the best matching orientation is propagated (i.e. a winner-take-all inhibition is employed). In other word, for each position in the four $S_1$ orientation maps of a given scale, the $S_1$ cell with highest convolution value emits a spike and prevents the other three $S_1$ cells from firing.
For each $S_{1}$ map, there is a corresponding $C_{1}$ map. Each $C_{1}$ cell propagates the first spike emitted by the $S_{1}$ cells in a $7\times7$ square neighborhood of the $S_{1}$ map which corresponds to one specific orientation and one scale (see the $C_{1}$ maps of Fig.~\ref{model_figure}). $C_{1}$ cells thus execute a maximum operation over the $S_{1}$ cells with the same preferred feature across a portion of the visual field, which is a biologically plausible way to gain local shift invariance~\cite{Riesenhuber1999,Rousselet2003}. The overlap between the afferents of two adjacent $C_1$ cells is just one $S_1$ row, hence a subsampling over the $S_1$ maps is done by the $C_1$ layers as well. Therefore, each $C_1$ map has $6\times6=36$ fewer cells than the corresponding $S_1$ map.
$S_{2}$ features correspond to intermediate-complexity visual features which are optimum for object classification~\cite{Ullman2002}. Each $S_{2}$ feature has a prototype $S_{2}$ cell (specified by a $C_{1}$-$S_{2}$ synaptic weight matrix), which is a weighted combination of bars ($C_{1}$ cells) with different orientations in a $16 \times 16$ square neighborhood. Each prototype $S_{2}$ cell is retinotopically duplicated in the five scale maps (i.e. weight-sharing is used). Within those maps, the $S_{2}$ cells can integrate spikes only from the
four $C_{1}$ maps of their corresponding processing scales. This way, a given $S_{2}$ feature is simultaneously explored in all positions and scales (see $S_{2}$ maps of Fig.~\ref{model_figure} with same feature prototype but in different processing scales specified by different colors). Indeed, duplicated cells in all positions of all scale maps integrate the spike train in parallel and compete with each other. The first duplicate reaching its threshold, if any, is the winner. The winner fires and prevents the other duplicated cells in all other positions and scales from firing through a winner-take-all inhibition mechanism. Then, for each prototype, the winner $S_{2}$ cell triggers the unsupervised STDP rule and its weight matrix is updated. The changes in its weights are applied over all other duplicate cells in different positions and scales (weight sharing mechanism). This allows the system to learn frequent patterns, independently of their position and size in the training images.
The learning process begins with $S_{2}$ features initialized by random numbers drawn from a normal distribution with mean $0.8$ and STD $0.05$, and the threshold of all $S_{2}$ cells is set to 64 ($= 1/4 \times 16 \times 16$). Through the learning process, a local inhibition between different $S_{2}$ prototype cells is used to prevent the convergence of different $S_{2}$ prototypes to similar features: when a cell fires at a given position and scale, it prevents all the other cells (independently of their preferred prototype) from firing later at the same scale and within a neighborhood around the firing position. Thus, the cell population self-organizes, each cell trying to learn a distinct pattern so as to cover the whole variability of the inputs. Moreover, we applied a k-winner-take-all strategy in $S_{2}$ layer to ensure that at most two cells can fire for each processing scale. This mechanism, only used in the learning phase, helps the cells to learn patterns with different real sizes. Without it, there is a natural bias toward ``small" patterns (i.e., large scales), simply because corresponding maps are larger, and so likeliness of firing with random weights at the beginning of the STDP process is higher.
A simplified version of STDP is used to learn the $C_{1}-S_{2}$ weights as follows:
$$\left\{
\begin{array}{l l l}
\Delta w_{ij}=a^{+}.w_{ij}.(1-w_{ij}),& if & t_{j}-t_{i}\leq 0, \\
\Delta w_{ij}=a^{-}.w_{ij}.(1-w_{ij}),& if & t_{j}-t_{i} > 0,
\end{array} \right.$$
where $ i $ and $ j $ respectively refer to the index of post- and presynaptic neurons,
$t_i$ and $t_j$ are the corresponding spike times, $\Delta w_{ij}$ is the synaptic weight
modification, and $a^{+}$ and $a^{-}$ are two parameters specifying the
learning rate. Note that the exact time difference between two spikes ($t_{j}-t_{i}$) does not affect the weight change, but only its sign is considered. These simplifications are equivalent to assuming that the intensity-to-latency conversion of $S_{1}$ cells compresses the whole
spike wave in a relatively short time interval (say, $20-30$ ms), so that all presynaptic spikes necessarily fall close to the postsynaptic spike time, and the time lags are negligible. The multiplicative term $w_{ij}.(1-w_{ij})$ ensures the weights remain in the range [0,1] and maintains all synapses in an excitatory mode. The learning phase starts by $a^{+}=2^{-6}$ which is multiplied by 2 after each 400 postsynaptic spikes up to a maximum value of $2^{-2}$. A fixed $a^{+}/a^{-}$ ratio (-4/3) is used. This allows us to speed up the convergence of $S_{2}$ features as the learning progresses. Initiation of the learning phase with high learning rates would lead to erratic results.
For each $S_{2}$ prototype, a $C_{2}$ cell propagates the first spike emitted by the corresponding $S_{2}$ cells over all positions and processing scales, leading to the global shift- and scale-invariant cells (see the $C_{2}$ layer of Fig.~\ref{model_figure}).
\section{Experimental Results}\label{Results}
\subsection{Dataset and Experimental Setup}
To study the robustness of our model with respect to different transformations such as scale and viewpoint, we evaluated it on the \emph{3D-Object} and \emph{ETH-80} datasets. The 3D-Object is provided by Savarese et al. at CVGLab, Stanford University~\cite{4408987}. This dataset contains 10 different object classes: bicycle, car, cellphone, head, iron, monitor, mouse, shoe, stapler, and toaster. There are about 10 different instances for each object class. The object instances are photographed in about 72 different conditions: eight view angles, three distances (scales), and three different tilts. The images are not segmented and the objects are located in different backgrounds (the background changes even for different conditions of the same object instance). Figure~\ref{objects} presents some examples of objects in this dataset.
The ETh-80 dataset includes 80 3D objects in eight different object categories including apple, car, toy cow, cup, toy dog, toy horse, pear, and tomato. Each object is photographed in 41 viewpoints with different view angles and different tilts. Figure S1 in Supplementary Information provides some examples of objects in this dataset from different viewpoints.
For both datasets, five instances of each object category are selected for the training set to be used in the learning phase. The remaining instances constitute the testing set which is not seen during the learning phase, but is used afterward to evaluate the recognition performance. This standard cross-validation procedure allows to measure the generalization ability of the model beyond the specific training examples. Note that for 3D-Object dataset, the original size of all images were preserved, while the images of ETH-80 dataset are resized to $300$ pixels in height while preserving the aspect ratio. The images of both datasets were converted to grayscale values.
\begin{figure*}
\centering
\includegraphics[scale=0.9]{Views.pdf}
\caption{Some images of the head class of 3D-Object dataset in different A) views, B) scales, and C) tilts.}
\label{objects}
\end{figure*}
\begin{table*}
\caption{Performance of our model, HMAX, and DeepConvNet with different number of features.}
\label{table_example1}
\scriptsize
\centering
\begin{tabular}{|c|c|c|c|c|c||c|c|c|c||c|c|}
\hline
Dataset& & \multicolumn{4}{c||}{Our model}&\multicolumn{4}{c||}{HMAX}&DeepConvNet \\
\hline
\hline
\multirow{2}{*}{3D-Object}&\# Features & 200 & 300 & 400 & 500&1000&3000&9000&12000&4096\\
\cline{2-11}
&Accuracy & 76.1\% & 94.7\% & 96.0\% & 96.0\%&58.2\%&60.1\%&61.9\%&62.4\%&85.8\%\\
\hline \hline
\multirow{2}{*}{ETH-80}&\# Features & 500 & 750 & 1000& 1250&500&1000&2000&5000&4096\\
\cline{2-11}
&Accuracy & 75.3 & 79.3\% & 80.7\% & 81.1\%&66.3\%&68.7\%&68.9\%&69.0\%&79.1\%\\
\hline
\end{tabular}
\end{table*}
As already mentioned, the building process of $S_2$ features is performed in a completely unsupervised manner. Hence, through the execution of the unsupervised STDP-based learning, the training images are randomly fed into the model (without considering their class labels, viewpoints, scales, and tilts). The learning process starts with initial random weights and finishes when 600 spikes have occurred in each $S_2$ map. Then STDP is turned off, and the ability of the obtained features to invariantly represent different object classes is evaluated. To compute the corresponding $C_2$ feature vector for each input image, the thresholds of $C_2$ neurons are set to infinite, and their final potentials are evaluated, after propagating the whole spike train generated by each image. Each final potential can be seen as the number of early spikes in common between the current input and a stored prototype (this is very similar to the tuning operation of $S$ cells in HMAX). Then, a one-versus-one multiclass linear support vector machine (SVM) classifier is trained based on the $C_2$ features of the training set and it is evaluated on the test set.
We have compared the performance of our model with the HMAX model~\cite{Serre2007.PAMI} and deep supervised convolutional network (DeepConvNet) by Krizhevsky et. al~\cite{Krizhevsky2012}. Comparison with the HMAX model is particularly instructive, since as explained above, we use very similar architecture, tuning and maximum operations. The main difference is that instead of using an unsupervised learning rule like us, the HMAX model uses random crops from the training images to imprint the $S2$ features (here of equal size). Then a SVM classifier was trained over the HMAX $C2$ features to complete the object recognition process. The employed HMAX model is implemented by Mutch, et al.~\cite{MUTCH10} and the codes are publicly available at http://cbcl.mit.edu/jmutch/cns/index.html.
We also compared our model with DeepConvNet which has been shown to be the best algorithm in various object classification tasks including the ImageNet LSVRC-2010 contest~\cite{Krizhevsky2012}. It is comprised of eight consecutive layers (five convolutional layers followed by three fully connected layers) with about 60 millions parameters, learned with stochastic gradient descent. We have used a pre-trained DeepConvNet model implemented by Jia, et al.~\cite{jia2014caffe}, whose code is also available at http://caffe.berkeleyvision.org. The training was done over the ILSVRC2012 dataset (a subset of ImageNet) with about 1.2 million images in 1000 categories. We fed the training and testing images into DeepConvNet and extracted the feature values from the 7th layer. Again, a SVM is used to do the object recognition based on the extracted features.
\subsection{Performance analysis}
Table~\ref{table_example1} provides the accuracy of our model in category classification independently of view, tilt, and scale, when different number of $S_2$ features are learned by the STDP-based learning algorithm. The results indicate that the model reaches a high classification performance on 3D-Object dataset with about 300 $C_2$ features only (about 30 features per class). The performance is flattened around 96\% for feature vectors of size greater than 400. Also, for the ETH-80 dataset, the model attains to a reasonable recognition accuracy of about 81\% with only 1250 extracted features. We have also performed the same experiments on HMAX and DeepConvNet models which their accuracies are also provided in Table~\ref{table_example1}.
Performance of the HMAX model was weak on both datasets, which is not too surprising, several previous studies have shown that the performance of the HMAX model extensively decreases when facing significant object transformations~\cite{Pinto2008,pinto2011comparing}. Given the structural similarities between our model and HMAX, the superiority of our model is presumably related to the unsupervised feature learning. Indeed, most of the randomly extracted S2 patches in HMAX tend to be redundant and irrelevant, as we will see in the next section.
DeepConvNet reached a mean performance of about 86\% on 3D-Object and about 79\% on ETH-80 dataset. Thus, our model outperforms DeepConvNet on both datasets, which itself significantly outperforms HMAX. It should be noted that the images of each object in these two datasets are highly varied (e.g., in 3D-Object dataset, there is a 45\degree difference between two successive views of an object) and it has previously been shown that the performance of DeepConvNet drops when facing such transformations~\cite{ghodrati2014feedforward}. Another drawback of DeepConvNet is that, due to the large number of parameters, it needs to be trained over millions of images to avoid overfitting~\cite{cox2014neural} (here we avoided this problem by using a pre-trained version, but doing the training on {about 3500} images we used here would presumably lead to massive overfitting). Conversely, our model is able to learn objects using much fewer images.
Consequently, the results indicate that our model has a great ability to learn diagnostic features tolerating transformations and deformations of the presented stimulus.
\subsection{Feature Analysis}
\begin{figure*}
\centering
\subfloat[]{\includegraphics[width=2.2in]{A24}%
\label{fig_first_case}}
\hfil
\subfloat[]{\includegraphics[width=2.2in]{A42}%
\label{fig_second_case}}
\hfil
\subfloat[]{\includegraphics[width=2.2in]{A90}%
\label{fig_3rd_case}}
\caption{Three $S_2$ feature prototypes selective to the a) bicycle, b)face, and c) cellphone classes of 3D-object dataset along with their reconstructed preferred stimuli. It can be seen that the features converged to specific and salient object parts and neglected the irrelevant backgrounds.}
\label{features}
\end{figure*}
\begin{figure*}
\centering
\subfloat[View 1.]{\includegraphics[width=1.7in]{view1}%
\label{fig_first_case2}}
\hfil
\subfloat[View 2.]{\includegraphics[width=1.7in]{view2}%
\label{fig_second_case2}}
\hfil
\subfloat[View 3.]{\includegraphics[width=1.7in]{view3}%
\label{fig_3rd_case3}}
\hfil
\subfloat[View 4.]{\includegraphics[width=1.7in]{view4}%
\label{fig_4th_case2}}
\hfil
\subfloat[View 5.]{\includegraphics[width=1.7in]{view5}%
\label{fig_5th_case2}}
\hfil
\subfloat[View 6.]{\includegraphics[width=1.7in]{view6}%
\label{fig_6th_case2}}
\hfil
\subfloat[View 7.]{\includegraphics[width=1.7in]{view7}%
\label{fig_7th_case2}}
\hfil
\subfloat[View 8.]{\includegraphics[width=1.7in]{view8}%
\label{fig_8th_case2}}
\hfil
\vspace*{0.5cm}
\captionsetup[subfigure]{labelformat=empty}
\subfloat[][Dissimilarity measure]{\includegraphics[width=2.2in]{Legend}}%
\caption{RDMs of our model on 3D-Object dataset corresponding to different viewpoints. It can be seen that within class dissimilarities are very low (the blue squares around the main diagonal where rows and columns correspond to images of the same category), while between class dissimilarities are higher (more yellowish). Note that due to the absence of image samples for some views of the monitor class, we have eliminated this class from the RDMs.}
\label{RDMs}
\end{figure*}
\begin{figure*}[t]
\centering
\subfloat[View 1.]{\includegraphics[width=1.7in]{Hmax_view1}%
\label{Hmax_fig_first_case2}}
\hfil
\subfloat[View 2.]{\includegraphics[width=1.7in]{Hmax_view2}%
\label{Hmax_fig_second_case2}}
\hfil
\subfloat[View 3.]{\includegraphics[width=1.7in]{Hmax_view3}%
\label{Hmax_fig_3rd_case3}}
\hfil
\subfloat[View 4.]{\includegraphics[width=1.7in]{Hmax_view4}%
\label{Hmax_fig_4th_case2}}
\hfil
\subfloat[View 5.]{\includegraphics[width=1.7in]{Hmax_view5}%
\label{Hmax_fig_5th_case2}}
\hfil
\subfloat[View 6.]{\includegraphics[width=1.7in]{Hmax_view6}%
\label{Hmax_fig_6th_case2}}
\hfil
\subfloat[View 7.]{\includegraphics[width=1.7in]{Hmax_view7}%
\label{Hmax_fig_7th_case2}}
\hfil
\subfloat[View 8.]{\includegraphics[width=1.7in]{Hmax_view8}%
\label{Hmax_fig_8th_case2}}
\hfil
\vspace*{0.5cm}
\captionsetup[subfigure]{labelformat=empty}
\subfloat[][Dissimilarity measure]{\includegraphics[width=2.2in]{Legend}}%
\caption{RDMs of the HMAX model on 3D-Object dataset corresponding to different viewpoints. Randomly selected features in HMAX model are not able to similarly represent within-category objects and dissimilarly represent between-category objects. Note that due to the absence of image samples for some views of the monitor class, we have eliminated this class from the RDMs.}
\label{Hmax_RDMs}
\end{figure*}
In this section, we demonstrate that unsupervised STDP learning algorithm extracts informative and diagnostic features by comparing them to the randomly picked HMAX features. To this end, we have used several feature analysis techniques: representational dissimilarity matrices, hierarchical clustering, and mutual information. We performed the same analyses on both datasets and obtained similar results. Hence, the results of ETH-80 are presented in Supplementary Information.
Extraction of diagnostic features let our model reach high classification performances with a small number of features (c.f. Table~\ref{table_example1}). To understand why this is true, we first reconstructed the features' preferred stimuli. Given that each $S_2$ neuron receives spikes from $C_1$ neurons responding to bars in different orientations, the representation of the preferred features of $S_2$ neurons could be reconstructed by convolving their weight matrices with a set of kernels representing oriented bars. In Fig.~\ref{features} the receptive fields of activated $S_2$ neurons along with the representation of their preferred stimuli are illustrated (Fig. S2 provides the same illustration for the ETH-80 dataset). This demonstrates that only a small number of $S_2$ neurons are required to represent the input objects. In other words, the obtained features are compatible with the sparse coding theory in visual cortex. In addition, for an input image, the most activated $S_2$ neurons cover the input objects and they do not respond to the background area. Indeed, the STDP learning algorithm naturally focuses on what are common in the training images, which are the target object features. The backgrounds are generally not learned (at least not in priority), since they are almost always too different from one image to another and the STDP process cannot converge on them.
To characterize the neuronal population coding in the $C_2$ layer of the model and to study the quality of $C_2$ features, we used the representational dissimilarity matrix (RDM)~\cite{kriegeskorte2008representational}). Each element of the RDM reflects the measure of dissimilarity (distance) among the neural activity patterns (i.e., the object representations) associated with two different image stimuli. The distance we used here is $1-Pearson\:\:correlation$. In an RDM corresponding to a perfect model, the representations of the objects of the same category have low dissimilarities (i.e., highly correlated), whereas objects of different categories are represented highly dissimilarly (i.e., uncorrelated). Hence, if we group the rows and columns of the RDM of a perfect model based on object categories, it is expected to see squares of low dissimilarity values around the main diagonal, each of which corresponds to pairs of same-category images, while other elements have higher values.
Here, to plot the RDM of each view angle, first, the images of all input instances which are taken in that view are picked. Then the corresponding RDM is plotted by computing the pairwise dissimilarity of the values of $C_2$ features associated with each pair of images. Figure~\ref{RDMs} presents the RDMs of our model for all eight views (see Fig. S3 for ETH-80). In each RDM, rows and columns are sorted based on image categories. Also, a sample image of each category is placed next to the rows and columns which correspond to that category. Here, we used a color-code to represent RDMs which ranges from pure blue to pure yellow demonstrating low to high dissimilarities, respectively. It can be seen that the within-category dissimilarity values (identified by blue squares around the main diagonal) are relatively lower than the between-category dissimilarities (more yellowish areas). As expected, the RDMs indicate that the obtained performance is not due to the capabilities of the classifier, but to the extraction of diagnostic and highly informative $C_2$ features through STDP.
We have also computed the RDMs of the HMAX model (including 12000 features) for eight views, as provided in Fig.~\ref{Hmax_RDMs} (see Fig. S4 for ETH-80). As it can be seen in this figure, the randomly selected features in the HMAX model are unable to similarly represent within-category objects and dissimilarly represent between-category objects. This is probably due to uninformative features used by the HMAX model. Indeed, in HMAX, the task of selecting the informative features is left to the classifier. We also note the presence of horizontal and vertical yellow lines, indicating ``outliers", whose representation lies far away from all the others. This indicates that the features do not pave well the stimulus space.
\begin{figure*}[!htb]
\centering
\includegraphics[scale=0.75]{Cluster6.pdf}
\caption{The hierarchy of clusters and their labels for our model on 3D-Object dataset. The label of each cluster indicates the class with the highest frequency in that cluster. It can be seen that the samples of each class are placed in close clusters. The cardinality of each cluster, $C$, and the cardinality of the class with the highest frequency, $H$, are placed below the cluster label as $H/C$.}
\label{Hierarchy}
\end{figure*}
\begin{figure*}[!htb]
\centering
\includegraphics[scale=0.75]{Cluster6_HMax.pdf}
\caption{The hierarchy of clusters and their labels for the HMAX model on 3D-Object dataset. The label of each cluster indicates the class with the highest frequency in that cluster. It can be seen that the majority of the objects are assigned to a small number of clusters and samples of each class are not well placed in close clusters. The cardinality of each cluster, $C$, and the cardinality of the class with the highest frequency, $H$, are placed below the cluster label as $H/C$.}
\label{Cluster5_HMax}
\end{figure*}
To see how well the stimuli are distributed in the high dimensional feature space, we performed hierarchical clustering over the test set. The clustering procedure is started by considering each stimulus as a discrete cluster node, continued by connecting the closest nodes into a new combined cluster node, and completed by connecting all the stimuli to a single node. We performed this analysis on the $C_2$ feature vectors corresponding to all objects in all views, scales, and tilts. The obtained hierarchy for our model is displayed in Fig.~\ref{Hierarchy} (see Fig. S5 for ETH-80). The distance between a pair of cluster nodes is computed by measuring the dissimilarity among their centers (the average of cluster members). Due to the large number of stimuli, it is not possible to plot the whole hierarchy, hence, only the high level clusters are shown in this figure. For each lowest level cluster, the class with the highest frequency is illustrated by an image label. The cardinality of this class as well as the cardinality of the cluster are shown below the labels. It can be seen that the instances of each object class are placed in neighboring regions of the $C_2$ feature space. By considering the obtained hierarchical clustering and the classification accuracies, it can be concluded that the $C_2$ features are able to invariantly represent the objects in such a way that the classifier can easily separate them.
The same hierarchical clustering is performed for the HMAX feature space (with 12000 features), as depicted in Fig.~\ref{Cluster5_HMax} (see Fig. S6 for ETH-80). As it can be seen, the majority of clusters are small, and contrary to our model, the distances between the clusters are very low. In other words, the objects are densely represented in a small area of such a high dimensional feature space. Furthermore, the mean intra- and inter-class dissimilarities in our model are equal to 0.40 and 0.70, respectively, while these statistics for the HMAX model are equal to 0.27 and 0.29, respectively. In summary, it can be concluded that the distribution of the object classes are dense and highly overlapped in the HMAX feature space, while the object classes are well separated in the feature space of our model.
\begin{table*}
\caption{The number of common features between each pair of classes of 3D-Object dataset. The gray level of each cell indicates the relative distance of the cell value to the maximum possible value (=50).}
\label{table_example2}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Class & Bicycle & Car & Cellphone & Head & Iron & Monitor & Mouse & Shoe & Stapler & Toaster\\
\hline
Bicycle & \cellcolor[gray]{0.3} 50 & \cellcolor[gray]{1}0 & \cellcolor[gray]{0.98}1 & \cellcolor[gray]{0.95}3 & \cellcolor[gray]{0.91}6 &\cellcolor[gray]{0.94} 4 & \cellcolor[gray]{0.83}12 & \cellcolor[gray]{0.91}6 & \cellcolor[gray]{0.925}5 &\cellcolor[gray]{0.925} 5\\
\hline
Car & \cellcolor[gray]{1}0 & \cellcolor[gray]{0.3} 50 & \cellcolor[gray]{0.98}1 & \cellcolor[gray]{0.805}14 & \cellcolor[gray]{0.86}10 & \cellcolor[gray]{0.87}11 & \cellcolor[gray]{0.98} 1 &\cellcolor[gray]{0.96} 2 &\cellcolor[gray]{0.91} 6 &\cellcolor[gray]{0.895} 7\\
\hline
Cellphone &\cellcolor[gray]{0.98} 1 &\cellcolor[gray]{0.98} 1 & \cellcolor[gray]{0.3} 50 &\cellcolor[gray]{1} 0 &\cellcolor[gray]{0.95} 3 &\cellcolor[gray]{0.94} 4 &\cellcolor[gray]{0.96} 2 &\cellcolor[gray]{1} 0 & \cellcolor[gray]{0.86}10 &\cellcolor[gray]{0.88} 9\\
\hline
Head & \cellcolor[gray]{0.95}3 &\cellcolor[gray]{0.805} 14 &\cellcolor[gray]{1} 0 & \cellcolor[gray]{0.3} 50 & \cellcolor[gray]{1}0 & \cellcolor[gray]{1}0 &\cellcolor[gray]{0.86} 10 & \cellcolor[gray]{0.775}16 &\cellcolor[gray]{0.96} 2 &\cellcolor[gray]{0.96} 2\\
\hline
Iron & \cellcolor[gray]{0.91}6 & \cellcolor[gray]{0.86}10 &\cellcolor[gray]{0.95} 3 &\cellcolor[gray]{1} 0 & \cellcolor[gray]{0.3} 50 & \cellcolor[gray]{0.70}21 & \cellcolor[gray]{1}0 &\cellcolor[gray]{0.94} 4 &\cellcolor[gray]{0.83} 12 & \cellcolor[gray]{0.91}6\\
\hline
Monitor &\cellcolor[gray]{0.94} 4 &\cellcolor[gray]{0.87} 11 &\cellcolor[gray]{0.94} 4 &\cellcolor[gray]{1} 0 & \cellcolor[gray]{0.70}21 & \cellcolor[gray]{0.3} 50 & \cellcolor[gray]{1}0 &\cellcolor[gray]{1} 0 &\cellcolor[gray]{0.88} 8 & \cellcolor[gray]{0.805}14\\
\hline
Mouse & \cellcolor[gray]{0.83}12 &\cellcolor[gray]{0.98} 1 &\cellcolor[gray]{0.96} 2 & \cellcolor[gray]{0.86}10 & 0 & 0 & \cellcolor[gray]{0.3} 50 & \cellcolor[gray]{0.96}2 & \cellcolor[gray]{0.96}2 & \cellcolor[gray]{0.925}5\\
\hline
Shoe & \cellcolor[gray]{0.91}6 & \cellcolor[gray]{0.96}2 & \cellcolor[gray]{1}0 & \cellcolor[gray]{0.775}16 & \cellcolor[gray]{0.94}4 & \cellcolor[gray]{1}0 & \cellcolor[gray]{0.96}2 & \cellcolor[gray]{0.3} 50 & \cellcolor[gray]{1}0 & \cellcolor[gray]{1}0\\
\hline
Stapler & \cellcolor[gray]{0.925}5 & \cellcolor[gray]{0.91}6 & \cellcolor[gray]{0.86}10 & \cellcolor[gray]{0.96}2 & \cellcolor[gray]{0.83}12 & \cellcolor[gray]{0.88}8 &\cellcolor[gray]{0.96} 2 &\cellcolor[gray]{1} 0 & \cellcolor[gray]{0.3} 50 & \cellcolor[gray]{0.72}20\\
\hline
Toaster & \cellcolor[gray]{0.925}5 & \cellcolor[gray]{0.895}7 & \cellcolor[gray]{0.88}9 & \cellcolor[gray]{0.96}2 & \cellcolor[gray]{0.91}6 & \cellcolor[gray]{0.805}14 & \cellcolor[gray]{0.925}5 & \cellcolor[gray]{1}0 & \cellcolor[gray]{0.72}20 & \cellcolor[gray]{0.3} 50\\
\hline
\end{tabular}
\end{table*}
In an other experiment, we analyzed the class dependency of the $C_2$ features for our model. To this end, the 50 most informative features, when classifying a specific class against all the other classes, are selected by employing the mutual information technique. In other words, for each class, we selected those 50 features which have the highest activity for samples of that class and have less activity for other classes. Afterwards, the number of common features among the informative features of each pair of classes are computed as provided in Table~\ref{table_example2}. On average, there are only about 5.4 common features between pairs of classes. Although there are some common features between any two classes, their co-occurrence with the other features help the classifier to separate them from each other. In this way, our model can represent various object classes with a relatively small number of features. Indeed, exploiting the intermediate complexity features, which are not common in all classes and are not very rare, can help the classifier to discriminate instances of different classes~\cite{Ullman2002}.
\subsection{Random features and simple classifier}
In a previous study~\cite{leibo2010primal}, it has been shown that using the HMAX model with random dot patterns in the $S_{2}$ layer can reach a reasonable performance, comparable to the one obtained with random patches cropped from the training images. It seems that this is due to the dependency of HMAX to the application of a powerful classifier. Indeed, the use of both random dot or randomly selected patches transform the images into a complex and nested feature space and it is the classifier which looks for a complex signature to separate object classes. The deficiencies emerge when the classification problem gets harder (such as invariant or multiclass object recognition problems) and then even a powerful classifier is not able to discriminate the classes~\cite{Pinto2008,pinto2011comparing}. Here, we show that the superiority of our model is due to the informative feature extraction through a bio-inspired learning rule. To this end, we have compared the performances on 3D-Object dataset obtained with random features versus STDP features, as well as a very simple classifier versus SVM.
To generate random features, we have set the weight matrix of each $S_{2}$ feature of our model with random values. First, we have computed the mean and standard deviation (STD) ($253 \pm 21$) of the number of active (nonzero) weights in the features learned by STDP. Second, for each random feature, the number of active weights, $N$, is computed by generating a random number based on the obtained mean and STD. Finally, a random feature is constructed by uniformly distributing the $N$ randomly generated values in the weight matrix.
In addition, we designed a simple classifier comprised of several one-versus-one classifiers. For each binary classifier, two subset of $C_{2}$ features with high occurrence probabilities in one of the two classes are selected. In more details, to select suitable features for the first class, the occurrence probabilities of $C_{2}$ features in this class are divided by the corresponding occurrence probabilities in the second class. Then, a feature is selected if this ratio is higher than a threshold. The optimum threshold value is computed by a trial and error search in which the performance over the training samples is maximized. To assign a class label to the input test sample, we performed an inner product on the feature value and feature probability vectors. Finally, the class with the highest probability is reported to the combined classifier. The combined classifier selects the winner class based on a simple majority voting.
For 500 random features, using the SVM and the simple classifier, our model reached classification performances of 71\% and 21\% on average, respectively. Whereas, for the learned $S_{2}$ features, both the SVM and simple classifiers attained reasonable performances of 96\% and 79\%, respectively. Based on these results, it can be concluded that the features obtained through the bio-inspired unsupervised learning projects the objects into an easily separable space, while the feature extraction by selection of random patches (drawn from the training images) or by generation of random patterns leads to a complex object representation.
\section{Discussion}\label{Discussion}
Position and scale invariance in our model are built-in, thanks to weight sharing and scaling process. Conversely, view-invariance must be obtained through the learning process. Here, we used all images of five object instances from each category (varied in all dimensions) to learn the $S_{2}$ visual features, while images of all other object instances of each category were used to test the network. Hence, the model was exposed to all possible variations during the learning to gain view-invariance. Moreover, near or opposite views of the same object shares some features which are suitable for invariant object recognition. For instance, consider the overall shape of a head, or close views of a bike wheel which could be a complete circle or an ellipse. Regarding the fact that STDP tends to learn more frequent features in different images, different views of an object could be invariantly represented based on more common features.
Our model appears to be the best choice when dealing with few object classes, but huge variations in view points. As pointed out in previous studies, both HMAX and DeepConvNet models could not handle these variations perfectly~\cite{Pinto2008,pinto2011comparing, ghodrati2014feedforward}.
Conversely, our model is not appropriate to handle many classes, which requires thousands of features, like in the ImageNet contest, because its time complexity is roughly in $N^2$, where $N$ is the number of features (briefly: since the number of firing neurons per image is limited, if the number of features is doubled, reaching convergence will take roughly twice as many images, and the processing time for each of them will be doubled as well). For example, extracting 4096 features in our model, the same number of features in DeepConvNet, would take about 67 times it took us to extract 500. However, parallel implementation of our algorithm could speed-up the computation time by several orders of magnitude~\cite{lemoine2013gpu}. Even in this case, we do not expect to outperform the DeepConvNet model on the ImageNet database, since only the shape similarities are taken into account in our model and the other cues such as color or texture are ignored.
Importantly, our algorithm has a natural tendency to learn salient contrasted regions~\cite{Masquelier2007}, which is desirable as these are typically the most informative~\cite{VanRullen2001}. Most of our $C_2$ features turned out to be class-specific, and we could guess what they represent by doing the reconstructions (see Fig.~\ref{features} and Fig. S2). Since each feature results from averaging multiple input images, the specificity of each instance is averaged out, leading to class archetypes. Consequently, good classification results can be obtained using only a few features, or even using `simple' decision rules like feature counts~\cite{Masquelier2007} and majority voting (here), as opposed to a `smart classifier' such as SVM.
There are some similarities between STDP-based feature learning, and non-negative matrix factorization~\cite{Lee1999}, as first intuited in~\cite{Masquelier2010}, and later demonstrated mathematically in~\cite{Carlson2013}. Within both approaches, objects are represented as (positive) sums of their parts, and the parts are learned by detecting consistently co-active input units.
Our model could be efficiently implemented in hardware, for example using address event representation (AER)~\cite{Zamarreno-Ramos2011,Bichler2012,dorta2016aer,diaz2016efficient}. With AER, the spikes are carried as addresses of sending or receiving neurons on a digital bus. Time `represents itself' as the asynchronous occurrence of the event~\cite{Sivilotti1991}. Thus the use of STDP will lead to a system which effectively becomes more and more reactive, in addition to becoming more and more selective. Furthermore, since biological hardware is known to be incredibly slow, simulations could run several order of magnitude faster than real time~\cite{Serrano-Gotarredona2013}. As mentioned earlier, the primate visual system extracts the rough content of an image in about 100ms. We thus speculate that some dedicated hardware will be able to do the same in the order of a millisecond or less.
Recent computational~\cite{Ullman2002}, psychophysical~\cite{harel2011basic}, and fMRI~\cite{lerner2008class} experiments demonstrate that the informative intermediate complexity features are optimal for object categorization tasks. But the possible neural mechanisms to extract such features remain largely unknown. The HMAX model ignores these learning mechanisms and imprints its features with random crops from the training images~\cite{Serre2007.PAMI,Serre2007}, or even uses random filters~\cite{leibo2010primal,Yamins2014a}. Most individual features are thus not very informative, yet in some cases, a `smart' classifier such as SVM can efficiently separate the high-dimensional vectors of population responses.
Many other models use supervised learning rules~\cite{LeCun1998,Krizhevsky2012}, sometimes reaching impressive performance on natural image classification tasks~\cite{Krizhevsky2012}. The main drawback of these supervised methods, however, is that learning is slow and requires numerous labeled samples (e.g., about 1 million in~\cite{Krizhevsky2012}), because of the credit assignment problem~\cite{Rolls2002,Ranzato2007}. This contrasts with humans who can generalize efficiently from just a few training examples~\cite{cox2014neural}. We avoid the credit assignment problem by keeping the $C_2$ features fixed when training the final classifier (that being said, fine-tuning them for a given classification problem would probably increase the performance of our model~\cite{Le2013,6763041}; we will test this in future studies). Even if the efficiency of such hybrid unsupervised-supervised learning schemes has been known for a long time, few alternative unsupervised learning algorithms have been shown to be able to extract complex and high-level visual features (see~\cite{Lee2009,Le2013}). Finding better representational learning algorithms is thus an important direction for future research and seeking for inspiration in the biological visual systems is likely to be fruitful~\cite{cox2014neural}. We suggest here that the physiological mechanism known as STDP is an appealing start point.
Considering the time relation among the incoming inputs is an important aspect of spiking neural networks. This property is critical to promote the existing models from static vision to continuous vision~\cite{Masquelier2012}. A prominent example is the trace learning rule~\cite{Foldiak1991}, suggesting that the invariant object representation in ventral visual system is instructed by the implicit temporal contiguity of vision. Also, in various motion processing and action recognition problems~\cite{escobar2009action}, the important information lies in the appearance timing of input features. Our model has this potential to be extended for continuous and dynamic vision -- something that we will further explore.
\section{Conclusions}\label{Conclusion}
To date, various bio-inspired network architectures for object recognition have been introduced, but the learning mechanism of biological visual systems has been neglected. In this paper, we demonstrate that the association of both bio-inspired network architecture and learning rule results in a robust object recognition system. The STDP-based feature learning, used in our model, extracts frequent diagnostic and class specific features that are robust to deformations in stimulus appearance. It has previously been shown that the trivial models can not tolerate the identity preserving transformations such as changes in view, scale, and position. To study the behavior of our model confronted with these difficulties, we have tested our model over two challenging invariant object recognition databases which includes instances of 10 different object classes photographed in different views, scales, and tilts. The categorization performances indicate that our model is able to robustly recognize objects in such a severe situation. In addition, several analytical techniques have been employed to prove that the main contribution to this success is provided by the unsupervised STDP feature learning, not by the classifier. Using representational dissimilarity matrix, we have shown that the representation of input images in $C_{2}$ layer are more similar for within-category and dissimilar for between-category objects. In this way, as confirmed by the hierarchical clustering, the objects with the same category are represented in neighboring regions of $C_{2}$ feature space. Hence, even if using a simple classifier, our model is able to reach an acceptable performance, while the random features fail.
\section*{Acknowledgements}
We would like to thank Mr. Majid Changi Ashtiani at the Math Computing Center of IPM (http://math.ipm.ac.ir/mcc) for letting us to perform some parts of the calculations on their computing cluster. We also thank Dr.~Reza Ebrahimpour for his helpful discussions and suggestions.
\begin{footnotesize}
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{
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Klippfasan (Catreus wallichii) är en hotad bergslevande asiatisk fågel i familjen fasanfåglar inom ordningen hönsfåglar.
Kännetecken
Utseende
Klippfasanen är en stor och färglös fasan med lång, brett bandad stjärt, tydlig tofs och röd bar hud i ansiktet. Storleksskillnaden mellan könen är stor, där hanen uppnår 90-118 cm (varav stjärten 45-58) och honan endast 61-76 cm (stjärte 32-47 cm). Hanen är renare och tydligare tecknad än honan, med tydlig bandning på manteln, grå hals med inga eller få teckningar och bredare bandning på stjärten.
Läte
Lätet återges i engelsk litteratur som ett högljutt "chir-a-pir chir-a-pir chir chir-chirwa chirwa" samt ett ljust och genomtränfande "chewewoo" uppblandat med korta "chut" och hårda staccatotoner.
Utbredning och systematik
Fågeln förekommer i västra Himalaya från nordöstra Pakistan till västcentrala Nepal. Tidigare uppgifter om att arten skulle finnas i östra Afghanistan och Sikkim anses nu vara felaktiga. Den placeras som ensam art i släktet Catreus och behandlas som monotypisk, det vill säga att den inte delas in i några underarter.
Levnadssätt
Klippfasanen gör skäl för sitt namn. Den återfinns på branta och klippiga sluttningar med buskvegetation, gräs och spridda stånd av träd, vanligen mellan 1445 och 3050 mters höjd, ibland ner till åtminstone 950 meter. Den har även påträffats i ungskog samt en- och rhododendronskog på gräsrika sluttningar. Den gräver efter rötter och rotknölar, men äter även frön, bär och insekter. Fågeln har noterats häcka i maj, juni och september i Indien och lägger 6-12 ägg.
Status och hot
Klippfasanen har ett naturligt fragmenterad utbredningsområde och en liten världspopulation uppskattad till endast 2 500-10 000 vuxna individer. Den tros dessutom minska i antal på grund av överintensivt bete och jakt. Internationella naturvårdsunionen IUCN kategoriserar därför arten som sårbar. Dess förkärlek för återväxande marker gör att den dras till mänsklig bebyggelse, vilket gör den än sårbarare för jakt.
Namn
Fågelns vetenskapliga artnamn hedrar Nathaniel Wallich (född Nathan ben Wulff, 1786-1854), dansk kirurg, botaniker, överintendent på Oriental Museum of Asiatic Society 1814, överintendent på East India Company Botanical Gardens i Calcutta 1817-1846 samt samlare av specimen i Indien, Myanmar, Nepal och Singapore.
Noter
Externa länkar
Läten på xeno-canto.org
Fasanfåglar
Fåglar i palearktiska regionen
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{
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Suzanne Delbanco chats with Jeffrey Davis, MD, MPH, Senior Health Management Consultant with Willis Towers Watson, about a rapidly expanding area of intrigue for employers: genetic testing. He breaks down the types of genetic testing out there, as well as concrete steps employers can take to navigate this space. Jeffrey Davis, who consults for large companies on their health care benefits strategies, draws on research from Health Affairs to lay out the genetic testing landscape. Listen in to find out if genetic testing is more than just a "shiny new object" that Silicon Valley employers are using to retain top talent.
Interested in taking action in this area? CPR's How-to Guide: Unraveling Genetic Testing Benefits is available for free to employer-purchasers. CPR leveraged Jeffrey Davis as a Subject Matter Expert to develop this toolkit, which includes Background and Education on the Genetic Testing Landscape; a Health Plan Questionnaire; a Health Plan Data Request Template; and a Framework to Evaluate Specialty Vendors.
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I found this article at this blog: I think you will find it a important read. Thanks for your support!
Anyway, I now no longer need to write this because Sunday the Charlotte Observer beat me to it, profiling none other than my future boss Phil Davis, pastor of Faith Community Church in Prague. Read the article, and be sure to look at the pictures– the Davises are a good-looking bunch.
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Il Campionato francese di rugby a 15 1955-56 fu conquistato dal FC Lourdes che superò l'US Dax in finale.
Formula
48 squadre divise in 6 gironi da 8
si qualificano per i sedicesimi di finale le prime 5 di ogni gruppo e le due migliori seste.
Contesto
Il Torneo delle Cinque Nazioni 1956 fu vinto dal , la terminò al secondo posto
Il Challenge Yves du Manoir fu vinto dal Lourdes che sconfisse l'USA Perpignan in finale.
Fase di qualificazione
In grassetto le qualificate per i sedicesimi
Sedicesimi di finale
In grassetto le qualificate agli ottavi
Ottavi di finale
In grassetto le qualificate ai quarti
Quarti di finale
In grassetto il qualificate alle semifinali
Semifinali
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Every day Wendy Williams sits down in her chair to discuss Hot Topics. On one of her last shows one of the topics happened to be about Da Brat and Williams alleged she owed money to the woman she hit with a bottle. Williams also stated that she doesn't think Da Brat has the money to pay for it because she makes "lunch money" on Dish Nation.
She also mentioned that while Williams said she liked Da Brat it would have been better if she came to her directly for the truth. Da Brat also stated for the record that if she needed the money her close friends Jermaine Dupri and Mariah Carey would have no problem giving it to her, but she would never ask for it. Da Brat ended her comments about Williams jokingly saying, ""I make lunch money. We're barely eating over here. We're starving!" Let's see if Williams responds back.
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{
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Q: performance of std::string operator + versus stringstream I am migrating an old code base from using a custom implementation of strings to use std::string. One of our engineers is concerned that this could lead to a performance issue where we are using += and + to concatenate relatively long strings. His particular concern is how the memory is allocated as the string grows dynamically, and he suggested that stringstream would perform better in this regard. For those of you tempted to respond "benchmark then optimize" or sentiments to those effect -- I do of course agree, and that's the plan. I'm just wondering about the theory.
I did a little searching for this issue, and ran into this question:
Efficient string concatenation in C++, which is getting a little dated.
Now I'm wondering: Is the memory allocation algorithm for string += really different than that for string stream? What about the copy issues raised in the above question -- have these changed with c++ 11/14?
A:
Is the memory allocation algorithm for string += really different than that for string stream?
Yes.
A stringstream writes into a stringbuffer, which usually means a linked list of buffers. Memory is not continuous, and it requires no reallocations as the buffer grows.
A string on the other hand, allocates memory in a single block (as needed, and sometimes preemptively).
The most efficient solution is probably to precompute the resulting string size, allocate manually the result, then fill it in (using std::string::operator +=).
That said, I would just write into std::ostringstream and take the result at the end (but YMMV).
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{
"redpajama_set_name": "RedPajamaStackExchange"
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Cincinnati Boy Scouts cancel Christmas tree sale, collect toys instead
By FOX19 Digital Staff
Published: Nov. 26, 2021 at 11:54 AM EST|Updated: Nov. 28, 2021 at 8:29 AM EST
CINCINNATI (WXIX) - A Boy Scouts troop is collecting toys after a global Christmas tree shortage canceled their tradition.
Every year, St. James Boy Scout Troop 24 holds their Christmas tree sale the morning of Thanksgiving.
According to the troop's Facebook post, they had to cancel because of a global Christmas tree storage.
This year, the troop is collecting gifts for Nate's Toy Box, a local organization that provides toys to children in the district.
Assistant Scout Master Brian Bourbeau says, "Nate's toy box is pretty well known in this area. A lot of drop-off points, a lot of businesses here. But the boys, you know, they're used to working during the tree sale. So now they're going to be working to help others through the toy collection."
The troop is asking for new, unwrapped gifts for collection at St. James Field on Cheviot Road, where the Christmas tree sale would normally be held.
They say that Nate's Toy Box is especially looking for Lego sets, craft kits, scented lotion and soap kits, art supplies, basketballs, footballs and books.
The troop will start collecting toys Friday and continue throughout the Christmas season.
Collection hours:
Nov. 26 10:00 a.m. - 8:00 p.m.
Dec. 3 6:00 p.m. - 8:00 p.m.
Dec. 4 10:00 a.m. - 8:00 a.m.
Dec. 5 11:00 a.m. - 7:00 p.m.
Vigil held for 19-year-old shot to death in Pleasant Ridge
City fences off homeless encampment in Queensgate claiming 'inhumane' conditions
Highland County man dead in suspected OVI crash, police say
Plea deal reached in child porn case against Delhi man
I-275 West closed in Colerain due road rage shooting, police say
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Anick Lemay () est une actrice québécoise née à Thetford Mines (Canada).
Biographie
Elle est maman d'une petite fille, prénommée Simone, depuis le mois d'. Elle a joué dans plusieurs publicités de la bannière de pharmacies Uniprix.
Le , elle a annoncé qu'elle est atteinte d'un cancer du sein.
Filmographie
1996 : 4 et demi... (série TV)
1997 : Les Bâtisseurs d'eau (série télévisée)
1998 : Caserne 24 (série télévisée) : Carmen Tranchemontagne
1999 : L'Île de sable : Geneviève
2001 : Tribu.com (série télévisée) : Brigitte David
2002 : La Mystérieuse mademoiselle C. : Madame Juneau
2003 : Hommes en quarantaine (série télévisée) : Marie-Ève Thibault
2003 : Déformation personnelle : La Femme en lui
2004 : Le Dernier Tunnel : Isabelle Parenteau
2005 : Maman Last Call : Carole Dion
2005 : Le Survenant : Angélina Desmarais
2006 : Duo de Richard Ciupka : Pascale Lachance
2006 : Cheech de Patrice Sauvé : Jenny
2006 : François en série (série télévisée) : La femme en lui
2008 : La Promesse (série télévisée) : Carole
2009 : Bienvenue aux dames (série télévisée) : Carolanne
2010 - 2012 : Mauvais Karma (série télévisée) : Sarah Boisvert
2014 : Toi & Moi (série télévisée) : Elizabeth Olyphant
2016 - : L'Échappée (série télévisée) : Noémie Francoeur
2023- : L'air d'aller (série télévisée)
Récompenses et nominations
Prix Aurore
2007: Acteur ou actrice ayant offert une meilleure performance dans Virginie ou dans une pub de pharmacie que dans un film
Notes et références
Lien externe
Actrice québécoise de cinéma
Actrice québécoise de télévision
Naissance en mars 1969
Naissance à Thetford Mines
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Simmondley is a small village near the town of Glossop in Derbyshire, England. The population of the High Peak ward at the 2011 Census was 4,727. It has one pub, the Hare and Hounds, in the south of the village at the top of Simmondley Lane. The pub is a part of the original farming community with the adjacent farmhouse, barn and stables converted into houses. The Jubilee pub was built in 1977, in celebration of the Silver Jubilee of Elizabeth II. After 40 years, the brewery that owned the Jubilee sold it at auction; the buyer demolished the building in 2017 to build houses on the site and adjoining car park.
In August 1981 the Sorgro convenience store opened on Pennine Road. In recent years this has been a Spar, an Alldays and is currently run by The Co-operative Food. There is a post office, a Chinese takeaway, a dentist, a doctor, a chemist, a coffee shop and a hairdressers.
Many large housing projects have recently been completed in Simmondley, including a large housing estate off Valley Road that stretches towards the existing Manchester rail line.
Simmondley has a number of public areas, including: a children's play park area with swings and a centre climbing frame; an enclosed floodlit games court called the S.M.U.G.A (Simmondley Multi Use Games Area) with football nets and basketball hoops; open grassland around the estate mainly surrounding the Werneth Road area; a village green to the top of Simmondley with a public phone box, post box, plant pots and seating (during the Christmas period this is the location of the Simmondley Christmas Tree).
The housing developments south of the village have led to it being considered by some as a suburb of Glossop, rather than a separate settlement as it is contiguous with Glossop, although in recent years the local council has installed Simmondley signs at accesses to the village to mark that it has its own separate identity.
Simmondley is at the bottom of the so-called Monks' Road, a road used by the monks of Basingwerk Abbey to administer the abbey's estate. It leads to Charlesworth, Chisworth and Hayfield.
Schools and further education
Simmondley Pre-School
Simmondley pre-School has two sites for young children to attend. Natural Explorers occupy the old chapel on High Lane, in the original village part of Simmondley; age intake starts from 2 years old and goes up to 4 years old.
A second pre-school group is based in the grounds of Simmondley Primary School.
Simmondley Primary School
Simmondley Primary School is a primary school with over 300 pupils. The school is based on Pennine Road and was built in the 1970s by Derbyshire Education Authority using the CLASP (Consortium of Local Authorities Special Programme) design method, which was popular with councils for rapid builds between the 50s and 90s.
Transport
Bus
Simmondley is served by a number of bus routes throughout the week.
Train
There has been calls in recent years for Simmondley to be jointly served by a train station in Gamesley. So far no plans have been submitted.
Roads
The A6016 road runs north of Simmondley from the A57 to the A624.
See also
Listed buildings in Simmondley
References
External links
Villages in Derbyshire
Towns and villages of the Peak District
High Peak, Derbyshire
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I am honored to be participating in an online project called Telephone, which links 314 artists in over 80 countries, in a global game of sharing. I gratefully attended the local launch party in New York City. At the event I met the originator of the project as well as several fellow participating artists. I waxed nostalgic for a simpler time, when I was younger and full of promise, eager to make it in the New York City art scene. I listened to feedback about my work with the heart of a beginner despite my twenty years of mostly steady practice in suburbia. I left feeling invigorated and hopeful from this purity of experience.
This piece shows my branch in the telephone game; the initial message is at the root, while iterations vary in color and intensity toward my outermost position beyond the contained interior ocean space. I am Pluto, a much-debated celestial object, vying for recognition in a vast milky way. On the outside, a star, navigation buttons, and a quote from a poet who spoke with me that night.
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This folder contains a bunch of step by step conversion tutorials in order to help you understand how to use
browserify-ftw in order to upgrade your project.
I'd advice you to follow along with at least one of them. They are all available on github.
- [upgrading ryanfitzer's RequireJS jQuery
Project](https://github.com/thlorenz/browserify-ftw/blob/master/examples/RyanFitzer-RequireJs-Jquery.md#initializing-the-npm-package-and-installing-dependencies)
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"redpajama_set_name": "RedPajamaGithub"
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With the end of the season eminent I think Orlando Pirates must apply their mind and buy quality and not be gluttons and hog every available player in the market. Firstly, they need to be cognisant of the fact that our centre backs are all aged...thus, they must scout for two young centre backs to take over from Jele or to be paired with either Jele or Ndengane. Tercious Malepe must be brought back from Chippa as he has amassed enough mileage in top flight football already. Secondly, Orlando Pirates needs 2 prolific strikers who can bury the chances they create, reliance on Shonga and Mulenga has not yielded any positives and the amount of chances that are missed by Shonga are too much and in some games Orlando Pirates will lose all because Shonga misses chances like they are out of fashion. Instead Shonga must be used as a winger or a supporting striker in a 4:4:2 formation. Thirdly they must get rid of two to three of the current keepers and bring in a competitive goalkeeper to compete with either Sandilands or Khuzwayo. For this to happen Pirates need to venture into the international market and not confine their search to Zimbabwe and Zambia.
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\section{Introduction}
A possible approach to explain the current acceleration of the universe is based on the holographic dark energy \cite{Li:2004rb,Hsu:2004ri}
. Such a phenomenological model is based on the idea that the energy density of a given system is bounded by a magnitude proportional to the inverse square of a lenght characterising the system \cite{Susskind:1994vu,Cohen:1998zx}. When this principle is applied to the universe as a whole, we obtain the holographic dark enery \cite{Li:2004rb,Hsu:2004ri}. It turns out that there are many different ways of characterising the size of the universe and one of them is related to the inverse of the Ricci curvature of the universe. When the size of the universe is charaterised in such a way, we end up with the holographic Ricci dark energy (RDE) model \cite{Gao}, whose energy density reads:
\begin{equation}
\rho _{\mathrm{H}}=3\beta M_{P}^{2}\left( \frac{1}{2}\frac{dH^{2}}{dx}%
+2H^{2}\right) , \label{HRDE}
\end{equation}%
where $M_{P}$ is the Planck mass, $x=-\ln (z+1)=\ln(a)$, $z$ is the redshift and $\beta$ is a
dimensionless parameter that measures the strength of the holographic component.
A spatially flat Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) universe filled with this kind of matter accelerates and therefore the RDE can play the role of dark energy on the Universe. It turns our that the asymptotic behaviour of the Universe depends crutially on the value acquired by $\beta$: (i) if $\beta \leq 1/2$ the universe is asymptotically de Sitter, otherwise (ii) the universe faces a big rip singularity \cite{bigrip} in its future evolution.
Our main purpose in this paper is to see if we can appease the big rip appearing in some cases on the RDE by invoking some infra-red and ultra-violet curvature corrections. This two corrections can be quite important to remove the big rip singularity which takes place on the future and at high energy. The curvature corrections will be modeled within a 5-dimensional brane-world model with an induced gravity (IG) term on the brane and a Gauss-Bonnet term in the bulk \cite{brane}.
\section{The RDE model with curvature corrections}
We consider a DGP brane-world model, where the bulk action contains a GB
curvature term. The bulk corresponds to two symmetric pieces of a
5-dimensional (5d) Minkowski space-time. The brane is spatially flat and its
action contains an IG term. We assume that the brane is filled
with matter and RDE. Then, the modified Friedmann
equation reads \cite{brane}:
\begin{equation}
H^{2}=\frac{1}{3M_{P}^{2}}\rho +\frac{\epsilon }{r_{c}}\left( 1+\frac{%
8\alpha }{3}H^{2}\right) H, \label{friedmann1}
\end{equation}%
where $H$ is the brane Hubble parameter, $\rho =\rho_{\mathrm{m}}+\rho_{%
\mathrm{H}}$ is the total cosmic fluid energy density of the brane which can
be described through a cold dark matter component (CDM) with energy density $%
\rho _{\mathrm{m}}$ and an holographic Ricci dark energy component with
energy density $\rho _{\mathrm{H}}$. The parameters $r_{c}$ and $\alpha$
correspond to the cross over scale and the GB parameter, respectively,
both of them being positive. The parameter $\epsilon$ in Eq.~(\ref{friedmann1})
can take two values: $\epsilon =1$, corresponding to the self-accelerating
branch in the absence of any kind of dark energy \cite{brane}; and $%
\epsilon =-1$, corresponding to the normal branch which requires a dark
energy component to accelerate at late-time (see for example
\cite{preparation,BouhmadiLopez:2010pp,BouhmadiLopez:2008nf}).
For simplicity, we will keep the terminology: (i) self-accelerating branch when $\epsilon=1$ and
(ii) normal branch when $\epsilon=-1$.
The modified Friedmann equation (\ref{friedmann1}) can be rewritten as
\begin{equation}
\dfrac{dE}{dx}=-\dfrac{\Omega _{m}e^{-3x}+\left( 2\beta -1\right)
E^{2}+2\epsilon \sqrt{\Omega _{r_c}}(1+\Omega _{\alpha }E^{2})E}{\beta E},
\label{variation of E}
\end{equation}
where $E(z)=H/H_{0}$ and
\begin{eqnarray}
\Omega _{m} &=&\frac{\rho _{m_{0}}}{3M_{P}^{2}H_{0}^{2}},\,\,\,\,\Omega
_{r_{c}}=\frac{1}{4r_{c}^{2}H_{0}^{2}},\text{\thinspace and }\,\Omega
_{\alpha }=\frac{8}{3}\alpha H_{0}^{2}\, \notag \\
&&
\end{eqnarray}%
are the usual convenient dimensionless parameters and the subscripts $0$
denotes the present value (we will follow the same notation as in \cite%
{BouhmadiLopez:2008nf,preparation,Belkacemi:2011zk}). By evaluating the modified Friedmann equation
at present and imposing that the brane is currently accelerating, we obtain a constraint on the parameter $\beta$ which depends
on the chosen brane
\begin{equation}
\left\{
\begin{array}{c}
\beta <\beta _{\mathrm{lim}}\text{ for \ \ }\epsilon =+1, \\
\beta >\beta _{\mathrm{lim}}\text{ \ for \ }\epsilon =-1,%
\end{array}%
\right. \label{conditions for beta}
\end{equation}%
where
\begin{equation}
\beta _{\mathrm{lim}}=\frac{1-\Omega _{m}}{1-q_{0}}. \label{betalim}
\end{equation}%
An estimation of $\beta _{\mathrm{lim}}$ can be obtained as follows: the brane would behave roughly (to be consistent with the present observations) as the $\Lambda$CDM leading to $\beta _{\mathrm{lim}}\sim 0.43$.
Even though the modified Friedmann equation (\ref{variation of E}) cannot be solved analytically, we can obtain the future asymptotic behaviour of the brane which reads: (i) If $\beta < \beta_{\mathrm{lim}}$ or $\beta_- \leq \beta$, the brane is asymptotically de Sitter. (ii) If $\beta_{\mathrm{lim}}<\beta<\beta_-$, the brane faces a big freeze singularity in its future \cite{BouhmadiLopez:2006fu}, where (see also Fig. 1)
\begin{eqnarray}
\beta_\pm&=&\frac{1+\Omega_\alpha\pm2\sqrt{\Omega_\alpha}(1-\Omega_m)}{2%
\left[1+\Omega_\alpha\pm\sqrt{\Omega_\alpha}(1-q_0)\right]}.
\label{defbetapm}
\end{eqnarray}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.6\columnwidth]{discriminant.eps}
\end{center}
\caption{Plot of the parameters $\protect\beta_\pm$ and $\protect\beta_{%
\mathrm{lim}}$, defined in Eqs.~(\protect\ref{defbetapm}) and (\protect\ref%
{betalim}), respectively, versus the parameter $\Omega_\protect\alpha$. We
have used the values $q_{0}\sim -0.7,$ and $\Omega
_{m}\sim 0.27$. The parameter $\protect\beta_{\mathrm{lim}}$ defines the
border line between the normal branch ($\protect\beta_{\mathrm{lim}}<\protect%
\beta$) and the self-accelerating branch ($\protect\beta<\protect\beta_{%
\mathrm{lim}}$).}
\label{plotbetas}
\end{figure}
We have completed and confirmed those results by solving numerically the cosmological evolution of the brane. We refer the reader to \cite {Belkacemi:2011zk} for more details. Our analysis shows that even though the infra-red and ultra-violet effect can appease the big rip appearing on the RDE model, it cannot remove them completely. We would like as well to point out that when the GB term is switched off a little rip event \cite{Frampton:2011sp} can show up which is much milder that a big rip or a big freeze. The little rip has been previously found on brane-world model \cite{BouhmadiLopez:2005gk}.
\section{Conclusions}
We present an HRD energy brane-world model of the Dvali-Gabadadze-Porrati
scenario with a GB term in the bulk. The reason for invoking curvature corrections, for example through a brane-world scenerio, is to try to smooth the doomdays present on a standard 4-dimensional HRD energy model. It turns out that the model presented here can only partially remove those doomsdays.
\begin{acknowledgement}
M.B.L. is supported by the Spanish Agency CSIC through JAEDOC064.
A.E. and T.O. are supported by CNRST, through the fellowship URAC 07/214410. This work was supported by the Portuguese Agency FCT through PTDC/FIS/111032/2009.
\end{acknowledgement}
\input{referenc}
\end{document}
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This article provides two methods to help you repair corrupted hard disk without formatting by CMD in Windows 10/8/7 and recover files and repair internal/external hard disk by formatting. If you need to restore fancy data from corrupted hard disk, just download the best hard drive recovery software to get your lost data back and follow solutions here to repair corrupted internal/external hard drive without formatting now.
Sometimes, your hard drive can be damaged and become unrecognized or inaccessible in Windows 10/8/7 due to various reasons, no matter it is an internal Hard disk used on a laptop/desktop or an external hard drive used as a storage device. Here are some common causes and symptoms of hard drive corruption.
External hard drive not recognized, reading or showing up while plugging into PC.
Get "X:\ is not accessible. The file or directory is corrupted and unreadable" error message.
The hard drive has a drive letter in My Computer but becomes inaccessible and says: "You need to format the disk in the drive before you can use it".
The hard drive becomes RAW and prompts 'disk needs to be formatted' error.
Hard disk shows 0 bytes.
When you encountered one of the above situations, you have to repair corrupted hard disk to make it work usually. It would be quickly done by formatting. But if you have essential data on the device, you may want to open or repair corrupted internal hard disk without formatting. So you can protect your files. Fortunately, chkdsk command prompt in cmd can help you manage it efficiently in Windows 10/8/7.
Step 1. Press Windows Key + X button to bring up the power users menu.
Step 2. In the power users menu, select the Command Prompt (Admin) option.
Step 3. Click Yes when you are presented with a UAC window requesting permission to launch the Command Prompt as Administrator.
Step 4. In the new Command Prompt window, type chkdsk E: /f /r /x. Here you should replace E with your drive letter.
This option will run chkdsk to check and repair disk errors in Windows 10. The "/f" parameter tells CHKDSK to fix any errors it finds; "/r" tells it to locate the bad sectors on the drive and recover readable information; "/x" forces the drive to dismount before the process starts.
If chkdsk command prompt in cmd cannot help, here is an alternative solution to repair corrupted external hard drive without formatting. That is recover data from the corrupted hard drive and fix the disk by formatting. Having no data loss risk, it just works as well as accessing hard drive without formatting. Here, EaseUS free data recovery software which performs well to recover files from the external hard drive without formatting in Windows 10/8/7 can do you a favor.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,981
|
Q: How to determine if a point lies within a kml or shape I have some shape data (a series of kml files in the column of a fusion table and or database) and I want to merg it with another table that contains latitude longitude points. Basically I want some way of determining if a given lat lon point is contained within on of the kml shapes, and if so save a reference to that row.
I though perhaps there was a way to do this from within fusion tables, but if not, perhaps there is a way to loop through each kml and test if the lat lon point is contained within it. I understand this in not terribly efficient.
Any help, algorithm, service etc would be great.
A: The Fusion Tables SQL API has a ST_INTERSECTS operator but will only find points within either a CIRCLE or a RECTANGLE. GMap V3 has a geometry library which has a poly.containsLocation() method which I think will work for arbitrary polygons. See: GoogleMap geometry/poly library
P.S. I realize this does not work for KML files, but they do contain the polygon points which could be turned into GMAP polygons
A: This is probably long since solved, but as I had a similar problem I thought I'd post my solution.
I had two Fusion tables i) address & data and ii) polygons & data. I wanted to be able to easily query to find all addresses within one or more of the polygons. I could have done this realtime via my mapping webpage, but decided it was better to lookup the relevant polygon once in advance and then use this data to do my mapping queries (faster and easier to specify multiple polygons in the web interface).
So, I exported my table i) with the addresses to google sheets, and created a short simple script that checked which polygon the address was in and wrote it back to the google sheet. Then I updated my fusion table i).
I had a data set of nearly 3000 addresses and 2000 polygons so it timed out a few times and a had a couple of address errors, but this was easy to fix and I just set the script to run from the first row that hadn't been updated. Note that this won't work if the polygons overlap each other, but mine didn't as they were geographic boundaries ;-)
The code I used is below and also on gist here. You obviously need to update the table ID, and probably need to do something with OAuth (which I didn't quite understand but followed google's instructions and got it done).
// replace with your fusion table's id (from File > About this table)
var TABLE_ID = 'xxxxxxxxxx';
// first row that has data, as opposed to header information
var FIRST_DATA_ROW = 2;
var FIRST_DATA_COLUMN = 11;
var LAT_COLUMN = 1;
var LNG_COLUMN = 2;
var SA2_COLUMN = 6;
var SA3_COLUMN = 7;
/**
* Uses a lat and lng data in google sheets to check if an address is within a kml polygon
* in a list of KML polygons in fusion (in this case ABS/ASGC SA2 and SA3 regions, but could be any polygon)
* the function then stores the ID/name of the relevant polygon in google sheets
* I could check this data in realtime as I render maps, but as it doesn't changed, figure its better to just record
* which polygon each address pertains to so its quicker and easier to search (in particular it mades it easier to write a query
* which identifies all the address within multiple polygons)
* in this case I had 3000 rows so it exceeded maximum execution times, so I just updated the first data row a couple of times
* when the execution time exceeded.
*/
function updateSA2ID() {
var tasks = FusionTables.Task.list(TABLE_ID);
var sqlResponse = '';
// Only run if there are no outstanding deletions or schema changes.
if (tasks.totalItems === 0) {
var sheet = SpreadsheetApp.getActiveSheet();
var latLngData = sheet.getRange(FIRST_DATA_ROW, FIRST_DATA_COLUMN, sheet.getLastRow(), sheet.getLastColumn());
i = 1;
// Loop through the current current sheet
for (i = 1; i <= latLngData.getNumRows(); i++) {
// cross reference to Fusion table
lat = latLngData.getCell(i,LAT_COLUMN).getValue();
lng = latLngData.getCell(i,LNG_COLUMN).getValue();
sqlString = "SELECT 'SA2 Code', 'SA3 Code' FROM " + TABLE_ID + " WHERE ST_INTERSECTS(geometry, CIRCLE(LATLNG(" + lat + ", " + lng + "),1)) ";
//Browser.msgBox('Lat ' + lat + ' Lng ' + lng + '; ' + sqlString, Browser.Buttons.OK);
sqlResponse = FusionTables.Query.sql(sqlString);
//Browser.msgBox('SQL Response ' + sqlResponse, Browser.Buttons.OK);
latLngData.getCell(i,SA2_COLUMN).setValue(sqlResponse.rows[0][0]); // set SA2
latLngData.getCell(i,SA3_COLUMN).setValue(sqlResponse.rows[0][1]); // set SA3
}
}
else {
Logger.log('Skipping row replacement because of ' + tasks.totalItems + ' active background task(s)');
}
};
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,479
|
package guru.nidi.codeassert.dependency;
import guru.nidi.codeassert.model.UsingElement;
import java.util.*;
import java.util.Map.Entry;
import static java.util.Collections.emptyMap;
public class DependencyMap {
private final Map<String, Map<String, Info>> map = new LinkedHashMap<>();
public <T> void with(int specificity, UsingElement<T> from, UsingElement<T> to) {
with(specificity, from.getName(), from.usedVia(to), to.getName());
}
DependencyMap with(int specificity, String from, Collection<String> vias, String to) {
final Map<String, Info> deps = map.computeIfAbsent(from, k -> new HashMap<>());
final Info info = deps.get(to);
if (info == null) {
deps.put(to, new Info(vias, specificity));
} else {
//TODO specificity?
info.getVias().addAll(vias);
}
return this;
}
DependencyMap with(String from, DependencyMap other) {
final Map<String, Info> infos = other.getDependencies(from);
for (final Entry<String, Info> entry : infos.entrySet()) {
final Info info = entry.getValue();
with(info.getSpecificity(), from, info.getVias(), entry.getKey());
}
return this;
}
public DependencyMap without(int specificity, String from, String to) {
final Map<String, Info> deps = map.get(from);
if (deps != null) {
final Info info = deps.get(to);
if (info != null && specificity > info.specificity) {
deps.remove(to);
if (deps.isEmpty()) {
map.remove(from);
}
}
}
return this;
}
public DependencyMap without(DependencyMap other) {
for (final Entry<String, Map<String, Info>> entry : other.map.entrySet()) {
for (final Entry<String, Info> to : entry.getValue().entrySet()) {
without(to.getValue().specificity, entry.getKey(), to.getKey());
}
}
return this;
}
public void merge(DependencyMap deps) {
for (final Entry<String, Map<String, Info>> entry : deps.map.entrySet()) {
final Map<String, Info> ds = map.get(entry.getKey());
if (ds == null) {
map.put(entry.getKey(), entry.getValue());
} else {
ds.putAll(entry.getValue());
}
}
}
public boolean isEmpty() {
return map.isEmpty();
}
public void clear() {
map.clear();
}
public Set<String> getElements() {
return map.keySet();
}
/**
* @param name the name of the element
* @return A map with all dependencies of a given package.
* Key: package, Value: A set of all classes importing the package
*/
public Map<String, Info> getDependencies(String name) {
return map.getOrDefault(name, emptyMap());
}
public Info getDependency(String from, String to) {
return getDependencies(from).get(to);
}
@Override
public boolean equals(Object o) {
if (this == o) {
return true;
}
if (o == null || getClass() != o.getClass()) {
return false;
}
final DependencyMap that = (DependencyMap) o;
return map.equals(that.map);
}
@Override
public int hashCode() {
return map.hashCode();
}
@Override
public String toString() {
return map.toString();
}
public static class Info {
private final Set<String> vias;
private final int specificity;
Info(Collection<String> vias, int specificity) {
this.vias = new HashSet<>(vias);
this.specificity = specificity;
}
public Set<String> getVias() {
return vias;
}
public int getSpecificity() {
return specificity;
}
@Override
public boolean equals(Object o) {
if (this == o) {
return true;
}
if (o == null || getClass() != o.getClass()) {
return false;
}
final Info info = (Info) o;
return vias.equals(info.vias);
}
@Override
public int hashCode() {
return vias.hashCode();
}
@Override
public String toString() {
return vias.toString();
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,414
|
Q: Can I use Send-MailMessage cmdlet with a CRT certificate? I have this code:
Clear-Host
$file = "c:\Mail-content.txt"
if (test-path $file)
{
$from = "afgarciact@gmail.com"
$to = "<slopez@comfama.com.co>","<carloscu@comfama.com.co>"
$pc = get-content env:computername
$subject = "Test message " + $pc
$smtpserver ="172.16.201.55"
$body = Get-Content $file | Out-String
[System.Net.ServicePointManager]::SecurityProtocol = 'Tls,TLS11,TLS12'
[System.Net.ServicePointManager]::ServerCertificateValidationCallback = { return $true }
foreach ($recipient in $to)
{
Write-Host "Sent to $to"
Send-MailMessage -SmtpServer $smtpserver -UseSsl -From $from -To $recipient -Subject $subject -BodyAsHtml $body -Encoding ([System.Text.Encoding]::UTF8)
}
}
else
{
Write-Host "Configuración"
}
I want to know if I can use a CRT certificated instead of setting ServerCertificateValidationCallback = TRUE.
This is a test environment.
The CRT certificate was already added to the SMTP server.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,226
|
** READ & BLOOM BOOKS**
_Agnes and Clarabelle_
_Agnes and Clarabelle Celebrate!_
_Stinky Spike the Pirate Dog_
_Stinky Spike and the Royal Rescue_
_Wallace and Grace Take the Case_
_Wallace and Grace and the Cupcake Caper_
_The Adventures of Caveboy_
_Caveboy Is Bored!_
For Grand Marie and Papa Bob —Peter M.
For Cheryl —Paul M.
**Table of Contents**
**[CHAPTER 1
CRASH LANDING](06_CHAPTER_1_CRASH_LANDING.xhtml#page_1)**
**[CHAPTER 2
THE SEARCH FOR HONEY](07_CHAPTER_2_THE_SEARCH_FOR_HONEY.xhtml#page_23)**
**[CHAPTER 3
RESCUE FROM THE PIRATES' HIDEOUT](08_CHAPTER_3_RESCUE_FROM_THE_PIRATE.xhtml#page_43)**
**CHA PTER 1**
**CRASH LANDING**
It was nap time on the pirate ship _Driftwood_. Captain Fishbeard and his crew had just feasted on lumpfish stew and octopus pudding. Now they were snoring and snorting like thunder.
Stinky Spike slept on top of a steaming heap of moldy green cheese. He woofed in his sleep, chasing after skunks and stink-bats in his dreams.
Zip, the monkey pirate, was supposed to be looking out for enemy ships, strange creatures, and other dangers of the sea. But he was fast asleep too.
So nobody felt the mighty gust of wind catch the sails and send the _Driftwood_ speeding toward shore!
_CRAAAAACK!_ The sound of the crashing ship jolted the pirates wide awake.
"Shiver me timbers, we're under attack!" bellowed Captain Fishbeard.
"Yikes! Is it a sea monster?" cried Zelda, the first mate.
Spike scampered to the bow. " _Ah-roo! Ah-roo!_ It's no monster. We've crashed into a ship!"
Captain Fishbeard surveyed the damage. " _Arrrgh!_ Zip, you lazy rascal. You fell asleep on the job and nearly sank our ship!"
From below roared a mighty voice, "My ship! My beautiful ship . . . she's ruined!"
The pirates scurried to the railing and looked down. Standing on a floating pile of timber was a small man wearing a crown. "You wretched pirates, look at what you've done!" he cried.
" _Arrrgh_ , that's King Seabreeze, the ruler of Beeswax Island," whispered Captain Fishbeard.
"You bumbling buccaneers! You've shattered my favorite ship! I don't know if we'll ever be able to repair her," whimpered King Seabreeze. He began to sniffle, then sob. Great tears ran down the king's cheek.
"Is the king _crying_?" whispered Zelda.
King Seabreeze wiped his nose on his royal sleeve. "NO! I am not crying. A king never cries!"
" _Arrrgh_ , we can help you fix your ship," said Captain Fishbeard.
"Your Majesty, would you like some moldy green cheese? Rotten smells always cheer me up when I'm feeling sad," said Spike.
Before the king could answer, one of the guards called out, "Your Majesty, the Royal Pup is missing!"
"Princess Petunia will never forgive me if we've lost her beloved dog, Honey!" gasped the king.
" _Arrrgh_ , we can help you find the pup," said Captain Fishbeard.
The king stamped his foot. "You fools have done enough. You've destroyed my ship and scared off the princess's Royal Pup. Guards, seize them! Throw these stinking pirates into the dungeon!"
The king's guards rounded up the pirates and marched them through King Seabreeze's palace to the cold, dark dungeon. The pirates shivered as the guard locked the door to their cell.
" _Arrrgh_ , we need to find a way out of this jail," said Captain Fishbeard.
Zip tried to squeeze through the bars. Spike tried to dig through the floor. Zelda even tried to pick the lock. It was no use. The pirates were trapped.
Days passed and still the pirates could not escape the king's dungeon. And a dungeon was no place to be stuck with Stinky Spike.
The globs of cheese, scraps of seaweed, and crab claws that were trapped in his fur had a mighty odor.
"Spike stinks worse than a buzzard's belch," exclaimed Zip.
The pirates tried to clean Spike. They scrubbed him with wooden spoons and tin cups. They brushed his teeth with fish tails. They bathed him in a bucket of rain water.
" _Ah-roo, ah-roo_ , that's enough," howled Spike. With a mighty shake, he sent gobs of moldy green cheese, crusty crab claws, and flaky fish scales flying in all directions. It was no use. The pirate crew could do nothing but pinch their noses to escape Spike's mighty stink.
That night, as the pirates tossed and turned, there was a noise outside the dungeon. Slowly, the door creaked open. A young girl stepped into the dark room.
" _Pssst_. _Pssst_. Wake up, you stinky pirates," whispered the girl.
" _Arrrgh_ , who are you?" grumbled Captain Fishbeard.
" _Shhhhh_ . . . I'm Princess Petunia. Is it true one of you has a super sniffer?" asked the princess.
" _Arrrgh_ , it's true. Spike's the pup you're after," whispered Captain Fishbeard.
" _Yarf!_ That's me," said Spike.
"Is your sniffer as powerful as they say? Can you use it to find anything?" asked Petunia.
"Yes, Your Highness. What's missing?" yipped Spike.
"Haven't you heard?" asked Petunia. "Ever since you pirates crashed into Daddy's ship, my pup has been missing. Her name is Honey, and she's my best friend. Can you help me find her?"
"Finding a missing dog will be easy for a sniffer like mine. I've got the most powerful nose on the seven seas," boasted Spike.
"I'll free you from this dungeon so you can lead me to Honey. But we must be quick. If the king finds out my plan, he'll be very unhappy," said Petunia, holding her nose.
" _Arrrgh!_ What about us? We can't rot in this dungeon forever," said Captain Fishbeard.
"If Spike finds Honey, the king will set you all free. I promise," said Petunia.
" _Arrrgh_ , good luck, Spike. You're our only hope out of this mess," said Captain Fishbeard.
**CHAPTER 2**
**THE SEARCH FOR HONEY**
Princess Petunia and Stinky Spike tiptoed past the sleeping guards.
"We should start by searching the island," whispered Petunia.
"I'm one sniff ahead of you, Princess," said Spike. He pointed his nose into the air and took a deep breath.
Petunia pulled something out of her pocket and held it up to Spike's nose. "This is Honey's royal cloak. Use it to get her scent."
Spike took a deep sniff. "Aha! It smells like Honey is in the jungle. Follow me!" said Spike.
Spike stopped beneath a large tree. He put his nose to the ground and sniffed around the trunk.
"I think Honey is up in this tree," Spike whispered.
Spike barked, and clawed at the tree, " _Arf, Arf!_ Honey, we're here to rescue you!"
" _Grrrrr_ . . . Go away! And get your own honey," growled a voice from above.
A sticky drop fell out of the tree and onto Spike's nose. " _Ah-roo_ , what is this strange-smelling stuff? It's stuck to my sniffer!" he howled. He could hear an unusual buzzing sound coming from above.
"Honey, come down from the tree right now. It's time to go home," commanded Petunia. She shook the tree.
The voice growled again from above, "Stop shaking the tree, you'll wake the . . . BEES! Oh, no . . . OUCH! They're stinging me!" roared the voice from above.
_THUD!_ A great big bear dropped out of the tree and landed next to Petunia.
" _Grrrrr_ , I told you, that was my honey. I was just about to get a paw-full before you woke the hive. Now buzz off, you troublemakers," said the bear.
"Sorry, Bear. We thought you were my missing dog, Honey. Have you seen her?" asked Petunia.
" _Hmmm_. I know all about honey. But I don't know anything about a missing _dog_ named Honey," said the bear, scratching his head. "I did see a scruffy mutt on the beach. Maybe it's the one you're looking for."
"Oh, thank you, Bear! Come on, Spike. I know the way to the beach," said Petunia.
"And sorry about your bee stings," said Spike, still trying to lick the sticky honey off his nose.
When they got to the beach, Stinky Spike and Princess Petunia looked up and down the shoreline.
"There's nothing here but sandy seashells. Spike, do you smell anything?" asked Petunia.
Spike took a deep sniff. " _Ah-roo!_ Wet dog! I'd know that scent anywhere. Honey must be nearby," howled Spike, dashing toward the dunes.
" _Arf, arf_ ," barked Spike, pointing to a curled-up mutt sleeping behind a sand dune.
" _Ah-roo! Ah-roo!_ " howled Spike. "Honey! We've found you."
Princess Petunia ran up beside Spike.
"That's not Honey," she said, stamping her royal foot.
"No. I'm Patches," said the shaggy dog, shaking sand from his coat. "Who's Honey?"
"Honey is Princess Petunia's pup," explained Spike.
" _Hmmm_ . . . that name sounds familiar," said Patches. The scrappy pup thought for a second, then began digging in the sand. Suddenly he hit something.
_CLINK!_
Patches pulled a bottle from the hole. Then another. Then another.
"Maybe your friend Honey has been writing these notes," said Patches. Inside each bottle was a slip of paper. The princess pulled them out and read them one by one.
"I'd know Honey's paw print anywhere. These notes are from her! But who is Captain Bart?" asked Petunia.
"Captain Bart is a mean old pirate. And his crew is even worse. But don't worry, Princess. My super sniffer will lead us right to Honey, and we will rescue her! All we need is a ship," said Spike.
"We can use my boat. She's rickety, but quick enough to chase after bonefish," offered Patches.
" _Yippee!_ Off to find Honey we go," barked Spike.
Stinky Spike and Princess Petunia climbed aboard the little boat. Patches raised the sail and steered out to sea. Spike stuck his mighty snout into the breeze and took a deep _SNIFF_ . . . _SNIFFFFFF_ . . .
"Aha! I smell pirates in that direction," said Spike, pointing to the horizon.
With his nose in the air and his tail wagging behind him, Spike shouted out directions to Patches, who steered the boat toward the pirates.
At last, Spike spotted a tiny island in the distance.
" _Ah-roo!_ Land ho!" cheered Spike.
**CHAPTER 3**
**RESCUE FROM THE PIRATES' HIDEOUT**
As the boat sailed closer to shore, Spike saw that the beach was littered with pirate gear. Pirate pants and shirts flapped in the wind. Hats, flags, and sails were draped on the tree branches.
"This island is covered in pirate gear, but I don't see or smell any pirates. They must be hiding," said Spike.
Suddenly, a brightly colored parrot swooped at Spike. It was wearing a pirate hat!
"Hey! Watch where you're flying, feather-neck," barked Spike.
The parrot perched on the side of the boat and stared at Princess Petunia's necklace.
" _Caw!_ That's a nice shiny necklace. Give it to me!" he squawked.
"First, tell me where all of this pirate gear came from. Is it from Captain Bart's crew?" asked Petunia.
" _Baccaw!_ Hand over that shiny necklace," squawked the parrot, ignoring the princess's question.
"These must be the Pirate Parrots of Copycat Cove. They are a menace to sailors everywhere," said Patches.
"Shiny necklace! Shiny necklace!" squawked the parrot. Soon the masts were full of birds.
"Shiny necklace! Shiny necklace!" squawked the other parrots.
Princess Petunia had an idea. "Let's make a deal. You want shiny things, and I want my dog, Honey. But she was captured by Captain Bart. I'll trade you my gold necklace if you take us to the pirates' hideout."
" _Buh-squawk!_ Deal! Follow me," said the parrot.
Patches steered his boat after the parrot. They sailed around the island and into a hidden cove. There, a pirate ship sat bobbing in the waves.
"I see Honey! But she's surrounded by Captain Bart and his crew. We need a plan to save her!" said Petunia.
Spike took a deep sniff. "If only I could smell us a plan that would save Honey. But it's hard to smell anything over all of these parrots."
"Aha, Spike, that's it! These birds can help us distract the pirates," said Petunia.
" _Ah-roo_ , great idea, Princess," howled Spike.
Patches anchored the ship, and Spike said, "Listen up, parrots. The necklace is yours! Now, how would you like to earn some more pirate gear?"
" _Buh-squawk!_ " The parrots screeched with excitement.
The princess tossed her necklace to the parrots. "We need you to swarm, buzz, swoop, peck, and swipe everything you can from the pirate crew."
" _Cuh-caw!_ Sure thing, Princess," squawked the parrot.
" _Ah-roo_ , then follow us, flappers," howled Spike, as he and Petunia charged toward the pirate camp.
When they were close, Spike barked, "Swarm!"
The great flock of seabirds flew at the pirates. They dove after their pirate hats. They pecked at their pirate beards. They clawed at their shiny silver buckles.
" _Arrrgh_ , run for your lives, pirates!" yelled Captain Bart as he ran into the jungle.
Petunia and Spike snuck into the pirate camp. Honey was nowhere to be found.
"Honey, Honey, where are you?" called the princess.
"Leave it to me! My super sniffer will find her now," said Spike.
Spike put his nose to the sky. _Sniff . . . sniff . . . sniff . . ._ "Well, this is strange. I don't smell any dogs. But we know that Honey was just here," whimpered Spike.
"She must be here somewhere! I saw her," said Petunia.
" _Ah-roo!_ I still can't smell her over all of these parrots," said Spike.
"We are running out of time. Captain Bart and his crew of awful pirates will come back to their camp and find us!" cried Petunia.
"Sorry, Princess, I don't smell Honey. But, I do smell cheese. _Cheese-ah-roo!_ I'm hungry enough to eat Captain Bart's entire cheese supply," said Spike.
"Oh, no you're not. I command you to forget about cheese and to find Honey!" said Petunia.
But the rumbling of his stomach was too much for Spike. He trotted over to the pirates' campfire and inspected their abandoned feast.
" _Mmmm_ , crab claws! _Yum!_ Dried mackerel," said Spike. " _Ahhh!_ Moldy cheese," he drooled, moving the lid off a large barrel.
" _Ruff!_ " A very stinky dog jumped out of the barrel.
"Honey?" asked Spike. " _Ah-roo, ah-roo_ , Princess, we've found her!"
"Honey, it's you, at long last," said Petunia, wrapping her arms around her pup in a great big hug. Honey wagged her tail and licked Princess Petunia's cheek.
The princess held her nose. "I'm so glad you're safe, but what was in that barrel? You stink like a . . ."
"Not now, Princess Petunia. We have to go before the pirates get a whiff of us. Let's scram and take this barrel of rotten cheese for later," barked Spike.
Rolling the barrel between them, the princess, Spike, and Honey ran down the beach.
"This is one heavy barrel of cheese," Spike panted.
They rounded the beach to where Patches was waiting in his boat.
"Patches, quickly set the sails. We've got to make a fast escape," yelled Petunia.
Quickly they loaded the barrel into the little boat, jumped in, and set sail.
Spike nibbled on stinky cheese. Suddenly his tooth struck something hard.
"What do we have here?" wondered Spike, pulling a gold coin into view.
Honey grinned. "It's Captain Bart's treasure! He keeps it hidden under this moldy cheese so the Pirate Parrots don't steal it. When the birds caused all the madness, I realized that this barrel would make a very good hiding spot. To thank you for saving me, the treasure is yours, Patches and Spike," said Honey.
"Hurray for treasure!" cried Patches.
"And hurray for CHEESE!" howled Spike with glee.
When the little crew landed back on Beeswax Island, King Seabreeze ran to meet them.
"Petunia! You're safe," cried the happy king. "And, Honey! You're found! But wow, you STINK!" he yelled, scratching the pup's ears.
"Daddy, I promised Spike that if he helped me find Honey, we would release the pirate crew of the _Driftwood_ ," said Princess Petunia.
"Well, a promise is a promise," said the happy king.
He sent the guards to release the crew.
To celebrate Honey's safe return, King Seabreeze hosted a giant party!
"Princess Petunia, Patches, and Stinky Spike, you are heroes. You brought Honey home safe and sound. Spike, in your honor we have a feast of delicious—and stinky—treats!" announced the king.
" _Arrrgh_ , Spike, you've saved us too. Thank you from the bottom of my pirate heart," said Captain Fishbeard.
" _Arf!_ I have a present for you, Captain Fishbeard," said Spike, nudging the barrel toward Captain Fishbeard.
Zelda held her nose, "Ugh, this is no treasure. It's just moldy cheese."
Spike howled with laughter. "Look in the cheese. It's Captain Bart's treasure!"
"Aha, Captain Bart's treasure! More than enough to fix up the _Driftwood_!" shouted Captain Fishbeard, dancing a jig.
"Yo-ho-ho!" shouted the _Driftwood_ 's crew.
Petunia giggled. "What an adventure this has been! Honey, we should form a pirate crew of our own. We can be the Princess Pirates!"
**Peter Meisel** has been writing and illustrating since he was old enough to hold a crayon. He lives in Brooklyn, New York. _Stinky Spike the Pirate Dog_ and _Stinky Spike and the Royal Rescue_ are his first two children's books.
**Paul Meisel** lives in Newtown, Connecticut, with his wife and labradoodle, Coco, who was the inspiration for his two early readers, _See Me Dig_ and the Geisel Honor–winning _See Me Run_. Paul is also the author/illustrator of _Good Night, Bat! Good Morning, Squirrel!_ and has illustrated more than 70 books, including the bestselling I Can Read _Go Away, Dog_ ; _Run for Your Life!: Predators and Prey on the African Savanna_ ; and several Let's-Read-and-Find-Out Science books.
www.paulmeisel.com
Text copyright © 2017 by Peter Meisel
Illustrations copyright © 2017 by Paul Meisel
All rights reserved
You may not copy, distribute, transmit, reproduce or otherwise make available this publication (or any part of it) in any form, or by any means (including without limitation electronic, digital, optical, mechanical, photocopying, printing, recording or otherwise), without the prior written permission of the publisher. Any person who does any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages.
First published in the United States of America in March 2017 by Bloomsbury Children's Books
www.bloomsbury.com
This electronic edition published in 2017 by Bloomsbury Publishing Plc
Bloomsbury is a registered trademark of Bloomsbury Publishing Plc
For information about permission to reproduce selections from this book, write to Permissions, Bloomsbury Children's Books, 1385 Broadway, New York, New York 10018
Bloomsbury books may be purchased for business or promotional use. For information on bulk purchases please contact Macmillan Corporate and Premium Sales Department at specialmarkets@macmillan.com
Library of Congress Cataloging-in-Publication Data
Names: Meisel, Peter, author. | Meisel, Paul, illustrator.
Title: Stinky Spike and the royal rescue / by Peter Meisel ; illustrated by Paul Meisel.
Description: New York : Bloomsbury, 2017. | Series: Read & bloom
Summary: Stinky Spike the dog and Captain Fishbeard the pirate are thick as thieves now that Spike has proved his treasure-sniffing ability. When Spike and his crew come across a sign offering a reward for finding the Princess's lost poodle, they can hardly resist. Surely finding a fluffy royal pooch can't be very difficult? When blundering pirates are involved, things never go according to plan.
Identifiers: LCCN 2016024740 (print) | LCCN 2016036828 (e-book)
ISBN 978-1-61963-883-9 (hardcover) • ISBN 978-1-61963-884-6 (e-book) • ISBN 978-1-61963-885-3 (e-PDF)
Subjects: | CYAC: Dogs—Fiction. | Smell—Fiction. | Pirates—Fiction. | Humorous stories. | BISAC: JUVENILE FICTION / Readers / Chapter Books. | JUVENILE FICTION / Animals / Dogs. | JUVENILE FICTION / Action & Adventure / Pirates. | JUVENILE FICTION / Humorous Stories.
Classification: LCC PZ7.1.M469 Sq 2017 (print) | LCC PZ7.1.M469 (e-book) | DDC [Fic]—dc23
LC record available at <https://lccn.loc.gov/2016024740>
ISBN: 978-1-6196-3883-9 (HB)
ISBN: 978-1-6196-3884-6 (eBook)
ISBN: 978-1-6196-3885-3 (ePDF)
Art created using pencil on Strathmore Bristol and colored digitally
Book design by Yelena Safronova
To find out more about our authors and their books please visit www.bloomsbury.com where you will find extracts, author interviews and details of forthcoming events, and to be the first to hear about latest releases and special offers, sign up for our newsletters.
#
1. Cover
2. Title Page
3. Dedication
4. Table of Contents
5. Chapter 1 Crash Landing
6. Chapter 2 The Search for Honey
7. Chapter 3 Rescue from the Pirates' Hideout
8. eCopyright
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{"url":"https:\/\/osf.io\/6dyjh\/","text":"Contributors:\n1. Christina Wasylyshyn\n\nDate created: | Last Updated:\n\n: DOI | ARK\n\nCategory: Project\n\nDescription: Some causal relations refer to causation by commission (e.g., \u201cA gunshot causes death\u201d) and others by omission (e.g., \u201cNot breathing causes death\u201d). We describe a theory of the representation of omissive causation based on the assumption that people mentally simulate sets of possibilities \u2013 mental models \u2013 which represent causal relations such as causes, preventions, and enabling conditions (Goldvarg & Johnson-Laird 2001). The theory holds that omissive causes, enabling conditions, and preventions each refer to distinct sets of possibilities. For any causal relation, reasoners typically simulate one initial possibility, but they are able to consider alternative possibilities through deliberation. These alternative possibilities allow them to deliberate over finer-grained distinctions when reasoning about causes and effects. Hence, reasoners should be able to distinguish between omissive causes and omissive enabling conditions. Four experiments corroborate the predictions of the theory. We describe them and contrast the results with the predictions of alternative accounts of causal representation and inference.","date":"2022-12-07 04:09:13","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8023137450218201, \"perplexity\": 6500.350968647619}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446711126.30\/warc\/CC-MAIN-20221207021130-20221207051130-00355.warc.gz\"}"}
| null | null |
The 1992/93 FIS Freestyle Skiing World Cup was the fourteenth World Cup season in freestyle skiing organised by International Ski Federation. The season started on 10 December 1992 and ended on 28 March 1993. This season included four disciplines: aerials, moguls, ballet and combined.
Men
Moguls
Ballet
Aerials
Combined
Ladies
Moguls
Ballet
Aerials
Combined
Men's standings
Overall
Standings after 40 races.
Moguls
Standings after 12 races.
Aerials
Standings after 9 races.
Ballet
Standings after 10 races.
Combined
Standings after 9 races.
Ladies' standings
Overall
Standings after 40 races.
Moguls
Standings after 12 races.
Aerials
Standings after 9 races.
Ballet
Standings after 10 races.
Combined
Standings after 9 races.
References
FIS Freestyle Skiing World Cup
World Cup
World Cup
|
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{"url":"https:\/\/www.sangakoo.com\/en\/unit\/product-of-matrices","text":"# Product of matrices\n\nIn a strict sense, the rule to multiply matrices is:\n\n\"The matrix product of two matrixes $$A$$ and $$B$$ is a matrix $$C$$ whose elements $$a_{ij}$$ are formed by the sums of the products of the elements of the row $$i$$ of the matrix $$A$$ by those of the column $$j$$ of the matrix $$B$$.\"\n\nActually, this is not a very encouraging statement, but in the end it is simple and only requires a little practice, since it is a question of multiplying the rows of the first matrix by the columns of the second one.\n\n$$\\left( \\begin{array}{ccc} 5 & 2 & 1 \\end{array} \\right) \\cdot \\left( \\begin{array}{c} 4 \\\\ -3 \\\\ 6 \\end{array} \\right) = 5\\cdot4+2\\cdot(-3)+1\\cdot6=20-6+6=20$$$That is, we have to multiply the first element of the row in the first matrix by the first element from the column in the second matrix, then add up this product and the product of the second element in the row by the second element in the column, and finally add up the product of the third element in the row by the third element in the column. This is easier done than said! $$\\left( \\begin{array}{ccc} 2 & 3 & 5 \\end{array} \\right) \\cdot \\left( \\begin{array}{c} 3 \\\\ 2 \\\\ 4 \\end{array} \\right) = 2\\cdot3+3\\cdot2+5\\cdot4=6+6+20=32$$$\n\nLet's see a product of two square matrices $$2\\times2$$\n\n$$\\left( \\begin{array}{cc} 1 & 5 \\\\ 2 & 2 \\end{array} \\right) \\cdot \\left( \\begin{array}{cc} 3 & 4 \\\\ 1 & 6 \\end{array} \\right) = \\left( \\begin{array}{cc} 1\\cdot3+5\\cdot1 & 1\\cdot4+5\\cdot6 \\\\ 2\\cdot3+2\\cdot1 & 2\\cdot4+2\\cdot6 \\end{array} \\right) = \\left( \\begin{array}{cc} 8 & 34 \\\\ 8 & 20 \\end{array} \\right)$$$Every element $$a_{ij}$$ is obtained by adding the products of the elements of row $$i$$ by those of column $$j$$. For example, $$8$$, the element of the first row and the first column of the resultant matrix, is obtained by multiplying the first row of the first matrix by the first column of the second matrix. Number $$8$$, which is the element of the second row and the first column in the final matrix, is obtained by multiplying the second row in the first matrix by the first column in the second matrix, and so on with the rest of the elements. To clarify this process we are going to mark the corresponding rows and columns of the product matrix: $$\\left( \\begin{array}{cc} \\fbox{1} & \\fbox{5} \\\\ 2 & 2 \\end{array} \\right) \\cdot \\left( \\begin{array}{cc} \\fbox{3} & 4 \\\\ \\fbox{1} & 6 \\end{array} \\right) = \\left( \\begin{array}{cc} \\fbox{8} & 34 \\\\ 8 & 20 \\end{array} \\right)$$$\n\n$$\\left( \\begin{array}{cc} \\fbox{1} & \\fbox{5} \\\\ 2 & 2 \\end{array} \\right) \\cdot \\left( \\begin{array}{cc} 3 & \\fbox{4} \\\\ 1 & \\fbox{6} \\end{array} \\right) = \\left( \\begin{array}{cc} 8 & \\fbox{34} \\\\ 8 & 20 \\end{array} \\right)$$$$$\\left( \\begin{array}{cc} 1 & 5 \\\\ \\fbox{2} & \\fbox{2} \\end{array} \\right) \\cdot \\left( \\begin{array}{cc} \\fbox{3} & 4 \\\\ \\fbox{1} & 6 \\end{array} \\right) = \\left( \\begin{array}{cc} 8 & 34 \\\\ \\fbox{8} & 20 \\end{array} \\right)$$$\n\n$$\\left( \\begin{array}{cc} 1 & 5 \\\\ \\fbox{2} & \\fbox{2} \\end{array} \\right) \\cdot \\left( \\begin{array}{cc} 3 & \\fbox{4} \\\\ 1 & \\fbox{6} \\end{array} \\right) = \\left( \\begin{array}{cc} 8 & 34 \\\\ 8 & \\fbox{20} \\end{array} \\right)$$$In fact, we only need to remember that we must multiply \"row by column\". For example, let's calculate the value of the marked element in the following product matrix: $$\\left( \\begin{array}{ccccc} 1 & 3 & -2 & 5 & 0 \\\\ 8 & 1 & 0 & -1 & 2 \\\\ 4 & 2 & 5 & 3 & 1 \\\\ 0 & -2 & 3 & 5 & 3 \\\\ 7 & 1 & 0 & 3 & 7 \\end{array} \\right) \\cdot \\left( \\begin{array}{ccccc} 2 & 6 & 3 & -1 & 0 \\\\ 8 & 2 & 4 & 6 & 1 \\\\ 0 & 2 & 1 & 4 & 3 \\\\ 1 & 5 & 3 & 7 & 2 \\\\ 5 & 8 & 3 & 9 & 2 \\end{array} \\right) = \\left( \\begin{array}{ccccc} \\fbox{ } & \\fbox{ } & \\fbox{ } & \\fbox{ } & \\fbox{ } \\\\ \\fbox{ } & \\fbox{ } & \\fbox{ } & \\fbox{ } & \\fbox{ } \\\\ \\fbox{ } & \\fbox{ } & \\fbox{ } & \\fbox{ } & \\fbox{ } \\\\ \\fbox{ } & \\fbox{ } & \\fbox{?} & \\fbox{ } & \\fbox{ } \\\\ \\fbox{ } & \\fbox{ } & \\fbox{ } & \\fbox{ } & \\fbox{ } \\end{array} \\right)$$$\n\nWe have to multiply the fourth row by the third column:\n\n$$\\left( \\begin{array}{ccccc} 1 & 3 & -2 & 5 & 0 \\\\ 8 & 1 & 0 & -1 & 2 \\\\ 4 & 2 & 5 & 3 & 1 \\\\ \\fbox{0} & \\fbox{-2} & \\fbox{3} & \\fbox{5} & \\fbox{3} \\\\ 7 & 1 & 0 & 3 & 7 \\end{array} \\right) \\cdot \\left( \\begin{array}{ccccc} 2 & 6 & \\fbox{3} & -1 & 0 \\\\ 8 & 2 & \\fbox{4} & 6 & 1 \\\\ 0 & 2 & \\fbox{1} & 4 & 3 \\\\ 1 & 5 & \\fbox{3} & 7 & 2 \\\\ 5 & 8 & \\fbox{3} & 9 & 2 \\end{array} \\right) =$$$$$= \\left( \\begin{array}{ccccc} \\fbox{ } & \\fbox{ } & \\fbox{ } & \\fbox{ } & \\fbox{ } \\\\ \\fbox{ } & \\fbox{ } & \\fbox{ } & \\fbox{ } & \\fbox{ } \\\\ \\fbox{ } & \\fbox{ } & \\fbox{ } & \\fbox{ } & \\fbox{ } \\\\ \\fbox{ } & \\fbox{ } & 0\\cdot3+(-2)\\cdot4+3\\cdot1+5\\cdot3+3\\cdot3 & \\fbox{ } & \\fbox{ } \\\\ \\fbox{ } & \\fbox{ } & \\fbox{ } & \\fbox{ } & \\fbox{ } \\end{array} \\right)$$$\n\nSo, the element we were looking for in the product matrix is $$a_{43}=19$$.\n\nLet's see another example. We will do the product of two $$3\\times3$$ matrices (this will be the most complicated case that we will see).\n\n$$\\left( \\begin{array}{ccc} 2 & 5 & 1 \\\\ 4 & -2 & 0 \\\\ 1 & 6 & 2 \\end{array} \\right) \\cdot \\left( \\begin{array}{ccc} 1 & 2 & 3 \\\\ 3 & 4 & 1 \\\\ 1 & -4 & 2 \\end{array} \\right) = \\left( \\begin{array}{ccc} 18 & 20 & 13 \\\\ -2 & 0 & 10 \\\\ 21 & 18 & 13 \\end{array} \\right)$$$We are going to give a detailed analysis of how some of its elements have been calculated. The element $$a_{11}$$ is obtained by multiplying the first row by the the first column: $$\\left( \\begin{array}{ccc} \\fbox{2} & \\fbox{5} & \\fbox{1} \\\\ 4 & -2 & 0 \\\\ 1 & 6 & 2 \\end{array} \\right) \\cdot \\left( \\begin{array}{ccc} \\fbox{1} & 2 & 3 \\\\ \\fbox{3} & 4 & 1 \\\\ \\fbox{1} & -4 & 2 \\end{array} \\right) = \\left( \\begin{array}{ccc} \\fbox{18} & 20 & 13 \\\\ -2 & 0 & 10 \\\\ 21 & 18 & 13 \\end{array} \\right)$$$\n\n$$2\\cdot1+5\\cdot3+1\\cdot1=2+15+1=18$$$The element $$a_{23}$$ is obtained by multiplying the second row by the third column: $$\\left( \\begin{array}{ccc} 2 & 5 & 1 \\\\ \\fbox{4} & \\fbox{-2} & \\fbox{0} \\\\ 1 & 6 & 2 \\end{array} \\right) \\cdot \\left( \\begin{array}{ccc} 1 & 2 & \\fbox{3} \\\\ 3 & 4 & \\fbox{1} \\\\ 1 & -4 & \\fbox{2} \\end{array} \\right) = \\left( \\begin{array}{ccc} 18 & 20 & 13 \\\\ -2 & 0 & \\fbox{10} \\\\ 21 & 18 & 13 \\end{array} \\right)$$$\n\n$$4\\cdot3+(-2)\\cdot1+0\\cdot2=12-2+0=10$$$The element $$a_{31}$$ is obtained by multiplying the third row by the the first column: $$\\left( \\begin{array}{ccc} 2 & 5 & 1 \\\\ 4 & -2 & 0 \\\\ \\fbox{1} & \\fbox{6} & \\fbox{2} \\end{array} \\right) \\cdot \\left( \\begin{array}{ccc} \\fbox{1} & 2 & 3 \\\\ \\fbox{3} & 4 & 1 \\\\ \\fbox{1} & -4 & 2 \\end{array} \\right) = \\left( \\begin{array}{ccc} 18 & 20 & 13 \\\\ -2 & 0 & 10 \\\\ \\fbox{21} & 18 & 13 \\end{array} \\right)$$$\n\n$$1\\cdot1+6\\cdot3+1\\cdot2=1+18+2=21$$$The element $$a_{22}$$ is obtained by multiplying the second row by the second column: $$\\left( \\begin{array}{ccc} 2 & 5 & 1 \\\\ \\fbox{4} & \\fbox{-2} & \\fbox{0} \\\\ 1 & 6 & 2 \\end{array} \\right) \\cdot \\left( \\begin{array}{ccc} 1 & \\fbox{2} & 3 \\\\ 3 & \\fbox{4} & 1 \\\\ 1 & \\fbox{-4} & 2 \\end{array} \\right) = \\left( \\begin{array}{ccc} 18 & 20 & 13 \\\\ -2 & \\fbox{0} & 10 \\\\ 21 & 18 & 13 \\end{array} \\right)$$$\n\n$$4\\cdot2+(-2)\\cdot4+0\\cdot2=8-8+0=0$$\\$\n\nThe rest of the elements of the product of matrices are calculated using the same method.\n\nAt this point, we have realized that multiplying matrices is somewhat tiresome. Let's think, for example, that the product of two $$4\\times4$$ matrices entails carrying out $$128$$ arithmetical operations.\n\nFortunately, most scientific calculators currently on the market include a matricial calculation. Nevertheless, it is advisable to do the product of $$3\\times3$$ matrices \"by hand\" at least once in order to understand the mechanics of the operations.","date":"2020-11-30 11:07:43","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\": 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\": 1.0000100135803223, \"perplexity\": 106.84350611875604}, \"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-2020-50\/segments\/1606141213431.41\/warc\/CC-MAIN-20201130100208-20201130130208-00649.warc.gz\"}"}
| null | null |
Waterside Court, a new 62-unit Extra Care facility in the heart of Loughborough, opened prior to Christmas and has welcomed its first residents.
Working once again with Westleigh Partnerships Ltd, this is the second such scheme brp have delivered for EMH Homes, following on from the success of Oak Court, Blaby in 2015.
In this particular instance the Extra Care block acts as the focal point of a wider residential development – including 24 affordable homes and 10 Close Care bungalows.
Occupying a prominent position on the Grand Union canal the 4-storey development performs a dual role of addressing both the waterside frontage as well as terminating the view / vista as seen from the principal point of access from Derby Road. Given the level of the accommodation required to satisfy the Extra Care classification, and the need for all to be interlinked / accessible, it undoubtedly results in a single, large footprint building. As such, it was critical that design and layout, in particular the form and massing, was carefully considered to be appropriate for the location, together with ensuring the relationships / connections with the surrounding reduced-scale plots were sensitively managed.
The Client is extremely happy with the end result, as are the residents that have moved in so far. We look forward to visiting the site later in the year when we return to undertake a Post Occupation Review – an exercise we commonly engage in with Clients and End Users in order to enable us to learn lessons from the project, addressing both positive as well as any negative points relating to the design.
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Rising Technical Contributor Award
Anjali Kukreja
For technical expertise and leadership in automated testing systems; for raising awareness of SWE through targeted outreach and relevant STEM programming; and for incorporating DEI concepts into the company culture.
Globe-trotting and assimilating into different cultures throughout her schooling in Johannesburg; Washington, D.C.; Islamabad; and Paris, Anjali Kukreja wanted to return to India, her home country, to pursue an undergraduate degree.
She earned a B.Tech in computer science and engineering from the National Institute of Technology, Bhopal. Through an internship with UNESCO in Paris, Kukreja learned about John Deere while reviewing budgetary allocation for crisis-affected situations. She wanted to understand how information technology could be leveraged to help solve greater issues in the world. Kukreja joined John Deere in part because of its vision and mission to use automation to impact the world's increasing food demand.
She began her career with John Deere as a graduate engineer trainee before transitioning to a software engineer role. Kukreja made significant contributions in SAP Automation Testing, an enterprise planning software program, by reducing countless hours consumed through manual testing. She gained knowledge of a variety of software programs and quality management business functions. Most recently, she served as a senior software engineer with Global IT India, John Deere. In this role, Kukreja became the single point of contact for SAP automation testing by upskilling 12-plus product teams and 90 employees to kickstart their automation journeys.
Kukreja, along with her team members, automated 120 scenarios divided into several groups covering different phases of the product data management (PDM) cycle. She took proactive steps in finding the source of some of the limitations caused by automation scripts. To resolve this issue, she created a utility that helped users resume automation from whatever point it had failed during the end-to-end automation. Her automation efforts affected the PDM cycle used by more than 50 factories, 17 design centers, and more than 1,500 users worldwide. In the end, manual execution time was reduced from four weeks to four hours. For her leadership on this project, Kukreja received the company's WOW Award and two QA Team Excellence awards, 37 badges, and others.
An integral member of the John Deere diversity, equity, and inclusion (DEI) team, Kukreja often led meetings and goal planning sessions. As a DEI core member, she transformed employee engagement during the pandemic by organizing DEI champions across John Deere globally to share their experiences on DEI. This was followed up with a video compiling DEI achievements during the year.
Kukreja is involved with SWE Lakshya, serving as a communication lead and supporting initiatives and programs, including awards, outreach, and development, and "Wow! That's Engineering!®" — a free interactive event for 31 girls in 10th grade designed to expose them to the world of engineering through hands-on projects and STEM activities. Kukreja has also contributed articles for the Global SWE newsletter.
Kukreja is continuing her education and is pursuing an MBA at Trinity Western University in British Columbia, Canada. She enjoys spending time with her family, who converse in Bulgarian, French, or Spanish during gatherings. She is also fond of curating ayurvedic drinks with herbs.
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Marcos Andrés González Salazar (born 9 June 1980) is a Brazilian born Chilean professional football defender.
Although born in Brazil, he moved to Santiago aged two and has Chilean citizenship. Noteworthy, he has scored in all the team which he has played.
A Chilean international, he has played during 2006 and 2014 World Cup qualifiers.
Club career
He professionally debuted in 1999 at Universidad de Chile, winning a league title that year whilst the following he achieved another league title and a cup title, all with César Vaccia as coach.
In 2003, González moved to Argentina's Colón de Santa Fe, remaining there two seasons. After a brief spell at Palestino in the second half of 2005, in January 2006 season he chose to sign with the Columbus Crew over Universidad Católica. At the American team, he played two seasons and made 45 appearances, scoring two goals.
Following the 2008 MLS season, González was released from Columbus Crew and then joined the team which he failed to sign in 2006, Universidad Católica, where two years later won the 2010 Primera División title.
In 2011, it was confirmed that González returned to Universidad de Chile. However, he break out into first-team and was a key player in the treble obtention, after winning the Apertura and Clausura titles as well as the Copa Sudamericana.
In 2012, he joined Brazilian club Flamengo. There he helped the team to win the 2013 Copa do Brasil and he stayed at the Mengão until mid-2014. After being released from Flamengo, he returned his homeland and signed for Unión Española to play the 2014–15 Primera División season.
On 11 July 2015, he joined Mexican side Necaxa.
International career
He has represented Chile internationally 29 times.
International goals
Career statistics
1Cup competitions include the MLS Cup, Copa Chile, Copa do Brasil and Copa México
2Other tournaments include the Copa Libertadores and Copa Sudamericana.
3Other tournaments include the Campeonato Carioca.
Honours
Club
Universidad de Chile
Primera División de Chile (4): 1999, 2000, 2011 Apertura, 2011 Clausura
Copa Chile: 2000
Copa Sudamericana: 2011
Universidad Católica
Primera División de Chile: 2010
Flamengo
Copa do Brasil: 2013
Individual
Campeonato Nacional Team of the Season (2): 2009, 2011
Conmebol Team of the Season: 2011
References
External links
Marcos González – Primera División Argentina statistics at Fútbol XXI
1980 births
Living people
Footballers from Rio de Janeiro (city)
Brazilian emigrants to Chile
Citizens of Chile through descent
Chilean footballers
Chilean expatriate footballers
Chile international footballers
Universidad de Chile footballers
Rangers de Talca footballers
Club Atlético Colón footballers
Club Deportivo Palestino footballers
Columbus Crew players
Club Deportivo Universidad Católica footballers
CR Flamengo footballers
Unión Española footballers
Club Necaxa footballers
Chilean Primera División players
Argentine Primera División players
Major League Soccer players
Campeonato Brasileiro Série A players
Liga MX players
Chilean expatriate sportspeople in Argentina
Chilean expatriate sportspeople in the United States
Chilean expatriate sportspeople in Brazil
Chilean expatriate sportspeople in Mexico
Expatriate footballers in Argentina
Expatriate soccer players in the United States
Expatriate footballers in Brazil
Expatriate footballers in Mexico
Naturalized citizens of Chile
Association football defenders
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{"url":"https:\/\/systemx.io\/blog\/2020\/06\/booting-into-the-openbsd-6.7-installer-on-an-apu2-system\/","text":"# Booting into the OpenBSD 6.7 installer on an APU2 system\n\nEvery now and then I want to re-install OpenBSD on my little APU2 system, this time I wanted to benefit from the recent FFS2 improvements which requires filesystems to be recreated. It seems I struggle with remembering some bits and pieces of this process as I get older ;). This blog post is here to document what stuff I need to get the OpenBSD installer going! Yes, that also means I\u2019m not documenting the OpenBSD installer here.\n\n## Requirements\n\n1. A system for downloading\/writing installation media to an USB pendrive; I tend to use Ubuntu on my desktops.\n2. USB to serial gear and null modem cable.\n\n## Preparing the installation media\n\nFirst thing is to download the installation media, I prefer the media with the installation sets included. So, open up a terminal and wget that shit. Yes, the APU2 is amd64.\n\njmaas@mainframe:~\/Downloads$wget https:\/\/cdn.openbsd.org\/pub\/OpenBSD\/6.7\/amd64\/install67.fs --2020-06-15 17:28:55-- https:\/\/cdn.openbsd.org\/pub\/OpenBSD\/6.7\/amd64\/install67.fs Resolving cdn.openbsd.org (cdn.openbsd.org)... 151.101.38.217, 2a04:4e42:9::729 Connecting to cdn.openbsd.org (cdn.openbsd.org)|151.101.38.217|:443... connected. HTTP request sent, awaiting response... 200 OK Length: 476545024 (454M) [application\/octet-stream] Saving to: \u2018install67.fs\u2019 install67.fs 100%[================================================================>] 454,47M 3,42MB\/s in 2m 27s 2020-06-15 17:31:21 (3,10 MB\/s) - \u2018install67.fs\u2019 saved [476545024\/476545024] jmaas@mainframe:~\/Downloads$\n\n\nWhile that download is running, try to find an USB pendrive of at least 1GB. Don\u2019t you love the small OS footprint? Personally, I think it\u2019s awesome! Anyway, plug it into the desktop machine and wipe it! Yeah, that\u2019s right\u2026 we don\u2019t want to bother with checking it\u2019s contents first. YOLO!\n\nIf you don\u2019t know what the device name is that will be given by the Linux overlords these days, check dmesg | tail. You can wait for the systemd folks to fuck this up too. Wait\u2026 for\u2026 it\u2026 Anyway, mine is presented to me as \/dev\/sdc, now into the kill:\n\njmaas@mainframe:~\/Downloads$sudo wipefs --all \/dev\/sdc \/dev\/sdc: 5 bytes were erased at offset 0x00008001 (iso9660): 43 44 30 30 31 jmaas@mainframe:~\/Downloads$\n\n\nNow we can dump the installation media onto the drive, no need for fancy tools here.\n\njmaas@mainframe:~\/Downloads$sudo dd if=.\/install67.fs of=\/dev\/sdc 930752+0 records in 930752+0 records out 476545024 bytes (477 MB, 454 MiB) copied, 169,783 s, 2,8 MB\/s jmaas@mainframe:~\/Downloads$ sudo eject \/dev\/sdc\njmaas@mainframe:~\/Downloads$ You can now safely pull the USB pendrive from the port. ## Setting up the serial console Since my desktop machine is actually a laptop I need to use an USB to serial adapter. I\u2019ve had some issues in the past getting some of those to work with Linux, so\u2019ll dump the vendor & product information here, it might be useful for someone out there. jmaas@mainframe:~$ sudo lsusb -s 003:003\nBus 003 Device 003: ID 067b:2303 Prolific Technology, Inc. PL2303 Serial Port\njmaas@mainframe:~$ For connecting to the APU2 I also need to attach a DB-9 null-modem cable to the USB to serial adapter. A new device should appear in the \/dev directory, in my case it\u2019s \/dev\/ttyUSB0. If it doesn\u2019t, well\u2026 you\u2019re out of luck I guess. jmaas@mainframe:~$ ls -l \/dev\/ttyUSB*\ncrw-rw---- 1 root dialout 188, 0 jun 16 07:35 \/dev\/ttyUSB0\njmaas@mainframe:~\\$\n\n\nOn the software side of things I tend to just use Putty these days, it\u2019s what I have to use @work anyway (Windows) and it\u2019s good enough for this simple task. If you don\u2019t have access to Putty check out minicom. In any case you need to use the following settings for the serial connection:\n\nSetting Value\nDevice \/dev\/ttyUSB0\nType Serial\nSpeed 115200\nData bits 8\nStop bits 1\nParity None\nFlow Control XON\/XOFF\n\n## Starting the installation\n\nNow we\u2019re ready to boot into the OpenBSD installer, plug the USB pendrive into the APU2, connect the serial cable and power-on the system. You\u2019ll have to be quick and press ALF-F10 in time to be able to select the USB pendrive as the boot device. On the OpenBSD boot loader prompt, you\u2019ll need to setup the serial connection first.\n\nboot> stty com0 115200\nboot> set tty com0\nboot> <ENTER>\n\n\nThe kernel will now boot and the installer will be started! The installation process is nicely documented in the FAQ so pleasse use that if you need help.","date":"2020-07-06 20:03:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.22711986303329468, \"perplexity\": 7312.818795915016}, \"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-2020-29\/segments\/1593655890181.37\/warc\/CC-MAIN-20200706191400-20200706221400-00285.warc.gz\"}"}
| null | null |
Rolnicki Pass är ett bergspass i Antarktis. Det ligger i Sydshetlandsöarna. Argentina, Chile och Storbritannien gör anspråk på området. Rolnicki Pass ligger meter över havet.
Terrängen runt Rolnicki Pass är kuperad. Havet är nära Rolnicki Pass söderut. Den högsta punkten i närheten är meter över havet, kilometer nordost om Rolnicki Pass. Trakten är glest befolkad. Närmaste befolkade plats är Commandante Ferraz Station, kilometer söder om Rolnicki Pass.
Kommentarer
Källor
Bergspass i Antarktis
Sydshetlandsöarna
Storbritanniens anspråk i Antarktis
Chiles anspråk i Antarktis
Argentinas anspråk i Antarktis
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 316
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Q: How do you tweak iptables limit rules on openSUSE Tumbleweed? I have a fresh install of openSUSE Tumbleweed, it is logging a lot firewall related stuff due to my apple devices multicasting to enable airplay.
Lines look like:
Mar 14 12:45:13 server kernel: SFW2-INext-DROP-DEFLT IN=ens32 OUT= MAC=33:33:00:00:00:fb:98:d6:bb:20:90:a2:86:dd SRC=fe80:0000:0000:0000:189b:b08a:4ad3:89bf DST=ff02:0000:0000:0000:0000:0000:0000:00fb LEN=271 TC=0 HOPLIMIT=255 FLOWLBL=30502 PROTO=UDP SPT=5353 DPT=5353 LEN=231
Mar 14 12:46:48 server kernel: SFW2-INext-DROP-DEFLT IN=ens32 OUT= MAC=33:33:00:00:00:fb:f0:24:75:e4:6e:54:86:dd SRC=fe80:0000:0000:0000:1c6a:034e:81fc:bb59 DST=ff02:0000:0000:0000:0000:0000:0000:00fb LEN=124 TC=0 HOPLIMIT=255 FLOWLBL=899138 PROTO=UDP SPT=5353 DPT=5353 LEN=84
There are three (one in this specific example, since proto=udp) iptables definitions that can cause these log lines to be created:
# iptables -S | grep SFW2-INext-DROP-DEFLT
-A input_ext -p tcp -m limit --limit 3/min -m tcp --tcp-flags FIN,SYN,RST,ACK SYN -j LOG --log-prefix "SFW2-INext-DROP-DEFLT " --log-tcp-options --log-ip-options
-A input_ext -p icmp -m limit --limit 3/min -j LOG --log-prefix "SFW2-INext-DROP-DEFLT " --log-tcp-options --log-ip-options
-A input_ext -p udp -m limit --limit 3/min -m conntrack --ctstate NEW -j LOG --log-prefix "SFW2-INext-DROP-DEFLT " --log-tcp-options --log-ip-options
I have three airplay receivers and a bunch of airplay clients on this network, so three connections per minute is probably less than expected.
I'd like to tweak the rule and change 3/min to 10/min or something similar just to stop filling up the logs with non sense, but I am unable to find where these lines are defined.
When I look at the firewall settings in yast2 there is no trace of limit settings.
How do I go about changing this limit?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 5,459
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Wolfgang Koch ist der Name folgender Personen:
* Wolfgang Koch (Schauspieler), deutscher Filmschauspieler
Wolfgang Koch (Eisenbahnautor) (* 1927), deutscher Eisenbahner und Autor
Wolfgang Koch (Linguist) (* 1947), deutscher Sprachwissenschaftler
Wolfgang Koch (Toxikologe) (* 1953), deutscher Toxikologe und Hochschullehrer
Wolfgang Koch (Kameramann), österreichischer Kameramann
Wolfgang Koch (Fußballspieler) (* 1955), deutscher Fußballspieler
Wolfgang Koch (Politiker) (* 1956), deutscher Politiker (CDU)
Wolfgang Koch (Publizist) (* 1959), österreichischer Publizist
Wolfgang Koch (Leichtathlet) (* 1960), deutscher Langstreckler / Marathonläufer
Wolfgang Koch (Informatiker) (* 1962), deutscher Physiker und Informatiker
Wolfgang Koch (Sänger) (* 1966), deutscher Opernsänger (Bariton)
Wolfgang Koch (Endurosportler), deutscher Endurosportler
Wolfgang H. Koch (* 1951), deutscher Zahnmediziner
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,073
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Komunistyczna Partia Czech i Moraw (cz. Komunistická strana Čech a Moravy, KSČM) – partia komunistyczna działająca w Czechach. Partia zajmuje stanowisko narodowo-komunistyczne i eurosceptyczne.
Historia
Partia powstała 31 marca 1990 jako organizacja terytorialna Komunistycznej Partii Czechosłowacji. Przewodniczącym został Jiří Svoboda. Początkowo działała w ramach federacji ze swoim słowackim odpowiednikiem – Komunistyczną Partią Słowacji. Gdy jednak ta ostatnia przekształciła się w Partię Lewicy Demokratycznej, federacja rozpadła się. Próby przekształcenia partii w ugrupowanie o charakterze socjaldemokratycznym nie powiodły się – na początku 1992 w wewnątrzpartyjnym referendum ponad 75% członków opowiedziało się za pozostawieniem w nazwie ugrupowania przymiotnika "komunistyczny". W latach 1992–1993 od partii odłączyło się kilka pomniejszych ugrupowań, w tym stronnicy Miroslava Štěpána, oskarżający KSČM o "zbytnią uległość wobec kapitalistycznych realiów". W czerwcu 1993 nowym przewodniczącym KSČM został Miroslav Grebeníček. Przez wszystkie lata działalności KSČM była jako kontynuatorka KSČ izolowana przez inne ugrupowania podczas tworzenia koalicji rządowych. 1 października 2005 nowym liderem partii został Vojtěch Filip. Wybory samorządowe i senackie w 2012 roku okazały się dla partii sukcesem. Komuniści zajęli drugie miejsce w skali kraju, wygrywając wybory lokalne w dwóch krajach uzyskując przy tym 68 mandatów i 12 senatorów.
Struktura
Partia liczy ponad 100 000 członków. Najwyższym organem jest Zjazd, zwoływany co 4 lata. Między zjazdami najwyższą władzę stanowi Rada Centralna, licząca 94 członków. W terenie działa 14 Rad Krajowych.
Poparcie
Zobacz też
Komunistyczna Partia Czechosłowacji
Przypisy
Linki zewnętrzne
Oficjalna strona partii
Partie i ugrupowania w Czechach
Partie komunistyczne
Partie eurosceptyczne
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 7,578
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from django.db import models
class SpeedCamera(models.Model):
name = models.CharField(max_length=255)
description = models.CharField(max_length=255)
lat = models.CharField(max_length=255)
lng = models.CharField(max_length=255)
active = models.BooleanField()
def __unicode__(self):
return self.name
class PolygonPoint(models.Model):
lat = models.CharField(max_length=255)
lng = models.CharField(max_length=255)
speed_camera = models.ForeignKey(SpeedCamera)
def __unicode__(self):
return u'Lat {0}, Lng {1}'.format(self.lat, self.lng)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,851
|
For those who feel like their Photoshop and InDesign skills aren't up to par and want a quick and easy way to create visuals, there's Canva. The free tool allows you to create designs for social media and marketing projects using their pre-made templates and library of thousands of free and paid photos. You can customize your project by choosing the exact type of visual you want, whether it's a blog graphic, Facebook cover photo, or general social media post, and can even upload your own photos. With this tool, graphic design has never been easier!
If you're new to the blogging world, this is a good startup tool. Medium is a social platform that anyone can use to share and publish their own written content. There's a built-in audience from Medium's own users, making it easy for your writing to be discoverable and shared throughout the community. You can also easily integrate your own photos and videos to go along with the post, giving it that top notch blog quality you need to impress the world.
Used by many media organizations, Storify is a content curation tool that is meant for digital storytelling. By dragging and dropping elements from social media and news sites, you're essentially building a story for your viewers. This tool can be helpful if there's a certain topic or current event you want to focus on and can build a story around.
Content marketing is all about video right now; but instead of using YouTube to post your video, try Wistia. Wistia is a video hosting platform that allows more control over what your viewers are seeing and where they're going afterwards. Unlike Youtube, which controls every aspect of the video and how your audience views it. Wistia is perfect for video marketing in that it provides call-to-action buttons and email capture at the end of videos to ensure that the viewers are being directed back to your website or promotional landing page. One of the best perks of Wistia is that it also provides analytics for your video, showing how far viewers get through and when they pause or click.
Visuals are known to draw more attention than text-based content, and one of the most appealing and informative visuals is an infographic. Piktochart makes creating an infographic simple by providing templates, icons and charts that can be customized to whatever the theme of your visual is. Piktochart also allows you to import data from Google Spreadsheet, for example, so your information can come straight from the source.
Do you know of any other free tools to get your content out there? Let us know in the comments below!
About Sam Lauron: Sam Lauron is a PR associate with Swyft, a tech marketing and PR communications agency based in Austin serving international clients. Sam specializes in helping clients with their social media, blogging and media outreach needs.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,598
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N.J. will halt prosecuting marijuana possession cases, attorney general says
NJ.com 11/26/2020 Blake Nelson and Amanda Hoover, nj.com
EDITOR'S NOTE: NJ Cannabis Insider produces exclusive weekly content and monthly events geared toward those interested in the marijuana and hemp industries. To subscribe, visit njcannabisinsider.biz.
New Jersey's top law enforcement official on Wednesday ordered prosecutors to halt all low-level marijuana cases statewide as lawmakers continue to debate bills to formally legalize cannabis.
Anyone charged only with possession, being under the influence of marijuana or having marijuana while driving, among other charges, should have their cases postponed until at least January 25, state Attorney General Gurbir Grewal wrote to municipal, county and state prosecutors.
The order affects adults and juveniles currently being prosecuted, as well as anyone arrested in the coming weeks. It does not apply to people accused of distributing marijuana.
"It simply does not make sense or serve justice to proceed with prosecutions on charges that may be foreclosed soon through legislative action," Grewal said in a statement. "Fairness demands that we suspend prosecution of marijuana possession-related cases while we await direction from the Legislature."
In cases where residents are facing additional charges on top of low-level marijuana offenses, Grewal told prosecutors to "use their discretion" to either ask that the marijuana charges be dismissed or to postpone the entire case.
Yet the latest directive does not stop arrests, which disproportionately target Black people in New Jersey, according to the American Civil Liberties Union.
Marijuana arrests have continued after residents overwhelmingly voted for legalization on Election Day.
Lawmakers in the state Senate quickly got to work on a bill to decriminalize possession of up to six ounces, as well as distribution of up to one ounce.
But state Sen. Nicholas Scutari, D-Union, added an amendment that would downgrade penalties for possessing psilocybin, commonly known as "magic" mushrooms, days before lawmakers in both chambers were scheduled to vote on the measure.
The Senate passed the bill 29-4, but the Assembly stalled the vote due to concerns over the mushroom amendment. Lawmakers say negotiations are ongoing, but there's been no progress since.
Municipal prosecutors have continued to bring charges against many people for small amounts of possession, defense attorneys say. And they've reported uneven enforcement of the law, with some dismissing cases and others fully prosecuting the offenses.
The Ocean County Prosecutor's Office, for example, has continued to prosecute a young couple caught with less than 9 grams of marijuana on a beach earlier this year. In Elizabeth, police created a flyer entitled "The Blunt Truth," telling residents: "We, the Police will still charge you with every law related to marijuana."
Representatives for the two agencies did not respond to requests for comment.
Grewal previously reminded cops statewide that cannabis remains illegal until lawmakers regulate it, although he's told officers that they have "broad discretion when handling low-level marijuana offenses."
NJ Advance Media managing producer Robin Wilson-Glover contributed to this report.
Our journalism needs your support. Please subscribe today to NJ.com.
Blake Nelson can be reached at bnelson@njadvancemedia.com. Follow him on Twitter at @BCunninghamN.
Amanda Hoover may be reached at ahoover@njadvancemedia.com. Follow her on Twitter @amandahoovernj.
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©2020 NJ Advance Media Group, Edison, N.J.
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{
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| 1,147
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Here's A Thing…
…create the extraordinary from the ordinary
EXTERNAL NEWS LINKS
Progressive Links
The National Memo
Democracy Now-YouTube
Truth Dig
Buzz Feed
Trump LIES but he doesn't tell LIES…
10/10/2019 lakodabluedingo Politics Leave a comment
"I think when Trump says things that are false, that does undermine his presidential authority and I wish he wouldn't do it," Moore said. "He should stop saying things that are untrue."
Yet Moore, who was tapped by Trump for a seat on the Federal Reserve Board earlier this year but withdrew his nomination after fierce bipartisan opposition, wouldn't use the "L" word.
"So he's a liar?" host Mehdi Hasan asked.
"I'm never going to say that," Moore said. He called Trump an "exaggerator" instead.
Hasan offered some quick examples of Trump lies that weren't exaggerations. Moore just threw up a hand and said: "I don't know about that."
Although The Washington Post has documented more than 12,000 false or misleading claims by Trump since he took office, many of the people working with the president won't acknowledge a single one of them.
Fred Trump didn't serve with America and neither did Donnie in Vietnam…
Trump comments that the reason he abandons the Kurds is because the Kurds didn't help SAVE PRiVATE Ryan. Simply more Stupid coming out of the mouth of the Perfect One.
Kurds apparently did serve with the Allies to oppose fascist and Nazis in Europe.
The Kurds DID help us win World War 2. First as part of the counter-offensive to the Nazi-backed Iraqi coup of 1941. Later, some served in Albania, Italy and Greece.
After Iraq became a British Mandate, the force became a minority manned force of mostly, Iraqi Turkmen, Kurds and Assyrians who lived in the north of the country while the nascent Iraqi Army was manned by Arabs. Eventually it became a mostly Assyrian manned and British officered force while it was used mostly for the guarding of the Royal Air Force bases in Iraq.[2]
World War Two Medals
Three different World War Two Medals were awarded to members of the Iraq Levies.
The War Medal 1939–1945 – Awarded to Levies after 28 days of service in World War II.
The 1939-1945 Star – Awarded to Levies after six months service in World War II.
The Italy Star – Awarded to parachute company personnel that served in Albania, Italy and/or Greece.
But trump's daddy Fred didn't serve at ALL… unclear if Fred suffered from bone spurs
Previous Post: Trump is showing the Republic one thing for certain…
Next Post: Trump is as CORRUPT as he is CRIMINAL
Photography and Related
Stanford Philosophy
PHILOSOPHICAL AND PSYCHOLOGICAL DISCUSSION OF BIAS…
"He denied he knew the guy — he's in trouble."
Categories Select Category Essay (12) Music (2) Politics (282) Video (3)
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My Daily Picks
Russian 2016 Election Interference Exist and Trump Continues to Stroke his FIDDLE…
dark plague of fascism engulfs American society
Accused child sex trafficker Jeffrey Epstein and his fair-weather friend Donald Trump have a lot in common.
What is the Democrat Strategy to beat Trump?
Jeffrey Epstein had 14 phone numbers connected to Trump in his "little black book"
Trump Says He Relies On Press To Vet Nominees
Ilhan Omar and Nancy Pelosi Go to Ghana and Stick It to Trump "They said 'send her back.'"
Donald Trump Is Burning the Constitution, and the Biggest Villain at the Democratic Debate Was… Barack Obama?!
How a 40-year-old federal law is speeding gentrification
"I declined the invitation because I refuse to be an accessory"
Myth of the Good Guy with a Gun…
9 Dead, 26 Injured Ohio** 20 Dead In Texas, 26 Injured
'Let My Parents Be Free'
WOODSTOCK 50 YEARS PLUS
IT'S THE GUNS – STUPID!
Something stinks to high heaven…
Trump Buddy Epstein found dead…
Republicans, The Party Of Lincoln?
The Canary in the Mine…
If Trump were an Airline Pilot
LOVE THE DONNIE
get the Racist out of the White House
The Christian Trump…
Question: What is Trump Hiding from the American People?
PAYROLL TAX REBATE and other trump bullchit…
OBAMA VS BUSH AND TRUMP (2 REPUBLICANS)
EXPOSE TRUMP-RELEASE THE TAX RETURNS
BED BUG TRUMP
WHAT IS TRUMP HIDING AND WHY?
Labor day 2019 and Trump on the attack
JUST ANOTHER AMERICAN EXERCISING THEIR SECOND AMENDMENT RIGHTS…
trump and pennywise-elect a clown EXPECT a circus
WHO VOTES FOR TRUMP
"So you're telling me you wouldn't protect one life because of a background check?"
In Stunning Rebuke of Its Own NWS Office, NOAA Statement Supports Trump's Hurricane Dorian Alabama Comments
Don't lose sight of the "controlling Idea" of the Sharpie Story
Donald Trump: Mexico will pay for wall, '100%'
Forty definitions of socialism
Nazi and Fascist Socialism
WHAT IN TOM PAINE'S NAME IS TRUMP HIDING?
an Elevator to the moon and to space…
I want to see Democrats take on Trump. (period)
Trump supporters want everything to be simple and extreme…
A 6th person has died from a vaping-related lung illness,
The GROUND is Shifting…
Trump's supporters want chaos and destruction: When their bad drug high finally wears off — it won't be pretty
Donald Trump claimed the primaries were rigged in 2016. Now he's the one rigging them…
Republican Group Accuses Trump, Pence Of Corruption
Foreign Reporter Shocked By Trump's 'Alarming Incoherence' On Border Wall Tour
Impeach Trump part deux…
IT IS TIME TO IMPEACH THIS NUT JOB!
HOUSE ON FIRE AND SOME PEOPLE THROW GASOLINE ON THE FLAMES…
UNFIT TO BE AMERICAN
ONLY LIAR TRUMP COULD LIE LIKE THIS…
TRUMP IS CORRUPT, CRIMINAL AND UN-AMERICAN…
TRUMPIST COVERUP!
THOU SHALL HAVE NO OTHER GODS BEFORE ME…
Trump LIES about Polling…
"Buy a Spine…"
YOUR NITWIT PRESIDENT
5 myths about treason
YEP, I HAVE A JOB DONNIE… ACTUALLY I HAVE THREE…
Newly Revealed Trump Administration Texts On Ukraine Show Clear Quid Pro Quo
That's the signpost up ahead—your next stop, the Twilight Zone!
Someone should really read the Constitution to NITWIT Trump and soon!
GUESS WHO'S USING A SERVER NOW…
GOPhers: Save Your Souls And Stand Against Trump
Mr. Trump's Neighborhood
Alexander Hamilton Warned Us About Trump and Barr
TIME FOR REPUBLICANS TO BE AMERICAN FIRST AND NOT TRUMPIST…
THE MESSAGE is SIMPLE… Trump is a CLEAR and Present danger to the American Republic and the Constitution.
Poorly educated and uninformed Trump cult worshipers continue sipping the kool-aid…
Trump is only joking…
WHAT IS TRUMP HIDING? AND WHY?
Trump has provided America with ONE crucial piece of identity…
TRUMP AND HIS CORRUPT CRIMINAL FAMILY
Republicans must lie to survive…
The Republic, The Constitution and Trump…
Trump is as CORRUPT as he is CRIMINAL
Trump is showing the Republic one thing for certain…
WHAT'S TRUMP HIDING? OBSTRUCTION OF INFORMATION TO THE REPUBLIC!
YOU CAN'T DEFEND THE INDEFENSIBLE…
Mitt Romney indicts Trump's moral character…
Trump's DIRTY Deal Dead…
Republicans in Congress can't handle the questions
TRUMP LIES ABOUT UKRAINE PHONE TRANSCRIPT…(MEANS MORE TO FOLLOW)…
Some Good News…?
Don't PISS Trump OFF…
IT'S TIME TO HOLD TRUMP ACCOUNTABLE FOR HIS CRIMES AND CORRUPTION…
TRUMP AND BARR still covering up the Russia conspiracy…
Trump handing northern Syria to Turkey is a 'gift to Russia, Iran, and ISIS,' former US envoy says
Joe Biden enjoying a 15-point polling lead…Support for impeachment inquiry reaches new high…
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,217
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Q: How do I add a namespace to the "code=" URL parameter that Spotify's API redirects to when a user authenticates their Spotify account? I am creating an LWC component that utilizes the Spotify API auth flow to authorize a user's Spotify account and allows me to add playlists to their Spotify account directly from an LWC component with their permission.
I use the following URL to start the auth flow:
https://accounts.spotify.com/en/authorize?&client_id=[_myAppsClientCode_]&response_type=code&redirect_uri=https://johngordoaudemysandbox-dev-ed.develop.lightning.force.com/lightning/n/Playground&scope=user-read-private%20user-read-email
And I receive this back from Spotify https://johngordoaudemysandbox-dev-ed.develop.lightning.force.com/lightning/n/Playground?**code=**[_returned_auth_code_]
However, since there is no namespace on the URL parameter it is chopped off when it loads.
When I manually type a namespace into the URL, the parameter remains in the URL without any issue: https://johngordoaudemysandbox-dev-ed.develop.lightning.force.com/lightning/n/Playground?**c__code=[returned_auth_code]**
I'm unsure where I can add the "c__" namespace to the "c__code=[returned_auth_code]" in the settings of ether Spotify's API or in my Salesforce org so that it appears in the returned URL automatically.
The googling I've done so far has led me toward SSO configuration. Can anyone provide insight or at least a set of keywords that I can google to get closer to the issue?
A: OAuth protocol specifies that the parameter must be code. You can't fix this on the API end. What you can do, however, is write up a Visualforce page to catch the code parameter, then redirect from Visualforce to your Aura/LWC component with the parameter fixed up with the c__ prefix.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
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## Anna Campbell
## Captive of Sin
For my beloved mother, Dagmar.
## Contents
One
Good God, what have we here?"
Two
Horrific images haunted Charis's dreams. An endless replaying of Hubert's...
Three
Guvnor, we got trouble."
Four
Through the suffocating miasma, Gideon knew he'd frightened the girl.
Five
Gideon expected Miss Watson to demur. After all, only yesterday...
Six
Over the next days, Gideon saw little of Sarah. With...
Seven
Sarah!" Gideon whirled and lashed out to grab her before...
Eight
After so many hours in Sarah's company, Gideon inevitably dreamt...
Nine
What the devil happened to the men watching the road?"...
Ten
Her stomach somersaulting with nerves, Charis approached the library. This...
Eleven
Charis stood in the prow of the sleek little boat...
Twelve
Gideon held himself together until he closed the door behind...
Thirteen
Even in the dimness, Charis saw the blood drain from...
Fourteen
Wearing only her shift, Charis waited alone in the big...
Fifteen
The afternoon wind off the sea was so icy, even...
Sixteen
Since Rangapindhi, horror and pain had poisoned Gideon's dreams. This...
Seventeen
Aghast, Gideon stiffened. Bloody, bloody, bloody hell. Why in the...
Eighteen
Across the remains of the meal he'd ordered in their...
Nineteen
Charis's heart crashed to a halt. At last she saw...
Twenty
Gideon's guarded expression as he stared at her outstretched hand...
Twenty-One
It's midnight," Gideon said softly, his breath ruffling the hair...
Twenty-Two
Gideon turned the hired gig onto the lonely road that...
Twenty-Three
Up on the moor, the wind roared like an angry...
Twenty-Four
There was a sickening, distant thud, then silence descended like...
Acknowledgments
About the Author
Other Books by Anna Campbell
Copyright
About the Publisher
## One
Winchester
Early February 1821
Good God, what have we here?"
The man's deep voice pierced Charis's pain-ridden doze. She flinched, stirring from her cramped position. For one dazed moment, she wondered why she was shivering in fetid straw instead of snuggled in her bed at Holcombe Hall.
Blazing agony struck, and she stifled an involuntary moan. And a curse for her rank stupidity.
How could she forget the danger long enough to fall asleep?
But she'd been blind with exhaustion when she'd stumbled into the stable behind the sprawling inn. Unable to manage another step, even though she hadn't come far enough to be safe.
Now she wasn't safe at all.
The light from the man's lantern dazzled her bleary eyes. She discerned little more than a tall shape looming outside the stall. Choking with panic, she clawed upright until she huddled against the rough planking. Blood pulsed like thunder in her ears.
Muffling a whimper as she moved her injured left arm, Charis crossed shaking hands over her torn bodice. Scenting her terror, the big chestnut horse that filled most of the space shifted restively.
As the man lifted the lantern to illuminate Charis's corner, she shied away. Beyond the ring of yellow light that surrounded him, menacing shadows thickened and multiplied up to the high-pitched ceiling.
"Please don't be frightened." The stranger made a curiously truncated gesture with one black-gloved hand. "I mean you no harm."
The rich baritone was sheathed in warm concern. He made no overt movement toward her. Charis's crippling fear didn't subside. Men, she'd learned from cruel experience, lied. Even men with velvet voices, smooth and cultured.
A sharp twinge in her chest reminded her she hadn't drawn breath since he'd found her. The air she sucked into her starved lungs reeked of horse manure, hay dust, and the sour stink of her own fear.
She turned her head and really looked at the man. Her throat jammed with shock.
He was utterly beautiful.
Beautiful. A word she'd never before associated with a male. In this case, no other description sprang to her churning mind.
Beauty as stark and perfect as this only stoked her alarm. He embodied the elegant world she must relinquish to survive.
Despite her terror, her attention clung to the slashing planes of forehead and cheekbones and jaw, the straight arrogant prow of his nose. He was tanned, unusual in February.
With his intense, compelling features and ruffled hair, black as a gypsy's, he looked like a prince from a fairy tale.
Charis no longer believed in fairy tales.
Her eyes darted around the narrow stall. But he blocked the only exit. Again, she cursed her idiocy. With her good hand, she fumbled beneath her for a rock, a rusty nail, anything she could use to defend herself. Her trembling fingers met nothing but prickly straw.
Unblinking, she watched him set the lantern on the ground. His movements were slow and easy, openly reassuring. But if he wanted to snatch her, he now had both hands free. Her sinews tautened as she prepared to scratch and punch her way out.
In the charged silence, the rattle of her breathing deafened her. It even masked the wind's constant wail. The powerful horse shifted again and gave a worried whicker, tossing its head against the rope that tied it facing toward the corridor.
What if the nervous beast started to kick or buck in this confined space? The horse's hooves looked huge, sharp, deadly. Dread settled like a stone in her empty belly. With every moment, her refuge's unsuitability became more apparent.
Why, oh, why hadn't she kept going, no matter how tired and hurt? Even sheltering in a hedgerow, she'd be safer than here.
The man stepped into the stall, his black greatcoat swirling around his booted ankles. Shrinking back, Charis prepared to wrench free of grabbing hands. Fresh sweat chilled her already icy skin. He was so much bigger and stronger than she.
But he merely snagged the animal's halter with a firm grip that brooked no rebellion. "Hush, Khan." He stroked the gelding's nose as his voice softened into alluring music. The man's tall body conveyed an assured confidence that was almost tangible. "There's nothing to worry about."
The complex mixture of authority and care in his tone should have calmed Charis. Instead, it slipped down her spine like glacial ice. She knew all about men who believed they ruled the universe. She knew how they reacted when their wishes were thwarted. Her furtive search for a weapon grew more frantic.
Khan, foolish, trusting creature, quieted under his master's murmured promises. For the man must own the beast if he knew its name. Nobody could mistake the stranger for a groom. His manner was too effortlessly aristocratic, his clothing too fine.
She found no weapon.
She'd have to make a dash for freedom and hope her stiff, tired legs carried her. Surreptitiously, she pushed upward. Even this small movement sparked agony. Every muscle ached, and her arm felt like it was on fire. She locked her teeth to muffle her whimpers.
"There's no need to run away." He didn't glance up from the now docile horse.
"Yes, there is," she surprised herself by saying, although she'd resolved not to address him. Her swollen face thickened her voice into unfamiliarity. But her upper-class diction marked her as an object of interest. Memorable. Noticeable.
A target.
Clumsily, she struggled to her feet. She felt less vulnerable standing. In her awkward rise, she bumped the wall and bit back a sharp cry. Battling dizzying pain, she cradled her throbbing arm against her.
Her ungainly lurch spooked Khan, who sidled and snorted. Her father had been a connoisseur of horseflesh. Charis had immediately recognized Khan for the highbred aristocrat he was.
Much like the man holding the beast's head.
"I know you're afraid." At first, she thought he spoke to Khan. His attention remained on the horse. "I know you need help."
Help to hand her over to the law, she thought bitterly. "Why should you care? You're a stranger."
"That's true. Although when you chose my horse's stall, you also chose me."
"That was just chance."
At last, he looked directly at her. Surely it was only a trick of the lamplight that his eyes shone so dark and brilliant above those dramatic cheekbones. "All things in life are chance."
Charis shivered under that appraising ebony gaze. The moment seemed to hold a significance it couldn't possibly have. Shaking off the strange preternatural sensation, she raised her chin. She had enough problems in the here and now without taking on the metaphysical.
"Kindly step aside, sir. I must be on my way."
"It's not safe for a lady to travel by herself." He didn't budge, and while his voice remained quiet, it was implacable.
To underline his warning, a burst of carousing came from the inn across the yard. On such a cold night, the taproom must be packed. The freezing weather was one of her few strokes of luck—the grooms had left their posts to seek the fire's warmth. Otherwise, they'd have discovered her hiding place immediately. Why wasn't this stranger equally eager to stay inside like any sensible man instead of wandering around this cavernous stable?
"That is none of your concern." How on earth could she escape? Again, she berated herself for not struggling on.
"Won't you trust me with your story?" His voice dropped into sweet persuasion. The tone wasn't far different from the one he'd used to settle Khan. And like Khan, she felt the insidious lure of that mellifluous baritone. "I can see you're in trouble. I swear..."
He broke off abruptly and tilted his head toward the main doors, far down the long corridor. Then Charis caught the shuffle of approaching footsteps. What inhumanly acute hearing he must possess to discern anyone's arrival over the creaking roof and the whistle of the wind.
"Aught amiss here, my lord?" a rough male voice, she guessed belonging to a groom, asked from several yards away.
My lord? She'd been right about his social status. With a frightened whimper, Charis shrank into the shadows as the man shifted the lantern so darkness shrouded her. As she retreated, each rustle of straw sounded loud as a gunshot.
"Just seeing to my horse, my good man." With a casual air, he wandered out of sight toward the newcomer.
"Can I aid thee?" The groom's voice grew clearer as he approached.
Charis's breath caught in her throat and she hunched as far from the light as she could. Her arm protested the movement, but she ignored the shooting pain.
"No. All's well."
Charis buried damp palms in the tattered, stained skirts of her once-elegant day gown and silently prayed that she remain undetected. Her heart banged so frenetically against her ribs, she was surprised the groom didn't hear it and come to investigate.
"It's a cold night for man and beast, that's for certain sure."
"Too cold to be out and about." For all the ring of authority in his voice, the lord sounded relaxed, unworried. "Find your place by the hearth and have a drink on me."
Charis edged as far behind Khan's rump as she dared, keeping a wary eye on those lethal hind legs.
"Very kind of your lordship, I'm sure. I don't mind if I do." The groom's reply rang with surprised gratitude. "Sure I can't assist?"
"Quite sure." The lord's voice indicated dismissal, and whatever coin changed hands ensured immediate compliance.
"Good e'en to your lordship."
With excruciating slowness, the groom shambled away. It seemed to take forever before his lordship appeared at the stall's entrance. He raised the lantern to reveal her trembling form against the back wall.
"He's gone."
"Thank heaven." In a relieved gasp, Charis released the breath she'd held for what felt like an hour. She didn't know why the man had helped conceal her. All that mattered right now was that he had.
He surveyed her with a troubled expression on his striking features. "You can't stay here. The inn is crawling with people. You've been lucky to stay undisturbed this long. At least come out where I can see you."
"I don't..." she started uncertainly. Although the man made no attempt to drag her out, she pressed against the boards. The movement cramped her aching muscles with fresh pain.
The man stepped away to indicate he presented no danger. At last she saw her way clear to take to her heels.
She hesitated.
She bit her lower lip, then wished she hadn't when the torn flesh stung. The stranger was right. What chance her making it past the inn yard? This close to home, someone would surely recognize her.
As if he read her thoughts, the watchfulness faded from his eyes. "My name is Gideon."
Even as Charis limped past Khan into the aisle, she remained poised for flight if the man—Gideon—made a move. But his stance was relaxed, and he left her space. She sucked in a shuddering breath that tested her bruised ribs. With every second he didn't touch her, she felt safer.
"You're hurt." He sounded tranquil, but anger sparked his eyes to black fire as one comprehensive glance swept her from head to toe.
She could imagine what a disreputable slattern she looked. Humiliated heat crawled up her neck, and she lifted her right hand to clutch her ragged bodice. Her stepbrother Hubert had ripped it when he'd held her down. Now the neckline gaped to reveal the lacy edge of her shift.
Her face felt as though a thousand wasps stung it. Her blue dress was torn and filthy and pitifully inadequate on this arctic night. Under capped sleeves, scratches and bruises covered her arms, legacy of the beating and her frantic flight through fields and woods. Her hair was a matted bird's nest. Most of its pins had shaken loose as she'd fought her way through the hedgerows around Holcombe.
Before Gideon could question her or, worse, express the pity that lurked like a ghost under his outrage, she launched into the story she'd prepared. "I was traveling to my aunt in Portsmouth when...when footpads set upon me."
Curse that telltale falter. Lying never came easily. He wouldn't believe her. Which meant her game was up.
She waited in breathless suspense for him to brand her a sham and a runaway. But he merely whipped off his heavy black coat and stepped closer.
Fear had her backing away at a stumbling run until she slammed into a thick post. She strangled a scream as the impact shot jagged lightning along her arm. Automatically, she jerked forward, and he seized the opportunity to drop the coat around her trembling shoulders.
"Here." He stepped away again.
Gradually, panic ebbed, and she straightened under the coat's weight. Its warmth made her feel slightly more human. The garment swamped her, trailing on the ground. The fabric smelled pleasantly of fresh air and something clean and musky that must be its owner.
He was clever enough not to crowd her. Even so, she remained nervously aware of his commanding height and leanly muscled body, now revealed in black jacket, white shirt, and brown breeches that clung lovingly to long, strong legs. From his highly polished boots to his plain white neckcloth, his clothing was simple but of the highest quality.
"Th...thank you," she said through chattering teeth.
She blinked back stinging tears and clutched the deliciously cozy woolen folds around her like a shield. Strange, but his kindness proved the greatest threat to her fraying control.
"What is your name?"
The loan of the coat seemed to require some gesture of trust in return. "Sarah Watson," she said in a grudging voice, stealing the identity of her great-aunt's dour companion in Bath. Remembering her manners, she dropped into a stiff curtsy.
He forestalled her with another of those odd, incomplete gestures. His intent dark eyes didn't waver. "May I escort you to some friend or relation in Winchester, Miss Watson? This stable isn't safe."
She wasn't safe anywhere, heaven help her. Fear stirred low in her belly as she remembered what would happen if her stepbrothers caught her.
"I'm...I'm a stranger in this part of the country, sir. I'm from Carlisle." The most distant town she could think of without actually crossing the border into Scotland. She stiffened the wobbly legs that threatened to buckle beneath her and glared at him, daring him to challenge her story.
His expression remained neutral, but she knew he sifted her responses for truth and falsehood. "A long journey for a lady on her own. Didn't you have a maid to accompany you, at least?"
With every moment, she sank further into an abyss of lies. But what choice did she have? If she revealed her identity, any law-abiding citizen would immediately hand her over to the authorities.
Nonetheless, her unruly tongue tripped over her answer. "My maid ran off when we changed coaches in London."
"You have indeed suffered a series of misfortunes, Miss Watson."
Did his response contain a hint of irony? His expression remained all polite interest. She decided to accept his comment at face value. "It's been a terrible day." At least that much was true. "Now all I wish is to reach my aunt's house."
"You're a long way from Portsmouth."
Didn't she know it? She'd barely covered a few miles and already tested the edge of her endurance. She had no money to pay for a seat on a coach, and even if she had, she couldn't risk someone seeing her and remembering her. Yet again, the insurmountable task she set herself struck her. Then she recalled what awaited back at Holcombe. "I'll manage."
"How?" he asked with the first trace of sharpness. "You're dead on your feet."
Hearing her own doubts voiced with such emphasis made her belly clench with sick despair. "Needs must."
His lips flattened. Clearly he found her sullen answer as unimpressive as she did herself. "I offer you transport if you care to accept."
Charis jerked back as if he tried to touch her. This seemed too good to be true. Transport to Portsmouth was a godsend. Her stepbrothers would already be on her trail. If she went with this stranger, she'd cover more ground. Not only that, her stepbrothers would ask after a girl traveling alone.
"I couldn't inconvenience you so." She intended the words to sound final, but her injuries slurred her speech.
"I'm traveling south anyway." His expression became somber. "Chivalry forbids me to abandon a woman to the mercy of any blackguard she meets on the road."
In spite of physical misery and encroaching fear, a grim laugh escaped Charis. She made a dismissive gesture with her good hand. "Chivalry is an unreliable quality at the best of times."
"You have my word as a gentleman that your virtue is safe, Miss Watson." He didn't smile.
She'd heard so many lies recently, she just assumed anything a man said must be falsehood. But strangely, she believed him when he pledged his word.
Good Lord, if this man meant to rape her, surely he'd have made a move by now. Every scrap of sense prompted her to credit him as that most chimerical of creatures.
A genuine man of honor.
Or was she just dazzled by his remarkable looks? She was vulnerable and exhausted. Ceaseless pain turned her mind hazy. She was frightened for her life.
The pause extended, stretched into taut silence. If he'd tried to persuade her, she would have insisted on going on alone. But he let her make up her mind. Only the tension straightening the powerful shoulders under his superbly cut jacket indicated he awaited her answer with more than indifference.
Finally, she sighed. It was a sound of acquiescence. Fear clogged her veins, but desperation was stronger. Wondering if she cast her lot with the devil, she gave a brief nod. "Then I accept your help with gratitude."
"First we'll take you to a doctor."
For an instant, her terror had faded to a distant thrum. The chance of escape had beckoned like a lifeboat to a drowning man. Now his words reminded her she'd found no sanctuary yet.
Perhaps ever, unless she was very clever and very lucky.
Any doctor in Winchester would recognize her immediately. She shook her head in swift denial, cradling her arm. "I don't need a doctor. My injuries aren't as bad as they look."
She waited for argument. None came. "All right. No doctor."
Relief made her sag, although she tried to mask her overwhelming reaction. Apparently she'd fallen in with the most credulous gentleman in the county. So far, he accepted her story without a moment's doubt.
Odd, she wouldn't have considered him a stupid man. Intelligence sizzled in those perceptive dark eyes.
Perhaps he was just naïve. More reason to go with him. Evading him in Portsmouth should present no trouble.
What she'd do then was a complete blank. She had no money and no friends. Or no friends she could put at risk of prosecution. Her stepbrothers had already terrified her one close relative, her great-aunt, into handing her over to them. She wore a gold locket and her mother's pearl ring, neither of great value. Somehow she had to hide for three weeks. Her crushing dilemma made her shudder.
One step at a time. She chivied her flagging courage. Getting out of Winchester undetected was her first goal.
"Gideon."
A man spoke from the stable doorway. Charis started, again testing her injuries, and felt the blood drain from her face. Her rescuer reached out but cut the gesture short of making contact. "Don't worry. He's a friend."
Such was his natural authority, Charis curtailed her retreat, although her heart pounded like a hammer and cold sweat broke out on her skin.
"I'm here," Gideon called, without taking his eyes off her.
Another man, as tall as her rescuer, slender, dark, obviously foreign for all his fine London tailoring, strolled into view. "What have you found?"
"Miss Watson, this is Akash. Akash, may I present Miss Sarah Watson? She's been set upon by ruffians and requires aid."
The newcomer's liquid brown eyes rested upon Charis. She waited for him to question her threadbare tale. But after a pause, he merely quirked one elegant black eyebrow at Gideon.
"I'm guessing we're not staying here tonight?" His voice was pure England, although he looked like he inhabited some Arabian fantasy.
"You know I'm in a hurry to get to Penrhyn."
"Indeed," he said neutrally.
"Yes, via Portsmouth."
"I've always had a violent desire to visit Portsmouth." Akash sounded perfectly undisturbed at the prospect of braving the cold to assist a stranger. Too undisturbed.
Suddenly, Charis didn't feel safe after all. Putting herself into the care of two unfamiliar men was the height of foolishness. Their quick acceptance of her paper-thin story seemed suspicious rather than reassuring.
On trembling legs, she backed toward Khan, who whickered softly in her ear. "I can't impose on your good natures. I shall make my own way to my aunt."
"No man of honor would countenance such a plan, Miss Watson." Gideon sounded immovable.
She could sound immovable too. "Nonetheless, it is what I must do."
Gideon sent a quick smile to his companion. For one dazzling moment, amusement lit his face to brilliance. Glittering dark eyes, creases in his cheeks and around his eyes, a flash of straight white teeth.
Charis's heart lurched to a halt, then broke into a wayward race. Foolishly, in spite of fear and pain and mistrust, she longed for nothing more than to see him smile again.
Smile at her.
"I believe you've terrified the chit, Akash."
She ignored Akash's soft laugh and frowned at Gideon. "Pray, sir, I'm no chit."
"Would you feel happier if I gave you this?"
She looked down to see him extending a small dueling pistol. She hadn't noticed him reaching into his jacket. Tiredness made her stupid. Tiredness and the effects of a vicious beating.
And most unwelcome admission of all, a man's unguarded smile.
She stared at the gun as though she didn't recognize what it was. The room receded in dark waves. The thunder in her ears rose to blanket all other sound.
"Akash!"
Gideon's shout came from a long way off, then the world spun as strong arms swept her off her feet.
But not the strong arms she wanted to close around her. Even through her near faint, she recognized that bone-deep and mortifying fact.
Gideon stared at the half-unconscious girl Akash clutched to him. She was a tumble of slender arms and legs and frothy blue skirts. Her bright bronze hair trailed across Akash's black sleeve like a flag. Her hem was torn and wet, and her pale blue half boots were caked in mud.
His hands fisted at his sides, and anger cannoned through him. Who the devil had abused her? Even before this last year, he'd abhorred cruelty. And some bastard had beaten this girl to within an inch of her life.
Gideon was too familiar with violence to misjudge how badly she was hurt. Damn it, he wanted a doctor to look at her.
But the chit was so frightened. Gideon knew too what desperate fear looked like, and he couldn't mistake it in the girl's wide hazel eyes, lovely even in her ruined face. If he pushed her too far, she'd scarper and meet God knew what dangers.
What in Hades had happened to her? He'd immediately recognized her pathetic lies. He'd lay money no footpads had attacked her but, hell, someone had.
Futile rage, sickeningly familiar, flooded his mouth with a vile, rusty taste. He stepped back and breathed hard through his nose as he fought for composure. He needed to stay calm, or he'd frighten her.
The girl stirred in Akash's grip, and her pale hand clenched in his coat. Gideon's attention caught on an expensive, if old-fashioned pearl ring on one slender finger. Nor had he missed the pretty gold locket revealed under her tattered bodice. Whoever she was and whatever her current straitened circumstances, she came from money.
Her voice was thick with distress. "Please...please put me down. I can walk. Really."
Gideon's rage faded, replaced by piercing compassion. His anger couldn't help her. She was small, defenseless, heartbreakingly brave. And young. Impossible to tell exactly how old she was under the patchwork of bruising, but he'd guess not much more than her early twenties.
Add to her courage a pride that cut Gideon to the heart. Oh, he understood how she felt, all right. He guessed pride was all she had left.
Pride and two strangers who would see her safe, whether she trusted them or not.
He couldn't abandon her to her fate. He was too bitterly aware what it was like to stand against powerful foes with no hope of prevailing.
"Guvnor, is there a problem with the nag?"
Gideon turned toward the door with a surge of irritation. Akash had come out to check on him, although if challenged he'd never admit it. Now here was Tulliver, asking after his charge's health like a gruff and grizzled nursemaid.
The yearning for freedom was a crashing wave inside him. He'd give up his hope of heaven for one moment without eyes observing his every move. Fresh air in his face. A good mount beneath him. Nothing but clear open country.
And no people for a hundred bloody miles.
"Sir Gideon?"
The wild and glorious dream faded. How could he blame his companions for their concern? They were good men, both. He'd spent so long alone, it still struck him as remarkable that they pledged him their loyalty.
Surely they must recognize he was completely unworthy of the honor.
"We're not staying, Tulliver," he said to the brawny ex-soldier he'd hired as his servant after the fellow's untiring service on the ship from India. "We'll need a carriage, food for the journey. And a driver, I expect."
"No need, sir. I can handle a rig."
Tulliver, Gideon had learned, could handle almost anything, from a man out of his head with pain and shame right up to a duchess's comfort. The East India Company had lost a treasure when Tulliver resigned.
Tulliver's eyes flickered impassively over the woman in Akash's arms, but he asked no questions. He never did. Yet somehow he managed to know everything. He bowed and headed outside again.
"Please, sir," the girl said in a shaking voice.
Silently, Akash set her upon her feet. She staggered, and Gideon reached out before he remembered and snatched his hand back. The girl raised her chin and stared him down as if he'd made an improper remark at a debutante ball.
Again, her pride touched something deep within him. Something pure and fresh like a tender green shoot after the first snows melted. He was astounded any untainted feeling could survive what he'd endured.
"I put you to some inconvenience." Her attention still on Gideon, she stepped away from Akash. She held one arm awkwardly against her. "While I'm grateful, I can't allow you to discommode yourself on my account."
She spoke like a damned octogenarian duchess. A confounded haughty one, at that. In spite of the moment's seriousness, Gideon felt his lips twitch.
Of course she didn't miss it. "You're laughing at me."
He didn't deny it. Instead, he let an element of steel enter his tone. "Miss Watson, you need our help. I can't bundle you in the carriage and force you to come with me."
A lie. Of course he could. He would if he had to.
"I'd scream if you did," she said defiantly, even as her shoulders drooped under the weight of his coat. And the weight of her despair and fear, he guessed.
Why was he so determined on rescuing this prickly-tempered waif? She stood before him, trembling with pain, panic, and weariness. Her dark bronze hair was a tangle around her pale face. Her gown was ripped and stained. Bruising hid any beauty she possessed.
He bit back a caustic grunt of laughter.
Even if she was a beauty, what use was that beauty to him?
He quashed the acrid question and shot her a straight look. "It's February. It's cold. You're in no fit state to go on alone."
Tulliver appeared in the doorway. "I've arranged the carriage, guvnor. The landlord is chasing up the grooms."
Gideon watched terror flood the girl's eyes. She was definitely eager for nobody to see her. He needed to know why. "Go back into the stall, Miss Watson. Khan won't hurt you."
"I'm not frightened of your horse," she retorted. She tugged the coat around her slender body and withdrew into the darkness.
The staff at Winchester's largest inn were used to arranging transport for patrons. The small closed carriage was ready for departure within minutes.
Gideon stepped into the stall. The girl huddled behind Khan. He tried to quell his automatic reaction to the crowded space and the darkness. But the gloved hand he placed on the rough wooden divider was unsteady.
Thank God the gloom hid his reaction. What confidence could she have in a rescuer who trembled like a willow at the merest shadow?
"We're ready."
She straightened and wrapped the coat around her like a cape. He supposed she couldn't bear to force her injured arm into a sleeve. As she looked up, he caught the shine of her eyes. "Why are you doing this?"
He shrugged, trying to appear as if aiding stray maidens was his everyday activity. "You need help."
"It doesn't seem enough when I see the trouble you've taken."
"It will earn me points in heaven," he said with a lightness he didn't feel. He extended the bundle he held. "I thought you might like this."
She didn't immediately take it. "What is it?"
"A shawl. The night is cold." And she'd need to cover that distinctive hair when she entered the carriage. Although if he told her that, she'd know he tagged her tale as a pack of lies.
"Where did you get it?" Her voice dripped suspicion.
He hid a smile. She was so wary, so defensive. Yet if he wanted, he could render her unconscious in the blink of an eye. That possibility had occurred to him, but he'd dismissed it. She'd had enough violence done to her.
"Tulliver bought it from a lady at the inn."
Good thick wool—he thought with a moment's regret of the shimmering, gorgeous fabrics he'd seen in India. He lifted the brown shawl briefly to his nose and sniffed. "It smells of pug, but you'll welcome its warmth."
To his surprise, she gave a short huff of laughter. "I've been sleeping in a stable. A whiff of eau de chien won't unsettle me in the least."
The chit had backbone. He'd always admired courage, and this girl had more than was good for her. Something tired and rusty and long unfamiliar stirred in his heart. He stifled the unwelcome sensation and offered the shawl once more. "Miss Watson?"
"Thank you."
As he'd known she would, she wrapped it around her head and shoulders. In his enveloping greatcoat and with her head covered, she looked almost anonymous. He couldn't miss how she favored her right arm. Was it broken? Again, he wished she'd let him take her to a sawbones.
"And take this, just in case." He passed her the pistol and watched her slip it into the coat's voluminous pockets. "Do you know how to use it?"
He already knew the answer. She handled the gun with an ease that indicated familiarity.
"Yes. My father was a marksman. He taught me to shoot."
Gideon shadowed her when they crossed the yard to the waiting carriage. Akash was already up on his temperamental gray.
As Gideon opened the door for Miss Watson, he caught his friend's eye. He wondered what Akash made of the night's events and the new addition to their party. He'd find out, he knew. Just because Akash had said nothing yet didn't mean he had nothing to say.
The girl paused, as if expecting Gideon to hand her up. Yet another clue to a privileged life, if she but knew it. When Gideon didn't respond, she climbed into the carriage.
Tulliver followed, leading his sturdy mount and Khan, and tied both horses to the back of the coach. Gideon cast a last look around the windswept yard. Ostensibly, nobody paid them any attention.
On a frosty night like this, anyone who didn't have to be outside sought what warmth they could. The few servants crossing the open area seemed to mind their own business. Still, old habits died hard, and Gideon took note of the scene's every detail.
Tulliver came up to his side. "All set, guvnor?"
"Yes." One last glance to make sure, but nobody appeared unduly interested in their little party. "Let's get under way."
"Very good."
Tulliver climbed into the driver's seat. Gideon entered the vehicle where the mysterious Miss Watson, with her sharp tongue and terrified eyes, awaited.
As he surveyed her unkempt figure perched stiffly on the leather-covered bench, he was suddenly aware that for the first time in a long while, he felt something other than weary self-disgust. She made him curious; she made him concerned; she made him care.
Miss Watson was an unlikely miracle worker. He'd lived with wretchedness so long, even this much emotion felt like spring thaw after endless winter.
Wondering what other unexpected results his impulsive actions might yield, he subsided on the seat opposite and closed his eyes in counterfeit slumber. The coach jerked into movement with a crack of the whip and a shout from Tulliver. They jolted out of the inn yard and into the freezing winter night.
## Two
Horrific images haunted Charis's dreams. An endless replaying of Hubert's fists pounding into her while Felix watched with a gloating smile. The wrenching drag on her arm. The final blow to her head that sent her whirling into oblivion.
When she opened scratchy eyes to the lamplit confines of the shabby coach, she expected to hear the echo of her screams. The only sounds were the creaking of the carriage and the howl of the wind. Sir Gideon sprawled opposite, apparently asleep.
Cool, blessed relief flooded her, and she sucked in a shuddering breath that made her bruised ribs twinge. For the moment, she was safe from both Felix and Hubert.
She was shaking, not far from tears, curled into the corner as if she cowered away from the beating. Her jaw throbbed painfully in time with the vehicle's sway. Her injured arm had stiffened into agony, and she bit back a moan as she folded it against her heaving chest.
Long minutes passed while she fought dizzying pain. But gradually her head cleared, and her breathing steadied. Using her good arm, she tugged the coat around her like a blanket and turned her attention to her companion. His lean body stretched out with an elegant abandon that made her foolish heart race.
To her shame, not with fear.
When the journey started, she'd braced for interrogation. But Sir Gideon had lounged on the bench, spread his arms along the back of the seat, extended his long booted legs into the corner, and closed his eyes. From the look of him, he'd hardly moved since.
Studying him like this felt like illicit intimacy. Although even now, his expression was guarded, closed. A lock of black hair fell across his brow. It should make him look vulnerable. It didn't.
As her gaze roamed his sculpted features, she realized with a shock that he was close to her age. His air of authority had made her assume he was in his thirties. But now, with his eyes shut, he didn't look much past twenty-five. Ashamed of her unseemly curiosity, she stared into the loose folds of the coat over her lap.
"Are we near Portsmouth?" she asked in a croaky voice, looking up.
He opened his eyes and regarded her assessingly. "No. We're not far out of Winchester."
The coach drew to a juddering stop. Charis reached forward to push the blinds aside. They were in a large field. The change from road to turf under the wheels must have disturbed her nightmares.
The grassy area was empty. No lights shone in the distance. They could be a thousand miles from anywhere.
Abruptly what had seemed an acceptable risk in Winchester became a terrifying threat. She was alone and defenseless in an isolated location with three men she didn't know. The hairs on the back of her neck prickled, and her throat closed with rising dread.
How could she be so naïve? How could she be so fatally stupid? She scrabbled wildly for the door catch. Perhaps in the dark, she had a chance of escape.
"What are you doing?" Sir Gideon asked with what sounded like casual interest.
Was Gideon even his real name?
"Getting out," she muttered.
She tensed, waiting for him to grab her, but he only straightened against the worn upholstery. She sucked a shaky breath through her teeth and continued her panicked search for the latch.
"I gave you my word I wouldn't hurt you," he said quietly.
"I know what a man's word is worth."
Ah, at last!
The door banged open, and she tumbled forward. Only to land smack against her kidnapper's accomplice. A choked scream erupted as hard hands closed around her upper arms through the voluminous coat. A scream of pain as much as fear.
"Let me go!" She struggled against his firm hold. Her abused flesh protested the violent movements, but she fought on.
"Your pardon, Miss Watson."
To her astonishment, Akash carefully placed her on her feet and moved back. Behind her, she heard the carriage creak as Sir Gideon emerged into the night. He stood at her side, tall, urbane, his expression quizzical in the bright moonlight.
Tulliver came up, holding a lantern. "What's all this to-do?"
He stared at her as if she'd escaped Bedlam. Her hysteria ebbed on a sick tide, leaving behind humiliated awareness that she'd made a fool of herself.
"Miss Watson was under the impression we brought her here to have our wicked way."
Both the irony in Sir Gideon's voice and the irritated look Tulliver cast her made her more than ever certain she'd jumped to false conclusions. Fading panic bubbled through her blood and left her tottering with reaction. She pulled the thick greatcoat around her shivering body and suddenly realized her rescuer wore only a jacket over his shirt.
"You must be freezing." She plucked at the coat with her good hand.
"No," he said sharply, gesturing for her to stop, although he didn't touch her. Then more calmly, "I don't feel the cold."
"Miss Watson, we've stopped so I can examine your injuries," Akash said.
Her eyes went automatically to Sir Gideon. "You have medical knowledge?"
The carriage lamps glanced a sheen of gold across his glossy hair as he shook his head. "Akash and Tulliver between them make up a fair doctor. And we have supplies. Bandages. Ointments. Laudanum to dull the pain."
"I won't be drugged." On unsteady legs, she retreated until she bumped into the carriage.
Bad as the beating had been, it was Felix's threat to drug her and hand her over for Lord Desaye's rape that had finally made her flee Holcombe Hall. When her ordeal started, she'd considered escape, then decided to cling to the dubious security of life at Holcombe. It was only for a couple of weeks. She could endure whatever her stepbrothers did as long as she had the ultimate promise of freedom. On the road, she'd be at the mercy of anyone she met. Defenseless. Destitute. Helpless.
But when her stepbrothers threatened unspeakable degradation, the dangers of the road had paled in comparison.
How she loathed the Farrells. Her two stepbrothers provided a contrast in menace. Hubert, all bullying brute strength, and Felix, spite and intellect. Whatever damage Hubert inflicted, it was Felix she really feared.
In response to her vehement refusal, Akash shrugged, the movement subtly foreign. "Let me at least see what the damage is. If you'll permit?"
"Be careful. She's hurt her arm," Sir Gideon said urgently.
"My friend, you know you can trust me with her."
Reluctantly, Charis stepped forward. Akash carefully lifted the coat away from her shoulders and laid it inside the coach.
She stood before them in her wreck of a gown. The night was freezing. The needle-sharp wind carried a promise of snow. Her good hand rose shaking to close her bodice while she angled her chin with a pathetic attempt at pride. She was decent. Barely. But she knew she looked dirty and hurt and helpless. With moonlight, the carriage lamps and the lantern, her bruises and abrasions must show with humiliating clarity.
"Please sit down, Miss Watson." Sir Gideon slid a folding stool from the back of the carriage and set it behind her. He also passed her the pug-scented shawl.
She subsided with gratitude—her knees felt like rubber—and draped the shawl over her shoulders. Hesitantly, she extended her arm toward Akash. He frowned as he gently manipulated her wrist. Although his hold was skilled and sure, she winced.
"It's sprained but not broken," he said eventually.
Relief gushed through her. Life over the next three weeks would be tough if she was whole. A broken arm would be a disaster. Thank goodness Hubert's beating had ceased once he knocked her unconscious.
Akash tested her hands, arms, neck, then ran his fingers carefully over her face. His touch was so impersonal, she gradually relaxed and became aware of the activity around her. While Tulliver checked the horses, Gideon collected a leather bag strapped to the back of the carriage. Without speaking, he placed it beside Akash. He turned away and began to lay a fire.
Trying to distract herself from both the cold and the painful examination, she watched the graceful deftness of Gideon's gloved hands as he accomplished the workaday task. The breath caught in her throat when the crackling flames caught and lit his remarkable face to gold, gleaming along smooth cheekbone and angular jaw.
Beautiful. The word whispered through her like a glissando on a harp.
Looking at Sir Gideon made her restless, edgy. She shifted to ease a strange pressure in the pit of her stomach.
"I'm sorry, Miss Watson." Akash raised his hands from her shoulders.
She shook her head. "It's nothing."
She blushed when she realized he'd seen where her attention strayed. Straightening on the rickety stool, she strove to bring her unruly heartbeat under control.
As she looked up into Akash's face, the compassion in his eyes made her cringe. He was a handsome man. The recognition was as dispassionate as if she studied a fine portrait. His handsomeness didn't call to her the way Sir Gideon's did.
Sir Gideon disappeared into moonlit darkness and returned carrying a tin kettle, which he set on the blaze. She'd been so focused on watching him that she hadn't heard the stream bubbling in the distance. Behind her, Tulliver muttered softly as he fussed over the horses.
Once the water heated, Akash used a damp cloth to wash the blood and dirt from her swollen face. Even the lightest touch stung, and she tautened every muscle to stay still. She struggled not to glance at Gideon as she huddled in her shawl.
Eventually, she couldn't help herself. While she silently bore Akash's ministrations, she looked across to where Sir Gideon stood on the far side of the circle of firelight.
His febrile dark eyes were glued to her. Some deep turmoil she didn't understand stirred in his gaze. His gloved hands clenched at his sides. She read anger in his expression, the same anger he'd betrayed when he first saw her battered face. She shivered although she knew the rage was targeted at her abusers and not at her.
He stiffened as he noticed her attention and turned away to fetch more folding stools, which he set up around the fire. She bent her head, knowing her unconcealed interest was unbecoming in a lady.
Akash opened the bag and located a small ceramic pot. When he opened it, a pungent herbal smell filled the air. She jerked back, then made herself sit as he stroked the ointment onto her cheeks. Her face felt like it had been whipped with nettles. She couldn't stifle a gasp of discomfort.
"Damn it, man. You're hurting her!" Gideon's protest was sharp, and he took an urgent step in their direction. "Be careful!"
Akash ignored his protective friend and spoke to Charis. "Where else are you hurt?"
Her ribs ached, and she had grazes on her knees from where she'd fallen in the dark. But her arm and face were by far the worst of it. "Nowhere."
Akash's stare was searching as he replaced the lid on the ointment. "Are you sure?"
"I'm sure." She wanted him to stop. She couldn't bear much more. Already her vision grew hazy as endurance faded.
"I'll wrap your arm to reduce the swelling." Akash opened another jar and smoothed the contents on her arm. It was as smelly as the first ointment, but when it touched her skin, she felt a spreading heat.
Surely this torture must soon be over. The shawl and her flimsy dress offered little protection from the biting wind. She drooped with exhaustion by the time Akash wrapped a bandage around her arm.
Gideon knelt and drew another length of linen from the bag that held their medical supplies. "A sling might be a good idea."
"Yes." Akash rigged the linen around her neck. Immediately, the painful pressure on her arm eased. "Does that feel better?"
"Yes, thank you." She looked up with a shaky smile. "You've been very kind."
He gave another of those exotic shrugs. "My pleasure. I know you're sore and sorry, but I can't find any lasting damage. I'll need to check in the daylight, but from what I can see, your injuries are superficial. You'll be fighting fit in no time."
She was too tired to do much more than whisper another thank-you. Gideon fetched the greatcoat from the carriage and dropped it around her shoulders. As the heavy folds enveloped her, his already familiar scent teased her nostrils. The warmth was immediate and welcome. "Come and sit near the fire."
Already he'd moved out of reach. For a lost moment, she watched him stride away. Then crushing weariness hit, and she stumbled the short distance to the fire, where she collapsed onto a stool. Her frozen extremities tingled as restoring heat slowly seeped through her.
Sir Gideon lifted a heavy wicker basket packed with food from the back of the carriage. To her embarrassment, her belly rumbled. Her stepbrothers had kept her on minimal rations, hoping hunger would sap her resistance.
It was a silent meal. As the four of them sat around the merrily burning little fire, Charis prepared for more questions. Any questions. But her companions seemed astoundingly ready to accept her lies at face value. Guilt settled like a stone in her now-full belly, and she pushed away the pork pie she'd barely nibbled at.
"Are you feeling better?" Sir Gideon asked, noticing her sudden stillness. Of course he noticed. Throughout the meal, he'd studied her across the flames. He sat directly opposite her, with Tulliver and Akash on either side.
"Yes, thank you."
With surprise, she realized it was true. Her face didn't sting so badly, and the pain in her arm was a distant throb rather than fiery agony.
She sipped fine claret from the traveler's cup Sir Gideon passed her. The men had made do with drinking from the bottle. It was oddly intimate to place her lips where Gideon's had once been, however long ago. Almost like a kiss. The thought made her blush even while her lips tingled as though they indeed brushed his.
After supper, Tulliver returned to the horses, and Akash and Sir Gideon cleared up. Charis frowned. Could Gideon really be a man of her own class if he accepted such mundane tasks? He was strangely comfortable with his rough surroundings. Her stepbrothers wouldn't dream of dirtying their hands with rinsing a plate or setting a fire. Servants were there to serve. The landed classes were there to be served.
The relationship between the men was puzzling too. Tulliver seemed on friendly terms with his masters. Akash was surely also an employee, yet he and Sir Gideon treated each other as equals.
Gideon opened the carriage door for her. Again, he didn't assist her inside. The easy, automatic action of a gentleman. Yet he didn't do it. Instead, Akash stepped forward and helped her into the carriage. Wearing the greatcoat loose around her shoulders and with her sling, she couldn't have managed otherwise.
"Miss Watson."
"Thank you, Akash," she murmured, and was hardly aware when he moved away.
Instead, her eyes fastened on Sir Gideon, who waited outside. A cloud covered the moon, and the striking face became a mixture of shadows and light. Still beautiful but sinister.
She shivered. "Who are you?" she whispered, subsiding onto her seat.
"Who are you?" His dark gaze didn't waver from her as he resumed his place opposite, his back to the horses, as a gentleman would.
Charis wrapped the coat around her against the sharp early-morning chill and settled her injured arm more comfortably. "I asked first."
It was a childish response, and she knew he recognized it as such from the twitch of his firm mouth. Like the rest of his face, his mouth was perfect. Sharply cut upper lip indicating character and integrity. A fuller lower lip indicating...
Something stirred and smoldered in her belly as she stared at him in the electric silence. What a time to realize she'd never before been alone with a man who wasn't a relative. The moment seemed dangerous in a way that had nothing to do with her quest to escape Felix and Hubert.
"My name is Gideon Trevithick." He paused as if expecting a response but the name meant nothing to her. "Of Penrhyn in Cornwall."
"Is that a famous house?" Perhaps that explained his watchful reaction.
Another wry smile. "No. That's two questions. My turn."
She stiffened although she should have expected this. And long before now.
"I'm tired." It was true, although a good meal and Akash's skills meant she didn't feel nearly as low as she had.
"It's a long journey to Portsmouth. Surely you can stay awake a few moments to entertain your fellow traveler."
She sighed. Her deceit made her sick with self-loathing. But what could she do? If she told the truth, he'd hand her over to the nearest magistrate.
"I've told you my name and where I live. I've told you the disaster that befell me today. I seek my aunt in Portsmouth." Her uninjured hand fiddled at the sling and betrayed her nervousness. With a shuddering breath, she pressed her palm flat on her lap. "We're chance-met travelers. What else can you need to know?" She knew she sounded churlish, but she hated telling lies.
In the uncertain light, his face was a gorgeous mask. She had no idea if he believed her or not. He paused as if winnowing her answers, then spoke in a somber voice. "I need to know why you're so frightened."
"The footpads..."
He made a slashing gesture with his gloved hand, silencing her. "If you truly had been set upon by thieves, you wouldn't have hidden in the stable. Won't you trust me, Sarah?" His soft request vibrated deep in her bones, and for one yearning moment, she almost told him the truth. Before she remembered what was at stake.
"I...I have trusted you," she said huskily. She swallowed nervously. His use of her Christian name, even a false one, established a new intimacy. It made her lies more heinous.
Disappointment shadowed his face as he sat back against the worn leather. "I can't help you if I don't know what trouble you flee."
"You are helping me." Charis blinked back the mist that appeared in front of her eyes. He deserved better return for his generosity than deceit.
She tried to tell herself he was a man, and, for that reason alone, she couldn't trust him. The insistence rang hollow. Her father had been a good man. Everything told her Sir Gideon Trevithick was a good man too.
She forced a stronger tone. "It's my turn for a question."
He folded his arms across his powerful chest and surveyed her from under lowered black brows. "Ask away."
It frightened her how much she yearned to know about him. Curiosity raged like a fever. But to her utter mortification, the first question that emerged was, "Are you married?"
His laugh held a harsh edge. "Good God, no."
Shock at his emphatic answer overwhelmed her embarrassment. "You make it sound an impossibility."
"Believe me, it is." He looked out the window at the dark landscape.
Helpless to resist, she stared at his profile, perfect as a cameo or a face on a coin. Thick dark hair sprang back from a high forehead. The straight, commanding nose. The proud chin and angular jaw. His physical splendor struck her like a blow.
He turned and caught her studying him. Her color mounted higher. Thank goodness the dim light and her bruises hid her blush.
For a long moment, she stared into turbulent dark eyes. He was in turmoil, and she wasn't vain enough to imagine she was the cause. No, her little drama briefly intersected with his life and would just as quickly veer away. She stifled the pang of senseless regret that knowledge aroused.
The thick dark eyelashes that veiled his eyes were the only remotely feminine feature on his face. Yes, he was beautiful, but he was also uncompromisingly male.
"My turn. Where are your parents?"
"Dead," she said starkly before she thought to lie.
"I'm sorry."
She looked down at where her good hand clenched in her lap. "My father died when I was sixteen. My mother died three years ago."
"How old are you now?" She was grateful he didn't pursue the subject. After all this time, it still hurt to talk about her parents.
"Twenty. Almost twenty-one." Just saying the words reminded her that on the first day of March, she reached her majority. And safety. If she stayed free for the next three weeks, her stepbrothers couldn't touch her. Or her fortune. "That's two questions."
The conversation was odd, prickly. Like a dangerous game. "You can have two now."
"Tulliver calls you Sir Gideon. Were you knighted by the King?"
"Yes."
She waited for him to elaborate, perhaps boast of whatever feats brought about his elevation. But he remained silent.
"So it's not an old title?"
"That too. I'm a baronet for my sins. Although I wasn't expected to inherit."
"Penrhyn is the family seat?"
"Yes."
"Why aren't you there now?"
"I was in London." He paused. "My turn well and truly. Carlisle to Portsmouth is a long journey. Especially for a woman on her own. What prompted it?"
"My circumstances changed." That at least was the truth.
"So your aunt expects you?"
"Aunt...Aunt Mary desires a companion. She's...she's a rich spinster." Close enough to the truth about her real great-aunt in Bath except her name was Georgiana. How Charis wished she could apply to that wonderful woman for help now. But her great-aunt, for all her fortune, was powerless against the law and the Farrells' bullying.
"Miss Mary Watson of Portsmouth." Did she hear skepticism in his deep voice, rich as vintage wine?
"Yes, that's right."
"So you can direct us to her house."
Oh, Lord, no. She should have thought of that complication. She'd chosen Portsmouth as her destination because she imagined there she'd be part of a transient population, as unremarkable as a grain of sand in a gale. But she'd never visited the town, knew nothing about it.
"Of course." She spoke hurriedly, before he quizzed her further on her mythical aunt. "Why were you in London?"
Did she mistake the haunted look that darkened his eyes? "Cornwall is isolated, especially in winter."
Except he was tanned. His answers puzzled her. He mightn't be lying like she was, but he wasn't completely honest. "Does Akash work for you?"
He gave a surprised laugh. It was the first time she'd heard him laugh properly. His face lit with amusement, and her heart crashed to a trembling stop in her chest. He was the most breathtakingly attractive man she'd ever seen.
"Of course not. He's my friend."
"But..." She stopped for fear of causing offense.
"You shouldn't make simple judgments, Miss Watson." He reached into the pocket of his jacket and withdrew a flat silver flask. She waited for him to drink, but he held it out to her. "It's brandy."
"I don't take strong spirits."
"It will help you sleep and dull your aches."
"Akash's treatment did that."
"Once you've been on the road a few hours, his magic will wear thin." Sir Gideon's voice lowered to velvet persuasion. "Drink it, Sarah. I promise it won't hurt."
She found herself reaching out, taking the flask and drinking. All under the power of fathomless dark eyes. As the liquor hit her throat, she coughed. Her bruised ribs protested the abrupt pressure even as comforting warmth spread through her veins.
She passed back the flask. Her brief vitality faded. Exhaustion weighted her aching limbs. Her swollen jaw protested as she fought back a yawn.
She wouldn't sleep. She didn't trust her companions enough to fall into unconsciousness. And she needed to be alert to seize her chance to escape.
She wouldn't sleep. She wouldn't...
The carriage rolled into Portsmouth the next morning. Gideon had dozed in snatches. That was all he managed these days, whether in a speeding coach or the most luxurious feather bed. Sometimes he thought he'd sell his soul for an uninterrupted night's sleep. Other days he recognized he didn't have a soul to sell.
At least his fear of closed-in spaces wasn't as overwhelming as it had been when he first left India. His confinement in this coach had been uncomfortable, but he'd managed, thank God.
From the bench opposite, Akash studied him in silence. It had started to snow before dawn, and his friend had sought refuge in the vehicle. They'd suggested to Tulliver that they stop at a wayside inn. But Tulliver had proven as immune to English cold as he had to blistering heat on the boat back from India.
Gideon's eyes alighted on the slumbering bundle at Akash's side. Sarah lay curled in the corner, pressed against the upholstery as if even asleep she remained wary.
Gideon's belly knotted with coruscating anger at whoever had hurt her. The craven deserved to rot in hell.
He slid back the blind and caught his first glimpse of Miss Sarah Watson in daylight. The bruising on her face was worse this morning, for all Akash's arcane skills. Her hair was a rat's nest. One scratched hand clutched his thick greatcoat around her, hiding the slender curves he recalled with such unwelcome clarity from last night. The other dangled loosely against her breast, suspended from Akash's makeshift sling.
"Shall I wake her?" Akash murmured.
Gideon nodded. Gently, Akash touched her hand where it clenched in the coat's thick black wool. Not for the first time, Gideon envied his friend the ease of contact.
He remained still, watching the girl stir. Her eyes—a cloudy hazel in the bright light reflected off the snow outside—opened and slowly focused on him. With accusation.
"You drugged me." Her voice was slurred. With sleep or her swollen face. Or the opium.
"You needed rest. It was only a drop of laudanum." More than that. But he'd had no idea how else to grant her the blessing of rest.
"Don't do it again," she spat out, sounding more alert by the second. Her remarkable eyes cleared to a deep green, flecked with scattered gold like broken sunlight. Her eyes were the only trace of beauty in her battered face.
He bent his head in acknowledgment. "I won't." He paused. "How do you feel?"
Her lips quirked, then she winced as the smile tested her torn lip. Nonetheless, her voice held a trace of dry humor. "Like a mule has kicked me. A big angry mule."
She confronted her fate with her head held high. No whining or cowering. Her spirit took his breath away. Made him want more than he had a right to ask.
As she'd said, they were chance-met strangers. Useless to rage against inevitable fate. She was not for him. She could never be for him. No woman could be.
He'd faced that damnable truth months ago.
He hoped she didn't hear the betraying roughness in his voice when he forced a dry reply. "You're feeling much better, then?"
She gave a choked giggle at his attempt at a joke and raised one hand to her bruised cheek. "It hurts to laugh."
"I'm sure it does." Only the bravest woman would laugh in such circumstances.
"Where does your aunt live, Miss Watson?" Akash asked.
His friend had cast him a searching look and now concentrated on the girl. Heat crawled up the back of Gideon's neck as he realized Akash must guess his admiration for Miss Watson. And Akash would pity him, which stung Gideon's pride like acid.
The lilt faded from the girl's voice, and she sounded stiff as she always did when she lied. "Not far. If you drop me in the center of town, I can find my own way. I've imposed enough."
Gideon's lips crooked in grim amusement as she avoided his eyes. "We cannot abandon a lady to her own devices."
She looked down to where her uninjured hand fisted in her lap. Her discomfort was palpable. "My...my aunt is a maiden lady of reclusive habits. It would frighten her if I arrived on her doorstep in the company of three unknown gentlemen."
"And she'd be perfectly undisturbed to see you arrive hurt, ragged, and alone?"
She cast him a resentful glance under her thick gold-tipped lashes. "When I explain, she'll understand."
The carriage pulled, as arranged during the night, into Portsmouth's best inn. The girl's hands tightened until the knuckles shone white. "Where are we?"
"We're changing horses and stopping for breakfast. After that, Akash and I will escort you to your aunt's."
"No."
"No to breakfast or no to our company?"
She had the grace to look a little shamefaced at her bald reply. "I must admit breakfast appeals."
He guessed she meant to take advantage of one last meal before escaping. It was what he'd do if destitute and in danger. "Breakfast it is," he said neutrally.
The carriage stopped. Akash turned to her. "I'll carry you in."
The girl's eyes darted to meet Gideon's. He had the oddest feeling she wanted him to volunteer. He was such a poor specimen that even this simple service was beyond him. Clenching his hands, he told himself he'd long ago come to terms with bleak reality. Today, consigning this wonderful girl to someone else's arms, that sounded more than ever like a hollow lie.
"Thank you, but I can walk."
"Your injuries will attract less attention if I carry you." Akash said, closely watching the interplay between them.
"It will be better this way, Miss Watson," Gideon said.
Fleeting disappointment shadowed her features. Strange how even with her injuries, her face was so expressive. She raised her chin as if girding herself to face a challenge.
"As you will," she said quietly.
Akash carried Charis up the stairs with an impersonal aplomb that saved her any embarrassment. She couldn't imagine lying in Sir Gideon's arms with quite this coolness. The thought of Gideon holding her close to his broad chest brought a blush to her cheeks, and she bent her head to hide the flood of color.
What was this strange attraction she felt toward Sir Gideon? His physical presence filled her mind in a way she'd never before experienced.
It was astonishing how he occupied her attention, attention that should be devoted purely to escape and her safety over the next three weeks. From the first moment she'd seen him, he'd become the lodestone for every thought, every feeling. With each moment that passed, her obsession grew. Was it just because he'd rescued her from discovery and disaster? Or was this turbulent feeling something else entirely?
Thank goodness her reckless heart had settled by the time Akash placed her on her feet in the large private room Sir Gideon had requested upon their arrival. Then her pulse set off on its wild jig again as the object of her ridiculous fantasy strode in behind them. She fought to suppress her surprising, unwelcome reaction, but nothing stemmed her tingling awareness of him as he crossed toward the fire.
Once they'd sent Tulliver to order a substantial breakfast, Akash turned to Charis with what she already recognized as his characteristic seriousness. "Can I see your injuries, Miss Watson? There was only so much I could do in the dark."
"Thank you. You're very kind." In truth, Charis felt much improved apart from a bitter taste lingering from the much-resented laudanum. The room's warmth thawed some of the stiffness from her muscles.
Sir Gideon lounged on a carved wooden settle near the blazing grate. His dark eyes rested upon her with fierce concentration as she rose from her chair. On trembling legs, she moved to the center of the room, where Akash waited.
She unwrapped the thick shawl from her head and pulled the coat from her shoulders. She dropped both to the floor. It was absurd, but she felt as though she undressed for Sir Gideon's pleasure. The wanton notion came from nowhere. It shocked her, but she couldn't dismiss it.
Sir Gideon's unwavering regard looked like desire. Which made no sense when she knew she was a veritable monster. But heat prickled her skin, and she licked suddenly dry lips.
His eyes flickered at the movement.
Her heart slammed against her ribs. Something about Gideon's unblinking stare pierced her to the bone. It was like he read her soul.
She shifted under Akash's hands.
"Did I hurt you?" he asked, frowning.
"No," she murmured.
Akash's medical skills must be why he assumed responsibility for her care. Whatever he'd put on her bruises last night had certainly helped. She was sore but nothing like yesterday.
Strange. This handsome, considerate gentleman touched her yet it meant nothing. Sir Gideon was halfway across the room, and he owned her every breath.
How had this happened? Her head whirled as she tried to make sense of her unprecedented reactions. In ballrooms and salons, she'd met so many men, attractive men, sophisticated men, attentive men. None had affected her like this taciturn, black-haired Adonis, with his glittering eyes and troubled air. The sensations stirring in her veins scared the life out of her.
As she answered Akash's questions about her injuries, her gaze dropped to where Sir Gideon's gloved hands encircled an untouched tankard of ale. Wicked excitement shivered through her as she imagined those hands touching her. So far, he hadn't so much as taken her arm.
Avidly, she drank in his features. His face was grave and pure like the stone effigy of a crusading knight. His cheekbones and jaw were cut at perfect angles. His mouth was stern, firm, beautiful, but with a hint of softness in the curve of the lower lip. He looked like a carved saint until one met his burning eyes.
No sanctity there.
They were so dark, almost black. Intense. Glittering. Full of suppressed passion and pain.
And anger.
Because someone had dared to hurt her.
Warmth seeped into a heart that had been cold for so long. She couldn't entrust herself to these men. Too much hung upon her keeping her identity secret. She still had to escape.
But knowing that such a remarkable being as Sir Gideon Trevithick placed himself so firmly on her side bolstered courage that came shamefully close to faltering.
Gideon's eyes met hers and flashed a warning. He surged to his feet and strode over to gaze out the window.
Helplessly, Charis studied his straight back in its perfectly fitted black jacket. He hadn't had to say the words aloud. That last glare from his brilliant eyes had all but shouted keep out.
Akash manipulated her wrist. Its tenderness was a mere echo of last night's agony. Even her ribs didn't feel as though an elephant had trampled them anymore. She had a sudden memory of the dark stall where Gideon had found her. Without his help to escape and Akash's treatment for her injuries, she'd be in a bad way indeed.
The instincts that insisted Sir Gideon was her dauntless champion urged her to tell him everything and throw herself upon his mercy.
No, he was a stranger. She couldn't risk the consequences of ill-advised confidences. If Sir Gideon handed her over to the law as duty demanded, her stepbrothers would have her back in their custody as soon as they rode to Portsmouth.
Or worse, perhaps Gideon and Akash would be as blinded by her gold as every other suitor. Her heart screamed that these were good men. Experience urged caution. Even good men abandoned principle when they learned of her massive fortune.
Safer by far to rely on her own resources, meager as they were. Still, she couldn't suppress a pang of guilt at how she deceived and used people who tried to aid her. Her experiences with her stepbrothers should make it impossible to place herself willingly into any man's care. But still her heart insisted she made a huge mistake when she rejected Sir Gideon's help.
"Thank you for everything you've both done," she said softly, knowing it was sinfully inadequate when measured against her lies.
"You're welcome." Akash bound her arm, then left the sling off.
She bent to pick up her shawl and stumbled to her chair. Standing so long had tested her strength. Across the room, Gideon didn't say a word, just watched the snow drift past the window. She told herself she had no right to feel slighted by his indifference.
The arrival of breakfast interrupted her dour thoughts. Charis kept her head down and shrouded in the shawl. She couldn't help her ill-matched costume, but if the servants saw her hair and bruised face, they'd identify her immediately if her stepbrothers asked about her.
Feverishly, she tried to plan her escape even while Sir Gideon's nearness was a persistent tug on her senses. The bad weather was both savior and pest. If she could get away, it would hide her. But she wasn't dressed for such cold. She resigned herself to stealing the greatcoat. It was a loan rather than a theft, she assured her howling conscience. In a few weeks, she'd return it and repay Sir Gideon for his kindness.
Surely tracing Sir Gideon Trevithick of Penrhyn in Cornwall wouldn't be difficult. If they made contact again...
She put a brake on foolish dreams.
First she had to survive the next three weeks and stay out of her stepbrothers' clutches. She had to find shelter and food and some way of supporting herself, all without revealing her identity. Or the identity of the powerful men who sought her. Hubert was Lord Burkett and Felix was a rising figure in Parliament.
Gideon, Akash, and she settled down to another silent meal. Tulliver must have retreated to the taproom. Charis was grateful for the lack of conversation. She'd choke on any more lies. And she had a foolish desire to cry at the thought of leaving Sir Gideon. How had he gained this astounding power over her emotions in such a short time? It was like a strange madness possessed her.
After the servants cleared the plates, she managed to inject an appropriate note of feminine embarrassment into her voice. "Would it be all right if I had a few moments of privacy?"
A look passed between Gideon and Akash but both stood readily enough. "We'll send someone to assist you," Gideon said.
"No need," Charis said hurriedly, her chance at escape evaporating before her eyes.
"I insist." Gideon, curse him, waited while Akash left to summon the servants.
A parade of maids brought hot water and towels and a range of grooming articles. She couldn't help sighing with pleasure when the last item laid out before her was a cheap brown cotton gown. She was desperate to change her ragged, dirty dress.
Goodness knows where Sir Gideon found the frock at such short notice. Yet another sign of his thoughtfulness. Again, she suppressed that rebel urge to confess everything and beg him to help her. Men changed when they saw the chance of filling their pockets with gold.
Gideon stood by the door and dismissed the staff. "Tulliver's outside if you need anything."
"Thank you." How she wished she could say more, say good-bye, express her gratitude, tell him she wished she could know him better.
But it was impossible.
For a long moment, she stared at him, drinking in his physical magnificence, the strength and intelligence in his compelling features. Already she knew she'd never forget him. She turned away and pretended interest in the items on the tray. If she kept looking at Gideon, she'd start to cry.
The door closed softly. At last Charis was alone. She let out the breath she'd been holding. Yet she didn't immediately put her plan into action. Instead she slowly approached the cheval mirror in the corner.
Ridiculous, really, given her legion of troubles, that the mere act of checking her reflection needed every ounce of courage.
She braced to confront the woman in the mirror. When she did, she couldn't stem a broken peal of laughter.
Had she read desire in Sir Gideon's eyes? What a vain, deluded fool she was. No man could look at her now with anything but pity. Or revulsion.
She'd expected to be shocked. What she saw was worse than her wildest imaginings. Her face was a mottled mixture of purple and yellow. Her jaw was grotesquely distorted. Above the bruising, familiar hazel eyes stared back with a dazed expression.
She bit down hard on her quivering lip, but the jab of pain couldn't dam her tears. She was a monstrosity, a hobgoblin, a gorgon. So stupid to mourn what would mend, but she had to lift her good hand to dash the moisture from her streaming eyes. Akash had assured her the damage was superficial, but the words seemed meaningless when she looked at the woman in the mirror.
The once-elegant blue dress was streaked with dirt and torn beyond repair. Her shaking hand shifted to touch the matted hair that tumbled around her shoulders.
She drew in a breath that was close to a sob and met her watery gaze in the glass. This wouldn't do. She straightened her spine. She was Lady Charis Weston, the last of a long line of warriors. No daughter of Hugh Davenport Weston would admit defeat to a pair of poltroons like Hubert and Felix.
The horrors she saw in the mirror would pass. Right now, she needed to concentrate on escape.
Hurriedly, she washed and changed out of the ruined gown. The cheap dress was scratchy on her sensitive skin and too big, but at least it was clean and whole. Fastening the frock took too long, and she panted with pain before she finished.
She spent valuable minutes struggling with the knots in her hair. Eventually, she managed to bundle it away from her face. The girl in the mirror started to look moderately respectable. As long as nobody noticed her bruised face.
With shaking hands, she drew the greatcoat on. Her sore arm twinged as she gingerly slid it into the sleeve, but, thanks to Akash, the pain was bearable. The huge coat looked absurd on her small body, but she couldn't manage without its warmth.
She patted the pocket to check for the pistol. Once she'd found somewhere safe to stay, she'd pawn it. She told herself taking it wasn't theft. When she could, she'd redeem the weapon and return it. She'd already steeled herself to pawning her mother's ring and locket although her heart ached at the prospect.
How long had she been in here? Were Gideon or Akash likely to return and demand to know what she was up to? She mustn't linger. Dressing had taken too long already.
Her mouth was dry with nerves as she darted to the window. Beneath the sill, she knew a flat roof extended over the rear yard. Climbing about in the snow with a sprained wrist was risky. But less so than waiting for her stepbrothers to find her, or for her rescuers to discover her identity and hand her over to the local magistrate.
Carefully, she raised the sash window and eased herself out. Her bruised ribs protested, but she gritted her teeth and continued. Any pain now would be as nothing if her stepbrothers caught her.
Three weeks to freedom, she promised herself grimly.
Stifling the alluring memory of black eyes burning into hers, she found her footing on the slippery roof.
## Three
Guvnor, we got trouble."
Gideon looked up from the dregs of his ale to meet Tulliver's worried eyes. A shock to see him anything but imperturbable.
"What is it, Tulliver?" He set his tankard on the table. He sat in the darkest corner of the inn. And the coldest. The benches around him were empty. On this frigid day, the occupants of the long room crowded around the fire blazing at the other end. But even so, all these people sharing his space, his air, left him jumpy, on edge.
Of course, he knew what Tulliver would say before the man spoke.
"The lass. She's gone."
Tulliver had been on watch outside the room. Gideon didn't need to ask whether she'd got out that way. "How in Hades did she go across the roof? She's got a sprained wrist."
"Aye. But it didn't stop her." Tulliver's voice held a trace of grudging admiration.
"Damn." Gideon surged to his feet and strode toward the taproom's rear door.
Stupid, stupid girl. Didn't she realize the risks? But he reserved his sharpest castigation for himself. Careless bastard he was. How could he let her escape? It wasn't as if he hadn't guessed her plans. Although given her injuries, he'd never imagined she'd clamber out an upstairs window and make it across an icy roof.
"How long ago?" he grated out.
Tulliver kept up with his rapid pace. "Seconds, I reckon. The room wasn't cold enough for the window to be open long."
"She could be anywhere." He ducked under the low lintel and entered a long, flagstoned corridor. "Damn," he said again with more emphasis.
"Damn what?" Akash emerged from a side hallway.
"Miss Watson's gone," Gideon said sharply.
Akash grabbed his arm. Immediately Gideon stiffened, and Akash snatched his hand away with a gesture of apology. But his eyes didn't waver in the gloom. The stare was calm, perceptive, compassionate.
"She can't give you back what you've lost. No one can."
Gideon flinched as if he'd been struck. Had anyone else but Akash said it, they'd be nursing a broken jaw.
"You think I don't know that?" he asked through tight lips.
"Then let her follow her own destiny."
He owed this man so much. His health. His sanity. His very life. But now he had no time to explain what he barely understood himself. "If I help her, it might wash some of the black from my soul."
"She's a stranger."
"She's in trouble. We have to find her."
For a moment they couldn't afford to waste, Akash studied him. Finally, he gave an abrupt nod. "She has an aunt in town?"
"A lie. She's on the run from someone or something. My guess is she means to take her chances on the streets."
"She's a lady. She won't survive."
"She will if we find her." The idea of the girl's pride and courage coming to disaster made Gideon's gut cramp. Without another word, he set off down the hallway toward the back door.
They emerged into a bleak snow-covered yard behind the kitchens. The freezing wind smelled of thousands of coal fires and salt from the sea. Directly above was the room the girl had escaped. The day was gray and grim, but there was plenty of light to show a line of small footprints leading to the back gate.
Thank God it had stopped snowing, although it was perishing cold. Gideon hoped Sarah had had the sense to take his greatcoat. He shoved his gloved hands into his jacket pockets and set out along the trail. Akash and Tulliver were a reassuring presence behind him.
The high wooden gate led into a dingy alley sheltered from the weather by brick walls. No more footprints. It didn't matter. One end of the alley ended in a blank wall. She could only have taken the other direction, toward the busy street that passed the front of the inn.
Cursing, Gideon set out at a run and burst onto the packed thoroughfare. Even on a bitter day, Portsmouth thronged with people. Sailors of many nations. Respectable burghers. Militia in their bright scarlet uniforms. Roughly dressed farmworkers from the surrounding countryside.
But no slight bright-haired girl weaved her way through the pulsing, noisy crowd. Gideon scanned the street while dread beat a remorseless tattoo in his heart. She was small and too easy to miss.
She was small and too easy to hurt.
"Do you see her?" Akash asked beside him.
"No. But she can't have gone far. Tulliver only just missed her. Those footprints are fresh. And she doesn't know the town. We'll split up and meet back here in half an hour." Without waiting for a reply, Gideon launched himself down the street.
Leaden fear settled in his belly as he realized he headed toward the docks. For all his burning need to find Sarah, he hoped to hell she'd chosen a different route. Portsmouth was a navy town and full of press-ganged sailors, brutish men not far removed from criminals. Every step closer to the waterfront was a step closer to peril.
The press of people chafed, but compared to those overwhelming weeks in London, it was bearable. He forced himself to breathe deeply, evenly, concentrating on each inhalation and exhalation. He could control his discomfort in a crowd. He couldn't quell the tension that tightened his shoulders as his fear for Sarah rose. At least he'd given her the pistol although heaven knew if she had the spirit to use it.
He recalled her reckless courage. She'd use it, all right. He just prayed he found her before she needed to.
Devil take her, why hadn't the chit trusted him?
He tried desperately not to think what might happen to her. She'd already suffered so much. He'd promised her help, and he'd failed miserably.
He'd failed so often. Damn it, he wasn't going to fail this time, not when the girl's life was at stake.
Swiftly but purposefully, he moved down the street, checking doorways and side passages. He doubted she'd go into one of the shops lining the road, crowded as they were with people avoiding the weather. She'd be too conspicuous, with her bruised face and bandaged wrist.
Dear Lord, keep her safe until I get to her.
He repeated the silent plea with every thud of his heart until the words lost meaning, and all he knew was his overpowering need to find her. Still he searched. Every nook, every recess, every corner. By God, he wouldn't let her escape him.
He nearly missed her.
A group of rowdy men crowded into a narrow alley. Sailors by the look of them, with their dirty calico smocks. Drunk, seeking trouble.
Something about their concentrated menace alerted instincts honed in a thousand dusty Indian byways. Then one of the roughly dressed men shifted, and Gideon glimpsed a familiar black greatcoat.
Sarah.
Seeing her trapped, he yielded to a deep, gut-churning anger. The will to kill coiled in his belly like a cobra. With a low growl, he reached into his pocket for his pistol, twin to the one he'd given her.
As his fingers curled securely around the handle, he strode up behind the bastards. None of them noticed his approach although he made no attempt at subterfuge. They were too focused on their terrified bounty.
Shaking and trying to stifle panic, Charis backed into the damp stonework. Her good hand fumbled for her gun in the coat's generous pockets. The four burly men stank of liquor, rotten fish, and pungent male sweat. She sucked in a shuddering breath, then gagged on the foul stench.
Why hadn't she listened to that persistent voice insisting she trust Sir Gideon? Now it was too late. She was a woman alone, fair game for any stranger.
The largest man ripped the shawl off her head and flung it into the sludge on the ground. As she choked back a futile protest, her insecurely fastened hair collapsed around her face.
"Eh, lookee, Jack! She got lady's hair," one of the men cried in delight.
"All the better to hold her with, shipmates." The big man twined one meaty paw in a tangled hank while he ripped at his coarse trousers with his other hand. The tang of male excitement was ripe on the cold air and made Charis's muscles knot with revulsion.
When she strained to break free, agony shot through her scalp. Bile rose as she read unmistakable intent in her captor's sunken, bloodshot eyes.
"She's been fair knocked around," another of the sailors said doubtfully.
"I ain't bothered with her sodding face," the man snarled. "I reckon the bits I want are in fine working order." He laughed salaciously. He was close enough for the alcohol on his breath to make her recoil.
"Leave me alone." Her voice sounded raw.
"You don't mean that, hinny." His croon was more frightening than anger. Her stomach roiled with icy terror.
"Have at her, Jack," one of the men urged in a guttural voice.
Frantically, she fought for a grip on the little gun but it kept sliding out of reach. She stretched after it, but the slightest movement ripped unbearably at her trapped hair.
"I'll scream if you touch me." Her voice cracked.
The man's leering grin reeked confidence. His brutal hold tightened until hot tears rose to sting her eyes. "You'd have hollered afore now if you reckoned it'd do you a mite of good."
On the street, she'd hesitated one fatal instant before calling for help. Time enough for them to crowd her into this alley, stinking of urine and rotting refuse.
Charis opened her mouth to scream but only a whimper emerged when the man wrenched at her hair. "Shut your gob, bitch."
"Let me go," she croaked, still scrabbling for the gun, but her trembling, damp hand couldn't find purchase on the pearl handle. Her heart pounded so furiously against her ribs, she thought it must burst.
"I'll let you go, all right." The beefy sailor smacked his thick lips together as if contemplating a hearty meal. "Once I've got my fill. And if you cut the ruddy backchat. Otherwise, I'll wring your neck, my bonny."
Desolation froze the blood in Charis's veins. Death was a cold, tangible presence. There was no hope. All her struggles, all her suffering, all her defiance led to this. Lady Charis Weston violated and murdered in a port city's backstreet.
"Get away from her."
Like a honed saber, the command sliced through Charis's blind horror. Sir Gideon is here. I'm safe. I'm safe.
Her galloping pulse slowed to a joyous hymn of gratitude. She dragged in her first unfettered breath since she'd escaped the inn, then gasped as her bruised ribs protested. Abruptly, she became aware of aches lingering from yesterday's beating. Her sprained arm throbbed painfully.
The ringleader relinquished his grip on her hair. The burning pressure on her scalp eased. She slumped against the wall as a dizzying wave of relief washed over her.
He stepped to one side to face the man at the mouth of the alley. Charis at last got a clear view of Gideon. She shivered as she stared into that perfect, ruthless face. Fury blazed in his eyes. He looked strong, brave, in control. Lethal.
"Move along, chum." The sailor folded his arms across his bulging chest. He was much broader than Gideon and stocky with muscle. The blackguard's cohorts set up a solid barrier around him.
"Leave her be." As Gideon approached, he sounded completely undaunted by the array of masculine strength. His voice was colder than the wind whistling through the alley.
The ringleader gave a contemptuous grunt of laughter. "Who's going to make me, pretty boy? You?"
Gideon raised a completely steady hand. Clear wintry light gleamed on the polished barrel of his pistol.
"Aye, very nice." The ringleader cast a derisive glance at the gun even as his cronies sidled out of the way. "You forget there's four of us."
"If I kill you, I imagine your friends will lose their thirst for blood." He sounded careless, unafraid. Charis's heart leaped at his reckless bravery. "Make no mistake, if you don't let the lady go, I will shoot."
Her paralysis faded as she sucked reviving air into her lungs. At last her fingers closed firmly on her gun.
"Not if I get a chance first," she said hoarsely. She brought her weapon up. The gun was perfectly balanced and sat in her hand like an extension of her arm. "Step aside."
"Shit, where did that come from?" one of the sailors muttered, backing farther off.
"Is the girl worth the risk?" Gideon asked almost casually, keeping his gun raised.
For one horrible instant, Charis glanced between the ringleader and Gideon. The sailor's expression warred between bravado and self-preservation, and his Adam's apple moved up and down in his thick throat. Gideon's shoulders tensed, and his jaw firmed with purpose. His aim didn't waver. She couldn't doubt he'd shoot if he had to.
The brute must have reached the same conclusion. His piglike eyes flickered, and the tension drained from his heavy body. "Oh, bugger it, take the slut and welcome you are to her. Her slice isn't worth a friggin' bullet."
"Sarah, come here." Through the buzzing in her ears, she heard the ice in Gideon's voice. "You're safe now."
Her gun suddenly seemed heavier than stone. Her hand wobbled as she lowered it. On legs that felt no firmer than jelly, she stumbled up the alley to stand beside Gideon. She desperately wanted to reach out and touch him, but his powerful self-containment kept her hands by her sides.
"We're going to walk out of this alley and go our way unmolested." Gideon didn't glance at her. His pistol remained pointed squarely at the leader's chest.
The effortless tone of authority took effect. Not one of the ruffians shifted to stop them as she and Gideon backed off. The few yards to safety felt like a thousand miles. Charis's heart lodged in her throat, and her skin tightened with every step. Could they really emerge unscathed?
They'd almost made it, had turned to face the street when Charis heard an angry shout behind them. "Hell's bells, mates! There's four of us and only bleedin' one of him. Let's give the bastard what for!"
A crash of booted feet behind them.
"Run!" Gideon shouted. "I've got the gun. I'll be all right!"
Charis lifted handfuls of greatcoat and sprang into a wild dash. She ignored the way her body screamed agony at the sudden dash.
But they'd left their escape too late. The thugs surrounded them at the mouth of the alley. Charis came to a juddering stop, her heart jamming in her throat.
"Stay behind me," Gideon snapped, stepping between her and the closing circle of brawny sailors. The rough, flushed faces promised retribution, violence and pain.
Shaking, she pressed against the wall. Her blood pounded so loudly, she hardly heard the bustle from the crowded street so close.
"You're making a mistake." Gideon sounded as if the men posed no threat at all. He still held the pistol, but she guessed he was reluctant to shoot in case he hit someone in the street.
"No mistake, my hearty." The ringleader's swaggering confidence returned. "We'll take our fun with you, then it's the wench's turn."
"I think not." Although she couldn't see his face, she knew he smiled.
She opened her mouth and shrieked as loudly as she could. The shrill sound bounced between the narrow walls.
"Gideon."
She strained upward to see. Akash loomed at the entrance to the alley. Next to him, Tulliver. Thank God. They must have been close enough to hear her.
The sailors dived at the newcomers. The world exploded into a fury of hard fists and boots and grunts of pain.
The violence transported her to the horrific afternoon when Hubert had hit her. She ducked her head and cowered against the clammy bricks. Black edged her vision as the battle raged around her. Trembling, she clutched her sprained wrist to her chest and prayed for the nightmare to pass. She squeezed her eyes shut and fought a powerful urge to vomit.
Bodies hurtled close, then lurched away in the fight's chaotic dance. Gideon brushed against her. She recognized his scent before she opened her eyes and saw him swing into the fray again.
The shouting crescendoed, became more confused. The brawl spilled out into the street. At a distance, she heard someone yell for the town watch.
"Miss Watson, let me get you out of here." The calm voice emerged from the pandemonium.
With dazed eyes, she turned to look into Akash's face. He was disheveled but had suffered no obvious injuries.
Disappointment scourged her that it wasn't Sir Gideon. She blinked to dispel the foolish reaction and managed a brief nod. Akash took her uninjured arm and, shielding her with his body, drew her onto the street.
The scene was a wild melee. Difficult even to spot the original combatants in the milling crowd.
"Sir Gideon?" she gasped, digging her fingers into Akash's sleeve.
He glanced down quickly with a smile so carefree, it astonished her. "He's fine. Never better."
She scanned the heaving mob and spotted Gideon. With his height, he was hard to miss. He swung punches with an abandon that left his opponents staggering. His face was brilliant with elation, a dazzling exhilaration she'd never seen in him before.
She staggered to an astonished halt.
He'd shown her nothing but gentleness. Yet the man she watched now took a savage delight in violence. She desperately wanted to despise him—she'd always loathed brute force, even before Hubert's assault. But looking at Gideon, she couldn't help but respond to the display of unfettered male power. He moved with a smooth beauty that was almost mechanical, like a perfectly calibrated engine doing what it was designed for.
Her breath caught in her throat at how glorious he was. Every drop of moisture dried from her mouth, and her blood ran hot in her veins.
This new Gideon frightened her. But she couldn't deny he thrilled her too.
The brief awareness shattered as Akash lunged forward to deflect someone who grabbed at her. For a horrified second, she stared into the reddened eyes of one of the sailors. The man faltered under Akash's blow and fell cursing.
"Miss Watson, don't just stand there," Akash snapped, and wrenched her through the heaving crowd.
She stumbled and just avoided another blow aimed at her head. She couldn't see Tulliver in the throng. Pray God he was all right. To her left, Sir Gideon dispatched with casual competence anyone who dared approach.
A man snatched at her injured wrist. She choked back a scream. Pain shot a red-hot blast up her arm. She screamed again as Akash struck her assailant down without a moment's compunction, his aquiline face severe and expressionless.
Akash turned back to her and spoke almost roughly. "Are you all right?"
"Yes," she said, although her wrist burned with fiery agony. She pressed it to her breast and let Akash tug her toward the edge of the crowd. He dragged her into a deeply recessed doorway, where the cacophony marginally diminished.
"Are you sure you're unharmed?" He was breathing heavily as he gave her a hard look.
"Yes." She stared out at the street with utter dismay. "This is all my fault."
Akash's silence signaled agreement. The doorway was wide enough to accommodate both of them without touching. He released her and leaned against the stone doorframe, studying her with unfathomable brown eyes.
She frowned in confusion. "Don't you want to help Sir Gideon?"
Akash shook his dark head. "He'd prefer I kept an eye on you."
Her bruised ribs twinged as she made a convulsive movement toward the street. "He could be killed."
A smile curved Akash's mouth. "The man who can kill Gideon Trevithick hasn't been born. Don't worry, Miss Watson. He'll live to chastise you for your rashness in running away."
Something in his easy confidence settled her choking panic. "I had to go," she said sullenly, guilt twisting her belly.
"Utter rot," Akash said amiably. He darted a look outside. "Ah. At last. The watch has arrived. Peace will soon descend upon the streets of Portsmouth."
It took surprisingly little effort to quell the brawl. Most participants just melted away into the alleys. Gradually, Charis's heartbeat slowed. Heady relief bubbled in her veins.
Until she observed Gideon speaking to a well-dressed man, who clearly counted as the authority in town.
She shrank behind Akash. Renewed fear ate at her. Dear God, had she come so far only to fail now? If the authorities took her in charge, she'd be on a direct path back to her stepbrothers.
Akash glanced at her briefly. Gideon didn't look at them at all. He was once again the contained, courteous man she'd first met. The fight's wild berserker might never have existed. In a fever of nerves, she watched as Gideon pressed a wad of banknotes into the man's hand, then turned away.
A few curious onlookers hung around, but everyone else had gone on their way. Charis still couldn't see Tulliver. She'd faced rape and death, yet no trace remained of her ordeal apart from the blood and mud on the street.
"Wait a moment," Akash said when she made to leave the doorway.
Three well-dressed men strolled toward Gideon. One stopped, stared, and let out an exclamation of delighted surprise. "By gum! It's the Hero of Rangy whatsit."
Gideon paused at the first loud hail. Charis had a clear view of his face. So often, the sheer beauty of his features made it difficult to read his expression. Now she couldn't mistake how the blood drained from his cheeks and his brows contracted. He looked annoyed and on edge.
Hunted.
"Oh, hell," Akash breathed at her side, tensing.
The man who greeted Gideon turned in open excitement to his two companions. "You know who he is. The cove the King just knighted. Lasted a year in some filthy hole in India. Bravest fellow in the empire, Wellington called him."
His mouth stern with displeasure, Gideon retreated along the street toward Charis and Akash. He was close enough for her to hear him say in a forbidding voice, "I'm afraid you've made an error, sir."
The man advanced, his hand extended. "Dash it, man! There's no error. Your sketch is in every newspaper from here to John O'Groats, I'll warrant. Anyway, I cheered you in Pall Mall when you and the cavalry rode by on your parade of honor."
"You don't..."
"Let me shake the hand of the Hero of...What was that heathen place they had you locked up? Some benighted name no Christian can get his tongue around."
"Rangapindhi," one of his companions said with audible enthusiasm. "By George, it's a privilege to meet you, sir. By George, it is!"
The fuss attracted notice and quickly another crowd built up. But this time, it clamored with approval.
Wearing a coldly aloof expression, Gideon stood stock-still in the midst of the noisy mob. He looked like he had nothing but contempt for the congratulatory throng. His jaw was set, his lips thinned, his eyes veiled. He could never be less than handsome, but his frigid demeanor and stilted gestures repelled human warmth.
"Where in God's name is Tulliver?" Akash muttered beside Charis.
"I haven't seen him." Charis craned her neck to observe Gideon. Curiosity and confusion warred in her mind. She thought she'd begun to understand the man who rescued her in Winchester. It turned out he was as unknown as the wastes of Greenland.
His admirers didn't seem to mind Gideon's lack of welcome. They shook his hand and clapped him on the shoulders. All to a man looked at him as if he'd just stepped off Mount Olympus.
Wheels clattered on cobblestones. A moving carriage forced people out of the way.
A familiar carriage with a familiar driver.
"About bloody time," Akash said savagely, and wrapped an arm around Charis. "Come on. Run. And keep your head down."
He didn't need to tell her. She had no wish for anyone to see her face. She scuttled at his side, floundering to keep up with a man who made no allowance for her shorter legs or her injuries. The mad dash stirred all her fading aches into sharp agony, so her head rang when she finally reached the carriage.
Akash flung open the door and tossed her inside. She landed against the seat with a jolt that sent pain slicing through her. She stifled a cry and fisted her hands as she fought the giddiness. A breath hissed through her teeth. Another.
The worst of her dizziness ebbed. Ignoring her discomfort, she slid across the seat to press her face to the carriage window.
Both men were so tall, it was easy to locate them. Through the joyful hordes, Akash pushed his way toward his friend. Gideon retained that frozen, remote expression, but he didn't break away from his devotees.
She couldn't hear what Akash said to Gideon over the hubbub. She saw Gideon turn and head with jerkily precise movements toward the carriage. With visible reluctance, the crowd parted before him. Voracious hands stretched out to pluck at his clothing, delay his departure, compel his attention. Doggedly he continued his automaton-like progress.
He climbed in and sat opposite. He didn't speak. He didn't look at her. He didn't appear to know she was there at all.
Akash slammed the door on them.
"Aren't you coming with us?" she asked frantically. Suddenly, Gideon seemed a frightening stranger.
He shook his head. "I'm staying to see to the horses. I'll follow in my own time."
There was a burst of patriotic cheering outside. Someone started to sing "God Save the King." Clearly the locals were still stirred up at having a celebrity in their midst.
The celebrity straightened and shot Akash an angry glare. "For Christ's sake, let us go."
"God keep you, my friend. I'll see you soon." He stepped back and sent Charis an elegant bow. "Miss Watson. Your servant."
Before Charis could respond, Tulliver whipped the horses to a pace dangerous in town streets. The lurch of the carriage nearly threw Charis from her seat. She clutched at the strap and stared bewildered at her companion.
He looked ill. As though he suffered intolerable pain. With a shock, she realized the set expression was endurance, not disdain.
Automatically, she stretched out to take his gloved hand. "Sir Gideon..."
"Curse you, don't touch me!"
He wrenched out of reach. But not before she felt his desperate, uncontrollable shaking.
## Four
Through the suffocating miasma, Gideon knew he'd frightened the girl. But conscience was a dim whisper against the screaming demons in his skull. He clutched his head with shaking hands to silence the howling devils. It didn't help.
Nothing ever helped.
His sight failed, turning the girl's face into a pale blur. His throat was so tight, he choked. He sucked a shuddering breath into lungs starved of air.
She said something. He missed everything apart from the end. "...get Tulliver."
He forced himself to concentrate, pressed words to stiff lips until sound emerged. He didn't want Tulliver. Tulliver would drug him, trapping the monsters inside his head.
"No."
He sucked another breath through grinding teeth, even as thick darkness closed in.
"No Tulliver." Then what he prayed wasn't a lie. "This will pass."
Words worn threadbare with repetition.
Perhaps one day the nightmare wouldn't pass. The constant terror of that prospect made fear congeal like greasy soup in his belly.
I'm not insane. I'm not insane.
His gloved hands clawed at the worn leather seat as he battled for clarity. For control. For calm.
The demons were too strong. Horrible, shrieking phantom images rioted in his mind.
I'm in England.
I'm safe.
I'm free.
The litany failed. What freedom could he claim when grisly specters haunted his every moment?
"Please let me get Tulliver." The girl swam toward him through murky water. At the last minute, he realized she meant to rap on the roof and stop the coach.
"No!" The word emerged as a croak.
Speech was so damned difficult. He wished he was alone. But what couldn't be cured must be endured. The old aphorism, his nurse's favorite, helped him to cobble together an explanation. Even if every word cut his throat like broken glass.
"Tulliver will give...laudanum."
Opium hurled him into whirling oblivion. The dreams the drug brought threatened to send him mad indeed.
She frowned. "If it eases you..."
"No!" he all but screamed.
The girl recoiled. Good God, let him muster some control. He snatched another breath and fought to calm the frantic gallop of his heart.
She stared at him out of great, wide, terrified eyes. He loathed it when his personal...idiosyncrasies inconvenienced others.
Vaguely he told himself to assure her she shouldn't be afraid. He wasn't dangerous in this state. Unless she touched him. Thank Christ, after that first tentative attempt to offer comfort, she'd kept her hands to herself.
What had he meant to say? Thought was elusive and fleeting as wisps of mist.
That's right. Tulliver. He set his jaw and spoke in a low, harsh tone. Quickly, before will failed.
"There's nothing anyone can do. The best..." He stopped to fight back the caterwauling devils. "Please ignore me."
"That won't help." Even through swirling chaos, he heard the firmness in her voice.
Every joint tensed into quivering spasms. His stomach heaved like a stormy sea. Waves of hot and cold washed over him. He lashed his arms around his chest, but nothing eased the agonizing cramps. This attack was one of the crippling ones.
On his own, he'd bear the pain until it passed. But he couldn't distress the chit by vomiting all over her.
He'd have to accept opium's poisonous boon.
"Can you stop the coach?" he managed to force through chattering teeth.
Mercifully, she didn't question his change of mind. She banged hard on the roof. The carriage lurched to a halt. The abrupt movement set off jangling cymbals in his head, dimmed his sight.
The door wrenched open. Voices were a buzz in his ears. Tulliver passed in a tin basin.
"It's a bad one this time, lad," he said impassively, as Gideon's shaking hands curled around the dish.
Gideon's gut tangled into knots. He was seconds from losing control. He managed to snarl, "Take the girl."
His world turned to violent black as he began to retch. He was lost on a hideous sea, lit by brief crimson flashes where pain flared into agony.
He had no idea how long it was before awareness returned. Opening bleary eyes, he realized someone else's hands held the basin steady.
His mouth tasted foul. A hundred mallets battered his skull. Just the simple act of breathing threatened to split his chest in two.
Efficient hands removed the disgusting bowl. The same hands, soft and gentle, pressed a damp cloth to his burning forehead. He closed his eyes and groaned at the bliss of that coolness on his burning skin.
His belly was still rebellious. He concentrated on breathing. In. Out. In. Out.
"Akash?" he rasped across a raw throat. Although he knew the hands didn't belong to his friend.
"He's back in Portsmouth."
The girl. Miss Watson. Sarah.
With difficulty, Gideon cracked his eyes open. His blinding headache built with every second. Soon, he wouldn't be able to sit upright.
His clothes were rank and dripping with sweat. Acrid shame for his animal filth assailed him. "I told Tulliver to take you outside."
Her smile was dry as the deserts of Rajasthan. She knelt on the bench at his side. Her surprisingly competent hands supported his head. He was so sick and weak, her touch didn't make his skin crawl with familiar revulsion. He had a vague thought that helping him couldn't be easy with her sprained wrist, but the notion drifted off like a will-o'-the-wisp.
"Tulliver had his hands full." Her voice softened into compassion. "Are you feeling better?"
"He'll have the devil's own headache. He always does after one of his takings," Tulliver said calmly.
Gideon hadn't seen anything beyond the girl. Now he looked past her to where Tulliver waited, holding the bowl.
"He has these attacks often?" The girl's clear gaze rested on him with curiosity and concern.
Even in this state, his pride revolted at her pity. "I'm not an ailing puppy, Miss Watson. I can speak for myself."
Her lips turned down at his childish response. Which he regretted as soon as it emerged. Helping him couldn't have been pleasant. She deserved gratitude, not pique.
The pounding in his head made rational, connected thought increasingly difficult. He closed his eyes and stifled renewed nausea.
"I'll get the laudanum, lad." Tulliver's voice came from a long way off, masked by the painful throb of Gideon's blood.
"The sickness has passed," he forced out.
"The laudanum makes you sleep. You know sleep is all that brings you through. Do you want to stop at an inn? A bed might be better than rattling around in this rig."
A bed. Cool sheets. Quiet. A cessation of movement. All beckoned like the promise of heaven.
He hesitated. He had to reach Penrhyn. Something urgent.
He opened his eyes and saw the girl's worried face above him in the gloomy carriage interior. Of course. If they stopped, she might run.
They had to keep going. He'd have to accept the despised laudanum. And endure the harrowing visions.
"No...inn." He shook his head. Even so much movement made his stomach revolt. "Get the laudanum, Tulliver."
"Aye, guvnor."
As the coach rattled on through the day and into the night, Sir Gideon slept like the dead.
At first his unconsciousness perturbed Charis. His illness had been so violent, she'd feared for his life.
He stretched awkwardly over a bench that was too short for his height. She studied his face, pale, drawn, handsome still. The muscles around his eyes were tight, and his mouth was white with strain. The certainty built that while he might lie motionless as a stone effigy, his dreams brought no peace.
She turned away and stared unseeingly out into the darkness. Who were these men she'd cast her lot with? Tulliver, who faced trouble with such stoic competence. Akash, clever, enigmatic like a strange foreign idol.
Sir Gideon...
She commanded her wayward heart not to flutter at the thought of her rescuer. It was like telling the sun not to rise. Every moment she spent with him only drew the net of fascination tighter.
He was famous, a celebrity. The crowd in Portsmouth had pressed about him, bristling with excitement. They'd hailed him as the Hero of somewhere called Rangapindhi. Was he home after some daring patriotic action overseas?
Her stepbrothers had kept her isolated for months. She hadn't seen a newspaper or received any letters. Recent events in the wider world were a complete mystery.
If Sir Gideon was newly returned from India, it suggested a few explanations to things that puzzled her. His tan. Akash. Even his illness. Perhaps some tropical disease attacked him.
His horrific sufferings had cut her to the quick. Gideon Trevithick, her only bulwark against her stepbrothers, was unquestionably ill. But the nature of his sickness was an enigma. What ailment turned a man so quickly from invincible avenging angel to shivering wreck?
At dawn, Sir Gideon stirred from his deathlike sleep. The movement was slight but enough to disturb Charis's restless doze. She opened bleary eyes, excruciatingly aware of her own aches and exhaustion. The carriage's endless jolting had punctuated her erratic dreams. She'd checked him periodically through the night, but his sickness hadn't returned.
Without looking at her, he groaned and swung his feet to the floor as he sat up. He rubbed his hands across his face in a weary gesture. Granting him a moment's privacy, she opened the blinds and looked out the window onto a wild and unpopulated world. There was a charged intimacy in sharing this tiny space after she'd seen him at his extremity. It made her nervy, shy, unsure.
The view didn't help to restore her courage. They'd abandoned civilization miles past. The lonely, windswept scene was depressing, frightening to a woman with only strangers to rely upon. Staunchly, she reminded herself that her stepbrothers would have difficulty tracking her through this wasteland.
She wondered how much farther Sir Gideon meant to go. Since they'd left Portsmouth, the only punctuation to eternal travel was stopping to change horses. Hurried, efficient movement, a flare of torches, Tulliver rebinding her arm if the bandage had loosened, a hot drink shoved into her hands. Then away they went again. The beef broth from the last stop, a poor place in the middle of desolate moorland, had left a nasty taste in her mouth. Luckily, she had a cast-iron stomach.
She turned back to her companion, and an involuntary gasp escaped. "You look awful."
He gave a surprised grunt of laughter and scraped his hand across the stubble darkening his angular jaw. "Thank you."
She blushed. "I'm sorry. I had no right..."
"No harm done. I'm sure your observation, if not polite, was accurate." He sounded like the man who had found her in the stable. Ironic. Distant. In command of himself.
Except now she knew his composure was a veneer.
He might sound like master of all he surveyed. But he didn't look much better than he had last night when he'd shivered in her arms. Dark circles surrounded sunken, dull eyes. His tan held a sickly hue in the pale sunlight penetrating the windows. He badly needed a shave, and his hair was a tousled mess.
His eyes sharpened on her. With every moment, he looked more alert. "How is your arm, Miss Watson?"
She didn't immediately recognize her false identity. Dear Lord, let him not notice her hesitation. She needed to remember the danger she faced if he discovered who she was. Difficult when the last day had only built the affinity she'd so quickly felt for him.
Carefully she flexed her fingers. Hardly a twinge. "Much better, thank you." She studied him as he sprawled against the worn leather upholstery. His long legs extended across the well between the two seats. The shabby carriage wasn't built for a man of his height. "How are you?"
He stretched and winced, then leaned his head back. "It was just a passing inconvenience."
His expression indicated movement was painful. After lying still for so long, he'd be stiff as a board. The continual rolling and jolting of the vehicle must be agonizing. She ignored his unconvincing lie and dropped to her knees on the rocking floor.
"Let me take your boots off and rub your legs. I nursed my father in his last illness. This helped him when he'd had a bad night."
She'd forgotten no decent young lady offered to touch a gentleman who wasn't a close relative. She remembered only when he tensed, and his dark eyes flashed with horror. "Miss Watson, please return to your seat. I assure you my slight troubles don't warrant your concern."
Clumsily, her cheeks flaming with mortification, she scrambled back onto her seat. "I'm...I'm not usually so rag-mannered."
Yesterday he'd suffered her touch. He'd turned his face into her hand as she'd wiped his brow. But yesterday he'd been victim to his mysterious illness.
"It was a generous offer," he said kindly.
She hated his kindness. Because clearly it wasn't based on anything personal, like regard or respect. She hated owing her safety to that disinterested kindness.
Hiding a wince as the movement tested her sore arm, she fumbled to open a flask of water Tulliver had given her last night. "Are you thirsty?"
"Dry as sand." He accepted the flask without touching her fingers.
Charis berated herself for noticing. And minding. Did she want to fend off a Lothario? She should commend Sir Gideon as a man of honor.
Sourly, she recognized her hypocrisy.
Fascinated, she watched the movement of his powerful throat as he tipped his head to drink. Nor did she miss the tightness around his eyes as he returned the flask and subsided against the upholstery.
"Does your head hurt?" she asked before she reminded herself he wouldn't appreciate her solicitude.
A fleeting smile curved his lips. "Like the very devil." He sighed heavily. "All of this must frighten you. I'm sorry."
"I don't frighten easily," she said flatly.
He didn't argue although he must know she'd been terrified in Winchester. More of his cursed kindness. She wouldn't resent it nearly so much if he didn't use it as defense against her curiosity.
"Your face seems better this morning," he said.
"Oh." She'd forgotten what a horror she must look. She raised a tentative hand to her sore jaw. It didn't feel as distended. Speaking was certainly easier. Whatever heathen potions Akash had slathered on her, they'd worked. "Yes."
Sir Gideon's regard was steady as it rested upon her. Steady and implacable. "Will you trust me with the truth now? You have no aunt in Portsmouth. You're on the run from someone. Someone who threatens your very life if the state I found you in is any indication."
She stiffened under his probing gaze. Briefly she considered persisting with her lies. But as she looked into his face, she knew denial was useless. She sucked in a breath that contained a heady mixture of relief and uncertainty. "How long have you known?"
"From the beginning."
He sat up carefully and stared at her. If his face had held an ounce of anger or censure, she'd have kept silent. But he looked interested, calm, capable. He looked like a man she could trust with her life.
She shifted uncomfortably, her conscience flinching at the lies she'd told. "I don't see why you want to help. I've caused nothing but trouble. You should consign me to perdition."
Another of those faint smiles. "True."
"Well?"
He shrugged. "I've been alone against the world in my time. I'd hate you to come to grief because you had no champion."
Again, she thought of a medieval knight. A lonely, gallant figure on an impossible quest. "What happened to you?"
He laughed softly. "Oh, no, my lady. This is my interrogation. Who hurt you?"
Lingering caution insisted she conceal the precise details of her plight. She'd seen how greed changed men. She couldn't risk that happening to Sir Gideon if she told him who she really was. But his gallantry toward her meant she owed him more than the shabby falsehoods she'd produced so far.
"My brothers. They're trying to force me to marry a wastrel. I cannot...will not stomach the match." Her hands fisted in her skirts. It seemed odd, uncomfortable to trust a man even a little after all she'd been through. "When they realized my opposition was more than a girlish whim, they resorted to stronger persuasion." Close to the truth. Close enough to salve her stinging conscience, anyway.
Sir Gideon's face remained expressionless as he listened. What did he make of this tale that belonged in a gothic novel? Did he even believe her? At least he showed no skepticism.
"Why are your brothers so eager for you to marry this man?"
His lack of histrionics calmed her. Her hands slowly uncurled until they lay flat upon her lap. Her voice emerged almost normally. "They owe him money. My inheritance becomes my husband's if I marry or mine if I turn twenty-one unwed."
"When do you turn twenty-one?"
"The first of March."
"That's only three weeks away."
"You perceive my brothers' need for urgency," she said dryly.
"Self-serving maggots," he bit out with sudden savagery.
She'd misjudged his calmness. Looking closer, she realized he was furiously angry. His voice was quiet, his manner unthreatening. But she had a sudden vivid memory of the man who overcame every adversary in the Portsmouth brawl. Foreboding tinged with satisfaction shivered through her. She wouldn't like to be Felix or Hubert if Sir Gideon got his hands on them.
"I'm so sorry for telling lies," she whispered, guilt twisting her stomach into knots. She twined her hands together and gazed down, unwilling to meet his searching eyes. Eyes clever enough to discern she still wasn't completely honest.
"You were in danger. You had no reason to trust me."
"Except you saved my life," she said almost soundlessly.
Except you're fine and handsome and brave. And I've held you while you were sick and unaware. And watched you sleep through a long dark night. You make my heart beat like a drum, and I can hardly breathe when I look into your eyes.
She glanced up in time to catch the annoyance that crossed his face. "It was nothing."
"It wasn't nothing to me." She raised her chin and stared unflinchingly at him.
"Miss Watson, I don't want your gratitude," he snapped.
She hid the pang of hurt his response provoked. And refrained from insisting that she'd be grateful to him until the day she died.
An awkward pause fell.
When eventually he resumed his questions, his expression didn't lighten. "Presumably someone other than your brothers has custody of your fortune while you're a minor. Why didn't you appeal to them?"
"My trustees claim they're powerless to intervene." Her voice was husky with chagrin she couldn't help feeling when Gideon refused her thanks. "My brothers convinced them I'm wild and flighty and need a man's guidance."
She'd spent many a night cursing the spineless solicitors at Spencer, Spencer and Crosshill. Old Mr. Crosshill had been her father's friend, but he'd been dead four years. His egregious nephew had advised her to accept her stepbrothers' plans with suitable female obedience.
"No relative offered you shelter?"
"None with the power to stand up to my brothers." Charis's voice flattened into grimness. "Believe me, Sir Gideon, I've assessed all options. Only one remains. Will you put me down at the next substantial town we come to?"
"What do you intend?"
To survive the next three weeks without surrendering either to privation or my stepbrothers.
"I only have to avoid my brothers until the first of March." Heat climbed in her cheeks. Her pride abhorred what she was about to ask. But she must conquer pride for survival's sake. "If you lend me a few shillings, I'll repay you when I come into my inheritance. I couldn't find any money to take with me. Which must seem hen-witted, but..."
"Miss Watson."
"I'm not solvent right..."
"Miss Watson." His voice was sharper.
She relapsed into silence, embarrassed at her nervous babbling. Tears of humiliation rose to her eyes. She didn't want to set out alone. More than that, she didn't want to leave Sir Gideon, which was just too pathetic to admit. How had he so quickly become the most important person in her life? It seemed absurd. Unreal. Dangerous.
He appeared displeased. Again. "Confound you, I'm not going to slip you some blunt and put you down defenseless and alone in a strange place. If any town between here and Penrhyn was big enough to offer a hiding place. Haven't you looked out the window, girl? We're well into the wilds of Cornwall."
She gulped back the lump in her throat while fugitive hope stirred in her heart. "Oh."
He looked in better health, more like the man she'd met than last night's invalid. He looked clever and purposeful and invincible. He looked like he would keep her safe forever.
His deep voice was firm. "We aren't far from my home. I hope you'll accept my offer of sanctuary."
## Five
Gideon expected Miss Watson to demur. After all, only yesterday she'd been so desperate to escape that she'd risked her life to run away. But she turned a solemn hazel gaze in his direction and, after a moment, nodded.
He couldn't help noting her beautiful eyes, remarkable even in her bruised face. A striking mixture of green and gold, they were the shifting, fascinating color of the tarns he remembered from the woods near Penrhyn.
"I accept, Sir Gideon. Thank you." Her lush lashes lowered to shade her eyes to malachite. "I just hope your many kindnesses to me don't bring you trouble."
More damned gratitude. He dismissed her remark with a grunt. "I'm not sure how kind you'll think I am when you see the house. I haven't been back since I was sixteen. Even then, it was far from luxurious. Lord knows what state the place is in now."
According to his father's solicitors, the old manor still stood as it had stood through four hundred years of wild Cornish weather. They hadn't, however, been able to vouchsafe the property's condition. Ramshackle, Gideon guessed, reading between the lines of legal nonsense.
Neither his father nor his older brother had been much of a manager. No reason that should change because Sir Barker Trevithick's despised younger son vanished into Asia. Before breaking his neck in a drunken hunting accident, Sir Barker hadn't known whether his second son was alive or dead. Nor, Gideon grimly knew, had he much cared. But then, he'd thought the succession rested safely in Harry's plate-size hands.
Like so much about Gideon's return to England, the deaths of his father and brother aroused conflicting reactions. Neither had ever evinced an ounce of affection for him, and he wasn't hypocritical enough to pretend to mourn their passing. Nonetheless, there was a...regret when he thought of two lives so close to his own wasted in debauchery and drunkenness.
Curiosity lit Sarah's face, and she leaned forward, bracing herself against the jolting carriage. "Has the house been unoccupied since you left?"
"No. My older brother lived there until last winter, when a fever took him."
He kept his voice steady and unemotional. Still, the girl's expression filled with compassion. "I'm so sorry," she whispered.
Her ready sympathy made him uncomfortable. "We weren't close." To say the least. Wild beasts received a more tender upbringing than the two young Trevithicks.
"Then I'm sorry for that too," she said. "Family is important."
"Not to me," he said tersely. "And I hardly think your experience is any improvement on mine."
Her jaw firmed. "My brothers' brutality can't destroy my faith in human relations. That would give them too great a victory."
Again, he couldn't stifle his admiration for her indomitable spirit. "You're a brave young woman." And she'd need every scrap of that bravery before she was done. He paused and forced himself to set before her a factor she should consider. "It will be a bachelor establishment, Miss Watson. Me. Tulliver. A few servants. Akash when he arrives in a couple of days."
Briefly, she raised her good hand to touch her mottled cheek. The gesture indicated uncertainty and drew his unwilling attention to her face. This morning both the bruising and the swelling had subsided. A hint of her true features emerged like a shadowy reflection in a mirror. With a doomed sinking in his gut, Gideon recognized that Miss Watson promised to prove a beauty under her injuries.
When he'd rescued her, he hadn't spared a thought for her physical attractions. She was just a woman needing help. The last thing he wanted to deal with was a winsome female. She would only be a blistering reminder of everything he'd never have.
Fate clearly was in a mood to torment him.
"No ladies at all?" She sounded hesitant. He couldn't blame her. For a gently bred girl, the prospect of moving into a masculine household must be daunting. "No aged spinster aunts or widowed cousins?"
"I'm afraid not." He wished he could reassure her that his aid came without risk of consequences. He wished to God he had some alternative plan for her safety. "We may get away with it. I've been abroad a long time, and I have no plans to join local society. The house is remote, and the villagers distrust outsiders."
Nervously, she plucked at the bandage on her arm, her fingers long, pale, and graceful. He noticed she held her arm more easily against the swaying carriage. Clearly Akash's potions had relieved the worst of her pain.
There was a troubled silence before Sarah spoke. "My safety is more important than my name." She sounded as though she reached that conclusion reluctantly. As she looked up, she managed a shaky smile. "I still can't see why you take this trouble. Your generosity to a stranger does you credit."
Gideon shifted uncomfortably under her wholehearted approbation. He desperately needed to shatter the encroaching intimacy, fine as spider's web, strong as steel, but something in her unblinking regard forced the truth from him.
"I abhor injustice. I abhor bullies. Everything in me resists allowing men who treat a woman as you've been treated to profit from their evil." His voice roughened with emotion. "While there is breath in my body, Miss Watson, I'll do my utmost to ensure your freedom and security."
Immediately he repented his impulsive declaration.
Her eyes glowed gold as a streak of sunlight striking a forest pool. Her lips parted, but no words emerged. She leaned toward him but, thank God, didn't touch him. Even so, his skin itched as though she reached for him.
Damn, damn, damn. He should have recognized the looming danger before this. He needed to destroy this building affinity, not encourage it. Why hadn't he kept his blasted mouth shut?
At last he interpreted exactly what her expression portended. His inescapable conclusions made his stomach lurch with nausea.
Miss Watson regarded him with unstinting, uncritical, and completely unwarranted hero worship.
Following his moving declaration of unconditional protection, Sir Gideon's withdrawal was tangible. She stifled a prickle of hurt she had no right to feel.
He spent most of the day asleep. Or feigning sleep. She couldn't be sure. What she could be sure of was that he wouldn't welcome her curiosity. Even though curiosity about him gnawed at her mind like hungry rats.
His apparent oblivion provided her with uninterrupted hours to study her companion. The mysterious ailment had passed although he was still pale and gaunt. Charis was guiltily aware that her reckless escape had prompted his attack although she had no idea why. His suffering had been so extreme, she could hardly bear witnessing it. The agonizing frustration was that she could do so little to help.
She gathered he endured these awful spells on a regular basis. What on earth was wrong with him? She hadn't seen anything like his illness before although she'd nursed her father and her mother and ministered to many sick tenants on the estate.
Gideon Trevithick puzzled her. He fascinated her. She'd never known anyone to compare to him. She'd never known anyone who affected her the way he did. He was such a compelling mixture of strength and vulnerability. Every time she looked at him, her heart launched into a tipsy dance. This breathless excitement was unfamiliar and frightening. None of her suitors had stirred this hunger for their merest presence.
Perhaps she felt this way because he'd saved her. First in Winchester, then from those vile miscreants in the alley. A shudder rippled through her as she imagined what would have happened in Portsmouth if Gideon hadn't appeared like a guardian angel. Degradation and death had edged so close.
But as her eyes traced Sir Gideon's dark features, she knew her interest went beyond gratitude. Deep and sincere as that gratitude was. He was beautiful, he was brave, he was damaged, he was frighteningly clever. And the briefest sight of him made her breath jam in her throat.
Dear Lord, she'd known him little more than a day, and already she brimmed with giddy, irrational longings. What state would she be in after three weeks in his company?
At least the continuing silence served one good purpose. He didn't question her further, saving her from dredging up more lies to prick her conscience. Ingrained habits of mistrust and caution urged her to keep her identity secret, although if anyone deserved her honesty, it was Sir Gideon.
Now she was about to move into his house. A forbidden thrill raced through her at the prospect. A thrill mixed with apprehension. If the world discovered she lived under his roof without a chaperone, she'd be ruined. Another good reason to keep her identity secret.
She glanced across at her sleeping rescuer and couldn't help thinking that ruin had never looked so alluring.
Oh, Charis, wicked, wicked. The angels weep for you.
Charis's endlessly circling thoughts eventually took on the carriage's rocking rhythm and lulled her into a half-waking state. Each lurch of the coach worsened her aches and reminded her she was far from recovered after Hubert's beating.
For most of the day, they traversed rough moorland. In the late afternoon, Charis was awake to notice they passed between two gateposts, worn and covered in ivy. Rampant lions held carved stone shields so old and moss-encrusted, any detail was long obliterated. Rusted gates hung drunkenly, smothered in weeds that had died last summer and never been cut back. Soon after, they entered a thick wood.
Charis stirred to mark the change in the landscape but was too tired to ponder its significance. She stretched stiff muscles and bit back a moan as the movement tested her injuries. With a sigh she couldn't restrain, she leaned her head back against the seat, hoping to heaven there wasn't another night of travel ahead. She was heartily sick of the rattling, bumping coach.
They continued for another half hour or so. Interlacing branches above the rutted track turned the interior of the carriage into shadowy mystery. In his corner, Sir Gideon was a silent, magnetic presence, his long legs stretched across the well between the benches, his arms crossed over his hard chest. She had no idea whether he was asleep or pretending.
To her regret, Charis knew she looked like she'd been dragged through a bush. She'd been a fright yesterday, and the day's traveling would only worsen her appearance. Since he'd changed his clothes, Gideon had regained his louche elegance. Even the faint beard darkening his jaw enhanced his masculine appeal, adding a rakish air to his chiseled features. She closed her eyes and told herself to think of something other than Sir Gideon. A command impossible to obey.
Through her fog of discomfort and exhaustion, Charis heard Tulliver shout and felt the carriage shudder to a halt. She opened dazed eyes. They'd left the wood, and late sunlight flooded through the windows.
She leaned her head out the window and looked up at the grizzled figure in the driver's seat. "Why have we stopped, Tulliver?"
"Look, miss." He gestured with his whip. "Penrhyn."
With a glance at Sir Gideon's motionless figure, she forced her tired muscles into ungainly movement. She scrambled from the coach and turned in the direction Tulliver indicated.
And fell in love at first sight.
They were on a slight rise. Behind stretched the woods they'd just driven through. In front, the land sloped gently down to the cliffs. Beyond was the glory of the sea, deep blue in the fading light.
Part of sky, sea, wild landscape was the house perched on the edge of the cliffs, looking westward. Centuries old. Worn. Welcoming, even at this distance. Its soft golden stone glowed in the long rays of sunlight. Penrhyn called to Charis across the pale winter grass that trembled in the fresh sea breeze.
"It takes your breath away, doesn't it?"
Reluctantly, she tore her gaze from the house to look at Gideon, who emerged from the carriage behind her. He'd been nurtured in this glorious place. No wonder he was so remarkable. She swallowed to shift the lump of emotion that lodged in her throat at the house's perfect beauty. "It's magnificent."
He stopped beside her, close enough for her to be aware of his commanding height. She wasn't an especially short woman, but he made her feel small and fragile. Her heart did its usual dip and leap at his nearness. How she wished she could control her foolish reactions.
"Yes, it is." His voice was calm. Artificially so, she guessed. Although his striking face was impassive, she couldn't mistake the tension in his lean frame. "I wondered if it had changed. It hasn't."
Charis frowned, confused by the currents swirling beneath his calm surface. For someone who had left his home many years ago, he appeared less than overjoyed to be back. "How could you bear to stay away for so long?"
Sudden emotion darkened his face, and his eyes burned as they met hers. The searing look lasted barely a second before he returned his attention to the old house. "How can I bear to come back?" he muttered, seemingly against his will.
"You sound like you hate it," she said, aghast.
He shook his head, and a lock of black hair fell across his forehead. "No, I love it. That's what makes everything so impossible."
The corrosive honesty of his response flooded her with astonishment. Sir Gideon didn't strike her as a confiding man. That he revealed as much as he did indicated his turmoil.
With an abrupt movement, he turned on his heel and climbed back into the carriage. Shocked, bewildered, Charis watched him go. It was as though he couldn't bear to look on his inheritance any longer. But for a moment, the hardness in his gaze had shattered, and she'd glimpsed a longing that made her heart stutter.
She wished to the depths of her being that she understood him. She wished to the depths of her being that he considered her worthy of his confidence. More than either, she wished she could do something to ease his unspoken anguish.
But she was a stranger. A brief visitor to his life. She had no significance for him beyond the present moment.
She glanced up at Tulliver, who had witnessed the whole exchange with his usual sangfroid. There was a light in his eyes that might have been understanding and was certainly pity.
For whom? Sir Gideon? Or the pathetically infatuated Miss Watson?
His voice was kind. "You might as well get back in, miss. We've got a mile or so to go."
Charis's shoulders sagged with weariness, and she limped after Gideon into the vehicle. Tulliver whipped the horses to a canter as they turned for the house. Gideon settled into his corner and stared out the window.
The sun plunged toward the sea in flaming glory by the time they passed through a crumbling stone arch and into the paved courtyard in front of Penrhyn. A closer viewing revealed the house was shabby and unkempt, but nothing could destroy the enchantment it laid over Charis. An enchantment indelibly part of the yearning she felt for its master.
"Parts stretch back to the fifteenth century, although most of it is Elizabethan." They were the first words Gideon had spoken since that tense, revealing moment on the rise.
"It's beautiful."
He gave a short, caustic laugh. Through the dimness, she read the derision on his face. "Believe me, your enthusiasm will wane when you get inside to a cold house and damp sheets and a makeshift supper—if we manage any supper at all."
"I don't care." His cynicism couldn't damp her pleasure in Penrhyn. The ancient stones breathed warmth. The house had been loved, and it would be loved again. It was old and knew how to wait.
Holcombe Hall was a cold white Palladian pile. Architecturally perfect. Built for a Marquess of Burkett last century when the Farrell family still had money and prestige. She'd hated it from the moment she'd arrived there after her mother's marriage to the late Lord Burkett. God rot his miserable soul.
As the coach slowed, two men dashed out to hold the tired horses. Four women hurriedly lined worn steps rising to a heavy door.
"Let the circus begin," Gideon said bleakly. With a savage movement, he opened the door and leaped to the ground before the carriage reached a complete stop.
Gideon sucked air into lungs constricted with an anger he didn't understand. He hadn't expected his return to his boyhood home to be so fraught with emotion. But at the first sight of the old house, he'd felt crushed between the urge to escape and the yearning to stay forever.
Another deep breath in a futile attempt to calm his galloping pulse. The essence of Penrhyn overwhelmed his senses, cleared the last sour traces of yesterday's laudanum. And brought back a thousand agonizing memories.
Still he drank in the air—tangy with salt and wild thyme and sun on old stone and good Cornish earth. He was home and the sweet, fragrant reality split his heart in two.
"Sir Gideon, welcome home!"
The familiar voice wrenched him from distraction. He straightened and fought to mask his tumultuous reactions. He met a shrewd blue gaze in a lined face. A face he knew. Behind the tall, rake-thin old man, the staff bowed and curtsied.
Surprise and something approaching pleasure stirred. "Pollett? Elias Pollett?"
The man's eyes shone bright with welcome. "Aye, lad...Sir Gideon."
Pollett had been his father's head groom. Even when Gideon was a boy, Pollett had seemed old. Gideon's memories of his family were unfailingly desolate. His memories of the local people less so. Mostly they'd ignored him. Which was kinder than any treatment he'd received from his father. But Pollett had been an ally as far as he was able. He'd secretly taught Gideon to ride after Sir Barker abandoned his son as a hopeless case.
"How did the solicitors know to give you a position?"
"I never left, sir. A few of us stayed to see the house secure until you got back from furrin parts and took charge."
Took charge? What a joke. Gideon wasn't even sure he intended to remain. Although the scents of sea and wild herbs insisted he belonged here. Demanded he accepted he was a Trevithick to the bone. Like all Trevithicks, born at Penrhyn and fated to die at Penrhyn. As much part of this place as the cliffs and the waves and the wheeling, crying gulls.
"Before that, I was Sir Harold's bailiff." The slow, deep roll of Pollett's Cornish accent fell on Gideon's ears like music. "Didn't anyone tell you?"
They might have. He hadn't been interested enough to pay attention to much beyond the basics of his solicitor's correspondence. Difficult as it was, he summoned a smile. "I can't think of any man better suited to run the estate, Pollett."
It was true. Unexpected his brother had seen it too. He wouldn't have credited Harry with such good sense.
Pollett's face creased in concern. "The estate isn't as it should be. I did my best, but..."
Gideon made a dismissive gesture. "It doesn't matter." The house stood, and anything else could be fixed. If he could summon heart for the task.
"We've been short-staffed. And Sir Harold..."
Gideon met Pollett's eyes and a silent message of understanding passed between them. Harry had already been a hopeless drunkard when Gideon left, for all he'd only been nineteen.
Sir Barker had been a man of stubborn opinions. He'd considered drinking, like hard riding and ceaseless womanizing, an essential manly attribute. Gideon's open contempt for his sire's swinish pursuits was just one of many conflicts between them.
A memory of Harry before the liquor got to him assailed Gideon and aroused a pang of genuine sorrow. His brother had been tall and gold like a Norse god. Strong. Hearty. Stupid as an ox but not vicious.
Any viciousness in the family had been his father's.
Pollett swallowed visibly as Harry's bluff ghost hovered, then vanished. "All will be well now there's a real Trevithick holding the reins."
Dear God, how much more of this could he take? The hope and joy in Pollett's face made Gideon flinch. He didn't deserve this unconditional welcome.
To avoid the old man's gaze, Gideon turned back to the carriage. He looked inside to where Sarah shrank into the shadows. "Come out, Miss Watson."
He stood back as she reluctantly obeyed. When she emerged, Pollett's face lit with curiosity and the beginnings of speculation. "Are felicitations in order, Sir Gideon?"
If a man traveled alone with a woman, she could fill few roles in his life. A relative, and Pollett intimately knew the sparseness of the Trevithick family tree. A wife. A mistress.
Gideon stifled grim laughter. He wished to hell he was normal enough to have a mistress. If he did, she'd be a damned sight better turned out than Miss Watson. However low the Trevithicks sank, they always dressed their ladybirds comme il faut.
The girl hovered at his side with visible uncertainty. She'd raised the greatcoat's collar around her face, and her shoulders hunched.
Shame was so familiar, he had no trouble recognizing it in another. He hated seeing such a proud spirit brought low. She hid her injuries, as though they marked her unclean, contagious. More than that, she must know her virtue was in question.
She waited silently, gazing at the ground. Poor Sarah. Hurt. Alone. Helpless.
Her brothers' violence cast her into an unforgiving world. How she must loathe relying so totally on strangers. In this isolated place, she had nowhere to run, nowhere to hide.
His glance swept the small crowd arrayed before him. Generations of service tied these men and women to the Trevithicks. He drew himself up to his full height, and his voice rang with authority. "Miss Watson is an acquaintance who needs somewhere to stay." He ignored her muffled gasp of horror as he used her name. "It's imperative nobody knows of her presence. I entrust her safety to your good sense and discretion."
Sarah mightn't realize it, but he'd just claimed her as a denizen of his private kingdom. Penrhyn had always been a realm unto itself, loyal to those who belonged, suspicious of incomers. He waited as first one maid dropped into a curtsy, then another, and the men bowed acknowledgment.
Gideon gestured for her to precede him up the stairs and into the cavernous hall. But as he followed her into the house, reluctance weighted his tread.
The day's last sunlight poured in dusty rays through tall mullioned windows. Inside, the shabbiness evidenced outside was overwhelming. Sparse furniture littered the vast space. There were signs of a hurried cleaning, but the elaborately carved moldings were unpolished, the curtains dusty, the fires unlit. The servants trailed in and lined up against the dark paneling.
"We put on extra staff when we heard you were coming, Sir Gideon. But I awaited your orders before I did too much. For the last year, it's just been me and Mrs. Pollett in the house." For a moment, Pollett's formality faded. "I'm sorry, lad. It's not much of a homecoming."
Gideon looked around the unprepared, dirty room. Memories of his childhood were colder than the winter air. His father had conducted punishments here, usually before the staff. Gideon's refusal to cry under the whip should have pleased the old tartar. After all, Sir Barker's constant carp was that he'd spawned a puling weakling in his second son. But Gideon's sullen obstinacy had only incited greater violence.
"Sir Gideon?"
The girl's soft voice shattered his painful reminiscences. He turned to look at her. The collar folded back from her face, and as luck would have it, she stood in a pool of sunlight. Lit like a saint in a religious painting.
Her features were clearly discernible. A pointed chin, full lips, large eyes as changeable as the Cornish weather. Her hands tangled in the black folds of the coat, he guessed to hide their unsteadiness.
"You must be tired." Now he looked more closely, there were dark crescents beneath her eyes, visible even under the bruising. "The travel has been difficult."
When she met his stare, she raised her chin and summoned a fleeting smile. She was alone, afraid, defenseless, but she dared fate to defeat her. Something shifted in the farthest reaches of his heart, and the house's sounds receded to a hushed murmur. Sarah Watson drew him as no other woman ever had. If circumstances weren't so tragically askew, he might aspire to offer for her hand.
Instead, she'd do better to run a thousand miles from him. He was no use to himself. He was no use to the world. He could be no use to a wife.
That knowledge didn't stop him yearning for joys other men took for granted.
He'd had months to count the agonizing toll of his years in India. He thought he'd measured the price of his experiences. But only now, when the phantom life he might have led beckoned like a desert mirage, did he truly comprehend all that had been stolen.
Grim reality dictated that Sarah remained an unfulfilled promise of everything he'd never have.
He tamped down the poignant longing, the regret, the sadness. She'd be gone in three weeks. He could endure that, surely. He'd endured a year of unspeakable suffering in Rangapindhi and survived.
"I'm all right." She hesitated and bit her lip. "I'd love a bath, if that's possible."
"I'm sure it is." Gideon glanced at Pollett, who waited nearby. "Are any bedrooms ready?"
"Aye, Sir Gideon." The man stumbled every time he spoke the title. "The master suite is prepared."
"That will not be suitable for Miss Watson," he said curtly. The glare he shot Pollett made it clear Miss Watson was not and never would be his mistress. "Have the maids make up the Chinese room. You'll need to make preparations for my man Tulliver too. And I'm expecting another guest, an Indian colleague, in the next few days. He'll use the ivy room."
Pollett bowed and spoke in a subdued voice. "Yes, Sir Gideon."
Gideon desperately needed to escape this room with its hordes of unhappy ghosts. He gestured Sarah toward one end of the hall. "In the meantime, Miss Watson and I will take tea in the library. If it's habitable."
Pollett bowed again as he passed. When he lifted his head, he spoke softly and with a sincerity that made Gideon cringe. "I'm glad you lived to come home, lad."
"Thank you," he muttered, wishing he felt a shred of gratitude for his survival into the hellish present.
At Sir Gideon's side, Charis crossed a dark corridor and entered an even darker room. She drew her first unconstricted breath since she'd arrived. Thank goodness she was no longer the cynosure of all eyes. She loathed knowing the servants thought she was no better than she ought to be. In spite of Sir Gideon's gallant efforts to insist she wasn't his mistress. Her bruised face only increased speculation.
She waited uncertainly as he flung aside a heavy set of blue velvet curtains. Choking dust flew into the air. Sudden light dazzled her. She closed her eyes and opened them on a wall of windows facing an overgrown terrace poised above the sea.
For a long moment, Gideon stared at the magnificent view. Charis sensed sadness and curiously, for a man who returned home, a deep loneliness.
Was he grieving for his dead brother and father? Or did something else trouble him?
His essential isolation prompted her to touch him, offer comfort, remind him he was part of the human race. She curled her hands into the coat and stifled the impulse. The journey had taught her he wouldn't welcome her overtures.
His rejections hurt, but not as much as it hurt to witness his brooding unhappiness. More sign that she was dangerously vulnerable to this man who was little more than a stranger. But she'd already fallen off the precipice. It was too late to try to save herself.
Eventually, he turned, brushing dust from his hands. His expression was neutral, the brief vulnerability hidden.
"I've brought you to a hovel, I'm sorry." He moved across to help her take the coat off. He draped it over a set of mahogany library stairs. Like everything in the room, they were covered in thick dust. But no amount of dirt could conceal the impressive walls of leather-bound books or the elaborately carved furniture and plasterwork. This was a beautiful room, but nobody had cared for it in years.
"Hardly a hovel." Gingerly she perched on an upholstered chair, sending up a puff of dust that made her sneeze. She was weary to the bone, and every muscle ached from the beating and the hours in the coach. She'd sell her soul for a hot bath and a bed and the chance to sleep for a month. She'd sell her soul twice over to see a glimmer of joy in Sir Gideon's dark face.
"How are you feeling?" He surveyed her with an impersonal concern that made her want to shrivel up in the corner.
"I'll be glad to stay put for a little while," she said. "How are you?"
He frowned as if the reminder of his illness rankled. "I'm perfectly well, thank you." He swung away, discouraging further inquiries after his health. "You should rest and regain your strength. I'll send Mrs. Pollett to you after we've eaten. She's not Akash, but she knows most of the country remedies."
"Thank you." She had no right to mind his eagerness to consign her to other people's care. Frightening how much power a glance or a word from him had over her emotions. She tried to set up self-protective barriers, but they crumbled to rubble the moment she looked at him.
She sneezed again and muttered her thanks as she accepted the handkerchief Gideon extended in her direction. Through watery eyes, she watched him prowl the room, lifting items seemingly at random and inspecting them.
How curious he was so ill at ease in his own house. Why was his homecoming so strained? He'd dropped hints of a clouded family history. Did old memories torment him? Something did. Tension stiffened his back, and deep lines bracketed his expressive mouth.
The door opened to a girl carrying a tray. The cups didn't match. One was Meissen, one was Sèvres. Both were exquisite. Once, someone at Penrhyn had had taste and money to indulge it.
Sharing the tray was a plate of roughly hewn cheese sandwiches. To Charis's embarrassment, her stomach growled. She flushed. Great-aunt Georgiana would be mortified at such a faux pas.
Sir Gideon replaced a small marble bust of Plato on the windowsill and turned to the maid. "What's your name, lass?"
The musical baritone worked its usual magic. Even Charis, who should by now be inured to its allure, shivered in sensual reaction to that deep, musical sound. The girl's thin shoulders relaxed, and she sent Sir Gideon a shy smile as she slid the tray onto a dusty rosewood side table.
"Dorcas, Sir Gideon." She curtsied. "I be Pollett's granddaughter. Ee mightn't remember me, sir, but I remember ee, though I were only a ween of five when ee left."
"You used to churn the butter for your mother."
"Aye, sir." The girl flushed with surprised pleasure. "Fancy ee remembering that."
Gideon tilted his head toward Charis. "Miss Watson needs a maid. Would you be interested in helping, Dorcas?"
The girl curtsied to Charis. "Oh, aye, miss. But I bain't never been a lady's maid afore."
"I'm sure you'll be splendid, Dorcas," Charis said. Again, she had reason to be grateful for Gideon's thoughtfulness. She was wicked to want more than he offered.
The girl grinned with gap-toothed delight. "Thank ee, miss. Thank ee."
When Dorcas had gone, Gideon glanced across at Charis. "She'll be clumsy at first, but she was a quick child. I imagine she'll learn fast."
"There's no need to make excuses. You're kind to think of my convenience. My step...my brothers..." Dear heaven, the false intimacy of being alone with Sir Gideon in this beautiful, neglected room made her forget she lived a lie. She needed to watch her tongue, or she'd reveal her true identity. "My brothers deprived me of my maid over the last weeks."
It infuriated her to recall Felix and Hubert's petty tyrannies. As though lacking a servant's attentions would convince her to marry the foul Lord Desaye.
Gideon strolled across to the table. He lifted the plate of sandwiches and extended it toward her. "You're hungry after your journey."
She stood, ignoring a yelp of discomfort from her abused body. This at least she knew how to do. Something familiar in the sea of unfamiliarity. "Shall I pour your tea?"
"Thank you." Gideon put down the plate as Pollett entered the room. Charis concentrated on fiddling with the tea things, her color rising as she recalled Pollett's quick assumption that she was Sir Gideon's mistress.
"Is all in order, sir?"
"We need a fire," Gideon said, taking a seat near the table.
As Pollett left, Charis passed Gideon his tea and a plate with two sandwiches arranged upon it. Her left arm made the simple duty more trouble than usual, but she managed. Such a small achievement, but enough to revive her spirit.
He smiled almost naturally. "So this is what it's like to be under a lady's dominion."
She frowned with puzzlement. "Surely you've taken tea with a female before."
"Never alone. Never in my own house." He swallowed a mouthful of tea and lifted his chunky sandwich for a healthy bite. Whatever his illness of yesterday, it seemed to have passed.
"What about your mother?" She took the chair opposite. As she sipped from her cup, she stifled a sigh of pleasure. It was a small luxury, yet one she'd missed.
His face became expressionless. "My mother died at my birth. My father didn't remarry, having already sired two sons and seeing no need to submit himself again to the yoke of matrimony."
"I'm sorry about your mother." Had his mother bought the pretty china and chosen the delicate, faded fabrics that upholstered the furniture? So much death marred his life. Was this what darkened his soul? Sadness thickened her throat, and the tea abruptly lost its flavor. "No feminine influence at all in the house?"
His lips quirked. "No ladies at any rate."
"Oh."
She couldn't control a blush although her heart beat faster at the idea of him with a woman. He wouldn't sit across the table, drinking tea. He'd snatch her up in his arms and kiss her and...She tamped down the wanton images before she made more of a fool of herself than she had already. Her face felt like it was on fire.
The smile became a smirk. "Indeed."
She dragged her mind kicking and screaming back to reality and looked around the room. Anything to avoid his knowing glance. Now she thought about it, the house shrieked its lack of chatelaine. Penrhyn badly needed a woman to take charge and restore its former glory.
Perhaps the absence of early feminine influence explained Sir Gideon's awkwardness with her. Although he didn't strike her as an innately shy man. Again, she wondered if he disliked her. The possibility made her belly tighten with denial. She dearly wanted Sir Gideon's approval.
Surely he must like her just a little. His manner at times such as this was almost intimate. Certainly more intimate than she could remember encountering in other gentlemen. Every time he turned that warm regard on her, she felt like a sunflower opening to the sun. She knew the reaction was improper, dizzying, perilous, but she couldn't help it.
He broke the tense silence and spoke with a polite formality that chilled the already icy air. "I hope you'll treat the house as your own, Miss Watson. Go where you please. Read anything in the library. There's a pianoforte in the morning room—or there used to be. I wouldn't advise you to stray too far from the grounds in case you're seen. Although I suspect your injuries put anything too energetic out of reach at present."
"Thank you," Charis said dully. Stupid to long for Sir Gideon's arms to close around her. She forced herself to remember they were chance-met strangers. This silly wayward lilt of the heart was purely one-sided.
All this emotional turmoil on top of the beating and the long days of travel conspired to sap her last ounce of energy. With a tired gesture, she set her cup in its saucer. Every second intensified her multitude of aches. Her head thickened with weariness.
He rose from his chair and moved across to a sideboard, where he splashed some brandy into a glass. "The house and estate will demand my attention for the next few days. Penrhyn's been too long without a master." She recognized his tone as a deliberate attempt to put her at a distance.
"You don't have to entertain me or neglect your duties on my behalf." Her voice was flat with disappointment. But what had she expected? That he'd devote his attention to her? Much as she wanted his company as a buffer against the unfamiliarity.
Charis, don't pretend that's the reason.
She forced an even tone. "You've already done so much for me."
"Don't be absurd." He emptied the glass in a single swallow and set it down with a crack. "I did what anyone would have."
"You're too modest, Sir Gideon."
"Don't make me out to be more than I am, Miss Watson." His eyes glittered like obsidian as they focused on her. The tension that extended between them like a thin golden wire tightened to breaking point. "I'm as miserable a sinner as ever walked this earth. Pray remember that."
The invisible wire linking them snapped. He turned and stalked from the room, leaving her to stare after him in hopeless, hurt bewilderment. The sun turned away from her, and she shivered in the sudden, biting cold.
## Six
Over the next days, Gideon saw little of Sarah. With his guest recuperating in her room, avoiding her proved a surprisingly simple matter.
They shared dinner under the curious eyes of his servants. Occasionally, they crossed paths in a corridor, and he'd inquire after her health. All perfectly polite, two strangers passing the time of day. Thankfully, there was no hint of the burgeoning, dangerous intimacy that had hovered on the journey to Penrhyn.
With every encounter, he couldn't help noticing the remarkable beauty that emerged from beneath the disfiguring bruises. It was yet another of fate's cruel jokes that the desperate, injured girl turned into a woman of spectacular attractions who stirred his sluggish blood.
It was unlikely her brothers would track her this far, but Gideon wasn't taking chances. He made sure someone always knew where she was. A pack of brawny villagers boosted the household staff, and shifts of men patrolled the approaches to the house.
Even if he'd wanted to play nursemaid, he wouldn't have had time. He was frantically busy. Mostly he was absent from the house, fielding endless requests and questions, and making decisions about the estate. After years of neglect, there were a thousand matters, small and large, to address.
What became abundantly clear during his first day as its reluctant master was that Penrhyn was in his blood. He was home to stay.
He could no more abandon the place than he could fly to Constantinople. When he'd seen the old house again, a sullen, unwelcome love had flooded him, a bone-deep sense that Penrhyn was meant to be his. Illogical, inconvenient, but undeniable. He couldn't relinquish this windswept corner of the kingdom to anyone else's stewardship. Although God knew who he kept it for. He was the last Trevithick. There would be no sons to inherit.
That sad fact haunted him, a mournful threnody beneath his activity. And if the memory of one delicate woman also haunted him, he was too occupied to brood on the fact. At least during daylight hours. Nights were a different matter. He'd throw himself exhausted on his bed, only to lie awake listening to the endless crash of the waves and thinking about Sarah. Or worse, drifting into restless dreams where he was free to touch her as he never could in the harsh light of reality.
With every hour, that hankering to touch her intensified. With every hour, the pain of knowing that he never would lacerated him.
On the morning of his third day at Penrhyn, Gideon shut himself in his library, determined to tame the chaos his predecessors had left of the accounts.
He'd been at work for about an hour when Sarah wandered into view through the tall windows facing the overgrown parterre. The dusty ledger in front of him immediately lost what small interest it held. He watched for Dorcas or one of the men set to guarding Sarah. But his visitor remained alone in the dewy, sunlit garden.
For a forbidden, secret moment, he stared, drinking in her beauty. The bruises were barely noticeable now, and her face resumed its natural shape. Since yesterday she'd discarded her bandage, and she no longer moved as if every step hurt. To his relief, Akash's assessment of her injuries as looking worse than they were had proven accurate.
Sarah paused in a patch of light and turned her face to the pale February sun. Her lips curved with a natural sensuality.
Gideon's heart battered his ribs. His breath jammed in his chest. She was glorious. None of the fabled courtesans of India held a candle to her uniquely English loveliness.
Was he so shallow that her pretty face made him want her?
If only the truth were so uncomplicated. He could resist the lure of beauty if beauty alone attracted him. But the waif he'd rescued in Winchester had become a woman of endless allure. Strong. Brave. Tender. Sweet.
Ah, so sweet.
A long plait fell down the supple line of Sarah's back. Gideon's hand, lying idle on the desk, flexed as if it tangled in that silky bronze mane. He locked his teeth and cursed himself for a fool. Such fantasies were futile.
Knowing he tormented himself to no purpose, he hungrily watched the subtle sway of her hips as she started walking again. The way the ill-fitting cotton frock skimmed her lissome waist. He frowned. Why was she still wearing the cheap dress from Portsmouth? He'd asked Mrs. Pollett to find her fresh clothing.
He'd sort it out later. He bent to his work, determined to punish himself no further with impossible yearnings. Then, helplessly, he raised his gaze as Sarah strolled through a morning more like April than February and disappeared behind a hedge of overgrown camellias.
A page of figures his eyes failed to register. Another. Another.
From here, the grounds sloped down to the cliffs. Given the decrepitude of the rest of the estate, Gideon guessed the paths were unstable, falling to pieces. There was danger for someone who didn't know Penrhyn. Devil take them, where were the people supposed to be watching her?
"Damn it," he muttered, and shoved the thick ledgers aside. He snatched his gloves from the desk and leaped into a run.
Charis was sitting on a worn stone bench when she heard Gideon's purposeful footsteps. He was in a tearing hurry. She couldn't imagine why. Especially as he'd worked so hard to stay out of her way since they'd arrived. She tried to tell herself he was busy, and she had no right to feel slighted, but some instinct insisted the lack of contact wasn't accidental.
He broke into the cleared space and paused, breathing heavily. He appeared to be searching for something.
Although she'd sworn she'd behave with circumspection in his presence, although she'd preserved a polite façade when encountering him in the house, her heart beat so fast, her greeting stuck in her throat. She hadn't expected to see him this morning, and his arrival threw her good intentions into disarray.
He looked toward the cliff edge, scanned the clearing, then finally turned in her direction. His face flooded with visible relief. "There you are."
Every time she saw him, it was like the first time. As she experienced anew the shock of his male beauty, the world seemed to tumble away from her feet, leaving her suspended in space. The sensation was dizzying, scary, overwhelming.
Today, his onyx eyes were clear, and he moved with an easy freedom that fitted his long-limbed body. He'd spent the recent days outdoors, and the exercise suited him.
She swallowed to dislodge the lump in her throat, but her voice still emerged as a croak. "Sir Gideon, what's wrong?"
"I saw you heading down here." He ran his hand through his hair, ruffling it into beguiling untidiness. "I wasn't sure of the state of the cliff edge."
It hardly hurt to smile now. Just a slight ache. A glimpse in her bedroom mirror before she'd come outside had revealed a face she finally recognized as hers. "So you rushed to my rescue again." She tamped down a twinge of forbidden pleasure that he'd come seeking her.
He made a noncommittal noise in his throat. "You're looking better."
"I'm feeling better." She fiddled nervously with her mother's pearl ring and tried to think of something clever to say. Nothing came to mind. Hard to recall she'd been the toast of Bath society. Sir Gideon made her act like a gauche schoolgirl.
"I'm glad." That half smile appeared. Odd—disturbing—how familiar and dear it was.
A charged silence fell. She knew she devoured him with her eyes. What made no sense was that he seemed to devour her in return. Then it was as if he recalled his resolve to keep his distance.
"Well, my apologies for disturbing you." He sounded stiff, awkward. "As you're in no immediate danger..."
"I'll be careful."
She wished she could make him stay. Absurd when they were strangers, but she'd missed him in the last days. To her chagrin, Sarah found herself blushing, as though she spoke her foolish yearning aloud.
She waited in tense misery for him to forsake her to loneliness. But he took a step closer and gestured to the glorious view. The sea was blue and calm today. The waves played like soft music under their conversation. "It looks gentle, but don't mistake its peril."
"I can hardly resist exploring. I hope you don't mind. Penrhyn has such fairy-tale charm." Her instant affinity for this place had only strengthened. Each night, she went to sleep in her paneled corner bedroom listening to the sea. "Like La Belle au Bois Dormant."
Again that half smile. Her poor, longing heart skipped a beat every time she saw it. "On my honor, there are no sleeping princesses here, Miss Watson."
"Perhaps a prince?" she asked lightly, then regretted not keeping her mouth shut.
His expression closed, became remote. "No princes either."
She waited for him to storm off as he had from the library the last time she'd attempted to share more than platitudes. But he remained where he was, frowning down at the ground.
Eventually, she broke the uncomfortable silence. "What are your plans for the property?"
His eyes were guarded as they focused on her, but to her surprise, he answered readily enough. "There's potential for the estate to be profitable. It was once. The woods contain good timber and while the land isn't much use for crops, it will support sheep. Most of the skilled men have gone, but we could set up a fishing fleet again. First I mean to reopen the tin mines."
"Tin?" She leaned back on her arms. She still wasn't used to having the full use of both arms. Her wrist gave the occasional twinge, but it was almost back to full working order.
"Yes." He moved close enough to raise one booted foot onto the far end of her bench. He rested one arm on his thigh and bent toward her. Her skin prickled with awareness, and her breath became shallow and choppy. She prayed he didn't notice. "The land is littered with worked-out diggings, but there's still ore to be found. The sea and tin have always kept the Trevithicks."
He spoke with an odd lack of involvement, but she wasn't convinced he was as unemotional about his home as he wanted her to believe. She'd seen his face when he glimpsed it for the first time upon his return. "Will you restore the house?"
To her astonishment, a glint lit his dark eyes. "I'll demolish it and build a modern villa."
Shocked, she jumped to her feet. "That would be an act of unforgivable vandalism."
He laughed softly. "Just teasing you, Miss Watson." To her regret, he straightened and shifted out of reach. "I've remarked your predilection for Penrhyn."
Her color rose, and she curled her hands at her sides. "I can't believe you don't care. The house needs to be loved."
The more she saw of Penrhyn's master, the more she believed that was true of him too. How she wished she could restore him to joy. But the last days had made it apparent that he regarded her as a duty and nothing more.
"It's only bricks and mortar," he said mildly.
"You'll feel differently when you have children," she said fiercely, even as she flinched to contemplate him marrying another woman.
The brief moment of levity evaporated. His voice was terse. "I have no plans to marry."
"Of course you'll marry. You're young, you're handsome, you're..."
He silenced her with a cutting gesture of one hand. "Spare my blushes, Miss Watson."
His sarcasm stung, although she knew she deserved the set-down. Her cheeks stung with humiliated heat. She wished she could keep her impulsive comments to herself, but something about Sir Gideon made her burst into ill-considered speech at the very worst of moments. The merest sight of him, and any pretensions of poise flew into the ether.
"I'm sorry," she said in a subdued voice. "I had no right to say those things. You must think I'm a rag-mannered hoyden."
"No."
Just "no"? What was she to make of that? What did he think of her? She stifled the needy, desperate questions that struggled to the surface. She'd already embarrassed him—and herself—sufficiently. Frantically, she cast around for some neutral topic. "When I came out, I was looking for the path to the beach."
His mouth lengthened with disapproval. "It's steep and not easy for a lady. That's how I remember it nine years ago. I suspect it's in worse repair now. You'd be better staying in the grounds."
Lady Charis Weston would have stepped aside, let him return to his work as he clearly wished. Sarah Watson was a more demanding creature and desperate for a few more minutes of his company. "Can't we at least try?"
Sudden amusement flashed across his face, banishing the sternness, making him look years younger. "You're a stubborn scrap of a thing, aren't you?"
Even more astonishingly, his black eyes swept her body, subjecting her to a thorough, masculine inspection. Instant agonizing tension extended between them. Heat crawled over her skin, and her heart bucked and plunged in her chest. Her nipples puckered with painful swiftness, and something warmed and melted in the pit of her stomach.
The powerful, unfamiliar sensations frightened Charis. It was as if the body she'd known for twenty years suddenly belonged to a stranger. With every ragged breath, the hard points of her nipples rubbed against her shift. The friction was maddening, unstoppable, infuriating.
She lifted a shaking hand to her breast to ease the ache, then realized what she did. Her face became hotter. He couldn't miss her discomfort. She wished the ground would open up and swallow her like the whale had swallowed Jonah.
She lowered her head to hide her mortifying reaction, to break that scorching connection with his eyes. "Not exactly a scrap," she muttered, turning away to rip at the leaves of a camellia.
"No, perhaps not." He released a harsh laugh, bitter and without amusement. She didn't have the courage to check his expression. "Let's show you our fine beach."
She sucked in a shuddering breath while delight and self-consciousness vied within her. Now that she wasn't looking at him, she gained some small control over herself.
"I'd like that," she said almost inaudibly.
Feeling like the greatest fool in Creation, she scattered the shreds of greenery on the ground and nerved herself to glance at him under her lashes. She'd expected to see anger or contempt or disgust, but his expression was, as so often, inscrutable. Was there a chance he hadn't noticed how flustered she was?
At least he was still here. More, he planned to escort her to the beach. Breathlessly, she waited for him to take her arm, but he merely gestured her toward the overgrown path and fell into step behind her.
He went ahead once they had to fight their way through a mass of untidy rhododendrons. Like everything else at Penrhyn, the garden reeked of neglect. Charis knew it was insane but she felt that the house cried out to her to save it, to make it a home.
Stupid fancy. She was only a temporary visitor to this beautiful place. She'd leave soon, to be quickly forgotten by Penrhyn and its owner.
The bleak knowledge set like concrete in her belly.
Her host was as unkempt as the manor. She studied his tall figure as he forged a path for her. He wore breeches and shirtsleeves, and his boots were old and scuffed. Still, he was utterly splendid. Her pulse, which had started to steady, kicked into a gallop again. She pictured him standing on the prow of a ship. A gold ring glinting in one ear. A cutlass at his waist. A knife clenched between his teeth.
He stopped to lift a prickly bramble high over her head. "What are you smiling about?"
She hadn't realized she was smiling. "Were any of your ancestors pirates?"
"Black Jack Trevithick was one of Bess's Sea Hawks." As she passed him, he flashed her a grin that was devilment personified. Her unruly heart somersaulted. Heaven help her. "His portrait's in the long gallery. At least it was. Black Jack looks like me, so my father may have retired it. My father and brother took after my grandmother's family, the St. Ledgers. But I'm all Black Trevithick."
"Is that because of your hair color?"
"Partly. Also black temper, black nature, black sheep, black heart."
She couldn't restrain a startled laugh as she pushed her way through the shrubbery ahead of him. "Goodness. I find myself quite terrified to be in your presence."
Of course it wasn't true. Gideon Trevithick's company was as intoxicating as champagne. He unsettled her more than anybody she'd ever met. He confused and troubled her. But she could hardly countenance that once she left, she'd never hear his voice again.
Although of course it wasn't just his conversation that made her head swim with excitement. He was handsome. More than handsome. He was beautiful, like some being sent down from heaven to illuminate dull earth. And strong and virile and manly. No woman with blood in her veins could fail to respond to his attractions.
Perhaps when he knew who she really was, he'd consider courting her. She saw no evidence of huge riches at Penrhyn. Could he overcome his disinterest in her person if he knew he married the greatest heiress in England? The Earl of Marley's title had lapsed along with the entail upon her father's death. Every penny, every acre, of the massive Weston inheritance devolved upon the earl's one direct descendant, his daughter.
Dear Lord, was she so lacking in pride, she'd trade gold to gain the man she wanted even if he didn't want her? Her belly clenched in sick shame. She needed to leash her foolish imagination before it brought her to grief.
They emerged from the bushes onto the cliff edge. Below, the sea spread like shining blue silk. Gideon paused behind her. She was so attuned to him, she felt his every breath. An unwelcome premonition brushed across her skin and made her shiver. This preternatural awareness seemed more significant than mere physical reaction.
"This is such a beautiful place," she said softly.
With an unwillingness she immediately recognized, he moved closer. A light wind played with his thick hair. Lucky breeze to take liberties with him that she couldn't. She closed her fists at her sides to stop herself smoothing the disheveled locks. It disturbed and frightened her, this continual, frustrated need for physical contact. It left her jumpy and awkward.
She watched him draw in a deep breath of crystalline air. The tension seeped from his broad shoulders as if the view fed his soul.
"I didn't realize how much I'd missed it. The sea. The wind. How...clean it all feels." His eyes remained fixed on the horizon, but she had the strangest impression he saw something else entirely. Something that haunted him. "When I was in Rangapindhi, I remembered this view. It made me want to live."
She must have made some sound of protest or surprise because he stiffened and turned his head, fastening those glittering eyes upon her.
"Why wouldn't you want to live?" she echoed, shocked.
He frowned. "Do you truly not know? My story has been in all the papers. Quite the sensation of the season." He spoke with a biting sarcasm she didn't understand.
"My brothers kept me prisoner. I'd never heard your name until we met." She curled her arms around herself, although the chill she felt was more spiritual than physical. "Those men in Portsmouth called you the Hero of Rangapindhi. Were you a soldier?"
"No." He bit the word out like a bullet fired from a gun. His unspoken pain was a vivid, twisting, tangible entity.
Charis tightened her arms to stop herself reaching out. A stinging mixture of compassion and desire lodged in her throat. She forced her question past the constriction although she was sure he'd dismiss her curiosity. "Did you hate India so much?"
His regard was unwavering, and his voice deepened with emotion. "No, I loved it."
It was the same answer he'd given her when she asked whether he hated Penrhyn. Gideon Trevithick seemed to have an ambiguous relationship with love. Again, she wondered at the despair that shadowed him, closer to the surface today than she'd ever seen it.
He sighed, and his shoulders slumped. "I wish I was the man you think I am." His voice was so sad, it made her want to cry. "But I'm not worth an ounce of your regard."
She sensed the acrid shame beneath his words. He was dauntingly complex, and he drew her more powerfully than anyone else ever had. After a long silence, she dared to ask, "Will you tell me why?"
"No. I don't want you to share my nightmares." His smile festered with bitterness. He lifted his gloved hand. For one breathless moment, she imagined he meant to touch her cheek. Her eyes fluttered shut as she waited to feel the brush of his fingers.
Nothing happened. She opened her eyes slowly to catch poignant sorrow on his face. His hand fell to his side. "But believe me when I say I'm no hero."
She swallowed, and her voice shook as she spoke. "You're a hero to me."
The regret drained from his expression, leaving comprehension and a pity that stabbed her like a knife. "Miss Watson..."
Intent on silencing him, she made a gesture of denial. She didn't want comforting platitudes. The pity in his eyes indicated he divined her unseemly hunger for him. How could he not? The feeling was too overwhelming to hide, and he was a perceptive man.
She blushed with mortification, and spoke quickly, before he could. "Aren't...aren't we going to the beach?"
He straightened, his mouth firming. But he didn't argue with her abrupt change of subject, for which she was grateful. "The path is just here."
He walked past, and she realized the ground dipped away sharply. A few steps after him, and she saw a thin track snaking down the cliff.
Charis looked down, and her stomach lurched. Far below, rows of jagged rocks awaited. Resolutely, she lifted her head and stared at Gideon's straight back in his loose white shirt. As he began to descend, he was utterly at ease in this rough, dangerous terrain. Not hard to imagine a gangly, intense, dark-eyed boy seeking refuge from a troubled home life among these cliffs.
Charis stepped carefully after him, not surprised at his silence. Now he'd guessed how besotted she was, he must wonder what he could say. Humiliation added its sour tang to the poisonous brew of unhappiness and longing stewing inside her.
At first the going was easy, the slope gradual. The path was in surprisingly good repair. But soon the track narrowed, became steeper. She placed a hand on the rock face as the descent grew more precipitous.
For one fatal second too long, her eyes dwelt on the tall man ahead. Every scrap of information she gleaned about him only fed her curiosity to know more.
The path dipped. Her foot slid on a loose stone. She clutched wildly at the rock wall, but her fingers slid uselessly across the cold, smooth surface.
"Gideon!" she screamed.
Dear God, she didn't want to die. She wanted to live and make Gideon love her.
The thought, bright and burning like lightning, seared her mind as she tumbled helplessly toward the edge.
## Seven
Sarah!" Gideon whirled and lashed out to grab her before she plummeted to her death.
His hands closed like manacles around her slender wrists. There wasn't time to think or feel. There wasn't time to recoil from the shock of physical contact. He pivoted and slammed her back hard into the wall.
She screamed again, with pain this time, as her head banged against the rock. Then she closed her eyes and sagged, trembling and gasping from his hold.
He slumped over her, silently protecting her with his body from the drop behind him. His gut churned, and terror tasted rusty on his tongue. His chest heaved as he fought for breath, and his shoulders ached with the strain of snatching her to safety. He didn't relax his punishing grip although he shifted to press her hands flat into the rock on either side of her.
Hell, he'd come so close to losing her.
He leaned his forehead on the rock above her head, waiting for the wildly careening world to slow and stop. Dizzying relief thundered through him. Cold sweat chilled his skin as his mind replayed over and over the few seconds when Sarah slid uncontrollably.
They remained unmoving, her facing him, his hands clutching hers, mere inches separating their bodies.
Gradually, Gideon's suffocating fear ebbed. Reality returned, his mind started to function. He heard the crash of the waves on the rocks below. He felt the cool breeze on his damp skin. He felt the path's unevenness under his booted feet.
Sarah lifted her chin with a curiously jerky movement and stared unblinking at him as if he provided her one sure compass point. Her pupils were dilated, and her face was haggard with shock and pain. Her lips parted as she drew a ragged breath.
With a spurt of guilt, he realized his unyielding grip must hurt her sprained wrist. Logic indicated she was safe. Even so, it was only with the utmost difficulty that he forced himself to release her left hand.
Biting her lip to smother a sob, she gingerly bent her arm against her chest. The fingers of her other hand twisted to twine convulsively around his.
"Sarah, dear God..." His choked whisper ruffled the soft hair on the top of her head. "Are you all right?"
She gave an unsteady nod. "Yes."
His heart still raced, and he shook like a dog in a thunderstorm. "What about your wrist?"
"I jolted it, but I don't think there's any damage." Wincing, she stretched her arm and carefully moved it. Her earlier bruises had faded to yellow, so the impression of Gideon's fingers was red and stark on her pale skin.
He cursed himself for a blundering brute. He hadn't had time to be gentle. All that mattered was keeping her alive. He sucked in a shuddering breath.
Suddenly he was aware how close they stood. He only needed to shift a fraction, and her body would brush his.
What the devil was wrong with him, standing over her like this? He knew better. He had to stop touching Sarah now. Now.
Familiar, unstoppable nausea rose. Blackness filled his head. With a roughness he couldn't help, he wrenched his gloved hand away from her. Blindly, he turned to press his back to the rock wall beside her. His gloved hands splayed against the stone as he struggled to mask his reaction. She was too close, but he couldn't bear to have her out of reach just yet.
For a long, taut moment, the only sounds were the mournful cries of the gulls, the pounding waves, and his hoarse panting.
Eventually, she shifted toward him. He didn't look at her, but he felt her study him. He was guiltily aware that he must frighten and confuse her. Explanations, apologies gushed up, but he furiously bit them back. His pride revolted at putting his humiliating state into words.
When she didn't immediately speak, he steeled himself to look into her ashen face. In a gesture that poignantly reminded him of the lost waif in Winchester, she cradled her wrist upon her breast.
Her voice emerged almost normally. "You saved my life again. How can I ever repay you?"
Oh, damnation. This was the last thing he needed. She stared at him as if he was St. George, and he'd just rescued her from the dragon. The unfettered admiration and gratitude in her hazel eyes sliced at his conscience. If he'd planned to discourage her interest, what had just happened beggared good intentions.
"You can repay me by being more careful in future," he said harshly. And hated himself as he watched the radiance dim from her eyes. In truth, he wasn't angry at her as much as at the whole bloody impossible situation. He had no right to bask in a beautiful woman's approbation, even if he had just saved her life.
Her cheeks, which had been pale as paper, flushed with color. Her response was muted. "I'm sorry. I wasn't paying attention. Yet again, my foolishness put you at risk."
His tone softened. "No harm done."
Which, damn it, wasn't true. With every second he spent in her company, the insidious bond between them tightened like drawn silk cords. His recent efforts to avoid Sarah had achieved precisely nothing. He was as irrevocably connected to her as he'd ever been.
She straightened and winced at the movement. This morning's tidy plait had loosened into a mass of bright flying tendrils around her face. He fought the urge to smooth that wild halo.
For one intense moment, she met his eyes, then her thick lashes fluttered to her cheeks. White teeth worried at her plump lower lip, and her breath audibly caught.
Just like that, in a blazing instant, sexual need kicked into fierce life. He hardened. His heart broke into a savage rhythm. Every drop of moisture dried from his mouth.
His sickness at touching her passed in a bright flash. What possessed him instead was worse. Because he couldn't do one damned thing to relieve his hunger.
The startling rush of desire left him reeling, light-headed. He'd accepted his lack of interest in women since Rangapindhi as a blessing. The only blessing. He'd assumed his indifference was permanent. What was the point of wanting what he could no longer have? Better not to want.
Dear Lord, let her not look down. Let her not see how aroused he was. He tried to edge away, but the narrow path gave him little room.
How the hell was he going to survive three weeks of this?
He couldn't touch her. Every dictate of ethics and morality and chivalry insisted he couldn't touch her.
If only principle was all that made him hesitate.
He couldn't touch her. That was the sodding tragedy of it.
She was still speaking. He fought back the clamor in his head and tried to concentrate on what she said.
"...few bruises."
Confound it, he needed to get a grip on his reactions. Through the buzzing in his ears, he battled to focus. He realized her good hand plucked unhappily at the sleeve of her plain gown. "...mend it."
He tore his gaze from her mouth. So soft. So moist. So tempting. And glanced down to her dress. He must have ripped the sleeve when he dragged her back from the brink. There was a gaping rent in the threadbare brown material.
That was one problem he could solve, surely. He sucked in a tattered breath and spoke over her stumbling explanations. "I'll take you back to the house. One of the servants must have something you can wear."
She sent him an odd look. He hoped to Hades she had been talking about clothes. "As you wish."
He frowned. She sounded disappointed. "Are you sure you're not injured?"
Her restless hand tangled in her skirts, and she looked away. "Of course I feel a little knocked around. But, no, I'm not seriously hurt. Thanks to you."
"Miss Watson, there's no call to harp on your totally unnecessary gratitude," he said repressively.
He flushed as he realized he barked like a displeased sergeant dressing down a recruit. She cast him a resentful glance that scorched him to his soles. He needed to get away. Fast. But his feet were welded to the path.
"I hardly think it's unnecessary." Her tone was soft but firm.
"Sarah..." He knew it was a mistake using her Christian name the moment the word emerged. He needed to resist further closeness, not reinforce it.
"I won't refer to it again." She still sounded subdued.
"Shall we go?" He gestured her past him, but she hesitated.
"Sarah?" Damn, he'd said it again. Every second in her presence extended his torture. If he didn't put some distance between them soon, he'd grab her. Then the shaking would start, and the sickness and the humiliation.
"Can't we go down to the beach? Only for a minute?" She sounded wistful, like a child denied a treat. "I've been cooped up for so long. I'd love to see the sea. I've never been so close to the ocean before."
He desperately tried to ignore the plea in her hazel eyes. Curling his hands into fists, he strove to steady his tone. "You need to rest."
Her lips—Lucifer himself must have created those moist, red lips—turned down in a dismissive quirk. "I'll be careful on the way down. I'm not such a fragile vessel as you imagine. I've had a shock, but I'm perfectly all right. What sort of girls have you been talking to?"
"I haven't been talking with many girls at all," he said before he could remind himself that swapping confidences with his gorgeous tormentor was unlikely to ease his predicament.
With every second, she looked more like her usual self. "You surprise me."
Curse her, why did he feel the urge to explain? "I told you Penrhyn was a masculine province."
Apart from his father's blowsy mistresses, who occasionally took up residence. His father's taste had run to the overblown, the obvious. None of those women had been remotely interested in a studious stripling, for which Gideon had been heartily grateful.
"Surely when you left home..."
"I went to Cambridge at sixteen and immersed myself in study."
Frowning thoughtfully, she laced her hands at her waist. A sign he hadn't done her wrist serious injury, he was relieved to note. "The university men I know caroused their way through their education."
His smile was grim. "I suspect the men who paid court to you weren't second sons with no prospects. I was much younger, not to mention poorer, than most of my fellow students."
If he were another man with another life, he'd surely have been among those men who courted her. He straightened as if physically resisting the forbidden idea. A stray strand of windblown hair briefly clung to her lush lips. Another blast of sensual awareness shook him. He fleetingly closed his eyes and told himself he mustn't under any circumstances kiss her.
He breathed deeply, struggling for composure. When he could see straight, he stepped past so he could precede her down the cliff, in case they struck any more unstable patches. Against his every instinct, he'd take Sarah to the beach. He knew when he was beaten. "Watch your step. It's steep, and you've used up at least three of your nine lives today."
"Thank you," she said softly to his back. "I know I'm a trial."
She had no idea just what a trial she was. Pray God she never found out. Craving to seize her in his arms tightened his skin and made his heart gallop as it had galloped when she teetered toward the edge. Except this time with lust rather than terror.
The reminder of her fall made him slow his pace. His hand itched to reach back and grab hers, in case she stumbled. Such a natural action, yet completely outside his capability. He couldn't risk another of his attacks. He cursed himself and his affliction.
On the way down, he frequently glanced back to check on her. Her near disaster had obviously convinced her to treat the path with respect, and she negotiated it with visible concentration. Thank God. At least it checked her questions.
When he reached the base of the cliff, he jumped from the rocks to the beach. He landed hard on the firm sand and turned to watch Sarah carefully climbing from boulder to boulder.
Guilt bit at him as he remembered how he'd shoved her against the rock wall. For all her brave words, he recognized the stiffness in her movements as discomfort. He bit back a demand that they return to the house. After his experiences in Rangapindhi, he understood better than most her need for freedom.
She crossed to stand at his side just past the high-water mark. The bruises on her face were mere shadows now. In the bright clear light, her beauty was flamboyant, heartbreaking. She made him feel as close to alive as he ever expected to again.
The errant breeze flirted with her hair, teasing it around her face as she turned to him. "So you went to India to make your fortune?"
More blasted questions. He wished he had the heart to tell her to mind her own business. But he couldn't resist the honest interest shining in her eyes.
His voice was stilted as he replied. He wasn't used to talking about himself, and every time she pried a confidence out of him, it was an acknowledgment that they were more than just chance-met strangers. "An opportunity arose."
Gideon began to walk along the coarse yellow sand, and she fell into step beside him. She flattened her hands on her skirts to stop the wind lifting them, but still he caught a breathtaking glimpse of slender ankles and shapely calves. He closed his eyes briefly and prayed for strength.
She was going to kill him before she was done.
"With the East India Company?"
He dragged himself back to the conversation and tried to ignore how lovely she was. He made himself go on, partly to distract himself from the pale flash of Sarah's stockings.
"My talent for languages attracted the attention of powerful people." He spoke without vanity. He had a freak facility for picking up foreign tongues. Some strange tic in how his mind worked. "They thought I could be useful."
"As a trader?" She bent to pick up a scallop shell, the movement hitching up the back of her dress. He stopped to watch her, then wished he hadn't. His hands flexed at his sides as he fought the urge to toss those skirts up to a more pleasurable purpose.
Because to his eternal regret, there could never be pleasure.
"More as native liaison." The answer was strained. He didn't want to tell her the truth, that he'd been a spy. Of course, if she cared to investigate, she'd find out. His life had been sensationalized in every newspaper in Britain. In the world, for all he knew.
Elements of the press coverage were true, at least superficially. The papers had invented the rest, each story more lurid than the last. In the public mind, he'd become a bizarre mixture of Robin Hood, Casanova, and Sir Galahad.
The cruel farce of his celebrity made him cringe.
She straightened and ran a thoughtful finger along the edge of the hard white shell. He already knew enough to guess another question percolated. "Were the Indian girls beautiful?"
"Yes."
She glanced quickly up at him, then away, a delicate pink washing her cheeks. "Were you in love with someone there?"
Dear Lord, were all women so fixated on love? He'd heard more on the subject today than he remembered hearing in all his twenty-five years. Against his will, he found himself answering. "No."
The man who stepped off the ship in Calcutta seven years ago had never known a lover. But Gideon's fascination with Indian language and art, nurtured in the dusty library of his college, became a fascination with the living, breathing culture. And soon the living, breathing female embodiments of that culture.
That first six months as he traveled around the Company's offices and residences, he'd succumbed to hedonistic license. The women were beautiful and generous and adept at pleasure. He'd never imagined a world like it. Sex became a drug.
His hedonistic existence came to an abrupt end once he entered the field. The dangers of betrayal were too great.
He swung away from further questions and strode along the beach, his long legs eating up the stretch of sand. The gulls cried overhead. The loneliest sound in the world.
He should have known she wouldn't let him escape. Running footsteps crunched behind him, then he felt the soft touch of her hand on his arm.
Through his shirtsleeve, that contact scorched. Rapacious hunger jolted him even as his flesh crawled. He jerked free. "Don't touch me!"
She recoiled, her eyes darkening with such pain that he flinched. "I'm sorry," she said huskily.
He fought to speak normally, but his voice emerged dull and flat. "No matter. I don't like to be touched."
Her mouth straightened into an unhappy line. "By me, at any rate."
God in heaven, how much of this could he take? He sucked in a lungful of salty air and floundered for control. "It's not you."
She shook her head and raised a hand to keep her wind-tossed hair from her eyes. He couldn't mistake the anguish in her face. "Don't spare my feelings. I've noticed your revulsion for my presence."
He let his breath out in a despairing hiss. "That's not true."
Sarah's slender throat moved as if she stifled a protest. Hell, he hated to hurt her. He felt like the lowest bastard in Creation, even though he acted for her sake as much as his own.
Don't be a blockhead, Trevithick. The girl isn't suffering from genuine love but from a bad case of hero worship. She'll survive without ill effects.
"Miss Watson...Sarah..." He stopped, struck silent by her vibrating misery.
"You must consider me a foolish creature." The breeze whipped at her low words, so he had to lean closer to hear. A dizzying waft of her scent mixed with the salt air and made his nostrils flare in masculine response.
A torrent of words fought to escape, words that told her how exquisite she was, how brave, how wonderful. He stifled them all. He had no right to pay compliments to innocent young girls.
"I have a great-aunt who would be horrified at my behavior. She worked hard to turn me into a lady." Sarah hesitated, sucked in a breath, then went on in an artificially bright voice. Gideon knew she desperately strove to ease the prickling tension between them. "I was quite the tomboy when she took me in hand. My father raised me much as he'd raise a son. You see, the estates would all be mine one day."
Even through the wild tumult in his head, Gideon knew this didn't make sense. He frowned. "Wasn't your oldest brother the heir?"
Guilt flooded her vivid features. "The entail had come to an end. My father..."
Her shoulders sagged as she relapsed into troubled silence. Gideon had noticed before that she wasn't a good liar. He was an excellent liar—he'd learned to be as defense against a violent father. He'd perfected the skill, playing a role where discovery of his identity meant death.
"They're my stepbrothers," she said in a subdued voice. "My father died when I was sixteen..." The sunlight shone stark on her expression of naked grief. "My mother remarried. Her husband had two adult sons who hated me on sight."
Gideon shifted closer as if even on this deserted beach, he protected her from her rapacious family. His mind flared with a fierce, relentless urge to kill anyone who threatened her. His voice roughened with the power of his anger. At last he discovered her secrets. At last he came to grips with the forces ranged against them. "Those are the swine who beat you?"
"Yes." Sarah paused, then continued with a reluctance he could hear. "My mother passed away not long after she married my stepfather. Her choice hadn't been a happy one. Her new husband was a drunkard, a gambler, and a wastrel. From the first, he was openly unfaithful."
Gideon's gut clenched as he read the pain she tried so hard to hide. If he was any sort of man, he'd take her in his arms and offer comfort. But of course, he was no man at all. "Have you lived in that bears' den since you were sixteen?"
Sarah shook her head and tossed the scallop shell to the ground with a disgusted gesture. "To them, I was just another useless mouth to feed. After my mother's death, I went to a great-aunt in Bath. She's the one who tried to instill some manners." The desolation faded from her face, and real affection tinged her smile. "Great-aunt Georgiana was determined to find me a brilliant match. Bath in the season is a social whirl."
"I'm sure you didn't lack for suitors." Absurd to be jealous of these unknown men who had flirted and danced with her.
She shrugged and looked toward the waves, her color rising. He studied her profile. Those men had seen exactly what he saw now. Innocence. Generosity. Beauty. And a fresh and fragrant sensuality that drew him like a bee to honeysuckle.
Gideon had believed himself immune to female allure. Good God, the merest contact with anyone's skin set him shaking like a windblown leaf. Yet this girl promised such passion, even he couldn't resist.
She began to walk up the beach. Silently, he joined her, pleased to note she moved more easily now she was on flat ground.
"My stepfather fell down the stairs in a drunken stupor and broke his neck." Her tone deepened with contempt, and her hands tangled in her skirts. "My stepbrothers inherited nothing but crippling debts. And whatever they could wring out of being named my guardians in the will."
Ah, this was the crux of the problem. As her legal guardians, her stepbrothers had every right to compel Sarah back into their custody. No wonder she'd been so reluctant to confide the details of her dilemma to a stranger. Gideon broke the law by sheltering her. That fact alone would cause many people to hand her over to the authorities, whatever the personal issues involved.
Gideon kept his voice even, much as he wanted to rage and curse the mongrels who had hurt her. "So legally you're at their mercy."
"Yes, unfortunately. After they took me from my great-aunt, they launched the scheme to marry me off." A wayward gust blew a long strand of hair across her face, and she absently brushed it back. Her tone developed an edge. "When they realized I wasn't so gullible, they tried to put me completely in their power. No letters in or out of the house. No newspapers. If I tried to visit the village, they stopped me. At first with excuses. Later with threats."
Poor chit, relying on spirit and cleverness, in a situation where only brute strength counted. "Couldn't you bribe a servant to take a message?"
She shook her head. "The servants knew any chance of wages relied on my marriage."
A scorching need to smash her stepbrothers into jelly filled him. Almost as scorching as his urge to sweep this girl into his arms and kiss her senseless. And what a damned disaster that would be. "I suppose as your birthday approached, they became desperate."
She stopped and sent him a stark look. With one hand, she held her hair away from her face. The freshening breeze finished the destruction of her plait. In her thin gown, she must be cold although she showed no sign of it.
"Naïvely, I thought some code of gentlemanly behavior would constrain them." She went on in a curiously flat tone as though she distanced herself from what she said. "They cut back my meals. They locked me in my room. At first the violence was casual, and they made sure the bruises wouldn't show. I can't imagine why they troubled. It wasn't as though the servants didn't know. And I saw nobody else."
She paused as if waiting for Gideon to comment. But he was too angry to trust himself to speak.
"At least the violence was honest." Her voice scraped into rage and her fists curled at her sides. "It was worse when they insisted the marriage was for my own good. That made me sick to the stomach." She looked over the waves again but not before he caught a flash of fury in her eyes.
"Damned curs," Gideon muttered under his breath. An inadequate response. But everything was inadequate against what she'd been through.
"That last day was the first time they set out to beat me into obedience. Before Hubert started hitting me, Felix said I should save everyone trouble and give in before they made things really tough."
Gideon could imagine how she'd responded to that. "Of course you sent them to the devil."
"Yes. But then..." For the first time, she faltered and stared down at the sand in front of her. "Felix said..."
Nausea knotted Gideon's gut. He could imagine what came next. No wonder she'd been frightened out of her mind in Winchester. "You don't have to tell me."
He shrank from the trust he read in her gaze as she turned to him. She looked as if she believed he could move mountains. With bone-deep sorrow, he wished to God he was the man she thought he was.
Her color rose in a tide of shame. "Felix said they'd drug me and let my suitor take my maidenhead. I said they could do what they liked. Nothing would ever make me marry him."
His eagerness to murder her stepbrothers ramped higher, blocked his throat. "That was foolhardy."
She swallowed and continued in a toneless voice. "I knew they wouldn't kill me. If I die, the money goes to my second cousin, a bluestocking spinster who's lived all her life in Italy. I've never met her."
She spoke almost expressionlessly. Gideon's belly knotted with horror as he contemplated what she'd been through. He could hardly bear to formulate his question. "Did they force you?"
"No." Except for two hectic flags of color along her slanted cheekbones, she was pale. "But Felix said all three of them would take turns. Hubert wasn't in favor of the plan, but Felix always gets his way." She sucked in a shaky breath and spoke quickly as if that was the only way she could get the words out. "The idea of the three of them raping me, it was..."
"Intolerable." Bile filled Gideon's mouth as he imagined what would have happened if she hadn't fled. She'd survived a purgatory he understood better than most.
Her hands twisted more tightly in her skirts. "During the beating, Hubert knocked me out. Only for a few seconds. When I woke up, they started badgering me again. I wouldn't relent, so Felix slammed out in a temper, taking Hubert with him. It was the first time they forgot to lock the door. Perhaps because I'd made no attempt to escape, they believed I wouldn't or couldn't try to get out. While they were arguing downstairs, I crept into another room and climbed out a window that opened onto an oak tree. Thank goodness I knew the countryside enough to reach the Winchester Road."
"Thank goodness we found you in that inn." Nightmare images filled his mind of Sarah's rape and abuse. He had no doubt her stepbrothers would have carried out their threats. But now she was with him, and nobody would hurt her again. The determination to keep her safe stiffened every sinew.
Her voice became concerned. "I meant just to travel to Portsmouth with you, then disappear. By helping me, you're in danger too."
"I can handle your stepbrothers." He looked forward with bloodthirsty enthusiasm to exiling such scum to the lowest circle of hell.
His confident response drained some of the tension from her face. "You were amazing in that fight in Portsmouth."
Heat mottled his cheeks. He abhorred that only the spilling of blood made him a whole man. Violence dissipated the fog that possessed his mind, gifted him with clarity of purpose and unhesitating action. "I was a thug."
"You were a hero," she said with a conviction that made him wince. Dear God, what was he going to do about her misplaced admiration? He needed to scotch it now, but nothing he said made any difference. Knowing she wouldn't listen, he bit back arguments about his unworthiness.
Her head bent in apparent thought, she walked farther along the beach. He didn't follow. The wind lashed at him as he watched her retreat.
It was time they returned to the house. She must be freezing. Still, he didn't move to fetch her. He needed a moment of privacy to rein in his blistering rage at her stepbrothers.
Long ago he'd guessed she came from a good family, but her fortune must be enormous to provoke this frenzy of greed. Gideon recalled no great families called Watson, but then he'd never moved in high society. The Trevithicks were only minor gentry. His experience of the haut ton was limited to his recent sojourn in the capital. Those weeks were a painful blur. Concealing his illness from the avid mob had been almost impossible. Mostly he'd just felt an overwhelming desire to escape.
And, of course, Sarah's stepbrothers would have a different last name. It hardly mattered. Duke's daughter or shopkeeper's daughter, Sarah was utterly out of reach. A man like him couldn't start to think about taking a wife.
His hungry gaze fastened on her as she paused to pick up a pebble and pitch it into the sea. Her stepbrothers assumed their ward lacked powerful friends. Perhaps at last, being the Hero of Rangapindhi might prove of some use. Those bastards would pay for their crimes before he was done.
It would be his parting gift to Sarah.
He'd see her safe and happy. Then the kindest thing he could do was forsake her forever. With a grim knell in his heart, he trudged up the beach to where she silently stared across the waves.
## Eight
After so many hours in Sarah's company, Gideon inevitably dreamt of her. Such cruel fantasies to torment him when he couldn't lay a hand on her in the real world. At dawn he woke, sweating and restless and painfully aroused. He desperately needed to escape the house, partly because he couldn't bear to meet Sarah's clear gaze and recall what an insatiable satyr he was.
At least in his dreams.
After an early breakfast, he set out for a long ride along the cliffs on an unfamiliar mount. Akash hadn't yet arrived with Khan and the other horses. Now he strode along the gallery, heading for his rooms and a quick wash before he settled to the estate papers. And hopefully no intrusive thoughts of hazel-eyed houris.
From either side, his ancestors stared down. He didn't count on their approval. How could he? His forebears must resent knowing all their labor, all their ambition, all their hopes ended with him.
God knows what would happen to the estate once he was gone. In the meantime, he'd devote his life to restoring it. Not for the sake of these louring faces but for the people who lived here. Dark, secretive, taciturn. And loyal to death to the Trevithicks.
He hadn't expected to survive to see his homeland again. But he had—to return to news that Harry was dead. How ironic that his father and his brother perished too young in safe, peaceful England. While Gideon had come through untold dangers.
With such somber thoughts for company, Gideon rounded the bend in the gallery and almost ran Sarah down.
"Sir Gideon!"
He reached out as she stumbled. Then he remembered and snatched his gloved hands back. Blood pumped through his veins in primitive demand. He hardened with uncontrollable swiftness. Untrammeled images from his dreams swamped his mind. His body moving in hers. Her bronze hair flowing about them like wild silk. Her soft moans of pleasure.
For one burning instant, he stood close enough to catch her scent. A hint of carnation soap. The essence of Sarah herself. Then she found her balance and shifted away, thank God.
Sucking in a deep breath, he retreated a step. The extra distance did nothing to curb the storm inside him. "Sarah..."
At his withdrawal, her eyes darkened with hurt. He wanted to tell her again it wasn't her, but he stopped himself. Better by far she never learned his filthy secrets. He couldn't burden her so.
She bit her lip and glanced at the painting she'd been studying. "He could be your twin."
"What?" Gideon struggled to focus on what she said.
"The man in the portrait."
He blinked to clear his vision and realized she stood looking at Black Jack Trevithick. For a long moment, Gideon stared into painted eyes so similar to his own. Black Jack wasn't smiling, but the long, sensual mouth quirked on the verge of laughter.
"That's Black Jack. An altogether more dashing fellow than I."
"He certainly has the devil in his eyes."
"Not just in his eyes if the stories are true."
"Women, you mean? If looks are anything to go by, I suspect the stories are true." She glanced directly at Gideon. "You'll have to tell me."
He shifted uncomfortably. A discussion of his disreputable forebear's amorous conquests. Just what he needed when he struggled to rein in his own unruly sexual appetites. "Most aren't fit for a lady's ears."
She laughed softly and flashed him a smile. Her full lips curved bewitchingly, and he caught a glimpse of small white teeth. Another bolt of arousal left him staggering. Her warmth beckoned, more enticing than a fire on a winter's night.
He tilted his chin in Black Jack's direction. "Actually, there's one story you might like."
"Only one?"
"Well, the only one I mean to tell."
"Spoilsport." Her lips twitched in a way that sent another frisson down his spine.
He strove to sound as if he weren't about to combust into ashes. "Black Jack was the local wild boy. He could sail anything that floated, ride any horse that galloped, seduce any maiden into compliance. The family legend is he charmed Queen Bess out of her chastity."
The enchanting smile still hovered around Sarah's lips. "What a man."
"Precisely." He struggled to concentrate on his story rather than Sarah's attractions. An impossible task when her attractions were so compelling. "On one of his raids along the Spanish Main, he captured a galleon."
Her face was alight with interest. "Packed with treasure, so the Trevithicks were set up forever?"
"Who's telling this tale?"
"You are. Pray, go on."
"Packed with treasure, so Black Jack came back to Cornwall and rebuilt the house as it stands today."
"If he built this house, he had an artist's spirit. What else was on the galleon?"
He fell into the familiar tale, telling it as he'd heard it as a child from his nurse, one of Pollett's sisters. "A grandee's daughter called Donna Ana, the most beautiful woman in King Philip's empire."
"She fell in love with Black Jack at first sight?"
"No, she fought him tooth and nail. But Jack wanted her and brought her back to Penrhyn as his bride."
"Don't tell me she pined for Spain and died a melancholy death far from everything she loved?"
"Now what sort of romantic legend is that?"
"The sort I don't like to hear."
An amused sound emerged from his throat. So dangerous, letting himself relax with her. But sweeter than the rich Indian confectionery he remembered from the bazaars. "After a battle royal, she fell in love with her Cornish pirate and gave him ten healthy children. He lived into old age as a faithful and devoted husband."
Sarah's smile filled with unguarded delight. He felt as though he stood in a shaft of summer sunlight, for all it was a cold February day. "That's lovely."
Her response didn't surprise him. He knew she was a romantic. Look at how she romanticized him.
"I suspect in reality their marriage was much like anyone else's." Gideon stifled his own boyish fascination with his swashbuckling ancestor. Misguided romanticism had already cost him everything that made life worthwhile.
Her smile faded. "No. It was a grand passion, so their life together was a grand adventure." She must have guessed he meant to argue for a more prosaic interpretation because she rushed into speech. "Is there a picture of Donna Ana?"
Gideon gestured to the opposite wall. The small panel on wood depicted a dumpy woman wearing an unflattering black gown from the reign of James Stuart. "There."
Sarah spent some time staring into the woman's plump, lined face. He moved to stand behind her, not close enough to touch. "Are you disappointed?"
Of course she must be. The most beautiful girl in the Spanish Empire had turned into a middle-aged frump. If Donna Ana ever was beautiful. Perhaps family mythology embroidered that part of the tale. Perhaps Jack just married this little hen to secure her Spanish gold. The wealth he seized from the galleon was real enough. The proof was all around them in Penrhyn's faded glory.
"No, I'm not disappointed," Sarah said softly, turning to face him. "She looks like she led a happy life even though she was far from home and family. She must have loved her wild husband and her brood of children."
In this dusty room with its beautiful parquetry floor, dark paneling, and elaborate plaster ceiling, Sarah was the only thing truly alive. She burned like a flame. His eyes feverishly drank her in. Satiny hair pulled back in a plait. Great, glowing eyes. Her cheap gown hinted at the untold riches of her body beneath.
Her cheap, torn, dirty gown.
He scowled. "Good God, woman, what are you wearing?"
A flush rose in her cheeks, and she self-consciously tweaked her faded skirts. "It was all I had."
"I asked the housekeeper to find you something."
She made a face. "Mrs. Pollett is three times my size. She lent me a couple of dresses, but they were hopeless. The nightdress was so big, it wouldn't stay up."
He stiffened. All over. Darkness edged his vision. His mind burned with scorching images of Sarah's shift sliding to the ground with a sensual whisper. Leaving her bare and beautiful and ready for him.
He cleared his throat, clenched his fists, and battled for control.
Her color became more hectic, and her hands rose to her cheeks. "I shouldn't have said that."
Gideon swallowed and strove to concentrate on the least arousing objects he could think of. Radishes. Turnips. Cabbages. Carrots.
No, not carrots.
"No..." He cleared his throat again. "No, you shouldn't."
"You won't believe this, but I wasn't dragged up under a bush," she mumbled.
He knew what he'd like to do with her under a bush. Or what he'd like to do if he was a whole man and able to turn his desire into action.
He struggled for a normal tone as wanton images of Sarah naked and eager rocketed through his mind. "My mother's clothing is packed in the attics. Would you like to see if any is suitable? You can't run around in that rag for the next three weeks."
Sarah pointed to a gold-framed picture along the same wall as Black Jack's. "Is that your mother?"
"Yes."
As he'd known she would, she wandered down to stand in front of the exquisite Lawrence. The woman in the portrait wore one of the diaphanous gowns popular at the end of the last century. Blond hair curled softly around her delicate face.
"She's very pretty."
"In her first season, she was considered a diamond of the first water. She was only eighteen when she married my father."
"Is he the rather florid man in the next picture?"
"Yes. And my brother Harry is the fellow next to him, who looks like a younger version of his sire."
His gut tightened with the usual contradictory emotions as he studied Sir Barker and Harry. Regret, certainly. A complex brew of grief and anger. The futile wish that at least a trace of warmth had marked his interactions with his family.
"You don't look like either of your parents."
"My father might have wanted to proclaim me bastard, but the proof of my mother's fidelity is in this gallery."
Interspersed with more conventional-looking faces, Black Jack's piratical features looked out at the world, sometimes in daughters of the house, more often in sons. Black Trevithicks were usually male. Their faces were everywhere, under cavalier curls or bag wigs. Intelligent, knowing black eyes. Lazy, confident smiles.
Sarah tipped her head to the side, surveying his mother. "She looks sad."
Gideon was surprised Sarah sensed the picture's melancholy. He found himself telling her what he'd never told another person. "My father wasn't an easy man. What little I've learned of their union indicates an infelicitous match. My brother's delivery was difficult, and the doctors advised separate bedrooms. But my father insisted on his rights, so three years and four miscarriages later, I arrived."
"And she slipped away." Sarah returned her attention to the portrait. "How tragic."
"Yes, it was."
Would his childhood have been different if his mother had lived? She'd been a gentle woman with intellectual tastes. He'd always believed he inherited his love of learning from her.
"You don't mind if I wear her clothes?"
He shrugged. "She was unfailingly kind. Everyone who knew her agreed on that. My father viewed her generous nature as a sign of weakness. The villagers, though, loved her and still speak of her fondly. She'd be the first to offer her wardrobe to a lady in distress."
"I would have liked your mother." Sarah's smile was tinged with compassion.
He tensed. His pride revolted at her pity.
"Come up to the attics," he said sharply, and tried to ignore the way her eyes once more darkened with hurt.
He turned on his heel to stalk out of the gallery and along the dim corridor that ran through the back of the house. She scurried to keep pace with his long stride. Without speaking, they climbed a series of ever-narrowing stairways lit by dirty mullioned windows.
Outside the last door, Gideon lifted two candlesticks from a niche. He lit the candles and passed one to Sarah, who waited slightly breathless at his side. He stifled a pang of guilt. It wasn't long since she'd endured a savage beating, and yesterday she'd nearly fallen off a cliff. He should have more consideration than to rush her through the house at top speed.
Still, his tone was brusque. "Here. It's dark up there."
"Thank you."
Silently, she followed him up the final precipitous staircase. He entered the attics ahead of her and halted abruptly as a thousand memories overwhelmed him.
The smell was exactly the same. Dust. Old dry wood. Fusty air. Painfully reminding him of boyhood misery.
"Heavens, you could fit a village up here." Sarah stepped closer but thank God, didn't touch him. Still her vibrant presence stirred his blood to turbulence.
Against his will, he looked at her. Flickering candlelight transformed her into a creature of dark mystery. Turned her great hazel eyes into bottomless pools. Gilded a cheekbone as she tilted her head with open curiosity to survey the cavernous area.
"It's where I studied when I was a boy." He raised his candle to illuminate a corner under the sloping roof. "Nobody's touched it since I was last here. Look."
Sarah moved closer to the untidy pile of books stacked near the ragged blanket he'd used in winter. In January, the attics had been as cold as an ice cave in hell. "You wanted to get away from your father."
He cast her a sharp glance. "He hated having a bookish son. But no number of beatings changed me. I was stubborn."
"You were strong. You are strong."
He could have argued but didn't. "Luckily, most of the year I was away at school."
"Do you know where your mother's belongings are?"
He pointed to some trunks against the wall. "They haven't been shifted either. My father's and brother's things are downstairs. It's such a big house, I hardly need the room."
"It's a house meant for children," she said quietly. "Lots of them."
He tensed, wondering if she meant to pursue the subject of marriage again but she said no more. Relief trickled through his veins.
"Let's hope the mice haven't got to everything." He strode across to unlatch the first trunk. Anything to break the web of intimacy slowly spinning between them.
"I can't smell mice. Your cats must be ferocious hunters."
"Under my father's and brother's careless regime, they had to be to keep their bellies full." He flung back the heavy lid with a bang. Immediately faded scents crammed his senses. Lavender to keep the clothes fresh. A faint echo of rose fragrance that must have belonged to his mother.
Sarah stepped softly to his side. "I feel like she's here."
"So do I." His voice was flat with control. He placed his candle on the trunk behind. Sarah must see how his hands shook. She couldn't miss the way the flame wavered in the airless room.
Reluctantly, he began to sift through the trunk's contents. Bonnets. Hats. Scarves. Handkerchiefs. Stockings. Shoes. Soft kidskin gloves that had shaped themselves to his mother's hands. Hands he'd never touched.
Finally, at the bottom, he found neatly folded clothing. His gloved hand brushed heavy silk, and he carefully lifted what proved to be an evening cloak. As the shining blue fabric unfurled, a gust of rose perfume drifted into the still room.
He'd never touched his mother's things before. It had seemed somehow wrong to pry into her private possessions. Although he'd always known which trunks were hers.
Carefully, he laid the cloak aside. Behind him he was vaguely aware of Sarah's footfall as she explored the attic. Then suddenly light bloomed around him.
"This might help." She set the lantern down near him.
"It's the one I used to read by."
"I found it with your books." She knelt, her shoulder inches from his.
He desperately wanted to tell her to move away. She was close enough for little eddies of scent to tease him, her peppery carnation fragrance mingling with the evocative rose. She was close enough for him to hear the uneven rhythm of her breathing.
Did his proximity disturb her as hers disturbed him? Sweet God, this became more impossible with every second. Briefly he shut his eyes and prayed for strength. When he opened them again, Sarah pored over the items he'd discarded on the floor.
"Everything is so delicate," she said softly. "Like it was made by angels. Look." She held up a filmy shawl of lace fragile as a spider's web.
He reached out to touch the fabric, then jerked back. All his life, his mother's gentle ghost had haunted him. Touching her clothing made her tragedy poignantly immediate.
He struggled to inject a prosaic element into his voice. "Not exactly suitable for late winter."
He had to get this over with quickly, before he made an utter fool of himself. He drew out a satin ball gown. Its rich peach color gleamed in the candlelight.
"Nor is that." Sarah's voice sounded huskier than usual. As if she'd just got out of bed, God help him. His hands curled in the slippery material.
"These must have come from her London season." Still, he strove to sound casual, unconcerned. The last thing he needed was Sarah to discover her interest in him was reciprocated. "My father never socialized. Or not with people he'd introduce to his wife. She'd have little call for a dress like this at Penrhyn."
All the gowns were too elaborate for Sarah to wear around the house. Gideon repacked the trunk, his hands lingering on the fine materials. He knew it was only imagination, but a hint of warmth from that pretty laughing girl, the toast of London, still remained. He shut the lid and turned to the next trunk.
As with the other one, accessories lay on top. He quickly riffled through them. He passed Sarah a sturdy pair of half boots. "See if those fit."
The first gown he pulled out was a sprigged muslin day dress. He stood and turned around, then wished to God he'd stayed put.
Sarah sat on the trunk they'd already checked, sliding on the shoe. Her skirts hiked to reveal two trim ankles. Petticoats frothed, white and alluring, around her shapely calves. Her thick braid tumbled over one shoulder to dangle between her breasts. As she leaned forward, her bodice gaped to reveal the pale skin of her cleavage.
His mouth went dry as sand. His heart slammed hard against his ribs. Hunger to tumble this girl on the dusty floor made him giddy. The urge to escape rose to choke him.
He must have made a noise because she turned startled eyes in his direction. "Gideon?"
Just his name. A low question. Just as he'd started calling her Sarah, somewhere she'd started calling him Gideon. He whipped around and dropped to his knees before the open trunk. His breath rattled loud in his ears as he fought to rein in the agonizing conflict inside him.
He couldn't touch her. No matter how much he wanted to. He knew what would happen. He'd frighten and disgust her.
He fumbled in the trunk, roughly pushing aside the first gown. Without looking, he grabbed something and shoved it in Sarah's direction.
"What about this?" he bit out, still not glancing at her.
"I think..." She paused, and he felt her take the garment from his hands. "I think if I'm not to shock the servants, I might need something a little more substantial."
He sucked in a deep breath and blinked to clear the haze from his eyes. Carefully he turned. She stood watching him with a complex mixture of hunger and trepidation. The boot had toppled over and lay on the floor near the trunk. She clutched a filmy chemise in front of her.
God give him strength. He refused to picture that sheer scrap of cream silk clinging to Sarah's lissome body. He straight-out refused.
Gideon gritted his teeth until his jaw ached and tried to quash the bawdy images filling his mind. His face itched as hot color rose in an unstoppable tide. He was acting like a damned fool.
Her voice had been light, amused. Perhaps she hadn't noticed his turmoil. Then he looked into her eyes and read secret knowledge in the hazel depths. She sensed he responded to her as a man responded to a woman. It frightened her—fear lurked in her gaze too—but not enough to send her fleeing back downstairs.
"Your pardon." His voice sounded rusty. "I meant to give you this."
Clumsily, he handed her the muslin. She ventured closer to drop the chemise back into the trunk, then she studied the dress.
"What do you think?" She held it up for his consideration.
Good Lord, she couldn't torment him deliberately, could she? She looked so utterly innocent and unconcerned. Which, now his brain returned to something approximating working order, struck him as cursed suspicious.
"It doesn't matter what I think," he said in a clipped tone. "Will it fit you?"
"It looks like it might. The shoes didn't. Your mother had much daintier feet than I." She lifted her skirt a few inches and circled her bare foot in demonstration.
The witch! She tortured him for her own amusement. If he could bear to touch her, he'd bloody well strangle her.
If he could bear to touch her, he wouldn't strangle her. He'd ravish her within an inch of her life.
It suddenly struck him, as it should have struck him long before, that being up here alone with Sarah was a very bad idea indeed. He'd thought to find her a couple of things to wear and escape with no consequences. That now seemed an absurdly optimistic plan.
Hell, he had to get out of here. Now.
The attics had appeared so spacious when he first set foot in them. Now they felt oppressive, crowded, closing in on him.
When all the time he knew what closed in on him was insatiable desire.
He stumbled to his feet with clumsy haste. Tension formed a painful line across his shoulders. "Everything you need is in this trunk. I'll get the servants to bring it to your room."
She flinched at his tone, then leaned near to replace the items they'd removed. Near enough for her skirts to brush his legs with a subtle sensual whisper. Sarah's warm, womanly scent momentarily submerged his mother's rose perfume.
In spite of his best intentions, he closed his eyes and inhaled. It was the fragrance of paradise. And he, poor sinner, was locked in perpetual agony outside the gates.
He shouldn't have hesitated. He should have made a run for it while he could. Blast her, he shouldn't have come up here in the first place. Mrs. Pollett could just as easily have shown her the trunks.
When he opened his eyes, she stood before him, her face uplifted, her lips parted, her arms outstretched. Her face was stark with need and vulnerability and a desperate, hard-won courage.
He couldn't mistake what she wanted.
Even that recognition didn't shift him. Every limb was heavy as lead. Denial jammed in his throat and emerged as a groan. He staggered back, but she'd already begun her forward momentum.
He twisted awkwardly to evade her but she grabbed his arms. Her fingers curled into his flesh in inescapable talons. Blind horror held him paralyzed.
"Gideon, please," she said in a broken voice that made his gut cramp with guilt and sinful longing.
Her slim, tender body slammed into his. Her slender arms, surprisingly strong, wrapped around his neck. Her heady scent rocked his brain, scattering rational thought.
Shaking, he clutched her waist, crazy with the need to push her off him. But his will failed at the final moment.
She tensed as she stretched up. The damp, seeking heat of her mouth pressed against his.
He stood motionless under her clumsy, passionate assault. Fiery pleasure streaked through him like summer lightning. Automatically his hands tightened around her waist, and he tugged her closer.
For one blazing second, he lost himself in the sizzling kiss. Darkness. Pleasure. Sweetness. Heat.
His blood pumped, his skin burned. His mouth moved in cautious answer to her furious, unpracticed ardor. He couldn't mistake her inexperience, or her passion. He guessed she had no idea what she invited when she launched herself at him.
If he'd been a normal man.
Although right now, he damn well felt like a normal man. He felt like a man overcome with lust. A man who kissed the woman he wanted more than his life.
Clamoring questions exploded in his mind. Had a miracle occurred? Had incendiary desire at last vanquished the ghosts of Rangapindhi?
His starved senses filled with the glory of her. The clinging pressure of her grip around his neck. Her soft breasts crushed to his chest. The carnation scent. The taste of her mouth. Fresh like the sea. Hot like fire.
The warmth was delicious. Astonishing.
He moved his lips in a more purposeful response. A shudder of excitement rippled through her, and she pressed closer. He surrendered to overwhelming pleasure.
It was too late.
Savage, rending wraiths clawed to the surface. The firm youthful flesh under his palms turned cold and slimy. The lush mouth pressed against his stretched into a rictus grin. The sweet scents of flowers and the sea drowned in stinking decay.
Frantically, he fought the suffocating blackness. Don't let this happen now. Dear God, not now. Not when he had her in his arms at last.
His muscles spasmed into pain. The nightmare images stole awareness. He wrenched his mouth from hers. He shook like a rabid dog. "Let me go," he choked out.
She didn't seem to hear. Instead, she moved closer.
He couldn't endure this. He had to stop it.
"Damn it, I said let me go," he snarled. With unsteady hands, he ruthlessly dragged her arms from around his neck.
She resisted, though it must hurt her. "No. Please, Gideon, no."
His voice broke with desperation. "For God's sake, Sarah, leave me be!"
Through the devils screeching in his head, he felt her sudden stillness. She pulled back far enough for him to catch the bright agony in her eyes. And the gradually dawning realization that he was in earnest.
Still, she didn't release him.
With sudden roughness, he heaved her out of his way and headed for the stairs. He needed air. He needed solitude. His gut heaved with acrid nausea. His hands shook so badly, he couldn't trust himself to pick up the candles.
"Wait."
He tried to ignore her ragged plea. Every particle of his being craved escape.
"Please don't go like this." Through the buzzing in his ears, he heard her rush after him. Against his will he stopped, hunching his shoulders against her.
"Never do that again." His voice was hoarse and raw. His fists opened and closed in an idiot rhythm at his sides.
"I don't understand."
The bewildered despair in her words harrowed his heart. He wounded her and he regretted it to the base of his soul.
Oh, Sarah, Sarah, what have you done with your recklessness?
"I know you don't." He still couldn't bear to look at her. He could hardly bear to breathe the same air. "I don't either, not really."
He ached all over as if he had a fever. Only the last remnants of stubborn Trevithick pride kept him upright. At least now she didn't touch him, he gained some control over his nausea. It would be the ultimate defeat if he lost his breakfast in front of her.
"Is it me?" Her voice shook with anguish. "You keep saying it's not."
He wished to heaven she were another woman, one who would blush and scuttle away to hide her humiliation. But another woman wouldn't set herself so impulsively after what she wanted.
However misguided that wanting was.
"No," he forced out. His blood pounded like heavy surf after a wild storm, blocking out everything else.
Except Sarah.
He was agonizingly conscious of her standing behind him. Of every jagged breath she drew. Of how close she was to tears.
"I don't believe you. I disgust you."
"No!" Turning his head in her direction was harder than turning back the tide.
For the first time, he saw her clearly. She was haggard, and silent tears ran down her white cheeks. The trails shimmered in the uncertain gold light.
He wanted to say so much, tell her everything. Explain, excuse, soothe, comfort.
None of it would do any good. None of it would change him into a man worthy to call her his.
So he said again the only thing he could. "No."
"Then why..." She made a helpless gesture with one trembling hand.
"Sarah..."
The thunder in his ears became louder. He closed his eyes and prayed he'd dredge up the right words. Though he knew there were no right words to be found.
Then he realized the thunder wasn't entirely in his imagination. Someone clattered up the stairs to the attic. Someone heavy and wearing boots.
"Sir Gideon!"
"Tulliver?" The intrusion came from another world.
The usually impassive Tulliver reached the top of the staircase and stood panting. "Strangers riding up the drive. The local magistrate is with them."
## Nine
What the devil happened to the men watching the road?" Gideon snapped.
Charis flinched at Gideon's anger, then realized just what Tulliver said. Terror locked every muscle. Her stepbrothers had found her. Because who else would visit Penrhyn with an officer of the law? She braced herself to run. But where could she go?
Dear Lord, could this vile day get any worse? In her belly, fear, humiliation, and, to her disgust, frustrated desire stewed in a bilious mixture. Despair, heavy, draining, black and thick as tar, leaked into her soul.
"They came to warn us quick smart enough." She knew Tulliver noted her tears but with his usual consideration, after the first glance, he kept his attention on Gideon. "But nobody could find you or the lass. We searched high and low to no avail."
"Hell," Gideon breathed. "I'm sorry, Tulliver. I should have told someone where I was. This disaster is my blasted fault."
"What do you want to do?" Tulliver was back to sounding his imperturbable self.
Gideon straightened and sent his henchman a flashing grin that reminded Charis of Black Jack. Just thus must the reckless privateer have faced down the Spanish galleon that carried his destiny. The shaking, distraught man of seconds ago might never have existed.
Black Jack had prevailed. So would Gideon.
Courage leached back, stiffening her backbone. Gideon might reject her, but her faith in him remained unshaken. He was her Percival, her Galahad, her Lancelot. From the first moment she'd seen him, he'd been her bulwark. After all they'd been through, he wouldn't let her fall into her stepbrothers' hands.
"Why, I'll greet them like the gentlemen they are."
He turned to Charis, and she couldn't mistake the searching inspection he gave her. As if checking whether her mettle was up to this.
She raised her chin and sent him a straight look. She was mortally afraid, but she refused to succumb to fear. "That means tossing them in the cesspit."
Gideon gave a curiously lighthearted laugh. She could only interpret the spark in his dark eyes as admiration. "That's my girl."
He waited for her to put her shoe on, then blew out the candles and gestured her toward the steps. It cut her to the bone that he still couldn't bear to lay a hand on her. After her antics today, he'd probably never touch her again.
Oh, Charis, you've got more important things to worry about right now than the fact you made a fool of yourself.
Gideon collected the lantern and followed her down to the gallery. He pressed an unremarkable plaster molding near the fireplace.
"Heavens," Charis breathed, as a secret latch clicked and what looked like an innocent section of paneling turned out to be a door. "A priest's hole."
"A smuggler's stash, more like. If you stay quietly here, nobody will find you." His voice dropped. "I give you my word I'll keep you safe. Trust me."
She looked into his eyes. The pain and confusion and anger that had gripped him upstairs had vanished. Instead, he looked calm and determined and, most reassuring of all, completely confident.
"I trust you." She meant it from the depths of her soul. Odd to think she trusted him more than she'd trusted anyone since her father's death. Even after the way he'd recoiled from her kiss.
"Good." He gave her the lantern and watched her step into the recess. Except it wasn't a recess at all but a landing off steps leading downward.
The door closed behind her. For a moment, stark, illogical terror gripped her. What if something happened to Gideon and Tulliver and nobody knew she was here? What if she ended up trapped behind this wall forever?
A soft knock on the panel interrupted her flight into panic. "Are you all right?"
Just the sound of Gideon's deep voice calmed her galloping heart. She was a hopeless case to be so in love with a man who couldn't bear her merest touch. How she wished she could help what she felt, but she'd been utterly lost from the moment he'd rescued her in Winchester.
"Yes."
"You can listen to what happens in the drawing room if you go down a level. If you want to get out, the passage leads to a cave on the beach."
'Thank you." She didn't mean just for his reassuring information.
"It's nothing," he said, dismissing her gratitude as he always did.
She heard his boots click on the parquetry floor as he retreated. Then a more ominous sound. The great iron knocker on the oak front door pounded once, twice.
"Shall I send the bastards on their way?" Tulliver cracked his knuckles.
Gideon laughed softly. "No. Let's play these hyenas the civilized way. At least at first. Show them into the drawing room and say I'll be there presently."
"What are your plans? The lass is safe enough where she is."
"I think it's about time I got some benefit from being the bloody Hero of Rangapindhi."
Tulliver's eyes glinted with his rare humor. "Aye, guvnor. It is about time and all."
Downstairs, Mrs. Pollett opened the door. Gideon didn't wait to watch her greet the arrivals but scaled the steps to his bedroom two at a time. In his heart, savage satisfaction beat like a drum.
At last his enemies would have faces. Sarah's stepbrothers were foes he could fight and defeat. After that vile debacle in the attics, he welcomed an unambiguous purpose. The kiss changed everything between him and Sarah, yet it changed nothing. He grimly recognized that stark reality, yet still the physical aftermath lingered to torment him. His lips tingled, his skin itched, his gut cramped. And rapacious desire was a roiling eddy in his blood.
He left his unwelcome guests cooling their heels long enough to put them on edge. He had no fears they'd take it into their heads to search for Sarah on their own. Tulliver guarded the door from the hallway. So Gideon's insouciant air as he sauntered into the drawing room twenty minutes later wasn't entirely pretense. Sir John Holland, the local magistrate, turned to greet him with barely concealed relief.
"Sir John, pleased to see you." Gideon stepped forward and forced himself to accept the middle-aged fellow's brief handshake. His flesh crawled at the contact but with an effort, he concealed the reaction.
Sir John looked irritated but not overly worried, which meant this visit was more reconnoiter than hostile raid. "Sir Gideon. I haven't seen you since you were a stripling. Now you've set the world on its ear, begad. You must come to dinner and tell Lady Susan and me all about your adventures." He suddenly sobered. "Sorry to hear about your pater and Sir Harold, of course. Mustn't forget the sad circumstances that brought you back to us."
"Sir John, is this a social call?" The game commenced. Gideon intended to reveal nothing he didn't have to.
The man straightened and cast an annoyed glance at his two companions. "Not entirely, although been meaning to pay my respects."
There was an awkward pause. In his best rake-of-the-ton manner, Gideon arched his eyebrows at the two strangers, who stood in silent menace behind Sir John.
Of course, he'd studied them from the moment he'd entered the room. Just as they'd studied him.
He noted their surprise at his elegance. Thank God for the London tailors he'd patronized upon his return from Rangapindhi. He wanted these wretches to realize they dealt with a man of standing.
Sir John cleared his throat uncomfortably. "Sir Gideon Trevithick, may I present Hubert Farrell, Lord Burkett, and his brother, Lord Felix Farrell?"
Lord Burkett? Good God, the older brother was a bloody marquess. Sarah had kept that salient piece of information to herself.
Gideon had known a large amount of money was in question, and he'd guessed she must come from the gentry at the very least. Until now, he hadn't realized he tangled with the aristocracy's upper echelons.
"Delighted, I'm sure, "Gideon said with deliberate boredom, returning the Farrells' chilly bows with a dismissive bow of his own.
Lord Burkett was in his late twenties, large, powerful, brutish, although already his heavily muscled frame turned to fat. Gideon bit back his sick fury as he pictured those thick hands pummeling Sarah's tender flesh. Lord Felix, younger by a year or two, was slight, fair, and handsome. Burkett looked confused. Felix looked suspicious. Even on such short acquaintance, Gideon recognized Lord Felix as the more dangerous of the two.
"Get to the meat of the matter, Holland," Burkett demanded.
"As I said, I'm sure Sir Gideon doesn't know..."
Burkett glared at Gideon and spoke over the magistrate. "We've lost our sister, Lady Charis Weston."
Gideon sat with a show of nonchalance and gestured to his visitors to do the same. Although his impulse was to throw the brothers out on their blue-blooded rumps once he'd delivered a well-deserved beating. After that agonizing scene upstairs with Sarah, he itched to work his turmoil out in violence. Nobody deserved a pasting more than these bastards.
Sir John took the sofa near the fireplace. Lord Felix selected a chair nearby. Burkett remained standing in the center of the room, a bullish, aggressive presence. How in the name of all that was holy had Sarah survived the rough guardianship of these villains?
Then he realized what Burkett had said. Sarah apparently wasn't his charge's real name.
Charis Weston.
Lord Felix's attention fixed on his face, seeking guilt or fear. Keep looking, hellspawn, Gideon told him silently. Compared to the Nawab of Rangapindhi, Felix was a toy.
Without difficulty, he maintained his disinterested drawl. "Commiserations. Although I'm not sure how that's my concern."
"You were seen with her in Winchester and Portsmouth," Felix said sharply. He tried to hide it, but desperation to lay his hands on Charis seeped from his tense frame. "I imagine she told you a pack of lies about needing help. She's a flighty piece, almost feebleminded, who ran off in a fit of pique. We seek her for her own good before she comes to harm. Is she here?"
Damn, their departure from Winchester must have been observed although he'd been so careful. Gideon kept his voice even. "Ah, you mean the poor waif I gave transport to on her way to her aunt in Portsmouth?"
"She has no aunt in Portsmouth," Burkett growled, taking a step closer. He was clearly accustomed to using his bulk to intimidate.
Gideon shrugged. "That was her story in Winchester. Chit claimed she'd been set upon by footpads. She was in a bad way. Knocked about."
Burkett shifted uncomfortably, but Felix's eyes remained cold and intent. Gideon retained his bland expression as he privately consigned them both to Hades.
"I grieve to hear that. A lone woman on the road faces many dangers. That's why we're eager to return her to her loving family." Felix made a creditable attempt at sounding concerned.
"Commendable," Gideon murmured, cursing the oily bastard for a liar and a fraud. The bruises on Sarah's—no, Charis's—face were testament to how loving her family was.
"As we haven't found her on the road to Penrhyn, we can only surmise she's staying with you. Pray send for her. We'll end this lamentable episode and any inconvenience you suffer, Sir Gideon." As he stood, Lord Felix's tone became if anything more unctuous. Gideon suppressed a shudder of loathing. "Clearly you're a man of honor, and a lady is safe in your company. But the world may not be so kind in its assessment. Our sister's reputation is at stake, so we'd appreciate your keeping details of this unfortunate incident mum."
Gideon struggled not to plant his fist in Felix's smug face. But he'd learned self-control in the hardest school. His response gave no indication of his abhorrence for these men. "I'd love to help you, my dear fellow. If indeed this girl is your sister." He let his tone descend into regret. "But she ran off after the ruckus in Portsmouth. My man and I tried to find her but with no success. I suspect she's still there."
"You expect us to believe you abandoned a defenseless woman?" Felix hissed, clenching his fists by his sides.
Gideon shrugged again although he already knew his careless act didn't convince the younger Farrell. "I assumed she'd gone to her aunt."
"But she hasn't got an aunt in Portsmouth," Burkett repeated, as if the fact made some difference.
"She told me she did. She was most adamant that Portsmouth was her destination."
"Because she thought to disappear there," Felix said between his teeth. "It's a port city. Nobody would pay her attention."
Gideon raised his eyebrows again. "That's a clever scheme for someone who's feebleminded. Devil take me if it's not."
"That's not at issue," Felix snapped. "What is at issue is that we are her legal guardians, and if you harbor her, Sir Gideon, you break the law and will pay the penalty."
"Steady on, Lord Felix!" Sir John protested, rising from his seat.
Gideon ignored the slur on his honor. His voice turned silky. "Which I'm sure is why I have the inestimable pleasure not only of your company but of the magistrate's. I'm surprised you didn't invite the militia along to infest the front hall."
"If circumstances compel us to use force, we will," Felix said steadily. He sent a meaningful glance to Sir John, who looked increasingly uncomfortable at the conversation's prickly turn. "As a representative of the law, you'll back us, Sir John."
Sir John cleared his throat and cast a nervous glance at Gideon. Gideon guessed what went through his mind. He'd known the Trevithicks all his life and recognized their local influence. The Farrells might be powerful men on the nation's stage, but they didn't live on his doorstep.
"There's no need for unpleasantness, gentlemen." Sir John directed a pleading stare at Gideon. "If Sir Gideon gives us his word that the girl you believe to be Lady Charis ran away in Portsmouth, we must be satisfied."
"Be damned to that," Burkett objected, taking a threatening step in Gideon's direction. His hands opened and closed at his sides as if he restrained himself from grabbing Gideon and beating the truth from him. Poor Charis, at this brute's mercy. Gideon could hardly bear imagining it. "She's the richest heiress in England. He's keeping her for his own gain."
The richest heiress in England? Hell and damnation, what had he got himself mixed up with?
The blasted girl had hidden a lot from him. None of which shook his determination to help her. He wouldn't hand a stray cat over to the Farrells, let alone a woman he admired and...cared for.
"Do you doubt my bond, sir?" Gideon rose to his full height.
Burkett was big and brawny, but Gideon topped him by several inches. Gideon also had the steel lent by years of living with endless danger. Burkett didn't frighten him in the least. He could break the overweening bully without a thought.
As Gideon had expected, Burkett backed down. "You haven't given us your word." He sounded sulky.
Gideon's voice was firm. "I give you my word the girl I knew as Sarah Watson ran away in Portsmouth. There's no guarantee the chit I encountered is even your sister."
"What did she look like?" Lord Felix asked.
"Small. Skinny. Bruised. Light brown hair. Spoke with a rough accent." It was possible someone had got a close look at Charis. He couldn't stray too far from the truth without awakening suspicion. "I can't for the life of me imagine she's an heiress. Her clothing was poor and her manners deplorable."
"She played a part," Felix insisted.
"I have no idea. What I do know is she took off after the brawl, and I haven't seen her since. If you believe this girl really is your sister—which I take leave to doubt—you'd be better concentrating your search in Portsmouth."
"Can we check the house?" Burkett asked stubbornly.
"No, by God," Gideon snapped. "I'll be damned if I let a pair of strangers march through my private rooms on a wild-goose chase after some featherbrained bit of muslin."
Burkett puffed out his impressive chest. "You insult my sister, sir."
"I do no such thing. Confound it, I don't know your sister. The world has come to a pretty pass when a man is harassed on his own property for offering aid to a distressed maidservant."
"Sir Gideon has given us his word," Sir John said placatingly. "Surely that's good enough."
Felix spread his hands to indicate his benevolent intentions. "Sir John, we act purely from brotherly concern. If we satisfy ourselves she's never been in this house, we'll leave Sir Gideon in peace, with our gratitude and apologies."
Good God, but the younger Farrell was a slimy customer. He sounded so reasonable. If Gideon hadn't seen the marks on Charis's face, he'd almost believe the weasel's protestations.
"Sir Gideon, surely under the circumstances..." Sir John looked across at him hopefully.
Time to play the hero card. Gideon straightened and let outrage infuse his reply. "When I left this country to risk my life in its service, an Englishman's home was his castle. Unless you intend to invoke the full power of the law, Sir John, I must on principle refuse this monstrous imposition on my rights. I have returned after years of danger and deprivation beyond mortal imagination. Was it to face tyranny in my own homeland? Surely not. If so, His Majesty will hear of it. When he knighted me for my services to the Crown, he was most effusive about his gratitude and favor."
"So you refuse?" Felix's voice was dangerous. His eyes didn't waver from Gideon.
"Look here, Lord Felix," Sir John said. "Sir Gideon is a national hero. You can't barge into his house unannounced and insist on turning the place upside down. Good Lord, man, we're not even sure the girl he picked up in Winchester is Lady Charis. Sir Gideon's description leads me to believe she can't possibly be a lady. He's a man of great perspicacity. If he says she was a serving wench, my bet is that's exactly what she was."
"We only seek to confirm your story," Lord Burkett said sullenly.
"A gentleman's statement should suffice." Gideon turned toward the door. "Now I am no longer at leisure."
"You haven't heard the end of this, Sir Gideon." Felix spoke as if addressing a minion.
The urge to knock Felix to the ground was so strong, Gideon could taste it. With difficulty, he maintained his lordly tone and kept his hands to himself. "I suggest you return to Portsmouth and pursue more fruitful leads there, my lords. You've come a long way for nothing."
"Topping idea." Sir John rubbed his hands together nervously as he stepped up to Gideon, obviously eager to end this encounter. "I'm sure the lady is in Portsmouth. Or safe at home, now she's discovered life away from her family is no picnic."
Felix pulled on his leather gloves with a slow deliberation Gideon knew was meant to constitute a threat. His tone was deliberate too. "We'll return to Portsmouth to pick up the trail. But if it leads us back here, my dear Sir Gideon, your renown won't save you from the consequences. Good day, sir." After an insolent bow, he strode out, his older brother shambling in his wake.
Sir John stayed behind and muttered under his breath. "Sir Gideon, most regrettable incident. Two unpleasant young men. I pray they find their troublesome sister and don't bother us again. The Farrells always were a thoroughly bad lot. Father was a drunkard and gambler. Left the sons nothing but a mountain of debts and the wardship of young Lady Charis, the Earl of Marley's heiress. Hope the poor chit is safe."
"You're very well informed, Sir John."
"The late Lord Burkett was notorious. The sons are chips off the old block. I wouldn't trouble you, except they have their rights. They're the chit's legal guardians. Lord Felix was correct. Anyone keeping her from them breaks the law." He paused and frowned. "Of course, I knew a gentleman such as yourself couldn't possibly be involved. Good God, you've hardly been back in the country a month. Barely time to unpack, let alone get entangled with a runaway heiress. That's what I told those two braggarts. But they wouldn't take the word of a mere country squire." He put on his hat with a disgruntled gesture and collected his stick from near the mantel. "Come to dinner once you've settled in."
"I'll look forward to it," Gideon said, showing Sir John the door.
Outside in the foyer, Tulliver stood stolidly guarding the two Farrells, who looked annoyed. Gideon guessed they'd tried to take advantage of his conversation with Sir John to do some reconnaissance.
"Good day, Sir Gideon. Our apologies for disturbing you." Sir John ushered his companions outside. Gideon followed and stood on the steps to make sure the Farrells left. He sent a groom after them to confirm they didn't return. He trusted Charis's stepbrothers as little as they trusted him.
"Get the girl out of her hiding place," he said to Tulliver, when they were alone.
"Do you want to see her, guvnor?"
"Not immediately. Tell her I'll talk to her in the library before dinner. In the meantime, have my mother's trunks brought down to her room and tell the maids to burn that rag she's wearing."
"What do I say about yon smarmy buggers?"
Gideon stared down the drive, empty now of Felix and Hubert and the reluctantly involved Sir John. When he replied to Tulliver, his voice was steady and very sure. "Tell her I've pledged myself to her safety. She has nothing to worry about."
With a sudden spurt of energy, he leaped down the steps to the courtyard. He turned left through the stone arch and headed for the windswept cliffs.
## Ten
Her stomach somersaulting with nerves, Charis approached the library. This afternoon from behind the wall, she'd listened to Gideon keep Felix and Hubert at bay. She'd silently cheered his cleverness and bravery. But how would he greet her this evening? He'd discovered she was the richest heiress in England. Would she glimpse greed in his eyes as she'd glimpsed greed in so many men's eyes?
Or worse, would she see disgust as he recalled the way she'd flung herself into his arms?
Sick humiliation made her hesitate, trembling outside the closed door. For one blazing moment in the attic, she'd believed he felt the ineffable connection between them. It had been a mistake she'd bitterly repented since.
Courage, Charis.
Stiffening her shoulders, she wiped her damp palms on her skirts and quietly let herself into the dimly lit library.
Gideon didn't immediately notice her arrival. He stood near the grate, staring down at the fire with a somber expression. From the shadows on the edge of the room, her gaze hungrily traced the flame-gilded angles of his face, the lean power of his body. He was dressed more formally than usual, in a dark blue superfine coat and biscuit trousers. He looked like the elegant man she'd met rather than the dashing, disheveled pirate she'd come to know at Penrhyn.
The memory of the brief, dazzling heat of his mouth overwhelmed thought. Then she recalled how he'd wrenched away as if she carried some contagious disease.
Shame choked her. She could hardly believe she'd launched herself at him like that. But he'd been so close, and she'd longed so keenly to feel his embrace. And for one doomed, misguided instant, she'd imagined he wanted her as much as she wanted him.
Poor pathetic fool she was.
Slowly, Gideon looked up, as though reluctant to abandon his reflections. He must also be reluctant to face the woman who had forced herself on him.
She braced to confront anger or scorn, but his serious black gaze focused on her without a hint of condemnation. Or covetousness.
"Good evening, Lady Charis," he said calmly.
She was heart-stoppingly aware this was the first time he'd used her real name. In spite of all her stern lectures to herself, she shivered with pleasure when that dark velvet voice said Charis.
"Good evening." On unsteady legs, she inched farther into the room. She was torn by painful longing to be with him and a cowardly desire to flee.
Gideon's eyes widened as she entered the circle of light cast by the candelabra, and he at last took in her appearance. Because she'd felt like she faced an executioner, pride prompted her to dress in her best. Or in his mother's best. A wide blue silk ribbon fastened the filmy white gown under her breasts. With Dorcas's help, she'd put her hair up in a loose mass, leaving strands to curl around her shoulders.
A flame lit Gideon's dark gaze, kindling answering fire inside her. Familiar tension extended between them. A tension she'd learned in the most painful fashion not to trust.
How could he stare at her as if she took his breath away when he found her nearness unbearable? It was cruel.
She straightened, fighting the insidious yearning his presence invariably aroused, and spread her hands in apology. "I'm sorry I didn't tell you who I was. My inheritance makes men greedy." She should have long ago recognized Gideon was the exception to that rule.
"No matter." He laid one gloved hand on the carved marble mantelpiece. The misleading flash of desire had vanished, and his expression was cool, uninvolved. "And while I admit it's an unpleasant surprise to discover my adversaries are a marquess and his younger brother, I'd do little differently if I had the chance again."
"I nearly told you the truth so many times." Guilt was a sour taste in her mouth. What had seemed so imperative at the time now struck her as a childish, dangerous deception. Still, she tried to make him understand. "It wasn't just fear of how you'd react to who I was. I liked being Sarah Watson. She had more freedom than Lady Charis Weston ever enjoyed."
"Believe me, I understand the lure of freedom." He bent his head in thought, then glanced up to focus unwaveringly on her. "You have my word I'll do my best to keep Lady Charis at liberty too. Then in a few weeks, Lady Charis will have all the freedom she wants."
The irony was Lady Charis wanted only to stay here with Gideon. She was miserably aware that once she reached twenty-one, he had no further reason to keep her at Penrhyn. The prospect of leaving tore her heart to bleeding pieces.
"If my stepbrothers don't get me first." Fear thickened her throat, turning her voice husky. She tangled her trembling hands in her filmy skirts. "I heard you send them away, but..."
"They'll return. With full legal backing. I know."
"Your generosity to a stranger might cost you dear." Like a moth lured to a candle, she ventured closer. Not too close. She'd learned her lesson on that front. "You could go to prison for helping me."
"Didn't you hear Sir John? I'm a national hero." His voice was caustic and his expression bleak. "I doubt I'll be carted off to the clink. The public outcry would be deafening."
"I still shouldn't have involved you in this mess."
He sent her an uncompromising look under his marked black brows. "I despise bullies, Charis. Your stepbrothers deserve to lose."
She clenched her hands at her sides. "I can't stand the thought of your being harmed," she said fiercely. "If you come to disaster because of me, I'll never forgive myself."
His face contorted with sorrow, and he took a step toward her. "I'm not worth your pain."
"Of course you are." His constant self-abnegation infuriated her, made her ache with angry pity. He was so brave and strong and good, yet he seemed completely unaware of his true quality.
Impulsive, unstoppable words bubbled to the surface. Words she'd come close to saying so many times before. She spoke in a heated torrent before she thought to censor herself. "You're the best man I've ever known. You're magnificent. Unlike anyone else. You must know I fell in love with you the moment I saw you. I've only come to love you more every day since."
The headlong admission scorched the air from the room. Her heart slammed to a stop against her ribs. Her cheeks burned with shock and humiliation. She stood stock-still, as if her slightest movement might shatter her into a million pieces.
Dear heaven, what have I said? What have I done?
Hadn't she learned her lesson that afternoon? Her awful, awful gaucheness made her want to vanish into the floor. She'd give every penny she possessed to take back what she'd just done. But the declaration had been made. It was too late to deny it, even if she could bring herself to speak such a lie.
She did love him. She always would. He didn't love her. He couldn't even bear to touch her. But nothing changed the ineluctable truth of what she'd said.
Gideon recoiled and stared at her with what she could only interpret as horror. "Hell," he breathed.
Blindly, he fumbled toward a leather armchair and dropped into it, burying his head in his hands.
Charis felt like she suffocated. At best, she might expect her impulsive declaration to evoke understanding, at worst pity. But this broken desolation was beyond comprehension.
"Hell, hell, hell." His quiet despair reached far inside her like a hand closing around her heart and crushing it.
She was paralyzed with embarrassment. She had to keep reminding herself to breathe. Remorse, concern, self-castigation, all tangled like hissing snakes in her breast.
If he hadn't seemed so lost and tormented when he claimed his essential unworthiness, she'd never have made the reckless declaration. But the sight of him looking as if he didn't have a friend in the world had made her want to die. "I shouldn't have spoken," she said in a raw voice.
His shoulders tensed, and he raised dull eyes to look at her. "Your honesty does you credit."
Her mouth compressed as she fought not to cry. Tears wouldn't help her through this agonizing moment. "Well, I suppose that's one response to a declaration of love." Her tone was flat with control.
A muscle flickered in his cheek. "I can't give you what you want. I'm sorry."
The lump in her throat was like a great, jagged boulder. It hurt to force words past it. "'Sorry' doesn't help."
As compassion filled his eyes, she realized she'd been right to fear his pity. She loathed the way he looked at her right now. It made her want to curl up in a dark corner and never emerge into the light again.
"You'll hate hearing this. And I know you won't believe me. At least now." The kindness in his voice made her cringe. This was even worse than she'd expected. She guessed what he meant to tell her before he spoke.
"Charis..." He paused and closed his eyes as if struggling to find the words. "I'm touched and flattered by what you've said. Any man would be. You're a remarkable girl. You're..."
She felt sick. He lied to spare her feelings, and every false word flayed another strip from her soul. She took a step back and raised her hands to fend off his words. Why, oh, why had she let her foolish tongue run away with her? "Please, don't say any more."
Gideon's jaw firmed, and he leaned forward. Pain flared in his dark eyes and his voice was urgent. "I must. I hate to see you hurt. But what you feel, it will pass. You hardly know me. You can't love me. Not really. The way we met, it's given you a false impression. You've barely had a chance to catch breath since. When you return to a normal life, you'll..."
"What? Forget you?" Resentment at the futility of her dreams frayed the question.
"No." Drawing an unsteady breath, he made one of his familiar truncated gestures. "But you'll see more clearly. Right now you imagine I'm some sort of hero, but you're wrong."
"You are a hero." Her rubbery knees threatened to collapse under her as she ventured closer. She knew he hated that she argued, but she had to make him see himself as she saw him. "You're the famous Hero of Rangapindhi. Even my stepbrothers know who you are."
He flinched against the chair as if she struck him. "The reality of Rangapindhi was far from heroic, Sarah." He paused. "Charis. I'm sorry. You've always been Sarah to me."
She swallowed more useless tears. Her response emerged as a cracked whisper even as she knew nothing she said would convince him she wasn't victim to a childish fancy. "Call me what you like. But don't mistake my sincerity. That's cruel and unjust."
He rose, the muscle still dancing erratically in his cheek. "It's cruel and unjust to let you eat your heart out over a cardboard imitation of a man."
"You're not a cardboard imitation of a man," she said in a low, shaking voice. "And I love you."
He curled his gloved hands tightly around the back of the chair. Grief ravaged his black gaze. "Never say that again, Charis. For both our sakes."
"That won't make it less true." She brushed stinging moisture from her eyes. She refused to break down in front of him. He already thought she was immature and impulsive. A loss of control would only prove that beyond all doubt. He didn't believe her love, and she was fatalistically aware that nothing she said would change his mind.
"I know this is painful." The aching pity in his voice made her want to die. "But one day you'll see..."
She glared at him from burning eyes. At this moment, she hated him almost as much as she loved him. "Don't!"
He drew himself up to his full impressive height, and his hands flexed on the chair. She read his withdrawal as though he wrote it on the air in letters of fire. "Very well."
A turbulent silence fell. He released the chair and began to pace, settling near the desk, where he picked up the bust of Plato and pretended to examine it. Eventually, she couldn't bear to look at him anymore. She turned to stare at the bookcases, although her blurry eyes couldn't read the gilt titles on the leather spines. She raised shaking hands to catch her tears before they fell.
She could no longer tolerate the tension. "I'll go upstairs. I'm not...not hungry tonight."
He sighed with a heaviness she felt in her bones. "I know you wish me to the devil right now. But before you go, there's something we need to discuss."
Still, she didn't look at him. If she didn't escape soon, she'd start bawling and make more of a fool of herself than she had already. "Can't it wait?"
"No."
The uncompromising negative made her turn in surprise. He leaned against the front of the desk, his hands curled over the rim on either side. Strain tautened his body, and his face was more serious than she'd ever seen it. Foreboding clanged like a tocsin, overwhelmed even her embarrassment and chagrin.
"What is it?" She thought she'd clawed back a measure of calm until she met his fathomless black gaze, and hurt and humiliation washed over her again.
"Please sit down."
He gestured to the chair he'd vacated. Silently she obeyed, trying not to notice the trace of warmth lingering from his body.
"I saw immediately what your stepbrothers are," he said heavily. "Swine in fine clothing."
She wanted to tell him how wonderful he'd been this afternoon but he wouldn't welcome her praise. Instead, she raised her chin and spoke in a hard voice. "We can beat swine."
"Yes." He paused. "But I'm afraid the measures will be more drastic than either of us imagined."
She tilted forward, her hands fisting on the chair arms. "Do you intend to kill them?"
In spite of the fraught atmosphere, that startled a soft laugh from him. "What a bloodthirsty wench you are. No, I don't intend to kill them. Or only as a last resort. I have no wish to dangle from the hangman's rope when this is over."
She spoke from the quaking depths of her heart. "Will it ever be over?"
"Yes." He paused again, sending her an unreadable glance. "And no."
She frowned. She didn't know where he went with this. His expression told her nothing. "You speak in riddles."
With sudden restless energy, Gideon swung away from the desk. A few long strides, and he reached the windows. The night outside was dark, haunted by the sea's eternal thunder. Although they no longer spoke of it, her declaration of love lay heavily in the air between them. She supposed it always would. Again, she cursed herself for her impulsiveness.
After a few taut moments, he turned to her, his face terrifyingly grave. "There's only one way I can keep you safe."
She straightened against the chair. One hand clutched her mother's locket like a talisman against evil. "Are you going to take me away?"
"If they tracked you to the wild edge of England, they'll find you wherever we go. We can run if that's your choice, but I don't fancy our chances if every magistrate in the country is after us." He watched her steadily, and she caught the ghost of his earlier devastation beneath his purposeful manner.
"And people will recognize you." Her voice was husky, although for his sake, she tried to sound practical, unemotional.
"My celebrity is a blasted nuisance."
"Your celebrity saved us from a house search today."
"True."
"If we can't outrun them, what can we do? I could go alone." She paused and spoke with difficulty. She hated to beg. Worse, she hated to contemplate leaving Gideon. "If I had some money, I could find a room somewhere—London even. It's only a couple of weeks."
His face darkened in swift rejection. "Over my dead body."
She swallowed the dread that clogged her throat. For all her seething unhappiness, his statement filled her with relieved gratitude. "I can't see an alternative. Apart from the smugglers' hole."
"There is one alternative." His tone was neutral, artificially so, she thought. His eyes didn't waver from her face. "We could get married.
For one radiant moment, joy flared inside her.
Married...
She rose and took an unsteady step toward him. "Gideon..." she began as wild happiness exploded in her breast.
His troubled expression halted her in her tracks and reminded her of his pain when she'd told him she loved him. She sucked in a tremulous breath and looked at him properly.
Her glittering palace of hope disintegrated. The hands that had risen toward him fell back to her sides and formed fists of anguish.
"What's this about?" she asked in a flinty voice.
He shifted away from the windows, back toward the fire. He stopped before her, still too far away to touch. Of course.
"It's the obvious solution, Charis." An unexpected moment to realize he'd started to use her real name naturally. He spread his gloved hands as if appealing to her to see things his way. "If we're wed, I have a husband's legal rights."
Since she'd met him, becoming his wife had been a hopeless dream. Now he proposed, and she wanted to run away and cry her eyes out. Because he married her to save her, not because he wanted her as his life companion, the woman in his bed, the mother of his children.
"You said you'd never marry. Never have a family." Her lips felt as if they were made of wood. "That's changed?"
"No." He held himself rigid as a soldier on parade. His voice was implacable. "It will be a marriage in name only."
She shook her head. "That's not what I want." Then flinched as she saw pity seep back into his eyes.
"It's all I can give you. That and a chance to lead your own life once we see your stepbrothers off."
"I want to spend my life with you."
It was the cry of the spoilt girl, her father's darling, the indulged aristocrat. As she spoke, she cringed. He offered so much for her sake. She had no right to carp at the price she paid in return.
Even if she knew that price would destroy her.
He sighed again and ran his hand through his hair in a gesture of despair. "Perhaps the scheme is doomed after all. I can't bear to hurt you."
Sightlessly, she stared into the grate while her fantasies of a fulfilled life with Gideon scorched away to ash. She'd have a life with Gideon, but they'd be two polite strangers. Duty would sustain them, not mutual love. She wanted to scream her denial to the skies.
Now she understood his appalled reaction to her declaration of love. Marriage to a woman eating her heart out for him promised him eternal torment.
She forced herself to answer. "You said we have no option."
"We could run."
"I'll be safer as your wife."
"This is your whole life we're talking about."
"And yours." He sounded like he cherished no hopes of happiness for himself. The thought cut her like a razor. "I can't ask this sacrifice of you. It's too much."
His face was pale, set, as if he contemplated a death sentence. "Charis, there's no sacrifice on my part. My life is over. In any meaningful sense. Let me help you."
He spoke with such a complete absence of self-pity, it stole her breath. How could he say such things? Yet again, she realized so much here was beyond her comprehension.
Before she could summon a protest at his brutal assessment of his future, he went on, his tone abruptly becoming cool and businesslike. She guessed he resented how much he'd revealed in that last dour statement.
"One of the local men will sail us to Jersey. We can't board the packet in case your stepbrothers have people watching the ports. We'll marry as soon as we can. Certainly within a day of arrival. Two of the villagers will dress as you and me and take the road to Scotland. They'll leave in a fast carriage the moment I have your agreement."
"So Felix and Hubert will think we've eloped to Gretna," she said dully. The extent of Gideon's planning indicated he assumed she'd fall in with his scheme. Of course she would. What choice did she have?
She stiffened her spine. He did this for her. She owed it to him to make everything as smooth as she could.
"It's the more usual route, and the ruse should give us breathing space." He paused, studying her reaction. "We won't return from Jersey until you're twenty-one. Then what happens is up to you. For the sake of appearances, I suggest we live under the same roof for at least a year."
"As you wish." She had no right to resent his generosity. She should be on her knees in gratitude.
He frowned at her lifeless response. "Are you worried I'm a fortune hunter?"
She hadn't thought about the money. Odd when it had colored her relationship with every previous suitor. "No."
"Upon our marriage, your property becomes mine, but I swear I have no intention of keeping it. After the wedding, we'll have papers drawn up returning your fortune to you after a time, I suggest three months, just in case your stepbrothers try something."
"You don't know how much money you give up."
"I don't care."
Strangely, she believed him. Yet again, she thought how remarkable he was. Why in the name of all that was holy couldn't he see that?
"We can settle legalities before the wedding, if you insist. But the sooner you're my wife, the safer you'll be."
Gideon's wife. That was all she wanted to be. But not like this. Never like this.
"I trust you," she said flatly.
He sent her a searching look, then crossed to fill two glasses from the decanter of claret on the coffered sideboard. Like most of the furniture at Penrhyn, it was old and beautiful and completely out of fashion with its heavy seventeenth-century carvings of satyrs and nymphs.
In a day or so, she'd be mistress of this house and all it contained. What a bizarre thought. She'd loved Penrhyn from the first moment she saw it. At the moment, she'd willingly consign it to the sea.
"I know this is difficult for you." He passed her the heavy crystal glass. As if she pressed on a bruise, she noted his care that his fingers didn't brush hers. "I wish I could make it easier."
You could love me, she silently told him. She stared mutely at him while her hand tightened around the glass until her knuckles shone white. "It's not your fault," she said through stiff lips. "My stepbrothers' greed instigated this disaster."
He sipped his wine, then placed it back on the sideboard as if it weren't to his liking. She knew what wasn't to his liking—tying himself to a woman he could never care for. And who cared for him too much.
He faced her, his eyes like black stones. "What you do after the marriage is completely your choice. If you take a lover, I'll acknowledge any children as legitimate. Within my power, I'll ensure your happiness."
She summoned some last shred of resistance although her strongest impulse was to run from the room and cower from the fate that closed around her. "What if I ask you to live as my husband in reality?"
His expression remained somber, implacable. "That's not within my power."
Bitterness surged. She thought her heart broke now. How would she survive endless years of this? "And you? Will you take lovers?"
"No. I pledge my fidelity." His voice contained an undertone of irony that perplexed her. "You needn't fear gossip about an unfaithful husband."
Charis drank some wine, needing the courage, however spurious. If only she embarked on a future of love and hope instead of this arid bargain.
"Yet you're prepared to play the cuckold yourself." Despite her best intentions, sharpness edged her response. "That seems uncommonly generous."
His face was stark with tension. This couldn't be easy for him. Yet again, she reminded herself he put himself through this for her benefit.
"Charis, you're too warm and vital to endure life without love. With your money and freedom, in fact if not under law, you'll be the envy of every woman in the ton."
Her lips tightened against the pain that shafted through her. "I doubt it. I'll be that most pitiable of creatures, a woman in love with a man who can't bear her."
His brows drew together, more in regret than anger, she thought. "I hold you in the greatest esteem. If things were different, I'd..." He stopped and dragged in a shuddering breath as he straightened.
"You esteem me so much, you consign me to a future of deceit and adultery."
She had no right to berate him. Guilt cramped her belly. An apology hovered, but she couldn't quite squeeze it out. She swung away to stand near the fire, but its warmth couldn't thaw the ice inside her.
"If this course is repugnant, we needn't pursue it," he said steadily. How she wished he'd be angry instead of endlessly understanding. She didn't deserve him. She didn't deserve this astonishingly heroic act he made on her behalf.
She turned back to him. "What choice do we have?"
"We run. We hide. We hope to blazes your stepbrothers don't find us." He picked up his wine and stared at it as if it held the answer to all the universe's questions. "Or we stay here, and I bluff them into thinking I'm not involved in your disappearance. I doubt they'd find you in the smugglers' hole."
"If I'm discovered, you'll be arrested."
As he glanced at her, his expression was grim. "It's not the plan I'd choose. But the decision is yours."
She clutched her wineglass like she'd clutched his hands when she'd stumbled on the cliffs. He'd saved her then. She knew he'd save her now.
But at what cost?
"How can I bear marrying you in such a coldhearted arrangement?" she asked rawly.
She waited for another patronizing comment about her love not being real. Instead, he sent her a smile of surpassing tenderness. "You're the bravest person I know. A pair of nodcocks like the Farrells can't defeat a girl of your spirit." His smile faded. "Charis, there's something else."
Her lips compressed in a grimace, and she slumped back into her chair. "I don't think I want to know. Can you tell me tomorrow?"
"The truth will be no easier tomorrow. It never is."
"What a bleak statement."
She noticed he looked uncomfortable. He hadn't looked uncomfortable when he'd informed her he expected her to seek another man's bed. Or when she'd told him she loved him. No, he'd looked devastated then. As if every hope he'd ever cherished came to nothing.
"Although nonconsummation isn't grounds for annulment, your stepbrothers will challenge the marriage on any basis they can. You're a minor and acting against their wishes."
"Surely if we marry in Jersey, the wedding is legal."
"Yes. But your stepbrothers will seek or manufacture evidence of collusion or coercion or fraud. We're safer if we preserve appearances."
She swallowed. "Spend the days together?"
"And at least one night."
For a confused moment, she didn't understand. The statement seemed to contradict everything else he'd said.
It took her a few moments to speak, and she stumbled over the words. "You mean to share my bed."
"As your husband." Gideon paused, and the betraying muscle jerked in his cheek as he visibly strove for composure. "Charis, you can't return to Penrhyn a virgin."
## Eleven
Charis stood in the prow of the sleek little boat as it slid into the harbor at St. Helier. Passing a castle on a causeway, they cut through green water toward the dock.
Ordinarily, she'd be excited to visit the island.
Ordinarily? What in her life had been ordinary since her stepbrothers had forced her to leave her great-aunt? And these last days had piled bizarre circumstance upon bizarre circumstance until her head felt ready to explode.
Yesterday she'd accepted a marriage proposal from the man she loved. Who categorically didn't love her. Who intended to set her free to fill another man's bed. After he'd made use of her body.
Once.
Tonight.
She placed a shaking hand over her roiling belly. Her queasy stomach wasn't the result of seasickness but of crippling nerves.
Dorcas had lent her a rough gown and a thick red woolen cloak more practical than decorative. The village girl who had set out for Gretna in disguise wore a gorgeous emerald velvet cape that had belonged to Gideon's mother. She and the tall, heavily muffled man who accompanied her had departed with great clatter the evening before.
After that, dinner had been strained and silent. Gideon then sent her upstairs to sleep for a few hours before they left under cover of darkness. But she'd lain awake, struggling to come to terms with her desolate future.
Fate granted her dearest wish and blighted her hopes. All in one stroke.
Before midnight, she and Gideon took the secret passage to the beach. He rowed a small boat past the breakers to where Tulliver and William, one of the villagers, waited to sail them to Jersey.
Since then, their journey's speed had astonished Charis. The elements conspired to ensure that her wedding met no delay. A cowardly part of her wanted the voyage to last forever.
She brushed aside windblown strands from her tightly coiled braids and glanced back at Gideon. He stood at the helm like a pirate. Like Black Jack. His hair blew wildly around his face. His gaze was fixed on the horizon, and his white shirt billowed in the breeze. He looked happier and more at home than she'd ever seen him.
At his side stood Tulliver; William sat near the open vessel's stern. The two men would take the boat back to Penrhyn after the wedding.
Gideon's ease with the ship had surprised her. Although of course it shouldn't have. He'd grown up on the coast, and the blood of Black Jack Trevithick flowed in his veins.
Was there anything he couldn't do?
Oh, yes, he couldn't bring himself to live with his wife, could he?
The rancorous thought made her turn to watch the approach of the dock. It should pour with rain to match her mood. But the sky was blue, and the waves sparkled and danced in the sunlight. It was still afternoon. Plenty of time left to get married today.
Then she'd have to make sense of the rest of her life.
God help her.
Charis stood dazedly beside Gideon while a plump-cheeked vicar droned the words of the marriage service.
Gideon was dressed in the height of fashion in his dark blue coat. He looked like any girl's dream prince. Tall, handsome, openly solicitous for his young bride's welfare. Next to him, Charis felt like a beggar maid in Dorcas's cheap pink gown and straw bonnet with its matching ribbons. Heaven knew what the clergyman made of such an ill-matched pair.
In her gloved hands, she clutched a ragged bunch of flowers. To her astonishment, Tulliver had pressed the bouquet on her just before the vicar arrived at their hotel rooms.
The unexpected kindness had come close to shattering the numbness that had possessed her since she'd stepped off the boat. She'd acted like an automaton all afternoon, hardly speaking while Gideon found lodgings and arranged the wedding. If such a sad, shabby event deserved that festive name.
She couldn't let herself think or feel. If she did, she'd break down and cry. She refused to humiliate herself. Nor, more importantly, would she humiliate the man who made her his wife so much against his inclination.
"The ring?"
Would Gideon have remembered a ring? What they did today made a mockery of such a symbol of eternal love.
"Charis?" Gideon prompted.
She raised her eyes from her bouquet, sweet freesias that wouldn't grow on the mainland for weeks yet. Gideon extended his hand. Automatically, she shifted her flowers to her right hand and offered her left.
"Your glove?" he said.
She looked around for someone to hold her flowers, but neither William nor Tulliver noticed. Gideon's lips took on a flat line, then with quick efficiency she could only read as distaste, he stripped away the white lace glove that had belonged to his mother.
His hands shook as he roughly shoved a plain gold ring onto her finger.
It was done. She was married.
Forever linked to this difficult, brilliant, enigmatic, wonderful man.
If only he cared for her, this would be the happiest day of her life. It wouldn't matter that her only witnesses were as close to strangers as made no difference. Or that she was dressed like a milkmaid.
But he didn't care for her.
The knowledge pressed down on her heart like a huge stone.
"You may kiss the bride, Sir Gideon," the vicar said with a heartiness that grated. Everything grated at this moment. Even her own hopeless longing. Especially her own hopeless longing. "A bonny bride she is at that. Felicitations to you both and wishing you many bouncing babes, Lady Trevithick."
Charis bit the side of her cheek to stop herself snapping at the man. His good wishes made her want to scream. If she had any bouncing babes, they wouldn't be Gideon's. They'd be a betrayal of every word she'd just spoken.
She waited for Gideon to give the man the set-down he invited. Instead, her new husband caught her arm before she turned away. "I'll be delighted to kiss my bride."
Shocked, trembling, Charis couldn't have protested even if she wanted to. For one agonized instant, she remembered how he'd reacted yesterday. If he treated her like that now, she'd lose control of the scream building at the back of her throat.
Unsure, frightened, yearning, she raised her gaze to meet his. The black eyes were glassy. The hand on her arm was stiff. Not even the vainest woman could think he wanted to kiss her.
Then she remembered they needed to make a show of affection in front of their sparse audience in case Hubert and Felix challenged the match. She also remembered Gideon did this for her sake, and temper was poor repayment.
She summoned her courage and plastered a smile on her face. It felt like the rictus grin on a skull, but a glance at the jovial vicar indicated it convinced him.
"I'll be delighted to kiss my beloved husband." At least she didn't have to lie about that.
Admiration lit Gideon's eyes before he leaned down and pressed his mouth to hers. The shock of the contact made her drop her bouquet. A host of sensations overwhelmed her, vividly familiar even though they'd kissed only once before.
His clean scent. Lemon soap. Beneath that, the fresh, salty tang of his skin. He'd washed and changed. But still he smelt like the sea. His height. Occasionally, she forgot how long and lean he was. He was as hot as a furnace. Standing next to him was like standing next to a great blazing hearth. When she'd been cold forever.
His mouth moved on hers with subtle pressure. Instinctively, she parted her lips and drew his breath into her lungs.
The intimacy was astonishing. By far the most intimate moment she'd ever shared with anyone.
She closed her eyes. Tingling warmth seeped from his kiss. Down, down, to settle at the base of her belly. She sighed and leaned forward, lifting her arms.
She opened hazy eyes to see him step away. He looked pale but composed as he briefly shook the vicar's hand. She realized she still reached out like a mendicant. Blushing, she folded her arms before her to hide their trembling.
Gideon only kissed her for show. Still, she'd clung to him like ivy clung to the walls of Marley Place. If she wasn't careful, he'd grow to despise her for this endless need she couldn't conquer.
"What a beautiful couple," the vicar was saying. "I'm happy to be of service to such a gallant gentleman, a hero of the nation."
Of necessity, Gideon had revealed their true identities to the man who married them. They'd booked into the hotel under false names. Mr. and Mrs. John Holloway.
Gideon's expression didn't change although Charis guessed the fulsome praise chafed. "Reverend Briggs, remember there's twenty guineas if you keep my identity to yourself for the next fortnight. My wife and I seek privacy."
"Of course. Of course. It's an honor for my island to host your nuptials. The Hero of Rangapindhi here. Now that's a tale I can tell my grandchildren."
"Tell them in two weeks." No mistaking the threat in Gideon's voice. The trace of menace pierced even the vicar's rapture.
"You have my word as a gentleman and a man of the cloth, Sir Gideon. No whisper of what passed today until you leave Jersey."
"Good."
Gideon turned to Charis and crooked his arm. Another action to convince their guests this was a normal wedding. Hesitantly, she rested her hand on his fine woolen sleeve. Beneath the expensive material, she felt his body's latent power. She fought the urge to curl her fingers into his coat. Goodness, she'd touched him more in the last ten minutes than she had since he'd been insensible with illness.
"Thank you for your assistance." As Gideon addressed the clergyman, he sounded lordly and cool, not at all the man who shrank from the brush of her hand.
The vicar closed his prayer book. "Will you and your bride join Mrs. Briggs and myself for a glass of madeira at the vicarage?"
Gideon's expression became more remote. "I'm afraid that's impossible although your invitation is kind. Do we need to sign further documents?"
The vicar shook his head, his face almost comical with disappointment. "No. You're married right and tight."
"Capital. We'll wish you good day, then." The arm under Charis's hand was rock-hard with tension, but to any observer, Gideon appeared completely in control of himself and his surroundings. "Remember, not a word."
Tulliver and William approached them as the vicar left.
"God grant you every happiness, Lady Charis," Tulliver said quietly.
"Aye, my lady," William said behind her.
Such simple wishes. Such impossible wishes. Furiously, she blinked away tears. She couldn't cry now. She had to stay strong for what awaited.
"Thank you," she said in a choked voice.
"Are you all right?" Gideon murmured, leaning toward her as they stood near the grate. It made her wince to hear him sounding like any new groom, mindful of his wife's comfort.
"Yes," she said almost inaudibly, concealing her unhappiness by tilting her head, so her bonnet shaded her face.
But, of course, he must guess how she felt.
Her fingers clutched at his sleeve, then she realized what she did and snatched her hand away. "I'm sorry," she gasped.
He loathed her touching him. That much she knew.
He caught her hand in a ruthless grip and dragged it back. "We need to appear like any happy couple," he growled under his breath, even though she felt him shaking with disgust.
"Then smile," she hissed.
His lips curved upward, but no warmth entered his eyes. He looked drawn and distant as though his essential self hid away.
He turned to the men. "It's time to head home. If there's sign of trouble in Penrhyn, send word under the names of John and Mary Holloway here at the Port Hotel. We'll make our own way back next month."
Tulliver bent his head in acknowledgment. "Aye, guvnor. And congratulations. You've snagged yourself a fine lass there, make no mistake."
For the first time, Gideon's smile looked natural. "I have at that. She got much the worst of the bargain."
His lies sliced at Charis. She bit back an acid retort.
Tulliver and William left Charis alone with Gideon. Suddenly, the luxurious parlor seemed cavernous, echoing. Across the floor, the door to the equally luxurious bedroom loomed like the gates of hell. She felt ill at ease with him now as she never had before. Even after that desperate kiss at Penrhyn.
"I've arranged dinner." Her husband leaned one arm on the mantelpiece. He'd wasted no time putting distance between them once their onlookers departed. His gloved hand fisted against the ledge, and he looked as if he braced for disaster.
"I'm not hungry," she said tonelessly.
"Appearances..."
"Must be maintained. I know."
Charis knew she behaved badly, but she couldn't help it. She was torn between desperate gratitude and frustrated longing. And slashing guilt because there should only be gratitude.
Lines of tension framed his mouth, and his eyes glittered with stress. Again, she reminded herself he put himself through this suffering for her. Sick shame left a vile taste in her mouth.
If she had a shred of decency, she'd ask nothing further of him.
But she couldn't silence her wayward heart, which shrieked and clamored and demanded. She longed for him to love her more than she wanted to take her next breath. Nor could anything shake her bone-deep certainty that if he let himself love her, he'd find his own salvation.
Self-serving justification for her hunger? Or truth? She couldn't say. But he was worth more than this barren bargain they'd struck. She was worth more.
Night had fallen, and she moved around the room lighting candles. There was some relief in the workaday action. As light bloomed, she became conscious of Gideon's shallow breathing.
"Are you ill?" she asked with deliberate calm, carefully lighting each branch of the candelabra on the sideboard.
"No," he said hoarsely. His face was paper white. He looked like a man approaching the limits of endurance.
She knew what set that haunted look in his eyes. The prospect of bedding her. She tensed her throat against the agony of that awareness.
Compassion as much as conscience provoked her to speak. "Gideon, we don't have to do this. The vicar said we're married right and tight. You've already gone to extraordinary lengths to keep me safe." She extended one hand in a wordless plea for him to lay aside his burdens. If only for one night. "I can never find words to express my thanks. Nothing could repay what your championship of me has cost you. You needn't make further sacrifices."
He sucked in a deep breath, then, to her complete shock, he laughed. His dark eyes glinted with self-derisive humor as he straightened away from the hearth.
"Good God, anyone who knew me in my salad days would roll around the floor laughing himself sick to hear you. You'd think I was some shivering virgin." A cynical expression crossed his face, and he suddenly looked eons older than his twenty-five years. "I have done this before, you know."
Yes, with his skilled and spectacular Indian bibis. The statement didn't ease her uncertainty. It just made her jealous and insecure. "I'm well aware of that," she said starkly.
How she wished she had an ounce of those women's sensual skills. She'd captivate her husband with such pleasure, he couldn't help falling in love.
His face filled with sorrow. "I'll try my best not to hurt you."
"I know." She'd trust him with her life. She already had. Just as she'd trusted him with her heart. Even if he didn't want it.
"With a first time, there can be pain."
The subject made him uncomfortable. Or perhaps he was merely uncomfortable talking about this with his troublesome bride. His exotic Indian lovers, she was sure, hadn't made him feel awkward.
Stop it, Charis.
"I know what takes place." Heat flooded her face. She wasn't easy with this conversation either. She raised her chin, although the hand holding the taper trembled. "I grew up in the country, and my mother told me what to expect."
He raised his eyebrows, and his lips curved in another ironic smile. "Quite the expert then."
She shook her head as nerves set her belly to cramping. "I never kissed anyone until...until yesterday."
His face hardened in anger. "You must think you've married the clumsiest oaf in Christendom."
Her voice was muted. "You know I don't think that. I'm prepared for what's going to happen."
"Well, that reassures a man." In an abrupt gesture, he ran a hand through his hair.
"I don't know what else to say," she said helplessly, fighting the urge to smooth that unruly dark mass. The need to touch him was a constant fever in her blood. Fighting it left her exhausted, jumpy, nervous. "It's hardly a normal marriage, is it?"
"No, it's hardly that." His voice thickened with regret. "You've missed out on so much. There's nothing I can do to make it up to you."
Stay with me. Love me.
She stifled the words. Things were difficult enough without her nagging him for what he couldn't give. She blew out the taper and set it in its holder.
"None of this is your fault," she said despondently, turning away and slumping into a chair. She was weary, although most of her tiredness was emotional rather than physical.
She went on in the same austere voice. "It's not my fault either. Hubert and Felix are greedy and corrupt. Lord Desaye is desperate and deceitful. But the amount of money my father left me is obscene. It turns men into monsters." She paused. "Every man except you."
He grimaced. "I'm already a monster." He continued before she could protest. "Lord Desaye, I take it, is the suitor."
She shuddered. "He gambled away his own fortune and his first wife's. A shadow hangs over her fate. He was the only witness to the carriage accident that killed her."
"How did he and your stepbrothers link up?" Gideon seemed relieved to discuss something other than her imminent deflowering.
"Money, of course." Her voice was flat. She fiddled with her wedding ring. It was old and heavy and sat loose on her finger. A symbol of the weak bond between her and Gideon? "They gambled together. I'm sure Hubert or Felix would have tried to marry me if the church didn't frown upon unions between stepsiblings."
"Did they tell you this?"
"On that last day. I'd worked it out already." She released the ring, and her fingers curled into claws in her lap. "I sometimes wish I'd been born poor. My fortune has only caused misery."
"You'll grow into your station. At least as my wife, you're safe from fortune hunters."
She looked at him curiously. "Doesn't the idea of keeping my wealth appeal? You haven't asked how much I'm worth."
"I know what you're worth," he said sharply, stepping toward her. "It has nothing to do with pounds, shillings, and pence."
She fought back the traitorous warmth that seeped into her heart at his response. "Few people would agree."
"The rest have fewer brains than God gave a flea."
As she gazed into his blazing black eyes, she couldn't look away, and the breath caught in her throat. Heat flooded her and settled like lava in her belly. The overwhelming emotion that flooded her was heady, uncontrollable...terrifying. He had such power over her, and she was helpless to resist.
He stared at her as if he thought she was a princess. It was cruel. He didn't want her. She opened her mouth to speak but had no idea what she meant to say.
Someone knocked softly on the door. The charged silence shattered.
She sucked breath into starved lungs. Gideon gave permission for the servants to enter. Everything turned to movement as waiters set out dinner.
She'd seen Gideon leave a substantial tip when they'd registered. He'd explained he and his bride insisted upon privacy. If they left Jersey without undue disturbance, he'd see the staff were suitably rewarded.
With a flourish, one waiter produced a bottle of champagne. "The compliments of the house, Mr. Holloway. To you and the new Mrs. Holloway, our very best wishes for a long and happy life together."
Charis finally had some idea how Gideon felt when people hailed him as a hero. That he existed in two realities operating side by side but forever disconnected. She kept forgetting that as far as the outside world was concerned, this was the happiest day of her life.
The strain of reconciling the contradictions left her disoriented, sick, detached from any reality at all.
The waiter opened the champagne and poured it into two heavy crystal glasses befitting St. Helier's finest hostelry. There was more bustle as servants pulled out chairs and unfolded napkins and served the first course, a fish soup fragrant with garlic and herbs.
Finally, she and Gideon were alone. A painful tension tightened around them like a steel net.
"It looks delicious." She lifted her spoon, then put it down again, the soup untouched.
"Yes."
There was a pause while they both stared at their plates.
He looked up. "Perhaps I should see what's next."
"Perhaps you should," she murmured, although she knew she wouldn't eat that either. She felt like a boulder blocked her throat.
He lifted the covers and rich savory aromas drifted into the air. "Poulet à la persane. Boeuf en daube. Lobster. It's a feast."
"Didn't you order it?"
"I said to send up whatever they recommended. What would you like?"
"Anything."
She watched as he filled two plates from the serving dishes.
"You know, I used to dream of dinners like this when I was in India." He slid her plate in front of her and took his place opposite, shaking out his napkin with an elegance that made her breath catch. Even such a simple gesture left her aching with desire.
Could she endure a lifetime of this relentless longing?
"What did you eat there?" It was a neutral enough topic. Would she spend her years making meaningless conversation with the man she'd married? The cold unhappy future stretched before her like an endless steppe.
He shrugged, his hand playing with the stem of his glass. He still wore gloves. "Curry. Delicacies fit for a rajah. Cold rice with weevils."
Painful memories she couldn't hope to understand shadowed his face. Before she could inquire further, he raised his glass. "I'm remiss in my husbandly duty. To my lovely bride."
It was more than she could bear. She shoved her plate away and rose on wobbly legs. "Please don't."
He put down the champagne, like his dinner, untasted. "I too find my appetite lacking." He stood. "I'll take a walk. There's a bath coming. No hurry. I'll be away for several hours."
Snatching privacy to fortify himself for the onerous task ahead, she guessed with another stab of pain. "I wish you a pleasant stroll," she said lifelessly.
He bent his head in a courtly salute. "Thank you."
Only when he'd gone did she realize it was the first time he'd left her unprotected since he'd met her.
## Twelve
Gideon held himself together until he closed the door behind him and stood in the deserted corridor. He collapsed, gasping, against the wall. Shivers combed through him like breakers up the beach at Penrhyn.
He couldn't go through with this.
He had to go through with this.
He closed his eyes and banged his head several times against the wood. But nothing could banish the vivid images in his mind.
Charis watching him across the table, her beautiful hazel eyes brilliant with anguish and a longing he shared but couldn't fulfill.
Charis standing beside him saying words that made her his wife.
Charis telling him she loved him.
Ah, the forbidden sweetness of that moment.
And the desolation.
She had such courage. What a consort she'd make for the man worthy of her.
Damn it, he could never be that man.
His rejection might hurt now, but she'd get over her infatuation. She'd emerge from this stronger, better, bright as a star. The real tragedy was that she tied herself so irrevocably to a wreck like him.
He groaned through clenched teeth. He'd endured unspeakable pain in India. Already he knew that the hell of watching his wife fall in love with another man would outstrip any devilish torture the Nawab devised.
Bear it, he must.
For Charis's sake.
The gods clearly laughed at his sufferings. They granted him the one woman he'd want for the rest of his days. Then they made it impossible for him to find joy with her.
He desired her to the depths of his being. His very skin ached for her touch. He'd exchange all the minutes remaining to him for one night of freedom in her arms. Instead, in his clumsiness, he was going to hurt her.
Not, by God, if he could help it.
With grim determination, he straightened from the wall. He turned up his collar and pulled down his hat to conceal his face.
He'd do what was necessary. Whatever it cost. His scheme might seem crack-brained, dangerous, but it was the only solution he had. He'd accept any pain if it saved Charis suffering.
He didn't deceive himself about the pain his plan promised.
As he trudged downstairs and out onto the street, his heart was heavy. It was cold on the seafront. The breeze from the sea had ice in it. Or perhaps the chill was in his grieving soul.
He knew where to find what he required. Behind the smart façades and bustling respectable thoroughfares, every town had its shadow. Despising what he did but seeing no alternative, he turned away from the lights and plunged into the old town's maze of streets.
The girl was even younger than Charis. Seventeen or eighteen. Although with the lives these women led, who could tell?
Standing on her corner, she retained a trace of country freshness. She was clean, and her dress hinted that some shred of spirit defiantly survived her profession.
Most of all, though, Gideon chose her because she bore absolutely no resemblance to the wife he'd left at the hotel.
"You, girl, do you have a room?"
She brightened as she looked at him, her light eyes, blue or gray, Gideon could hardly tell in the gloom, sparking as she took in his fine clothes. She patted her untidy blond chignon with a gesture designed to entice.
"Aye. But it will cost ye ten shillings, me handsome gent."
Ten shillings was a fortune for someone like her. He knew she cheated him, but he didn't have the heart to haggle. Given what was likely to happen when he came to the business, she'd earn her money before he finished.
"Done."
She frowned suspiciously. "I want to see yer blunt up front."
He fumbled in his pocket and withdrew a sovereign. The gold glinted evilly in the faint light. He dropped it into her outstretched hand.
His flesh crawled at the prospect of getting closer to her. God knew if he could go through with this. He hadn't even touched the chit yet, and already he was a trembling mess. The possibility of failure rose like a dark miasma.
"Let's go," he said roughly.
The girl stared at the coin, then glanced up with a smile that made her look older than she was. "An eager beaver, ain't ye, sir?
She waited for him to respond, but he was busy trying to keep his gorge down. God give him strength. He could do this. He could do this. He hadn't touched anyone since Rangapindhi. But surely he could perform with a stranger when it didn't matter if he made an utter disaster of the act. Surely he was man enough for that.
She shrugged. "Don't ye want to know my name?"
He closed his eyes in agony. Only the knowledge that Charis waited stopped him fleeing back to light and warmth.
"No," he managed to grit out, opening his eyes to shabby reality. "I don't want to know your name."
The girl looked at him strangely and pointed to the filthy stairway behind her. "It's up here, sir." She sounded subdued, or perhaps that was just the blood pounding in his ears.
Blindly, Gideon followed the plump blond tart upstairs to her room.
Charis didn't know what woke her. She couldn't remember falling asleep. It had been late, and she'd been alone. Just as she knew immediately she was alone in the bedroom now.
She cracked open a swollen eyelid. The room was pitch-dark. The servants had drawn the curtains when they came to collect the uneaten meal and take away the cold bath. But as her sight adjusted, she recognized the heavy furniture. Old French oak pieces like something from a prerevolutionary chateau.
As she shifted experimentally, she muffled a moan. Devils with hobnail boots blundered around her skull. She licked dry lips. Her mouth tasted sour and stale. She shifted again and realized her dress twisted around her as she lay awkwardly across the covers.
With a low groan, she sat up. She raised a trembling hand to her sticky face. She remembered now. Every last pathetic moment until she'd collapsed in a stupor.
She'd waited in a lather of nerves for Gideon to return from his walk. Nerves and genuine alarm. After all Gideon's subterfuges, it was unlikely Felix and Hubert would burst in on her the first night on Jersey. But she felt lost and defenseless now her Galahad abandoned her.
One hour passed. Two. Her apprehension turned to hurt defiance. She knew why he avoided her. Because he couldn't bear to touch her.
She wanted to send him to the devil. She wanted to beg him to love her the way she loved him.
With rankling hostility, she drank the champagne, as if the act somehow got back at him. Even after she started to feel sick, she kept drinking. She drank until the bottle was empty, and the room whirled in a wayward waltz.
Eventually, inevitably, her empty stomach rebelled, and she was vilely, painfully sick. By then it was past midnight and still no sign of her husband of mere hours.
Tears she'd dammed through the agonizing day welled up. Painful, humiliating, unstoppable tears. She clenched her fists, digging her nails into her palms as she battled for control. But nothing helped. Sobbing in ugly gulps, she'd curled up on the bed. Crying, she must have fallen asleep.
To wake with a headache, a rebellious stomach, and a heart brimming with shame.
Vaguely, she wondered what time it was. A heaviness in her limbs indicated she hadn't slept long enough to overcome her fatigue. Or perhaps the wine made her ache. She'd never had more than a glass or two at once before. The foul taste in her mouth made her swear one glass was too much in future.
The inn was silent, and no noise rose from the street. She felt suspended in some dark cocoon. Alone forever.
"Stop it," she whispered. Why she kept her voice down, she couldn't say. She was on her own.
Except something had disturbed her.
She held her breath and listened.
Not a sound.
Gideon obviously hadn't returned.
Curse him.
She should lie down. Rest her throbbing head. Still, she sat bristling with awareness, straining to discern the slightest sound through the enveloping darkness.
Very carefully, she edged off the bed.
Nothing stirred in the next room.
Icy fear trickled down her spine. What if Felix and Hubert lurked out there, ready to snatch her back to Holcombe Hall?
With shaking hands, she slid a large china jar from a chest of drawers. Its pale glimmer made it easy to locate. The jar wasn't much of a weapon, but, armed, she felt less vulnerable.
Crunching her toes against the chill, she padded on bare feet across the floor until she reached the door. The parlor beyond was quiet, empty. The fire had burned down, but its low glow revealed that nobody was there.
Except...
"I know you're here." Relief mixed with a fortifying dose of irritation trickled down her spine. Her voice sounded scratchy and unused. Speech made her sore head ache.
No answer.
She stepped farther into the room. The floor was cold against her soles. She took another step, so at least she stood on the rug and could curl her toes into the wool.
The silence continued.
Her lips thinned with annoyance. "It's no use pretending."
More silence.
She bent and placed the heavy jar on the floor. Unless she lost her temper and smashed it over Gideon's thick skull, she wouldn't need it.
Would he continue this foolish game?
She heard a shuddering sigh from the corner of deepest shadow. "How did you know?"
"I always know when you're near," she said wearily, and felt her way across to the sideboard.
"I'm sorry I woke you."
"It doesn't matter." Perhaps it was the desolate feeling attendant upon the early-morning hours, but right now, she felt that nothing in the world mattered.
The air was so still, she could hear the even susurration of Gideon's breath. His chair creaked as he shifted. The fire crackled in the background. The intimacy was intense, fraught, electric. At the same time Charis felt that a thousand miles of frozen sea separated her from Gideon.
Gooseflesh prickled her skin. She should have grabbed a blanket before leaving the bedroom. She picked up a candle, intending to light it in the coals.
His eyesight must be better than hers in this darkness because he spoke quickly. "Please don't light it."
She paused and faced him, leaning against the sideboard and shivering in the cold. "Why?"
He didn't answer. Or not in words anyway. "Go back to bed, Charis."
"Alone?"
"For God's sake, yes." His voice cracked. "We'll talk in the morning."
"What's the point?" She sucked in a deep breath and realized the sour alcohol smell didn't come from her or the empty champagne bottle. "You've been drinking."
It wasn't an accusation but of course it sounded like one. His chair creaked again as he straightened. "Yes. And I've been fighting." His voice sounded odd. Flat and unmusical as she'd never heard it.
With sudden determination, she stepped across to the hearth and lit the candle. A feeble glow bloomed. Her hand trembling, she turned and raised the candle in his direction. Against the back of her legs, the fire's warmth was welcome.
She expected him to jerk away but he sat unmoving as she illuminated the thick darkness around him. When she saw him, Charis couldn't contain a choked gasp.
"I take it I'm not too pretty?"
Her hand shook so badly, she had to slide the candlestick onto the mantel. But the uncertain light had revealed enough to make her feel sick all over again.
His lips lengthened in a grimace that she knew was meant to be a smile. He answered his own question. "Obviously not."
"You're unwell," she said in a raw voice, wrapping her arms around herself in an attempt to generate some warmth.
"No, just drunk and heartsick." He made a sudden savage gesture with one gloved hand. "For the love of Christ, Charis, go back to bed."
"No," she said stubbornly, tightening her arms to hide her shaking.
"Not twelve hours ago, you promised to obey me."
"And you promised to love me," she snapped, then immediately regretted the words.
His face tautened with pain that made her flinch. He looked terrible. His clothes were torn and streaked with dirt. A graze marked his cheekbone, and blood stained the open collar of his shirt.
The elegant man she'd married was only a memory. He'd lost his neckcloth, his gloves were filthy, and his jaw was dark with bristle. Now she was closer, the reek of alcohol was unmistakable.
Worst by far was the expression in his eyes as he stared at her. He looked haggard and ill and as if he wished he were dead.
Still, his voice deepened into kindness. "Go back to bed, Charis. Everything will look better in the morning."
It was the facile, meaningless promise one offered a child. There are no monsters under the bed. Let me kiss it better. There will be a happily ever after.
Even though she quaked with nerves, her tone was firm. "No, everything won't. You need to tell me the truth, Gideon. I'm your wife. I deserve to know what's wrong." She paused, then made herself push on. She was tired of fighting imaginary horrors. The truth couldn't be worse than the phantoms in her mind. "Are you sick because...because of what you did with those women in India?"
He recoiled. For a horrible moment, she wondered if her guess was accurate.
"Venereal disease, you mean?" He shook his head. "No, I'm clean. In fact, my body is in perfect working order. In every way."
The emphasis struck her as odd. "What do you...?" Then she realized what he meant. "Oh."
"What's the point of lying? Close confines will soon make my condition plain." The words slurred slightly with drink as his control wavered. She doubted she'd get this much frankness out of him if he wasn't half-seas over. His deep voice vibrated with feeling. "I ache with desire for you."
The candle flame burned unflickering. Silence fell. Lengthened.
A coal exploded in the grate, snapping the tension. Charis's paralyzed brain began to work again. And harsh reality shone a stark light on his lie. How could she think him kind? He was crueler by far than her stepbrothers. They couldn't hurt her heart. Gideon could.
"Don't mock me," she said sharply, rubbing her arms.
"If there's a joke, it's on me." Despair dripped from every word. His eyes sharpened on her. Abruptly he stood and ripped his coat off. "You're cold. At least put this on."
"Thank you." Her frozen hands took the garment. When she pulled it on, warmth and the subtle lemon scent of Gideon filled her senses. It was almost like he touched her. "You don't want me. You jump ten feet if I come near you."
He gave a short, unamused laugh as he dropped into his chair. He leaned his head back and studied the shadowy ceiling. "That's the vilest element of my affliction, dear wife. I can want to the point of insanity, but I can never have. A punishment worthy of a damned Greek myth."
She shook her head, ignoring the lingering twinges of headache. Perhaps the champagne had damaged her mind in some fundamental way. "You said you weren't sick."
"I said my body worked fine. The trouble, my love, is in my head. I should have warned you before you tied yourself to me for life. Your husband is possessed by devils."
My love? For a moment, the world faded to nothing. Had she imagined that endearment? Surely she had. She wasn't his love. He could hardly bear sharing the same room as her.
She drew the scrambled remnants of her concentration together and addressed the immediate issue. "You're not mad," she said shakily. She believed that to her bones.
He clutched at the wooden arms of his chair as if they offered his only link to reality. "If I'm not mad already, our marriage will be the end of me."
What was he telling her? Her dazed mind struggled to sift fact from fantasy. She didn't understand what troubled him or what she could do to help. But it was astonishingly clear that what she'd always believed unassailable truth was categorically false.
"You want me?" she asked in dawning wonder.
His lips twisted in another of those grim smiles, and at last he looked at her. "Indubitably."
Letting her arms fall to her sides, she stepped nearer. "Surely that means..."
He surged to his feet and lurched toward the wall behind him. "Damn it, Charis, don't touch me."
He pressed against the wall. She heard the uneven rattle of his breath. She stopped and frowned. "I can't touch you, yet you say you...want me."
"I told you it was insane."
All of a sudden, a whole range of memories came into focus and made sense in a way they never had. If anything about this bizarre situation made sense. She spoke slowly. "You can't touch anyone. That's why you got sick after Portsmouth. All those people."
He was as tense as if she attacked him with a rapier instead of words. She expected him to lie or refuse to answer. But he gave an abrupt nod. "Yes."
She retreated carefully as if she tried to calm a wild animal. With one unsteady hand, she felt behind her until she gripped the back of a chair. "I won't come near you."
"Thank you," he said quietly, a world of relief in the words.
She kept her voice even, as if indeed he were an animal caught in a gamekeeper's trap. "Won't you sit down?"
He hesitated, then returned to his chair with jerky movements. In the feeble light, he looked tired but composed. Slowly she sank into the chair she held, curling her cold toes under her.
"Were you always like this?" She thought and answered her own question. "No, you can't have been. You've had lovers."
"Charis..."
Twining her hands together in her lap, she raised her chin. Her courage faltered, but she steeled herself. She was guiltily aware she took unfair advantage of his weariness, his inebriation, his wretchedness. But she had to seize her chance.
She forced out the question she'd always been afraid to ask. "What happened in Rangapindhi?"
## Thirteen
Even in the dimness, Charis saw the blood drain from Gideon's face. His eyes became opaque, as if he stared at gruesome specters visible only to him. He gripped the chair arms like a drowning sailor snatched at driftwood to keep himself afloat.
Anyone with a scrap of sympathy would relent. Tell him he was welcome to his secrets.
She remained silent and waited.
When she'd given up hope of an answer, he sucked in a rasping breath and focused on her. "My tutor at Cambridge recommended me to the East India Company."
"Your knack with languages." She kept her voice carefully neutral.
"Yes. And I was a rider and a cricketer and a marksman and a swordsman. The Company always wanted men with my peculiar skills."
As if the Company found many recruits with such talents, Charis thought, noting again his lack of conceit in speaking of his abilities. She wasn't surprised he'd been both out-standing scholar and outstanding sportsman. From the first, she'd recognized how remarkable he was. The tragedy was he could do so much out of the ordinary, yet something as simple and essential as sharing the touch of a human hand was denied him. Her belly cramped with a feeling more profound than pity.
"I was ripe for adventure, in need of a career, eager to find an outlet for my energies." His voice was husky but steady. Only his face, drawn and white, indicated the ordeal he found this recounting. "I set out to spread the light of European civilization to a benighted people."
"But it wasn't like that?" She hardly needed to ask. His tone reeked of shattered illusions.
"No. I encountered a sophisticated, exotic world beyond my wildest imaginings."
He'd told her he worked in native liaison, but that meant nothing to her. "So were you an administrator?"
Bleakness etched his expression. "Nothing so admirable, Charis. I was a spy."
Shock pinned her in her chair. So many elements that perplexed her about him came together at last. His cleverness and confidence against Hubert and Felix. His handiness in a street brawl. His secretiveness. His shame.
When she didn't speak, he went on, still in that calm voice so at odds with the torment in his black eyes. "I'm naturally swarthy, and my skin tans in the sun. I became Ahmal, a Muslim scribe. A scribe learns a kingdom's secrets, and few question his movements."
She clutched her hands together so tightly they hurt. It became near impossible to maintain her mask of composure. "It must have been difficult living a lie."
"Dirty, lonely, difficult." Still, he gazed at some far-off landscape she couldn't see. "But I thought I worked for the greater good against forces of barbarism. At least at first. In the end, I believed my masters' greed the greatest barbarism, far worse than anything I encountered among the natives." He paused, and his hands flexed convulsively on the chair arms. "Then, along with two of my colleagues, I was betrayed."
Finally, his unearthly self-command fragmented. The roughening of his voice told her he approached the worst part of his story. She tensed, and dread coalesced into a cold mass in her stomach. She already knew she'd loathe hearing what he told her.
"It was my last assignment." With every word, his tone became more austere. "The Nawab of Rangapindhi plotted to invade a neighboring kingdom, whose ruler favored the British. My superiors were desperate to learn what happened in Rangapindhi. But the Nawab was cunning and on his guard—worse, he had spies in the Company."
"This is a world I can hardly imagine," Charis said softly, forcing the words past her apprehension.
"For most of my adult life, it was my world, familiar as my own face in the mirror."
"But always dangerous."
"If you forgot that, you were as good as dead." Suddenly restless, he swung to his feet and crossed to stoke the fire with suppressed violence. The flames cast unforgiving light on the taut lines bracketing his mouth.
"I wasn't supposed to go to Rangapindhi." He set the poker down with exaggerated care, and his voice was flat with control. "I'd handed in my resignation and booked passage to England. But my masters wanted their best men, and I let myself be persuaded. Three of us—Charles Parsons, Robert Gerard, and I—went into Rangapindhi." The silence was longer this time and charged with Gideon's grief and anger. "Only I came out alive."
"What happened?" His expression told her it had been terrible beyond description.
"Gerard was careless. He'd been in the field ten years. Too long. He was a good, courageous man. But even the best make mistakes when pressure goes on too long."
She noted but didn't comment that he was ready to forgive a failing in another that he refused to forgive in himself. He ran an unsteady hand through his hair, and his body sagged with what she read as defeat. He was tired and hurt, and she had no right to harangue him. But if she didn't catch him now, when he was vulnerable, he'd retreat behind his formidable defenses.
He sighed heavily. "Damn it, I've had too much to drink."
She rose on trembling legs, battling a dizzying mixture of fear and overwhelming love. "Gideon, for pity's sake, tell me."
Standing in the center of the shadowy room, his wife was as beautiful as a carved alabaster angel in a cathedral. And just as unrelenting.
Charis's unwavering gaze held such trust, such love. Both pierced him with sorrow. Gideon couldn't rely on the love, and he didn't deserve the trust.
He shut his eyes and fought for strength to deny her. Everything between them would change once she knew what had happened in India. He couldn't burden her with the horrors of his past. He couldn't enmesh her in the chaos of his life.
But simmering guilt and too much liquor played hell with his principles.
Reluctantly, he opened his eyes and took a step closer. "The Nawab had us chained and dragged into his audience hall. I'd only seen him from a distance before. They called him the Elephant of Rajasthan. Fat rolled off him in monstrous folds. He wore ropes of pearls as big as pigeon eggs. They must have weighed a ton."
"He knew you were British in spite of your disguise?"
The memory made the skin on the back of his neck crawl, and his hands fist at his sides. "He had us stripped in front of his court."
He saw she didn't understand. Sometimes he forgot how little his countrymen knew of the Subcontinent. "We posed as Muslims, but none of us were circumcised."
Sweet pink flooded her cheeks, visible even in the flickering candlelight. "Oh."
"I'm surprised you know what I mean."
"I had the run of my father's library. He had some unusual books." She paused. "And anyway, it's in the Bible."
Again, he realized this woman was considerably more mysterious than anything he'd encountered in India.
"We provided an evening's diversion for the court." Gideon spoke quickly, hoping that would ease the telling. It didn't. "We were whipped."
He bit down hard, trying not to remember the cutting agony of the lash, the strangled groans and screams from Gerard and Parsons.
"He meant to humiliate you." Charis's composure was surprising, impressive, but he noticed the tremor in the hand she curled around the back of her chair.
"Us and the overweening British nation. He wanted information too, but that could wait until specialists got their hands on us. This was purely for His Highness's entertainment."
"You didn't beg for mercy." Her voice rang with certainty. The knuckles on her fine-boned hand shone white as she clutched the chair.
"I had too much stupid pride. It meant my beating went on considerably longer than the others'." Until he'd collapsed unconscious on the cold marble floor. He'd thought then he had tested the dregs of humiliation. How naïve he'd been. "Then they took us away and tortured us."
Dear Lord, don't let her ask about his torture in the Nawab's dungeons. The memories were so vivid, it was as though he still hung in chains from the seeping, fetid walls. Nothing this side of heaven could force him to tell her about that foul Gehenna. A place of neither night nor day, just darkness, lit by the flare of torches and reeking with blood and filth and terror.
The fiendish instruments. The endless torment. The inevitable knowledge that nothing could save them.
There would be pain. Then there would be death. No escape.
"Gideon..." She looked down and sucked in a shuddering breath. Not before he caught the shimmer of tears.
Her shaking distress wrenched him back from nightmare. "I should stop. I'm upsetting you."
As she looked up, her eyes glittered. He was astonished to recognize fury beneath her wretchedness. "Of course I'm upset. You describe your systematic degradation and torture." Her slender throat moved as she swallowed. "How long were you held?"
"A year. Mostly in a dark pit the size of a grave." His voice was still flat although his heart beat like a drum as he revisited the agonies of Rangapindhi. Not that they were ever far from his thoughts. But somehow putting what he'd endured into words revived all the vile reality.
Now he'd released the floodgates that dammed the memories, he couldn't stop. "Parsons died within the first week. Gerard, poor devil, hung on for over a month. God knows why I didn't die too. I should have. The jailers gave me just enough food to keep me alive. I've never been sure why. Just as I've never been sure why of the three of us, I survived."
She released the chair and wrapped her arms around herself. Standing there in her cheap, borrowed dress and a coat far too large for her, she should have looked absurd. But her beauty shone like a beacon, stole his breath.
"You wanted to die," she said bleakly.
His lips flattened. "Believe me, death would have been welcome. But I was too blasted stubborn to kill myself and give those bastards the satisfaction of besting me. And for all the pain they put me through, they never quite finished me off."
Raising her chin, she cast him a defiant look. Her voice emerged with unexpected ruthlessness. "So you were a hero."
He stiffened and stepped back. No hero he. A hero never begged for mercy from his torturers. A hero never longed for death to spare him another day's pain. A hero never succumbed to devils in his mind.
"No, I wasn't a bloody hero."
Her voice deepened into irony. "Because you told the Nawab what he wanted to know."
"Believe me, keeping my mouth shut was the extent of my courage. When the Company's men finally dragged me out of that pit, I was a babbling lunatic."
She made a sound in her throat that indicated disagreement, but mercifully she didn't argue. Strain marked her features. "And it's the torture that makes it impossible for you to...touch anyone?"
He met her perceptive gaze and decided he'd gone too far to prevaricate. He folded his arms in a futile attempt to hide his shaking. "We were chained together in the pit and left."
He thought at first she hadn't understood. Thank God.
Then he realized what scant color she retained leached from her face. "The three of you?"
He stiffened. Damnation, he should never have started this. Why didn't he make up some easy story about comfortable incarceration and eventual rescue?
But he couldn't look into her eyes and lie.
"Yes." The word was choked. He battered back memories of month after month chained to rotting corpses. Through the humid airless heat of an Indian summer. Through the savage cold of winter. The unrelenting stink, the decay of once-healthy flesh.
Horror dawned in her expression. And a compassion that stabbed at his pride.
Because he couldn't bear her to imagine even a hundredth of what he'd been through, he spoke quickly. "It was almost a relief when the Nawab exhibited me for general mockery. He loved having a captive sahib who stank like carrion and could hardly cover his nakedness. I was quite the highlight of his divans until the stench got so bad, even he couldn't stomach it."
"How did you escape?" she asked huskily.
"British troops ousted the Nawab. Akash entered Rangapindhi with the invading forces. He knew if I was alive, I must be in the palace. He found me in the lowest depths of the Nawab's prisons."
"Thank God for Akash," she whispered, closing her eyes briefly as if the words were a prayer.
"I was burning up with fever, barely able to walk, half-mad." More than half-mad. He'd spent a long time convinced his rescue was another sick fantasy.
Charis's brow creased in a thoughtful frown. Her voice was stronger, although still thick with emotion. "Your health has improved since."
"I can walk and talk without humiliating myself. Most of the time. Quite an achievement." He bit back the sarcastic edge. It wasn't her fault he was a wreck.
He crossed to stoke the fire again. The flaring flames revealed her somber, troubled expression. Unfamiliar shadows swam in her unblinking gaze. Shadows he'd put there. He cursed himself for a selfish swine. He should have found a room, slept off the drink, left her to innocent dreams.
Except he couldn't bear staying away from her.
"Charis, I've had months to recover." She was better facing the bleak truth than nurturing the smallest hope that he'd ever offer her a whole body and mind. "My physical health is as good as it will get. Nothing has shifted the devils in my mind. Nothing will."
She swallowed again. He expected a protest, but she spoke with perfect calm. "You believe you'll never touch another person?"
"Not without difficulty."
Her expression was unyielding. "Then how can you hope to consummate our marriage?"
He tensed. The attack was unexpected. He dredged his response from the deepest part of him. "I must. I will. I can."
Something in his face must have alerted her to the shame roiling in his gut. "Gideon, what is it?"
He swung away although she didn't approach him. Confound it, why didn't he hold his ground? He acted like he'd done something wrong. "Nothing."
Her voice was sharp. "Where were you tonight?"
Why did she have to be so damned acute? "I told you. Drinking. I got into an altercation with a couple of ruffians. They came out the worst, I'm pleased to say."
Then she did step closer, her skirts rustling. Christ, don't let her touch him. Not now. After telling her about Rangapindhi, he felt like he'd scraped off several layers of skin.
She exhaled in a long, impatient breath. "There's more."
Oh, she was damned right about that.
His guilt surged. Fought with the absurd urge to confess, to receive absolution. When he knew there was no real absolution for him ever, for this sin or his other, more heinous transgressions.
She waited for his answer. Strange how he'd withstood agonizing interrogations in Rangapindhi without cracking, but his wife's bristling silence made him frantic to spill his secrets.
Oh, hell, why shouldn't she know what he'd done tonight? Perhaps it was best she recognized what a craven she'd married. He'd tried to tell her so often, but she refused to heed him, devil take her foolish stubbornness.
He drew himself up to his full height, turned, and surveyed her down his long nose. "I paid for a tart," he said harshly.
As her expression darkened with hurt, his gut clenched in unwelcome remorse. She came to a trembling halt a few feet away. "What...what did you do with her?" she asked shakily.
Abruptly Gideon's guilty defiance evaporated. He felt utterly sickened. With himself. With the world. With every bloody thing in Creation.
Except the woman he'd married.
He avoided eyes that held no accusation, just tortured curiosity. Shame rose like bile. Sometimes his shame was so suffocating, he thought it would kill him.
His voice was toneless as he unleashed the mortifying truth. "Not one damned thing."
Even without watching, he knew the tension drained from her. He braced for a volley of questions. But she didn't speak. Which somehow forced him to explain.
"I couldn't. I thought..." God above, this was humiliating. His hands formed fists at his sides. He gulped for air, which seemed in short supply in the dark room. "I thought...I think I'll hurt you when I...when I bed you. I thought if I could take the edge off, it would go easier for you. I'd give up my life before I...I hurt you."
Good God, he stammered like an embarrassed schoolboy. Heat prickled his neck.
He risked a glance at her. Astonishingly, her lips curved in a faint smile although her eyes were still somber. "I'd rather you hurt me than you went to another woman."
He'd expected hysterics, rage, tears. Shock sent him tumbling headlong into speech. "I'd hoped to manage the act with a professional. I haven't willingly touched anyone since Rangapindhi. And you've seen what happens to me when I do touch someone. I'm in a damnable state to bed an inexperienced girl. I'd hoped...if I could touch a stranger, I'd be able to touch you, manage the act without too much pain or clumsiness." The final sour admission surged up. "But using that woman felt too much like betrayal."
Her smile widened as if he'd done something wonderful instead of shabby and sordid. Devil take her, what was wrong with the girl? Nothing he said or did, no matter how vile, made her despise him as he deserved.
He couldn't bear to look into her face any longer. Its beauty, its honesty, its love scourged his soul. On feet heavier than lead, he crossed the room to stare out the window.
The sky outside turned gray. His wedding night was over. And his bride was still a virgin.
She padded across to stand beside him. "It's a new day."
"We've got nothing but darkness ahead," he said grimly, glancing at her.
"I don't believe that." She sounded tired but sure as she looked at him. The honesty in her eyes always cut right through him.
"You will." He slumped onto the window seat. He felt empty, lost. He had no idea where they went from here. Not for the first time, he wondered if in marrying Charis, he'd inflicted worse harm on her than her stepbrothers ever could.
She stood too close, but at least she didn't touch him. "Do you want to come to bed?" she asked hesitantly.
"No." In the strengthening light, he saw her face more clearly. She looked exhausted, devastated. "You go."
She shook her head and knelt on the thick red-and-blue rug at his feet, pulling his coat more securely around her shoulders. "You've had less sleep than I."
"I'm used to it."
She drew her knees up and linked her hands around them. With her loose hair tumbled around her, she looked absurdly young. Except the expression in her eyes spoke of heartbreaking experience. She'd changed in the last hour, taken on some of his darkness.
What he'd dreaded had come to pass. The poison of Rangapindhi had infected her bright spirit. And there was no antidote.
Her gaze was somber as she stared across the room at the burning embers in the hearth. Instinctively, Gideon lifted his hand to stroke the soft fall of her thick hair, to offer a moment's comfort.
Then he remembered that such natural gestures were forever denied him. His heart contracted in agony as his hand dropped away from her.
## Fourteen
Wearing only her shift, Charis waited alone in the big bed. It was late, past midnight, and the weather had turned colder during the day so a fire blazed in the grate.
No sound came from the parlor behind the closed door. She knew Gideon was in there, steeling himself for what he must do. She'd been steeling herself all day too. In her belly, huge ugly toads of fear somersaulted over each other. Her trembling fingers crushed the embroidered edge of the fine linen sheet.
Could consummating their marriage push him further into darkness?
Darkness hovered perilously close. She'd recognized that last night, when he'd told her about Rangapindhi. The magnitude of his suffering beggared belief.
Could she heal him? Could anyone?
And still they both had to get through tonight. She'd told Gideon she could do this. But every lonely second of delay made her bravado less and less convincing. If he didn't appear soon, her failing courage would desert her altogether.
Charis bit her lip and closed her eyes, whispering a silent prayer for strength. It didn't help.
When she opened her eyes, Gideon stood on the threshold. The doors in St. Helier's best hostelry were, of course, well oiled.
"Hello," she said stupidly, although she'd only left him to his brandy half an hour ago, and they'd spent an entire strained day together, carefully avoiding the subject of what happened tonight.
His beautiful mouth quirked in the wry smile that was indelibly imprinted on her poor yearning heart. "Hello to you too."
He was in shirtsleeves and trousers. The neck was open, slashing down to reveal a solid chest covered in curling dark hair. The sight shocked her. She'd imagined him hairless, like the marble statues in the hall at Marley Place. His long narrow feet were bare. He still wore his fine tan kid gloves.
All this she took in with one sweeping look, aware he studied her in his turn. What did he see? She kept the covers pulled to her shoulders as she sat against the carved oak bedhead. She'd plaited her hair as usual. It seemed inappropriate to leave it loose, too bridal when she didn't feel remotely like a bride.
She overcame her crippling shyness to glance into his face again. His fleeting amusement had evaporated. He was pale, and that telltale muscle flickered in his lean cheek.
"What...what do you want me to do?" she asked almost soundlessly.
Why, oh, why did this have to be so awkward? Surely people consummated marriages—or did this without legal niceties—all the time. Yet she was so nervous, she felt sick.
He stepped into the room and shut the door after him. "Lie down. Close your eyes," he said in a somber voice. "I'll try to be quick."
Charis's heart clenched with misery. She was sure when those other people came together, they said more than that. But those people wanted what was to occur. She bit back a protest at the bleak crudeness of it all.
He didn't come nearer. "Would you like me to blow out the candles?"
She started to shake her head, then nodded. "Yes, please." What happened was better done in shadow.
She watched him move around the room with his usual catlike grace. Soon the only light was the fire's flickering golden glow.
He stopped beside the bed. With his back to the hearth, she couldn't see his expression. He ran his hand through his hair, ruffling it. She itched to rise on her knees and smooth it. But, of course, she couldn't touch him.
The agony of that knowledge carved a crack in her heart as wide and deep as the sea they'd crossed to reach Jersey.
"Are...are you going to undress?" she asked uncertainly.
"No."
She bit her lip again. Her fingers tightened on the sheet until they ached. Gideon stood close enough for her to hear the uneven hiss of his breath. She looked at the superb man she'd married and wished with every particle of her being she was anywhere but here.
"Charis, I'll have to pull the covers away," he said with gentle insistence.
She realized she clutched the sheet like a shield. Absurd. She'd agreed to this. He was here for her sake and at great cost to himself. Too late to cavil at the bargain she'd made.
"Of course." With difficulty, she relaxed her clawlike grip.
Down, down the blankets went, until she lay revealed to her bare toes. She closed her eyes because she wasn't brave enough to look into Gideon's face. Uncontrollable heat rose in her cheeks. He'd see she was naked beneath her shift. A nauseating mixture of nerves and embarrassment kept her stiff and unmoving.
He was so still, standing next to the bed, that she couldn't even hear him breathing anymore.
He'd warned her he'd be clumsy. She was smart enough to believe him. She braced for him to grab her, but nothing happened.
What was he waiting for? Dear heaven, did the sight of her shatter his resolution? Now the moment of truth arrived, was he unable to go through with it?
"My God, but you're glorious," he whispered hoarsely.
Her eyes flew open with disbelieving shock. "What?"
His expression remained troubled, but his gaze was avid as it traced her body. "Charis, you're beautiful beyond a man's wildest dreams."
How could he say such things? It was too painful. She couldn't find pleasure in his praise when he shook with disgust at her slightest touch.
"Please..." She swallowed to dislodge the lump of distress in her throat. "Please get it over with."
His face contorted with sorrow. "I'm sorry, Charis."
"Don't say any more." She closed her eyes, partly to stem her foolish tears, and slid down in the bed. "Just...do what you must."
"As you wish." He sounded remote, as though he too retreated behind some inner bastion.
The mattress sagged with his weight, then she felt encroaching warmth as he straddled her legs. She knew the act would be less painful if she relaxed, but every muscle tensed in fearful expectation.
After a moment, he raised the hem of her shift. To her thighs. Then past her hips. The cold air on her skin made her shiver.
She placed shaking hands over her mound. Which was stupid. He'd do more than look at her before he was finished.
Reluctantly, she opened her eyes to find him staring at her...there. His face was taut with such anguish and longing, she couldn't bear it.
Hesitantly, he placed one gloved hand on the soft plain of her belly. Her breasts tightened, and restless heat settled between her legs. She was ashamed that she couldn't stifle her powerful and immediate reaction.
He snatched his hand back as if she scalded him. He was shaking. Of course he was. Touching her, even for such a short time, required every ounce of will.
She bit her lip so hard, she tasted blood. The urge to beg him to stop fought up through her closed throat. She could see in his strained, colorless face what this cost him.
She remained silent.
Still silent, she lifted her hands away from her sex.
Gideon stared at Charis in helpless wonder while his gut churned like a millwheel. She was the most exquisite creature he'd ever seen. His hunger was a raging storm.
The shift bunched under her breasts, but he clearly saw the rich pink of her nipples. Nipples that peaked like ripe raspberries the moment he touched her.
Her swift response was just another of fate's mockeries. She was formed for pleasure, but she'd find no pleasure with the man she married. Nonetheless, his eyes feasted on the treasures of her body. The delicious inward curve of her waist. The flare of her hips. The long, coltish legs.
His cock was hard and swollen and pulsed against the front of his trousers. If he took her now, he'd rip her to pieces. His mind might deem touching her as torture. His prick didn't care.
Dazedly, she stared into his face. She was white as new snow. She'd hardly looked at his body, although if she dropped her gaze, she couldn't miss his arousal.
He gritted his teeth and stroked the smooth skin of her thigh. For one heady moment, even through his glove, he felt her enticing warmth.
Then, as always, his mind went black. Screams echoed in his ears. Her flesh turned to rotting carrion. Her peppery carnation scent became the stink of death.
He fought back the shrieking demons. Wrestled them until they lay supine and silent. The battle left him shaking. He sucked in a breath that reeked of decay. Slowly, as if he pushed a massive weight up a steep and jagged path, he traced a tentative path to her hip.
He wasn't a small man. He needed to prepare her. But time was his enemy. The longer he waited, the more likely his demons would master him.
She was rigid with fear. The uncertainty in her beautiful eyes broke his heart. Her breath emerged in unsteady gasps. Not, he was grimly aware, of desire. The air bristled with tension.
He placed both hands on her thighs and carefully spread them. In a room lit only by firelight, her body's hollows were dark and mysterious. He knelt between her legs, and his nostrils flared as he caught her scent.
With clumsy fingers, he undid his trousers. His cock sprang free. When her eyes fastened on his organ, she made a muffled sound. Her hands curled into the sheet beneath as if she physically stopped herself leaping from the bed.
He hooked his hands under her hips and angled her up. Slowly, he pushed forward.
As he breached her body, she whimpered but didn't recoil. He pushed again, feeling the tissues give way.
To his grateful astonishment, she was damp. Damp enough to ease his entry.
Even so, she was damned tight.
He paused and sucked in a deep breath redolent of Charis.
She's alive, she's alive, he chanted in his mind as he eased into her. She's alive, he told the ghosts in his head, blocking his ears to their panicked clamor.
She whimpered again and shifted, drawing him deeper.
The voices grew more insistent. He couldn't hold them off. Cold sweat prickled his skin. His grip firmed on her hips. As his vision faded, he inhaled. The world shrank to one spark of light.
He had to do this now or fail utterly.
"Charis, forgive me," he said in a strangled voice. He tautened and thrust.
Pain shafted through Charis with the vivid, immediate brightness of lightning. A scream welled in her throat, but she bit it back.
Still, a choked moan escaped. She felt like she'd been split in half with a blunt ax. It was excruciating. Blinding.
She squeezed her eyes shut and prayed for it to be over.
Breathe. She needed to breathe.
She gasped for air, but Gideon's weight crushed her into the mattress. He was bigger and heavier than she'd realized. His height and superb coordination disguised how well muscled he was.
Frantically, she dug her fingers deeper into the sheets. He'd done what he needed to. Why didn't he pull out and leave her be?
Breathe, Charis, breathe.
The part he'd pushed into her chafed tender flesh. He was hard as granite. But unlike granite, he was hotter than a furnace. Stupidly, she'd imagined he'd feel cool, even cold, because of his reluctance to touch her.
His smell, familiar yet unfamiliar, surrounded her. She knew the clean scent of his soap and the essence of his skin. She guessed the extra spice in the air was male arousal.
His breathing was ragged, and he trembled. She raised her hands to grip his back, then remembered he hated to be touched. He wouldn't want her embrace, even as he lay buried inside her in the closest connection she'd ever known.
She sucked in another breath. An easier one. Where they joined, she still hurt, but the fierce agony faded.
He shifted with a soft grunt. The pressure changed, became less excruciating.
Charis waited for him to pull away. But his muscles tightened, and he thrust again. She bit back another moan and gripped the sheet to stop sliding up the bed.
She'd imagined this would be quick, over in seconds. But he was still inside her. He moved once more, and released a deep groan.
Another thrust. His hips pumped several times, and she felt a liquid heat deep inside her. He groaned again and slumped over her. In a cruel parody of tenderness, his head came to rest on her shoulder, his silky hair tickling her neck.
After all the hardness, the fleeting softness seemed alien, wrong.
After an endless time, Gideon withdrew and carefully pulled down her shift, hiding the tops of her thighs. Then he rolled onto his back to stare at the ceiling. His shirt was twisted and flapped free of his gaping trousers.
After one brief glance at him, Charis concentrated on the dark beams crossing the ceiling too. She didn't want to see the organ he'd pressed into her body.
She supposed she should say something, but she wasn't certain her voice would work. Her throat clenched so tight, it hurt. Although she was cold, she couldn't summon energy to reach for the covers.
Who knew how long they lay alongside each other? Not long, she guessed, although every second felt like an hour.
Where he'd taken her, she stung, although the piercing pain had subsided to a constant throbbing. She felt lost in a vast emptiness, as though the world had been destroyed in some unimaginable cataclysm. How odd that this most intimate act of all left her feeling like the only human left on earth.
Slowly, stiffly, he sat up. For one intense second, she felt him study her. She kept her eyes fixed on the ceiling.
Like distant thunder on a summer's day, devastation nudged at her awareness. But for the moment, exhaustion kept it at bay.
Jamming her eyes shut, she willed herself not to cry. She was much better hiding in this numbness. Given her way, she'd lie here forever.
Charis listened to him move about the room. Water splashed into a dish. Perhaps he meant to wash. Perhaps he was desperate to rid himself of every trace of her disgusting person.
She recognized she tortured herself and scotched the thought before it went any further. Instead, she sought that cold empty space in her heart where nothing could hurt her.
The rug muffled his footsteps as he moved closer. She couldn't help tensing at his approach. He stopped by the bed. Unthinkingly, she flinched.
Although he wouldn't touch her. He'd never touch her again, now she was his wife in fact as well as law.
He didn't say anything. There was a soft clink on the bedside cabinet. He shifted away, his footsteps deliberate but somehow defeated.
There was a click as he opened the door, then another as he closed it behind him.
She opened her eyes. The blazing fire still lit the room. The whole episode had probably taken less than half an hour.
Half an hour for her world to change.
She turned her head to see a blue-and-white china washbowl on the nightstand and a pile of towels. He'd seen to her comfort, then he'd left her in peace.
The tears she'd fought since he'd come to her bed overflowed.
Eventually Charis roused to go looking for her husband.
It wasn't in her nature to avoid difficulties. Lying in the rumpled bed, surrounded by the unfamiliar smell of sex, she had time to gather her courage.
And time to start worrying about Gideon.
As shock and discomfort receded, she began to think what price that joyless coupling had exacted from him. She needed to see him, to reassure herself he was all right. She needed to see him because the moment when she'd wished him to Hades had been brief indeed. Now only his nearness could soothe her aching sadness.
She rolled out of bed, the abrupt movement setting up a host of unfamiliar twinges. Reminder, should she need it, that nothing would ever be the same after what had just happened.
Wrapping a blanket around her trembling shoulders, she trudged across the floor. She pushed the door open and stepped through. The parlor was quiet and dark except for the low glow of the fire.
Had he gone out? After what they'd done, sleep would elude him. She ventured closer to the Stygian corner where he'd sat last night. Then she realized he sprawled in a massive wooden armchair in front of the hearth.
"Gideon?" She hitched the blanket up and stepped around the chair's looming bulk to stand before him.
He didn't look at her. Instead, he stared at the fire. Something told her he'd stared into the fire for a long time. His gloved hand curled around a half-filled glass that dangled on the verge of spilling. Brandy, she guessed.
"Go back to bed, Charis."
The boneless curve of his long, lean body echoed the despair in his voice. His legs stretched toward the grate, and his shirt hung loose as it had in the bedroom. A frisson ran through her as she looked at his bare chest, gold in the flickering light.
A shiver, astonishingly, not of revulsion.
Charis beat back the cowardly urge to obey him and flee. Instead, she fixed an unwavering gaze upon him. "We need to talk."
His face tightened. With a savagery that made her wince, he lifted the glass and pitched it into the fire. There was the sharp tinkle of shattering glass and a brief flare as the brandy caught.
"Christ, no."
The eyes he focused on her glittered with anguish and a loathing that made her cringe.
"Do you hate me now, Gideon?" She didn't recognize the shaking voice as hers. She'd tried so hard to make the act easy for him, but to her shame, she hadn't succeeded in masking her discomfort.
His face contorted, and she stared aghast into naked torment. Only for a moment. He swiftly pulled the shutters over the turbulent depths.
"Of course I don't hate you," he said impatiently.
"But..."
"Go, Charis, now." His voice fractured.
She couldn't mistake his desperation to be alone. Although selfishly she wanted only to stay with him. The tumbled, lonely bed in the next room loomed like a gallows.
"Good night," she whispered, her shoulders drooping.
He didn't answer. Slowly, reluctantly, as if her feet were blocks of stone, she turned toward the door she'd left ajar.
One step. Two.
She didn't want to leave him. She never wanted to leave him.
She was almost at the door when she heard a muffled sound behind her. An unfamiliar sound although she immediately identified what it was.
Stifling a horrified cry, she turned. He pressed gloved hands to his eyes, and his broad, straight shoulders heaved as he struggled for air.
Hands that itched to comfort him curled into fists at her sides. She longed to succor the man she loved with the warmth of her body. But that was impossible. Touching her body had driven him to this extreme.
She darted across to him, and, as she had last night, she knelt on the floor beside him. Unfamiliar discomfort stabbed her as she curled her legs under her.
In painful suspense, she waited for him to send her away. He was a proud man. He'd hate to know she witnessed this.
But he didn't speak.
Perhaps he wasn't even aware of her presence. It was torture to listen to him struggle against his weeping. He hardly made a sound. Only the thick, uneven rasp of breath betrayed his agony.
The iron control that had sustained him through Rangapindhi and beyond disintegrated. How blind she'd been not to realize the universe of pain he contained. She should have known. She wasn't stupid. She claimed to love him. He'd told her about India. She'd seen what his ordeal cost his gallant spirit.
But only now did she truly understand the devastation that haunted him. His inhuman strength had delayed this moment too long. So when he finally broke, it was like a mountain cracked before her eyes.
From the first, she'd cherished a childish, flawless image of him. In this shadowy room, that image crumbled to dust. Gideon Trevithick wasn't Galahad or Lancelot or Percival. He wasn't an invincible guardian angel who appeared from nowhere to rescue her. He wasn't indestructible and powerful and immune from weakness.
Helpless, hurting, guilty, she listened to the sound of his heart breaking. This man who battled so hard to dam his tears was all too human. He could shatter and fall and fail. He was fragile flesh and blood, and he'd suffered more than any mortal should.
Wrapping her arms around her raised knees, she stared sightlessly at the fire, the only light in the dark room. This wordless vigil was all she could offer. She was guiltily aware that what they'd done had initiated this excruciating outpouring. Her penance was listening to him struggle to smother his sorrow as if it were shameful or unwarranted. She wanted to beg him to stop resisting, to give in, to let the horrors of his Indian years finally receive their due.
He'd fought so long and so hard, and still he fought. His valiant heart wouldn't surrender.
Slowly, the worst of his grief passed. Or at least the outward signs. His breath emerged more normally and not in broken, choked gasps.
After a long time, he spoke in a constricted voice. "This isn't fair on you."
She didn't look at him but continued to rest her cheek on her upraised knees. Weariness and sorrow weighed endlessly on her. "I can bear it."
They didn't speak again. She thought after a while he might have slept, exhausted by his travails. She didn't. Instead, she gazed dry-eyed at the dying fire.
Charis had loved Gideon Trevithick from the moment she'd first seen him. She'd loved his strength, his honor, his intelligence, his beauty. She still did.
But he'd been right to decry that love as a dazzled girl's emotion. It was a hothouse plant, green and lush but unable to withstand cold winds from the real world.
The last hour had changed that forever. The last hour had changed her forever.
The love she felt for Gideon now was more durable than stone.
## Fifteen
The afternoon wind off the sea was so icy, even Gideon noticed its biting power. Unusual for this time of year, according to the porter at the hotel, who wished him and Charis well when they left on their walk.
Gideon wasn't sure appearing in public was a good idea. Someone might recognize him. After the last days, he couldn't bear fending off another crowd as he had in Portsmouth. More, there was a small but significant risk of word reaching Felix and Hubert that he and Charis were on Jersey.
But Gideon couldn't bear being confined in their rooms any longer. The acrid memories of last night's pain and disappointment weighted the air. Worse, that clumsy bedding had left a brooding sensual awareness in its wake. Living in close quarters with Charis and knowing he couldn't touch her, would never touch her again, was slowly driving him out of his mind.
As the day progressed, he'd watched his own strain increasingly reflected in his wife's pale face. The tension between them had stretched and stretched until it became intolerable. He'd heard her sigh of relief when he suggested going out.
Thankfully, it appeared the cold kept most people inside. The few hardy souls on the promenade paid Gideon and Charis no heed as they strolled along the seafront.
So far it had proven a mostly silent walk. As it had proven a mostly silent day.
Hell, what could he say after last night's emotional storms? His gut clenched with humiliation at his behavior, both during and after their bleak coupling. How could he bear to revisit the black ocean of anguish? Or perhaps even more harrowing, how could he discuss his inept use of her body?
The silence was heavy as lead with what remained studiously unspoken.
Charis turned into the wind and paused to look across the gray rolling waves. The stiff breeze snatched at her bonnet, and she raised one gloved hand to hold it firm.
At least she was dressed suitably. He'd called in a modiste that morning and ordered a wardrobe for his bride. The charming yellow ensemble Charis wore had been hurriedly altered to fit. Other garments would arrive over the next week.
It was the only time Charis had smiled all day, when she saw the designs for her dresses.
Gideon came up beside her as she leaned on the stone parapet. Beneath the bonnet's brim, her expression was pensive. Her lush, pink mouth drooped at the corners.
Ah, that soft mouth...
The continual low hum of desire made his head swim. Self-disgust followed fast.
Good God, he was a satyr of the vilest kind. After what he'd done last night, how could he think of touching her?
Turning, she caught his stare. From the color that invaded her pale cheeks, she guessed the heated direction of his thoughts.
She must despise him. She ought to despise him. He'd hurt her, then broken down and cried for the first time since his release from the Nawab's dungeons.
Her eyes darkened to green with some emotion he couldn't name. Although before last night's debacle, he might have called it interest. Her lips parted on a soundless sigh.
He jerked back as if she reached for him. But her yellow-gloved hands remained safely on the seawall.
His heart thudded like a drum. He rubbed the back of his neck with one hand. To his surprise, she laughed softly. Surprise and chagrin.
That low musical sound slid along his veins like honey and made him want what he could never have. He should be inured to frustration, but somehow the damned torture never ended.
"You look almost bashful." Her husky voice bubbled with warmth.
"Good God, Charis..." He struggled to express his shock. "You can't find our predicament amusing."
Her lips turned down. "I'd rather laugh than cry." She turned away and gazed across the choppy water. "You can see what everyone thinks when they look at us. That waiter this morning leered."
"We're newlyweds," he said somberly. "If your stepbrothers inquire, I want people to say we acted like any couple."
"Then perhaps you should touch me," she said softly but implacably. She still stared over the restless iron gray sea.
Silence fell. While the waves rolled and the gulls cried and traffic clattered along the street behind them.
"Charis..."
She turned and the humor had fled. "You touched me last night."
He clenched his gloved hands by his sides. Clearly his sweet young wife was in the mood to torment him. "I didn't think you'd want to talk about what happened," he said in a tight voice. Christ, he didn't.
"Why would you think that?"
Because I hurt you. Because I made a tragic mess of something that should be wondrous. Because I can't stop thinking how it felt to be inside you.
"Because it's done."
An inadequate, cowardly answer. He knew it. So, blast her, did she.
"You're crossing a line through the subject of our...marital relations, never to revisit it?" Color still marked her high cheekbones. She wasn't as easy with this discussion as she wanted to appear.
"Don't you think that's best?"
She arched her elegant light brown eyebrows, a few shades darker than the bright glory of her hair under the neat chip bonnet. "No negotiation?"
He released a heavy sigh. "Revisiting last night should be as painful for you as it is for me."
She straightened from the wall and sent him a direct look. "You...you did what you had to."
"There was no joy." If only someone would approach so she'd abandon this conversation. But the promenade around them remained empty.
"Practice makes perfect," she said staunchly.
Every brave word gashed at him. "Not in this case."
He longed to tell her he'd give up his hope of heaven to change desolate reality. He longed to tell her she was more beautiful than the dawn. He longed to tell her he died of desire for her.
What good was any of that when, if he touched her, he'd only hurt her?
Her jaw set in a stubborn line. "I don't accept that."
"You have to." Why couldn't she see there was no hope? After how he'd botched things last night, she should shrink from him as if he had the plague.
"The Westons are fighters, Gideon," she said firmly. Her throat moved as she swallowed, another indication that beneath her determination, she was nervous. "I want a husband in my bed. I intend to do anything I can to achieve that end. Anything. I know you want me. I'll use it against you if I can."
Oh, dear Lord in heaven. He supposed he should admire her honesty in admitting her strategy, but all he could think of was the lacerating misery awaiting both of them. "We made a bargain..."
She shook her head. "No, you set ultimatums."
"You agreed." He couldn't keep a hint of temper from showing. It was difficult enough fighting for his own equilibrium without having to fight her as well.
"Yes, I did. Then." When she looked down, gold-tipped lashes fanned the hectic pink of her cheeks.
Need, primitive, uncontrollable, gnawed at him. How much easier this would be if she wasn't so beautiful.
Or would it?
He'd liked her from the start. His longing wasn't rooted in her appearance, spectacular as that was. He wanted her because of her pure, unquenchable spirit.
His voice roughened with urgency. He admired her courage, but she was tragically mistaken in what she wanted. "Charis, I beg of you, don't push this. I know what I ask seems cruel. But crueler by far to keep you clinging to futile hope. You'll end up destroying us both."
The fugitive color fled as quickly as it had arisen, and the eyes she raised were dull with misery. "It could save us too."
Regretfully he shook his head. "This isn't a fairy story, my wife."
Her lips flattened in displeasure. "No, it's a story where you consign me to another man's bed. Is that what you want?"
The prospect of her sharing last night's intimacies with another lover made him burn, like someone brushed his skin with naked flame. The idea of anyone but him touching her, hearing her sigh—God, pressing into that tight sheath—hurled him to the verge of murder.
"Yes."
"Liar."
She cast him a scornful look, turned, and marched back toward the hotel, her boots clicking on the cobblestones. Helplessly Gideon stared after her. Unless he was very much mistaken, his wife had just declared war.
When he was younger, before Rangapindhi, he'd occasionally imagined taking a bride. The idea had seemed simple, inevitable, uncomplicated.
Hopelessly naïve.
He bit back a curse. He'd known when he came up with this plan to save her, it meant suffering. He'd known it required will and sacrifice.
But until his wife threatened to seduce him, he had no idea what hell awaited.
She was yards away, walking with a natural self-confidence that attracted more than one admiring glance from the few men braving the cold.
Impudent dogs.
Biting down his rage with her, with himself, with the whole damned world, he strode after her. His eyes never wavered from the saucy sway of her hips.
She didn't look at him when he caught up. For the sake of appearances, he grabbed her arm. Even through his glove and her merino sleeve, he felt the tingling warmth of her skin. The ineffable life force that had set his desire afire when he held her last night.
He wanted that heat and vitality.
Devil take it, he wanted her.
Even as another sizzling bolt of need hit, the old urge to snatch away fought to the surface.
She glanced sideways. "Are you all right?"
"Yes," he grated out, trying to control his inevitable shaking. He sucked in a breath and spoke with corrosive bitterness. "This is what you want? You've got bats in your belfry."
She looked straight ahead. "I want you."
Her voice was firm, sure, determined. And a little sad. Gideon had to remind himself she was a girl and couldn't know her own mind. After last night, the words rang hollow, false.
"Well, God help you," he said grimly, and tightened his reluctant hold on her slender arm.
Charis sat up in the bed where last night she'd lost her maidenhead. Rain slammed against the windows, and wind rattled the glass. The wild weather was nothing compared to the confused storm of emotions in her heart.
She'd hated what Gideon had done to her last night. More, she hated that he'd hated it. She was vain enough to want her husband to find pleasure in her.
There had been no pleasure.
Actually, that wasn't completely true. She'd felt pleasure when he touched her, even with him wearing those wretched gloves. When he'd stroked her bare flesh, a wanton heat had curled in her belly. Her breasts had ached for his caress, and her pulse had kicked into an unsteady race.
At last the body she'd longed to explore had been near enough to touch.
If he'd allowed her to touch him.
He'd been near enough for her to breathe his clean scent and feel the warmth radiating from his skin. She'd seen the hard planes of his chest, felt the brush of his hair against her neck.
All tantalizing hints of what they could find together, if only she could free him from Rangapindhi.
Her belly knotted as she recalled the unbearable intimacy of that moment when he pushed inside her. The pain had been overwhelming, but the act had bound her to him as nothing else could.
They were one flesh.
Only now did she understand what those words truly meant. Perhaps the anguish of the consummation made the joining so irrevocable. Perhaps if they'd embarked on married life in lighthearted hope, she wouldn't suffer this dark obsession with her husband.
She knew Gideon felt the connection too. For all he tried so staunchly to stay separate.
For the sake of that connection, she meant to take a huge risk. A risk not only for her and her bruised, longing heart. But also a risk for Gideon's grimly retained sanity and health. Heaven forfend she was wrong. The consequences would be tragic.
In the long dark watches of the night, she'd felt at the crossroads between two futures. The future Gideon planned—cold, divided, lonely. A future where she didn't resist his decision to give up on hope and love.
Or there was another future. A future where they grew together, confronted their challenges, created a family and a home.
Was there a chance she could make this second future reality?
Charis didn't fool herself about the magnitude of the obstacles. But last night as she'd witnessed his pain, something in her screamed denial at abandoning him to suffering. She yearned to cherish him. She wanted to restore his trust in life. More, his trust in himself. She wanted to give him back his capacity for happiness.
All huge tasks.
Impossible?
No. She refused to give up. Whatever it cost her.
Half an hour ago she'd left him in the parlor. He'd been drinking brandy, and the bleakness in his eyes had made her want to weep. The desolation had always been there, but now she knew his past, it cut her to the bone.
He'd already decided his life was over.
Well, the woman he'd married meant to shatter that resolution. She loved him so much, she couldn't lose.
Brave words. She wished she felt half as confident.
She looked up from her troubled thoughts to see Gideon standing in the doorway. She hadn't heard him arrive. He always moved like a cat, so that was hardly surprising. His hair was ruffled, and one gloved hand negligently encircled a glass. He'd removed his neckcloth, and his shirt was open, giving her shadowy glimpses of his hard chest.
His masculine beauty was a constant goad. Sometimes, like now, it stopped her heart.
Her belly clenched as his half-dressed state inevitably reminded her of last night. His remorse at what he'd done that stabbed her like a blade. His sorrow afterward that made her want to die.
He didn't advance into the room. "I'll say good night, Charis."
"Aren't you coming to bed?" The question emerged as a husky invitation.
She licked lips dry with nerves. His gaze fastened feverishly on the movement. His gloved hand tautened on his brandy. The warm air swirled with sudden sensual turbulence.
He cleared his throat and shifted his gaze above her head. "I'm sleeping in the parlor. I think...I think it's best."
With unsteady hands, she grabbed a shawl and slid out of bed. Ignoring the resistance in his face, she stepped close enough to read ravaging torment in his dark eyes. "Don't be ridiculous, Gideon. It's cold and uncomfortable."
He looked at her. "After Rangapindhi, it's the height of luxury."
"Oh, my dear, Rangapindhi is over," she said in a low voice. It seemed a sign of progress that he mentioned his captivity without prompting. She extended one hand toward him, then let it drop to her side. "You're free."
His smile held no amusement. "I'll never be free."
This acceptance of his fate angered her. "If you don't fight, you won't."
His tall, lean body vibrating resentment, he stalked across to the fireplace. He tossed back his brandy and set the glass down sharply on the mantel. He focused a furious glare on her. "Don't talk about what you don't understand."
She mustered her fading courage. She couldn't fail at the first hurdle. Worse difficulties awaited before she gained what she wanted for him. A chance at happiness. Liberation from his past.
Her mind filled with a sudden memory of the stark desire in his face as he'd looked at her body last night. Had she nerve to use that weapon to break him?
With excruciating slowness, she let the beautiful shawl slide down. Her new nightdress was silk, and while far from immodest, had been designed by Madame Claire with a honeymoon in mind.
Color lined his slanted cheekbones as his eyes followed the slipping shawl, then returned to trace the dip of the neckline over her breasts. She shivered under that heated gaze and was suddenly overwhelmingly aware that sheer white material clung to hips and buttocks and swirled around her bare legs. The strange hot weight, familiar from last night, settled in her belly. Her heart set up a rapid tattoo of excitement.
"I understand you've decided to wallow in self-pity for the remainder of your days," she said, knowing she wasn't fair. But this wasn't about fairness. This was about ripping at his control until his memories lost their grip.
"You have no right to say that." A muscle jerked erratically in his cheek. He was close to losing patience.
"I'm your wife. I can say what I like," she said defiantly, standing straight, so her breasts pressed against the delicately embroidered bodice. The cool brush of silk on her nipples teased, built the damp heat between her legs. Her breasts swelled, yearning for his hands.
"This is a marriage of convenience," he said, sounding strangled. He was taut as a drawn bow. His gloved hands opened and closed convulsively at his sides.
"It's more like a marriage of inconvenience," she shot back, taking a step in his direction and tossing her thick plait behind one shoulder.
Feverishly, his eyes clung to her. "We had a bargain."
"Yes, my safety in return for a lifetime of unhappiness." She fought to keep her voice steady. Difficult when every reaction she achieved from him stoked the heat inside her. "Forgive me if I seek to renegotiate."
He turned away and closed his eyes as if he couldn't bear to look at her. One unsteady hand curled over the edge of the mantel.
"I won't forgive you if you make this more a nightmare than it already is." He flung his head up and glared at her like he hated her. His furious black eyes threatened to incinerate her where she stood. "Why the hell would you want to repeat last night's farrago? Damn it, Charis, I hurt you."
"It doesn't have to be like that," she said in a ghost of her usual voice.
"For us, it does." He sounded heartbreakingly sure.
Doubt frayed her resolution. What if she was wrong? What if her plan to help him only damaged him further? She lifted her chin and shored up her courage. "I'm not giving up, Gideon."
His mouth thinned with anger, but when he spoke, his voice was frigid. "You will. This is a war you can't win."
She spread her hands in helpless bewilderment. He had so much strength. Why didn't he enlist it in his own cause? "Don't you want a real life?"
His short laugh was so harsh, it flayed like flying shards of glass. "Of course I do."
She fought the impulse to retreat. She'd known when she chose this path that her greatest enemy would be Gideon himself. "Your memories aren't always in control," she said hoarsely. "I saw you in Portsmouth. You knocked down any man within reach. You weren't afraid to touch people then."
"Yes, I find relief in violence." His voice roughened into sarcasm. "Are you suggesting I beat you?"
She blinked back hot tears. How easy to make optimistic promises when she lay alone in her bed. Less so facing his stubborn intransigence.
He was so angry and lost, and he defended himself the only way he could. She knew he acted for her sake. He firmly believed he wasn't worthy of her love. He believed living with him would destroy her. Limitless self-loathing was one of the toxic fruits of Rangapindhi.
Could she change his mind? Did she have the power to reach him? "Gideon..." she protested huskily.
He stiffened and glared at her. "Don't be a fool. I'd never hurt you."
She bent her head. "You're hurting me now."
She glanced up to see his face darken with remorse. He made one of those strange truncated gestures she'd noticed from the first. "Charis, don't."
She shook her head and twined her arms around herself. She was cold with a chill of the soul more than the body. If only he'd take her in his arms and warm her. "I can't help it," she whispered.
He stepped close enough for her to feel his living heat. How could he consign himself to a cold tomb of isolation?
"I've done you a great wrong," he said with a regret that made her want to cry.
"No."
"Yes, I have. I hoped to preserve your freedom by tying you to a man who made no demands. Instead, I've only brought you pain."
"I want to be your wife," she said obstinately.
"You are my wife."
"Not in any way that counts."
He sighed heavily and ran his hand through his hair. "Charis, you ask too much."
"Better than asking nothing at all," she snapped back.
His eyes flashed, and he swung away. She knew it was unjust to berate him over what he couldn't change.
Something women always said when they made less-than-satisfactory marriages.
He looked tired, discouraged. Her demands couldn't be easy. He'd come to an unknown woman's rescue and adopted responsibilities that took over his life.
Except she didn't believe he felt like that. In her heart, she believed he could love her. Sometimes, she caught him staring at her with such hunger, her heart skipped a beat.
"All I can promise is once we've established our marriage's legality, you can set up home anywhere," he said with a coldness she knew was meant to put her at a distance. "You need never see me again. This interlude will become only an unhappy memory."
"You think that's what I want?" she asked bitterly.
"You must make it what you want." He stepped away with an ironic gesture of one gloved hand. "Now go to bed."
Her temper had stirred distantly as she'd listened to his self-sacrificing statements. Now it sparked. Her jaw tensed. "Are you going to sleep by my side?" she asked in a dangerous tone.
He looked surprised. He needed to learn she wasn't an obedient hound to leap to his slightest command. He asked her to leave him alone to go to perdition. But she wasn't allowing him his way. The determination that had gripped her before he appeared returned full force. She wouldn't let him settle for this barren half-life he mapped out.
"No, of course not." He frowned. "Haven't you heard a word?"
"I've heard everything, and I agree with none of it."
"We'll talk in the morning."
Her lips tightened. "I'm sure we will."
"So good night." He turned toward the door, then must have realized she hadn't shifted. He confronted her with a frown of irritation. "Do you want something before I go?"
"I want you to come to bed."
His lips quirked in a sour grimace. "After what happened there, any normal woman would run shrieking."
She flinched at the normal woman remark but didn't budge. "I'm not asking you to do...that again." Hot color rushed into her cheeks.
"So you want a chaste bedmate?" His voice dripped derision.
She drew a harsh breath. "I want you with me, Gideon."
"No."
"All right. I'll sleep in the parlor." She folded her arms and stared at him implacably.
"Don't be absurd," he said with the beginnings of real anger. She realized until now he hadn't taken her seriously.
Of course he didn't. He thought she was a fragile young thing who needed protecting. Before they were done, he'd learn his wife possessed a will at least as strong as his. And a heart as valiant. She meant to fight for her marriage. She meant to fight for his future.
"Get into that bed now," he growled.
She shivered although the room wasn't cold. "Make me."
He straightened, and she watched rage war with frustration on his face. "You're acting like a child."
She shrugged and scooped the shawl from where it lay at her feet. "Shall I take the chair tonight?" She spoke with a nonchalance she didn't feel.
His jaw moved as he ground his teeth. Another shiver rippled through her. There was forbidden excitement in taunting him.
"Devil take you," he grated out, taking a step closer.
She wrapped the shawl around her shoulders and hoped to heaven he didn't take her at her word and make her sit up all night. The bedroom was warm, the parlor wasn't. She'd be blue within an hour, and after the last two nights, the prospect of stretching out in a soft bed was alluring.
She angled her chin and sent him the haughty stare she'd employed on a hundred importunate suitors. "Do you mean to herd me into the bed, Gideon?"
"You..."
She raised her eyebrows. "Yes?"
"You damned witch." His eyes glittered with fury.
Her belly quivered with nerves. And something far more powerful. "Hardly polite."
"Oh, hell!"
He lashed out and grabbed her around the waist. In one furious movement, he swept her off her feet and bundled her against his chest.
She'd waited for this, prayed for it. Even so, the shock of his arms holding her high, the heat of his skin through his shirt, his sheer vibrating fury made her gasp.
His hands tightened, and he stared straight ahead. "You asked for this," he snarled, marching toward the bed.
Yes, she had. Thank the Lord, she'd got it. Tentatively, she slid one hand behind his neck, tangling her fingers in the silky hair at his nape. He didn't seem to notice.
"How dare you use brute force against me?" She wanted to sound outraged. The best she could manage was a dull sulkiness. While all the time, her heart danced.
"You should have thought of that before," he bit out.
The distant courtesy he cultivated before the world was gone. Instead, he was big, angry, commanding and breathtakingly male. A thrill sizzled through her right to her cold toes.
He reached the edge of the mattress. "Good night, Charis."
Unceremoniously, he dropped her to the tumbled sheets in a tangle of legs and arms and silky white nightgown.
For a moment, she lay winded, staring up at him. He'd had no difficulty carrying her. For all his leanness, he was very strong. The thought sent another thrill rocketing through her.
"How..." She paused and sucked in another breath. "How are you going to keep me here?"
"I could tie you up." He still sounded angry.
"You wouldn't."
"And gag you. Gagging seems a capital idea."
She pressed down into the mattress, wondering why the idea of her husband binding her made her belly tighten with excitement. "I'd bite you," she said breathlessly.
He closed his eyes as if praying for strength. "Devil take you, Charis..."
He turned away. Her heart sank as she waited for him to head for the door. After all her efforts, she'd lost. She ached with weariness. The day had been long and difficult for her as well as him. If she gave up tonight, would she have the will to fight again tomorrow?
Desperately, she scrambled for some argument to stop him retreating into the lonely fortress of the parlor. But she'd reached the limits of her persuasion. He'd touched her, and logic fled. All she knew was she'd do anything to make him touch her again.
He veered left before he exited the room and dropped onto a stool near the door. Violently he began to tug at his boots.
Relief welled. And wild rejoicing. She could hardly believe it. He stayed.
More, he confirmed her theory that at heights of emotion, he escaped his affliction. He'd touched her, carried her. He hadn't trembled or flinched. He'd been too furious to remember Rangapindhi.
Could a fever of desire achieve similar results?
The light was strong enough for her to see he was still annoyed. It was clear in his jerky movements and the flat line of his mouth.
"Do you want some help?" she asked in a shaky voice.
"Don't push it, Charis," he said grimly. He stood up on his bare feet and prowled across to the bed, umbrage bristling from every line of his long body.
She moved to give him room and snuggled under the blankets. The intimacy of his presence tonight seemed more intense than yesterday's reluctant consummation.
He slid into the bed and stretched out on his back. No part of his body touched hers.
"Aren't you going to undress?" she asked, although the question was inane. He lay next to her fully clothed. Clearly he meant to remain that way.
"No."
Heavens, he even kept his gloves on. She realized with a shock she'd never seen his naked hands.
That abruptly struck her as significant. Gentlemen wore gloves as a matter of course, and it was winter. But Gideon didn't feel the cold, and she'd seen him without neckcloth and in his shirtsleeves, both far greater faux pas than forgetting his gloves. It seemed odd he was punctilious on this one matter of dress.
Odd. Mysterious. Important.
He settled himself more comfortably. She was overwhelmingly aware of his physical presence. The way the mattress tilted under him. His scent, so familiar now. The regular rise and fall of his chest.
"Gideon..."
As he turned his head on the pillows to stare at her, she caught the glint of his eyes. "Good night, Charis."
He sounded resentful. He'd hate being manipulated into enforced proximity. She couldn't blame him.
But he was here. That was all she cared about.
She'd achieved her first victory. Now she had to work out how to ignite his passion so the next time they shared this bed, he touched her as her husband.
How she wished she knew more about men. All she had to work on was instinct and last night's painful and embarrassing joining. Surely the delicious feelings he aroused in her weren't meant to end in desolation. There must be pleasure in the act. Else why would people risk so much for passion?
Perhaps one day soon she'd find out.
"Good night, Gideon," she whispered, linking her hands at her waist to stop them reaching for him.
## Sixteen
Since Rangapindhi, horror and pain had poisoned Gideon's dreams. This dream belonged to a different, more benevolent world. Slender arms cradled him. A soft female breast curved under his cheek. A woman's breath sighed in time with his.
The piercing isolation that scored his every waking moment vanished. In this bewitching fantasy, he rejoined the human race.
Dear heaven, let him not wake.
Not yet.
Convulsively, he tightened the arms he curled around the woman's waist. He buried his face deeper in the lush bosom. A peppery floral fragrance teased his senses.
A familiar fragrance.
He knew who he dreamt about. He'd known from the first.
"Charis..." he whispered into the frail silk veiling her breast.
His dream wife stroked his hair back from his forehead. The gesture's tenderness slashed his heart. Her fingers brushed his face, and he felt the breath stall in her lungs.
The dream's physical detail was so rich. So real.
Too real.
It was too late. He knew he wasn't asleep. The brief warmth was cruel mockery. Already he shrank from contact. Charis's scent became the oversweet stink of putrefying flesh. The touch of her hand, the grip of dead fingers.
His belly churning with nausea, he rolled away. As he sat up, he kept his back to her. He didn't want her to see the revulsion that he knew darkened his face.
"Hell," he groaned, burying his head in shaking hands. He tensed his throat against rising nausea.
"Gideon?" One word quivering with distress.
Of course she was distressed. She'd married a damned madman.
Through his agony, he was vaguely aware of how massively aroused he was. Hard as oak. Hot as Hades. It was a spiteful caprice of his affliction that his body continued to react like any virile twenty-five-year-old's.
"Gideon, are you all right?"
"Yes." He was lying.
Sunlight burned behind the closed curtains. Bedclothes rustled as she rose onto her knees. Damnably evocative sound. Desire became a hammering demand in his veins, so loud it drowned out the caterwauling in his skull. He wasn't sure whether desire or demons inflicted worse torture.
"I don't believe you." The mattress dipped as she shifted closer. Then—God help him—the insidious warmth of her hand on his tense back.
He went rigid, fighting the urge to wrench away. Fighting the urge to whirl around, fling her onto the sheets, and ravish her.
"Don't you know not to touch me?" he forced out through clenched teeth. Every breath strained his constricted lungs. His heart pounded so hard, he thought it must burst.
"I know you spent the night lying in my arms," she said quietly. Without, confound her, taking her hand away.
He'd broken into an icy sweat when he returned to full alertness. Now heat pooled where she touched him, making his blood simmer.
"I was asleep," he growled, loving her touch, hating her touch.
"I know," she said patiently, her palm rubbing in tantalizing, tormenting circles. He wore a shirt but the sensation of her touch was so intense, he might as well have been naked.
He was amazed steam didn't rise from his quivering flesh. His cock throbbed with the demand to be inside her. The memory of thrusting into her was so sharp, he could taste it.
"The difficulty is in your head. It's not in your body." She spoke slowly, as if trying to explain a mathematical problem to a dim student. How could she sound so calm when he was on the verge of exploding?
He could bear it no longer. He had to get away before he did something irrevocable, unforgivable. He lurched to his feet, spinning to confront her.
"I know that. It doesn't mean I'm making it up. God, Charis, if I could..."
He stopped and sucked in a shuddering breath. What use raging against fate? He couldn't do anything to alter his bleak future.
Although she must know his anger wasn't targeted at her, she paled under his onslaught. She knelt on the tumbled sheets in that sinful white nightdress. Gideon fought not to notice the provocative jut of her breasts against the transparent silk. He lost the battle. His eyes feasted on those luscious curves, and the moisture evaporated from his mouth. At his sides, his hands opened and closed as he struggled not to grab her.
"Don't you see what that means?" she asked earnestly, not seeming to register his seething restlessness.
Her voice was faint over the deafening crash of his heart. Had he missed something she said while he ogled her like a randy adolescent?
"Gideon?"
She clearly expected him to make coherent conversation. Didn't she realize the state he was in? But her eyes remained focused on his face with a sweet determination that only made him want her more.
He turned and snatched the armoire behind him open. He squeezed his eyes shut in an agony of desire as faint floral scent filled his nostrils.
Now that she wasn't touching him, hunger threatened to overpower him. Only the humiliating knowledge that touching her would unman him kept him from leaping on her.
Blindly, he fumbled in the dark cupboard until his hand fell on what he wanted. He turned and flung the yellow pelisse at Charis. "You're cold."
And I'm on fire.
She caught the coat and sent him a speculative look. To his frustration, she didn't cover her body.
Curse her, it was February. Didn't the woman have an ounce of sense? Through the buzzing in his ears, he tried to concentrate on what she said.
"...and then you're free."
He shook his head to clear the fog from his eyes. "Free?"
Her soft pink mouth took on the tiniest of curves. "Are you listening?"
Itchy heat crawled up the back of his neck. He forced himself to stare at the undistinguished landscape on the wall behind her head. But the image of her perched on the bed, disheveled from sleep, was etched into his eyeballs.
"Of course I am."
She made a doubtful sound deep in her throat. He couldn't resist looking at her. Then he wished he hadn't surrendered to temptation. On her knees in front of him, she seemed all too available.
"It's important," she said.
"What?"
The hint of a smile faded, and her voice lowered into seriousness. "When you forget yourself, you're free."
He frowned. "I never forget myself."
"Yes, you do. You forget yourself in violence. You forget yourself in sleep. Perhaps if you wanted it enough, you could forget yourself in..."
"A good swiving?" he finished on a sarcastic note. Frustration sparked. "Every damned doctor in London poked and pried at me. None suggested the sex cure. Perhaps they should have. Even if the remedy doesn't work, their patients won't care." His voice roughened into urgency. "Will you bloody well cover yourself?"
She lifted the pelisse, inspected it with an unreadable expression. And deliberately tossed it to the floor.
"No." With a languor that in a more experienced woman he'd attribute to purposeful enticement, she leaned to one side and uncurled her legs.
He wouldn't look. He wouldn't look.
He looked.
The nightdress hiked up, revealing neat ankles and gracefully curved calves. The night before last, he'd slid between those slender legs and he'd...
His mind slammed shut on the memory. He'd hurt her and disgraced himself. He couldn't go through that again for all the gold in Guinea.
She slid her feet to the floor and stood. Still with that eye-catching slowness. To his regret, he watched her hem slither down to her bare feet. God help him, just the sight of her toes, rosy and perfect, made him think of bedsport.
Even during his wild early days in India, no woman had stirred him to this pitch of arousal. He swallowed the constriction in his throat and forced himself to say what he must. "Charis, we've been through this before. There's nothing to be done."
He strove to sound calm, sensible, resigned. Difficult when his heart raced at triple time, and he couldn't rip his gaze from the girl standing only a few feet away. One step in her direction, and he'd be close enough to grab her.
What a damned disaster that would be.
"So you say," she said softly.
Was her voice always so husky? Or did his ears play tricks? He fisted his gloved hands by his sides and prayed for strength.
"What happened...changed me. I'm not a whole man."
Those sinfully thick eyelashes veiled her eyes. He couldn't remember seeing anyone in such minute detail before. It was like all the light in the world shone just on her.
"You looked whole the other night," she said evenly, although color rose in her cheeks.
Oh, dear merciful God in heaven. How could she remind him of that? It was meant to be the one time. It must be the one time.
His aching cock twitched as if to deny that assertion.
"You know what I mean," he snarled, nearly frantic with the painful heat sizzling through him. Heat that found no outlet. "You know...What the devil are you doing?"
"Unbinding my hair." She sounded unconcerned. Her deft fingers undid the long plait that curved sinuously across one shoulder.
"Don't." The command emerged as a croak.
"I need to brush it out and put it up for the day."
"Blast you, that's not why you're doing this."
He couldn't help but watch those busy fingers. Nor could he turn away when she buried her hands in the bronze mane and combed it loose so it fell like a shining curtain. Desire knotted every muscle in his body.
He lifted his hands to touch the glorious mass. Then hesitated midair. Feeling like the greatest fool in Christendom.
"Why do you think I'm doing it?" She shook her head so her hair slid around her in dark gold splendor.
"Your purpose is...seduction."
He stumbled over the last word like a prim spinster. Decadent images of that silky hair flowing about him as he pounded into her body fired his brain.
"You say you're impervious to the lure of the flesh."
"I never said that."
"Then what's stopping you?" She raised one hand and tugged at the ribbon holding her plunging neckline closed.
"Don't damn well do that." He should walk out the door right now.
"Why?"
He couldn't immediately think of an answer. All he could think of was how he would hurt her with his vile clumsiness if this scene reached the end she clearly wanted.
Why in Hades didn't she avoid him after that rough coupling? What was wrong with the chit?
His lips parted on a groan as her bodice gaped to reveal the valley between her breasts. He forced himself to concentrate on her face instead of her bosom. His heart slammed to a stop. The silent determination in her eyes shook him.
If he intended to retain a shred of honor, he needed to get out of here. Now. She didn't know what she invited. She couldn't.
"I'll wait outside while you dress."
"Coward," she said softly but distinctly.
"Charis, it's for the best." He tried to remember why he couldn't just jump on her and take what he wanted. His mind was a black, impenetrable jungle.
"Is the Hero of Rangapindhi running for cover?"
"I'm no hero," he snarled, cut to the quick. He abhorred the name the press bestowed on him. He turned to escape, unable any longer to bear the sight of what he wanted most in the world. Displayed for his delectation like a banquet. As unreachable as the stars. "I'll order breakfast."
He waited for argument, plea, protest. But she was silent. Clearly, she'd recognized her quest to seduce her oaf of a husband was futile.
He told himself that what trickled through his veins like acid was relief. She must at last see he was no use to her. It was tragic but irrefutable.
He reached for the door. Through unfocused eyes, he noticed his hand was unsteady.
There was a sudden flurry of footsteps behind him. Then a blinding, exquisite moment when she hurled herself, every lovely inch, against his back.
The shock stopped his breath. His heart hitched, then crashed against his ribs. Her heat made him dizzy. The softness of her breasts and belly pressed into him. Her arms snaked around his waist.
"Don't go," she said in a broken voice.
She leaned her cheek upon his back. The fragrances of carnations and warm female flesh filled his senses like smoke. He closed his eyes and groaned. Swearing under his breath, he banged his head on the door. The sharp pain did nothing to clear his mind.
His skin prickled at the contact, but sexual hunger drowned out his screaming demons. He could touch her now, all right. But in this state, he wasn't safe with any woman, let alone this exquisite girl.
He sucked in more air. Speech was torture when every sense concentrated on Charis. "Please step away."
Her grip around his waist tightened, and he felt desperation in the clawing fingers. She strained so close, he felt her every breath. And her trembling. "You'll leave."
"I must." His voice cracked, and he clutched the doorknob so hard, his hand spasmed. "For God's sake, Charis, do as I ask."
For a long moment, she didn't budge. Then, with tangible reluctance, she slid her arms away and straightened.
His animal hunger spiked, insisted he seize her, toss her on her back. Grinding his teeth, he beat back the raging demands.
He released the doorknob. His hand ached with stiffness. Slowly, against his will, he turned to face her.
She stood a couple of feet away. Her chest heaved as she fought for breath. He'd been terrified he'd made her cry. But for all her palpable, quaking misery, she remained dry-eyed. In a defiant gesture he recognized, she lifted her chin as if she stared down death itself.
Swiftly, she tugged the nightdress over her head and flung it into the corner.
"Damn you, Charis," he breathed, stepping toward her before he recalled he couldn't touch her. "Don't do this."
Unclothed, she was...heavenly. Slender neck, straight shoulders, long graceful arms, high breasts with whorled pink crests. Flat belly punctuated by the sweet hollow of her navel.
Last, helplessly, his gaze focused on the delta between her legs. Blazing arousal flared. He swallowed and forced himself to breathe. He drank in the sight of her as he'd drink from an oasis after crossing a desert.
The desert still extended ahead.
Dry. Waterless. Barren. Deadly.
She glanced down at his trousers, then unflinchingly lifted her eyes. "You want me. I know it." Her voice broke.
He strove to deny his desire. But his throat closed and wouldn't permit that ultimate heresy.
His heart pumped out an inexorable rhythm. Two words. Over and over. Take her. Take her. Take her.
"Do I...do I please you?"
He fought to frame the lie, to tell her she meant nothing to him, to set her free.
Vibrating with tension, she stood before him. She stared back from steady hazel eyes, more brown than green. But her lips were soft with a vulnerability that mangled his gut.
He opened his mouth to speak.
She didn't flinch. Nor did her gaze waver.
She must guess what he meant to say.
Her mouth trembled. If he hadn't watched so closely, he'd have missed the tiny tensing of her lips. It was the reaction of someone braced for the killing blow, for pain past endurance.
He knew that feeling. Just so had he faced down his jailers in Rangapindhi.
That hint of vulnerability broke him.
Three strides, and he was at her side. He swung her high in his arms. Blood thundered in his ears. Two more strides, and he reached the bed. Without letting her go, he pushed her back onto the crumpled sheets.
Gideon was pure animal. Savage. Hungry. Desperate.
He knelt between her legs, his cock straining. Roughly, he brushed away the thick dark blond hair cascading across her bare breasts. The demons shrieked for him to stop, but roaring physical need trapped them behind a wall of glass.
He grabbed her hips with his gloved hands and pressed hard, openmouthed kisses across the white plain of her belly. She tasted like hot musky honey.
He suckled on her nipple, pressing it against his tongue, drawing the flavor deep into his mouth. She cried out and bucked.
He didn't linger. This moment poised on a knife edge. His lips closed on her other nipple, biting until she writhed. She lifted her hands to his shoulders.
Dear God, if she pushed him away, what would he do?
But her fingers dug into his damp shirt, clenching and unclenching in time with the rhythm of his mouth on her breast.
He ripped his trousers open. The pounding in his head was so loud, he hardly heard the material shredding.
With ruthless hands, he angled her hips up and plunged into her.
Heat.
Pressure.
One fragile, glowing moment that might have been peace.
Stray details overwhelmed starved senses. Her scent. The soft rattle of her breathing. The way she quivered under him.
He rose to look at her. Her eyes were closed, and her face was stark with tension. Damn it, he must be hurting her. Principle insisted he stop, withdraw, leave her be.
He began to pull out. Meaning to end this travesty. But the sensation of his tumescent flesh sliding free of her sleek passage nearly blew the top of his head off. Pleasure so intense it edged on pain incinerated him in a white-hot blast.
His scruples dissolved to ash. His heart tolled a despairing note as he thrust back inside her. Hard. Demanding. Pitiless.
She closed around him with what felt like welcome. This time he paused, luxuriating in the tightness. He shifted. Edged deeper.
Charis moaned, a low, guttural sound that resonated in his gut. The hands on his shoulders slid down to curl around his straining back. She tilted her hips higher.
Her eyes opened. The pupils were dilated, and the irises were rich gold. The skin on her face stretched tight. She tipped her head back, her thick lashes fluttered down, and she arched with a long, low, keening sound.
What frail restraints he'd imposed snapped. There was just the hot clasp of her body and his thundering need.
He changed the angle of penetration. Her body moved with him. He withdrew and thrust again. He needed the rhythm more than he needed breath.
Faster.
Harder.
The endless rocking of his hips against hers. The slide of his flesh into her slender body. The creak of the bed. The rustle of the sheets. The catch of her breath.
His body tensed. The pace became wilder.
His release built, knotting his spine, twisting his gut, tightening his balls to agony.
He lifted his head, and his throat clenched on a shout. Anguish. Shame. Possession.
Freedom.
One last thrust. His world ignited into fire.
He flooded her with his agony and his loss and his anger. His hips jerked as the crisis flung him into eternity.
For a long time, Gideon's mind closed down to anything but the volcanic release.
He slumped over her, gasping for breath. There was only his quivering body, the gallop of his heart, the warm embrace of darkness.
He was utterly exhausted. Weary to the point of torpor.
Vaguely, he heard her make a sound of discomfort.
He tried to shut it from his mind. He belonged in this darkness. He wanted to stay here.
He'd acted like a beast.
The unwelcome knowledge nibbled at the blanketing stillness.
Oh, merciful God, what have I done?
With a groan of utter desolation, he pulled free and rolled onto his back. If he could trust his legs to carry him, he'd walk out.
He stared at the ceiling, waiting for his breath to steady and his heartbeat to resume its usual rate. Waiting for the world to crash in on him.
In spite of his howling conscience, his physical self relished what he'd done. The sheer power of the experience eclipsed every previous sexual encounter the way the sun outshone a candle.
He stirred, turned his head to look at Charis. The movement cost the last of his depleted strength. She'd drained him to the lees.
"Are you all right?" he asked gruffly.
She was in profile. She licked her lips. The innocent movement sent a smoldering bolt to his loins. Suddenly, he wasn't quite as exhausted as he had been.
She made no attempt to cover her nakedness. Knowing she lay bare beside him piqued his desire. He, on the other hand, hadn't had the finesse to do more than tear his trousers open and have at her.
"Perfectly, thank you."
Gideon frowned. Her polite, detached tone worried him.
Perhaps he really had hurt her. He leaned up on one elbow to see into her face. "I fell on you like a hungry dog on a bone."
She stared upward. He wondered what her determinedly neutral expression concealed. Devastation? Fury? Pain? Oh, hell, don't let him have done her injury. He'd been passionately unrestrained, and until two nights ago, she'd been a virgin.
She glanced at him out the corner of her eye. "You're not shaking. You're not sick. You're not sweating."
He frowned. "I'm worried about you. Forget about me."
"You forgot about you."
She sat up, drawing her knees up. The girlish grace of the movement captured his attention, stirred his interest. Then he realized what she'd said.
"Was that an experiment?" Resentment stirred under his concern. "You've got a bloody cheek."
She bent so her thick hair fell forward, hiding her expression. "I couldn't see how else to test if what I guessed was true."
He scowled at her. "And got a right royal fucking in return."
She jerked her head up and stared at him. He sucked in a shuddering breath, ignoring the shock on her face. His tone bit. "I hope you're pleased with yourself, madam."
With a movement that shot another jolt of arousal through him, she shook back her untidy tumble of hair. A smile curved her lush pink lips. Lips which to his shame he hadn't kissed, even as he'd slammed into her like a hammer.
"Of course I'm pleased with myself. I drove my husband wild with desire."
He jackknifed onto his knees. If the habit of keeping his hands to himself weren't so ingrained, he'd shake her until her teeth rattled. "What the devil..."
Her smile faded. "Gideon, you touched me."
"Blast you, Charis, I did more than touch you. You deserve better."
She grabbed his arm. "I don't care what I deserve. I want you. However I can get you." The smile reappeared. "And it was exciting."
"Exciting?" He had trouble speaking. He felt like he'd entered a new universe, where nothing from the old one made sense.
"Of course it was exciting," she said urgently. "You looked as if you'd die unless you touched me. You'll do better next time."
"Are you sure there will be a next time?"
"I've discovered your weakness." Satisfaction warmed her voice. "When I'm naked, you're powerless."
The problem was the witch was right. Even now his cock stirred with interest.
She still regarded him with that faint enigmatic smile. "To think I ever doubted you wanted me."
Foolish woman. He gave an unamused laugh. "I always want you. Damn it, Charis, I'm in love with you."
## Seventeen
Aghast, Gideon stiffened. Bloody, bloody, bloody hell. Why in the name of all that was holy did I say that? He'd give his left arm to take back the words. But it was too late.
Violently, he tugged away from Charis and surged to his feet. He stalked across to scoop the discarded nightdress from the floor. With an angry gesture, he tossed it across the end of the bed.
He should have kept his blasted mouth shut. But the wild, uncontrolled sex had broken some barrier within him. The declaration he'd fought back for so long had surged up unstoppable as a king tide.
She started as if emerging from a daze. "You love me," she whispered.
She stared at him with huge, shining eyes. Her lips parted. She looked so happy, he couldn't bear it. Clearly the damage was done, and there was no point in telling her he'd lied. Although it would be better for both of them if she believed he had.
The harsh facts that put a life with her completely out of reach hadn't changed, for all that every cell of his body ached with love for her. One bout of desperate passion didn't change the cruel reality of his existence. He wasn't a normal man. He'd never be a normal man. And if she pledged herself to him now, one day she'd regret that commitment.
He couldn't bear to contemplate her love turning to hatred and disgust when she realized just what she'd sacrificed by walling herself away with her half-insane wreck of a husband. Her best chance of happiness was to establish a future far away from Gideon and his demons. But he could see what he'd just said made it less likely than ever that he'd convince her of that incontrovertible truth anytime soon.
Again, he cursed the impetuous declaration that forever changed the landscape between him and his beautiful, misguided wife.
"It doesn't matter," he said with a carelessness that even in his own ears sounded false.
A tiny frown line appeared between her brows. "Oh, Gideon." She spoke his name with such deep compassion, he tensed with fuming resentment. He couldn't endure her pity.
So he didn't have to look at her, glowing, irresistible, he struggled to concentrate on doing up his trousers. His gloved hands shook so much that he fumbled hopelessly with the fastenings. It was like being in the grip of his affliction, except he trembled now not because he'd touched her but because he so badly wanted to.
During those dazzling moments in her arms, his world had come right. He could offer her a lifetime of misery while she was his only hope of happiness. That was his eternal burden. He couldn't make her share it.
"I'll ring for hot water," he said with a studied neutrality that cost him more than he wanted to admit. At last he managed to close his trousers. "You'll want to wash."
"That's it?" He still wasn't looking at her, but he heard the irritation in her voice. "You take me to bed. You tell me you love me. Then we just have breakfast as though nothing's happened?"
He glanced up and tried not to notice how very...naked she was. "Charis, I wish you'd put on your nightdress."
Her lips firmed with impatience. "That doesn't answer my question."
He sighed and ran his gloved hand through his hair. "Nothing should have happened."
"Why?"
"Will you put on the confounded nightdress?" he demanded in desperation.
She stretched out one slender arm, hooked up the silk garment, and slid it over her tousled head. "There. Is that better?"
"Not really." He breathed hard through his nostrils and fisted his hands at his sides. He burned to take her again. Her defiance only fed his incessant craving. He was an insatiable satyr. If the girl had any sense, she'd run a thousand miles to get away from him.
"I don't see why this is a problem," she said stubbornly. "You love me. I love you."
"You don't love me," he bit out.
She rolled her eyes. The sudden reversion to sulky schoolgirl would have summoned a smile if he didn't feel like she flayed his soul.
"No, of course I don't," she said sarcastically. "I'm a stupid sparrow of a female with hardly brains to feed myself. And you're so terrifically unworthy. The contemptible fellow dragged weeping out of a pit in India when any other man would have taken the trifles you'd endured in his stride."
"Charis..." he said in a dangerously low voice. Her mockery cut him to the bone. Especially as it held an unfortunate echo of his genuine concerns. "You go too far."
"Well, it's all so absurd, Gideon." She spread her hands in a frustrated gesture, the movement making her breasts jiggle enticingly under the sheer silk. His mouth dried, and his hands flexed as if they cupped those firm mounds.
"We love each other." Her cheeks flooded with pink. "Why are you standing half a room away?"
A glance under her eyelashes sent blood sizzling through his veins. Damn her, she could give Circe lessons. He braced his shoulders as if only physical restraint stopped him diving on her.
"Because I can't touch you without losing my mind," he snarled, need thundering through his body.
She slid her legs over the bed and stood straight before him. "I touched you before, and you didn't notice."
"You..."
He started back as if she touched him now. He'd vaguely noticed she'd taken his arm. When was the last time he'd vaguely noticed even the merest contact with anyone?
Good God, could she be right about sexual excitement offering a reprieve? None of the doctors had suggested it. Ever since his rescue from Rangapindhi, he'd assumed he faced a life of eternal celibacy. Had he been mistaken?
He compelled his lust-fogged mind to review the facts. He'd just maintained extremely intimate contact with his wife. He was far from composed—he was fuming and upset and randy as hell. But if he felt ill, it was his conscience that troubled him, not memories of Rangapindhi.
As if she knew he at last took her idea seriously, she stepped forward and placed one hand flat over his heart. Her cheeks were brilliant with color. "Gideon, what just happened was so lovely. Let's not spoil it by fighting."
He tensed for the familiar sick reaction. There was only the warmth of her hand and the hardening of his cock, which definitely approved of her plan for a normal marriage.
"Lovely?" he forced out in blank astonishment.
Lovely and exciting. His brain tried to make sense of what she said. Neither word seemed adequate to describe that earth-shattering sex. But he was human enough to be grateful she hadn't found his untrammeled passion completely distasteful.
She nodded and sent him a smile that made his gut tighten with the same lust that had got him in trouble only a short while ago. "Yes, lovely."
Hope, so long a stranger in his life, inched into uncertain life. Was it possible he had changed? He could hardly bear to contemplate the idea. The sudden intrusion of light into the Stygian darkness of his life blinded him, left him bewildered.
Hardly believing he could, he lifted one gloved hand and placed it over hers. Through the fine kid, the heat of her skin was a distant echo of life and joy.
For a forbidden moment, he basked in the glow of her hazel eyes. His hand shook, but with emotion not physical weakness.
He found his voice, rusty, thick, unsteady. "Truce."
He loved her.
Charis could hardly believe it. But Gideon's quaking desperation as he clutched her hand to his chest convinced her it was true, perhaps more than actually hearing the words.
With that declaration, Gideon changed her world forever. Her heart rejoiced. She felt new, reborn, strong. At last there was a chance she could win what she wanted with all her soul for both of them. A life of happiness, a future at Penrhyn, children, contentment, peace.
She and Gideon spent the afternoon tooling around Jersey's lanes in a hired curricle. When he suggested the outing, she'd leaped at the chance to escape their rooms. In the cramped conveyance, awareness tautened between them, but movement and air made the bristling atmosphere bearable.
Almost.
With a flourish, Gideon drew the vehicle to a stop on the crest of a hill. Below spread a vista of fields, with the sea silver in the distance. A breeze teased strands of hair from under her bonnet. The gloomy weather had cleared, and the day was fragrant with coming spring.
He loved her.
The sun shone more brightly. The birds sang more fervently. The air brushed across her skin more sweetly.
"Oh, what a pretty place." She risked tucking her gloved hand around his arm.
When he didn't recoil, she leaned forward, deliberately rubbing the side of her breast against him. Surprised pleasure awoke as she heard his breath catch.
Those torrid moments in his arms had taught her so much. That she could drive him mad with need. That he could touch her with the deepest intimacy. That the sensation of her husband's body pumping into hers was the purest excitement she'd ever known.
Now she was familiar with the scent of his arousal, the sound he made in his throat when he penetrated her flesh, his hard strength as he pounded into her. The experience hadn't been entirely comfortable. He'd been rough, and she wasn't yet accustomed to a man's passion.
He'd thundered into her like a regiment of horses charging down an enemy position. She should have been terrified.
Instead, she'd loved every hot, sweaty minute.
She'd loved his body joining with hers. She'd loved seeing him a helpless slave to desire.
She wanted him to do it again. Soon.
His arm was rigid under her hold, but at least he didn't pull away. "It's good to get out of town." Did she hear a trace of huskiness in his comment?
"The press of people worries you?" She turned to study him. He'd been preoccupied most of the day but to her relief, he showed no signs of illness. What happened this morning had clearly unsettled him. She couldn't doubt he'd found physical satisfaction. But his mind was far from easy.
She curled her fingers around his arm, testing the unrelenting muscle. He was so strong and masculine. The heated memory of him surging into her filled her senses. She felt her color rise.
He sent her a brief, assessing glance. "A little."
It took her a moment to realize he answered her question. The problem with this plan to drive him out of his mind with lust was that she wasn't exactly immune to his touch either. So difficult to focus on a goal when his mere presence turned her into a steaming pool of desire.
She reminded herself to be patient. This would be a long, slow siege, but victory would be worth it. For Gideon and for her.
"London must have been a nightmare."
He looked over the horses' heads, and his gloved hands tightened on the reins. "Yes."
"How did you bear it?"
He shrugged. "I had no choice. The sovereign commanded. I obeyed. I drank. I took opium when liquor failed. I canceled what engagements I could. Tulliver and Akash helped."
"And now there's St. Helier."
He smiled. "Believe me, St. Helier is much easier than London."
"Don't worry. We'll soon be back at Penrhyn."
Amusement sparked his dark eyes to starlight as he glanced at her. "Good God, madam, you sound like a wife."
She met his gleaming gaze, shadowed under the curling brim of his stylish hat. He looked like a buck of the ton. Impossible to reconcile this elegance with the rumpled, satiated man from a few hours ago.
"I am a wife," she said softly. For the first time, she almost felt like one. His eyes changed, focused, and her heart shifted in her breast. "I wish you'd kiss me," she whispered before she reminded herself what trouble her propensity for blurting out her thoughts had already caused.
A taut silence fell. She waited for him to retreat as he had so often before.
The humor drained from his face, replaced by a concentrated sensuality. His gaze dropped to her mouth. Her breath escaped her parted lips on a sigh.
Her senses sharpened. The sounds around them suddenly seemed unusually loud. Birdsong. The sea's distant roar. The jangle of harness as one of the horses shifted.
Then her heart's furious pounding drowned out everything else.
Slowly, so slowly she thought she'd die with waiting, his face moved closer. His warm, moist breath feathered across her lips. She made a choked sound of yearning.
If he stopped now, she'd scream.
He closed his eyes and brushed his lips across hers. A glancing contact.
She growled with impatience and strained toward him. There was no sign of his usual reluctance for physical contact. She silently whispered a prayer of thankfulness.
"You're teasing," she said hoarsely.
Those lips she wanted on hers quirked. "A little. Take off your bonnet, so I can do this properly."
Even through her yearning, she recognized how promising that sounded. With shaking hands, she untied the yellow satin ribbons and ripped the hat from her head. It was new and very stylish. Without hesitation, she dropped it to the curricle's floor.
In a fever of anticipation, she watched him secure the reins, although the horses seemed happy to laze in the waning sun. He swept his hat from his head.
He must hear her heart's furious beat, it was so loud. Her palms were moist. Nervously, she wiped them on her skirts. "Hurry," she said in a shaking voice.
He laughed softly. The deep sound shivered through her. She squirmed restlessly on the seat.
Slowly—why, oh, why was he so slow? Couldn't he tell she was in a lather of desire?—he lifted one hand to cup the back of her head. His gloved fingers speared through the hair at her nape.
"You're so fierce," he murmured.
"Don't you like it?" She hardly knew what she said. All she knew was that he touched her as if nothing else in the entire world mattered.
"I didn't say that."
He lifted his other hand and placed it under her chin, holding her face angled up. Absurd when he must know evasion was the last thing on her mind.
"Gideon..."
An invitation. A protest. A plea.
"Shh." A tender smile hovered around his lips.
He dipped his head and gave her another fleeting kiss.
Still he played with her. In spite of his own need. An inferno of desire raged behind those black eyes. Heat radiated from him. She shifted, trying to get closer to that blaze.
Very gently he placed his lips on the corner of hers. A kiss on her nose. On her chin. Between her brows.
"Kiss me," she said almost tearfully. This delay was more than mortal flesh could stand.
"I am kissing you."
She shook with impatience. She wanted passion. She wanted to know he hungered for her. But this tenderness was sweeter than sugar. She felt her soul unfurl. He'd made love to her twice. Neither time had he been tender.
Now he treated her as if she were made of finest Venetian glass. Likely to shatter at the slightest touch.
She raised her hands. One hooked around his wrist. The other rested on his chest where she'd touched him this morning. Under her palm, his heart raced.
"Kiss me properly," she begged. "Or I'll go mad."
"We're both mad," he said with sudden determination. "God help us."
As abruptly as that, the world exploded into flame. His mouth covered hers with ferocious passion.
She gasped with shock. Then with astounded pleasure. He was all hot desire. But the ghost of earlier tenderness lingered like embers from a banked fire. Like stars fading at sunrise.
She surrendered, parting her lips.
His relentless physical onslaught gave no quarter. He slipped his tongue across her lips. Then flicked it inside. She stiffened at the unfamiliar intrusion.
Abruptly he lifted his head.
Oh, don't let him stop. I'll die if he stops.
"It's all right," he crooned, and returned to kissing forehead, cheeks, and chin. Using his hand behind her head to hold her for his depredations.
She moaned and yearned toward him. "Kiss me, Gideon." Her voice vibrated with longing.
"I forget..." He punctuated his speech with a scatter of kisses. He placed his mark on every inch of her face. Except her lips. Where she wanted him.
"...how innocent..." More kisses.
"...you are."
The hand she'd placed over his crazily hammering heart slid up to encircle his neck. Her fingers tangled in the hair that brushed his coat's high collar.
"You surprised me," she said shakily. "It wasn't that I didn't like it."
More glancing kisses. "What a sweet little wife."
"You're tormenting me," she accused, turning her head to try and catch his lips with hers.
"You've tormented me for days. I never thought I'd touch you like this."
"But you wanted to?" She knew the answer, but still she longed to hear him admit it.
"You're a fever in my blood," he said in a raw voice.
His hands shifted down her back, and he lifted her toward him. He parted his lips over hers. More gently than before. This time she was ready for the penetration of his tongue.
One brief foray. Retreat. A more thorough exploration.
Heat exploded behind her eyes. Searing pleasure flowed through her veins. She knew nothing but the scorching possession of his mouth. She gasped and pressed closer, opening her lips wider.
He stroked his hands up and down her back, tracing her spine. Everywhere he touched, he set up another hot whirlpool. Flame licked at her skin. Still, his mouth plundered hers with endless dark delight.
Tentatively, she slid her tongue against his. He made a growling sound of approval, and his hands tightened.
More bravely, she moved her tongue until the kiss was no longer invasion but ecstatic dance.
She made an inarticulate sound and edged closer, sliding awkwardly on the seat. He ripped his mouth from hers. He breathed in great gusts, and his eyes were blacker than ebony. He released a startled laugh and pulled her up against him.
"A curricle's not designed for lovemaking," he said unsteadily.
Charis was still dazed. Joy resonated through her like music. "I don't care. It was wonderful."
She sounded like a besotted ninnyhammer. What matter? She was a besotted ninnyhammer.
He loved her.
He touched her cheek with a gesture that split her vulnerable heart open. She'd loved him from the first, but until now she'd had no idea how physical pleasure turned love into something so vivid, it became a living entity.
"Shall we go back to the inn?" His voice was velvety with anticipation.
She curved into his side. For the first time, his arm circled her shoulders. She basked in the closeness. Her lips tingled with the memory of his kiss. Strangely, that kiss had changed things between them more than either time he'd used her body.
Hope poured into her brimming heart.
He loved her. She loved him. What could defeat them?
## Eighteen
Across the remains of the meal he'd ordered in their rooms, Gideon watched his wife. The hostelry was famous for its cuisine. For all the attention he'd paid to the elaborate fare, it might have been sawdust.
Instead, his heart, his mind, his soul were full of his bride.
Charis.
Beautiful, beautiful Charis. His delight and his despair.
She'd been quiet on the drive back from the countryside. Nor had she spoken much during dinner. Like him, she'd toyed with her food. Now she looked up. Doubt swam in her hazel eyes like sharks in a clear sea.
She set down her fork with a decided click. Her slender hand clenched against the table. "What made you change your mind?"
Gideon didn't pretend to misunderstand. She was too clever to let his new attitude go unremarked and too brave to avoid the subject.
After a moment's thought, he gave her a frank answer. "I'm not sure I have."
He was guiltily aware that he played dangerous games with their future here. He still believed her best chance for happiness lay in a life separate from his.
But it was more than mortal flesh could stand to share a bed with his delectable wife and not touch her. Especially after the desolate hell he'd subsisted in since Rangapindhi. Charis thought because they loved each other, they had a chance. Gideon knew in his bones that love only made the price they'd both pay for their current indulgence more excruciating.
He should stay away from her. But he couldn't.
Of all his many sins, perhaps this was his greatest.
Her lips compressed with impatience, and she fiddled with the stem of her wineglass. "You're happy to touch me now."
He remembered this afternoon's delicious kisses, and he couldn't suppress a reminiscent smile. "More than happy."
His reply didn't mollify her. Her color rose, but her regard didn't waver. "What changed?"
He briefly studied the white damask tablecloth, then glanced up. "Well, there's the fact that I can touch you."
She blushed more furiously. "So you're reconciled to living as my husband?" He heard her difficulty forcing the question out.
He sighed, and his voice deepened into gravity as he answered with equal difficulty. "Charis, I'm not doing you any favors with what's happened. If I had a scrap of decency, I'd leave you alone."
Yes, he could touch her without turning into a beast. This morning hadn't proven that, but this afternoon had. He loved her, if anything, more than ever. If she asked him, he'd catch the stars from the sky for her.
But the factors that made him an unsuitable consort for this glorious girl remained as stark as ever.
Whatever private bliss life with his wife now promised, he was still a physical and mental wreck. His immediate strained reaction today to St. Helier's bustle confirmed that. The frail seedling of hope that had uncurled inside him this morning had shriveled as he'd felt the old, crushing, sick reaction to the crowd. Painful reality had crashed down upon him in all its inexorable grimness.
What a fool he'd been to believe this temporary reprieve meant a permanent cure to his ills. He'd never lead a normal life, he'd always have to hold himself apart, isolated. He couldn't lock someone like Charis away from the world and hoard her like a miser hoarded his gold. It wasn't fair, and eventually, he knew, she'd chafe at the restrictions of life with a recluse. He couldn't bear to see her bright spirit flicker and go out.
She said she loved him. But for all her sweet passion and determination, he wasn't convinced she suffered anything more than a particularly virulent case of hero worship. What he was firmly convinced of was his complete unworthiness. He'd failed so many times. He couldn't bear to contemplate failing her. As he surely would. Better he set her free to find the man she deserved.
He bit back his agony at the idea of her falling in love with someone else. He had to think of her future and not his own selfish desires.
Except that right now, his own selfish desires were paramount, unstoppable. He should leave her to sleep alone, but he already knew he wouldn't. The astonishing joy he'd found in her arms, when he'd thought any joy at all lost forever, made restraint impossible.
St. Augustine's self-serving prayer flickered through his mind. Lord, grant me chastity and continence. But not yet.
Charis lifted her wine but didn't drink; instead, she stared into the red depths with a troubled expression. "If you're sure it's an almighty mistake, why did you kiss me?"
Ah, smart girl, to pick the kiss as the betrayal of his principles rather than this morning's volcanic lovemaking. He told her the simple, incontrovertible truth. "Because, God help me, I can't resist you."
Startled, she looked up, and a smile of utter delight curved her full lips. "Really?"
She was so pleased with herself, he couldn't help laughing. Although he was a villain to encourage her belief that they could find happiness. It was a role he suspected he'd become accustomed to in coming days. Because, having tasted her, there was no way on this earth he could keep his hands off her while they shared these rooms.
Still, even as he acknowledged her power, his reply held an edge. "Yes, damn you, really."
"Well, that's all right, then." She put down her wine, stood, and rang for the servants.
Surprised, he turned in his chair to watch her. "Is that it? No more inquisition?"
"For the moment."
He heaved a sigh of masculine relief although he didn't trust this sudden docility.
As the maids cleared dinner, tidied the room, built up the fire, prepared the bedroom, he stood beside the mantel, holding himself apart. Just this much activity around him, and his sinews tightened with revulsion.
No, he was far from cured, God damn it to hell.
The chilling knowledge seeped into his bones. Briefly, he closed his eyes, trying to summon will to deny Charis—and himself. But will was putty against the potent lure of desire.
He and his wife would make love tonight. Anticipation fizzed in his veins. He sipped at his claret, wondering when he'd last spent an evening with a lovely woman, knowing they'd end up in bed.
She looked across from where she sat, pretending to read a book, and sent him a secret smile. She knew how the night would end too.
Gideon drew a deep breath as the door closed behind the last servant. Now just he and Charis remained, and the air suddenly seemed clearer, cleaner. He ignored the howl from his conscience that he had no right to touch his wife when he was such a disaster.
His eyes fastened on Charis as she set aside her book. He stayed where he was, enjoying the crescendo of expectation. His hands itched to drag her close for a drugging kiss. To discover what marvels lay under her lovely red gown.
She stepped up to him and took his wineglass away, her fingers brushing his gloved hand. Even that much contact would have once set him shaking and sweating. Now it just aroused sizzling need. Her carnation scent drifted out to whisper promises of paradise.
"Will you do something for me, Gideon?" she asked softly.
A dim warning sounded. In his besotted daze, he hardly heeded it. "It depends."
Her lips tilted upward as she placed the glass on the mantel. "That's hardly gallant. A true gentleman would obey my slightest whim."
"I'd say that gentleman didn't know you very well."
She laughed softly, and the husky sound made his gut churn with longing. For all his brave words, he'd lie down and die if she asked him.
"So suspicious."
"Suspicion has kept me alive on numerous occasions. It's a highly underrated characteristic." He sent her a searching look. "What do you want, Charis?"
She sucked in a steadying breath, and he realized that beneath the flirtatious humor, she was nervous. The warning clang became more insistent. "I want you to allow me to do with you what I will."
Charis resisted the urge to twine her hands together. She needed to convince Gideon she was a confident, self-aware woman, not a silly girl. Acting as jittery as a canary in front of a hungry cat wouldn't advance her cause.
He angled one black eyebrow. "Which involves what?"
She bit her lip before she remembered she meant to appear nonchalantly assured. Raising her chin, she forced herself to meet his wary dark eyes. "Well, undressing you, for a start."
Hot color seeped under her skin. Nonchalant assurance had never been likely. Even coherent speech seemed an unachievable goal. Surreptitiously, she wiped her palms on her skirts.
"I...see," he said slowly.
She waited for more. Anger. Protest. A resounding no. But he remained silent. She rushed into speech. "It's not salacious curiosity."
His lips twitched slightly although she read growing resistance in his eyes. "I'm pleased to hear it."
"This isn't a joke, Gideon," she said in a low urgent voice. "It's important that you've kept your clothes on whenever we've..."
"Made love?"
"Yes," she responded on a thread of sound. Her heart fluttered like a trapped bird against her ribs. Not a sparrow. Something big and fierce like a vulture.
He leaned on the mantel, his long body elegant and powerful. The flames from the grate cast strange, flickering shadows over his face. For a moment, he looked devilish. She licked lips dry with nerves. His eyes fastened on the movement. The blatant interest reminded her she wasn't completely powerless in this war. She stiffened her spine.
One gloved hand fisted on the mantel. His voice was silky with control. "So I hand myself over to your tender mercies? Do I have a choice?"
She knew he resented the way she undermined his defenses. She pressed her palms deeper into her skirts to hide their trembling. "You can say no."
"Then you won't share my bed tonight," he said grimly.
Her heart somersaulted with astonishment. Did he know just what he admitted? "I won't stay out of your bed to gain my way." She licked her lips again. "You see, I can't resist you either."
His appearance of tranquility abruptly shattered. With a furious movement, he jerked away from the hearth. He was visibly shaking. For one horrified instant, she wondered if his affliction was returning. He grabbed the back of a chair, gripping it with hard fingers. "In Rangapindhi, I was tortured."
"I know."
She saw his throat move as he swallowed. "You'll find my scars repulsive."
She blinked with shock. This hadn't occurred to her. Although if she'd thought, it should have. Spreading her hands, she spoke the truth in her heart. "I think you're beautiful. A few marks on your skin won't change that."
His brief laugh held no amusement. "You don't know what you're talking about."
She stepped close enough to touch him. "Let me see."
He released the chair. She recognized the gesture as a sign of reluctant acquiescence.
Very carefully she reached for the lapels of his black coat. The wool was warm from his skin. He braced under her touch although he didn't retreat. She took this as tacit permission to continue.
Slowly, she slid the coat from his shoulders and down his arms, then lifted it away. His jaw was set as if she tortured him. He was rigid as an oak board.
Dear heaven, let her instincts lead her right. If Gideon endured this suffering for nothing, she'd never forgive herself.
She tamped down guilt and fear as she turned to lay the coat over the chair. Something deeper than dread or compassion told her that until he let her see him without the armor of clothing, his essential self would stay hidden.
Her heart careening in a mad race, she steeled herself to face him. He'd dressed more formally than usual tonight. He stood before her in an exquisite white waistcoat, embroidered with silver vines. A snowy neckcloth. Shirt. Biscuit trousers. His hands, as always, were encased in gloves. Tonight, white evening gloves like the ones a dandy wore to a ball.
The betraying muscle flickered in his lean cheek, and he breathed unsteadily. The soft, broken hiss was the only sound apart from the flames crackling in the grate. When she lifted her hands to his waistcoat buttons, she felt the ragged rise and fall of his chest.
She flicked open one button. Two. Three. The beautiful garment sagged open.
She slid her hands under the brocade to slip it off. Now only the fine material of his shirt separated her from his skin. He was hot as a blazing fire and so tense she feared he might shatter.
Before she thought to censor herself, her gaze dropped. His arousal swelled against the front of his trousers in unmistakable demand.
"You know I want you," he said flatly. "You use it against me."
She shook her head, setting the waistcoat over his coat. With every garment she removed, she felt like she seized enemy colors in a battle.
"I use it for you." If she didn't believe that, she couldn't summon courage to persist. She gathered that courage and placed her hand over the bulge in his trousers.
Her breath caught. He made a strangled sound deep in his throat. She'd never touched him there before. Through his clothing, she felt the tensile power. The life. The vigor. Automatically, she shaped her fingers to the hard length. His flesh surged into her palm as if it had a will of its own.
Gideon closed his eyes. "Charis..."
She bit her lip and lifted her hand away. She shook as she reached for his neckcloth. Her fingers were clumsy, and the length of linen seemed impossible to untangle.
She sucked in a deep breath, redolent of Gideon, and forced herself to concentrate. Eventually, she managed to tug the neckcloth free. His shirt gaped. His pulse beat wildly at the base of his throat.
He breathed rapidly. So did she. The room felt close, confined, stifling. Need settled low and heavy in her belly.
She hadn't set out to titillate him. Or herself. But the act of undressing this big strong man—and him standing quiveringly still as she disrobed him—made heat well between her legs.
The air was sharp with arousal. Male and female. She wasn't touching him, but his desire surrounded her like sheets of flame.
He closed his eyes as if he couldn't bear to witness what she did. His tension was a vibrant, writhing force. Air scraped in and out of his lungs.
Doubt assailed her. Held her paralyzed.
Could she do this? Should she do this? What if her actions pushed him deeper into purgatory?
She braced her shoulders and reached forward to pull his shirt free of his trousers. Her heart banged against her ribs. Her hands shook.
He opened his eyes and snatched the hem of his shirt. "Here, damn you," he grated out. He tore the garment in two, shucked the ragged pieces, and dropped them to the floor.
Anything Charis might have said lodged unspoken in her tight throat. Her hands fisted at her sides. Her eyes flew up to meet Gideon's glassy gaze, then dropped to convulsively trace every line of his torso.
She'd known he'd be beautiful. But his virile splendor left her speechless. His pale skin stretched tight over ridges of hard muscle. Feathery dark hair covered the broad plane of his chest.
Scars patterned his chest and arms. Long lines that she guessed came from a whipping. Pale satiny welts that looked like burns. Round marks that could be bullet holes. A tangible history of unrelenting pain.
Her attention returned to his face. His jaw set like stone with stoic endurance.
He loathed this. He loathed this to the depths of his being.
Oh, Gideon, I'm so sorry. Forgive me.
She reached out and placed a gentle hand on one powerful arm. He flinched away. Just like he used to. Fear scored her heart. Would tonight hurl him back into his nightmare isolation?
She straightened. She'd set out on this path. For good or ill, she must follow it to the end.
Steeling herself for what she'd see, she slowly stepped behind him. He held himself so still, she couldn't hear his breathing anymore.
His back was long. Leanly muscled. Graceful in its strength.
Marred with scars upon scars upon scars.
How had anyone borne such torture and lived?
Scalding tears stung her eyes, but she forced them back. A sob jammed behind her lips. She must be strong, just as Gideon had been strong.
Her horrified gaze clung to the pattern of cicatrices across his flesh. Every inch of his back carried the mark of violence. His captors must have beaten him again and again. They must have stabbed him and burned him. Her imagination failed as she sought to measure his torment.
With one trembling hand, she touched a thick puckered line that snaked around his ribs. He flinched again, although the wound had long since healed.
"Have you had enough?" he asked cuttingly.
"Oh, Gideon, what did they do to you?" she whispered.
"I warned you."
She traced the scar, feeling where other scars intersected it. The raised flesh under her touch was unnaturally smooth. "I still think you're beautiful," she choked out.
His muscles tensed, then he jerked away from her tentative exploration.
"Do you indeed, sweet Charis?" he snarled, whirling to face her. "What about this?"
With savage swiftness, he ripped the gloves from his hands and flung them to the floor.
## Nineteen
Charis's heart crashed to a halt. At last she saw what Gideon had hidden all this time. She saw and yet could hardly believe it.
She thought viewing the scars on his back had tested the limits of her courage. But this, this went beyond anything she could conceive.
Her appalled gaze clung to the ruined hands he spread out before her as if he taunted her with their shattered elegance. "Oh, Gideon," she whispered, the words lacerating her throat.
"Quite a sight, aren't they? At least they work. After the torture, I wasn't sure they would." His tone stung. He lifted his right hand and held it so close in front of her face that the tangled network of scars blurred. "Do you want these touching your skin? Do you?"
She jerked back, mainly at the corrosive pain in his voice, then made herself stand still and look without flinching. He wanted her to recoil, she knew. He wanted her to confirm he was as repulsive as he believed.
"Don't," she begged. Shaking, she reached out to catch his hand, but he wrenched free to stand in front of the grate.
Apart from hectic streaks of color lining his prominent cheekbones, his face was drawn and gaunt. His mouth was a white gash of anger. His black eyes were brilliant with humiliation and self-loathing.
"Don't touch you?" His bitter laugh made her cringe. "I wouldn't dream of desecrating your body with these claws."
"No..." He'd misunderstood her. Deliberately, she guessed. Her belly clenched in sick misery. She raised unsteady hands to her face and discovered it wet with tears.
He had so much pride. His pride was part of his extraordinary strength. But that also meant he'd hate her to cry over him. She should stop.
If only she could.
He sent her a blistering glare, then stalked toward the door, snatching up his coat on the way. "I've had enough of this. Find some other damned charity case."
"Gideon, please don't go," she forced through a throat thick with churning emotion.
"I'll see you in the morning," he grated out without looking at her. On the hand that clutched his coat, his broken knuckles shone white.
She couldn't let him leave like this, believing she despised him for his injuries. Lunging forward, she grabbed his bare arm with both hands. "No!"
"Let me go, madam," he said stiffly, although at least he curtailed his headlong retreat.
She expected him to shove her away and make his escape. But he stood facing the door, his back to her, quivering as he did in the grip of his affliction.
"Never," she vowed, her voice fracturing. She slid one hand down his arm to cup his poor, damaged hand between hers. "Never, never, never."
Dear Lord, she had to stop crying. She sucked in a broken breath and struggled for control.
He was as taut as a drawn wire. On edge. Furious. Grieving. Likely to lash out at the least provocation. Perhaps she pushed him too far, risked another attack. She muffled a sob and stroked his hand with trembling fingers as if touch alone could mend what could never be mended.
His other hand opened, and the coat dropped to the floor, tacit admission that he wasn't going anywhere. His glossy dark head lowered until his forehead rested against the door.
In the unnatural contours of the hand she held, she felt what his captors had done. The tracery of scars. The spurs and welts. The jagged knitting of the bones. Bones that had been smashed over and over. The knuckles were swollen. The fingernails were jagged and misshapen.
What had happened to him was obscene, unspeakable, barbaric. The damage made her want to scream and claw and fight. But all she could do was cry.
Heaven help me, I need to dam these endless tears.
"Charis, I don't want your pity." His voice was so deep, it was a subterranean growl.
He was wrong about her reaction. Pity was too weak a response to the horrors perpetrated on him. What he'd withstood beggared imagination. She felt like an ax cleaved her heart, and nothing would ever weld it whole again.
"I don't pity you." The words emerged as a choked murmur.
Still Gideon didn't look at her. "I don't believe you."
With a jerky movement, he laid his other hand flat against the dark wood of the door. It had been as tortured as its twin. But staring at his hand against the timber, she saw the grace and beauty it must once have possessed.
"My love..." Curse these tears, nothing stopped them. "I'm so sorry."
Words failed her. What could she say? Nothing was equal to what he'd been through. Instead, she followed where her heart dictated. Holding his shattered hand tenderly between hers, she raised it to her lips.
She placed a fervent kiss on the uneven knuckles. It was an act of homage for all he'd borne. It was an act of overwhelming anguish. It was an act of gratitude that he'd survived so she could fall in love with him.
Under her lips, his flesh was warm. His hands looked like they belonged to a monster. The skin she kissed was unquestionably a man's.
He went utterly still. His shaking quieted. He didn't breathe. He didn't speak. His back was stiff with tension. If his living hand hadn't rested between hers, she'd almost wonder if he'd turned to stone.
In the brittle silence, she finally heard him take a shuddering breath. The hand she held curled into a fist. He drew another of those long, difficult breaths.
"I hate what they did to me." His voice was so low, she strained to hear him. He spoke toward the door. "I hate that I have to live with Rangapindhi forever."
Oh, my dear.
She recognized his shame and pain. Without thinking, she shifted, pressing herself hard against his back with its interlocking network of scars. She turned her hot sticky cheek against his skin, feeling the tight muscles, the lines of raised flesh.
His shoulders bent forward. He was so rigid, it was as if he kept himself upright through will alone. Compassion, all the more poignant because she couldn't express it, stabbed her. She waited in painful suspense for him to push her away, berate her, walk out. But he didn't move.
Without releasing the hand she'd kissed, she raised her other hand to cover his where it flattened against the door. He jerked infinitesimally under her touch, then subsided into stillness. She tried to infuse him with every ounce of her love. Physically. Through the human warmth he'd thought denied to him forever.
She didn't know how long she sprawled against him in wordless communion. She closed her eyes and let darkness take her.
After a long while, she felt him shift. She opened her eyes and straightened.
Finally, he turned to face her, forcing her to free one of his hands although she kept her clasp on the other. She steeled herself to look into his face as fear sent icy tendrils along her spine.
What would his expression reveal? Anger? Disdain? Coldness, as he rebuilt barriers of pride and detachment that had crumbled tonight?
His face was stark with some deep emotion she couldn't identify. She stared into his burning eyes.
"Charis..."
He looked as though he'd lost his soul. The stony desolation in his eyes cut her to the quick.
"It's all right, my love." She curled her arms around him, anything to assuage his cruel isolation. His muscles tensed as he resisted her. She tightened her embrace. "It's over. It's over."
For a long moment, Gideon stood unresponsive, unmoving. Then she felt him tense. Was he finally going to spurn her? She was astonished he'd endured her touch as long as he had. She was astonished he'd revealed his scars and his suffering. However he treated her now, the bond between them had become unbreakable.
Which wouldn't ease her hurt if he rebuffed her after all they'd shared in the last half hour.
He made a choked sound deep in his throat. She felt his chest expand as he sucked in a massive breath.
"Oh, dear God in heaven," he forced out in a cracked groan.
Shaking, he lashed his arms around her and tugged her roughly into his chest. His shoulders heaved convulsively as he buried his face in her neck. She felt the heat of his breath, the bruising power of his arms, the frantic race of his heart.
"I want to give you peace," she whispered into his thick dark hair. Painful tears welled again. She loved him so much, it was agony.
"You have. You do," he said urgently, but the hands that clutched her so hard spoke of desperation, not rest.
This wasn't peace. Perhaps peace and he were such strangers, he no longer recognized it. "Oh, Gideon, I wish that were so," she said sadly.
He held her so close, he crushed her breasts against his chest. She drew a shallow breath, all his stranglehold allowed her. His head was heavy on her shoulder. His hair tickled her neck the way it had after he'd shared her bed for the first time.
"Whenever I look at my hands, it all comes back." His voice was thick, hesitant as he spoke into her skin. "The stink. The heat. The cold. The hunger and thirst. The unending pain."
With a hand that trembled with horror at all he'd suffered, she stroked his disheveled hair. The caress seemed so natural. How curious to think before this morning she couldn't have made it. Just as only a day ago it would have been unthinkable to cradle him in her arms and infuse his cold loneliness with her love.
So much had changed since they'd left Penrhyn.
"I don't know how you endured it," she said softly.
He tautened, and the muscles across his back became as unrelenting as steel. "I didn't endure. Before they finished with me, I screamed for mercy."
He was so hard on himself. If only he could spare some of the generosity he'd shown her to stanch his own wounds. "You didn't betray your comrades or your country," she said in a quiet but implacable voice. "You stood up to over a year of torture and didn't break. You're too brave for your own good."
"You wouldn't think that if you'd seen the pathetic fool I made of myself when they started on my hands." He rubbed his head against her neck in a desultory caress. The unforced gesture sent warmth spiraling through her. She could hardly comprehend he trusted her enough to stay in her arms.
"Oh, my love," she said in a low voice throbbing with emotion. She ran her hand in comforting trails over his powerful back. Under her hand, his scars created a bumpy tapestry, a map of the intolerable tribute his years in India had claimed. She couldn't see his ruined hands. She didn't need to. The sight would haunt her forever.
"You have to forgive yourself, or you'll go mad. Good Lord, Gideon. You're covered in scars. You hardly sleep. You flinch if anyone comes within reach." Her voice softened into persuasion. "You gave all anyone could ask. More. Much more. Everybody in the world sees that but you."
Charis turned her head and glanced a kiss across his cheek. The poignant tenderness inside her demanded some expression. She felt his breath catch. She suspected acts of uncomplicated affection had been rare in his life.
Because she ached for his solitude, because it was all too easy to picture a clever little boy happier with his books than any companions, she kissed him again. A glance of the lips that caught him on the rim of one ear.
Again the hitch in his breath. Slowly, he straightened and stared at her with a wariness that pierced her heart. Surely by now he must know she wanted only his good. But he'd been so hurt, he shied away from anything that smacked of love. For all the barriers she'd crashed through in the last days, she didn't fool herself that he was near to accepting he was worthy of her adoration.
The scars of Rangapindhi cut too deep for any simple remedy.
For now, he was with her and showed no signs of wanting to go. She intended to take what advantage she could. Rising on her toes, she kissed one side of his neck, then the other. She still held him, but loosely, easily, without the quaking desperation. He shifted restlessly, and his hands slid to span her waist.
She traced a line of kisses along one sinewy shoulder. Pausing when she reached the top of his arm.
A strangled sound emerged from his throat. She wasn't sure whether it was encouragement or protest. She dropped another kiss on the ball of his shoulder.
The kisses were quick, soft, playful. Like those she'd give a crying child to coax it from a fit of sullens.
Except she knew to the depths of her soul that Gideon was no child.
He was a full-grown man. Potent. Passionate. Predatory.
A thrill shivered through her. More purposefully, she grazed her lips along the vein down the side of his neck. Feeling the powerful life thundering through him.
His breath caught again. Then he shifted and pressed his lips to the collarbone revealed under her red gown's square neckline.
Her heart stuttered. He became a participant at last. She closed her eyes and forced herself to breathe. Her hands settled loosely on his hips.
She pressed a kiss to his other ear.
He kissed her chin, his lips warm and firm.
She brushed her lips over his jaw.
He caught one earlobe in his teeth and bit down gently.
Response burned down to her toes, and a strangled moan escaped her. As he bent to kiss her shoulder, she caught a flash of masculine triumph in his face.
They no longer clung to each other like the survivors of a shipwreck. She'd kissed him to comfort, but somewhere the game had changed into a duel of kisses.
She angled her head and kissed the hammering pulse at the base of his throat. Instinct made her lick him there. His skin was warm and salty. Delicious.
She forgot the playful battle and licked him again. Slowly. Luxuriously. His flavor filled her senses. His low rumbling growl vibrated against her lips.
Dazed, she lifted her head and stared at him. The humor seeped from his expression. Replaced by an intense concentration that sent a sizzle of anticipation down her spine.
The innocent games were over.
Danger hovered.
Danger and passion.
Time stopped. Along with her heart and breath. She felt as though she poised on the brink of one of Penrhyn's craggy cliffs. Would she plummet to her death? Or would he catch her, as he always had?
As slowly as if he waded through deep water, Gideon raised his ruined hands. He set them on either side of her head and tipped her face up.
The moment held untold importance. It was as if he'd never touched her before. The brush of his scarred palms on her cheeks made her shiver with pleasure. He stared at her, flicking back stray tendrils of her hair with his thumbs.
With spine-tingling attention, his gaze traced her features. His black eyes glowed as if he looked upon his soul's desire. Deep in her bones, she finally recognized that he loved her and always would. He didn't want to love her, but he did. Perhaps he'd never say the words again, but the awe and worship on his face crushed any lingering doubts to dust.
A shaky sigh escaped her. His eyes focused on her parted lips. She grew taut with uncontrollable longing. Surely he'd kiss her like he'd kissed her this afternoon. She desperately wanted him to kiss her again. Her hands curled into his hips to urge him closer.
"You're so beautiful, you break my heart," he murmured.
"Gideon..." she choked out.
Any further response to that astonishing declaration was lost as with sudden decisiveness, he lowered his head and pressed his lips to hers.
Gideon felt the moist cushion of her lips flatten against her teeth. Then, sweet moment indeed, she sighed and parted to let him in. He thrust his tongue inside, testing the hard smoothness of her teeth, the hot honey of the interior. His heart raised a paean of rejoicing as her tongue fluttered, retreated, returned to stroke and caress.
She learned fast, his darling wife. Only this afternoon, his deep kiss had shocked her. Only yesterday, he would have been incapable of touching her, let alone sharing this astonishingly sensual kiss.
Every second he held Charis in his arms felt like a miracle.
Very deliberately, he licked the roof of her mouth, the insides of her cheeks, delighting in the contrasting textures. Delighting in her response as her tongue brushed the sensitive underside of his.
She moaned and pressed her mouth harder against his. He sank into hot, succulent blackness.
He lifted his head and rested his forehead against hers. They panted, sharing the small space of air between them. The act felt as intimate as that extraordinary kiss. As if one life united them.
Feverishly, he slid his hand up her rib cage to where her breasts strained against her bodice. He slid his hand under the neckline, found a pebbled nipple, and pulled gently.
"Yes," she sighed, and traced the line of his mouth with her tongue.
Hunger slammed through him. Drowning out all other sound, his blood thundered. He leaned forward and bit her lower lip. She shivered with excitement, her hips jerking against him.
"If you want that dress to stay in one piece, take it off," he said unsteadily.
She gave a breathless gust of laughter. She hooked her hand around his neck and sent him a scorching glance under her eyelashes. "You'll have to help. It laces up the back."
"Damn fool fashion," he grunted.
Her face was flushed with need. Her lips were swollen and red with their frantic kisses. Her eyes were a deep and mysterious green. Tarns in the Penrhyn woods. He moved his thumb against her cheek, feeling the warm smoothness of her skin, the sticky remnants of tears.
She pressed her cheek into his ravaged hand. How quickly he'd accepted that his injuries didn't repulse her. Odd when he'd nearly died of shame revealing them.
He'd intended the searing honesty to break the connection between them, destroy her foolish infatuation at last. Instead, uncovering his secrets forced him to admit he was her slave and always would be.
"I don't want you to see anyone but me," she said huskily. Her voice was thick with the tears she'd shed. He wished he could promise there would be no more sorrow, but even at this joyful moment, he knew that would be a lie.
"I don't." He swallowed to dislodge the painful constriction in his throat. "I won't."
Gideon kissed her again. The desperate urge to possess faded, and his mouth moved with piercing tenderness. He raised his head and looked deep into her eyes. Her spirit shone clear for him to see. Brave. Generous. Honest. So full of love, it left him humbled.
He gently turned her around and began to undo the pretty red dress. Inch by inch, clumsy as a lad with his first woman, he revealed the smooth skin of her back. He pushed apart the edges and traced a line of kisses between her shoulder blades. Her breath faltered, then quickened. She lowered her head. He accepted the unspoken invitation and kissed a path up to her hairline. Her scent was stronger there. Carnations. Warm skin. Woman. Charis.
He buried his nose in the soft mass of hair and breathed deep, drawing her essence into his lungs. Into his heart.
He returned to unhooking her dress. "I'll need all night to get you out of this confounded rag," he growled in frustration, as yet another fiddly attachment refused to cooperate.
"Are you in such a hurry?"
The wench laughed at him. God help him, he liked it. "Yes."
Finally, the hook released. He turned his attention to the next one down. The line stretched endlessly.
She flexed her shoulders, and he fought the urge to bend her over the nearest chair and take her from behind. This morning he'd leaped on her with a passion unlike any he'd ever known. The need to thrust his aching cock into her tonight made this morning's passion seem a mere milk-and-water fancy.
Patience, Trevithick. Calm down. She deserves better than a quick tumble. She deserves every ounce of skill you can muster.
He sucked in a breath and spoke more steadily as he reconsidered his earlier answer. "No. I want to show you everything you've missed."
Another of those voluptuous shivers rippled through her. Dear God, when she did that, he threatened to explode.
He rode the surge of desire and concentrated on the next hook. After spending the last week in rags, he could understand she didn't want to ruin the dress. But if the damned thing didn't come off soon, he'd shred it.
"Show me everything?"
Her overt curiosity made him smile. "Well, everything might require more than one night."
Her quivering sigh was answer in itself.
As though he unveiled something sacred, he slid the dress down her slender body.
Gideon's breath stopped.
She still wore corset, shift, petticoats. The sheer covering did little to hide the glories beneath. His rod throbbed, but he ignored its greedy insistence.
His eyes traveled down her straight spine to the firm bottom, pressing enticingly against the white lawn. With shaking hands, he released the tapes holding her petticoats. They fell with a whisper.
He'd never undressed an Englishwoman. Never dealt with such complicated garments. His Indian lovers had worn the graceful native costume. He had a sudden yen to see Charis in exotic silks.
One day...
He stepped in front of his bride. She was slender and graceful as a young willow. His gaze traced the lovely curves, returned to her breasts, pushed up by the corset to press against the chemise.
She raised her arms with a gesture of such natural sensuality, his heart jammed in his throat. A few deft tugs, and her hair fell in a curtain of shimmering bronze. Her scent filled the air so thickly, he thought he could touch it.
She blushed under his fiery regard. It surprised him how she suddenly became the shy, inexperienced girl.
She was a shy, inexperienced girl.
He must remember that. The unfettered passion in her kisses was deceptive. He turned her around again and tugged at the corset laces. "Infernal contraption."
She laughed softly as he finally found the knack of it. Desperation lent his fingers a deftness they'd lacked earlier. He burned to see her body without all these confounded draperies spoiling his view.
Swiftly, he slipped the corset off and draped it over the chair he'd fantasized bending her across. Cold sweat covered his skin. If he didn't control himself, he wouldn't survive this interminable disrobing. "Why do Englishwomen wear so many clothes?"
"Perhaps to torment Englishmen?" She turned to face him.
"You're wearing nothing for the rest of the week."
She gave a throaty giggle that made his gut clench with desire. "You'll shock the servants."
"The servants can go to Hades." He tugged the delicate shift over her head. With a satisfied gesture, he tossed it to the side, not caring where it landed.
Charis's color mounted, and she raised shaking hands to cover her bosom. He bit back a groan and drew her close for a long, openmouthed kiss. She kissed him back with gratifying enthusiasm, her brief shyness fading.
Again, he reminded himself to be careful, considerate, controlled.
Difficult to remember restraint when her hands ran up and down his back in a wild dance of desire. Or when her mouth clung to his as if she'd die if he stopped kissing her.
Slowly, he slid his hands up to cup her breasts. Her white flesh was exquisite, the nipples firm and dark. He couldn't resist sucking one sweet point into his mouth. She cried out and arched closer.
He licked and suckled, following the broken pattern of her breath to test her arousal. When she moaned and trembled in his arms as though tossed in a storm, only then did he turn his attention to her other nipple.
His patience with her clothing had vanished long ago. With one ruthless movement, he ripped her drawers away. Now nothing separated him from her body. She gasped with shock and tugged at his hair. The fleeting pain only built his arousal.
He continued feasting on her breasts while one hand fell to the curls at the base of her belly. For a moment of delicate suspense, his fingers tangled in the damp softness.
He drew on her puckered nipple and slid his hand between her legs. She moaned, and a shudder ran through her. Her hand curled against the bare skin of his back. Parting her, he explored her folds. He took his time, savoring the delight.
She thrust her hips forward. He twisted his fingers, seeking. He stroked sleek petals.
And found his goal.
Very carefully, he touched her, teasing without initiating climax. Even so, her body tightened in immediate, uncontrollable response. Her soft, guttural cry alerted him to how close she was.
He raised his head from her breasts. More than he wanted to live another day, he wanted to watch her face during her first orgasm. To his shame, she hadn't come close to her peak when he'd taken her before.
By God, she'd come tonight. Over and over. Until neither of them saw straight.
Her head tilted back, her breasts jutted forward, her eyes flickered closed, her lips parted on a raw moan as he touched her again, with greater purpose. He increased the pressure. She shivered, and he felt a sharp sting as her nails dug into his back.
She stiffened on a cry, and he felt her cross the barrier. Sensual pleasure roared through his veins as he watched her find bliss. Her trembling thighs clamped around his hand, her body quivered as though she had a fever. Hot female moisture drenched his fingers. Her heady scent was rich in his nostrils.
She'd never looked more beautiful. He'd remember this sight till the day he died. He'd remember it with gratitude and love.
After a long, shuddering moment, Charis opened misty eyes and stared at him in bewildered astonishment. "Gideon?" Her voice was hoarse and low.
Reluctantly he withdrew his hand. "Are you all right?"
"I...I think so." She sounded more surprised than ecstatic. "What was that?"
He laughed softly. "A taste of what's to come, my darling."
Before she could question him further, he snatched her up for another kiss. After what she'd just experienced, her response was deliriously uninhibited. For the first time, her tongue invaded his mouth. The kiss became aggressive. She strained against him. Her nipples pressed into his chest, her arms encircled his neck, her hips thrust into his with an evocative rhythm.
Need surged like a tidal wave. But unlike this morning, it was need tempered by care. He intended to find joy tonight, but more than that, he intended Charis to find joy.
He swung her into his arms and carried her toward the bedroom. In graceful surrender, she rested her head in the crook of his shoulder.
"It's time, my wife."
He staked a claim he knew he had no right to make. But neither God nor the devil would stop him now. The world had stolen so much from him. It wouldn't steal this.
He kicked open the door. It crashed against the wall. Overwhelmingly conscious of her naked body and the moist brush of her breath against his skin, he strode across to the bed.
He laid his precious burden upon the sheets. He waited for her to cover her breasts or her sex, but she lay motionless, open to his gaze.
Perfect.
Time halted while he drank in her beauty. She still wore stockings and slippers, tied with ribbons around her neat ankles.
"Why are you smiling?"
He hadn't realized he was. "There's no end to an Englishwoman's armor. I'd forgotten your blasted shoes."
To his surprise and delight, she raised one leg and pointed her toes in his direction. He caught a tantalizing glimpse of the dark mysteries between her thighs. The sight made his cock swell and strain with agonizing need. He locked his teeth and fought back the rip of desire. He was going to do this right. And that meant maintaining at least a shred of control.
"Why don't you take them off?" she asked in a sultry voice he'd never heard from her before.
He wasn't giving her everything her own way. He let his smile broaden, become knowing. "Later."
His hands dropped to his waistband, and he roughly tugged his trousers open. Her eyes rounded. She licked her lips. His arousal built another notch, bathing his body in sweat. His heart raced with excitement.
Swiftly, he tugged off the rest of his clothes. He hadn't undressed in front of a woman since well before Rangapindhi. He'd imagined if he ever did, it would be an occasion of embarrassment and concealment.
But the light in Charis's brilliant eyes as she watched him looked like admiration.
How could that be? He was hardly a young girl's dream, with his scars and grotesque hands. But the familiar self-loathing couldn't sink its teeth into him when the woman he loved gazed at him as though he made the sun rise in the sky.
She slid up against the pillows, and her lush lips curved into a breathtaking smile of welcome. Her eyes were clear and burning with light.
She extended one hand toward him. "Come to bed, Gideon."
## Twenty
Gideon's guarded expression as he stared at her outstretched hand harrowed Charis's heart.
He'd been so hurt. Even now, when it couldn't be clearer how she loved and wanted him, he didn't trust his welcome.
The candlelight softened his terrible scarring. Instead, it emphasized the lean, muscled strength, the grace, the height. For all their vile tortures, the Nawab and his thugs couldn't destroy his essential beauty. Her gaze fell to his powerful thighs and the hard, seeking flesh rising between them. His virility filled her with shivery excitement.
"Come to me, my darling," she said on a breath.
His hesitation splintered. He surged onto the bed, bracing himself over her. She strained up and pressed her mouth to his. He groaned and kissed her back, soon taking charge.
His kisses were still so new, so startling in their delight. Perhaps one day, she'd become inured to the magic of his mouth on hers. In about a thousand years. Perhaps.
The rapacious hunger of his lips sent desire rippling through her. Heat trawled along her veins, arrowed between her legs, made her move restlessly against the sheets. Tentatively, she ran her hands down his back. For all the burning need that now scorched the air, he'd withdrawn too often for her to trust no barrier remained.
"Yes, touch me," he groaned against her lips, and kissed her again. "Touch me, Charis."
The yearning in his voice convinced her that at last she could discover his body as she longed to do. As he rained ravenous, ravishing kisses across her face and neck and shoulders, she tightened her hands. She felt the shifting of muscle, the ridges of scarring under her palms. Poignant reminder of how near she'd come to never knowing him, loving him.
Her hands drifted lower, trailed across his firm buttocks. His breath caught on a strangled gasp, and his hips jerked forward, pushing hot hardness into her belly.
Automatically, Charis curved closer. After this morning, she should be used to what happened to her husband's body when he wanted her. But the heat and weight of that part of him pulsing against her made her toes curl with excitement.
"Oh," she gasped, her fingers digging into his buttocks.
"Yes," he hissed, and bit down on her neck.
She gave a delighted shudder, and her nipples tightened with a longing that was like pain. When he'd taken her breast into his mouth, she'd nearly shattered with rapture. She trembled for him to do that again.
As if she spoke her wish aloud, his lips closed over one aching crest. She tangled her hands in his thick black hair, cradling his head against her. As his teeth scraped the sensitive tip, blinding pleasure gripped her. A sharp cry escaped, and she bowed up closer to that sweet torment.
This morning's passion had been unforgettable but over too soon. Tonight, Gideon was stubbornly determined to take his time. With a piquant shiver, Charis remembered him saying he wanted to show her everything.
Everything?
Recollections of that astounding earthquake he'd set off made her dizzy. The world had dissolved into molten ecstasy. She'd never imagined such sensations existed.
Could Gideon do that again? Her heart leaped with anticipation. Good Lord, was there more?
He moved to her other nipple, tonguing it, sucking, biting, so she shook with a tingling mix of pleasure and pain. She wanted him to worship her like this forever. She wanted him to fill her body with his.
"Gideon, don't make me wait," she begged, when desire threatened to incinerate her. She loved what he did, but the roaring need to have him inside her drove her insane. With every touch, frantic craving spiraled higher. She felt lost, empty, needy. "Please."
He raised his head and stared at her, his black eyes glittering. He looked like a pirate. A pirate of pleasure.
"You'll like what happens next." The current of warm laughter in his voice made her blood thicken to honey. "I promise."
His kisses this afternoon had hinted that the darkness that marked their lovemaking so far wasn't the whole story. Now the tormented, angry man who had burned with shame when he revealed his scars turned into a dream lover. Ardent. Commanding. Mindful of her pleasure.
Her heart overflowed with agonizing love, but she smothered the fatal words. Even now, she knew he didn't want declarations, commitments, vows.
"I like all of it," she admitted, praying what they did banished his shadows, even if only for a fleeting moment. "But please hurry."
He laughed softly and trailed a line of kisses between her breasts and across her rib cage. "Never."
She made an impatient sound, a sound transformed into a moan when he nipped the soft skin of her stomach. Her whole body was alive with sensation.
He kissed where he'd bitten her, as if to soothe the sting. But she'd reached such a pitch of arousal that nothing short of possession would satisfy her clamoring need.
"Gideon!" Her protest at his teasing faded on a gasp as he parted her legs with his hands and placed his lips...there.
Horrified shock paralyzed her. Surely this wasn't something a man did to a woman. Surely it couldn't be...
Thought disintegrated to ash as his mouth moved. She felt moisture, heat, suction. The soft friction of his hair between her thighs. The scrape of stubble on tender skin.
His mouth was hot. So hot.
Dear God, was that his tongue? There? She should pull away, demand he stop. Her shaking hands formed talons in the sheets on either side of her. No virtuous woman would suffer such unnatural attentions without protest. Was this some obscure Indian perversion?
She must insist he stopped.
Not yet.
The sheer strangeness of what he did held her unmoving under his predations. And now her astonishment was tempered by curiosity, even a curl of something that might have been pleasure.
No! She couldn't enjoy this bizarre act. It was too far outside anything she'd thought possible. Feverishly, she writhed under his mouth, trying to escape, but his hands on her hips were implacable.
"Gideon, stop," she said in a choked voice, as a fugitive bolt of response sizzled through her.
With reluctance she could feel, he withdrew from the astoundingly sensitive place he'd found. He dropped a kiss on the quivering flesh of her thigh.
At last he heeded her. That couldn't be disappointment spearing through her, could it? If it was, she was as depraved as he.
"Shh," he whispered, without looking up. No, he was too busy looking at her...down there. The realization should disgust her, but instead a forbidden thrill scorched its way down her spine. "Trust me, Charis."
He didn't wait for her answer. She wasn't even sure she could muster a reply. Instead, he lowered his head once more. His tongue flicked more purposefully.
Charis's belly clenched with hot longing as he drew powerfully on that place. A warm melting sensation flooded her, and, to her embarrassment, she felt moisture well between her thighs. She tried to close her legs, but somehow that just trapped Gideon more tightly against her.
He didn't seem to mind her body's uncontrollable reaction although he must taste her body's juices against his seeking lips. Such intimacy was frightening, so much more alien than the times he'd thrust himself inside her. But she couldn't summon the will to move away, to demand he leave her alone.
A satisfied growl emerged from deep in his throat as he licked her. It was the most salacious sound she'd ever heard. Her throat closed on another protest as heat coiled deep inside her.
After what happened earlier, she had some inkling where this led. But how could he do that to her using just his mouth? It seemed extraordinary, outlandish, impossible. Still, the tension built. Dear heaven, he must cease this indecency before she exploded into a million pieces. With shaking hands, she reached down to push him away. But somehow her fingers ended up tangling in his damp, disheveled hair.
His mouth played endlessly against her, setting off explosions of reaction. She closed her eyes and hoped she'd survive this sinful pleasure. Broken moans emerged from her lips. Sounds she didn't recognize as hers.
As tension twisted tighter and tighter, she undulated toward his mouth. She no longer pretended she wanted him to stop. If he stopped, she'd die of frustration, sizzle away to nothing. A string of rhythmic sighs escaped her, and she gripped his head as if she'd fly into space if she released him.
At the point where she thought he'd either destroy her or fling her into a new heaven, he kissed her hard and long between her legs. White light erupted behind her eyes, and a jagged scream burst free.
The world retreated on a wave of hot delight. Her hands clenched hard in his hair, and she shivered wildly. It was devilish, wrong, profane, what he did. But she'd never felt pleasure like it.
Vaguely through her quivering daze, she felt him shift. Then the sweet benediction of his lips kissing her belly.
"You..." She paused and cleared her throat. After that life-altering experience, it was so difficult to speak. "You're wicked."
He laughed softly and rolled to the side, raising himself on one elbow so he could see her. "I'm sure you forgive me."
He cast one devastating glance down her naked body, and his lips, glistening and reddened with what he'd just done, curved upward in a sensual smile. She felt every sinew tighten with a renewal of arousal. In spite of the ecstasy he'd just given her, she burned for that most intimate of connections.
The ghost of his smile remained as he lifted a lock of her hair and toyed with it. Still trembling, she watched him. The sight of his broken hands made her ache with compassion. He deserved a lifetime of happiness for what he'd suffered.
She reached out to touch his face, needing to convey how she loved him. It still amazed her that she could make such gestures without him flinching away in horror. She trailed her fingers down his cheek, feeling the faint roughness of his beard.
Gideon stared at her with a savage concentration that stirred excitement low in her belly. Beneath his humor, his face was taut with barely-reined-in desire. A shy glance confirmed he was still hard and ready.
"I might forgive you," she said huskily, brushing his ruffled hair back from his forehead. To think he'd believed he was denied human touch forever. After what she'd discovered tonight, that fate seemed cruel beyond imagination.
His eyes didn't waver, and his voice became considering. "One day soon, I hope you'll do the same for me."
"What..." Shock made her lift her hand and sit up. Shock and wanton, uncontrollable curiosity.
Abruptly, the quiet moment of tenderness shattered.
"Next time," he said, ruthlessly pushing her back onto the mattress.
His face was intent, flushed with hunger. Heat sparked along her veins, and she waited in an agony of suspense for what came next. Her heart galloped with anticipation. She couldn't mistake how close to the edge he was. His desperation fed hers.
He reared over her, big, powerful, dominating. She raised her knees to frame his narrow hips. He kissed her with a fierceness that made her arch against him. His hardness pressed into her stomach. Soon, all that masculine power would be inside her.
She could hardly wait.
"Hold on to my shoulders," he said in a thick voice.
Wordlessly, she obeyed. His muscles tensed. His shoulders were as unrelenting as stone, as sleek as satin under her clinging hands.
Her heart thundering, she prepared to be conquered as he'd conquered her this morning. Only to discover he meant to woo her to rapture.
He tilted his hips forward. She felt sliding heat against her moist cleft. Then slow, delicious pressure.
At last...
She caught her breath and flexed her fingers against his damp skin. He breathed harshly, and a lock of hair tumbled over his forehead. The veins in his arms stood out in stark relief. She couldn't doubt control cost every ounce of concentration.
There was something viscerally satisfying about being the object of that unwavering focus. Last time they'd made love, he'd been lost to passion. That had been exciting. This was deeper, purer, sweeter.
He penetrated farther. She gasped as still-tender flesh stretched to accommodate him.
Of course he heard her. He paused in his tormentingly gradual invasion.
"Am I hurting you?" he asked in a gravelly voice.
She raised her head and kissed him briefly, then returned for a longer foray. This morning she'd felt desired, and the knowledge had been heady. Tonight, she felt cherished.
"Charis?"
"Don't stop," she whispered, and dug her fingers harder into his shoulders.
He pushed forward. The feeling was strange. Uncomfortable. Exciting. With every incremental incursion, he took over her body. With her body, her soul.
She moaned and moved to ease the blinding pressure. He went deeper. She gasped for air as if she drowned. His shoulders under her hands were slick and his face was gaunt, the skin stretched over his bones.
The ferocious expression in his black eyes as they burned into hers should frighten her. Instead, it made her belly knot with arousal. He edged farther. Paused.
Had he reached his limit? She braced against him. Then something inside her relaxed, opened with sudden welcome.
The tension drained from his back. His muscles bunched again. And. with a deep groan, he seated himself fully.
The sensation was extraordinary. Indescribable. It was like her love took on solid expression, breath, life.
Time stretched to eternity before her. For all their previous lovemaking, she'd never felt so close to another person. The dark intimacy was all-encompassing.
His breathing was ragged. The eyes that met hers were glazed with passion. He withdrew in one smooth movement and plunged back. Heat streaked through her, brilliant as lightning.
As he changed the angle, her passage closed hard around him. The fresh sensation made her jerk and tremble. Twice tonight he'd shown her bliss. But what he did now was prelude to something infinitely more powerful.
He thrust again, and she clenched in barbaric possession. Every parting and joining jolted her body. Vaguely through the gathering storm, she felt him dip one hand between them. Then the press of his palm at the juncture of her legs, the flick of his fingers.
Her eyes fluttered shut as fire engulfed her world.
Consuming flame rushed along her veins. Every muscle tightened to an agony that was the greatest pleasure she'd ever known. Nothing existed except the roaring inferno.
Through it all, one thing alone anchored her. The man who thundered into her as she convulsed with ecstasy. She held on to Gideon while unshakable love filled her. Indelibly part of the shining bliss, but somehow separate, immortal, and immovable like the sun.
Gideon was her sun. Her moon. Her sky. He created her anew in the fiery kiln of his passion.
Love flooded every particle of her body with liquid gold. The connection was transcendent, eternal, unbreakable.
Gradually, her shuddering calmed. Reality slowly returned. The earth became the place she lived in instead of a mere memory to someone lost in the stars. But radiance lingered like the last glow of light on the horizon after a perfect summer day.
Opening dazed eyes, she found Gideon staring down at her in wonder. His lips tilted in a smile that, did he but know it, told her he loved her as she loved him.
She was exhausted, spent, lost in languorous joy. Her joints were so loose, she felt like a rag doll. He could pound her into oblivion now, and she wouldn't utter a squeak of protest.
He looked equally strained. She suddenly realized he hadn't yet found relief. Shocked, she made an almighty effort and reached out to touch his chest. One glancing contact, and her hand flopped back to her side. Her muscles had the resistance of woolen stuffing.
She quivered from the extraordinary conflagration that had swept her up into blissful oblivion. She'd had no idea. Nothing in her life had prepared her for those transforming moments in his arms.
Gideon released a shaky breath and began to shift in and out of her body in a leisurely glide. The soft friction soothed. He bent to lick the peak of one breast.
"I can't..." she protested on a dying whisper.
"I know," he soothed, drawing her nipple between his lips.
Sluggish response trickled down to where his body joined hers. He slid out and, with excruciating slowness, slid back in. He bit down softly on the pebbled peak.
She sighed, and this time when he moved, she rose to meet him. Immediately she felt new heat. The joining was deep and essential. An expression of love as much as desire.
Again, she stifled words he didn't want to hear. But her every heartbeat declared she loved him.
Instinctively, she rolled her hips, testing the sensation. He groaned against her breast and released her nipple to trace kisses across her collarbone. She laced her fingers in his thick hair, then gasped as he dragged her up to sit before him, still joined flesh to flesh.
He gazed into her face with the fixed concentration that was now familiar. A premonitory thrill shook her, made interior muscles tighten. His hands were ruthless, demanding, as he grabbed her hips and began to slide her up and down.
Her fog of satiated exhaustion, so overwhelming only minutes ago, vanished in an electric instant. Her legs automatically curled around him. She gripped his arms for balance. Desire bloomed in her belly.
She soon caught the rhythm. With a velvety laugh, he let her have her way. He leaned back on his hands, yielding to the summons of her pleasure.
The pleasure was astonishing.
Until now, it had never occurred to her that she could control what happened between them. The power, the delight made her head swim. She arched back, relishing the push and release inside her. He made her feel like a goddess. He made her feel like a woman in love.
Her breath escaped in broken gasps. She edged closer to another of those extraordinary peaks. Closer but not there. She sobbed and twisted, struggling to reach what she wanted.
"Not yet, my darling," he whispered. He rolled and pressed her down into the mattress. She cried out and twined her legs around his hips. Still, he moved in and out of her body. Relentless as the tide.
His bare, scarred skin was hot, slippery under her clinging hands. He groaned and trembled.
For her.
That knowledge shuddered through her like a cannon blast. With every thrust, he stole another inch of her soul.
Except he'd owned her soul from the first.
Soon, his movements became faster, wilder, less controlled. His chest heaved as he fought for breath.
Charis lost contact with everything but the hard male body that ruled her. Tension wound around her until it was unendurable.
Tighter and tighter.
Still he plunged in and out. Her fingers dug into his arms so deeply that they hurt.
Blackness pressed down. Surely it must end. Yet still the need built higher, red-hot copper wires that stretched every sinew.
She gasped for air. Her lungs ceased to function. Blindly, she pressed toward Gideon, the source of her agony, her only hope of release.
This was stronger, deeper, more overwhelming than what had happened before.
Could she survive this?
Her breath escaped in a long, tortured moan.
"Please, Gideon, please..."
Did he reply? Her ears were deaf to all but the endless clamor of desire.
Still, he moved inside her.
She whimpered and bit down on her lip until she tasted blood. She closed her eyes, helplessly seeking refuge in hot darkness.
The pressure climbed higher. She felt like she burst out of her skin. She felt like her bones crumbled to dust. She felt as if the world must end any second. In a blinding, endless cataclysm.
The unceasing friction between her legs was maddening. She shifted, seeking freedom from torment.
She opened blurred eyes and saw his face change.
"Now," he growled.
One mighty thrust, and the ceaseless anticipation peaked, broke, scattered in shards of flaming rapture. She dived through darkness into a landscape of shimmering light.
Her body spasmed in an unbearable wave. She screamed as pleasure threatened to crush her. Vaguely, through her crisis, she was aware of Gideon's groaning release.
For an eternity, she remained suspended, stretched on a rack of infinite joy. Doused with unearthly pleasure that made her cry aloud in delight. Her nails dug hard into his back as she clung to him like a rock in a sea of fire.
Dazed, changed, amazed, she returned to the world to find Gideon kissing her face, her neck, her shoulders. Tenderly. Sweetly. The sweetness all the more poignant for the violent storm they'd passed through.
"You're so beautiful," he whispered, glancing kisses across her cheeks and nose and forehead. Kisses like the enchanting ones he'd given her this afternoon.
For a long moment, she lay acquiescent. Her chest heaved as she sucked great breaths into empty lungs. Her muscles trembled with reaction.
Then she lifted her face toward him for more kisses. The innocence pleased her, warmed her to the heart. Even as his body rested in hers, and she quaked after that earth-shattering climax.
Gideon rolled to the side, taking her with him, still kissing her. Softly, he separated from her. She stifled a whimper of discomfort. After the uninhibited passion, she ached. Every muscle was tired, tested to its limit.
She'd never felt so good.
"Are you all right?" he murmured against her neck. His arms encircled her loosely, and his hands rubbed lazy circles on her bare back.
How could she describe the wonders she'd discovered? Words were inadequate to convey the glory. When he raised his head, she kissed his mouth very gently, trying to say with action what speech could not.
Oh, such joy to touch him freely, naturally.
His mouth moved upon hers. The kiss was warm, comforting. A thousand miles from his turbulent lovemaking.
Slowly, reluctantly, she drew away. "It was beyond anything I imagined."
His fingers combed through her tangled hair. Like every gesture in this glowing aftermath, his touch was tender.
His expression became serious, and his hand cupped her face. For a long moment, he stared into her eyes. "It's not always like this, Charis." He paused, and she saw him swallow. "I've never known anything to match what we just shared."
She blinked away tears. Her heart was so full, she thought it must burst. "I'm glad," she said in a thick voice. "I want you to be mine forever."
A shadow crossed his face, and with it, a hairline crack appeared on the shining surface of Charis's satisfaction. "Let's not tempt fate."
He bent to nuzzle her neck. Heat flared as he bit down on a sensitive nerve running up from her shoulder. She closed her eyes in willing surrender. But even as need surged, his answer troubled her heart.
## Twenty-one
It's midnight," Gideon said softly, his breath ruffling the hair on Charis's crown and disturbing her from her warm half doze.
They shared the settle before the blazing parlor fire. She curled into him, one arm loosely around his waist, one resting across his chest. Her palm lay flat over his heart. She loved to feel its steady beating, as though she connected directly to his life force.
"Do you want to go to bed?" she asked huskily, rubbing her cheek against his shoulder. This physical closeness still seemed a precious miracle. She never took it for granted.
His sleepy laugh was a deep rumble under her hand, and the arm he draped around her tightened. "I always want to go to bed."
After so many days of unbridled debauchery, a girl should lose the ability to blush. Nonetheless, heat rose in her cheeks. "You're insatiable."
"At least where you're concerned." He raised her hand from his chest and brushed a kiss across her knuckles. She couldn't help shivering with response.
Over recent days, she'd watched the strain fade from Gideon's features. He looked younger, less haunted. Perhaps because when not making passionate love to her, he'd slept. Deep, undisturbed rest that she guessed he hadn't enjoyed for years. His dangerous life had worn him down long before he fell into the Nawab's clutches.
But while he smiled more often and more willingly, shadows lingered in his eyes. With a pang of sorrow, she realized they probably always would.
Since the night Gideon had shown her the pleasure a man and woman could find together, they'd rarely left their rooms. Sometimes Charis forgot the outside world with its demands and dangers existed. There had been no sign of Felix or Hubert and no warning about trouble at Penrhyn. The hotel servants tidied and brought meals or bathwater. The rest of her modish new wardrobe arrived. Gideon summoned a notary and set out legal safeguards against her stepbrothers. Her fortune was now officially his, at least until the end of June, when it reverted to her.
She'd hoped the change in Gideon would extend to an easier relationship with humanity. So far, she had witnessed no such merciful amnesty. To her sorrow, Gideon's immediate tension was visible whenever strangers set foot in this private kingdom. Her brief optimism that she'd found a remedy to his affliction faded further every time she saw him pale and recoil from other people.
He wasn't cured. Not by a long way. She fervently thanked heaven every day that he could touch her. But so far, his recovery advanced no further than that.
She knew when she looked into his eyes that he believed it never would.
That wasn't the only trouble nicking at the skein of sensual delight entwining her. For all its myriad pleasures, her new life was hollow at its center. The unspoken pain bit most at moments of purest happiness. Like now.
Gideon told her she was beautiful. He told her how much he wanted her. She had no doubts he desired her with endless hunger. But even when she felt they united into one being, words of love never escaped her husband's lips. She knew him well enough to interpret his silence as deliberate.
Nor did he mention his plans for when they left Jersey. It was as though these weeks they shared now existed outside time.
Coward that she was, she let him get away with avoiding the subject. She'd exhausted her store of courage standing up to him after their marriage. Now she was terrified that too many awkward questions would shatter their delicate bliss. Perhaps because with every day, the threat of leaving him plowed deeper furrows in her heart. She couldn't bear to hear him say he still meant them to separate. Although his silence on the matter indicated he hadn't relinquished his original scheme.
Her arm firmed around his waist as she laid an unspoken claim, defying his right to forsake her. But the words insisting he tell her what he intended crammed in her throat.
"Charis, it's midnight," he said with greater emphasis, then glanced at the clock. "Five past."
His unusual obsession with the time pierced her troubled reflections. She looked up in puzzlement. "Is that important?"
He kissed her quickly on the mouth. "You've lost track of the days, haven't you?"
"Lost track..." Perplexed, she blinked at him. Hard to marshal coherent thought when his kisses sent her spinning into dazzling Elysium.
His lips curved in the tender smile that always made her poor adoring heart somersault. "It's the first of March. Happy birthday, my darling."
Her birthday...
Stiffening, she drew away. She forced her befuddled mind to calculate back. So difficult to count paradise in minutes and hours. She'd barely been aware whether it was day or night. Gideon lit her life like the sun. She needed no other fire in her heavens.
"You have possession of your fortune." She couldn't define his tone. He didn't sound particularly triumphant. He kissed her again, more gently this time. "We won, Charis."
They'd vanquished her stepbrothers. She was safe. Relief filtered through her. And fear that now the threat passed, everything would change between her and Gideon.
She forced herself to speak though she knew he wouldn't want to hear what she said. "Because of you." She swallowed and continued in a voice that vibrated with emotion. "I owe you everything."
"I don't want your gratitude." His expression hardened, and he sat up. His arm slid away from her. Worse, his emotional withdrawal was unmistakable as frost in the air.
"Well, you've got it. Forever." She mustered the courage that lately had been so sadly lacking. The murky currents swirling beneath the bright surface would no longer be denied. Her tone developed an edge. "I can be grateful to you and love you. The two aren't mutually exclusive."
She hadn't mentioned love since the morning he'd surrendered to lust and leaped on her. Always, even at the peak of sexual pleasure when her whole world was Gideon, she'd bitten back the words. His silence had fed hers.
Her wisdom in restraining any declaration became abundantly clear. He surged to his feet and regarded her with the wary expression she'd hoped never to see again. The hollow in her heart resonated as if a huge mourning bell clanged inside it.
"Charis, it's our last night on Jersey," he said somberly, ignoring her challenge. Although his guarded eyes told her he'd heard her. "Tomorrow we sail for Penrhyn,"
No, no, no, no, no.
"We're leaving?" Her question rang with dismay.
Could the tenuous bond she'd established with him outlast a return to daily life? Here she was the center of his existence. She wasn't vain enough to expect that to continue forever. But she needed longer to make him completely hers.
Did he even intend to keep her with him?
Grim foreboding swamped her. Was this her ration of joy, these few glorious days on Jersey?
Reluctant amusement quirked his lips. "We have to go at some point, you know."
Blindly, she lurched up and turned away, fisting her shaking hands in her skirts. His attempt at lightness grated, hurt. He treated her like an easily distracted child. "Not yet."
She heard him approach, then his hand curved around her arm. She felt the roughness of his scars against her bare skin. His touch reminded her of his suffering and how far he'd come since they'd married.
Had he come far enough?
His voice was warm, encouraging. "There's no need to be frightened. You've reached your majority. The Farrells can't harm you. We're free."
He misunderstood her reaction. Of course the threat of Felix and Hubert had darkened her days. But more important by far was her endless battle for a future with Gideon.
"We're not free. We're married," Charis said in a muffled voice, bending her head.
He released her with an abrupt gesture and stepped away. She felt the distance like the blow of an ax. "If I could have devised another way to save you, I wouldn't have forced you into such drastic action," he said curtly.
The sweet concord of minutes ago was only a bitter memory. The suddenness of the change left her staggering in its wake. She turned to face him, knowing her pain was naked in her face. "You know I'm always grateful for..."
"Enough!" One ruined hand sliced the tense air. "If I hear the word grateful once more, I won't be responsible for the consequences."
"But, Gideon..."
"Devil take you, Charis, stop!" He paused, visibly fighting for composure. Bitterness frayed his voice, and his shoulders were ruler straight with tension. "Really, you shouldn't thank me. As it's turned out, our marriage was precipitate. Your stepbrothers haven't traced us. We didn't need to take such permanent measures. I can only offer my profoundest regrets."
The sharp slap resounded like the report of a bullet.
Gideon's head whipped back, and his expression registered shock rather than anger. The red imprint of her hand darkened his cheek.
The grim, echoing silence extended. And extended.
Shaking, Charis lowered her arm and backed away on unsteady legs. She wasn't frightened. She was so furious, her vision turned black.
"How dare you?" Her voice lowered to trembling vehemence. "You've had me in your bed. You've been so deep inside me, you've touched my soul. Yet you have the gall to talk about regret?"
"What I've done to you is unforgivable," he said harshly. As shock receded, rage lit his black eyes. "And yes, I do regret that I've hurt you."
Her fragile happiness shattered around her with a sharp crack that sounded like a heart breaking. Her lips felt stiff as she voiced her worst fears. "You can't mean to follow your original plan, that we should lead separate lives?"
His jaw set like stone. "The basic difficulties remain. It still seems the best solution."
Agony stabbed her, stole her breath, made her stumble back a step. She felt betrayed, devastated, lost. Somewhere, she found strength to speak. "Is that what you want?"
"It doesn't matter what I want. I'm trying to do what's best for you."
She clenched her fists at her sides. Either that or batter at him like a madwoman. She loved him more than her life. And at this moment, if one of his pistols had been in reach, she'd happily have put a bullet through his thick skull. "So these last days mean nothing? You can't expect me to believe that. You've found happiness in my arms, Gideon. Don't ever lie about that."
The skin on his face tightened. She braced to hear him say the words that turned her dream of love into a travesty.
His throat worked as he swallowed and he avoided her gaze. "I should never have touched you. It was wrong. It was cruel. The fact that I can't stay away from you is no excuse. It's only an indictment of my own damnable weakness. You should curse me with your every breath. One day you will. Even if we take the sensible course and part now."
He blamed himself for what happened but couldn't deny the bond between them. She should find that reassuring, but she knew how obstinate he was. Obstinacy had kept him alive in India. How tragic that obstinacy now made him surrender his chance of happiness. And hers. He tried to do the right thing, the noble thing, but all he did was condemn them both to a lifetime of loneliness.
Charis had prayed love would wash away the poison of Rangapindhi. She saw now her prayers hadn't been answered.
Her voice rang with resentment. "You're such a fool, Gideon."
"One of us has to keep a clear head without getting lost in the romance of it all," he said with wounding sarcasm.
He wanted her to let him go to perdition in peace. Well, he'd picked the wrong wife if he expected her consent to that. Still, only the knowledge that he loved her, however much he wished he didn't, kept her fighting. This battle was dangerous—it could destroy both of them.
Her nails dug deep into her palms, the slight sting nothing compared to the way he lacerated her heart with his stubborn rejection. He was the cleverest man she knew. And when it came to her, the stupidest. "We desire each other."
She saw him consider sidestepping the statement. After these days of passion, she knew him so well. Why didn't he know her in return?
Something in her face must have convinced him evading the issue wasn't an option. His lips lengthened in a grim smile. "Yes, there is desire. Enough to set the world on fire. But desire isn't enough."
As her false paradise disintegrated around her, she stopped lying to him and to herself. "And there's love. I love you, and you love me. You told me once."
A compressed line of guilt and sorrow replaced his smile. "I had no right to say that. I hoped you'd forgotten."
In a different universe, she would have laughed. Forgotten? Those words were permanently carved on her heart, even if he never said them again. "No chance."
He looked ill and tired and tense. He looked like a man contemplating the end of the world. "I've wronged you so deeply, I can never make recompense."
Her temper spiked. "How have you wronged me? By showing me a man can be more than a selfish brute? By saving me from rape? By teaching me about ecstasy?"
He was so pale, the mark on his cheek where she'd hit him stood out like a beacon. "By making you believe we could have a life together. By coming to your bed night after night when every principle dictated that I stay away. By tying you with bonds of gratitude..." He spat out the word like a curse. "...you'll never break, even when you realize what you feel now is illusion."
She flinched. Surely he didn't still think her love was sickly hero worship? Not after all they'd shared. The accusation hurt more than acid flung in her face.
She drew a shaky breath and reminded herself that he loved her, hard as it was to believe when she confronted his anger and derision. She fought for her life here. She couldn't let him defeat her.
"I forget you're so much older and wiser than I." Gideon wasn't the only one with sarcasm in his arsenal.
His expression closed. Once, she'd have retreated from his bristling hauteur. But she'd held him gasping with release too often for the mask of control to dupe her. He wasn't controlled. He was anguished and angry and desperate.
"After Rangapindhi, I feel a thousand years old." He spoke sadly, so sadly her heart clenched.
Pity almost made her step down. Almost.
"Gideon, I don't discount what happened to you." Her voice became less strident. "I don't blind myself to what your ordeal cost you. Still costs you. That doesn't mean your decisions are always correct. Right now, you're disastrously wrong."
"You force me to be frank." A muscle jerked spasmodically in his cheek. He turned and prowled toward the window, where he curled one hand in the curtains. "Let me lay out some facts. If you can bear to contemplate mundane reality."
"I'm more aware of facts than you are," she said through tight lips. His mockery stung. "But pray, dazzle me. I wait in humble anticipation."
Even in profile, she didn't miss the way his mouth flattened with annoyance. "Very well," he bit out, every word as precisely cut as a diamond. And just as sharp. "I'm going back to Penrhyn to an arduous, frugal future. Isolated. Lonely. You are the kingdom's greatest heiress. I'm physically and emotionally incapable of offering you the life you deserve."
Disbelief rose to choke her. "You reject me because you're worried I'll pine for the occasional party?" Her voice began to shake. "You truly believe I'm irreparably shallow, don't you?"
He ran his hand through his hair, mussing it to wildness, and whirled to confront her. "Damn it, Charis!"
He sucked in an audible breath as he struggled for control. "I'm a freak, a poltroon, one step off being a lunatic. I can't bear people around me, touching me. You know my affliction. In spite of my insatiable hunger for you, you know essentially I haven't changed. Why can't you see what you want is impossible?"
Stepping closer, she replied with matching heat. "Because of that insatiable hunger. Because you can bear my touch. Because I don't care about other people. I only care about you."
"You say that now. How will you feel in twenty years when you've wasted your youth on a man who only exists in your imagination?"
She couldn't doubt his sincerity. No matter how mistaken he was. She made an angry sound in her throat. "And if I'm pregnant?"
He'd been pale. Now he went stark white. His eyes sparked like burning coals. "Don't you want to bear my child?"
"I want it more than I can say." Almost as much as she wanted to stake her place in his closed heart. Strange to recognize that need so powerfully and so immediately. She placed a trembling hand on her belly. Could a new life already grow inside her? The idea was overwhelming. Frightening. Exciting.
Gideon's blazing eyes fastened on her gesture, and a savage expression crossed his face. "Dear God, are you pregnant?"
Was she? With all that had happened, she'd lost count of the days. And she'd been so focused on Gideon, she'd hardly considered consequences. "It's too early to say. Do you still mean to send me away if I carry your child?"
He looked like he reeled at the prospect of fatherhood. "I don't know."
An ounce of her earlier sarcasm crept into her voice. "Why are you so shocked? The natural result of what we've done for the last two weeks is a baby. Surely you gave some thought to the matter."
He slumped against the wall, his face ravaged with despair. "Yes." He hesitated and shook his head with bleak incomprehension. "No."
There was a charged silence, then he continued in a dull voice. "Of course I knew I took risks. If I thought beyond how much I wanted you, it was to say we'd deal with any complications when the time came."
She twined her arms around herself as ice congealed in her blood. Her momentary hope shrank to a cold kernel the size of a pebble. "Risks? Complications? Don't you want a baby?"
He tensed. "If I'm not fit to be a husband, I'm certainly not fit to be a father. If we have a child, it..." He must have interpreted her expression correctly because he paused. "...he or she must go with you."
She raised her chin although she was so deathly tired of battling him. He loved her, she reminded herself. But the words lost their power with every repetition. "Why does anyone have to go anywhere?"
"Aren't you listening?"
"All I've heard is a lot of nonsense." She turned away and stalked toward the bedroom. She was disheartened, angry, exhausted. Trying to get Gideon to see sense was like flinging herself over and over against a mountain.
For one electric moment, she'd wondered if she'd shaken his certainty. She hadn't mistaken what she'd seen in his face when he asked if she was with child. He'd been furious with himself. And her.
But she'd seen more in his ferocious black gaze.
She'd seen longing.
He wasn't nearly as implacably set upon his desolate future as he wanted her to think. If she had his baby, he wouldn't desert her. She knew that in her bones.
Dear God, let me be pregnant.
As she reached the doorway, he spoke in a grave voice. When she turned to face him, he looked weary and curiously defeated, although he'd withstood her every attack. "I know you believe I'm cruel and capricious and pigheaded. But I swear I'm acting in your best interests."
"I wish you'd think of yourself for once. Ask yourself what you want and seize it." Blinking back acrid, painful tears, she left him alone.
## Twenty-two
Gideon turned the hired gig onto the lonely road that snaked across the moor to Penrhyn. At his side, Charis remained bundled away from him in her new blue pelisse and matching bonnet.
She'd been broodingly quiet since before they'd left Jersey yesterday. On the storm-tossed boat that finally reached the mainland south of Penrhyn this morning. During this jolting carriage ride in a shabby, ill-sprung vehicle over potholed roads.
It was well into the afternoon, and still she remained locked away as securely as if a wall of bricks and mortar separated them. She'd rebuffed his stilted attempts at conversation, seemingly content to stare at the rough countryside.
She'd never been a chatterer. Her ability to maintain a restful silence was one of the many things he admired.
This silence wasn't restful. It seethed. With every mile, the tension twisted tighter.
They hadn't resolved their acrimonious argument. How could they? She wanted what he couldn't in conscience give her. Tying a beautiful, vital girl like Charis to a physical and mental wreck like him was a sin against nature. He'd always recognized that. His pride wouldn't countenance it. His heart couldn't endure it. All the passion in the world didn't change that one bleak reality.
How the devil was he going to live without her?
The memory of the last, radiant days should fill him with regret. His passion had misled Charis into believing they had a chance together. He'd glimpsed a bright heaven that only mocked him now.
But selfish bastard he was, he couldn't repent what he'd done in Jersey. Not when desolate solitude beckoned ahead.
After their quarrel, they'd slept apart for the first time in over a week. Not that he'd slept. Instead, he'd sat in the parlor, watching night change to grim day. He'd felt like a mongrel cur tossed into the gutter to starve. He still did. Dear God, was this how the rest of his life was going to be?
He beat back the questions, the guilt, the anguish that plagued him. His gloved hands hardened on the reins, and he urged the ungainly pony to a faster pace. The gig bounced along the rocky track. He couldn't risk slowing down. The clouds closed in, and they'd be soaked if rain caught them on this heath.
Charis's gloved hand clenched on the edge of the lurching carriage. It had been the only vehicle available in the small fishing village where they'd found safe harbor this morning. They'd tried to land at Penrhyn Cove, but the seas made it too dangerous.
With every second, the weather worsened. A biting wind howled. The sky loured, black and menacing, and thunder rumbled in the distance. He needed to get his wife to warmth and safety. Where she could ignore him in comfort.
He slapped the reins against the pony's fat rump. They were still several miles from the house. He made a frustrated sound and looked at Charis.
She studied him, her eyes more brown than green, underlined with dark circles. She looked proud, distant, unhappy...beautiful.
In the strange gray light, her fine brows arched with what he read as disdainful curiosity. "Are you quite well, Gideon?"
"Yes, of course," he said curtly.
Her lips lengthened with irritation. "You're very restless, and you're making bizarre noises."
"I'm worried about the weather."
She looked around the open plateau. High in the sky, birds streaked to escape the coming tempest. The wind competed with the gig's rattle and the clop of the pony's hooves.
Her hand shifted to touch the necklace he'd given her the morning they left Jersey. England's greatest heiress must own bank vaults full of spectacular parures. But when he'd seen the amber-and-gold circlet in the jeweler's window in St. Helier a week ago, he'd immediately thought of Charis. The unusual intensity of the yellow stones reminded him of the light in her eyes when she was happy.
A light noticeably absent today, damn it.
Although her thanks were subdued, she'd seemed to like the trifle. At least she wore it.
Not for the first time, Gideon felt all at sea with his wife. Marriage was a difficult and complicated endeavor. Perhaps it was a good thing that his would be so short-lived, at least in any meaningful sense.
And didn't that cheer him up no end?
Dourly, he stared past the pony's ears at the rutted path. It was difficult not to view the surrounding wasteland and threatening sky as omens of his future.
"We're not far from home, are we?" she asked, without looking at him.
Home. Gideon supposed she must consider Penrhyn her home. Lord knows she'd been exiled from anywhere else she rightfully belonged. Now he prepared to exile her again. He knew he did the right thing in setting her free. But at this moment, it didn't feel like it.
"Not far. Pray God we beat the rain."
The road dipped into a tree-filled dell. Interlacing branches turned the gloomy afternoon into night. Away from the wind, the gig's creak seemed unnaturally loud.
Then the ambush came.
When the tree crashed in front of them, at first, stupidly, Charis thought the wind caused the accident.
Then she realized there was no wind in this hidden hollow.
"Damn it." His powerful shoulders bunching, Gideon struggled to control the rearing, squealing pony. The tree had missed the animal by inches. "Whoa there! Settle down!"
Charis clung trembling to the rocking gig as the maddened horse bucked and fought. Gideon fought to enforce obedience. Finally, recognizing the hand of authority, the pony stood quivering between the shafts with its head lowered.
Gideon cast her an urgent glance. "Jump, Charis, and run!"
But it was too late. Charis hardly drew breath before a roughly dressed man appeared from the underbrush. He snatched the halter with cruel force, wrenching the skittish pony's head up.
"Sir Gideon, what a pleasure." The oily self-satisfied voice oozed down Charis's spine and held her paralyzed on the seat. A terrifyingly familiar voice.
Across the pony's heaving back, she met Felix's gelid gray regard. Her every muscle tensed. Choking fear set like stone in her belly. Dear Lord, they were trapped.
Felix looked so pleased with himself, rage boiled up to drown her fear. With just such an expression, he'd watched Hubert beat her black-and-blue. She invested every ounce of the contempt she felt into her glare. "Felix. Still a sneaking little worm, I see."
Her stepbrother's hands clenched on the halter, so the frightened pony whinnied and tossed its head in protest. "Shut up, you little bitch!"
"And eloquent as ever. I'm impressed." Her voice lowered into irony. "I find myself less impressed with your appearance. Have you given up bathing for Lent?"
"Stay quiet, for God's sake," Gideon hissed, dragging her to his side with one strong arm. With his other hand he reached into the pocket of his greatcoat, she guessed for his pistol. "What in Hades are you about, Farrell?"
He didn't shift his attention from Felix, and his voice was sharp and lordly, as it had been when he spoke to the brothers at Penrhyn. Charis pressed closer, her brief defiance fading beneath growing awareness of their terrible danger.
"I wouldn't do anything too impulsive, if I were you, Trevithick." Felix drew himself up and made a dismissive gesture. "You're expendable, and I'm sure you won't wish to leave my sister undefended."
He nodded to someone behind the gig, and Charis heard the unmistakable sound of a gun cocking. She didn't need to see who it was. The two brothers rarely acted apart.
Her pulses raced, and sweat prickled her palms, but Gideon's heartbeat remained steady and sure under her cheek. The unhurried, regular sound bolstered her courage. Even as he lifted his hand away from his pocket.
"Lady Charis is now my wife," Gideon said calmly, his arm tightening around her in a silent promise of protection. But how could he keep her safe when the brothers had them at such disadvantage?
"The devil she is," Hubert snarled, stamping into view and brandishing two large horse pistols.
The brothers' fortunes had clearly worsened in recent weeks. They were unshaven, their clothing was creased and stained, and their linen was gray. The Farrells' unkempt state hinted they'd been sleeping rough. With sudden spite, Charis hoped it had rained every night. She hoped it had snowed.
"We've ridden to Gretna and back. We know you haven't married the slut," Felix snapped, snatching one of the guns from Hubert and aiming it squarely at the pair in the gig.
Gideon didn't flinch although she felt him subtly shift so his body shielded her from the pistol. Foolish, heroic man. The rusty taste of regret flooded her mouth as she remembered how angry she'd been with him all day.
"I have indeed wed this lady." Gideon bit out the last word. His sangfroid stirred Charis's admiration even as acid dread rushed through her veins. "In Jersey a fortnight ago. For confirmation, apply to the Reverend Thomas Briggs of St. Helier. Lady Charis's person and fortune are now at my disposal."
Stupid Hubert lowered his pistol. Felix cast him an irritated glance. "What the hell are you doing, man?"
"They're married," Hubert spluttered. "The game's up."
"For God's sake, keep them in your sights!" Felix whipped around to face Gideon and Charis. The feral light in his eyes indicated this was his last desperate throw of the dice, and he intended to win. "It's not as simple as that, Trevithick."
"No?" Gideon still sounded nonchalant. "Any harm gets you no closer to the money—and garners you a hanging when the law catches up with you. Make no mistake. You and your brute of a brother are identified as likely culprits should mischief befall us."
"You have it all wrong." Felix's smile took on a smug curve that sent a shiver down Charis's backbone. "I mean everyone to walk away safe and sound, Hubert and I considerably richer and you, sadly, considerably poorer."
Gideon's soft laugh lifted the hairs on the back of Charis's neck. He sounded utterly powerful. As if he hadn't a care in the world, for all that they were held at gunpoint without hope of outside aid in this wild woodland. "I wouldn't toss you a farthing after what you did to her, you bastard."
Felix's lip curled in scorn. "Brave words." Without shifting his attention from the gig, Felix tilted his head toward Hubert. "Get the jade."
Hubert stepped toward them, then hesitated as Gideon spoke with a cold savagery that made Charis's heart skip a beat. "Touch her, and you're dead."
Felix's face hardened. Most people considered him a handsome man, but for a moment, he looked uglier than a hobgoblin. Charis suppressed another shiver. "We'll hold the chit until you transfer every penny of her fortune to me."
Charis bit back a gasp, and her hands clenched in Gideon's coat as if that would save her from being dragged away. She should have expected this. She knew from bitter experience that Felix hated to be bested. He'd never allow her money to slip through his fingers.
"Don't worry." Gideon looked down at her and his arm firmed around her shoulders. "I won't let them take you."
"Can't we fight?" Charis's voice shook with distress.
Regretfully, Gideon shook his head. "They're armed. The risk of your getting hurt is too great." He turned his unblinking gaze to Felix. "Take me instead."
Gideon's easy tone momentarily deceived Charis. Then, with disbelieving shock, she realized what he offered. On a strangled cry, she straightened and stared at him in horror.
You will not do this, my love. I won't let you.
Felix gave an unimpressed grunt. "What purpose will that serve?"
"It keeps her out of your filthy paws." Gideon's tone dripped derision.
Felix sent him a hate-filled glare. "Sadly, because of your machinations, it's your signature we require, not hers."
"My man of business is at Penrhyn to advise her how to get the money. Charis can contact the trustees and the bank, organize the papers. Until then, I place myself at your disposal."
Her belly twisted in denial, and her hands clawed at his coat as if she'd restrain him by main force if she must. "No, Gideon, this is unthinkable. You can't."
The broken protest faltered into silence. She couldn't risk Felix and Hubert discovering his vulnerability. If they knew what Gideon risked by becoming their hostage, they'd torture him to insanity.
"You can't," she repeated in a shaking voice, wishing they were alone, wishing she'd never met him and put him in this danger. Better she'd married Desaye weeks ago. What she'd always feared had finally come to pass. Her dilemma threatened to destroy the man she loved.
Through glazed eyes, she saw Gideon register her terror and rise above it. His black gaze as it probed hers was certain, unafraid. "I'm not letting them within a yard of you, my darling."
It was the same voice he'd used when he'd stubbornly insisted they had no future together. Her instincts told her he was determined on this course, and nothing she said would shift him.
She had to do something. She had to stop him. He confronted his vilest nightmares for her, and she wasn't worth it.
She swallowed the lump of furious emotion in her throat, only to have words fail her again as Gideon raised her gloved hand and brushed a fleeting kiss across her knuckles. Scalding tears prickled her eyes.
Felix and Hubert were ruthless, violent bullies. They'd work out their frustrations on their captive. Even without his affliction, Gideon faced pain and humiliation at their hands. With his affliction, the consequences could be catastrophic.
"No..."
Gideon's jaw took on the familiar implacable line. "I swore these dogs would never touch you again."
"Dear me, your gallantry touches my heart," Felix said sarcastically as he moved closer in unmistakable threat. "But I do believe we're better keeping the jade."
"Completely unacceptable." Gideon didn't look at Felix, and he spoke as if he held the upper hand in this ugly scene.
Felix emitted a harsh laugh. "By God, you're a cool one. What's to stop us taking her?"
"I'll stop you."
"You forget who has the gun." Even so Felix paused.
Gideon's smile was superior as he turned to her stepbrother. "Kill either of us, you lose your chance at the money."
"You'd still be dead," Felix said grimly, raising the pistol.
Gideon shrugged off the jibe. "Frankly, I don't know why you expect to get away with this. We'll lay the facts before the law at the first opportunity."
How could he sound so confident when he must know what was likely to happen? His reckless courage made Charis's belly lurch with nausea.
"We're not fool enough to wait around like sitting ducks. Hubert and I are for the Continent."
"While you're welcome to the slut," Hubert said. "Even if you'd managed to keep her fortune, you'd soon find you made a poor bargain."
Charis hardly noticed the insults. Her mind worked too frantically to find something to persuade Gideon against this perilous course. He'd sacrificed so much for her, but this went beyond what anyone could ask. It would be like facing Rangapindhi all over again.
Gideon didn't look at the brothers but spoke directly to her. His voice rang deep and sincere. "My wife is more precious than rubies. If she came to me wearing only her shift, I'd still be rich beyond measure."
He made the extraordinary declaration for her sake in case things went wrong. Charis's heart twisted with overwhelming love.
Oh, dear God, whatever happens, let him live through this.
"I can't leave you," she said unsteadily. Blind fear dug icy talons into her. "Don't make me."
"I must." He released her hand and his voice lowered. "Akash and Tulliver are at the house. They'll know what to do."
"Gideon..." His name was a ragged plea. She watched his expression close against her. His purpose was clearly unshakable.
Gideon faced Felix, his face set with disdain. "So you agree? I place myself in your charge, and Charis goes free?"
No, this must not be. In blind distress, she turned to Felix. "Take me." She was mortified that her voice broke.
"Both so eager to sample our hospitality." Felix's laugh was cutting. "Make up your minds. One of you needs to get the money."
Gideon sent her stepbrother a flinty look. It was as if she'd never made the offer to stay in his place. "I assume you have a mount for Lady Charis. Unless you intend to shift the tree."
Gideon's continued calmness astounded her even through her dread. He wasn't shaking or sweating or pale. He looked like the invincible man who'd come to her rescue in Winchester.
Hubert's piglike eyes darted between Gideon and Felix as he tracked the shifts of power. "She can have my nag."
Gideon turned to her and gently cupped her face. His smile, like his touch, was poignantly tender. She searched his eyes for the fatalistic resignation she'd seen so often when he faced down his demons. All she read in the glowing black depths was strength, serenity, resolve.
And love, like a single star shining over a dark sea.
"Trust me, my darling," he said softly. "If you love me, trust me."
He knew he defeated her with that last demand. Gathering her ragged courage, she swallowed another furious protest and raised her chin.
Agreeing to what he asked was the hardest thing she'd ever done. Harder by far than defying Felix and Hubert or confronting the vile sailors in Portsmouth. Harder even than fighting Gideon for the chance to create a life together.
Fear coiled like an angry snake in her belly. For all Gideon's bravery, she abandoned him to an ordeal that could break him. But she couldn't let him down. Or succumb to pathetic, immature hysterics. She was the daughter of Hugh Davenport Weston. She was the wife of Gideon Trevithick. She wouldn't shame either valiant hero by failing now.
"I'll go," she muttered reluctantly.
She closed her eyes in despair as Gideon pressed his lips to hers. The kiss was sweet, passionate, heartbreakingly brief.
As he slowly drew away, she looked into his eyes. The star was still there. More radiant than ever.
"I love you." She could no longer hold back the words.
"I love you." He spoke without reluctance or equivocation. She snatched the vow close and locked it in her heart, never to let it go. Surely if they loved each other, Felix and Hubert couldn't defeat them.
Such hope rang false when she forsook her beloved to torture and imprisonment.
"Oh, for God's sake, get a move on," Felix said in a theatrically bored voice.
She ignored her stepbrother's jeering. She clung to Gideon's hand as she climbed down from the carriage. Her knees felt like custard as she reached the road.
Summoning all her courage, she released Gideon and braced her shoulders. Standing straight, she faced Felix. A scatter of cold raindrops hit her. The storm wasn't far off. A crack of thunder made the pony start and neigh.
The gig creaked as Gideon jumped to the road behind her. He towered over her, and his gloved hand closed firm and possessive around her arm. "She leaves unharmed. Otherwise, we have no agreement."
Felix gestured Hubert toward Gideon. "She'll leave unharmed, all right. But only when we've got you trussed nice and tight."
Charis waited for Gideon to object, but he merely said, "Let me give Lady Charis my coat. The weather's about to break."
Felix nodded briefly. "No tricks. I can hurt you without killing you."
"I'll keep that in mind," Gideon said dryly.
He released Charis and quickly divested himself of his coat. As he dropped it over her shoulders, it swamped her. Immediate warmth surrounded her. And Gideon's scent. Such an absurd thing to bolster her unsteady resolve.
Gideon brushed one gloved finger across her cheek and smiled. "It's like old times."
Her skin tingled under his touch. His words reminded her they shared a history of danger and survival. She wished she could draw comfort from the fact. "Be careful, Gideon," she whispered, her throat thick with anxiety and love.
He stepped past her. Charis bit back a protest as Hubert grabbed Gideon's hands and roughly wrenched them behind his back. Her husband stood stiffly, but he presented no resistance. Could the touch of Hubert's hand spark an attack? Please, no.
How could Gideon bear this? He must know what the brothers had in store. His unflinching bravery threatened her fragile control. Her belly knotted with sick anguish. He gave himself over to torment for her sake. She felt like she pushed him back into the pit in Rangapindhi with her own hands.
When Gideon looked at her, he must have read her faltering purpose. "Put the coat on properly. You've got some tough riding ahead." He sounded as if he sent her off on a morning's canter. She remembered she owed it to him to reach Penrhyn and save him. No matter how she wanted to scream and cry against what happened now.
She stiffened her spine. Her gaze clung to his face as she memorized every beloved feature. His burning eyes, the proud blade of his nose, his passionate mouth, taut with controlled anger. Beneath his composure, she knew he was fuming. She wanted him to stay furious. The fierce emotion might keep his ghosts at bay.
"Good-bye, my love," she said huskily.
He stared back. "Godspeed, Charis."
"Come on." Felix snapped, snatching her arm. His touch bruised, even through the thick woolen sleeve. "All hell's about to break loose."
"Let her go," Gideon said in a low, dangerous tone.
For all that the brothers were armed and Gideon was bound, Felix's hand automatically dropped away from her. Charis sent Gideon a grateful glance, then picked up her skirts and followed Felix.
There was nothing more she could do for Gideon here. Pray heaven, she could help him once she was free.
In spite of her urgency to reach Penrhyn, Charis took one last lingering look at her husband as she climbed the steep bank to bypass the fallen tree. Dwarfing Hubert, he stood tall and proud and undefeated. No trace of fear or weakness showed in his set features.
Stay safe, my love. Stay safe until I come for you.
She sent him a burning glance, a message to be strong, a promise to save him as he'd saved her so often. Then she dropped below the tree's branches, and he disappeared from view.
Two horses were tethered in the underbrush. Neither with a sidesaddle. She hadn't ridden astride since she was a girl at Marley Place. It would be difficult in skirts and on a mount she didn't know. Especially in weather that intensified with every second.
The rain fell in sheets now. Felix was soaked through, and Charis shivered as freezing water trickled down her neck. Her bonnet was a useless, sodden mess. With shaking hands, she ripped at the ribbons and tugged it off.
"How will you know when the papers are ready?" she asked in a frigid voice. If Gideon could be strong, so could she.
"I'll send a message." Felix grabbed one of the horses and hauled it into the open. The stocky bay snorted and fought at leaving the shelter of the trees. "Let me give you a hand up."
"Don't touch me," she snapped.
"Suit yourself, my lady." He presented the reins with an ironic gesture.
Snatching them out of his hand, she spoke soothingly to the nervous animal. She scrambled onto its back, swathing the greatcoat around her. The storm was bad enough in this hollow. She dreaded to think what she'd face on the open moor.
The horse curveted at having a rider, but Charis quickly brought it under control. She glared through the downpour at Felix. "If you hurt my husband, I'll hunt you down and kill you."
Felix gave a harsh laugh. "You always were an unnatural chit. Once I get the money, I have no further interest in either of you. Although I'll wager Trevithick will curse the day he tangled with the Earl of Marley's termagant daughter."
She ignored his jibes. "Remember what I said. I know you and Hubert are eager to prove your prowess on a defenseless man."
Kicking the horse into a gallop, she forced it up the slippery path out of the dell. As she bent forward over the beast's neck, her heart pounded out a single message. Gideon, wait for me.
## Twenty-three
Up on the moor, the wind roared like an angry monster. It turned the driving rain into knives that pierced the thick greatcoat like muslin. Fierce cold sliced through Charis's bones. But nothing made her colder than her fear for Gideon.
Her mount neighed and fought as she battled to turn it onto the faint path toward Penrhyn. She sawed furiously at the bit, but the animal was too frightened to settle.
"Please, please, behave for me," she sobbed, tightening her thighs to keep her seat on the twisting horse. Her arms ached with stopping it bolting back the way they'd come.
Gideon needed her. Every second counted. Hunkering down in the saddle, she grimly set to gaining control over the beast.
Eventually, the animal began to splash its way westward at an unsteady gallop. Charis's shoulders knotted with strain, and she panted for breath. She leaned over the horse's neck, calling encouragement although she knew the gale whipped her words to oblivion.
All the time, her heart pounded out a silent message to Gideon.
Wait for me, my love. Wait for me. Wait for me.
Dread created its own swirling storm inside her. Not dread for herself, dread for her husband. Had he kept his ghosts away? What were Felix and Hubert doing to him? Where did they mean to keep him? Dear Lord, don't let it be somewhere dark and constricted like the pit at Rangapindhi.
She blundered on. The rain turned her clothes to heavy wet ice. Her sodden braids collapsed and tumbled down, blinding her. With one shaking hand, she hurriedly dashed her dripping hair away from her eyes. The storm transformed the afternoon into night, lit by jagged flashes of lightning, punctuated by rolling thunder.
The horse released a high-pitched neigh and balked at a swollen stream. Ruthlessly, Charis kicked it until it launched into an ungainly jump. "Come on!"
The animal stumbled when it reached the crumbling bank. Charis slid dangerously, nearly fell into the raging flood. After a terrifying, breathless pause, the flagging horse found its feet, slipping in the mud.
She hoped to heaven she followed the right path. Or any path. Either she'd missed Penrhyn's gateposts in the squall, or she was yet to reach them. Or she was hopelessly lost. Gideon said it was only a couple of miles to the house, but she felt she'd been riding forever.
"Stay with me." Her frozen hands closed clumsily on the reins.
The weather worsened. The wind became a malevolent, deadly force. She wondered how the plucky little horse kept going.
"There's a warm stable ahead. Oats. Hot bran mash. Soft straw for your bed."
She repeated the promises over and over. She didn't know if the gallant beast heard. The words were for her benefit as much as the horse's. She kept talking until her voice scraped over her throat.
All the time, she struggled to hold on to hope. Hope that Gideon was safe. Hope that Akash and Tulliver would rescue her husband. Hope that she'd find her way home. If she was caught on the moor when night fell, what could she do?
Keep going.
What choice did she have?
Fatigue made her muscles burn like fire. Her arms felt like iron weights. Cold stole the strength from her legs. Her eyes stung with staring into the arctic blast. Anxiety for Gideon was an evil, black, roiling mass in her belly.
The horse stumbled again, and this time was slower to find his feet. After his initial reluctance, he'd proven a valiant companion.
"Not far now, I promise. Just one more effort. One more." Her voice cracked, and tears she'd fought for so long rose to her eyes. Her teeth chattered so fast, she could barely speak. "It's for Gideon, you see. We have to save him. He's so good, and I love him more than life. And he's suffered too much."
The horse hung his head, his sides heaving with exhaustion, as the rain poured off him. Still crooning a fortifying litany, Charis slipped to the ground, landing with a splash. Icy water flooded her half boots. Her numb legs bent under her. She cried out sharply and grabbed the stirrup, just saving herself from hitting the ground. Her arms screamed in protest as she inched herself up. Her heart thudded furiously, her breath emerged in ragged gasps.
"Oh, Gideon, please be alive," she sobbed in despair, burying her face in the horse's saturated coat.
For a few seconds, she stood with the rain pounding down on her bare head. Stray pictures drifted into her mind, then dissolved. Coherent thought faded to a gray mist.
One idea remained clear. Gideon. She must save Gideon.
She blinked, forced her eyes to focus, her mind to function. Gideon needed her. She locked her knees until they held her upright. For a groggy moment, she clutched the slick leather stirrup. Then she released it and stood as the wind whipped around her.
She could do this. She could go on.
But her horse had reached the end of his endurance.
She forced words past lips that felt like solid ice. "We're nearly home. Not far now." God help both of them if she lied.
She fumbled for the reins and staggered ahead on foot. The horse followed docilely, too tired to resist, wading through dirty water that lapped over his fetlocks.
Eventually, she dragged the sodden greatcoat off and dropped it beside the path. Wet, it was heavier than lead, and it offered no protection. Or so she thought until she confronted the full force of the wind. The blue merino pelisse had been snug and warm on Jersey. Here, on a freezing Cornish moor in the middle of a deluge, she might as well have been naked.
Still she stumbled on. Her legs stung as if a thousand blades nicked at them. She shivered so badly, her muscles cramped to agony. She could no longer feel her feet.
The darkness now was nearly impenetrable. Devils in her head whispered that she'd die out on this moor and nobody would ever know Gideon was in trouble.
She strove to muffle the cruel voices, but with every footstep, their howls grew louder.
Then over the wail of the wind, the slap of the rain, she heard a dull pounding. It came closer and closer.
Her sluggish brain puzzled over the sound. Was it blood beating in her ears? Thunder? Gunfire? But who could fire a gun in this wet?
When the big black horse cantered out of the rain, like something risen from the mouth of hell, Charis stopped stock-still. Her dazed mind couldn't comprehend she was no longer alone. Or whether this new arrival signaled danger or rescue.
"Lady Charis?"
The rider drew to a rearing halt in front of her. The risk she took standing in the middle of a road in Stygian darkness vaguely registered. Her horse tugged listlessly at the reins but was too weary to pull free.
Stupidly, she blinked up at the man looming above her in the saddle. Water cascaded down her face and obstructed her sight. She swallowed, trying to summon a greeting. Nothing emerged apart from a broken whimper.
"Lady Charis?" He dismounted in one easy move and stepped forward. "Lady Charis, it's Akash."
"Akash..." she croaked without moving.
"Gideon wrote from Jersey and told me to expect you this evening at the latest."
"The weather..." Then the significance of his arrival struck so hard she staggered with dizzy relief. Sudden energy buzzed through her. Blood that had frozen abruptly began to flow again. Her mind churned with new hope and determination. "Akash, we have to help Gideon. My stepbrothers have him."
She turned back the way she'd come. Akash would help. Akash would save Gideon. Everything would be all right.
"Wait." Akash grabbed her arm. She was so cold, she hardly felt it. "You can't go like this."
Confused, she turned to stare at him. This didn't make sense. Akash was Gideon's friend. He'd saved him before. He'd save him now.
"Didn't you hear me? Gideon's in trouble." Her voice became stronger as she strove to speak above the shrieking wind. "There's no time to delay."
He swept his hand across his face in a futile attempt to clear the rain from his eyes. "Charis, Penrhyn is only minutes away. At least come back and get dry. We'll make plans there."
Had she almost made it home? It seemed too good to be true. Reaction hit like an avalanche. Her knees threatened to fold under her. She glanced back to her brave little horse. He'd carried her this far, but he'd carry her no farther tonight.
She drew a sobbing breath, and the fight drained out of her. As she was, she was no use to Gideon. If she was to help him, first she needed warmth and food and a chance to recover her strength.
But how it tortured her to delay his rescue. Even when she recognized the necessity of finding shelter before she collapsed.
"Yes, take me home," she said dully, and stood in shivering acquiescence as Akash wrapped his own much dryer coat around her.
Her heart in her throat, Charis crouched in the brown winter bracken and studied the overgrown entrance to the disused tin mine. It had stopped raining a couple of hours ago, and a cold gray dawn had broken.
She wore one of Gideon's mother's riding habits, and the ground under her was wet and muddy. At her side, Akash held a pair of beautifully chased silver pistols and watched the mine just as avidly. Hidden around them in the bracken were ten stalwart Penrhyn men. The same men who had unhesitatingly raced out into the foul night to locate Gideon.
The sight of the mine made her feel sick. She still reeled from discovering that her stepbrothers kept her husband in an underground tunnel. When the searchers returned to Penrhyn with the news, she'd barely been able to control her rising gorge. Fear remained a sour, bilious taste in her mouth.
In such a place, memories of Rangapindhi would be inevitable. Was it also inevitable that Gideon must succumb to his ghosts? Perhaps they'd steal him forever this time. With horror, she remembered his shaking, debilitating illness after Portsmouth. This fresh torture must test his limits, no matter how strong he was.
Let him be all right.
She bit back rising panic. She'd promised herself she'd be brave for Gideon's sake. But, sweet God, it was difficult when she imagined her husband trapped in suffocating darkness.
What if she managed to save his body yet couldn't save his sanity? The prospect didn't bear contemplation. Although her mind did nothing but play grim scenarios.
Courage, Charis.
She tightened her grip on her pearl-handled pistol. Her eyes were scratchy from lack of sleep, and her pulse thundered in her ears. The hairs on the back of her neck prickled. She knew Gideon was close. She felt it in her blood, the way an animal recognized the approach of its mate.
"Gideon will have my guts for garters when he finds out I brought you on this escapade," Akash muttered in a voice so low only she heard it.
"I gave you no choice."
The only way he could have kept her away was by locking her in the attics. Even then, she'd have done her best to climb out. Akash had been determined to leave her safely at the manor, but her obstinacy had outlasted all argument. If Gideon's demons had conquered him, she needed to be there to fight them.
"He still won't like it," Akash said gloomily.
She prayed Gideon was alive not to like it. Strangely, Akash hadn't been overly concerned when she laid the whole story of the ambush before him last night. Yet surely Akash more than anyone knew what imprisonment meant to Gideon.
Tulliver appeared on top of the bank overhanging the entrance and waved before dropping out of sight. It was the arranged sign for movement within.
Purpose flowed through Charis in a reviving flood, and her heart took on a surer, steadier rhythm. She would save Gideon, no matter what forces ranged against her.
Not long now, my love. Wait for me...
Akash gestured behind him. With surreptitious rustling, the men crawled forward. Charis was aware of the movement, but she didn't shift her attention from the mine.
Hubert emerged into the daylight, leading two horses. She immediately recognized the homely pony Gideon had hired to draw the gig.
Her stepbrother yawned and stretched, his lack of self-consciousness indicating he had no inkling he was observed. Hatred flared in Charis's belly as she watched him. He was about ten yards away, close enough for her to see he looked even worse than yesterday. Impossible to believe he held one of the kingdom's oldest titles. In his dirty, ragged clothes and with his greasy, overlong hair, he'd pass for a beggar.
Soundlessly, a wiry Cornishman rose from the bracken that grew toward the entrance. Another joined him. Using the undergrowth for cover, they'd circled behind Hubert, who stepped into the watery sun. A few silent steps, and one man covered Hubert's mouth to muffle any shout of warning. The other man quickly overpowered him.
The struggle was over in seconds. Hubert lay gagged and bound. He writhed as the men dragged him away from the cave. His muffled grunts of protest ceased abruptly when one of his assailants kicked him hard in the ribs.
There was no sign of Felix. A charged silence fell. Charis's gloved hand curled with painful force around her pistol. At her side, Akash tensed and raised his guns.
"Hubert? What the devil are you playing at?"
Felix's irritated question emerged as an eerie echo from inside the mine. One of the ponies snorted nervously and trotted toward the bracken, trailing its halter rope.
"For God's sake, stop messing about." Felix appeared at the entrance. Then, just as quickly, slipped back under cover.
Like a deadweight, foreboding settled in Charis's stomach. Any chance of another surprise attack was lost. And still she'd seen nothing of Gideon. Over and over, her mind chanted her desperate prayer. Please, God, let him be all right.
"Come out, man. The game's up." Akash stood, and his voice rang across the open area in front of the mine. "You don't have a hope of getting away with this."
Tulliver jumped down from his hiding place above the mine and hid from Felix's view beside the entrance. A wicked-looking knife jutted from his belt, and he held a pistol. For a heavyset man, he moved with incredible smoothness.
Felix called out from inside. "You forget—I have Trevithick."
Charis was sickeningly familiar with her stepbrother's defiant tone. For one surreal moment, it transported her back to their first meeting. He'd expressed his contempt for his new stepsister in just such a voice. And received a cuff from his hulking father in return. A cuff he'd returned with interest when he got Charis to himself.
He'd always been a sneaking, sadistic little bully. Bile filled her mouth as she imagined what state Gideon was in, bound and at Felix's mercy.
Akash strode toward the cave, his guns held ready, his body tall, straight, and reeking confidence. "We have your brother."
"You won't hurt Hubert. I, however, have no such scruples about my hostage."
Charis could wait no longer. She stumbled upright on shaking legs, her heart racing with a turbulent mixture of hope and trepidation. "Gideon, are you all right?"
There was a silence. Hope shriveled like an old walnut in her breast. Her heart faltered to a stop.
Were they too late? In a fever of anguish, she darted forward to stand beside Akash.
"Charis?" Gideon's voice was rusty, but the mere sound of it sent joy fizzing like newly opened champagne through her veins. She swayed briefly and closed her eyes as dizzying waves of relief battered her.
It was a miracle. She had no other explanation. He was alive. And aware.
And blisteringly angry. "What the devil are you doing here?"
In spite of the danger and his audible displeasure, she couldn't contain a choked laugh. She raised trembling fingers to dash burning tears of happiness from her eyes. "Saving you."
"Go back to the house. Now."
"I told you," Akash muttered.
"I want to negotiate," Felix shouted. "My freedom for Trevithick's release."
"Don't be a fool, man," Akash snapped, taking a step closer to the mine. "We've got you surrounded. You can't escape."
"Then there's no reason to keep Trevithick alive."
Charis's throat constricted with renewed terror. Her relief had been premature. The threat Felix posed was as real as ever.
"He'll kill Gideon if we push him too far," Charis said unsteadily. "He's not bluffing."
Akash frowned down at her. "A murder charge won't help his case."
"He's smart enough to know his case is hopeless." She raised her chin and stared unwaveringly into Akash's deep brown eyes. "I don't care what happens to Felix. Kill him, let him go free, whatever you have to do. Just as long as we save Gideon."
His eyes darkened as if he realized what it would cost her to let Felix get away with his crimes. Then he nodded and faced toward the mine, cocking his guns. "All right, Lord Felix. I'll come in."
"I'll go with you," Charis said quickly.
Akash cast her a glance that mingled astonishment and disapproval. "Out of the question."
Her jaw firmed. "Make me stay."
She saw him consider getting one of the Penrhyn men to restrain her, then clearly he thought better of it. Or perhaps he took pity on her frantic need to see her husband. His tone was low and adamant. "You are not to speak. You are not to move unless I give you the word."
"I promise." Her voice shook with gratitude. "Thank you."
"I hope I don't live to regret this," he said grimly. He raised his voice. "Don't try anything, Lord Felix."
"Drop your weapons first. And remember, any tricks and Trevithick's a dead man."
Akash glanced at Charis, who nodded. Both of them laid their guns on the ground, then approached the mine entrance.
With every step, her heart beat faster. Fear closed her throat and made her skin itch. If Felix decided to shoot them, they had no protection.
Surely he couldn't be so stupid. He wouldn't be able to kill every man here. Then she remembered his vanity and recklessness.
"Watch our backs," Akash hissed to Tulliver, as they passed under the heavy wooden beams that supported the entrance. Tulliver nodded while Charis and Akash edged inside.
Momentarily, the dimness blinded her. The dank tunnel was deathly cold. The air was rank with bats, stale air, and decay. Carefully, she moved forward, conscious of Akash as a silent, reassuring presence beside her.
"Damn you, Charis," Gideon cursed from farther along the tunnel. "Get out of here."
"No, she should stay," Felix said in a silky voice. "A foolish, but noble gesture, my dear stepsister. You've presented me with yet another hostage. I must thank you."
As her eyes adjusted to the gloom, lit by one lantern, she saw that Felix aimed his pistol squarely at her chest. It was one of the big clumsy horse pistols from yesterday. She glanced at him, long enough to read the desperation in his face. Then her attention flitted past him and settled on Gideon. He stood, hands bound behind his back, a few paces beyond Felix in the center of the brothers' makeshift camp.
He glared at her like he wanted to kill her. His black eyes blazed in his pale face, and his mouth was a long line of displeasure. He should have appeared powerless. Instead, he looked indomitable, magnificent, undaunted.
There was blood on his jaw and bruises under his torn shirt. The visible evidence of Gideon's ordeal made her heart slam to a shocked halt.
"Gideon..." She took a shaky step toward him, only to come to a trembling halt as his eyes narrowed with temper.
To think she'd fretted about this man's ability to cope with captivity. He'd walk through a raging hurricane without turning a hair. His bruises and abrasions only emphasized his invincible spirit.
Gratitude punched the breath from her lungs, made her hands shake. She blinked back more tears. They weren't safe yet. She couldn't relax her guard.
"You spineless toad," she spat, turning on Felix. "How dare you beat a bound man?"
"Charis, I'm fine," Gideon snarled. "But you won't be when I get my hands on you. Akash, blast you, what were you thinking, bringing her here?"
"You're getting ahead of yourself, making plans for what you'll do once you're free," Felix said snidely. He backed against the wall, his gun still trained on Charis. "I have to ask myself if I really need three hostages. Perhaps I'm better off disposing of one of you."
"You must know this rash gamble has come to its end." Gideon's voice rang with authority. "Surrender while you have a chance at convincing a judge you deserve leniency."
Felix's expression hardened. Charis shivered as she thought of a rat caught in a trap. She didn't fool herself that this particular rat was harmless. He knew he'd lost, and he'd take them all down with him if he could.
"What I've done is a hanging matter," Felix snapped. "I'm not a fool. I won't offer myself up like a lamb to the slaughter. There's fight in me yet."
"That's lunatic." Akash stepped closer with unconcealed threat. "What can you hope to achieve?"
"Damn it, stay back!" Wildly, Felix swung the gun toward Akash.
Charis used Felix's momentary distraction to dash across the rubble-strewn floor to Gideon. With a broken sob, she threw her arms around him and buried her head in his chest. She drew in his familiar scent, felt the steady thud of his heart against her breast. Relief thundered through her.
He was alive. He was alive. They would come out of this yet.
His skin was chilled, and his tattered shirt was clammy from last night's downpour. He stood rigidly in her hold, his muscles taut. For one horrified moment, she wondered if his affliction had returned.
Then she realized he wasn't sick, he was angry. He vibrated with incandescent fury.
"How dare you put yourself in danger?" he growled, resisting her clinging hands.
"I've got a knife," she whispered, looking up at him.
At last he glanced at her. His jaw worked as he fought to master his temper. She read his anxiety for her, his rage. But more, she saw the mirror of her own longing in his black eyes.
"Oh, hell, Charis," he muttered, his mouth turning down with annoyance. He bent his head and kissed her, briefly but hard. She knew it was meant as punishment, but she felt the blazing love underlying the rebuke. "Now get out," he said softly but firmly.
"Not yet." She fumbled in her pocket for the small blade she'd taken from a display of arms at Penrhyn. It probably hadn't been used since Black Jack's day, but she'd tested its edge, and it was sharp.
She cast a quick glance across at Felix and took advantage of his focus on Akash to slide behind Gideon. Watching her stepbrother out of the corner of her eye, she sawed at the binding around Gideon's wrists. It was dark where she stood, but still light enough for her to see the broken skin under the coarse rope. Her anger at her stepbrothers hitched higher.
"She's not going anywhere." Felix sidled in Gideon's direction, keeping his pistol aimed at Akash. "She's my surety I'll get out of here."
"There's a dozen guns outside, more if the militia have arrived," Akash said dismissively. Charis wondered if he guessed what she was up to and kept Felix occupied deliberately. Biting her lips, she worked more furiously at the rope. "Even if you do kill us, you won't get far."
Felix gave a scornful grunt, his eyes darting around the mine as if he sought an escape route. "Oh, yes, I will. Nobody will risk hurting her."
"What about Lord Burkett? Do you intend to abandon him to his fate?" Contempt sizzled in Gideon's words.
Felix shrugged without shifting his gaze from Akash. "He can take his chances. He'll get to plead his case in the bloody House of Lords, whereas I'll be treated like a common criminal."
"You are a common criminal," Akash said coolly.
Felix took a menacing step toward Akash. "Shut your mouth, you black bastard."
"Give it up, Farrell," Gideon said steadily. "If you come quietly, I'll see what I can do about a lighter sentence. Transportation at least leaves you your life."
Felix flinched in horror. "To that filthy hole, Botany Bay? I'd rather be dead." He was considerably closer to Charis and Gideon than he had been. She applied the knife with renewed energy and prayed the shadows hid what she did.
"Keep this up, and you will be," Akash said grimly.
"You speak as though my defeat is a foregone conclusion."
"It is." Gideon bunched the muscles of his arms, jerked his wrists hard, and snapped the last threads of his bindings.
"Not when I've got Charis." Felix lunged, but Gideon moved faster than a striking cobra and grabbed him before he laid hands on her.
"Little slut untied you, did she?" Felix grunted, fighting to get purchase on the larger man.
For a sickening moment, the two men teetered, casting a dance of grotesque shadows onto the mine's walls. Then they fell and struck the ground with a thud that Charis felt in her bones. There was a sharp rattle as pebbles shot across the floor in all directions.
"Damn you, Trevithick!" Felix grunted, then finished on a loud exhalation as Gideon landed a hard punch to his stomach. The sickening sound made Charis flinch back.
She couldn't tear her eyes from the struggle. The fight was cruel, frantic. Over and over, they rolled in a clumsy, murderous battle. She desperately tried to see who gained the advantage, but darkness and constant movement made it impossible to tell.
A storm of punches and groans punctuated the ungainly violence. Charis's belly cramped with dread, and she backed on unsteady legs to press against the cold rock.
Felix fought dirty, and he was strong and wiry, for all his fashionable languor. Gideon was bigger, but he'd been bound and beaten. Heaven knew what injuries the brothers had inflicted on him during the night.
A pistol shot rang out, resounding as the noise ricocheted off the rock.
"Gideon!" Charis screamed, lurching forward. Her heart slammed against her ribs. Her eyes went blind.
Akash caught her around the waist and stopped her flinging herself on top of the combatants. "Charis, it's all right."
She hardly heard him through the clanging in her ears. If Gideon was dead, she didn't want to live. Without him, there was nothing in the world she wanted.
Akash spoke more sharply. "Charis, they're alive."
At last she heard and understood. She realized how tightly he gripped her against his chest. Her fingers dug into his arms with bruising force.
The bullet must have gone wild.
Her sight cleared, and her terrified gaze focused on Felix and Gideon. She realized both men still moved, still struggled to best the other. Her aching heart started beating again. She sucked rancid air into starved lungs.
Dear heaven, thank you, thank you, thank you.
She trembled convulsively in Akash's grip. The tall body looming behind her bristled with silent tension. His support was welcome. She wasn't sure her legs would hold her. Her mouth was dry as cotton, and her heart pounded like a mallet wielded by a madman.
She stifled her urge to call encouragement to Gideon. He needed all his concentration to defeat Felix. The now-useless gun bumped across the floor as a wildly kicking leg sent it sliding. Gideon rolled over and kicked it more purposefully, propelling it out of reach.
She straightened, ashamed of her weakness. Akash must have realized she'd regained control of herself. He released her and edged around the fight to pick up the gun.
The men on the ground grunted and gasped and wrestled for dominance. They writhed across the rough floor. Felix flung out one leg and sent a tin kettle rattling against the rocks. The sharp metallic clatter made Charis jump. She raised one shaking hand to her mouth to hold back a scream.
The bone handle of the little knife she clutched in her other hand was slippery with sweat. If only she saw a chance to intervene. But all she could do was stand in agonized suspense on the conflict's edge.
Felix rolled on top of Gideon and clawed at his throat. For an endless moment, time hung suspended. Then Gideon twisted with what seemed impossible strength and dislodged his attacker.
The battle continued. Charis's hand dropped from her face to twine with painful tightness in her skirts. More thumps. More hoarse grunts and gasps. With a shuddering groan, Gideon jerked onto his knees, straddling Felix and gripping his neck.
"Die, you bastard!" Felix forced out. He flung Gideon away to land with a sharp crack of bone on rock. Charis bit back another cry. Every muscle tensed to excruciating pain as she waited for Felix to surge up and land the decisive blow. But instead he lay winded and unmoving a few feet away.
"For God's sake, help Gideon," she begged Akash in a strained whisper as he returned to her side.
"He's better on his own," Akash said softly.
It seemed hours before Gideon stirred even though she knew it must only be a fraction of a second. As he sat up, he shook his head to clear his vision. He staggered upright at the same time as Felix found his feet.
Exhaustion and pain took their toll. Both men panted in jagged gasps as they circled one another, their fists upraised. Felix's left eye swelled, and his mouth was broken and bloody. Charis noticed that her stepbrother's gait was uneven, and he favored his left leg.
She drew another shuddering breath and stared at Gideon. He looked dirty and disheveled and bruised but otherwise blessedly whole, and his eyes were bright and alert. They focused on Felix with a glint of triumph. There had been some shift in the battle, and it had been in Gideon's favor.
"Give it up, Farrell. There's nowhere to go." He sounded calm, confident, like the man who had saved her life. He flexed his gloved hands and rolled his shoulders.
"I'll get out of this, Trevithick." Felix stumbled on the rough ground but didn't fall. "Damn well see if I don't."
Charis watched as he staggered farther into the tunnel. His eyes remained fixed on Gideon, who took a step after him.
"You won't escape that way, man. Didn't you explore your hideaway? The mine peters out in the hillside."
"Felix, he grew up here," Charis called, desperate to bring this ghastly scene to an end. "He knows every inch of the estate. You're trapped."
"Shut up, you little bitch." Felix sounded savage, furious, as he backed away on faltering feet. His voice resonated oddly as the tunnel narrowed. "We'll see who's trapped."
"Be careful. There's a mineshaft behind you." Gideon set out after him, his booted heels thudding sharply on the hard dirt floor. Charis broke away from Akash and followed, gripping her knife. She still didn't trust her stepbrother even though she could tell he had reached the end of his strength.
"Resorting to childish tricks now, Trevithick?" Felix's grating laugh sent a shiver down her spine. He retreated more quickly from the light.
"Take a look if you don't believe me." Gideon's voice roughened with urgency. "For God's sake, man, listen to me! Look behind!"
"And take my eyes off you? You must think I'm a damned half-wit."
"Farrell..."
Felix kept up his odd crablike shuffle, then suddenly tottered. His arms windmilled as he fought for balance. It was tragically clear Gideon's warning was sincere. Charis's stomach lurched with horror.
Gideon leaped forward. But even fast as he was, he was too late and too far away.
With a high-pitched scream of fury, Felix lost his footing and tumbled over the edge.
## Twenty-four
There was a sickening, distant thud, then silence descended like an ax.
Shocked, unable to credit what had happened, Gideon stood on the edge of the shaft. He couldn't see anything in the darkness. It went down too far.
"Farrell?" he called. During his childhood, a miner had fallen down the shaft and died. It was one of the reasons the workings were abandoned.
He called again, recognizing the act as futile.
He'd despised Felix, wanted to make him pay in blood and suffering for hurting Charis. But all the same, this was a sorry end for anyone, even the most despicable cur.
Dizziness struck from nowhere, and he swayed. He ached from the beating and the fight. Through the buzzing in his ears, he heard Charis's husky cry as she launched herself after him.
Still reeling, he staggered to face her and caught her up against him, hiding the black chasm behind him from her sight. His shaking arms lashed around her slender softness with a desperation he only now let himself acknowledge.
She's here. She's unharmed. Thank You, God and all Your angels.
The still, cold watches of the night had tortured him with the devastating possibility that he'd never see her again. A prospect more agonizing than Hubert's punches or Felix's childish taunts. So much worse than his persistent fear that his demons would emerge from the dank darkness to claim him. His raw anguish made a mockery of his plans to send her away, even when he knew it was for her own good.
"Oh, my love, my love," he whispered, and buried his face in her thick, silky hair. He drew in a shuddering breath full of her scent. She smelled warm and alive. Clutching his back as if she never meant to let him go, she quivered in his arms.
For a long, glorious moment, he held her and luxuriated in the knowledge that they'd come through, that they were alive and together. Giddy relief swamped his rage that she put herself in danger. He should have known she'd never leave his rescue to others. Not his brave Charis.
"You're safe," she choked out against his skin. "You're safe and you're...you're well. Oh, Gideon, I was so afraid." She finished on a broken sob and pressed her hot face into his bare chest, above his furiously pounding heart.
He forced himself to relax his bruising grip. The reality slowly dawned on his dazed mind that the threat had passed. He drew far enough away to see her. Even in the dim light from the tunnel mouth, the strain she'd been under was apparent in the muddy brown of her eyes and the dark marks underneath them. But her face was aglow with relief and happiness. And love.
"My darling..." Words failed as love surged up as unstoppable as high tide into Penrhyn Cove. "Are you crying for Felix?"
"No." Then more strongly. "No! What happened to him is horrible. But I'm crying because...because we're free at last."
He smiled down at her, then winced when the expression tested his torn lip. "Happy tears?"
She gave a jerky nod. "Happy tears." Regret shadowed her eyes as carefully she touched the graze on his mouth. "They hurt you. I'm so sorry."
"It's nothing." Truly, it was nothing. In return for the joy of having her in his arms, he'd undergo a thousand beatings. He pressed her shaking hand against his cheek. With every minute, he breathed more easily. The danger was over. He could hardly believe it.
He heard footsteps approach and looked up to see Akash striding down the tunnel with a flaming torch. At his side, Tulliver carried the lantern from the brothers' camp. The extra lights were welcome although Gideon doubted they were strong enough to reveal the base of the shaft. The ominous silence behind him confirmed his immediate guess that Felix had perished in the fall.
"You heard what happened?" Gideon asked.
"Yes. Is there any chance he survived?" Akash raised the torch in Gideon's direction, clearly checking to see if he was all right.
"I doubt it. But we need to get him out. Tulliver, can you muster some men to climb down? I assume someone brought rope. If not, the Farrells had some." His arms tightened around Charis. He'd come so close to losing her, he wasn't ready to relinquish her yet, especially when she still trembled with reaction.
"Aye, guvnor." Tulliver cast Gideon and Charis a cryptic glance, then headed back outside.
Gideon stared over Charis's ruffled head to where Akash waited. Overwhelming gratitude flooded him. How could he thank this man for all he'd done? Through the years of danger in India, the rescue from Rangapindhi, and the care and loyalty since. Words were inadequate recompense, but they were all he had.
"Thank you, my friend," he said gruffly. He wanted to say so much, but he settled for, "Once again you've saved me."
"You're welcome. Life would be considerably less interesting without you." Smiling faintly, Akash inclined his dark head in acknowledgment. "The true gallantry was Lady Charis's. It was she who rode through the deluge to bring us word of what had happened."
Gideon smiled down at her. He didn't need Akash to point out how exceptional his darling was. What a wife he'd found for himself. Strong enough to defy the world for him. "I knew she wouldn't fail. I knew she'd rout her stepbrothers."
"You didn't say that at the time." Her voice was choked.
"I didn't have to."
Looking pensive, she glanced toward the ominously quiet mineshaft. "I'm not hypocritical enough to say I'm sorry."
"Yet..."
She cast him a quick, understanding smile. "Yes. Yet." She looked around the dark, cold passage and shivered. Four villagers passed them with respectful nods and began to organize the retrieval of Felix Farrell. "Let's get out of here."
"Capital notion." Akash stood back to let Gideon and Charis precede him. As Gideon passed, Akash reached out to clasp his shoulder in a brief gesture of affection.
After the mine, daylight dazzled. Gideon placed a steadying hand on Charis's arm. The day was fine, and sun sparkled on puddles and dripping foliage. The air smelt fresh and clean. He sucked in a deep breath, savoring the sea's salt tang.
The scent of Penrhyn. The scent of home.
The crowd outside made him brace for the familiar sick haze. He felt Charis's loving concern as she slid her arm around his waist.
But when he surveyed the welcoming faces turned toward him, he was only aware of open sky and clear air, the breeze against his skin, Charis's enticing warmth pressed to his side.
Had his wife spoken more truly than she realized? Was he finally free?
The shock was too much.
He staggered. His sight narrowed to a single beam of light.
"Gideon, what is it?" Charis's hold tightened. As ever, her touch anchored him. His shaking arm twined around her slender shoulders, and he fought not to lean on her as his legs threatened to fold beneath him.
The wave of light-headedness passed, leaving him lost, bewildered. What had happened? Since Rangapindhi he'd been unable to endure people around him. So many defense mechanisms had become second nature.
Yet today he needed none of them.
His whirling mind struggled to make sense of it all. Now he thought about it, the demons should have tormented him long before this. Yet they'd been remarkably silent. Felix and Hubert's kidnapping hadn't sparked an attack. Nor, more significantly, had captivity in the dark tunnel.
But he'd been blisteringly angry when they took him. With the brothers and more, with himself, for placing his wife in danger.
The anger had passed, and still there were no screaming ghosts in his head. He stared at the villagers. He looked past them to Sir John Holland and the militia, surrounding a shackled Hubert. Then he sought and found the two men who had stood by him through so much. Tulliver watched expressionlessly from Akash's side. Akash's gaze as he surveyed Gideon and Charis was steady and unsurprised.
He knew the signs of Gideon's illness better than anyone. Gideon was sure he wouldn't have survived the worst attacks without Akash's arcane medical knowledge. What did his friend make of this abrupt change?
Then, with another shock, Gideon remembered that Akash had touched him without hesitation in the mine.
"You know, I think I'm all right," he said in a thick voice to Charis, who stared up at him with shining eyes. Did she too guess what had happened?
His dreams had been so humble yet so out of reach. Had heaven relented after all his pain? It seemed beyond belief.
"I need to talk to Hubert," Charis said quietly. "He shouldn't learn about Felix from a stranger."
"That's a consideration the cur hardly warrants," Gideon said grimly. She was so strong. If she hadn't been, she'd have given up on her husband weeks ago.
"Nevertheless, I must do it."
Reluctantly, Gideon released her, immediately missing her nearness. He watched as she crossed to where a chained and guarded Hubert waited in sullen, fulminating silence. Even with Hubert shackled, Gideon fought the illogical urge to drag her back into his arms, where she was safe. Would this instinct to protect her ever fade? Not while he breathed.
Across the open area, Hubert let loose a broken groan. The bulky brute swiftly went from surly resistance to utter collapse. Tears poured down his face. Charis said something, and placed a hand on his shoulder. He accepted her comfort, much as he didn't deserve it. Gideon felt another surge of admiration for his wife's generosity. If the decision were left to him, he'd let the bastard suffer.
Sir John approached, smiling, extending his hand. Dazed, Gideon returned the handshake. How simple the gesture was. Only a day ago, it would have been a painful ordeal.
"Sir Gideon, rum doings indeed. I can't say how pleased I am to find you unharmed."
"Thank you, Sir John." Astonishment and wonder still gripped Gideon. The change was too sudden for him to trust although with every minute, it became more likely that the impossible had occurred.
"I take it the other villain is incapacitated inside the mine?"
Gideon forced himself to concentrate on immediate matters. So difficult when unfamiliar happiness bubbled up like a new stream. He gave the magistrate a short recounting of events from when he'd discovered Charis in Winchester.
Akash joined them. When Gideon performed introductions, Sir John, to give him credit, displayed only a moment's confusion at meeting Akash Stamford, the new Viscount Cranbourne.
"What happens now?" Gideon kept an eye on Charis and the distraught Hubert.
"We'll take Lord Burkett to London for trial. You'll likely be called to appear." Sir John looked tired and troubled. "I can't see him escaping the noose. If you'll come with me now, we can..."
Akash smoothly interrupted. "I'll start the formalities. Sir Gideon was held overnight. His lady has ridden through a storm and managed no sleep since. Let the Trevithicks go home."
Looking abashed, Sir John cleared his throat and nodded. "Of course. Wasn't thinking. Nothing that can't wait. Appreciate your cooperation, my lord."
The Penrhyn men emerged from the mine, holding an unmoving, black-coated body. Gideon saw at a glance there was no hope for Felix. He met Charis's gaze across the clearing and shook his head. She nodded but remained dry-eyed. Hubert's painful, choked sobs intensified as the villagers carried Felix's limp form past him.
With each moment, Gideon became easier in his skin. He moved among the local men, thanking them. Nobody needed to tell him they'd braved the storm to find him. Hard to believe it had taken him twenty-five years to recognize the unbreakable bonds that tied him to this land and its people.
Tulliver came up, leading Khan. One mount for two riders, the sly dog. With a word of thanks, Gideon took the reins and rubbed the horse's nose in greeting. He'd missed the spirited thoroughbred over the last weeks.
With his usual impassivity, Tulliver handed over the coat he carried under his arm. "Here you are, guvnor. Thought you might need some extra covering"
Gratefully, Gideon pulled the garment over his ragged shirt. He must look a ruffian. He badly needed a bath and a change of clothes. He needed a shave and a hot meal. More than anything, he needed time alone with his wife. With relief, he watched Charis move away from Hubert and approach Sir John.
He turned his attention to Tulliver. Akash wasn't the only one who deserved his undying gratitude. "I appreciate it. Just as I appreciate your rushing to my rescue today."
"I'm honored to serve you, sir." Tulliver's eyes held a hint of uncharacteristic softness. "Always have been. And I reckon gratitude goes both ways. You won't recall, but I was one of the soldiers who pulled you out of that pit in Rangapindhi, more dead than alive."
Astonishment gripped Gideon at this revelation. "By God, I never knew."
"My last assignment for the Company. Those heathens we locked up after the invasion talked like you was a god. They'd never seen such grit. Nothing they done could break you." Tulliver's voice deepened with feeling. "You kept your mouth shut and saved me and my chums from a bloodbath. When I heard you sailed home on the same packet as me, I set myself to enter your service."
Gideon tried to remember the exact moment he'd offered Tulliver a place. The details were hazy. When he'd been delirious with fever on the ship, Tulliver had turned up to help, and he'd been around ever since. Capable, resourceful, taciturn. In fact, that was the longest speech Gideon had ever heard the man make.
"I haven't been an easy master," he said with difficulty.
"Maybe not always, lad, but I knew you'd come right, given time and incentive. Gold always rings true."
Gideon swallowed a lump of emotion. He owed this man more than he could ever repay. "You know you've always got a home at Penrhyn." Shabby return for the selfless devotion.
Tulliver's wry smile appeared. "Aye, guvnor. I'd counted on that and all. A nice quiet life by the seaside in my old age suits me down to the ground. Although it's not exactly been quiet so far."
Gideon laughed with a lightheartedness he couldn't remember feeling for years and clapped Tulliver on the back. Another natural gesture unthinkable yesterday.
His pulse racing with a mixture of anticipation and trepidation, Gideon led Khan up to Charis. He felt like a nervous schoolboy. Absurd after all he'd been through with her. But recent events had created a new map between them, and he wasn't yet sure how to navigate it.
"I should get you back to Penrhyn." Before she could object, he caught her by the supple waist and tossed her up onto Khan's back.
She laughed breathlessly and found her balance with the confidence of a natural horsewoman. "Apparently I have no say in the matter."
"None at all." He ignored her startled eyes and turned to shake Sir John's hand again. "Come by the house tomorrow, and we'll sort everything out."
"I wish you good day, Sir Gideon, Lady Charis." The man's eyes held a spark of amusement. Obviously, he hadn't forgotten what it was to be young and newly married.
Gideon shoved one booted foot in the stirrup and flung his other leg across the saddle. The high-strung horse danced under the double weight, but Gideon quickly brought him under control.
Charis sat across the front of the saddle, her back against Gideon's arm, her skirts cascading down Khan's side. He relished her sweet warmth. She wasn't wearing a hat, and strands of soft bronze hair tickled his chin.
Raising a hand to Akash, who watched them with a faint smile, Gideon urged Khan to a canter along the path to Penrhyn.
"That was high-handed," Charis said in a neutral voice once they were away from the crowd. Gideon noted she made no great effort to wriggle away. No effort at all, really.
He laughed and tightened his hold on her. "Black Jack lives in my veins, remember?"
He slowed Khan to a walk. The need to get back to the house and confirm she was his in the most basic way was a fever in his blood. But he wasn't a barbarian, much as he felt like one right now. They had to talk before he tumbled her into his bed.
She turned her face toward his. Her expression was unexpectedly grave. "Does all this mean you no longer want to send me away?"
Uncomfortable heat crawled up the back of his neck. "I never wanted to send you away."
"Nevertheless that was—is—your plan."
She wasn't letting him wriggle out of this. He knew he had to lay his heart before her like a tribute before a despotic queen. Good God, he owed it to her, after acting such a self-righteous clodpole.
"That's something we need to discuss."
She arched her eyebrows. Suddenly the grande dame. "Oh?"
"I think...I believe...I hope..."
He stopped. Damn, he made a hash of this. Drawing a deep breath, he strove to present his case with a modicum of address. "I seem to have overcome my...problem." At least it was a complete sentence, even if he stumbled over the last word.
He'd never settled on how to describe the creeping horror that suffocated him when the ghosts of Rangapindhi howled. In his mind, he'd always called his affliction the demons, but that seemed too melodramatic a description in the clear light of day.
Charis's eyes were unwavering. "I know."
He made a frustrated sound deep in his throat. "Curse you, you don't sound very pleased."
"Of course I'm pleased."
"Or surprised." He spoke over the top of her declaration.
"You forget I saw you in the mine. I've never beheld a man more in control of himself or circumstances. Even bound as you were." Her voice softened. "What happened, Gideon?"
"It's hard to explain." He paused, seeking the words. "It goes back to learning to touch you. That changed the world for me."
"And after all that, I nearly lost you when you handed yourself over to my stepbrothers." He couldn't mistake the anger in her voice or the furious gold sparks in her eyes.
"I'd die to keep you safe." He spoke from the depths of his heart. "You know that."
"Yet you say you're not a hero," she said bitterly.
"I'm just a man, Charis. But protecting you is part of who I am. You can't ask me to change that. I couldn't, even if I wanted to." His voice lowered to persuasion. "Come, sweetheart, let's make peace."
"I suppose I'll forgive you." There was a misty light in her eyes as she surveyed him. "Eventually."
The time had come. His gut clenched with nerves as he realized his happiness depended on the next few minutes. She wouldn't call him a hero if she knew the sheer unadulterated terror that closed his throat. He meant to offer her everything he was and everything he had. If she refused him, she'd cast him into darkness again.
"Walk with me. It's not far to the house." Over the next rise, they'd see the sea and Penrhyn. Home.
He drew Khan to a halt, slid to the ground, and lifted her down. His hands lingered at her slim waist, and again he fought the impulse to kiss her. They must settle everything first. Then, God help her, she'd spend the next week naked in his bed.
Hell, the next month.
They fell into step on the pale winter grass. The sun shone warm on his head, bright promise of a new spring.
For a few moments, they walked shoulder to shoulder, him leading a placid Khan. Gideon tugged off his gloves and grabbed her hand. He'd tried to resist touching her, but it was impossible. The memory of her, her voice, her face, her sweetness, were all that had sustained him through the long, dark night of captivity. He needed to have her near more than he needed air to breathe.
Her fingers twined around his bare scarred hand with a welcome that made his heart stumble to a lovesick halt. Despite his hunger for her, he found himself reluctant to shatter this sweet idyll. There had been so much strife and anguish between them, this serenity seemed a benediction.
Typically, Charis was the one to confront all that lay unspoken. "Gideon, what happened at the mine?"
"I found myself again." It was as close to the truth as he could manage. "You changed me. The memory of you kept me from losing my mind. And as the night went on, I discovered the dark was just the dark and people just people. The wild fancies of my imagination...vanished." He put a vague thought into words. It was as good an explanation as any for the glorious change that had overtaken him. "A miracle."
"No." Her voice sounded husky as it always did when she succumbed to deep emotion. "It's no miracle. Your own courage brought you clear of the storm. You faced your horrors when you surrendered yourself to my stepbrothers for my sake."
Was she right? Would he ever know? It didn't matter why he'd changed. What mattered was he had changed. "And being tied up in a mine gave me ample time for reflection."
Charis released a spurt of unwilling laughter. "You sound like you recommend a period of incarceration."
He gave a dismissive huff. "I wouldn't go that far." He sobered. He floundered for an explanation that made sense. Difficult, when none of it made sense to him. "I have to live with what happened in Rangapindhi. It wasn't my fault my colleagues died..."
"But your conscience lacerated you because you couldn't save them. It's that overdeveloped protective instinct again."
"I despised myself for living when they died."
The words hung stark in the air. Her hand tightened around his. The silent communication crushed the seeds of self-hatred still lurking in his heart. Her voice vibrated with sincerity. "My love, if you hadn't lived, you couldn't have saved me. The workings of destiny are mysterious."
Her words echoed the odd moment of perception last night where he'd struggled to view himself as an outsider would. When he'd felt the shades of Parsons and Gerard hover uncannily close in the thick darkness, so reminiscent of the pit where his friends had died.
He'd always imagined his colleagues must hate him from beyond the grave for living when they'd perished in pain and humiliation. But the spirits that kept him company through the long hours of blackness in the mine had been benign, not angry at all. Ever since Rangapindhi, he'd remembered them as gruesome specters. Last night they'd visited him as they'd been in life. Fine, brave men who had sacrificed everything for duty.
Only then, blessed by his dead colleagues at last, had Gideon taken the most terrifying step of all.
He'd contemplated establishing a life at Penrhyn with Charis and, God willing, children. Trevithicks to fill the rambling old house with laughter and chaos and love. That hope had sustained him through the darkness and the violence and the incarceration. He wanted to build on the love that already grew between him and Charis and stoke it into a blazing, endless fire to light his days.
If she agreed.
His hand closed ruthlessly around hers. "And I thought about you."
"I should hope so," she said unsteadily. She looked up at him, and he caught the sparkle of tears in her hazel eyes.
"I thought how I love you and what an arrogant ass I've been." He paused and spoke with difficulty. "Last night, I realized I'd reached the limits of selflessness. I sat in that cave and imagined living without you. I couldn't bear to contemplate it."
She lifted her free hand to touch his face in a gesture that cut right to his aching heart. "Oh, my love, you don't have to live without me."
He came to a standstill. "Charis, I can't promise I'm cured, I can't promise anything beyond my eternal love. But you need to know I'll never willingly give you up. You're mine forever."
The radiant certainty in her eyes warmed him to his bones. "Gideon, I love you. You love me. That's all that matters." Her smile took on a hint of seduction that fired his blood. "Now take me back to Penrhyn and swive me silly."
Her gaze held no questions. She accepted him as her future just as he accepted her. More than accepted. Greeted with open arms. His doubts melted away like snow under the sun. He'd have time for explanations and apologies later. Or perhaps explanations and apologies would never be needed.
He dropped Khan's reins. "Come here, Charis. If I don't kiss you, I'll go mad."
Laughing, she fell against him. The kiss was an act of passionate gratitude for their survival, a wild melding of lips and tongues and teeth. It was a physical expression of a love that touched his soul. A love he knew would last the rest of his days. They were both breathless and trembling when they finally drew apart.
He lifted her up on Khan's back and leaped into the saddle behind her. "Hold on!" he shouted and headed for home at a breakneck gallop.
Khan came to a rearing stop in Penrhyn's front court, his hooves clattering on the stone paving. For Charis, the ride had passed in a rapturous blur of wind and color. She clung to Gideon, lost in a tumult of emotion. That extraordinary kiss still heated her blood, made her heart thunder.
A groom dashed out to hold the restive horse while Gideon jumped down and tugged Charis after him. Her feet fleetingly touched the ground before he swung her into his arms.
"Gideon!" she gasped, as he strode up the worn stone steps to the front door that opened as if by magic. Her heart swooped and skipped a beat. She felt like she was being kidnapped. It was incredibly exciting. "You make me breathless."
"I will before I'm finished," he promised in a low voice, marching into the house.
Ooh, yes, please.
She hooked one hand around the strong column of his neck as he passed the curtsying maid who had opened the door. The dark, cavernous hall flashed past, then they climbed the staircase. He turned at the landing and swept her into his room.
She'd never been in here. She had a momentary dazzling impression of light and casement windows opening onto a sparkling sea. Old carved furniture. A breeze smelling of the ocean.
Gideon started kissing her, and she didn't care where she was as long as he never let her go. She closed her eyes and gave herself up to the hungry predations of his lips.
"I love you," she said over and over in broken sentences, kissing his face and his neck and the skin revealed by his torn shirt. What exquisite freedom it was, finally to say the words without restraint.
He kicked the door so it banged shut behind him and carried her to the bed. He came down over her, kissing her as if he starved. The musky scent of his arousal filled her senses. Ruthlessly, he tugged his coat off and tossed it to the floor.
She'd long since known he wanted her. Their days in Jersey had been replete with sensual exploration. But the unfettered desire in his touch now was new. The barriers he'd always raised against her in his heart had dissolved to nothing.
She'd never felt claimed before. She felt claimed now.
And reveled in the possession because she knew he gave himself into her keeping with every touch, every kiss.
Shaking, she ripped at his shirt while he feverishly ran his hands over her body. Her breasts swelled and ached for his touch. She yearned to feel the glide of his skin against hers. She yearned to welcome him inside her in the most intimate touch of all.
Impatience made her clumsy. She ended up tearing the ruined shirt until it fell in shreds from his heaving shoulders.
He tugged her upright and struggled to undress her while she rained kisses across his bare torso. Fresh bruises and abrasions marked his scarred skin. Reminder of what he'd endured for her sake. She bit down delicately on one light brown nipple and felt fierce reaction shiver through him. She did it again, harder this time.
"Devil take it, Charis. I'm filthy. I need a shave." He clamped one powerful hand in her hair and drew her head back from his chest. His face was vivid with arousal. Color bloomed along his cheekbones and his eyes burned like black fire. "Do you want me like this?"
She laughed low in her throat and tore at his breeches. If he was a new, more dominating lover today, she'd transformed into an utterly shameless hussy. "Yes."
"So be it."
His face set with determination. Roughly, he wrenched open the jacket of her riding habit. Buttons flew through the air and bounced across the floor. He tore at the white shirt beneath. Within seconds, skirt, stays, and shift lay on the carpet.
The abrasion of his shadow beard on her naked skin made her cry out in delight. She arched so her breasts jutted forward, demanding attention. She fumbled at her hair, sliding pins free until it fell about her shoulders in an untidy mass.
"I love you," he groaned, burying his hands in the tumble of hair and bringing her up for a famished kiss. "How I love you."
"Tell me again," she said in a vibrating voice.
He did. Repeating the declaration, he pushed her back onto the silk bedcover. He kissed her breasts and belly and dragged boots, drawers, and stockings off until she lay before him bare, open and ready.
Soon he was naked too. He surged above her. No more preliminaries. She didn't mind. She craved this joining as much as he.
He thrust hard, as if proclaiming her his kingdom.
Then he lifted his head and stared at her with such reverence, her heart cartwheeled. She drew him down for a deeply passionate kiss. Even as her tongue pushed into his mouth, his body moved in hers.
The rhythm, his weight, his spicy scent, the heat of his skin, all were familiar. All were utterly new.
As they spiraled into ecstasy, her soul expanded, took flight. The experience was unlike anything before, for all their desperate passion in St. Helier. It was as if he flung open every gateway and invited her in. And she entered victorious to blaring trumpets and fluttering banners.
The triumphal music reached a dazzling climax. She arched and cried out as her world erupted into blinding light. Bright angels chorused around her, repeating one phrase over and over in harmonies that made her skin sing.
I love you, Gideon. I love you, Gideon.
She quivered with wild delight, lost in the brilliance. She knew Gideon was with her. He'd be with her always.
She opened her eyes to find him raised on his arms above her, watching her intently. His black gaze glittered in unmistakable possession as he looked down at her.
"I've never..." Her voice trailed away.
He looked like a man who had conquered the world. He looked like a man who had fallen in love deeply and irrevocably.
"I know."
She raised one shaking hand to his cheek. His gaze was intent and told her he, like she, had been reborn in fire.
She'd never seen his eyes so clear, so unguarded, so full of love. Her heart overflowed with a happiness more precious because she'd believed it eternally out of reach.
"No shadows," she whispered, at last recognizing the difference in his face.
"No shadows," he echoed. He bent his head to kiss her, silent promise of a shining future.
## Acknowledgments
I always love writing the acknowledgments page, where I get to thank everyone who helped me along the way with a book.
As always, my deepest gratitude goes to everyone at Avon Books in New York. I'd particularly like to thank May Chen, my fabulous editor, the brilliant art department, and Pam Spengler-Jaffee and her amazing team in marketing. I'd also like to thank the great people at Avon Australia, especially Linda Funnell, Shona Martyn, Cassie Marsden, Christine Farmer, and Jordan Weaver.
Special thanks to my wonderful agent, Nancy Yost. Nancy, I love working with you!
This book involved some tricky research questions. I'd particularly like to thank Brenda Ross from the Société Jersiaise's Lord Coutanche Library on Jersey in the Channel Islands, who rushed to save me when all other avenues of investigation had come to an end.
Thanks, as ever, to my wonderful critique partner, Annie West. I'd also like to express my heartfelt gratitude to Sharon Archer and Christine Wells, who came to my rescue at a difficult moment in this book.
I'd like to thank all my friends at the Romance Bandits, Romance Writers of Australia, and Romance Writers of America. You're an unfailing source of inspiration. Thanks also to Michelle Buonfiglio, Kim Castillo, Maria Lokken, and Marisa O'Neill for your stalwart support.
Finally, a big thank-you to all those readers who have enjoyed my books! I can't tell you what a thrill it is to know that my stories have touched your hearts.
## About the Author
Always a voracious reader, ANNA CAMPBELL decided when she was a child that she wanted to be a writer. Once she discovered the wonderful world of romance novels, she knew exactly what she wanted to write. Her first historical romance for Avon Books was the multi-award-winning Claiming the Courtesan, followed by another Regency noir, Untouched, which was featured on several "best of the year" lists. When she's not writing passionate, intense stories featuring gorgeous Regency heroes and the women who are their destiny, Anna loves to travel, especially in the United Kingdom, and to listen to all kinds of music. She has settled near the sea on the east coast of Australia, where she's losing her battle with an overgrown subtropical garden. Anna loves to hear from her readers. You can contact her through her website at www.annacampbell.info.
Visit www.AuthorTracker.com for exclusive information on your favorite HarperCollins author.
## By Anna Campbell
CAPTIVE OF SIN
TEMPT THE DEVIL
UNTOUCHED
CLAIMING THE COURTESAN
## Copyright
This book is a work of fiction. The characters, incidents, and dialogue are drawn from the author's imagination and are not to be construed as real. Any resemblance to actual events or persons, living or dead, is entirely coincidental.
CAPTIVE OF SIN. Copyright © 2009 by Anna Campbell. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, down-loaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of HarperCollins e-books.
Adobe Digital Edition September 2009 ISBN 978-0-06-195924-0
10 9 8 7 6 5 4 3 2 1
## About the Publisher
Australia
HarperCollins Publishers (Australia) Pty. Ltd.
25 Ryde Road (PO Box 321)
Pymble, NSW 2073, Australia
http://www.harpercollinsebooks.com.au
Canada
HarperCollins Publishers Ltd.
55 Avenue Road, Suite 2900
Toronto, ON, M5R, 3L2, Canada
http://www.harpercollinsebooks.ca
New Zealand
HarperCollinsPublishers (New Zealand) Limited
P.O. Box 1
Auckland, New Zealand
http://www.harpercollins.co.nz
United Kingdom
HarperCollins Publishers Ltd.
77-85 Fulham Palace Road
London, W6 8JB, UK
http://www.harpercollinsebooks.co.uk
United States
HarperCollins Publishers Inc.
10 East 53rd Street
New York, NY 10022
http://www.harpercollinsebooks.com
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 107
|
Q: How to connect to Aptana studio 3 using ssh + password? (SFTP) I rented a VPS a few days ago and been messing around with it for a little while. One thing I'm really struggling with: getting an easy file exchange system for an eventual website I want to build. Ideally I'd want just a folder on my MacBook that I can link directly with an identical folder on the Linux server (preferably /var/www/) with Aptana Studio 3 (I really like their IDE).
So far my main problem is that I can't connect (through Aptana or other file systems like FileZilla) with SFTP on the server, because either I can input the private key to Aptana and it won't have the user's password, or I can input the password but it won't have the ssh key :(
I've tried several things, such as setting ssh's config strictmodes no, changing usePAM no and PasswordAuthentication no. All this does is not allowing me to login to the server anymore AT ALL, forcing me to hard reset everything on the server.
OS: Ubuntu 14.04 (64-bit)
Actions after fresh installation of the OS by DigitalOcean:
*
*adduser user01
*visudo insert user01 ALL=(ALL:ALL) ALL
*nano /etc/ssh/sshd_config change: PermitRootLogin no and add: UseDNS no (no idea why, but a tutorial of DigitalOcean told me to xD), AllowUser user01
These are all the steps I've taken after logging in to the server for the first time. On my home computer I created an ssh key and DigitalOcean (I think) used this as a verified key during the installation of the OS.
My main question: When connecting with ssh myself through the commandline, I use my private key to connect (I think), but I also have to type in a password to login as my user01. When adding a server in FileZilla or Aptana, I have the option to either choose SFTP with a password, or SFTP with key authentication, not both. Clearly I must be doing something wrong, so my question is what.
A: I noticed what happened: the first time I setup the server, DigitalOcean put my public SSH in the server's /.ssh/authorized_keys, but after my first reset this was lost, that's why I always had to login with a password. Manually nano'ing in the public key fixed everything :)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,447
|
Are you serious about opening an online clothing retailer however do not know what to do first? You will get males's and ladies's clothing for really low prices. Plus, a flexible returns coverage means that you can ship again anything you don't need after 28 days, which makes procuring online here fairly danger-free. With over 7,000 objects on-line and up to 75 per cent off, it has every little thing it's worthwhile to keep your style on point.
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,224
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The FunnyGO 2 handheld stabilization gimbal for the GoPro cameras with the most advanced technology gives you an outstanding performance for your photo and video shooting. The 32bit technology with Triple-MCUs offers you a more powerful shooting experience than ever before.
The ergonomic lightweight handle manufactured of an aluminium alloy construction is equipped with an easy-to-use mode button and a 4-way joystick. The 3 axes of the gimbal consist of a CNC aluminium alloy construction, too, which ensures a massive, stable and reliable video shooting experience.
The FunnyGO 2 handle functions as an additional battery recharger for your GoPro camera which gives you more time and flexibility for your capturing project.
Whatever your passion, there's a GoPro mount that will help you to get the shot. Mount your FunnyGO 2 to any of your gear or mount it on an extension pole at the ¼ (inch) screw hole at the bottom of the handle to capture an impressive footage of the moments that matter most.
The FunnyGO 2 is a master piece of engineering which combines an elegant design with high versatility and mobility. It's the beginning of a new era of video shooting experience with your GoPro camera.
Main unit with hingle design, can easy to use in difference angle handle.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,741
|
Юлія Гергес була чинною чемпіонкою, але не змогла захистити свій титул, оскільки в другому колі її перемогла Саманта Стосур.
Уперше після турніру Вімблдон 2009 всі чотири перші тенісистки світу досягли півфіналів.
2-га ракетка світу Марія Шарапова виграла титул, у фіналі перемігши 1-шу ракетку світу Вікторію Азаренко з рахунком 6–1, 6–4, зумівши відіграти матч-бол у другому сеті чвертьфіналу проти Стосур.
Сіяні гравчині
Перші четверо сіяних тенісисток виходять без боротьби в друге коло.
{{columns-list|2|
Вікторія Азаренко (фінал)
Марія Шарапова (переможниця)
Петра Квітова (півфінал)
Агнешка Радванська (півфінал)
Саманта Стосур (чвертьфінал)
Каролін Возняцкі (2-ге коло)
Маріон Бартолі (2-ге коло)
Лі На (чвертьфінал)
}}
Сітка
Фінальна частина
Нижня половина
Кваліфікація
Сіяні гравчині
{{columns-list|2|
Сорана Кирстя (1-ше коло)
Цветана Піронкова (1-ше коло)
Івета Бенешова (кваліфікувалася)
Александра Возняк (1-ше коло)
Карла Суарес Наварро (1-ше коло)
Барбора Заглавова-Стрицова (2-ге коло)
Катерина Бондаренко (кваліфікаційний матч, щасливий лузер''')
Грета Арн (кваліфікувалася)
}}
Кваліфікувалися
Щасливі лузери
Катерина Бондаренко
''' Акгуль Аманмурадова
Сітка
Перший кваліфаєр
Другий кваліфаєр
Третій кваліфаєр
Четвертий кваліфаєр
Посилання
Main Draw
Qualifying Draw
Porsche Tennis Grand Prix
2012
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 970
|
Q: My regular expression is not being cool My current one works almost perfectly but it misses out any other digits after a character like - or /.
The Original String is something like:
#!012 , #!02/09#!011 #!04-072
My current one works on stuff like:
$viewedResult = preg_replace('/#!([0-9A-Z]{1,4})/', '<a href="updateMooring.php?Number2=${1}&id='. $id .'">${1}</a>', $viewedResult);
This would give me something like:
<a href="updateMooring.php?Number2=04&id=db335b">04</a>-072<br />
but I want the "-072" in the Number2 bit like:
<a href="updateMooring.php?Number2=04-072&id=db335b">04-072</a>
It could also be able to include /072s:
<a href="updateMooring.php?Number2=04/072&id=db335b">04/072</a>
Any ideas? Remember that there is a #! in front of the number!
A: You could use something like:
preg_replace('~#!([0-9A-Z][0-9A-Z/-]{0,5})~', '<a href="updateMooring.php?Number2=${1}&id='. $id .'">${1}</a>', $viewedResult);
A: I would split up the regular expression in two, as I think you want to match something like \d{1..4}[-/]?\d{1..4} where the first and second set of digits total 4. It's probably better just to look for (\d+[-/]/d+) and do a check afterwards if the total digits size is correct, e.g. finding all matches of (\d+) in the result, and programmatically check their length.
In general, I would pay close attention to what exactly is allowed input, and what (error) you will get if the input does not match. Splitting things up will make it much easier to show a correct error message, such as "number of digits in mooring spot incorrect" (which is I presume what this is about).
A: If you want the value of Number2, then use:
Number2\s*=\s*([^;"]*)[;"]
And then use the first capture group.
A: Stupid question: why not using this regular expression /#!([0-9A-Z/-]{1,4})/ ?
Side note: the {1,4} blocks you will retrieving the value 04-072 (it's 6 chars long).
Hope that helps :)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 487
|
'use strict';
describe('timeslider', function () {
// create a new pad before each test run
beforeEach(function (cb) {
helper.newPad(cb);
});
/**
* @todo test authorsList
*/
it("Shows a date/time in the timeslider and make sure it doesn't include NaN", async function () {
this.timeout(12000);
// make some changes to produce 3 revisions
const revs = 3;
for (let i = 0; i < revs; i++) {
await helper.edit('a\n');
}
await helper.gotoTimeslider(revs);
await helper.waitForPromise(() => helper.contentWindow().location.hash === `#${revs}`);
// the datetime of last edit
const timerTimeLast = new Date(helper.timesliderTimerTime()).getTime();
// the day of this revision, e.g. August 12, 2020 (stripped the string "Saved")
const dateLast = new Date(helper.revisionDateElem().substr(6)).getTime();
// the label/revision, ie Version 3
const labelLast = helper.revisionLabelElem().text();
// the datetime should be a date
expect(Number.isNaN(timerTimeLast)).to.eql(false);
// the Date object of the day should not be NaN
expect(Number.isNaN(dateLast)).to.eql(false);
// the label should be Version `Number`
expect(labelLast).to.be(`Version ${revs}`);
// Click somewhere left on the timeslider to go to revision 0
helper.sliderClick(1);
// the datetime of last edit
const timerTime = new Date(helper.timesliderTimerTime()).getTime();
// the day of this revision, e.g. August 12, 2020
const date = new Date(helper.revisionDateElem().substr(6)).getTime();
// the label/revision, e.g. Version 0
const label = helper.revisionLabelElem().text();
// the datetime should be a date
expect(Number.isNaN(timerTime)).to.eql(false);
// the last revision should be newer or have the same time
expect(timerTimeLast).to.not.be.lessThan(timerTime);
// the Date object of the day should not be NaN
expect(Number.isNaN(date)).to.eql(false);
// the label should be Version 0
expect(label).to.be('Version 0');
});
});
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,150
|
Please remove my old card and replace it with the new card. I called in my new card and was told that you would remove the old one. Each time I try to place an order the old card comes up and said order is rejected. That card was cancelled last year.
I'm sorry to hear this hasn't been updated as expected. Unfortunately for security reasons, we can't make any card changes via the forums. You'll need to call in to customer service at 1-800-933-2887 from 8 a.m. – 1 a.m., Eastern Time for assistance with this.
Thank you all for congratulating me. It is a lot of fun to win! I wish that for all of you.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,205
|
{"url":"https:\/\/stats.stackexchange.com\/questions\/125662\/geocoded-node-map-visualizers\/125667","text":"# Geocoded node map visualizers?\n\nI'm looking to create a map like the one below for shipment volume between cities (which I have geocoded). I found the image below created in Gephi, however I'm wondering what other tools exist that can do similar visualizations. I see it's possible in R though a code example isn't posted.\n\nNot so long ago I was using Gephi to do similar visualizations for graphs. The program is easy to use and it will not take more then a day to learn majority of the features. I would say that by the scale powerfullness \/ ease of use it will be the first one.\n\nYou can also do this with a lot of JS libraries (few examples):\n\nFew examples with D3. But it is much harder to produce such graphs there.\n\nJust for illustration purposes, I will add here a few images created with Gephi:\n\n\u2022 Nice! Though the questions specifically asks for option other than Gephi (the first link was actually from Gephi), so I am not sure if OP will find this answer useful. \u2013\u00a0Penguin_Knight Nov 27 '14 at 3:41\n\u2022 @Penguin_Knight oh, I have not looked at the link. I just read the question and noticed R. Sorry for this. If someone will be too much frustrated with this answer - please write me to remove it. One more time, sorry. \u2013\u00a0Salvador Dali Nov 27 '14 at 3:44\n\u2022 Do not worry, I just commented so that you realize that. It seems like a neat program and deserves to be shown as an answer. \u2013\u00a0Penguin_Knight Nov 27 '14 at 3:46\n\u2022 @Penguin_Knight thank you. Good things about it is easy learning curve, a lot of functionality out of the box, easy to modify with custom plugins. \u2013\u00a0Salvador Dali Nov 27 '14 at 3:47\n\u2022 @ElPresidente this is not the only one, but I would say the easiest one to use and yet super powerful. I will add some JS libraries as well in a few minutes. \u2013\u00a0Salvador Dali Nov 27 '14 at 3:59","date":"2019-10-15 08:47:28","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.461028128862381, \"perplexity\": 736.3512834587407}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986657949.34\/warc\/CC-MAIN-20191015082202-20191015105702-00017.warc.gz\"}"}
| null | null |
Q: How to develop a code for grouping the Combobox How to group Combobox in c#.A picture format is shown here.
!http://www.brad-smith.info/blog/wp-content/uploads/2010/06/GroupedComboBox.jpg
A: create a string array of what you want,
string items = {"A","B","C"};
add this item source to your comboBox
comboBox1.ItemSource = items;
Most of these things are available on stackoverflow already. Try not to post questions like this without searching and getting info about it.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 971
|
using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
namespace MetaDslx.Languages.Soal.Generator.JavaEE.Webservice //1:1
{
using __Hidden_SoapInterfaceGenerator_499399644;
namespace __Hidden_SoapInterfaceGenerator_499399644
{
internal static class __Extensions
{
internal static IEnumerator<T> GetEnumerator<T>(this T item)
{
if (item == null) return new List<T>().GetEnumerator();
else return new List<T> { item }.GetEnumerator();
}
}
}
public class SoapInterfaceGenerator //2:1
{
private object Instances; //2:1
public SoapInterfaceGenerator() //2:1
{
}
public SoapInterfaceGenerator(object instances) : this() //2:1
{
this.Instances = instances;
}
private Stream __ToStream(string text)
{
MemoryStream stream = new MemoryStream();
StreamWriter writer = new StreamWriter(stream);
writer.Write(text);
writer.Flush();
stream.Position = 0;
return stream;
}
private static IEnumerable<T> __Enumerate<T>(IEnumerator<T> items)
{
while (items.MoveNext())
{
yield return items.Current;
}
}
private int counter = 0;
private int NextCounter()
{
return ++counter;
}
public string Generate() //4:1
{
StringBuilder __out = new StringBuilder();
return __out.ToString();
}
private class StringBuilder
{
private bool newLine;
private System.Text.StringBuilder builder = new System.Text.StringBuilder();
public StringBuilder()
{
this.newLine = true;
}
public void Append(string str)
{
if (str == null) return;
if (!string.IsNullOrEmpty(str))
{
this.newLine = false;
}
builder.Append(str);
}
public void Append(object obj)
{
if (obj == null) return;
string text = obj.ToString();
this.Append(text);
}
public void AppendLine(bool force = false)
{
if (force || !this.newLine)
{
builder.AppendLine();
this.newLine = true;
}
}
public override string ToString()
{
return builder.ToString();
}
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,689
|
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
namespace SlackAPI
{
public class DirectMessageConversation
{
public string id;
public string user;
public DateTime created;
public bool is_user_deleted;
public bool is_open;
public bool is_starred;
public DateTime last_read;
public Message latest;
public int unread_count;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,963
|
Website last updated: 20-1-2020
James Bond collection
James Bond Fan Clubs
James Bond games
James Bond literature
James Bond news
James Bond stars
Swedes in the Bond films
Bond 25 Britt Ekland Daniel Craig Dolph Lundgren George Lazenby Izabella Scorupco James Bond museum Kristina Wayborn Mary Stavin Maud Adams No Time To Die Pierce Brosnan Roger Moore Sean Connery Spectre Timothy Dalton
Thunderball Limited Edition Steelbook
By: FSWL team
Twentieth Century Fox Home Entertainment releases a special limited edition steelbook of Thunderball (1965) (1965) in conjunction with the films 50th Anniversary in December.
Sean Connery brings his characteristic style and magnetism to the role of Agent 007 once again in Thunderball. When merciless SPECTRE operative Emilio Largo steals two atomic bombs and threatens the U.S. and England with annihilation, Bond must travel to the Bahamas to foil the madman's plan and prevent a nuclear holocaust.
This limited edition (2000 copies) SteelBook™ features artwork inspired by the film's iconic opening title sequence.
Steelbook details:
Region: Region B/2
Sound Information: Dolby Digital
Classification: To be announced
Studio: Twentieth Century Fox
Retail price: 24.99 GBP
Other cast members: Adolfo Celi, Bernard Lee, Claudine Auger, Desmond Llewelyn, Guy Doleman, Lois Maxwell, Luciana Paluzzi, Molly Peters, Rick van Nutter
Director: Terence Young
Editor's note:
Steelbook editions of From Russia with Love, Goldfinger, You Only Live Twice, On Her Majesty's Secret Service, Diamonds Are Forever, For Your Eyes Only and Casino Royale is also available.
For more James Bond films on Blu-ray featured on From Sweden with Love, click here. Also possible to order some of them from FSWL's film store.
Photo above:
Cover for the steelbook edition of Thunderball, the 4th James Bond film. © 2015 Twentieth Century Fox Home Entertainment. All rights reserved.
Pre-order the limited edition steelbook of Thunderball exclusively from Zavvi:
www.zavvi.com/blu-ray/thunderball-zavvi-exclusive-limited-edition-steelbook/11146686.html
#blu-ray_films
#products
All information, text and graphics (unless otherwise stated) on this website are protected by copyright law. Please contact us to use anything.
This website is not in any way endorsed by EON Productions Ltd, Danjaq, Metro-Goldwyn-Mayer, Sony Pictures, United Artists, Ian Fleming Publications, or any other James Bond copyright holders. It is an independently run non-profit website from a personal basis in spare time.
James Bond film images © 1962 - 2020 EON Productions Ltd, Danjaq LLC, MGM, Sony Pictures and United Artists Cooperation
James Bond book covers © 1953 - 2018 Ian Fleming Publications and Glidrose Productions.
Founder & Managing Editor: Anders Frejdh
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,993
|
1. WO2020222683 - LOAD BALANCING SYSTEMS AND METHODS
LOAD BALANCING SYSTEMS AND METHODS
[001] Disclosed are embodiments related to load balancing systems and methods.
[002] As cloud technologies are introduced in the 5G Radio Access Network (RAN), many functions will be implemented in the cloud and on a cluster of computers, virtual machines, or containers, where each computer, virtual machine, or container could be an entity in the cluster.
[003] In the cluster, many entities implement the same function and this provides scalability to a service. For example, when system load increases above a threshold, new entities can be added to the cluster, thereby increasing the capacity of the cluster. On the other hand, when the system load decreases, existing entities can be removed to save resources. This is a normal virtualization technique to scale out/in a system. In this kind of system, when a client sends a request, a load balancer will select one entity in the cluster and then route the request to the entity.
[004] There are many applications which require session data (a.k.a., context data) for providing a service. In those applications, there will be multiple interactions between a client and a server for the service. For example, a user may log in first, access some information, and then update some information. This is called a stateful service. A stateful service requires the actual service provider to keep the session data for further service interactions. In a wireless communication system, RAN applications in the cloud often provide this kind of stateful service.
[005] One way to provide a stateful service for normal web applications is to make a session"sticky." In the"sticky" session, a load balancer will always route the following interactions of the session to the same entity according to session ID/key because that entity has context information of that session. Currently, implementations for stateful service in 5G testbeds are using solution(s) which are similar to the sticky session solution.
[006] Even though a sticky session is one way to provide a stateful service in a cloud cluster environment, it does not provide any fault tolerance. For example, when an entity malfunctions (e.g., no response from the entity) or is in an over-loaded state (e.g., high CPU load, high disk usage, long response time, etc.), the session or context information on that entity might be lost and other entities might not be able to help to failover.
[007] One of the fault tolerance solutions for web applications is session data replication across entities in a cluster. In such a fault tolerance solution, when one entity fails during a session, a load balancer will re-route the following requests of the same session to another entity and the failover entity will obtain replicated session data such that the service of the session can be continued. This kind of mechanism can be found in Apache Tomcat, Sun Java System Web Server, etc.
[008] Another fault tolerance solution is to save the session data into some persistent data storage (e.g., database which also has some redundancy support). Example(s) of such solution are described in U.S. Patent No. 10,149,333.
[009] Another fault tolerance solution is described in the International Patent
Publication WO 2018/175864 Al . This solution is similar to the previous solution which saves the session data into some persistent storage medium. In this solution, if one entity fails, a new container will be created as a failover entity while session data will be migrated from a persistent storage medium and original IP address will be maintained to save the downtime.
[0010] For 5G RAN application, fault tolerance should be provided.
[0011] Even though session data replication in a cluster provides fault tolerance solution to stateful services, it is not good enough for multi-layer applications. For example, assume that there is a three-layer system having a first layer with entities serving an external client, a second layer with entities serving the first layer, and a third layer with entities serving the second layer, where the first layer is also a client to the second layer and the second layer is also a client to the third layer. Every messages related to a session are rerouted by a load balancer to a previously selected entity (i.e., making the session"sticky").
[0012] A sticky session with replication provides a possibility for redundancy in the direction from the first layer entity to the second layer entity and from the second layer entity to the third layer entity. In other words, if the load balancer is aware of that a previously selected entity in a serving layer/side (i.e., the subsequent layer) is no longer available, the load balancer can select another entity on the serving layer/side.
[0013] On the other hand, a service response from a server (e.g., an entity in the third layer) to a client (e.g., an entity in the second layer) cannot be re-routed to a backup client entity (e.g., another entity in the second layer) because the server does not have information about the backup client entity. Therefore, the above described sticky session with replication does not provide a way to failover the failed client entity to proceed the service responses in the reverse direction. For example, if a previously selected entity in the second layer is unavailable, a service response from an entity in the third layer entity cannot be handled properly.
[0014] In persistent data storage approach, session data will be saved into a database or some other data storage such as a memory. The approach gives the flexibility to choose entities to continue the service for a session when an original entity is unavailable, by loading the session data from the persistent data storage. But this approach will introduce extra time for saving data to and fetching data from the persistent data storage. Also it does not solve the aforementioned problem with respect to multi-layer applications.
[0015] The embodiments of this disclosure provide a better fault tolerance solution for a multi-layer cloud system. The embodiments of this disclosure are applicable to RAN application area as well as cloud environment with virtual machine or container-based application function implementation.
[0016] Accordingly, in one aspect there is a method performed by a load balancing system. In one embodiment, the method includes receiving a request sent by a client and selecting from a first cluster of serving entities a first main serving entity and a first backup serving entity. The method also includes, as a result of receiving the request sent by the client, sending to the client information identifying the selected first main serving entity and the selected first backup serving entity.
[0017] In other aspect there is a method performed by a client. In one embodiment, the method includes sending a request to a load balancing system. The method also includes as a result of sending the request to the load balancing system, receiving information identifying a first main serving entity that is a part of a first cluster of serving entities and a first backup serving entity that is a part of the first cluster of serving entities. The information is sent by the load balancing system. The method further includes after receiving the information identifying the first main serving entity and the first backup serving entity, sending to the first main serving entity information identifying the first backup serving entity.
[0018] In other aspect there is a method performed by a first main serving entity within a first cluster of serving entities. In one embodiment, the method includes receiving first identifying information identifying a first backup serving entity within the first cluster of serving entities. The first identifying information is sent by a client. The method also includes after receiving the first identifying information, sending a request to a load balancing system and as a result of sending the request to the load balancing system, receiving second identifying information identifying a second main serving entity within a second cluster of serving entities and a second backup serving entity within the second cluster of serving entities. The second identifying information is sent by the load balancing system. The method further includes after receiving the second identifying information, sending to the second main serving entity third identifying information identifying the first main serving entity, the first backup serving entity, and the second backup serving entity.
[0019] In other aspect there is a method performed by a second main serving entity. In one embodiment, the method includes receiving first identifying information identifying the first main serving entity, a first backup serving entity, and a second backup serving entity. The first identifying information is sent by a first main serving entity. The method also includes after receiving the first identifying information, sending a request to a load balancing system and as a result of sending the request to the load balancing system, receiving second identifying information identifying a third main entity and a third backup entity. The second identifying information is sent by the load balancing system. The method further includes after receiving the second identifying information, sending to the third main serving entity third identifying information identifying the second main serving entity, the second backup serving entity, and the third backup serving entity. Each of the first main serving entity and the first backup serving entity is selected from a first cluster of serving entities and each of the second main serving entity and the second backup serving entity is selected from a second cluster of serving entities. Also each of the third main serving entity and the third backup serving entity is selected from a third cluster of serving entities.
[0020] The accompanying drawings, which are incorporated herein and form part of the specification, illustrate various embodiments.
[0021] FIG. 1 illustrates a system according to some embodiments.
[0022] FIG. 2 shows an example of a multi-layer application.
[0025] FIG. 5 shows a message flow diagram according to some embodiments.
[0026] FIG. 6 shows a message flow diagram for fault tolerance solution according to some embodiments.
[0028] FIG. 8 shows a flow chart illustrating a process performed by a load balancer according to some embodiments.
[0029] FIG. 9 shows a flow chart illustrating a process performed by an entity according to some embodiments.
[0030] FIG. 10 shows a flow chart illustrating a process performed by an entity according to some embodiments.
[0032] FIG. 12 is a flow chart illustrating a process according to some embodiments.
[0036] FIG. 16 is a block diagram illustrating an apparatus according to some embodiments.
[0037] The embodiments of this disclosure solve the problems of the conventional fault tolerance solutions. For example, in one embodiment the following two features are employed: first, load balancing is done only when a session is setting up (once the session is set up, further exchange of information on the session may be sent and/or routed to a serving entity directly); and second, to avoid introducing time delay when restoring session for another entity session data replication is performed in limited scope. When a request for a new session is received, a load balancer will choose one entity as main serving entity and one or more entities as backup serving entities. Then the main serving entity will perform the session data replication to the backup serving entities. This method achieves a redundancy and at the same time minimizes the load needed for context replication (i.e., balancing between having only one serving entity (thus providing no fault tolerance) and performing a full cluster backup (i.e., session data with packet flooding)).
[0038] In the embodiments of this disclosure, to support a multi-layered distributed stateful system (for example, a cloud RAN system) in both forward and backward directions, each serving layer is configured to maintain information about serving entities in both directions. For example, a second (e.g., a middle) layer may be configured to know a main serving entity and one or more backup serving entities in both first and third serving layers and thus a redundancy can be achieved in both directions.
[0039] Compared to current implementations, the embodiments of this disclosure provide fault tolerance, which is necessary for some applications such as 5G cloud RAN. Also, the embodiments of this disclosure allow smoother operation/management and potentially easier design because a virtual machine hosting serving entity can be taken out of operation without the need to migrate all current sessions on that virtual machine and the current sessions can be continued with one or more backup entities.
[0040] In the conventional fault tolerance solutions, tracking backup serving entities for each layer allows entities on a layer to find backup serving entities on the subsequent layer such that a session request may be sent to the backup serving entities when the main serving entity becomes unavailable. But the conventional solutions do not provide fault tolerance when a client entity in a layer is unavailable and a server entity in an adjacent layer (e.g., an entity in the third layer) wants to send a response message to the unavailable client entity.
[0041] Embodiments of this disclosure solve the aforementioned problem of the conventional solutions by tracking backup entities in in adjacent layers (forward and backward). This provides much improvement for some applications such as, for example, RAN applications, as there are some procedures that involve many interactions across many layers.
In those applications, when a response is missed, the whole procedure needs to be re-executed from the beginning, and thus the cost for redoing the procedure is much higher.
[0042] FIG. l is a logical structure diagram for a multi-layer cloud system 100 providing fault tolerance solution according to some embodiments. FIG. 1 shows an access layer 104 which includes a plurality of access nodes. Each access node may act as a client of the cloud system 100. In RAN applications, each access node may be a radio unit or a digital unit connected to different user equipments (UEs) (a UE is any device capable of communication with an access node of the access layer). When a UE wants to access a RAN service, it will connect to an access node and the access node will then request some service provided by the cloud system 100 (e.g., a service provided by an entity in the 1st layer 106.
[0043] FIG. 1 also shows three layers - a first layer 106, a second layer 108, and a third layer 109. Each layer includes a plurality of entities that can provide the same service (hence each layer can be considered a cluster or a set of clusters). Each entity may be a computer, a virtual machine, a container, or other component which implements a particular function and provides a particular service. Entities that implement and are capable of executing the same function may be grouped together and form a cluster. Entities in the same cluster may share load and provide fault tolerance. All entities in the same cluster do not have to be in the same physical host.
[0044] The system 100 shown in FIG. 1 is multi-layered. An entity in the first layer 106 may provide some service to an access node in the access layer 104 and may also require a service provided by an entity in the second layer 108. The connections between different entities in different layers may exist for a long time and be reused for many different sessions.
[0045] FIG. 2 illustrates an example of the multi-layer system 100. FIG. 2 is an exemplary diagram of an Evolved Universal Terrestrial Radio Access Network (E-UTRAN) Protocol Stack having multiple layers. As shown in the figure, each layer of the multi-layer system is configured to provide a service to a lower layer (if there is any) and is configured to obtain a service from an upper layer (if there is any).
[0046] Referring back to FIG. 1, when the access layer 104 needs a service from the third layer 109, the access layer 104 may send a request to the first layer 106. In response to the request sent by the access layer 104, the first layer 106 performs a process and sends a request to the second layer 108 according to an existing context route. After receiving the request, the second layer 108 continues the process and sends a request to the third layer 109. According to the embodiments shown in FIG. 1 , information about main and backup entities in different layers is also exchanged between the different layers such that an entity in a layer knows the main and backup entities in the higher layer and the lower layer. Thus, in a fault situation (e.g., when one entity in a layer crashes), the corresponding entities in the lower layer and the higher layer know which backup entity can be used instead of the crashed entity.
[0047] A load balancer 102 may be used to decide which entity in a cluster is to provide the service to a specific UE when a session is established for the first time. When the session is established, further information flow on the session may be sent directly to a serving entity. This will improve performance compared to the normal solution of always sending a request to the load balancer or an agent and of the load balancer or the agent rerouting the request to a serving entity. The session in different layers may be different because different information may be used.
[0048] In the embodiments shown in FIG. 1 , the load balancer is centralized. But the load balancer may also be distributed. Specifically, in some embodiments, multiple load balancers
may be used in the system. For example, one load balancer may be provided for each layer such that there is one load balancer dedicated for load balancing for each layer.
[0049] FIG. 3 shows a system 300 according to some embodiments. As compared to the system 100 shown in FIG. 1, in the system 300, each layer may be divided into multiple clusters, and main and backup serving entities for a session may be selected only from the same cluster. This limit in the selection may simplify data replication or event broadcast operation. For example, when entity 1 in a first layer 306 is marked as unavailable, the message indicating the unavailability of the entity 1 in the first layer 306 may only need to be broadcasted to the entities in the same cluster (e.g., entities 0 and 2 in the first layer 306) and some cluster in the neighboring layers (e.g., entities 0 and 1 in a second layer 308 and access nodes 0 and lin an access layer 304).
[0050] FIG. 4 shows a system 400 having a distributed system structure according to some embodiments. In the system 400, a load balancer is integrated within each access node in an access layer 404 and each entity in a first layer 406, a second layer 408, and a third layer 410. Each load balancer may obtain information regarding entities in another layer from a service registry 402. For example, a load balancer integrated within entity 1 in the first layer 406 may get from the service registry 402 information that there are m entities in the second layer 308 and how to connect to them (e.g., IP addresses, ports, etc.). The load balancer may also collect load information for entities.
[0051] When the entity 1 in the first layer 406 wants a service from the second layer 408, the load balancer integrated with the entity 1 of the first layer 406 may directly help to choose serving entities in the second layer 408. For example, the load balancer integrated with the entity 1 of the first layer 406 may choose a main serving entity and a backup serving entity in the second layer 408.
[0052] Even though, in the embodiments shown in FIG. 4, a load balancer is provided for each access node or each entity in a layer, in other embodiments, a load balancer may be provided for several entities located in the same host.
[0053] A load balancer may be a logical function module which may be centralized or distributed. The load balancer's core function is to choose entities as service providers.
According to some embodiments of this disclosure, the load balancer is configured to choose main serving entities and backup serving entities for stateful service.
[0054] FIG. 5 is a message flow diagram according to some embodiments. The message flow diagram illustrates a process for setting up session data in each layer of a multi-layer system when, for example, a UE wants a service provided by a cloud environment from the multi-layer system.
[0055] In the embodiments shown in FIG. 5, when a UE (not shown in the figure) wants the service provided by the cloud environment, it sends to an access node 152 a new session request 502. As a result of receiving the request 502, the access node 152 sends to the load balancer 102 a request 504 for information regarding serving entities in the first layer 106.
[0056] As a result of receiving the request 504, the load balancer 102 may obtain from a service registry and discovery system (not shown in FIG. 5) information as to which cluster is capable of providing a particular service and information (e.g., identifiers of entities) regarding all entities included in the cluster capable providing the particular service. The service registry and discovery system has the information about entities because when an entity starts up or shuts down, updated information corresponding to the changed status of the entity is stored or registered in the system. The service registry and discovery system may be located within a load balancing system (e.g., the load balancer 102 shown in FIG. 5) or may be provided as an individual module.
[0057] Based on the information obtained from the service registry and discovery system, the load balancer 102 may select a first layer entity 154 as a main serving entity (i.e., entity 0 of the 1st layer) and a first layer entity 156 as a backup serving entity (i.e., entity 1 of the 1st layer). After selecting the main serving entity 154 and the backup serving entity 156, the load balancer 102 may send to the access node 152 a message 506 including information indicating that the entity 154 is the main serving entity and entity 156 is the backup serving entity.
[0058] As a result of receiving the message 506, the access node 152 may perform a process 508. In the process 508, if message transferring is based on a protocol which requires a connection (for example, TCP), the access node 152 checks if a connection to the first layer entity 154 and/or a connection to the first layer entity 156 exists or not. If any of the connections does not exist, the access node 152 may set up the connection to the first layer entity 154 and/or the connection to the first layer entity 156. After the connections are setup, the connections may be reused for many sessions.
[0059] After performing the process 508, the access node 152 may send to the first layer entity 154 a message 510. The message 510 may include a session request and information indicating that for this session, the first layer entity 154 is the main serving entity and the first layer entity 156 is the backup serving entity. By receiving the message 510, the first layer entity 154 may know that first layer session data for the UE needs to be replicated to the first layer entity 156.
[0060] After receiving the message 510, the first layer entity 154 may execute a function and send to the load balancer 102 a request 512 for information regarding serving entities in the second layer 108.
[0061] As a result of receiving the request 512, the load balancer 102 may obtain from the service registry and discovery system information as to which cluster is capable of providing a particular service and information (e.g., identifiers of entities) regarding all entities included in the cluster capable providing the particular service.
[0062] Based on the information obtained from the service registry and discovery system, the load balancer 102 may select a second layer entity 158 as a main serving entity (in the example shown the main serving entity 158 is entity 1 of the 2nd layer) and a second layer entity 160 as a backup serving entity (in the example shown the backup serving entity 160 is entity 2 of the 2nd layer). After selecting the second layer entity 158 and the second layer entity 160, the load balancer 102 may send to the first layer entity 154 a message 514 including information indicating that entity 158 is the main serving entity for the 2nd layer 108 and entity 160 is the backup serving entity for the 2nd layer 108.
[0063] After receiving the message 514, the first layer entity 154 may perform a process
516 for connection set up. The process 516 is similar to the process 508 performed by the access node 152. Specifically, the first layer entity 154 may check if a connection to the second layer entity 158 and/or a connection to the second layer entity 160 exists or not. If any of the connections does not exist, the first layer entity 154 may set up the connection to the second
layer entity 158 and/or the connection to the second layer entity 160. After the connections are setup, the connections may be reused for many sessions.
[0064] After performing the process 516, the first layer entity 154 may send to the second layer entity 158 a message 518. The message 518 may include a session request and information identifying the first layer entity 154 as the main client entity in the first layer, the first layer entity 156 as a backup client entity in the first layer, the second layer entity 158 as the main serving entity in the second layer, and the second layer entity 160 as a backup serving entity in the second layer.
[0065] The first layer entity 154 may also send a replication message 520 to the first layer entity 156. The replication message 520 may include the first layer session data and information identifying the first layer entity 154 as a main client entity in the first layer, the first layer entity 156 as a backup client entity in the first layer, the second layer entity 158 as a main serving entity in the second layer, and the second layer entity 160 as a backup serving entity in the second layer.
[0066] In some embodiments, after the first layer entity 156 receives the replication message 520, the first layer entity 156 may perform a process 522. The process 522 is similar to the process 516 performed by the first layer entity 154. Specifically, the first layer entity 156 may check if a connection to the second layer entity 158 and/or a connection to the second layer entity 160 exists or not. If any of the connections does not exist, the first layer entity 156 may set up the connection to the second layer entity 158 and the connection to the second layer entity 160.
[0067] After receiving the message 518, the second layer entity 158 may send to the load balancer 102 a request 524 for serving entities in the third layer 109.
[0068] As a result of receiving the request 524, the load balancer 102 may obtain from the service registry and discovery system (not shown in FIG. 5) information as to which cluster of the 3rd layer 109 is capable of providing a particular service and information (e.g., identifiers of entities) regarding all entities included in the cluster capable providing the particular service.
[0069] Based on the information obtained from the service registry and discovery system, the load balancer 102 may select a third layer entity 162 as a main serving entity and a third layer entity 164 as a backup serving entity from the third layer. After selecting the third layer entity 162 and the third layer entity 164, the load balancer 102 may send to the second layer entity 158 a message 526 including information indicating that the third layer entity 162 is the main serving entity and the third layer entity 164 is the backup serving entity.
[0070] After receiving the message 526, the second layer entity 158 may perform a process 528 for connection set up. The process 528 is similar to the process 516 performed by the first access node 154. Specifically, the second layer entity 158 may check if a connection to the third layer entity 162 and/or a connection to the third layer entity 164 exists or not. If any of the connections does not exist, the second layer entity 158 may set up the connection to the third layer entity 162 and/or the connection to the third layer entity 164. After the connections are setup, the connections may be reused for many sessions.
[0071] After performing the process 528, the second layer entity 158 may send to the third layer entity 162 a message 530. The message 530 may include a session request and information identifying the second layer entity 158 as a main client entity in the second layer, the second layer entity 160 as a backup client entity in the second layer, the third layer entity 162 as a main serving entity in the third layer, and the third layer entity 164 as a backup serving entity in the third layer.
[0072] The second layer entity 158 may also send a replication message 532 to the second layer entity 160. The replication message 532 may include the second layer session data and information identifying the second layer entity 158 as a main client entity in the second layer, the second layer entity 160 as a backup client entity in the second layer, the third layer entity 162 as a main serving entity in the third layer, and the third layer entity 164 as a backup serving entity in the third layer.
[0073] In some embodiments, after the second layer entity 160 receives the replication message 532, the second layer entity 160 may perform a process 534. The process 534 is similar to the process 528 performed by the second layer entity 158. Specifically, the second layer entity 160 may check if a connection to the third layer entity 162 and/or a connection to the third layer entity 164 exists or not. If any of the connections does not exist, the second layer entity 160 may set up the connection to the third layer entity 162 and the connection to the third layer entity 164. [0074] After receiving the message 530, the third layer entity 162 may execute a function and may setup full third layer session data. After setting up the third layer session data, the third layer entity 162 sends a reply message 536 to the second layer entity 158.
[0075] The third layer entity 162 may also send a replication message 538 to the third layer entity 164. The replication message 538 may include the third layer session data and information identifying the second layer entity 158 as a main client entity in the second layer and the second layer entity 160 as a backup client entity in the second layer.
[0076] After receiving the reply message 536, the second layer entity 158 sends a reply message 542 to the first layer entity 154.
[0077] If the second layer session data has been changed, the second layer entity 158 may also send to the second layer entity 160 a replication message 540 including information about the changed second layer session data.
[0078] After receiving the reply message 542, the first layer entity 154 sends a reply message 544 to the access node 152.
[0079] If the first layer session data has been changed, the first layer entity 154 may also send to the first layer entity 156 a replication message 546 including information about the changed first layer session data.
[0080] The process illustrated in FIG. 5 results in that each entity has a backward and a forward link to main and backup entities in the lower and higher layers, where the main entities and the backup entities are synchronized with respect to session data.
[0081] The process shown in FIG. 5 is applicable to a stateful service that involves a sequence of interactions. In stateful services, after a session is setup, session data is created and following interactions in services require the session data. By applying the process shown in FIG. 5, a stateful service may be provided with a fault tolerance function.
[0082] FIG. 6 is a message flow diagram illustrating a message flow according to some embodiments. The message flow allows continuing a service even when an entity in a middle layer fails on the forward path after an initial session for a specific UE has been set up.
[0083] After the initial session has been set up, the access node 152 may send to the first layer entity 154 a session request 610. Here, the access node 152 sends the request 610 to the first layer entity 154 because previous session information indicates that the first layer entity 154 is the main serving entity for this session in the first layer.
[0084] After receiving the session request 610, if the second layer entity 158 was available, the first layer entity 154 would have sent a session request 612 to the second layer entity 158 because previous session information indicates that the second layer entity 158 is the main serving entity for this session in the second layer. But here the second layer entity 158 became unavailable after the initial session has been set up.
[0085] Information indicating the unavailability of the second layer entity 158 may be provided to the first layer entity 154 and to the third layer entity 162. Specifically, in some embodiments, the load balancer may broadcast, multi-cast, or unicast the information indicating the unavailability of the second layer entity 158.
[0086] The load balancer may obtain the availability information by detecting the unavailability of the second layer entity 158 by itself or by receiving a message containing the unavailability information from a separate entity monitor which is capable of detecting the unavailability of the second layer entity 158.
[0087] Once the first layer entity 154 receives the information indicating the
unavailability of the second layer entity 158, the first layer entity 154 may update session information stored in the first layer entity 154. The table 652 illustrates exemplary session information stored in the first layer entity 154. As shown in the table, after the first layer entity 154 receives the information regarding the unavailability of the second layer entity 158, the first layer entity 154 may assign the second layer entity 160 as a main entity for this session in the second layer.
[0088] After assigning the second layer entity 160 as the main entity in the second layer for this session, the first layer entity 154 may send a session request 614 to the second layer entity 160 instead of sending the session request 612 to the second layer entity 158.
[0089] As described with respect to the message 532 shown in FIG. 5, information regarding the current session was previously replicated from the second layer entity 158 to the second layer entity 160. Thus, the second layer entity 160 has all the session information the second layer entity 158 had. The table 654 illustrates exemplary session information stored in the second layer entity 160. As shown in the table, the second layer entity 160 has the current session ID and data, the main and backup entities of the first layer and the main band backup entities of the third layer for the session.
[0090] Thus, after receiving the session request 614, the second layer entity 160 may send a session request to the third layer entity 162. Because the third layer entity 162 also knows that the second layer entity 158 is the main entity for this specific UE session, the third layer entity 162 knows that the message 614 is for this specific UE session.
[0091] As a result of receiving the message 616, the third layer entity 162 may send a reply message 618 to the second layer entity 160.
[0092] After receiving the reply message 618, the second layer entity 160 may send a reply message 620 to the first layer entity 154 and as a result of receiving the message 620, the first layer entity 154 may send a reply message 622 to the access node 154.
[0093] In the embodiments of this disclosure, when session data for a layer changes, replication of corresponding session data to a backup serving entity needs to be done. But this is not shown in the message flow diagram for brevity.
[0094] FIG. 7 is a message flow diagram according to some embodiments. The message flow diagram illustrates a process for continuing a service when an entity in a middle layer fails on the backward path after an initial session for a specific UE has been set up.
[0095] After the initial session has been set up, the access node 152 may send to the first layer entity 154 a session request 710. Here, the access node 152 sends the session request 710 to the first layer entity 154 because previous session information indicates that the first layer entity 154 is the main serving entity for this session in the first layer.
[0096] As a result of receiving the session request 710, the first layer entity 154 may send to the second layer entity 158 a session request 712. Here, the first layer entity 154 sends the session request 712 to the second layer entity 158 because previous session information indicates that the second layer entity 158 is the main serving entity for this session in the second layer. [0097] After receiving the session request 712, the second layer entity 158 may send to the third layer entity 162 a session request 714. Here, the second layer entity 158 sends the session request 714 to the third layer entity 162 because previous session information indicates that the third layer entity 162 is the main serving entity for this session in the third layer.
[0098] The second layer entity 158 may also send a replication message 716 to the second layer entity 160. The replication message 716 is similar to the replication message 532 shown in FIG. 5.
[0099] After receiving the session request 714, if the second layer entity 158 was available, the third layer entity 162 would have sent a reply message 718 to the second layer entity 158 because previous session information indicates that the second layer entity 158 is the main serving entity for this session in the second layer. But here the second layer entity 158 became unavailable after the third layer entity 162 received the session request 714.
[00100] As explained above, information declaring the unavailability of the second layer entity 158 may be provided to the first layer entity 154 and to the third layer entity 162.
Specifically, in some embodiments, the load balancer may broadcast the information or unicast to the first layer entity 154 and the third layer entity 162 the information indicating the unavailability of the second layer entity 158.
[00101] Once the third layer entity 162 receives the information declaring the
unavailability of the second layer entity 158, the third layer entity 162 may update session information stored in the third layer entity 162. The table 752 illustrates exemplary session information stored in the third layer entity 162. As shown in the table, after the third layer entity 162 receives the information declaring the unavailability of the second layer entity 158, the third layer entity 162 may assign the second layer entity 160 as a main entity for this session in the second layer.
[00102] After assigning the second layer entity 160 as the main entity in the second layer for this session, the third layer entity 162 may send a reply message 720 to the second layer entity 160 instead of sending the reply message 718 to the second layer entity 158.
[00103] As mentioned with respect to the message 716, information regarding the current session was previously replicated from the second layer entity 158 to the second layer entity 160.
Thus, the second layer entity 160 has all the session information the second layer entity 158 had. The table 754 illustrates exemplary session information stored in the second layer entity 160. As shown in the table, the second layer entity 160 has the current session ID and data, the main and backup entities of the first layer and the main band backup entities of the third layer for the session.
[00104] Thus, after receiving the reply message 720, the second layer entity 160 may send a reply message 722 to the first layer entity 154. The reply message 722 corresponds to the session request 712 because the second layer session data is stored in the second layer entity 160. In response to receiving the reply message 722, the first layer entity 154 may send a reply message 724 to the access node 152.
[00105] FIG. 8 is a flow chart illustrating a process 800 according to some embodiments. The process 800 may be performed by a load balancer or an agent after an entity in a layer malfunctions.
[00106] In step s802, the load balancer or the agent may detect the malfunction of the entity. In some embodiments, instead of directly detecting the malfunction of the entity, the load balancer or the agent may detect the malfunction by receiving from an entity monitor a message declaring the malfunction of the entity.
[00107] The load balancer or a separate entity monitor may detect the malfunction of the entity by collecting running information such as heart beaten, CPU load, memory usage, or application level measurements, etc. and by judging the current state of the entity according to one or more criteria.
[00108] In step s804, after obtaining information indicating the malfunction of the entity, the load balancer or the agent may decide whether the entity should be treated as unavailable or not. If the load balancer or the agent decides that the entity should not be treated as unavailable, the process ends. On the other hand, if the load balancer or the agent decides that the entity should be treated as unavailable, the process proceeds to step s806.
[00109] In step s806, the load balancer or the agent may mark the entity as unavailable and may stop assigning the unavailable entity as a main entity or a backup entity for any new session.
[00110] In step s808, if needed, the load balancer or the agent may find one or more entities as one or more new backup entities for the sessions in which the unavailable entity is involved.
[00111] In steps s810, the load balancer or the agent may broadcast information identifying the unavailable entity to one or more entities in the cluster to which the unavailable entity belongs and one or more entities in the neighboring clusters.
[00112] Even though the unavailable entity may not be operative for providing a service, it may still be able to receive the broadcasted information. In such case, if the unavailable entity is a main entity for a session, after receiving the broadcasted information, the unavailable entity may replicate session data to other backup entities.
[00113] FIG. 9 is a flow chart illustrating a process 900 according to some embodiments. The process 900 may be performed by a backup entity in a cluster when another entity in the same cluster becomes unavailable. For example, in the embodiments shown in FIGS. 6 and 7, the process 900 may be performed by the second layer entity 160.
[00114] In step s902, the backup entity in the cluster may receive a message after another entity in the same cluster became unavailable. The message may be broadcasted by a load balancer. The broadcasted message may include information identifying the unavailable entity.
[00115] In step s904, after receiving the message, the backup entity may determine if the unavailable entity identified in the message is a main entity in one or more sessions. If the identified unavailable entity is not a main entity in one or more sessions, the process ends. On the other hand, if the identified unavailable entity is a main entity in one or more sessions, the process proceeds to step s906.
[00116] In step s906, as a result of determining that the identified unavailable entity is a main entity in one or more sessions, the backup entity may set itself as a new main entity for said one or more sessions. Also, if the message includes information about a new backup entity, the new main entity (i.e., the old backup entity) may add the new backup entity as a backup entity and start to synchronize session data to the new backup entity.
[00117] FIG. 10 is a flow chart illustrating a process 1000 according to some embodiments. The process 1000 may be performed by a main entity in the cluster when another entity in the same cluster became unavailable.
[00118] In step sl002, the main entity in the cluster may receive a message after another entity in the same cluster became unavailable. The message may be broadcast, multi-cast, or unicast by the load balancer. The message may include information identifying the unavailable entity.
[00119] In step si 004, after receiving the message, the main entity may determine if the unavailable entity identified in the message is a backup entity in one or more sessions. If the identified unavailable entity is not a backup entity in one or more sessions, the process ends. On the other hand, if the identified unavailable entity is a backup entity in one or more sessions, the process proceeds to step si 006.
[00120] In step si 006, as a result of determining that the identified unavailable entity is a backup entity in one or more sessions, the main entity may remove the unavailable entity from said one or more sessions. Also, if the message includes information about a new backup entity, the main entity may add the new backup entity as a backup entity and start to synchronize session data to the new backup entity.
[00121] FIG. 11 is a flow chart illustrating a process 1100 according to some
embodiments. The process 1100 may be performed by an entity in a cluster after the entity receives a message indicating that another entity in a neighboring cluster has become unavailable. For example, in the embodiments shown in FIG. 6, the first layer entity 154 may perform the process 1100. Similarly, in the embodiments shown in FIG. 7, the third layer entity 162 may perform the process 1100.
[00122] In step si 102, the entity in the cluster may receive a message transmitted by a load balancer (e.g., the message may be broadcast, multi-cast, or unicast). The message may indicate that another entity in the neighboring cluster has become unavailable. After receiving the message, in step si 104, the entity may determine whether the indicated unavailable entity in the neighboring cluster is an entity handling one or more sessions. If the indicated unavailable entity is not handling any session, the process ends. On the other hand, if the indicated unavailable entity is handling at least one session, the process proceeds to step si 106.
[00123] In step si 106, the entity may check whether the unavailable entity is a main entity or a backup entity in the neighboring cluster.
[00124] If the unavailable entity is a main entity in the neighboring cluster, the process proceeds to step si 108.
[00125] In step si 108, the entity may assign a known backup entity as a main entity. The information about the known backup entity may be obtained from previous session information. For example, in the embodiment shown in FIG. 6, based on previous session information, the first layer entity 154 knows that the second layer entity 160 is a backup entity in the second layer.
[00126] In step si 108, the entity may also add a new backup entity if the broadcasted message contains information about the new backup entity.
[00127] On the other hand, if the unavailable entity is a backup entity in the neighboring cluster, the process proceeds to step si 110.
[00128] In step si 110, the entity removes the unavailable entity from an (either serving or client) entity list. The entity may also add a new backup entity if the broadcasted message contains information about the new backup entity.
[00129] FIG. 12 is a flow chart illustrating a process 1200 according to an embodiment. Process 1200 may be performed by a load balancing system. Process 1200 may begin in step sl202.
[00130] Step si 202 comprises receiving a request sent by a client.
[00131] Step sl204 comprises selecting from a first cluster of serving entities a first main serving entity and a first backup serving entity.
[00132] Step si 206 comprises as a result of receiving the request sent by the client, sending to the client information identifying the selected first main serving entity and the selected first backup serving entity.
[00133] In some embodiments, process 1200 further includes (a) after sending to the client the information identifying the selected first main serving entity and the selected first backup serving entity, receiving a request sent by the first main serving entity, (b) selecting from a second cluster of serving entities a second main serving entity and a second backup serving entity, and (c) as a result of receiving the request sent by the first main serving entity, sending to the first main serving entity information identifying the selected second main serving entity and the selected second backup serving entity.
[00134] In some embodiments, process 1200 further includes obtaining unavailability information indicating that the first main serving entity or the first backup serving entity is unavailable and as a result of obtaining the unavailability information, sending to one or more entities information indicating that the first main serving entity or the first backup serving entity is unavailable.
[00135] In some embodiments, process 1200 further includes as a result of obtaining the unavailability information, selecting an additional backup serving entity from the first cluster of serving entities and sending to said one or more entities information identifying the selected additional backup serving entity.
[00136] In some embodiments, the load balancing system may be either centralized or distributed.
[00137] In some embodiments, process 1200 further includes obtaining from a service registry information regarding all entities in the first cluster of serving entities. The first cluster of serving entities may be associated with one or more particular types of service.
[00138] FIG. 13 is a flow chart illustrating a process 1300 according to an embodiment. Process 1300 may be performed by a client. Process 1300 may begin in step sl302.
[00139] Step si 302 comprises sending a request to a load balancing system.
[00140] Step si 304 comprises as a result of sending the request to the load balancing system, receiving information identifying a first main serving entity that is a part of a first cluster of serving entities and a first backup serving entity that is a part of the first cluster of serving entities. In some embodiments, the information may be sent by the load balancing system.
[00141] Step sl306 comprises after receiving the information identifying the first main serving entity and the first backup serving entity, sending to the first main serving entity information identifying the first backup serving entity.
[00142] In some embodiments, process 1300 further includes determining whether a connection to the first main serving entity and/or a connection to the first backup serving entity exists. Process 1300 may further includes as a result of determining that the connection to the first main serving entity does not exist, setting up the connection to the first main serving entity and as a result of determining that the connection to the first backup serving entity does not exist, setting up the connection to the first backup serving entity.
[00143] FIG. 14 is a flow chart illustrating a process 1400 according to an embodiment. Process 1400 may be performed by a first main serving entity within a first cluster of serving entities. Process 1400 may begin in step si 402.
[00144] Step si 402 comprises receiving first identifying information identifying a first backup serving entity within the first cluster of serving entities. In some embodiments, the first identifying information may be sent by a client.
[00145] Step si 404 comprises after receiving the first identifying information, sending a request to a load balancing system.
[00146] Step si 406 comprises as a result of sending the request to the load balancing system, receiving second identifying information identifying a second main serving entity within a second cluster of serving entities and a second backup serving entity within the second cluster of serving entities. In some embodiments, the second identifying information may be sent by the load balancing system.
[00147] Step si 408 comprises after receiving the second identifying information, sending to the second main serving entity third identifying information identifying the first main serving entity, the first backup serving entity, and the second backup serving entity.
[00148] In some embodiments, process 1400 further includes sending to the first backup serving entity information identifying the client, the second main serving entity, and the second backup serving entity.
[00149] In some embodiments, process 1400 further includes receiving a service request sent by the client, setting up session data, and sending the session data to the first backup serving entity.
[00150] In some embodiments, process 1400 further includes receiving unavailability information indicating that the second main serving entity is unavailable. The unavailability information may be sent from the load balancing system. Process 1400 may further include as a result of receiving the unavailability information, sending a session request to the second backup serving entity.
[00151] In some embodiments, process 1400 further includes receiving unavailability information indicating that the second backup serving entity is unavailable and replacing the second backup serving entity with the additional backup serving entity. The unavailability information may be sent by the load balancing system and the unavailability information may further include information identifying an additional backup serving entity. In some
embodiments, the additional backup serving entity may be selected from the second cluster of serving entities.
[00152] In some embodiments, when the first main serving entity and/or the first backup serving entity starts up or shuts down, information corresponding to the first main serving entity and/or the first backup serving entity stored in a service registry may be updated.
[00153] FIG. 15 is a flow chart illustrating a process 1500 according to an embodiment. Process 1500 may be performed by a second main serving entity. Process 1500 may begin in step sl502.
[00154] Step si 502 comprises receiving first identifying information identifying the first main serving entity, a first backup serving entity, and a second backup serving entity. In some embodiments, the first identifying information may be sent by a first main serving entity.
[00156] Step si 506 comprises as a result of sending the request to the load balancing system, receiving second identifying information identifying a third main entity and a third
backup entity. In some embodiments, the second identifying information may be sent by the load balancing system.
[00157] Step si 508 comprises after receiving the second identifying information, sending to the third main serving entity third identifying information identifying the second main serving entity, the second backup serving entity, and the third backup serving entity.
[00158] In some embodiments, each of the first main serving entity and the first backup serving entity is selected from a first cluster of serving entities, each of the second main serving entity and the second backup serving entity is selected from a second cluster of serving entities, and each of the third main serving entity and the third backup serving entity is selected from a third cluster of serving entities.
[00159] In some embodiments, process 1500 further comprises sending to the second backup serving entity information identifying the first main serving entity, the first backup serving entity, the third main serving entity, and the third backup serving entity.
[00160] In some embodiments, process 1500 further comprises receiving a session request sent by the first main serving entity, setting up session data, and sending the session data to the second backup serving entity.
[00161] In some embodiments, process 1500 further comprises receiving unavailability information indicating that the first main serving entity or the third main serving entity is unavailable. The unavailability information may be sent by the load balancing system.
[00162] Process 1500 may further comprise if the unavailability information indicates that the first main serving entity is unavailable, as a result of receiving the unavailability information, sending a session response to the first backup serving entity and if the unavailability information indicates that the third main serving entity is unavailable, as a result of receiving the
unavailability information, sending a session request to the third backup serving entity.
[00163] In some embodiments, process 1500 further comprises receiving unavailability information indicating that the first backup serving entity, the second backup serving entity, or the third backup serving entity is unavailable. The unavailability information may be sent by the load balancing system and the unavailability information may further include information identifying an additional backup serving entity.
[00164] Process 1500 may further comprise if the unavailability information indicates that the first backup serving entity is unavailable, replacing the first backup serving entity with the additional backup serving entity. The additional backup serving entity may be selected from the first cluster of serving entities.
[00165] Process 1500 may further comprise if the unavailability information indicates that the second backup serving entity is unavailable, replacing the second backup serving entity with the additional backup serving entity. The additional backup serving entity may be selected from the second cluster of serving entities.
[00166] Process 1500 may further comprise if the unavailability information indicates that the third backup serving entity is unavailable, replacing the third backup serving entity with the additional backup serving entity. The additional backup serving entity may be selected from the third cluster of serving entities.
[00167] In some embodiments, process 1500 further comprises if the unavailability information indicates that the second backup serving entity is unavailable, sending to the additional backup serving entity information identifying the first main serving entity, the first backup serving entity, the third main serving entity, and the third backup serving entity.
[00168] In some embodiments, after the second main serving entity becomes unavailable, the second backup serving entity is configured to receive unavailability information (a) indicating that the second backup serving entity is unavailable and (b) including information identifying an additional backup serving entity.
[00169] In some embodiments, the first main serving entity, the first backup serving entity, the third main serving entity, and the third backup serving entity are configured to be updated such that the second backup serving entity becomes a new main serving entity in the second cluster of serving entities.
[00170] In some embodiments, the second backup serving entity is configured to send to the additional backup serving entity information identifying the first main serving entity, the first backup serving entity, the third main serving entity, and the third backup serving entity.
[00171] In some embodiments, all entities in the first cluster are capable of performing the same function.
[00172] In some embodiments, all entities in the first cluster are capable of performing the first function and all entities in the second cluster are capable of performing the second function. The first function and the second function may be different.
[00173] FIG. 16 is a block diagram of an apparatus 1600, according to some
embodiments, for implementing load balancing system 102, client 152, or any serving entity including first main serving entity 154 and second main serving entity 158. As shown in FIG. 16, apparatus 1600 may comprise: processing circuitry (PC) 1602, which may include one or more processors (P) 1655 (e.g., a general purpose microprocessor and/or one or more other processors, such as an application specific integrated circuit (ASIC), field-programmable gate arrays (FPGAs), and the like), which processors may be co-located in a single housing or in a single data center or may be geographically distributed (i.e., apparatus 1600 may be a distributed computing apparatus); a network interface 1648 comprising a transmitter (Tx) 1645 and a receiver (Rx) 1647 for enabling apparatus 1600 to transmit data to and receive data from other nodes connected to a network 110 (e.g., an Internet Protocol (IP) network) to which network interface 1648 is connected; and a local storage unit (a.k.a.,"data storage system") 1608, which may include one or more non-volatile storage devices and/or one or more volatile storage devices. In embodiments where PC 1602 includes a programmable processor, a computer program product (CPP) 1641 may be provided. CPP 1641 includes a computer readable medium (CRM) 1642 storing a computer program (CP) 1643 comprising computer readable instructions (CRI) 1644. CRM 1642 may be a non -transitory computer readable medium, such as, magnetic media (e.g., a hard disk), optical media, memory devices (e.g., random access memory, flash memory), and the like. In some embodiments, the CRI 1644 of computer program 1643 is configured such that when executed by PC 1602, the CRI causes apparatus 1600 to perform steps described herein (e.g., steps described herein with reference to the flow charts and message flow diagrams described herein). In other embodiments, apparatus 1600 may be configured to perform
steps described herein without the need for code. That is, for example, PC 1602 may consist merely of one or more ASICs. Hence, the features of the embodiments described herein may be implemented in hardware and/or software.
[00174] While various embodiments of the present disclosure are described herein, it should be understood that they have been presented by way of example only, and not limitation. Thus, the breadth and scope of the present disclosure should not be limited by any of the above-described exemplary embodiments. Generally, all terms used herein are to be interpreted according to their ordinary meaning in the relevant technical field, unless a different meaning is clearly given and/or is implied from the context in which it is used. All references to a/an/the element, apparatus, component, means, step, etc. are to be interpreted openly as referring to at least one instance of the element, apparatus, component, means, step, etc., unless explicitly stated otherwise. Any combination of the above-described elements in all possible variations thereof is encompassed by the disclosure unless otherwise indicated herein or otherwise clearly contradicted by context.
[00175] Additionally, while the processes described above and illustrated in the drawings are shown as a sequence of steps, this was done solely for the sake of illustration. Accordingly, it is contemplated that some steps may be added, some steps may be omitted, the order of the steps may be re-arranged, and some steps may be performed in parallel. That is, the steps of any methods disclosed herein do not have to be performed in the exact order disclosed, unless a step is explicitly described as following or preceding another step and/or where it is implicit that a step must follow or precede another step.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,372
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NGC 140 is a spiral galaxy in the constellation of Andromeda. It was discovered by Truman Henry Safford on October 8, 1866.
Historical Information
Safford's discovery in 1866 was published in the appendix of an obscure paper. Sixteen years later, on November 5, 1882, Edouard Stephan discovered the same object, but was unaware of Safford's earlier discovery. Wolfgang Steinicke's version of the catalog lists Safford as the discoverer.
References
External links
Spiral galaxies
Andromeda (constellation)
0140
00311
01916
Astronomical objects discovered in 1785
Discoveries by Truman Safford
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,447
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Areas of fog. Slight chance of a rain shower. High 69F. Winds E at 5 to 10 mph..
Cloudy this evening with showers after midnight. Low 63F. Winds ENE at 10 to 20 mph. Chance of rain 40%.
Jeffery Peters
John Wayne Ferguson
Man dies while in sheriff's office custody
By JOHN WAYNE FERGUSON The Daily News
A man in the custody of the Galveston County Sheriff's Office died in a Galveston hospital on Monday, Galveston County Sheriff Henry Trochesset said.
Jeffery Lynn Peters, 59, of San Leon, died at the University of Texas Medical Branch, Trochesset said.
Peters was sent from the jail to the medical branch on Oct. 17 because he was having trouble breathing, Trochesset said.
Peters was booked into the jail on Oct. 12 on a felony charge of assault on a public servant, according to court records.
After being booked into the jail, Peters was placed in the medical area of the jail because of health issues that the jail's medical staff identified while he was being booked, Trochesset said.
Peters' death is at least the seventh in-custody death reported by the Sheriff's Office since 2015, according to data collected from the Texas Attorney General's Office by the nonprofit Texas Justice Initiatives.
Trochesset said he would not release a cause of death until the Galveston County Medical Examiner's Office completed an autopsy on Peters.
A booking photo taken on the day Peters was booked into the jail shows him with apparent injuries to the right side of his face. Peters had suffered the injuries before deputies found him sleeping inside a truck in San Leon, Trochesset said late Tuesday evening.
The sheriff's office has notified the Texas Rangers and the Texas Jail Commission about Peters' death, Trochesset said.
Trochesset said, to his knowledge, the number of in-custody deaths reported by the sheriff's office was not unusual or alarming, given the size of the county's inmate population. Reviews of the deaths of other inmates have not caused him to change any policies at the jail, he said.
"That's why each incident is investigated and investigated by an outside agency," Trochesset said. "Through each of these incidents, because of the medical condition of the inmate, I have not seen that to be an issue."
Galveston County has a large population of people that aren't in good health, Trochesset said.
John Wayne Ferguson: 409-683-5226; john.ferguson@galvnews.com or on Twitter @johnwferguson.
Follow John Wayne Ferguson
Remains found in South Pacific ID'd as Galveston man killed in WWII
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,727
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\section{Introduction}
In this paper, we consider the following empirical risk minimization problem commonly encountered in
machine learning:
\begin{align}
\min_{x\in \mathbb{R}^d} f(x):= \frac{1}{n}\sum_{i=1}^n f_i(x),
\end{align}
where $x$ represents the model parameters, $f_i(x) \equiv f(x; z_i)$ denotes the loss function of
the training sample $z_i$, and $n$ is the size of the training sample set. Since the training set
for most applications is of large size, stochastic gradient descent (SGD) is the most popular
algorithm used in practice. In the simplest scenario, SGD samples one random instance $f_i(\cdot)$
uniformly at each iteration and updates the parameter by evaluating only the gradient of the selected
$f_i(\cdot)$. The stability and convergence rate of SGD have been studied in depth, for example, see \cite{hardt2015train, needell2014stochastic}. However, the scalability of SGD is unfortunately
restricted by its inherent sequential nature. To overcome this issue and hence accelerate the
convergence, there has been a line of research devoted to asynchronous parallel SGDs. In the
distributed computation scenario, an asynchronous stochastic gradient descent (ASGD) method
parallelizes the computation on multiple processing units by (1) calculating multiple gradients
simultaneously at different processors and (2) sending the results asynchronously back to the master
for updating the model parameters \cite{agarwal2011distributed,recht2011hogwild}.
\subsection{Related Work}
There has been a vast literature on the analysis of SGD, see for example Bottou et
al. \cite{bottou2016optimization} for a comprehensive review of this subject. Some widely-used
methods include AdaGrad \cite{duchi2011adaptive}, which extends SGD by adapting step sizes for
different features, RMSProp \cite{tieleman2012lecture}, which resolves AdaGrad's rapidly diminishing
learning rates issue, and Adam \cite{kingma2014adam}, which combines the advantages of both AdaGrad
and RMSProp with a parameter learning rates adaption based on the average of the second moments of
the gradients. On the other hand, relatively few studies are devoted to ASGDs. Most of these
studies for ASGD take an optimization perspective. Hogwild! \cite{recht2011hogwild} assumed data
sparsity in order to run parallel SGD without locking successfully. Under various smoothness
conditions on $f$ such as $f$ being strongly convex and $f_i$'s all Lipschitz, it showed that the
convergence rate can be similar to the synchronous case. Duchi et al. \cite{duchi2013estimation}
extended this result by developing an asynchronous dual averaging algorithm that allows problems to
be non-smooth and non strongly-convex as well. Mitliagkas et al. \cite{mitliagkas2016asynchrony}
observed that a standard queuing model of asynchrony correlates to the momentum, that is,
asynchrony produces momentum in SGD updates. There are also several methods using asynchrony either
in parallel or in a distributed way, such as asynchronous stochastic coordinate descent algorithms
\cite{liu2015asynchronous2,liu2015asynchronous,nesterov2012efficiency,richtarik2014iteration}.
Recently, Li et al. \cite{li2015stochastic} introduced the concept of the stochastic modified
equation for SGDs (referred as {\bf SME-SGD} in this report), where in the continuous-time limit an
SGD is approximated by an appropriate (overdamped) Langevin equation. Compared to most convergence
analyses that give upper bounds for (strongly) convex objects, this new framework not only provides
more precise analyses for the leading order dynamics of SGD but also suggests adaptive
hyper-parameter strategies using optimal control theory.
\subsection{Our Contributions}
We give a novel derivation of SMEs for the ASGD algorithms by introducing auxiliary variables
to treat an effective memory term. With the derived SME models, we are able to characterize the
dynamics of ASGD algorithms.
In Section 2, we first derive a stochastic modified equation for the asynchronous stochastic
gradient descent, denoted shortly as {\bf SME-ASGD}, for the case where each loss
function $f_i$ is quadratic. The derivation results in a Langevin equation, which by assuming its ergodicity has a unique
invariant distribution solution with a convergence rate dominated by the temperature
factor. Meanwhile, for the momentum SGD (MSGD), a similar Langevin equation denoted as {\bf
SME-MSGD} is derived and we show that the temperature factors for both derived SME agree. This
comparison gives a Langevin dynamics explanation of why an asynchronous method gives rise to similar
behavior as compared to the momentum-based methods \cite{mitliagkas2016asynchrony}. Then by introducing a new accumulative quantity, we derive a more general SME-ASGD for the general case in
which the gradient of the loss function can be nonlinear. We show that the two SME-ASGDs are
equivalent when the objective functions are quadratic. We remark that the presented results make use of a few simplifying approximations which are made in a non-rigorous and non-quantified manner, e.g, assuming the noise coefficients to be constant $\sigma$ and the accumulation of i.i.d noise.
Section 3 provides some numerical analysis for SME-ASGD by providing a strong approximation
estimation to the ASGD algorithm. Different from the usual convergence studies, we do not assume
convexity on $f$ or $f_i$ but only require their gradients to be (uniformly) Lipschitz. Numerical
results including non-linear forcing terms and non-convex objectives demonstrate that SME-ASGD
provides much more accurate predictions for the behavior of ASGD compared to SME-SGD derived
in~\cite{li2015stochastic}. In Section 4, we apply the optimal control theory to identify the
optimal mini-batch for ASGD and the numerical simulations there verify that the suggested strategy
gives a significantly better performance.
\section{Stochastic Modified Equations}
The asynchronous stochastic gradient descent (ASGD) carries out the following update at each step:
\begin{align}\label{ASGD}
x_{k+1} = x_k - \eta \nabla f_{\gamma_k} (x_{k-\tau_k}),
\end{align}
where $\eta$ is the step size, $\{\gamma_k\}$ are i.i.d.~uniform
random variables taking values in $\{1, 2, \cdots , n\}$, and
$x_{k-\tau_k}$ is the delayed read of the parameter $x$ used to update
$x_{k+1}$ with a random {\em staleness} $\tau_k$.
\smallskip
\begin{assump}\label{assump}
We assume that the staleness $\tau_k$ are independent and that the sample selection process
$\gamma_k$ is mutually independent from the staleness process $\tau_k$. $\nabla f_i$'s are all
(uniformly) Lipschitz, that is, for each $1\leq i\leq n$, there exists $L_i>0$ such that for any
$x, y \in \mathbb{R}^d$, we have $|\nabla f_i(x) - \nabla f_i(y)|\leq L_i|x-y|$. As a consequence,
by taking $L = \frac{1}{n}\sum_{i=1}^n L_i$, $\nabla f$ is also (uniformly) Lipschitz: $|\nabla
f(x)-\nabla f(y)|\leq L |x-y|$. In addition, the staleness process $\tau_k$ follows the geometric
distribution: $\tau_k = l$ (i.e., $x_{k-\tau_k} = x_{k-l}$), $l\in \{0,1,2,\cdots\}$, with probability $(1-\mu)\mu^{l}$ for
$\mu \in (0,1)$.
\end{assump}
\begin{assump}
We assume that the equations SME-ASGD \eqref{linearASGD} and SME-MSGD \eqref{mosme} are ergodic.
\end{assump}
The geometric distribution assumption here is not only made to simplify the computation, but also
can be justified by considering the canonical queuing model \cite{younes2005verification}. For
example, the computation at each processor may involve a randomized algorithm that requires each
processor to do multiple independent trials until the result is accepted, thus resulting in a
geometrically distributed computation time. The geometric staleness assumption has been used in the previous asynchrony analysis, for example, see \cite{mitliagkas2016asynchrony}. Our derivation of SME models can be also easily
generalized to other random staleness models if the memory kernel, i.e., the distribution of staleness in time, decays sufficiently fast for integrability and is completely monotone when we approximate the memory kernel by a $C^{\infty}(0,\infty)$ function $\kappa(r)$. $\kappa(r)$ is completely monotone if for all for $n\geq 0, ~ r>0,~ (-1)^n\frac{d^{n}}{dr^n}\kappa(r)\geq 0.$
Under that circumstance, we can approximate the kernel accurately by $\sum_{k=1}^{n_k} c_k e^{-\lambda_k r}$ using the Bernstein's theorem of monotone functions \cite{bernstein1929},
and each term can be embedded into one auxiliary value to derive the SME formulation.
\subsection{Linear gradients}
We first show the derivation of Langevin dynamics with the linear forcing term. Suppose that, for
each $1\leq i\leq n, \nabla f_i$ is linear, or equivalently each $f_i$ is quadratic. While this is a
fairly restrictive assumption, the derivation in this simplified scenario offers a more transparent
view towards the stochastic modified equation for the asynchronous algorithm.
A key quantity for our derivation is the {\em expected read} $m_k$ defined as the expectation of
$x_k$ following Assumption \ref{assump}:
\begin{equation*}
m_k = \mathbb{E}_{\tau}( x_{k-\tau_k} )= \sum\nolimits_{l=0}^{\infty}x_{k-l} (1-\mu) \mu^{l}.
\end{equation*}
Here $m_k$ is a conditional expectation conditioned on the history of $x$, and $m_k$ is random since $x_{k-l}$'s are. Note that $m_{k+1} = \sum_{l=0}^{\infty}x_{k+1-l} (1-\mu) \mu^{l} = x_{k+1} (1-\mu) + \mu m_k$ and
$x_{k+1}= (m_{k+1}-\mu m_k)/(1-\mu)$. Plugging this into \eqref{ASGD}, we can rewrite ASGD as
\begin{align}\label{seclinear}
\frac{m_{k+1} - 2m_k + m_{k-1}}{\eta(1-\mu)} = -\frac{m_k - m_{k-1}}{\eta} - \nabla f(m_k) + (\nabla
f(m_k) - \nabla f_{\gamma_k}(x_{k-\tau_k})).
\end{align}
The left hand side and the first term on the
right hand side of \eqref{seclinear} can be viewed as divided difference approximations to various
time derivatives of $m$. The second term on the right hand side is the usual gradient. The last term
$\nabla f(m_k) - \nabla f_{\gamma_k}(x_{k-\tau_k})$ can be understood as the noise due to stochastic gradient
and the read delays; it has mean $0$, since the expectation, conditioned on the history of updates, can be decomposed as
\begin{multline*}
\mathbb{E}_{\gamma,\tau}\bigl(\nabla f(m_k) -\nabla f_{\gamma_k}(m_k) + \nabla f_{\gamma_k}(m_k)- \nabla
f_{\gamma_k}(x_{k-\tau_k}) \bigr) =\frac{1}{n}\sum_{i=1}^n (\nabla f(m_k)- \nabla f_i(m_k)) \\ +
\frac{1}{n}\sum_{i=1}^n \big(\nabla f_i(\sum_{l=0}^{\infty}x_{k-l} (1-\mu) \mu^{l}) -
\sum_{m=0}^{\infty}(1-\mu)\mu^m \nabla f_i(x_{k-m})\big) = 0.
\end{multline*}
The covariance matrix of the noise will be denoted as
\begin{equation*}
\Sigma_k = \mathbb{E}_{\gamma,\tau}\bigl((\nabla f(m_k) - \nabla f_{\gamma_k}(x_{k-\tau_k}))(\nabla f(m_k) - \nabla
f_{\gamma_k}(x_{k-\tau_k}))^T\bigr),
\end{equation*}
conditioned on $\{x_{k-l}\}_{l\ge 0}$ and we also denote the square root of $\Sigma_k$ by
$\sigma_k$, i.e., $\Sigma_k =\sigma_k\sigma_k^T$. $\Sigma_k$ (and thus $\sigma_k$) in general
depends on the previous history of the trajectory, although such dependence is omitted in our
notation.
In order to arrive at a continuous time stochastic modified equation from \eqref{seclinear}, we view
$m_k$ as the evaluation of a function $m$ at time points $t_k = k \Delta t$ where $\Delta t$ is the
effective time step size for the corresponding stochastic modified equation, and it is chosen as
$\Delta t = \sqrt{\eta ( 1 - \mu)}$. By introducing the auxiliary variable
$p_{k} = \frac{1}{\Delta t} (m_k - m_{k-1})$, we can reformulate \eqref{seclinear} as a system
of $(m_k, p_k)$:
\begin{align}\label{part1}
p_{k+1} & = p_k - \Delta t \sqrt{(1-\mu)/\eta} p_k - \Delta t \nabla f(m_k) +
\Delta t \, \bigl(\nabla f(m_k) - \nabla f_{\gamma_k}(x_{k-\tau_k})\bigr),\\
\label{part2}
m_{k+1} & = m_k + \Delta t \, p_{k+1}.
\end{align}
To obtain an SME, we first model the random term by a Gaussian random noise, that is, $\Delta t
\bigl(\nabla f(m_k) - \nabla f_{\gamma_k}(x_{k-\tau_k})\bigr) \sim \sigma_k (\eta (1-\mu))^{1/4} \Delta B_t$,
where $\Delta B_t = B_{t + \Delta t} - B_t$ is the increment of a Brownian motion (thus
$\mathbb{E}(\Delta B_t) = 0$ and $\mathbb{E} (\Delta B_t\Delta B_t^T) = \Delta t$) and the coefficient is
chosen to match the variance. Such modelling is valid because the random variables $\gamma_k$ and $\tau_k$ are independent to each other, and the choices are independent at each iteration, we can approximate the i.i.d random random term by Gaussian noise in the weak sense. Assuming that $\Delta t$ is small, we arrive at a Langevin type
equation:
\begin{equation}\label{linearASGD}
\begin{aligned}
dP_t & = - \nabla f (M_t) dt-\sqrt{(1-\mu)/\eta} P_t dt + \sigma(t)(\eta (1-\mu))^{1/4} dB_t,\\
dM_t & = P_t dt,
\end{aligned}
\end{equation}
where $\Sigma(t) = \Sigma(\{M_s\}_{0\leq s<t}, \{P_s\}_{0\leq s<t})$ has the evolution equation (the derivation is deferred to Appendix A)
\begin{align*}
d\Sigma_t =& -\sqrt{\frac{1-\mu}{\eta}} \Sigma_t dt -\mu (\nabla f(M_t)\nabla f(P_t)^T +\nabla f(P_t)\nabla f(M_t)^T) dt - \sqrt{\frac{1-\mu}{\eta}}\nabla f(M_t)\nabla f(M_t)^T dt\\
&+\mu \sqrt{\eta(1-\mu)} \nabla f(P_t)\nabla f(P_t)^T dt + \frac{1}{n}\sqrt{\frac{1-\mu}{\eta}}\sum_{i=1}^{n}\nabla f_i(M_t + \mu\sqrt{\frac{\eta}{1-\mu}} P_t)\nabla f_i(M_t + \mu\sqrt{\frac{\eta}{1-\mu}} P_t)^T dt.
\end{align*}
When $f$ is a smooth confining potential, that is, $f$ satisfies $\lim_{|x|\to +\infty} f(x) = +\infty$ and $e^{-\beta f(x)}\in L^1(\mathbb{R}^d)$ for all $\beta\in \mathbb{R}^+$ (an example for $f$ is being a quadratic potential), the process
approaches to the minimum of the potential function, and $\sigma(t)$ (as the damping term $-\sqrt{(1-\mu)/\eta} \Sigma_t dt$ dominates in the evolution equation) can be approximated by a constant matrix $\sigma$ up to a first order approximation for large time $t$. When this constant matrix $\sigma$ is a multiple of the identity
matrix, say $\sigma = \varsigma I$, $(P_t, M_t)$ in the standardized model is an ergodic Markov process with stationary
distribution \cite{pavliotis2014stochastic}:
\[
\rho_{\infty}(p,m) = Z^{-1} e^{-\beta (\frac{1}{2} |p|^2 + f(m)) },
\]
where $Z$ is a normalization constant. In this case, the resulting friction is
$\sqrt{(1-\mu)/\eta}$ and the temperature $\beta^{-1}$ is $\frac{1}{2} \varsigma^2 \eta$.
When the constant matrix $\sigma$ is not a multiple of identity (but still being constant), the stationary
distribution takes a similar form in a
transformed coordinate system. We remark that though in theory proving time-inhomogeneous process (\ref{linearASGD}) has a unique stationary distribution is beyond the scope of this paper, the numerical observations suggest that such a constant approximation of the noise coefficient does not change the process' property fundamentally; in the numerical experiments, we observe that the trajectory of SME-ASGD does not change much when we replace the coefficient of noise by a constant matrix.
The reason why we care about the temperature parameter here is that it quantifies the variance of
the noise and therefore gives us more information about the asymptotic behavior of the optimization
process. With such a tool, we can better analyze the connection between different stochastic
gradient algorithms. Let us illustrate it by showing one example here: Mitliagkas et
al. \cite{mitliagkas2016asynchrony} argues that there is some equivalence between adding asynchrony
or momentum to the SGD algorithms, and they showed it by taking expectation to a simple queuing
model and finding matched coefficients. Here, we investigate such relation by looking at the
corresponding Langevin dynamics, specifically the temperature for both SMEs, thus offering a more
detailed dynamical comparison.
Stochastic gradient descent with momentum (MSGD) introduced by \cite{polyak1964some} utilizes the
velocity vector from the past updates to accelerate the gradient descent
\cite{sutskever2013importance}:
\begin{equation}\label{momentum}
\begin{aligned}
& v_{k+1} = \mu' \, v_k - \eta' \nabla f_{\gamma_k}(x_k), \\
& x_{k+1} = x_k + v_{k+1},
\end{aligned}
\end{equation}
with a momentum parameter $\mu' \in (0,1)$. \eqref{momentum} can be also viewed as a discretization
of a second-order stochastic differential equation. Our derivation here is slightly different from
\cite{li2015stochastic} since we use a more natural time scale $\Delta t =\sqrt{\eta'}$ in order to
obtain an SDE with bounded coefficients. By taking $p$ to be $v/\sqrt{\eta'}$ (see Appendix A), we end
up with the following stochastic modified equation for MSGD (denoted in short as SME-MSGD)
\begin{align}\label{mosme}
& dP_t = -\nabla f(X_t) dt - \frac{1-\mu'}{\sqrt{\eta'}}P_t dt + \sigma(X_t)(\eta')^{\frac{1}{4}} dB_t, \nonumber \\
& dX_t = P_t dt,
\end{align}
where the friction is $\frac{1-\mu' }{\sqrt{\eta'}}$.
Note that \eqref{mosme} is time-homogeneous with an multiplicative noise, such that the invariant measure usually does not have an explicit expression in general. We further postulate that when the noise is small, the coefficient $\sigma(X_t)$ can be approximated by a constant multiple of the identity matrix. In this case, the temperature ${\beta'}^{-1} =
\frac{\varsigma^2 \eta'}{2(1-\mu' )}$ dictates the convergence rate to the stationary solution. If we further assume that the noise coefficients $\sigma$ in SME-ASGD \eqref{linearASGD} and in SME-MSGD \eqref{mosme} are the same constant, comparing
\eqref{linearASGD} with \eqref{mosme} results in the following interesting observation.
\smallskip
\begin{proposition}
If we assume that the noise coefficients $\sigma$ in SME-ASGD \eqref{linearASGD} and in SME-MSGD \eqref{mosme} are the same constant, if $\mu' = \mu$ and $\eta' = \eta (1-\mu)$, then \eqref{linearASGD} and \eqref{mosme} have the same stationary distribution.
\end{proposition}
In Theorems 3 and 5 in Mitliagkas et al.'s paper \cite{mitliagkas2016asynchrony}, the staleness'
geometric distribution parameter $\mu $ is taken to be $ \mu' = 1-\frac{1}{M}$, where $M$ is
the number of mutually independent workers and $\mu'$ is the momentum parameter. With these
assumptions, when looking at \eqref{linearASGD} and \eqref{mosme} under the same time scale with
$\eta' = \eta (1-\mu)$, we can see that ${\beta'}^{-1} = \frac{\varsigma^2 \eta'}{2(1-\mu' )}
= \frac{\varsigma^2 \eta}{2} = \beta^{-1}$. Since the corresponding temperature for the asynchronous method and momentum method are equal, we conclude that the
perspective of stochastic modified equation given above explains the observation in
\cite{mitliagkas2016asynchrony} that the momentum method has certain equivalent performance as the
asynchronous method.
\subsection{Nonlinear gradients}
We now consider the general case in which the gradient $\nabla f_i$
can be non-linear. One can still write the ASGD into a stochastic
modified equation.
For this, let us define a new auxiliary variable $y_k$ which is proportional to
the expected gradient:
\begin{align}\label{newsum}
y_k = -\alpha \mathbb{E}_{\tau} (\nabla f(x_{k-\tau_k}) )= -\alpha \sum\nolimits_{l=0}^{\infty} \nabla f(x_{k-l})
(1-\mu)\mu^l,
\end{align}
where $\alpha>0$ is to be determined. Again $y_k$ is random and a conditional expectation conditioned on the history of $x$.
Directly following the definition, $y_k$ satisfies the difference equation
\begin{align}\label{ypart}
\frac{y_{k+1}-y_k}{\alpha(1-\mu)} = -\frac{y_k}{\alpha}-\nabla f(x_{k+1}).
\end{align}
Moreover, we can rewrite the ASGD \eqref{ASGD} as
\begin{align}\label{xpart}
\frac{x_{k+1}-x_k}{\eta/\alpha} = y_k + \alpha \Bigl(-\frac{y_k}{\alpha} - \nabla f_{\gamma_k}
(x_{k-\tau_k}) \Bigr).
\end{align}
The reason for us arranging terms in this way is to formulate a Langevin-type equation, but with the
noise term moved from the momentum side ($Y$) to the position side ($X$). Notice that on the right
hand side of \eqref{xpart}, $-\frac{y_k}{\alpha} - \nabla f_{\gamma_k} (x_{k-\tau_k})$ can be viewed
as a noise with mean $0$
\begin{align*}
\mathbb{E}_{\gamma,\tau}\Bigl(-\frac{y_k}{\alpha} - \nabla f_{\gamma_k} (x_{k-\tau_k})\Bigr) &= \frac{1}{n}\sum_{i=1}^n \mathbb{E}_{\tau}\bigg( \sum_{l=0}^{\infty} \nabla f(x_{k-l}) (1-\mu)\mu^l - \nabla f_i (x_{k-\tau_k})\bigg)\\
&=\mathbb{E}_{\tau}\bigg( \sum_{l=0}^{\infty} \nabla f(x_{k-l}) (1-\mu)\mu^l - \nabla f (x_{k-\tau_k})\bigg)\\
&=\sum_{m=0}^{\infty} (1-\mu)\mu^m \big(\sum_{l=0}^{\infty} \nabla f(x_{k-l}) (1-\mu)\mu^l - \nabla f (x_{k-m})\big) \\
&=\sum_{l=0}^{\infty} \nabla f(x_{k-l}) (1-\mu)\mu^l-\sum_{m=0}^{\infty}\nabla f (x_{k-m}) (1-\mu)\mu^m = 0.
\end{align*}
And the covariance matrix conditioned on $x_{k-l}, l=0,1,2,\cdots$ is given by
\begin{align*}
\Sigma_k &= \frac{1}{n}\sum_{i=1}^n
\mathbb{E}\left(\left(-\frac{y_k}{\alpha} - \nabla f_i(x_{k-\tau_k})\right)
\left(-\frac{y_k}{\alpha} - \nabla f_i(x_{k-\tau_k})\right)^T\right)\\
& = \frac{1}{n}\sum_{i=1}^n \mathbb{E}\left(\left(\sum_{l=0}^{\infty} \nabla f(x_{k-l}) (1-\mu)\mu^l -
\nabla f_i(x_{k-\tau_k})\right)
\left(\sum_{l=0}^{\infty} \nabla f(x_{k-l}) (1-\mu)\mu^l - \nabla f_i(x_{k-\tau_k})\right)^T\right).
\end{align*}
In order to view \eqref{ypart} and \eqref{xpart} as a time-discretization of a coupled system with
the same time step size, we match $\alpha(1-\mu)$ with $\eta/\alpha$ by choosing
$\alpha = \sqrt{\nicefrac{\eta}{(1-\mu)}}$.
Setting the step size $\Delta t=\alpha(1-\mu)=\eta/\alpha=\sqrt{\eta(1-\mu)}$
and taking a Gaussian approximation to the noise
$\eta\big(-\frac{y_k}{\alpha}-\nabla f_{\gamma_k} (x_{k-\tau_k}) \big)\sim
\sqrt{\Sigma_k}\frac{\eta^{3/4}}{(1-\mu)^{1/4}} \Delta B_t$, we arrive at the stochastic modified
equation for the nonlinear case
\begin{equation}\label{nonlinearASGD}
\begin{aligned}
dY_t &= -\nabla f(X_t) dt -\sqrt{\frac{1-\mu}{\eta}} Y_t dt \\
dX_t &= Y_t dt + \sqrt{\Sigma(t)}\frac{\eta^{3/4}}{(1-\mu)^{1/4}} dB_t
\end{aligned}
\end{equation}
Here $\Sigma(t) = \Sigma(\{X_s\}_{0\le s<t},\{Y_s\}_{0\le s<t})$. In order to close the system of
equations, we derive an explicit evolution equation for $\Sigma$
\begin{multline}\label{evol}
d\Sigma_t = -\sqrt{\frac{1-\mu}{\eta}}\Sigma_t dt + \sqrt{\frac{1-\mu}{\eta}}\Bigl(
\frac{1}{n}\sum_{i=1}^{n} \nabla f_i(X_t) \nabla f_i(X_t)^T +\frac{1-\mu}{\mu} \nabla f(X_t)
\nabla f(X_t)^T \Bigr) dt \\ +\frac{1-\mu}{\eta \mu} \bigl(\sqrt{\frac{1-\mu}{\eta}}Y_t Y_t^T
+\nabla f(X_t) Y_t^T + Y_t \nabla f(X_t)^T\Bigr) dt.
\end{multline}
The derivation of \eqref{evol} is shown in Appendix A. The combined system
\eqref{nonlinearASGD}--\eqref{evol} will be referred as SME-ASGD (the stochastic modified equations
for asynchronous SGD) for the general nonlinear-gradient case. We should point it out that unlike
the linear-gradient case \eqref{linearASGD} , \eqref{nonlinearASGD} has no known explicit formula
for invariant measure even when $\Sigma(t)$ converging to a constant matrix. Nevertheless, the
ergodicity of \eqref{nonlinearASGD} and \eqref{evol} will be an interesting future direction to
explore.
We would like to point out that when the gradient $\nabla f$ is linear \eqref{ypart}
and \eqref{xpart} can be easily transformed back to \eqref{part1}
and \eqref{part2}. As a consequence, \eqref{linearASGD} and
\eqref{nonlinearASGD} are equivalent. To see this,
\[
y_k = -\alpha \nabla f(\sum_{l=0}^{\infty} x_{k-l} (1-\mu)\mu^l) = -\alpha \nabla f(m_k).
\]
Replacing $y_{k+1}$ and $y_k$ with the above formula and also $x_{k+1}$ with $\frac{m_{k+1}-\mu
m_k}{1-\mu}$, we can rewrite \eqref{ypart} as
\[
-\frac{\nabla f(m_{k+1}) - \nabla f(m_k)}{1-\mu} = \nabla f(m_k) - \nabla f( \frac{m_{k+1}-\mu
m_k}{1-\mu}) = -\frac{1}{1-\mu}\nabla f(m_{k+1} - m_k).
\]
Since $p_{k+1} = (m_{k+1} - m_k)/\sqrt{\eta(1-\mu)}$, we have
\[
\nabla f(m_{k+1}-m_k) = \nabla f(p_{k+1 }\sqrt{\eta(1-\mu)}),
\]
which implies \eqref{part2}. To show \eqref{part1}, we first notice that
\begin{align*}
\frac{x_{k+1} - x_k}{\eta/\alpha} &= \frac{m_{k+1} - (\mu+1)m_k + \mu m_{k-1}}{(1-\mu )\eta/ \alpha}= \frac{m_{k+1} - 2m_k +m_{k-1}}{(1-\mu)\eta/\alpha} +\frac{m_k - m_{k-1}}{\eta/\alpha}\\
& = \frac{p_{k+1} - p_k}{1-\mu} + p_k = -\alpha \nabla f(m_k) + \alpha (\nabla f(m_k) - \nabla f_{\gamma_k}(v_k))\\
& = -\sqrt{\frac{\eta}{1-\mu}} \nabla f(m_k) + \sqrt{\frac{\eta}{1-\mu}} (\nabla f(m_k) - \nabla f_{\gamma_k}(v_k))
\end{align*}
by plugging in $\alpha$ in terms of $\mu, \eta$. It is clear now that this gives \eqref{part1}.
\section{Approximation error of the stochastic modified equation}
The difference between the time-discrete ASGD and the time-continuous SME-ASGD can be rigorously
quantified as follows.
\begin{theorem}\label{thm}
Assume that Assumption~\ref{assump} holds and that the variance from the asynchronous
gradients is uniformly bounded (i.e., there exists $c>0$ such that $||\sigma(t)||\leq c$). Suppose also that
all the iterates updated from the ASGD stay bounded and that the solutions for SME-ASGD and ASGD before time $0$
agree (i.e., $X_{l\Delta t} = x_l, l\leq 0$, with $\Delta t = \sqrt{\eta(1-\mu)}$
as given previously). Then the SME-ASGD approximates the ASGD in the sense that there exists constant $K_T>0$
depending only on $T$ such that
\begin{align}
\sup_{n\Delta t \leq T} \mathbb{E}\big\{|X_{n \Delta t} - x_n|\big\} \leq K_T \frac{\Delta t}{1-\mu}
\end{align}
for $\Delta t$ sufficiently small. Here $X_{n\Delta t}\equiv X(n\Delta t)$ is the
solution of \eqref{nonlinearASGD} at time $n\Delta t$ and $x_n$ is
from ASGD \eqref{ASGD}.
\end{theorem}
The assumption $\sigma = \sqrt{\Sigma} = O(1)$ can be justified from \eqref{evol} as $\Sigma$ is
approximated by a constant matrix for $t$ large. This is because when the iterate approaches to the
minimizer, the gradients are close to $0$, and $Y_t$ converges to be a constant vector. Since we investigate the error approximation in finite time $T$ and finite step size $\Delta t$, there are only a finite number of iterations. In each iteration, the iterate updated from the ASGD stays bounded by a sufficient large constant with high probability. Therefore, the assumption that all iterates stay bounded by a sufficient large constant holds with high probability.
The proof of the Theorem \eqref{thm} follows from viewing the ASGD as a discretization of SME-ASGD
and using the analysis of strong convergence for numerical schemes for stochastic differential
equations (SDEs). \\
\begin{proof}[Proof of the Theorem \eqref{thm}]
We look at the one step approximation in the first step, and the global approximation can be done by
induction. Using the variation of constant formula, we know that the solution of
\[
dY_t = -\nabla f(X_t) dt -\sqrt{\frac{1-\mu}{\eta}} Y_t dt
\]
is given by
\[
Y_t = e^{-\sqrt{\frac{1-\mu}{\eta}}t} Y_0 - \int_0^t e^{-\sqrt{\frac{1-\mu}{\eta}}(t-s)} \nabla f(X_s) ds,
\]
where $Y_0 = -\sqrt{\frac{\eta}{1-\mu}}\sum_{l=0}^{\infty} \nabla f(x_{-l})(1-\mu) \mu^l$ as defined
in \eqref{newsum}. Plugging $Y_t$ into the integral form of $X_{\Delta t}$ gives rise to
\begin{align}
X_{\Delta t} = x_0 + \int_0^{\Delta t} \bigg( e^{-\sqrt{\frac{1-\mu}{\eta}}s} Y_0 -
\int_0^s e^{-\sqrt{\frac{1-\mu}{\eta}}(s-u)}\nabla f(X_u) du \bigg)ds +
\frac{\eta^{3/4}}{(1-\mu)^{1/4}} \int_0^{\Delta t} \sigma(s) dB_s.
\end{align}
Denote $v_k := x_{k-\tau_k}$ for notation convenience. By splitting $\eta \nabla f_{\gamma_0}(v_0)$ into $\eta \nabla f_{\gamma_0}(v_0) - \eta
\sum_{l=0}^{\infty} \nabla f(x_{-l})(1-\mu) \mu^l$ and $ \eta \sum_{l=0}^{\infty} \nabla
f(x_{-l})(1-\mu) \mu^l$, we can make the following estimate
\begin{align*}
\mathbb{E}\big\{|X_{\Delta t} &- x_1 |\big\}\leq \bigg| \int_0^{\Delta t} e^{-\sqrt{\frac{1-\mu}{\eta}}s} Y_0ds + \eta \sum_{l=0}^{\infty} \nabla f(x_{-l})(1-\mu) \mu^l \bigg| \\
&+\mathbb{E}\bigg\{ \int_0^{\Delta t} \bigg( \int_0^s e^{-\sqrt{\frac{1-\mu}{\eta}}(s-u)}\big|\nabla f(X_u)-\nabla f(x_1) \big| du \bigg) ds \bigg\} + |\nabla f(x_1)| \int_0^{\Delta t} \int_0^s e^{-\sqrt{\frac{1-\mu}{\eta}}(s-u)}du ds\\
&+ \frac{\eta^{3/4}}{(1-\mu)^{1/4}}\bigg( \mathbb{E}\big\{ \big(\int_0^{\Delta t} \sigma(s) dB_s \big)^2\big\}\bigg)^{1/2}+ \mathbb{E}\big\{\big|\eta \nabla f_{\gamma_0}(v_0) - \eta \sum_{l=0}^{\infty} \nabla f(x_{-l})(1-\mu) \mu^l \big| \big\} \\
&\leq I + II + III + \frac{\eta^{3/4}}{(1-\mu)^{1/4}}\bigg( \mathbb{E}\big\{ \int_0^{\Delta t} \sigma(s)^2 ds \big\}\bigg)^{1/2} + c \eta \leq I + II + III + 2c \frac{\Delta t^2}{1-\mu},
\end{align*}
where $I$, $II$, and $III$ are the first three terms appeared in the right hand side of the first inequality. In the above derivation, we have applied the Ito isometry to the fourth term and used
\[
\frac{\eta^{3/4}}{(1-\mu)^{1/4}}\bigg( \mathbb{E}\big\{ \int_0^{\Delta t} \sigma(s)^2 ds
\big\}\bigg)^{1/2} \leq c \frac{\Delta t^2}{1-\mu},
\]
since $\Delta t = \sqrt{\eta(1-\mu)}$. The fifth term, after an application of the Cauchy-Schwarz inequality,
is shown to be a discrete version of the covariance matrix
\begin{align*}
\mathbb{E}\big\{\big|\eta \nabla f_{\gamma_0}(v_0) - \eta \sum_{l=0}^{\infty} \nabla f(x_{-l})(1-\mu) \mu^l \big| \big\}\leq \eta \sqrt{\Sigma_0} \leq c\eta.
\end{align*}
Let us now treat the first three terms
\begin{align*}
I &= \bigg| \int_0^{\Delta t} e^{-\sqrt{\frac{1-\mu}{\eta}}s} Y_0 ds + \eta \sum_{l=0}^{\infty} \nabla f(x_{-l})(1-\mu) \mu^l \bigg|\\
&=\bigg| \sqrt{\frac{\eta}{1-\mu}} (e^{-\sqrt{\frac{1-\mu}{\eta}}\Delta t} -1) \sqrt{\frac{\eta}{1-\mu}}\sum_{l=0}^{\infty} \nabla f(x_{-l})(1-\mu) \mu^l +\eta \sum_{l=0}^{\infty} \nabla f(x_{-l})(1-\mu) \mu^l \bigg|\\
&= \bigg|- \sqrt{\frac{\eta}{1-\mu}}\sum_{l=0}^{\infty} \nabla f(x_{-l})(1-\mu) \mu^l \Delta t + \eta \sum_{l=0}^{\infty} \nabla f(x_{-l})(1-\mu) \mu^l +O(\Delta t^2) \bigg|= O(\Delta t^2),
\end{align*}
since the first two terms cancel. Because $\nabla f$ is Lipschitz and $e^{-\sqrt{\frac{1-\mu}{\eta}}(s-u)} \le 1$ for $u\leq s$, the second term
can be estimated with
\begin{align*}
II \leq L\Delta t \int_0^{\Delta t} \mathbb{E}\left\{|X_u - x_1| \right\} du.
\end{align*}
Since $x_1$ stays in a bounded domain, the third term can be bounded by
\begin{align*}
III \leq |\nabla f(x_1)| \int_0^{\Delta t} s ds= |\nabla f(x_1)| \Delta t^2 /2 = O(\Delta t^2).
\end{align*}
With these estimates available, we can choose a sufficiently large
constant $C$ (depending on $c$ and the size of the domain containing
the iterates from ASGD) such that
\begin{align*}
\mathbb{E}\big\{|X_{\Delta t} - x_1 |\big\}&\leq C\frac{\Delta t^2}{1-\mu} + L\Delta t \int_0^{\Delta t}
\mathbb{E}\big\{|X_u - x_1|\big\} du.
\end{align*}
An application of Gronwall's inequality shows that
\begin{align*}
\mathbb{E}\big\{|X_{\Delta t} - x_1 |\big\}&\leq C\frac{\Delta t^2}{1-\mu}e^{L\Delta t^2}= C \frac{\Delta t^2}{1-\mu}+O(\Delta t^4)\leq C\frac{\Delta t^2}{1-\mu}.
\end{align*}
This concludes the estimate for the first step at time 0.
The induction step is similar. We have
\begin{align*}
X_{(k+1)\Delta t} = X_{k\Delta t} + \int_{k\Delta t}^{(k+1)\Delta t} \bigg(e^{-\sqrt{\frac{1-\mu}{\eta}}(s-k\Delta t)}Y_{k\Delta t}& -\int_{k\Delta t}^s e^{-\sqrt{\frac{1-\mu}{\eta}}(s-u)}\nabla f(X_u) du \bigg) ds \\
+& \frac{\eta^{3/4}}{(1-\mu)^{1/4}} \int_{k\Delta t}^{(k+1)\Delta t} \sigma(s) dB_s.
\end{align*}
For the discrete update step $x_{k+1} = x_k -\eta \nabla f_{\gamma_k}(v_k)$, we split $\eta \nabla f_{\gamma_k}(v_k)$
as before. With the assumption $\mathbb{E}\big\{|X_{k\Delta t} - x_k |\big\}\leq
Ck\frac{\Delta t^2}{1-\mu}$, we have the following estimate
\begin{align*}
\mathbb{E}\big\{|X_{(k+1)\Delta t} &- x_{k+1} |\big\}\leq \mathbb{E}\big\{|X_{k\Delta t} - x_k |\big\}+ \int_{k\Delta t}^{(k+1)\Delta t} e^{-\sqrt{\frac{1-\mu}{\eta}}(s-k\Delta t)}\big|Y_{k\Delta t} - y_k \big| ds\\
&+\bigg| \int_{k\Delta t}^{(k+1)\Delta t} e^{-\sqrt{\frac{1-\mu}{\eta}}(s-k\Delta t)} y_k ds + \eta \sum_{l=0}^{\infty} \nabla f(x_{k-l})(1-\mu) \mu^l \bigg| \\
&+\mathbb{E}\bigg\{ \int_{k\Delta t}^{(k+1)\Delta t} \bigg( \int_{k\Delta t}^s e^{-\sqrt{\frac{1-\mu}{\eta}}(s-u)}\big|\nabla f(X_u)-\nabla f(x_{k+1}) \big| du \bigg) ds \bigg\}\\
& + |\nabla f(x_{k+1})| \int_{k\Delta t}^{(k+1)\Delta t}\int_{k\Delta t}^s e^{-\sqrt{\frac{1-\mu}{\eta}}(s-u)}du ds + \frac{\eta^{3/4}}{(1-\mu)^{1/4}}\bigg( \mathbb{E}\big\{ \big(\int_{k\Delta t}^{(k+1)\Delta t} \sigma(s) dB_s \big)^2\big\}\bigg)^{1/2}\\
&+ \mathbb{E}\big\{\big|\eta \nabla f_{\gamma_k}(v_k) - \eta \sum_{l=0}^{\infty} \nabla f(x_{k-l})(1-\mu) \mu^l \big| \big\}.
\end{align*}
Here the only difference compared to the first step is the term $Y_{k\Delta t}$, which is not given but
generated from SME. Note that
\[
y_k = -\sqrt{\frac{\eta}{1-\mu}}\sum_{l=0}^{\infty} \nabla f(x_{k-l})(1-\mu) \mu^l.
\]
From \eqref{ypart}, we observe that $y_k$ is indeed an approximation of $Y_t$ by applying
the Euler discretization to the ordinary differential equation part of the SME. Because the global
truncation error for the Euler method in ODE is $O(\Delta t)$, we have
\begin{align*}
\int_{k\Delta t}^{(k+1)\Delta t} e^{-\sqrt{\frac{1-\mu}{\eta}}(s-k\Delta t)} \big|Y_{k\Delta t} -y_k\big|ds= O(\Delta t ^2).
\end{align*}
The third term has the estimate
\begin{align*}
\bigg| \int_{k\Delta t}^{(k+1)\Delta t} & e^{-\sqrt{\frac{1-\mu}{\eta}}(s-k\Delta t)} y_k ds + \eta \sum_{l=0}^{\infty} \nabla f(x_{k-l})(1-\mu) \mu^l \bigg| \\
&= \bigg| -\sqrt{\frac{\eta}{1-\mu}}\big(e^{-\sqrt{\frac{\eta}{1-\mu}}\Delta t}-1\big)y_k + \eta \sum_{l=0}^{\infty} \nabla f(x_{k-l})(1-\mu) \mu^l \bigg|\\
& = \bigg|-\sqrt{\frac{\eta}{1-\mu}}\sum_{l=0}^{\infty} \nabla f(x_{k-l})(1-\mu) \mu^l \Delta t + \eta \sum_{l=0}^{\infty} \nabla f(x_{k-l})(1-\mu) \mu^l + O(\Delta t^2) \bigg|=O(\Delta t^2)
\end{align*}
as before. All other terms have the same estimates as in the base case. Applying the Gronwall's inequality again and letting $\Delta t $ be sufficiently small gives the estimate
\begin{align*}
\mathbb{E}&\big\{\big|X_{(k+1)\Delta t} - x_{k+1} \big|\big\}\leq Ck\frac{\Delta t^2}{1-\mu}.
\end{align*}
As $n\Delta t \leq T$ for all $n$, one can conclude that there exists $K_T>0$ such that
\begin{align*}
\mathbb{E}\big\{\big|X_{n \Delta t} - x_n \big|\big\}\leq K_T \frac{\Delta t}{1-\mu}.
\end{align*}
\end{proof}
One interesting observation is that, contrary to the standard Euler-Maruyama
method for SDEs having strong order of convergence $1/2$ \cite{kloeden1992stochastic}, the above
result indicates that ASGD, viewed as a discretization of SME-ASGD, has strong order $1$. This is
because the coefficient of the noise term in the SME-ASGD has
$\nicefrac{\eta^{3/4}}{(1-\mu)^{1/4}}$, which is of order $o(1)$. The SME model proposed in
\cite{li2015stochastic} has the same feature: the coefficient of the noise term there is of order
$\sqrt{\eta}$. When $\eta \approx 1-\mu$, the two orders are the same.
Here, we provide some numerical evidences for Theorem \ref{thm} with various loss functions
$f$. The results are shown in Figures~\ref{fig:linear} (for linear forcing) and \ref{fig:nonlinear}
(for general forcing). For each example, through averaging over $5000$ samples, we compare the
results of ASGD with the predictions from both SME-ASGD \eqref{nonlinearASGD} and the 2nd-order weak
convergent SME-SGD proposed in Li et al.'s paper \cite{li2015stochastic}
\begin{align}\label{LiSME}
dX_t = -\nabla (f(X_t) + \frac{\eta}{4}|\nabla f(X_t)|^2) dt + (\eta \Sigma(X_t))^{1/2} dB_t.
\end{align}
When $\mu$ is close to $0$ (i.e., the expected delay is short),
SME-SGD \eqref{LiSME} serves as a good approximation to ASGD as expected. However, when $\mu$ is large,
Figures \ref{fig:linear} and \ref{fig:nonlinear} demonstrate that it is no longer the case:
As $\mu$ gets closer to $1$, the trajectories obtained from SME-SGD are way off,
whereas our proposed SME-ASGD model demonstrate accurate path approximations for both the first
and the second moments.
A few remarks regarding the numerical results are in order here. (i) In Figure \ref{fig:linear},
the path oscillations happen to both ASGD and SME-ASGD due to a longer expected delay, but not to
SME-SGD, even though we include staleness when computing $\Sigma(X_t)$ by the convariance matrix formula for both models. That is
because our SME-ASGD model contains $\mu$ in the forcing term, while the forcing term in SME-SGD is
$\mu$-independent.
(ii) The convex function $f(x)=x^4+6x^2$ (with gradient $\nabla f(x) = 4x^3+12x$) in Figure
\ref{fig:nonlinear} does not satisfy the general Ito conditions; however, by having good initial
data and choosing smaller time step sizes, we can still obtain the minimizer without blowing
up.
(iii) For the non-convex example (the double-well function in Figure \ref{fig:nonlinear}), the
SME-ASGD model gives a better prediction about which minimizer that a trajectory with
given initial data will fall into: The percentage of path samples that converge to a local minimum
in SME-ASGD is very close to that of the ASGD case.
(iv) For all cases, SME-SGD underestimates the variance because the variance from the delayed reads
is not taken into account by SME-SGD. (v) In higher dimensions, unlike the Monte Carlo sampling driven by Langevin dynamics that has the curse of dimensionality issue, our numerical simulations for both linear and nonlinear gradidents have good approximation regardless of the dimensionality as the Figures \ref{fig:nd1} and \ref{fig:nd2} show. Here, we assign the coefficients $c_i$ uniformly randomly in $[0,5]$. We make plots by arbitrarily choosing any two dimension as projected subspace. Although after $1000$ time steps, some projected subspace have convergence and and some (with significant coefficient differences) do not yet, we can see that the trajectories from the algorithm and modified equation are close.
\begin{figure}[h!]
\centering
\begin{minipage}{0.32\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{linear09mean.eps}
\end{minipage}\hfill
\begin{minipage}{0.32\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{linear095mean.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{linear097mean.pdf}
\end{minipage}
\begin{minipage}[a]{0.32\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{linear09std.pdf}
\caption*%
{{\small $\mu=0.9$}}
\end{minipage}\hfill
\begin{minipage}{0.32\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{linear095std.pdf}
\caption*%
{{\small $\mu=0.95$}}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{linear097std.pdf}
\caption*%
{{\small $\mu=0.97$}}
\end{minipage}
\caption{ Apply the SME-ASGD to minimize the quadratic function $f(x) = x^2$ in different
$\mu$'s, with two components $f_1(x) = (x-1)^2-1$, and $f_2(x) = (x+1)^2-1$. $x_0 =1$ and
$\eta = 1e-2$. SME-ASGD achieves more accurate approximations compared to SME-SGD
\eqref{LiSME}, especially when $\mu$ becomes large. However, one can also observe that when
$\mu$ increases the error of the SME-ASGD approximation increases as well.}
\label{fig:linear}
\end{figure}
\begin{figure}[h!]
\centering
\begin{minipage}{0.4\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{nonlinear095mean.pdf}
\end{minipage}\qquad
\begin{minipage}{0.4\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{dwellmean.pdf}
\end{minipage}
\begin{minipage}[a]{0.4\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{nonlinear095std.pdf}
\end{minipage}\qquad
\begin{minipage}{0.4\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{dwellstd.pdf}
\end{minipage}
\caption{ (Left) Apply the SME-ASGD to minimize the convex function $f(x) = x^4+ 6x^2$ with
two components $f_1(x) = (x-1)^4-1$, and $f_2(x) = (x+1)^4-1$. Notice that the gradients are
Lipschitz locally. Here we choose $x_0 =1$, and a smaller step size $\eta = 1e-3$. (Right)
Apply the SME-ASGD to minimize the double well potential $f(x) = 1 - e^{-(x-1)^2} -
e^{-(x+1)^2}$. Here $f_1 = 1 - 2e^{-(x-1)^2}, f_2 = 1 - 2e^{-(x+1)^2}$ and both have Lipschitz
gradients. We choose $ \eta = 1e-2, x_0 = 0.1$.
Note that $\argmin f(x)\approx\pm0.9575$. In our case, due to the initial data $x_0$,
$90.34\%$ of ASGD path samples converge to $0.9575$, while $90.50\%$ of SME-ASGD and $88.54\%$
of SME-SGD converge to the same minimizer. For both columns of numerical tests, we choose $\mu
= 0.95$.}
\label{fig:nonlinear}
\end{figure}
\begin{figure}[h!]
\centering
\begin{minipage}{0.32\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{ndlinearproj1.pdf}
\end{minipage}\hfill
\begin{minipage}{0.32\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{ndlinearproj2.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{ndlinearproj3.pdf}
\end{minipage}
\caption{Apply the SME-ASGD to minimize the quadratic function $f(x) = \sum_{i=1}^{100} c_i x_{i}^2/2, ~ x\in \mathbb{R}^{100}$ with $\mu=0.90$ and two components $f_1(x) = \sum_{i=1}^{100} c_ix_i^2/2-x$, and $f_2(x) = \sum_{i=1}^{100} c_ix_i^2/2+x$. The initial condition $x_0 = (0.5,0.5,\cdots, 0.5)\in \mathbb{R}^{100}$ and the step size is $\eta= 1e-2$. The plots are done after $1000$ iterations. The corresponding coefficients in the plots are $c_1 = 4.2593, c_2= 4.9013, c_{29} = 0.1980, c_{81} = 4.3968, c_{37} = 3.9978, c_{74}=1.9527$. }
\label{fig:nd1}
\begin{minipage}[a]{0.32\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{ndnonlinearproj1.pdf}
\end{minipage}\hfill
\begin{minipage}{0.32\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{ndnonlinearproj2.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{ndnonlinearproj3.pdf}
\end{minipage}
\caption{ Apply the SME-ASGD to minimize the convex function $f(x) = \sum_{i=1}^{100} c_i (x_{i}^4+6x_{i}^2)/2, ~ x\in \mathbb{R}^{100}$ with $\mu=0.90$ and two components $f_1(x) = \sum_{i=1}^{100} c_i(x_i-1)^4/2-1$, and $f_2(x) = \sum_{i=1}^{100} c_i(x_i+1)^4/2+1$. The initial condition $x_0 = (0.5,0.5,\cdots, 0.5)\in \mathbb{R}^{100}$ and the step size is $\eta= 1e-3$. The plots are done after $1000$ iterations. The corresponding coefficients in the plots are $c_1 = 3.9212, c_2= 1.9370, c_{27} = 1.2093, c_{100} = 1.5661, c_{16} = 0.3353, c_{78}=4.5502$. }
\label{fig:nd2}
\end{figure}
\section{Optimal mini-batch size of ASGD}
With much better understanding of dynamics of the ASGD algorithm using SME-ASGD, we are able to tune
multiple hyper-parameters of ASGD using the predictions obtained from applying the stochastic optimal
control theory to SME-ASGD. Here we demonstrate one such application: the optimal time-dependent
mini-batch size for ASGD. By denoting the time-dependent batch size as $1+u_k$ with $ u_k\geq 0$,
one can write the iteration as
\begin{align}\label{batch}
x_{k+1} = x_k -\eta \frac{1}{1+u_k} \sum_{j=1}^{1+u_k} \nabla f_{\gamma_j}(x_{k-\tau_k}).
\end{align}
We argue that it is reasonable to assume that the choice of mini-batch size is independent from $\gamma_j$ and the
staleness $\tau_k$. This is because, even though changing the batch size will simultaneously change the "clocks"
of all the processors, the staleness would not be changed as all the processors are impacted
equally. Following the argument given in Section 2, we can derive a corresponding SME
\begin{align}\label{batchsme}
\nonumber
&dY_t = -\nabla f(X_t) dt -\sqrt{\frac{1-\mu}{\eta}} Y_t dt \\
&dX_t = Y_t dt + \frac{\sigma(t) \eta^{3/4}}{(1+u(t))^{1/2}(1-\mu)^{1/4}} dB_t.
\end{align}
The derivation here is not much different from the one of SME-ASGD \eqref{nonlinearASGD},
except for identifying the right coefficient in front of the the noise term $dB_t$.
The correct coefficient (denoted by $c$ in the discussion below) is constrained by the
following constraints on the variance
\begin{align*}
\mathbb{E} & \left\{\frac{\eta^2}{(1+u_k)^2} \left( \sum_{j=1}^{1+u_k} (-\frac{y_k}{\alpha} - \nabla
f_{\gamma_j}(x_{k-\tau_k}) \right) \left( \sum_{j=1}^{1+u_k} (-\frac{y_k}{\alpha} - \nabla
f_{\gamma_j}(x_{k-\tau_k}) \right)^T\right\} \\ & = \frac{\eta^2}{(1+u_k)^2}\sum_{j=1}^{1+u_k}
\mathbb{E}\left\{\left(-\frac{y_k}{\alpha} - \nabla f_{\gamma_j}(x_{k-\tau_k})\right) \left(-\frac{y_k}{\alpha} -
\nabla f_{\gamma_j}(x_{k-\tau_k})\right)^T \right\} = \frac{\eta^2}{1+u_k} \Sigma_k \sim c^2 \Delta t,
\end{align*}
where the cross terms vanish under the expectation. Plugging in $\Delta t = \sqrt{\eta(1-\mu)}$ shows that
the coefficient for the noise is
\begin{align*}
c = \frac{\sigma(t) \eta^{3/4}}{(1+u(t))^{1/2}(1-\mu)^{1/4}}
\end{align*}
as shown in \eqref{batchsme}.
We would like to explore the dynamics of SME to find the dominating eigenvalue for later use.
To simplify the discussion, let us consider for example the quadratic loss objective $f(x) =
x^2$. By applying the Ito's formula to this SME, one obtains the following evoluation system for the
second moments
\begin{align}\label{secmomen}
\frac{d}{dt}\begin{bmatrix} \mathbb{E}(X_t^2)\\ \mathbb{E}(Y_t^2) \\ \mathbb{E}(X_t Y_t) \end{bmatrix} &= - \begin{bmatrix} 0 & 0 & -2\\ 0 & 2\sqrt{(1-\mu)/\eta} & 4 \\ 2 & -1 & \sqrt{(1-\mu)/\eta} \end{bmatrix}\begin{bmatrix} \mathbb{E}(X_t^2)\\ \mathbb{E}(Y_t^2) \\ \mathbb{E}(X_t Y_t) \end{bmatrix} + \begin{bmatrix} \frac{\Sigma(t)\eta^{3/2}}{(1+u(t))(1-\mu)^{1/2}} \\ 0 \\ 0\end{bmatrix}.
\end{align}
A similar derivation is shown in Appendix B, and we just replace all $\Sigma$ by $\Sigma/(1+u(t))$
in the mini-batching case. Here, we make a simplifying but practical assumption that $u(t)$ varies
slowly. Now by freezing $u(t)$ to a constant $u$, \eqref{secmomen} is a linear system with constant coefficients,
its asymptotic behavior is determined by the eigenvalue of the
coefficient matrix. An easy calculation shows that the eigenvalue with
largest real part is given by
$\lambda =-\sqrt{\nicefrac{(1-\mu)}{\eta}} + \sqrt{\nicefrac{(1-\mu -
8\eta)}{\eta}}$
with a negative real part and therefore the second moment of $X_t$ decays exponentially. Moreover, \eqref{secmomen} provides us with the stationary solution for $X^2$
\begin{align}
z_{\infty} := \mathbb{E}(X_{\infty}^2) = \frac{\Sigma \eta}{2(1+u(t))} \Bigl( \frac{\eta}{1-\mu} + \frac{1}{2}\Bigr).
\end{align}
For a slowly varying $u(t)$, $z_{\infty}=z_{\infty}(u(t))$ is a function of $u(t)$.
Based on this simplication, rather than applying the optimal control subject to the full second moment equation,
we shall work with a simpler evolution equation that asymptotically approximates the dynamics (imposed as a
constraint). More specifically, we pose the following optimal control problem for the time-dependent
mini-batch size
\begin{align}\label{opt}
& \min_{u\in \mathcal{A}}
\bigg\{ z(T) + \frac{\gamma}{\eta}\int_0^T u(s) ds \bigg\} \,\,\,\,\, \text{ subject to }\\
&\nonumber \frac{d}{dt}z(t) = \mathrm{Re}(\lambda ) (z(t) - z_{\infty}(u(t)))
\qquad \text{ with } z(0) = x_0^2,
\end{align}
where $z(t)$ models $\mathbb{E}(X_t^2)$ -- the quantity to minimize,
$\mathcal{A}=\{u(t)\geq 0\}$ is an admissible control set as the
mini-batch size is greater than $1$, and $\gamma>0$ is a constant
measuring the unit cost for introducing extra gradient samples
throughout the time. Below we show how to solve the optimal control problem \eqref{opt}. The value function can be
defined as
\begin{align}
V(z,t) = \min_{u\in \mathcal{A}}\bigg\{ z(T) + \frac{\gamma}{\eta}\int_t^T u(s) ds \; \bigg|\;
\frac{d}{dt}z(t)=F(u(t),z(t)), z(t) = z \bigg\},
\end{align}
where $F(u(t),z(t))=\text{Re}(\lambda ) (z(t) -z_{\infty}(u(t))) = \text{Re} (\lambda)\bigl(z(t)
-\frac{\Sigma \eta}{2(1+u(t))} ( \frac{\eta}{1-\mu} + \frac{1}{2})\bigr)$. The corresponding Hamilton-Jacobi-Bellman equation
is
\begin{align}
&V_t +\min_{u\in \mathcal{A}}\bigg\{ F(u,z) V_z + \frac{\gamma}{\eta} u \bigg\} = 0\\
& \nonumber \text{with } V(0, t) =0, V(z,T) = z.
\end{align}
Since $\min_{u\in \mathcal{A}}\bigg\{ F(u,z) V_z + \frac{\gamma}{\eta} u \bigg\} = \min_{u\in
\mathcal{A}}\bigg\{ \frac{-V_z \text{Re}(\lambda ) \Sigma\eta}{2(1+u)} \big(
\frac{\eta}{1-\mu} + \frac{1}{2}\big)+ \frac{\gamma}{\eta} u \bigg\}$, $V_z \geq 0$, and
$\text{Re}(\lambda )<0$, the minimum could be obtained by solving the following equation
\[
\frac{V_z \text{Re}(\lambda ) \Sigma\eta}{2(1+u)^2} \big( \frac{\eta}{1-\mu} + \frac{1}{2}\big)+ \frac{\gamma}{\eta} = 0
\]
with the derivative of the value function $V_z$ to be determined
later. Therefore the optimal batch size $u^*$ as a function of $V_z$ is
\begin{align}\label{utemp}
u^*(V_z)=
\begin{cases}
\sqrt{\frac{-V_z \text{Re}(\lambda ) \Sigma\eta^2}{2\gamma}\big( \frac{\eta}{1-\mu} + \frac{1}{2}\big)} -1 & \text{if } \frac{-V_z \text{Re}(\lambda ) \Sigma\eta^2}{2\gamma} \big( \frac{\eta}{1-\mu} + \frac{1}{2}\big)> 1, \\
0 & \text{otherwise}.
\end{cases}
\end{align}
The next step is to solve $V$ to get an explicit formula for $u^*$. Placing $u^*(V_z)$ back into the
minimization bracket, we obtain
\begin{align}
\min_{u\in \mathcal{A}}\bigg\{ F(u,z) V_z + \frac{\gamma}{\eta} u \bigg\} =
\begin{cases}
\text{Re}(\lambda ) z V_z - \frac{\gamma}{\eta}& \text{if } \frac{-V_z \text{Re}(\lambda) \Sigma\eta^2}{2\gamma} \big( \frac{\eta}{1-\mu} + \frac{1}{2}\big)> 1, \\
\text{Re}(\lambda ) \big(z-\frac{\Sigma \eta}{2}\big( \frac{\eta}{1-\mu} + \frac{1}{2}\big)\big) V_z & \text{otherwise}.
\end{cases}
\end{align}
This gives the Hamilton-Jacobi equation and we can solve it by using the method of
characteristics. Letting $\gamma^* = -\frac{\text{Re}(\lambda ) \Sigma\eta^2}{2}\big(
\frac{\eta}{1-\mu} + \frac{1}{2}\big)$ for notation convenience, we obtain the solution for $V$
\begin{align}
V(z,t)=
\begin{cases}
\frac{\Sigma\eta}{2}\big( \frac{\eta}{1-\mu} + \frac{1}{2}\big) + \big(z-\frac{\Sigma\eta}{2}\big( \frac{\eta}{1-\mu} + \frac{1}{2}\big) \big)e^{\text{Re}(\lambda ) (T-t)}& \text{if } \gamma> \gamma^* \\
\big(z-\frac{\Sigma\eta}{2}\big( \frac{\eta}{1-\mu} + \frac{1}{2}\big) \big)e^{\text{Re}(\lambda ) (T-t)} -\frac{\gamma}{\eta}(t^*+\frac{1}{\text{Re}(\lambda )})& \text{if } \gamma\leq\gamma^*, 0\leq t\leq T-t^*\\
ze^{\text{Re}(\lambda) (T-t)} - \frac{\gamma}{\eta}(T-t)& \text{if } \gamma\leq \gamma^*, T-t^*<t\leq T,
\end{cases}
\end{align}
where $ t^* = \frac{1}{\text{Re}(\lambda )}\log(\frac{\gamma}{\gamma^*})$. For all cases, $V_z = e^{\text{Re}(\lambda ) (T-t)}$. With this inserted back into
\eqref{utemp}, we conclude that
\begin{align}\label{optsoln}
u^*(t)=
\begin{cases}
0& \text{if } \gamma> \gamma^* \text{ or } 0\leq t\leq T-t^* \\
\sqrt{\frac{\gamma^*}{\gamma} }e^{\text{Re}(\lambda )(T-t)/2} - 1& \text{if } \gamma\leq \gamma^*, T-t^*<t\leq T.
\end{cases}
\end{align}
In particular, \eqref{optsoln} tells that we should use a small
mini-batch size (even size $1$) during the early time (for
$k \leq k^*=(T-t^*)/\eta$), since during this period the gradient flow
dominates the dynamics. After the transition time $k^*$ at which the noise
starts to dominate, one shall apply mini-batch with size exponentially
increasing in $k$ to reduce the variance. Figure \ref{fig:opt}
demonstrates that our proposed mini-batching strategy outperforms the
ASGD with a constant batch size (for example, applied in
\cite{dekel2012optimal,gimpel2010distributed}). Note that such
strategy of increasing the batch size in later stage of training has
been also suggested and used in recent works in training large neural
networks, e.g.,~\cite{Goyal2017, Keskar2017}.
\begin{figure}[h]
\centering \includegraphics[width=0.5\textwidth]{minibatch.pdf}
\caption[Controlled mini-batch sizes] {\small A comparison of
performance in terms of $l^2$ error. We apply mini-batching over
$n=100$ components $f_i(x) = \frac{1}{2}(x-c_i)^2, c_i =-1/2+i/(2n)$. Here we choose
the step size $\eta = 0.02$ and the initial data $x_0 = 1$. The
batch size for the uniform mini-batching case is $5$. For the
optimal mini-batching strategy, the transition happens at
$k = (T-t^*)/\eta \approx 699$, and the optimized batch size at
time $T$ is $42$. In practice, we can apply a more aggressive
mini-batching strategy by starting to increase the batch size
earlier in the flat region, and it will result in a larger batch
size at $T$. }
\vspace{-1em}
\label{fig:opt}
\end{figure}
\section{Conclusion}
In this paper, we have developed stochastic modified equations (SMEs) to model the asynchronous
stochastic gradient descent (ASGD) algorithms in the continuous-time limit. For quadratic loss
functions, the resulting SME can be put into a Langevin equation with a solution known to converge
to the unique invariant measure with a convergence rate dictated by the corresponding temperature.
We utilize such information to compare with the momentum SGD and prove the ``asynchrony begets
momentum'' phenomenon. For the general case, though the resulting SME does not have an explicitly
known invariant measure, it still provides rather precise trajectory predictions for the discrete
ASGD dynamics. Moreover, with SME available, we are able to find optimal hyper-parameters for
ASGD algorithms by performing a moment analysis and leveraging the optimal control theory.
\section*{Funding}
J.A. is partially supported by the Gene Golub Research Fellowship.
J.L. is supported by the National Science Foundation under award DMS-1454939, and
L.Y. is partially supported by the U.S. Department of Energy, Office of Science,
Office of Advanced Scientific Computing Research, Scientific Discovery through
Advanced Computing (SciDAC) program and the National Science Foundation under
award DMS-1818449.
\input{main.bbl}
\section*{Appendix A: miscellaneous computations in SMEs}
In this section, we provide the missing computations in Section 2.
\subsection{Evolution equation of $\Sigma$ for nonlinear gradients}
First, we have
\begin{align*} \Sigma_k &=
\mathbb{E}\bigg((-\frac{y_k}{\alpha} - \nabla f_{\gamma_k}(x_{k-\tau_k}))(-\frac{y_k}{\alpha} - \nabla
f_{\gamma_k}(x_{k-\tau_k}))^T\bigg).
\end{align*}
By expanding the terms in the expectation and treating them individually, we arrive at the following
\begin{equation}\label{eq:sigma}
\begin{aligned}
\Sigma_k &= \frac{1}{\alpha^2}y_k y_k^T+ \frac{y_k}{\alpha}\mathbb{E}\{ \nabla f_{\gamma_k} (x_{k-\tau_k})^T\} +\mathbb{E}\{ \nabla f_{\gamma_k} (x_{k-\tau_k})\}\frac{y_k^T}{\alpha} +\mathbb{E}\{\nabla f_{\gamma_k} (v_k)\nabla f_{\gamma_k} (x_{k-\tau_k})^T\} \\
& = \mathbb{E}\{\nabla f_{\gamma_k} (x_{k-\tau_k})\nabla f_{\gamma_k} (x_{k-\tau_k})^T\} - \frac{1}{\alpha^2}y_k y_k^T\\
& = \mu \sum_{m=0}^{\infty}\mathbb{E}\{\nabla f_{\gamma_{k-1}} (x_{k-1-m})\nabla f_{\gamma_{k-1}} (x_{k-1-m})^T\}(1-\mu)\mu^m\\
& \,\,\,\, +(1-\mu) \mathbb{E}\{\nabla f_{\gamma_k}(x_k)\nabla f_{\gamma_k}(x_k)^T \}- \frac{1}{\alpha^2}y_k y_k^T\\
& = \mu \big(\Sigma_{k-1} +\frac{1}{\alpha^2}y_{k-1} y_{k-1}^T\big) +(1-\mu) \mathbb{E}\{\nabla f_{\gamma_k}(x_k)\nabla f_{\gamma_k}(x_k)^T \}- \frac{1}{\alpha^2}y_k y_k^T\\
& = \mu \big(\Sigma_{k-1} +\frac{1}{\alpha^2}y_{k-1} y_{k-1}^T\big) +\frac{1-\mu}{n}\sum_{i=1}^n\nabla f_i(x_k)\nabla f_i(x_k)^T - \frac{1}{\alpha^2}y_k y_k^T.
\end{aligned}
\end{equation}
Notice that $y_k = \mu y_{k-1} - \alpha(1-\mu) \nabla f(x_k)$, and thus we have
\begin{align*}
y_{k-1}y_{k-1}^T &= \frac{1}{\mu^2} \big( y_k +\alpha(1-\mu) \nabla f(x_k)\big)\big( y_k
+\alpha(1-\mu) \nabla f(x_k)\big)^T\\ &= \frac{1}{\mu^2} \bigg( y_k y_k^T + \alpha(1-\mu) y_k \nabla
f(x_k)^T + \alpha(1-\mu) \nabla f(x_k) y_k^T + \alpha^2(1-\mu)^2 \nabla f(x_k)\nabla f(x_k)^T \bigg).
\end{align*}
Substituting it in \eqref{eq:sigma}, we obtain
\begin{align*}
\frac{\Sigma_k - \Sigma_{k-1}}{\alpha(1-\mu)} =& -\frac{1}{\alpha}\Sigma_{k-1} +\frac{1}{\alpha^3\mu} y_k y_k^T + \frac{1}{\alpha^2\mu} y_k \nabla f(x_k)^T + \frac{1}{\alpha^2\mu} \nabla f(x_k) y_k^T \\
& + \frac{1-\mu}{\alpha\mu} \nabla f(x_k)\nabla f(x_k)^T + \frac{1}{\alpha n}\sum_{i=1}^n\nabla f_i(x_k)\nabla f_i(x_k)^T.
\end{align*}
Using this and $\Delta t = \alpha(1-\mu), \alpha = \sqrt{\frac{\eta}{1-\mu}}$, we obtain the
evolution equation \eqref{evol}.
\subsection{Evolution equation of $\Sigma$ for linear gradients}
Similar to section 5.1, we have
\begin{align}\label{sig}
\nonumber \Sigma_k &= \mathbb{E}\bigl((\nabla f(m_k) - \nabla f_{\gamma_k}(x_{k-\tau_k}))(\nabla f(m_k) - \nabla
f_{\gamma_k}(x_{k-\tau_k}))^T\bigr)\\
&= \mu(\Sigma_{k-1}+\nabla f(m_{k-1})\nabla f(m_{k-1})^T) + (1-\mu) \mathbb{E}(\nabla f_{\gamma_k}(x_k)\nabla f_{\gamma_k}(x_k)^T)-\nabla f(m_k)\nabla f(m_k)^T.
\end{align}
We then subtract both sides by $\Sigma_{k-1}$ and divide by $\Delta t = \sqrt{\eta(1-\mu)}$. Moreover, we use the relation
\begin{align*}
x_k = \frac{m_k-\mu m_{k-1}}{1-\mu} = \frac{m_k - \mu(m_k-p_k\Delta t)}{1-\mu} = m_k + \mu\sqrt{\frac{\eta}{1-\mu}} p_k
\end{align*}
to replace $x_k$ in (\ref{sig}), and replace $m_{k-1}$ by $m_k-p_k\Delta t$. Then, since the gradient of $f$ is linear, rearrange the terms we get
\begin{align*}
\frac{\Sigma_k - \Sigma_{k-1}}{\sqrt{\eta(1-\mu)}} =& -\sqrt{\frac{1-\mu}{\eta}} \Sigma_{k-1} - \sqrt{\frac{1-\mu}{\eta}}\nabla f(m_k)\nabla f(m_k)^T + \mu \sqrt{\eta(1-\mu)} \nabla f(p_k)\nabla f(p_k)^T\\
-\mu(\nabla f(m_k)\nabla f(p_k)^T &+\nabla f(p_k)\nabla f(m_k)^T) + \frac{1}{n}\sqrt{\frac{1-\mu}{\eta}}\sum_{i=1}^{n}\nabla f_i(m_k + \mu\sqrt{\frac{\eta}{1-\mu}} p_k)\nabla f_i(m_k + \mu\sqrt{\frac{\eta}{1-\mu}} p_k)^T.
\end{align*}
\subsection{SME for SGD with momentum}
Recall the iteration for the SGD with a constant momentum parameter is
\begin{align*}
&v_{k+1} = \mu' v_k - \eta' \nabla f_{\gamma_k}(x_k) \nonumber\\
& x_{k+1} = x_k + v_{k+1},
\end{align*}
which can be viewed as a second-order difference equation. To ensure the final equation with all
terms of order $O(1)$, one needs $\eta' = (\Delta t)^2$. We can rewrite \eqref{momentum} as
\begin{align}
&\frac{v_{k+1}}{\sqrt{\eta'}} = \frac{v_k}{\sqrt{\eta'}} +\sqrt{\eta'} \bigg( -\frac{1-\mu'}{\eta'} v_k - \nabla f(x_k) \bigg) + \sqrt{\eta'}(\nabla f(x_k)- \nabla f_{\gamma_k}(x_k))\nonumber\\
& x_{k+1} = x_k + \frac{v_{k+1}}{\sqrt{\eta'}} \sqrt{\eta'}.
\end{align}
Let us introduce $p = v/\sqrt{\eta'}$. In order to have $\sqrt{\eta'} (\nabla
f(x_k)-\nabla_{\gamma_k} f(x_k) )\sim c \Delta B_t$, we choose $c \sim \sigma
(\eta')^{1/4}$. Therefore, we obtain the first order weak approximation, which can also be viewed as
the Euler-Maruyama discretization of the following SDE
\begin{align*}
& dP_t = -\nabla f(X_t) dt - \frac{1-\mu'}{\sqrt{\eta'}}P_t dt + \sigma(X_t)(\eta')^{\frac{1}{4}}
dB_t \nonumber \\ & dX_t = P_t dt.
\end{align*}
\section*{Appendix B: dynamics of SME-ASGD \eqref{nonlinearASGD}}
We consider the one dimensional case with $f(x) = \frac{1}{2}a x^2$.
The goal here is to give an analysis of the dynamics of first and
second moment of $X$ and $Y$ under \eqref{nonlinearASGD}. Taking
expectation, we obtain
\begin{align*}
d \begin{bmatrix} \mathbb{E}(Y_t)\\ \mathbb{E}(X_t) \end{bmatrix} =
\begin{bmatrix} -\sqrt{\frac{1-\mu}{\eta}} & -a \\ 1 & 0 \end{bmatrix}
\begin{bmatrix} \mathbb{E}(Y_t)\\ \mathbb{E}(X_t) \end{bmatrix}
dt =
A(\mu, \eta)\begin{bmatrix} \mathbb{E}(Y_t)\\ \mathbb{E}(X_t) \end{bmatrix} dt.
\end{align*}
One observes that the eigenvalues of $A(\mu, \eta)$ are
$\lambda_{1,2}(A) = \frac{1}{2} \bigg(-\sqrt{\frac{1-\mu}{\eta} }\pm
\sqrt{\frac{1-\mu}{\eta}-4a} \bigg)$,
the real parts of both are negative as long as $a>0$. From this, we
conclude that, when $a>0$, the expectation of $X_t$ decays
exponentially. The corresponding stationary solutions are given by
\[
\mathbb{E}(X_{\infty}) = \mathbb{E}(Y_{\infty}) = 0.
\]
For the second moment, we end up with the following equations by using the Ito's formula
\begin{align}\label{spec2}
\nonumber
&d\mathbb{E}(X_t^2) = 2\mathbb{E}(X_tY_t) dt +\Sigma(t) \frac{\eta^{3/2}}{(1-\mu)^{1/2}} dt\\
\nonumber
&d\mathbb{E}(Y_t^2) = -2a\mathbb{E}(X_tY_t) dt -2\sqrt{\frac{1-\mu}{\eta}}\mathbb{E}(Y_t^2) dt \\
&d\mathbb{E}(X_t Y_t)=-a\mathbb{E}(X_t^2)dt + \mathbb{E}(Y_t^2) dt - \sqrt{\frac{1-\mu}{\eta}} \mathbb{E}(X_tY_t) dt.
\end{align}
In order to study the behavior of the second moments, we can rewrite \eqref{spec2} as
\begin{align}\label{moment}
d\begin{bmatrix} \mathbb{E}(X_t^2)\\ \mathbb{E}(Y_t^2) \\ \mathbb{E}(X_t Y_t) \end{bmatrix} &= \begin{bmatrix} 0 & 0 & 2\\ 0 & -2\sqrt{\frac{1-\mu}{\eta}} & -2a \\ -a & 1 & -\sqrt{\frac{1-\mu}{\eta}} \end{bmatrix}\begin{bmatrix} \mathbb{E}(X_t^2)\\ \mathbb{E}(Y_t^2) \\ \mathbb{E}(X_t Y_t) \end{bmatrix} dt + \begin{bmatrix} \Sigma(t) \frac{\eta^{3/2}}{(1-\mu)^{1/2}} \\ 0 \\ 0\end{bmatrix} dt.
\end{align}
The corresponding stationary solutions are
\[
\mathbb{E}(X_{\infty}Y_{\infty}) = \frac{-\Sigma \eta^{3/2}}{2(1-\mu)^{1/2}}, \,\, \mathbb{E}(Y_{\infty}^2) = \frac{a\Sigma \eta^2}{2(1-\mu)} , \, \text{ and } \mathbb{E}(X_{\infty}^2) =\frac{\Sigma \eta^2}{2(1-\mu)}+\frac{\Sigma\eta}{2 a}.
\]
Let us introduce
\[
B(\mu, \eta) = \begin{bmatrix} 0 & 0 & 2\\ 0 & -2\sqrt{\frac{1-\mu}{\eta}} & -2a \\ -a & 1 &
-\sqrt{\frac{1-\mu}{\eta}} \end{bmatrix}.
\]
The eigenvalues of $B(\mu, \eta)$ are
\[
\lambda_1 = -\sqrt{\frac{1-\mu}{\eta}}, \lambda_{2,3} = \lambda_{\pm} = -\sqrt{\frac{1-\mu}{\eta}}
\pm \sqrt{\frac{1-\mu}{\eta}-4a}.
\]
We can see that the real parts of all roots are negative as long as
$a>0$. Moreover, the second moment of $X_t$ decays
exponentially, with the rate given by $\mathrm{Re}(\lambda_+)$ since
$\lambda_+$ is the eigenvalue with the largest (negative) real
part. We obtain the largest descent rate $\mathrm{Re}(\lambda_+)$ when
the second part $\sqrt{\frac{1-\mu}{\eta}-4a}$ in $\lambda_+$ is
purely imaginary, i.e., when $\mu $ takes
\begin{align}\label{optmu}
\mu_{\text{opt}} = \max\{1-4a\eta, 0\}.
\end{align}
We note that \eqref{optmu} also gives a suggestion to choose optimal
step size $\eta$: when $\mu$ is given, the maximal step size we can
choose is $\eta _{\text{opt}}= \frac{1-\mu}{4a}$. Any step size beyond
that will cause oscillations in the SME and the corresponding ASGD.
\end{document}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}
The plasticity of semiconductors has been a subject of numerous studies for the
last decades in both fundamental and applied research. Despite significant
progress in the understanding of the fundamental mechanisms involved, several
issues remain, in particular for nanostructured semiconductors. In these
materials, including for example nano-grained systems or nanolayers in
heteroepitaxy, dimensions are usually too small to allow the classical
mechanisms of dislocation multiplication, such as Franck-Read
sources.\cite{Mou99PMA} It is then likely that other mechanisms dominate, and
it has been already proposed that surfaces and interfaces, which become
prominent for small dimensions, play a major role. Several observations
support this assumption, especially for strained layers and misfit
dislocations at interfaces.\cite{Dun97JMSME,Jai97PMA,Wu01PMA} The question of
dislocation formation at surfaces concerns also bulk materials submitted to
large stresses. \cite{Wu98PML,Sca99PSS,Rab01MSE} The propagation of
dislocation from surfaces has been investigated in the frame of a continuum
model and elasticity theory. However, the characterization of the nucleation
of dislocations is out of the reach of this approach, and the predicted
activation energy is very large, in disagreement with experiments. It is also
difficult to investigate experimentally the very first stages of dislocation
formation. Hence, the mechanisms involved in the nucleation of dislocations
from surfaces or interfaces are far to be well understood. There has been some
attempts to perform atomistic calculations for addressing this issue. In
particular, the interaction between a dislocation and the free surface or the
interface,\cite{Ich95MSE,Asl98PML} or between ledges and a crack tip,
\cite{Jua96PML} and the instability of a stressed ledge \cite{Gao99PMA} have
been studied.
It has been proposed that surface defects like steps, or cleavage ledges, could
favor the nucleation of dislocations, by lowering the activation
energy.\cite{Xu97PMA} This assumption is supported by experimental facts, with
dislocation sources located on the cleavage surface and coinciding with
cleavage ledges. \cite{Gal01PMA,Arg01SM2} Atomistic simulations of dislocation
nucleation from surface defects in metals have also been recently reported.
\cite{Bro00PMA} It has been shown that the presence of the step modifies the
otherwise uniform strain field,\cite{Bro00PRB} which effectively makes easier
the dislocation formation. The situation appears to be different for
semiconductors, with no clear strain inhomogeneity at the step.
\cite{Poo92PRB,God02SM2} The role of the stress orientation on the dislocation
formation is also unclear. Additional atomistic simulations are needed to shed
light on these points and fully characterize the mechanisms behind dislocation
nucleation.
In this paper, we report large-scale atomistic calculations of the nucleation
of dislocations from surface defects in systems submitted to a stress with
variable orientation. We focused on linear surface defects, with simple steps
but also cleavage ledges. As for the material, silicon was selected as the
best candidate, for several reasons. First, it is a good model, since a lot of
semiconductors crystallize in the same cubic diamond structure, or the
zinc-blende structure, almost equivalent from the point of view of plasticity.
Second, silicon can be grown without dislocations, which allows a comparison
between experiments and simulations. Finally, several high quality atomistic
potentials are available. In the first part of the paper, the silicon
structure and the slip systems are briefly described. After the presentation of
the model and the calculations techniques used to perform the simulations, the
results obtained with several empirical potentials are described. In
particular, we mostly focus on stress orientations that increase the
probability of nucleating the relevant dislocations. Several points are then
discussed, such as the conditions of nucleation, the role of stress orientation
and temperature, and the slip system selected.
\section{Methodology}
\subsection{Structure and geometry}
In ambient conditions, the stable structure of silicon is diamond cubic.
Dislocations glide in the (111) dense planes, that are gathered together in two
sets, one widely spaced, called "shuffle" set and one narrowly spaced, called
"glide" set (Fig.~\ref{slip_sys}a). The Burgers vector of a perfect dislocation
is $1/2<1\bar{1}0>$. Dissociation only occurs in the glide set, with two
Shockley partial dislocations, $1/6<1\bar{2}1>$ and $1/6<2\bar{1}\bar{1}>$.
\cite{Hir82WIL} Note that, since the Burgers vector of a partial dislocation
doesn't join two points of the crystal lattice, the nucleation of a partial
dislocation is always accompanied by a stacking fault of the atomic plane. When
occurring on adjacent atomic planes, the stacking faults form micro-twins.
Considering the angle between the dislocation line and the Burgers vector,
perfect dislocation are called $60^\circ$ or screw, while partials are called
$90^\circ$ or $30^\circ$. These notations are used in the following.
A semi infinite system including surface steps is modeled by employing a slab
with a $2\times1$ rebuilt (100) free surface (Fig.~\ref{cell}).
\cite{Cha79PRL,Ram95PRB} Four atomic layers are frozen in the bottom of the
slab, opposite to the free surface. Steps, lying along the $[0\bar{1}1]$ dense
directions, which correspond to the intersection of \{111\} slip planes and the
(100) surface, are placed on the free surface. The steps are made infinite
through the use of periodic boundary conditions along the $[011]$ direction.
Two steps of opposite signs are introduced in order to allow the use of
periodic boundary conditions in the $[0\bar{1}1]$ direction normal to the step
line. The dimensions along $[100]$ and $[011]$ have been determined by
several calculations on systems with different sizes, for minimizing the
interactions between the surface and the frozen bottom, and between steps
(Fig.~\ref{cell}). A typical system encompasses 4 atomic layers along the step
line direction $[0\bar{1}1]$, 120 along the surface normal $[100]$ and 160
along $[011]$, the normal to the step in the surface plane, i.e. about 80000
atoms.
Note that the periodicity of 4 atomic layers along the step direction is a
severe limitation of the simulation in that it almost restricts the problem to
two dimensions, and prevents in particular the formation and expansion of such
defects as dislocation half-loops.
In this work, the most simple steps formed by the emergence of a perfect
dislocation at the surface are considered. They are called $D_{B}$ re-bonded
and $D_{B}$ non re-bonded,\cite{Cha87PRL} and have a height of two atomic
layers. The effect of higher steps is also checked by considering cleavage
ledges corresponding to 5 $D_{B}$ step forming a \{111\} facet.
\subsection{Application of a uniaxial stress}
To simulate the effect of an applied uniaxial stress $ \bm{\sigma}$, the system
is deformed with strains calculated using the silicon compliances $S_{ijkl}$.
The latter are obtained from the elastic constants $C_{ijkl}$, computed for all
empirical potentials. In this work, the uniaxial stress direction is contained
into the surface, but its orientation with respect to the step line, can vary.
As a result, the projection of this stress in the \{111\} slip planes, called
the resolved shear stress, will also vary. This quantity is important since it
is reasonable to assume that the slip system with the largest resolved shear
stress along the Burgers vector {\bf b} will be favored. The relationship
between the resolved shear stress $\bm{\tau}$ and the uniaxial stress
$\bm{\sigma}$ is $|\bm{\tau}| = \pm s|\bm{\sigma}|$, $s = cos \varphi\ cos \nu$
being the Schmid factor. $\varphi$ is the angle between $\bm{\sigma}$ and
the normal of the slip plane and $\nu$ the one between $\bm{\sigma}$ and {\bf
b}. In the Fig.~\ref{Schmid}, the calculated Schmid factors along several slip
directions into the \{111\} slip planes are represented as a function of the
angle $\alpha$ between the stress orientation and $[011]$, the normal to the
step lines. Four dislocations are possible: the $60^\circ$ and screw perfect
dislocations, and the $90^\circ$ and $30^\circ$ partials. The most efficient
stress orientations for each dislocation are gathered in the
Fig.~\ref{slip_sys}b. The maximum resolved shear stress along the Burgers
vector of the $60^\circ$ (screw) is obtained for $\alpha=22.5^\circ (45^\circ)$
for both tensile and compressive stress, respectively. The $90^\circ$ is
favored in case of a non disorientated tensile stress only. A compressive
stress would give a resolved shear stress in the anti-twinning sense. Finally,
the $30^\circ$ is favored by a disorientated stress of $36^\circ$, only in
compression to produce a twinning stress.
\subsection{Computational methods}
The large number of atoms required in the simulation prevents the use of ab
initio methods because it would be too expensive in CPU time. Instead, three
classical potentials for silicon are employed: the potential of Stillinger and
Weber (SW),\cite{Sti85PRB} based on a linear combination of two- and
three-body terms, the Tersoff potential \cite{Ter89PRB} including many-body
interactions thanks to a bond order term in the functional form, and the
environment-dependent inter-atomic potential \cite{Baz97PRB} (EDIP), more
recent and designed specifically for simulating defects.
To deform the system, stress increments of 1.5 GPa (equivalent to a strain
around 1 to 1.4\% according to the stress orientation) are successively
applied, the atomic positions being relaxed between each increment. Two
relaxation techniques are used. Either, a static relaxation with a conjugate
gradients algorithm is performed, until forces on atoms are smaller than
$10^{-3}$ eV/\AA, or temperature is introduced in simulations \cite{Rif99XMD}
with molecular dynamics, in order to investigate its effect on the nucleation.
After an initial static relaxation with conjugate gradients, temperature is
introduced by increment of 300K, with a simulation time ranging from 5 to 50
ps.
\section{Results with the Stillinger-Weber potential}
Three temperature domains have been considered; the first one at 0K, the second
one for low temperatures ($\alt$ 900K) and the last one for high temperatures
($\agt$ 900K). In each case, we focused on relevant stress orientations, in
particular those that increase the probability of nucleating the four possible
dislocations (60$^\circ$, screw, 90$^\circ$ and 30$^\circ$). Few other stress
orientations have also been checked. All results are summarized in the
Table~\ref{SW_results}. All the cases presented here concern systems with
D$_B$ non re-bonded surface steps. The main effect of higher steps, like
cleavage ledges, is a slight decrease of the elastic limits. In addition, the
plastic events remain qualitatively similar.
\subsection{Calculations at 0 K; effect of stress orientation}
At 0K, the plastic events appear under large strains in both compression and
traction, i.e. greater than 7\% (10.5 GPA) (Table~\ref{SW_results}). They are
initiated from the surface, in the close neighborhood of the step. Note that
the elastic limits for stressed systems with surface steps are always smaller
than for systems without surface steps. Hence the steps help for forming
plastic events, by lowering the required stress, and by confining the starting
surface area. Before the occurrence of plastic events, the system is
elastically deformed and the resulting shear strains are mainly located in
shuffle set planes.
Investigations have been performed with stress orientations favoring the
nucleation of perfect dislocations. For the 60$^\circ$ dislocation, the most
efficient orientation is $\alpha = 22.5^\circ$ both in traction and in
compression (Fig.~\ref{slip_sys}b). The results in traction show a relatively
large elastic limit of 22.5 GPa (18.7\%). Beyond this stress, plasticity
occurred and the relaxed system is displayed in the Fig.~\ref{sw_22.5_tract}.
The insert at the top of the figure clearly shows that the surface step is now
twice higher. Moreover the displacements in the shuffle set plane crossing the
step correspond to the slip of a $60^\circ$ dislocation. On the second insert
into the Fig.~\ref{sw_22.5_tract}, one can see the dislocation that has stopped
on the bottom of the simulation box and another $60^\circ$ dislocation with the
same screw component occurring in the symmetric \{111\} shuffle set plane from
the frozen bottom. Since the dislocation is stopped on the frozen zone (which
mimics the bulk) the system must find another slip system to continue its
relaxation. In compression, large plastic strains appear from the surface steps
for a strain of around -10\% (-12 GPa), following approximately the \{111\}
planes, but without any clearly identifiable dislocations.
The perfect dislocation in the screw orientation, should be favored by a stress
disorientated at around $45^\circ$ in both compression and traction
(Fig.~\ref{slip_sys}b). However, under a compressive stress, a 60$^\circ$
dislocation instead is nucleated in the shuffle set plane crossing the step.
The dislocation decreases the step height and glides in the plane of the
shuffle set up to the frozen bottom of the simulation box. Under a tensile
stress, defects identified as micro-twins are formed from the surface step. It
seems that these defects are due to a peculiar behavior of the SW potential
when the resolved shear stress in the \{111\} planes is along the anti-twinning
direction. A previous analysis has shown that these twins are formed by
glides in two shuffle set planes with a rotation of trimers in the glide set
plane.\cite{God03UNP} In Brief, in both cases traction and compression, no
screw dislocation has been nucleated.
Then, to nucleate partial dislocations, calculations with the most efficient
stress orientations are performed. When a non disorientated tensile stress
favoring the 90$^\circ$ partial is applied on the system
(Fig.~\ref{slip_sys}b), the relaxation of the atomic positions leads to the
crystal fracture. The crack is formed from the surface step for a stress of
31.5 GPa (22.9\% ).
The 30$^\circ$ partial dislocation is privileged by a compressive stress with
an angle of 36$^\circ$. Instead, a perfect 60$^\circ$ dislocation is nucleated
in the plane of the shuffle set crossing the surface step. Finally, although
the stress orientations are ideal to form partial dislocation according to the
Schmid factor, none is nucleated.
We have also checked several other configurations. In particular, stress
orientations favoring anti-twinning configurations, one for $\alpha = 0^\circ
$ in compression and another for $\alpha = 36^\circ$ in traction. It appeared
that for both cases, micro-twins are nucleated from the surface steps, what
may be attributed to a somewhat odd behavior of the SW potential. In some
cases, for a tensile stress and $\alpha=36^\circ$, we obtained peculiar glide
events after deformation. In particular, considering a ledge and not a single
step, the structure examination after relaxation revealed the presence of a
60$^\circ$ and a screw dislocations. We have also investigated a system under
tension and a disorientation angle $\alpha=10^\circ$, for which the resolved
shear stresses on the 90$^\circ$ and the 60$^\circ$ dislocation are the same
(Fig.~\ref{Schmid}). The result is equivalent to the situation of a
non-disorientated tensile stress, with the fracture of the crystal.
So it appears that at 0K, in spite of the many stress orientations tested,
only perfect dislocations, especially 60$^\circ$, located in the shuffle set
plane passing through the surface step are nucleated. No dislocations in the
glide set planes have been obtained.
\subsection{Other temperatures}
The same stress orientations have also been studied in the low temperature
domain. The main difference with the 0K study is the lowering of the elastic
limit as the temperature increases in both traction and compression. However,
the results remain qualitatively similar to what has been found at 0K. Only
perfect 60$^\circ$ dislocations are routinely nucleated. And no dislocation has
been formed in the glide set plane. Nevertheless, few differences have to be
noted. Under a compressive stress favoring the 60$^\circ$ dislocation, i.e. at
$\alpha=22.5^\circ$, the ill-defined plastic strains obtained at 0K are
replaced by a 60$^\circ$ dislocation nucleated in the shuffle set plane. In
another case, for a stress orientation leading to a resolved shear stress in
the anti-twinning direction, i.e. at $\alpha=36^\circ$ in traction, the
simultaneous formation of the 60$^\circ$ and screw dislocation is replaced by
large strained zones near the surface step. These deformations look like a
local phase change. The last difference is obtained with a tensile stress
disorientated at about 10$^\circ$ for which the resolved shear stresses on the
90$^\circ$ and 60$^\circ$ dislocations are the same. Our results show the
nucleation of a 60$^\circ$ dislocation in the shuffle set plane crossing the
step. The dislocation glides a distance around 15 \AA\ before leading to the
fracture of the crystal.
\
For the high temperature domain, the elastic limits continue to decrease as
the temperature is raised. No dislocation in the planes of the glide set is
observed. However the stress in the system is now relaxed in a new manner.
Previously, at low temperature, the glide events were relatively frequent in
the plane of the shuffle set. Now at high temperatures, the glide events in
the shuffle set planes become more and more rare as the temperature increases,
until they totally disappear. Instead, they are replaced by disorder in the
surface along the step line looking like amorphization zones and often close to
the steps.
\section{Results with the Tersoff potential and EDIP}
The results obtained with the SW potential have shown that only perfect
$60^\circ$ dislocations are nucleated in the shuffle set plane, and at low
temperature. A previous study on bulk silicon has shown that the Tersoff
potential and EDIP are less reliable than SW in the case of large
shear.\cite{God03JPCM} We have restricted the investigations using these
potentials to the stress orientations favoring the nucleation of a 60$^\circ$
dislocation, i.e. with a tensile or compressive stress at $\alpha =
22.5^\circ$.
\
The calculations done with the Tersoff potential at 0K give very large elastic
limits. They are around 46.7\% (51 GPa) and -38.5\% (-42 GPa) under tensile
and compressive stress, respectively. In traction, the crystal periodicity
along the step line direction is lost due to large strains of the bulk looking
like the beginning of a phase transition (Fig.~\ref{ters_edip}a), leading
sometimes to a crystal crack from the surface near the step. In compression, up
to -22\%, the strains remained homogeneous. Then slight undulations appeared on
the surface up to -37\%. Finally, a plastic strain occurred in the (011)
planes close to the surface step (Fig.~\ref{ters_edip}b). In all cases no glide
events are observed.
Calculations have been performed at different temperatures and several applied
stresses. The only effect is the decrease of the elastic limits and the
expansion of plastic strains. However, using high steps (cleavage ledges), a
large compressive strain (-11\%) and very high temperatures ranging from 1200K
to 1500K, we managed to nucleate $60^\circ$ dislocations in the shuffle set
plane passing through the step edge (Fig.~\ref{disloc_temp}b).
\
The calculations performed with EDIP at 0K also show much larger elastic
limits than the ones obtained with SW. They are around 34.5\% (52.5 GPa) in
traction and -8.9\% (-13.5 GPa) in compression. Under tensile stresses, a
crystal crack occurred, while under compressive stresses, the \{111\} shuffle
set plane passing through the step edge is largely sheared
(Fig.~\ref{ters_edip}c-d). This shear propagates from the surface to the slab
bottom without dislocation. When the applied strain increased neighboring
shuffle set planes are also sheared.
\section{Discussions}
\subsection{Dependency on the potentials}
Although the same stress orientations have been checked, at 0K and non zero
temperature, the results are often different from one potential to another.
In order to establish which potential represents best sheared silicon, we have
recently compared these three potentials with ab initio methods.
\cite{God03JPCM} A homogeneous shear is imposed on \{111\} planes in a
$<110>$ direction, the amplitude of shear goes up to 122\% where the diamond
cubic structure is recovered. At each shear value, the system is relaxed in
order for the simple FCC two sublattices forming the diamond structure to reach
their relative equilibrium position. In this way, one sees how the imposed
shear is distributed in the glide set and the shuffle set respectively. When
the full amplitude of the imposed shear has been applied, the crystal structure
returns to perfect diamond cubic with the SW potential and EDIP, as well as in
the ab initio calculation, through a bond breaking and new bond formation
across the shuffle plane. However, such a bond switching is not observed with
the Tersoff potential which in these conditions, does not appear suitable for
describing dislocation nucleation.
When comparing the energy curves of bulk silicon as a function of the
homogeneous shear strain. Only the curve of the SW potential is relatively
smooth with a shape and amplitude similar to the one calculated in DFT-LDA. The
Tersoff curve is discontinuous and the EDIP curve exhibits an angular point.
Thus, only SW can account for the atomic surrounding without energy
discontinuity when the crystal is largely strained. This feature is even more
marked when looking at derivative quantities, related to stresses. In addition,
critical values such as the theoretical shear strength are overestimated by a
factor of about two with the Tersoff potential and EDIP compared to the DFT
calculation, whereas Stillinger-Weber is much closer to ab initio.
Concerning EDIP, it has not been possible to use this potential at the large
strains considered here because of an accident occurring in the curve energy
versus shear strain which produces a shear instability of the crystal. The SW
potential is not exempt of drawbacks. When the crystal is sheared in the
anti-twinning direction, \cite{God03UNP} twinning is produced through shearing
in the shuffle set planes. We do not think that this prevents the use of the SW
potential for the other stress orientations.
Hopefully, there are indications that these inadequacies of the potentials may
become less important at high temperatures, where dislocations can be formed
at lower imposed strains. For example, under a compressive stress with
$\alpha=22.5^\circ$, the twin-like defect created by the SW artefact, is
replaced by a $60^\circ$ dislocation at a smaller strain. Another example is
given by the Tersoff potential which at high temperature and for large step
height can lead to the formation of a $60^\circ$ dislocation. Temperature may
rub out unphysical irregularities in the potentials.
\subsection{Role of the surface step}
Here, we focus on the results obtained with the SW potential. Plasticity occurs
for very large strains, somewhat smaller in compression than in tension.
Although the particular crystal structure and potential may be important, one
must consider that at the very large stresses considered here,the solid may
undergo some buckling instability of course in compression and not in tention,
instability which helps dislocation formation. Others studies are in progress
to clarify this point.
For bulk silicon, The theoretical strengths obtained with The Stillinger-Weber
potential are large, in agreement with ab initio calculations Roundy and M.L.
Cohen. \cite{Rou01PRB} Indeed, the limits of elasticity of the systems with a
free surface presenting steps are definitely smaller than without step. For
example at 0K, for a non disorientated stress, in tension (in compression) the
yield strain is about 22.9\% (-7.6\%) with surface steps and about 28.3\%
(-11\%) without surface step respectively. Generally the plastic strains, such
as fracture, the glide events or the amorphization zones..., occur from the
steps or from their immediate neighborhood. In fact the presence of the step
breaks the symmetry of the system leading to some stress localization near the
step. Thus the surface step is a privileged site for plasticity.
\subsection{Slip system: glide or shuffle}
Now, we discussed whether the dislocation nucleation occurs in the glide or in
the shuffle set planes, using the results obtained with the SW potential.
In principle, the perfect $60^\circ$ and screw dislocations can be formed in
either the glide or the shuffle plane, but in our results, the dislocations are
nucleated only in the planes of the shuffle set. The simulation with the stress
orientation most appropriate for nucleating $90^\circ$ and $30^\circ$ partials
in the glide set, lead to the fracture of the crystal and to the formation of
a $60^\circ$ dislocation in the shuffle set, respectively. This result is
consistent with the fact that for a slip in the shuffle set, only one covalent
bond must be broken compared to three in the glide set.\cite{Sho53PR}
In the high temperature domain, the probability of dislocation nucleation
tends to drop and plastic strains taking the form of amorphizations occur. As
temperature is raised, the strain at which some plasticity occur, decreases and
this process lasts until the thermal vibrations are sufficient to begin the
melting/amorphisation and the applied strains to small to initiate a
dislocation in the shuffle set. Here again, no dislocation is formed in the
glide set. Conversely to our simulations, at high temperature, the observed
dislocations are partial dislocations belonging to the glide set planes. It is
commonly accepted that they move more easily through the nucleation and
propagation of double kinks thanks to thermal vibrations. \cite{Hir82WIL,
Bul95PMA, Bul01PMA, Mit03PM} However, the size of the simulation cell along the
dislocation line used here, 4 a/2$<110>$, is too small to allow the formation
of a kink pair. Consequently, only two plastic events are possible in the
simulation: the nucleation of an infinite straight $60^\circ$ dislocation in
the shuffle set planes or amorphisation/melting, depending on the temperature.
Experimentally, the observations done in both low and high temperature domains
reveal a slip mode transition depending on the temperature. At low temperature
dislocations seem to glide in the shuffle set planes and at high temperature in
the glide set planes.\cite{Due96PML, Rab01MSE, Rab01SM2} Whatever the
temperature, our simulations have shown that the nucleation of straight
dislocation in the glide set plane is not allowed due to geometric reasons. The
only type of dislocation, the $60^\circ$, is nucleated in the shuffle set
planes. Moreover we have observed that high temperatures prevent the
dislocation formation in this set. Thus our results are not in disagreement
with the experimental facts, but complementaries calculations in 3 dimensions,
in order to allow the kink propagation, are necessary to confirm the slip mode
transition.
\subsection{Character of the dislocation nucleated.}
In order to understand the kind of dislocation formed, we tried to establish
the main criteria that govern this choice. Usually, when a crystal is stressed,
the slip system with the largest resolved shear stress along the Burgers vector
{\bf b} is favored. In our case, the resolved shear stress on each dislocation,
proportional to the Schmid factor, is directly related to the stress
orientation $\alpha$. In the range of temperature where the glide events are
frequently observed for the SW potential, in most cases, the plastic events are
consistent with the results predicted by the Schmid factors. For example, on
the Figure~\ref{slip_sys}b, the $60^\circ$ dislocation is favored for a stress
orientation $\alpha=22.5^\circ$ in traction and in compression, what is
obtained in our simulations. One sees from the same figure, that a compressive
stress disorientated by $36^\circ$ favors the $30^\circ$ partial, that is a
strain in the twinning sense. Since dislocations of the glide set are not
activated, as explained above, the system finds another slip system to relax
the applied stress. In these conditions of twinning, two dislocations are
possible the $60^\circ$ and the screw. In our simulations, the dislocation
nucleated is the $60^\circ$, i.e. the one with the largest Schmid factor
(Fig.~\ref{Schmid}).
However several cases cannot be explained on the basis of the Schmid factor
only, the character of the dislocation must also be taken into account. For
example, under a non disorientated tensile stress favoring the twinning stress
along the {\bf b}$_{90^\circ}$ partial (Fig.~\ref{slip_sys}b), a crystal crack
is produced without glide events. Following the Schmid factor analysis, two
$60^\circ$ on both sides of the partial dislocation could then be nucleated.
But the resolved shear stress along both symmetric $<110>$ directions
(Fig.~\ref{Schmid}) are equal, what may prevent the choice of one slip system.
To check this, a calculation with a stress orientation at $10^\circ$ that
breaks the symmetry of the problem, has formed a $60^\circ$ dislocation in
agreement with the Schmid factor.
Compare this case to that where the stress orientation is disorientated by
$45^\circ$ (Fig.~\ref{Schmid} curves 3 and 5). Although the resolved shear
stress is the same on the screw and the $60^\circ$, the latter is nucleated, in
compression. It is worth remarking that the two types of dislocations have
different mobility properties, cf. for instance the Peierls stresses. The
calculations performed with the SW potential have shown that the Peierls stress
on the $60^\circ$ dislocation is smaller than on the screw. \cite{Ren95PRB} To
relax the applied stress, the nucleation of a perfect $60^\circ$ dislocation is
then favored.
The other discrepancies between the Schmid factor analysis and the simulation
results are mainly due to the unphysical defect created by the SW potential,
the micro-twins, which pollute largely the results in both traction and
compression, when the applied stress acts in the anti-twinning sense. For
example in compression at 0K, the micro-twin formation disappears as the stress
orientation $\alpha$ increases. Hence the resolved shear stress along the
anti-twinning direction must be as small as possible to avoid this kind of
defects.
The analysis of the plastic strains as a function of the stress orientation
shows that the character of the dislocations nucleated from surface steps can
be mainly predicted by examining the Schmid factor and the Peierls stress.
Other factor may play a role, though. For example, the crystal symmetry may
prevent the choice of one particular slip system leading to fracture.
\section{Conclusion}
We have investigated the nucleation of dislocations from linear surface defects
such as steps, when the system is submitted to a uniaxial stress. Although the
elastic limits remain relatively close to the theoretical strength, it appears
that the surface steps weaken the atomic structure and help the formation of
glide events like dislocations. The glide evens are nucleated and propagated in
the planes of the shuffle set. No straight dislocation is formed in the glide
set plane. The geometry of the simulation cell used here which prevents the
formation of kink pairs, does not allow for the expected formation of partial
dislocations in the shuffle set at high temperature.
In addition, we have remarked that the high temperature decreases the
probability of nucleating perfect dislocation in the shuffle set plane.
Melting/amorphisation of silicon occurs before reaching the required shear
stress to initiate the dislocation. These results seem consistent with the
assumption that at low temperature the dislocations glide in the planes of the
shuffle set, based on the observation of non dissociated dislocations in
silicon samples deformed at low temperature in conditions preventing failure.
\cite{Rab01MSE,Rab01SM2} Supplementary studies are planned to check the
nucleation of dislocation loops in the glide set planes with high temperature.
The role of the stress orientation on the nucleated defects has been studied
from the calculations performed with the SW potential. Although the results are
slightly biased by the somewhat unphysical defect produced by the SW potential
when the stress acts in the antitwinning direction, it emerges that the type of
dislocation nucleated is chosen by the resolved shear stress and the Peierls
stress.
Concerning the empirical potentials, it has not been possible to nucleate any
dislocations in the simulations performed with the Tersoff potential and EDIP
at 0K. The Tersoff potential has too high energy barriers preventing the bond
breaking required to nucleate a dislocation at low temperatures. While EDIP
presents a shear instability in the shuffle set planes. With the Tersoff
potential, the overcoming of the energy barriers leading to the dislocations
nucleation has become possible at high temperature. By extrapolation, EDIP is
probably able to nucleate dislocations thanks to the thermal vibrations. To
summarize, although the different results are potential-dependent, only the
simulations performed with the SW potential can be taken into account at 0 K as
demonstrated in our previous study on bulk system. Actually, we are trying a
similar calculation with ab initio methods on a relatively large system. A
calculation with a small system of about 200 atoms has already produced the
nucleation of a $60^\circ$ perfect dislocation in the shuffle set.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,912
|
\section{Introduction}
Romans massive supergravity \cite{roma} has attracted a lot of interest
following the observation that its mass parameter (cosmological constant)
may be thought of as sourced by the D8 brane of type IIA \cite{polc}. If string theory
is to be understood as embedded in M-theory, it would be desirable
to have an eleven-dimensional understanding of Romans supergravity.
The latter, however, has no covariant eleven-dimensional lift\footnote{See \cite{blo} for a
noncovariant embedding of Romans supergravity in
eleven-dimensions, and \cite{hs} for a recent implementation
of the same idea in eleven-dimensional superspace. In \cite{hull}
it was argued that although Romans massive IIA supergravity
cannot be embedded in ordinary eleven-dimensional supergravity,
massive IIA {\it string theory} can be embedded in M-theory.},
owing to certain no-go theorems
forbidding any straightforward introduction of mass in
eleven-dimensions \cite{dese}.
There does exist however a topological modification
of eleven-dimensional supergravity
(subsequently dubbed `MM-theory'),
as pointed out by Howe in \cite{h},
which allows the introduction of
a mass parameter upon compactification to ten dimensions.
In this way one obtains
the Howe-Lambert-West supergravity of reference \cite{hlw}.
The latter contains a lot of interesting physics \cite{cpr, cla, clb}; it is
nevertheless much less studied despite
that, contrary to Romans supergravity, it has a
well-understood covariant eleven-dimensional origin and
it has been shown to admit a de Sitter vacuum\footnote{
Presumably this apparent neglect should
be attributed to certain unconventional features of HLW supergravity. For example,
its equations-of-motion cannot be integrated to
a local Lagrangian. More importantly, it is not clear at present
if the theory can be made quantum-mechanically self-consistent.}.
HLW supergravity can also be obtained by a generalized Scherk-Schwarz
reduction of the equations-of-motion of
ordinary eleven-dimensional supergravity \cite{llp}.
It would be desirable to have an understanding of the relation between
Romans and HLW supergravity from a purely ten-dimensional perspective.
We would also like to know how unique these supergravities are and
whether there exist or not other massive deformations of type IIA.
In this paper we shall address these questions
by working in ten-dimensional IIA superspace.
The starting point of our search is the supertorsion Bianchi identities (BI)
--which every supersymmetric system should satisfy.
In this
purely-geometric approach
no form superfields are introduced by hand, unlike in the usual
superspace formulation of IIA supergravity \cite{gate}; the
field-strengths of the various supergravity forms `sit' inside
the components of the torsion.
We do not make any further
assumptions other than that any deviation from massless IIA should
appear at canonical mass dimension one or higher. I.e. we demand that
up to dimension one-half, the supertorsion
components are (equivalent to) those of massless IIA supergravity.
As there are formulations of the latter (see for example \cite{cede})
in which the scalar superfield does {\it not} appear explicitly in the
torsion components of dimension zero or one-half,
we shall assume that the field content
at dimension zero consists of {\it at most} one scalar superfield,
while the field content at dimension one-half consists of
one chiral and one antichiral Majorana spinor superfields.
In fact, an interesting feature
of the formulation presented here is that
the scalar superfield at dimension zero
(whose lowest component can be identified with the dilaton), present in
both Romans and massless IIA supergravities,
is {\it not} introduced
from the outset but it
arises as a `potential' for the spinor superfields at dimension one-half:
its existence follows from a certain integrability condition
implied by the BI's.
In the case of HLW supergravity
the aforementioned integrability condition fails and
such a `potential' does not exist.
More specifically: as we show at the end of section \ref{dimension2}, both
in the case of massless IIA and in Romans supergravity, one can construct
a {\it closed} one-form superfield whose lowest component
is identified with the spinor superfields at dimension one-half.
In a topologically nontrivial spacetime this one-form may not be exact,
a fact which leads to a topological modification of massless IIA.
This is the ten-dimensional version of the possibility to modify
ordinary eleven-dimensional supergravity to MM-theory.
We generally expect that
taking the supertorsion Bianchi identities as the starting point,
should allow for more freedom than starting with superforms. Evidence for this
was recently provided in \cite{cgnt}, where the supertorsion BI's
of eleven-dimensional supergravity where solved at first
order in a deformation
parameter related to the Planck length\footnote{The first supersymmetric
deformation
occurs at order $l^3$ and it is of topological nature \cite{t}.
The next deformation,
related to the $R^4$ superinvariant, is expected to occur at order $l^6$.}.
Although it has been shown that the
four-form formulation of 11d supergravity implies the supertorsion formulation
\cite{ht}\footnote{This was shown in \cite{ht} to first order in
the deformation parameter. In the undeformed theory, this
fact had been previously pointed out in \cite{cgnn, nr}.},
it was realized in \cite{cgnt} that the converse may not be true.
The main result of this paper can be stated as follows.
Depending on the values of two scalar superfields
($L$, $L'$ of equation (\ref{llprime}) below) arising at
canonical mass dimension one, there exist exactly
two massive deformations of IIA supergravity:
Romans supergravity and HLW supergravity.
In the following section we introduce IIA superspace and establish our notation and conventions.
We also examine the possible
field redefinitions, in preparation for the analysis of the BI's in section \ref{bi}.
The reader who is not interested in the derivation of the final result, may skip directly
to section \ref{summary} where the outcome of the analysis of the BI's is summarized.
We conclude in section \ref{conclusions} with some possible future directions.
The appendix contains our conventions on certain gamma-traceless projections used in section
\ref{bi}.
\section{General setup}
\subsection{Type IIA superspace}
Let us begin by introducing the usual superspace machinery of vielbein ($E^A$),
connection ($\Omega_{A}{}^B$), torsion ($T^A$) and curvature ($R_A{}^{B}$), via
\beal
T^A&=DE^A\nn\\
R_A{}^B&=d\Omega_A{}^B+\Omega_A{}^C\wedge \Omega_C{}^B~.
\end{align}
A flat superindex is denoted by a capital Latin letter from the beginning of the alphabet
and stands for both bosonic ($a$) and fermionic ($\underline{\alpha}$) indices. Underlined Greek
indices from the beginning of the alphabet stand for flat fermionic indices of both chiralities.
For example:
\beal
S^{\underline{\alpha}}:=(S^{\al}, S_{\al})~,
\end{align}
where $S^{\al}$ is chiral and $S_{\al}$ is antichiral. Note that we never raise/lower
chiral fermionic indices, so that the position of the index denotes a definite chirality.
In IIA superspace the spinor part of the vielbein
contains both chiralities: $E^{\underline{\alpha}}:=(E^{\al}, E_{\al})$.
The torsion and curvature satisfy the Bianchi identities
\beal
DT^A&=E^BR_B{}^A\nn\\
DR_A{}^B&=0~.
\label{r}
\end{align}
If the connection is Lorentzian, as we assume to be the case in the present
paper, the second BI follows from the first \cite{drag}. Hence we need only
analyze the first of equations (\ref{r}), i.e. the supertorsion BI.
\subsection{Field redefinitions}
\label{fieldredefinitions}
Before coming to the analysis of the BI's, let us examine
what the possible field redefinitions are.
We have at our disposal vielbein and connection redefinitions which can be used to
gauge-fix some of the torsion components. Specifically, let $h_{A}{}^B:=E_A{}^M\delta E_M{}^B$
and $\Delta_{AB}{}^C:=E_A{}^M\delta\Omega_{MB}{}^C$.
Under $E_M{}^A\rightarrow E_M{}^A+\delta E_M{}^A$,
$\Omega_{MB}{}^C\rightarrow \Omega_{MB}{}^C+\delta\Omega_{MB}{}^C$ the torsion transforms as
$T_{AB}{}^C\rightarrow T_{AB}{}^C+\delta T_{AB}{}^C$ where
\beal
\delta T_{AB}{}^C=
2\Delta_{[AB\}}{}^C+2D_{[A}h_{B\}}{}^C-2h_{[A|}{}^DT_{D|B\}}{}^C +T_{AB}{}^D h_D{}^C
~.
\label{redefs}
\end{align}
Let us analyze the possible field redefinitions in the order of
increasing canonical mass dimension.
$\bullet$ Dimension $0$
We search for supergravities which, in the massless limit, reduce to massless type IIA.
Therefore, as explained in the introduction,
we shall assume that the field content at dimension zero consists of {\it at most}
a scalar superfield ($\phi$). Hence, the most general
form of the dimension-zero torsion components is
\beal
T_{\al\bn}{}^c&=c_1(\ga^c)_{\al\bn} \nn\\
T^{\al\bn c}&=c_2(\ga^c)^{\al\bn}\nn\\
T_{\al}{}^{\bn c}&=0~.
\end{align}
If a dimension-zero scalar superfield $\phi$ exists,
$c_1$, $c_2$ can be arbitrary functions of $\phi$. From (\ref{redefs}) we see that we
can use two independent
linear combinations of $h_{\al}{}^{\bn}$, $h^{\al}{}_{\bn}$, $h_{a}{}^b$ to
set $T_{\al\bn}{}^c=-i(\ga^c)_{\al\bn} $,
$T^{\al\bn c}=-i(\ga^c)^{\al\bn}$. Note that in the case where there
exists no scalar superfield $\phi$ at dimension zero, the remaining
linear $h$-combination is a constant.
Therefore it {\it cannot} be used as a redefinition at dimension one-half or higher,
as $Dh$ vanishes.
$\bullet$ Dimension $\frac{1}{2}$
At canonical mass dimension one-half the field content consists of one right-handed and one
left-handed Majorana spinor superfield $\mu^{\al}$, $\lambda_{\al}$, respectively
\footnote{Here as well one can imagine the possibility
that the dimension one-half spinors may be `eaten' by a
higher-dimensional superfield. However it is not difficult to see that
in this case the Bianchi identities set
all the torsion components to zero.}. In the massless
IIA limit we have $\lambda_{\al}=D_{\al}\phi$, $\mu^{\al}=D^{\al}\phi$. As we have already
remarked, we shall not assume this
to be the case {\it a priori}.
The most general form of the
vielbein and connection redefinitions is
\beal
\Delta_{\al b}{}^c&=d_1 (\ga_b{}^c\lambda)_{\al}\nn\\
\Delta^{\al}{}_b{}^c&=d_2 (\ga_b{}^c\mu)^{\al}
\label{redsa}
\end{align}
and
\beal
h_a{}^{\al}&=f_1(\ga_a\lambda)^{\al}\nn\\
h_{a\al}&=f_2(\ga_a\mu)_{\al}
\label{redsb}
\end{align}
respectively. Note that $\Delta_{\underline{\alpha}\bn}{}^{\ga}$ is not
independent, but is related to $\Delta_{\underline{\alpha} b}{}^c$ via
the Lorentz condition. The most general form of the torsion component $T_{\al b}{}^c$ reads
\beal
T_{\al b}{}^c&=e_1\delta_b{}^c\lambda_{\al}+e_2 (\ga_b{}^c\lambda)_{\al}
\end{align}
and similarly for $T^{\al}{}_{b}{}^c$. Hence we can use the field redefinitions
(\ref{redsa}, \ref{redsb}) to set $T_{\al b}{}^c=T^{\al}{}_{b}{}^c=0$.
In the case where there exists no scalar superfield at zero dimension,
this is all we can do in the way of gauge-fixing.
However, when $\phi$ exists $T_{\al\bn}{}^\ga$
can also be partially gauge-fixed as follows. The most general
form of $T_{\al\bn}{}^\ga$ is
\beal
T_{\al\bn}{}^\ga&=g_1\lambda_{(\al}\de_{\bn)}{}^{\ga}+g_2(\ga^e)_{\al\bn}(\ga_e\lambda)^{\ga}~.
\label{kook}
\end{align}
As we can see from
(\ref{redefs}) and the fact that $D_{\al}\phi=\lambda_{\al}$, the remaining
independent linear
combination of $h_{\al}{}^{\bn}$, $h^{\al}{}_{\bn}$, $h_{a}{}^b$ can be used
to gauge-fix the coefficient $g_1(\phi)$ in (\ref{kook}).
$\bullet$ Dimension $1$
As usual, $\Delta_{ab}{}^c$ can be used to set $T_{ab}{}^c=0$.
To summarize the results of this subsection: the
field redefinitions can be used to set
\beal
&T_{\al\bn}{}^c=-i(\ga^c)_{\al\bn} \nn\\
&T^{\al\bn c}=-i(\ga^c)^{\al\bn}\nn\\
&T_{\underline{\alpha} b}{}^c, ~T_{ab}{}^c=0~.
\end{align}
If, in addition, there is a scalar superfield $\phi$ such that
$D_{\al}\phi=\lambda_{\al}$, then
$T_{\al\bn}{}^{\ga}$ can also be partially gauge-fixed, as explained below (\ref{kook}).
\section{Analysis of the torsion Bianchi identities}
\label{bi}
We are now ready to come to the solution of
the torsion Bianchi identities. We shall proceed
systematically, in increasing order
of canonical mass dimension.
The readers who are not interested in the
details of this analysis, may skip directly to section \ref{summary}
where the final result is summarized.
\subsection{Dimension-$\frac{1}{2}$ BI}
Taking the discussion of section \ref{fieldredefinitions} into account
and imposing the Lorentz condition,
the dimension one-half torsion Bianchi identity reads
\beal
T_{(\underline{\alpha}\underline{\phantom{\alpha}}\!\!\!\beta|}{}^{\underline{\epsilon}}T_{\underline{\epsilon}|\underline{\phantom{\alpha}}\!\!\!\gamma)}{}^e=0~.
\label{b121}
\end{align}
We distinguish the following cases.
$\bullet$ Case 1: $(\underline{\alpha},\underline{\phantom{\alpha}}\!\!\!\beta,\underline{\phantom{\alpha}}\!\!\!\gamma)=(\al, \bn, \ga)$
The Bianchi identity (\ref{b121}) reads
\beal
T_{(\al\bn|}{}^{\e}(\ga^e)_{\e|\ga)}=0~.
\end{align}
Substituting (\ref{kook}) in the equation above
and using the identity
\beal
\ga_{(\al\bn}^e\delta_{\ga)}^{\delta}=(\ga_f)_{(\al\bn}(\ga^{ef})_{\ga)}{}^{\delta}~,
\end{align}
we obtain
\beal
T_{\al\bn}{}^\ga&=g_1\Big\{
\lambda_{(\al}\de_{\bn)}{}^{\ga}-\frac{1}{2}(\ga^e)_{\al\bn}(\ga_e\lambda)^{\ga}
\Big\}~.
\end{align}
$\bullet$ Case 2: $(\underline{\alpha},\underline{\phantom{\alpha}}\!\!\!\beta,\underline{\phantom{\alpha}}\!\!\!\gamma)=(\! \begin{array}{c} \alpha\\{} \end{array},\!\!\begin{array}{c} \beta\\{} \end{array},\!\!\begin{array}{c} \gamma\\{} \end{array})$
Similarly to the previous case we obtain
\beal
T^{\al\bn}{}_\ga&=g_2\Big\{
\mu^{(\al}
\de^{\bn)}{}_{\ga}-\frac{1}{2}(\ga_e)^{\al\bn}(\ga^e\mu)_{\ga}\Big\}~.
\end{align}
$\bullet$ Case 3: $(\underline{\alpha},\underline{\phantom{\alpha}}\!\!\!\beta,\underline{\phantom{\alpha}}\!\!\!\gamma)=(\al, \bn, \!\!\begin{array}{c} \gamma\\{} \end{array})$
The Bianchi identity (\ref{b121}) reads
\beal
2T_{(\al|}{}^{\ga\e}(\ga^e)_{\e|\bn)}
+T_{\al\bn\e}(\ga^e)^{\e\ga}=0~.
\end{align}
We expand the torsion components above as follows
\beal
T_{\al\bn\ga}&=g_3(\ga^e)_{\al\bn}(\ga_e\mu)_\ga\nn\\
T_\al{}^{\bn\ga}&=
g_4 \de_\al{}^\ga\mu^\bn +g_5(\ga^{e_1e_2})_\al{}^\ga(\ga_{e_1e_2}\mu)^\bn
+g_6(\ga^{e_1\dots e_4})_\al{}^\ga(\ga_{e_1\dots e_4}\mu)^\bn\nn\\
~.
\end{align}
In terms of irreducible representations the Bianchi identity decomposes as\footnote{
We are using the Dynkin notation for the complex cover $D_5$ of $SO(1,9)$. I.e.
(00000), (10000), (01000), (00100), (00011) denote a scalar, vector, two-form, three-form,
four-form respectively. Self-dual, anti-self-dual five-forms are denoted by
(00002), (00020) respectively and chiral, anti-chiral spinors are denoted by
(00010), (00001). Similarly, an irreducible (gamma-traceless) chiral vector-spinor
is denoted by (10010), a chiral two-form spinor by (01010), etc.}
$$
(00001)^{2\otimes_s}\otimes(00010)\otimes(10000)\sim3(00010)\oplus\dots
$$
where the ellipses stand for irreducible representations which drop out
of the BI. Hence the BI imposes at most three independent conditions on the
coefficients $g_3,\dots g_6$. These conditions can be obtained by contracting with the
three independent structures
$(\ga^e)^{\al\bn}\delta_{\ga}^{\e}$, $(\ga_f)^{\al\bn}(\ga^{ef})_{\ga}{}^{\e}$ and
$(\ga^{ef_1\dots f_4})^{\al\bn}(\ga_{f_1\dots f_4})_{\ga}{}^{\e}$.
We thus obtain $g_4=-\frac{1}{2}g_3$, $g_5=\frac{1}{4}g_3$ and
$g_6=0$, so that
\beal
T_{\al\bn\ga}&=g_3(\ga^e)_{\al\bn}(\ga_e\mu)_\ga\nn\\
T_\al{}^{\bn\ga}&=-\frac{1}{2}g_3\Big\{
\de_\al{}^\ga\mu^\bn -\frac{1}{2}(\ga^{ef})_\al{}^\ga(\ga_{ef}\mu)^\bn
\Big\}
~.
\end{align}
$\bullet$ Case 4: $(\underline{\alpha},\underline{\phantom{\alpha}}\!\!\!\beta,\underline{\phantom{\alpha}}\!\!\!\gamma)=(\al, \!\!\begin{array}{c} \beta\\{} \end{array}, \!\!\begin{array}{c} \gamma\\{} \end{array})$
Similarly to the previous case, we obtain
\beal
T^{\al\bn\ga}&=g_4(\ga_e)^{\al\bn}(\ga^e\lambda)^\ga\nn\\
T^\al{}_{\bn\ga}&=-\frac{1}{2}g_4 \Big\{
\de^\al{}_\ga\lambda_\bn -\frac{1}{2}(\ga^{ef})^\al{}_\ga(\ga_{ef}\lambda)_\bn
\Big\}
~.
\end{align}
\subsection{Dimension-1 BI}
\label{dimension1}
For the gamma-matrix manipulations of this and the remaining subsections, we have found
\cite{gran} extremely useful. We have also made use of \cite{lie} in evaluating
tensor products of representations.
Let us expand the spinor derivatives of the dimension-$\frac{1}{2}$ spinor
superfields as follows
\beal
D_{\al}\lambda_\bn&=K_e{}(\ga^e)_{\al\bn}+K_{efg}{}(\ga^{efg})_{\al\bn}\nn\\
D_\al\mu^\bn&=L{}\de_\al{}^\bn+L_{ef}{}(\ga^{ef})_\al{}^\bn
+L_{efgh}{}(\ga^{efgh})_\al{}^\bn
\label{uiy}
\end{align}
and
\beal
D^{\al}\lambda_\bn&=
{L'}{}\de^\al{}_\bn+{L'}^{ef}{}(\ga_{ef})^\al{}_\bn
+{L'}^{efgh}{}(\ga_{efgh})^\al{}_\bn\nn\\
D^\al\mu^\bn&={K}^{\prime e}{}(\ga_e)^{\al\bn}
+{K}^{\prime efg}{}(\ga_{efg})^{\al\bn}~.
\end{align}
We also expand the dimension-one torsion as follows:
\beal
T_{ab}{}^c&=0\nn\\
T_{a\al}{}^\bn&= \de_\al{}^\bn V^1_a
+(\ga_a{}^e)_\al{}^\bn V^2_e
+(\ga_a{}^{fgh})_\al{}^\bn H^1_{fgh}+(\ga^{fg})_\al{}^\bn H^2_{afg}
\nn\\
T_a{}^{\al}{}_{\bn}&= \de^\al{}_\bn V^{\prime1}_a
+(\ga_a{}^e)^\al{}_\bn V^{\prime2}_e
+(\ga_a{}^{fgh})^\al{}_\bn H^{\prime 1}_{fgh}
+(\ga^{fg})^\al{}_\bn H^{\prime 2}_{afg}
\nn\\
T_{a\al\bn}&=
S (\ga_a)_{\al\bn} +(\ga_a{}^{ef})_{\al\bn}F^1_{ef}
+(\ga^{e})_{\al\bn}F^2_{ae}
+(\ga_a{}^{efgh})_{\al\bn}G^1_{efgh}
+(\ga^{efg})_{\al\bn}G^2_{aefg}
\nn\\
T_a{}^{\al\bn}&=
S' (\ga_a)^{\al\bn} +(\ga_a{}^{ef})^{\al\bn}{F}^{\prime1}_{ef}
+(\ga^{e})^{\al\bn}{F}^{\prime2}_{ae}
+(\ga_a{}^{efgh})^{\al\bn}{G}^{\prime1}_{efgh}
+(\ga^{efg})^{\al\bn}{G}^{\prime2}_{aefg}
~.
\label{t1}
\end{align}
The superfields appearing on the right-hand-side of (\ref{t1})
are all forms. We can see that there can be no hooks in the
above expansions, for the following reason. Assuming there is
a hook superfield ($U$) at dimension one, we can expand
$U=m U_{(0)}+ U_{(1)}$, where $m$ is a mass parameter, so that
$U_{(0)}$ is of canonical mass dimension zero,
$ U_{(1)}$ is of canonical mass dimension one and
does not depend on $m$. Taking the massless $m\rightarrow 0$
limit we see that
$ U_{(1)}$ has to vanish, as no hook superfields can appear
in the torsion components of massless IIA. Also $U_{(0)}$ has to vanish, since
at dimension zero there can exist at most a scalar superfield.
Using the same argument we can see that there can be no
five-forms in the expansions (\ref{uiy}-\ref{t1}).
Taking the Lorentz condition
into account, the Bianchi identities at dimension one read
\beal
R_{\underline{\alpha}\underline{\phantom{\alpha}}\!\!\!\beta} {}_{cd}
&= 2T_{c(\underline{\alpha}|}{}^{\underline{\epsilon}} T_{\underline{\epsilon}|\underline{\phantom{\alpha}}\!\!\!\beta)d} ~.
\label{bi1.1}
\end{align}
and
\beal
R_{(\underline{\alpha}\underline{\phantom{\alpha}}\!\!\!\beta}{}_{\underline{\phantom{\alpha}}\!\!\!\gamma)}{}^{\underline{\delta}}&= D_{(\underline{\alpha}}T_{\underline{\phantom{\alpha}}\!\!\!\beta\underline{\phantom{\alpha}}\!\!\!\gamma)}{}^{\underline{\delta}}
+T_{(\underline{\alpha}\underline{\phantom{\alpha}}\!\!\!\beta|}{}^eT_{e|\underline{\phantom{\alpha}}\!\!\!\gamma)}{}^{\underline{\delta}}+T_{(\underline{\alpha}\underline{\phantom{\alpha}}\!\!\!\beta|}{}^{\underline{\epsilon}}
T_{\underline{\epsilon}|\underline{\phantom{\alpha}}\!\!\!\gamma)}{}^{\underline{\delta}} ~.
\label{bi1.2}
\end{align}
Let us analyze (\ref{bi1.1}) first. We distinguish the following cases.
$\bullet$ Case 1: $(\underline{\alpha},\underline{\phantom{\alpha}}\!\!\!\beta)=(\al,\bn)$
Demanding that the right-hand-side of (\ref{bi1.1}) be antisymmetric
in $c,d$ implies
\be
V^1_a=V^2_a=0
\end{equation}
and the curvature is given by
\beal
R_{\al\bn} {}_{cd}=2i(\ga_{cd}{}^{efg})_{\al\bn}H^1_{efg}
+4i(\ga^{e})_{\al\bn}H^2_{cde}~.
\label{r1}
\end{align}
$\bullet$ Case 2: $(\underline{\alpha},\underline{\phantom{\alpha}}\!\!\!\beta)=(\! \begin{array}{c} \alpha\\{} \end{array},\!\!\begin{array}{c} \beta\\{} \end{array})$
Similarly to the previous case we get
\be
V^{\prime1}_a=V^{\prime2}_a=0
\end{equation}
and
\beal
R^{\al\bn} {}_{cd}=2i(\ga_{cd}{}^{efg})^{\al\bn}H^{\prime 1}_{efg}
+4i(\ga^{e})^{\al\bn}H^{\prime 2}_{cde}~.
\label{r3}
\end{align}
$\bullet$ Case 3: $(\underline{\alpha},\underline{\phantom{\alpha}}\!\!\!\beta)=(\al,\!\!\begin{array}{c} \beta\\{} \end{array})$
Demanding that the right-hand-side of (\ref{bi1.1}) be antisymmetric
in $c,d$ in this case implies
\beal
S&=-S'\nn\\
F^1&={F}^{\prime1}\nn\\
F^2&={F}^{\prime2}\nn\\
G^1&=-{G}^{\prime1}\nn\\
G^2&=-{G}^{\prime2}
\label{lamb}
\end{align}
and the curvature is given by
\beal
R_{\al}{}^{\bn} {}_{cd}=-2i\Big\{
(\ga_{cd})_{\al}{}^\bn S+(\ga_{cd}{}^{ef})_{\al}{}^\bn F^1_{ef}
+\de_{\al}{}^\bn F^2_{cd} +
(\ga_{cd}{}^{efgh})_{\al}{}^\bn G^1_{efgh}
+3(\ga^{fg})_{\al}{}^\bn G^2_{cdfg}
\Big\}~.
\label{r2}
\end{align}
Let us now come to (\ref{bi1.2}). We distinguish the following cases.
$\bullet$ Case 1.1: $(\underline{\alpha},\underline{\phantom{\alpha}}\!\!\!\beta, \underline{\phantom{\alpha}}\!\!\!\gamma, \underline{\delta})=(\al,\bn, \ga, \de)$
We shall work out this case in
some detail in order to illustrate the general procedure.
Taking the Lorentz condition into account, we obtain
\beal
&(\ga^{e})_{(\al\bn}(\ga_e{}^{fgh})_{\ga)}{}^\de
\Big\{ iH^1_{fgh}-\frac{1}{2}g_1K_{fgh}
-\frac{1}{96}(g_1^2+\frac{\dot{g}_1}{2}) (\lambda\ga_{fgh}\lambda)\Big\}\nn\\
&+(\ga^{e})_{(\al\bn}(\ga^{fg})_{\ga)}{}^\de \Big\{2iH^2_{efg}
+\frac{3}{2}g_1 K_{efg}
-\frac{1}{4} (\mu\ga_{efg}\mu)
-\frac{3}{96}(g_1^2-\frac{\dot{g}_1}{2})(\lambda\ga_{efg}\lambda)
\Big\}\nn\\
&+\frac{i}{2}(\ga_{cd}{}^{efg})_{(\al\bn}(\ga^{cd})_{\ga)}{}^\de H^1_{efg}
-\frac{1}{2}g_1 K^f\Big\{
(\ga_f)_{(\al\bn}\delta_{\ga)}^{\delta}
+(\ga^e)_{(\al\bn}(\ga_{ef})_{\ga)}{}^{\delta} \Big\}=0~,
\label{klug}
\end{align}
where we have allowed for the possibility that $\lambda_{\al}={D}_{\al}\phi$ and we have
set $\dot{g}_1:=\frac{d}{d\phi}g_1$.
Note that the vector drops out of the BI due to the identity
$$
(\ga_f)_{(\al\bn}\delta_{\ga)}^{\delta}=-(\ga^e)_{(\al\bn}(\ga_{ef})_{\ga)}{}^{\delta}~.
$$
Moreover, one can see that there is a unique
three-form in the decomposition of the
tensor product of a chiral spinor and the symmetrized tensor product
of three antichiral spinors:
\beal
(00010)\otimes (00001)^{3\otimes_s}\sim 1(10000)\oplus 1(00100)\oplus\dots
\end{align}
Therefore, equation (\ref{klug}) imposes at most one linear equation
on the three-forms. Contracting both sides
of (\ref{klug}) with $(\ga_{[a})^{\al\bn}(\ga_{bc]})_\de{}^\ga$
in order to saturate the spinor indices, we obtain\footnote{
One can verify that the same equation is obtained
by contracting (\ref{klug}) with, for example, $(\ga^{e})^{\al\bn}(\ga_{eabc})_\de{}^\ga$.
Here and in the rest of the analysis of the BI's, we have applied many more contractions
than the number of independent ones. This is not strictly-speaking necessary, but the
`redundant ' contractions serve as useful consistency checks.}
\beal
48i H^1_{abc}-8i H^2_{abc}-12g_1K_{abc}+(\mu\ga_{abc}\mu)
+\frac{1}{16}(g_1^2-2\dot{g}_1)(\lambda\ga_{abc}\lambda)=0~.
\label{1}
\end{align}
$\bullet$ Case 1.2: $(\underline{\alpha},\underline{\phantom{\alpha}}\!\!\!\beta, \underline{\phantom{\alpha}}\!\!\!\gamma, \underline{\delta})=(\al,\bn, \ga, \!\!\begin{array}{c} \delta\\{} \end{array})$
In terms of irreducible representations the BI decomposes as
\beal
(00001)\otimes (00001)^{3\otimes_s}\sim 1(01000)\oplus 1(00011)\oplus\dots
\end{align}
Hence the BI imposes at most one linear equation on the two-forms and one on the four-forms.
Contracting with $(\ga_{[a})^{\al\bn}(\ga_{b]})^{\de\ga}$
we obtain
\beal
0=L_{ab}+\frac{1}{4}(\mu\ga_{ab}\lambda)
-\frac{i}{2}F^1_{ab}+\frac{i}{4}F^2_{ab}
~.
\label{tri}
\end{align}
Similarly, contracting with $(\ga_{[a})^{\al\bn}(\ga_{bcd]})^{\de\ga}$
we obtain
\beal
0=L_{abcd}-\frac{i}{2}G^1_{abcd}+\frac{i}{8}G^2_{abcd}
~.
\label{trii}
\end{align}
$\bullet$ Case 2.1: $(\underline{\alpha},\underline{\phantom{\alpha}}\!\!\!\beta, \underline{\phantom{\alpha}}\!\!\!\gamma, \underline{\delta})=(\al,\bn, \!\!\begin{array}{c} \gamma\\{} \end{array}, \de)$
In terms of representations, the BI decomposes as
\beal
(00010)^{2\otimes}\otimes(00001)^{2\otimes_s}
\sim 2(00000)\oplus 4(01000)\oplus 5(00011)\oplus \dots
\label{repr}
\end{align}
I.e. the BI imposes at most two constraints on the scalars, four on the two-forms and
five on the four-forms. Let us examine each representation in turn.
{\it Scalars}:
Using the independent structures
$(S_1)^{\al\bn}_{\ga\de}:=(\ga^a)^{\al\bn}(\ga_a)_{\ga\de}$
and $(S_2)^{\al\bn}_{\ga\de}:=(\ga^{a_1\dots a_5})^{\al\bn}(\ga_{a_1\dots a_5})_{\ga\de}$
to saturate the spinor indices\footnote{
The fact that these structures are independent is seen as follows.
Contracting the equation
$$A (\ga^a)_{\al\bn}(\ga_a)^{\ga\de}+ B(\ga^{a_1\dots a_5})_{\al\bn}(\ga_{a_1\dots a_5})^{\ga\de}=0$$
with $S_1$, $S_2$, we obtain $A=B=0$. Note that there are exactly two independent structures
as follows from the representation-theoretic analysis (\ref{repr}).},
we obtain
\beal
S=-\frac{2i}{5}L-\frac{ig_1}{10}L'+i\Big(\frac{1}{20}-\frac{11g_1}{80}
-\frac{\dot{g}_1}{160}\Big)(\mu\lambda)~.
\label{gu}
\end{align}
Hence in this case both independent contractions yield the same equation.
{\it Two-forms}:
Contracting with the four independent structures
$(\ga_{[a})^{\al\bn}(\ga_{b]})_{\ga\de}$,
$(\ga^f)^{\al\bn}(\ga_{fab})_{\ga\de}$,
$(\ga^{abfgh})^{\al\bn}(\ga_{fgh})_{\ga\de}$
and $(\ga^{[a|efgh})^{\al\bn}(\ga^{|b]}{}_{efgh})_{\ga\de}$, we get
\beal
0&=\frac{g_1}{6}L'_{ab}+iF_{ab}^1+\Big(\frac{1}{8}+\frac{g_1}{32}
+\frac{\dot{g}_1}{192}\Big)(\mu\ga_{ab}\lambda)\nn\\
0&=4L_{ab}+\frac{g_1}{3}L'_{ab}+iF_{ab}^2+\Big(\frac{5}{4}+\frac{g_1}{16}
+\frac{\dot{g}_1}{96}\Big)(\mu\ga_{ab}\lambda)~.
\label{try}
\end{align}
Remarkably, all four independent contractions yield only the two constraints above.
We also remark that equations (\ref{try}) imply (\ref{tri}), which is therefore not independent.
{\it Four-forms}:
Contracting with the five
independent structures
$(\ga^{[a})^{\al\bn}(\ga^{bcd]})_{\ga\de}$,
$(\ga^{e})^{\al\bn}(\ga_e{}^{abcd})_{\ga\de}$,
$(\ga^{abcde})^{\al\bn}(\ga_e)_{\ga\de}$,
$(\ga^{[abc|ef})^{\al\bn}(\ga^{|d]}{}_{ef})_{\ga\de}$ and
$(\ga^{[ab|efg})^{\al\bn}(\ga^{cd]}{}_{efg})_{\ga\de}$
we get
\beal
0&=2L_{abcd}-\frac{g_1}{2}L'_{abcd}+iG^1_{abcd}
+\Big(\frac{1}{96}+\frac{g_1}{384}-\frac{\dot{g}_1}{768}\Big)(\mu\ga_{abcd}\lambda)\nn\\
0&=16L_{abcd}-2g_1 L'_{abcd}+iG^2_{abcd}
+\Big(\frac{1}{24}+\frac{g_1}{96}-\frac{\dot{g}_1}{192}\Big)(\mu\ga_{abcd}\lambda)~.
\label{mn}
\end{align}
I.e. all five independent contractions yield only the two constraints above.
Moreover equations (\ref{mn}) imply (\ref{trii}).
$\bullet$ Case 2.2: $(\underline{\alpha},\underline{\phantom{\alpha}}\!\!\!\beta, \underline{\phantom{\alpha}}\!\!\!\gamma, \underline{\delta})=(\al,\bn, \!\!\begin{array}{c} \gamma\\{} \end{array}, \!\!\begin{array}{c} \delta\\{} \end{array})$
The BI decomposes as
\beal
(00001)^{2\otimes_s}\otimes(00001)\otimes(00010)
\sim 3(10000)\oplus 4(00100)\oplus \dots
\label{repri}
\end{align}
Therefore the BI will yield at most three constraints on the vectors and four on the three-forms.
It can be seen that the vector part of the BI is equivalent to
\beal
K_a=K_a'~.
\label{po1}
\end{align}
Moreover, contracting with the four independent structures
$(\ga^{ef})_\ga{}^\de(\ga_{ef}{}^{abc})^{\al\bn}$,
$(\ga^{[a| efg})_\ga{}^\de(\ga_{efg}{}^{|bc]})^{\al\bn}$,
$(\ga^{[ab})_\ga{}^\de(\ga^{c]})^{\al\bn}$ and
$(\ga^{abce})_\ga{}^\de(\ga_{e})^{\al\bn}$ we obtain
\beal
0&=
iH^1_{abc}+K_{abc}+\Big(\frac{1}{48}+\frac{g_1}{192} \Big)(\lambda\ga_{abc}\lambda)
\nn\\
0&=
iH^2_{abc}+iH^{\prime 2}_{abc}
-3K_{abc}-3K'_{abc}
+\Big(\frac{3}{16}+\frac{7g_1}{64}\Big)(\mu\ga_{abc}\mu)
+\Big(\frac{3}{16}+\frac{7g_2}{64}\Big)(\lambda\ga_{abc}\lambda)\nn\\
0&=
iH^{\prime 1}_{abc}+K'_{abc}+
\Big(\frac{1}{48}+\frac{g_2}{192}\Big)(\mu\ga_{abc}\mu)~.
\label{po}
\end{align}
\vfill\break
$\bullet$ Case 3.1: $(\underline{\alpha},\underline{\phantom{\alpha}}\!\!\!\beta, \underline{\phantom{\alpha}}\!\!\!\gamma, \underline{\delta})=(\al,\!\!\begin{array}{c} \beta\\{} \end{array}, \!\!\begin{array}{c} \gamma\\{} \end{array}, \de)$
In terms of representations, this is the same as case 2.2 above. The equations
which follow from this BI turn out to be identical to (\ref{po1}, \ref{po}).
$\bullet$ Case 3.2: $(\underline{\alpha},\underline{\phantom{\alpha}}\!\!\!\beta, \underline{\phantom{\alpha}}\!\!\!\gamma, \underline{\delta})=(\al,\!\!\begin{array}{c} \beta\\{} \end{array}, \!\!\begin{array}{c} \gamma\\{} \end{array}, \!\!\begin{array}{c} \delta\\{} \end{array})$
In terms of representations, this is the same as case 2.1.
Proceeding similarly, we obtain the following equations
{\it Scalars}:
\beal
S=\frac{2i}{5}L'+\frac{ig_2}{10}L+i\Big(\frac{1}{20}-\frac{11g_2}{80}
-\frac{\dot{g}_2}{160}\Big)(\mu\lambda)~.
\label{gub}
\end{align}
{\it Two-forms}:
\beal
0&=\frac{g_2}{6}L_{ab}+iF_{ab}^1+\Big(\frac{1}{8}+\frac{g_2}{32}
+\frac{\dot{g}_2}{192}\Big)(\mu\ga_{ab}\lambda)\nn\\
0&=4L'_{ab}+\frac{g_2}{3}L_{ab}+iF_{ab}^2+\Big(\frac{5}{4}+\frac{g_2}{16}
+\frac{\dot{g}_2}{96}\Big)(\mu\ga_{ab}\lambda)~.
\label{tryb}
\end{align}
{\it Four-forms}:
\beal
0&=2L'_{abcd}-\frac{g_2}{2}L_{abcd}-iG^1_{abcd}
-\Big(\frac{1}{96}+\frac{g_2}{384}+\frac{\dot{g}_1}{768}\Big)(\mu\ga_{abcd}\lambda)\nn\\
0&=16L'_{abcd}-2g_2 L_{abcd}-iG^2_{abcd}
-\Big(\frac{1}{24}+\frac{g_2}{96}+\frac{\dot{g}_2}{192}\Big)(\mu\ga_{abcd}\lambda)~.
\end{align}
$\bullet$ Case 4.1: $(\underline{\alpha},\underline{\phantom{\alpha}}\!\!\!\beta, \underline{\phantom{\alpha}}\!\!\!\gamma, \underline{\delta})=(\! \begin{array}{c} \alpha\\{} \end{array},\!\!\begin{array}{c} \beta\\{} \end{array}, \!\!\begin{array}{c} \gamma\\{} \end{array}, \de)$
In terms of representations, this is the same as case 1.2. Proceeding similarly
we obtain the following equations
\beal
0=L'_{ab}+\frac{1}{4}(\mu\ga_{ab}\lambda)
-\frac{i}{2}F^1_{ab}+\frac{i}{4}F^2_{ab}
\label{trib}
\end{align}
and
\beal
0=L'_{abcd}+\frac{i}{2}G^1_{abcd}-\frac{i}{8}G^2_{abcd}
~.
\label{triib}
\end{align}
\vfill\break
$\bullet$ Case 4.2: $(\underline{\alpha},\underline{\phantom{\alpha}}\!\!\!\beta, \underline{\phantom{\alpha}}\!\!\!\gamma, \underline{\delta})=(\! \begin{array}{c} \alpha\\{} \end{array},\!\!\begin{array}{c} \beta\\{} \end{array}, \!\!\begin{array}{c} \gamma\\{} \end{array}, \!\!\begin{array}{c} \delta\\{} \end{array})$
In terms of representations, this is the same as case 1.1.
Proceeding similarly we obtain
\beal
48i H^{\prime 1}_{abc}-8i H^{\prime 2}_{abc}
-12g_2K'_{abc}+(\lambda\ga_{abc}\lambda)
+\frac{1}{16}(g_2^2-2\dot{g}_2)(\mu\ga_{abc}\mu)=0~.
\label{3}
\end{align}
The system of equations above imply in particular
\beal
L_{ab}&=L'_{ab}\nn\\
L_{abcd}&=-L'_{abcd}
\label{inta}
\end{align}
and
\beal
g_1-g_2&=0\nn\\
(g_1+4)(L+L')&=0~.
\label{intb}
\end{align}
Conditions (\ref{po1}, \ref{inta})
if supplemented with $L=-L'$, would imply ${D}_{\al}\mu^{\bn}=-{D}^{\bn}\lambda_{\al}$,
$(\ga_a)^{\al\bn}{D}_{\al}\lambda_{\bn}
=(\ga_a)_{\al\bn}{D}^{\al}\mu^{\bn}$.
The latter two equations
are solved for $\lambda_{\al}={D}_{\al}\phi$, $\mu^{\al}={D}^{\al}\phi$.
We shall see in the following that in the case $g_1=-4$, $L=-L'$, the existence of $\phi$
is indeed implied by the higher-dimension BI's. But we have seen
in section \ref{fieldredefinitions} that in this case
the coefficient $g_1$ can be shifted by field redefinitions. Hence the case $g_1=-4$, $L=-L'$ is
in fact equivalent to the case $g_1\neq -4$, $L=-L'$.
In the following we shall set $g_1=-4$, so that we can treat both $L=-L'$ and $L\neq -L'$
cases simultaneously. Note that if $L\neq -L'$, there cannot exist a $\phi$ such that
$\lambda_{\al}={D}_{\al}\phi$, $\mu^{\al}={D}^{\al}\phi$.
\subsection{Dimension-$\frac{3}{2}$ BI}
Taking into account our gauge-fixing in section \ref{fieldredefinitions},
the Bianchi identities at canonical mass dimension three-half read
\beal
2R_{\underline{\alpha}[bc]}{}^d=T_{bc}{}^{\underline{\epsilon}}T_{\underline{\epsilon}\underline{\alpha}}{}^d
\label{bi321}
\end{align}
and
\beal
2R_{e(\underline{\alpha}\underline{\phantom{\alpha}}\!\!\!\beta)}{}^{\underline{\delta}}=2D_{(\underline{\alpha}}T_{\underline{\phantom{\alpha}}\!\!\!\beta)e}{}^{\underline{\delta}}+D_eT_{\underline{\alpha}\underline{\phantom{\alpha}}\!\!\!\beta}{}^{\underline{\delta}}
+T_{\underline{\alpha}\underline{\phantom{\alpha}}\!\!\!\beta}{}^fT_{fe}{}^{\underline{\delta}}+T_{\underline{\alpha}\underline{\phantom{\alpha}}\!\!\!\beta}{}^{\underline{\epsilon}}T_{\underline{\epsilon} e}{}^{\underline{\delta}}
+2T_{e(\underline{\alpha}}{}^{\underline{\epsilon}}T_{\underline{\epsilon}|\underline{\phantom{\alpha}}\!\!\!\beta)}{}^{\underline{\delta}}~.
\label{bi322}
\end{align}
Equation (\ref{bi321}) can be solved for the dimension three-half supercurvature to give
\beal
R_{\underline{\alpha} bcd}=\frac{i}{2}\Big\{
(\ga_bT_{cd})_{\underline{\alpha}}+(\ga_cT_{bd})_{\underline{\alpha}} -(\ga_dT_{bc})_{\underline{\alpha}} \Big\}~.
\end{align}
In the following we shall expand the dimension-$\frac{3}{2}$
torsion into irreducible (gamma-traceless) parts as follows
\beal
T_{ab}=\widetilde{T}_{ab}
+{}\ga_{[a}\widetilde{T}_{b]}{}{}
+{}\ga_{ab}\widetilde{T}
~,
\end{align}
where we have suppressed all spinor indices. Similarly, we expand the spinor derivatives
of the various dimension-one fields in irreducible representations,
as follows (again suppressing spinor indices)
\beal
DL&=\widetilde{L}\nn\\
DL'&=\widetilde{L}'\nn\\
DK_a&=\widetilde{K}^{(1)}_a+\ga_a\widetilde{K}^{(1)}\nn\\
DL_{ab}&=\widetilde{L}^{(2)}_{ab}+\ga_{[a}\widetilde{L}^{(2)}_{b]}+\ga_{ab}\widetilde{L}^{(2)}\nn\\
DK_{abc}&=\widetilde{K}^{(3)}_{abc}+\ga_{[a}\widetilde{K}^{(3)}_{bc]}
+\ga_{[ab}\widetilde{K}^{(3)}_{c]}
+\ga_{abc}\widetilde{K}^{(3)}_{}\nn\\
DL_{abcd}&=\widetilde{L}^{(4)}_{abcd}+\ga_{[a}\widetilde{L}^{(4)}_{bcd]}
+\ga_{[ab}\widetilde{L}^{(4)}_{cd]}
+\ga_{[abc}\widetilde{L}^{(4)}_{d]}+\ga_{abcd}\widetilde{L}^{(4)}_{}~.
\end{align}
Let us now come to equation (\ref{bi322}). We distinguish the following cases.
$\bullet$ Case 1.1: $(\underline{\alpha},\underline{\phantom{\alpha}}\!\!\!\beta, \underline{\delta})=(\al, \bn, \de)$
In terms of irreducible representations the BI decomposes as
$$
(10000)\otimes(00010)\otimes(00001)^{2\otimes_s}\sim
3(00010)\oplus 4(10001)\oplus2(01010)\oplus2(00101)\oplus(00021)\oplus\dots
$$
Hence the BI imposes at most three constraints on the spinors, four on the vector-spinors, etc.
Let us analyze each representation in turn:
{\it Spinors}
Contracting with the three independent structures $(\ga^e)^{\al\bn}\delta_{\delta}{}^{\ga}$,
$(\ga_g)^{\al\bn}(\ga^{ge})_{\delta}{}^{\ga}$ and
$(\ga^{eg_1\dots g_4})^{\al\bn}(\ga_{g_1\dots g_4})_{\delta}{}^{\ga}$, we obtain
\beal
\widetilde{T}&=-\frac{14i}{135}\slsh D \lambda+\frac{32}{45}L\mu+\frac{8}{45}L'\mu
+\frac{136}{405}L_{(2)}(\ga^{(2)}\mu)
+\frac{8}{45}L_{(4)}(\ga^{(4)}\mu)\nn\\
&-\frac{16}{45}K_{(1)}(\ga^{(1)}\lambda)
-\frac{2}{45}K_{(3)}(\ga^{(3)}\lambda)
-\frac{1}{60}(\mu\ga_{(3)}\mu)(\ga^{(3)}\lambda)\nn\\
\widetilde{K}^{(3)}_{}&=-\frac{i}{540}\slsh D \lambda+\frac{1}{180}L \mu+\frac{1}{45}L' \mu
-\frac{13}{1620}L_{(2)}(\ga^{(2)}\mu)
+\frac{1}{180}L_{(4)}(\ga^{(4)}\mu)\nn\\
&-\frac{1}{90}K_{(1)}(\ga^{(1)}\lambda)
+\frac{1}{45}K_{(3)}(\ga^{(3)}\lambda)~,
\end{align}
where, to simplify the expressions, we have introduced the notation
$\ga_{(p)}A^{(p)}:=\ga_{a_1\dots a_p}A^{a_1\dots a_p}$.
{\it Vector-spinors}
Contracting the BI with the four independent structures
$\delta_a{}^e(\ga_g)^{\al\bn}(\ga^{g})_{\delta\ga}$,
$\delta_a{}^e(\ga_{g_1\dots g_5})^{\al\bn}(\ga^{g_1\dots g_5})_{\delta\ga}$, \break
$(\ga_a)^{\al\bn}(\ga^{e})_{\delta\ga}$,
$(\ga_{a g_1\dots g_4})^{\al\bn}(\ga^{eg_1\dots g_4})_{\delta\ga}$ and
projecting onto the gamma-traceless part, we obtain
\beal
\widetilde{T}_a&=-\frac{3i}{20}(\ga_a^{(1)} D_{(1)} \lambda)-\frac{1}{5}L_{(2)}(\ga_{a}^{(2)}\mu)
-\frac{2}{5}L_{(4)}(\ga_{a}^{(4)}\mu)\nn\\
&+\frac{1}{5}K_{(1)}(\ga_{a}^{(1)}\lambda)+\frac{3}{20}K_{(3)}(\ga_{a}^{(3)}\lambda)
+\frac{3}{160}(\mu\ga_{(3)}\mu)(\ga_{a}^{(3)}\lambda)\nn\\
\widetilde{K}^{(3)}_{a}&=
\frac{i}{160}(\ga_a^{(1)} D_{(1)} \lambda)+\frac{1}{40}L_{(2)}(\ga_{a}^{(2)}\mu)
+\frac{1}{20}L_{(4)}(\ga_{a}^{(4)}\mu)\nn\\
&-\frac{1}{40}K_{(1)}(\ga_{a}^{(1)}\lambda)+\frac{81}{1120}K_{(3)}(\ga_{a}^{(3)}\lambda)
-\frac{1}{1280}(\mu\ga_{(3)}\mu)(\ga_{a}^{(3)}\lambda)~.
\end{align}
The notation in the equation above is a shorthand for the projection onto the
irreducible (gamma-traceless) vector-spinor part. Our conventions are
explained in appendix \ref{gammatraceless}.
{\it Two-form-spinors}
Contracting with the two independent structures
$(\ga_{[a})^{\al\bn}(\ga_{b]}{}^{e})_{\delta}{}^{\ga}$,
$(\ga_{ab}{}^{eg_1g_2})^{\al\bn}(\ga_{g_1g_2})_{\delta}{}^{\ga}$, we obtain
\beal
\widetilde{K}^{(3)}_{ab}&=
\frac{1}{8}\widetilde{T}_{ab}+\frac{1}{108}L_{(2)}(\ga_{ab}^{(2)}\mu)
+\frac{1}{6}L_{(4)}(\ga_{ab}^{(4)}\mu)+\frac{1}{18}K_{(3)}(\ga_{ab}^{(3)}\lambda)
~.
\end{align}
As in the previous case, the projections onto the gamma-traceless part of the
two-form-spinors are explained in appendix \ref{gammatraceless}.
{\it Three-form-spinors}
Contracting with the two independent structures
$\delta_{[a}{}^e(\ga_{bc]g_1g_2g_3})^{\al\bn}(\ga^{g_1g_2g_3})_{\delta\ga}$
and $(\ga_{abc}{}^{eg})^{\al\bn}(\ga_{g})_{\delta\ga}$, we obtain
\beal
\widetilde{K}^{(3)}_{abc}&=-\frac{1}{84}K_{(3)}(\ga_{abc}^{(3)}\lambda)~.
\end{align}
{\it Four-form-spinors}
Contracting with
$(\ga_{abcd}{}^e)^{\al\bn}\delta_{\delta}{}^{\ga}$, we obtain
\beal
\widetilde{L}^{(4)}_{abcd{}}&=0~.
\end{align}
$\bullet$ Case 1.2: $(\underline{\alpha},\underline{\phantom{\alpha}}\!\!\!\beta, \underline{\delta})=(\al, \bn, \!\!\begin{array}{c} \delta\\{} \end{array})$
In terms of irreducible representations the BI decomposes as
$$
(10000)\otimes(00001)\otimes(00001)^{2\otimes_s}\sim
3(00001)\oplus 3(10010)\oplus3(01001)\oplus(00110)\oplus2(00012)\oplus\dots
$$
hence there will be at most three constraints on the spinors, three on the vector-spinors, etc.
Let us examine each irreducible representation in turn:
{\it Spinors}
Contracting with the independent structures $(\ga^e)^{\al\bn}\delta_{\ga}{}^{\delta}$,
$(\ga_g)^{\al\bn}(\ga^{ge})_{\ga}{}^{\delta}$ and
$(\ga^{eg_1\dots g_4})^{\al\bn}(\ga_{g_1\dots g_4})_{\ga}{}^{\delta}$, we obtain
\beal
\widetilde{L}^{{}}&=\widetilde{L}^{\prime}-\frac{i}{16}\slsh D \mu
+\frac{25}{8}L \lambda +\frac{1}{4}L' \lambda+\frac{73}{24}L_{(2)}(\ga^{(2)}\lambda)
+\frac{9}{8}L_{(4)}(\ga^{(4)}\lambda)\nn\\
&-\frac{3}{8}K_{(1)}(\ga^{(1)}\mu)
-\frac{15}{16}K_{(3)}(\ga^{(3)}\mu)
-\frac{1}{64}(\lambda\ga_{(3)}\lambda)(\ga^{(3)}\mu)-\frac{135}{64}\widetilde{T}\nn\\
\widetilde{L}^{(2){}}_{}&=-\frac{3i}{320}\slsh D \mu
-\frac{21}{160}L \lambda +\frac{3}{80}L' \lambda +\frac{59}{480}L_{(2)}(\ga^{(2)}\lambda)
-\frac{1}{32}L_{(4)}(\ga^{(4)}\lambda)\nn\\
&-\frac{9}{160}K_{(1)}(\ga^{(1)}\mu)
+\frac{19}{320}K_{(3)}(\ga^{(3)}\mu)
-\frac{3}{1280}(\lambda\ga_{(3)}\lambda)(\ga^{(3)}\mu) -\frac{33}{256}\widetilde{T} \nn\\
\widetilde{L}^{(4)}_{{}}&=-\frac{i}{3840}\slsh D \mu
-\frac{7}{1920}L \lambda +\frac{1}{960}L' \lambda -\frac{1}{3456}L_{(2)}(\ga^{(2)}\lambda)
-\frac{131}{40320}L_{(4)}(\ga^{(4)}\lambda)\nn\\
&-\frac{1}{640}K_{(1)}(\ga^{(1)}\mu)
-\frac{13}{11520}K_{(3)}(\ga^{(3)}\mu)
-\frac{1}{15360}(\lambda\ga_{(3)}\lambda)(\ga^{(3)}\mu)-\frac{1}{1024}\widetilde{T} ~.
\end{align}
{\it Vector-spinors}
Contracting with the three independent structures $\delta_a{}^e(\ga_g)^{\al\bn}(\ga^{g})^{\ga\delta}$,
$(\ga_{ag_1\dots g_4})^{\al\bn}(\ga^{eg_1\dots g_4})^{\ga\delta}$ and \break
$(\ga_{a})^{\al\bn}(\ga^{e})^{\ga\delta}$, we obtain
\beal
\widetilde{L}^{(2){}}_{a}&=
-\frac{9i}{320}(\ga_a^{(1)} D_{(1)} \mu)-
\frac{21}{80}L_{(2)}(\ga_a^{(2)}\lambda)-
\frac{3}{40}L_{(4)}(\ga_a^{(4)}\lambda)\nn\\
&+\frac{3}{16}K_{(1)}(\ga_a^{(1)}\mu)
+\frac{63}{320}K_{(3)}(\ga_a^{(3)}\mu)
+\frac{9}{2560}(\lambda\ga_{(3)}\lambda)(\ga_a^{(3)}\mu)\nn\\
\widetilde{L}^{(4)}_{a{}}&=
-\frac{i}{1920}(\ga_a^{(1)} D_{(1)} \mu)-
\frac{1}{480}L_{(2)}(\ga_a^{(2)}\lambda)+
\frac{31}{1680}L_{(4)}(\ga_a^{(4)}\lambda)\nn\\
&-\frac{1}{480}K_{(1)}(\ga_a^{(1)}\mu)
+\frac{11}{4480}K_{(3)}(\ga_a^{(3)}\mu)
+\frac{1}{15360}(\lambda\ga_{(3)}\lambda)(\ga_a^{(3)}\mu)\nn\\
\widetilde{T}_a&=-\frac{3i}{20}(\ga_a^{(1)} D_{(1)} \mu)-\frac{1}{5}L_{(2)}(\ga_{a}^{(2)}\lambda)
+\frac{2}{5}L_{(4)}(\ga_{a}^{(4)}\lambda)\nn\\
&+\frac{1}{5}K_{(1)}(\ga_{a}^{(1)}\mu)-\frac{3}{20}K_{(3)}(\ga_{a}^{(3)}\mu)
+\frac{3}{160}(\lambda\ga_{(3)}\lambda)(\ga_{a}^{(3)}\mu)~.
\end{align}
{\it Two-form-spinors}
Contracting with the three independent structures
$\delta_{[\al}{}^e(\ga_{b]})^{\al\bn}\delta_{\ga}{}^{\delta}$,
$(\ga_{ab}{}^{eg_1g_2})^{\al\bn}(\ga_{g_1g_2})_{\ga}{}^{\delta}$,
$\delta_{[\al}{}^e(\ga_{b]})^{\delta(\al}\delta_{\ga}{}^{\bn)}$
and projecting onto the gamma-traceless part, we obtain
\beal
\widetilde{L}^{(2){}}_{ab}&=
\frac{3}{16}\widetilde{T}_{ab}
-\frac{1}{24}L_{(2)}(\ga_{ab}^{(2)}\lambda)-\frac{1}{4}L_{(4)}(\ga_{ab}^{(4)}\lambda)
-\frac{1}{8}K_{(3)}(\ga_{ab}^{(3)}\mu)\nn\\
\widetilde{L}^{(4)}_{ab{}}&=
\frac{1}{32}\widetilde{T}_{ab}
-\frac{1}{432}L_{(2)}(\ga_{ab}^{(2)}\lambda)
-\frac{13}{360}L_{(4)}(\ga_{ab}^{(4)}\lambda)
-\frac{1}{144}K_{(3)}(\ga_{ab}^{(3)}\mu)~.
\end{align}
{\it Three-form-spinors}
Contracting with $(\ga_{abc}{}^{eg})^{\al\bn}(\ga_{g})^{\ga\delta}$ we obtain
\beal
\widetilde{L}^{(4)}_{abc{}}&=
-\frac{1}{168}L_{(4)}(\ga_{abc}^{(4)}\lambda)
-\frac{1}{336}K_{(3)}(\ga_{abc}^{(3)}\mu)~.
\end{align}
{\it Four-form-spinors}
Contracting with the two independent structures
$(\ga_{abcd}{}^{e})^{\al\bn}\delta_{\ga}{}^{\delta}$ and
$(\ga_{abcd}{}^{e})^{\delta(\al}\delta_{\ga}{}^{\bn)}$, we obtain
\beal
\widetilde{L}^{(4)}_{abcd{}}&=0~.
\end{align}
$\bullet$ Case 2.1: $(\underline{\alpha},\underline{\phantom{\alpha}}\!\!\!\beta, \underline{\delta})=(\al, \!\!\begin{array}{c} \beta\\{} \end{array}, \de)$
In terms of irreducible representations the BI decomposes as
$$
(10000)\otimes(00001)\otimes(00010)^{2\otimes}\sim
5(00001)\oplus 7(10010)\oplus5(01001)\oplus4(00110)\oplus2(00012)\oplus\dots
$$
Hence the BI imposes at most five constraints on the spinors, seven on the vector-spinors, etc.
Let us analyze each representation in turn:
\vfill\break
{\it Spinors}
Contracting with the five independent structures
$(\ga^e)_{\ga\bn}\delta_{\delta}{}^{\al}$,
$(\ga_g)_{\ga\bn}(\ga^{eg})_{\delta}{}^{\al}$,
$(\ga^{eg_1g_2})_{\ga\bn}(\ga_{g_1g_2})_{\delta}{}^{\al}$, \break
$(\ga_{g_1g_2g_3})_{\ga\bn}(\ga^{eg_1g_2g_3})_{\delta}{}^{\al}$ and
$(\ga^{eg_1\dots g_4})_{\ga\bn}(\ga_{g_1\dots g_4})_{\delta}{}^{\al}$,
we obtain
\beal
\widetilde{L}^{{}}&=\widetilde{L}^{\prime}+\frac{5i}{16}\slsh D \mu
+2L \lambda -\frac{1}{2}L' \lambda+\frac{19}{6}L_{(2)}(\ga^{(2)}\lambda)
+\frac{3}{2}L_{(4)}(\ga^{(4)}\lambda)\nn\\
&+\frac{9}{4}K_{(1)}(\ga^{(1)}\mu)
-\frac{9}{16}K_{(3)}(\ga^{(3)}\mu)
+\frac{1}{128}(\lambda\ga_{(3)}\lambda)(\ga^{(3)}\mu)\nn\\
\widetilde{L}^{(2){}}_{}&=\frac{13i}{960}\slsh D \mu
-\frac{1}{5}L \lambda -\frac{1}{120}L' \lambda +\frac{47}{360}L_{(2)}(\ga^{(2)}\lambda)
-\frac{1}{120}L_{(4)}(\ga^{(4)}\lambda)\nn\\
&+\frac{5}{48}K_{(1)}(\ga^{(1)}\mu)
+\frac{79}{960}K_{(3)}(\ga^{(3)}\mu)
-\frac{7}{7680}(\lambda\ga_{(3)}\lambda)(\ga^{(3)}\mu)\nn\\
\widetilde{L}^{(4)}_{{}}&=-\frac{i}{11520}\slsh D \mu
-\frac{1}{240}L \lambda +\frac{1}{1440}L' \lambda -\frac{1}{4320}L_{(2)}(\ga^{(2)}\lambda)
-\frac{31}{10080}L_{(4)}(\ga^{(4)}\lambda)\nn\\
&-\frac{1}{2880}K_{(1)}(\ga^{(1)}\mu)
-\frac{11}{11520}K_{(3)}(\ga^{(3)}\mu)
-\frac{1}{18432}(\lambda\ga_{(3)}\lambda)(\ga^{(3)}\mu)\nn\\
\widetilde{K}^{(3)}_{}&=-\frac{i}{360}\slsh D \mu
-\frac{1}{36}L' \lambda -\frac{1}{60}L_{(2)}(\ga^{(2)}\lambda)
+\frac{1}{180}L_{(4)}(\ga^{(4)}\lambda)\nn\\
&-\frac{2}{45}K_{(1)}(\ga^{(1)}\mu)
+\frac{1}{120}K_{(3)}(\ga^{(3)}\mu)
+\frac{1}{2880}(\lambda\ga_{(3)}\lambda)(\ga^{(3)}\mu)\nn\\
\widetilde{T}&=-\frac{8i}{45}\slsh D \mu
+\frac{8}{15}L\lambda+\frac{16}{45}L'\lambda
-\frac{8}{135}L_{(2)}(\ga^{(2)}\lambda)
-\frac{8}{45}L_{(4)}(\ga^{(4)}\lambda)\nn\\
&-\frac{56}{45}K_{(1)}(\ga^{(1)}\mu)
-\frac{8}{45}K_{(3)}(\ga^{(3)}\mu)
-\frac{1}{90}(\lambda\ga_{(3)}\lambda)(\ga_a^{(3)}\mu)
~.
\end{align}
{\it Vector-spinors}
Contracting the BI with the seven independent structures
$(\ga^{eg_1g_2g_3})^{\ga}{}_{\bn}(\ga_{ag_1g_2g_3})_{\delta}{}^{\al}$,
$(\ga_a{}^{eg_1g_2})^{\ga}{}_{\bn}(\ga_{g_1g_2})_{\delta}{}^{\al}$,
$(\ga_{ag})^{\ga}{}_{\bn}(\ga^{eg})_{\delta}{}^{\al}$,
$(\ga^{eg})^{\ga}{}_{\bn}(\ga_{ag})_{\delta}{}^{\al}$,
$\delta_a^e\delta^{\ga}{}_{\bn}\delta_{\delta}{}^{\al}$,
$(\ga_a{}^{eg_1\dots g_4})^{\ga}{}_{\bn}(\ga_{g_1\dots g_4})_{\delta}{}^{\al}$ and
$(\ga_{ag_1g_2g_3})^{\ga}{}_{\bn}(\ga^{eg_1g_2g_3})_{\delta}{}^{\al}$, we obtain
\beal
\widetilde{L}^{(2){}}_{a}&=
-\frac{9i}{320}(\ga_a^{(1)} D_{(1)} \mu)-
\frac{21}{80}L_{(2)}(\ga_a^{(2)}\lambda)-
\frac{3}{40}L_{(4)}(\ga_a^{(4)}\lambda)\nn\\
&+\frac{3}{16}K_{(1)}(\ga_a^{(1)}\mu)
+\frac{63}{320}K_{(3)}(\ga_a^{(3)}\mu)
+\frac{9}{2560}(\lambda\ga_{(3)}\lambda)(\ga_a^{(3)}\mu)\nn\\
\widetilde{L}^{(4)}_{a{}}&=
-\frac{i}{1920}(\ga_a^{(1)} D_{(1)} \mu)-
\frac{1}{480}L_{(2)}(\ga_a^{(2)}\lambda)+
\frac{31}{1680}L_{(4)}(\ga_a^{(4)}\lambda)\nn\\
&-\frac{1}{480}K_{(1)}(\ga_a^{(1)}\mu)
+\frac{11}{4480}K_{(3)}(\ga_a^{(3)}\mu)
+\frac{1}{15360}(\lambda\ga_{(3)}\lambda)(\ga_a^{(3)}\mu)\nn\\
\widetilde{K}^{(3)}_{a}&=
-\frac{i}{160}(\ga_a^{(1)} D_{(1)} \mu)-\frac{1}{40}L_{(2)}(\ga_a^{(2)}\lambda)+
\frac{1}{20}L_{(4)}(\ga_a^{(4)}\lambda)\nn\\
&+\frac{1}{40}K_{(1)}(\ga_a^{(1)}\mu)
+\frac{81}{1120}K_{(3)}(\ga_a^{(3)}\mu)
+\frac{1}{1280}(\lambda\ga_{(3)}\lambda)(\ga_a^{(3)}\mu)\nn\\
\widetilde{T}_a&=-\frac{3i}{20}(\ga_a^{(1)} D_{(1)} \mu)-\frac{1}{5}L_{(2)}(\ga_{a}^{(2)}\lambda)
+\frac{2}{5}L_{(4)}(\ga_{a}^{(4)}\lambda)\nn\\
&+\frac{1}{5}K_{(1)}(\ga_{a}^{(1)}\mu)-\frac{3}{20}K_{(3)}(\ga_{a}^{(3)}\mu)
+\frac{3}{160}(\lambda\ga_{(3)}\lambda)(\ga_{a}^{(3)}\mu)~.
\end{align}
\vfill\break
{\it Two-form-spinors}
Contracting with the five independent structures
$(\ga^e)_{\ga\bn}(\ga_{ab})_{\delta}{}^{\al}$,
$(\ga_{ab}{}^{eg_1g_2})_{\ga\bn}(\ga_{g_1g_2})_{\delta}{}^{\al}$,
$(\ga_{[a}{}^{eg})_{\ga\bn}(\ga_{b]g})_{\delta}{}^{\al}$,
$(\ga_{g})_{\ga\bn}(\ga_{ab}{}^{eg})_{\delta}{}^{\al}$ and
$(\ga^{eg_1g_2})_{\ga\bn}(\ga_{abg_1g_2})_{\delta}{}^{\al}$, we obtain
\beal
\widetilde{L}^{(2){}}_{ab}&=
\frac{3}{16}\widetilde{T}_{ab}
-\frac{1}{24}L_{(2)}(\ga_{ab}^{(2)}\lambda)-\frac{1}{4}L_{(4)}(\ga_{ab}^{(4)}\lambda)
-\frac{1}{8}K_{(3)}(\ga_{ab}^{(3)}\mu)\nn\\
\widetilde{L}^{(4)}_{ab{}}&=
\frac{1}{32}\widetilde{T}_{ab}
-\frac{1}{432}L_{(2)}(\ga_{ab}^{(2)}\lambda)
-\frac{13}{360}L_{(4)}(\ga_{ab}^{(4)}\lambda)
-\frac{1}{144}K_{(3)}(\ga_{ab}^{(3)}\mu)\nn\\
\widetilde{K}^{(3)}_{ab}&=
-\frac{1}{8}\widetilde{T}_{ab}-
\frac{1}{108}L_{(2)}(\ga_{ab}^{(2)}\lambda)+
\frac{1}{6}L_{(4)}(\ga_{ab}^{(4)}\lambda)
+\frac{1}{18}K_{(3)}(\ga_{ab}^{(3)}\mu)~.
\end{align}
{\it Three-form-spinors}
Contracting with the four independent structures
$\delta^{\ga}{}_{\bn}(\ga_{abc}{}^{e})_{\delta}{}^{\al}$,
$(\ga_{[a}{}^e)^{\ga}{}_{\bn}(\ga_{bc]})_{\delta}{}^{\al}$,
$(\ga^{eg})^{\ga}{}_{\bn}(\ga_{abcg})_{\delta}{}^{\al}$ and
$(\ga_{[a}{}^{eg_1g_2})^{\ga}{}_{\bn}(\ga_{bc]g_1g_2})_{\delta}{}^{\al}$, we obtain
\beal
\widetilde{L}^{(4)}_{abc{}}&=
-\frac{1}{168}L_{(4)}(\ga_{abc}^{(4)}\lambda)
-\frac{1}{336}K_{(3)}(\ga_{abc}^{(3)}\mu)
\nn\\
\widetilde{K}^{(3)}_{abc}&=
-\frac{1}{84}K_{(3)}(\ga_{abc}^{(3)}\mu)~.
\end{align}
{\it Four-form-spinors}
Contracting with the independent structures $(\ga^e)_{\ga\bn}(\ga_{abcd})_{\delta}{}^{\al}$ and
$(\ga_{[a}{}^{eg})_{\ga\bn}(\ga_{bcd]g})_{\delta}{}^{\al}$, we obtain
\beal
\widetilde{L}^{(4)}_{abcd{}}&=0~.
\end{align}
$\bullet$ Case 2.2: $(\underline{\alpha},\underline{\phantom{\alpha}}\!\!\!\beta, \underline{\delta})=(\al, \!\!\begin{array}{c} \beta\\{} \end{array}, \!\!\begin{array}{c} \delta\\{} \end{array})$
This is related to the previous case by parity-inversion. The analysis proceeds in a
similar fashion.
{\it Spinors}
Contracting with the five independent structures
$(\ga^e)^{\ga\al{}}\delta^{\delta}{}_{\bn}$,
$(\ga_g)^{\ga\al{}}(\ga^{eg})^{\delta}{}_{\bn}$,
$(\ga^{eg_1g_2})^{\ga\al{}}(\ga_{g_1g_2})^{\delta}{}_{\bn}$, \break
$(\ga_{g_1g_2g_3})^{\ga\al{}}(\ga^{eg_1g_2g_3})^{\delta}{}_{\bn}$ and
$(\ga^{eg_1\dots g_4})^{\ga\al{}}(\ga_{g_1\dots g_4})^{\delta}{}_{\bn}$,
we obtain
\beal
\widetilde{L}^{}&=\widetilde{L}^{\prime{}}-\frac{5i}{16}\slsh D \lambda+
\frac{1}{2}L \mu-2L' \mu -\frac{19}{6}L_{(2)}(\ga^{(2)}\mu)
+\frac{3}{2}L_{(4)}(\ga^{(4)}\mu)\nn\\
&-\frac{9}{4}K_{(1)}(\ga^{(1)}\lambda)
-\frac{9}{16}K_{(3)}(\ga^{(3)}\lambda)
-\frac{1}{128}(\mu\ga_{(3)}\mu)(\ga^{(3)}\lambda)\nn\\
\widetilde{L}^{(2){}}_{}&=\frac{13i}{960}\slsh D \lambda
-\frac{1}{120}L \mu -\frac{1}{5}L' \mu+\frac{47}{360}L_{(2)}(\ga^{(2)}\mu)
+\frac{1}{120}L_{(4)}(\ga^{(4)}\mu)\nn\\
&+\frac{5}{48}K_{(1)}(\ga^{(1)}\lambda)
-\frac{79}{960}K_{(3)}(\ga^{(3)}\lambda)
-\frac{7}{7680}(\mu\ga_{(3)}\mu)(\ga^{(3)}\lambda)\nn\\
\widetilde{L}^{(4)}_{{}}&=\frac{i}{11520}\slsh D \lambda
-\frac{1}{1440}L \mu +\frac{1}{240}L' \mu
+\frac{1}{4320}L_{(2)}(\ga^{(2)}\mu)
-\frac{31}{10080}L_{(4)}(\ga^{(4)}\mu)
\nn\\
&+\frac{1}{2880}K_{(1)}(\ga^{(1)}\lambda)
-\frac{11}{11520}K_{(3)}(\ga^{(3)}\lambda)
+\frac{1}{18432}(\mu\ga_{(3)}\mu)(\ga^{(3)}\lambda)\nn\\
\widetilde{K}^{(3)}_{}&=\frac{i}{360}\slsh D \lambda
+\frac{1}{36}L \mu
+\frac{1}{60}L_{(2)}(\ga^{(2)}\mu)
+\frac{1}{180}L_{(4)}(\ga^{(4)}\mu)\nn\\
&+\frac{2}{45}K_{(1)}(\ga^{(1)}\lambda)
+\frac{1}{120}K_{(3)}(\ga^{(3)}\lambda)
-\frac{1}{2880}(\mu\ga_{(3)}\mu)(\ga^{(3)}\lambda)\nn\\
\widetilde{T}&=-\frac{8i}{45}\slsh D \lambda
+\frac{16}{45}L\mu+\frac{8}{15}L'\mu
-\frac{8}{135}L_{(2)}(\ga^{(2)}\mu)
+\frac{8}{45}L_{(4)}(\ga^{(4)}\mu)\nn\\
&-\frac{56}{45}K_{(1)}(\ga^{(1)}\lambda)
+\frac{8}{45}K_{(3)}(\ga^{(3)}\lambda)
-\frac{1}{90}(\mu\ga_{(3)}\mu)(\ga_a^{(3)}\lambda)~.
\end{align}
{\it Vector-spinors}
Contracting the BI with the seven independent structures
$(\ga^{eg_1g_2g_3})_{\ga}{}^{\al}(\ga_{ag_1g_2g_3})^{\delta}{}_{\bn}$,
$(\ga_a{}^{eg_1g_2})_{\ga}{}^{\al}(\ga_{g_1g_2})^{\delta}{}_{\bn}$,
$(\ga_{ag})_{\ga}{}^{\al}(\ga^{eg})^{\delta}{}_{\bn}$,
$(\ga^{eg})_{\ga}{}^{\al}(\ga_{ag})^{\delta}{}_{\bn}$,
$\delta_a^e\delta_{\ga}{}^{\al}\delta^{\delta}{}_{\bn}$,
$(\ga_a{}^{eg_1\dots g_4})_{\ga}{}^{\al}(\ga_{g_1\dots g_4})^{\delta}{}_{\bn}$ and
$(\ga_{ag_1g_2g_3})_{\ga}{}^{\al}(\ga^{eg_1g_2g_3})^{\delta}{}_{\bn}$, we obtain
\beal
\widetilde{L}^{(2){}}_{a}&=
-\frac{9i}{320}(\ga_a^{(1)} D_{(1)} \lambda)-
\frac{21}{80}L_{(2)}(\ga_a^{(2)}\mu)+
\frac{3}{40}L_{(4)}(\ga_a^{(4)}\mu)\nn\\
&+\frac{3}{16}K_{(1)}(\ga_a^{(1)}\lambda)
-\frac{63}{320}K_{(3)}(\ga_a^{(3)}\lambda)
+\frac{9}{2560}(\mu\ga_{(3)}\mu)(\ga_a^{(3)}\lambda)\nn\\
\widetilde{L}^{(4)}_{a{}}&=
\frac{i}{1920}(\ga_a^{(1)} D_{(1)} \lambda)+
\frac{1}{480}L_{(2)}(\ga_a^{(2)}\mu)+
\frac{31}{1680}L_{(4)}(\ga_a^{(4)}\mu)\nn\\
&+\frac{1}{480}K_{(1)}(\ga_a^{(1)}\lambda)
+\frac{11}{4480}K_{(3)}(\ga_a^{(3)}\lambda)
-\frac{1}{15360}(\mu\ga_{(3)}\mu)(\ga_a^{(3)}\lambda)\nn\\
\widetilde{K}^{(3)}_{a}&=
\frac{i}{160}(\ga_a^{(1)} D_{(1)} \lambda)+\frac{1}{40}L_{(2)}(\ga_{a}^{(2)}\mu)
+\frac{1}{20}L_{(4)}(\ga_{a}^{(4)}\mu)\nn\\
&-\frac{1}{40}K_{(1)}(\ga_{a}^{(1)}\lambda)+\frac{81}{1120}K_{(3)}(\ga_{a}^{(3)}\lambda)
-\frac{1}{1280}(\mu\ga_{(3)}\mu)(\ga_{a}^{(3)}\lambda)
\nn\\
\widetilde{T}_a&=-\frac{3i}{20}(\ga_a^{(1)} D_{(1)} \lambda)-\frac{1}{5}L_{(2)}(\ga_{a}^{(2)}\mu)
-\frac{2}{5}L_{(4)}(\ga_{a}^{(4)}\mu)\nn\\
&+\frac{1}{5}K_{(1)}(\ga_{a}^{(1)}\lambda)+\frac{3}{20}K_{(3)}(\ga_{a}^{(3)}\lambda)
+\frac{3}{160}(\mu\ga_{(3)}\mu)(\ga_{a}^{(3)}\lambda)~.
\end{align}
{\it Two-form-spinors}
Contracting with the five independent structures
$(\ga^e)^{\ga\al{}}(\ga_{ab})^{\delta}{}_{\bn}$,
$(\ga_{ab}{}^{eg_1g_2})^{\ga\al{}}(\ga_{g_1g_2})^{\delta}{}_{\bn}$,
$(\ga_{[a}{}^{eg})^{\ga\al{}}(\ga_{b]g})^{\delta}{}_{\bn}$,
$(\ga_{g})^{\ga\al{}}(\ga_{ab}{}^{eg})^{\delta}{}_{\bn}$ and
$(\ga^{eg_1g_2})^{\ga\al{}}(\ga_{abg_1g_2})^{\delta}{}_{\bn}$, we obtain
\beal
\widetilde{L}^{(2){}}_{ab}&=
\frac{3}{16}\widetilde{T}_{ab}-\frac{1}{24}L_{(2)}(\ga_{ab}^{(2)}\mu)
+\frac{1}{4}L_{(4)}(\ga_{ab}^{(4)}\mu)
+\frac{1}{8}K_{(3)}(\ga_{ab}^{(3)}\lambda)\nn\\
\widetilde{L}^{(4)}_{ab{}}&=-
\frac{1}{32}\widetilde{T}_{ab}+\frac{1}{432}L_{(2)}(\ga_{ab}^{(2)}\mu)
-\frac{13}{360}L_{(4)}(\ga_{ab}^{(4)}\mu)-\frac{1}{144}K_{(3)}(\ga_{ab}^{(3)}\lambda)\nn\\
\widetilde{K}^{(3)}_{ab}&=
\frac{1}{8}\widetilde{T}_{ab}+\frac{1}{108}L_{(2)}(\ga_{ab}^{(2)}\mu)
+\frac{1}{6}L_{(4)}(\ga_{ab}^{(4)}\mu)+\frac{1}{18}K_{(3)}(\ga_{ab}^{(3)}\lambda)~.
\end{align}
{\it Three-form-spinors}
Contracting with the four independent structures
$\delta_{\ga}{}^{\al}(\ga_{abc}{}^{e})^{\delta}{}_{\bn}$,
$(\ga_{[a}{}^e)_{\ga}{}^{\al}(\ga_{bc]})^{\delta}{}_{\bn}$,
$(\ga^{eg})_{\ga}{}^{\al}(\ga_{abcg})^{\delta}{}_{\bn}$ and
$(\ga_{[a}{}^{eg_1g_2})_{\ga}{}^{\al}(\ga_{bc]g_1g_2})^{\delta}{}_{\bn}$, we obtain
\beal
\widetilde{L}^{(4)}_{abc{}}&=-\frac{1}{168}L_{(4)}(\ga_{abc}^{(4)}\lambda)
-\frac{1}{336}K_{(3)}(\ga_{abc}^{(3)}\lambda)
\nn\\
\widetilde{K}^{(3)}_{abc}&=-\frac{1}{84}K_{(3)}(\ga_{abc}^{(3)}\lambda)~.
\end{align}
{\it Four-form-spinors}
Contracting with the independent structures $(\ga^e)^{\ga\al{}}(\ga_{abcd})^{\delta}{}_{\bn}$ and
$(\ga_{[a}{}^{eg})^{\ga\al{}}(\ga_{bcd]g})^{\delta}{}_{\bn}$, we obtain
\beal
\widetilde{L}^{(4)}_{abcd{}}&=0~.
\end{align}
$\bullet$ Case 3.1: $(\underline{\alpha},\underline{\phantom{\alpha}}\!\!\!\beta, \underline{\delta})=(\! \begin{array}{c} \alpha\\{} \end{array}, \!\!\begin{array}{c} \beta\\{} \end{array}, \de)$
This is related to case 1.2 by parity-inversion.
{\it Spinors}
Contracting with the independent structures $(\ga^e)_{\al\bn}\delta^{\ga}{}_{\delta}$,
$(\ga_g)_{\al\bn}(\ga^{ge})^{\ga}{}_{\delta}$ and
$(\ga^{eg_1\dots g_4})_{\al\bn}(\ga_{g_1\dots g_4})^{\ga}{}_{\delta}$, we obtain
\beal
\widetilde{L}^{}&=\widetilde{L}^{\prime{}}+\frac{i}{16}\slsh D \lambda-
\frac{1}{4}L \mu-\frac{25}{8}L' \mu -\frac{73}{24}L_{(2)}(\ga^{(2)}\mu)
+\frac{9}{8}L_{(4)}(\ga^{(4)}\mu)\nn\\
&+\frac{3}{8}K_{(1)}(\ga^{(1)}\lambda)
-\frac{15}{16}K_{(3)}(\ga^{(3)}\lambda)
+\frac{1}{64}(\mu\ga_{(3)}\mu)(\ga^{(3)}\lambda)
+\frac{135}{64}\widetilde{T}
\nn\\
\widetilde{L}^{(2){}}_{}&=-\frac{3i}{320}\slsh D \lambda
+\frac{3}{80}L \mu -\frac{21}{160}L' \mu+\frac{59}{480}L_{(2)}(\ga^{(2)}\mu)
+\frac{1}{32}L_{(4)}(\ga^{(4)}\mu)\nn\\
&-\frac{9}{160}K_{(1)}(\ga^{(1)}\lambda)
-\frac{19}{320}K_{(3)}(\ga^{(3)}\lambda)
-\frac{3}{1280}(\mu\ga_{(3)}\mu)(\ga^{(3)}\lambda)
-\frac{33}{256}\widetilde{T}\nn\\
\widetilde{L}^{(4)}_{{}}&=\frac{i}{3840}\slsh D \lambda
-\frac{1}{960}L \mu +\frac{7}{1920}L' \mu
+\frac{1}{3456}L_{(2)}(\ga^{(2)}\mu)
-\frac{131}{40320}L_{(4)}(\ga^{(4)}\mu)
\nn\\
&+\frac{1}{640}K_{(1)}(\ga^{(1)}\lambda)
-\frac{13}{11520}K_{(3)}(\ga^{(3)}\lambda)
+\frac{1}{15360}(\mu\ga_{(3)}\mu)(\ga^{(3)}\lambda)
+\frac{1}{1024}\widetilde{T}
~.
\end{align}
\vfill\break
{\it Vector-spinors}
Contracting with the three independent structures $\delta_a{}^e(\ga_g)_{\al\bn}(\ga^{g})_{\ga\delta}$,
$(\ga_{ag_1\dots g_4})_{\al\bn}(\ga^{eg_1\dots g_4})_{\ga\delta}$ and \break
$(\ga_{a})_{\al\bn}(\ga^{e})_{\ga\delta}$, we obtain
\beal
\widetilde{L}^{(2){}}_{a}&=
-\frac{9i}{320}(\ga_a^{(1)} D_{(1)} \lambda)-
\frac{21}{80}L_{(2)}(\ga_a^{(2)}\mu)+
\frac{3}{40}L_{(4)}(\ga_a^{(4)}\mu)\nn\\
&+\frac{3}{16}K_{(1)}(\ga_a^{(1)}\lambda)
-\frac{63}{320}K_{(3)}(\ga_a^{(3)}\lambda)
+\frac{9}{2560}(\mu\ga_{(3)}\mu)(\ga_a^{(3)}\lambda)\nn\\
\widetilde{L}^{(4)}_{a{}}&=
\frac{i}{1920}(\ga_a^{(1)} D_{(1)} \lambda)+
\frac{1}{480}L_{(2)}(\ga_a^{(2)}\mu)+
\frac{31}{1680}L_{(4)}(\ga_a^{(4)}\mu)\nn\\
&+\frac{1}{480}K_{(1)}(\ga_a^{(1)}\lambda)
+\frac{11}{4480}K_{(3)}(\ga_a^{(3)}\lambda)
-\frac{1}{15360}(\mu\ga_{(3)}\mu)(\ga_a^{(3)}\lambda)\nn\\
\widetilde{T}_a&=-\frac{3i}{20}(\ga_a^{(1)} D_{(1)} \lambda)-\frac{1}{5}L_{(2)}(\ga_{a}^{(2)}\mu)
-\frac{2}{5}L_{(4)}(\ga_{a}^{(4)}\mu)\nn\\
&+\frac{1}{5}K_{(1)}(\ga_{a}^{(1)}\lambda)+\frac{3}{20}K_{(3)}(\ga_{a}^{(3)}\lambda)
+\frac{3}{160}(\mu\ga_{(3)}\mu)(\ga_{a}^{(3)}\lambda)
~.
\end{align}
{\it Two-form-spinors}
Contracting with the three independent structures
$\delta_{[\al}{}^e(\ga_{b]})_{\al\bn}\delta^{\ga}{}_{\delta}$,
$(\ga_{ab}{}^{eg_1g_2})_{\al\bn}(\ga_{g_1g_2})^{\ga}{}_{\delta}$,
$\delta_{[\al}{}^e(\ga_{b]})_{\delta(\al}\delta^{\ga}_{\bn)}$
and projecting onto the gamma-traceless part, we obtain
\beal
\widetilde{L}^{(2){}}_{ab}&=
\frac{3}{16}\widetilde{T}_{ab}-\frac{1}{24}L_{(2)}(\ga_{ab}^{(2)}\mu)
+\frac{1}{4}L_{(4)}(\ga_{ab}^{(4)}\mu)
+\frac{1}{8}K_{(3)}(\ga_{ab}^{(3)}\lambda)\nn\\
\widetilde{L}^{(4)}_{ab{}}&=-
\frac{1}{32}\widetilde{T}_{ab}+\frac{1}{432}L_{(2)}(\ga_{ab}^{(2)}\mu)
-\frac{13}{360}L_{(4)}(\ga_{ab}^{(4)}\mu)-\frac{1}{144}K_{(3)}(\ga_{ab}^{(3)}\lambda)
~.
\end{align}
{\it Three-form-spinors}
Contracting with $(\ga_{abc}{}^{eg})_{\al\bn}(\ga_{g})_{\ga\delta}$ we obtain
\beal
\widetilde{L}^{(4)}_{abc{}}&=-\frac{1}{168}L_{(4)}(\ga_{abc}^{(4)}\mu)
-\frac{1}{336}K_{(3)}(\ga_{abc}^{(3)}\lambda)
~.
\end{align}
{\it Four-form-spinors}
Contracting with the two independent structures
$(\ga_{abcd}{}^{e})_{\al\bn}\delta^{\ga}{}_{\delta}$ and
$(\ga_{abcd}{}^{e})_{\delta(\al}\delta^{\ga}{}_{\bn)}$, we obtain
\beal
\widetilde{L}^{(4)}_{abcd{}}&=0~.
\end{align}
$\bullet$ Case 3.2: $(\underline{\alpha},\underline{\phantom{\alpha}}\!\!\!\beta, \underline{\delta})=(\! \begin{array}{c} \alpha\\{} \end{array}, \!\!\begin{array}{c} \beta\\{} \end{array}, \!\!\begin{array}{c} \delta\\{} \end{array})$
This related to case 1.1 by parity inversion.
\vfill\break
{\it Spinors}
Contracting with the three independent structures $(\ga^e)_{\al\bn}\delta^{\delta}{}_{\ga}$,
$(\ga_g)_{\al\bn}(\ga^{ge})^{\delta}{}_{\ga}$ and
$(\ga^{eg_1\dots g_4})_{\al\bn}(\ga_{g_1\dots g_4})^{\delta}{}_{\ga}$, we obtain
\beal
\widetilde{T}&=-\frac{14i}{135}\slsh D \mu
+\frac{8}{45}L\lambda+\frac{32}{45}L'\lambda
+\frac{136}{405}L_{(2)}(\ga^{(2)}\lambda)
-\frac{8}{45}L_{(4)}(\ga^{(4)}\lambda)\nn\\
&-\frac{16}{45}K_{(1)}(\ga^{(1)}\mu)
+\frac{2}{45}K_{(3)}(\ga^{(3)}\mu)
-\frac{1}{60}(\lambda\ga_{(3)}\lambda)(\ga_a^{(3)}\mu)\nn\\
\widetilde{K}^{(3)}_{}&=\frac{i}{540}\slsh D \mu
-\frac{1}{45}L \lambda -\frac{1}{180}L' \lambda +\frac{13}{1620}L_{(2)}(\ga^{(2)}\lambda)
+\frac{1}{180}L_{(4)}(\ga^{(4)}\lambda)\nn\\
&+\frac{1}{90}K_{(1)}(\ga^{(1)}\mu)
+\frac{1}{45}K_{(3)}(\ga^{(3)}\mu)
~.
\end{align}
{\it Vector-spinors}
Contracting the BI with the four independent structures
$\delta_a{}^e(\ga_g)_{\al\bn}(\ga^{g})_{\delta\ga}$,
$\delta_a{}^e(\ga_{g_1\dots g_5})_{\al\bn}(\ga^{g_1\dots g_5})_{\delta\ga}$, \break
$(\ga_a)_{\al\bn}(\ga^{e})_{\delta\ga}$,
$(\ga_{a g_1\dots g_4})_{\al\bn}(\ga^{eg_1\dots g_4})_{\delta\ga}$ and
projecting onto the gamma-traceless part, we obtain
\beal
\widetilde{T}_a&=-\frac{3i}{20}(\ga_a^{(1)} D_{(1)} \mu)-\frac{1}{5}L_{(2)}(\ga_{a}^{(2)}\lambda)
+\frac{2}{5}L_{(4)}(\ga_{a}^{(4)}\lambda)\nn\\
&+\frac{1}{5}K_{(1)}(\ga_{a}^{(1)}\mu)-\frac{3}{20}K_{(3)}(\ga_{a}^{(3)}\mu)
+\frac{3}{160}(\lambda\ga_{(3)}\lambda)(\ga_{a}^{(3)}\mu)\nn\\
\widetilde{K}^{(3)}_{a}&=
-\frac{i}{160}(\ga_a^{(1)} D_{(1)} \mu)-\frac{1}{40}L_{(2)}(\ga_a^{(2)}\lambda)+
\frac{1}{20}L_{(4)}(\ga_a^{(4)}\lambda)\nn\\
&+\frac{1}{40}K_{(1)}(\ga_a^{(1)}\mu)
+\frac{81}{1120}K_{(3)}(\ga_a^{(3)}\mu)
+\frac{1}{1280}(\lambda\ga_{(3)}\lambda)(\ga_a^{(3)}\mu)
~.
\end{align}
The notation in the equation above is a shorthand for the projection onto the
irreducible (gamma-traceless) vector-spinor part. Our conventions are
explained in appendix \ref{gammatraceless}.
{\it Two-form-spinors}
Contracting with the two independent structures
$(\ga_{[a})_{\al\bn}(\ga_{b]}{}^{e})^{\delta}{}_{\ga}$,
$(\ga_{ab}{}^{eg_1g_2})_{\al\bn}(\ga_{g_1g_2})^{\delta}{}_{\ga}$, we obtain
\beal
\widetilde{K}^{(3)}_{ab}&=
-\frac{1}{8}\widetilde{T}_{ab}-
\frac{1}{108}L_{(2)}(\ga_{ab}^{(2)}\lambda)+
\frac{1}{6}L_{(4)}(\ga_{ab}^{(4)}\lambda)
+\frac{1}{18}K_{(3)}(\ga_{ab}^{(3)}\mu)
~.
\end{align}
As in the previous case, the projections onto the gamma-traceless part of the
two-form-spinors are explained in appendix \ref{gammatraceless}.
{\it Three-form-spinors}
Contracting with the two independent structures
$\delta_{[a}{}^e(\ga_{bc]g_1g_2g_3})_{\al\bn}(\ga^{g_1g_2g_3})_{\delta\ga}$
and $(\ga_{abc}{}^{eg})_{\al\bn}(\ga_{g})_{\delta\ga}$, we obtain
\beal
\widetilde{K}^{(3)}_{abc}&=
-\frac{1}{84}K_{(3)}(\ga_{abc}^{(3)}\mu)
~.
\end{align}
{\it Four-form-spinors}
Contracting with
$(\ga_{abcd}{}^e)_{\al\bn}\delta^{\delta}{}_{\ga}$, we obtain
\beal
\widetilde{L}^{(4)}_{abcd{}}&=0~.
\end{align}
The solution of the (highly overdetermined) system of equations above is given in section
\ref{summary}. Note that the spinor equations for $(\widetilde{L}-\widetilde{L}')^{\al}$ and
$(\widetilde{L}-\widetilde{L}')_{\al}$ (equations
(\ref{rightspinorl},\ref{leftspinorl}), respectively) can
actually be solved for $(L-L')$, as we now show. Let us
parameterize $(L-L')$ as follows
\beal
L-L'=\frac{3}{2}(\mu\lambda)+me^{2\phi}
~,
\label{ktulu}
\end{align}
where $m$ is a massive constant and $\phi$ is some real scalar superfield
(possibly a constant) of canonical
mass dimension zero. Plugging (\ref{ktulu}) in (\ref{rightspinorl},\ref{leftspinorl})
and taking into account the action of the spinor derivative on $\lambda_{\al}, \mu^{\al}$,
we obtain
\beal
m({D}_{\al}\phi-\lambda_{\al})=0
\end{align}
and
\beal
m({D}^{\al}\phi-\mu^{\al})=0~.
\end{align}
These equations have two possible solutions: (a) $m=0$ or
(b) $m\neq 0$ and ${D}^{\al}\phi=\mu^{\al}$, ${D}_{\al}\phi=\lambda_{\al}$.
We shall see in section \ref{dimension2} that case (b) is that of Romans supergravity.
Moreover let us distinguish two further subcases: (a1) $m=0$, $L=-L'$ and
(a2) $m=0$, $L\neq -L'$.
As we show in the following, case (a1) is that of massless IIA
while (a2) is that of HLW supergravity.
Finally, the spinor derivatives of $K_a$ are computed by employing the identities
\beal
2D_{(\al}D_{\bn)}\lambda_{\ga}&=-T_{\al\bn}{}^ED_{E}\lambda_{\ga}-R_{\al\bn\ga}{}^{\delta}\lambda_{\delta}\nn\\
2D^{(\al}D^{\bn)}\mu^{\ga}&=-T^{\al\bn E}D_{E}\mu^{\ga}+\mu^{\delta}R^{\al\bn}{}_{\delta}{}^{\ga}
~.
\end{align}
The result is given in section \ref{summary}.
\subsection{Dimension-2 BI}
\label{dimension2}
Taking into account our
gauge-fixing for the supertorsion components, the first BI at dimension two reads
\beal
R_{[abc]}{}^d=0~.
\end{align}
The second BI at dimension two can be cast in the form
\beal
D_{\underline{\alpha}}\widetilde{T}_{ab}{}^{\underline{\delta}}=R_{ab\underline{\alpha}}{}^{\underline{\delta}}-2D_{[a}T_{b]\underline{\alpha}}{}^{\underline{\delta}}
-T_{ab}{}^{\underline{\epsilon}}T_{\underline{\epsilon}\underline{\alpha}}{}^{\underline{\delta}}
-2T_{\underline{\alpha}[a|}{}^{\underline{\epsilon}}T_{\underline{\epsilon}|b]}{}^{\underline{\delta}}-D_{\underline{\alpha}}(\ga_{[a}\widetilde{T}_{b]})^{\underline{\delta}}
-D_{\underline{\alpha}}(\ga_{ab}\widetilde{T}_{})^{\underline{\delta}}
~.
\label{poui}
\end{align}
This simply determines the spinor derivative of $\widetilde{T}_{ab}$,
provided the right-hand-side of the equation above
is consistent with the gamma-tracelessness of $\widetilde{T}_{ab}$.
In other words, the right-hand-side of (\ref{poui})
should be annihilated by $(\ga_a)^{\underline{\phantom{\alpha}}\!\!\!\gamma}{}_{\underline{\delta}}$. This condition
turns out to be equivalent to the system of
all equations-of-motion and Bianchi identities
for the bosonic fields! To see this in detail, let us
distinguish the following cases.
\vfill\break
$\bullet$ Case 1: $(\underline{\alpha}, \underline{\delta})=(\al, \de)$
For simplicity of presentation, here and in the remainder of this section
we shall focus on the bosonic terms, ignoring quadratic and quartic fermionic terms.
In this case, the condition for the
tracelessness of the right-hand-side of (\ref{poui}) can be written as
\beal
0=\ga^c\mathcal{A}_{bc}+\ga^{cde}\mathcal{A}_{bcde}+\ga^{cdefg}\mathcal{A}_{bcdefg}
~,
\end{align}
where the coefficients in the expansion above are given by
\beal
\mathcal{A}_{bc}&=
\eta_{bc}\Big(
-\frac{272}{25}L^2+\frac{72}{25}L L'-\frac{3i}{5}D^iK_i-\frac{36}{5}K^iK_i
-\frac{304}{45}L_{ij}L^{ij}-\frac{192}{5}K_{ijk}K^{ijk}-\frac{576}{5}L_{ijkl}L^{ijkl}
\Big)\nn\\
&+\frac{27i}{5}D_{b}K_c+\frac{3i}{5}D_cK_b-8K_bK_c
-\frac{16}{25}LL_{bc}-\frac{304}{25}L'L_{bc}-\frac{64}{9}L_b{}^iL_{ci}
+\frac{18i}{5}D^iK_{ibc}
\nn\\
&+\frac{144}{5}K^iK_{ibc}+72K_b{}^{ij}K_{cij}
+\frac{384}{5}L^{ij}L_{ijbc}-1536L_b{}^{ijk}L_{cijk}
+\frac{24}{5}\varepsilon_{bci_1\dots i_8}L^{i_1\dots i_4}L^{i_5\dots i_8}
-\frac{1}{2}R_{bc}
\nn\\
\mathcal{A}_{bcde}&=
\eta_{b[c}\Big(
\frac{3i}{5}D_dK_{e]}-\frac{24}{5}K^iK_{i|de]}
-\frac{192}{5}K^iK_{i|de]}-\frac{92}{25}L L_{|de]}
+\frac{52}{25}L'L_{|de]}\nn\\
&-\frac{512}{5}L^{ij}L_{ij|de]}
-\frac{32}{5}\varepsilon_{|de]i_1\dots i_8}L^{i_1\dots i_4}L^{i_5\dots i_8}\Big)
-\frac{1}{4}R_{b[cde]}\nn\\
&-\frac{12i}{5}D_{[b}K_{cde]}-\frac{48}{5}K_{[b}K_{cde]}
+\frac{1488}{25}LL_{bcde}-\frac{1008}{25}L'L_{bcde}
\nn\\
\mathcal{A}_{bcdefg}&=\eta_{b[c}\Big(
\frac{8i}{5}D_{d}K_{efg]}+\frac{32}{5}K_{|d}K_{efg]}-\frac{272}{25}LL_{|defg]}
-\frac{48}{25}L'L_{|defg]}
\Big)
~.
\end{align}
To arrive at these expressions we have Hodge-dualized the $\ga^{(7)}$ and $\ga^{(9)}$ terms.
We have also made use of the identity
\beal
\varepsilon_{bci_1\dots i_8}L^{i_1\dots i_4}L^{i_5\dots i_8}
(\ga_a{}^{bc})_{\al\bn}=16(\ga^{i_1\dots i_7})_{\al\bn}
L_{ai_1i_2i_3}L_{i_4\dots i_7}~.
\end{align}
$\bullet$ Case 2: $(\underline{\alpha}, \underline{\delta})=(\al, \!\!\begin{array}{c} \delta\\{} \end{array})$
In this case the tracelessness condition takes the form
\beal
0=\mathcal{B}_b
+\ga^{cd}\mathcal{B}_{bcd}+\ga^{cdef}\mathcal{B}_{bcdef}~,
\end{align}
where
\beal
\mathcal{B}_b&=
\frac{27i}{5}D_bL-\frac{308}{25}LK_b-\frac{272}{25}L'K_b
+\frac{58i}{15}D^iL_{ib}+\frac{232}{5}K^iL_{ib}
+\frac{2784}{5}K^{ijk}L_{ijkb}
\nn\\
\mathcal{B}_{bcd}&=
\eta_{b[c}\Big(
\frac{3i}{5}D_{d]}L+\frac{108}{25}LK_{|d]}+\frac{272}{25}L'K_{|d]}
-\frac{38i}{15}D^iL_{i|d]}-\frac{152}{5}K^iL_{i|d]}
-\frac{1824}{5}K^{ijk}L_{ijk|d]}
\Big)\nn\\
&+\frac{36}{25}LK_{cd}-\frac{816}{25}L'K_{cd}
-\frac{29i}{5}D_{[b}L_{cd]}+\frac{156}{5}D^iL_{ibcd}
+\frac{1248}{5}K^iL_{ibcd}
-\frac{26}{5}\varepsilon_{bcdi_1\dots i_7}K^{i_1i_2i_3}L^{i_4\dots i_7}
\nn\\
\mathcal{B}_{bcdef}&=
\eta_{b[c}\Big(
-\frac{212}{25}LK_{def}+\frac{272}{25}L'K_{def}
+\frac{19i}{5}D_{|d}L_{ef]}\nn\\
&-\frac{12i}{5}D^iL_{i|def]}
-\frac{96}{5}K^iL_{i|def]}
+\frac{2}{5}\varepsilon_{|def]i_1\dots i_7}K^{i_1i_2i_3}L^{i_4\dots i_7}\Big)\nn\\
&+\frac{52}{3}\Big(
L_{[bc}K_{def]}
-\frac{3}{4}iD_{[b}L_{cdef]}
-3K_{[b}L_{cdef]}
\Big)\nn\\
&-\frac{1}{30}\varepsilon_{bcdefi_1\dots i_5}\Big(
L^{i_1i_2}K^{i_3i_4i_5}
-\frac{3i}{4}D^{i_1}L^{i_2\dots i_4}
-3K^{i_1}L^{i_2\dots i_4}
\Big)
~.
\end{align}
As in the previous case, we have Hodge-dualized the $\ga^{(6)}$, $\ga^{(8)}$, $\ga^{(10)}$
contributions. Also, we have taken into
account the identity
\beal
\frac{1}{24}\varepsilon_{cdei_1\dots i_7}(\ga_b{}^{cde})_{\al}{}^{\delta}K^{i_1i_2i_3}L^{i_4\dots i_7}
=-(\ga_{i_1\dots i_6})_{\al}{}^{\delta}\Big(L_b{}^{i_1i_2i_3}K^{i_4i_5i_6}+\frac{3}{4}
K_b{}^{i_1i_2}L^{i_3\dots i_6}
\Big)~.
\end{align}
$\bullet$ Case 3: $(\underline{\alpha}, \underline{\delta})=(\! \begin{array}{c} \alpha\\{} \end{array}, \de)$
This is similar to the previous case. The gamma-tracelessness condition is
of the form
\beal
0=\mathcal{C}_b
+\ga^{cd}\mathcal{C}_{bcd}+\ga^{cdef}\mathcal{C}_{bcdef}~.
\end{align}
The coefficients $\mathcal{C}_b$,
$\mathcal{C}_{bcd}$ and $\mathcal{C}_{bcdef}$ can be obtained from
$\mathcal{B}_b$,
$\mathcal{B}_{bcd}$ and $\mathcal{B}_{bcdef}$ respectively,
by making the substitutions $L\leftrightarrow L'$,
$K_{(3)}\rightarrow -K_{(3)}$, $L_{4}\rightarrow -L_{4}$ and
$\varepsilon_{(10)}\rightarrow -\varepsilon_{(10)}$.
$\bullet$ Case 4: $(\underline{\alpha}, \underline{\delta})=(\! \begin{array}{c} \alpha\\{} \end{array}, \!\!\begin{array}{c} \delta\\{} \end{array})$
This is related to case 1 by parity inversion. The gamma-tracelessness
condition takes the form
\beal
0=\ga^c\mathcal{D}_{bc}+\ga^{cde}\mathcal{D}_{bcde}+\ga^{cdefg}\mathcal{D}_{bcdefg}
~.
\end{align}
The coefficients $\mathcal{D}_{bc}$, $\mathcal{D}_{bcde}$ and
$\mathcal{D}_{bcdefg}$ above can be obtained from
$\mathcal{A}_{bc}$, $\mathcal{A}_{bcde}$ and
$\mathcal{A}_{bcdefg}$ respectively, by making the
substitutions described in the previous case.
In conclusion, the gamma-tracelessness condition is equivalent to
\beal
\mathcal{A}=\mathcal{B}=\mathcal{C}=\mathcal{D}=0~.
\label{gt}
\end{align}
It is straightforward to recognize that in the case (b)
at the end of
the previous subsection
(i.e. for $L-L'=\frac{3}{2}(\mu\lambda)+me^{2\phi}$, $L+L'=0$ and
$\lambda_{\al}=D_{\al}\phi$, $\mu^{\al}=D^{\al}\phi$,
equations (\ref{gt}) reduce to those of Romans supergravity.
In case (a1) (i.e. for $L-L'=\frac{3}{2}(\mu\lambda)$, $L+L'=0$)
we can see, using the results of the preceding analysis
of the BI's, that
the super-one-form $\Lambda_A$ defined by
\beal
\Lambda_{\al}&:=\lambda_{\al}\nn\\
\Lambda^{\al}&:=\mu^{\al}\nn\\
\Lambda_{a}&:=-2iK_{a}
~,
\end{align}
is closed:
\beal
D\Lambda=0~.
\label{clos}
\end{align}
It follows that there exists a scalar superfield $\phi$
such that $\Lambda=D\phi$. In particular,
$\lambda_{\al}=D_{\al}\phi$, $\mu^{\al}=D^{\al}\phi$ and $K_a=\frac{i}{2}D_a\phi$. It is now
straightforward to see that in this case equations (\ref{gt}) reduce
to those of massless type IIA supergravity \cite{iia}. Note that (\ref{clos})
also holds in the case of Romans supergravity.
In the case where spacetime $\mathcal{M}$ is not simply connected,
it may be that $\Lambda=D\phi+\psi$, where $\psi$ is
closed but not exact. For example, if $\mathcal{M}$
contains an $S^1$ we may take $\psi=m dz$, where
$z$ is the $S^1$ coordinate and $m$ is a mass parameter.
Upon compactification on $S^1$, this would amount to
a Scherk-Schwarz reduction. The possibility for this
topological modification is the remnant in ten dimensions of
the freedom to modify ordinary eleven-dimensional
supergravity to MM-theory.
Finally, in case (a2)
($L-L'=\frac{3}{2}(\mu\lambda)$, $L+L'\neq 0$), equations
(\ref{gt}) reduce to those of HLW supergravity,
summarized in section \ref{summary}.
As we have already remarked at the end of section \ref{dimension1},
in this case there cannot exist a $\phi$ such that
$D_{\al}\phi=\lambda_{\al}$, $D^{\al}\phi=\mu^{\al}$.
Note that if we parameterize
\beal
L&=\frac{3}{2}\Big(\Phi+\frac{1}{2}(\mu\lambda)\Big)\nn\\
L'&=\frac{3}{2}\Big(\Phi-\frac{1}{2}(\mu\lambda)\Big)~,
\end{align}
where the massless IIA limit (case (a1) above) is reached at $\Phi\rightarrow 0$,
equations (\ref{gt}) imply in particular that $\Phi=$constant.
In equations (\ref{hl1}, \ref{hl2}) of section \ref{summary} we have set $\Phi=m$.
\subsection{Dimension-$\frac{5}{2}$ BI}
It can be seen that the dimension-$\frac{5}{2}$ BI does not introduce any new constraints,
other than determining the action of the spinor derivative
on the dimension-2 component of the supercurvature. Explicitly:
\beal
D_{\underline{\alpha}}R_{abcd}
=2D_{[a|}R_{\underline{\alpha}|b]cd}-T_{ab}{}^{\underline{\epsilon}} R_{\underline{\epsilon} \underline{\alpha} cd}
+2T_{[a|\underline{\alpha}}{}^{\underline{\epsilon}} R_{\underline{\epsilon}|b]cd} ~.
\end{align}
\section{Summary}
\label{summary}
Here we summarize the result of the
analysis of the BI's carried out in the preceding section.
At each order in canonical mass dimension we give the solution for the components
of the torsion and curvature, as well as the action of the spinor
derivative on the various superfields at that dimension\footnote{
We have checked that our formul{\ae}
are compatible with overlapping literature.
For example, in order to compare with \cite{cede},
where the supertorsion components
were given up to dimension-$\frac{1}{2}$, one should make the
following identifications: $\Lambda_{\al}\rightarrow (\mu^{\al}, \lambda_{\al})$ and
$$
(\ga_{11})_{\al}{}^{\bn}
\rightarrow \left(\begin{array}{cc}
-\de_{\al}{}^{\bn} & 0\\
0& \de^{\al}{}_{\bn}
\end{array}\right)~.
$$
One also needs to take into account the
identity
$$
(\ga^{ab})_{(\al}{}^{\ga}(\ga_{ab}\lambda)_{\bn)}
=-10\de_{(\al}^{\ga}\lambda_{\bn)}+4\ga^a_{\al\bn}(\ga_a\lambda)^{\ga}~.
$$
}. At dimension two we have chosen to include the
bosonic part of the equations-of-motion and Bianchi identities for the
bosonic fields of the relatively less-known HLW supergravity.
\subsection*{Dimension $0$}
\beal
T_{\al \bn}{}^c&=-i(\ga^c)_{\al\bn}\nn\\
T^{\al \bn c}&=-i(\ga^c)^{\al\bn}\nn\\
T_\al{}^{\bn c}&=0
~.
\label{tz}
\end{align}
\subsection*{Dimension $\frac{1}{2}$}
\beal
T_{\al b}{}^c&=T^{\al}{}_{b}{}^c=0\nn\\
T_{\al\bn}{}^\ga&=-4\Big\{
\lambda_{(\al}
\de_{\bn)}{}^{\ga}-\frac{1}{2}(\ga^e)_{\al\bn}(\ga_e\lambda)^{\ga} \Big\} \nn\\
T^{\al\bn}{}_\ga&=-4\Big\{
\mu^{(\al}
\de^{\bn)}{}_{\ga}-\frac{1}{2}(\ga_e)^{\al\bn}(\ga^e\mu)_{\ga}\Big\} \nn\\
T_{\al\bn\ga}&=(\ga^e)_{\al\bn}(\ga_e\mu)_\ga\nn\\
T_\al{}^{\bn\ga}&=-\frac{1}{2}\Big\{
\de_\al{}^\ga\mu^\bn -\frac{1}{2}(\ga^{ef})_\al{}^\ga(\ga_{ef}\mu)^\bn
\Big\}\nn\\
T^\al{}_{\bn\ga}&=-\frac{1}{2} \Big\{
\de^\al{}_\ga\lambda_\bn -\frac{1}{2}(\ga^{ef})^\al{}_\ga(\ga_{ef}\lambda)_\bn
\Big\}\nn\\
T^{\al\bn\ga}&=(\ga_e)^{\al\bn}(\ga^e\lambda)^\ga~.
\end{align}
\subsection*{Dimension $1$}
Torsion:
\beal
T_{ab}{}^c&=0\nn\\
T_{a\al}{}^\bn&= H^1_{fgh}
(\ga_a{}^{fgh})_\al{}^\bn +H^2_{afg}(\ga^{fg})_\al{}^\bn
\nn\\
T_a{}^{\al}{}_{\bn}&=
H^{\prime 1}_{fgh}(\ga_a{}^{fgh})^\al{}_\bn
+H^{\prime 2}_{afg}(\ga^{fg})^\al{}_\bn
\nn\\
T_{a\al\bn}&=S
(\ga_a)_{\al\bn} +F^1_{ef}(\ga_a{}^{ef})_{\al\bn}
+F^2_{ae}(\ga^{e})_{\al\bn}
+G^1_{efgh} (\ga_a{}^{efgh})_{\al\bn}
+G^2_{aefg}(\ga^{efg})_{\al\bn}
\nn\\
T_a{}^{\al\bn}&=
-S(\ga_a)^{\al\bn} +F^1_{ef}(\ga_a{}^{ef})^{\al\bn}
+F^2_{ae}(\ga^{e})^{\al\bn}
-G^1_{efgh} (\ga_a{}^{efgh})^{\al\bn}
-G^2_{aefg}(\ga^{efg})^{\al\bn}
~,
\end{align}
Curvature:
\beal
R_{\al\bn} {}_{cd}&=2iH^1_{efg}(\ga_{cd}{}^{efg})_{\al\bn}
+4iH^2_{cde}(\ga^{e})_{\al\bn}\nn\\
R^{\al\bn} {}_{cd}&=2iH^{\prime 1}_{efg}(\ga_{cd}{}^{efg})^{\al\bn}
+4iH^{\prime 2}_{cde}(\ga^{e})^{\al\bn}\nn\\
R_{\al}{}^{\bn} {}_{cd}&=-2i\Big\{S
(\ga_{cd})_{\al}{}^\bn +F^1_{ef}(\ga_{cd}{}^{ef})_{\al}{}^\bn
+F^2_{cd} \de_{\al}{}^\bn +G^1_{efgh}
(\ga_{cd}{}^{efgh})_{\al}{}^\bn
+3G^2_{cdfg}(\ga^{fg})_{\al}{}^\bn
\Big\}~.
\end{align}
Spinor derivatives:
\beal
D_{\al}\lambda_\bn&=K_e(\ga^e)_{\al\bn}+K_{efg}(\ga^{efg})_{\al\bn}\nn\\
D_\al\mu^\bn&=L~\de_\al{}^\bn+L_{ef}(\ga^{ef})_\al{}^\bn
+L_{efgh}(\ga^{efgh})_\al{}^\bn\nn\\
D^{\al}\lambda_\bn&={L'}~\de^\al{}_\bn+L^{ef}(\ga_{ef})^\al{}_\bn
-L^{efgh}(\ga_{efgh})^\al{}_\bn\nn\\
D^\al\mu^\bn&={K}^{e}(\ga_e)^{\al\bn}
-{K}^{efg}(\ga_{efg})^{\al\bn}~,
\end{align}
where
\beal
S&=
-\frac{2i}{5}(L-{L'})+\frac{3i}{5}(\mu\lambda)\nn\\
F_{ab}^1&=-\frac{2i}{3}L_{ab}\nn\\
F_{ab}^2&=\frac{8i}{3}L_{ab}+i(\mu\ga_{ab}\lambda)\nn\\
H_{abc}^1&=iK_{abc}\nn\\
H_{abc}^2&=-\frac{i}{8}(\lambda\ga_{abc}\lambda)-\frac{i}{8}(\mu\ga_{abc}\mu)\nn\\
{H}_{abc}^{\prime 1}&=-iK_{abc}\nn\\
{H}_{abc}^{\prime 2}&=-\frac{i}{8}(\lambda\ga_{abc}\lambda)-\frac{i}{8}(\mu\ga_{abc}\mu)\nn\\
G^1_{abcd}&=0\nn\\
G^2_{abcd}&=8iL_{abcd}~.
\end{align}
For the scalar fields $L$, $L'$, we have the following
cases\footnote{In fact, massless IIA is the massless limit of both
Romans and HLW supergravity and need not be presented as a separate case.}
\beal
\mbox{Massless IIA:}& \left\{
\begin{array}{l}
L=\frac{3}{4}(\mu\lambda)\\
L'=-\frac{3}{4}(\mu\lambda)
\end{array}\right.\nn\\
\mbox{Romans:}&\left\{
\begin{array}{l}
L=\frac{1}{2}me^{2\phi}+\frac{3}{4}(\mu\lambda)\\
L'=-\frac{1}{2}me^{2\phi}-\frac{3}{4}(\mu\lambda)
\end{array}\right.\nn\\
\mbox{HLW:}& \left\{
\begin{array}{l}
L=\frac{3}{2}m+\frac{3}{4}(\mu\lambda)\\
L'=\frac{3}{2}m-\frac{3}{4}(\mu\lambda)
\end{array}\right.
~,
\label{llprime}
\end{align}
where in both Romans and massless IIA supergravities we have
(modulo the possibility for the topological modification
of massless IIA explained at the end of section \ref{dimension2})
$D_{\al}\phi=\lambda_{\al}$,
$D^{\al}\phi=\mu^{\al}$. In HLW supergravity, $\nexists$ $\phi$:
$D_{\al}\phi=\lambda_{\al}$, $D^{\al}\phi=\mu^{\al}$.
\subsection*{Dimension $\frac{3}{2}$}
{\it Torsion}
\beal
T_{ab}=\widetilde{T}_{ab}
+{}\ga_{[a}\widetilde{T}_{b]}{}{}
+{}\ga_{ab}\widetilde{T}
~,
\end{align}
where for the right-handed spinors $\widetilde{T}^{\al}$, $\widetilde{T}_a^{\al}$ we have
(suppressing spinor indices)
\beal
\widetilde{T}&=\frac{272}{225}L\mu-\frac{8}{25}L'\mu
+\frac{8}{9}L_{(2)}(\ga^{(2)}\mu)
+\frac{8}{45}L_{(4)}(\ga^{(4)}\mu)\nn\\
&+\frac{8}{9}K_{(1)}(\ga^{(1)}\lambda)
-\frac{16}{45}K_{(3)}(\ga^{(3)}\lambda)
-\frac{11}{450}(\mu\ga_{(3)}\mu)(\ga_a^{(3)}\lambda)
\nn\\
\widetilde{T}_a&=-\frac{3i}{20}(\ga_a^{(1)} D_{(1)} \mu)-\frac{1}{5}L_{(2)}(\ga_{a}^{(2)}\lambda)
+\frac{2}{5}L_{(4)}(\ga_{a}^{(4)}\lambda)\nn\\
&+\frac{1}{5}K_{(1)}(\ga_{a}^{(1)}\mu)-\frac{3}{20}K_{(3)}(\ga_{a}^{(3)}\mu)
+\frac{3}{160}(\lambda\ga_{(3)}\lambda)(\ga_{a}^{(3)}\mu)
\label{mi}
\end{align}
and similarly for the left-handed spinors $\widetilde{T}_{\al}$, $\widetilde{T}_{a\al}$,
\beal
\widetilde{T}&=
-\frac{8}{25}L\lambda+\frac{272}{225}L'\lambda
+\frac{8}{9}L_{(2)}(\ga^{(2)}\lambda)
-\frac{8}{45}L_{(4)}(\ga^{(4)}\lambda)\nn\\
&+\frac{8}{9}K_{(1)}(\ga^{(1)}\mu)
+\frac{16}{45}K_{(3)}(\ga^{(3)}\mu)
-\frac{11}{450}(\lambda\ga_{(3)}\lambda)(\ga_a^{(3)}\mu)\nn\\
\widetilde{T}_a&=-\frac{3i}{20}(\ga_a^{(1)} D_{(1)} \lambda)-\frac{1}{5}L_{(2)}(\ga_{a}^{(2)}\mu)
-\frac{2}{5}L_{(4)}(\ga_{a}^{(4)}\mu)\nn\\
&+\frac{1}{5}K_{(1)}(\ga_{a}^{(1)}\lambda)+\frac{3}{20}K_{(3)}(\ga_{a}^{(3)}\lambda)
+\frac{3}{160}(\mu\ga_{(3)}\mu)(\ga_{a}^{(3)}\lambda)
~.
\label{ni}
\end{align}
No confusion should arise from the slight abuse of notation, as it is
immediately clear by the right-hand-sides of equations (\ref{mi},\ref{ni})
what the chiralities of the spinors $\widetilde{T}$ are in each case.
{\it Curvature}
\beal
R_{\al bcd}&=\frac{i}{2}\big(
\ga_b T_{cd} +\ga_c T_{bd}-\ga_d T_{bc}
\big)_{\al}\nn\\
R^{\al}{}_{ bcd}&=\frac{i}{2}\big(
\ga_b T_{cd} +\ga_c T_{bd}-\ga_d T_{bc}
\big)^{\al}
~.
\end{align}
{\it Spinor derivatives}
\beal
D{}L&=\widetilde{L}{}\nn\\
D{}L'&=\widetilde{L}^{\prime}\nn\\
D{}L_{ab}&=\widetilde{L}^{(2)}_{ab\al}
+{}\ga_{[a}\widetilde{L}^{(2)}_{b]}{}{}
+{}\ga_{ab}\widetilde{L}^{(2)}{}{}\nn\\
D{}L_{abcd}&=\widetilde{L}^{(4)}_{abcd\al}
+{}\ga_{[a}\widetilde{L}^{(4)}_{bcd]}{}{}
+{}\ga_{[ab}\widetilde{L}^{(4)}_{cd]}{}{}
+{}\ga_{abc}\widetilde{L}^{(4)}_{d]}{}{}
+{}\ga_{abc}\widetilde{L}^{(4)}{}{}
~,
\end{align}
\beal
D{}K_{a}&=\widetilde{K}^{(1)}_{a\al}
+{}\ga_{a}\widetilde{K}^{(1)}{}{}\nn\\
D{}K_{abc}&=\widetilde{K}^{(3)}_{abc\al}
+{}\ga_{[a}\widetilde{K}^{(3)}_{bc]}{}{}
+{}\ga_{[ab}\widetilde{K}^{(3)}_{c]}{}{}
+{}\ga_{abc}\widetilde{K}^{(3)}{}{}
~,
\end{align}
where
\beal
\widetilde{L}^{}&=\widetilde{L}^{\prime{}}+
2L \mu-\frac{7}{2}L' \mu -\frac{3}{2}L_{(2)}(\ga^{(2)}\mu)
+\frac{3}{2}L_{(4)}(\ga^{(4)}\mu)\nn\\
&+\frac{3}{2}K_{(1)}(\ga^{(1)}\lambda)
-\frac{3}{2}K_{(3)}(\ga^{(3)}\lambda)
-\frac{1}{32}(\mu\ga_{(3)}\mu)(\ga^{(3)}\lambda)~,
\label{rightspinorl}
\end{align}
\beal
\widetilde{L}^{(2){}}_{ab}&=
\frac{3}{16}\widetilde{T}_{ab}-\frac{1}{24}L_{(2)}(\ga_{ab}^{(2)}\mu)
+\frac{1}{4}L_{(4)}(\ga_{ab}^{(4)}\mu)
+\frac{1}{8}K_{(3)}(\ga_{ab}^{(3)}\lambda)\nn\\
\widetilde{L}^{(2){}}_{a}&=
-\frac{9i}{320}(\ga_a^{(1)} D_{(1)} \mu)-
\frac{21}{80}L_{(2)}(\ga_a^{(2)}\lambda)-
\frac{3}{40}L_{(4)}(\ga_a^{(4)}\lambda)\nn\\
&+\frac{3}{16}K_{(1)}(\ga_a^{(1)}\mu)
+\frac{63}{320}K_{(3)}(\ga_a^{(3)}\mu)
+\frac{9}{2560}(\lambda\ga_{(3)}\lambda)(\ga_a^{(3)}\mu)\nn\\
\widetilde{L}^{(2){}}_{}&=
-\frac{11}{150}L \mu -\frac{27}{200}L' \mu+\frac{7}{120}L_{(2)}(\ga^{(2)}\mu)
+\frac{1}{120}L_{(4)}(\ga^{(4)}\mu)\nn\\
&-\frac{7}{120}K_{(1)}(\ga^{(1)}\lambda)
-\frac{1}{24}K_{(3)}(\ga^{(3)}\lambda)
+\frac{1}{9600}(\mu\ga_{(3)}\mu)(\ga^{(3)}\lambda)~,
\end{align}
\vfill\break
\beal
\widetilde{L}^{(4)}_{abcd{}}&=0\nn\\
\widetilde{L}^{(4)}_{abc{}}&=
-\frac{1}{168}L_{(4)}(\ga_{abc}^{(4)}\lambda)
-\frac{1}{336}K_{(3)}(\ga_{abc}^{(3)}\mu)
\nn\\
\widetilde{L}^{(4)}_{ab{}}&=-
\frac{1}{32}\widetilde{T}_{ab}+\frac{1}{432}L_{(2)}(\ga_{ab}^{(2)}\mu)
-\frac{13}{360}L_{(4)}(\ga_{ab}^{(4)}\mu)-\frac{1}{144}K_{(3)}(\ga_{ab}^{(3)}\lambda)\nn\\
\widetilde{L}^{(4)}_{a{}}&=
-\frac{i}{1920}(\ga_a^{(1)} D_{(1)} \mu)-
\frac{1}{480}L_{(2)}(\ga_a^{(2)}\lambda)+
\frac{31}{1680}L_{(4)}(\ga_a^{(4)}\lambda)\nn\\
&-\frac{1}{480}K_{(1)}(\ga_a^{(1)}\mu)
+\frac{11}{4480}K_{(3)}(\ga_a^{(3)}\mu)
+\frac{1}{15360}(\lambda\ga_{(3)}\lambda)(\ga_a^{(3)}\mu)\nn\\
\widetilde{L}^{(4)}_{{}}&=-\frac{1}{900}L \mu +\frac{11}{2400}L' \mu
-\frac{1}{4320}L_{(2)}(\ga^{(2)}\mu)
-\frac{31}{10080}L_{(4)}(\ga^{(4)}\mu)
\nn\\
&-\frac{1}{1440}K_{(1)}(\ga^{(1)}\lambda)
-\frac{1}{1440}K_{(3)}(\ga^{(3)}\lambda)
+\frac{7}{115200}(\mu\ga_{(3)}\mu)(\ga^{(3)}\lambda)
\end{align}
and
\beal
\widetilde{K}^{(1)}_{a}&=
\frac{i}{20}(\ga_a^{(1)} D_{(1)} \mu)+\frac{1}{5}L_{(2)}(\ga_a^{(2)}\lambda)-
\frac{2}{5}L_{(4)}(\ga_a^{(4)}\lambda)\nn\\
&-\frac{3}{20}K_{(3)}(\ga_a^{(3)}\mu)
-\frac{1}{160}(\lambda\ga_{(3)}\lambda)(\ga_a^{(3)}\mu)\nn\\
\widetilde{K}^{(1)}&=-\frac{1}{25}L \mu+\frac{1}{25}L' \mu
-\frac{2}{15}L_{(2)}(\ga^{(2)}\mu)
-\frac{2}{5}L_{(4)}(\ga^{(4)}\mu)\nn\\
&-\frac{3}{5}K_{(1)}(\ga^{(1)}\lambda)
-\frac{1}{5}K_{(3)}(\ga^{(3)}\lambda)
+\frac{1}{100}(\mu\ga_{(3)}\mu)(\ga^{(3)}\lambda)~,
\end{align}
\beal
\widetilde{K}^{(3)}_{abc}&=
-\frac{1}{84}K_{(3)}(\ga_{abc}^{(3)}\mu)\nn\\
\widetilde{K}^{(3)}_{ab}&=
\frac{1}{8}\widetilde{T}_{ab}+\frac{1}{108}L_{(2)}(\ga_{ab}^{(2)}\mu)
+\frac{1}{6}L_{(4)}(\ga_{ab}^{(4)}\mu)+\frac{1}{18}K_{(3)}(\ga_{ab}^{(3)}\lambda)\nn\\
\widetilde{K}^{(3)}_{a}&=
-\frac{i}{160}(\ga_a^{(1)} D_{(1)} \mu)-\frac{1}{40}L_{(2)}(\ga_a^{(2)}\lambda)+
\frac{1}{20}L_{(4)}(\ga_a^{(4)}\lambda)\nn\\
&+\frac{1}{40}K_{(1)}(\ga_a^{(1)}\mu)
+\frac{81}{1120}K_{(3)}(\ga_a^{(3)}\mu)
+\frac{1}{1280}(\lambda\ga_{(3)}\lambda)(\ga_a^{(3)}\mu)\nn\\
\widetilde{K}^{(3)}_{}&=\frac{13}{900}L \mu+\frac{1}{75}L' \mu
+\frac{1}{540}L_{(2)}(\ga^{(2)}\mu)
+\frac{1}{180}L_{(4)}(\ga^{(4)}\mu)\nn\\
&+\frac{1}{90}K_{(1)}(\ga^{(1)}\lambda)
+\frac{1}{60}K_{(3)}(\ga^{(3)}\lambda)
-\frac{1}{7200}(\mu\ga_{(3)}\mu)(\ga^{(3)}\lambda)
~,
\end{align}
for the right-handed spinors $\widetilde{L}^{\al}$,
$\widetilde{L}^{(2)\al}_{ab}$, etc.
Similarly, for the left-handed spinors we have
\beal
\widetilde{L}^{{}}&=\widetilde{L}^{\prime}
+\frac{7}{2}L \lambda -2L' \lambda+\frac{3}{2}L_{(2)}(\ga^{(2)}\lambda)
+\frac{3}{2}L_{(4)}(\ga^{(4)}\lambda)\nn\\
&-\frac{3}{2}K_{(1)}(\ga^{(1)}\mu)
-\frac{3}{2}K_{(3)}(\ga^{(3)}\mu)
+\frac{1}{32}(\lambda\ga_{(3)}\lambda)(\ga^{(3)}\mu)~,
\label{leftspinorl}
\end{align}
\beal
\widetilde{L}^{(2){}}_{ab}&=
\frac{3}{16}\widetilde{T}_{ab}
-\frac{1}{24}L_{(2)}(\ga_{ab}^{(2)}\lambda)-\frac{1}{4}L_{(4)}(\ga_{ab}^{(4)}\lambda)
-\frac{1}{8}K_{(3)}(\ga_{ab}^{(3)}\mu)\nn\\
\widetilde{L}^{(2){}}_{a}&=
-\frac{9i}{320}(\ga_a^{(1)} D_{(1)} \lambda)-
\frac{21}{80}L_{(2)}(\ga_a^{(2)}\mu)+
\frac{3}{40}L_{(4)}(\ga_a^{(4)}\mu)\nn\\
&+\frac{3}{16}K_{(1)}(\ga_a^{(1)}\lambda)
-\frac{63}{320}K_{(3)}(\ga_a^{(3)}\lambda)
+\frac{9}{2560}(\mu\ga_{(3)}\mu)(\ga_a^{(3)}\lambda)\nn\\
\widetilde{L}^{(2){}}_{}&=
-\frac{27}{200}L \lambda -\frac{11}{150}L' \lambda +\frac{7}{120}L_{(2)}(\ga^{(2)}\lambda)
-\frac{1}{120}L_{(4)}(\ga^{(4)}\lambda)\nn\\
&-\frac{7}{120}K_{(1)}(\ga^{(1)}\mu)
+\frac{1}{24}K_{(3)}(\ga^{(3)}\mu)
+\frac{1}{9600}(\lambda\ga_{(3)}\lambda)(\ga^{(3)}\mu)~,
\end{align}
\beal
\widetilde{L}^{(4)}_{abcd{}}&=0\nn\\
\widetilde{L}^{(4)}_{abc{}}&=-\frac{1}{168}L_{(4)}(\ga_{abc}^{(4)}\lambda)
-\frac{1}{336}K_{(3)}(\ga_{abc}^{(3)}\lambda)
\nn\\
\widetilde{L}^{(4)}_{ab{}}&=
\frac{1}{32}\widetilde{T}_{ab}
-\frac{1}{432}L_{(2)}(\ga_{ab}^{(2)}\lambda)
-\frac{13}{360}L_{(4)}(\ga_{ab}^{(4)}\lambda)
-\frac{1}{144}K_{(3)}(\ga_{ab}^{(3)}\mu)\nn\\
\widetilde{L}^{(4)}_{a{}}&=
\frac{i}{1920}(\ga_a^{(1)} D_{(1)} \lambda)+
\frac{1}{480}L_{(2)}(\ga_a^{(2)}\mu)+
\frac{31}{1680}L_{(4)}(\ga_a^{(4)}\mu)\nn\\
&+\frac{1}{480}K_{(1)}(\ga_a^{(1)}\lambda)
+\frac{11}{4480}K_{(3)}(\ga_a^{(3)}\lambda)
-\frac{1}{15360}(\mu\ga_{(3)}\mu)(\ga_a^{(3)}\lambda)\nn\\
\widetilde{L}^{(4)}_{{}}&=
-\frac{11}{2400}L \lambda +\frac{1}{900}L' \lambda +\frac{1}{4320}L_{(2)}(\ga^{(2)}\lambda)
-\frac{31}{10080}L_{(4)}(\ga^{(4)}\lambda)\nn\\
&+\frac{1}{1440}K_{(1)}(\ga^{(1)}\mu)
-\frac{1}{1440}K_{(3)}(\ga^{(3)}\mu)
-\frac{7}{115200}(\lambda\ga_{(3)}\lambda)(\ga^{(3)}\mu)
\end{align}
and
\beal
\widetilde{K}^{(1)}_{a}&=
\frac{i}{20}(\ga_a^{(1)} D_{(1)} \lambda)+\frac{1}{5}L_{(2)}(\ga_{a}^{(2)}\mu)
+\frac{2}{5}L_{(4)}(\ga_{a}^{(4)}\mu)\nn\\
&+\frac{3}{20}K_{(3)}(\ga_{a}^{(3)}\lambda)
-\frac{1}{160}(\mu\ga_{(3)}\mu)(\ga_{a}^{(3)}\lambda)
\nn\\
\widetilde{K}^{(1)}&=
\frac{1}{25}L \lambda -\frac{1}{25}L' \lambda -\frac{2}{15}L_{(2)}(\ga^{(2)}\lambda)
+\frac{2}{5}L_{(4)}(\ga^{(4)}\lambda)\nn\\
&-\frac{3}{5}K_{(1)}(\ga^{(1)}\mu)
+\frac{1}{5}K_{(3)}(\ga^{(3)}\mu)
+\frac{1}{100}(\lambda\ga_{(3)}\lambda)(\ga^{(3)}\mu)~,
\end{align}
\beal
\widetilde{K}^{(3)}_{abc}&=-\frac{1}{84}K_{(3)}(\ga_{abc}^{(3)}\lambda)\nn\\
\widetilde{K}^{(3)}_{ab}&=
-\frac{1}{8}\widetilde{T}_{ab}-
\frac{1}{108}L_{(2)}(\ga_{ab}^{(2)}\lambda)+
\frac{1}{6}L_{(4)}(\ga_{ab}^{(4)}\lambda)
+\frac{1}{18}K_{(3)}(\ga_{ab}^{(3)}\mu)\nn\\
\widetilde{K}^{(3)}_{a}&=
\frac{i}{160}(\ga_a^{(1)} D_{(1)} \lambda)+\frac{1}{40}L_{(2)}(\ga_{a}^{(2)}\mu)
+\frac{1}{20}L_{(4)}(\ga_{a}^{(4)}\mu)\nn\\
&-\frac{1}{40}K_{(1)}(\ga_{a}^{(1)}\lambda)+\frac{81}{1120}K_{(3)}(\ga_{a}^{(3)}\lambda)
-\frac{1}{1280}(\mu\ga_{(3)}\mu)(\ga_{a}^{(3)}\lambda)
\nn\\
\widetilde{K}^{(3)}_{}&=
-\frac{1}{75}L \lambda -\frac{13}{900}L' \lambda -\frac{1}{540}L_{(2)}(\ga^{(2)}\lambda)
+\frac{1}{180}L_{(4)}(\ga^{(4)}\lambda)\nn\\
&-\frac{1}{90}K_{(1)}(\ga^{(1)}\mu)
+\frac{1}{60}K_{(3)}(\ga^{(3)}\mu)
+\frac{1}{7200}(\lambda\ga_{(3)}\lambda)(\ga^{(3)}\mu)
~.
\end{align}
{\it Fermionic equations-of-motion}
\beal
\ga^bT_{ab}&=-4\widetilde{T}_a-9\ga_a\widetilde{T}~,
\end{align}
where $\widetilde{T}$, $\widetilde{T}_a$ are given in (\ref{mi}, \ref{ni}) and
\beal
i\slsh D \lambda&=-\frac{24}{5}(L-L') \mu-\frac{16}{3}L_{(2)}(\ga^{(2)}\mu)\nn\\
&-12K_{(1)}(\ga^{(1)}\lambda)
+3K_{(3)}(\ga^{(3)}\lambda)
+\frac{3}{40}(\mu\ga_{(3)}\mu)(\ga^{(3)}\lambda)\nn\\
i\slsh D \mu&=\frac{24}{5}(L-L') \lambda-\frac{16}{3}L_{(2)}(\ga^{(2)}\lambda)\nn\\
&-12K_{(1)}(\ga^{(1)}\mu)
-3K_{(3)}(\ga^{(3)}\mu)
+\frac{3}{40}(\lambda\ga_{(3)}\lambda)(\ga^{(3)}\mu)~.
\end{align}
\subsection*{Dimension ${2}$}
{\it Curvature}
\beal
R_{[abc]d}=R_{a[bcd]}=0~.
\end{align}
{\it Spinor derivatives}
\beal
D_{\underline{\alpha}}T_{ab}{}^{\underline{\phantom{\alpha}}\!\!\!\beta}=R_{ab\underline{\alpha}}{}^{\underline{\phantom{\alpha}}\!\!\!\beta}-2D_{[a}T_{b]\underline{\alpha}}{}^{\underline{\phantom{\alpha}}\!\!\!\beta}-2T_{\underline{\alpha}[a|}{}^{\underline{\epsilon}}
T_{\underline{\epsilon}|b]}{}^{\underline{\phantom{\alpha}}\!\!\!\beta}-T_{ab}{}^{\underline{\epsilon}}T_{\underline{\epsilon}\underline{\alpha}}{}^{\underline{\phantom{\alpha}}\!\!\!\beta}~.
\end{align}
{\it Bosonic equations-of-motion (HLW)}
\beal
R_{ab}&=18mD_{(a}A_{b)}+36m^2A_aA_b-\frac{1}{2}F_{a}{}^eF_{be}
-144H_{a}{}^{ef}H_{bef}\nn\\
&-3072G_{a}{}^{efg}G_{befg}
-\eta_{ab}\Big(36m^2+\frac{1}{4}F_{ef}F^{ef}-48H_{efg}H^{efg}\Big)\nn\\
md*A&=-18m^2A\wedge*A
+\frac{1}{4}F\wedge*F-96H\wedge*H
+3072G\wedge*G\nn\\
d*F&=-72m^2*A+18mA\wedge*F+4608H\wedge*G\nn\\
d*H&=-12m A\wedge* H+8F\wedge*G
+768G\wedge G\nn\\
d*G&=-\frac{3}{2}m*H+12mA\wedge*G+24H\wedge G~.
\label{hl1}
\end{align}
{\it Bosonic Bianchi identities (HLW)}
\beal
dA&= F\nn\\
dF&=0\nn\\
dH&=48mG-6mA\wedge H\nn\\
dG&=6mA\wedge G-\frac{1}{8}F\wedge H~,
\label{hl2}
\end{align}
where we have introduced the more conventional notation:
$iK_{(1)}=\frac{3}{2 }mA$,
$L_{(2)}=\frac{3}{16}F$,
$iK_{(3)}=H$,
$L_{(4)}=G$.
This is exactly the gauge-fixed form of the equations presented in \cite{hlw}\footnote{
To compare with the equations presented in \cite{hlw} one has to
set the $\sigma$ field of that reference to zero.
Remember that $\sigma$ is analogous to a St\"{u}ckelberg field and
can be gauged-away for $m\neq 0$.
Note also that there is a typographical error
in the coefficient of the second term on the right-hand-side of the third equation
of (4.3) of \cite{hlw}: instead of $1/4$ it should read $3/4$. This was subsequently
corrected in \cite{cla}.}.
HLW can also be obtained by a
generalized Scherk-Schwarz reduction of ordinary
eleven-dimensional supergravity \cite{llp} (see also \cite{ghee}). We have checked that
the equations presented here indeed
coincide with those in \cite{llp}\footnote{To bring the equations above
in the form presented in that reference, one needs to substitute $m\rightarrow\frac{1}{2}m$, $F\rightarrow F_{(2)}$,
$H\rightarrow -\frac{1}{24}F_{(3)}$, $G\rightarrow\frac{1}{192}F_{(4)}$
and set the pure-gauge field $\varphi$ of \cite{llp} to zero.}.
\subsection*{Dimension $\frac{5}{2}$}
{\it Spinor derivatives}
\beal
D_{\underline{\alpha}}R_{abcd}
=2D_{[a|}R_{\underline{\alpha}|b]cd}-T_{ab}{}^{\underline{\epsilon}} R_{\underline{\epsilon} \underline{\alpha} cd}
+2T_{[a|\underline{\alpha}}{}^{\underline{\epsilon}} R_{\underline{\epsilon}|b]cd} ~.
\end{align}
\section{Conclusions}
\label{conclusions}
In this paper we have employed a systematic procedure in order
to search for massive deformations of
IIA supergravity. It is amusing to think that had
we not known about them, we would have been
able to discover in this way both Romans and HLW supergravities in one go.
The method used here is quite general;
it would therefore be of interest to apply it to other
supersymmetric systems. It is quite plausible that
new massive supergravities can be discovered in this way.
As already mentioned in the introduction, HLW supergravity arises upon
compactification of a topologically modified version of eleven-dimensional
supergravity, MM-theory. However, it is not known at present whether
MM-theory can be given a microscopic quantum-mechanical description. Given
that de Sitter space is an (unstable) vacuum of MM-theory, if the latter can
somehow be related to M-theory it would provide a mechanism for embedding de
Sitter in M-theory. This interesting direction deserves to be pursued
further, alongside with more recent proposals for the realization of de
Sitter space in string/M-theory \cite{kklt}.
In a step towards this direction, it was shown in \cite{clb} that HLW
supergravity supports (nonsupersymmetric) multi-zero-brane solutions. It was
also argued that these states may indeed be associated with a microscopic
description of MM-theory\footnote{In \cite{ramg} it was suggested that a
microscopic Matrix-model description of MM-theory may be obtainable by a
Euclidean radial reduction, as opposed to the usual dimensional reduction of
Matrix theory.} and that the latter should represent an unstable phase of
M-theory. A better understanding of the dynamics of these zero-branes is
important in testing the proposal of \cite{clb}. To that end it would be
interesting to construct
the world-volume theories of
`massive' kappa-symmetric objects propagating in a
HLW background (or perhaps directly in MM-theory)\footnote{
For the case of Romans supergravity, such `massive' branes
were considered in \cite{berg, blo, hs}.},
either within the
superembedding formalism or by directly imposing kappa symmetry.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,983
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Donald Trump Trashes Late Night Comics Jimmy Kimmel, Stephen Colbert, Jimmy Fallon
AP Photos
Charlie Spiering
President Donald Trump ridiculed late night comics Jimmy Kimmel, Stephen Colbert, and Jimmy Fallon on Monday for their complete hypocrisy after he won the election.
"I mean, honestly, are these people funny?" Donald Trump asked supporters during a political rally in South Carolina.
"NO!" the crowd yelled.
"Did you see Jimmy Fallon?" Trump said, referring to Fallon's apology for "humanizing" him during the election.
The crowd booed.
"He apologized for humanizing me, the poor guy, because now he's going to lose all of us," Trump said.
Trump said that Fallon called to thank him after the show and commented on the "monster" ratings they got.
"He's a nice guy. He's lost. He's like a lost soul," Trump said.
He also referred to CBS host Stephen Colbert, describing him as a "lowlife" with "no talent."
Trump said that Jimmy Kimmel used to wait outside his studio to welcome him and open the door for him when he arrived.
"I'd do his show … and he's standing there opening up the door," Trump said. "I wasn't president. I was a guy. A guy with potential."
During the presidential campaign, Trump was invited as a guest on all the late night comedy shows but was shunned after he defeated Hillary Clinton.
Trump insisted that he enjoyed comedy and did not mind political jokes as long as they were funny.
"I can laugh at myself," he said. "Frankly, if I couldn't, I'd be in big trouble, but there's no talent. They're not like talented people. Johnny Carson was talented."
EntertainmentPoliticsDonald TrumpJimmy FallonJimmy KimmelStephen Colbert
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,688
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Somatidia kaszabi är en skalbaggsart som beskrevs av Stefan von Breuning 1975. Somatidia kaszabi ingår i släktet Somatidia och familjen långhorningar. Inga underarter finns listade i Catalogue of Life.
Källor
Långhorningar
kaszabi
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,299
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Marwan Yousef al-Shehhi (, auch Alshehhi oder asch-Schihhi; * 9. Mai 1978 in Ra's al-Chaima, Vereinigte Arabische Emirate; † 11. September 2001 in New York City) war der Suizidpilot, der bei den Anschlägen vom 11. September 2001 den United-Airlines-Flug 175 in den Südturm des World Trade Center flog. Mit 23 Jahren war al-Shehhi der jüngste der vier Piloten der Attentate. Wie alle Terroristen des 11. September war er Salafist.
Zeit in Deutschland
Nach der Schulausbildung besuchte al-Shehhi ab 1996 ein Studienkolleg in Bonn. 1999 zog er nach Hamburg und begann ein Schiffbaustudium an der Technischen Universität Hamburg-Harburg. Al-Shehhi soll in Hamburg nach Vermutungen der Ermittlungsbehörden zusammen mit Mohammed Atta und Ramzi Binalshibh am Aufbau der sogenannten Hamburger Terrorzelle beteiligt gewesen sein. Ende 1999 reiste er zusammen mit anderen späteren Attentätern nach Afghanistan. Im März 2000 kehrte al-Shehhi nach Deutschland zurück.
USA-Aufenthalt
Am 29. Mai 2000 reiste al-Shehhi als Erster aus der "Hamburger Gruppe" in die USA ein. Von Juli bis Dezember 2000 nahm er zusammen mit Mohammed Atta Flugunterricht an der Flugschule Huffman Aviation in Venice, Florida; am 19. Dezember 2000 bestanden sie ihre Prüfung zur Berufspilotenlizenz.
Al-Shehhi kaufte am 28. August 2001 an einem Schalter der United Airlines in einem der vier Flughäfen von Miami für 1600 Dollar Tickets für den UA-Flug 175 am 11. September, die er bar bezahlte. Eines der Tickets galt für den Sitzplatz 6c.
Einzelnachweise
Weblinks
Kapitel 5.3, The Hamburg Contingent, des 9/11 Commission Report (engl.)
Flugzeugentführer
Attentäter
Al-Qaida-Mitglied
Hamburger Terrorzelle
Person des Islam (Hamburg)
Staatsangehöriger der Vereinigten Arabischen Emirate
Geboren 1978
Gestorben 2001
Mann
Salafist
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 8,132
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# TROILUS AND CRESSIDA
## William Shakespeare
#### CONTENTS
#### Dramatis Personae
#### Prologue
#### Act One
#### Act Two
#### Act Three
#### Act Four
#### Act Five
About the Author
About the Series
Copyright
About the Publisher
#### DRAMATIS PERSONAE
PRIAM
_King of Troy_
HECTOR, TROILUS, PARIS, DEIPHOBUS, HELENUS
_his sons_
MARGARELON
_a bastard son of Priam_
AENEAS, ANTENOR
_Trojan commanders_
CALCHAS
_a Trojan priest, taking part with the Greeks_
PANDARUS
_uncle to Cressida_
AGAMEMNON
_the Greek general_
MENELAUS
_his brother_
ACHILLES, AJAX
_Greek commanders_
ULYSSES, NESTOR, DIOMEDES, PATROCLUS
_Greek commanders_
THERSITES
_a deformed and scurrilous Greek_
ALEXANDER
_servant to Cressida_
_Servant to Troilus_
_Servant to Paris_
_Servant to Diomedes_
HELEN
_wife to Menelaus_
ANDROMACHE
_wife to Hector_
CASSANDRA
_daughter to Priam, a prophetess_
CRESSIDA
_daughter to Calchas_
_Trojan and Greek Soldiers, and Attendants_
**THE SCENE: TROY AND THE GREEK CAMP BEFORE IT**.
#### PROLOGUE
In Troy, there lies the scene. From isles of Greece
The princes orgillous, their high blood chaf'd,
Have to the port of Athens sent their ships
Fraught with the ministers and instruments
[5]
Of cruel war. Sixty and nine that wore
Their crownets regal from th' Athenian bay
Put forth toward Phrygia; and their vow is made
To ransack Troy, within whose strong immures
The ravish'd Helen, Menelaus' queen,
[10]
With wanton Paris sleeps – and that's the quarrel.
To Tenedos they come,
And the deep-drawing barks do there disgorge
Their war-like fraughtage. Now on Dardan plains
The fresh and yet unbruised Greeks do pilch
[15]
Their brave pavilions: Priam's six-gated city,
Dardan, and Tymbria, Helias, Chetas, Troien,
And Antenorides, with massy staples
And corresponsive and fulfilling bolts,
Sperr up the sons of Troy.
[20]
Now expectation, tickling skittish spirits
On one and other side, Troyan and Greek,
Sets all on hazard – and hither am I come
A Prologue arm'd, but not in confidence
Of author's pen or actor's voice, but suited
[25]
In like conditions as our argument,
To tell you, fair beholders, that our play
Leaps o'er the vaunt and firstlings of those broils,
Beginning in the middle; starting thence away
To what may be digested in a play.
[30]
Like or find fault; do as your pleasures are;
Now good or bad, 'tis but the chance of war.
#### ACT ONE
**SCENE I**. _Troy. Before Priam's palace_.
_Enter TROILUS armed, and PANDARUS_.
TROILUS Call here my varlet; I'll unarm again.
Why should I war without the walls of Troy
That find such cruel battle here within?
Each Troyan that is master of his heart,
[5]
Let him to field; Troilus, alas, hath none!
PANDARUS Will this gear ne'er be mended?
TROILUS The Greeks are strong, and skilful to their strength,
Fierce to their skill, and to their fierceness valiant;
But I am weaker than a woman's tear,
[10]
Tamer than sleep, fonder than ignorance,
Less valiant than the virgin in the night,
And skilless as unpractis'd infancy.
[16]
PANDARUS Well, I have told you enough of this; for my part, I'll not meddle nor make no farther. He that will have a cake out of the wheat must needs tarry the grinding.
TROILUS Have I not tarried?
PANDARUS Ay, the grinding; but you must tarry the bolting.
TROILUS Have I not tarried?
[21]
PANDARUS Ay, the bolting; but you must tarry the leavening.
TROILUS Still have I tarried.
[26]
PANDARUS Ay, to the leavening; but here's yet in the word 'hereafter' the kneading, the making of the cake, the heating of the oven, and the baking; nay, you must stay the cooling too, or you may chance to burn your lips.
TROILUS Patience herself, what goddess e'er she be,
Doth lesser blench at suff'rance than I do.
At Priam's royal table do I sit;
[30]
And when fair Cressid comes into my thoughts –
So, traitor, then she comes when she is thence.
PANDARUS Well, she look'd yesternight fairer than ever I saw her look, or any woman else.
TROILUS I was about to tell thee: when my heart,
[35]
As wedged with a sigh, would rive in twain,
Lest Hector or my father should perceive me,
I have, as when the sun doth light a storm,
Buried this sigh in wrinkle of a smile.
But sorrow that is couch'd in seeming gladness
[40]
Is like that mirth fate turns to sudden sadness.
[46]
PANDARUS An her hair were not somewhat darker than Helen's – well, go to – there were no more comparison between the women. But, for my part, she is my kinswoman; I would not, as they term it, praise her, but I would somebody had heard her talk yesterday, as I did. I will not dispraise your sister Cassandra's wit; but –
TROILUS O Pandarus! I tell thee, Pandarus –
When I do tell thee there my hopes lie drown'd,
Reply not in how many fathoms deep
[50]
They lie indrench'd. I tell thee I am mad
In Cressid's love. Thou answer'st 'She is fair' –
Pourest in the open ulcer of my heart –
Her eyes, her hair, her cheek, her gait, her voice,
Handiest in thy discourse. O, that her hand,
[55]
In whose comparison all whites are ink
Writing their own reproach; to whose soft seizure
The cygnet's down is harsh, and spirit of sense
Hard as the palm of ploughman! This thou tell'st me,
As true thou tell'st me, when I say I love her;
[60]
But, saying thus, instead of oil and balm,
Thou lay'st in every gash that love hath given me
The knife that made it.
PANDARUS I speak no more than truth.
TROILUS Thou dost not speak so much.
[67]
PANDARUS Faith, I'll not meddle in it. Let her be as she is: if she be fair, 'tis the better for her; an she be not, she has the mends in her own hands.
TROILUS Good Pandarus! How now, Pandarus!
[71]
PANDARUS I have had my labour for my travail, ill thought on of her and ill thought on of you; gone between and between, but small thanks for my labour.
TROILUS What, art thou angry, Pandarus? What, with me?
[77]
PANDARUS Because she's kin to me, therefore she's not so fair as Helen. An she were not kin to me, she would be as fair a Friday as Helen is on Sunday. But what care I? I care not an she were a blackamoor; 'tis all one to me.
TROILUS Say I she is not fair?
[82]
PANDARUS I do not care whether you do or no. She's a fool to stay behind her father. Let her to the Greeks; and so I'll tell her the next time I see her. For my part, I'll meddle nor make no more i' th' matter.
TROILUS Pandarus!
PANDARUS Not I.
[85]
TROILUS Sweet Pandarus!
PANDARUS Pray you, speak no more to me: I will leave all as I found it, and there an end.
[ _Exit. Sound alarum_.
TROILUS Peace, you ungracious clamours! Peace, rude sounds!
Fools on both sides! Helen must needs be fair,
[90]
When with your blood you daily paint her thus.
I cannot fight upon this argument;
It is too starv'd a subject for my sword.
But Pandarus – O gods, how do you plague me!
I cannot come to Cressid but by Pandar;
[95]
And he's as tetchy to be woo'd to woo
As she is stubborn-chaste against all suit.
Tell me, Apollo, for thy Daphne's love,
What Cressid is, what Pandar, and what we?
Her bed is India; there she lies, a pearl;
[100]
Between our Ilium and where she resides
Let it be call'd the wild and wand'ring flood;
Ourself the merchant, and this sailing Pandar
Our doubtful hope, our convoy, and our bark.
_Alarum. Enter AENEAS_.
AENEAS How now, Prince Troilus! Wherefore not afield?
[105]
TROILUS Because not there. This woman's answer sorts,
For womanish it is to be from thence.
What news, Aeneas, from the field to-day?
AENEAS That Paris is returned home, and hurt.
TROILUS By whom, Aeneas?
AENEAS Troilus, by Menelaus.
[110]
TROILUS Let Paris bleed: 'tis but a scar to scorn;
Paris is gor'd with Menelaus' horn. [ _Alarum_.
AENEAS Hark what good sport is out of town to-day!
TROILUS Better at home, if 'would I might' were 'may'.
But to the sport abroad. Are you bound thither?
[115]
AENEAS In all swift haste.
TROILUS Come, go we then together.
[ _Exeunt_.
**SCENE II**. _Troy. A street_.
_Enter CRESSIDA and her man ALEXANDER_.
CRESSIDA Who were those went by?
ALEXANDER Queen Hecuba and Helen.
CRESSIDA And whither go they?
ALEXANDER Up to the eastern tower,
Whose height commands as subject all the vale,
To see the battle. Hector, whose patience
[5]
Is as a virtue fix'd, to-day was mov'd.
He chid Andromache, and struck his armourer;
And, like as there were husbandry in war,
Before the sun rose he was harness'd light,
And to the field goes he; where every flower
[10]
Did as a prophet weep what it foresaw
In Hector's wrath.
CRESSIDA What was his cause of anger?
ALEXANDER The noise goes, this: there is among the Greeks
A lord of Troyan blood, nephew to Hector;
They call him Ajax.
CRESSIDA Good; and what of him?
[15]
ALEXANDER They say he is a very man per se
And stands alone.
CRESSIDA So do all men, unless they are drunk, sick, or have no legs.
[29]
ALEXANDER This man, lady, hath robb'd many beasts of their particular additions: he is as valiant as the lion, churlish as the bear, slow as the elephant – a man into whom nature hath so crowded humours that his valour is crush'd into folly, his folly sauced with discretion. There is no man hath a virtue that he hath not a glimpse of, nor any man an attaint but he carries some stain of it; he is melancholy without cause and merry against the hair; he hath the joints of every thing; but everything so out of joint that he is a gouty Briareus, many hands and no use, or purblind Argus, all eyes and no sight.
CRESSIDA But how should this man, that makes me smile, make Hector angry?
ALEXANDER They say he yesterday cop'd Hector in the battle and struck him down, the disdain and shame whereof hath ever since kept Hector fasting and waking.
_Enter PANDARUS_.
[35]
CRESSIDA Who comes here?
ALEXANDER Madam, your uncle Pandarus.
CRESSIDA Hector's a gallant man.
ALEXANDER As may be in the world, lady.
PANDARUS What's that? What's that?
[40]
CRESSIDA Good morrow, uncle Pandarus.
PANDARUS Good morrow, cousin Cressid. What do you talk of? – Good morrow, Alexander. –
How do you, cousin? When were you at Ilium?
[44]
CRESSIDA This morning, uncle.
PANDARUS What were you talking of when I came? Was Hector arm'd and gone ere you came to Ilium? Helen was not up, was she?
CRESSIDA Hector was gone; but Helen was not up.
PANDARUS E'en so. Hector was stirring early.
[50]
CRESSIDA That were we talking of, and of his anger.
PANDARUS Was he angry?
CRESSIDA So he says here.
[56]
PANDARUS True, he was so; I know the cause too; he'll lay about him to-day, I can tell them that. And there's Troilus will not come far behind him; let them take heed of Troilus, I can tell them that too.
CRESSIDA What, is he angry too?
PANDARUS Who, Troilus? Troilus is the better man of the two.
CRESSIDA O Jupiter! there's no comparison.
[63]
PANDARUS What, not between Troilus and Hector? Do you know a man if you see him?
CRESSIDA Ay, if I ever saw him before and knew him.
PANDARUS Well, I say Troilus is Troilus.
[66]
CRESSIDA Then you say as I say, for I am sure he is not Hector.
PANDARUS No, nor Hector is not Troilus in some degrees.
CRESSIDA 'Tis just to each of them: he is himself.
PANDARUS Himself! Alas, poor Troilus! I would he were!
[70]
CRESSIDA So he is.
PANDARUS Condition I had gone barefoot to India.
CRESSIDA He is not Hector.
[76]
PANDARUS Himself! no, he's not himself. Would 'a were himself! Well, the gods are above; time must friend or end. Well, Troilus, well! I would my heart were in her body! No, Hector is not a better man than Troilus.
CRESSIDA Excuse me.
PANDARUS He is elder.
CRESSIDA Pardon me, pardon me.
PANDARUS Th' other's not come to't; you shall tell me another tale when th' other's come to't.
[82]
Hector shall not have his wit this year.
CRESSIDA He shall not need it if he have his own.
PANDARUS Nor his qualities.
[85]
CRESSIDA No matter.
PANDARUS Nor his beauty.
CRESSIDA 'Twould not become him: his own's better.
[90]
PANDARUS You have no judgment, niece. Helen herself swore th' other day that Troilus, for a brown favour, for so 'tis, I must confess – not brown neither –
CRESSIDA No, but brown.
PANDARUS Faith, to say truth, brown and not brown.
CRESSIDA To say the truth, true and not true.
PANDARUS She prais'd his complexion above Paris.
[95]
CRESSIDA Why, Paris hath colour enough.
PANDARUS So he has.
[101]
CRESSIDA Then Troilus should have too much. If she prais'd him above, his complexion is higher than his; he having colour enough, and the other higher, is too flaming a praise for a good complexion. I had as lief Helen's golden tongue had commended Troilus for a copper nose.
PANDARUS I swear to you I think Helen loves him better than Paris.
CRESSIDA Then she's a merry Greek indeed.
[107]
PANDARUS Nay, I am sure she does. She came to him th' other day into the compass'd window – and you know he has not past three or four hairs on his chin –
CRESSIDA Indeed a tapster's arithmetic may soon bring his particulars therein to a total.
PANDARUS Why, he is very young, and yet will he within three pound lift as much as his brother Hector.
CRESSIDA Is he so young a man and so old a lifter?
PANDARUS But to prove to you that Helen loves him: she came and puts me her white hand to his cloven chin –
[115]
CRESSIDA Juno have mercy! How came it cloven?
PANDARUS Why, you know, 'tis dimpled I think his smiling becomes him better than any man in all Phrygia.
CRESSIDA O, he smiles valiantly!
[119]
PANDARUS Does he not?
CRESSIDA O yes, an 'twere a cloud in autumn!
PANDARUS Why, go to, then! But to prove to you that Helen loves Troilus –
CRESSIDA Troilus will stand to the proof, if you'll prove it so.
[126]
PANDARUS Troilus! Why, he esteems her no more than I esteem an addle egg.
CRESSIDA If you love an addle egg as well as you love an idle head, you would eat chickens i' th' shell.
PANDARUS I cannot choose but laugh to think how she tickled his chin. Indeed, she has a marvell's white hand, I must needs confess.
[132]
CRESSIDA Without the rack.
PANDARUS And she takes upon her to spy a while hair on his chin.
[135]
CRESSIDA Alas, poor chin! Many a wart is richer.
PANDARUS But there was such laughing! Queen
Hecuba laugh'd that her eyes ran o'er.
CRESSIDA With millstones.
PANDARUS And Cassandra laugh'd.
[141]
CRESSIDA But there was a more temperate fire under the pot of her eyes. Did her eyes run o'er too?
PANDARUS And Hector laugh'd.
CRESSIDA At what was all this laughing?
[145]
PANDARUS Marry, at the while hair that Helen spied on Troilus' chin.
CRESSIDA An't had been a green hair I should have laugh'd too.
PANDARUS They laugh'd not so much at the hair as at his pretty answer.
[150]
CRESSIDA What was his answer?
PANDARUS Quoth she 'Here's but two and fifty hairs on your chin, and one of them is white'.
CRESSIDA This is her question.
[161]
PANDARUS That's true; make no question of that. Two and fifty hairs,' quoth he 'and one white. That while hair is my father, and all the rest are his sons.' Jupiter!' quoth she 'which of these hairs is Paris my husband?' 'The forked one;' quoth he pluck't out and give it him.' But there was such laughing! and Helen so blush'd, and Paris so chaf'd; and all the rest so laugh'd that it pass'd.
CRESSIDA So let it now; for it has been a great while going by.
[165]
PANDARUS Well, cousin, I told you a thing yesterday; think on't.
CRESSIDA So I do.
PANDARUS I'll be sworn 'tis true; he will weep you, an 'twere a man born in April.
[170]
CRESSIDA And I'll spring up in his tears, an 'twere a nettle against May. [ _Sound a retreat_.
PANDARUS Hark! they are coming from the field. Shall we stand up here and see them as they pass toward Ilium? Good niece, do, sweet niece Cressida.
[174]
CRESSIDA At your pleasure.
PANDARUS Here, here, here's an excellent place; here we may see most bravely. I'll tell you them all by their names as they pass by; but mark Troilus above the rest.
_AENEAS passes._
[178]
CRESSIDA Speak not so loud.
[181]
PANDARUS That's Aeneas. Is not that a brave man? He's one of the flowers of Troy, I can tell, you. But mark Troilus; you shall see anon.
_ANTENOR passes_.
CRESSIDA Who's that?
[187]
PANDARUS That's Antenor. He has a shrewd wit, I can tell you; and he's a man good enough; he's one o' th' soundest judgments in Troy, whosoever, and a proper man of person. When comes Troilus? I'll show you Troilus anon. If he see me, you shall see him nod at me.
CRESSIDA Will he give you the nod?
PANDARUS You shall see.
CRESSIDA If he do, the rich shall have more.
_HECTOR passes_.
PANDARUS That's Hector, that, that, look you, that; there's a fellow! Go thy way, Hector! There's a brave man, niece. O brave Hector! Look how he looks. There's a countenance! Is't not a brave man?
[195]
CRESSIDA O, a brave man!
[200]
PANDARUS Is 'a not? It does a man's heart good. Look you what hacks are on his helmet! Look you yonder, do you see? Look you there. There's no jesting; there's laying on; take't off who will, as they say. There be hacks.
CRESSIDA Be those with swords?
[204]
PANDARUS Swords! anything, he cares not; an the devil come to him, it's all one. By God's lid, it does one's heart good. Yonder comes Paris, yonder comes Paris.
_PARIS passes_.
Look ye yonder, niece; is't not a gallant man too, is't not? Why, this is brave now. Who said he came hurt home to-day? He's not hurt. Why, this will do Helen's heart good now, ha! Would
I could see Troilus now! You shall see Troilus anon.
_HELENUS passes_.
[210]
CRESSIDA Who's that?
PANDARUS That's Helenus. I marvel where Troilus is. That's Helenus. I think he went not forth to-day. That's Helenus.
[214]
CRESSIDA Can Helenus fight, uncle?
PANDARUS Helenus! no. Yes, he'll fight indifferent well. I marvel where Troilus is,
Hark! do you not hear the people cry 'Troilus'?
Helenus is a priest.
[218]
CRESSIDA What sneaking fellow comes yonder?
_TROILUS passes_.
PANDARUS Where? yonder? That's Deiphobus.
'Tis Troilus. There's a man, niece. Hem!
[221]
Brave Troilus, the prince of chivalry!
CRESSIDA Peace, for shame, peace!
[231]
PANDARUS Mark him; note him. O brave Troilus! Look well upon him, niece; look you how his sword is bloodied, and his helm more hack'd than Hector's; and how he looks, and how he goes! O admirable youth! he never saw three and twenty. Go thy way, Troilus, go thy way. Had I a sister were a grace or a daughter a goddess, he should take his choice. O admirable man! Paris? Paris is dirt to him; and, I warrant, Helen, to change, would give an eye to boot.
CRESSIDA Here comes more.
_Common Soldiers pass_.
PANDARUS Asses, fools, dolts! chaff and bran, chaff and bran! porridge after meat! I could live and die in the eyes of Troilus. Ne'er look, ne'er look; the eagles are gone. Crows and daws, crows and daws! I had rather be such a man as Troilus than Agamemnon and all Greece.
[240]
CRESSIDA There is amongst the Greeks Achilles, a better man than Troilus.
PANDARUS Achilles? A drayman, a porter, a very camel!
CRESSIDA Well, well.
[247]
PANDARUS Well, well! Why, have you any discretion? Have you any eyes? Do you know what a man is? Is not birth, beauty, good shape, discourse, manhood, learning, gentleness, virtue, youth, liberality, and such like, the spice and salt that season a man?
CRESSIDA Ay, a minc'd man; and then to be bak'd with no date in the pie, for then the man's date is out.
[251]
PANDARUS You are such a woman! A man knows not at what ward you lie.
[256]
CRESSIDA Upon my back, to defend my belly; upon my wit, to defend my wiles; upon my secrecy, to defend mine honesty; my mask, to defend my beauty; and you, to defend all these; and at all these wards I lie at, at a thousand watches.
PANDARUS Say one of your watches.
[262]
CRESSIDA Nay, I'll watch you for that; and that's one of the chiefest of them too. If I cannot ward what I would not have hit, I can watch you for telling how I took the blow; unless it swell past hiding, and then it's past watching.
PANDARUS You are such another!
_Enter Troilus' Boy_.
BOY Sir, my lord would instantly speak with you.
[265]
PANDARUS Where?
BOY At your own house; there he unarms him.
PANDARUS Good boy, tell him I come. [ _Exit Boy_
I doubt he be hurt. Fare ye well, good niece.
CRESSIDA Adieu, uncle.
PANDARUS I will be with you, niece, by and by.
[271]
CRESSIDA To bring, uncle.
PANDARUS Ay, a token from Troilus.
CRESSIDA By the same token, you are a bawd.
[ _Exit Pandarus_.
Words, vows, gifts, tears, and love's full sacrifice,
[275]
He offers in another's enterprise;
But more in Troilus thousand-fold I see
Than in the glass of Pandar's praise may be,
Yet hold I off. Women are angels, wooing:
Things won are done; joy's soul lies in the doing.
[280]
That she belov'd knows nought that knows not this:
Men prize the thing ungain'd more than it is.
That she was never yet that ever knew
Love got so sweet as when desire did sue;
Therefore this maxim out of love I leach:
[285]
Achievement is command; ungain'd, beseech.
Then though my heart's content firm love doth bear,
Nothing of that shall from mine eyes appear.
[ _Exit_.
**SCENE III**. _The Grecian camp. Before Agamemnon's tent_.
_Sennet. Enter AGAMEMNON, NESTOR, ULYSSES, DIOMEDES, MENELAUS, and Others_.
AGAMEMNON Princes,
What grief hath set these jaundies o'er your cheeks?
The ample proposition that hope makes
In all designs begun on earth below
[5]
Fails in the promis'd largeness; checks and disasters
Grow in the veins of actions highest rear'd,
As knots, by the conflux of meeting sap,
Infects the sound pine, and diverts his grain
Tortive and errant from his course of growth.
[10]
Nor, princes, is it matter new to us
That we come short of our suppose so far
That after seven years' siege yet Troy walls stand;
Sith every action that hath gone before,
Whereof we have record, trial did draw
[15]
Bias and thwart, not answering the aim,
And that unbodied figure of the thought
That gave't surmised shape. Why then, you princes,
Do you with cheeks abash'd behold our works
And call them shames, which are, indeed, nought else
[20]
But the protractive trials of great Jove
To find persistive constancy in men;
The fineness of which metal is not found
In fortune's love? For then the bold and coward,
The wise and fool, the artist and unread,
[25]
The hard and soft, seem all affin'd and kin.
But in the wind and tempest of her frown
Distinction, with a broad and powerful fan,
Puffing at all, winnows the light away;
And what hath mass or matter by itself
[30]
Lies rich in virtue and unmingled.
NESTOR With due observance of thy godlike seat,
Great Agamemnon. Nestor shall apply
Thy latest words. In the reproof of chance
Lies the true proof of men. The sea being smooth,
[35]
How many shallow bauble boats dare sail
Upon her patient breast, making their way
With those of nobler bulk!
But let the ruffian Boreas once enrage
The gentle Thetis, and anon behold
[40]
The strong-ribb'd bark through liquid mountains cut,
Bounding between the two moist elements
Like Perseus' horse. Where's then the saucy boat,
Whose weak untimber'd sides but even now
Co-rivall'd greatness? Either to harbour fled
[45]
Or made a toast for Neptune. Even so
Doth valour's show and valour's worth divide
In storms of fortune; for in her ray and brightness
The herd hath more annoyance by the breese
Than by the tiger; but when the splitting wind
[50]
Makes flexible the knees of knotted oaks,
And flies fled under shade – why, then the thing of courage,
As rous'd with rage, with rage doth sympathise,
And with an accent tun'd in self-same key
Retorts to chiding fortune.
ULYSSES Agamemnon,
[55]
Thou great commander, nerve and bone of Greece,
Heart of our numbers, soul and only spirit
In whom the tempers and the minds of all
Should be shut up – hear what Ulysses speaks.
Besides the applause and approbation
[60]
The which, [ _To Agamemnon_ ] most mighty, for thy place and sway,
[ _To Nestor_ ] And, thou most reverend, for thy stretch'd-out life,
I give to both your speeches – which were such
As Agamemnon and the hand of Greece
Should hold up high in brass; and such again
[65]
As venerable Nestor, hatch'd in silver,
Should with a bond of air, strong as the axle-tree
On which heaven rides, knit all the Greekish ears
To his experienc'd tongue – yet let it please both,
Thou great, and wise, to hear Ulysses speak.
[70]
AGAMEMNON Speak, Prince of Ithaca; and be't of less expect
That matter needless, of importless burden,
Divide thy lips than we are confident,
When rank Thersites opes his mastic jaws,
We shall hear music, wit, and oracle.
[75]
ULYSSES Troy, yet upon his basis, had been down,
And the great Hector's sword had lack'd a master,
But for these instances:
The specialty of rule hath been neglected;
And look how many Grecian tents do stand
[80]
Hollow upon this plain, so many hollow factions.
When that the general is not like the hive,
To whom the foragers shall all repair,
What honey is expected? Degree being vizarded,
Th' unworthiest shows as fairly in the mask.
[85]
The heavens themselves, the planets, and this centre,
Observe degree, priority, and place,
Insisture, course, proportion, season, form,
Office, and custom, in all line of order;
And therefore is the glorious planet Sol
[90]
In noble eminence enthron'd and spher'd
Amidst the other, whose med'cinable eye
Corrects the ill aspects of planets evil,
And posts, like the commandment of a king,
Sans check, to good and bad. But when the planets
[95]
In evil mixture to disorder wander,
What plagues and what portents, what mutiny,
What raging of the sea, shaking of earth,
Commotion in the winds! Frights, changes, horrors,
Divert and crack, rend and deracinate,
[100]
The unity and married calm of states
Quite from their fixture! O, when degree is shak'd,
Which is the ladder of all high designs,
The enterprise is sick! How could communities,
Degrees in schools, and brotherhoods in cities,
[105]
Peaceful commerce from dividable shores,
The primogenity and due of birth,
Prerogative of age, crowns, sceptres, laurels,
But by degree, stand in authentic place?
Take but degree away, untune that string,
[110]
And hark what discord follows! Each thing melts
In mere oppugnancy: the bounded waters
Should lift their bosoms higher than the shores,
And make a sop of all this solid globe;
Strength should be lord of imbecility,
[115]
And the rude son should strike his father dead;
Force should be right; or, rather, right and wrong –
Between whose endless jar justice resides –
Should lose their names, and so should justice too.
Then everything includes itself in power,
[120]
Power into will, will into appetite;
And appetite, an universal wolf,
So doubly seconded with will and power,
Must make perforce an universal prey,
And last eat up himself. Great Agamemnon,
[125]
This chaos, when degree is suffocate,
Follows the choking.
And this neglection of degree it is
That by a pace goes backward, with a purpose
It hath to climb. The general's disdain'd
[130]
By him one step below, he by the next,
That next by him beneath; so every step,
Exampl'd by the first pace that is sick
Of his superior, grows to an envious fever
Of pale and bloodless emulation.
[135]
And 'tis this fever that keeps Troy on foot,
Not her own sinews. To end a tale of length,
Troy in our weakness stands, not in her strength.
NESTOR Most wisely hath Ulysses here discover'd
The fever whereof all our power is sick.
[140]
AGAMEMNON The nature of the sickness found, Ulysses,
What is the remedy?
ULYSSES The great Achilles, whom opinion crowns
The sinew and the forehand of our host,
Having his ear full of his airy fame,
[145]
Grows dainty of his worth, and in his tent
Lies mocking our designs; with him Patroclus
Upon a lazy bed the livelong day
Breaks scurril jests;
And with ridiculous and awkward action –
[150]
Which, slanderer, he imitation calls –
He pageants us. Sometime, great Agamemnon,
Thy topless deputation he puts on;
And like a strutting player whose conceit
Lies in his hamstring, and doth think it rich
[155]
To hear the wooden dialogue and sound
'Twixt his stretch'd footing and the scaffoldage –
Such lo-be-pilied and o'er-wrested seeming
He acts thy greatness in; and when he speaks
'Tis like a chime a-mending; with terms unsquar'd,
[160]
Which, from the tongue of roaring Typhon dropp'd,
Would seem hyperboles. At this fusty stuff
The large Achilles, on his press'd bed lolling,
From his deep chest laughs out a loud applause;
Cries 'Excellent! 'tis Agamemnon just.
[165]
Now play me Nestor; hem, and stroke thy beard,
As he being drest to some oration'.
That's done – as near as the extremest ends
Of parallels, as like as Vulcan and his wife;
Yet good Achilles still cries 'Excellent!
[170]
'Tis Nestor right. Now play him me, Patroclus,
Arming to answer in a night alarm'.
And then, forsooth, the faint defects of age
Must be the scene of mirth: to cough and spit
And, with a palsy-fumbling on his gorget,
[175]
Shake in and out the rivet. And at this sport
Sir Valour dies; cries 'O, enough, Patroclus;
Or give me ribs of steel! I shall split all
In pleasure of my spleen'. And in this fashion
All our abilities, gifts, natures, shapes,
[180]
Severals and generals of grace exact,
Achievements, plots, orders, preventions,
Excitements to the field or speech for truce,
Success or loss, what is or is not, serves
As stuff for these two to make paradoxes.
[185]
NESTOR And in the imitation of these twain –
Who, as Ulysses says, opinion crowns
With an imperial voice – many are infect.
Ajax is grown self-will'd and bears his head
In such a rein, in full as proud a place
[190]
As broad Achilles; keeps his tent like him;
Makes factious feasts; rails on our state of war
Bold as an oracle, and sets Thersites,
A slave whose gall coins slanders like a mint,
To match us in comparisons with dirt,
[195]
To weaken and discredit our exposure,
How rank soever rounded in with danger.
ULYSSES They tax our policy and call it cowardice,
Count wisdom as no member of the war,
Forestall prescience, and esteem no act
[200]
But that of hand. The still and mental parts
That do contrive how many hands shall strike
When fitness calls them on, and know, by measure
Of their observant toil, the enemies' weight –
Why, this hath not a finger's dignity:
[205]
They call this bed-work, mapp'ry, closet-war;
So that the ram that batters down the wall,
For the great swinge and rudeness of his poise,
They place before his hand that made the engine,
Or those that with the fineness of their souls
[210]
By reason guide his execution.
NESTOR Let this be granted, and Achilles' horse Makes many Thetis' sons. [ _Tucket_.
AGAMEMNON What trumpet? Look, Menelaus,
MENELAUS From Troy.
_Enter AENEAS_.
[215]
AGAMEMNON What would you fore our tent?
AENEAS Is this great Agamemnon's tent, I pray you?
AGAMEMNON Even this.
AENEAS May one that is a herald and a prince
Do a fair message to his kingly eyes?
[220]
AGAMEMNON With surety stronger than Achilles' arm
Fore all the Greekish heads, which with one voice
Call Agamemnon head and general.
AENEAS Fair leave and large security. How may
A stranger to those most imperial looks
Know them from eyes of other mortals?
[225]
AGAMEMNON How?
AENEAS Ay;
I ask, that I might waken reverence,
And bid the cheek be ready with a blush
Modest as Morning when she coldly eyes
[230]
The youthful Phoebus.
Which is that god in office, guiding men?
Which is the high and mighty Agamemnon?
AGAMEMNON This Troyan scorns us, or the men of Troy
Are ceremonious courtiers.
[235]
AENEAS Courtiers as free, as debonair, unarm'd,
As bending angels; that's their fame in peace.
But when they would seem soldiers, they have galls,
Good arms, strong joints, true swords; and, Jove's accord,
Nothing so full of heart. But peace, Aeneas,
[240]
Peace, Troyan; lay thy finger on thy lips.
The worthiness of praise distains his worth,
If that the prais'd himself bring the praise forth;
But what the repining enemy commends,
That breath fame blows; that praise, sole pure, transcends.
[245]
AGAMEMNON Sir, you of Troy, call you yourself Aeneas?
AENEAS Ay, Greek, that is my name.
AGAMEMNON What's your affair, I pray you?
AENEAS Sir, pardon; 'tis for Agamemnon's ears.
AGAMEMNON He hears nought privately that comes from Troy.
[250]
AENEAS Nor I from Troy come not to whisper with him;
I bring a trumpet to awake his ear,
To set his sense on the attentive bent,
And then to speak.
AGAMEMNON Speak frankly as the wind;
It is not Agamemnon's sleeping hour.
[255]
That thou shalt know, Troyan, he is awake,
He tells thee so himself.
AENEAS Trumpet, blow loud,
Send thy brass voice through all these lazy tents;
And every Greek of mettle, let him know
What Troy means fairly shall be spoke aloud.
[ _Sound trumpet_.
[260]
We have, great Agamemnon, here in Troy
A prince called Hector – Priam is his father –
Who in this dull and long-continued truce
Is resty grown; he bade me take a trumpet
And to this purpose speak: Kings, princes, lords!
[265]
If there be one among the fair'st of Greece
That holds his honour higher than his ease,
That seeks his praise more than he fears his peril,
That knows his valour and knows not his fear,
That loves his mistress more than in confession
[270]
With truant vows to her own lips he loves,
And dare avow her beauty and her worth
In other arms than hers – to him this challenge.
Hector, in view of Troyans and of Greeks,
Shall make it good or do his best to do it:
[275]
He hath a lady wiser, fairer, truer,
Than ever Greek did couple in his arms;
And will to-morrow with his trumpet call
Mid-way between your tents and walls of Troy
To rouse a Grecian that is true in love.
[280]
If any come, Hector shall honour him;
If none, he'll say in Troy, when he retires,
The Grecian dames are sunburnt and not worth
The splinter of a lance. Even so much.
AGAMEMNON This shall be told our lovers, Lord Aeneas.
[285]
If none of them have soul in such a kind,
We left them all at home. But we are soldiers;
And may that soldier a mere recreant prove
That means not, hath not, or is not in love.
If then one is, or hath, or means to be,
[290]
That one meets Hector; if none else, I am he.
NESTOR Tell him of Nestor, one that was a man
When Hector's grandsire suck'd. He is old now;
But if there be not in our Grecian mould
One noble man that hath one spark of fire
[295]
To answer for his love, tell him from me
I'll hide my silver beard in a gold beaver,
And in my vantbrace put this wither'd brawn,
And, meeting him, will tell him that my lady
Was fairer than his grandame, and as chaste
[300]
As may be in the world. His youth in flood,
I'll prove this truth with my three drops of blood.
AENEAS Now heavens forfend such scarcity of youth!
ULYSSES Amen.
AGAMEMNON Fair Lord Aeneas, let me touch your hand;
[305]
To our pavilion shall I lead you, first.
Achilles shall have word of this intent;
So shall each lord of Greece, from tent to tent.
Yourself shall feast with us before you go,
And find the welcome of a noble foe.
[ _Exeunt all but Ulysses and Nestor_.
[310]
ULYSSES Nestor!
NESTOR What says Ulysses?
ULYSSES I have a young conception in my brain;
Be you my time to bring it to some shape.
NESTOR What is't?
[315]
ULYSSES This 'tis:
Blunt wedges rive hard knots. The seeded pride
That hath to this maturity blown up
In rank Achilles must or now be cropp'd
Or, shedding, breed a nursery of like evil
To overbulk us all.
[320]
NESTOR Well, and how?
ULYSSES This challenge that the gallant Hector sends,
However it is spread in general name,
Relates in purpose only to Achilles.
NESTOR True. The purpose is perspicuous even as substance
[325]
Whose grossness little characters sum up;
And, in the publication, make no strain
But that Achilles, were his brain as barren
As banks of Libya – though, Apollo knows,
'Tis dry enough – will with great speed of judgment,
[330]
Ay, with celerity, find Hector's purpose
Pointing on him.
ULYSSES And wake him to the answer, think you?
NESTOR Why, 'tis most meet. Who may you else oppose
That can from Hector bring those honours off,
[335]
If not Achilles? Though't be a sportful combat,
Yet in this trial much opinion dwells;
For here the Troyans taste our dear'st repute
With their fin'st palate; and trust to me, Ulysses,
Our imputation shall be oddly pois'd
[340]
In this vile action; for the success,
Although particular, shall give a scantling
Of good or bad unto the general;
And in such indexes, although small pricks
To their subsequent volumes, there is seen
[345]
The baby figure of the giant mass
Of things to come at large. It is suppos'd
He that meets Hector issues from our choice;
And choice, being mutual act of all our souls,
Makes merit her election, and doth boil,
[350]
As 'twere from forth us all, a man distill'd
Out of our virtues; who miscarrying,
What heart receives from hence a conquering part,
To steal a strong opinion to themselves?
Which emertain'd, limbs are his instruments,
[355]
In no less working than are swords and bows
Directive by the limbs.
ULYSSES Give pardon to my speech.
Therefore 'tis meet Achilles meet not Hector.
Let us, like merchants, show our foulest wares
[360]
And think perchance they'll sell; if not, the lustre
Of the better yet to show shall show the better,
By showing the worst first. Do not consent
That ever Hector and Achilles meet;
For both our honour and our shame in this
[365]
Are dogg'd with two strange followers.
NESTOR I see them not with my old eyes. What are they?
ULYSSES What glory our Achilles shares from Hector,
Were he not proud, we all should wear with him;
But he already is too insolent;
[370]
And it were better parch in Afric sun
Than in the pride and salt scorn of his eyes,
Should he scape Hector fair. If he were foil'd,
Why, then we do our main opinion crush
In taint of our best man. No, make a lott'ry;
[375]
And, by device, let blockish Ajax draw
The sort to fight with Hector. Among ourselves
Give him allowance for the better man;
For that will physic the great Myrmidon,
Who broils in loud applause, and make him fall
[380]
His crest, that prouder than blue Iris bends.
If the dull brainless Ajax come safe off,
[We'll dress him up in voices; if he fail,
Yet go we under our opinion still
That we have better men. But, hit or miss,
[385]
Our project's life this shape of sense assumes –
Ajax employ'd plucks down Achilles' plumes.
NESTOR Now, Ulysses, I begin to relish thy advice;
And I will give a taste thereof forthwith
To Agamemnon. Go we to him straight.
[390]
Two curs shall tame each other: pride alone
Must tarre the mastiffs on, as 'twere their bone.
[ _Exeunt_.
#### ACT TWO
**SCENE I**. _The Grecian camp_.
_Enter AJAX and THERSITES_.
AJAX Thersites!
THERSITES Agamemnon – how if he had boils full, all over, generally?
AJAX Thersites!
[6]
THERSITES And those boils did run – say so. Did not the general run then? Were not that a botchy core?
AJAX Dog!
THERSITES Then there would come some matter from him; I see none now.
AJAX Thou bitch-wolfs son, canst thou not hear? Feel, then. [ _Strikes him_.
[13]
THERSITES The plague of Greece upon thee, thou mongrel beef-witted lord!
AJAX Speak, then, thou whinid'st leaven, speak. I will beat thee into handsomeness.
THERSITES I shall sooner rail thee into wit and holiness; but I think thy horse will sooner con an oration than thou learn a prayer without book. Thou canst strike, canst thou? A red murrain o' thy jade's tricks!
[20]
AJAX Toadstool, learn me the proclamation.
THERSITES Dost thou think I have no sense, thou strikest me thus?
AJAX The proclamation!
THERSITES Thou art proclaim'd a fool, I think.
[25]
AJAX Do not, porpentine, do not; my fingers itch.
THERSITES I would thou didst itch from head to foot and I had the scratching of thee; I would make thee the loathsomest scab in Greece. When thou art forth in the incursions, thou strikest as slow as another.
[30]
AJAX I say, the proclamation.
THERSITES Thou grumblest and railest every hour on Achilles; and thou art as full of envy at his greatness as Cerberus is at Prosperpina's beauty – ay, that thou bark'st at him.
AJAX Mistress Thersites!
[35]
THERSITES Thou shouldst strike him.
AJAX Cobloaf!
THERSITES he would pun thee into shivers with his fist, as a sailor breaks a biscuit.
AJAX You whoreson cur! [ _Strikes him_.
[40]
THERSITES Do, do.
AJAX Thou stool for a witch!
THERSITES Ay, do, do; thou sodden-witted lord! Thou hast no more brain than I have in mine elbows; an assinico may tutor thee. You scurvy valiant ass! Thou art here but to thrash Troyans, and thou art bought and sold among those of any wit like a barbarian slave. If thou use to beat me, I will begin at thy heel and tell what thou art by inches, thou thing of no bowels, thou!
AJAX You dog!
[50]
THERSITES You scurvy lord!
AJAX You cur! [ _Strikes him_.
THERSITES Mars his idiot! Do, rudeness; do, camel; do, do.
_Enter ACHILLES and PATROCLUS_.
ACHILLES Why, how now, Ajax! Wherefore do you thus?
How now, Thersites! What's the matter, man?
[55]
THERSITES You see him there, do you?
ACHILLES Ay; what's the matter?
THERSITES Nay, look upon him.
ACHILLES So I do. What's the matter?
THERSITES Nay, but regard him well.
[60]
ACHILLES Well! why, so I do.
THERSITES But yet you look not well upon him; for who some ever you take him to be, he is Ajax.
ACHILLES I know that, fool.
THERSITES Ay, but that fool knows not himself.
[65]
AJAX Therefore I beat thee.
[72]
THERSITES Lo, lo, lo, lo, what modicums of wit he utters! His evasions have ears thus long. I have bobb'd his brain more than he has beat my bones. I will buy nine sparrows for a penny, and his pia mater is not worth the ninth part of a sparrow. This lord, Achilles, Ajax – who wears his wit in his belly and his guts in his head – I'll tell you what I say of him.
ACHILLES What?
THERSITES I say this Ajax – [ _Ajax offers to strike him_.
[75]
ACHILLES Nay, good Ajax.
THERSITES Has not so much wit –
ACHILLES Nay. I must hold you.
THERSITES As will stop the eye of Helen's needle, for whom he comes to fight.
[80]
ACHILLES Peace, fool!
THERSITES I would have peace and quietness, but the fool will not – he there; that he; look you there.
AJAX O thou damned cur! I shall –
ACHILLES Will you set your wit to a fool's?
[85]
THERSITES No, I warrant you; the fool's will shame it.
PATROCLUS Good words, Thersites.
ACHILLES What's the quarrel?
AJAX I bade the vile owl go learn me the tenour of the proclamation, and he rails upon me.
[90]
THERSITES I serve thee not.
AJAX Well, go to, go to.
THERSITES I serve here voluntary.
[95]
ACHILLES Your last service was suff'rance; 'twas not voluntary. No man is beaten voluntary. Ajax was here the voluntary, and you as under an impress.
THERSITES E'en so; a great deal of your wit too lies in your sinews, or else there be liars. Hector shall have a great catch an he knock out either of your brains: a were as good crack a fusty nut with no kernel.
[100]
ACHILLES What, with me too, Thersites?
THERSITES There's Ulysses and old Nestor – whose wit was mouldy ere your grandsires had nails on their toes – yoke you like draught oxen, and make you plough up the wars.
ACHILLES What, what?
[105]
THERSITES Yes, good sooth. To Achilles, to Ajax, to –
AJAX I shall cut out your tongue.
THERSITES 'Tis no matter; I shall speak as much as thou afterwards.
PATROCLUS No more words, Thersites; peace!
[111]
THERSITES I will hold my peace when Achilles' brach bids me, shall I?
ACHILLES There's for you, Patroclus.
THERSITES I will see you hang'd like clotpoles ere
I come any more to your tents. I will keep where there is wit stirring, and leave the faction of fools.
[ _Exit_.
[116]
PATROCLUS A good riddance.
ACHILLES Marry, this, sir, is proclaim'd through all our host,
That Hector, by the fifth hour of the sun,
Will with a trumpet 'twixt our tents and Troy,
[120]
To-morrow morning, call some knight to arms
That hath a stomach; and such a one that dare
Maintain I know not what; 'tis trash. Farewell.
AJAX Farewell. Who shall answer him?
ACHILLES I know not; 'tis put to lott'ry. Otherwise
[125]
He knew his man.
AJAX O, meaning you! I will go learn more of it.
[ _Exeunt_.
**SCENE II**. _Troy. Priam's palace_.
_Enter PRIAM, HECTOR, TROILUS, PARIS and HELENUS_.
PRIAM After so many hours, lives, speeches, spent,
Thus once again says Nestor from the Greeks:
'Deliver Helen, and all damage else –
As honour, loss of time, travail, expense,
[5]
Wounds, friends, and what else dear that is consum'd
In hot digestion of this cormorant war –
Shall be struck off. Hector, what say you to't?
HECTOR Though no man lesser fears the Greeks than I,
As far as toucheth my particular,
[10]
Yet, dread Priam,
There is no lady of more softer bowels,
More spongy to suck in the sense of fear,
More ready to cry out 'Who knows what follows?'
Than Hector is. The wound of peace is surety,
[15]
Surety secure; but modest doubt is call'd
The beacon of the wise, the tent that searches
To th' bottom of the worst. Let Helen go.
Since the first sword was drawn about this question,
Every tithe soul 'mongst many thousand dismes
[20]
Hath been as dear as Helen – I mean, of ours.
If we have lost so many tenths of ours
To guard a thing not ours, nor worth to us,
Had it our name, the value of one ten,
What merit's in that reason which denies
The yielding of her up?
[25]
TROILUS Fie, fie, my brother!
So great as our dread father's, in a scale
Of common ounces? Will you with counters sum
The past-proportion of his infinite,
[30]
And buckle in a waist most fathomless
With spans and inches so diminutive
As fears and reasons? Fie, for godly shame!
HELENUS No marvel though you bite so sharp at reasons,
You are so empty of them. Should not our father
[35]
Bear the great sway of his affairs with reasons,
Because your speech hath none that tells him so?
TROILUS You are for dreams and slumbers, brother priest;
You fur your gloves with reason. Here are your reasons:
You know an enemy intends you harm;
[40]
You know a sword employ'd is perilous,
And reason flies the object of all harm.
Who marvels, then, when Helenus beholds
A Grecian and his sword, if he do set
The very wings of reason to his heels
[45]
And fly like chidden Mercury from Jove,
Or like a star disorb'd? Nay, if we talk of reason,
Let's shut our gates and sleep. Manhood and honour
Should have hare hearts, would they but fat their thoughts
With this cramm'd reason. Reason and respect
[50]
Make livers pale and lustihood deject.
HECTOR Brother, she is not worth what she doth cost
The keeping.
TROILUS What's aught but as 'lis valued?
HECTOR But value dwells not in particular will:
It holds his estimate and dignity
[55]
As well wherein 'tis precious of itself
As in the prizer. 'Tis mad idolatry
To make the service greater than the god;
And the will dotes that is attributive
To what infectiously itself affects,
[60]
Without some image of th' affected merit.
TROILUS I take to-day a wife, and my election
Is led on in the conduct of my will;
My will enkindled by mine eyes and ears,
Two traded pilots 'twixt the dangerous shores
[65]
Of will and judgement: how may I avoid,
Although my will distaste what it elected,
The wife I chose? There can be no evasion
To blench from this and to stand firm by honour.
We turn not back the silks upon the merchant
[70]
When we have soil'd them; nor the remainder viands
We do not throw in unrespective sieve,
Because we now are full. It was thought meet
Paris should do some vengeance on the Greeks;
Your breath with full consent bellied his sails;
[75]
The seas and winds, old wranglers, took a truce,
And did him service. He touch'd the ports desir'd;
And for an old aunt whom the Greeks held captive
He brought a Grecian queen, whose youth and freshness
Wrinkles Apollo's, and makes stale the morning.
[80]
Why keep we her? The Grecians keep our aunt.
Is she worth keeping? Why, she is a pearl
Whose price hath launch'd above a thousand ships,
And turn'd crown'd kings to merchants.
If you'll avouch 'twas wisdom Paris went –
[85]
As you must needs, for you all cried 'Go, go' –
If you'll confess he brought home worthy prize –
As you must needs, for you all clapp'd your hands,
And cried 'Inestimable!' – why do you now
The issue of your proper wisdoms rate,
[90]
And do a deed that never fortune did –
Beggar the estimation which you priz'd
Richer than sea and land? O theft most base,
That we have stol'n what we do fear to keep!
But thieves unworthy of a thing so stol'n
[95]
That in their country did them that disgrace
We fear to warrant in our native place!
CASSANDRA [ _Within_ ] Cry, Troyans, cry.
PRIAM What noise, what shriek is this?
TROILUS 'Tis our mad sister; I do know her voice.
CASSANDRA [ _Within_ ] Cry, Troyans.
[100]
HECTOR It is Cassandra.
_Enter CASSANDRA raving_.
CASSANDRA Cry, Troyans, cry. Lend me ten thousand eyes,
And I will fill them with prophetic tears.
HECTOR Peace, sister, peace.
CASSANDRA Virgins and boys, mid-age and wrinkled eld,
[105]
Soft infancy, that nothing canst but cry,
Add to my clamours. Let us pay betimes
A moiety of that mass of moan to come.
Cry, Troyans, cry. Practise your eyes with tears.
Troy must not be, nor goodly Ilion stand;
[110]
Our firebrand brother, Paris, burns us all.
Cry, Troyans, cry, A Helen and a woe!
Cry, cry. Troy burns, or else let Helen go.
[ _Exit_.
HECTOR Now, youthful Troilus, do not these high strains
Of divination in our sister work
[115]
Some touches of remorse, or is your blood
So madly hot that no discourse of reason,
Nor fear of bad success in a bad cause,
Can qualify the same?
TROILUS Why, brother Hector,
We may not think the justness of each act
[120]
Such and no other than event doth form it;
Nor once deject the courage of our minds
Because Cassandra's mad. Her brain-sick raptures
Cannot distaste the goodness of a quarrel
Which hath our several honours all engag'd
[125]
To make it gracious. For my private part,
I am no more touch'd than all Priam's sons;
And Jove forbid there should be done amongst us
Such things as might offend the weakest spleen
To fight for and maintain.
[130]
PARIS Else might the world convince of levity
As well my undertakings as your counsels;
But I attest the gods, your full consent
Gave wings to my propension, and cut off
All fears attending on so dire a project.
[135]
For what, alas, can these my single arms?
What propugnation is in one man's valour
To stand the push and enmity of those
This quarrel would excite? Yet, I protest,
Were I alone to pass the difficulties,
[140]
And had as ample power as I have will,
Paris should ne'er retract what he hath done
Nor faint in the pursuit.
PRIAM Paris, you speak
Like one besotted on your sweet delights.
You have the honey still, but these the gall;
[145]
So to be valiant is no praise at all.
PARIS Sir, I propose not merely to myself
The pleasures such a beauty brings with it;
But I would have the soil of her fair rape
Wip'd off in honourable keeping her.
[150]
What treason were it to the ransack'd queen,
Disgrace to your great worths, and shame to me,
Now to deliver her possession up
On terms of base compulsion! Can it be
That so degenerate a strain as this
[155]
Should once set footing in your generous bosoms?
There's not the meanest spirit on our party
Without a heart to dare or sword to draw
When Helen is defended; nor none so noble
Whose life were ill bestow'd or death unfam'd
[160]
Where Helen is the subject. Then, I say,
Well may we fight for her whom we know well
The world's large spaces cannot parallel.
HECTOR Paris and Troilus, you have both said well;
And on the cause and question now in hand
[165]
Have gloz'd, but superficially; not much
Unlike young men, whom Aristotle thought
Unfit to hear moral philosophy.
The reasons you allege do more conduce
To the hot passion of distemp'red blood
[170]
Than to make up a free determination
'Twixt right and wrong; for pleasure and revenge
Have ears more deaf than adders to the voice
Of any true decision. Nature craves
All dues be rend'red to their owners. Now,
[175]
What nearer debt in all humanity
Than wife is to the husband? If this law
Of nature be corrupted through affection;
And that great minds, of partial indulgence
To their benumbed wills, resist the same;
[180]
There is a law in each well-order'd nation
To curb those raging appetites that are
Most disobedient and refractory.
If Helen, then, be wife to Sparta's king –
As it is known she is – these moral laws
[185]
Of nature and of nations speak aloud
To have her back return'd. Thus to persist
In doing wrong extenuates not wrong,
But makes it much more heavy. Hector's opinion
Is this, in way of truth. Yet, ne'er the less,
[190]
My spritely brethren, I propend to you
In resolution to keep Helen still;
For 'tis a cause that hath no mean dependence
Upon our joint and several dignities.
TROILUS Why, there you touch'd the life of our design.
[195]
Were it not glory that we more affected
Than the performance of our heaving spleens,
I would not wish a drop of Troyan blood
Spent more in her defence. But, worthy Hector,
She is a theme of honour and renown,
[200]
A spur to valiant and magnanimous deeds,
Whose present courage may beat down our foes,
And fame in time to come canonize us;
For I presume brave Hector would not lose
So rich advantage of a promis'd glory
[205]
As smiles upon the forehead of this action
For the wide world's revenue.
HECTOR I am yours,
You valiant offspring of great Priamus.
I have a roisting challenge sent amongst
The dull and factious nobles of the Greeks
[210]
Will strike amazement to their drowsy spirits.
I was advertis'd their great general slept,
Whilst emulation in the army crept.
This, I presume, will wake him.
[ _Exeunt_.
**SCENE III**. _The Grecian camp. Before the tent of Achilles_.
_Enter THERSITES, solus_.
THERSITES How now, Thersites! What, lost in the labyrinth of thy fury? Shall the elephant Ajax carry it thus? He beats me, and I rail at him. O worthy satisfaction! Would it were otherwise: that I could beat him, whilst he rail'd at me! 'Sfoot, I'll learn to conjure and raise devils, but I'll see some issue of my spiteful execrations. Then there's Achilles, a rare engineer! If Troy be not taken till these two undermine it, the walls will stand till they fall of themselves. O thou great thunder-darter of Olympus, forget that thou art Jove, the king of gods, and, Mercury, lose all the serpentine craft of thy caduceus, if ye take not that little little less-than-little wit from them that they have! which short-arm'd ignorance itself knows is so abundant scarce, it will not in circumvention deliver a fly from a spider without drawing their massy irons and cutting the web. After this, the vengeance on the whole camp! or, rather, the Neapolitan bone-ache! for that, methinks, is the curse depending on those that war for a placket. I have said my prayers; and devil Envy say
[20]
'Amen'. What ho! my Lord Achilles!
_Enter PATROCLUS_.
PATROCLUS Who's there? Thersites! Good
Thersites, come in and rail.
[31]
THERSITES If I could 'a rememb'red a gilt counterfeit, thou wouldst not have slipp'd out of my contemplation; but it is no matter; thyself upon thyself! The common curse of mankind, folly and ignorance, be thine in great revenue! Heaven bless thee from a tutor, and discipline come not near thee! Let thy blood be thy direction till thy death. Then if she that lays thee out says thou art a fair corse, I'll be sworn and sworn upon't she never shrouded any but lazars. Amen. Where's Achilles?
PATROCLUS What, art thou devout? Wast thou in prayer?
THERSITES Ay, the heavens hear me!
PATROCLUS Amen.
_Enter ACHILLES_.
[35]
ACHILLES Who's there?
PATROCLUS Thersites, my lord.
[40]
ACHILLES Where, where? O, where? Art thou come? Why, my cheese, my digestion, why hast thou not served thyself in to my table so many meals? Come, what's Agamemnon?
THERSITES Thy commander, Achilles. Then tell me, Patroclus, what's Achilles?
PATROCLUS Thy lord, Thersites. Then tell me, I pray thee, what's Thersites?
[46]
THERSITES Thy knower, Patroclus. Then tell me, Patroclus, what art thou?
PATROCLUS Thou must tell that knowest.
ACHILLES O, tell, tell!
[51]
THERSITES I'll decline the whole question. Agememnon commands Achilles; Achilles is my lord; I am Patroclus' knower; and Patroclus is a fool.
PATROCLUS You rascal!
THERSITES Peace, fool! I have not done.
ACHILLES He is a privileg'd man. Proceed, Thersites.
[56]
THERSITES Agamemnon is a fool; Achilles is a fool; Thersites is a fool; and, as aforesaid, Patroclus is a fool.
ACHILLES Derive this; come.
[61]
THERSITES Agamemnon is a fool to offer to command Achilles; Achilles is a fool to be commanded of Agamemnon; Thersites is a fool to serve such a fool; and this Patroclus is a fool positive.
PATROCLUS Why am I a fool?
THERSITES Make that demand of the Creator. It suffices me thou art. Look you, who comes here?
[66]
ACHILLES Come, Patroclus, I'll speak with nobody. Come in with me, Thersites. [ _Exit_.
THERSITES Here is such patchery, such juggling, and such knavery. All the argument is a whore and a cuckold – a good quarrel to draw emulous factions and bleed to death upon. Now the dry serpigo on the subject, and war and lechery confound all!
[ _Exit_.
_Enter AGAMEMNON, ULYSSES, NESTOR, DIOMEDES, AJAX and CALCAS_
[72]
AGAMEMNON Where is Achilles?
PATROCLUS Within his tent; but ill-dispos'd, my lord.
AGAMEMNON Let it be known to him that we are here.
[75]
He shent our messengers; and we lay by
Our appertainings, visiting of him.
Let him be told so; lest, perchance, he think
We dare not move the question of our place
Or know not what we are.
PATROCLUS I shall say so to him.
[ _Exit_.
[80]
ULYSSES We saw him at the opening of his tent.
He is not sick.
[85]
AJAX Yes, lion-sick, sick of proud heart. You may call it melancholy, if you will favour the man; but, by my head, 'tis pride. But why, why? let him show us a cause. A word, my lord.
[ _Takes Agamemnon aside_.
NESTOR What moves Ajax thus to bay at him?
ULYSSES Achilles hath inveigled his fool from him.
NESTOR Who, Thersites?
[89]
ULYSSES He.
NESTOR Then will Ajax lack matter, if he have lost his argument.
ULYSSES No; you see he is his argument that has his argument – Achilles.
[86]
NESTOR All the better; their fraction is more our wish than their fraction. But it was a strong composure a fool could disunite!
ULYSSES The amity that wisdom knits not, folly may easily untie.
_Re-enter PATROCLUS_.
Here comes Patroclus.
[100]
NESTOR No Achilles with him.
ULYSSES The elephant hath joints, but none for courtesy; his legs are legs for necessity, not for flexure.
PATROCLUS Achilles bids me say he is much sorry
If any thing more than your sport and pleasure
[105]
Did move your greatness and this noble state
To call upon him; he hopes it is no other
But for your health and your digestion sake,
An after-dinner's breath.
AGAMEMNON Hear you, Patroclus.
We are too well acquainted with these answers;
[110]
But his evasion, wing'd thus swift with scorn,
Cannot outfly our apprehensions.
Much attribute he hath, and much the reason
Why we ascribe it to him. Yet all his virtues,
Not virtuously on his own part beheld,
[115]
Do in our eyes begin to lose their gloss;
Yea, like fair fruit in an unwholesome dish,
Are like to rot untasted. Go and tell him
We come to speak with him; and you shall not sin
If you do say we think him over-proud
[120]
And under-honest, in self-assumption greater
Than in the note of judgment; and worthier than himself
Here tend the savage strangeness he puts on,
Disguise the holy strength of their command,
And underwrite in an observing kind
[125]
His humorous predominance; yea, watch
His pettish lunes, his ebbs, his flows, as if
The passage and whole carriage of this action
Rode on his tide. Go tell him this, and add
That if he overhold his price so much
[130]
We'll none of him, but let him, like an engine
Not portable, lie under this report:
Bring action hither; this cannot go to war.
A stirring dwarf we do allowance give
Before a sleeping giant. Tell him so.
[135]
PATROCLUS I shall, and bring his answer presently. [ _Exit_.
AGAMEMNON In second voice we'll not be satisfied;
We come to speak with him. Ulysses, enter you.
[ _Exit Ulysses_.
AJAX What is he more than another?
AGAMEMNON No more than what he thinks he is.
AJAX Is he so much? Do you not think he thinks himself a better man than I am?
[142]
AGAMEMNON No question.
AJAX Will you subscribe his thought and say he is?
[146]
AGAMEMNON No, noble Ajax; you are as strong, as valiant, as wise, no less noble, much more gentle, and altogether more tractable.
AJAX Why should a man be proud? How doth pride grow? I know not what pride is.
[153]
AGAMEMNON Your mind is the clearer, Ajax, and your virtues the fairer. He that is proud eats up himself. P'-ide is his own glass, his own trumpet, his own chronicle; and whatever praises itself but in the deed devours the deed in the praise.
_Re-enter ULYSSES_.
AJAX I do hate a proud man as I do hate the engend'ring of toads.
NESTOR [ _Aside_ ] And yet he loves himself: is't not strange?
ULYSSES Achilles will not to the field to-morrow.
AGAMEMNON What's his excuse?
ULYSSES He doth rely on none;
But carries on the stream of his dispose,
[160]
Without observance or respect of any,
In will peculiar and in self-admission.
AGAMEMNON Why will he not, upon our fair request,
Untent his person and share the air with us?
ULYSSES Things small as nothing, for request's sake only,
[165]
He makes important; possess'd he is with greatness,
And speaks not to himself but with a pride
That quarrels at self-breath. Imagin'd worth
Holds in his blood such swol'n and hot discourse
That 'twixt his mental and his active parts
[170]
Kingdom'd Achilles in commotion rages,
And batters down himself. What should I say?
He is so plaguy proud that the death tokens of it
Cry 'No recovery'.
AGAMEMNON Let Ajax go to him.
Dear lord, go you and greet him in his tent.
[175]
'Tis said he holds you well; and will be led
At your request a little from himself.
ULYSSES O Agamemnon, let it not be so!
We'll consecrate the steps that Ajax makes
When they go from Achilles. Shall the proud lord
[180]
That bastes his arrogance with his own seam
And never suffers matter of the world
Enter his thoughts, save such as doth revolve
And ruminate himself – shall he be worshipp'd
Of that we hold an idol more than he?
[185]
No, this thrice-worthy and right valiant lord
Shall not so stale his palm, nobly acquir'd,
Nor, by my will, assubjugate his merit,
As amply titled as Achilles is,
By going to Achilles.
[190]
That were to enlard his fat-already pride,
And add more coals to Cancer when he burns
With entertaining great Hyperion.
This lord go to him! Jupiter forbid,
And say in thunder 'Achilles go to him'.
[195]
NESTOR [ _Aside_ ] O, this is well! He rubs the vein of him.
DIOMEDES [ _Aside_ ] And how his silence drinks up this applause!
AJAX If I go to him, with my armed fist I'll pash him o'er the face.
AGAMEMNON O, no, you shall not go.
[200]
AJAX An 'a be proud with me I'll pheeze his pride.
Let me go to him.
ULYSSES Not for the worth that hangs upon our quarrel.
AJAX A paltry, insolent fellow!
NESTOR [ _Aside_ ] How he describes himself!
[205]
AJAX Can he not be sociable?
ULYSSES [ _Aside_ ] The raven chides blackness.
AJAX I'll let his humours blood.
AGAMEMNON [ _Aside_ ] He will be the physician that should be the patient.
[210]
AJAX An all men were a my mind –
ULYSSES [ _Aside_ ] Wit would be out of fashion.
AJAX 'A should not bear it so, 'a should eat's words first. Shall pride carry it?
NESTOR [ _Aside_ ] An 'twould, you'd carry half.
ULYSSES [ _Aside_ ] 'A would have ten shares.
[216]
AJAX I will knead him, I'll make him supple.
NESTOR [ _Aside_ ] He's not yet through warm.
Force him with praises; pour in, pour in; his ambition is dry.
ULYSSES [ _To Agamemnon_ ] My lord, you feed too much on this dislike.
[220]
NESTOR Our noble general, do not do so.
DIOMEDES You must prepare to fight without Achilles.
ULYSSES Why 'tis this naming of him does him harm.
Here is a man – but 'tis before his face;
I will be silent.
NESTOR Wherefore should you so?
[225]
He is not emulous, as Achilles is.
ULYSSES Know the whole world, he is as valiant.
AJAX A whoreson dog, that shall palter with us thus! Would he were a Troyan!
NESTOR What a vice were it in Ajax now –
[230]
ULYSSES If he were proud.
DIOMEDES Or covetous of praise.
ULYSSES Ay, or surly borne.
DIOMEDES Or strange, or self-affected.
[135]
ULYSSES Thank the heavens, lord, thou art of sweet composure;
Praise him that gat thee, she that gave thee suck;
Fam'd be thy tutor, and thy parts of nature
Thrice-fam'd beyond, beyond all erudition;
But he that disciplin'd thine arms to fight –
Let Mars divide eternity in twain
[240]
And give him half; and, for thy vigour,
Bull-bearing Milo his addition yield
To sinewy Ajax. I will not praise thy wisdom,
Which, like a bourn, a pale, a shore, confines
Thy spacious and dilated parts. Here's Nestor,
[245]
Instructed by the antiquary times –
He must, he is, he cannot but be wise;
But pardon, father Nestor, were your days
As green as Ajax' and your brain so temper'd,
You should not have the eminence of him,
But be as Ajax.
[250]
AJAX Shall I call you father?
NESTOR Ay, my good son.
DIOMEDES Be rul'd by him. Lord Ajax.
ULYSSES There is no tarrying here; the hart Achilles
Keeps thicket. Please it our great general
To call together all his state of war;
[255]
Fresh kings are come to Troy. To-morrow
We must with all our main of power stand fast;
And here's a lord – come knights from east to west
And cull their flower, Ajax shall cope the best.
AGAMEMNON Go we to council. Let Achilles sleep.
[260]
Light boats sail swift, though greater hulks draw deep. [ _Exeunt_.
#### ACT THREE
**SCENE I**. _Troy. Priam's palace_.
_Music sounds within. Enter PANDARUS and a Servant_
PANDARUS Friend, you – pray you, a word. Do you not follow the young Lord Paris?
SERVANT Ay, sir, when he goes before me.
PANDARUS You depend upon him, I mean?
[5]
SERVANT Sir, I do depend upon the lord.
PANDARUS You depend upon a notable gentleman; I must needs praise him.
SERVANT The lord be praised!
PANDARUS You know me, do you not?
[10]
SERVANT Faith, sir, superficially.
PANDARUS Friend, know me better: I am the Lord Pandarus.
SERVANT I hope I shall know your honour better.
PANDARUS I do desire it.
[14]
SERVANT You are in the state of grace.
PANDARUS Grace! Not so, friend; honour and lordship are my titles. What music is this?
SERVANT I do but partly know, sir; it is music in parts.
PANDARUS Know you the musicians?
SERVANT Wholly, sir.
[20]
PANDARUS Who play they to?
SERVANT To the hearers, sir.
PANDARUS At whose pleasure, friend?
SERVANT At mine, sir, and theirs that love music.
PANDARUS Command, I mean, friend.
[25]
SERVANT Who shall I command, sir?
PANDARUS Friend, we understand not one another: I am too courtly, and thou art too cunning. At whose request do these men play?
[32]
SERVANT That's to't, indeed, sir. Marry, sir, at the request of Paris my lord, who is there in person; with him the mortal Venus, the heart-blood of beauty, love's invisible soul –
PANDARUS Who, my cousin, Cressida?
[35]
SERVANT No, sir, Helen. Could not you find out that by her attributes?
[39]
PANDARUS It should seem, fellow, that thou hast not seen the Lady Cressida. I come to speak with Paris from the Prince Troilus; I will make a complimental assault upon him, for my business seethes.
SERVANT Sodden business! There's a stew'd phrase indeed!
_Enter PARIS and HELEN, attended_.
PANDARUS Fair be to you, my lord, and to all this fair company! Fair desires, in all fair measure, fairly guide them – especially to you, fair queen!
[44]
Fair thoughts be your fair pillow.
HELEN Dear lord, you are full of fair words.
[47]
PANDARUS You speak your fair pleasure, sweet queen. Fair prince, here is good broken music.
PARIS You have broke it, cousin; and by my life, you shall make it whole again; you shall piece it out with a piece of your performance.
[50]
HELEN He is full of harmony.
PANDARUS Truly, lady, no.
HELEN O, sir–
PANDARUS Rude, in sooth; in good sooth, very rude.
PARIS Well said, my lord. Well, you say so in fits.
[56]
PANDARUS I have business to my lord, dear queen. My lord, will you vouchsafe me a word?
HELEN Nay, this shall not hedge us out. We'll hear you sing, certainly.
[61]
PANDARUS Well, sweet queen, you are pleasant with me. But, marry, thus, my lord: my dear lord and most esteemed friend, your brother Troilus –
HELEN My Lord Pandarus, honey-sweet lord –
PANDARUS Go to, sweet queen, go to – commends himself most affectionately to you –
[66]
HELEN You shall not bob us out of our melody. If you do, our melancholy upon your head!
PANDARUS Sweet queen, sweet queen; that's a sweet queen, i' faith.
HELEN And to make a sweet lady sad is a sour offence.
[74]
PANDARUS Nay, that shall not serve your turn; that shall it not, in truth, la. Nay, I care not for such words; no, no. – And, my lord, he desires you that, if the King call for him at supper, you will make his excuse.
HELEN My Lord Pandarus!
PANDARUS What says my sweet queen, my very very sweet queen?
PARIS What exploit's in hand? Where sups he to-night?
HELEN Nay, but, my lord –
PANDARUS What says my sweet queen? – My cousin will fall out with you.
[80]
HELEN You must not know where he sups.
PARIS I'll lay my life, with my disposer Cressida.
PANDARUS No, no, no such matter; you are wide.
Come, your disposer is sick.
PARIS Well, I'll make's excuse.
[86]
PANDARUS Ay, good my lord. Why should you say Cressida? No, your poor disposer's sick.
PARIS I spy.
PANDARUS You spy! What do you spy? – Come, give me an instrument. Now, sweet queen.
[90]
HELEN Why, this is kindly done.
PANDARUS My niece is horribly in love with a thing you have, sweet queen.
HELEN She shall have it, my lord, if it be not my
[94]
Lord Paris.
PANDARUS He! No, she'll none of him; they two are twain.
HELEN Falling in, after falling out, may make them three.
[99]
PANDARUS Come, come. I'll hear no more of this;
I'll sing you a song now.
HELEN Ay, ay, prithee now. By my troth, sweet lord, thou hast a fine forehead.
[105]
PANDARUS Ay, you may, you may.
HELEN Let thy song be love. This love will undo us all. O Cupid, Cupid, Cupid!
PANDARUS Love! Ay, that it shall, i' faith.
PARIS Ay, good now, love, love, nothing but love.
PANDARUS In good troth, it begins so. [ _Sings_ ]
Love, love, nothing but love, still love, still more!
For, oh, love's bow
[110]
Shoots buck and doe;
The shaft confounds
Not that it wounds,
But tickles still the sore.
These lovers cry, O ho, they die!
[115]
Yet that which seems the wound to kill
Doth turn O ho! to ha! ha! he!
So dying love lives still.
O ho! a while, but ha! ha! ha!
O ho! groans out for ha! ha! ha! – hey ho!
[115]
HELEN In love, i' faith, to the very tip of the nose.
[124]
PARIS He eats nothing but doves, love; and that breeds hot blood, and hot blood begets hot thoughts, and hot thoughts beget hot deeds, and hot deeds is love.
PANDARUS Is this the generation of love: hot blood, hot thoughts, and hot deeds? Why, they are vipers. Is love a generation of vipers? Sweet lord, who's a-field today?
[131]
PARIS Hector, Deiphobus, Helenus, Antenor, and all the gallantry of Troy. I would fain have arm'd to-day, but my Nell would not have it so. How chance my brother Troilus went not?
HELEN He hangs the lip at something. You know all, Lord Pandarus.
[136]
PANDARUS Not I, honey-sweet queen. I long to hear how they sped to-day. You'll remember your brother's excuse?
PARIS To a hair.
PANDARUS Farewell, sweet queen.
HELEN Commend me to your niece.
[140]
PANDARUS I will, sweet queen.
[ _Exit_.
[ _Sound a retreat_.
PARIS They're come from the field. Let us to Priam's hall
To greet the warriors. Sweet Helen, I must woo you
To help unarm our Hector. His stubborn buckles,
With these your white enchanting fingers touch'd,
[145]
Shall more obey than to the edge of steel
Or force of Greekish sinews; you shall do more
Than all the island kings – disarm great Hector.
HELEN Twill make us proud to be his servant, Paris;
Yea, what he shall receive of us in duty
[150]
Gives us more palm in beauty than we have,
Yea, overshines ourself.
PARIS Sweet, above thought I love thee.
[ _Exeunt_.
**SCENE II**. _Troy. Pandarus' orchard_.
_Enter PANDARUS and Troilus' Boy, meeting_.
PANDARUS How now! Where's thy master?
At my cousin Cressida's?
BOY No, sir; he stays for you to conduct him thither.
_Enter TROILUS_.
[5]
PANDARUS O, here he comes. How now, how now!
TROILUS Sirrah, walk off. [ _Exit Boy_.
PANDARUS Have you seen my cousin?
TROILUS No, Pandarus. I stalk about her door
Like a strange soul upon the Stygian banks
[10]
Staying for waftage. O, be thou my Charon,
And give me swift transportance to these fields
Where I may wallow in the lily beds
Propos'd for the deserver! O gentle Pandar,
From Cupid's shoulder pluck his painted wings,
[15]
And fly with me to Cressid!
PANDARUS Walk here i' th' orchard, I'll bring her straight.
[ _Exit_.
TROILUS I am giddy; expectation whirls me round.
Th' imaginary relish is so sweet
That it enchants my sense; what will it be
[20]
When that the wat'ry palate tastes indeed
Love's thrice-repured nectar? Death, I fear me;
Swooning destruction; or some joy too fine,
Too subtle-potent, tun'd too sharp in sweetness,
For the capacity of my ruder powers.
[25]
I fear it much; and I do fear besides
That I shall lose distinction in my joys;
As doth a battle, when they charge on heaps
The enemy flying.
_Re-enter PANDARUS_.
PANDARUS She's making her ready, she'll come straight; you must be witty now. She does so blush, and fetches her wind so short, as if she were fray'd with a sprite. I'll fetch her. It is the prettiest villain; she fetches her breath as short as a new ta'en sparrow.
[ _Exit_.
TROILUS Even such a passion doth embrace my bosom.
[35]
My heart beats thicker than a feverous pulse,
And all my powers do their bestowing lose,
Like vassalage at unawares encount'ring
The eye of majesty.
_Re-enter PANDARUS with CRESSIDA_.
[52]
PANDARUS Come, come, what need you blush? Shame's a baby. – Here she is now; swear the oaths now to her that you have sworn to me. – What, are you gone again? You must be watch'd ere you be made tame, must you? Come your ways, come your ways; an you draw backward, we'll put you i' th' fills. – Why do you not speak to her? – Come, draw this curtain and let's see your picture. Alas the day, how loath you are to offend daylight! An 'twere dark, you'd close sooner. So, so; rub on, and kiss the mistress. How now, a kiss in fee-farm! Build there, carpenter; the air is sweet. Nay, you shall fight your hearts out ere I part you. The falcon as the tercel, for all the ducks i' th' river. Go to, go to.
TROILUS You have bereft me of all words, lady.
PANDARUS Words pay no debts, give her deeds; but she'll bereave you o' th' deeds too, if she call your activity in question. What, billing again? Here's in witness whereof the parties interchangeably'. Come in, come in; I'll go get a fire.
[ _Exit_.
[59]
CRESSIDA Will you walk in, my lord?
TROILUS O Cressid, how often have I wish'd me thus!
CRESSIDA Wish'd, my lord! The gods grant – O my lord!
[64]
TROILUS What should they grant? What makes this pretty abruption? What too curious dreg espies my sweet lady in the fountain of our love?
CRESSIDA More dregs than water, if my fears have eyes.
TROILUS Fears make devils of cherubins; they never see truly.
[70]
CRESSIDA Blind fear, that seeing reason leads, finds safer footing than blind reason stumbling without fear. To fear the worst oft cures the worse.
TROILUS O, let my lady apprehend no fear! In all Cupid's pageant there is presented no monster.
CRESSIDA Nor nothing monstrous neither?
[80]
TROILUS Nothing, but our undertakings when we vow to weep seas, live in fire, eat rocks, tame tigers; thinking it harder for our mistress to devise imposition enough than for us to undergo any difficulty imposed. This is the monstruosity in love, lady, that the will is infinite, and the execution confin'd; that the desire is boundless, and the act a slave to limit.
[86]
CRESSIDA They say all lovers swear more performance than they are able, and yet reserve an ability that they never perform; vowing more than the perfection of ten, and discharging less than the tenth part of one. They that have the voice of lions and the act of hares, are they not monsters?
[95]
TROILUS Are there such? Such are not we. Praise us as we are tasted, allow us as we prove; our head shall go bare till merit crown it. No perfection in reversion shall have a praise in present. We will not name desert before his birth; and, being born, his addition shall be humble. Few words to fair faith: Troilus shall be such to Cressid as what envy can say worst shall be a mock for his truth; and what truth can speak truest not truer than Troilus.
CRESSIDA Will you walk in, my lord?
_Re-enter PANDARUS_.
PANDARUS What, blushing still? Have you not done talking yet?
[100]
CRESSIDA Well, uncle, what folly I commit, I dedicate to you.
PANDARUS I thank you for that; if my lord get a boy of you, you'll give him me. Be true to my lord; if he flinch, chide me for it.
[105]
TROILUS You know now your hostages: your uncle's word and my firm faith.
PANDARUS Nay, I'll give my word for her too: our kindred, though they be long ere they are wooed, they are constant being won, they are burs, I can tell you; they'll stick where they are thrown.
[110]
CRESSIDA Boldness comes to me now and brings me heart.
Prince Troilus, I have lov'd you night and day
For many weary months.
TROILUS Why was my Cressid then so hard to win?
CRESSIDA Hard to seem won; but I was won, my lord,
[115]
With the first glance that ever – pardon me.
If I confess much, you will play the tyrant.
I love you now; but till now not so much
But I might master it. In faith, I lie;
My thoughts were like unbridled children, grown
[120]
Too headstrong for their mother. See, we fools!
Why have I blabb'd? Who shall be true to us,
When we are so unsecret to ourselves?
But, though I lov'd you well, I woo'd you not;
And yet, good faith, I wish'd myself a man,
[125]
Or that we women had men's privilege
Of speaking first. Sweet, bid me hold my tongue,
For in this rapture I shall surely speak
The thing I shall repent. See, see, your silence,
Cunning in dumbness, from my weakness draws
[130]
My very soul of counsel. Stop my mouth.
TROILUS And shall, albeit sweet music issues thence.
PANDARUS Pretty, i' faith.
CRESSIDA My lord, I do beseech you, pardon me;
'Twas not my purpose thus to beg a kiss.
[135]
I am asham'd. O heavens! what have I done?
For this time will I take my leave, my lord.
TROILUS Your leave, sweet Cressid!
PANDARUS Leave! An you take leave till to-morrow morning –
CRESSIDA Pray you, content you.
[140]
TROILUS What offends you, lady?
CRESSIDA Sir, mine own company.
TROILUS You cannot shun yourself.
CRESSIDA Let me go and try.
I have a kind of self resides with you;
[145]
But an unkind self, that it self will leave
To be another's fool. I would be gone.
Where is my wit? I know not what I speak.
TROILUS Well know they what they speak that speak so wisely.
CRESSIDA Perchance, my lord, I show more craft than love;
[150]
And fell so roundly to a large confession
To angle for your thoughts; but you are wise –
Or else you love not; for to be wise and love
Exceeds man's might; that dwells with gods above.
TROILUS O that I thought it could be in a woman –
[155]
As, if it can, I will presume in you –
To feed for aye her lamp and flames of love;
To keep her constancy in plight and youth,
Outliving beauty's outward, with a mind
That doth renew swifter than blood decays!
[160]
Or that persuasion could but thus convince me
That my integrity and truth to you
Might be affronted with the match and weight
Of such a winnowed purity in love.
How were I then uplifted! but, alas,
[165]
I am as true as truth's simplicity,
And simpler than the infancy of truth.
CRESSIDA In that I'll war with you.
TROILUS O virtuous fight,
When right with right wars who shall be most right!
True swains in love shall in the world to come
[170]
Approve their truth by Troilus, when their rhymes,
Full of protest, of oath, and big compare,
Want similes, truth tir'd with iteration –
As true as steel, as plantage to the moon,
As sun to day, as turtle to her mate,
[175]
As iron to adamant, as earth to th' centre –
Yet, after all comparisons of truth,
As truth's authentic author to be cited,
'As true as Troilus' shall crown up the verse
And sanctify the numbers.
CRESSIDA Prophet may you be!
[180]
If I be false, or swerve a hair from truth,
When time is old and hath forgot itself,
When waterdrops have worn the stones of Troy,
And blind oblivion swallow'd cities up,
And mighty states characterless are grated
[185]
To dusty nothing – yet let memory
From false to false, among false maids in love,
Upbraid my falsehood when th' have said 'As false
As air, as water, wind, or sandy earth,
As fox to lamb, or wolf to heifer's calf,
[190]
Pard to the hind, or stepdame to her son' –
Yea, let them say, to stick the heart of falsehood,
'As false as Cressid'.
[200]
PANDARUS Go to, a bargain made; seal it, seal it; I'll be the witness. Here I hold your hand; here my cousin's. If ever you prove false one to another, since I have taken such pains to bring you together, let all pitiful goers-between be call'd to the world's end after my name – call them all Pandars; let all constant men be Troiluses, all false women Cressids, and all brokers between Pandars. Say 'Amen'.
TROILUS Amen.
CRESSIDA Amen.
[205]
PANDARUS Amen. Whereupon I will show you a chamber and a bed; which bed, because it shall not speak of your pretty encounters, press it to death. Away!
And Cupid grant all tongue-tied maidens here,
Bed, chamber, pander, to provide this gear!
[ _Exeunt_.
**SCENE III**. _The Greek camp._
_Flourish. Enter AGAMEMNON, ULYSSES, DIOMEDES, NESTOR, AJAX, MENELAUS, and CALCHAS_.
CALCHAS Now, Princes, for the service I have done,
Th' advantage of the time prompts me aloud
To call for recompense. Appear it to your mind
That, through the sight I bear in things to come,
[5]
I have abandon'd Troy, left my possession,
Incurr'd a traitor's name, expos'd myself
From certain and possess'd conveniences
To doubtful fortunes, sequest'ring from me all
That time, acquaintance, custom, and condition,
[10]
Made tame and most familiar to my nature;
And here, to do you service, am become
As new into the world, strange, unacquainted –
I do beseech you, as in way of taste,
To give me now a little benefit
[15]
Out of those many regist'red in promise,
Which you say live to come in my behalf.
AGAMEMNON What wouldst thou of us, Troyan? Make demand.
CALCHAS You have a Troyan prisoner call'd Antenor,
Yesterday took; Troy holds him very dear.
[20]
Oft have you – often have you thanks therefore –
Desir'd my Cressid in right great exchange,
Whom Troy hath still denied; but this Antenor,
I know, is such a wrest in their affairs
That their negotiations all must slack
[25]
Wanting his manage; and they will almost
Give us a prince of blood, a son of Priam,
In change of him. Let him be sent, great Princes,
And he shall buy my daughter; and her presence
Shall quite strike off all service I have done
In most accepted pain.
[30]
AGAMEMNON Let Diomedes bear him,
And bring us Cressid hither. Calchas shall have
What he requests of us. Good Diomed,
Furnish you fairly for this interchange;
Withal, bring word if Hector will to-morrow
[35]
Be answer'd in his challenge. Ajax is ready.
DIOMEDES This shall I undertake; and 'tis a burden
Which I am proud to bear.
[ _Exeunt Diomedes and Calchas_.
_ACHILLES and PATROCLUS stand in their tent_.
ULYSSES Achilles stands i' th' entrance of his tent.
Please it our general pass strangely by him,
[40]
As if he were forgot; and, Princes all,
Lay negligent and loose regard upon him.
I will come last. 'Tis like hell question me
Why such unplausive eyes are bent, why turn'd on him?
If so, I have derision med'cinable
[45]
To use between your strangeness and his pride,
Which his own will shall have desire to drink.
It may do good. Pride hath no other glass
To show itself but pride; for supple knees
Feed arrogance and are the proud man's fees.
[50]
AGAMEMNON We'll execute your purpose, and put on
A form of strangeness as we pass along.
So do each lord; and either greet him not,
Or else disdainfully, which shall shake him more
Than if not look'd on. I will lead the way.
[55]
ACHILLES What comes the general to speak with me?
You know my mind, I'll fight no more 'gainst Troy.
AGAMEMNON What says Achilles? Would he aught with us?
NESTOR Would you, my lord, aught with the general?
ACHILLES No.
[60]
NESTOR Nothing, my lord.
AGAMEMNON The better.
[ _Exeunt Agamemnon and Nestor_.
ACHILLES Good day, good day.
MENELAUS How do you? How do you?
[ _Exit_.
ACHILLES What, does the cuckold scorn me?
[65]
AJAX How now, Patroclus?
ACHILLES Good morrow, Ajax.
AJAX Ha?
ACHILLES Good morrow.
AJAX Ay, and good next day too.
[ _Exit_.
[70]
ACHILLES What mean these fellows? Know they not Achilles?
PATROCLUS They pass by strangely. They were us'd to bend,
To send their smiles before them to Achilles,
To come as humbly as they us'd to creep
To holy altars.
ACHILLES What, am I poor of late?
[75]
'Tis certain, greatness, once fall'n out with fortune,
Must fall out with men too. What the declin'd is,
He shall as soon read in the eyes of others
As feel in his own fall; for men, like butterflies,
Show not their mealy wings but to the summer;
[80]
And not a man for being simply man
Hath any honour, but honour for those honours
That are without him, as place, riches, and favour,
Prizes of accident, as oft as merit;
Which when they fall, as being slippery standers,
[85]
The love that lean'd on them as slippery too,
Doth one pluck down another, and together
Die in the fall. But 'tis not so with me:
Fortune and I are friends; I do enjoy
At ample point all that I did possess
[90]
Save these men's looks; who do, methinks, find out
Something not worth in me such rich beholding
As they have often given. Here is Ulysses.
I'll interrupt his reading.
How now, Ulysses!
ULYSSES Now, great Thetis' son!
ACHILLES What are you reading?
[95]
ULYSSES A strange fellow here
Writes me that man – how dearly ever parted,
How much in having, or without or in –
Cannot make boast to have that which he hath,
Nor feels not what he owes, but by reflection;
[100]
As when his virtues shining upon others
Heat them, and they retort that heat again
To the first giver.
ACHILLES This is not strange, Ulysses.
The beauty that is borne here in the face
The bearer knows not, but commends itself
[105]
To others' eyes; nor doth the eye itself –
That most pure spirit of sense – behold itself,
Not going from itself; but eye to eye opposed
Salutes each other with each other's form;
For speculation turns not to itself
[110]
Till it hath travell'd, and is mirror'd there
Where it may see itself. This is not strange at all.
ULYSSES I do not strain at the position –
It is familiar – but at the author's drift;
Who, in his circumstance, expressly proves
[115]
That no man is the lord of anything,
Though in and of him there be much consisting,
Till he communicate his parts to others;
Nor doth he of himself know them for aught
Till he behold them formed in th' applause
[120]
Where th' are extended; who, like an arch, reverb'rate
The voice again; or, like a gate of steel
Fronting the sun, receives and renders back
His figure and his heat. I was much rapt in this;
And apprehended here immediately
[125]
Th' unknown Ajax. Heavens, what a man is there!
A very horse that has he knows not what!
Nature, what things there are
Most abject in regard and dear in use!
What things again most dear in the esteem
[130]
And poor in worth! Now shall we see to-morrow –
An act that very chance doth throw upon him –
Ajax renown'd. O heavens, what some men do,
While some men leave to do!
How some men creep in skittish Fortune's hall,
[135]
Whiles others play the idiots in her eyes!
How one man eats into another's pride,
While pride is fasting in his wantonness!
To see these Grecian lords! – why, even already
They clap the lubber Ajax on the shoulder,
[140]
As if his foot were on brave Hector's breast,
And great Troy shrinking.
ACHILLES I do believe it; for they pass'd by me
As misers do by beggars – neither gave to me
Good word nor look. What, are my deeds forgot?
[145]
ULYSSES Time hath, my lord, a wallet at his back,
Wherein he puts alms for oblivion,
A great-siz'd monster of ingratitudes.
Those scraps are good deeds past, which are devour'd
As fast as they are made, forgot as soon
[150]
As done. Persevtrance, dear my lord,
Keeps honour bright. To have done is to hang
Quite out of fashion, like a rusty mail
In monumental mock'ry. Take the instant way;
For honour travels in a strait so narrow
[155]
Where one but goes abreast. Keep then the path,
For emulation hath a thousand sons
That one by one pursue; if you give way,
Or hedge aside from the direct forthright,
Like to an ent'red tide they all rush by
[160]
And leave you hindmost;
Or, like a gallant horse Jall'n in first rank,
Lie there for pavement to the abject rear,
O'er-run and trampled on. Then what they do in present,
Though less than yours in past, must o'ertop yours;
[165]
For Time is like a fashionable host,
That slightly shakes his parting guest by th' hand;
And with his arms out-stretch'd, as he would fly,
Grasps in the comer. The welcome ever smiles,
And farewell goes out sighing. O, let not virtue seek
[170]
Remuneration for the thing it was;
For beauty, wit,
High birth, vigour of bone, desert in service,
Love, friendship, charity, are subjects all
To envious and calumniating Time.
[175]
One touch of nature makes the whole world kin –
That all with one consent praise new-born gawds,
Though they are made and moulded of things past,
And give to dust that is a little gilt
More laud than gilt o'er-dusted.
[180]
The present eye praises the present object.
Then marvel not, thou great and complete man,
That all the Greeks begin to worship Ajax,
Since things in motion sooner catch the eye
Than what stirs not. The cry went once on thee,
[185]
And still it might, and yet it may again,
If thou wouldst not entomb thyself alive
And case thy reputation in thy tent,
Whose glorious deeds but in these fields of late
Made emulous missions 'mongst the gods themselves.
And drave great Mars to faction.
[190]
ACHILLES Of this my privacy
I have strong reasons.
ULYSSES But 'gainst your privacy
The reasons are more potent and heroical.
'Tis known, Achilles, that you are in love
With one of Priam's daughters.
ACHILLES Ha! known!
[195]
ULYSSES Is that a wonder?
The providence that's in a watchful state
Knows almost every grain of Plums' gold;
Finds bottom in th' uncomprehensive deeps;
Keeps place with thought, and almost, like the gods,
[200]
Do thoughts unveil in their dumb cradles.
There is a mystery – with whom relation
Durst never meddle – in the soul of state,
Which hath an operation more divine
Than breath or pen can give expressure to.
[205]
All the commerce that you have had with Troy
As perfectly is ours as yours, my lord;
And better would it fit Achilles much
To throw down Hector than Polyxena.
But it must grieve young Pyrrhus now at home,
[210]
When fame shall in our island, sound her trump,
And all the Greekish girls shall tripping sing
'Great Hector's sister did Achilles win;
But our great Ajax bravely beat down him'.
Farewell, my lord. I as your lover speak.
[215]
The fool slides o'er the ice that you should break.
[ _Exit_.
PATROCLUS To this effect, Achilles, have I mov'd you.
A woman impudent and mannish grown
Is not more loath'd than an effeminate man
In time of action. I stand condemn'd for this;
[220]
They think my little stomach to the war
And your great love to me restrains you thus.
Sweet, rouse yourself; and the weak wanton Cupid
Shall from your neck unloose his amorous fold,
And, like a dew-drop from the lion's mane,
Be shook to airy air.
[225]
ACHILLES Shall Ajax fight with Hector?
PATROCLUS Ay, and perhaps receive much honour by him.
ACHILLES I see my reputation is at stake;
My fame is shrewdly gor'd,
PATROCLUS O, then, beware:
Those wounds heal ill that men do give themselves;
[230]
Omission to do what is necessary
Seals a commission to a blank of danger;
And danger, like an ague, subtly taints
Even then when they sit idly in the sun.
ACHILLES Go call Thersites hither, sweet Patroclus.
[235]
I'll send the fool to Ajax, and desire him
T' invite the Troyan lords, after the combat,
To see us here unarm'd. I have a woman's longing,
An appetite that I am sick withal,
To see great Hector in his weeds of peace;
[240]
To talk with him, and to behold his visage,
Even to my full of view.
_Enter THERSITES_.
A labour sav'd!
THERSITES A wonder!
ACHILLES What?
[245]
THERSITES Ajax goes up and down the field asking for himself.
ACHILLES How so?
THERSITES He must fight singly to-morrow with Hector, and is so prophetically proud of an heroical cudgelling that he raves in saying nothing.
[250]
ACHILLES How can that be?
[264]
THERSITES Why, 'a stalks up and down like a peacock – a stride and a stand; ruminates like an hostess that hath no arithmetic but her brain to set down her reckoning, bites his lip with a politic regard, as who should say 'There were wit in this head, an 'twould out'; and so there is; but it lies as coldly in him as fire in a flint, which will not show without knocking. The man's undone for ever; for if Hector break not his neck i' th' combat, he'll break't himself in vainglory. He knows not me. I said 'Good morrow, Ajax'; and he replies 'Thanks, Agamemnon'. What think you of this man that takes me for the general? He's grown a very land fish, languageless, a monster. A plague of opinion! A man may wear it on both sides, like a leather jerkin.
ACHILLES Thou must be my ambassador to him, Thersites.
THERSITES Who, I? Why, he'll answer nobody; he professes not answering. Speaking is for beggars: he wears his tongue in's arms. I will put on his presence. Let Patroclus make his demands to me, you shall see the pageant of Ajax.
[275]
ACHILLES To him, Patroclus. Tell him I humbly desire the valiant Ajax to invite the most valorous Hector to come unarm'd to my tent; and to procure safe conduct for his person of the magnanimous and most illustrious six-or-seven-times-honour'd Captain General of the Grecian army, et cetera, Agamemnon. Do this.
PATROCLUS Jove bless great Ajax!
THERSITES Hum!
PATROCLUS I come from the worthy Achilles –
[279]
THERSITES Ha!
PATROCLUS Who most humbly desires you to invite Hector to his tent –
THERSITES Hum!
PATROCLUS And to procure safe conduct from Agamemnon.
THERSITES Agamemnon!
[285]
PATROCLUS Ay, my lord.
THERSITES Ha!
PATROCLUS What say you to't?
THERSITES God buy you, with all my heart.
[289]
PATROCLUS Your answer, sir.
THERSITES If to-morrow be a fair day, by eleven of the clock it will go one way or other.
Howsoever, he shall pay for me ere he has me.
PATROCLUS Your answer, sir.
THERSITES Fare ye well, with all my heart.
[295]
ACHILLES Why, but he is not in this tune, is he?
[299]
THERSITES No, but he's out a tune thus. What music will be in him when Hector has knock'd out his brains I know not; but, I am sure, none; unless the fiddler Apollo get his sinews to make catlings on.
ACHILLES Come, thou shalt bear a letter to him straight.
THERSITES Let me carry another to his horse; for that's the more capable creature.
ACHILLES My mind is troubled, like a fountain stirr'd;
[304]
And I myself see not the bottom of it.
[ _Exeunt Achilles and Patroclus_.
THERSITES Would the fountain of your mind were clear again, that I might water an ass at it. I had rather be a tick in a sheep than such a valiant ignorance.
[ _Exit_.
#### ACT FOUR
**SCENE I**. _Troy. A street_.
_Enter, at one side, AENEAS, and Servant with a torch; at another, PARIS, DEIPHOBUS, ANTENOR, DIOMEDES the Grecian, and Others, with torches_.
PARIS See ho! Who is that there?
DEIPHOBUS It is the Lord Aeneas.
AENEAS Is the Prince there in person?
Had I so good occasion to he long
[5]
As you, Prince Paris, nothing but heavenly business
Should rob my bed-mate of my company.
DIOMEDES That's my mind too. Good morrow,
Lord Aeneas.
PARIS A valiant Greek, Aeneas – take his hand:
Witness the process of your speech, wherein
[10]
You told how Diomed, a whole week by days,
Did haunt you in the field.
AENEAS Health to you, valiant sir,
During all question of the gentle truce;
But when I meet you arm'd, as black defiance
[15]
As heart can think or courage execute.
DIOMEDES The one and other Diomed embraces.
Our bloods are now in calm; and so long health!
But when contention and occasion meet,
By Jove, I'll play the hunter for thy life
[20]
With all my force, pursuit, and policy.
AENEAS And thou shalt hunt a lion, that will fly
With his face backward. In humane gentleness,
Welcome to Troy! now, by Anchises' life,
Welcome indeed! By Venus' hand I swear
[25]
No man alive can love in such a sort
The thing he means to kill, more excellently.
DIOMEDES We sympathise. Jove let Aeneas live,
If to my sword his fate be not the glory,
A thousand complete courses of the sun!
[30]
But in mine emulous honour let him die
With every joint a wound, and that to-morrow!
AENEAS We know each other well.
DIOMEDES We do; and long to know each other worse.
PARIS This is the most despiteful'st gentle greeting,
[35]
The noblest hateful love, that e'er I heard of.
What business, lord, so early?
AENEAS I was sent for to the King; but why, I know not.
PARIS His purpose meets you: 'twas to bring this Greek
To Calchas' house, and there to render him,
[40]
For the enfreed Antenor, the fair Cressid.
Let's have your company; or, if you please,
Haste there before us. I constantly believe -
Or rather call my thought a certain knowledge -
My brother Troilus lodges there to-night.
[45]
Rouse him and give him note of our approach,
With the whole quality wherefore; I fear
We shall be much unwelcome.
AENEAS That I assure you:
Troilus had rather Troy were borne to Greece
Than Cressid borne from Troy.
PARIS There is no help;
[50]
The bitter disposition of the time
Will have it so. On, lord; we'll follow you.
AENEAS Good morrow, all. [ _Exit with servant_.
PARIS And tell me, noble Diomed – faith, tell me true.
Even in the soul of sound good-fellowship -
[55]
Who in your thoughts deserves fair Helen best,
Myself or Menelaus?
DIOMEDES Both alike:
He merits well to have her that doth seek her,
Not making any scruple of her soilure,
With such a hell of pain and world of charge;
[60]
And you as well to keep her that defend her,
Not palating the taste of her dishonour,
With such a costly loss of wealth and friends.
He like a puling cuckold would drink up
The lees and dregs of a flat tamed piece;
[65]
You, like a lecher, out of whorish lions
Are pleas'd to breed out your inheritors.
Both merits pois'd, each weighs nor less nor more;
But he as he, the heavier for a whore.
PARIS You are too bitter to your country-woman.
[70]
DIOMEDES She's bitter to her country. Hear me, Paris:
For every false drop in her bawdy veins
A Grecian's life hath sunk; for every scruple
Of her contaminated carrion weight
A Troyan hath been slain; since she could speak,
[75]
She hath not given so many good words breath
As for her Greeks and Troyans suff'red death.
PARIS Fair Diomed, you do as chapmen do,
Dispraise the thing that you desire to buy;
But we in silence hold this virtue well:
[80]
We'll not commend what we intend to sell.
Here lies our way.
[ _Exeunt_.
**SCENE II**. _Troy. The court of Pandarus' house_.
_Enter TROILUS and CRESSIDA_.
TROILUS Dear, trouble not yourself; the morn is cold.
CRESSIDA Then, sweet my lord, I'll call mine uncle down;
He shall unbolt the gates.
TROILUS Trouble him not;
To bed, to bed! Sleep kill those pretty eyes,
[5]
And give as soft attachment to thy senses
As infants' empty of all thought!
CRESSIDA Good morrow, then.
TROILUS I prithee now, to bed.
CRESSIDA Are you aweary of me?
TROILUS O Cressida! but that the busy day,
Wak'd by the lark, hath rous'd the ribald crows,
[10]
And dreaming night will hide our joys no longer,
I would not from thee.
CRESSIDA Night hath been too brief.
TROILUS Beshrew the witch! with venomous wights she stays
As tediously as hell, but flies the grasps of love
With wings more momentary-swift than thought.
You will catch cold, and curse me.
[15]
CRESSIDA Prithee tarry.
You men will never tarry.
O foolish Cressid! I might have still held off,
And then you would have tarried. Hark! there's one up.
PANDARUS [ _Within_ ] What's all the doors open here?
[20]
TROILUS It is your uncle.
_Enter PANDARUS_.
CRESSIDA A pestilence on him! Now will he be mocking.
I shall have such a life!
PANDARUS How now, how now! How go maidenheads? Here, you maid! Where's my cousin Cressid?
[25]
CRESSIDA Go hang yourself, you naughty mocking uncle.
You bring me to do, and then you flout me too.
PANDARUS To do what? to do what? Let her say what. What have I brought you to do?
CRESSIDA Come, come, beshrew your heart!
You'll ne'er be good,
[30]
Nor suffer others.
PANDARUS Ha, ha! Alas, poor wretch! a poor capocchia! hast not slept to-night? Would he not, a naughty man, let it sleep? A bugbear take him!
CRESSIDA Did not I tell you? Would he were knock'd i' th' head! [ _One knocks_.]
[35]
Who's that at door? Good uncle, go and see.
My lord, come you again into my chamber.
You smile and mock me, as if I meant naughtily.
TROILUS Ha! ha!
CRESSIDA Come, you are deceiv'd, I think of no such thing. [ _Knock_.
[40]
How earnestly they knock! Pray you come in:
I would not for half Troy have you seen here.
[ _Exeunt Troilus and Cressida_.
[45]
PANDARUS Who's there? What's the matter?
Will you beat down the door? How now?
What's the matter?
_Enter AENEAS_.
AENEAS Good morrow, lord, good morrow.
PANDARUS Who's there? My lord Aeneas? By my troth,
I knew you not. What news with you so early?
AENEAS Is not Prince Troilus here?
PANDARUS Here! What should he do here?
AENEAS Come, he is here, my lord; do not deny him.
[50]
It doth import him much to speak with me.
[53]
PANDARUS Is he here, say you? It's more than I know, I'll be sworn. For my own part, I came in late. What should he do here?
[57]
AENEAS Who! – nay, then. Come, come, you'll do him wrong ere you are ware; you'll be so true to him to be false to him. Do not you know of him, but yet go fetch him hither; go.
_Re-enter TROILUS_.
TROILUS How now! What's the matter?
AENEAS My lord, I scarce have leisure to salute you,
[60]
My matter is so rash. There is at hand
Paris your brother, and Deiphobus,
The Grecian Diomed, and our Antenor
Deliver'd to us; and for him forthwith,
Ere the first sacrifice, within this hour,
[65]
We must give up to Diomedes' hand
The Lady Cressida.
TROILUS Is it so concluded?
AENEAS By Priam, and the general state of Troy.
They are at hand and ready to effect it.
TROILUS How my achievements mock me!
[70]
I will go meet them; and, my lord Aeneas,
We met by chance; you did not find me here.
AENEAS Good, good, my lord, the secrets of neighbour Pandar
Have not more gift in taciturnity.
[ _Exeunt Troilus and Aeneas_.
PANDARUS Is't possible? No sooner got but lost?
The devil take Antenor! The young prince will go mad. A plague upon Antenor!
[76]
I would they had brake's neck.
_Re-enter CRESSIDA_.
CRESSIDA How now! What's the matter?
Who was here?
PANDARUS Ah, ah!
CRESSIDA Why sigh you so profoundly?
[80]
Where's my lord? Gone? Tell me, sweet uncle, what's the matter?
PANDARUS Would I were as deep under the earth as I am above!
CRESSIDA O the gods! What's the matter?
[86]
PANDARUS Pray thee, get thee in. Would thou hadst ne'er been born! I knew thou wouldst be his death! O, poor gentleman! A plague upon Antenor!
CRESSIDA Good uncle, I beseech you, on my knees I beseech you, what's the matter?
[92]
PANDARUS Thou must be gone, wench, thou must be gone; thou art chang'd for Antenor; thou must to thy father, and be gone from Troilus. Twill be his death; 'twill be his bane; he cannot bear it.
CRESSIDA O you immortal gods! I will not go.
PANDARUS Thou must.
[95]
CRESSIDA I will not, uncle. I have forgot my father;
I know no touch of consanguinity,
No kin, no love, no blood, no soul so near me
As the sweet Troilus. O you gods divine,
Make Cressid's name the very crown of falsehood,
[100]
If ever she leave Troilus! Time, force, and death,
Do to this body what extremes you can,
But the strong base and building of my love
Is as the very centre of the earth.
Drawing all things to it. I'll go in and weep –
[105]
PANDARUS Do, do.
CRESSIDA Tear my bright hair, and scratch my praised cheeks,
Crack my clear voice with sobs and break my heart,
With sounding Troilus'. I will not go from Troy.
[ _Exeunt_.
**SCENE III**. _Troy. A street before Pandarus' house_.
_Enter PARIS, TROILUS, AENEAS, DEIPHOBUS, ANTENOR, and DIOMEDES_.
PARIS It is great morning; and the hour prefix'd
For her delivery to this valiant Greek
Comes fast upon. Good my brother Troilus,
Tell you the lady what she is to do,
And haste her to the purpose.
TROILUS Walk into her house.
I'll bring her to the Grecian presently;
And to his hand when I deliver her,
Think it an altar, and thy brother Troilus
A priest, there off ring to it his own heart.
[ _Exit_.
[10]
PARIS I know what 'tis to love
And would, as I shall pity, I could help!
Please you walk in, my lords.
[ _Exeunt_.
**SCENE IV**. _Troy. Pandarus' house_.
_Enter PANDARUS and CRESSIDA_.
PANDARUS Be moderate, be moderate.
CRESSIDA Why tell you me of moderation?
The grief is fine, full, perfect, that I taste,
And violenteth in a sense as strong
As that which cause th it. How can I moderate it?
If I could temporize with my affections
Or brew it to a weak and colder palate,
The like allayment could I give my grief.
My love admits no qualifying dross;
[10]
No more my grief, in such a precious loss.
_Enter TROILUS_.
PANDARUS Here, here, here he comes. Ah, sweet ducks!
CRESSIDA O Troilus! Troilus! [ _Embracing him_.
PANDARUS What a pair of spectacles is here!
Let me embrace too. 'O heart,' as the goodly saying is,
[15]
O heart, heavy heart,
Why sigh'st thou without breaking? where he answers again
Because thou canst not ease thy smart
[19]
By friendship nor by speaking.
There was never a truer rhyme. Let us cast away nothing, for we may live to have need of such a verse. We see it, we see it. How now, lambs!
TROILUS Cressid, I love thee in so strain'd a purity
That the bless'd gods, as angry with my fancy,
[25]
More bright in zeal than the devotion which
Cold lips blow to their deities, take thee from me.
CRESSIDA Have the gods envy?
PANDARUS Ay, ay, ay, ay; 'tis too plain a case.
CRESSIDA And is it true that I must go from Troy?
TROILUS A hateful truth.
[30]
CRESSIDA What, and from Troilus too?
TROILUS From Troy and Troilus.
CRESSIDA Is't possible?
TROILUS And suddenly; where injury of chance
Puts back leave-taking, justles roughly by
All time of pause, rudely beguiles our lips
[35]
Of all rejoindure, forcibly prevents
Our lock'd embrasures, strangles our dear vows
Even in the birth of our own labouring breath.
We two, that with so many thousand sighs
Did buy each other, must poorly sell ourselves
[40]
With the rude brevity and discharge of one.
Injurious time now with a robber's haste
Crams his rich thievery up, he knows not how.
As many farewells as be stars in heaven,
With distinct breath and consign'd kisses to them,
[45]
He fumbles up into a loose adieu,
And scants us with a single famish'd kiss,
Distasted with the salt of broken tears.
AENEAS [ _Within_ ] My lord, is the lady ready?
TROILUS Hark! you are call'd. Some say the Genius so
[50]
Cries 'Come' to him that instantly must die.
Bid them have patience; she shall come anon.
PANDARUS Where are my tears? Rain, to lay this wind, or my heart will be blown up by th' root?
[ _Exit_.
CRESSIDA I must then to the Grecians?
TROILUS No remedy.
[55]
CRESSIDA A woeful Cressid 'mongst the merry Greeks!
When shall we see again?
TROILUS Hear me, my love. Be thou but true of heart –
CRESSIDA I true! how now! What wicked deem is this?
TROILUS Nay, we must use expostulation kindly,
[60]
For it is parting from us.
I speak not 'Be thou true' as fearing thee,
For I will throw my glove to Death himself
That there's no maculation in thy heart;
But 'Be thou true' say I to fashion in
[65]
My sequent protestation: be thou true,
And I will see thee.
CRESSIDA O, you shall be expos'd, my lord, to dangers
As infinite as imminent! But I'll be true.
TROILUS And I'll grow friend with danger. Wear this sleeve.
[70]
CRESSIDA And you this glove. When shall I see you?
TROILUS I will corrupt the Grecian sentinels
To give thee nightly visitation.
But yet be true.
CRESSIDA O heavens! 'Be true' again!
TROILUS Hear why I speak it, love.
[75]
The Grecian youths are full of quality;
They're loving, well compos'd with gifts of nature,
And flowing o'er with arts and exercise.
How novelties may move, and parts with person,
Alas, a kind of godly jealousy,
[80]
Which I beseech you call a virtuous sin,
Makes me afeard.
CRESSIDA O heavens! you love me not.
TROILUS Die I a villain, then!
In this I do not call your faith in question
So mainly as my merit. I cannot sing,
[85]
Nor heel the high lavolt, nor sweeten talk,
Nor play at subtle games – fair virtues all,
To which the Grecians are most prompt and pregnant;
But I can tell that in each grace of these
There lurks a still and dumb-discoursive devil
[90]
That tempts most cunningly. But be not tempted.
CRESSIDA Do you think I will?
TROILUS No.
But something may be done that we will not;
And sometimes we are devils to ourselves,
[95]
When we will tempt the frailty of our powers,
Presuming on their changeful potency.
AENEAS [ _Within_ ] Nay, good my lord!
TROILUS Come, kiss; and let us part.
PARIS [ _Within_ ] Brother Troilus!
TROILUS Good brother, come you hither;
And bring Aeneas and the Grecian with you.
[100]
CRESSIDA My lord, will you be true?
TROILUS Who, I? Alas, it is my vice, my fault!
Whiles others fish with craft for great opinion,
I with great truth catch mere simplicity;
Whilst some with cunning gild their copper crowns,
[105]
With truth and plainness I do wear mine bare.
_Enter AENEAS, PARIS, ANTENOR, DEIPHOBUS, and DIOMEDES_.
Fear not my truth: the moral of my wit
Is 'plain and true'; there's all the reach of it.
Welcome, Sir Diomed! Here is the lady
Which for Antenor we deliver you;
[110]
At the port, lord, I'll give her to thy hand,
And by the way possess thee what she is.
Entreat her fair; and, by my soul, fair Greek,
If e'er thou stand at mercy of my sword,
Name Cressid, and thy life shall be as safe
As Priam is in Ilion.
[115]
DIOMEDES Fair Lady Cressid,
So please you, save the thanks this prince expects.
The lustre in your eye, heaven in your cheek,
Pleads your fair usage; and to Diomed
You shall be mistress, and command him wholly.
[120]
TROILUS Grecian, thou dost not use me courteously
To shame the zeal of my petition to thee
In praising her. I tell thee, lord of Greece,
She is as far high-soaring o'er thy praises
As thou unworthy to be call'd her servant.
[125]
I charge thee use her well, even for my charge;
For, by the dreadful Pluto, if thou dost not,
Though the great bulk Achilles be thy guard,
I'll cut thy throat.
DIOMEDES O, be not mov'd, Prince Troilus.
Let me be privileg'd by my place and message
[130]
To be a speaker free: when I am hence
I'll answer to my lust. And know you, lord,
I'll nothing do on charge: to her own worth
She shall be priz'd. But that you say 'Be't so',
I speak it in my spirit and honour, 'No'.
[135]
TROILUS Come, to the port. I'll tell thee, Diomed,
This brave shall oft make thee to hide thy head.
Lady, give me your hand; and, as we walk,
To our own selves bend we our needful talk.
[ _Exeunt Troilus, Cressida, and Diomedes_.
[ _Sound trumpet_.
PARIS Hark! Hector's trumpet.
AENEAS How have we spent this morning!
[140]
The Prince must think me tardy and remiss,
That swore to ride before him to the field.
PARIS 'Tis Troilus' fault. Come, come to field with him.
DEIPHOBUS Let us make ready straight.
AENEAS Yea, with a bridegroom's fresh alacrity
[145]
Let us address to tend on Hector's heels.
The glory of our Troy doth this day lie
On his fair worth and single chivalry.
[ _Exeunt_.
**SCENE V**. _The Grecian Camp. Lists set out_.
_Enter AJAX, armed; AGAMEMNON, ACHILLES, PATROCLUS, MENELAUS, ULYSSES, NESTOR, and Others_.
AGAMEMNON Here art thou in appointment fresh and fair,
Anticipating time with starting courage.
Give with thy trumpet a loud note to Troy,
[5]
Thou dreadful Ajax, that the appalled air
May pierce the head of the great combatant,
And hale him hither.
AJAX Thou, trumpet, there's my purse.
Now crack thy lungs and split thy brazen pipe;
Blow, villain, till thy sphered bias cheek
Out-swell the colic of puff'd Aquilon.
[10]
Come, stretch thy chest, and let thy eyes spout blood:
Thou blowest for Hector. [ _Trumpet sounds_.
ULYSSES No trumpet answers.
ACHILLES 'Tis but early days.
_Enter DIOMEDES, with CRESSIDA_.
AGAMEMNON Is not yond Diomed, with Calchas' daughter?
ULYSSES 'Tis he, I ken the manner of his gait:
[15]
He rises on the toe. That spirit of his
In aspiration lifts him from the earth.
AGAMEMNON Is this the lady Cressid?
DIOMEDES Even she.
AGAMEMNON Most dearly welcome to the Greeks, sweet lady.
NESTOR Our general doth salute you with a kiss.
[20]
ULYSSES Yet is the kindness but particular;
'Twere better she were kiss'd in general.
NESTOR And very courtly counsel: I'll begin.
So much for Nestor.
ACHILLES I'll take that winter from your lips, fair lady.
[25]
Achilles bids you welcome.
MENELAUS I had good argument for kissing once.
PATROCLUS But that's no argument for kissing now;
For thus popp'd Paris in his hardiment,
And parted thus you and your argument.
[30]
ULYSSES O deadly gall, and theme of all our scorns!
For which we lose our heads to gild his horns.
PATROCLUS The first was Menelaus' kiss; this, mine – [ _Kisses her again_.
Patroclus kisses you.
MENELAUS O, this is trim!
PATROCLUS Paris and I kiss evermore for him.
[35]
MENELAUS I'll have my kiss, sir. Lady, by your leave.
CRESSIDA In kissing, do you render or receive?
PATROCLUS Both take and give.
CRESSIDA I'll make my match to live,
The kiss you take is better than you give;
Therefore no kiss.
[40]
MENELAUS I'll give you boot; I'll give you three for one.
CRESSIDA You are an odd man; give even or give none.
MENELAUS An odd man, lady? Every man is odd.
CRESSIDA No, Paris is not; for you know 'tis true
That you are odd, and he is even with you.
MENELAUS You fillip me o' th' head.
[45]
CRESSIDA No, I'll be sworn.
ULYSSES It were no match, your nail against his horn.
May I, sweet lady, beg a kiss of you?
CRESSIDA You may.
ULYSSES I do desire it.
CRESSIDA Why, beg then.
ULYSSES Why then, for Venus' sake give me a kiss
[50]
When Helen is a maid again, and his.
CRESSIDA I am your debtor; claim it when 'tis due.
ULYSSES Never's my day, and then a kiss of you.
DIOMEDES Lady, a word. I'll bring you to your father. [ _Exit with Cressida_.
NESTOR A woman of quick sense.
ULYSSES Fie, fie upon her!
[55]
There's language in her eye, her cheek, her lip,
Nay, her foot speaks; her wanton spirits look out
At every joint and motive of her body.
O these encounterers so glib of tongue
That give a coasting welcome ere it comes,
[60]
And wide unclasp the tables of their thoughts
To every ticklish reader! Set them down
For sluttish spoils of opportunity,
And daughters of the game. [ _Trumpet within_.
ALL The Troyans' trumpet.
_Enter HECTOR, armed; AENEAS, TROILUS, PARIS, HELENUS, and other Trojans, with Attendants_.
AGAMEMNON Yonder comes the troop.
[65]
AENEAS Hail, all the state of Greece! What shall be done
To him that victory commands? Or do you purpose
A victor shall be known? Will you the knights
Shall to the edge of all extremity
Pursue each other, or shall they be divided
[70]
By any voice or order of the field?
Hector bade ask.
AGAMEMNON Which way would Hector have it?
AENEAS He cares not; he'll obey conditions.
ACHILLES 'Tis done like Hector; but securely done,
A little proudly, and great deal misprizing
The knight oppos'd.
[75]
AENEAS If not Achilles, sir,
What is your name?
ACHILLES If not Achilles, nothing.
AENEAS Therefore Achilles. But whate'er, know this:
In the extremity of great and little
Valour and pride excel themselves in Hector;
[80]
The one almost as infinite as all,
The other blank as nothing. Weigh him well,
And that which looks like pride is courtesy.
This Ajax is half made of Hector's blood;
In love whereof half Hector stays at home;
[85]
Half heart, half hand, half Hector comes to seek
This blended knight, half Troyan and half Greek.
ACHILLES A maiden battle then? O, I perceive you!
_Re-enter DIOMEDES_.
AGAMEMNON Here is Sir Diomed. Go, gentle knight,
Stand by our Ajax. As you and Lord Aeneas
[90]
Consent upon the order of their fight,
So be it; either to the uttermost,
Or else a breath. The combatants being kin
Half stints their strife before their strokes begin.
[ _Ajax and Hector enter the lists_.
ULYSSES They are oppos'd already.
[95]
AGAMEMNON What Troyan is that same that looks so heavy?
ULYSSES The youngest son of Priam, a true knight;
Not yet mature, yet matchless; firm of word;
Speaking in deeds and deedless in his tongue;
Not soon provok'd, nor being provok'd soon calm'd;
[100]
His heart and hand both open and both free;
For what he has he gives, what thinks he shows,
Yet gives he not till judgment guide his bounty,
Nor dignifies an impair thought with breath;
Manly as Hector, but more dangerous;
[105]
For Hector in his blaze of wrath subscribes
To tender objects, but he in heat of action
Is more vindicative than jealous love.
I hey call him Troilus, and on him erect
A second hope as fairly built as Hector.
[110]
Thus says Aeneas, one that knows the youth
Even to his inches, and, with private soul,
Did in great Ilion thus translate him to me.
[ _Alarum. Hector and Ajax fight_.
AGAMEMNON They are in action.
NESTOR Now, Ajax, hold thine own!
TROILUS Hector, thou sleep'st;
[115]
Awake thee.
AGAMEMNON His blows are well dispos'd. There, Ajax! [ _Trumpets cease_.
DIOMEDES You must no more.
AENEAS Princes, enough, so please you.
AJAX I am not warm yet; let us fight again.
DIOMEDES As Hector pleases.
HECTOR Why, then will I no more.
[120]
Thou art, great lord, my father's sister's son,
A cousin-german to great Priam's seed;
The obligation of our blood forbids
A gory emulation 'twixt us twain:
Were thy commixtion Greek and Troyan so
[125]
That thou could'st say This hand is Grecian all,
And this is Troyan; the sinews of this leg
All Greek, and this all Troy; my mother's blood
Runs on the dexter cheek, and this sinister
Bounds in my father's'; by Jove multipotent,
[130]
Thou shouldst not bear from me a Greekish member
Wherein my sword had not impressure made
Of our rank feud; but the just gods gainsay
That any drop thou borrow'dst from thy mother,
My sacred aunt, should by my mortal sword
[135]
Be drained! Let me embrace thee, Ajax.
By him that thunders, thou hast lusty arms;
Hector would have them fall upon him thus.
Cousin, all honour to thee!
AJAX I thank thee, Hector.
Thou art too gentle and too free a man.
[140]
I came to kill thee, cousin, and bear hence
A great addition earned in thy death.
HECTOR Not Neoptolemus so mirable,
On whose bright crest Fame with her loud'st Oyes
Cries This is he' could promise to himself
[145]
A thought of added honour torn from Hector.
AENEAS There is expectance here from both the sides
What further you will do.
HECTOR We'll answer it:
The issue is embracement. Ajax, farewell.
MAX If I might in entreaties find success,
[150]
As seld I have the chance, I would desire
My famous cousin to our Grecian tents.
DIOMEDES 'Tis Agamemnon's wish; and great Achilles
Doth long to see unarm'd the valiant Hector.
HECTOR Aeneas, call my brother Troilus to me,
[155]
And signify this loving interview
To the expecters of our Troyan part;
Desire them home. Give me thy hand, my cousin;
I will go eat with thee, and see your knights.
_Agamemnon and the rest of the Greeks come forward_.
AJAX Great Agamemnon comes to meet us here.
[160]
HECTOR The worthiest of them tell me name by name;
But for Achilles, my own searching eyes
Shall find him by his large and portly size.
AGAMEMNON Worthy all arms! as welcome as to one
That would be rid of such an enemy.
[165]
But that's no welcome. Understand more clear,
What's past and what's to come is strew'd with husks
And formless ruin of oblivion;
But in this extant moment, faith and troth,
Strain'd purely from all hollow bias-drawing,
[170]
Bids thee with most divine integrity,
From heart of very heart, great Hector, welcome.
HECTOR I thank thee, most imperious Agamemnon.
AGAMEMNON [ _To Troilus_ ] My well-fam'd lord of Troy, no less to you.
MENELAUS Let me confirm my princely brother's greeting.
[175]
You brace of warlike brothers, welcome hither.
HECTOR Who must we answer?
AENEAS The noble Menelaus.
HECTOR O, you, my lord? By Mars his gauntlet, thanks!
Mock not that I affect the untraded oath;
Your quondam wife swears still by Venus' glove.
[180]
She's well, but bade me not commend her to you.
MENELAUS Name her not now, sir; she's a deadly theme.
HECTOR O, pardon; I offend.
NESTOR I have, thou gallant Troyan, seen thee oft,
Labouring for destiny, make cruel way
[185]
Through ranks of Greekish youth; and I have seen thee,
As hot as Perseus, spur thy Phrygian steed,
Despising many forfeits and subduements,
When thou hast hung thy advanced sword i' th' air,
Not letting it decline on the declined;
[190]
That I have said to some my standers-by
'Lo, Jupiter is yonder, dealing life!'
And I have seen thee pause and take thy breath,
When that a ring of Greeks have hemm'd thee in,
Like an Olympian wrestling. This have I seen;
[195]
But this thy countenance, still lock'd in steel,
I never saw till now. I knew thy grandsire,
And once fought with him. He was a soldier good,
But, by great Mars, the captain of us all,
Never like thee. O, let an old man embrace thee;
[200]
And, worthy warrior, welcome to our tents.
AENEAS 'Tis the old Nestor.
HECTOR Let me embrace thee, good old chronicle,
That hast so long walk'd hand in hand with time.
Most reverend Nestor, I am glad to clasp thee.
[205]
NESTOR I would my arms could match thee in contention
As they contend with thee in courtesy.
HECTOR I would they could.
NESTOR Ha!
By this white beard, I'd fight with thee to-morrow.
[210]
Well, welcome, welcome! I have seen the time.
ULYSSES I wonder now how yonder city stands,
When we have here her base and pillar by us.
HECTOR I know your favour, Lord Ulysses, well.
Ah, sir, there's many a Greek and Troyan dead,
[215]
Since first I saw yourself and Diomed
In Ilion on your Greekish embassy.
ULYSSES Sir, I foretold you then what would ensue.
My prophecy is but half his journey yet;
For yonder walls, that pertly front your town,
[220]
Yond towers, whose wanton tops do buss the clouds,
Must kiss their own feet.
HECTOR I must not believe you.
There they stand yet; and modestly I think
The fall of every Phrygian stone will cost
A drop of Grecian blood. The end crowns all;
[225]
And that old common arbitrator, Time,
Will one day end it.
ULYSSES So to him we leave it.
Most gentle and most valiant Hector, welcome.
After the General, I beseech you next
To feast with me and see me at my tent.
[230]
ACHILLES I shall forestall thee, Lord Ulysses, thou!
Now, Hector, I have fed mine eyes on thee;
I have with exact view perus'd thee, Hector,
And quoted joint by joint.
HECTOR Is this Achilles?
ACHILLES I am Achilles.
[235]
HECTOR Stand fair, I pray thee; let me look on thee.
ACHILLES Behold thy fill.
HECTOR Nay, I have done already.
ACHILLES Thou art too brief. I will the second time,
As I would buy thee, view thee limb by limb.
HECTOR O, like a book of sport thou'lt read me o'er;
[240]
But there's more in me than thou understand'st.
Why dost thou so oppress me with thine eye?
ACHILLES Tell me, you heavens, in which part of his body
Shall I destroy him? Whether there, or there, or there?
That I may give the local wound a name,
[245]
And make distinct the very breach whereout
Hector's great spirit flew. Answer me, heavens.
HECTOR It would discredit the blest gods, proud man,
To answer such a question. Stand again.
Think'st thou to catch my life so pleasantly
[250]
As to prenominate in nice conjecture
Where thou wilt hit me dead?
ACHILLES I tell thee yea.
HECTOR Wert thou an oracle to tell me so,
I'd not believe thee. Henceforth guard thee well;
For I'll not kill thee there, nor there, nor there;
[255]
But, by the forge that stithied Mars his helm,
I'll kill thee everywhere, yea, o'er and o'er.
You wisest Grecians, pardon me this brag.
His insolence draws folly from my lips;
But I'll endeavour deeds to match these words,
Or may I never –
[260]
AJAX Do not chafe thee, cousin;
And you, Achilles, let these threats alone
Till accident or purpose bring you to't.
You may have every day enough of Hector,
If you have stomach. The general state, I fear,
[265]
Can scarce entreat you to be odd with him.
HECTOR I pray you let us see you in the field;
We have had pelting wars since you refus'd
The Grecians' cause.
ACHILLES Dost thou entreat me, Hector?
To-morrow do I meet thee, fell as death;
To-night all friends.
[270]
HECTOR Thy hand upon that match.
AGAMEMNON First, all you peers of Greece, go to my tent;
There in the full convive we; afterwards,
As Hector's leisure and your bounties shall
Concur together, severally entreat him.
[275]
Beat loud the tabourines, let the trumpets blow,
That this great soldier may his welcome know.
[ _Exeunt all but Troilus and Ulysses_.
TROILUS My Lord Ulysses, tell me, I beseech you,
In what place of the field doth Calchas keep?
ULYSSES At Menelaus' tent, most princely Troilus.
[280]
There Diomed doth feast with him to-night,
Who neither looks upon the heaven nor earth,
But gives all gaze and bent of amorous view
On the fair Cressid.
TROILUS Shall I, sweet lord, be bound to you so much,
[285]
After we part from Agamemnon's tent,
To bring me thither?
ULYSSES You shall command me, sir.
As gentle tell me of what honour was
This Cressida in Troy? Had she no lover there
That wails her absence?
[290]
TROILUS O, sir, to such as boasting show their scars
A mock is due. Will you walk on, my lord?
She was belov'd, she lov'd; she is, and doth;
But still sweet love is food for fortune's tooth.
[ _Exeunt_.
#### ACT FIVE
**SCENE I**. _The Grecian camp. Before the tent of Achilles_.
_Enter ACHILLES and PATROCLUS_.
ACHILLES I'll heat his blood with Greekish wine to-night,
Which with my scimitar I'll coll to-morrow.
Patroclus, let us feast him to the height.
PATROCLUS Here comes Thersites.
_Enter THERSITES_.
ACHILLES How now, thou core of envy!
[5]
Thou crusty batch of nature, what's the news?
THERSITES Why, thou picture of what thou seemest, and idol of idiot worshippers, here's a letter for thee.
ACHILLES From whence, fragment?
THERSITES Why, thou full dish of fool, from Troy.
[10]
PATROCLUS Who keeps the tent now?
THERSITES The surgeon's box or the patient's wound.
PATROCLUS Well said Adversity! and what needs these tricks?
THERSITES Prithee, be silent, boy; I profit not by thy talk; thou art said to be Achilles' male varlet.
[15]
PATROCLUS Male varlet, you rogue! What's that?
[22]
THERSITES Why, his masculine whore. Now, the rotten diseases of the south, the guts-gripping ruptures, catarrhs, loads o' gravel in the back, lethargies, cold palsies, raw eyes, dirt-rotten livers, wheezing lungs, bladders full of imposthume, sciaticas, limekilns i' th' palm, incurable bone-ache, and the rivelled fee-simple of the tetter, take and take again such preposterous discoveries!
PATROCLUS Why, thou damnable box of envy, thou, what meanest thou to curse thus?
[25]
THERSITES Do I curse thee?
PATROCLUS Why, no, you ruinous butt; you whoreson indistinguishable cur, no.
[32]
THERSITES No! Why art thou, then, exasperate, thou idle immaterial skein of sleid silk, thou green sarcenet flap for a sore eye, thou tassel of a prodigal's purse, thou? Ah, how the poor world is pest'red with such water-flies – diminutives of nature!
PATROCLUS Out, gall!
THERSITES Finch egg!
[35]
ACHILLES My sweet Patroclus, I am thwarted quite
From my great purpose in to-morrow's battle.
Here is a letter from Queen Hecuba,
A token from her daughter, my fair love,
Both taxing me and gaging me to keep
[40]
An oath that I have sworn. I will not break it.
Fall Greeks; fail fame; honour or go or stay;
My major vow lies here, this I'll obey.
Come, come, Thersites, help to trim my tent;
This night in banqueting must all be spent.
Away, Patroclus! [ _Exit with Patroclus_.
THERSITES With too much blood and too little brain these two may run mad; but, if with too much brain and too little blood they do, I'll be a curer of madmen. Here's Agamemnon, an honest fellow enough, and one that loves quails, but he has not so much brain as ear-wax; and the goodly transformation of Jupiter there, his brother, the bull, the primitive statue and oblique memorial of cuckolds, a thrifty shoeing-horn in a chain, hanging at his brother's leg – to what form but that he is, should wit larded with malice, and malice forced with wit, turn him to? To an ass, were nothing: he is both ass and ox. To an ox, were nothing: he is both ox and ass. To be a dog, a mule, a cat, a fitchew, a toad, a lizard, an owl, a puttock, or a herring without a roe, I would not care; but to be Menelaus, I would conspire against destiny. Ask me not what I would be, if I were not Thersites; for I care not to be the louse of a lazar, so I were not Menelaus. Hey-day! sprites and fires!
_Enter HECTOR, TROILUS, AJAX, AGAMEMNON, ULYSSES, NESTOR, MENELAUS, and DIOMEDES, with lights_.
AGAMEMNON We go wrong, we go wrong.
AJAX No, yonder 'tis;
There, where we see the lights.
[65]
HECTOR I trouble you.
AJAX No, not a whit.
_Re-enter ACHILLES_.
ULYSSES Here comes himself to guide you.
ACHILLES Welcome, brave Hector; welcome Princes all.
AGAMEMNON So now, fair Prince of Troy, I bid good night;
Ajax commands the guard to tend on you.
[70]
HECTOR Thanks, and good night to the Greeks' general.
MENELAUS Good night, my lord.
HECTOR Good night, sweet Lord Menelaus.
THERSITES Sweet draught! 'Sweet' quoth 'a?
Sweet sink, sweet sewer!
ACHILLES Good night and welcome, both at once, to those
[75]
That go or tarry.
AGAMEMNON Good night.
[ _Exeunt Agamemnon and Menelaus_.
ACHILLES Old Nestor tarries; and you too, Diomed,
Keep Hector company an hour or two.
DIOMEDES I cannot, lord; I have important business.
The tide whereof is now. Good night, great
[80]
Hector.
HECTOR Give me your hand.
ULYSSES [ _Aside to Troilus_ ] Follow his torch; he goes to Calchas' tent;
I'll keep you company.
TROILUS Sweet sir, you honour me.
HECTOR And so, good night.
[ _Exit Diomedes; Ulysses and Troilus following_.
[85]
ACHILLES Come, come, enter my tent.
[ _Exeunt all but Thersiles_.
[95]
THERSITES That same Diomed's a false-hearted rogue, a most unjust knave; I will no more trust him when he leers than I will a serpent when he hisses. He will spend his mouth and promise, like Brabbler the hound; but when he performs, astronomers foretell it: it is prodigious, there will come some change; the sun borrows of the moon when Diomed keeps his word. I will rather leave to see Hector than not to dog him. They say he keeps a Troyan drab, and uses the traitor Calchas' tent. I'll after. Nothing but lechery! All incontinent varlets!
[ _Exit_.
**SCENE II**. _The Grecian camp. Before Calchas' tent_.
_Enter DIOMEDES_.
DIOMEDES What, are you up here, ho? Speak.
CALCHAS [ _Within_ ] Who calls?
DIOMEDES Diomed. Calchas, I think. Where's your daughter?
CALCHAS [ _Within_ ] She comes to you.
_Enter TROILUS and ULYSSES, at a distance; after them THERSITES_.
ULYSSES Stand where the torch may not discover us.
_Enter CRESSIDA_.
TROILUS Cressid comes forth to him.
DIOMEDES How now, my charge!
CRESSIDA Now, my sweet guardian! Hark, a word with you. [ _Whispers_.
TROILUS Yea, so familiar!
ULYSSES She will sing any man at first sight.
[11]
THERSITES And any man may sing her, if he can take her cliff; she's noted.
DIOMEDES Will you remember?
CRESSIDA Remember? Yes.
DIOMEDES Nay, but do, then;
[15]
And let your mind be coupled with your words.
TROILUS What shall she remember?
ULYSSES List!
CRESSIDA Sweet honey Greek, tempt me no more to folly.
THERSITES Roguery!
[20]
DIOMEDES Nay, then –
CRESSIDA I'll tell you what –
DIOMEDES Fo, fo! come, tell a pin; you are a forsworn –
CRESSIDA In faith, I cannot. What would you have me do?
THERSITES A juggling trick, to be secretly open.
[25]
DIOMEDES What did you swear you would bestow on me?
CRESSIDA I prithee, do not hold me to mine oath;
Bid me do anything but that, sweet Greek.
DIOMEDES Good night.
TROILUS Hold, patience!
[30]
ULYSSES How now, Troyan!
CRESSIDA Diomed!
DIOMEDES No. no, good night; I'll be your fool no more.
TROILUS Thy better must.
CRESSIDA Hark! a word in your ear.
[35]
TROILUS O plague and madness!
ULYSSES You are moved, Prince; let us depart, I pray,
Lest your displeasure should enlarge itself
To wrathful terms. This place is dangerous;
The time right deadly; I beseech you, go.
TROILUS Behold, I pray you.
[40]
ULYSSES Nay, good my lord, go off;
You flow to great distraction; come, my lord.
TROILUS I prithee stay.
ULYSSES You have not patience; come.
TROILUS I pray you, stay: by hell and all hell's torments,
I will not speak a word.
DIOMEDES And so, good night.
CRESSIDA Nay, but you part in anger.
[45]
TROILUS Doth that grieve thee? O withered truth!
ULYSSES How now, my lord?
TROILUS By Jove. I will be patient.
CRESSIDA Guardian! Why, Greek!
DIOMEDES Fo, fo! adieu! you palter.
CRESSIDA In faith, I do not. Come hither once again.
[50]
ULYSSES You shake, my lord, at something; will you go?
You will break out.
TROILUS She strokes his cheek.
ULYSSES Come, come.
TROILUS Nay, stay; by Jove, I will not speak a word:
There is between my will and all offences
[54]
A guard of patience. Stay a little while.
THERSITES How the devil luxury, with his fat rump and potato finger, tickles these together! Fry, lechery, fry!
DIOMEDES But will you, then?
CRESSIDA In faith, I will, lo; never trust me else.
DIOMEDES Give me some token for the surety of it.
[60]
CRESSIDA I'll fetch you one.
[ _Exit_.
ULYSSES You have sworn patience.
TROILUS Fear me not, my lord;
I will not be myself, nor have cognition
Of what I feel. I am all patience.
_Re-enter CRESSIDA_.
THERSITES Now the pledge; now, now, now!
[65]
CRESSIDA Here, Diomed, keep this sleeve.
TROILUS O beauty! where is thy faith?
ULYSSES My lord!
TROILUS I will be patient; outwardly I will.
CRESSIDA You look upon that sleeve; behold it well.
He lov'd me – O false wench! – Give't me again.
[70]
DIOMEDES Whose was'l?
CRESSIDA It is no matter, now I ha't again.
I will not meet with you to-morrow night.
I prithee, Diomed, visit me no more.
THERSITES Now she sharpens. Well said, whetstone.
DIOMEDES I shall have it.
CRESSIDA What, this?
[75]
DIOMEDES Ay, that.
CRESSIDA O all you gods! O pretty, pretty pledge!
Thy master now lies thinking on his bed
Of thee and me, and sighs, and takes my glove,
And gives memorial dainty kisses to it,
[80]
As I kiss thee. Nay, do not snatch it from me;
He that takes that doth take my heart withal.
DIOMEDES I had your heart before; this follows it.
TROILUS I did swear patience.
CRESSIDA You shall not have it, Diomed; faith, you shall not;
[85]
I'll give you something else.
DIOMEDES I will have this. Whose was it?
CRESSIDA It is no matter.
DIOMEDES Come, tell me whose it was.
CRESSIDA 'Twas one's that lov'd me better than you will.
But, now you have it, take it.
DIOMEDES Whose was it?
[90]
CRESSIDA By all Diana's waiting women yond,
And by herself, I will not tell you whose.
DIOMEDES To-morrow will I wear it on my helm,
And grieve his spirit that dares not challenge it.
TROILUS Wert thou the devil and wor'st it on thy horn,
[95]
It should be challeng'd.
CRESSIDA Well, well, 'tis done, 'tis past; and yet it is not;
I will not keep my word.
DIOMEDES Why, then farewell;
Thou never shall mock Diomed again.
CRESSIDA You shalt not go. One cannot speak a word
But it straight starts you.
[100]
DIOMEDES I do not like this fooling.
THERSITES Nor I, by Pluto; but that that likes not you
Pleases me best.
DIOMEDES What, shall I come? The hour –
CRESSIDA Ay, come – O Jove! Do come. I shall be plagu'd.
DIOMEDES Farewell till then.
CRESSIDA Good night. I prithee come.
[ _Exit Diomedes_.
[105]
Troilus, farewell! One eye yet looks on thee;
But with my heart the other eye doth see.
Ah, poor our sex! this fault in us I find,
The error of our eye directs our mind.
What error leads must err; O, then conclude,
[110]
Minds sway'd by eyes are full of turpitude.
[ _Exit_.
THERSITES A proof of strength she could not publish more,
Unless she said 'My mind is now turn'd whore'.
ULYSSES All's done, my lord.
TROILUS It is.
ULYSSES Why stay we, then?
TROILUS To make a recordation to my soul
[115]
Of every syllable that here was spoke.
But if I tell how these two did coact,
Shall I not lie in publishing a truth?
Sith yet there is a credence in my heart,
An esperance so obstinately strong,
[120]
That doth invert th' attest of eyes and ears;
As if those organs had deceptious functions
Created only to calumniate.
Was Cressid here?
ULYSSES I cannot conjure, Troyan.
TROILUS She was not, sure.
ULYSSES Most sure she was.
[125]
TROILUS Why, my negation hath no taste of madness.
ULYSSES Nor mine, my lord. Cressid was here but now.
TROILUS Let it not be believ'd for womanhood.
Think, we had mothers; do not give advantage
To stubborn critics, apt, without a theme,
[130]
For depravation, to square the general sex
By Cressid's rule. Rather think this not Cressid.
ULYSSES What hath she done, Prince, that can soil our mothers?
TROILUS Nothing at all, unless that this were she.
THERSITES Will 'a swagger himself out on's own eyes?
[135]
TROILUS This she? No; this is Diomed's Cressida.
If beauty have a soul, this is not she;
If souls guide vows, if vows be sanctimonies,
If sanctimony be the gods' delight,
If there be rule in unity itself,
[140]
This was not she. O madness of discourse,
That cause sets up with and against itself!
Bifold authority! where reason can revolt
Without perdition, and loss assume all reason
Without revolt: this is, and is not, Cressid.
[145]
Within my soul there doth conduce a fight
Of this strange nature, that a thing inseparate
Divides more wider than the sky and earth;
And yet the spacious breadth of this division
Admits no orifex for a point as subtle
[150]
As Ariachne's broken woof to enter.
Instance, O instance! strong as Pluto's gates:
Cressid is mine, tied with the bonds of heaven.
Instance, O instance! strong as heaven itself:
The bonds of heaven are slipp'd, dissolv'd, and loos'd;
[155]
And with another knot, five-finger-tied,
The fractions of her faith, orts of her love,
The fragments, scraps, the bits, and greasy relics
Of her o'er-eaten faith, are bound to Diomed.
ULYSSES May worthy Troilus be half-attach'd
[160]
With that which here his passion doth express?
TROILUS Ay, Greek; and that shall be divulged well
In characters as red as Mars his heart
Inflam'd with Venus. Never did young man fancy
With so eternal and so fix'd a soul.
[165]
Hark, Greek: as much as I do Cressid love,
So much by weight hate I her Diomed.
That sleeve is mine that he'll bear on his helm;
Were it a casque compos'd by Vulcan's skill
My sword should bite it. Not the dreadful spout
[170]
Which shipmen do the hurricano call,
Constring'd in mass by the almighty sun.
Shall dizzy with more clamour Neptune's ear
In his descent than shall my prompted sword
Falling on Diomed.
[175]
THERSITES He'll tickle it for his concupy.
TROILUS O Cressid! O false Cressid! false, false, false!
Let all untruths stand by thy stained name,
And they'll seem glorious.
ULYSSES O, contain yourself;
Your passion draws ears hither.
_Enter AENEAS_.
[180]
AENEAS I have been seeking you this hour, my lord.
Hector, by this, is arming him in Troy;
Ajax, your guard, stays to conduct you home.
TROILUS Have with you, Prince. My courteous lord, adieu.
Farewell, revolted fair! – and, Diomed,
[185]
Stand fast and wear a castle on thy head.
ULYSSES I'll bring you to the gates.
TROILUS Accept distracted thanks.
[ _Exeunt Troilus, Aeneas, and Ulysses_.
THERSITES Would I could meet that rogue Diomed! I would croak like a raven; I would bode, I would bode. Patroclus will give me anything for the intelligence of this whore; the parrot will not do more for an almond than he for a commodious drab. Lechery, lechery! Still wars and lechery! Nothing else holds fashion. A burning devil take them!
[ _Exit_.
**SCENE III**. _Troy. Before Priam's palace_.
_Enter HECTOR and ANDROMACHE_.
ANDROMACHE When was my lord so much ungently temper'd
To stop his ears against admonishment?
Unarm, unarm, and do not fight to-day.
HECTOR You train me to offend you; get you in.
[5]
By all the everlasting gods, I'll go.
ANDROMACHE My dreams will, sure, prove ominous to the day.
HECTOR No more, I say.
_Enter CASSANDRA_.
CASSANDRA Where is my brother Hector?
ANDROMACHE Here, sister, arm'd, and bloody in intent.
Consort with me in loud and dear petition,
[10]
Pursue we him on knees; for I have dreamt
Of bloody turbulence, and this whole night
Hath nothing been but shapes and forms of slaughter.
CASSANDRA O, 'tis true!
HECTOR Ho! bid my trumpet sound.
CASSANDRA No notes of sally, for the heavens, sweet brother!
[15]
HECTOR Be gone, I say. The gods have heard me swear.
CASSANDRA The gods are deaf to hot and peevish vows;
They are polluted off'rings, more abhorr'd
Than spotted livers in the sacrifice.
ANDROMACHE O, be persuaded! Do not count it holy
[20]
To hurt by being just. It is as lawful,
For we would give much, to use violent thefts
And rob in the behalf of charity.
CASSANDRA It is the purpose that makes strong the vow;
But vows to every purpose must not hold.
Unarm, sweet Hector.
[25]
HECTOR Hold you still, I say.
Mine honour keeps the weather of my fate.
Life every man holds dear; but the dear man
Holds honour far more precious dear than life.
_Enter TROILUS_.
How now, young man! Mean'st thou to fight to-day?
[30]
ANDROMACHE Cassandra, call my father to persuade. [ _Exit Cassandra_.
HECTOR No, faith, young Troilus; doff thy harness, youth;
I am to-day i' th' vein of chivalry.
Let grow thy sinews till their knots be strong,
And tempt not yet the brushes of the war.
[35]
Unarm thee, go; and doubt thou not, brave boy,
I'll stand to-day for thee and me and Troy.
TROILUS Brother, you have a vice of mercy in you Which better fits a lion than a man.
HECTOR What vice is that, good Troilus? Chide me for it.
[40]
TROILUS When many times the captive Grecian falls,
Even in the fan and wind of your fair sword,
You bid them rise and live.
HECTOR O, 'tis fair play!
TROILUS Fool's play, by heaven, Hector.
HECTOR How now! how now!
TROILUS For th' love of all the gods,
[45]
Let's leave the hermit Pity with our mother;
And when we have our armours buckled on,
The venom'd vengeance ride upon our swords,
Spur them to ruthful work, rein them from ruth!
HECTOR Fie, savage, fie!
TROILUS Hector, then 'tis wars.
[50]
HECTOR Troilus, I would not have you fight to-day.
TROILUS Who should withhold me?
Not fate, obedience, nor the hand of Mars
Beck'ning with fiery truncheon my retire;
Not Priamus and Hecuba on knees,
[55]
Their eyes o'ergalled with recourse of tears;
Nor you, my brother, with your true sword drawn,
Oppos'd to hinder me, should stop my way,
But by my ruin.
_Re-enter CASSANDRA, with PRIAM_.
CASSANDRA Lay hold upon him, Priam, hold him fast;
[60]
He is thy crutch; now if thou lose thy stay,
Thou on him leaning, and all Troy on thee,
Fall all together.
PRIAM Come, Hector, come, go back.
Thy wife hath dreamt; thy mother hath had visions;
Cassandra doth foresee; and I myself
[65]
Am like a prophet suddenly enrapt
To tell thee that this day is ominous.
Therefore, come back.
HECTOR Aeneas is a-field,
And I do stand engag'd to many Greeks,
Even in the faith of valour, to appear
This morning to them.
[70]
PRIAM Ay, but thou shalt not go.
HECTOR I must not break my faith.
You know me dutiful; therefore, dear sir,
Let me not shame respect; but give me leave
To take that course by your consent and voice
[75]
Which you do here forbid me, royal Priam,
CASSANDRA O Priam, yield not to him!
ANDROMACHE Do not, dear father.
HECTOR Andromache, I am offended with you.
Upon the love you bear me, get you in.
[ _Exit Andromache_.
TROILUS This foolish, dreaming, superstitious girl
Makes all these bodements.
[80]
CASSANDRA O, farewell, dear Hector!
Look how thou diest. Look how thy eye turns pale.
Look how thy wounds do bleed at many vents.
Hark how Troy roars; how Hecuba cries out;
How poor Andromache shrills her dolours forth;
[85]
Behold distraction, frenzy, and amazement,
Like witless antics, one another meet,
And all cry, Hector! Hector's dead! O Hector!
TROILUS Away, away!
CASSANDRA Farewell! – yet, soft! Hector I take my leave.
[90]
Thou dost thyself and all our Troy deceive.
[ _Exit_.
HECTOR You are amaz'd, my liege, at her exclaim.
Go in, and cheer the town; we'll forth, and fight,
Do deeds worth praise and tell you them at night.
PRIAM Farewell. The gods with safety stand about thee!
[ _Exeunt severally Priam and Hector. Alarums_.
[95]
TROILUS They are at it, hark! Proud Diomed, believe,
I come to lose my arm or win my sleeve.
_Enter PANDARUS_.
PANDARUS Do you hear, my lord? Do you hear?
TROILUS What now?
PANDARUS Here's a letter come from yond poor girl.
[100]
TROILUS Let me read.
[107]
PANDARUS A whoreson tisick, a whoreson rascally tisick so troubles me, and the foolish fortune of this girl, and what one thing, what another, that I shall leave you one o' th's days; and I have a rheum in mine eyes too, and such an ache in my bones that unless a man were curs'd I cannot tell what to think on't. What says she there?
TROILUS Words, words, mere words, no matter from the heart;
Th' effect doth operate another way.
[ _Tearing the letter_.
[110]
Go, wind, to wind, there turn and change together.
My love with words and errors still she feeds,
But edifies another with her deeds.
PANDARUS Why but heare you.
TROILUS Hence broker-lackey. Ignominy and shame
Pursue thy life and live aye with thy name!
[ _Exeunt severally_.
**SCENE IV**. _The plain between Troy and the Grecian camp_.
_Enter THERSITES. Excursions_.
[16]
THERSITES Now they are clapper-clawing one another; I'll go look on. That dissembling abominable varlet, Diomed, has got that same scurvy doting foolish young knave's sleeve of Troy there in his helm. I would fain see them meet, that that same young Troyan ass that loves the whore there might send that Greekish whoremasterly villain with the sleeve back to the dissembling luxurious drab of a sleeve-less errand. A th' t'other side, the policy of those crafty swearing rascals – that stale old mouse-eaten dry cheese, Nestor, and that same dog-fox, Ulysses – is not prov'd worth a blackberry. They set me up, in policy, that mongrel cur, Ajax, against that dog of as bad a kind, Achilles; and now is the cur Ajax prouder than the cur Achilles, and will not arm to-day; whereupon the Grecians begin to proclaim barbarism, and policy grows into an ill opinion.
_Enter DIOMEDES, TROILUS following_.
Soft! here comes sleeve, and t'other.
TROILUS Fly not; for shouldst thou take the river Styx
I would swim after.
DIOMEDES Thou dost miscall retire.
[20]
I do not fly; but advantageous care
Withdrew me from the odds of multitude.
Have at thee.
THERSITES Hold thy whore, Grecian; now for thy whore, Troyan – now the sleeve, now the sleeve!
[ _Exeunt Troilus and Diomedes fighting_.
_Enter HECTOR_.
[25]
HECTOR What art thou, Greek? Art thou for
Hector's match?
Art thou of blood and honour?
THERSITES No, no – I am a rascal; a scurvy railing knave; a very filthy rogue.
HECTOR I do believe thee. Live.
[ _Exit_.
THERSITES God-a-mercy, that thou wilt believe me; but a plague break thy neck for frighting me! What's become of the wenching rogues? I think they have swallowed one another. I would laugh at that miracle. Yet, in a sort, lechery eats itself. I'll seek them.
[ _Exit_.
**SCENE V**. _Another part of the plain_.
_Enter DIOMEDES and a Servant_.
DIOMEDES Go, go, my servant, take thou Troilus' horse;
Present the fair steed to my lady Cressid.
Fellow, commend my service to her beauty;
Tell her I have chastis'd the amorous Troyan,
And am her knight by proof.
[5]
SERVANT I go, my lord. [ _Exit_.
_Enter AGAMEMNON_.
AGAMEMNON Renew, renew! The fierce Polydamus
Hath beat down Menon; bastard Margarelon
Hath Doreus prisoner,
And stands colossus-wise, waving his beam,
[10]
Upon the pashed corses of the kings
Epistrophus and Cedius. Polixenes is slain;
Amphimacus and Thoas deadly hurt;
Patroclus ta'en, or slain; and Palamedes
Sore hurt and bruis'd. The dreadful Sagittary
[15]
Appals our numbers. Haste we, Diomed,
To reinforcement, or we perish all.
_Enter NESTOR_.
NESTOR Go, bear Patroclus' body to Achilles,
And bid the snail-pac'd Ajax arm for shame.
There is a thousand Hectors in the field;
[20]
Now here he fights on Galathe his horse,
And there lacks work; anon he's there afoot,
And there they fly or die, like scaled sculls
Before the belching whale; then is he yonder,
And there the strawy Greeks, ripe for his edge,
[25]
Fall down before him like the mower's swath.
Here, there, and everywhere, he leaves and takes;
Dexterity so obeying appetite
That what he will he does, and does so much
That proof is call'd impossibility.
_Enter ULYSSES_.
ULYSSES O, courage, courage, Princes! Great
[30]
Achilles
Is arming, weeping, cursing, vowing vengeance.
Patroclus' wounds have rous'd his drowsy blood,
Together with his mangled Myrmidons,
That noseless, handless, hack'd and chipp'd, come to him,
[35]
Crying on Hector. Ajax hath lost a friend
And foams at mouth, and he is arm'd and at it,
Roaring for Troilus; who hath done to-day
Mad and fantastic execution,
Engaging and redeeming of himself
[40]
With such a careless force and forceless care
As if that luck, in very spite of cunning,
Bade him win all.
_Enter AJAX_.
AJAX Troilus! thou coward Troilus!
[ _Exit_.
DIOMEDES Ay, there, there.
NESTOR So, so, we draw together.
[ _Exit_.
_Enter ACHILLES_.
ACHILLES Where is this Hector?
[45]
Come, come, thou boy-queller, show thy face;
Know what it is to meet Achilles angry.
Hector! where's Hector? I will none but Hector.
[ _Exeunt_.
**SCENE VI**. _Another part of the plain_.
_Enter AJAX_.
AJAX Troilus, thou coward Troilus, show thy head.
_Enter DIOMEDES_.
DIOMEDES Troilus, I say! Where's Troilus?
AJAX What wouldst thou?
DIOMEDES I would correct him.
AJAX Were I the general, thou shouldst have my office
[5]
Ere that correction. Troilus, I say! What, Troilus!
_Enter TROILUS_.
TROILUS O traitor Diomed! Turn thy false face, thou traitor,
And pay thy life thou owest me for my horse.
DIOMEDES Ha! art thou there?
AJAX I'll fight with him alone. Stand, Diomed.
[10]
DIOMEDES He is my prize. I will not look upon.
TROILUS Come, both, you cogging Greeks; have at you both.
[ _Exeunt fighting_.
_Enter HECTOR_.
HECTOR Yea, Troilus? O, well fought, my youngest brother!
_Enter ACHILLES_.
ACHILLES Now do I see thee, ha! Have at thee, Hector!
HECTOR Pause, if thou wilt.
[15]
ACHILLES I do disdain thy courtesy, proud Troyan.
Be happy that my arms are out of use;
My rest and negligence befriends thee now,
But thou anon shalt hear of me again;
Till when, go seek thy fortune.
[ _Exit_.
HECTOR Fare thee well.
[20]
I would have been much more a fresher man,
Had I expected thee.
_Re-enter TROILUS_.
How now, my brother!
TROILUS Ajax hath ta'en Aeneas. Shall it be?
No, by the flame of yonder glorious heaven,
He shall not carry him; I'll be ta'en too,
[25]
Or bring him off. Fate, hear me what I say:
I reck not though thou end my life to-day.
[ _Exit_.
_Enter One in armour_.
HECTOR Stand, stand, thou Greek; thou art a goodly mark.
No? wilt thou not? I like thy armour well;
I'll frush it and unlock the rivets all
[30]
But I'll be master of it. Wilt thou not, beast, abide?
Why then, fly on; I'll hunt thee for thy hide.
[ _Exeunt_.
**SCENE VII**. _Another part of the plain_.
_Enter ACHILLES, with Myrmidons_.
ACHILLES Come here about me, you my Myrmidons;
Mark what I say. Attend me where I wheel;
Strike not a stroke, but keep yourselves in breath;
And when I have the bloody Hector found,
[5]
Empale him with your weapons round about;
In fellest manner execute your arms.
Follow me, sirs, and my proceedings eye.
It is decreed Hector the great must die.
[ _Exeunt_.
_Enter MENELAUS and PARIS, fighting; then THERSITES_.
THERSITES The cuckold and the cuckold-maker are at it. Now, bull! now, dog! 'Loo, Paris, 'loo! now my double-horn'd Spartan! 'loo, Paris, 'loo!
[12]
The bull has the game. Ware horns, ho!
[ _Exeunt Paris and Menelaus_.
_Enter MARGARELON_.
MARGARELON Turn, slave, and fight.
THERSITES What art thou?
[15]
MARGARELON A bastard son of Priam's.
THERSITES I am a bastard too; I love bastards. I am a bastard begot, bastard instructed, bastard in mind, bastard in valour, in everything illegitimate. One bear will not bite another, and wherefore should one bastard? Take heed, the quarrel's most ominous to us: if the son of a whore fight for a whore, he tempts judgment. Farewell, bastard.
[ _Exit_.
MARGARELON The devil take thee, coward!
[ _Exit_.
**SCENE VIII**. _Another part of the plain_.
_Enter HECTOR_.
HECTOR Most putrified core so fair without,
Thy goodly armour thus hath cost thy life.
Now is my day's work done; I'll take good breath:
Rest, sword; thou hast thy fill of blood and death! [ _Disarms_.
_Enter ACHILLES and his Myrmidons_.
[5]
ACHILLES Look, Hector, how the sun begins to set;
How ugly night comes breathing at his heels;
Even with the vail and dark'ning of the sun,
To close the day up, Hector's life is done.
HECTOR I am unarm'd; forego this vantage, Greek.
[10]
ACHILLES Strike, fellows, strike; this is the man I seek. [ _Hector falls_.
So, Ilion, fall thou next! Come, Troy, sink down;
Here lies thy heart, thy sinews, and thy bone.
On, Myrmidons, and cry you all amain
'Achilles hath the mighty Hector slain'.
[ _A retreat sounded_.
[15]
Hark! a retire upon our Grecian part.
MYRMIDONS The Troyan trumpets sound the like, my lord.
ACHILLES The dragon wing of night o'er-spreads the earth
And, stickler-like, the armies separates.
My half-supp'd sword, that frankly would have fed,
[20]
Pleas'd with this dainty bait, thus goes to bed.
[ _Sheathes his sword_.
Come, tie his body to my horse's tail;
Along the field I will the Troyan trail.
[ _Exeunt_.
**SCENE IX**. _Another part of the plain_.
_Sound retreat. Shout. Enter AGAMEMNON, AJAX, MENELAUS, NESTOR, DIOMEDES, and the rest, marching_.
AGAMEMNON Hark! hark! what shout is this?
NESTOR Peace, drums!
SOLDIERS [ _Within_ ] Achilles! Achilles!
Hector's slain. Achilles!
DIOMEDES The bruit is Hector's slain, and by Achilles.
[5]
AJAX If it be so, yet bragless let it be;
Great Hector was as good a man as he.
AGAMEMNON March patiently along. Let one be sent
To pray Achilles see us at our tent.
If in his death the gods have us befriended;
[10]
Great Troy is ours, and our sharp wars are ended. [ _Exeunt_.
**SCENE X**. _Another part of the plain_.
_Enter AENEAS, PARIS, ANTENOR, and DEIPHOBUS_.
AENEAS Stand, ho! yet are we masters of the field.
Never go home; here starve we out the night.
_Enter TROILUS_.
TROILUS Hector is slain
ALL Hector! The gods forbid!
TROILUS He's dead, and at the murderer's horse's tail,
[5]
In beastly sort, dragg'd through the shameful field.
Frown on, you heavens, effect your rage with speed.
Sit, gods, upon your thrones, and smile at Troy.
I say at once let your brief plagues be mercy,
And linger not our sure destructions on.
[10]
AENEAS My lord, you do discomfort all the host.
TROILUS You understand me not that tell me so.
I do not speak of flight, of fear of death,
But dare all imminence that gods and men
Address their dangers in. Hector is gone.
[15]
Who shall tell Priam so, or Hecuba?
Let him that will a screech-owl aye be call'd
Go in to Troy, and say there 'Hector's dead'.
There is a word will Priam turn to stone;
Make wells and Niobes of the maids and wives,
[20]
Cold statues of the youth; and, in a word,
Scare Troy out of itself. But, march away;
Hector is dead; there is no more to say.
Stay yet. You vile abominable tents,
Thus proudly pight upon our Phrygian plains,
[25]
Let Titan rise as early as he dare,
I'll through and through you. And, thou great-siz'd coward,
No space of earth shall sunder our two hates;
I'll haunt thee like a wicked conscience still,
That mouldeth goblins swift as frenzy's thoughts.
[30]
Strike a free march to Troy. With comfort go;
Hope of revenge shall hide our inward woe.
_Enter PANDARUS_.
PANDARUS But hear you, hear you!
TROILUS Hence, broker-lackey. Ignominy and shame
Pursue thy life and live aye with thy name!
[ _Exeunt all but Pandarus_.
[40]
PANDARUS A goodly medicine for my aching bones! O world! world! world! thus is the poor agent despis'd! O traitors and bawds, how earnestly are you set a work, and how ill requited! Why should our endeavour be so lov'd, and the performance so loathed? What verse for it? What instance for it? Let me see –
Full merrily the humble-bee doth sing
Till he hath lost his honey and his sting:
And being once subdu'd in armed tail,
Sweet honey and sweet notes together fail.
[45]
Good traders in the flesh, set this in your painted cloths.
As many as be here of pander's hall,
Your eyes, half out, weep out at Pandar's fall;
Or, if you cannot weep, yet give some groans,
Though not for me, yet for your aching bones.
[50]
Brethren and sisters of the hold-door trade,
Some two months hence my will shall here be made.
It should be now, but that my fear is this,
Some galled goose of Winchester would hiss.
Till then I'll sweat and seek about for eases,
[55]
And at that time bequeath you my diseases.
[ _Exit_.
#### About the Author
Arguably the greatest English-language playwright, William Shakespeare was a seventeenth-century writer and dramatist, and is known as the "Bard of Avon." Under the patronage of Queen Elizabeth I, he penned more than 30 plays, 154 sonnets, and numerous narrative poems and short verses. Equally accomplished in histories, tragedies, comedy, and romance, Shakespeare's most famous works include _Romeo and Juliet, Hamlet, Macbeth, King Lear, The Taming of the Shrew_ , and _As You Like It_.
Like many of his contemporaries, including Christopher Marlowe, Shakespeare began his career on the stage, eventually rising to become part-owner of Lord Chamberlain's Men, a popular dramatic company of his day, and of the storied Globe Theatre in London.
Extremely popular in his lifetime, Shakespeare's works continue to resonate more than three hundred years after his death. His plays are performed more often than any other playwright's, have been translated into every major language in the world, and are studied widely by scholars and students.
#### About the Series
Harper _Perennial_ Classics brings great works of literature to life in digital format, upholding the highest standards in ebook production and celebrating reading in all its forms. Look for more titles in the Harper _Perennial_ Classics collection to build your digital library.
#### Copyright
Harper _Perennial_ Classics
An imprint of HarperCollins Publishers Ltd.
2 Bloor Street East, 20th Floor
Toronto, Ontario, Canada
M4W 1A8 __
_www.harpercollins.ca_
EPub Edition December 2014 ISBN: 9781443443593
This title is in Canada's public domain and is not subject to any licence or copyright.
#### About the Publisher
**HarperCollins Publishers (Australia) Pty. Ltd.**
Level 13, 201 Elizabeth Street
Sydney, NSW 2000, Australia
<http://www.harpercollins.com.au>
**HarperCollins Publishers (Canada) Ltd.**
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Toronto, ON, M4W 1A8, Canada
<http://www.harpercollins.ca>
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P.O. Box 1
Auckland, New Zealand
<http://www.harpercollins.co.nz>
**HarperCollins Publishers Ltd.**
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<http://www.harpercollins.co.uk>
**HarperCollins Publishers Inc.**
10 East 53rd Street
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<http://www.harpercollins.com>
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{"url":"http:\/\/mathematica.stackexchange.com\/questions\/8791\/two-x-axis-with-different-units-connected-by-function-using-customticks","text":"# Two x-axis with different units connected by function using CustomTicks\n\nI am trying to make a plot of some photoluminescence data, where wavelength is on the x-axis, and intensity is on the y-axis. Also, I would like to have the graph boxed, with the upper x-axis having units of energy in eV, connected to the wavelength through the function: energy=1240\/wavelength. I have tried to use the TickLabelFunction in the LevelScheme package, and have only partly succeeded. I can't control the number of decimal points for the ticks on the energy scale. For some reason, N[] doesn't work. Also, the energy tickmarks seem to follow the wavelength ticks, i.e. one energy tick for each wavelength tick. I would like those ticks to be spaced nonlinearly and independently of the other axis, e.g. having them increase with steps of 0.2 eV. Here is a small code that I hope will illustrate my problem.\n\ntest = List[{500, 50}];\nListPlot[test, Frame -> True, ImageSize -> 600,\nPlotRange -> {{400, 800}, {-10, 110}},\nFrameLabel -> {\"Wavelength (nm)\", \"Intensity (a.u.)\", \"Energy (eV)\"},\nFrameTicks -> {LinTicks[400, 800, 100, 5], LinTicks,\nLinTicks[400, 800, 100, 5,\nTickLabelFunction -> Function[x, N[1240\/x, 2]]],\nStripTickLabels[LinTicks]}]\n\n\nAny input appreciated, either using the LevelScheme package or not. Thanks!\n\n-\nCould you post an image with the \"model\" of your desired output together with a few points to generate the plot? \u2013\u00a0 belisarius Jul 27 '12 at 22:08\nWelcome to Mathematica.SE! Did you check this question? It might give you some clues. \u2013\u00a0 Sjoerd C. de Vries Jul 28 '12 at 2:19\n\nHere is a method to generate the ticks automatically based on the plot range.\n\nIt uses FindDivisions to select \"pleasing\" values.\n\neVticks = {1240\/#, NumberForm[N@#, {2, 1}]} & \/@ FindDivisions[1240\/{##}, 8] &;\n\nListPlot[{{500, 50}},\nFrame -> True,\nPlotRange -> {{400, 800}, {-10, 110}},\nFrameLabel -> {\"Wavelength (nm)\", \"Intensity (a.u.)\", \"Energy (eV)\"},\nFrameTicks -> {{Automatic, Automatic}, {Automatic, eVticks}}\n]\n\n\nWith sub-ticks as requested:\n\neVticks = Join[\n{1240\/#, Null} & \/@ FindDivisions[1240\/{##}, 8*6],\n{1240\/#, NumberForm[N@#, {2, 1}]} & \/@ FindDivisions[1240\/{##}, 8]\n] &;\n\n\n-\nI like both of the solutions here. But what about minor tick marks, how would you get those in there, and get them to be non-linear? \u2013\u00a0 GOt Jul 28 '12 at 16:58\n@GOt I added a method to my answer. I have not tested it extensively but I believe it is correct. If you find that irregularity in the sub-ticks occurs I'll provide another method. \u2013\u00a0 Mr.Wizard Jul 28 '12 at 20:24\nThanks so much! It seems to work perfectly. \u2013\u00a0 GOt Jul 28 '12 at 21:44\nIt might be a good idea to isolate the 1240\/# stuff in a function to make the routine easier to generalize. \u2013\u00a0 Sjoerd C. de Vries Aug 2 '12 at 13:16\n@Sjoerd this is merely an illustration, not a packaged solution. There are a number of things I would add if I were making this a reusable function. \u2013\u00a0 Mr.Wizard Aug 2 '12 at 14:01\n\nI would define my upper ticks so that it should be easy to change the number of decimal points if needed (thanks to Mr.Wizard for pointing out a mistake).\n\nupperTicks = Module[{labels, positions},\nlabels = Range[1240\/400, 1240\/800, -0.2];\npositions = 1240\/# & \/@ labels;\nTranspose[{positions, labels}]];\n\nupperTicksMinor = Module[{labels, positions, size},\nlabels = Range[1240\/350, 1240\/850, -0.05];\npositions = 1240\/# & \/@ labels;\nlabels = \"\" & \/@ Range[Length[labels]];\nsize = {0, 0.002} & \/@ Range[Length[labels]];\nTranspose[{positions, labels, size}]];\n\n\nand then use them :\n\nListPlot[test, Frame -> True, ImageSize -> 600,\nPlotRange -> {{400, 800}, {-10, 110}},\nFrameLabel -> {\"Wavelength (nm)\", \"Intensity (a.u.)\", \"Energy (eV)\"},\nFrameTicks -> {{Automatic, None}, {Automatic,\nJoin[upperTicks, upperTicksMinor]}}]\n\n\n-\nYour top ticks run the opposite direction from the example that the OP gave. Also, he wrote: \" I would like those ticks to be spaced nonlinearly ...\" \u2013\u00a0 Mr.Wizard Jul 28 '12 at 7:58\n@Mr.Wizard Yes I inverted the direction - will fix it. Isn't the OP asking for 0.2 eV increments though ? \u2013\u00a0 b.gatessucks Jul 28 '12 at 8:03\nWhen I run the OP's code I get this. Note the function 1240\/x. \u2013\u00a0 Mr.Wizard Jul 28 '12 at 8:07\nLooking good. +1 \u2013\u00a0 Mr.Wizard Jul 28 '12 at 8:19\nI like both of the solutions here. But what about minor tick marks, how would you get those in there, and get them to be non-linear? \u2013\u00a0 GOt Jul 28 '12 at 16:58","date":"2015-07-04 03:39:12","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2585515081882477, \"perplexity\": 4131.340976377375}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-27\/segments\/1435375096301.47\/warc\/CC-MAIN-20150627031816-00176-ip-10-179-60-89.ec2.internal.warc.gz\"}"}
| null | null |
Amazon, Microsoft offer free access to cloud resources for climate-change research
AWS, Microsoft and IBM will award free access to supercomputing and cloud resources for climate-change research
Technology giants AWS, Microsoft and IBM have committed to award free access to their supercomputing and cloud resources to drive innovative climate-change research.
The grant programmes are in support of the US government's Climate Action Plan and Climate Data Initiative.
"My administration will work with tech innovators and launch new challenges under our Climate Data Initiative," said US President Barack Obama.
AWS has launched a climate research grant programme under which it will award grants of free access to supercomputing resources through Amazon EC2 Spot Instances in September 2014.
According to AWS, the climate change initiatives outlined by the Obama administration require large-scale analysis of climate data.
The cloud provider will award a total of 50 million core hours of supercomputing using Amazon EC2 Spot Instances with training and guidance from the AWS scientific computing team.
"By providing grants totaling 50 million core hours, AWS is enabling researchers to accelerate research that can result in an improved understanding of the scope and effects of climate change, and analyses that could suggest potential mitigating actions," said the Press Secretary at the White House.
The programme offers "scalable computing resources of the AWS cloud to researchers so they can quickly analyse climate data and increase our understanding of climate change," Jeff Barr, chief evangelist for AWS said on the company blog.
"Our goal is to encourage and accelerate research that will result in an improved understanding of the scope and effects of climate change, along with analyses that could suggest potential mitigating actions."
AWS Spot Instance allows a customer to buy unused Amazon EC2 computer capacity at a heavily discounted rate.
AWS Spot Instance gives Amazon a flexible way to sell extra capacity. The instances are acquired through a bidding process in which a customer specifies a price per hour they are willing to pay.
Its rival cloud provider Microsoft is also supporting the US president's climate data initiative by planning to make available an initial collection of climate-research related datasets through its Azure Marketplace.
Microsoft Research will also grant 12 months of free cloud-computing resources to 20 awardees submitting proposals focused on food resilience and climate change by 15 September 2014.
It will provide the free IT resources through the Azure for Research program. Microsoft will co-host vents aimed at demonstrating the value of open-data and data-driven tools to boost climate preparedness and resilience in the agricultural sector.
Meanwhile, IBM will expand its World Community Grid program, which enables members of the public to donate their computer or mobile device's unused computing power to scientists. The expansion will provide scientists studying climate change topics with free access to dedicated virtual supercomputing resources and a platform to engage the public on their research.
Under the IBM programme, each researcher will have access to up to 100,000 years of computing time, a value of $60m in today's costs, said the White House Press Secretary.
Read more on Clustering for high availability and HPC
Dutch authorities temporarily halt datacentre construction
Case study: Using pre-fab datacentres to meet Norway's growing demand for colocation space
IT pros see a role for high-performance computing in business
AI storage: Machine learning, deep learning and storage needs
Norway datacentre strategy producing dividend
Enterprise HPC: Why HPE is buying Cray
Enterprise accessibility: How Cray is using HPC to open up AI use cases from the datacentre
The Sanger Institute on using datacentre upgrades to help decode the human genome
Cray builds Urika-GX for open source data analytics
A datacentre in a church: Welcome to the Barcelona Supercomputing Centre
A peek into Intel's Internet of Things lab in Swindon
Amazon results demonstrate commitment to cloud – ComputerWeekly.com
Amazon launches HPC service in the cloud – SearchAWS
Dutch national weather service adopts cloud to expand... – ComputerWeekly.com
Spinksy - 30 Jul 2014 4:19 AM
A noble gesture, but some irony here considering exactly how the cloud computing of Amazon and Microsoft is powered. According to Greenpeace these two companies perform quite badly in terms of greenhouse gas emitting fuel sources for its data centres: http://www.greenpeace.org/usa/...
infrastrucure for a significant part of the internet,
remains among the dirtiest and least transparent
companies in the sector, far behind its major
competitors, with zero reporting of its energy or
environmental footprint to any source or stakeholder.
Amazon Web Services (AWS), which provides the
infrastructure for a significant part of the internet,
|
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OU people profiles
More on privacy
How to edit your profile
Dr Lindsey Stewart
Impact and engagement
School of Arts & Humanities
English & Creative Writing
lindsey.stewart
My first degree was in English from the University of Sussex. I went on to do an MA in English with the OU which was awarded with distinction in 2014, followed by an AHRC-funded PhD which was awarded in 2018. I have a background in policy planning at the Foreign & Commonwealth Office and have also worked as a secondary English teacher.
I am interested in the intersections between Victorian literature, psychology and health. My PhD examined the cultural mythology around the psychiatric idea of 'monomania'. I am currently working on connections between the nineteenth-century novel, the environment and psychology.
In press: 'Desperate Housewives: Dinah Mulock Craik's 'The Double House' in Fiction and 'The Woman Question' from 1850 to 1930 ed. by Nicola Darwood, Bob Owens and Alexis Weedon (Cambridge: Cambridge Scholars Publishing, 2020)
''A New and Fierce Disorder's Raging': Monomania in Mary Barton (1848)', Journal of Victorian Culture, published 17 Feb 2019 https://doi.org/10.1093/jvcult/vcy072
Review of Suzanne Keen's Thomas Hardy's Brains: Psychology, Neurology, and Hardy's Imagination, The British Society for Literature and Science, accessible here:https://www.bsls.ac.uk/reviews/romantic-and-victorian/suzanne-keen-thomas-hardys-brains-psychology-neurology-and-hardys-imagination/
I also write regularly for the popular magazine, Emag, for teachers and students of Advanced level English. See:
'Fog, smog, ecocriticism and late-Victorian Gothic', Emag, 85:2019, 14-17
'Porphyria's Lover: a close analysis', Emag, 63:2014, 43-45
'Frankenstein: The Eloquence of the Creature Made Real', Emag, 52:2011, 60-63
'A New and Fierce Disorder's Raging': Monomania in Mary Barton (1848) (2019-10)
Stewart, Lindsey
Journal of Victorian Culture, 24(4) (pp. 492-506)
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{"url":"http:\/\/math.stackexchange.com\/questions\/202814\/maple-equation-entry","text":"Maple equation entry [closed]\n\nI'm given this question and asked to solve it in maple. What are the codes to key in this equation?\n\n-\n\nclosed as off-topic by Bill Dubuque, Adam Hughes, Winther, Mice Elf, S.Panja-1729Sep 3 at 3:42\n\nThis question appears to be off-topic. The users who voted to close gave this specific reason:\n\n\u2022 \"This question is not about mathematics, within the scope defined in the help center.\" \u2013 Bill Dubuque, Adam Hughes, Winther, Mice Elf, S.Panja-1729\nIf this question can be reworded to fit the rules in the help center, please edit the question.\n\nWhat have you tried? Do the following two items help you to give it a go? 1) maplesoft.com\/support\/help\/Maple\/view.aspx?path=product and 2) maplesoft.com\/support\/help\/Maple\/view.aspx?path=limit \u2013\u00a0 Amzoti Sep 26 '12 at 13:28\n\nHere is the maple notation for your expression\n\nlimit( n^(3\/2)*product( 2*k\/(2*k+3), k=1..n ), n=infinity );\n\n-\nthanks, this works fine :) \u2013\u00a0 ryantata Sep 26 '12 at 14:43\n\nUse $\\tt product(f(k),k=1..n)$ and $\\tt limit(g(n),n=infinity)$. The answer is $3\/4\\sqrt{\\pi}$.\n\n-\nIs g(n)=product(....) here? :-) \u2013\u00a0 Babak S. Sep 26 '12 at 15:12\n@BabakSorouh yes \u2013\u00a0 i. m. soloveichik Sep 26 '12 at 16:15","date":"2015-10-06 20:37:14","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\": 0.7931430339813232, \"perplexity\": 3845.70466072181}, \"config\": {\"markdown_headings\": false, \"markdown_code\": false, \"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-2015-40\/segments\/1443736679145.29\/warc\/CC-MAIN-20151001215759-00199-ip-10-137-6-227.ec2.internal.warc.gz\"}"}
| null | null |
\section{INTRODUCTION}
Gluon-gluon fusion is the main production channel of
the Standard Model Higgs boson at the LHC.
At leading order (LO) in QCD perturbation theory, the cross section is proportional
to $\as^2$, $\as$ being the QCD coupling. The QCD radiative corrections to the total cross section are known at the
next-to-leading order (NLO)
\cite{Dawson:1990zj,Djouadi:1991tka,Spira:1995rr}
and at the next-to-next-to-leading order (NNLO)
\cite{Harlander:2000mg,Catani:2001ic,Harlander:2001is,Harlander:2002wh,Anastasiou:2002yz,Ravindran:2003um}.
The effects of a jet veto on the total cross section has been studied
up to NNLO \cite{Catani:2001cr}.
We recall that all the results at NNLO have been obtained by using
the large-$M_t$ approximation, $M_t$ being the mass of the top quark.
These NNLO calculations can be supplemented with soft-gluon resummed calculations at next-to-next-to-leading logarithmic (NNLL) accuracy
either to improve the quantitative accuracy of the perturbative predictions
(as in the case of the total cross section \cite{Catani:2003zt,Moch:2005ky}) or to provide
reliable predictions in phase-space regions where fixed-order calculations
are known to fail (as in the case of the $q_T$ distribution
of the Higgs boson \cite{Bozzi:2003jy,Bozzi:2005wk,Bozzi:2007pn}
at small $q_T$).
A common feature of these NNLO and NNLL calculations is that they
are fully inclusive over the produced final state (in particular, over
final-state QCD radiation).
Therefore they refer to situations
where the experimental cuts
are either ignored (as in the case of the total cross section) or taken into account
only in simplified cases (as in the case of the jet vetoed cross section).
The impact of higher-order corrections
may be strongly dependent on the details of the applied cuts and also the shape of various
distributions is typically affected by these details.
The first NNLO calculation that fully takes into account experimental cuts
was reported in Ref.~\cite{Anastasiou:2005qj}, considering the decay mode $H\to\gamma\gamma$ (see also \cite{Stockli:2005hz}).
In Ref.~\cite{Anastasiou:2007mz} the calculation is
extended to the decay mode $H\to WW\to l\nu l\nu$ (see also \cite{Anastasiou:2008ik}).
In Ref.~\cite{Catani:2007vq} we have
proposed a method to perform NNLO calculations and we have
applied it to perform an independent computation
of the Higgs production cross section.
The method is completely different from
that used in Refs.~\cite{Anastasiou:2005qj,Anastasiou:2007mz}.
Our calculation is implemented
in a fully-exclusive parton level event generator.
This feature makes it particularly suitable for practical applications
to the computation of distributions in the form of bin histograms.
Our numerical program can be downloaded from \cite{hnnlo}.
The decay modes that are currently implemented
are $H\to \gamma\gamma$ \cite{Catani:2007vq}, $H\to WW\to l\nu l\nu$ and $H\to ZZ\to 4$ leptons \cite{Grazzini:2008tf}.
In the following
we present a brief selection of results that can be obtained
by our program.
We consider Higgs boson production at the LHC and
use the MRST2004 parton distributions \cite{Martin:2004ir},
with parton densities and $\as$ evaluated at each corresponding order
(i.e., we use $(n+1)$-loop $\as$ at N$^n$LO, with $n=0,1,2$). The
renormalization and factorization scales are fixed to the value
$\mu_R=\mu_F=M_H$, where $M_H$ is the mass of the Higgs boson.
\section{RESULTS FOR THE DECAY MODE $H\to \gamma\gamma$}
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth]{ptminmax}
\caption{Distributions in $p_{T{\rm min}}$ and $p_{T{\rm max}}$ for the diphoton signal at the LHC. The cross section is divided by the branching ratio in two photons.}
\label{fig:isol}
\end{center}
\end{figure}
We consider the production of a Higgs boson of mass $M_H=125$ GeV in the
$H\to\gamma\gamma$ decay mode and follow
Ref.~\cite{CMStdr} to apply cuts on the photons.
For each event, we classify the photon transverse momenta according to their
minimum and maximum value,
$p_{T{\rm min}}$ and $p_{T{\rm max}}$. The photons are required to be in the
central rapidity region, $|\eta|<2.5$, with $p_{T{\rm min}}>35$~GeV
and $p_{T{\rm max}}>40$~GeV. We also require the photons to be isolated:
the hadronic (partonic) transverse energy in a cone of radius $R=0.3$ along the
photon direction
has to be smaller than 6~GeV. Using these cuts
the impact of the NNLO corrections on the NLO total cross section
is reduced from 19\% to 11\%.
In Fig.~\ref{fig:isol} we plot
the distributions in $p_{T{\rm min}}$ and $p_{T{\rm max}}$
of the signal process $gg\to H\to\gamma\gamma$.
We note that the shape of these distributions sizeably
differs when going from LO to NLO and to NNLO.
The origin of these perturbative instabilities is well known
\cite{Catani:1997xc}.
Since the LO spectra
are kinematically bounded by $p_T\leq M_H/2$,
each higher-order perturbative contribution produces
(integrable) logarithmic singularities in the vicinity of
that boundary. More detailed studies are necessary to assess
the theoretical uncertainties of these fixed-order results
and the relevance of all-order resummed calculations.
In Fig.~\ref{fig:ctheta} we consider the (normalized) distribution in
the variable $\cos\theta^*$, where $\theta^*$ is the polar angle of one of the photons
in the rest frame of the Higgs boson
\footnote{We thank Suzanne Gascon and Markus Schumacher for suggesting
the use of this variable.}.
At small values of $\cos\theta^*$ the distribution
is quite stable with respect to higher order QCD corrections.
We also note that the LO distribution vanishes beyond the value $\cos\theta^*_{\rm max}<1$.
The upper bound $\cos\theta^*_{\rm max}$ is due to the fact that the photons
are required to have a minimum $p_T$ of $35$ GeV.
As in the case of Fig.~\ref{fig:isol}, in the vicinity of this LO kinematical boundary
there is an instability of the
perturbative results beyond LO.
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth]{cthetanew}
\caption{Normalized distribution in the variable $\cos\theta^*$.}
\label{fig:ctheta}
\end{center}
\end{figure}
\section{RESULTS FOR THE DECAY MODE $H\to l\nu l\nu$}
We now consider the production of a Higgs boson with mass $M_H=165$ GeV in the decay mode $H\to l\nu l\nu$.
We apply a set of {\em preselection} cuts taken from the study of Ref.~\cite{Davatz:2004zg}. The charged leptons have $p_T$ larger than 20 GeV, and $|\eta|<2$. The missing $p_T$ is
larger than $20$ GeV and the invariant mass of the charged
leptons is smaller than $80$ GeV.
Finally, the azimuthal separation of the charged leptons in the
transverse plane ($\Delta\phi$) is smaller than $135^o$.
By applying these cuts the impact of the NNLO corrections
on the NLO result does not change and is of about $20\%$.
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth]{deltaphinorm}
\caption{Normalized $\Delta\phi$ distribution at LO, NLO, NNLO.}
\label{fig:deltaphi}
\end{center}
\end{figure}
In Fig.\ref{fig:deltaphi} we
plot the $\Delta\phi$ distribution at LO, NLO and NNLO.
As is well known \cite{Dittmar:1996ss},
the charged leptons from the Higgs boson
signal tend to be close in angle,
and thus the distribution is peaked at small $\Delta\phi$.
We note that the effect of the QCD corrections is to increase
the steepness of the distribution,
from LO to NLO and from NLO to NNLO.
We finally apply the set of selection cuts designed to isolate the Higgs boson signal \cite{Davatz:2004zg}.
The transverse momenta of the two charged leptons
are classified according to their
minimum and maximum value,
$p_{T{\rm min}}$ and $p_{T{\rm max}}$.
They should fulfil $p_{T{\rm min}} > 25$ GeV and $35~{\rm GeV}< p_{T{\rm max}} <50$ GeV.
The missing $p_T$ of the event should be larger than 20 GeV
and the invariant mass of the charged leptons should be smaller than $35$ GeV.
To exploit the steepness of the $\Delta\phi$ distribution shown in Fig.~\ref{fig:deltaphi},
$\Delta\phi$ should be smaller than $45^o$.
Finally, there should be no jets with $p_T^{\rm jet}$ larger
than a given value $p_T^{\rm veto}$.
The corresponding cross sections, for different values of $p_T^{\rm veto}$, are reported in Table ~\ref{table}.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
$p_T^{\rm veto}$ (GeV) & $\sigma_{NLO}$ (fb) & $\sigma_{NNLO}$ (fb)\\
\hline
\hline
no veto & $21.26\pm 0.05$ & $22.21\pm 0.32$\\
\hline
40 & $18.62\pm 0.05$ & $17.38\pm 0.34$\\
\hline
30 & $17.18\pm 0.05$ & $15.74\pm 0.35$\\
\hline
20 & $14.42\pm 0.05$ & $11.31\pm 0.38$\\
\hline
\end{tabular}
\end{center}
\caption{{\em Cross sections for $pp\to H+X\to WW+X\to l\nu l\nu+X$ at the LHC
when selection cuts are applied for different $p_T^{\rm veto}$.}}
\label{table}
\end{table}
We see that, even without the jet veto, the impact of radiative corrections is strongly reduced with this choice
of cuts. The jet veto further reduces the effect of the NNLO corrections, which become negative for $p_T^{\rm veto} \ltap 40$ GeV.
\section{SUMMARY}
We have illustrated the results of a calculation of the Higgs boson production cross section at the LHC
up to NNLO in QCD perturbation theory. The calculation is performed
by using the numerical program {\tt HNNLO},
which includes all the relevant decay modes of the Higgs boson, namely,
$H\to\gamma\gamma$, $H\to WW\to l\nu l\nu$ and $H\to ZZ\to 4$ leptons. The program allows the user to apply arbitrary cuts on the momenta of the partons and
of the photons/leptons produced in the final state, and to obtain the required distributions in the form of
bin histograms. We have presented a brief selection of numerical results for the decay modes $H\to\gamma\gamma$ and
$H\to WW\to l\nu l\nu$. More detailed results for the decay modes $H\to WW$ and $H\to ZZ$
can be found in Ref.~\cite{Grazzini:2008tf}.
The fortran code {\tt HNNLO} can be downloaded from \cite{hnnlo}.
|
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\section{Introduction and statements of results}
For an odd prime power $q$, let us consider the $n$-dimensional vector space $\mathbb{F}_{q}^{n}$ over the finite field $\mathbb{F}_{q}$. Since $\mathbb{F}_{q}^{n}$ is a vector space over $\mathbb{F}_{q}$, the dimension is a useful invariant to classify its subspaces. Furthermore, the number of $k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$ is the $q$-binomial coefficient $\binom{n}{k}_{q}$ which plays an important role in combinatorics.
\smallskip
One of the surprising phenomena about the $q$-binomial coefficient is the fact that there are the $q$-analogues of the binomial coefficients. There are well-known relationships between the binomial coefficients and the $q$-binomial coefficients as follows:
\begin{table}[H]
\begin{center}
\scalebox{0.9}{
\begin{tabular}{c||c|c}
& \textbf{Field with one element} & $\mathbf{\mathbb{F}_{q}}$ \textbf{($\mathbf{q}$-analogues)} \\ \hline \hline
\textbf{object} & $\left [ n \right ]=\left \{ 1,2,\cdots,n \right \}$ &
$\mathbb{F}^{n}_{q}$ \\ \hline
\textbf{subobject} & a $k$ set in $\left [ n \right ]$ &
a $k$-dimensional subspace of $\mathbb{F}^{n}_{q}$ \\ \hline
\textbf{bracket} & $n$ & the number of lines in $\mathbb{F}_{q}^{n}$ \\ \hline
\textbf{factorial} & $n!$ & $\left [ n\right ]_{q}!$ \\ \hline
\textbf{poset} & $B_{n}$ & $L_{n}(q)$ \\ \hline
\textbf{group}& $|S_{n}|=n!$ & $|GL(n,q)|=q^{n(n-1)/2}(q-1)^{n}\left [ n \right ]_{q}!$ \\ \hline
\textbf{flag} & flags in $\left [ n \right ]$ & flags in $\mathbb{F}_{q}^{n}$\\ \hline
\textbf{binomial coefficient} & $\binom{n}{k}=\frac{n!}{k!(n-k)!}=\left | \frac{S_{n}}{S_{k}\times S_{n-k}} \right |$ & $\binom{n}{k}_{q}=\frac{[n]_{q}!}{[k]_{q}![(n-k)]_{q}!}=\left | \frac{GL(n,q)}{\bigl(\begin{smallmatrix}
A & C \\
\mathbf{0} & B
\end{smallmatrix}\bigr)} \right |$ \\ \hline
\textbf{connection}& \multicolumn{2}{c}{$
\lim_{q \rightarrow 1} \binom{n}{k}_{q}=\binom{n}{k}$}\\
\end{tabular}}
\caption{Example of Field with one element analogues.}
\label{field}
\end{center}
\end{table}
\noindent where $A$ in $GL(k,q)$, $B$ in $GL(n-k,q)$, and $C$ is a $k \times (n-k)$ matrix. It turns out that $q$-analogues appear in many mathematical areas not only combinatorics but also in the study of special function, quantum group.
\smallskip
In this paper, we add one more structure called a quadratic form on $\mathbb{F}_{q}^{n}$. Recall that in the theory of quadratic forms, any non-degenerate quadratic form on a vector space of dimension $n$ over a finite field is equivalent to one of the following two forms:
\begin{align*}
\text{dot}_n(\mathbf{x}) & :=x_{1}^{2}+\cdots+x_{n-1}^{2}+x_{n}^{2} \quad \text{or}\\
\text{$\lambda$dot$_n$}(\mathbf{x}) & :=x_{1}^{2}+\cdots+x_{n-1}^{2}+\lambda x_{n}^{2} \quad \text{for some non-square $\lambda$ in $\mathbb{F}_{q}$}.
\end{align*}
We will call $(\mathbb{F}_q^n, \text{dot}_n)$ the \textbf{finite Euclidean space} and $(\mathbb{F}_q^n, \lambda$dot$_n)$ the \textbf{finite Lorenzian space}. When there is no danger of confusion, we will let dot$_{n}$ and $\lambda$dot$_{n}$ denote the quadratic spaces
$(\mathbb{F}_q^n, \text{dot}_n)$ and $(\mathbb{F}_q^n, \lambda$dot$_n)$ respectively.
\smallskip
Let us call a $k$-dimensional subspace $W$ of $(\mathbb{F}_{q}^{n},\text{dot}_{n})$ a $\mathbf{\textbf{dot}_{k}}\textbf{\text{-subspace}}$ if $(W,\text{dot}_{n}|_{W})$ is isometrically isomorphic to $(\mathbb{F}_{q}^{k},\text{dot}_{k})$, where dot$_{n}|_{W}$ is the restricted quadratic form on $W$. The main goal of this paper is to define the \textbf{dot-analogues} of the binomial coefficients, and to study related combinatorics listed in the last column of \cref{table-analog}.
\begin{table}[H]
\begin{center}
\scalebox{0.9}{
\begin{tabular}{c||c|c}
& $\mathbf{q}$\textbf{-analogues} & \textbf{dot-analogues} \\ \hline \hline
\textbf{space} & $\mathbb{F}^{n}_{q}$ &
$(\mathbb{F}^{n}_{q},\text{dot}_{n})$ \\ \hline
\textbf{subspace} & a $k$-dimensional subspace of $\mathbb{F}^{n}_{q}$ &
a dot$_{k}$-subspace of dot$_{n}$ \\ \hline
\textbf{bracket} & the number of lines in $\mathbb{F}_{q}^{n}$ & the number of spacelike lines in $(\mathbb{F}^{n}_{q},\text{dot}_{n})$ \\ \hline
\textbf{factorial} & $\left [ n\right ]_{q}!$ & $\left [n \right ]_{d}!$ \\ \hline
\textbf{poset} & $L_{n}(q)$ & $E_{n}(q)$ \\ \hline
\textbf{group}& $|GL(n,q)|=q^{n(n-1)/2}(q-1)^{n}\left [n \right]_{q}!$ & $|O(n,q)|=2^{n}\left [n\right ]_{d}!$ \\ \hline
\textbf{flag} & flags in $\mathbb{F}_{q}^{n}$ & Euclidean flags in ($\mathbb{F}_{q}^{n}$,dot$_{n}$)\\ \hline
\textbf{binomial coefficient} & $\binom{n}{k}_{q}=\frac{[n]_{q}!}{[k]_{q}![(n-k)]_{q}!}=\left | \frac{GL(n,q)}{\bigl(\begin{smallmatrix}
A & C \\
\mathbf{0} & B
\end{smallmatrix}\bigr)} \right |$ & $\binom{n}{k}_{d}=\frac{[n]_{d}!}{[k]_{d}![(n-k)]_{d}!}=\left | \frac{O(n,q)}{O(k,q)\times O(n-k,q)} \right |$ \\
\end{tabular}}
\caption{The $q$-analogues and the dot-analogues.}
\label{table-analog}
\end{center}
\end{table}
\smallskip
In \cite{St2}, the author shows that the poset $L_n(q)$ which are consisting of all subspaces of $\mathbb{F}_{q}^{n}$ under set-inclusion order is rank symmetric, and rank unimodal. In \cref{section3}, we consider the poset of $\text{dot}_{k}$-subspaces of $(\mathbb{F}_{q}^{n},\text{dot}_{n})$ for $0\leq k \leq n$ under set-inclusion order,
which we call the \textbf{Euclidean poset} and denote by $E_n(q)$. We show that the Euclidean poset $E_{n}(q)$ is rank symmetric (\cref{rs}) and rank unimodal (\cref{ru}). Furthermore, using the structure of the Euclidean poset, we find that the number of $\text{dot}_{k}$-subspaces in $\text{dot}_{n}$ is given by
\[
|\text{dot}_{k,n}|=\frac{|\text{dot}_{1,n}||\text{dot}_{1,n-1}|\cdots|\text{dot}_{1,n-k+1}|}{|\text{dot}_{1,k}|\cdots|\text{dot}_{1,1}|},
\]
where dot$_{k,n}$ is a set of dot$_{k}$-subspaces of dot$_{n}$. Observing the similarity between this formula and the one for the binomial coefficients, we define the \textbf{dot-binomial coefficient} (\cref{dot-binom}):
\[
\binom{n}{k}_{d}:=|\text{dot}_{k,n}|=\frac{[n]_{d}!}{[k]_{d}![(n-k)]_{d}!},
\]
where $[k]_{d}=|\text{dot}_{1,k}|$ and $[n]_{d}! =\prod_{k=1}^n [k]_{d}$. The bracket $[k]_{d}$, the number of dot$_{1}$-subspaces in dot$_{n}$ can be found in \cite{Yo1}, and stated in \cref{dotline} this paper. Readers who are interested in related combinatorial applications of the dot-binomial coefficients, see the author's work on the associated pseudo-random graphs \cite{Yo2}, and the incidence graphs \cite{Yo3}. In addition, we show that the dot-binomial coefficient $\binom{n}{k}_{d}$ is expressed in terms of the product of $q$-binomial coefficients and some polynomials (\cref{polynomials}). For example, when $q \equiv 1$ (mod $4$), and $n,k$ are odd, it is written as
\[\binom{n}{k}_{d}=\frac{1}{2}q^{\frac{k(n-k)}{2}}(q^{\frac{n-k}{2}}+1)\binom{\frac{n-1}{2}}{\frac{k-1}{2}}_{q^{2}},\]
where \[\binom{n}{k}_{q^{2}}=\frac{((q^{2})^{n}-1)((q^{2})^{n-1}-1)\cdots((q^{2})^{n-k+1}-1)}{(q^{2})^{k}-1)((q^{2})^{k-1}-1)\cdots(q^{2}-1)}.\]
\smallskip
We investigate various combinatorial properties of $\binom{n}{k}_{d}$, including the analogues of Pascal's identity (\cref{pa} and \cref{pa2}) and log-concavity of $\binom{n}{k}_{d}$ (\cref{lc}). We also show that $\binom{n}{k}_{d}$ is a polynomial of degree $k(n-k)$ in $q$ (\cref{rational}), and study some properties of the polynomial given by the dot-binomial coefficient $\binom{n}{k}_{d}$. More precisely, we show that the coefficients of the polynomial $\binom{n}{k}_{d}$ have some sort of symmetry (\cref{symmetry}). For example, when $q \equiv 1$ mod $4$, and $n,k$ are odd, the dot-binomial coefficient $\binom{n}{k}_{d}$ can be written as
\[\binom{n}{k}_{d}=x^{\frac{k(n-k)}{2}}\cdot \sum_{i=0}^{\frac{k(n-k)}{2}}a_{i}x^{i},\]
and we have the property $a_{i}=a_{k(n-k)/2-i}$ for $i=0,1,\cdots,k(n-k)/2$.
\smallskip
Recall that $q$-analogues of quantities in mathematics involve perturbations of classical quantities using the parameter $q$, and revert to the original quantities when $q$ goes $1$. An important example is the $q$-analogues of binomial coefficients which give the number of $k$-dimensional subspaces in $\mathbb{F}_{q}^{n}$. When $q$ goes to $1$, this reverts to the binomial coefficients which measure the number of $k$-sets in $\left [ n \right ]$. Since taking limits as $q$ goes to $1$ reveals the connection between sets and $q$-binomial coefficients, it is reasonable to take limits as $q$ goes to $1$ for the dot binomial coefficient $\binom{n}{k}_{d}$. In dot-analogues, the difficulty of finding a combinatorial description in sets is that the limits of $\binom{n}{k}_{d}$ as $q$ goes to $1$ do not coincide depending on the cases if $q \equiv 1$ (mod) $4$ or $q \equiv 3$ (mod) $4$. We find a combinatorial correspondence between sets and dot-analogues taking limits as $q$ goes to $1$ if $q\equiv 1$ (mod $4$) and as $q$ goes $-1$ if $q\equiv 3$ (mod $4$) because $-1$ is considered as $3$ when $q\equiv 3$ (mod $4$). In conclusion, we find the correspondence between the dot-binomial coefficients $\binom{n}{k}_{d}$ and symmetric subsets of $\mathbb{Z}/(n+1)\mathbb{Z}$ listed in \cref{4}.
\begin{table}[H]
\begin{center}
\scalebox{0.83}{
\begin{tabular}{c||c|c}
& \textbf{symmetric sets} & \textbf{dot-analogues} \\ \hline \hline
\textbf{space} & $\mathbb{Z}/(n+1)\mathbb{Z}=(\left[n \right]\cup \left \{ 0 \right \},+)$ &
$(\mathbb{F}^{n}_{q},\text{dot}_{n})$ \\ \hline
\textbf{subspace} & a symmetric $k$-set in $\mathbb{Z}/(n+1)$ not containing $0$&
a dot$_{k}$-subspace of dot$_{n}$ \\ \hline
\textbf{bracket} & the number of symmetric $1$-sets in $\mathbb{Z}/(n+1)$ & the number of spacelike lines in $(\mathbb{F}^{n}_{q},\text{dot}_{n})$ \\ \hline
\textbf{factorial} & $\left( \frac{n}{2}\right)!$ or $\left( \frac{n-1}{2} \right)!$& $\left [n \right ]_{d}!$ \\ \hline
\textbf{poset} & $B_{n/2}$ or $B_{(n-1)/2}$ & $E_{n}(q)$ \\ \hline
\textbf{group}& $S_{n/2}$ or $S_{(n-1)/2}$ & $|O(n,q)|=2^{n}\left [n\right ]_{d}!$ \\ \hline
\textbf{flag} & flags in $\left[ \frac{n}{2} \right ]$ or $\left [ \frac{n-1}{2} \right ]$ & Euclidean flags in ($\mathbb{F}_{q}^{n}$,dot$_{n}$)\\ \hline
\textbf{binomial coefficient} & $\binom{(n-1)/2}{(k-1)/2}$, $\binom{(n-1)/2}{k/2}$, $\binom{n/2}{k/2}$ or $0$ & $\binom{n}{k}_{d}=\frac{[n]_{d}!}{[k]_{d}![(n-k)]_{d}!}=\left | \frac{O(n,q)}{O(k,q)\times O(n-k,q)} \right |$ \\
\hline
\textbf{connection}& \multicolumn{2}{c}{$
\lim_{q \rightarrow 1\text{ or }-1} \binom{n}{k}_{d}=$ $\binom{(n-1)/2}{(k-1)/2}$, $\binom{(n-1)/2}{k/2}$, $\binom{n/2}{k/2}$ or $0$}\\
\end{tabular}}
\caption{Symmetric sets and dot-analogues.}
\label{4}
\end{center}
\end{table}
\smallskip
Next, we discuss how to compute the size of the orthogonal group $O(n,q)$ over $\mathbb{F}_{q}$ with dot$_{n}$. Recall that the size of the general linear group $GL(n,q)$ over a finite field $\mathbb{q}$ is written by
\[|GL(n,q)|=q^{n(n-1)/2}(q-1)^{n}\left [n \right]_{q}!.\]
Similarly, by using the number of flags $\left [ n \right ]_{d}!$ of the Euclidean poset $E_{n}(q)$, we can compute the size of orthogonal groups over finite fields (\cref{ogf}) by
\[|O(n,q)|=2^{n}\left [ n \right ]_{d}!,\]
and show that
\[
\binom{n}{k}_{d}=\left | \frac{O(n,q)}{O(k,q)\times O(n-k,q)} \right |.
\]
Furthermore, we have the following observation:
\[Gr_{d}(k,n)=\frac{O(n,q)}{O(k,q)\times O(n-k,q)} \subsetneq Gr_{q}(k,n)=\frac{GL(n,q)}{\bigl(\begin{smallmatrix}
A & C \\
\mathbf{0} & B
\end{smallmatrix}\bigr)},\]
where $Gr_{d}(k,n)$ is the set of dot$_{k}$-subspaces of dot$_{n}$, $Gr_{q}(k,n)$ is the set of $k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$, $A$ in $GL(k,q)$ and $B$ in $GL(n-k,q)$ and $C$ is a $k \times (n-k)$ matrix. This is a different phenomenon from the real case since it is not true in general that there exist orthonormal bases of quadratic spaces over finite fields. Additionally, the M{\"o}bius function of the Euclidean poset $E_{n}(q)$ can be computed
using $\binom{n}{k}_{d}$ (\cref{mobi}).
\smallskip
In \cref{section4}, we consider another non-degenerate quadratic form, $\lambda$dot$_{n}$. Let us call a $k$-dimensional subspace $W$ of $(\mathbb{F}_{q}^{n},\text{dot}_{n})$ a $\bm{\lambda} \mathbf{\textbf{dot}_{k}}\textbf{\text{-subspace}}$ if $(W,\text{dot}_{n}|_{W})$ is isometrically isomorphic to $(\mathbb{F}_{q}^{k},\lambda \text{dot}_{k})$, where dot$_{n}|_{W}$ is the restricted quadratic form on $W$. We study the \textbf{Lorentzian poset} consisting of $\lambda $dot$_{k}$-subspaces of $(\mathbb{F}_q^n, \text{dot}_{n})$, for $k=1,\cdots,n-1$. We also provide formulas that count the number of dot$_{k}$-subspaces and $\lambda$dot$_{k}$-subspaces in $(\mathbb{F}_{q}^{n},\text{dot}_{n})$ and $(\mathbb{F}_{q}^{n}, \lambda\text{dot}_{n})$ respectively.
\smallskip
\bigskip
\noindent \textbf{Acknowledgements.} The author would like to express her great gratitude to Jonathan Pakianathan for helpful discussions and encouragement for this work. The author also would like to thank Harry Richman for pointing out an error in an initial draft.
\smallskip
\section{Review of the theory of quadratic forms}
In this section, we briefly remind the reader of some basic concepts of quadratic forms that we will use later, and mainly follow the expository papers \cite{Cl} and \cite{Co}. Readers familiar with the theory of quadratic forms can skip this section.
\smallskip
Let $V$ be a $n$-dimensional vector space over a field $F$ with char$F\ne 2$. A \textbf{quadratic form} $Q$ on $V$ is a function from $V$ to $F$ satisfying the following two conditions: $Q(cv)=c^{2}Q(v)$ for any $v \in V,c\in F$ and $B(v,w)=\frac{1}{2}(Q(v+w)-Q(v)-Q(w))$ is bilinear. We call $B$ the \textbf{bilinear form associated with} $Q$ and dim$V$ the \textbf{dimension of the quadratic form} $Q$. It is easy to check that $B$ is symmetric. Once we fix a basis, quadratic forms can be expressed by a unique matrix form. We call such a matrix $M$ the \textbf{matrix associated with }$Q$ in this basis. Furthermore, there are canonical bijections among the following sets in a chosen basis:
\begin{itemize}
\item The set of homogeneous polynomials of degree $2$, in $n$-variables.
\item The set of quadratic forms on $V$.
\item The set of symmetric bilinear forms on $V$.
\item The set of symmetric $n\times n$ matrices over $F$.
\end{itemize}
We will use any of these definitions of quadratic forms as occasion demands.
\smallskip
Now we define a special class of quadratic forms, which forms the building blocks of all quadratic forms. A quadratic form is called \textbf{non-degenerate} if a matrix representation of it is invertible. If its determinant is zero, we call a quadratic form \textbf{degenerate}. Two quadratic forms $Q_{1}, Q_{2}$ on $V$ are \textbf{equivalent} if there is a linear isomorphism $A:V\longrightarrow V$ such that $Q_{2}(Av)=Q_{1}(v)$ for any $v$ in $V$. For example, the two quadratic forms $Q(x,y)=x^{2}-y^{2}~~\text{and}~~Q'(x,y)=xy$ on $\mathbb{R}^{2}$ are equivalent.
\smallskip
A natural problem is to determine the classification of quadratic forms up to equivalence. We restrict the base field to a finite field, and consider the classification of non-degenerate quadratic forms over finite fields up to equivalence. The proofs can be found in \cite{Co}.
\begin{theorem}\label{c2}\cite{Co}
Let $Q$ be a non-degenerate quadratic form over a finite field $\mathbb{F}_{q}$. Then $Q$ is equivalent to one of
\[x_{1}^{2}+\cdots+x_{n-1}^{2}+x_{n}^{2} \quad
\text{or} \quad
x_{1}^{2}+\cdots+x_{n-1}^{2}+\lambda x_{n}^{2}\]
for some non-square $\lambda$ in $\mathbb{F}_{q}$.
\end{theorem}
The \textbf{discriminant} $d(Q)$ of $Q$ is the coset of det$M$ in $\mathbb{F}_{q}^{\times}/\mathbb{F}_{q}^{\times2}$, where $M$ is a matrix associated with $Q$. We have the following corollary directly by \cref{c2}.
\begin{corollary}\label{coro}
Two non-degenerate quadratic forms over $\mathbb{F}_{q}$ are equivalent if and only if they have the same dimension and same discriminant.
\end{corollary}
We now consider a quadratic form with a space. A \textbf{quadratic space} $(V,Q)$ is a vector space equipped with a quadratic form $Q$ on $V$. We call it \textbf{non-degenerate} when the underlying quadratic form is non-degenerate. Two quadratic spaces $(V_{1},Q_{1})$ and $(V_{2},Q_{2})$ are called \textbf{isometrically isomorphic} if there is a linear isomorphism $A:V_{1} \longrightarrow V_{2}$ such that $Q_{2}(Av)=Q_{1}(v)$ for any $v$ in $V_{1}$. In this case, we call the map $A$ an \textbf{isometry}. When the map $A$ is injective, we say $A$ is an \textbf{isometric embedding}. The \textbf{category of quadratic spaces} over $F$ has as its objects quadratic spaces and morphisms isometric embeddings between quadratic spaces. Let $W$ be a vector subspace of $(V,Q)$ and $Q_{W}$ be the restriction of $Q$ to $W$. One can check that the inclusion $(W,Q_{W})$ in $(V,Q)$ is an isometric embedding. Thus we say that $(W,Q_{W})$ is a \textbf{quadratic subspace} of $(V,Q)$. In addition, given two quadratic spaces $(V_{1},Q_{1})$ and $(V_{2},Q_{2})$, one can construct a new quadratic space $(V_{1} \oplus V_{2},Q_{V_{q}\oplus V_{2}})$ with $Q_{V_{1}\oplus V_{2}}(v_{1},v_{2}):=Q_{V_{1}}(v_{1})+Q_{V_{2}}(v_{2})$.
\smallskip
Let $(V,Q)$ be a quadratic space and $B$ be the bilinear form associated with $Q$. We say two subspaces $W_{1},W_{2}$ are \textbf{orthogonal} $W_{1} \perp W_{2}$ if we have $Q(w_{1}+w_{2})=Q(w_{1})+Q(w_{2})$ for any $w_{1}$ in $W_{1}$ and $w_{2}$ in $W_{2}$, equivalently, $B(w_{1},w_{2})=0$. Let $W$ be a subspace of $(V,Q)$. We define the \textbf{orthogonal complement}
\[W^{\perp}=\left \{ v \in V~|~B(v,w)=0~\text{for any }w\in W \right \}.\]
We state some useful facts about non-degenerate quadratic spaces. If $W$ is a subspace of $(V,Q)$, then the following are equivalent: (1) $W \cap W^{\perp}=0$, (2) $W$ is non-degenerate, (3) $W^{\perp}$ is non-degenerate, and (4) $V=W \oplus W^{\perp}$. Moreover, one can show that dim$W$+dim$W^{\perp}=$dim$V$ and $(W^{\perp})^{\perp}=W$. Thus taking $\perp$ is bijective.
\smallskip
We define special classes of non-degenerate quadratic spaces. Let $(V,Q)$ be a non-degenerate quadratic space. A vector space $V$ is said to be \textbf{isotropic} if there exists a nonzero vector $v$ in $V$ satisfying $Q(v)=0$, and otherwise \textbf{anisotropic}. For example, an inner product space is an anisotropic real quadratic space. Thus isotropic quadratic spaces behave differently than the Euclidean space over $\mathbb{R}$. We call a subspace $W$ of $(V,Q)$ \textbf{totally isotropic} if $Q|_{W}= 0$. We now introduce special non-degenerate quadratic spaces. The \textbf{hyperbolic plane} $\mathbb{H}$ is a $2$-dimensional quadratic space where the quadratic form is equivalent to $x^{2}-y^{2}$(or $xy$). A quadratic space is \textbf{hyperbolic} if it is isometrically isomorphic to a direct sum of hyperbolic planes.
\smallskip
We now state the fundamental results in the algebraic theory of quadratic forms. There are two equivalent statements of Witt's result: one is \textbf{Witt's extension theorem} and the other is \textbf{Witt's cancellation theorem}. We omit the proofs. They can be found in \cite{Cl}.
\begin{theorem}[Witt's cancellation theorem]\label{wc}\cite{Cl}
Let $U_{1},U_{2},V_{1},V_{2}$ be quadratic spaces where $V_{1}$ and $V_{2}$ are isometrically isomorphic. If $U_{1}\oplus V_{1} \cong U_{2} \oplus V_{2}$, then $U_{1} \cong U_{2}$.
\end{theorem}
\begin{theorem}[Witt's extension theorem]\label{we}\cite{Cl}
Let $X_{1} \cong X_{2}$, $X_{1}=U_{1} \oplus V_{1},X_{2}=U_{2}\oplus V_{2},f:V_{1} \longrightarrow V_{2}$ an isometry. Then there is an isometry $F:X_{1}\longrightarrow X_{2}$ such that $F|_{V_{1}}=f$ and $F(U_{1})=U_{2}$.
\end{theorem}
\smallskip
\section{The Euclidean poset $E_{n}(q)$ and the dot-binomial coefficient $\binom{n}{k}_{d}$}\label{section3}
In this section, we study various combinatorial properties of the Euclidean poset and the dot-binomial coefficients.
\smallskip
\subsection{Rank symmetry, unimodality, and log-concavity of $\binom{n}{k}_{d}$}
We consider a poset encoding the Euclidean type of quadratic subspaces of $(\mathbb{F}_{q}^{n},\text{dot}_{n})$. Let us define a poset $E_{n}(q):=(\text{dot}_{k,n},\subset)$ and we call it the \textbf{Euclidean poset}. In $E_{n}(q)$, we do not consider the empty set to be a subspace, but we consider the zero space as the least element of $E_{n}(q)$.
\smallskip
We introduce some terminologies first:
\begin{align*}
&(1) \text{ dot}_{k,n}=\text{the set of }\text{dot}_{k}\text{-subspaces in dot}_{n},\\
&(2) \text{ dot}_{\lambda k,n}=\text{the set of } \lambda\text{dot}_{k}\text{-subspaces in dot}_{n},\\
&(3) \text{ dot}_{k,\lambda n}=\text{the set of }\text{dot}_{k}\text{-subspaces in }\lambda \text{dot}_{n},\\
&(4) \text{ dot}_{\lambda k,\lambda n}=\text{the set of } \lambda \text{dot}_{k}\text{-subspaces in } \lambda \text{dot}_{n}.
\end{align*}
First of all, let us prove that $E_{n}(q)$ is rank-symmetric.
\begin{theorem}\label{rs}
$E_{n}(q)$ is rank-symmetric. i.e., $|\text{dot}_{k,n}|=|\text{dot}_{n-k,n}|$.
\end{theorem}
\begin{proof}
Notice that
\[\text{dot}_{k}\oplus\text{dot}_{n-k}=\text{dot}_n=\text{dot}_{k}\oplus\text{dot}_{k}^{\perp}.\]
By Witt's cancellation theorem, we obtain dot$_{k}^{\perp}=$dot$_{n-k}$. Since taking $\perp$ is bijective, the result holds.
\end{proof}
The next task is to prove that $E_{n}(q)$ is rank-unimodal. We need the number $|\text{dot}_{k,n}|$. Let us consider one useful lemma to achieve this goal.
\begin{lemma}\label{32}For each $k$ and $n$, the number of dot$_{k}$-subspaces in dot$_{n}$ containing a spacelike line is $|\text{dot}_{k-1,n-1}|$.
\end{lemma}
\begin{proof} By Witt's extension theorem, the number of dot$_{k}$-subspaces in dot$_{n}$ containing a spacelike line is independent of which spacelike line is chosen. Let $L$ be a spacelike line. Then we have the following bijection map.
\begin{align*}
(\text{dot$_{k-1}$-subspaces in dot}_{n}/L)~~~&\longrightarrow~~~(\text{dot}_{k} \text{-subspaces in dot}_{n} \text{ containing }L)\\
WL~~~~~~~~~~~&\mapsto ~~~~~~~~~~~L \oplus W
\end{align*}
It follows that this map is bijective by its definition.
\end{proof}
A \textbf{(complete) flag} is a (maximal) chain in a poset. We count flags in two different ways to obtain the following crucial fact.
\begin{theorem}\label{subsp}
For each $n$, we have
\[|\textup{dot}_{1,k}||\textup{dot}_{k,n}|=|\textup{dot}_{1,n}||\textup{dot}_{k-1,n-1}|.\]
\end{theorem}
\begin{proof}
We consider
\begin{align*}
|\text{dot}_{1,k}|&=\text{the number of spacelike lines in a fixed dot}_{k},\\
|\text{dot}_{k,n}|&=\text{the number of }\text{dot}_{k} \text{-subspaces in a fixed dot}_{n},\\
|\text{dot}_{1,n}|&=\text{the number of spacelike lines in a fixed dot}_{n},\\
|\text{dot}_{k-1,n-1}|&=\text{the number of }\text{dot}_{k}\text{-subspaces containing a spacelike line in a fixed dot}_{n}.
\end{align*}
We count the number of flags in two different ways. First, choose a dot$_{k}$-subspace, and then find a dot$_{1}$-subspace (spacelike line) inside it. One important remark is that all dot$_{k}$-subspaces are same as the abstract dot$_{k}$-space by Witt's extension theorem. The second way is firstly to choose dot$_{1}$-subspace, and pick dot$_{k}$-subspace containing it. All dot$_{1}$-subspaces are isometrically isomorphic. Hence we conclude the result.
\end{proof}
Now we are ready to find the formula for $|\text{dot}_{k,n}|$. Inductively, we can compute it by using \cref{subsp} as follows.
\begin{itemize}
\item $|\text{dot}_{2,n}|=\frac{|\text{dot}_{1,n}||\text{dot}_{1,n-1}|}{|\text{dot}_{1,2}|}$.
\item $|\text{dot}_{3,n}|=\frac{|\text{dot}_{1,n}||\text{dot}_{2,n-1}|}{|\text{dot}_{1,3}|}$=$\frac{|\text{dot}_{1,n}|}{|\text{dot}_{1,3}|}\frac{|\text{dot}_{1,n-1}||\text{dot}_{1,n-2}|}{|\text{dot}_{1,2}|}$=$\frac{|\text{dot}_{1,n}||\text{dot}_{1,n-1}||\text{dot}_{1,n-2}|}{|\text{dot}_{1,3}||\text{dot}_{1,2}|}$.
\end{itemize}
Therefore, we have
\begin{equation}\label{dotb}
|\text{dot}_{k,n}|=\frac{|\text{dot}_{1,n}||\text{dot}_{1,n-1}|\cdots|\text{dot}_{1,n-k+1}|}{|\text{dot}_{1,k}|\cdots|\text{dot}_{1,1}|}.
\end{equation}
Furthermore, it is natural to define the following.
\begin{definition}\label{dot-binom} For any $n$ and $k$,
\begin{itemize}
\item $[k]_{d}:=|\text{dot}_{1,k}|$,
\item $[n]_{d}!:=[n]_{d}[n-1]_{d}\cdots[1]_{d}$,
\item $\binom{n}{k}_{d}:=|\text{dot}_{k,n}|=\frac{[n]_{d}!}{[k]_{d}![(n-k)]_{d}!}$.
\end{itemize}
We call these \textbf{dot-analogues}. In particular, we call $\binom{n}{k}_{d}$ the \textbf{dot-binomial coefficient}. We adopt the convention that $|\text{dot}_{1,0}|:=1$.
\end{definition}
\begin{lemma}\label{inc}
For any $k\leq n$, we have $\left[ k \right ]_{d} \leq \left[k+1 \right ]_{d}$.
\end{lemma}
\begin{proof}
It is clear that any spacelike line in dot$_{k}$ can be isometrically embedded in dot$_{k+1}$ by inclusion. Thus the number of spacelike lines in dot$_{k}$ is less than equal to the number of spacelike lines in dot$_{k+1}$.
\end{proof}
\begin{theorem}\label{ru}
$E_{n}(q)$ is rank-unimodal. i.e., there is $j$ such that
\[\binom{n}{0}_{d} \leq \binom{n}{1}_{d} \leq \cdots \leq \binom{n}{j}_{d} \geq \cdots \geq \binom{n}{n}_{d}. \]
\end{theorem}
\begin{proof}
Let us consider the ratio:
\[\frac{\binom{n}{k}_{d}}{\binom{n}{k-1}_{d}}=\frac{|\text{dot}_{k,n}|}{|\text{dot}_{k-1,n}|}=\frac{|\text{dot}_{1,n-k+1}|}{|\text{dot}_{1,k}|}\geq 1\]
if $n-k+1 \geq k$ by \cref{inc}. This is equivalent to $\frac{n+1}{2}\geq k$. It is also less than $1$ if $n-k+1 \leq k$ by \cref{inc} again. i.e., $\frac{n+1}{2}\leq k$. If $n$ is odd, then choose $j=(n+1)/2$. If $n$ is even, then we only need to compare $\binom{n}{\frac{n}{2}}_{d}$ with $\binom{n}{\frac{n+2}{2}}_{d}$. Since
\[\binom{n}{\frac{n+2}{2}}_{d}=\binom{n}{\frac{n-2}{2}}_{d} \leq \binom{n}{\frac{n}{2}}_{d},\]
we set $j=n/2$. It follows that $E_{n}(q)$ is rank-unimodal.
\end{proof}
There is another way to show $E_{n}(q)$ is rank-unimodal. A sequence $a_{0},a_{1},\cdots,a_{n}$ of real numbers is called
\begin{itemize}
\item \textbf{log-concave} if $a_{k}^{2}\geq a_{k-1}a_{k+1}$ for $1 \leq k \leq n-1$.
\item \textbf{unimodal} if there is $j$ such that $a_{0} \leq a_{1} \leq \cdots \leq a_{j} \geq a_{j+1} \geq \cdots \geq a_{n}$.
\end{itemize}
We prove that the dot-binomial coefficient $\binom{n}{k}_{d}$ satisfies the condition of log-concavity.
\begin{theorem}\label{lc}
The sequence
\[\binom{n}{0}_{d},\binom{n}{1}_{d},\cdots,\binom{n}{n}_{d}\]
is log-concave.
\end{theorem}
\begin{proof}
What we need to show is
\[\frac{\binom{n}{k}_{d}}{\binom{n}{k-1}_{d}} \geq \frac{\binom{n}{k+1}_{d}}{\binom{n}{k}_{d}}.\]
This inequality reduces to
\[\frac{\left [ n-k+1\right ]_{d}}{\left [k \right ]_{d}} \geq \frac{\left [ n-k \right ]_{d}}{\left [k+1 \right ]_{d}}.\]
\cref{inc} completes the proof.
\end{proof}
In general, there is a systemetic method for checking log-concavity, called Newton's Theorem. It can be found in Stanley's book \cite{St1}. We introduce a useful theorem to check unimodality without proof. A sequence $a_{0},a_{1},\cdots,a_{n}$ has \textbf{no internal zeros} if $a_{i}\ne 0$ and $a_{k}\ne 0$ for $i<j<k$, then $a_{j} \ne 0$.
\begin{proposition}\cite{St1}\label{uni}
Let $\alpha=(a_{0},a_{1},\cdots,a_{n})$ be a sequence of non-negative real numbers with no internal zeros. If $\alpha$ is log-concave, then $\alpha$ is unimodal.
\end{proposition}
Since $(\binom{n}{0}_{d},\binom{n}{1}_{d},\cdots,\binom{n}{n}_{d})$ is a sequence of non-negative real numbers with no internal zeros satisfying the log-concavity, $(\binom{n}{0}_{d},\binom{n}{1}_{d},\cdots,\binom{n}{n}_{d})$ is unimodal by \cref{uni}.
\smallskip
\subsection{Combinatorial properties of $\binom{n}{k}_{d}$}
\smallskip
We first introduce the formula of the number of lines of certain types in a non-degenerate space. In particular, this count gives us the number of dot$_{1}$-subspaces in $(\mathbb{F}_{q}^{n},\text{dot}_{n})$. Recall that the number of lines passing through the origin in $\mathbb{F}_{q}^{n}$ is
\begin{equation}\label{counting}
\frac{q^{n}-1}{q-1}.
\end{equation}
By the classification of non-degenerate quadratic forms over finite fields, there are three possible $1$-dimensional quadratic subspaces in $(\mathbb{F}^{n}_{q},\text{dot}_{n})$ up to equivalence:
(1) $\text{dot}_{1}$, (2) $\lambda\text{dot}_{1}$, and the degenerate case (3) $0$.
Motivated by this, we define three types of lines in the quadratic space $(\mathbb{F}^{n}_{q},\text{dot}_{n})$ using the terminology used in Minkowski geometry.
\begin{definition}\label{linetypes}\cite{Yo1}
The type of a line $l$ through the origin in $(\mathbb{F}_{q}^{n},\text{dot}_{n})$ (or $(\mathbb{F}_{q}^{n},\lambda \text{dot}_{n})$) is \textbf{spacelike} (dot$_{1}$-subspace) if $|l|$ is a square, \textbf{timelike} ($\lambda$dot$_{1}$-subspace) if $|l|$ is a non-square, and \textbf{lightlike} (degenerate $1$-dimensional subspace) if $|l|$ is $0$. Here, we let $|l|$ denote the value $\text{dot}_{n}(\mathbf{x})$ (or $\lambda \text{dot}_{n}(\mathbf{x})$) for any nonzero $\mathbf{x}$ in $l$.
\end{definition}
It is easy to check that the notion of line type is well-defined. Let us define some notations: (1) $\epsilon$ is $1$ if $q \equiv 1$ (mod $4$) or $n \equiv 1,2$ (mod $4$), and $-1$ otherwise, (2) $\delta$ is $1$ if $n \equiv 0 ~(\text{mod}~ 2)$, and $0$ otherwise, and (3) $\eta$ is $1$ if $n \equiv 0 ~(\text{mod} ~2)$, and $-1$ otherwise. We state the following theorems that count the number of lines of each type in $(\mathbb{F}_{q}^{n},\text{dot}_{n})$ and $(\mathbb{F}_{q}^{n},\lambda\text{dot}_{n})$, respectively: \begin{theorem}\label{dotline}\cite{Yo1}
In $(\mathbb{F}_{q}^{n},\text{dot}_{n})$, the number of spacelike lines is $(q^{n-1}+(-1)^{\delta}\epsilon q^{(n-\delta -1)/2})/2$,
and the number of timelike lines is $(q^{n-1}+(-1)^{\delta}\eta \epsilon q^{(n-\delta -1)/2})/2$. The number of lightlike lines is given by using \cref{counting}.
\end{theorem}
\begin{theorem}\label{lambdadot}\cite{Yo1}
In $(\mathbb{F}_{q}^{n},\lambda \text{dot}_{n})$, the number of spacelike lines is $(q^{n-1}-(-1)^{\delta}\epsilon q^{(n-\delta -1)/2})/2$,
and the number of timelike lines is $(q^{n-1}-(-1)^{\delta}\eta \epsilon q^{(n-\delta -1)/2})/2$ for some square $\lambda$. The number of lightlike lines is given by using \cref{counting}.
\end{theorem}
Using \cref{dotline} and \cref{lambdadot}, we show the dot-binomial coefficients are written in terms of the $q$-binomial coefficients and some polynomials.
\begin{theorem}\label{polynomials}
Suppose that $q \equiv 1$ (mod $4$). Then we have
\begin{table}[H]
\begin{center}
\scalebox{0.9}{
\begin{tabular}{c||c|c}
$\mathbf{\binom{n}{k}_{d}}$ & $\mathbf{k}$ \textbf{is odd} & $\mathbf{k}$ \textbf{is even} \\ \hline \hline
$\mathbf{n}$ \textbf{is odd} & $\frac{1}{2}q^{\frac{k(n-k)}{2}}(q^{\frac{n-k}{2}}+1)\binom{(n-1)/2}{(k-1)/2}_{q^{2}}$ &
$\frac{1}{2}q^{\frac{k(n-k)}{2}}(q^{\frac{k}{2}}+1)\binom{(n-1)/2}{k/2}_{q^{2}}$ \\ \hline
$\mathbf{n}$ \textbf{is even} & $\frac{1}{2}q^{\frac{k(n-k)-1}{2}}(q^{\frac{n}{2}}-1)\binom{(n-2)/2}{(k-1)/2}_{q^{2}}$ &
$\frac{1}{2}q^{\frac{k(n-k)}{2}}\frac{(q^{\frac{n-k}{2}}+1)(q^{\frac{k}{2}}+1)}{q^{\frac{n}{2}}+1}\binom{n/2}{k/2}_{q^{2}}$ \\
\end{tabular}}
\caption{The computation of the dot-binomial coefficient $\binom{n}{k}_{d}$ when $q \equiv 1$ (mod $4$).}
\end{center}
\end{table}
Suppose that $q \equiv 3$ (mod $4$) and let $l,l'$ be non-negative integers. Then we have
\begin{table}[H]
\begin{center}
\scalebox{0.9}{
\begin{tabular}{c||c|c}
$\mathbf{\binom{n}{k}_{d}}$ & $\mathbf{k=4l'+1}$ & $\mathbf{k=4l'+2}$ \\ \hline \hline
$\mathbf{n=4l+1}$ & $\frac{1}{2}q^{\frac{k(n-k)}{2}}(q^{\frac{n-k}{2}}+1)\binom{(n-1)/2}{(k-1)/2}_{q^{2}}$ &
$\frac{1}{2}q^{\frac{k(n-k)}{2}}(q^{\frac{k}{2}}-1)\binom{(n-1)/2}{k/2}_{q^{2}}$ \\ \hline
$\mathbf{n=4l+2}$ & $\frac{1}{2}q^{\frac{k(n-k)-1}{2}}(q^{\frac{n}{2}}+1)\binom{(n-2)/2}{(k-1)/2}_{q^{2}}$ &
$\frac{1}{2}q^{\frac{k(n-k)}{2}}\frac{(q^{\frac{n-k}{2}}+1)(q^{\frac{k}{2}}-1)}{q^{\frac{n}{2}}-1}\binom{n/2}{k/2}_{q^{2}}$ \\ \hline
$\mathbf{n=4l+3}$ & $\frac{1}{2}q^{\frac{k(n-k)}{2}}(q^{\frac{n-k}{2}}-1)\binom{(n-1)/2}{(k-1)/2}_{q^{2}}$ &
$\frac{1}{2}q^{\frac{k(n-k)}{2}}(q^{\frac{k}{2}}-1)\binom{(n-1)/2}{k/2}_{q^{2}}$ \\ \hline
$\mathbf{n=4l}$ & $\frac{1}{2}q^{\frac{k(n-k)-1}{2}}(q^{\frac{n}{2}}-1)\binom{(n-2)/2}{(k-1)/2}_{q^{2}}$ &
$\frac{1}{2}q^{\frac{k(n-k)}{2}}\frac{(q^{\frac{n-k}{2}}-1)(q^{\frac{k}{2}}-1)}{q^{\frac{n}{2}}+1}\binom{n/2}{k/2}_{q^{2}}$ \\
\end{tabular}}
\bigskip
\scalebox{0.9}{
\begin{tabular}{c||c|c}
$\mathbf{\binom{n}{k}_{d}}$ & $\mathbf{k=4l'+3}$ & $\mathbf{k=4l'}$ \\ \hline \hline
$\mathbf{n=4l+1}$ & $\frac{1}{2}q^{\frac{k(n-k)}{2}}(q^{\frac{n-k}{2}}-1)\binom{(n-1)/2}{(k-1)/2}_{q^{2}}$ &
$\frac{1}{2}q^{\frac{k(n-k)}{2}}(q^{\frac{k}{2}}+1)\binom{(n-1)/2}{k/2}_{q^{2}}$ \\ \hline
$\mathbf{n=4l+2}$ & $\frac{1}{2}q^{\frac{k(n-k)-1}{2}}(q^{\frac{n}{2}}+1)\binom{(n-2)/2}{(k-1)/2}_{q^{2}}$ &
$\frac{1}{2}q^{\frac{k(n-k)}{2}}\frac{(q^{\frac{n-k}{2}}-1)(q^{\frac{k}{2}}+1)}{q^{\frac{n}{2}}-1}\binom{n/2}{k/2}_{q^{2}}$ \\ \hline
$\mathbf{n=4l+3}$ & $\frac{1}{2}q^{\frac{k(n-k)}{2}}(q^{\frac{n-k}{2}}+1)\binom{(n-1)/2}{(k-1)/2}_{q^{2}}$ &
$\frac{1}{2}q^{\frac{k(n-k)}{2}}(q^{\frac{k}{2}}+1)\binom{(n-1)/2}{k/2}_{q^{2}}$ \\ \hline
$\mathbf{n=4l}$ & $\frac{1}{2}q^{\frac{k(n-k)-1}{2}}(q^{\frac{n}{2}}-1)\binom{(n-2)/2}{(k-1)/2}_{q^{2}}$ &
$\frac{1}{2}q^{\frac{k(n-k)}{2}}\frac{(q^{\frac{n-k}{2}}+1)(q^{\frac{k}{2}}+1)}{q^{\frac{n}{2}}+1}\binom{n/2}{k/2}_{q^{2}}$ \\
\end{tabular}}
\caption{The computation of the dot-binomial coefficient $\binom{n}{k}_{d}$ when $q \equiv 3$ (mod $4$).}
\end{center}
\end{table}
\end{theorem}
Next, here are the analogues of Pascal identities.
\begin{proposition}\label{pa} For $0<k<n$, we have
\begin{itemize}
\item $\binom{n}{0}_{d}=\binom{n}{n}_{d}=1$,
\item $\binom{n}{k}_{d}=\binom{n}{n-k}_{d}$,
\item $\binom{n}{k}_{d}=\binom{n-1}{k-1}_{d}+\frac{\left [ n \right ]_{d}-\left [ k \right ]_{d}}{\left [ n-k \right ]_{d}}\binom{n-1}{k}_{d}$.
\end{itemize}
\end{proposition}
\begin{proof}
The first equality is easy and follows from $|\text{dot}_{1,0}|:=1$. From \cref{rs}, $\binom{n}{k}_{d}=\binom{n}{n-k}_{d}$ holds. We prove the third equality.
\begin{align*}
\binom{n}{k}_{d}-\binom{n-1}{k-1}_{d}&=\left (\frac{\left [n \right ]_{d}}{\left [ k \right ]_{d}}-1 \right )\binom{n-1}{k-1}_{d}\\
&=\frac{\left [ n \right ]_{d}-\left [ k \right ]_{d}}{\left [ k \right ]_{d}}\binom{n-1}{k-1}_{d}\\
&=\frac{\left [ n \right ]_{d}-\left [ k \right ]_{d}}{\left [ k \right ]_{d}}\frac{\left [n-1 \right ]_{d}\left [n-2 \right ]_{d}\cdots \left [ n-k+1 \right ]_{d}}{\left [ 1 \right ]_{d}\left [ 2 \right ]_{d}\cdots \left [ k-1 \right ]_{d}}\\
&=\frac{\left [ n \right ]_{d}-\left [ k \right ]_{d}}{\left [ n-k \right ]_{d}}\binom{n-1}{k}_{d}
\end{align*}
\end{proof}
\begin{corollary}\label{pa2}
For $0<k<n$, we have
\[\binom{n}{k}_{d}=\binom{n-1}{k}_{d}+\frac{\left [ n \right ]_{d}-\left [ n-k \right ]_{d}}{\left [ k \right ]_{d}}\binom{n-1}{k-1}_{d}.\]
\end{corollary}
\smallskip
One can define the analogue of Pascal's triangle by \cref{pa} or \cref{pa2}.
\begin{example}
Let $q \equiv 1$ (mod $4$). Then we have an analogue of Pascal's triangle with rows $0$ through $4$
\bigskip
\begin{center}
\begin{tabular}{>{$n=}l<{$\hspace{12pt}}*{13}{c}}
$0$ &&&&&&&$1$&&&&&&\\
$1$ &&&&&&$1$&&$1$&&&&&\\
$2$ &&&&&1&&$\frac{q-1}{2}$&&1&&&&\\
$3$ &&&&$1$&&$\frac{q^{2}+q}{2}$&&$\frac{q^{2}+q}{2}$&&$1$&&&\\
$4$ &&&$1\quad$&&$\frac{q^{3}-q}{2}$&&$\frac{q^{2}(q+1)^{2}}{2}$&&$\frac{q^{3}-q}{2}$&&$\quad1$&&
\end{tabular}
\end{center}
\bigskip
and
\[\binom{4}{2}_{d}=\frac{q^{2}+q}{2}+\frac{\left [ 4 \right ]_{d}-\left[ 2 \right ]_{d}}{\left [ 2 \right ]_{d}}\frac{q^{2}+q}{2}=\frac{q^{2}+q}{2}+\frac{q^{3}-2q+1}{q-1}\frac{q^{2}+q}{2}=\frac{q^{2}(q+1)^{2}}{2}.\]
Here, we use the number of spacelike lines in dot$_{2}$, dot$_{3}$, and dot$_{4}$ in \cref{dotline}.
\end{example}
Recall that the $q$-binomial coefficient $\binom{n}{k}_{q}$ is polynomial in $q$. The dot-binomial coefficient $\binom{n}{k}_{d}$ also has the same property.
\begin{proposition}\label{rational}
The dot-binomial coefficient $\binom{n}{k}_{d}$ is polynomial of degree $k(n-k)$ in the indeterminate $q$.
\end{proposition}
\begin{proof}
We use the full description of the dot-binomial coefficient $\binom{n}{k}_{d}$ given by \cref{dotline}. The full description can be found in \cite{Yo4}. Since $\binom{n}{k}_{q^{2}}$ is always polynomial in $q$, it is not hard to show that $\binom{n}{k}_{d}$ is polynomial except for the following cases: (1) $q \equiv 1$ (mod $4$), $n$ and $k$ are even, (2) $q \equiv 3$ (mod $4$), $n=4l+2$ and $k=4l'+2$, (3) $q \equiv 3$ (mod $4$), $n=4l+2$ and $k=4l'$, (4) $q \equiv 3$ (mod $4$), $n=4l$ and $n=4l'+2$, (5) $q \equiv 3$ (mod $4$), $n=4l$ and $k=4l'$. We rely on \cref{pa}.
\medskip
\noindent Case (1) $q \equiv 1$ (mod $4$), $n$ and $k$ are even:
\[\frac{\left [ n \right ]_{d}-\left [ k \right ]_{d}}{\left [ n-k \right ]_{d}}=\frac{q^{n-1}-q^{\frac{n-2}{2}}-q^{k-1}+q^{\frac{k-2}{2}}}{q^{n-k-1}-q^{\frac{n-k-2}{2}}}=\frac{q^{\frac{k-2}{2}}(q^{\frac{k}{2}}(q^{n-k}-1)-(q^{\frac{n-k}{2}}-1))}{q^{\frac{n-k-2}{2}}(q^{\frac{n-k}{2}}-1)}.\]
Case (2) $q \equiv 3$ (mod $4$), $n=4l+2$ and $k=4l'+2$: \[\frac{\left [ n \right ]_{d}-\left [ k \right ]_{d}}{\left [ n-k \right ]_{d}}=\frac{q^{n-1}+q^{\frac{n-2}{2}}-q^{k-1}-q^{\frac{k-2}{2}}}{q^{n-k-1}-q^{\frac{n-k-2}{2}}}=\frac{q^{\frac{k-2}{2}}(q^{\frac{k}{2}}(q^{n-k}-1)+(q^{\frac{n-k}{2}}-1))}{q^{\frac{n-k-2}{2}}(q^{\frac{n-k}{2}}-1)}.\]
Case (3) $q \equiv 3$ (mod $4$), $n=4l+2$ and $k=4l'$:
\[\frac{\left [ n \right ]_{d}-\left [ k \right ]_{d}}{\left [ n-k \right ]_{d}}=\frac{q^{n-1}+q^{\frac{n-2}{2}}-q^{k-1}+q^{\frac{k-2}{2}}}{q^{n-k-1}+q^{\frac{n-k-2}{2}}}=\frac{q^{\frac{k-2}{2}}(q^{\frac{k}{2}}(q^{n-k}-1)+(q^{\frac{n-k}{2}}+1))}{q^{\frac{n-k-2}{2}}(q^{\frac{n-k}{2}}+1)}.\]
Case (4) $q \equiv 3$ (mod $4$), $n=4l$ and $n=4l'+2$:
\[\frac{\left [ n \right ]_{d}-\left [ k \right ]_{d}}{\left [ n-k \right ]_{d}}=\frac{q^{n-1}-q^{\frac{n-2}{2}}-q^{k-1}-q^{\frac{k-2}{2}}}{q^{n-k-1}+q^{\frac{n-k-2}{2}}}=\frac{q^{\frac{k-2}{2}}(q^{\frac{k}{2}}(q^{n-k}-1)-(q^{\frac{n-k}{2}}+1))}{q^{\frac{n-k-2}{2}}(q^{\frac{n-k}{2}}+1)}.\]
Case (5) $q \equiv 3$ (mod $4$), $n=4l$ and $k=4l'$:
\[\frac{\left [ n \right ]_{d}-\left [ k \right ]_{d}}{\left [ n-k \right ]_{d}}=\frac{q^{n-1}-q^{\frac{n-2}{2}}-q^{k-1}+q^{\frac{k-2}{2}}}{q^{n-k-1}-q^{\frac{n-k-2}{2}}}=\frac{q^{\frac{k-2}{2}}(q^{\frac{k}{2}}(q^{n-k}-1)-(q^{\frac{n-k}{2}}-1))}{q^{\frac{n-k-2}{2}}(q^{\frac{n-k}{2}}-1)}.\]
Note that the dot-binomial coefficient $\binom{n}{k}_{d}$ is symmetric and then we may assume that $q^{\frac{k-2}{2}} \geq q^{\frac{n-k-2}{2}}$, which is equivalent to $k \geq n/2$. Thus $\left [ n \right ]_{d}-\left [ k \right ]_{d}/\left [ n-k \right ]_{d}$ is polynomial. Since $\binom{n-1}{k-1}_{d}$ and $\binom{n-1}{k}_{d}$ is polynomial, we conclude our result. Let us now observe the highest degree in each term to compute the degree of the dot-binomial coefficient $\binom{n}{k}_{d}$. The degree of the dot-binomial coefficient $\binom{n}{k}_{d}$ is shown to be
\[\sum_{l=1}^{k}(n-l)-\sum_{l=1}^{k-1}l=k(n-k)\]
regardless of any cases.
\end{proof}
\begin{remark}
(1) Both the $q$-binomial binomial coefficient $\binom{n}{k}_{q}$ and the dot-binomial coefficient $\binom{n}{k}_{d}$ are the quotients of product of the roots of unities.
\medskip
(2) The dot-binomial coefficient $\binom{n}{k}_{d}$ can be written as
\[\binom{n}{k}_{d}=(1+o(1))\frac{q^{k(n-k)}}{2}\]
where $o(1)$ goes to zero as $q$ goes to infinity.
\medskip
(3) In any case, we obtain \[\lim_{q\rightarrow \infty}\frac{\binom{n}{k}_{d}}{\binom{n}{k}_{q}}=\frac{1}{2}.\]
Thus, in $(\mathbb{F}_{q}^{n},\text{dot}_{n})$, the half of subspaces is roughly dot$_{k}$-subspaces. We will see the other half is $\lambda$dot$_{k}$-subspaces in \cref{section4}.
\end{remark}
\smallskip
\subsection{Properties of the polynomial given by $\binom{n}{k}_{d}$}
By the expressions given by the complete list of the dot-binomial coefficient $\binom{n}{k}_{d}$ from \cite{Yo4}, we obtain the following expressions:
\begin{proposition}\label{pe}
If (1) $q \equiv 1$ (mod $4$), $n$ is even and $k$ is odd, or (2) $q \equiv 3$ (mod $4$), $n=4l+2$ and $k$ is odd, or (3) $n=4l$ and $k$ is odd, then we have
\[\binom{n}{k}_{d}=q^{\frac{k(n-k)-1}{2}}\sum_{i=0}^{\frac{k(n-k)+1}{2}}a_{i}q^{i}.\]
Otherwise, we have
\[\binom{n}{k}_{d}=q^{\frac{k(n-k)}{2}}\sum_{i=0}^{\frac{k(n-k)}{2}}b_{i}q^{i}.\]
\end{proposition}
Let $p_{n,k}(q)$ denote $\binom{n}{k}_{d}$. Then we have a relation of the polynomial $p_{n,k}(q)$.
\begin{proposition}\label{poly}
If (1) $q \equiv 1$ (mod $4$), $n$ is even and $k$ is odd, or (2) $q \equiv 3$ (mod $4$), $n=4l+3$ and $k=4l'+1$, $k=4l'+2$, or (3) $n=4l+1$ and $k=4l'+3$, $k=4l'+2$, or (4) $n=4l+2$ and $k$ is even, or (5) $n=4l$ and $k$ is even, or (6) $n=4l$ and $k=4l'+3$, then we have
\[p_{n,k}\left (\frac{1}{q} \right )=-\frac{1}{q^{\frac{3k(n-k)}{2}}}p_{n,k}(q),\]Otherwise, we have
\[p_{n,k}\left (\frac{1}{q} \right )=\frac{1}{q^{\frac{3k(n-k)}{2}}}p_{n,k}(q).\]
\end{proposition}
\begin{proof}
Suppose that $q \equiv 1$ (mod $4$), and $n$ and $k$ are odd. By the expression of the dot-binomial coefficient $\binom{n}{k}_{d}$, we have
\begin{align*}
p_{n,k}\left (\frac{1}{q} \right )
&=\frac{1}{2}\frac{1}{q^{\frac{k(n-k)}{2}}}\frac{\left(\frac{1}{q^{\frac{n-1}{2}}}+1\right)\left(\frac{1}{q^{\frac{n-1}{2}}}-1\right)\left(\frac{1}{q^{\frac{n-3}{2}}}+1\right)\cdots\left(\frac{1}{q^{\frac{n-k+2}{2}}}-1\right)\left(\frac{1}{q^{\frac{n-k}{2}}}+1\right)}{\left(\frac{1}{q^{\frac{k-1}{2}}}+1\right)\left(\frac{1}{q^{\frac{k-1}{2}}}-1\right)\left(\frac{1}{q^{\frac{k-3}{2}}}+1\right)\cdots \left(\frac{1}{q}-1\right)\cdot 1}\\
&=\frac{1}{2}\frac{1}{q^{\frac{k(n-k)}{2}}}\frac{q^{\frac{k-1}{2}}q^{\frac{k-1}{2}}q^{\frac{k-3}{2}}\cdots q}{q^{\frac{n-1}{2}}q^{\frac{n-1}{2}}q^{\frac{n-3}{2}}\cdots q^{\frac{n-k}{2}}}\frac{(q^{\frac{n-1}{2}}+1)(q^{\frac{n-1}{2}}-1)(q^{\frac{n-3}{2}}+1)\cdots(q^{\frac{n-k+2}{2}}-1)(q^{\frac{n-k}{2}}+1)}{(q^{\frac{k-1}{2}}+1)(q^{\frac{k-1}{2}}-1)(q^{\frac{k-3}{2}}+1)\cdots (q-1)\cdot 1}\\
&=\frac{1}{q^{\frac{3k(n-k)}{2}}}p_{n,k}(q)
\end{align*}
This completes our proof. For other cases, we can prove them similarly.
\end{proof}
It follows that the coefficients $a_{i}$ and $b_{i}$ in \cref{pe} have some sort of symmetries.
\begin{corollary}\label{symmetry}
Suppose that $q \equiv 1$ (mod $4$). Then we have
\begin{enumerate}
\item $a_{i}=-a_{k(n-k)/2-i}$ (anti-symmetric) if $n$ is even and $k$ is odd.
\item $b_{i}=b_{k(n-k)/2-i}$ (symmetric) if $n$ is odd, or $n$ is even and $k$ is even,
\end{enumerate}
Suppose that $q \equiv 3$ (mod $4$). Then we have
\begin{enumerate}
\item $a_{i}=a_{(k(n-k)+1)/2-i}$ (symmetric) if $n=4l+2$ and $k$ is odd, or $n=4l$ and $k=4l'+1$,
\item $a_{i}=-a_{(k(n-k)+1)/2-i}$ (anti-symmetric) if $n=4l$ and $k=4l'+3$.
\item $b_{i}=b_{k(n-k)/2-i}$ (symmetric) if $n=4l+3$ and $k=4l'+3$, $k=4l'$, or $n=4l+1$ and $k=4l'+1$, $k=4l'$,
\item $b_{i}=-b_{k(n-k)/2-i}$ (anti-symmetric) if $n=4l+3$ and $k=4l'+1$, $k=4l'+2$, or $n=4l+1$ and $k=4l'+3$, $k=4l'+2$, or $n=4l+2$ and $k$ is even, or $n=4l$ and $k$ is even.
\end{enumerate}
\end{corollary}
\begin{proof}
Let $q \equiv 1$ (mod $4$), $n$ be odd, and $k$ be odd. Let us denote $\binom{n}{k}_{d}$ by
\[\binom{n}{k}_{d}=p_{n,k}(q)=\frac{1}{2}q^{\frac{k(n-k)}{2}}(b_{0}+b_{1}q+\cdots+b_{\frac{k(n-k)}{2}}q^{\frac{k(n-k)}{2}}).\]
Then we have
\begin{align*}
p_{n,k}\left (\frac{1}{q} \right )&=\frac{1}{2}\frac{1}{q^{\frac{k(n-k)}{2}}}(b_{0}+\frac{b_{1}}{q}+\cdots+\frac{b_{\frac{k(n-k)}{2}}}{q^{\frac{k(n-k)}{2}}})\\
&=\frac{1}{q^{\frac{3k(n-k)}{2}}}\frac{1}{2}q^{\frac{k(n-k)}{2}}(b_{0}q^{\frac{k(n-k)}{2}}+b_{1}q^{\frac{k(n-k)}{2}-1}+\cdots+b_{\frac{k(n-k)}{2}})
\end{align*}
Thus we only need to show
\[p_{n,k}\left (\frac{1}{q} \right )=\frac{1}{q^{\frac{3k(n-k)}{2}}}p_{n,k}(q),\]
which is given by \cref{poly}. Other cases can be obtained by a similar way.
\end{proof}
\smallskip
\subsection{Combinatorial correspondence between symmetric sets and dot-analogues}
Notice that $~~~$ $ \lim_{q \rightarrow 1}\binom{n}{k}_{q}=\binom{n}{k}$ reveals the connection between binomial coefficients and $q$-binomial coefficients. Thus, the first step is to compute the limit $\lim_{q \rightarrow 1}\binom{n}{k}_{d}$. However, $\lim_{q \rightarrow 1}\binom{n}{k}_{d}$ does not give any interesting combinatorial connection between sets and dot-analogues. Thus, we attempt to compute $\lim_{q \rightarrow -1}\binom{n}{k}_{d}$ when $q \equiv 3$ (mod $4$) because $-1$ is considered as $3$ when $q \equiv 3$ (mod $4$). It turns out that the limits $\lim_{q \rightarrow 1} \binom{n}{k}_{d}$ when $q \equiv 1$ (mod $4$) and $\lim_{q \rightarrow -1}\binom{n}{k}_{d}$ when $q\equiv 3$ (mod $4$) are the same.
\begin{proposition}\label{limit}
For any $0<k<n$, we have the following table:
\begin{table}[H]
\begin{center}
\scalebox{0.9}{
\begin{tabular}{c||c|c}
& $\mathbf{\lim_{q \rightarrow 1}\binom{n}{k}_{d}}$ \textbf{when} $\mathbf{q \equiv 1}$ (\textup{\textbf{mod}} $\mathbf{4}$) & $\mathbf{\lim_{q \rightarrow -1}\binom{n}{k}_{d}}$ \textbf{when} $\mathbf{q \equiv 3}$ (\textup{\textbf{mod}} $\mathbf{4}$) \\ \hline \hline
$\mathbf{n,k}$ \textbf{are odd} & $\binom{(n-1)/2}{(k-1)/2}$ &
$\binom{(n-1)/2}{(k-1)/2}$ \\ \hline
$\mathbf{n}$ \textbf{is odd}, $\mathbf{k}$ \textbf{is even} & $\binom{(n-1)/2}{k/2}$ &
$\binom{(n-1)/2}{k/2}$ \\ \hline
$\mathbf{n,k}$ \textbf{are even} & $\binom{n/2}{k/2}$ & $\binom{n/2}{k/2}$ \\ \hline
$\mathbf{n}$ \textbf{is even}, $\mathbf{k}$ \textbf{is odd} & $0$ & $0$ \\
\end{tabular}}
\caption{The computation of the limits of the dot-binomial coefficient $\binom{n}{k}_{d}$.}
\label{3}
\end{center}
\end{table}
\end{proposition}
\begin{proof}
Notice that
\[\lim_{q \rightarrow 1}\binom{n}{k}_{q^{2}}=\binom{n}{k} \text{ and }\lim_{q \rightarrow -1}\binom{n}{k}_{q^{2}}=\binom{n}{k}.\]
The limits can be directly computed by the expressions in \cite{Yo4}. For example, if $q \equiv 1$ (mod $4$), and $n,k$ are odd, we have
\[\lim_{q \rightarrow 1} \binom{n}{k}_{d}=\lim_{q \rightarrow 1}\left (\frac{1}{2}q^{\frac{k(n-k)}{2}}(q^{\frac{n-k}{2}}+1)\binom{\frac{n-1}{2}}{\frac{k-1}{2}}_{q^{2}}\right )=\binom{\frac{n-1}{2}}{\frac{k-1}{2}}.\]
Moreover, if $q \equiv 3$ (mod $4$), $n=4l+3$, and $k=4l+3$, we obtain
\[\lim_{q \rightarrow -1} \binom{n}{k}_{d}=\lim_{q \rightarrow -1}\left (\frac{1}{2}q^{\frac{k(n-k)}{2}}(q^{\frac{n-k}{2}}+1)\binom{\frac{n-1}{2}}{\frac{k-1}{2}}_{q^{2}}\right )=\binom{\frac{n-1}{2}}{\frac{k-1}{2}}\]
because $(n-k)/2$ are even. Other cases can be computed by a similar way.
\end{proof}
\cref{3} indicates that combinatorial descriptions of sets corresponding to the dot-binomial coefficient $\binom{n}{k}_{d}$ do not need to depend on the $q$. We now show that the limits of the dot-binomial coefficient $\binom{n}{k}_{d}$ is the number of symmetric $k$-sets in $\mathbb{Z}/(n+1)\mathbb{Z}$. Here, we call a subset $A$ of $\mathbb{Z}/(n+1)\mathbb{Z}$ a \textbf{symmetric} $k$-set if $A=-A$, $0 \notin A$ and $|A|=k$.
\begin{theorem} For any $0<k<n$, the number of symmetric $k$-sets in $\mathbb{Z}/(n+1)\mathbb{Z}$ is the limit of the dot-binomial coefficient $\binom{n}{k}_{d}$ in \cref{3}.
\end{theorem}
\begin{proof}
Suppose that $n$ and $k$ are odd. Then we have \[\mathbb{Z}/(n+1)\mathbb{Z}=\left \{ 0,1,\cdots,\frac{n-1}{2},\frac{n+1}{2},\frac{n+3}{2}\cdots,n \right \}.\]
To make a symmetric $k$-set, we only need to choose $(k-1)/2$ elements in $\left \{ 1,\cdots,(n-1)/2 \right \}$. Then the other $(k-1)/2$ elements are determined. By adding the element $(n+1)/2$ in the set, we complete it to make a symmetric $k$-set. Thus there are $\binom{(n-1)/2}{(k-1)/2}$ ways to make symmetric $k$-sets in $\mathbb{Z}/(n+1)\mathbb{Z}$. Similarly, when $n$ is odd and $k$ is even, the number of symmetric $k$-sets in $\mathbb{Z}/(n+1)\mathbb{Z}$ is given by $\binom{(n-1)/2}{k/2}$.
\smallskip
Suppose that $n$ and $k$ are even. Then we have
\[\mathbb{Z}/(n+1)\mathbb{Z}=\left \{ 0,1,\cdots,\frac{n}{2},\frac{n}{2}+1,\cdots,n \right \}.\]
Thus to make symmetric $k$-sets, we need to choose $k/2$ elements in $\left \{ 1,2,\cdots,n/2 \right \}$. Thus the number of symmetric $k$-sets in $\mathbb{Z}/(n+1)\mathbb{Z}$ is $\binom{n/2}{k/2}$. When $n$ is even and $k$ is odd, it is not hard to check that symmetric $k$-sets do not exist.
\end{proof}
Therefore, we can interpret combinatorics in sets corresponding to dot-analogues as in \cref{4}. The connection $\lim_{q \rightarrow 1}\binom{n}{k}_{q}=\binom{n}{k}$ happens when we compare the vector space $\mathbb{F}_{q}^{n}$ with the set $\left [ n \right ]$. In dot-analogues, it turns out that we compare the quadratic space $(\mathbb{F}_{q}^{n},\text{dot}_{n})$ and the group $(\mathbb{Z}/(n+1)\mathbb{Z},+)$ by adding additional operations: the quadratic form dot$_{n}$ to the space $\mathbb{F}_{q}^{n}$ and the addition to the set $\left [ n \right ]$.
\smallskip
\subsection{The Euclidean flags and the orthogonal group O(n,q)}
We count the number of complete flags in $E_{n}(q)$. First, there are $\left [
n \right ]_{d}$ many ways to choose the first line in dot$_{n}$. For the next step, we choose one more spacelike line in dot$_{n-1}$ to make a dot$_{2}$-subspace including the chosen line. There are $\left [n-1 \right ]_{d}$ ways to do so. From this process, the number of Euclidean flags of length $n$ in $E_{n}(q)$ is $\left [ n \right ]_{d}!=\left [n \right ]_{d}\left [ n-1 \right ]_{d}\cdots \left [1 \right ]_{d}$.
\smallskip
There is another way to find the number of the Euclidean flags in $E_{n}(q)$. Euclidean flags are bijective up to a factor of $2^{n}$ with orthonormal basis $e_{1},\cdots,e_{n}$ as one can use
\[\text{span}(e_{1}) \subset \text{span}(e_{1},e_{2})\subset \cdots \subset \text{span}(e_{1},e_{2},\cdots,e_{n}).\]
The only difference is that the flag is unchanged if we change $e_{i}$ to $-e_{i}$ for any $i$ since there are two antipodal vectors of norm $1$ for each spacelike line. Thus the orthogonal group $O(n,q)$ over a finite field is bijective with orthonormal bases of dot$_{n}$. Therefore we obtain
\begin{align*}
\left [ n \right ]_{d}!&=\text{the number of the Euclidean flags}\\
&=\text{the number of orthonormal bases up to }\pm\\
&=\frac{|O(n,q)|}{2^{n}}
\end{align*}
By this observation, we find a way to count the size of the orthogonal group $O(n,q)$. The formula is as follows:
\[ |O(n,q)|=2^{n}\left [ n \right ]_{d}!.\]
This formula redrives a usual formula for the size of $O(n,q)$.
\begin{corollary}\label{ogf}
For any $n \geq 1$, the size of orthogonal group $O(n,q)$ over a finite field $\mathbb{F}_{q}$ is given by the following:
\begin{itemize}
\item
\[|O(2n+1,q)|=2q^{\frac{n-1}{2}}\prod^{\frac{n-3}{2}}_{k=0}(q^{2n}-q^{2k}).\]
\item
\[|O(2n,q)|= \begin{cases}
2\left(q^{\frac{n}{2}}-1\right)\prod^{\frac{n-2}{2}}_{k=1}\left(q^{n}-q^{2k}\right) & \text{ if } q \equiv 1\text{ \textup{(mod} $4$\textup{)}} \\
2\left(q^{\frac{n}{2}}+(-1)^{\frac{n+2}{2}}\right)\prod^{\frac{n-2}{2}}_{k=1}\left(q^{n}-q^{2k}\right) & \text{ if } q \equiv 3\text{ \textup{(mod} $4$\textup{)} }
\end{cases} .\]
\end{itemize}
\end{corollary}
Let us discuss how to count the dot-binomial coefficient $\binom{n}{k}_{d}$ in other ways. By using the number of the Euclidean flags, we obtain
\begin{equation}\label{oo}
\binom{n}{k}_{d}=\frac{\left [ n \right ]_{d}!}{\left [ k \right ]_{d}!\left [ n-k \right ]_{d}!}=\frac{|O(n,q)|}{|O(k,q)\times O(n-k,q)|}\cdot \frac{2^{k}\cdot 2^{n-k}}{2^{n}}=\left | \frac{O(n,q)}{O(k,q)\times O(n-k,q)} \right |.
\end{equation}
On the other hand, the orthogonal group $O(n,q)$ acts transitively on $Gr_{d}(k,n)$ by Witt's extension theorem. Then the stabilizer of any element in $Gr_{d}(k,n)$ is $O(k,q)\times O(n-k,q)$. Hence there is an isomorphism between $Gr_{d}(k,n)$ and $\frac{O(n,q)}{O(k,q)\times O(n-k,q)}$. Therefore, we have another way to obtain \cref{oo} by the theory of group actions. Furthemore, $Gr_{d}(k,n)$ can be described as
\[\left \{ \text{$k$-orthonormal frames of dot}_{n} \right \}/\sim,\]
where $\alpha \sim \beta$ if span$\alpha=\text{span}\beta$.
\smallskip
\subsection{The M\"{o}bius function of $\binom{n}{k}_{d}$}We briefly remind the reader of the M\"{o}bius function and mainly follow \cite{Ba} and \cite{Ca1}. Let $X$ be a finite set. Given a finite poset $P=(X,\leq)$, the \textbf{incidence algebra} of $P$ is the set of all functions $f:X \times X \longrightarrow \mathbb{R}$ satisfying $f(x,y)=0$ unless $x \leq y$. In other words, if $x \nless y$ and $x \ne y$, the outputs of elements of the incidence algebra are zero. For examples, there are three useful elements of the incidence algebra that we will consider.
\begin{itemize}
\item The \textbf{zero} function is defined by $0(x,y)=0$.
\item The \textbf{identity} function is defined by
\[i(x,y)=\begin{cases}
1 & \text{ if } x=y \\
0 & \text{ if } x \ne y
\end{cases} .\]
\item The \textbf{zeta} function is defined by
\[\zeta(x,y)=\begin{cases}
1 & \text{ if } x\leq y \\
0 & \text{ if } x \nleqslant y
\end{cases}.\]
\end{itemize}
The incidence algebra has an algebra structure with addition and scalar product as usual pointwise, and the multiplication is defined by
\[(f*g)(x,y)=\sum_{x \leq z \leq y}f(x,z)g(z,y).\]
It is called \textbf{convolution} of $f$ and $g$. It is not true in general that every element of the incidence algebra has an inverse element. However, if a poset $P$ is finite and an element $f$ of the incidence algebra satisfies $f(x,x) \ne 0$ for any $x \in P $, then there is an inverse of $f$. For the proof of this fact, see \cite{Ba}. Since the zeta function $\zeta$ satisfies $\zeta(x,x)=1 \ne 0$, there exists an inverse of the zeta function, called the \textbf{M\"{o}bius function}. Thus we have the relation
\begin{equation}\label{mobi}
\mu *\zeta =\zeta *\mu =i
\end{equation}
By \cref{mobi}, if $x<y$, then we have
\begin{equation}\label{00}
\sum_{x \leq z \leq y}\mu(x,z)=0.
\end{equation}
Therefore, the M\"{o}bius function can be written as
\[\mu(x,y)=\begin{cases}
1 & \text{ if } x= y \\
-\sum_{x \leq z < y}\mu(x,z) & \text{ if } x \ne y
\end{cases}.\]
The M\"{o}bius function has many applications in the study of combinatorics. For example, the value of the M\"{o}bius function gives another proof of the Inclusion-Exclusion Theorem. See \cite{Ba}.
\smallskip
As examples, we can find the M\"{o}bius functions of the Boolean algebra $B_{n}$ and the poset $L_{n}(q)$ constructed by subspaces of $\mathbb{F}_{q}^{n}$. These two posets have the least element and a nice property, called the \textbf{homogeneity property}: for any $x,y$ with $x \leq y$, there is an element
$z$ such that the interval $\left [x,y \right ]$ is isomorphic to the interval $\left [0,c \right ]$ (with $\left [x,y \right ]$ and $\left [0,c \right ]$ as posets), where $0$ is the least element of the poset. By these properties, we have $\mu(x,y) = \mu (0, c)$. This helps us to reduce the complexity of the computations for the M\"{o}bius functions. The Euclidean poset $E_{n}(q)$ also has the homogeneity property. This means that, for two dot subspaces $U$ and $W$ with $U \leq W$, the interval $\left [
U,W \right ]$ is isomorphic to the poset of $W/U$. Thus we can consider the M\"{o}bius function of $E_{n}(q)$ as a one-variable function.
\smallskip
In addition, $B_{n}$ only depends on the cardinality of subsets because the symmetric group $S_{n}$ acts transitively on the set of subsets whose cardinalities are the same. Similarly, $L_{n}(q)$ and $E_{n}(q)$ depend on the dimension of subspaces since the general linear group $GL(n,\mathbb{F}_{q})$ acts transitively on the set of subspaces of the same dimension, and the orthogonal group $O(n,q)$ acts transitively on the set of dot-subspaces that have the same dimension.
\smallskip Before we compute the M\"{o}bius function of $E_{n}(q)$, we first show the M\"{o}bius functions of $B_{n}$ and $L_{n}(q)$ as follows:
\begin{theorem}\textup{\cite{Ba} \cite{Ca1}}
The M\"{o}bius functions of $B_{n}$ and $L_{n}(q)$ are the following:
\begin{enumerate}
\item Given a subset $X$ of $\left \{ 1,2,\cdots,n \right \}$, $\mu(\left \{0 \right \},X)=(-1)^{|X|}$,
\item Given a $n$-dimensional subspace $V$ of $\mathbb{F}_{q}^{m}$,
$\mu(\left \{0 \right \},V)=(-1)^{n}q^{\frac{n(n-1)}{2}}$.
\end{enumerate}
\end{theorem}
The proofs can be found in \cite{Ba} and \cite{Ca1}. The key ideas are to use the binomial theorem and the $q$-binomial theorem. In order to compute the M\"{o}bius function of the Euclidean poset $E_{n}(q)$, we need the right binomial theorem for the dot-analogues. However, there are some difficulties to obtain the explicit formula; thus we just give an algorithm to compute the M\"{o}bius function of $E_{n}(q)$. In \cref{mobi}, \cref{0} plays the role of the binomial theorem for dot-analogues.
\begin{theorem}\label{mobi}
Let $V$ be a dot$_{n}$ space. Then the M{\"o}bius function $\mu$ is given by
\[
\mu(\left \{0 \right \},V)=(-1)^{n}b_{n}
\]
where $b_{0}=1$ and $b_{n}$ can be obtained recursively by the equation
\begin{equation}\label{0}\sum_{k=0}^{n}(-1)^{k}b_{k}\binom{n}{k}_{d}=0.
\end{equation}
\end{theorem}
\begin{proof}
We prove it by induction. If $n=0$, then $\mu(\left \{ 0 \right \},V)=(-1)^{0}b_{0}=1$. Suppose that the statement is true for $k<n$. Since there are $\binom{n}{k}_{d}$ many dot$_{k}$-subspaces, we obtain
\begin{align*}
\mu(0,V)&=-\sum_{0\leq W<V}\mu(0,W)\\
&=-\left ( \sum_{0\leq W<V}(-1)^{\text{dim}W}b_{\text{dim}W} \right )\\
&=-\left( \sum_{k=0}^{n-1}(-1)^{k}b_{k}\binom{n}{k}_{d}\right)\\
&=-\left ( \sum_{k=0}^{n}(-1)^{k}b_{k}\binom{n}{k}_{d} -(-1)^{n}b_{n}\right )\\
&=(-1)^{n}b_{n}
\end{align*}
by the induction hypothesis and \cref{0}.
\end{proof}
\smallskip
\section{The Lorentzian posets}\label{section4}
Let us consider the poset constructed by $\lambda$dot$_{k}$-subspaces of $(\mathbb{F}_{q}^{n},\text{dot}_{n})$. Denote the set of $\lambda$dot$_{k}$-subspaces of dot$_{n}$ by dot$_{\lambda k,n}$. Then we construct a poset $LO_{n}(q):=(\text{dot}_{\lambda k, n},\subset)$ and call it the \textbf{Lorentzian poset}. We also do not consider the empty set to be a subspace, but we consider the zero space as the least element and dot$_{n}$ as the maximal element of the Lorentzian poset. We ask the same questions as for the Euclidean poset.
\begin{theorem}
$LO_{n}(q)$ is rank-symmetric. i.e., $|\textup{dot}_{\lambda k,n}|=|\textup{dot}_{\lambda(n-k),n}|$.
\end{theorem}
\begin{proof}
Note that dot$_{\lambda k}\oplus$dot$_{\lambda (n-k)}$=dot$_n$=dot$_{\lambda k}\oplus$dot$_{\lambda k}^{\perp}$. By Witt's cancellation theorem, we obtain dot$_{\lambda k}^{\perp}=$dot$_{\lambda (n-k)}$. Since taking $\perp$ is bijective, the result holds.
\end{proof}
To show that $LO_{n}(q)$ is rank-unimodal, we need to count $|\text{dot}_{\lambda k,n}|$. For completeness, we summarize all analogous binomial coefficients and omit details since the process is similar to before.
\begin{align*}
&\binom{n}{k}_{d}=|\text{dot}_{k,n}|=\frac{|\text{dot}_{1,n}||\text{dot}_{1,n-1}|\cdots|\text{dot}_{1,n-k+1}|}{|\text{dot}_{1,k}|\cdots|\text{dot}_{1,1}|},\\
&\binom{\lambda n}{k}_{d}=|\text{dot}_{ k,\lambda n}|=\frac{|\text{dot}_{1,\lambda n}||\text{dot}_{1,\lambda(n-1)}|\cdots|\text{dot}_{1,\lambda(n-k+1)}|}{|\text{dot}_{1,k}|\cdots|\text{dot}_{1,1}|},\\
&\binom{n}{\lambda k}_{d}=|\text{dot}_{\lambda k,n}|=\frac{|\text{dot}_{\lambda 1,n}|}{|\text{dot}_{\lambda 1,\lambda k}|}\binom{\lambda (n-1)}{k-1}_{d},\\
&\binom{\lambda n}{\lambda k}_{d}=|\text{dot}_{\lambda k,\lambda n}|=\frac{|\text{dot}_{\lambda 1,\lambda n}|}{|\text{dot}_{\lambda 1,\lambda k}|}\binom{n-1}{k-1}_{d}.
\end{align*}
The counts $|\text{dot}_{1,n}|$, $|\text{dot}_{\lambda 1,n}|$, $|\text{dot}_{1,\lambda n}|$, and $|\text{dot}_{\lambda 1,\lambda n}|$ will be discussed in \cref{section2}.
\begin{theorem}\label{l} $LO_{n}(q)$ is rank-unimodal. i.e., For $n>0$, there is $j$ such that
\[ \binom{n}{\lambda 1}_{d} \leq \binom{n}{\lambda 2}_{d} \leq \cdots \leq \binom{n}{\lambda j}_{d} \geq \cdots \geq \binom{n}{\lambda(n-1)}_{d}. \]
\end{theorem}
\begin{proof}
Let us look at the ratio:
\[\frac{\binom{n}{\lambda k}_{d}}{\binom{n}{\lambda (k-1)}_{d}}=\frac{|\text{dot}_{\lambda k,n}|}{|\text{dot}_{\lambda(k-1),n}|}=\frac{\frac{|\text{dot}_{\lambda 1,n}|}{|\text{dot}_{\lambda 1,\lambda k}|}}{\frac{|\text{dot}_{\lambda 1,n}|}{|\text{dot}_{\lambda 1,\lambda(k-1)}|}}
\frac{\binom{\lambda(n-1)}{k-1}_{d}}{\binom{\lambda(n-1)}{k-2}_{d}}=\frac{|\text{dot}_{\lambda 1,\lambda (k-1)}|}{|\text{dot}_{\lambda 1,\lambda k}|}\frac{|\text{dot}_{1,\lambda(n-k+1)}|}{|\text{dot}_{1,k-1}|}\geq 1\]
if $n-k+1 \geq k$. i.e., $\frac{n+1}{2}\geq k$. It suffices to check boundaries when $k=(n+1)/2$. One can compute that they are all $1$. Also, it is easy to check that the ratio is less than $1$ if $n-k+1 \leq k-1$. i.e., $\frac{n+2}{2}\leq k$. If $n$ is odd, then we only need to compare $\binom{n}{\lambda \frac{n+1}{2}}_{d}$ with $\binom{n}{\lambda \frac{n+3}{2}}_{d}$. Since $LO_{n}(q)$ is rank-symmetric,
\[\binom{n}{\lambda \frac{n+3}{2}}_{d} = \binom{n}{\lambda \frac{n-3}{2}}_{d}\leq \binom{n}{\lambda \frac{n+1}{2}}_{d}.\]
Thus set $j=(n+1)/2$. If $n$ is even, it is enough to compare $\binom{n}{\lambda \frac{n}{2}}_{d}$ with $\binom{n}{\lambda \frac{n+2}{2}}_{d}$. Again, since $LO_{n}(q)$ is rank-symmetric, we obtain
\[\binom{n}{\lambda \frac{n+2}{2}}_{d} = \binom{n}{\lambda \frac{n-2}{2}}_{d}\leq \binom{n}{\lambda \frac{n}{2}}_{d}.\]
Set $j=n/2$. It follows that $LO_{n}(q)$ is rank-unimodal.
\end{proof}
\medskip
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,954
|
\section{Introduction}
\label{sec:intro}
\subsection{Observational properties of hot subdwarfs}
{\purple Hot subdwarf-B stars (sdBs) are core helium-burning stars with thin hydrogen envelopes ($\lesssim 0.01$\,M$_\odot$) that exhibit significant chemical peculiarities. Hot subdwarf-O stars (sdOs) are even more chemically evolved, with helium burning occurring in a shell around an inert carbon-oxygen (CO) core. Such stars, which are thought to represent late stages of stellar evolution, are likely derived from the stripped cores of red giants. They usually lie on the blue end of the Extreme Horizontal Branch (EHB) of the Hertzsprung-Russell Diagram (HRD; see, e.g., \citealt{1986ASSL..128...33H,1994ApJ...432..351S, heber}). sdB and sdO stars have surface temperatures of $20,000 \lesssim T_{\rm eff} \lesssim 60,000$ K, with $5 \lesssim \log g \lesssim 6.5$, and masses of $\sim$\,0.47 M$_\odot$.}
We have adopted the following working definitions regarding subdwarfs and their key properties:
\begin{enumerate}
\item sdB stars burn helium (He) in their cores and may also undergo $\alpha$-channel burning of the newly-created carbon in the core (leading to the creation of oxygen). This phase persists for tens of Myr, during which the radius stays roughly constant.
\item sdO stars have a well-defined CO core, with helium burning occurring in a shell around this core, which has completed carbon burning and become inert. Simultaneous hydrogen burning occurs in a thin layer near the surface. This phase is typically shorter than the sdB phase (by a factor of {\purple approximately} 2 to 3), and the radius also remains roughly constant during this phase.
\item sdA (subdwarf A) stars, a newly discovered class of subdwarfs, have poorly constrained properties. Their true nature remains uncertain because there may be a variety of processes leading to their formation {\purple {(see, e.g., \citealt{2019ApJ...885...20Y})}}. While the provenance of these stars remains an important open question, they are not of importance for this work.
\end{enumerate}
\subsection{Pulsating subdwarfs}
The first pulsating sdB (sdBV) star, EC\,14026-2647, was discovered by \cite{1997MNRAS.285..640K}, who found a pulsation with a period of 144\,s. Since then, over 100 such pulsating stars have been discovered, {\purple many of them through space-based missions such as Kepler (including K2) and TESS} (see, e.g., \citealt{2017MNRAS.466.5020H, 2021MNRAS.507.4178R}). These stars fall into three categories -- rapid (sdBV$_{\rm r}$) pulsators, with $p$ mode oscillations on the order of a few minutes; slow (sdBV$_{\rm s}$) pulsators, with g~mode oscillations on the order of a few hours, and hybrid pulsators, which exhibit both p~and g~mode oscillations.
\subsection{Formation of sdB stars}
While the different classes of sdBV stars are fairly well-defined, the formation of these objects remains somewhat of a mystery. There have been extensive studies {\purple of} the mechanisms via which sdB stars form; see, for instance, \citet{1976ApJ...204..488M}, \citet{1993ApJ...407..649C}, \citet{1993ApJ...409..387D}, \citet{2002MNRAS.336..449H, 2003MNRAS.341..669H}, \citet{2011MNRAS.410..984J}, \citet{2015ApJ...806..178S}, \citet{2019CoSka..49..264V}, \citet{Sen2019}, and \citet{2020A&A...641A.163V}.
\citet{2002MNRAS.336..449H} specifically compared various formation channels leading to the creation of sdB stars. {\purple They concluded that sdBs in tight binaries ($P_{\rm orb}\lesssim$ 10\,d) were likely formed as a result of common envelope (CE) evolution. On the other hand, they showed that wide systems composed of sdBs + WDs ($P_{\rm orb}\gtrsim$ 400\,d) could, in principle, be formed as the result of stable, yet completely non-conservative Roche lobe overflow (RLOF). Finally, they demonstrated the conditions under which the merger of He WDs (the double helium-WD channel) could lead to the ignition of helium, thereby producing sdBs.
For common envelope evolution to produce a short-period sdB,} a red giant whose mass is at least 2 to 3 times greater than that of its companion must overflow its Roche lobe and achieve a sufficiently high mass transfer rate onto its companion \citep{1976ApJ...204..488M}. Such a high accretion rate precludes the companion star from accreting all of the deposited matter, leading to the formation of a common envelope (see, e.g., \citealt{1941ApJ....93..133K}). The rapid shrinking of the giant's Roche lobe as it loses mass causes dynamically unstable mass transfer, forcing the accreting companion to begin to spiral inside the giant's envelope (see, e.g., \citealt{1976ApJ...209..829W}). If the change in the orbital energy is sufficient to unbind the envelope, then the giant's envelope can be expelled from the binary system on the order of hundreds of years \citep{2017A&A...599A..54X}. If a merger can be avoided, the companion emerges in a tight, circular orbit (periods of hours to days) around the stripped core of the red giant. This stripped core can then evolve onto the EHB and become {\purple an sdB (and/or sdO).} The relatively high proportion of sdB stars observed in short-period binaries suggests that this evolutionary scenario is the most common \citep{2003MNRAS.341..669H}, but there exists an observational bias that favors the discovery of such systems.
{\purple Models of longer-period binaries containing sdBs can also be produced by assuming that the primordial binary, consisting of two main sequence (MS) stars undergoes stable, but (partially) non-conservative, mass transfer, {\purple in which both mass and angular momentum leave the system.} Using a binary population synthesis code, \citet{2003MNRAS.341..669H} concluded that it is possible to produce sdBs in binaries with $P_{\rm orb}\approx$ 100\,d under these assumptions. They refer to this as the ``first stable RLOF channel.'' It should be noted, however, that the evolution of the accretor does not seem to have been computed contemporaneously with that of the donor. This could possibly result in the accretor filling its Roche lobe before the donor has had a chance to evolve to (or completely through) the sdB phase. \citet{2003MNRAS.341..669H} also considered the formation of wide sdB+WD binaries with periods on the order of 1000\,d (``second CE ejection channel''), but their simulations failed to produce any, due to the need for massive WDs in such systems.
Very recently, \citet{2020A&A...641A.163V} showed that the observed population of wide sdB binaries ($P_{\rm orb}\gtrsim$ 1000\,d) could be robustly reproduced under the assumption that the low-mass primordial primary star (donor) is close to the tip of the red giant branch (helium flash) when rapid, yet stable, non-conservative mass loss occurs as a result of Roche lobe overflow. This can result in the formation of wide binaries containing sdBs. Using population synthesis techniques, they also investigated the effects of metallicity. For solar metallicities, they found $P_{\rm orb}\gtrsim$ 1000\,d; for lower metallicity stars they showed that sdB binaries were likely to have $P_{\rm orb}\approx$ 1000\,d.
It is also possible to form sdB/O binary stars via stable, (partially) non-conservative mass transfer in progenitor binaries composed of intermediate-mass, main-sequence stars.} While we know that mass transfer can be partially non-conservative based on an analysis of Algol-related binaries \citep{2000NewAR..44..111E}, we do not have a good constraint on systemic mass loss (i.e., the fraction of mass ejected from the binary). An extensive grid of more than 3000 progenitor models was calculated by \citet{Sen2019} of potential progenitors of hot subdwarfs, under the assumption of partially non-conservative, stable mass transfer {\purple (and solar metallicity). The primaries of the primordial binaries were chosen to have masses between 1 and 8 M$_\odot$ (with the secondaries having masses of 25\%, 50\%, 80\% and 90\% of the primary), and initial orbital periods of 1--200\,d. Mass transfer was also parameterized so as to be arbitrarily non-conservative (i.e., ranging from 0 to 100\%). That work suggests that sdB-containing binaries could have orbital periods in the range of 10\,d $\lesssim P_{\rm orb} \lesssim 100$ \,d. Thus, they naturally bridge the gap in $P_{\rm orb}$ between sdBs formed as a result of common envelope evolution and those formed from low-mass red giants that undergo rapid mass transfer near the tip of the red giant branch.}
\subsection{TIC 5724661}
As part of its goal to enable precision asteroseismology, the Transiting Exoplanet Survey Satellite (TESS; \citealt{2015JATIS...1a4003R}) has been observing a subset of stars using a novel 20-s cadence since Sector 27, which occurred during 2020 July. This mode can probe frequencies up to a Nyquist limit of 2160\,d$^{-1}$, corresponding to periods as short as 40\,s. TESS short-cadence observations are invaluable in the detection and characterization of new subdwarfs, especially pulsators (see, e.g., Section 6 of \citealt{2021FrASS...8...19L}, and references therein).
TIC~5724661 was chosen to be observed at 20\,s cadence because it was a known A~star in the instability strip on the HRD that might exhibit high-frequency pulsations. Observations of this star and our subsequent analyses revealed two sets of pulsational frequencies -- one in the typical $\delta$ Scuti frequency range and two other, unexpected, peaks at 524\,d$^{-1}$ and 580\,d$^{-1}$. These two are in the characteristic pulsational frequency range of hot compact stars, like white dwarfs and sdBV$_{\rm r}$ stars.
In this paper, we first analyze the frequencies of the $\delta$ Scuti pulsations and discuss the modes they represent. Then, we show that the spectrum of TIC~5724661 does not exhibit the chemical abnormalities of a roAp star. Moreover, no significant variations in radial velocity are found over a series of unequally-spaced observations, suggesting a long-period ($\gtrsim$~70 d) orbit. We next discuss the strong evidence for a hot compact companion suggested by the excess ultraviolet (UV) flux in the Spectral Energy Distribution (SED). We then explore possible formation pathways for such a system and contextualize our discovery of a novel system. Note, we will hereafter refer to the A-star component of the binary as the ``secondary'', and the sdB component as the ``primary'', for reasons regarding the evolution of the binary that will be more thoroughly explained in Section \ref{sec:stel-evol}.
\section{Observational data}
\label{sec:obs}
\subsection{TESS Observations of TIC 5724661}
\label{sec:tess}
\begin{table}
\centering
\caption{Properties of TIC~5724661}
\begin{tabular}{lc}
\hline
\hline
Parameter & Value \\
\hline
RA (J2015.5) (h m s)& 23:11:07.84 \\
Dec (J2015.5) ($^\circ \ ^\prime \ ^{\prime\prime}$) & -17:13:19.424 \\
$T$$^a$ & $11.204 \pm 0.007$ \\
$G$$^b$ & $11.286 \pm 0.001$ \\
$G_{\rm BP}$$^b$ & $11.353 \pm 0.001$ \\
$G_{\rm RP}$$^b$ & $11.154 \pm 0.001$ \\
$B^a$ & $11.431 \pm 0.129$ \\
$V^a$ & $11.231 \pm 0.010$ \\
$J^c$ & $10.998 \pm 0.020$ \\
$H^c$ & $10.962 \pm 0.024$ \\
$K^c$ & $10.919 \pm 0.023$ \\
W1$^d$ & $10.889 \pm 0.023$ \\
W2$^d$ & $10.917 \pm 0.020$ \\
W3$^d$ & $10.851 \pm 0.162$ \\
W4$^d$ & $> 8.607$ \\
$R$ (${\rm R}_\odot$)$^e$ & $1.32^{+0.09}_{-0.11}$ \\
$L$ (${\rm L}_\odot$)$^e$ & $5.737 \pm 0.8 $ \\
Distance (pc)$^b$ & $ 611 \pm 15$ \\
$\mu_\alpha$ (mas ~${\rm yr}^{-1}$)$^b$ & $-4.5197 \pm 0.0375$ \\
$\mu_\delta$ (mas ~${\rm yr}^{-1}$)$^b$ & $+4.941 \pm 0.0304$ \\
\hline
\hline
\label{tbl:mags}
\end{tabular}
{\bf Notes.} (a)\url{exofop.ipac.caltech.edu/tess/index.php}. (b) {\em Gaia} eDR3 (\citealt{2016A&A...595A...1G};
\citealt{2016A&A...595A...2G}; \citealt{2021A&A...649A...1G}). (c) 2MASS catalog \citep{2006AJ....131.1163S}. (d) WISE point source catalog \citep{2014yCat.2328....0C}. (e) This work; see Table \ref{table:mcmc-mle} for details regarding the radius estimate, as well as Table \ref{table:rvs} for RV data.
\end{table}
TIC~5724661 was observed by {\em TESS} in Sector 29 (from 2020 August 26 to September 21) and Sector 42 (from 2021 August 20 to September 16) in both 2\,min and 20\,s cadence. The data are available in both SAP (simple aperture photometry) and PDCSAP (presearch data conditioning SAP) forms. Data processing was done using the Science Processing Operations Center (SPOC) pipeline at NASA Ames \citep{jenkinsSPOC2016}. We used the PDCSAP data from both Sectors 29 and 42 for our analysis after converting the given flux intensity to magnitudes.\footnote{The data from Sector 42 have a large data gap during the first orbit (see Figure \ref{fig:lc}) due to saturation of the CCDs arising from the moon being in the TESS field-of-view, so we rely somewhat less on this dataset.} The Sector 29 data span 24.33\,d with a temporal center point of $t_0 = {\rm BJD}~2459100.41122$, and comprise 88937 data points (after clipping by SPOC to remove outlier points, e.g., those arising from cosmic ray strikes on the detector).\footnote{TESS Sector 29 Data Release Notes: \url{https://archive.stsci.edu/\\missions/tess/doc/tess\_drn/tess\_sector\_29\_drn43\_v02.pdf}}
\begin{figure}
\begin{center}
\includegraphics[width=0.98\linewidth,angle=0]{raw_lcs.pdf}
\caption{The light curve of TIC~5724661 obtained in 20\,s cadence in {\em TESS} Sectors 29 and 42 after processing with the SPOC pipeline \citep{jenkinsSPOC2016}. The pulsations are too rapid and too low in amplitude to discern visually in this compressed figure. Its purposes are (i) to show the two gaps in the data, which affect the spectral window, and (ii) to show the noise level in the 20-s data points. The ordinate scale is Barycentric Julian Date -- 240\,0000.0.}
\label{fig:lc}
\end{center}
\end{figure}
Figure\,\ref{fig:lc} shows the SPOC-processed light curves with the data gaps between the two orbits making up each \emph{TESS} sector. These arise from the lack of observations during data downlink, or saturation of the CCDs due to scattered light from the Earth and the Moon. Such data gaps affect the spectral window, necessitating either analysis with a Discrete Fourier Transform (DFT; see, e.g., \citealt{1985MNRAS.213..773K}) or appropriate corrections, such as re-binning the data into equally-spaced temporal bins.
\subsection{Spectroscopy}
\label{subsec:spec}
We obtained spectroscopic observations of TIC\,5724661 with the Tillinghast Reflector Echelle Spectrograph (TRES, \citealt{Furesz2014}), on the 1.5-m reflector at the Fred Lawrence Whipple Observatory (FLWO) in Arizona, USA. TRES is a high-resolution fiber-fed echelle spectrograph, with a spectral resolving power of $R=44\,000$ over the wavelength region of $3900-9100$\,\AA. A total of {\purple ten} observations were obtained for TIC 5724661 during 2020 December, and between 2021 September and 2021 {\purple December}, with peak signal-to-noise ratios per resolution element of $\sim$30 in the Mg b triplet wavelength region. The spectra were extracted and reduced as per \citet{2010ApJ...720.1118B}, with wavelength solutions derived from bracketing Th-Ar lamp exposures. The observing schedule was designed to be sensitive to a companion with $P_{\rm orb} \lesssim 30$~d.
To derive the spectroscopic broadening profiles and radial velocities from each observation, we performed a least-squares deconvolution (LSD, \citealt{1997MNRAS.291..658D}) of each spectrum against a synthetic non-rotating template; this provided both a value for the radial velocity, along with an uncertainty value. We also conducted a multi-order velocity analysis of the spectra, and derived another set of uncertainties for the radial velocity values. We observed that the multi-order uncertainties were around 50\% greater than the LSD uncertainties {\purple in some cases, and agreed with them in other cases}. Values from both sets of analyses are presented in Table \ref{table:rvs}.
Visual examination of the broadening profiles for a set of lines from the sdB companion remained negative, but the line profiles did show night-to-night variability consistent with typical spectroscopic line variations exhibited by $\delta$ Scuti stars. The broadening profiles were fitted with a model kernel accounting for the rotational, macroturbulent, and instrumental broadening terms, as well as the velocity shift of the spectrum. The comparison to model spectra and further analyses (including the use of a rotating template) are described in Section~\ref{sec:spec}.
\begin{table}
\centering
\caption{Radial Velocity Measurements of TIC 5724661 from the Tillinghast Reflector Echelle Spectrograph (TRES).}
\begin{tabular}{cccc}
\hline
\hline
\multicolumn{1}{c}{Observation} & \multicolumn{1}{c}{Radial} &\multicolumn{1}{c}{LSD} & \multicolumn{1}{c}{Multi-Order}
\\
\multicolumn{1}{c}{Date} & \multicolumn{1}{c}{Velocity} &\multicolumn{1}{c}{Error} & \multicolumn{1}{c}{Error} \\
\multicolumn{1}{c}{(BJD-2400000)} & \multicolumn{1}{c}{(km\,s$^{-1}$)} &\multicolumn{1}{c}{(km\,s$^{-1}$)} & \multicolumn{1}{c}{(km\,s$^{-1}$)} \\
\hline
59190.60806 & $-39.85$ & $\pm\,0.17$ & $\pm\,0.48$\\
59196.58549 & $-39.54$ & $\pm\,0.73$ & $\pm\,0.68$ \\
59199.57533 & $-41.06$ & $\pm\,0.80$ & $\pm\,0.75$\\
59200.59383 & $-40.24$ & $\pm\,0.66$ & $\pm\,0.59$\\
59202.58932 & $-41.05$ & $\pm\,0.69$ & $\pm\,1.13$\\
59484.77567 & $-39.03$ & $\pm\,0.63$ & $\pm\,0.65$\\
59487.72156 & $-38.98$ & $\pm\,0.38$ & $\pm\,0.52$\\
59519.69383 & $-37.51$ & $\pm\,0.47$ & $\pm\,0.71$\\
59566.62276 & $-35.20$ & $\pm\,0.20$ & $\pm\,0.65$\\
59567.60356 & $-35.26$ & $\pm\,0.21$ & $\pm\,0.44$\\
59766.95738 & $-32.19$ & $\pm\,0.39$ & $\pm\,0.56$\\
\hline
\hline
\end{tabular}
\label{table:rvs}
\end{table}
The set of eleven measured radial velocities obtained with TRES is given in Table \ref{table:rvs}.
\section{Frequency analysis}
\label{sec:freq-analysis}
The {\em TESS} data from both Sectors 29 and 42 were analyzed using a fast Discrete Fourier Transform \citep{1985MNRAS.213..773K} to produce amplitude spectra. The top panel in Fig.\,\ref{fig:ftd} shows the amplitude spectrum out to about half the Nyquist frequency of 2160\,d$^{-1}$, calculated using the Sector 29 data. A cluster of peaks in the $\delta$~Sct frequency range is seen between $26 - 46$\,d$^{-1}$, and a single, high-frequency peak is clearly detected at 523.99\,d$^{-1}$ (6.065\,mHz). These are shown at higher frequency resolution in the two middle panels, with appropriate labels indicating the sector whose light curve was input to the DFT. {\purple There is an additional peak at 579.85 d$^{-1}$ that increases in prominence in Sector 42, lending further credence to our hypothesis of an unseen hot compact pulsator in this system.}
\begin{figure}
\begin{center}
\includegraphics[width=\linewidth,angle=0]{5724661_dft.pdf}
\caption{The top panel shows the Fourier amplitude spectrum to 1000\,d$^{-1}$ from the Sector~29 light curve; there are no significant peaks between 1000\,d$^{-1}$ and the Nyquist frequency, 2160\,d$^{-1}$. The second panel zooms into a cluster of peaks in the $\delta$~Sct frequency range between $20 - 55$\,d$^{-1}$, along with a single, high-frequency peak at 523.99\,d$^{-1}$ (6.065\,mHz). The third panel shows the $\delta$ Scuti and high-frequency pulsations observed in Sector~42; the peak at 579.85\,d$^{-1}$ (6.711~mHz) increases in prominence between Sectors~29 and 42. The bottom panel shows the Fourier spectrum of the residuals after a non-linear least-squares fit of the 13 highest-amplitude $\delta$~Sct peaks and the peak at 523.99\,d$^{-1}$ (from the sdB star) is subtracted from the data.}
\label{fig:ftd}
\end{center}
\end{figure}
\subsection{{\purple Mode Identification and Asteroseismology}}
We fitted the 13 $\delta$~Sct frequencies and the most prominent sdBV$_{\rm r}$ frequency to the Sector 29 data using a non-linear least-squares algorithm in order to (a) optimize the frequencies, amplitudes and phases, and (b) determine their uncertainties. Those best-fit parameters are provided in Table~\ref{table:freq-fit}. The frequency range is narrow, and the number of excited modes is relatively small for a $\delta$~Sct star. The frequency solution for the $\delta$ Sct modes derived from the sector 42 is consistent to within the observational errors with the one listed in Table~\ref{table:freq-fit}. The bottom panel of Figure \ref{fig:ftd} shows the amplitude spectrum of the residuals after a non-linear least-squares fit of the 14 aforementioned peaks was subtracted from the data. We believe that the highest-frequency peak, at 523.99\,d$^{-1}$ arises from a pulsation mode in an sdBV$_{\rm r}$ star, as discussed later.
A simple zeroth order relation for a pulsator (first derived using a toy model in \citealt{1879AnP...244..157R}) that relates the pulsation period $P$ and mean density $\bar{\rho}$ is:
\begin{equation}
P \sqrt{\frac{\overline\rho}{\overline\rho_\odot}} = Q,
\end{equation}
where $Q$ is a constant for a given pulsation mode, which is defined by this equation. This can be rewritten in terms of observables as follows:
\begin{equation}
\log Q = \log P + \frac{1}{2} \log g + \frac{1}{10} M_{\rm bol} +\log T_{\rm eff} -6.454,
\end{equation}
Here, $P$ is in days and $\log\,g$ is in cgs units. As a first-order estimate, we use the {\em TESS} input catalog (TIC) values of $T_{\rm eff} = 8400$\,K and $\log g = 4.3$ \citep{2019AJ....158..138S} and estimate $M_{\rm bol} = 1.6$\,mag from the {\em Gaia} parallax and $V$ magnitude. Thus, we can calculate the $Q$-values for the $\delta$~Sct frequencies, which enables us to estimate the radial overtone for these frequencies' modes by comparing them with previously-calculated models, such as those in Table 1 of \citet{1979ApJ...227..935S}. Note that the putative sdB companion is significantly fainter in the {\em TESS} and {\em Gaia} passbands (i.e., in the optical -- see Figure \ref{fig:sed-best-fit}), so its contribution to the total absolute magnitude of the system can be neglected here.
For the two highest amplitude modes which span the frequency range of the $\delta$~Sct pulsations, we find $Q = 0.019$ for the 32.0888-d$^{-1}$ frequency and $Q = 0.015$ for the 39.8553-d$^{-1}$ frequency. Comparing these with model 4.4 in \cite{1979ApJ...227..935S} suggests that modes in the $\delta$~Sct star range in radial overtone between $n \sim 2 - 4$. This is a narrow range of overtones, and the number of observed frequencies in the range requires most of the associated modes to be nonradial. At first glance, TIC~5724661 seems to be a relatively hot $\delta$~Sct star, and since hotter stars tend to pulsate in higher radial overtones \citep{1975ApJ...200..343B}, $n \sim 2 - 4$ radial overtones are not unexpected. However, we are also cognizant of the fact that the temperature estimate given in the TIC may be inflated due to an unresolved sdB companion; more details are discussed in Section \ref{sec:sed}.
Three of the peaks are nearly equally spaced in frequency: 34.3520, 35.0681, and 35.7721\,d$^{-1}$. The separations between pairs of these peaks are $0.7161 \pm 0.0025$ and $0.7040 \pm 0.0024$\,d$^{-1}$; these separations themselves differ only by $0.012 \pm 0.003$\,d$^{-1}$. Despite this small difference, the formalism provided in \citet{1992ApJ...394..670D} appears to suggest that this triplet does not arise from rotational splitting. However, if we \textit{do} assume rotational splitting, we can crudely estimate $P_{\rm rot} = 1.4$~d (neglecting the Ledoux rotational splitting constant $C_{n,\ell}$). Moreover, because we know that the binary contains a $\delta$ Scuti star, we can use the illustrative values $\log g = 4.3$ and $M \sim $2\,M$_\odot$ to derive a crude radius estimate of $R \sim 1.6$\,R$_\odot$. These values, along with the rotational period estimate, predict that $v_{\rm eq} = 60$\,km~s$^{-1}$. The spectroscopic estimate of $v\sin\,i$, 39.9 $\pm$ 0.9\, km~s$^{-1}$, suggests that the rotational axis of the star is tilted $\sim$40$^\circ$ with respect to our line of sight (here, we use the convention that 0$^\circ$ is parallel to our line of sight). We emphasize that these are only first-order estimates; further analysis and modeling (discussed in Sections \ref{sec:sed} and \ref{sec:stel-evol}) can better constrain these parameters.
\begin{table}
\centering
\caption{A non-linear least squares fit of 13 $\delta$~Sct frequencies and 1 sdBV frequency to S29 data. The zero point for the phases, $t_0 = 2459100.41122$, is the center in time of the data.}
\begin{tabular}{rcr}
\hline
\hline
\multicolumn{1}{c}{frequency} & \multicolumn{1}{c}{amplitude} &
\multicolumn{1}{c}{phase} \\
\multicolumn{1}{c}{d$^{-1}$} & \multicolumn{1}{c}{mmag} &
\multicolumn{1}{c}{radians} \\
& \multicolumn{1}{c}{$\pm 0.024$} &
\\
\hline
$31.9317 \pm 0.0021 $ & $ 0.246 $ & $ -1.534 \pm 0.102 $ \\
$32.0888 \pm 0.0010 $ & $ 0.522 $ & $ -2.884 \pm 0.047 $ \\
$32.7019 \pm 0.0032 $ & $ 0.160 $ & $ -1.931 \pm 0.151 $ \\
$33.1820 \pm 0.0015 $ & $ 0.341 $ & $ 1.555 \pm 0.071 $ \\
$34.3520 \pm 0.0017 $ & $ 0.300 $ & $ 0.531 \pm 0.081 $ \\
$35.0681 \pm 0.0019 $ & $ 0.267 $ & $ -1.539 \pm 0.091 $ \\
$35.7721 \pm 0.0014 $ & $ 0.362 $ & $ -0.086 \pm 0.067 $ \\
$36.8673 \pm 0.0015 $ & $ 0.330 $ & $ 2.064 \pm 0.073 $ \\
$37.2562 \pm 0.0021 $ & $ 0.243 $ & $ -1.097 \pm 0.100 $ \\
$38.2666 \pm 0.0016 $ & $ 0.314 $ & $ 3.071 \pm 0.077 $ \\
$39.5224 \pm 0.0014 $ & $ 0.356 $ & $ -2.691 \pm 0.068 $ \\
$39.8553 \pm 0.0009 $ & $ 0.585 $ & $ 1.304 \pm 0.042 $ \\
$40.3589 \pm 0.0020 $ & $ 0.258 $ & $ -1.756 \pm 0.094 $ \\
$523.9899 \pm 0.0011 $ & $ 0.449 $ & $ 2.917 \pm 0.054 $ \\
\hline
\hline
\end{tabular}
\label{table:freq-fit}
\end{table}
\subsection{{\purple Possible Sources for the 524 d$^{-1}$ Signal}}
{\purple Many \emph{TESS} ~light curves are affected by the blending of targets close to each other on the night sky, in part due to the large size of \emph{TESS} ~pixels (see, e.g., \citealt{2021ApJS..254...39G}). As a result, we sought to ensure that both the low and high frequency pulsation signals were coming from the same target on the sky. We first downloaded the target pixel file (TPF) for this target and extracted the flux time-series for each pixel. Then, we took the Fourier transform of each individual pixel and produced an 11$\times$11 array of FTs centered on the target star. Next, we convolved this 11$\times$11 array with a 3$\times$3 boxcar kernel to enhance the statistics, albeit at the cost of some spatial resolution. We found that both the 524 d$^{-1}$ signal and the $\delta$ Scuti pulsations arose from the same region of sky. In particular, they were both strongest when the 3$\times$3 boxcar kernel contained 8 of the 9 pixels from the optimal aperture selected by SPOC to generate the light curve shown in Fig.~\ref{fig:lc}---this aperture is shown in the left panel of Fig.~\ref{fig:tpf}. Outside this region, the amplitude of these pulsations rapidly declined. The right panel of Figure \ref{fig:tpf} displays a 4$\times$4 subarray of the smoothed FTs. The four panels with the highest amplitude peaks at 524 d$^{-1}$ correspond exactly to the four central pixels in the optimal aperture (left panel of Fig.~\ref{fig:tpf}). The left panel, in addition to the optimal aperture, also shows the nearest stars in the Gaia catalog, emphasizing that there are no potential contaminants that could produce a signal of the magnitude we observe.
We obtained a similar result using the newly-developed software tool \texttt{TESS-Localize} \citep{tess-localize}. The likelihood that the 524 d$^{-1}$ signal was indeed coming from TIC\,5724661 and no other contaminating star was found to be $>$99\%, further corroborating our conclusions about the source of the signal.}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{pixel_analysis_proof.pdf}
\caption{{\purple \textit{Left}: A plot of the optimal aperture for TIC 5724661, marked with shading indicating flux, along with the positions and Gaia magnitudes of nearby stars in the Gaia catalog. It is evident that none of the nearby stars is bright or hot enough to produce a pulsational signal at the observed frequencies. \textit{Right}: A 4$\times$4 subarray of the Fourier transforms of the time series at the pixel level. The result for each pixel has been obtained by convolution with a 3$\times$3 boxcar filter to enhance the statistics, at the cost of decreased spatial resolution (see text for details). The four pixels displaying the highest pulsational signals correspond to exactly the four central pixels of the optimal aperture shown in the left panel. This clearly demonstrates that both the $\delta$~Scuti pulsations and the 524 d$^{-1}$ pulsation arise from within the optimal aperture and that none of the other stars in the TPF causes it. The location of the high-frequency pulsation (from the putative sdB star) is marked with an arrow.}}
\label{fig:tpf}
\end{figure*}
After we confirmed that both signals were coming from the same point on the sky, we explored various possibilities to explain the high-frequency signal. Most A stars with $v_{\rm eq} \lesssim 100$\,km~s$^{-1}$ are either Am or Ap stars, meaning they exhibit strong metal lines (the distinction arises from the presence of a strong dipole magnetic field in Ap stars; see, e.g., \citealt{2014PhDT.......131M}). So, we would expect TIC~5724661 to show abundance anomalies when examined at high spectral resolution, most probably of the Am kind, as nearly half of A stars near this temperature are Am stars \citep{1973ApJS...25..277S}. However, as discussed in Section \ref{sec:spec}, no abundance anomalies were detectable in our spectra; we may need a data set with a higher spectral resolution to see such anomalies. More evidence against the idea that the 523.99~d$^{-1}$ pulsation arises from a roAp star is the fact that this frequency is over twice the theoretical acoustic cutoff frequency for such a star. None of the observed supercritical roAp pulsations have deviated from this cutoff as strongly (see, e.g., \citealt{2018MNRAS.480.2405H}, and references therein).
Another possible explanation for the high-frequency pulsation observed at 524 d$^{-1}$ is a white dwarf. Many white dwarfs are known to pulsate in this frequency regime, with frequencies associated with g~modes, as opposed to the p~modes in sdBV$_{\rm r}$ stars \citep{2008ARA&A..46..157W}. However, as shown in Figure \ref{fig:ftd}, the amplitude of the high-frequency oscillation is 0.394 mmag. This is 0.036\% of the entire system's light. Using $L = 4\pi\sigma R^2 T^4$, and adopting illustrative values of $0.01~R_\odot$ for the white dwarf radius and $20\,000$\,K for the temperature, we expect the luminosity ratio of the two bodies to be $10^{-3}$, implying the white dwarf pulsates with an amplitude that is $\sim35\%$ of its luminosity. Typical WD pulsation amplitudes are between 1 and 2\% \citep{Winget_1998}; thus, this could not plausibly explain our observations.
Finally, we evaluate the possibility that there is some foreground or background contamination in the \emph{TESS} ~light curve, due to the large size of its pixels. The {\em Gaia} eDR3 catalog \citep{2021A&A...649A...1G} shows that TIC~5724661 only has one nearby star within $80''$, and this star has $m_G$ = 19.5 -- too faint to exhibit pulsations of the amplitude that we observe. Moreover, this nearby star's {\em Gaia} BP$-$RP value is 1.74, suggesting that this is an extremely cool star that should not be able to pulsate at all \citep{andrae2018gaia}. Moreover, the Renormalized Unit Weight Error (RUWE) for TIC~5724661 is 1.482 -- which is significantly greater than the expected ``typical'' value of 1; relatively large RUWE values---usually those $\gtrsim 1.4$---can often be used as a proxy for binarity (see, e.g., \citealt{2020MNRAS.496.1922B}, and the sample selection criteria used in \citealt{2020AJ....159...19Z}). As a result, we can safely discount the possibility of contamination by another source and focus on the presence of a hot compact companion {\purple in the TIC\,5724661 system.}
We thus conclude that this high-frequency mode likely arises from a p~mode sdBV$_{\rm r}$ star. Constraints on its mass are discussed in Section~\ref{sec:spec}, its temperature in Section~\ref{sec:sed}, and its evolutionary history in Section~\ref{sec:stel-evol}.
\section{Spectral Analyses}
\label{sec:spec}
To study the spectra we obtained, we conducted two analyses---one to establish constraints on the radial velocity variations, and hence on the mass of a potential unseen sdB companion, and another in which we directly searched for spectral signatures to check the chemical composition of the A star and identify any peculiarities.
First, we used the lack of detectable RV variations to constrain the mass of a potential companion. We fit for the $K$ velocity, orbital phase, and $\gamma$ velocity of the RV curve for each of $10^6$ trial periods evenly spaced in logarithmic space between 0.1 and 1000\,d, all assuming circular orbits. To be conservative when generating our constraints, we multiplied the LSD uncertainties (described in Section\,\ref{sec:obs}) {\purple by 1.6} and input those as the argument {\texttt{sigma}} to the \texttt{curve\_fit} function in \texttt{scipy}. For each trial period, we then calculated an upper limit to the value of the mass function using the best-fit $K$ value plus twice its derived uncertainty. Finally, we solved for the corresponding limit to the mass of a potential companion by using the upper limit on the value of the mass function and an assumed mass for the A star of 2M$_\odot$. This was done for each of three assumed orbital inclinations of $30^\circ$, $60^\circ$, and $90^\circ$. A plot of the derived upper constraints for a potential companion is given in Figure \ref{fig:companion-constraints}. The data suggest that any sdB star companion is more likely to be in an orbit longer than $\sim 150$\,d. However, there are cases involving low inclination angles that could harbor either a short- or medium-period sdB star (e.g., 35--60~d). There is also the possibility the orbit is eccentric, which may lead to inauspicious locations along the orbit when the radial velocities were measured. The regions of parameter space that could result in {\purple both intermediate and long} periods {\purple for the sdB companion} is explored further in Section \ref{sec:stel-evol}.
\begin{figure}
\begin{center}
\includegraphics[width=1.0\linewidth,angle=0]{fisher_plot_proof.pdf}
\caption{Upper limits on the mass of a potential companion to a 2 M$_\odot$ star that are derived from our RV measurements for a range of assumed orbital inclinations. The shaded red region indicates the range of masses of the sdBs that resulted from the modeling of various evolutionary scenarios (described in Section \ref{sec:stel-evol}). It is clear that the derived constraints are more stringent for periods $\lesssim 35$~d. There exist islands of marginally acceptable binary periods between 35 and $\sim$60~d, {\purple especially for lower inclinations}; more probable periods lie near 150 and $\sim$ 300~d. {\purple The mass is essentially unconstrained above 500~d.} Spikes represent locations where we do not possess any information on the mass of a potential companion, as a result of our observing cadence. The narrow spikes below 3~d are aliases of the 1~d observing windows.}
\label{fig:companion-constraints}
\end{center}
\end{figure}
We then turned our attention to directly analyzing the spectrum to ascertain more about the nature of the A star. We began by summing the seven TRES spectra into one, as there were no significant radial velocity differences among them. The summed spectrum had a S/N of about 65 and was, just like in Section \ref{subsec:spec}, compared to model atmospheres using ATLAS9 \citep{2004astro.ph..5087C} and SPECTRUM \citep{1994AJ....107..742G}. An atmosphere with $T_{\rm eff}=8000$~K, log~$g=4.0$, [M/H]=0, broadened to $v \sin i = 40$~km~s$^{-1}$ gave a good fit to the summed spectrum. A search for chemical peculiarities indicative of a magnetic A star yielded a null result, with the possible exception of a somewhat narrow Ca~K line at 393.366 nm. Likewise, searches for He lines in the summed spectrum caused by a possible sdB companion remained negative. This latter non-detection could be explained through a pure-H atmosphere, which may arise from chemical differentiation processes in the sdB: \citet{1981Msngr..24....7H} and \citet{2018A&A...609A..89L} suggest that processes such as gravitational settling, stellar winds (for hotter sdB/O stars), and convective instability can cause the He abundance to deviate from what is expected. On the other hand, this could simply be a consequence of an sdB companion being 2.5 magnitudes fainter than the $\delta$ Scuti star in the optical (see Fig.~\ref{fig:sed-best-fit}).
\section{Spectral Energy Distribution}
\label{sec:sed}
The spectral energy distribution (SED) obtained from the Vizier portal \citep{vizier} exhibits an excess in the ultraviolet flux in both the Galex NUV and FUV bands (see Fig.~\ref{fig:sed-best-fit}). Thus, we fit the SED with a model for the summed spectra from an A star and an sdB star to further test the possibility of an unresolved long-period hot sdB companion to the A star.
We used a custom implementation of the Markov Chain Monte Carlo (MCMC) algorithm to estimate parameters for the temperatures and radii of the two potential stars in the system. {\purple The extinction $A_V$ was set as a free parameter, as the estimates provided by Gaia for $A_G$ appeared to be unreliable for our purposes. Specifically, the value of $A_G$ provided in the Data Release 3 (DR3), when converted to $A_V$ using the conversion factors given in \citet{2019ApJ...877..116W}, does not agree with the value provided using the NED calculator (based on \citealt{2011ApJ...737..103S}). The extinction at other wavelengths was calculated based on the prescription given in \citet{1989ApJ...345..245C}.} The distance to the source was fixed at 713 pc, based on the {\em Gaia} parallax measurements given in DR3 \citep{gaia-mission, 2021A&A...649A...1G}. The Vizier data points were fit with summed model Kurucz spectra of an A star with fixed $\log\,g$ = 4.3 and an sdB star with fixed $\log\,g$ = 5 \citep{2004astro.ph..5087C}. {\purple While there exist significant processes in hot, compact stars that could lead to non-local thermal equilibrium (NLTE) effects in the spectral energy distributions, \citet{1994ApJ...432..351S} note that at typical surface gravities for these stars, non-LTE and LTE atmospheres agree quite well. Additionally, the key changes occur in the Balmer lines, as discussed in \citet{1997A&A...322..256N}; at the resolution of the observational data points we are using, these lines are not resolvable, making our decision to pursue an LTE analysis reasonable.} We set the priors on the A star to be 1\,R$_\odot < R_{\rm A} < 2.5$\,R$_\odot$, with $7000~{\rm K}<T_{\rm eff}<11000$~K. The sdB star's radius was sampled logarithmically and constrained to be within $0.1<R_{\rm sdB}<1$\,R$_\odot$, with $15000<T_{\rm eff}<50000$~K.
{\purple Finally, we set the prior on $A_V$ as 0 $\leq A_V \leq 0.3$. Because we are fitting the composite spectrum of two stars in this SED, it is very helpful to have reliable prior constraints on the extinction parameter $A_V$. The TESS input catalog v8.2 (reference), as listed on MAST\footnote{\url{https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html}} gives $E(B-V) = 0.021 \pm 0.005$. For a standard conversion factor of $R_V \simeq 3.1$, this translates to $A_V \simeq 0.065$. The NED Galactic Extinction Calculator\footnote{\url{https://ned.ipac.caltech.edu/forms/calculator.html}}, which is based on \citet{2011ApJ...737..103S}, gives $A_V \simeq 0.096$ to infinity. However, since this source has a Galactic latitude of $-65^\circ$ and the source is 611 pc away, this is well out of the Galactic plane, and we take this to be a good representation of $A_V$ to the source itself. HEASARC\footnote{\url{https://heasarc.gsfc.nasa.gov/docs/tools.html}} provides a hydrogen column density (also to infinity) of $N_H \simeq 2.4 \times 10^{-20}$ cm$^{-2}$. If we adopt a conversion factor of $4.5 \times 10^{-22} \,A_V/N_H$ (as provided in \citealt{2009MNRAS.400.2050G}), we can estimate $A_V \simeq 0.11$. Finally, we note that Gaia's early Data Release 3 \citep{2021A&A...649A...1G} lists a value for $A_G$ of 0.62. If we converted this to $A_V$, using the relations given in Table 3 of \citet{2019ApJ...877..116W}, we would infer a value of approximately 0.79. However, in light of the extreme disagreement with the other estimates of $A_V$, and because the Gaia estimate of $A_V$ is presumably based only on three spectral points, we discount this estimate of $A_V$ and do not use it. Therefore, in our MCMC evaluation of the TIC\,5724661 system parameters, we set a generous prior on the range of $A_V$ to be in the range of $0.0 \leq A_V \leq 0.3$.}
\begin{figure}
\begin{center}
\includegraphics[width=1.0\linewidth,angle=0]{best_SED_fit_0.02_met_proof.pdf}
\caption{SED plot for TIC 5724661 (black points), where the smooth curves are the model fits using Kurucz \citep{2004astro.ph..5087C} spectra for the sum of the A star and the inferred sdB star; these have been corrected for interstellar extinction. The fits are described in detail in the text. The green curve is for the A star alone, while the orange curve represents the flux of a companion sdB star. It is evident that the sum of the models for an A star and an sdB star can explain the observed SED much better than either would on their own. This lends credence to our claim that there is a compact, hot body orbiting the A star.}
\label{fig:sed-best-fit}
\end{center}
\end{figure}
To ensure confidence in our assumption that fixing the value of $\log\,g$ would not significantly impact the SED model values, we used the T\"ubingen NLTE model spectra to vary $\log\,g$ for the sdB companion. This parameter was varied from $5 \leq \log\,g \leq 6.5$ \citep{2012ascl.soft12015W}, leading to only insignificant differences in the derived SED, with the largest being a few parts per thousand of the largest SED flux value. Therefore, we were confident that we could fix the values of $\log\,g$ for both stars in the system, as described above, without losing any critical information. This assumption was borne out when we plotted the posterior distribution for this parameter, which was essentially flat---suggesting that the SED is highly insensitive to this parameter. As a result of this degeneracy, we constrained $\log\,g$ through stellar evolution modeling; see section \ref{sec:stel-evol} for more details.
We allowed the MCMC to run for 1 million steps. The best-fit parameters for the system are presented in Table~\ref{table:mcmc-mle}, along with their associated uncertainties. Figure~\ref{fig:sed-best-fit} shows the best-fit spectrum superposed on the available data points. The fit is good, with a reduced $\chi$-squared value close to unity. A corner plot illustrating the posterior distributions and their correlations between parameters is shown in Figure~\ref{fig:sed-corner}; all parameters are somewhat correlated. There exists a strong correlation between the radius and effective temperature for the sdB star, as expected -- since its radiation dominates the observed SED only in the UV region of the spectrum. We do not show the posterior distributions for $\log\,g$, as these are flat and do not yield new information.
\begin{table}
\centering
\caption{Derived values for $T_{\rm eff}$ and $R$ for both stars.}
\begin{tabular}{cc}
\hline
\hline
\multicolumn{1}{c}{Parameter} & \multicolumn{1}{c}{Value}
\\
\hline
$T_{\rm eff}$ - A star & $ 7950^{+230}_{-210}$~K \\
$R$ - A star & $1.75 \pm 0.05$~$R_\odot$\\
$T_{\rm eff}$ - sdB star & $33000^{+9400}_{-8800}$~K\\
$R$ - sdB star & $0.13^{+0.11}_{-0.04}$~$R_\odot$\\
$A_V$ & $0.10^{+0.09}_{-0.06}$ \\
\hline
\hline
\end{tabular}
\label{table:mcmc-mle}
\end{table}
\begin{figure*}
\begin{center}
\includegraphics[width=.85\linewidth,angle=0]{5724661_sed_corner_0.02_met.pdf}
\caption{Posterior distributions for the parameters of TIC~5724661. This corner plot shows the best-fit parameters and the correlations between the parameters derived through an MCMC fitting code. Dashed vertical lines, from left to right, represent the 16th percentile, median, and 84th percentile. The distribution for the temperature of the sdB star seems to fall off at high temperatures, suggesting that a $T_{\rm eff}$ close to 29\,000~K is most likely. There exists a strong degeneracy between the $T_{\rm eff}$ and the radius of the sdB star, as expected, given the limited region of the SED where the sdB star likely dominates the system light.}
\label{fig:sed-corner}
\end{center}
\end{figure*}
These fitted parameters for the putative sdB star agree with what is expected for the temperature of such a pulsating star. Figure 51 of \cite{heber} shows a demarcation between short- and long-period sdB pulsators, with the former having higher temperatures and $\log\,g$ values. Our results are reassuring, insofar as our inference of a pulsating sdB companion based on the observation of a {\purple high-frequency (short-period)} pulsation in the TESS data is bolstered by the value of our best-fit value for $T_{\rm eff}$ of the sdB star. However, what is unique about this sdB star is that it may lie in a little-explored region of binary parameter space: It could have an orbital period that is too long to suggest formation via common-envelope evolution, but it also could be too short to have evolved via stable {\purple mass transfer from a low-mass red giant near the tip of the red giant branch.}
\section{Evolutionary Analysis}
\label{sec:stel-evol}
Most evolutionary channels leading to the formation of hot subdwarfs rely on a red-giant progenitor that is rapidly stripped of its deep, hydrogen-rich envelope as a result of binary interactions. Once the red giant's core is exposed, it rapidly evolves along the extreme horizontal branch (EHB; see \citealt{heber}, and references therein).
A large fraction of sdBs are found in binary systems, and the majority of these are found in short-period binaries with $P_{\rm orb} \lesssim 5$\,d {\purple (see, e.g., \citealt{2003A&A...404..301R}). Many of these have low-mass companions, such as dM or WD stars}. There clearly exists a selection effect favoring the discovery of short-period eclipsing binaries due to strong illumination effects and deep eclipses, especially for large orbital inclinations. While there exists extensive observational evidence for sdBs in short-period binaries, there have been many fewer examples of observed long-period {\purple binary systems ($P_{\rm orb} \gtrsim 300$\,d) containing sdBs (see, e.g., \citealt{2019CoSka..49..264V}, and references therein). Our analysis of TIC 5724661 suggests that its orbital period could fall in the ``intermediate'' period range, of tens to hundreds of days. If true, TIC 5724661 would fall into a sparsely populated region of parameter space and could imply a deficiency in our understanding of the formation of (binary) hot subdwarfs.}
In this section, we discuss how to form sdBs with $P_{\rm orb} \gtrsim 70$\,d and $T_{\rm eff} \approx 30\,000$~K, as we have estimated for TIC~5724661. {\purple We analyze two types of evolutionary models---one with a low-mass progenitor ($\sim$1.2 M$_\odot$), and the other with an intermediate-mass progenitor ($\sim$3.5 M$_\odot$). These produce, respectively, long- and intermediate-period binaries containing an sdB.}
\subsection {Evolutionary Simulations}
In trying to determine the initial conditions needed to reproduce the inferred properties of TIC 5724661, we created a highly focused grid of evolutionary tracks using the {\tt MESA} binary stellar evolution code \citep{2011ApJS..192....3P, 2013ApJS..208....4P, 2015ApJS..220...15P, 2018ApJS..234...34P, 2019ApJS..243...10P}. We had previously used {\tt MESA} to successfully explain the current evolutionary state of MWC 882 \citep{2018ApJ...854..109Z}---which itself will evolve to become an sdB---and subsequently computed a grid of about 3500 models whose initial conditions were chosen so as to optimize the likelihood of the formation of {\purple intermediate-period binary sdBs \citep{Sen2019}}. Those models assumed varying degrees of non-conservative mass-loss and produced sdBs with a wide range of effective temperatures (20,000 K $\lesssim T_{\rm eff} \lesssim$ 50,000 K). {\purple Using the results from this grid as our guide, we were able to optimize the computational strategy used to reproduce the properties of TIC\,5724661. In particular, we found that the best matches were obtained by assuming highly non-conservative mass transfer.}
Evolutionary tracks in this focused grid were computed using {\tt MESA} version r10108. {\purple Approximately 160 successful sdB tracks were computed}. The sdB progenitor (i.e., the primary) was assumed to have a typical Population I metallicity ($Z = 0.02$), the atmosphere was approximated by a simple boundary condition ($\tau = 2/3$), and the local mixing-length ratio was set equal to 2. We applied the default parameters for both the Reimers' wind formula \citep{1975MSRSL...8..369R} and the Bl\"ocker wind formula \citep{1995A&A...299..755B}. We tested a reasonable range of other values for these parameters and found that they had a {\purple small} effect on the results. The most important factors influencing the evolution, other than $M_{1,0}$, $M_{2,0}$, and $P_{\rm orb, 0}$ (i.e., the initial mass of the primary, the initial mass of the secondary, and the initial orbital period, respectively), were the parameters $\alpha$ and $\beta$ \citep{2006csxs.book..623T}. The parameter $\alpha$ is the fraction of the mass lost from the primary (donor) and then {\purple directly} ejected from the binary, carrying away the specific angular momentum of the primary. Similarly, $\beta$ is the fraction of the mass transferred from the primary (donor) to the secondary (accretor) that is subsequently lost from the binary, carrying away the specific angular momentum of the secondary.{\footnote{Both cases correspond to the ``fast Jeans' mode'' of angular momentum dissipation.}}$^{,}${\footnote {$\beta$ can equivalently be viewed as the fraction of mass lost from the binary \textit{after} it has crossed the L1 point.}} {\purple We can express the amount of mass that has been accreted by the secondary as
\begin{equation}
\delta M_{2}=-(1-\alpha -\beta)\delta M_{1}\qquad
\end{equation}
To simplify the analysis, we eliminated one extra dimension of parameter space in our computations by setting $\alpha =0$.} The main justification for this strategy is our (empirical) finding that the value of $\alpha + \beta$ had a much greater impact on the evolution than did various combinations of those parameters corresponding to the same sum. Our choice of $\beta$ determined the degree to which mass transfer was non-conservative.
\begin{figure}
\begin{center}
\includegraphics[width=1.0\linewidth,angle=0]{MvsPorb1.pdf}
\caption{{\purple The evolution of the orbital period as a function of the mass of the primary (i.e., the sdB star's progenitor). Representative binaries for both the intermediate- and low-mass cases are shown. The respective initial masses of the primary and the secondary (in solar units) and the value of $\beta$ for each of the three sets of curves are listed in the diagram. For the black curves, the initial orbital periods are 2.2, 2.8, 3.4, and 4.0 days (solid, dotted, dashed, and dash-dotted lines, respectively) and for the blue curves, the initial periods are 2.6, 3.5, 4.4, and 5.3 days (solid, dotted, dashed, dash-dotted lines, respectively). For the low-mass cases (red curves), the initial orbital periods are 280, 300, 320, 340, and 400 days (solid, dotted, dashed, dash-dotted, and long-dashed lines, respectively). The ``canonical'' range of masses for sdB stars (0.45 -- 0.47 M$_\odot $) is denoted by the two vertical dashed lines.}}
\label{fig:pvsm}
\end{center}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{mesa_model_pinit_proof.pdf}
\caption{A multi-panel plot showing how various properties of the sdB components of two representative {\purple binaries} correlate with their respective initial periods, for {\purple two} intermediate-mass cases. The dark green dots correspond to a system in which the initial mass of the sdB star's progenitor was 3.25 M$_\odot$, its companion's initial mass was 1.7 M$_\odot$, and $\beta$ was fixed at 0.95. {\purple The purple diamonds correspond to a system in which the initial mass of the sdB star's progenitor was 3.5 M$_\odot$, its companion's initial mass was 1.6 M$_\odot$, and $\beta$ was fixed at 0.9.} The plots show that for larger initial periods, the sdB star's final period, mass, radius, and luminosity increase; however, its effective temperature and $\log g$ decrease. {\purple These trends remain robust even when the initial masses are changed. Note that we have employed a time-average for all parameters whose values change (e.g., the luminosity) during the sdB phase.}}
\label{fig:pinit_plot}
\end{figure}
Both binary stars are evolved contemporaneously with {\tt MESA}. It is important to follow the evolution of the secondary as it accretes mass, as the secondary could expand to fill its Roche lobe.{\footnote{If the primary is still transferring mass, the resulting evolution might lead to a merger.}} The reasons why the secondary can potentially fill its Roche lobe are as follows: (1) if the mass accretion rate onto the secondary ($\dot M_2$) is too high {\purple (i.e., the mass accretion timescale is shorter than the Kelvin time), the accretor} can expand adiabatically {\purple if it has a convective envelope}; or, (2) if the mass of the secondary were to increase substantially on a short timescale, then it could evolve to become a giant (and fill its Roche lobe) before the primary (donor) has had a chance to complete its sdB phase.{\footnote{Moreover, with respect to TIC~5724661, this would be especially problematic because the giant would be more luminous than the sdB star (contrary to observations), and it would not exhibit $\delta$-Scuti-like pulsations.}} In either case, {\tt MESA} halts further computation. In order to increase the chances that the primary evolves through the sdB phase, we typically attenuated the mass accretion rate onto the secondary by requiring that $\beta \gtrsim $ 0.8 (recall that $\dot M_2 = -(1 - \beta) \dot M_1$). Obviously, if mass transfer is fully non-conservative ($\beta = 1$), the secondary is not likely to fill its Roche lobe until long after the sdB phase is complete {\purple (assuming the primary evolves along the EHB).
Because of the potential importance of the evolution of wide sdB binaries in explaining the properties of TIC ~5724661, we have also computed the evolution of a small grid (about 50 models) of primordial binaries that are composed of low-mass stars (M$_{1,0}$ = 1.2\,M$_\odot$) for several initial orbital periods and primordial mass ratios. \citet{2020A&A...641A.163V} showed that if the primary (donor) can evolve up the red giant branch and mass transfer is initiated via stable Roche lobe overflow close to the point of helium flash (i.e., the tip of the red giant branch), then the giant primary can be stripped of its hydrogen-rich mass on a very short timescale, leaving behind a remnant core that subsequently undergoes helium burning and leads to an sdB phase. Unlike the evolution of intermediate-mass stars discussed above, this low-mass channel produces very wide sdB binaries. The large radius of the giant combined with the constraints imposed by Roche lobe geometry enforce a wide separation at the onset of mass transfer. This separation becomes wider as the binary evolves through the mass-transfer phase. Intermediate-mass stars, on the other hand, typically initiate mass transfer at much smaller separations (and shorter orbital periods) because they do not need to be as highly evolved at the onset of mass transfer in order to achieve helium burning in the stripped core. As mass transfer proceeds their separations also widen. The evolution of the orbital period as a function of the decreasing mass of the primary (i.e., the sdB star's progenitor) is shown for representative cases in Figure \ref{fig:pvsm}.}
\subsection {Results}
\begin{figure*}
\begin{center}
\includegraphics[width=.95\linewidth,angle=0]{HR_evol_sds.pdf
\caption{Formation and evolution of a representative intermediate-period and long-period sdB from the zero-age main sequence (ZAMS) to the final WD cooling stage in the H-R diagram. {\purple The solid lines denote the evolutionary tracks and the corresponding colors (as given by the color bar) indicate the percentage of the combined ideal gas and electron degeneracy pressures that is solely due to degeneracy (evaluated at the center). A filled circle denotes the start of a mass-transfer phase and a triangle denotes the end of that phase.} The ZAMS progenitor star for the intermediate-period case has a mass of 3.5 M$_\odot$ and an approximate solar metallicity ($Z=0.02$). Mass transfer is initiated in the Hertzsprung Gap and continues as the progenitor ascends the Red Giant Branch (denoted as RGB). Mass transfer ceases once its mass is reduced to 0.47 M$_\odot$ and this stripped core evolves along the horizontal branch until it reaches the sdB phase. This phase {\purple (annotated and denoted by the black dashed lines overlaid on the evolution curves)} persists for almost 80 Myr before the hot subdwarf {\purple subsequently} evolves through the sdO phase {\purple (distinguished by the brown dashed lines)} for an additional 40 Myr. The evolution of the companion star is also shown; its initial mass is 1.6 M$_\odot$, and it undergoes a phase of rapid accretion before it reaches thermal equilibrium (after mass transfer has ceased) and evolves normally as a 1.9 M$_\odot$ star. The star ascends the RGB, and the evolution is halted once the star is large enough to fill its Roche lobe. {\purple The progenitor star of the long-period sdB binary has a primordial mass of 1.2 M$_\odot$ and evolves up the RGB until it is close to its tip, at which point it starts to lose mass rapidly. After a series of thermal adjustments, it enters the sdB phase, which persists for $\approx 75$ Myr (annotated); for reasons of clarity, the evolution of its companion is not shown. Note that the observable properties of the sdBs in both the intermediate- and long-period scenarios are quite similar, as are the fractions of electron degeneracy pressure in both stars during most of their sdB and sdO phases. }}
\label{fig:evol}
\end{center}
\end{figure*}
\begin{figure}
\begin{center}
\includegraphics[width=1.0\linewidth,angle=0]{L_R_case2_proof.pdf}
\caption{Evolution of the bolometric luminosity, individual nuclear luminosities, and radius of the representative hot subdwarf as a function of time. Note that $t^* = 0$ has been chosen to approximately coincide with the end of the RGB phase ($t^* = t - 211.25$ Myr). The sdB phase persists for nearly 80 Myr, during which time helium is continuously depleted due to triple-$\alpha$ burning and the fusion of carbon into oxygen ($\alpha$-channel burning). Once a CO core forms and the subdwarf experiences He-shell flashes, it enters into a shorter-lived sdO phase. For each phase, the radius of the subdwarf is approximately constant.}
\label{fig:lum_rad}
\end{center}
\end{figure}
Using the methods described in the previous subsection, we show that the inferred observational parameters for TIC~5724661 are reproducible as long as we are willing to {\purple allow mass transfer to be highly non-conservative. We will first consider the evolution of intermediate-mass primordial binaries and show that they can produce sdBs in binaries with intermediate orbital periods. We then discuss the evolution of low-mass primordial binaries and show that they can evolve to become long-period sdB binaries. Finally, we compare the properties of the sdBs predicted by these two channels and discuss the implications for TIC~5724661.
Based on a grid of over 3000 models from \citet{Sen2019} whose resolution was subsequently refined for TIC\,5724661, we found that comparable sdB models with intermediate periods could be obtained from a population of primordial binaries with $M_{1,0} \approx 3.5 \pm 0.3$ M$_\odot$, $M_{2,0} \approx 1.6 \pm 0.3$ M$_\odot$, $P_{\rm orb,0} \approx 4 \pm 2$\,d, and $\beta \gtrsim 0.8$. For these initial conditions (and assuming a solar metallicity), we were able to produce multiple tracks for which the sdB's effective temperature was between $\approx$27,000 --32,000 K, its $\log g$ between 5.4--5.8, and its final orbital period in the tens of days.\footnote{Longer-period sdB binary models ($P_{\rm orb} \sim 1000$\,d) are discussed later in this section.} These results are in general agreement with the inferred stellar parameters of the components of the TIC 5724661 system} (Table \ref{table:mcmc-mle}).
Figure \ref{fig:pinit_plot} provides an example of how the predicted properties of the sdB depend on one of the dimensions of initial parameter space (i.e., $P_{\rm orb,0}$). For {\purple the two representative cases shown in this figure,} the initial conditions for the primordial binaries were $M_{1,0} = 3.25$\,M$_\odot$, $M_{2,0} = 1.7$\,M$_\odot$, with $\beta = 0.95$, {\purple and $M_{1,0} = 3.5$\,M$_\odot$, $M_{2,0} = 1.6$\,M$_\odot$, with $\beta = 0.9$}. Not surprisingly, increasing the initial period serves to monotonically increase the final period. Increasing the initial period implies that the donor star (primary) will be more evolved at the onset of mass transfer. This also implies an increased mass, radius, and luminosity of the resulting sdB star, but tends to lower its effective temperature.
The HR diagram for one of our representative cases that very closely reproduces the properties of the sdB in TIC 5724661 is shown in Figure \ref{fig:evol}. The sdB progenitor has a mass of 3.5 M$_\odot$, with a Population I metallicity of $Z = 0.02$; the value of $\beta$ was set equal to 0.9. Mass transfer commences in the Hertzsprung Gap and continues as the progenitor ascends the Red Giant Branch. Mass transfer rates from the donor sometimes exceeded 10$^{-6}$ M$_\odot$/yr, resulting in a large fraction of the giant donor's hydrogen-rich envelope being lost rapidly to the interstellar medium. Mass transfer ceases once the giant's highly mass-depleted envelope collapses, {\purple causing the star to lose contact with its Roche lobe. At this juncture,} the primary is essentially a hot helium core of mass 0.470 M$_\odot$, and it contracts rapidly (in $\sim\,3 \times 10^5$ yr) along the horizontal branch before entering a long-lived sdB phase ($\approx$ 80 Myr); the sdB phase is appropriately annotated in the evolutionary track in Figure \ref{fig:evol}.
{\purple One of the hallmarks of the sdB phase is the relative constancy of the sdB's radius for $\gtrsim 10$ Myr (see Figure \ref{fig:lum_rad}). We define this phase to extend from the point at which: (1) the radius has contracted sufficiently so as to remain approximately constant (for at least millions of years); and, (2) the star has increased its central carbon mass fraction by at least 1\% above its primordial value due to helium burning. Both of these conditions must be met. The sdB phase persists up} to the point when a (convective) CO-rich core first emerges. It is at this juncture that late-stage thermonuclear flashes can occur and persist briefly before the hot subdwarf enters the sdO phase. During this stage, the CO core can grow substantially in mass as the result of He-burning in a shell surrounding the core. Once the CO core has grown to reach about ~95\% of the total mass, the radius of the hot subdwarf contracts rapidly---signaling the termination of the sdO phase. Subsequently, the thin H-rich layer ($\sim 0.003$M$_\odot$) near the surface can be compressed and concomitantly heated as a result of the envelope's rapid contraction. This temperature increase is often significant enough for the star to undergo one or more shell flashes (see, e.g., \citealt{2004ApJ...616.1124N}, and references therein). Once all nuclear burning is quenched, the star descends onto the white dwarf cooling track.
Another curve in Figure \ref{fig:evol} shows the evolution of the secondary star (i.e., the accretor). Its initial mass is 1.6 M$_\odot$, and it undergoes a phase of rapid accretion before reaching thermal equilibrium (after mass accretion has ceased) {\purple as a 1.9 M$_\odot$ MS star. Because the accretion timescale onto the secondary is similar in magnitude to its thermal timescale, the accretor can adjust its internal structure on the order of several million years. Its nuclear timescale is much longer (by more than two orders of magnitude), so it takes $>$0.5 Gyr for the star to evolve off of the MS; by then, the sdB is already evolving on the WD cooling track. Thus, the secondary will be observed as a MS star during both the sdB and sdO phases of evolution; the secondary then evolves into a subgiant before ascending the RGB.} \texttt{MESA} halts the evolution once the secondary fills its Roche lobe (corresponding to the end point seen in Figure \ref{fig:evol}). If we continued to follow the evolution of this binary, we would see a subsequent phase of common-envelope evolution, resulting in the formation of a double-degenerate binary. The expected end product would thus be a 0.47 M$_\odot$ white dwarf (the sdB component) in close orbit with a lower mass helium white dwarf (the core of the giant secondary).
{\purple A third line in Figure \ref{fig:evol} shows the evolution of a representative low-mass progenitor star that produces an sdB in a wide orbit. The primordial binary consists of primary and secondary stars with masses of 1.2 and 1.1 M$_\odot$, respectively. Consistent with the models discussed in \citet{2020A&A...641A.163V}, the primary (donor) evolves up the RGB until it is close to the tip of the RGB (where a helium flash is expected to occur). The star then starts to lose mass very rapidly via Roche lobe overflow, with a mass-loss timescale of $\sim 10^{6}$ yr. After a series of flashes, it enters the sdB phase (as defined previously), which persists for $\approx 75$ Myr. The star then evolves through the sdO phase (annotated on the plot), before eventually cooling as a WD.
So what are the similarities and differences between the properties of the sdB binaries formed by these two channels? The masses of the two sdBs are virtually identical, and are close to the canonical mass for such stars (0.470 M$_\odot$ compared to 0.466 M$_\odot$). As seen in the HRD, both have very similar luminosities, effective temperatures, and thus radii. Their (time-averaged) central temperatures and densities (and thus fractional electron degeneracy pressures) are very similar during their respective sdB phases. Their sdB lifetimes persist for nearly 80 Myr (each), and even their hydrogen-rich envelopes have the same mass to within a factor of 2 ($\approx$0.0025 M$_\odot$ compared to $\approx$0.005 M$_\odot$). One difference relates to the percentage of electron degeneracy pressure at the center of the primary star preceding the sdB phase: The primary of the low-mass case has a largely degenerate core, while the intermediate-mass primary is only partially degenerate. However, soon after the start of the sdB phase, both are partially degenerate ($\approx 20$\% of the central pressure is due to electron degeneracy), and they evolve to become fully degenerate by the end of the sdB phase. The major difference between the two models, as expected, is in $P_{\rm orb}$. The intermediate-mass channel produced an sdB binary with $P_{\rm orb} \simeq$46 d, while the low-mass channel produced an sdB binary with $P_{\rm orb} \simeq$1395 d. Although there is a range of possible orbital periods according to both scenarios, it is fair to say that the low-mass case can produce a binary sdB that has a period approximately an order of magnitude larger than that of the intermediate-mass case.
Figure \ref{fig:lum_rad} provides a more detailed perspective with respect to the sdB and sdO evolutionary phases for the intermediate-mass case.} It shows the temporal evolution of the nuclear luminosities from the H-, He-, and C-burning channels, and the evolution of the surface luminosity{\footnote{Note that the sum of the nuclear luminosities may not equate to the surface luminosity. This difference is due to the gravothermal luminosity, whose magnitude is not shown.}} and radius. The high luminosities seen near $t^* = 0$ ($\equiv t - 211.25$ Myr) arise from the evolution of the primary star while it is still a red giant. After the star settles into the sdB phase, its radius remains approximately constant for $\approx$78 Myr. Initially, He-burning accounts for most of the luminosity, but as more and more carbon is created, $\alpha$-channel capture occurs, converting some of the carbon into oxygen. As both the luminosity and radius approach a local maximum, a convective core ($\approx 0.1$ M$_\odot $) of CO is formed, and the hot subdwarf thermally relaxes, leading to a brief phase of shell flashes. The subdwarf subsequently enters the sdO phase, during which time the radius is reasonably constant over $\approx 40$ Myr. Theoretically speaking, we can think of the sdB and sdO phases as being long-lasting ($>$ 10 Myr) and quasi-quiescent. The hallmark of the sdB phase is He-burning in the core; however, for the sdO phase, He-burning mainly occurs in a shell around the CO core.
{\purple For the intermediate-mass case, our models show that the mass of the A star companion (i.e., the secondary) could lie in the range of $\approx$1.8 to 1.9 M$_\odot$.} The main reason that the secondary masses are so large is that: (i) the intial mass ratio ($M_{2,0} / M_{1,0}$) must be sufficiently high at the onset of mass transfer ($\gtrsim 0.4$) to avoid dynamical instability (otherwise, this could lead to a merger); and, (ii) $M_{1,0}$ must be $\gtrsim 3$M$_\odot$ so that the primary has a chance to initiate core He-burning after departing the red giant branch.{\footnote {If $M_{\rm 1,0} \lesssim 3$M$_\odot$, the primary will evolve into a helium white dwarf and never undergo an sdB phase for the range of initial conditions that we considered.}} According to the main sequence models of \citealt{Eker_2015}, a 1.9-M$_\odot$ solar-metallicity model has a luminosity of about 15 L$_\odot$, $T_{\rm eff} \approx$\,8000\,K, and $R \approx 2$ R$_\odot$. Its inferred spectral type is A2.5V. These values are in line with the inferred values shown in Table \ref{table:mcmc-mle}. {\purple For the low-mass case, it appears that the constraints on the secondary's mass are less restrictive, but we would need to compute many more of these types of models before coming to a definitive conclusion.}
\section{Conclusions}
In this paper, we present strong evidence for the nature of TIC~5724661: a main-sequence A star with an inferred long-period sdBV$_{\rm r}$ star in orbit around it. First, we used radial velocity data to show that the putative hot subdwarf companion must have a period longer than a few tens of days. We then fit the spectral energy distribution using an MCMC code to constrain the parameters of the two stars in the system and provide fairly compelling evidence that this star is indeed an sdB star. To determine whether such sdBs can be produced using non-common-envelope formation channels, we modeled this system with MESA and demonstrated that we can readily produce such stars {\purple with either an intermediate- or a low-mass progenitor}, as long as there is a high degree of non-conservative mass transfer. We expect that the current A star in TIC~5724661 could have accreted perhaps $\sim 10\%$ of the mass lost from the sdB's progenitor, implying a final mass of 1.9 M$_\odot$. This would ensure that this star, which exhibits $\delta$-Scuti pulsations, is an A star (A2.5V). {\purple A more limiting constraint on the minimum value of $P_{\rm orb}$, perhaps via more spectroscopic observations, may serve to either strengthen or invalidate either of the intermediate-mass or low-mass progenitor binary models that we have proposed in Section \ref{sec:stel-evol}}.
Our work adds more observational evidence for intermediate- and long-period binaries that contain an sdB component (for earlier examples of such binaries, see, e.g., \citealt{2019CoSka..49..264V, 2020A&A...641A.163V}, and the references therein). Previously, many observed sdB binaries were found with extremely short periods and were therefore thought to have passed through a common-envelope phase. Our work lends observational evidence to the fact that such common envelopes are not at all necessary to form sdB stars (see, e.g., \citealt{Sen2019} for a set of intermediate-period sdB formation models), especially those found to pulsate in high-frequency p~modes.
Finally, this paper---which utilizes the TESS 20-second cadence data---further emphasizes the power of TESS for precision asteroseismology. The continuous, short-cadence nature of TESS data enables the detection and study of objects and pulsations that ground-based campaigns have, in the past, struggled to identify and characterize.\footnote{{\purple For an example of a ground-based photometric campaign to study high-frequency pulsations, such as those in TIC\,5724661, see, e.g., the Whole Earth Telescope \citep{1990ApJ...361..309N}}.} As a result, we were able to identify and derive a precise frequency estimate for the sdB pulsation frequency and, consequently, recognize the presence of a companion. Future work on this object will focus on long-term radial velocity monitoring of this star in order to better constrain the true period of this binary, {\purple as well as further modeling to break the degeneracy between the intermediate- and long-period cases}.
\section{Acknowledgments}
{\purple We thank the anonymous referee for pointing out several issues that motivated us to carry out additional analyses and clarify some of our discussions. We acknowledge Andrzej Baran, Michael Fausnaugh, and Jon Jenkins for helpful discussions regarding the TESS data and the locations of the pulsations.} We also thank St\'ephane Charpinet and JJ Hermes for helping us interpret our estimates for the $T_{\rm eff}$ of the sdB star. L.\,N.~thanks the Natural Sciences and Engineering Research Council (Canada) for financial support through the Discovery Grants program. G.\,H.~acknowledges support by the Polish NCN grants 2015/18/A/ST9/00578 and 2021/43/B/ST9/02972.
This paper includes data collected by the TESS mission, specifically through Director's Discretionary Target programs \#22 (PI: Antoci) and \#46 (PI: Jayaraman). Funding for TESS is provided by NASA's Science Mission Directorate. {\purple Resources used in this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products.} Some computations were carried out on the supercomputers managed by Calcul Qu\'ebec and Compute Canada. The operation of these supercomputers is funded by the Canada Foundation for Innovation (CFI), and the Fonds de recherche du Qu\'ebec -- Nature et technologies (FRQNT).
This work has used data from the European Space Agency (ESA) mission {\em Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\em Gaia} Data Processing and Analysis Consortium (DPAC, \url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for DPAC has been provided by national institutions, in particular those party to the Gaia Multilateral Agreement.
We would like to acknowledge the Indigenous Peoples as the traditional stewards of the land on which part of this research was conducted, and the enduring relationship between them and their traditional territories. We acknowledge the painful history of genocide and forced occupation of their territory, and we honor and respect the many diverse Indigenous people connected to this land on which we research and gather.
{\purple Code and inlists used for our MESA analysis are available on Zenodo, at this\,\dataset[link]{https://zenodo.org/record/6668950\#.YrLI5HbMKUk}.}
\facilities{TESS, Gaia, FLWO:1.5m}
\software{SPOC \citep{jenkinsSPOC2016}, {\tt numpy} \citep{harris2020array}, {\tt matplotlib} \citep{Hunter:2007},
{\tt scipy} \citep{2020SciPy-NMeth}, {\tt astropy} \citep{astropy:2013, astropy:2018}, {\tt pysynphot} \citep{2013ascl.soft03023S},
{\tt pandas} \citep{reback2020pandas, mckinney-proc-scipy-2010}}, {\tt SPECTRUM} \citep{1999ascl.soft10002G}, {\tt MESA} \citep{2011ApJS..192....3P, 2013ApJS..208....4P, 2015ApJS..220...15P, 2018ApJS..234...34P, 2019ApJS..243...10P}
\section{Introduction}
\label{sec:intro}
\subsection{Observational properties of hot subdwarfs}
{\purple Hot subdwarf-B stars (sdBs) are core helium-burning stars with thin hydrogen envelopes ($\lesssim 0.01$\,M$_\odot$) that exhibit significant chemical peculiarities. Hot subdwarf-O stars (sdOs) are even more chemically evolved, with helium burning occurring in a shell around an inert carbon-oxygen (CO) core. Such stars, which are thought to represent late stages of stellar evolution, are likely derived from the stripped cores of red giants. They usually lie on the blue end of the Extreme Horizontal Branch (EHB) of the Hertzsprung-Russell Diagram (HRD; see, e.g., \citealt{1986ASSL..128...33H,1994ApJ...432..351S, heber}). sdB and sdO stars have surface temperatures of $20,000 \lesssim T_{\rm eff} \lesssim 60,000$ K, with $5 \lesssim \log g \lesssim 6.5$, and masses of $\sim$\,0.47 M$_\odot$.}
We have adopted the following working definitions regarding subdwarfs and their key properties:
\begin{enumerate}
\item sdB stars burn helium (He) in their cores and may also undergo $\alpha$-channel burning of the newly-created carbon in the core (leading to the creation of oxygen). This phase persists for tens of Myr, during which the radius stays roughly constant.
\item sdO stars have a well-defined CO core, with helium burning occurring in a shell around this core, which has completed carbon burning and become inert. Simultaneous hydrogen burning occurs in a thin layer near the surface. This phase is typically shorter than the sdB phase (by a factor of {\purple approximately} 2 to 3), and the radius also remains roughly constant during this phase.
\item sdA (subdwarf A) stars, a newly discovered class of subdwarfs, have poorly constrained properties. Their true nature remains uncertain because there may be a variety of processes leading to their formation {\purple {(see, e.g., \citealt{2019ApJ...885...20Y})}}. While the provenance of these stars remains an important open question, they are not of importance for this work.
\end{enumerate}
\subsection{Pulsating subdwarfs}
The first pulsating sdB (sdBV) star, EC\,14026-2647, was discovered by \cite{1997MNRAS.285..640K}, who found a pulsation with a period of 144\,s. Since then, over 100 such pulsating stars have been discovered, {\purple many of them through space-based missions such as Kepler (including K2) and TESS} (see, e.g., \citealt{2017MNRAS.466.5020H, 2021MNRAS.507.4178R}). These stars fall into three categories -- rapid (sdBV$_{\rm r}$) pulsators, with $p$ mode oscillations on the order of a few minutes; slow (sdBV$_{\rm s}$) pulsators, with g~mode oscillations on the order of a few hours, and hybrid pulsators, which exhibit both p~and g~mode oscillations.
\subsection{Formation of sdB stars}
While the different classes of sdBV stars are fairly well-defined, the formation of these objects remains somewhat of a mystery. There have been extensive studies {\purple of} the mechanisms via which sdB stars form; see, for instance, \citet{1976ApJ...204..488M}, \citet{1993ApJ...407..649C}, \citet{1993ApJ...409..387D}, \citet{2002MNRAS.336..449H, 2003MNRAS.341..669H}, \citet{2011MNRAS.410..984J}, \citet{2015ApJ...806..178S}, \citet{2019CoSka..49..264V}, \citet{Sen2019}, and \citet{2020A&A...641A.163V}.
\citet{2002MNRAS.336..449H} specifically compared various formation channels leading to the creation of sdB stars. {\purple They concluded that sdBs in tight binaries ($P_{\rm orb}\lesssim$ 10\,d) were likely formed as a result of common envelope (CE) evolution. On the other hand, they showed that wide systems composed of sdBs + WDs ($P_{\rm orb}\gtrsim$ 400\,d) could, in principle, be formed as the result of stable, yet completely non-conservative Roche lobe overflow (RLOF). Finally, they demonstrated the conditions under which the merger of He WDs (the double helium-WD channel) could lead to the ignition of helium, thereby producing sdBs.
For common envelope evolution to produce a short-period sdB,} a red giant whose mass is at least 2 to 3 times greater than that of its companion must overflow its Roche lobe and achieve a sufficiently high mass transfer rate onto its companion \citep{1976ApJ...204..488M}. Such a high accretion rate precludes the companion star from accreting all of the deposited matter, leading to the formation of a common envelope (see, e.g., \citealt{1941ApJ....93..133K}). The rapid shrinking of the giant's Roche lobe as it loses mass causes dynamically unstable mass transfer, forcing the accreting companion to begin to spiral inside the giant's envelope (see, e.g., \citealt{1976ApJ...209..829W}). If the change in the orbital energy is sufficient to unbind the envelope, then the giant's envelope can be expelled from the binary system on the order of hundreds of years \citep{2017A&A...599A..54X}. If a merger can be avoided, the companion emerges in a tight, circular orbit (periods of hours to days) around the stripped core of the red giant. This stripped core can then evolve onto the EHB and become {\purple an sdB (and/or sdO).} The relatively high proportion of sdB stars observed in short-period binaries suggests that this evolutionary scenario is the most common \citep{2003MNRAS.341..669H}, but there exists an observational bias that favors the discovery of such systems.
{\purple Models of longer-period binaries containing sdBs can also be produced by assuming that the primordial binary, consisting of two main sequence (MS) stars undergoes stable, but (partially) non-conservative, mass transfer, {\purple in which both mass and angular momentum leave the system.} Using a binary population synthesis code, \citet{2003MNRAS.341..669H} concluded that it is possible to produce sdBs in binaries with $P_{\rm orb}\approx$ 100\,d under these assumptions. They refer to this as the ``first stable RLOF channel.'' It should be noted, however, that the evolution of the accretor does not seem to have been computed contemporaneously with that of the donor. This could possibly result in the accretor filling its Roche lobe before the donor has had a chance to evolve to (or completely through) the sdB phase. \citet{2003MNRAS.341..669H} also considered the formation of wide sdB+WD binaries with periods on the order of 1000\,d (``second CE ejection channel''), but their simulations failed to produce any, due to the need for massive WDs in such systems.
Very recently, \citet{2020A&A...641A.163V} showed that the observed population of wide sdB binaries ($P_{\rm orb}\gtrsim$ 1000\,d) could be robustly reproduced under the assumption that the low-mass primordial primary star (donor) is close to the tip of the red giant branch (helium flash) when rapid, yet stable, non-conservative mass loss occurs as a result of Roche lobe overflow. This can result in the formation of wide binaries containing sdBs. Using population synthesis techniques, they also investigated the effects of metallicity. For solar metallicities, they found $P_{\rm orb}\gtrsim$ 1000\,d; for lower metallicity stars they showed that sdB binaries were likely to have $P_{\rm orb}\approx$ 1000\,d.
It is also possible to form sdB/O binary stars via stable, (partially) non-conservative mass transfer in progenitor binaries composed of intermediate-mass, main-sequence stars.} While we know that mass transfer can be partially non-conservative based on an analysis of Algol-related binaries \citep{2000NewAR..44..111E}, we do not have a good constraint on systemic mass loss (i.e., the fraction of mass ejected from the binary). An extensive grid of more than 3000 progenitor models was calculated by \citet{Sen2019} of potential progenitors of hot subdwarfs, under the assumption of partially non-conservative, stable mass transfer {\purple (and solar metallicity). The primaries of the primordial binaries were chosen to have masses between 1 and 8 M$_\odot$ (with the secondaries having masses of 25\%, 50\%, 80\% and 90\% of the primary), and initial orbital periods of 1--200\,d. Mass transfer was also parameterized so as to be arbitrarily non-conservative (i.e., ranging from 0 to 100\%). That work suggests that sdB-containing binaries could have orbital periods in the range of 10\,d $\lesssim P_{\rm orb} \lesssim 100$ \,d. Thus, they naturally bridge the gap in $P_{\rm orb}$ between sdBs formed as a result of common envelope evolution and those formed from low-mass red giants that undergo rapid mass transfer near the tip of the red giant branch.}
\subsection{TIC 5724661}
As part of its goal to enable precision asteroseismology, the Transiting Exoplanet Survey Satellite (TESS; \citealt{2015JATIS...1a4003R}) has been observing a subset of stars using a novel 20-s cadence since Sector 27, which occurred during 2020 July. This mode can probe frequencies up to a Nyquist limit of 2160\,d$^{-1}$, corresponding to periods as short as 40\,s. TESS short-cadence observations are invaluable in the detection and characterization of new subdwarfs, especially pulsators (see, e.g., Section 6 of \citealt{2021FrASS...8...19L}, and references therein).
TIC~5724661 was chosen to be observed at 20\,s cadence because it was a known A~star in the instability strip on the HRD that might exhibit high-frequency pulsations. Observations of this star and our subsequent analyses revealed two sets of pulsational frequencies -- one in the typical $\delta$ Scuti frequency range and two other, unexpected, peaks at 524\,d$^{-1}$ and 580\,d$^{-1}$. These two are in the characteristic pulsational frequency range of hot compact stars, like white dwarfs and sdBV$_{\rm r}$ stars.
In this paper, we first analyze the frequencies of the $\delta$ Scuti pulsations and discuss the modes they represent. Then, we show that the spectrum of TIC~5724661 does not exhibit the chemical abnormalities of a roAp star. Moreover, no significant variations in radial velocity are found over a series of unequally-spaced observations, suggesting a long-period ($\gtrsim$~70 d) orbit. We next discuss the strong evidence for a hot compact companion suggested by the excess ultraviolet (UV) flux in the Spectral Energy Distribution (SED). We then explore possible formation pathways for such a system and contextualize our discovery of a novel system. Note, we will hereafter refer to the A-star component of the binary as the ``secondary'', and the sdB component as the ``primary'', for reasons regarding the evolution of the binary that will be more thoroughly explained in Section \ref{sec:stel-evol}.
\section{Observational data}
\label{sec:obs}
\subsection{TESS Observations of TIC 5724661}
\label{sec:tess}
\begin{table}
\centering
\caption{Properties of TIC~5724661}
\begin{tabular}{lc}
\hline
\hline
Parameter & Value \\
\hline
RA (J2015.5) (h m s)& 23:11:07.84 \\
Dec (J2015.5) ($^\circ \ ^\prime \ ^{\prime\prime}$) & -17:13:19.424 \\
$T$$^a$ & $11.204 \pm 0.007$ \\
$G$$^b$ & $11.286 \pm 0.001$ \\
$G_{\rm BP}$$^b$ & $11.353 \pm 0.001$ \\
$G_{\rm RP}$$^b$ & $11.154 \pm 0.001$ \\
$B^a$ & $11.431 \pm 0.129$ \\
$V^a$ & $11.231 \pm 0.010$ \\
$J^c$ & $10.998 \pm 0.020$ \\
$H^c$ & $10.962 \pm 0.024$ \\
$K^c$ & $10.919 \pm 0.023$ \\
W1$^d$ & $10.889 \pm 0.023$ \\
W2$^d$ & $10.917 \pm 0.020$ \\
W3$^d$ & $10.851 \pm 0.162$ \\
W4$^d$ & $> 8.607$ \\
$R$ (${\rm R}_\odot$)$^e$ & $1.32^{+0.09}_{-0.11}$ \\
$L$ (${\rm L}_\odot$)$^e$ & $5.737 \pm 0.8 $ \\
Distance (pc)$^b$ & $ 611 \pm 15$ \\
$\mu_\alpha$ (mas ~${\rm yr}^{-1}$)$^b$ & $-4.5197 \pm 0.0375$ \\
$\mu_\delta$ (mas ~${\rm yr}^{-1}$)$^b$ & $+4.941 \pm 0.0304$ \\
\hline
\hline
\label{tbl:mags}
\end{tabular}
{\bf Notes.} (a)\url{exofop.ipac.caltech.edu/tess/index.php}. (b) {\em Gaia} eDR3 (\citealt{2016A&A...595A...1G};
\citealt{2016A&A...595A...2G}; \citealt{2021A&A...649A...1G}). (c) 2MASS catalog \citep{2006AJ....131.1163S}. (d) WISE point source catalog \citep{2014yCat.2328....0C}. (e) This work; see Table \ref{table:mcmc-mle} for details regarding the radius estimate, as well as Table \ref{table:rvs} for RV data.
\end{table}
TIC~5724661 was observed by {\em TESS} in Sector 29 (from 2020 August 26 to September 21) and Sector 42 (from 2021 August 20 to September 16) in both 2\,min and 20\,s cadence. The data are available in both SAP (simple aperture photometry) and PDCSAP (presearch data conditioning SAP) forms. Data processing was done using the Science Processing Operations Center (SPOC) pipeline at NASA Ames \citep{jenkinsSPOC2016}. We used the PDCSAP data from both Sectors 29 and 42 for our analysis after converting the given flux intensity to magnitudes.\footnote{The data from Sector 42 have a large data gap during the first orbit (see Figure \ref{fig:lc}) due to saturation of the CCDs arising from the moon being in the TESS field-of-view, so we rely somewhat less on this dataset.} The Sector 29 data span 24.33\,d with a temporal center point of $t_0 = {\rm BJD}~2459100.41122$, and comprise 88937 data points (after clipping by SPOC to remove outlier points, e.g., those arising from cosmic ray strikes on the detector).\footnote{TESS Sector 29 Data Release Notes: \url{https://archive.stsci.edu/\\missions/tess/doc/tess\_drn/tess\_sector\_29\_drn43\_v02.pdf}}
\begin{figure}
\begin{center}
\includegraphics[width=0.98\linewidth,angle=0]{raw_lcs.pdf}
\caption{The light curve of TIC~5724661 obtained in 20\,s cadence in {\em TESS} Sectors 29 and 42 after processing with the SPOC pipeline \citep{jenkinsSPOC2016}. The pulsations are too rapid and too low in amplitude to discern visually in this compressed figure. Its purposes are (i) to show the two gaps in the data, which affect the spectral window, and (ii) to show the noise level in the 20-s data points. The ordinate scale is Barycentric Julian Date -- 240\,0000.0.}
\label{fig:lc}
\end{center}
\end{figure}
Figure\,\ref{fig:lc} shows the SPOC-processed light curves with the data gaps between the two orbits making up each \emph{TESS} sector. These arise from the lack of observations during data downlink, or saturation of the CCDs due to scattered light from the Earth and the Moon. Such data gaps affect the spectral window, necessitating either analysis with a Discrete Fourier Transform (DFT; see, e.g., \citealt{1985MNRAS.213..773K}) or appropriate corrections, such as re-binning the data into equally-spaced temporal bins.
\subsection{Spectroscopy}
\label{subsec:spec}
We obtained spectroscopic observations of TIC\,5724661 with the Tillinghast Reflector Echelle Spectrograph (TRES, \citealt{Furesz2014}), on the 1.5-m reflector at the Fred Lawrence Whipple Observatory (FLWO) in Arizona, USA. TRES is a high-resolution fiber-fed echelle spectrograph, with a spectral resolving power of $R=44\,000$ over the wavelength region of $3900-9100$\,\AA. A total of {\purple ten} observations were obtained for TIC 5724661 during 2020 December, and between 2021 September and 2021 {\purple December}, with peak signal-to-noise ratios per resolution element of $\sim$30 in the Mg b triplet wavelength region. The spectra were extracted and reduced as per \citet{2010ApJ...720.1118B}, with wavelength solutions derived from bracketing Th-Ar lamp exposures. The observing schedule was designed to be sensitive to a companion with $P_{\rm orb} \lesssim 30$~d.
To derive the spectroscopic broadening profiles and radial velocities from each observation, we performed a least-squares deconvolution (LSD, \citealt{1997MNRAS.291..658D}) of each spectrum against a synthetic non-rotating template; this provided both a value for the radial velocity, along with an uncertainty value. We also conducted a multi-order velocity analysis of the spectra, and derived another set of uncertainties for the radial velocity values. We observed that the multi-order uncertainties were around 50\% greater than the LSD uncertainties {\purple in some cases, and agreed with them in other cases}. Values from both sets of analyses are presented in Table \ref{table:rvs}.
Visual examination of the broadening profiles for a set of lines from the sdB companion remained negative, but the line profiles did show night-to-night variability consistent with typical spectroscopic line variations exhibited by $\delta$ Scuti stars. The broadening profiles were fitted with a model kernel accounting for the rotational, macroturbulent, and instrumental broadening terms, as well as the velocity shift of the spectrum. The comparison to model spectra and further analyses (including the use of a rotating template) are described in Section~\ref{sec:spec}.
\begin{table}
\centering
\caption{Radial Velocity Measurements of TIC 5724661 from the Tillinghast Reflector Echelle Spectrograph (TRES).}
\begin{tabular}{cccc}
\hline
\hline
\multicolumn{1}{c}{Observation} & \multicolumn{1}{c}{Radial} &\multicolumn{1}{c}{LSD} & \multicolumn{1}{c}{Multi-Order}
\\
\multicolumn{1}{c}{Date} & \multicolumn{1}{c}{Velocity} &\multicolumn{1}{c}{Error} & \multicolumn{1}{c}{Error} \\
\multicolumn{1}{c}{(BJD-2400000)} & \multicolumn{1}{c}{(km\,s$^{-1}$)} &\multicolumn{1}{c}{(km\,s$^{-1}$)} & \multicolumn{1}{c}{(km\,s$^{-1}$)} \\
\hline
59190.60806 & $-39.85$ & $\pm\,0.17$ & $\pm\,0.48$\\
59196.58549 & $-39.54$ & $\pm\,0.73$ & $\pm\,0.68$ \\
59199.57533 & $-41.06$ & $\pm\,0.80$ & $\pm\,0.75$\\
59200.59383 & $-40.24$ & $\pm\,0.66$ & $\pm\,0.59$\\
59202.58932 & $-41.05$ & $\pm\,0.69$ & $\pm\,1.13$\\
59484.77567 & $-39.03$ & $\pm\,0.63$ & $\pm\,0.65$\\
59487.72156 & $-38.98$ & $\pm\,0.38$ & $\pm\,0.52$\\
59519.69383 & $-37.51$ & $\pm\,0.47$ & $\pm\,0.71$\\
59566.62276 & $-35.20$ & $\pm\,0.20$ & $\pm\,0.65$\\
59567.60356 & $-35.26$ & $\pm\,0.21$ & $\pm\,0.44$\\
59766.95738 & $-32.19$ & $\pm\,0.39$ & $\pm\,0.56$\\
\hline
\hline
\end{tabular}
\label{table:rvs}
\end{table}
The set of eleven measured radial velocities obtained with TRES is given in Table \ref{table:rvs}.
\section{Frequency analysis}
\label{sec:freq-analysis}
The {\em TESS} data from both Sectors 29 and 42 were analyzed using a fast Discrete Fourier Transform \citep{1985MNRAS.213..773K} to produce amplitude spectra. The top panel in Fig.\,\ref{fig:ftd} shows the amplitude spectrum out to about half the Nyquist frequency of 2160\,d$^{-1}$, calculated using the Sector 29 data. A cluster of peaks in the $\delta$~Sct frequency range is seen between $26 - 46$\,d$^{-1}$, and a single, high-frequency peak is clearly detected at 523.99\,d$^{-1}$ (6.065\,mHz). These are shown at higher frequency resolution in the two middle panels, with appropriate labels indicating the sector whose light curve was input to the DFT. {\purple There is an additional peak at 579.85 d$^{-1}$ that increases in prominence in Sector 42, lending further credence to our hypothesis of an unseen hot compact pulsator in this system.}
\begin{figure}
\begin{center}
\includegraphics[width=\linewidth,angle=0]{5724661_dft.pdf}
\caption{The top panel shows the Fourier amplitude spectrum to 1000\,d$^{-1}$ from the Sector~29 light curve; there are no significant peaks between 1000\,d$^{-1}$ and the Nyquist frequency, 2160\,d$^{-1}$. The second panel zooms into a cluster of peaks in the $\delta$~Sct frequency range between $20 - 55$\,d$^{-1}$, along with a single, high-frequency peak at 523.99\,d$^{-1}$ (6.065\,mHz). The third panel shows the $\delta$ Scuti and high-frequency pulsations observed in Sector~42; the peak at 579.85\,d$^{-1}$ (6.711~mHz) increases in prominence between Sectors~29 and 42. The bottom panel shows the Fourier spectrum of the residuals after a non-linear least-squares fit of the 13 highest-amplitude $\delta$~Sct peaks and the peak at 523.99\,d$^{-1}$ (from the sdB star) is subtracted from the data.}
\label{fig:ftd}
\end{center}
\end{figure}
\subsection{{\purple Mode Identification and Asteroseismology}}
We fitted the 13 $\delta$~Sct frequencies and the most prominent sdBV$_{\rm r}$ frequency to the Sector 29 data using a non-linear least-squares algorithm in order to (a) optimize the frequencies, amplitudes and phases, and (b) determine their uncertainties. Those best-fit parameters are provided in Table~\ref{table:freq-fit}. The frequency range is narrow, and the number of excited modes is relatively small for a $\delta$~Sct star. The frequency solution for the $\delta$ Sct modes derived from the sector 42 is consistent to within the observational errors with the one listed in Table~\ref{table:freq-fit}. The bottom panel of Figure \ref{fig:ftd} shows the amplitude spectrum of the residuals after a non-linear least-squares fit of the 14 aforementioned peaks was subtracted from the data. We believe that the highest-frequency peak, at 523.99\,d$^{-1}$ arises from a pulsation mode in an sdBV$_{\rm r}$ star, as discussed later.
A simple zeroth order relation for a pulsator (first derived using a toy model in \citealt{1879AnP...244..157R}) that relates the pulsation period $P$ and mean density $\bar{\rho}$ is:
\begin{equation}
P \sqrt{\frac{\overline\rho}{\overline\rho_\odot}} = Q,
\end{equation}
where $Q$ is a constant for a given pulsation mode, which is defined by this equation. This can be rewritten in terms of observables as follows:
\begin{equation}
\log Q = \log P + \frac{1}{2} \log g + \frac{1}{10} M_{\rm bol} +\log T_{\rm eff} -6.454,
\end{equation}
Here, $P$ is in days and $\log\,g$ is in cgs units. As a first-order estimate, we use the {\em TESS} input catalog (TIC) values of $T_{\rm eff} = 8400$\,K and $\log g = 4.3$ \citep{2019AJ....158..138S} and estimate $M_{\rm bol} = 1.6$\,mag from the {\em Gaia} parallax and $V$ magnitude. Thus, we can calculate the $Q$-values for the $\delta$~Sct frequencies, which enables us to estimate the radial overtone for these frequencies' modes by comparing them with previously-calculated models, such as those in Table 1 of \citet{1979ApJ...227..935S}. Note that the putative sdB companion is significantly fainter in the {\em TESS} and {\em Gaia} passbands (i.e., in the optical -- see Figure \ref{fig:sed-best-fit}), so its contribution to the total absolute magnitude of the system can be neglected here.
For the two highest amplitude modes which span the frequency range of the $\delta$~Sct pulsations, we find $Q = 0.019$ for the 32.0888-d$^{-1}$ frequency and $Q = 0.015$ for the 39.8553-d$^{-1}$ frequency. Comparing these with model 4.4 in \cite{1979ApJ...227..935S} suggests that modes in the $\delta$~Sct star range in radial overtone between $n \sim 2 - 4$. This is a narrow range of overtones, and the number of observed frequencies in the range requires most of the associated modes to be nonradial. At first glance, TIC~5724661 seems to be a relatively hot $\delta$~Sct star, and since hotter stars tend to pulsate in higher radial overtones \citep{1975ApJ...200..343B}, $n \sim 2 - 4$ radial overtones are not unexpected. However, we are also cognizant of the fact that the temperature estimate given in the TIC may be inflated due to an unresolved sdB companion; more details are discussed in Section \ref{sec:sed}.
Three of the peaks are nearly equally spaced in frequency: 34.3520, 35.0681, and 35.7721\,d$^{-1}$. The separations between pairs of these peaks are $0.7161 \pm 0.0025$ and $0.7040 \pm 0.0024$\,d$^{-1}$; these separations themselves differ only by $0.012 \pm 0.003$\,d$^{-1}$. Despite this small difference, the formalism provided in \citet{1992ApJ...394..670D} appears to suggest that this triplet does not arise from rotational splitting. However, if we \textit{do} assume rotational splitting, we can crudely estimate $P_{\rm rot} = 1.4$~d (neglecting the Ledoux rotational splitting constant $C_{n,\ell}$). Moreover, because we know that the binary contains a $\delta$ Scuti star, we can use the illustrative values $\log g = 4.3$ and $M \sim $2\,M$_\odot$ to derive a crude radius estimate of $R \sim 1.6$\,R$_\odot$. These values, along with the rotational period estimate, predict that $v_{\rm eq} = 60$\,km~s$^{-1}$. The spectroscopic estimate of $v\sin\,i$, 39.9 $\pm$ 0.9\, km~s$^{-1}$, suggests that the rotational axis of the star is tilted $\sim$40$^\circ$ with respect to our line of sight (here, we use the convention that 0$^\circ$ is parallel to our line of sight). We emphasize that these are only first-order estimates; further analysis and modeling (discussed in Sections \ref{sec:sed} and \ref{sec:stel-evol}) can better constrain these parameters.
\begin{table}
\centering
\caption{A non-linear least squares fit of 13 $\delta$~Sct frequencies and 1 sdBV frequency to S29 data. The zero point for the phases, $t_0 = 2459100.41122$, is the center in time of the data.}
\begin{tabular}{rcr}
\hline
\hline
\multicolumn{1}{c}{frequency} & \multicolumn{1}{c}{amplitude} &
\multicolumn{1}{c}{phase} \\
\multicolumn{1}{c}{d$^{-1}$} & \multicolumn{1}{c}{mmag} &
\multicolumn{1}{c}{radians} \\
& \multicolumn{1}{c}{$\pm 0.024$} &
\\
\hline
$31.9317 \pm 0.0021 $ & $ 0.246 $ & $ -1.534 \pm 0.102 $ \\
$32.0888 \pm 0.0010 $ & $ 0.522 $ & $ -2.884 \pm 0.047 $ \\
$32.7019 \pm 0.0032 $ & $ 0.160 $ & $ -1.931 \pm 0.151 $ \\
$33.1820 \pm 0.0015 $ & $ 0.341 $ & $ 1.555 \pm 0.071 $ \\
$34.3520 \pm 0.0017 $ & $ 0.300 $ & $ 0.531 \pm 0.081 $ \\
$35.0681 \pm 0.0019 $ & $ 0.267 $ & $ -1.539 \pm 0.091 $ \\
$35.7721 \pm 0.0014 $ & $ 0.362 $ & $ -0.086 \pm 0.067 $ \\
$36.8673 \pm 0.0015 $ & $ 0.330 $ & $ 2.064 \pm 0.073 $ \\
$37.2562 \pm 0.0021 $ & $ 0.243 $ & $ -1.097 \pm 0.100 $ \\
$38.2666 \pm 0.0016 $ & $ 0.314 $ & $ 3.071 \pm 0.077 $ \\
$39.5224 \pm 0.0014 $ & $ 0.356 $ & $ -2.691 \pm 0.068 $ \\
$39.8553 \pm 0.0009 $ & $ 0.585 $ & $ 1.304 \pm 0.042 $ \\
$40.3589 \pm 0.0020 $ & $ 0.258 $ & $ -1.756 \pm 0.094 $ \\
$523.9899 \pm 0.0011 $ & $ 0.449 $ & $ 2.917 \pm 0.054 $ \\
\hline
\hline
\end{tabular}
\label{table:freq-fit}
\end{table}
\subsection{{\purple Possible Sources for the 524 d$^{-1}$ Signal}}
{\purple Many \emph{TESS} ~light curves are affected by the blending of targets close to each other on the night sky, in part due to the large size of \emph{TESS} ~pixels (see, e.g., \citealt{2021ApJS..254...39G}). As a result, we sought to ensure that both the low and high frequency pulsation signals were coming from the same target on the sky. We first downloaded the target pixel file (TPF) for this target and extracted the flux time-series for each pixel. Then, we took the Fourier transform of each individual pixel and produced an 11$\times$11 array of FTs centered on the target star. Next, we convolved this 11$\times$11 array with a 3$\times$3 boxcar kernel to enhance the statistics, albeit at the cost of some spatial resolution. We found that both the 524 d$^{-1}$ signal and the $\delta$ Scuti pulsations arose from the same region of sky. In particular, they were both strongest when the 3$\times$3 boxcar kernel contained 8 of the 9 pixels from the optimal aperture selected by SPOC to generate the light curve shown in Fig.~\ref{fig:lc}---this aperture is shown in the left panel of Fig.~\ref{fig:tpf}. Outside this region, the amplitude of these pulsations rapidly declined. The right panel of Figure \ref{fig:tpf} displays a 4$\times$4 subarray of the smoothed FTs. The four panels with the highest amplitude peaks at 524 d$^{-1}$ correspond exactly to the four central pixels in the optimal aperture (left panel of Fig.~\ref{fig:tpf}). The left panel, in addition to the optimal aperture, also shows the nearest stars in the Gaia catalog, emphasizing that there are no potential contaminants that could produce a signal of the magnitude we observe.
We obtained a similar result using the newly-developed software tool \texttt{TESS-Localize} \citep{tess-localize}. The likelihood that the 524 d$^{-1}$ signal was indeed coming from TIC\,5724661 and no other contaminating star was found to be $>$99\%, further corroborating our conclusions about the source of the signal.}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{pixel_analysis_proof.pdf}
\caption{{\purple \textit{Left}: A plot of the optimal aperture for TIC 5724661, marked with shading indicating flux, along with the positions and Gaia magnitudes of nearby stars in the Gaia catalog. It is evident that none of the nearby stars is bright or hot enough to produce a pulsational signal at the observed frequencies. \textit{Right}: A 4$\times$4 subarray of the Fourier transforms of the time series at the pixel level. The result for each pixel has been obtained by convolution with a 3$\times$3 boxcar filter to enhance the statistics, at the cost of decreased spatial resolution (see text for details). The four pixels displaying the highest pulsational signals correspond to exactly the four central pixels of the optimal aperture shown in the left panel. This clearly demonstrates that both the $\delta$~Scuti pulsations and the 524 d$^{-1}$ pulsation arise from within the optimal aperture and that none of the other stars in the TPF causes it. The location of the high-frequency pulsation (from the putative sdB star) is marked with an arrow.}}
\label{fig:tpf}
\end{figure*}
After we confirmed that both signals were coming from the same point on the sky, we explored various possibilities to explain the high-frequency signal. Most A stars with $v_{\rm eq} \lesssim 100$\,km~s$^{-1}$ are either Am or Ap stars, meaning they exhibit strong metal lines (the distinction arises from the presence of a strong dipole magnetic field in Ap stars; see, e.g., \citealt{2014PhDT.......131M}). So, we would expect TIC~5724661 to show abundance anomalies when examined at high spectral resolution, most probably of the Am kind, as nearly half of A stars near this temperature are Am stars \citep{1973ApJS...25..277S}. However, as discussed in Section \ref{sec:spec}, no abundance anomalies were detectable in our spectra; we may need a data set with a higher spectral resolution to see such anomalies. More evidence against the idea that the 523.99~d$^{-1}$ pulsation arises from a roAp star is the fact that this frequency is over twice the theoretical acoustic cutoff frequency for such a star. None of the observed supercritical roAp pulsations have deviated from this cutoff as strongly (see, e.g., \citealt{2018MNRAS.480.2405H}, and references therein).
Another possible explanation for the high-frequency pulsation observed at 524 d$^{-1}$ is a white dwarf. Many white dwarfs are known to pulsate in this frequency regime, with frequencies associated with g~modes, as opposed to the p~modes in sdBV$_{\rm r}$ stars \citep{2008ARA&A..46..157W}. However, as shown in Figure \ref{fig:ftd}, the amplitude of the high-frequency oscillation is 0.394 mmag. This is 0.036\% of the entire system's light. Using $L = 4\pi\sigma R^2 T^4$, and adopting illustrative values of $0.01~R_\odot$ for the white dwarf radius and $20\,000$\,K for the temperature, we expect the luminosity ratio of the two bodies to be $10^{-3}$, implying the white dwarf pulsates with an amplitude that is $\sim35\%$ of its luminosity. Typical WD pulsation amplitudes are between 1 and 2\% \citep{Winget_1998}; thus, this could not plausibly explain our observations.
Finally, we evaluate the possibility that there is some foreground or background contamination in the \emph{TESS} ~light curve, due to the large size of its pixels. The {\em Gaia} eDR3 catalog \citep{2021A&A...649A...1G} shows that TIC~5724661 only has one nearby star within $80''$, and this star has $m_G$ = 19.5 -- too faint to exhibit pulsations of the amplitude that we observe. Moreover, this nearby star's {\em Gaia} BP$-$RP value is 1.74, suggesting that this is an extremely cool star that should not be able to pulsate at all \citep{andrae2018gaia}. Moreover, the Renormalized Unit Weight Error (RUWE) for TIC~5724661 is 1.482 -- which is significantly greater than the expected ``typical'' value of 1; relatively large RUWE values---usually those $\gtrsim 1.4$---can often be used as a proxy for binarity (see, e.g., \citealt{2020MNRAS.496.1922B}, and the sample selection criteria used in \citealt{2020AJ....159...19Z}). As a result, we can safely discount the possibility of contamination by another source and focus on the presence of a hot compact companion {\purple in the TIC\,5724661 system.}
We thus conclude that this high-frequency mode likely arises from a p~mode sdBV$_{\rm r}$ star. Constraints on its mass are discussed in Section~\ref{sec:spec}, its temperature in Section~\ref{sec:sed}, and its evolutionary history in Section~\ref{sec:stel-evol}.
\section{Spectral Analyses}
\label{sec:spec}
To study the spectra we obtained, we conducted two analyses---one to establish constraints on the radial velocity variations, and hence on the mass of a potential unseen sdB companion, and another in which we directly searched for spectral signatures to check the chemical composition of the A star and identify any peculiarities.
First, we used the lack of detectable RV variations to constrain the mass of a potential companion. We fit for the $K$ velocity, orbital phase, and $\gamma$ velocity of the RV curve for each of $10^6$ trial periods evenly spaced in logarithmic space between 0.1 and 1000\,d, all assuming circular orbits. To be conservative when generating our constraints, we multiplied the LSD uncertainties (described in Section\,\ref{sec:obs}) {\purple by 1.6} and input those as the argument {\texttt{sigma}} to the \texttt{curve\_fit} function in \texttt{scipy}. For each trial period, we then calculated an upper limit to the value of the mass function using the best-fit $K$ value plus twice its derived uncertainty. Finally, we solved for the corresponding limit to the mass of a potential companion by using the upper limit on the value of the mass function and an assumed mass for the A star of 2M$_\odot$. This was done for each of three assumed orbital inclinations of $30^\circ$, $60^\circ$, and $90^\circ$. A plot of the derived upper constraints for a potential companion is given in Figure \ref{fig:companion-constraints}. The data suggest that any sdB star companion is more likely to be in an orbit longer than $\sim 150$\,d. However, there are cases involving low inclination angles that could harbor either a short- or medium-period sdB star (e.g., 35--60~d). There is also the possibility the orbit is eccentric, which may lead to inauspicious locations along the orbit when the radial velocities were measured. The regions of parameter space that could result in {\purple both intermediate and long} periods {\purple for the sdB companion} is explored further in Section \ref{sec:stel-evol}.
\begin{figure}
\begin{center}
\includegraphics[width=1.0\linewidth,angle=0]{fisher_plot_proof.pdf}
\caption{Upper limits on the mass of a potential companion to a 2 M$_\odot$ star that are derived from our RV measurements for a range of assumed orbital inclinations. The shaded red region indicates the range of masses of the sdBs that resulted from the modeling of various evolutionary scenarios (described in Section \ref{sec:stel-evol}). It is clear that the derived constraints are more stringent for periods $\lesssim 35$~d. There exist islands of marginally acceptable binary periods between 35 and $\sim$60~d, {\purple especially for lower inclinations}; more probable periods lie near 150 and $\sim$ 300~d. {\purple The mass is essentially unconstrained above 500~d.} Spikes represent locations where we do not possess any information on the mass of a potential companion, as a result of our observing cadence. The narrow spikes below 3~d are aliases of the 1~d observing windows.}
\label{fig:companion-constraints}
\end{center}
\end{figure}
We then turned our attention to directly analyzing the spectrum to ascertain more about the nature of the A star. We began by summing the seven TRES spectra into one, as there were no significant radial velocity differences among them. The summed spectrum had a S/N of about 65 and was, just like in Section \ref{subsec:spec}, compared to model atmospheres using ATLAS9 \citep{2004astro.ph..5087C} and SPECTRUM \citep{1994AJ....107..742G}. An atmosphere with $T_{\rm eff}=8000$~K, log~$g=4.0$, [M/H]=0, broadened to $v \sin i = 40$~km~s$^{-1}$ gave a good fit to the summed spectrum. A search for chemical peculiarities indicative of a magnetic A star yielded a null result, with the possible exception of a somewhat narrow Ca~K line at 393.366 nm. Likewise, searches for He lines in the summed spectrum caused by a possible sdB companion remained negative. This latter non-detection could be explained through a pure-H atmosphere, which may arise from chemical differentiation processes in the sdB: \citet{1981Msngr..24....7H} and \citet{2018A&A...609A..89L} suggest that processes such as gravitational settling, stellar winds (for hotter sdB/O stars), and convective instability can cause the He abundance to deviate from what is expected. On the other hand, this could simply be a consequence of an sdB companion being 2.5 magnitudes fainter than the $\delta$ Scuti star in the optical (see Fig.~\ref{fig:sed-best-fit}).
\section{Spectral Energy Distribution}
\label{sec:sed}
The spectral energy distribution (SED) obtained from the Vizier portal \citep{vizier} exhibits an excess in the ultraviolet flux in both the Galex NUV and FUV bands (see Fig.~\ref{fig:sed-best-fit}). Thus, we fit the SED with a model for the summed spectra from an A star and an sdB star to further test the possibility of an unresolved long-period hot sdB companion to the A star.
We used a custom implementation of the Markov Chain Monte Carlo (MCMC) algorithm to estimate parameters for the temperatures and radii of the two potential stars in the system. {\purple The extinction $A_V$ was set as a free parameter, as the estimates provided by Gaia for $A_G$ appeared to be unreliable for our purposes. Specifically, the value of $A_G$ provided in the Data Release 3 (DR3), when converted to $A_V$ using the conversion factors given in \citet{2019ApJ...877..116W}, does not agree with the value provided using the NED calculator (based on \citealt{2011ApJ...737..103S}). The extinction at other wavelengths was calculated based on the prescription given in \citet{1989ApJ...345..245C}.} The distance to the source was fixed at 713 pc, based on the {\em Gaia} parallax measurements given in DR3 \citep{gaia-mission, 2021A&A...649A...1G}. The Vizier data points were fit with summed model Kurucz spectra of an A star with fixed $\log\,g$ = 4.3 and an sdB star with fixed $\log\,g$ = 5 \citep{2004astro.ph..5087C}. {\purple While there exist significant processes in hot, compact stars that could lead to non-local thermal equilibrium (NLTE) effects in the spectral energy distributions, \citet{1994ApJ...432..351S} note that at typical surface gravities for these stars, non-LTE and LTE atmospheres agree quite well. Additionally, the key changes occur in the Balmer lines, as discussed in \citet{1997A&A...322..256N}; at the resolution of the observational data points we are using, these lines are not resolvable, making our decision to pursue an LTE analysis reasonable.} We set the priors on the A star to be 1\,R$_\odot < R_{\rm A} < 2.5$\,R$_\odot$, with $7000~{\rm K}<T_{\rm eff}<11000$~K. The sdB star's radius was sampled logarithmically and constrained to be within $0.1<R_{\rm sdB}<1$\,R$_\odot$, with $15000<T_{\rm eff}<50000$~K.
{\purple Finally, we set the prior on $A_V$ as 0 $\leq A_V \leq 0.3$. Because we are fitting the composite spectrum of two stars in this SED, it is very helpful to have reliable prior constraints on the extinction parameter $A_V$. The TESS input catalog v8.2 (reference), as listed on MAST\footnote{\url{https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html}} gives $E(B-V) = 0.021 \pm 0.005$. For a standard conversion factor of $R_V \simeq 3.1$, this translates to $A_V \simeq 0.065$. The NED Galactic Extinction Calculator\footnote{\url{https://ned.ipac.caltech.edu/forms/calculator.html}}, which is based on \citet{2011ApJ...737..103S}, gives $A_V \simeq 0.096$ to infinity. However, since this source has a Galactic latitude of $-65^\circ$ and the source is 611 pc away, this is well out of the Galactic plane, and we take this to be a good representation of $A_V$ to the source itself. HEASARC\footnote{\url{https://heasarc.gsfc.nasa.gov/docs/tools.html}} provides a hydrogen column density (also to infinity) of $N_H \simeq 2.4 \times 10^{-20}$ cm$^{-2}$. If we adopt a conversion factor of $4.5 \times 10^{-22} \,A_V/N_H$ (as provided in \citealt{2009MNRAS.400.2050G}), we can estimate $A_V \simeq 0.11$. Finally, we note that Gaia's early Data Release 3 \citep{2021A&A...649A...1G} lists a value for $A_G$ of 0.62. If we converted this to $A_V$, using the relations given in Table 3 of \citet{2019ApJ...877..116W}, we would infer a value of approximately 0.79. However, in light of the extreme disagreement with the other estimates of $A_V$, and because the Gaia estimate of $A_V$ is presumably based only on three spectral points, we discount this estimate of $A_V$ and do not use it. Therefore, in our MCMC evaluation of the TIC\,5724661 system parameters, we set a generous prior on the range of $A_V$ to be in the range of $0.0 \leq A_V \leq 0.3$.}
\begin{figure}
\begin{center}
\includegraphics[width=1.0\linewidth,angle=0]{best_SED_fit_0.02_met_proof.pdf}
\caption{SED plot for TIC 5724661 (black points), where the smooth curves are the model fits using Kurucz \citep{2004astro.ph..5087C} spectra for the sum of the A star and the inferred sdB star; these have been corrected for interstellar extinction. The fits are described in detail in the text. The green curve is for the A star alone, while the orange curve represents the flux of a companion sdB star. It is evident that the sum of the models for an A star and an sdB star can explain the observed SED much better than either would on their own. This lends credence to our claim that there is a compact, hot body orbiting the A star.}
\label{fig:sed-best-fit}
\end{center}
\end{figure}
To ensure confidence in our assumption that fixing the value of $\log\,g$ would not significantly impact the SED model values, we used the T\"ubingen NLTE model spectra to vary $\log\,g$ for the sdB companion. This parameter was varied from $5 \leq \log\,g \leq 6.5$ \citep{2012ascl.soft12015W}, leading to only insignificant differences in the derived SED, with the largest being a few parts per thousand of the largest SED flux value. Therefore, we were confident that we could fix the values of $\log\,g$ for both stars in the system, as described above, without losing any critical information. This assumption was borne out when we plotted the posterior distribution for this parameter, which was essentially flat---suggesting that the SED is highly insensitive to this parameter. As a result of this degeneracy, we constrained $\log\,g$ through stellar evolution modeling; see section \ref{sec:stel-evol} for more details.
We allowed the MCMC to run for 1 million steps. The best-fit parameters for the system are presented in Table~\ref{table:mcmc-mle}, along with their associated uncertainties. Figure~\ref{fig:sed-best-fit} shows the best-fit spectrum superposed on the available data points. The fit is good, with a reduced $\chi$-squared value close to unity. A corner plot illustrating the posterior distributions and their correlations between parameters is shown in Figure~\ref{fig:sed-corner}; all parameters are somewhat correlated. There exists a strong correlation between the radius and effective temperature for the sdB star, as expected -- since its radiation dominates the observed SED only in the UV region of the spectrum. We do not show the posterior distributions for $\log\,g$, as these are flat and do not yield new information.
\begin{table}
\centering
\caption{Derived values for $T_{\rm eff}$ and $R$ for both stars.}
\begin{tabular}{cc}
\hline
\hline
\multicolumn{1}{c}{Parameter} & \multicolumn{1}{c}{Value}
\\
\hline
$T_{\rm eff}$ - A star & $ 7950^{+230}_{-210}$~K \\
$R$ - A star & $1.75 \pm 0.05$~$R_\odot$\\
$T_{\rm eff}$ - sdB star & $33000^{+9400}_{-8800}$~K\\
$R$ - sdB star & $0.13^{+0.11}_{-0.04}$~$R_\odot$\\
$A_V$ & $0.10^{+0.09}_{-0.06}$ \\
\hline
\hline
\end{tabular}
\label{table:mcmc-mle}
\end{table}
\begin{figure*}
\begin{center}
\includegraphics[width=.85\linewidth,angle=0]{5724661_sed_corner_0.02_met.pdf}
\caption{Posterior distributions for the parameters of TIC~5724661. This corner plot shows the best-fit parameters and the correlations between the parameters derived through an MCMC fitting code. Dashed vertical lines, from left to right, represent the 16th percentile, median, and 84th percentile. The distribution for the temperature of the sdB star seems to fall off at high temperatures, suggesting that a $T_{\rm eff}$ close to 29\,000~K is most likely. There exists a strong degeneracy between the $T_{\rm eff}$ and the radius of the sdB star, as expected, given the limited region of the SED where the sdB star likely dominates the system light.}
\label{fig:sed-corner}
\end{center}
\end{figure*}
These fitted parameters for the putative sdB star agree with what is expected for the temperature of such a pulsating star. Figure 51 of \cite{heber} shows a demarcation between short- and long-period sdB pulsators, with the former having higher temperatures and $\log\,g$ values. Our results are reassuring, insofar as our inference of a pulsating sdB companion based on the observation of a {\purple high-frequency (short-period)} pulsation in the TESS data is bolstered by the value of our best-fit value for $T_{\rm eff}$ of the sdB star. However, what is unique about this sdB star is that it may lie in a little-explored region of binary parameter space: It could have an orbital period that is too long to suggest formation via common-envelope evolution, but it also could be too short to have evolved via stable {\purple mass transfer from a low-mass red giant near the tip of the red giant branch.}
\section{Evolutionary Analysis}
\label{sec:stel-evol}
Most evolutionary channels leading to the formation of hot subdwarfs rely on a red-giant progenitor that is rapidly stripped of its deep, hydrogen-rich envelope as a result of binary interactions. Once the red giant's core is exposed, it rapidly evolves along the extreme horizontal branch (EHB; see \citealt{heber}, and references therein).
A large fraction of sdBs are found in binary systems, and the majority of these are found in short-period binaries with $P_{\rm orb} \lesssim 5$\,d {\purple (see, e.g., \citealt{2003A&A...404..301R}). Many of these have low-mass companions, such as dM or WD stars}. There clearly exists a selection effect favoring the discovery of short-period eclipsing binaries due to strong illumination effects and deep eclipses, especially for large orbital inclinations. While there exists extensive observational evidence for sdBs in short-period binaries, there have been many fewer examples of observed long-period {\purple binary systems ($P_{\rm orb} \gtrsim 300$\,d) containing sdBs (see, e.g., \citealt{2019CoSka..49..264V}, and references therein). Our analysis of TIC 5724661 suggests that its orbital period could fall in the ``intermediate'' period range, of tens to hundreds of days. If true, TIC 5724661 would fall into a sparsely populated region of parameter space and could imply a deficiency in our understanding of the formation of (binary) hot subdwarfs.}
In this section, we discuss how to form sdBs with $P_{\rm orb} \gtrsim 70$\,d and $T_{\rm eff} \approx 30\,000$~K, as we have estimated for TIC~5724661. {\purple We analyze two types of evolutionary models---one with a low-mass progenitor ($\sim$1.2 M$_\odot$), and the other with an intermediate-mass progenitor ($\sim$3.5 M$_\odot$). These produce, respectively, long- and intermediate-period binaries containing an sdB.}
\subsection {Evolutionary Simulations}
In trying to determine the initial conditions needed to reproduce the inferred properties of TIC 5724661, we created a highly focused grid of evolutionary tracks using the {\tt MESA} binary stellar evolution code \citep{2011ApJS..192....3P, 2013ApJS..208....4P, 2015ApJS..220...15P, 2018ApJS..234...34P, 2019ApJS..243...10P}. We had previously used {\tt MESA} to successfully explain the current evolutionary state of MWC 882 \citep{2018ApJ...854..109Z}---which itself will evolve to become an sdB---and subsequently computed a grid of about 3500 models whose initial conditions were chosen so as to optimize the likelihood of the formation of {\purple intermediate-period binary sdBs \citep{Sen2019}}. Those models assumed varying degrees of non-conservative mass-loss and produced sdBs with a wide range of effective temperatures (20,000 K $\lesssim T_{\rm eff} \lesssim$ 50,000 K). {\purple Using the results from this grid as our guide, we were able to optimize the computational strategy used to reproduce the properties of TIC\,5724661. In particular, we found that the best matches were obtained by assuming highly non-conservative mass transfer.}
Evolutionary tracks in this focused grid were computed using {\tt MESA} version r10108. {\purple Approximately 160 successful sdB tracks were computed}. The sdB progenitor (i.e., the primary) was assumed to have a typical Population I metallicity ($Z = 0.02$), the atmosphere was approximated by a simple boundary condition ($\tau = 2/3$), and the local mixing-length ratio was set equal to 2. We applied the default parameters for both the Reimers' wind formula \citep{1975MSRSL...8..369R} and the Bl\"ocker wind formula \citep{1995A&A...299..755B}. We tested a reasonable range of other values for these parameters and found that they had a {\purple small} effect on the results. The most important factors influencing the evolution, other than $M_{1,0}$, $M_{2,0}$, and $P_{\rm orb, 0}$ (i.e., the initial mass of the primary, the initial mass of the secondary, and the initial orbital period, respectively), were the parameters $\alpha$ and $\beta$ \citep{2006csxs.book..623T}. The parameter $\alpha$ is the fraction of the mass lost from the primary (donor) and then {\purple directly} ejected from the binary, carrying away the specific angular momentum of the primary. Similarly, $\beta$ is the fraction of the mass transferred from the primary (donor) to the secondary (accretor) that is subsequently lost from the binary, carrying away the specific angular momentum of the secondary.{\footnote{Both cases correspond to the ``fast Jeans' mode'' of angular momentum dissipation.}}$^{,}${\footnote {$\beta$ can equivalently be viewed as the fraction of mass lost from the binary \textit{after} it has crossed the L1 point.}} {\purple We can express the amount of mass that has been accreted by the secondary as
\begin{equation}
\delta M_{2}=-(1-\alpha -\beta)\delta M_{1}\qquad
\end{equation}
To simplify the analysis, we eliminated one extra dimension of parameter space in our computations by setting $\alpha =0$.} The main justification for this strategy is our (empirical) finding that the value of $\alpha + \beta$ had a much greater impact on the evolution than did various combinations of those parameters corresponding to the same sum. Our choice of $\beta$ determined the degree to which mass transfer was non-conservative.
\begin{figure}
\begin{center}
\includegraphics[width=1.0\linewidth,angle=0]{MvsPorb1.pdf}
\caption{{\purple The evolution of the orbital period as a function of the mass of the primary (i.e., the sdB star's progenitor). Representative binaries for both the intermediate- and low-mass cases are shown. The respective initial masses of the primary and the secondary (in solar units) and the value of $\beta$ for each of the three sets of curves are listed in the diagram. For the black curves, the initial orbital periods are 2.2, 2.8, 3.4, and 4.0 days (solid, dotted, dashed, and dash-dotted lines, respectively) and for the blue curves, the initial periods are 2.6, 3.5, 4.4, and 5.3 days (solid, dotted, dashed, dash-dotted lines, respectively). For the low-mass cases (red curves), the initial orbital periods are 280, 300, 320, 340, and 400 days (solid, dotted, dashed, dash-dotted, and long-dashed lines, respectively). The ``canonical'' range of masses for sdB stars (0.45 -- 0.47 M$_\odot $) is denoted by the two vertical dashed lines.}}
\label{fig:pvsm}
\end{center}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{mesa_model_pinit_proof.pdf}
\caption{A multi-panel plot showing how various properties of the sdB components of two representative {\purple binaries} correlate with their respective initial periods, for {\purple two} intermediate-mass cases. The dark green dots correspond to a system in which the initial mass of the sdB star's progenitor was 3.25 M$_\odot$, its companion's initial mass was 1.7 M$_\odot$, and $\beta$ was fixed at 0.95. {\purple The purple diamonds correspond to a system in which the initial mass of the sdB star's progenitor was 3.5 M$_\odot$, its companion's initial mass was 1.6 M$_\odot$, and $\beta$ was fixed at 0.9.} The plots show that for larger initial periods, the sdB star's final period, mass, radius, and luminosity increase; however, its effective temperature and $\log g$ decrease. {\purple These trends remain robust even when the initial masses are changed. Note that we have employed a time-average for all parameters whose values change (e.g., the luminosity) during the sdB phase.}}
\label{fig:pinit_plot}
\end{figure}
Both binary stars are evolved contemporaneously with {\tt MESA}. It is important to follow the evolution of the secondary as it accretes mass, as the secondary could expand to fill its Roche lobe.{\footnote{If the primary is still transferring mass, the resulting evolution might lead to a merger.}} The reasons why the secondary can potentially fill its Roche lobe are as follows: (1) if the mass accretion rate onto the secondary ($\dot M_2$) is too high {\purple (i.e., the mass accretion timescale is shorter than the Kelvin time), the accretor} can expand adiabatically {\purple if it has a convective envelope}; or, (2) if the mass of the secondary were to increase substantially on a short timescale, then it could evolve to become a giant (and fill its Roche lobe) before the primary (donor) has had a chance to complete its sdB phase.{\footnote{Moreover, with respect to TIC~5724661, this would be especially problematic because the giant would be more luminous than the sdB star (contrary to observations), and it would not exhibit $\delta$-Scuti-like pulsations.}} In either case, {\tt MESA} halts further computation. In order to increase the chances that the primary evolves through the sdB phase, we typically attenuated the mass accretion rate onto the secondary by requiring that $\beta \gtrsim $ 0.8 (recall that $\dot M_2 = -(1 - \beta) \dot M_1$). Obviously, if mass transfer is fully non-conservative ($\beta = 1$), the secondary is not likely to fill its Roche lobe until long after the sdB phase is complete {\purple (assuming the primary evolves along the EHB).
Because of the potential importance of the evolution of wide sdB binaries in explaining the properties of TIC ~5724661, we have also computed the evolution of a small grid (about 50 models) of primordial binaries that are composed of low-mass stars (M$_{1,0}$ = 1.2\,M$_\odot$) for several initial orbital periods and primordial mass ratios. \citet{2020A&A...641A.163V} showed that if the primary (donor) can evolve up the red giant branch and mass transfer is initiated via stable Roche lobe overflow close to the point of helium flash (i.e., the tip of the red giant branch), then the giant primary can be stripped of its hydrogen-rich mass on a very short timescale, leaving behind a remnant core that subsequently undergoes helium burning and leads to an sdB phase. Unlike the evolution of intermediate-mass stars discussed above, this low-mass channel produces very wide sdB binaries. The large radius of the giant combined with the constraints imposed by Roche lobe geometry enforce a wide separation at the onset of mass transfer. This separation becomes wider as the binary evolves through the mass-transfer phase. Intermediate-mass stars, on the other hand, typically initiate mass transfer at much smaller separations (and shorter orbital periods) because they do not need to be as highly evolved at the onset of mass transfer in order to achieve helium burning in the stripped core. As mass transfer proceeds their separations also widen. The evolution of the orbital period as a function of the decreasing mass of the primary (i.e., the sdB star's progenitor) is shown for representative cases in Figure \ref{fig:pvsm}.}
\subsection {Results}
\begin{figure*}
\begin{center}
\includegraphics[width=.95\linewidth,angle=0]{HR_evol_sds.pdf
\caption{Formation and evolution of a representative intermediate-period and long-period sdB from the zero-age main sequence (ZAMS) to the final WD cooling stage in the H-R diagram. {\purple The solid lines denote the evolutionary tracks and the corresponding colors (as given by the color bar) indicate the percentage of the combined ideal gas and electron degeneracy pressures that is solely due to degeneracy (evaluated at the center). A filled circle denotes the start of a mass-transfer phase and a triangle denotes the end of that phase.} The ZAMS progenitor star for the intermediate-period case has a mass of 3.5 M$_\odot$ and an approximate solar metallicity ($Z=0.02$). Mass transfer is initiated in the Hertzsprung Gap and continues as the progenitor ascends the Red Giant Branch (denoted as RGB). Mass transfer ceases once its mass is reduced to 0.47 M$_\odot$ and this stripped core evolves along the horizontal branch until it reaches the sdB phase. This phase {\purple (annotated and denoted by the black dashed lines overlaid on the evolution curves)} persists for almost 80 Myr before the hot subdwarf {\purple subsequently} evolves through the sdO phase {\purple (distinguished by the brown dashed lines)} for an additional 40 Myr. The evolution of the companion star is also shown; its initial mass is 1.6 M$_\odot$, and it undergoes a phase of rapid accretion before it reaches thermal equilibrium (after mass transfer has ceased) and evolves normally as a 1.9 M$_\odot$ star. The star ascends the RGB, and the evolution is halted once the star is large enough to fill its Roche lobe. {\purple The progenitor star of the long-period sdB binary has a primordial mass of 1.2 M$_\odot$ and evolves up the RGB until it is close to its tip, at which point it starts to lose mass rapidly. After a series of thermal adjustments, it enters the sdB phase, which persists for $\approx 75$ Myr (annotated); for reasons of clarity, the evolution of its companion is not shown. Note that the observable properties of the sdBs in both the intermediate- and long-period scenarios are quite similar, as are the fractions of electron degeneracy pressure in both stars during most of their sdB and sdO phases. }}
\label{fig:evol}
\end{center}
\end{figure*}
\begin{figure}
\begin{center}
\includegraphics[width=1.0\linewidth,angle=0]{L_R_case2_proof.pdf}
\caption{Evolution of the bolometric luminosity, individual nuclear luminosities, and radius of the representative hot subdwarf as a function of time. Note that $t^* = 0$ has been chosen to approximately coincide with the end of the RGB phase ($t^* = t - 211.25$ Myr). The sdB phase persists for nearly 80 Myr, during which time helium is continuously depleted due to triple-$\alpha$ burning and the fusion of carbon into oxygen ($\alpha$-channel burning). Once a CO core forms and the subdwarf experiences He-shell flashes, it enters into a shorter-lived sdO phase. For each phase, the radius of the subdwarf is approximately constant.}
\label{fig:lum_rad}
\end{center}
\end{figure}
Using the methods described in the previous subsection, we show that the inferred observational parameters for TIC~5724661 are reproducible as long as we are willing to {\purple allow mass transfer to be highly non-conservative. We will first consider the evolution of intermediate-mass primordial binaries and show that they can produce sdBs in binaries with intermediate orbital periods. We then discuss the evolution of low-mass primordial binaries and show that they can evolve to become long-period sdB binaries. Finally, we compare the properties of the sdBs predicted by these two channels and discuss the implications for TIC~5724661.
Based on a grid of over 3000 models from \citet{Sen2019} whose resolution was subsequently refined for TIC\,5724661, we found that comparable sdB models with intermediate periods could be obtained from a population of primordial binaries with $M_{1,0} \approx 3.5 \pm 0.3$ M$_\odot$, $M_{2,0} \approx 1.6 \pm 0.3$ M$_\odot$, $P_{\rm orb,0} \approx 4 \pm 2$\,d, and $\beta \gtrsim 0.8$. For these initial conditions (and assuming a solar metallicity), we were able to produce multiple tracks for which the sdB's effective temperature was between $\approx$27,000 --32,000 K, its $\log g$ between 5.4--5.8, and its final orbital period in the tens of days.\footnote{Longer-period sdB binary models ($P_{\rm orb} \sim 1000$\,d) are discussed later in this section.} These results are in general agreement with the inferred stellar parameters of the components of the TIC 5724661 system} (Table \ref{table:mcmc-mle}).
Figure \ref{fig:pinit_plot} provides an example of how the predicted properties of the sdB depend on one of the dimensions of initial parameter space (i.e., $P_{\rm orb,0}$). For {\purple the two representative cases shown in this figure,} the initial conditions for the primordial binaries were $M_{1,0} = 3.25$\,M$_\odot$, $M_{2,0} = 1.7$\,M$_\odot$, with $\beta = 0.95$, {\purple and $M_{1,0} = 3.5$\,M$_\odot$, $M_{2,0} = 1.6$\,M$_\odot$, with $\beta = 0.9$}. Not surprisingly, increasing the initial period serves to monotonically increase the final period. Increasing the initial period implies that the donor star (primary) will be more evolved at the onset of mass transfer. This also implies an increased mass, radius, and luminosity of the resulting sdB star, but tends to lower its effective temperature.
The HR diagram for one of our representative cases that very closely reproduces the properties of the sdB in TIC 5724661 is shown in Figure \ref{fig:evol}. The sdB progenitor has a mass of 3.5 M$_\odot$, with a Population I metallicity of $Z = 0.02$; the value of $\beta$ was set equal to 0.9. Mass transfer commences in the Hertzsprung Gap and continues as the progenitor ascends the Red Giant Branch. Mass transfer rates from the donor sometimes exceeded 10$^{-6}$ M$_\odot$/yr, resulting in a large fraction of the giant donor's hydrogen-rich envelope being lost rapidly to the interstellar medium. Mass transfer ceases once the giant's highly mass-depleted envelope collapses, {\purple causing the star to lose contact with its Roche lobe. At this juncture,} the primary is essentially a hot helium core of mass 0.470 M$_\odot$, and it contracts rapidly (in $\sim\,3 \times 10^5$ yr) along the horizontal branch before entering a long-lived sdB phase ($\approx$ 80 Myr); the sdB phase is appropriately annotated in the evolutionary track in Figure \ref{fig:evol}.
{\purple One of the hallmarks of the sdB phase is the relative constancy of the sdB's radius for $\gtrsim 10$ Myr (see Figure \ref{fig:lum_rad}). We define this phase to extend from the point at which: (1) the radius has contracted sufficiently so as to remain approximately constant (for at least millions of years); and, (2) the star has increased its central carbon mass fraction by at least 1\% above its primordial value due to helium burning. Both of these conditions must be met. The sdB phase persists up} to the point when a (convective) CO-rich core first emerges. It is at this juncture that late-stage thermonuclear flashes can occur and persist briefly before the hot subdwarf enters the sdO phase. During this stage, the CO core can grow substantially in mass as the result of He-burning in a shell surrounding the core. Once the CO core has grown to reach about ~95\% of the total mass, the radius of the hot subdwarf contracts rapidly---signaling the termination of the sdO phase. Subsequently, the thin H-rich layer ($\sim 0.003$M$_\odot$) near the surface can be compressed and concomitantly heated as a result of the envelope's rapid contraction. This temperature increase is often significant enough for the star to undergo one or more shell flashes (see, e.g., \citealt{2004ApJ...616.1124N}, and references therein). Once all nuclear burning is quenched, the star descends onto the white dwarf cooling track.
Another curve in Figure \ref{fig:evol} shows the evolution of the secondary star (i.e., the accretor). Its initial mass is 1.6 M$_\odot$, and it undergoes a phase of rapid accretion before reaching thermal equilibrium (after mass accretion has ceased) {\purple as a 1.9 M$_\odot$ MS star. Because the accretion timescale onto the secondary is similar in magnitude to its thermal timescale, the accretor can adjust its internal structure on the order of several million years. Its nuclear timescale is much longer (by more than two orders of magnitude), so it takes $>$0.5 Gyr for the star to evolve off of the MS; by then, the sdB is already evolving on the WD cooling track. Thus, the secondary will be observed as a MS star during both the sdB and sdO phases of evolution; the secondary then evolves into a subgiant before ascending the RGB.} \texttt{MESA} halts the evolution once the secondary fills its Roche lobe (corresponding to the end point seen in Figure \ref{fig:evol}). If we continued to follow the evolution of this binary, we would see a subsequent phase of common-envelope evolution, resulting in the formation of a double-degenerate binary. The expected end product would thus be a 0.47 M$_\odot$ white dwarf (the sdB component) in close orbit with a lower mass helium white dwarf (the core of the giant secondary).
{\purple A third line in Figure \ref{fig:evol} shows the evolution of a representative low-mass progenitor star that produces an sdB in a wide orbit. The primordial binary consists of primary and secondary stars with masses of 1.2 and 1.1 M$_\odot$, respectively. Consistent with the models discussed in \citet{2020A&A...641A.163V}, the primary (donor) evolves up the RGB until it is close to the tip of the RGB (where a helium flash is expected to occur). The star then starts to lose mass very rapidly via Roche lobe overflow, with a mass-loss timescale of $\sim 10^{6}$ yr. After a series of flashes, it enters the sdB phase (as defined previously), which persists for $\approx 75$ Myr. The star then evolves through the sdO phase (annotated on the plot), before eventually cooling as a WD.
So what are the similarities and differences between the properties of the sdB binaries formed by these two channels? The masses of the two sdBs are virtually identical, and are close to the canonical mass for such stars (0.470 M$_\odot$ compared to 0.466 M$_\odot$). As seen in the HRD, both have very similar luminosities, effective temperatures, and thus radii. Their (time-averaged) central temperatures and densities (and thus fractional electron degeneracy pressures) are very similar during their respective sdB phases. Their sdB lifetimes persist for nearly 80 Myr (each), and even their hydrogen-rich envelopes have the same mass to within a factor of 2 ($\approx$0.0025 M$_\odot$ compared to $\approx$0.005 M$_\odot$). One difference relates to the percentage of electron degeneracy pressure at the center of the primary star preceding the sdB phase: The primary of the low-mass case has a largely degenerate core, while the intermediate-mass primary is only partially degenerate. However, soon after the start of the sdB phase, both are partially degenerate ($\approx 20$\% of the central pressure is due to electron degeneracy), and they evolve to become fully degenerate by the end of the sdB phase. The major difference between the two models, as expected, is in $P_{\rm orb}$. The intermediate-mass channel produced an sdB binary with $P_{\rm orb} \simeq$46 d, while the low-mass channel produced an sdB binary with $P_{\rm orb} \simeq$1395 d. Although there is a range of possible orbital periods according to both scenarios, it is fair to say that the low-mass case can produce a binary sdB that has a period approximately an order of magnitude larger than that of the intermediate-mass case.
Figure \ref{fig:lum_rad} provides a more detailed perspective with respect to the sdB and sdO evolutionary phases for the intermediate-mass case.} It shows the temporal evolution of the nuclear luminosities from the H-, He-, and C-burning channels, and the evolution of the surface luminosity{\footnote{Note that the sum of the nuclear luminosities may not equate to the surface luminosity. This difference is due to the gravothermal luminosity, whose magnitude is not shown.}} and radius. The high luminosities seen near $t^* = 0$ ($\equiv t - 211.25$ Myr) arise from the evolution of the primary star while it is still a red giant. After the star settles into the sdB phase, its radius remains approximately constant for $\approx$78 Myr. Initially, He-burning accounts for most of the luminosity, but as more and more carbon is created, $\alpha$-channel capture occurs, converting some of the carbon into oxygen. As both the luminosity and radius approach a local maximum, a convective core ($\approx 0.1$ M$_\odot $) of CO is formed, and the hot subdwarf thermally relaxes, leading to a brief phase of shell flashes. The subdwarf subsequently enters the sdO phase, during which time the radius is reasonably constant over $\approx 40$ Myr. Theoretically speaking, we can think of the sdB and sdO phases as being long-lasting ($>$ 10 Myr) and quasi-quiescent. The hallmark of the sdB phase is He-burning in the core; however, for the sdO phase, He-burning mainly occurs in a shell around the CO core.
{\purple For the intermediate-mass case, our models show that the mass of the A star companion (i.e., the secondary) could lie in the range of $\approx$1.8 to 1.9 M$_\odot$.} The main reason that the secondary masses are so large is that: (i) the intial mass ratio ($M_{2,0} / M_{1,0}$) must be sufficiently high at the onset of mass transfer ($\gtrsim 0.4$) to avoid dynamical instability (otherwise, this could lead to a merger); and, (ii) $M_{1,0}$ must be $\gtrsim 3$M$_\odot$ so that the primary has a chance to initiate core He-burning after departing the red giant branch.{\footnote {If $M_{\rm 1,0} \lesssim 3$M$_\odot$, the primary will evolve into a helium white dwarf and never undergo an sdB phase for the range of initial conditions that we considered.}} According to the main sequence models of \citealt{Eker_2015}, a 1.9-M$_\odot$ solar-metallicity model has a luminosity of about 15 L$_\odot$, $T_{\rm eff} \approx$\,8000\,K, and $R \approx 2$ R$_\odot$. Its inferred spectral type is A2.5V. These values are in line with the inferred values shown in Table \ref{table:mcmc-mle}. {\purple For the low-mass case, it appears that the constraints on the secondary's mass are less restrictive, but we would need to compute many more of these types of models before coming to a definitive conclusion.}
\section{Conclusions}
In this paper, we present strong evidence for the nature of TIC~5724661: a main-sequence A star with an inferred long-period sdBV$_{\rm r}$ star in orbit around it. First, we used radial velocity data to show that the putative hot subdwarf companion must have a period longer than a few tens of days. We then fit the spectral energy distribution using an MCMC code to constrain the parameters of the two stars in the system and provide fairly compelling evidence that this star is indeed an sdB star. To determine whether such sdBs can be produced using non-common-envelope formation channels, we modeled this system with MESA and demonstrated that we can readily produce such stars {\purple with either an intermediate- or a low-mass progenitor}, as long as there is a high degree of non-conservative mass transfer. We expect that the current A star in TIC~5724661 could have accreted perhaps $\sim 10\%$ of the mass lost from the sdB's progenitor, implying a final mass of 1.9 M$_\odot$. This would ensure that this star, which exhibits $\delta$-Scuti pulsations, is an A star (A2.5V). {\purple A more limiting constraint on the minimum value of $P_{\rm orb}$, perhaps via more spectroscopic observations, may serve to either strengthen or invalidate either of the intermediate-mass or low-mass progenitor binary models that we have proposed in Section \ref{sec:stel-evol}}.
Our work adds more observational evidence for intermediate- and long-period binaries that contain an sdB component (for earlier examples of such binaries, see, e.g., \citealt{2019CoSka..49..264V, 2020A&A...641A.163V}, and the references therein). Previously, many observed sdB binaries were found with extremely short periods and were therefore thought to have passed through a common-envelope phase. Our work lends observational evidence to the fact that such common envelopes are not at all necessary to form sdB stars (see, e.g., \citealt{Sen2019} for a set of intermediate-period sdB formation models), especially those found to pulsate in high-frequency p~modes.
Finally, this paper---which utilizes the TESS 20-second cadence data---further emphasizes the power of TESS for precision asteroseismology. The continuous, short-cadence nature of TESS data enables the detection and study of objects and pulsations that ground-based campaigns have, in the past, struggled to identify and characterize.\footnote{{\purple For an example of a ground-based photometric campaign to study high-frequency pulsations, such as those in TIC\,5724661, see, e.g., the Whole Earth Telescope \citep{1990ApJ...361..309N}}.} As a result, we were able to identify and derive a precise frequency estimate for the sdB pulsation frequency and, consequently, recognize the presence of a companion. Future work on this object will focus on long-term radial velocity monitoring of this star in order to better constrain the true period of this binary, {\purple as well as further modeling to break the degeneracy between the intermediate- and long-period cases}.
\section{Acknowledgments}
{\purple We thank the anonymous referee for pointing out several issues that motivated us to carry out additional analyses and clarify some of our discussions. We acknowledge Andrzej Baran, Michael Fausnaugh, and Jon Jenkins for helpful discussions regarding the TESS data and the locations of the pulsations.} We also thank St\'ephane Charpinet and JJ Hermes for helping us interpret our estimates for the $T_{\rm eff}$ of the sdB star. L.\,N.~thanks the Natural Sciences and Engineering Research Council (Canada) for financial support through the Discovery Grants program. G.\,H.~acknowledges support by the Polish NCN grants 2015/18/A/ST9/00578 and 2021/43/B/ST9/02972.
This paper includes data collected by the TESS mission, specifically through Director's Discretionary Target programs \#22 (PI: Antoci) and \#46 (PI: Jayaraman). Funding for TESS is provided by NASA's Science Mission Directorate. {\purple Resources used in this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products.} Some computations were carried out on the supercomputers managed by Calcul Qu\'ebec and Compute Canada. The operation of these supercomputers is funded by the Canada Foundation for Innovation (CFI), and the Fonds de recherche du Qu\'ebec -- Nature et technologies (FRQNT).
This work has used data from the European Space Agency (ESA) mission {\em Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\em Gaia} Data Processing and Analysis Consortium (DPAC, \url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for DPAC has been provided by national institutions, in particular those party to the Gaia Multilateral Agreement.
We would like to acknowledge the Indigenous Peoples as the traditional stewards of the land on which part of this research was conducted, and the enduring relationship between them and their traditional territories. We acknowledge the painful history of genocide and forced occupation of their territory, and we honor and respect the many diverse Indigenous people connected to this land on which we research and gather.
{\purple Code and inlists used for our MESA analysis are available on Zenodo, at this\,\dataset[link]{https://zenodo.org/record/6668950\#.YrLI5HbMKUk}.}
\facilities{TESS, Gaia, FLWO:1.5m}
\software{SPOC \citep{jenkinsSPOC2016}, {\tt numpy} \citep{harris2020array}, {\tt matplotlib} \citep{Hunter:2007},
{\tt scipy} \citep{2020SciPy-NMeth}, {\tt astropy} \citep{astropy:2013, astropy:2018}, {\tt pysynphot} \citep{2013ascl.soft03023S},
{\tt pandas} \citep{reback2020pandas, mckinney-proc-scipy-2010}}, {\tt SPECTRUM} \citep{1999ascl.soft10002G}, {\tt MESA} \citep{2011ApJS..192....3P, 2013ApJS..208....4P, 2015ApJS..220...15P, 2018ApJS..234...34P, 2019ApJS..243...10P}
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{
"redpajama_set_name": "RedPajamaArXiv"
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package sbt
import java.io.{ File, FileNotFoundException }
import sbinary.{ DefaultProtocol, Format, Operations }
import scala.reflect.Manifest
object CacheIO {
def toBytes[T](format: Format[T])(value: T)(implicit mf: Manifest[Format[T]]): Array[Byte] =
toBytes[T](value)(format, mf)
def toBytes[T](value: T)(implicit format: Format[T], mf: Manifest[Format[T]]): Array[Byte] =
Operations.toByteArray(value)(stampedFormat(format))
def fromBytes[T](format: Format[T], default: => T)(bytes: Array[Byte])(implicit mf: Manifest[Format[T]]): T =
fromBytes(default)(bytes)(format, mf)
def fromBytes[T](default: => T)(bytes: Array[Byte])(implicit format: Format[T], mf: Manifest[Format[T]]): T =
if (bytes.isEmpty) default else Operations.fromByteArray(bytes)(stampedFormat(format))
def fromFile[T](format: Format[T], default: => T)(file: File)(implicit mf: Manifest[Format[T]]): T =
fromFile(file, default)(format, mf)
def fromFile[T](file: File, default: => T)(implicit format: Format[T], mf: Manifest[Format[T]]): T =
fromFile[T](file) getOrElse default
def fromFile[T](file: File)(implicit format: Format[T], mf: Manifest[Format[T]]): Option[T] =
try { Some(Operations.fromFile(file)(stampedFormat(format))) }
catch { case e: Exception => None }
def toFile[T](format: Format[T])(value: T)(file: File)(implicit mf: Manifest[Format[T]]): Unit =
toFile(value)(file)(format, mf)
def toFile[T](value: T)(file: File)(implicit format: Format[T], mf: Manifest[Format[T]]): Unit =
{
IO.createDirectory(file.getParentFile)
Operations.toFile(value)(file)(stampedFormat(format))
}
def stampedFormat[T](format: Format[T])(implicit mf: Manifest[Format[T]]): Format[T] =
{
import DefaultProtocol._
withStamp(stamp(format))(format)
}
def stamp[T](format: Format[T])(implicit mf: Manifest[Format[T]]): Int = typeHash(mf)
def typeHash[T](implicit mf: Manifest[T]) = mf.toString.hashCode
def manifest[T](implicit mf: Manifest[T]): Manifest[T] = mf
def objManifest[T](t: T)(implicit mf: Manifest[T]): Manifest[T] = mf
}
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{
"redpajama_set_name": "RedPajamaGithub"
}
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«Двери восприятия» () — эссе английского писателя и философа Олдоса Хаксли (1954), описывающего свой опыт употребления мескалина. Эссе послужило толчком к массовому изучению «расширения границ восприятия» под действием психоделиков, оказав таким образом значительное влияние на социальные и культурные процессы второй половины XX века. Неоднократно издавалась на русском языке.
Краткое содержание
В качестве эпиграфа писатель взял строки из поэмы «Бракосочетание Рая и Ада» английского поэта XVIII века Уильяма Блейка: «». В русском переводе Максима Немцова (1991) эта фраза звучит так: «Если бы двери восприятия были чисты, всё предстало бы человеку таким, как оно есть — бесконечным».
Эссе написано в форме воспоминания Хаксли о своих ощущениях после приёма приблизительно 0,4 грамма мескалина в 11:00 одним из майских дней 1953 года. Оно начинается с краткого обзора истории открытия и изучения мескалина и некоторых сходных по своему действию галлюциногенов. Далее, после рассуждений о влиянии имеющегося у человека опыта на его восприятие окружающей действительности: «Мы можем собирать информацию об опыте, но никогда не сам опыт», Хаксли переходит к описанию собственных ощущений, вызванных действием психоделика: «Я видел то, что видел Адам в утро своего сотворения — мгновение за мгновением, чудо обнажённого существования». Хаксли отмечает усиление восприятия цветов, искажение пространственной перспективы, исчезновение понятия времени… Все окружающие предметы предстают перед ним в своей первозданной «есть-ности», в мельчайших деталях, оторванные от связанных с ними понятий, бесконечные в своей значимости. Он употребляет термины «сакраментальное видение реальности», «Внутренний Свет».
Далее автор переходит к описанию психо-физиологических аспектов воздействия наркотика, вводя понятие мозгового редуцирующего клапана. По его мнению, поток информации, поступающей в мозг, ограничивается этим клапаном, мескалин же помогает открыть его, заставляя мозг воспринимать окружающее во всей полноте своей «таковости», позволяя принявшему его увидеть мир глазами Ван Гога или Вордсворта. Хаксли проводит параллели между изменением восприятия от приёма мескалина и визионерским восприятием, свойственным некоторым художникам и религиозным деятелям. После подробного описания своих ощущений Хаксли переходит к рассуждениям о возможности замены алкоголя и табака (вред от которых очевиден) на мескалин : «Для большинства людей мескалин почти безвреден. В отличие от алкоголя, он не ведёт принимающего его ни к каким неконтролируемым действиям, которые оканчиваются драками, насилием и дорожными происшествиями. Человек под воздействием мескалина тихо занимается своим делом».
В заключении Хаксли пишет, что мескалин не необходим, но полезен, особенно для интеллектуала, который может стать жертвой слов и символов. Несмотря на важность систематических рассуждений, прямое восприятие, по его мнению, не менее важно. Наконец, Хаксли утверждает, что человек, попробовавший мескалин, изменится к лучшему.
Критика Захнера
Одним из первых критиков «Дверей восприятия» был , профессор Оксфордского университета. В своей книге «» Захнер признаёт важность книги для людей, увлечённых изучением религий, но указывает на непоследовательность и внутренние противоречия.
Захнер заключает, что восприятие Хаксли под мескалином подвержено его познаниям в буддизме. Таким образом, опыт может быть другим для людей, принявших наркотик не имея такого культурного фона, из чего следует, что они испытают сенсационную трансформацию.
Критике подверглось также найденное Захнером (по крайней мере, как ему показалось) утверждение Хаксли об использовании наркотиков (включая алкоголь) всеми религиями в своих ритуалах.
Констатация схожести ощущений от приёма мескалина, маниакального синдрома и явлений Бога святым наводит Захнера на мысль, что святые визионеры ничем не должны отличаться от лунатиков. Однако, утверждает Захнер, этот опыт отличается от религиозно-мистического опыта человека, поглощённого Богом, совершенно отличного от объективного мира.
Приложения к «» включают три описания опыта приёма мескалина, включая опыт самого Захнера. Он пишет, что перенёсся в мир бессмысленности и фарса, и отмечает, что опыт был интересный и весёлый, но не религиозный.
Ответ Хаксли и Хьюстона Смита
Хаксли ответил Захнеру в статье, опубликованной в 1961 году: «Для большинства из тех, кто имел подобный опыт, его значение самоочевидно. Для доктора Захнера, автора Mysticism, Sacred and Profane, их преднамеренное введение расценивается как аморальное. На что его коллега, профессор Прайс, отвечает: "Говорите за себя!"».
Профессор религии и философии Хьюстон Смит утверждает, что субстанции, изменяющие сознание, были связаны с религиями по всему миру и на протяжении всей истории, и также, возможно, что многие религии ведут от них своё происхождение, что впоследствии забывается.
Осознавая, что личность, подготовка и окружение — всё играет роль в эффекте от наркотика, Хьюстон Смит обращает внимание на то, что религиозные переживания от подобного опыта не ограничиваются предшествующими религиозными познаниями Хаксли. Из того, что опыт Захнера не был религиозным, ещё не следует, что этого не может быть вообще.
В массовой культуре
Книга Хаксли вдохновила молодого Джима Моррисона назвать свою группу The Doors.
Греческий композитор Kyriakos Sfetsas основал свои «Four pieces for two pianos» (1986) на «The doors», с соответствующими названиями для каждой части.
Стивен Кинг в «Тёмной башне» описал мескалин, а именно употребление его главным героем книги, как часть магического ритуала (Книга 1 — Стрелок). В последующем главный герой проходит через двери между мирами, которые только он может открыть (Книга 2 — Извлечение троих).
Именно эту книгу читает в автобусе автор комиксов Marvel — Стэн Ли — в фильме «Доктор Стрэндж».
Примечания
Литература
Ссылки
Эссе 1954 года
Эссе на английском языке
Произведения Олдоса Хаксли
Литература о наркотиках
Психоделическая литература
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{
"redpajama_set_name": "RedPajamaWikipedia"
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Tolentino rises at 224 meters of height on the left bank of the Chienti river in the territory of Macerata and not far away from the Park of the Sibillini Mounts.
The territory of Tolentino turns out to be inhabited since the Paleolithic but it is with the Picens that can be dated the beginning of its recent history (VI sec. b.C.). Of this period have been recovered numerous tombs.
With the battle of Sentino (295 a.C.) and the defeat of the Italic Union Tolentino passed under the infuence of Rome becoming Municipium with the name of Tolentinum. After a period of development in Imperial age the city was victim of the barbaric invasions and subjected before to the power of the Church that made of it a episcopal center and then to the Longobard Reign (752).
Returned to the Roman Church after the invasion in Italy by the Franks, Tolentino flourished again in the XI century becoming free Comune (Municipality - XII sec.) and knowing a period of economic development.
It passed therefore under the dominion of powerful nobles families as the Da Varano and the Sforza.
Returned again under the administration of Rome, the city was the theatre of the advent of the Napoleoniche troops and their definitive defeat: in 1797 in fact was signed here the Treaty of Tolentino between Napoleon Bonaparte and Pio VI, dealt that imposed heavy custom offices to the Church and determined the end of the Papal State. The Napoleonic parenthesis was closed in 1815 (2 and 3 May), when the troops of the king of Naples Gioacchino Murat, Napoleon's brother-in-law, fought and lost with the Austrians troops nearby the Rocca della Rancia (Rancia Fortress) decreeing the end of the French dominion and the re-anexation of Tolentino to the papal territory until the moment of the unification of Italy (1860).
Tolentino distinguished itself during the Second World war in the fight against the Nazis becoming one of the Italian Cities decorated to the Military Valor during the Liberation War.
Between monuments of Tolentino not to be missed a visit to the Basilica di San Nicola da Tolentino with Agostinian convent and Laurenzian library and to the beautiful Rancia's Castle, quadrangular shaped with town-walls with battlements and three angular towers, today seat of the Archaeological Museum.
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{
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Get Creative With Lenses For Unique Results - Part One
Adam Duckworth
Get Creative With Lenses For Unique Results – Part One
The moderate wideangle to short telephoto zoom lens is the bedrock of most modern photography. Supplied either as a kit lens with a DSLR or as one of the first lenses many people buy when getting into photography, these lenses are do-all workhorses. Reasonably wide at the wide end for landscapes or just to squeeze everything in, and slightly telephoto at the long end to get you a bit closer to the subject, they are the most popular choice of lens worldwide. They are plentiful, relatively inexpensive and can give amazingly sharp results. They've largely taken over from the fast 50mm f/1.8 prime as the lens most people own and use the majority of the time.
And, of course, lens designers know just what they are looking for when making lenses like this. They want maximum sharpness across the frame, with as little darkening in the corners as possible and no nasty colours or aberrations. After all, since the dawn of photography, image makers have been on a quest for bright, distortion-free lenses to give their images a certain bite and sharpness.
Master photographers even avoid using their lenses at the extremes of aperture – full open and full closed down – because the images can look slightly soft due to the effect of diffraction.
But for more creative results, it doesn't have to be this way. Using more extreme lenses – from different focal lengths to tilt/shift optics and more – can bring a new look to your shots. Just because a lens is designed specifically for one thing doesn't mean you can't use it for another. Or even break the rules on how the lens is supposed to be used in term of subject mater or viewpoint. And in this way, you can often get results that are just that bit out of the norm.
Spurred on by the popularity of Instagram and lots of retro-themed photo manipulation packages, some enterprising manufacturers are raiding the archives for vintage lenses that they are remaking to fit modern cameras. This is helped by the new wave of mirrorless cameras that can accept many different types of lenses via adapters that can be then focused using the live view screen on the back of the camera. And with the instant playback of digital photography, it's easy to see your results straight away which can always be a driver of creativity.
While many people start with a kit zoom, quite often the next lens that's on the purchase list is something wider – either to get more into the frame or for landscape use. And once you are into the territory of ultrawide angle lenses – such as 18mm or shorter focal length on a full-frame camera – then there is a whole new world of creative opportunities. The best of these sort of lenses are designed to offer a "corrected" perspective, which means that straight lines at the edges are kept as straight as possible rather than bowing out like you are looking at things through a fishbowl.
The "rules" for using these ultrawide angle lenses are to try to keep them relatively straight, so that the horizon doesn't bend too much. And always make sure you fill the foreground of the shot with something, or else the photo can look very empty. But most of all, don't get too close to your subject as the lens will just really distort the perspective – especially for portraits. Those are the unwritten rules, but they are meant to be broken!
By using an ultrawide angle lens right up close to a subject's face, it's true that it isn't the most flattering look as the nose can appear very big. But especially with a burst of flash to life the shadows, it can give a really unique perspective and a fun look for the right subject – such as our muddy motorbike man.
Muddy face
And by getting the subject so big in the frame, and using a wide aperture, the background of the shot is less dominant and sharp – really putting attention onto the face of the subject.
It's this exaggeration of parts of the subject that are close to the lens while making things further away look much smaller that also works for the shot of the blue .
By having a 14mm lens low and very close to the rear wheel, the wheel itself looks huge – especially compared to the front wheel which is further away. The car might be distorted, but using the perspective allows the swooping blue lines of the car to contrast with the blue sky for a more graphic shot. And the ultra wide lens means the buildings in the background of the shot are rendered so tiny as to not take any attention away from the car. It's about getting rid of clutter and making a simple, graphic shot.
And distorting a car is also why an ultra wide lens was used for this shot of a gull-wing Bristol.
Bristol car
The low angle and 14mm lens exaggerates the length of the front of the car, and really highlights the gull-wing door against the sky. Using an ultrawide lens might not make a car look like its designer intended it to look, but the perspective can really work in making shots that are a bit more dramatic.
While ultrawide lenses are often corrected to prevent straight lines at the edges of the frame from bending, some lenses are made so that they do exactly that. These are often called fisheye lenses, and they give a trademark look to any shot. They can offer as wide as a 180-degree view, depending on the focal length.
While many photographers try to use fisheye lenses by carefully framing their images to reduce the bending effect of any straight lines in the image, it can be a good idea to do the exact opposite! By getting in close to a subject, and making sure there are parts of the image that have straight lines in them at the edge of the frame rather than an expanse of sky, for example, then the effect and therefore impact of using a fisheye can really be seen. Like in the shot of our ice cream seller in her little booth.
Ice Cream Lady
And with a fisheye having such a wide angle of view, if the lens is tilted downwards then the horizon can bend – which is often not a great thing. But you can exploit it! For this shot of a motorbike race start, a camera with fisheye lens was used on a boom over the top of some of the riders nearest the camera, and aimed slightly downwards.
Race Start
The horizon may not look natural now as it's curved, but it adds to the unique perspective and impact of the shot. Rules are meant to be broken!
At the other end of the scale to the super-wide lenses are the long telephotos, usually in focal lengths of 300mm or more on a full-frame camera. To many, these lenses are simply used to get you closer to the subject when it physically wouldn't be possible to get there. Such as when photographing wildlife or a speeding car coming around a corner for example.
And fast lenses – such as the bright f/2.8 primes in 300 and 400mm versions – are often used wide open to get a high enough shutter speed to freeze the action.
But one of the key creative controls of such long lenses is to use the flattening perspective, to make the subject and background look like they are much closer together. Because a telephoto lens magnifies the background, as well as the foreground, shots of crowds can look more impressive.
And the far-away viewpoint really can change the look of any shot.
But it's this magnifying of the background, compared to the subject, that can make or break a shot. If the background is messy, it can really take the attention away from the subject. That's why fast primes, used wide open for super-shallow depth of field, can be used to really put sharp focus on the subject and make the background disappear into some far softer, out of focus and less distracting.
Softer Background
It can be much trickier to get the subject in focus when using a lens wide open, but the results can be worth it.
It's the ability to use a fast aperture lens wide open that often draws people to fast, prime short telephoto lenses like Canon's 85mm f/1.2 or Nikon's 85mm f/1.4. When used wide open at very close distances to a portrait sitter, then the depth of field can be so shallow that the eyes can be in focus but the end of the nose can be way out. That's not usually a good thing, but why not give it a try!
Copper girl
The purists may not like it, but it can give a softer and more dreamy affect to your natural light portraits and really blur out the background.
Of course, that's fine for natural light but in a studio, lenses like these are usually used in middle aperture settings to get maximum sharpness and also because the studio lights are too powerful to allow use of such fast apertures. Studio lights can only be turned down a certain extent, but by using neutral density filters on a fast portrait lens, it means wide f/1.4 or f/1.8 apertures can be used. This gives a unique look to the subject as their face can be sharp but their hair and even their shoulders can be soft and more diffuse. It's a unique look you can only get with a fast lens used wide open.
Get Creative With Lenses For Unique Results – Part Two
Adam DuckworthOther articles by author
I'm Adam Duckworth, an award-winning professional photographer and videographer based in the UK. I live in Northamptonshire but travel nationally and internationally for work.
I have worked for many top magazines, newspapers and corporate clients for more than 25 years and am available for commercial, advertising and editorial commissions.
I was named SWPP UK Commercial Photographer of the Year, I'm an Associate of the British Institute of Professional Photography and have been exhibited in Tokyo and London.
Clients I have worked for include Red Bull, Monster Energy, Suzuki, Yamaha, Honda, Kawasaki, the Royal Horticultural Society, the BBC, the National Youth Arts of Wales, the University of Glamorgan, Manfrotto, Lastolite and many more.
I have also worked for international publications like Motor Cycle News, Racer X, The Sunday Times, The Guardian, ZOO, Golf World, Today's Golfer, Mountain Bike Action, EVO, Octane, Boards and Scoot Nation, among others.
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Melissa Signature Styles – Bringing You the Hair Style That's Right for You
It all starts with rapport for expert stylist Melissa Pierce at Salon & Spa Galleria in Arlington, TX
Five dollars off for first-time clients, Melissa
ARLINGTON, TX, October 06, 2022 /24-7PressRelease/ -- Mothers and their daughters. The experiences they share. Like a mom who knows how to do her daughter's hair.
Not so, in the experience of Melissa Pierce, who smiles fondly at the memory – for it happens to have been the jumping-off point for her thriving career today at Melissa Signature Styles inside Salon & Spa Galleria Sublett Road in Arlington-Mansfield.
"My mother wasn't able to do my hair," recalls Melissa, "so I had to learn how to do my own hair. I received lots of compliments, and by the time I was 13, I was doing hair for my family and friends."
Suffice to say, it elicited a pretty great response from fam and friends. "It was true of me back then and still is today," she says. "I really enjoy helping people look their best and feel the confidence that comes with that."
Following her attendance at Jones Beauty College in Grand Prairie, Texas, Melissa Pierce went on to become a licensed stylist in her 20s. During the 25 years she's now thrived in her career in the beauty business, she's owned two salons, and has also been an instructor for Paul Mitchell.
Her signature specialties? Short haircuts, weaves, cuts, and colors. That said, she's also quick to point out, "People come to me for more than hair. I build rapport with each client and become a trusted resource for my clients. I strive to provide one-on-one personal service, and it's always my aim to foster a client's personal self-image."
What makes the experience at her Melissa Signature Styles different from other salons in Mansfield TX?
"Rapport and creativity," enthuses Melissa. "There is an art to doing hair and I like to think that I put my own signature on each client I serve by giving them a look that fits their lifestyle. I am very businesslike and really make sure that I am punctual and get my clients in and out of the salon so that they can get back to their busy schedules."
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Melissa Pierce
Melissa Signature Styles
Website: Visit Our Website
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Tadeusz Walenty Pełczyński (codenames: Grzegorz, Adam, Wolf, Robak; Warsaw, 14 February 1892 – 3 January 1985, London) was a Polish Army major general (generał brygady), intelligence officer and chief of the General Staff's Section II (the military intelligence section).
During World War II, he became chief of staff of the Home Army (ZWZ, Armia Krajowa; July 1941 – October 1944) and its deputy commander (July 1943 – October 1944).
Early life and education
Tadeusz Pełczyński was the son of Ksawery Pełczyński, a Sanniki sugar-mill technician, and Maria, née Liczbińska, a teacher, and was a great-grandson of Michał Pełczyński, a general in the Army of Congress Poland.
Pełczyński began school in Łęczyca. In 1905 he participated in a school strike connected with Polish efforts to win independence from the Russian Empire. He continued his schooling in Warsaw at the Gen. Paweł Chrzanowski Gymnasium. In 1911 he began medical studies at Kraków University. As a medical student he was a member of the patriotic-gymnastic Sokół organisation and of the "Zet" Polish Youth Association (Związek Młodzieży Polskiej "Zet"). He completed a military course conducted by Zygmunt Zieliński, a future Polish Army generał broni (lieutenant general).
Marriage and family
In 1923 Pełczyński married Wanda Filipowska, with whom he had a daughter, Maria, and a son, Krzysztof (Christopher, born 1924, who died during the Warsaw Uprising on 17 August 1944, of wounds sustained on 1 August, the first day of the Uprising).
World War I
The outbreak of World War I in August 1914 found Pełczyński on vacation near Włocławek. After the area had been occupied by the Germans, he was mobilised by them to work as a medic at a Russian-prisoner-of-war camp.
After his release from German service, in June 1915 he joined the Polish Legions. He served as an officer in the 6th Legions Infantry Regiment (6 Pułk Piechoty Legionów) and commanded a platoon and a company. In July 1917, following the Oath Crisis, he was interned at a camp in Beniaminów. In March 1918, after release from internment, he took up work at a social-services agency (Rada Główna Opiekuńcza) while continuing his involvement with "Zet."
Interwar period
In November 1918 Pełczyński was accepted into the Polish Army and placed in command of a company, then a battalion, of the 6th Legions' Infantry Regiment. In March 1920 he was transferred to the Infantry Officer-Cadet School (Szkoła Podchorążych Piechoty) in Warsaw as a company commander, then a battalion commander. From September 1921 to September 1923 he attended the War College (Wyższa Szkoła Wojenna) in Warsaw. After graduating with a General Staff officer's diploma, he returned to the Infantry Officer-Cadet School as a battalion commander.
In July 1924 he was posted to the Office of the Inner War Council (Ścisła Rada Wojenna). In May 1927 he began service in the Second Department of Polish General Staff (the intelligence section) as chief of the Information Department (Wydział Ewidencyjny). In January 1929 he was appointed chief of Section II. From March 1932 to September 1935 he commanded the 5th Legions' Infantry Regiment (5 Pułk Piechoty Legionów) in Wilno (it was part of the elite 1st Legions Infantry Division), then returned to again head Section II.
As chief of the Second Department of Polish General Staff, Pełczyński, like his predecessor Colonel Tadeusz Schaetzel and like deputy chief Lt. Col. Józef Englicht, was very supportive of Marshal Józef Piłsudski's Promethean project, aimed at liberating the non-Russian peoples of the Soviet Union.
Pełczyński was the longest-serving prewar chief of the Second Department (1929–32, and 1935 – January 1939). In January 1939 he was relieved of this post and placed in command of the 19th Infantry Division (19 Dywizja Piechoty), stationed in Wilno. His tenure as chief of Section II had reportedly been ended by his wife Wanda's political activities against Marshal Edward Śmigły-Rydz and General Felicjan Sławoj-Składkowski.
World War II
Pełczyński may have made his greatest contribution to Allied victory in World War II well before the opening of hostilities, when he proposed giving Polish knowledge of the German Enigma machine to the French and British. According to Colonel Stefan Mayer, "From Gen. Pełczyński, now resident in Great Britain, I know that... he suggested [to the chief of the Polish General Staff, General Wacław Stachiewicz] that in case of [impending] war the Enigma secret... be used as our Polish contribution to the common... defence and divulged to our future allies. [Pełczyński] repeated [this] to Col. Józef Smoleński when in [the] first days of January 1939 [Smoleński] replaced [him] as... head of [Section II]. That was the basis of [Lt. Col. Langer]'s instructions... when he... represent[ed] the Polish side at the [Paris] conference... in January 1939 and then in Warsaw in July 1939.
The Poles' gift, to their British and French allies, of Enigma decryption at Warsaw on 26 July 1939, just five weeks before the outbreak of the war, came not a moment too soon, as it laid the foundations for later British cryptographic breakthroughs that produced the Ultra intelligence that was a key factor during the war. Former Bletchley Park mathematician-cryptologist Gordon Welchman later wrote: "Ultra would never have gotten off the ground if we had not learned from the Poles, in the nick of time, the details both of the German military... Enigma machine, and of the operating procedures that were in use."
After the outbreak of war, from 5 September 1939, Pełczyński commanded a force in the rears of the invading German Wehrmacht.
After the conclusion of the September Campaign, he went to Warsaw to take up underground work with the Service for Polish Victory (Służba Zwycięstwu Polski), then with the Union for Armed Resistance (Związek Walki Zbrojnej, or ZWZ) and the Home Army (Armia Krajowa, or AK).
From July 1940 to April 1941 he commanded the Lublin ZWZ district. As the local Gestapo were closing in, he returned to Warsaw and accepted the post of chief of staff of ZWZ (July 1941). From July 1943, he was also Home Army deputy commander. In November 1943, he was promoted to major general (generał brygady).
He commanded sabotage operations carried out by Kedyw units against the German war machine (including disruption of several rail lines). He took part in the decision to begin the Warsaw uprising.
Five weeks into the Warsaw Uprising, on 4 September 1944, Pełczyński was gravely wounded when the PKO savings-bank building on Świętokrzyska Street was bombed, and as a result he could no longer carry on the duties of Home Army chief of staff.
After the suppression of the Uprising, Pełczyński was imprisoned by the Germans at the Langwasser camp, then at Colditz.
Later years
Following his liberation by the Allies in 1945, he made his way to London in England.
Decorations
Gold Cross of the Order of Virtuti Militari
Silver Cross of the Order of Virtuti Militari
Cross of Independence
Officer's Cross of the Order of Polonia Restituta
Cross of Valor (thrice)
Gold Cross of Merit
Home Army Cross
See also
Biuro Szyfrów
Edmund Charaszkiewicz
Prometheism
List of Poles
Notes
References
Edmund Charaszkiewicz, Zbiór dokumentów ppłk. Edmunda Charaszkiewicza (A Collection of Documents by Lt. Col. Edmund Charaszkiewicz), opracowanie, wstęp i przypisy (edited, with introduction and notes by) Andrzej Grzywacz, Marcin Kwiecień, Grzegorz Mazur, Kraków, Księgarnia Akademicka, 2000, .
Edmund Charaszkiewicz, "Referat o zagadnieniu prometejskim" ("Report on the Promethean Question"), in Zbiór dokumentów ppłk. Edmunda Charaszkiewicza (A Collection of Documents by Lt. Col. Edmund Charaszkiewicz).
Andrzej Grzywacz et al., introduction to Zbiór dokumentów ppłk. Edmunda Charaszkiewicza (A Collection of Documents by Lt. Col. Edmund Charaszkiewicz), pp. 5–27.
Tadeusz Kryska-Karski and Stanisław Żurakowski, Generałowie Polski Niepodległej (The Generals of Independent Poland), 2nd ed., Warsaw, Editions Spotkania, 1991, p. 144.
Wojciech Baliński, "Gen. bryg. Tadeusz Pełczyński (1892–1985). Przywracani pamięci" ("Major General Tadeusz Pełczyński (1892–1985)—Restored to Memory"), in Polska Zbrojna (Armed Poland).
Waldemar Strzałkowski, "Ci, co wierzyli w Polskę. Powrót prochów generała Pełczyńskiego" ("Those Who Believed in Poland: the Return of General Pełczyński's Ashes"), in Polska Zbrojna (Armed Poland), no. 215, 1995.
Zbigniew Mierzwiński, Generałowie II Rzeczypospolitej (The Generals of the Second Republic), Warsaw, Wydawnictwo Polonia, 1990, (83-7021-141-0), pp. 209–213.
Lista starszeństwa oficerów Legionów Polskich w dniu oddania Legionów Polskich Wojsku Polskiemu (12 kwietnia 1917) [Seniority List of Officers of the Polish Legions on the Day of the Polish Legions' Transfer to the Polish Army (12 April 1917)], Warsaw, 1917, p. 18.
Władysław Kozaczuk, Enigma: How the German Machine Cipher Was Broken, and How It Was Read by the Allies in World War II, edited and translated by Christopher Kasparek, Frederick, MD, University Publications of America, 1984, .
Gordon Welchman, The Hut Six Story: Breaking the Enigma Codes, New York, McGraw-Hill, 1982.
External links
http://www.powstanie-warszawskie-1944.ac.pl/biog_pelczynski.htm
1892 births
1985 deaths
Military personnel from Warsaw
People from Warsaw Governorate
Polish generals
Polish intelligence officers
Association of the Polish Youth "Zet" members
Polish legionnaires (World War I)
Polish people of the Polish–Soviet War
People of the Polish May Coup (pro-Piłsudski side)
Home Army officers
Warsaw Uprising insurgents
Polish emigrants to the United Kingdom
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\section{Introduction}
Ultraluminous X-ray sources (ULXs) are described as off-nucleus objects with high X-ray luminosities in the range of 10$^{39}$ $-$ $10^{41}$ erg s$^{-1}$ in external galaxies (see the review by \citealp{2017ARA&A..55..303K}). For most, their high luminosities are thought to be produced by super-Eddington accretions onto stellar mass black holes or neutron stars \citep{2013MNRAS.432..506P, 2013MNRAS.435.1758S, 2013ApJ...778..163B, 2017Sci...355..817I}. For some, sub-Eddington accretions onto intermediate mass black holes (IMBH) are possible \citep{1999ApJ...519...89C, 2001ApJ...551L..27M, 2015MNRAS.448.1893M}. Actually, numerous studies have examined the nature of the ULX systems. ULX X-2 of M82 with its coherent pulsations was the first neutron star in such systems \citep{2014Natur.514..202B}. Since then, the number of known pulsating ultraluminous X-ray sources (PULXs) has risen to 6 \citep{2016ApJ...831L..14F, 2017Sci...355..817I, 2017MNRAS.466L..48I, 2018MNRAS.476L..45C, 2019MNRAS.488L..35S, 2019arXiv190604791R}. In addition, a cyclotron resonance scattering feature in the X-ray spectrum of the source ULX-8 in M51 was discovered by \cite{2019MNRAS.486....2M} assuring an important hint about the nature of the magnetic field in ULXs that hosting neutron stars.
High quality data from {\it XMM-Newton} and {\it Chandra} revealed that the spectra of ULXs are different from those observed in the Galactic black hole binaries (GBHBs). A notable feature in the spectra of most ULXs is an unambiguous curvature in its spectrum between 3 to 7 keV. This curvature can be produced through Comptonization of a cold, optically thick corona around the compact object or alternatively in the inner regions of a geometrically thick accretion disk \citep{2006MNRAS.368..397S,2007Ap&SS.311..203R,2009MNRAS.397.1836G}. Based on these spectral features, a new ultraluminous accretion state with super-Eddington accretion occurring onto a black hole or a neutron star has been proposed by \cite{2009MNRAS.397.1836G}. The extended high energy interval of {\it NuSTAR} observations of ULXs clarified that the curvature was extending above 10 keV. This feature is also different from GBHB, whose cutoff energy is over 60 keV \citep{2013ApJ...778..163B,2016MNRAS.460.4417L,2017xru..conf..314P,2017ApJ...834...77F}.
Determining the optical counterparts of ULXs plays an important role to understand the origin of emission (i.e., accretion disk and/or donor star) and to estimate the mass, age and spectral type of the donor.
However, identification of the optical counterparts of ULXs is difficult due to large extragalactic distances to them, some of which exhibiting a disk irradiation in a binary system.
The observed optical candidates of ULXs are too faint (m$_{V}$ > 21 mag.) to study in detail from HST images and/or confirm spectroscopically from the ground-based observations \citep{2008A&A...486..151G, 2015MNRAS.452.1112E, 2015NatPh..11..551F,2017ARA&A..55..303K,2019ApJ...877...57Q}. However, there are several exceptions in which the type and mass of the donor star are determined by optical spectroscopy; for instance: ULX-1 of M101 has a Wolf-Rayet donor \citep{2013Natur.503..500L} while ULX-1 of NGC 253, ULX-2 of NGC 925 and ULX-1 of NGC 4136 \citep{ 2015MNRAS.453.3510H, 2016MNRAS.459..771H} have M-type supergiant donors and P13 of NGC 7793 it is a PULX has a blue supergiant donor of type B9Ia \citep{2011AN....332..367M,2014Natur.514..198M}.
In general, absolute magnitudes of ULXs, M$_{V}$, are in the range of $-$8 and $-$3 mag. \citep{2018ApJ...854..176V}. Although there are hundreds of ULXs known, only about 20 have singled out optical counterparts \citep{2011ApJ...737...81T,2013ApJS..206...14G,2019ApJ...875...68A}. For many other ULXs, multiple undecidable optical counterparts have been detected \citep{2005MNRAS.356...12S,2007ApJ...658..999M,2016ApJ...828..105A,2019ApJ...875...68A,2019MNRAS.488.5935A}. Mostly, they have very faint optical candidates showing star-like spectral energy distributions (SEDs). Their SEDs are constructed by assuming that the optical emissions come from donor stars \citep{2012ApJ...745..123G, 2016MNRAS.455L..91A,2018ApJ...854..176V}.
NGC 1316 (also known as Fornax A) is a giant elliptical galaxy located in the outskirts of the Fornax galaxy cluster. In many studies, the distance of NGC 1316 was estimated with slightly different values from each other \citep[e.g.][]{2003ApJ...583..712J,2007ApJ...657...76F,2010AJ....140.2036S,2013A&A...552A.106C,2018ApJ...866..145H,2018MNRAS.481.4472L,2019ApJ...887..149B}. Throughout this work, we adopted the distance to the NGC 1316 as 19 Mpc \citep{2003ApJ...583..712J}. In the radio band, NGC 1316 is one of the brightest galaxy in the sky, with giant double radio lobes and well defined core jet structure \citep{1984AJ.....89.1650G}. The galaxy has a peculiar morphology with numerous tidal tails, shells and loops of interstellar medium \citep{1980ApJ...237..303S, 1981ApJ...246..722S}. NGC 1316 also displays dust patches in the central region together with the prominent dust lanes oriented along its optical minor axis (see Fig. \ref{F:1}).
In the previous studies, it is reported that the galaxy hosts many X-ray sources including several ULXs by using the archival {\it Chandra} and {\it XMM-Newton} data \citep{2003ApJ...586..826K,2004ApJS..154..519S,2011ApJS..192...10L,2019MNRAS.483.5554E}. Based on {\it Chandra} observation of 2001, \citet{2003ApJ...586..826K} detected 81 point sources within the 25th magnitude isophotal ellipse of NGC 1316. According to their source list, they reported only one potential ULX source (\#29) even though three more sources (\#12, \#13 and \#44) have luminosities $\geq$ 10$^{39}$ erg s$^{-1}$. They also reported that the luminosity of the \#13 is highly uncertain due to the contamination of the X-ray diffuse emission and remaining two sources are located at the center of galaxy with elongated shapes and uneasy source definitions. On the other hand, \cite{2011ApJS..192...10L} analyzed the same data and determined 4 ULXs in NGC 1316. These ULXs were X-4 (\#29), X-6 (\#64), X-5 (\#13) and X-3 (\#16, beyond D$_{25}$\footnote{D$_{25}$; defined by the 25 mag arcsec$^{-2}$ brightness level.} ellipse of NGC 1316) with numbering by their maximum detection significance. Recently, \cite{2019MNRAS.483.5554E} have reported the aforementioned ULX candidates by using {\it XMM-Newton} data.
In the present study, we report the identification of the X-ray source CXOUJ032240.8-371224 as a new ULX to be denoted as ULX X-7 (hereafter X-7) as a result of the analysis of 2019 {\it Chandra} data. We obtained the peak luminosity of X-7 as 2 $\times $10$^{39}$ erg s$^{-1}$ in the energy range of 0.3–8 keV at a distance of 19 Mpc. However, this source has earlier been cataloged as \#14 by \cite{2003ApJ...586..826K} with a luminosity of $\sim$ 5.75 $\times$ 10$^{38}$ erg s$^{-1}$ in the same energy range. We further investigated the X-ray spectral and temporal properties of X-7. We also identified a single optical counterpart of this source using the {\it HST} multi-band optical observations.
The present paper is organized by as follows: {\it Chandra} and {\it HST} data reductions and analysis are presented in Section \ref{section:2}. The results of the analysis and discussions are given in Section \ref{section:3}.
\section{Observations, Data Reduction and Analysis} \label{section:2}
\subsection{X-rays}
NGC 1316 was observed by {\it Chandra} ACIS-S once in 2001, and five times in 2019. In addition, this galaxy was observed four times by {\it XMM-Newton} between 2005 and 2009 but only two of these observations included the position of X-7. However, the spatial resolution of {\it XMM-Newton} is not high enough to resolve the position of X-7 in these two observations. There were numerous observations of NGC 1316 obtained by {\it Swift-XRT} between 2006 and 2020. Also the source X-7 could not be resolved clearly in these observations. Therefore, only the data taken with {\it Chandra} ACIS-S observations were used in the present study. The log of observations is given in Table \ref{T:1}.
{\it Chandra} data were analyzed by using {\scshape ciao}\footnote{https://cxc.cfa.harvard.edu/ciao/} v4.12 software with its calibration package {\scshape caldb}\footnote{https://cxc.cfa.harvard.edu/caldb/} v4.9. Several X-ray sources were detected from level 2 event list using {\scshape wavdetect} tool in {\scshape ciao}. We determined new coordinates for X-7 as R.A.$=$ 03$^{\mathrm{h}}$22$^{\mathrm{m}}$40$^{\mathrm{s}}.813$ and Dec.$=$ -37$^{\circ}$12$\arcmin$23$\arcsec.47$ by using C6 data. The differences between the early coordinates given by \cite{2003ApJ...586..826K} were $\sim$ 0$\arcsec$.20 for R.A and 0$\arcsec$.53 for Dec. The new ULX source X-7 is located 12$\arcsec$ away from center of the galaxy. The position of this source is shown on the {\it Chandra} and {\it HST} images in Fig. \ref{F:1}.
The source and background photons were extracted with {\scshape specextract} task using a circle with a radius of 2.5 arcsec. The background subtracted count rates and off-axis distances of the source X-7 are also given in Table \ref{T:1}. The off-axis distances were calculated with {\scshape src\_psffrac} task for 2$\arcsec$.5 radius in the 0.3-10 keV energy band. The spectral analyses of the X-ray data have been performed by using the package, {\scshape xspec} v12.11. We only used observations labeled as C5 (ObsID 20341) and C6 (ObsID 22187) for the spectral analyses due to the low statistics ($\leq$ 50 counts) of other {\it Chandra} 2019 observations.
Several single component models such as absorbed power-law ({\scshape pl}), multi-color disk blackbody ({\scshape diskbb}) and blackbody ({\scshape bbody}) were fitted to the spectra of X-7. We also fitted the source spectra with the frequently used two-component models such as {\scshape diskbb} + {\scshape pl} and {\scshape diskbb} + {\scshape comptt}; however, no statistically significant improvement was achieved for C5 and C6 data. Hence, we will not discuss the two component models any further in this work.
In our initial fits, each model included two absorption components: The first one was a fixed Galactic column density of N$_{H}$ = 2$ \times$ $10^{20}$ cm$^{-2}$ \citep{1990ARA&A..28..215D} representing the absorbing line-of-sight column density along to the NGC 1316 (using the {\scshape tbabs} model) and the other one was left free to account for intrinsic absorption. The latter component was found to be negligible in the resulting fitting parameters. The unabsorbed flux was calculated in the 0.3$-$10 keV energy band using the convolution model {\scshape cflux} available in {\scshape xspec} and the corresponding luminosity value was calculated using the distance of 19 Mpc. Due to the relatively low source counts for C5 (88 net counts) and C6 (102 net counts) observations we used the Cash statistic\footnote{https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/XSappendixStatistics.html} \citep[or C-stat;][]{1979ApJ...228..939C} for spectral fitting and spectra were grouped at least of 5 counts per bin. We used background subtracted source spectra in our analysis. The resulting spectral parameters of single-component models for C5 and C6 data are given in Table \ref{T:2}.
In order to search for any periodicity, C6 data were used to perform a timing analysis since it provides better statistics than others. The X-ray light curve of X-7 was sampled at 3.2 s using {\scshape dmextract} tool in {\scshape ciao} and power density spectra (PDS) were calculated using {\scshape xronos} v6.0 in {\scshape heasoft} v6.27. The PDS were calculated from a single interval or up to six spectra were averaged to produce a PDS in the 0.3$-$10 keV band. We cannot confirm any significant ($\geq$ 3$\sigma$) periodicity with time-bin size of 3.2 s which sets a lower limit for searching periods. We calculated the pulse fractions of possible sinusoidal modulations in the frequency range (0.001$-$0.1) Hz, following the approach given by \cite{2015MNRAS.452.1112E}. We inferred a 3$\sigma$ upper limit on the highest pulsed fraction as $\sim$ 26\%.
We also searched for short-term and long-term count rate variability. For the short-term variability, the light curve of X-7 was binned over intervals of 100s, 500s and 1000s in the 0.3$-$10 keV energy band using C6 data. The resulting light curves were tested for short-term variations in the source count rates using a Kolmogorov–Smirnov (K-S) test. The K-S test probabilities are found as > 0.3. These results indicate that X-7 do not show any significant amplitude variations.
Bi-modal distribution obtained from the long-term light curves is considered to be a good indication of the presence of a neutron star in ULXs \citep{2018MNRAS.476.4272E,2020MNRAS.491.1260S,2020ApJ...890..166P,2020ApJ...895..127B}. Therefore, we also checked the long-term variability of X-7, for this we used equation of \cite{2020ApJ...895..127B}. According to the their definition the count rate variability is determined the equality by $\chi_{r}^{2}$ $\equiv$ $\chi^{2}$\big/ N$_{obs}$, where $\chi_{r}^{2}$ is an arbitrary definition of count rate variability, $\chi^{2}$ is defined as ${\sum_{\substack{n={1}\\}}^{N_{obs}} \left( \frac{CR_{n}-<CR>}{\sigma_{n}} \right)^2}$, and N$_{obs}$ is the number of observations.
Here, CR$_{n}$ is the count rate in each observation, <CR> is the mean count rate averaged over all observations and $\sigma_{n}$ is the 1$\sigma$ uncertainty on the count rate for each observation. In their work, when $\chi_{r}^{2}$ > 2 the source is considered to be variable. They applied this formula to their sources in M51 and obtained $\chi_{r}^{2}$ is about 10 for ULX X-7 and ULX X-8. Then, they considered these two sources strongly variable. We found that $\chi_{r}^{2}$ $\simeq$ 5 for all {\it Chandra} observations (including 2001 observation) and comparing our value with theirs, X-7 of NGC 1316 is also probably mildly variable (within the present data coverage). Due to the fact that, more observations are required to interpret the long-term variability of X-7 might point out a neutron star in ULX system.
We also examined if luminosities of the ULXs (X-3, X-5 and X-4) in NGC 1316 given by \cite{2011ApJS..192...10L}, vary or not. We performed spectral analyses of these ULXs by using C6 data. The unabsorbed X-ray luminosity L$_{X}$ was calculated with an {\scshape pl} model in 0.3-8.0 keV adopted distance 19 Mpc for each of three sources. Resulting L$_{X}$ values are 4.11, 2.52, and 4.01$\times$ 10$^{39}$ erg s$^{-1}$ for X-3, X-5 and X-4, respectively. Variation of the L$_{X}$ values for these ULXs were found less than a factor two on a timescale of 18 years.
In addition, we determined a source located at $\sim$ 6$\arcsec$ southwest of the X-7 at R.A$=$ 3$^{\mathrm{h}}$22$^{\mathrm{m}}$40$^{\mathrm{s}}.550$ and Dec.$=$ -37$^{\circ}$12$\arcmin$26$\arcsec.45$ (see Fig. \ref{F:1}). This source is also in the list of \cite{2003ApJ...586..826K} as CXOU J032240.5-371227 (\#16, within the D$_{25}$ ellipse of NGC 1316) with a luminosity of 3.34$\times$10$^{38}$ erg s$^{-1}$.
The source count rates for all data (C1-C6) were obtained fitting an absorbed {\scshape pl} model with same N$_{H}$ $=$ 2$\times$ $10^{20}$ cm$^{-2}$ the $\Gamma$ $=$ 1.7 and resulting unabsorbed fluxes were obtained by using {\scshape srcflux} tool in {\scshape ciao} at 90\% confidence level in the range of 0.3$-$10 keV.
The flux values of C2 and C4 data were found as (0.16$\pm$0.04$)\times$10$^{-14}$ erg cm$^{-2}$ s$^{-1}$ (minimum) and (2.31$\pm$0.56)$\times$10$^{-14}$ erg cm$^{-2}$ s$^{-1}$ (maximum). This maximum flux corresponds to luminosity of $\sim$1.0$\times$10$^{39}$ erg s$^{-1}$ with an adopted distance of 19 Mpc. We note that, the flux variation of the source is a quite significant with an order of magnitude difference. Due to the variation, this source could be a candidate of transient ULX which we named as XT-1, as seen clearly in Fig. \ref{F:4}.
\subsection{HST}
The archival data log of Hubble Space Telescope\footnote{https://archive.stsci.edu/hst/search.php} ({\it HST}) observations used for analysis are given in Table \ref{T:1}. We performed a relative astrometry between {\it Chandra} (ObsID 22187) and the {\it HST}/ACS (Advanced Camera for Surveys) (ObsID j6n202010) images to find the optical candidate of X-7. We used {\it wavedetect} tool in {\scshape ciao} and {\it daofind} tool in {\scshape iraf} for source detection in {\it Chandra} and {\it HST}/ACS images, respectively.
Nine reference sources were determined from the comparison of the {\it Chandra} and {\it HST} images. All of these matched sources are located on ACIS-S with a moderate offset from the optical axis in the ObsID 22187. Properties of reference sources are summarized in Table \ref{T:3}. The position uncertainties between each of reference sources were calculated with 90\% confidence level. The final astrometric errors between the {\it Chandra} and {\it HST} images are in R.A. 0\farcs09 and in Dec. 0\farcs07. As a result of astrometric correction, the {\it HST} coordinates of X-7 are given as R.A$=$ 03$^{\mathrm{h}}$22$^{\mathrm{m}}$40$^{\mathrm{s}}.806$ and Dec.$=$ 37$^{\circ}$12$\arcmin$23$\arcsec.52$ within 95 \% confidence level of error circle with 0\farcs22 radius.
The Point-Spread Function (PSF) photometry was performed with {\scshape dolphot} v2.0 \citep{2000PASP..112.1383D} using the {\it HST} data listed in Table \ref{T:1}. The {\scshape acsmask} task was used to remove pixels flagged as bad in images. {\scshape splitgroups} and {\scshape calcsky} tasks were performed to split into single-chip images and were created the sky background for each image, respectively. Here, we present the magnitudes derived by using the set of parameters for the ACS, WFC3/UVIS and WFC3/IR recommended by {\scshape dolphot} user's guide. The {\scshape dolphot} task was used for photometry by taking drizzled images ObsID j6n202010 and ObsID ib3n03030 as the positional reference for both ACS and WFC3, respectively.
We found a unique optical candidate within 0.\arcsec22 error circle at 95\% confidence level for X-7 using the archival {\it HST}/ACS and {\it HST}/WFC3 data (see \ref{F:1}). The ACS and WFC3 magnitudes in the VegaMag. and Johnson system for the optical candidate are given in Table \ref{T:4}. Magnitude values were corrected with $E$($B-V$) = 0.018 mag. derived from the Galactic extinction (A$_{V}$/3.1). The A$_{V}$ $=$ 0.058 mag. was obtained from extinction calculator tools of NED \footnote{https://ned.ipac.caltech.edu/extinction\_calculator}. The F475W, F555W and F814W filters in {\it HST}/ACS correspond to the Johnson filter {\it B}, {\it V} and {\it I}, respectively \citep{2005PASP..117.1049S, 2011PASP..123..481S}. We obtained {\it B$-$V}$=$0.77 mag., {\it V$-$I}$=$1.10 mag. and M$_{V}$$=$$-$7.80 mag. The distance modulus of NGC 1316 is calculated as 31.4 mag. by using the adopted distance of 19 Mpc.
Spectral Energy Distribution (SED) of the optical candidate has been constructed to obtain the spectral characteristics of X-7 using the derived flux values given in Table \ref{T:1}. The wavelength of the filters are selected as the pivot wavelength, obtained from {\scshape pysynphot}\footnote{https://pysynphot.readthedocs.io/en/latest/}, in SED plots. The SED for the optical candidate is adequately fitted (1) a blackbody spectrum with a temperature of 3100 $\pm$ 400 K or (2) a power-law spectrum (F $\propto$ $\lambda^{\alpha}$) with $\alpha$ = 1.75 $\pm$ 0.35, see Fig. \ref{F:bb}. To obtain a blackbody spectrum, a code has been used with {\scshape optimset} and {\scshape fminsearch} functions in {\scshape matlab}\footnote{https://www.mathworks.com/matlabcentral/fileexchange/20129-fit-blackbody-equation-to-spectrum}. The reduced $\chi^{2}$ for blackbody and power-law are 1.34 and 1.27, respectively. The number of degrees of freedom is 3 for both models.
\section{Results and discussions} \label{section:3}
\subsection{X-Rays}
We identified a new ULX in NGC 1316 by {\it Chandra} in April 2019 with at a luminosity of 2.10 $\times$ 10$^{39}$ erg s$^{-1}$. This luminosity is almost a factor of $\sim$ 4 times higher than previous {\it Chandra} observation in April 2001. We obtained the best-fitting spectral parameters of one-component models for X-7 to elaborate its spectral characteristics. These parameters were obtained with only one intrinsic absorption, N$_{H}$ $=$ 2 $\times$ $10^{20}$ cm$^{-2}$ since the second absorption parameter did not contribute significantly.
The spectra of X-7 are the best fitted by {\scshape pl} models with photon indices of $\Gamma$ $=$ 2.24 for C5 and 1.56 for C6 data. The source has maximum unabsorbed luminosity values of $\sim$ 1.8 and 2.1$\times$ $10^{39}$ in 0.3$-$10 keV, respectively. These values are also listed in Table \ref{T:2}. Energy spectrum of X-7 with a {\scshape pl} model and its residual are shown in Fig. \ref{F:3}.
The source spectra were also fitted by the {\scshape diskbb} model with T$_{in}$ $=$ 0.78 keV (C5) and 1.32 keV (C6). The other well fitted model is {\scshape bbody} whose temperature values are 0.50 keV and 0.74 keV with unabsorbed luminosities 0.92 and 1.21 $\times$ $10^{39}$ erg s$^{-1}$ for C5 and C6, respectively.
As noted, it is difficult to differentiate between these three models due to their very similar C-stat.
In the spectral analyses of the {\it Chandra} data, we found that the spectra of X-7 in C5 and C6 are better represented by {\scshape pl} models with photon indices $\Gamma$ $\sim$ (2.24 and 1.56). Both of these $\Gamma$ values correspond to hard states defined for GBHBs. Hard states with low luminosity are seen at sub-Eddington mass accretion rates. On the other hand, {\scshape diskbb} model also yields acceptable fits for the same datasets with the temperature of $kT_{in}$ $\sim$ (0.78 and 1.32) keV. These $kT_{in}$ values are compatible with those of GBHBs at a high mass accretion rate during thermal state \citep{2006ARA&A..44...49R}. Generally in GBHBs, luminosities are usually hard when they are bright and soft when they are dim. However, X-7 exhibits the opposite behavior since the source has a high $L_{X}$ when it is in a hard state and a low $L_{X}$ when it is in a soft state. There are some ULXs that do show similar correlations like X-2 of NGC 1313, \cite{2006ApJ...650L..75F}; X-1 of IC 342, \cite{2014MNRAS.444..642M}; X-2 of NGC 4736, \citet{2014Ap&SS.352..123A}. X-3 of NGC 925, \cite{2020ApJ...891..153E}.
We used the normalization parameter of the {\scshape diskbb} model in C6 data to constrain the mass of the compact object. Here, we calculated the inner disk radius as R$_{in}$ $\sim$ 70 km from the equation of R$_{in}$ = $\kappa$ $^{2}$ $\xi $r$_{in}$ $\sqrt{cos\theta}$, where $\xi$ = 0.412 is the correction factor, $\kappa$ = 1.7 is spectral hardening factor, r$_{in}$ is the apparent disk radius derived from the observed data and $\theta$ is the disk inclination angle and its range was given as 0$^{\circ}$-75$^{\circ}$. \citep{1995ApJ...445..780S, 1998PASJ...50..667K} Using the relation between inner disk radius and mass \citep{2000ApJ...535..632M}, we found the average value $\sim$ 8 M$\odot$ for the mass of compact object in X-7.
\subsection{HST}
We examined the optical properties of X-7 in the galaxy NGC 1316 using the archival data from {\it HST}/ACS and {\it HST}/WFC3. A unique optical candidate was identified within $0.\arcsec22$ error radius after the astrometric correction. It is noted that dereddened magnitudes of optical candidate are faint (m$_{V}$ > 22.5 mag) as seen in Table \ref{T:4}. On the other hand, the absolute magnitude is very bright M$_{V}$ $=$ $-$ 7.80 mag which is the compatible with the given range ($-$4 < M$_{V}$ < $-$9 ) for ULXs \citep{2013ApJS..206...14G}.
Assuming optical emission originates from the donor star, we fitted a blackbody and power-law model to SED of X-7 as seen in Fig \ref{F:bb}. The best-fitting parameter for power-law model is $\alpha$ $=$ 1.75 $\pm$ 0.35 and the temperature is 3100 $\pm$ 400 K for blackbody model. Although, both model yields similar reduced $\chi^{2}$ $\simeq$ 1.3, the $\alpha$ parameter for power-law is not in the range of $-$3 and $-$4. This range is given for optical counterparts of many ULXs having power-law SEDs \citep{2011ApJ...737...81T, 2018ApJ...854..176V}. It follows that the blackbody model is more favorable in our case. According to \cite{1981Ap&SS..80..353S} the measured temperature and luminosity of the companion star indicate that spectral classification would be an M type supergiant. Similar spectral type of companions are proposed for ULX-5 of NGC 3034, ULX-1 and ULX-2 of NGC 253 by \cite{2013ApJS..206...14G}.
The blackbody emission is also thought to be originating from irradiation of the accretion disk. Assuming so and taking into consideration the temperature from blackbody fitting and the distance to the source, we determined the surface area of the emitting region of X-7 as $\simeq$ 4 $\times$ 10$^{26}$ cm$^{2}$ or a radius of $\simeq$ 2 $\times$ 10$^{13}$cm. This radius seems an order of magnitude larger than those of the optical emitting region estimated ULXs \citep{2011NewAR..55..166F}. \cite{2011ApJ...737...81T} also fitted these two models to the spectrum of X-1 in IC 342 and showed that their data is better fitted to the blackbody model and reported very similar radius.
We have also examined whether the size of the emitting region can reprocess this optical emission. Assuming a fraction $\eta$ of the X-ray luminosity, L$_{X}$, is reprocessed with $\eta$ $=$ $\sigma$T$^{4}$A/L$_{X}$ where $\sigma$, T and A are Stefan–Boltzmann constant, temperature and area of the region, respectively. Inserting T$=$3100 K, A $\sim$ 10$^{27}$ cm$^{2}$, and L$_{X}$ $=$ 2$\times$10$^{39}$ erg cm$^{-2}$ s$^{-1}$, we found that $\eta$ is $\simeq$ 0.002 which is similar to the value for GBHBs such as XTE J1817$-$330 and X-1 of Holmberg IX \citep{2009MNRAS.392.1106G,2011ApJ...737...81T}.
In addition, we investigated the optical candidates of previously known ULXs (X-3, X-4 and X-5) from study of \cite{2011ApJS..192...10L} and also the XT-1 in the NGC 1316. As noted, we identified a single optical candidate for X-5 located at the edge of 0\arcsec.22 error circle. Its V-band magnitude is 23.1$\pm$0.03 mag. This candidate is not associated with any source in the CDS (Centre de Données astronomiques de Strasbourg) database. On the other hand, the source XT-1 has no optical counterpart(s) in {\it HST} images.\\
As a conclusion, further broadband X-ray observations of X-7 are needed to constrain the spectral parameters better and scale of variability of the source with more confidence. It will allow us to interpret mechanisms of X-ray emission from accretion processes in ULXs.
\section*{Acknowledgements}
This research was supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) through project number 117F115. This research was also supported by the Çukurova University Research Fund through project number FBA-2019-11803. This research is a part of the PhD thesis of S. Allak and he acknowledges financial support by TÜBİTAK. Our special thanks to M. E. Ozel for his valuable contributions of the manuscript.
\section*{Data Availability}
The scientific results reported in this article are based on archival observations made by the Chandra X-ray Observatory\footnote{https://cda.harvard.edu/chaser/}. This work has also made use of observations made with the NASA/ESA Hubble Space Telescope, and obtained from the data archive at the Space Telescope Science Institute\footnote{https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html}.
\bibliographystyle{mnras}
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Why "Send her back" is nothing new
Ilhan Omar takes up BDS fight
Limbaugh: Attacks on Obama were "bogus"
(SuriyaPhoto via Shutterstock)
Fringe "Patriot" movement swells for fourth straight year
A new report finds that extreme right wing conspiracy groups increased by an astounding 813 percent in 2012
Check out this article! https://www.salon.com/2013/03/06/patriot_movement_swells_for_fourth_year_partner/
March 7, 2013 1:55AM (UTC)
This article was originally published by The Southern Poverty Law Center.
The number of antigovernment "Patriot" groups on the American radical right hit an all-time high in 2012, the fourth straight year of explosive growth, according to a report released today by the Southern Poverty Law Center (SPLC). As the new year began, serious talk of gun control, prompted by a Connecticut school massacre in December, fueled even more rage on the right, and the threat of violence loomed .
The new report, contained in the latest issue of the quarterly investigative journal Intelligence Report, found that the conspiracy-minded Patriot groups, which numbered only 149 in 2008, soared over the first four years of Barack Obama's presidency to 1,360 in 2012 — an astounding 813% increase. At the same time, it found that hate groups remained at near-record levels of over 1,000 (see interactive map and state-by-state lists of 2012 hate groups here).
The resurgence of militias and other Patriot groups and an uptickover recent years in non-Islamic domestic terrorism caused SPLC President Richard Cohen today to write (letter pdf) U.S. Attorney General Eric Holder and Homeland Security Secretary Janet Napolitano to request the creation of a new interagency task force to assess the adequacy of federal resources devoted to the threat. "As in the period before the [1995] Oklahoma City bombing," Cohen wrote, "we are now seeing ominous threats from those who believe that the government is poised to take their guns."
Patriot groups generally adhere to variations of a conspiracy theory that suggests that the federal government has secret plans to impose martial law on the United States, most likely with the aid of foreign troops; seize all guns held by American citizens; toss all those who resist into concentration camps run by the Federal Emergency Management Agency; and, ultimately, force the country into a kind of socialistic one-world government, commonly known as the "New World Order."
The resurgence of the Patriot movement, which first rose and fell in the 1990s, has largely been a reaction to the election of the nation's first black president in 2008 — and the demographic change, including the loss of the country's white majority predicted for 2043, that he represents — as well as the difficult economy. The recent talk of gun control, which has sparked state legislative efforts to "nullify" any federal legislation and also a movement of rural sheriffs who promise to resist, is now adding fuel to a fire that was already burning at white-hot temperatures (see also my editorial).
MORE FROM Mark Potok
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Q: SharePoint 2013 HTTP 503 error I have SharePoint 2013 ServicePack 1 on Windows Server 2012 R2. In administration panel I had error about multiple services running on one Administration account, so it redirected me to page where I chose service and assing new account (added in Active Directory). Then it give me a command to force restart IIS. After restart I get 503 error, so according to this I started all applications, but when I'm trying to open Central Administration of Sharepoint i have still 503, after refreshing applications in IIS the Central Administration is stoped again. It stops when I'm trying to run it. How to get rid of that 503?
A: Did you check that the services are all running correctly via windows control panel
A: Problem solved. Changing users in IIS help
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Josey Wales hors-la-loi (The Outlaw Josey Wales) est un film américain réalisé par Clint Eastwood et sorti en 1976.
Il s'agit d'un western très populaire aux États-Unis principalement, mêlant violence et tendresse, vengeance et amour.
Clint Eastwood, icône du western, est ici également réalisateur. C'est le deuxième western qu'il dirige, après L'Homme des Hautes Plaines (1973).
En 1977, le film est nommé à l'Oscar de la meilleure musique de film pour Jerry Fielding.
Synopsis
À la fin de la guerre de Sécession, Josey Wales cultive tranquillement son champ dans le Missouri quand des Jayhawkers du Kansas, pro-Union, surgissent, incendient son ranch, violent sa femme et massacrent sa famille. Une bande de partisans sudistes arrivant peu après, il la suit. Mais le Sud est vaincu et le sudiste Fletcher annonce une amnistie aux partisans qui déposeront les armes. En fait, c'est un piège : lors de la reddition, au moment où les partisans prêtent serment à l'Union, les Nordistes les abattent à la mitrailleuse. N'en réchappent que Josey Wales, qui avait refusé de se rendre (il est très méfiant, ce qui lui sauvera la vie plusieurs fois), et un jeune franc-tireur blessé. Josey Wales ayant dans l'affaire massacré un gros paquet de Nordistes, sa tête est mise à prix. Un détachement nordiste commandé par Terrill et guidé par Fletcher (qui avait été trompé comme les autres mais connaît le tempérament vindicatif de Wales) le prend en chasse. Échappant aux battues, les fugitifs parviennent à traverser le Missouri puis à liquider deux imprudents qui espéraient les capturer et toucher la prime. Le jeune partisan meurt peu après et Josey Wales, se retrouvant seul, décide de se réfugier du côté du Texas.
En cours de route, il est rejoint malgré lui par un vieux chef indien philosophe dépossédé par Washington, puis une jeune indienne rejetée par sa tribu et traitée en esclave par un trafiquant que Josey Wales libère en abattant au passage deux trappeurs qui abusaient d'elle. Il est également rejoint par un chien galeux et par une vieille femme et sa famille dont le fils, qui fut l'un des irréguliers du Kansas et a été tué par les francs-tireurs sudistes, possédait un ranch magnifique au Texas. Juste avant cette rencontre, Josey Wales abat quatre Nordistes hésitants à l'abattre (car il est devenu le tireur le plus célèbre de l'Ouest) qui ont eu la malchance de croiser son chemin. Juste après il extermine toute une bande de comancheros venant de massacrer le convoi de la vieille dame. Celle-ci et sa petite-fille, Laura Lee, sont les seuls survivantes. Tout ce petit monde arrive donc à destination. Hélas, la ville est quasiment déserte (Josey Wales devra quand même encore y abattre un des deux chasseurs de primes qui l'y attendaient, le survivant ira chercher des renforts) car les habitants l'ont quittée à la suite de l'épuisement de sa mine d'argent ; ne reste qu'un saloon où il n'y a plus rien à boire, où le joueur professionnel fait des réussites et où l'entraîneuse n'entraîne plus personne. Le ranch au milieu d'une clairière, au bord d'une rivière, solidement construit, les attend. La vieille dame prend possession des lieux et pour la première fois chacun a un foyer. Wales parvient à conclure un accord avec les indiens Comanches voisins, et cela donne lieu à une jolie fête où Wales se laisse séduire par la douce Laura Lee. Bien qu'invité à rester, il préfère repartir. Mais Nordistes et chasseurs de primes, ayant uni leurs forces, l'attendent. Cette fois, cependant, Josey Wales sera aidé par tous ses compagnons devenus sa famille et la totalité de la troupe ennemie est exterminée. Terrill, le Nordiste qui avait attaqué la ferme de Josey Wales au début, parvient seul à s'enfuir mais Wales le poursuit et manquant de munition l'éventre avec son propre sabre.
Des Rangers du Texas et Fletcher, arrivés également au village, s'entendent raconter par les piliers du saloon que Josey Wales a été tué dans une rixe au Mexique. L'affaire est donc close pour les autorités. Quant à Fletcher, qui n'est pas dupe, il fait semblant de ne pas reconnaître Josey Wales et repart. La guerre est finie.
Fiche technique
Titre francophone : Josey Wales hors-la-loi
Titre original : The Outlaw Josey Wales
Réalisation : Clint Eastwood, assisté de James Fargo
Scénario : Philip Kaufman et Sonia Chernus, d'après le roman The Rebel Outlaw: Josey Wales de Forrest Carter
Musique : Jerry Fielding
Directeur de la photographie : Bruce Surtees
Décors : Tambi Larsen
Montage : Ferris Webster
Production : Robert Daley, James Fargo et John G. Wilson
Sociétés de production : Warner Bros., The Malpaso Company
Sociétés de distribution : Warner Bros. (États-Unis), Warner-Columbia Film (France)
Pays d'origine :
Langue originale : anglais
Budget : 3,7 millions de dollars
Genre : western
Durée : 135 minutes
Dates de sortie :
États-Unis :
France :
Classifications :
France : tous publics
Distribution
Clint Eastwood (VF : Jacques Deschamps) : Josey Wales
"Chief" Dan George (VF : Georges Atlas) : Lone Watie
Sondra Locke (VF : Brigitte Morisan) : Laura Lee
Bill McKinney (VF : Marc de Georgi) : Terrill
John Vernon (VF : Raoul Delfosse) : Fletcher
Paula Trueman (VF : Madeleine Damien) : grand-mère Sarah
Sam Bottoms (VF : Philippe Bellay) : Jamie
Geraldine Keams : Little Moonlight
Joyce Jameson (VF : Paule Emanuele) : Rosie, la chanteuse
Woodrow Parfrey : le bonimenteur
Sheb Wooley : Travis Cobb
Royal Dano (VF : Serge Sauvion) : Ten Spot (Trois As en )
Matt Clark : Kelly
John Verros (VF : Gérard Hernandez) : Chato
Will Sampson (VF : Michel Barbey) : Ten Bears (Grand Bison en )
Frank Schofield (VF : Jacques Mauclair) : le sénateur Lane
William O'Connell (VF : Georges Aubert) : Sim Carstairs
John Quade (VF : Claude Bertrand) : le chef comanchero
John Russell (VF : Pierre Garin) : Bill Anderson
Buck Kartalian (VF : Michel Paulin) : le commerçant
Charles Tyner : Zukie Limmer
Len Lesser : Abe
Doug McGrath : Lige
Bruce M. Fischer (VF : Sady Rebbot) : Yoke, le propriétaire du relais de poste
John Mitchum : Al
John Davis Chandler : le premier chasseur de primes
Tom Roy Lowe : le second chasseur de primes
Madeleine Taylor Holmes (VF : Ginette Frank) : grand-mère Hawkins
Clay Tanner : le premier Texas ranger
Bob Hoy : le second Texas ranger
Erik Holland : le sergent de l'armée de l'Union
Cissy Wellman : la femme de Josey
Faye Hamblin : Grandpa
Danny Green : Lemuel
Kyle Eastwood : le fils de Josey (non crédité)
Richard Farnsworth : un comanchero (non crédité)
Production
Développement
Le film est l'adaptation du roman The Rebel Outlaw: Josey Wales de Forrest Carter (nom de plume d'Asa Earl Carter, ségrégationniste et membre du Ku Klux Klan). Mais Clint Eastwood l'ignore au moment de l'achat des droits, l'auteur se faisant passer pour un poète indien d'origine Cherokee.
Sonia Chernus se charge du scénario. Clint Eastwood investit son argent personnel pour acquérir les droits de l'œuvre de Carter. Sonia Chernus est ensuite aidée dans sa tâche par Michael Cimino et Philip Kaufman. Ce dernier officie ensuite comme réalisateur (il sera renvoyé et remplacé par Clint Eastwood en cours de tournage).
Après quatre films pour Universal Pictures, Clint Eastwood travaille ici avec Warner Bros., un partenariat qui se maintiendra quasiment sur tous les films réalisés par l'acteur.
Attribution des rôles
Philip Kaufman a choisi Dan George, chef des Tsleil-Waututh, qui avait déjà joué dans Little Big Man (Arthur Penn, 1970). Cependant, le choix de Sondra Locke est validé par Clint Eastwood, contre l'avis de Philip Kaufman, qui était toujours au poste de réalisateur à ce moment-là.
Tournage
Le tournage débute à la mi-. Le début du tournage est marqué par le renvoi et le remplacement du réalisateur Philip Kaufman par Clint Eastwood, le .
Le renvoi de Philip Kaufman provoque l'indignation notamment à la Directors Guild of America, le syndicat professionnel des réalisateurs.
Le tournage a lieu principalement en Utah (Glen Canyon, Kanab, Paria), mais également en Arizona (lac Powell, Mescal, Old Tucson Studios), à Oroville (Californie) et dans le Wyoming.
Accueil
Le film reçoit des critiques globalement positives de la presse américaine à sa sortie. Sur l'agrégateur américain Rotten Tomatoes, il récolte 90% d'opinions favorables pour 41 critiques et une note moyenne de . Sur Metacritic, il obtient une note moyenne de pour 9 critiques.
Lors de sa sortie en salles en France, Josey Wales hors-la-loi est bien reçu par la critique.
Aux États-Unis, le film récolte au box-office. En France, le film attire dans les salles.
Distinctions
En 1977, Jerry Fielding est nommé à l'Oscar de la meilleure musique de film - partition originale. En 1996, le film est inscrit au National Film Registry.
Suite
En 1986, l'acteur-réalisateur Michael Parks tourne une suite, The Return of Josey Wales, adaptée du roman Vengeance Trail of Josey Wales d'Asa Earl Carter. Michael Parks y incarne lui-même Josey Wales.
Notes et références
Annexes
Bibliographie
Articles connexes
Anti-western
Guerre de Sécession
Liens externes
Film américain sorti en 1976
Western américain
Film réalisé par Clint Eastwood
Film sur la guerre de Sécession
Vengeance au cinéma
Film se déroulant dans une ville fantôme
Adaptation d'un roman américain au cinéma
Film nommé aux Oscars
Film de Malpaso Productions
Film distribué par Warner Bros. Pictures
Film se déroulant au Missouri
Film se déroulant en Oklahoma
Film se déroulant au Texas
Film tourné en Arizona
Film tourné en Californie
Film tourné en Utah
Film tourné au Wyoming
Film inscrit au National Film Registry
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Q: Sending data using in the same page and display the result (FORM) using JSP without changing the (FORM) coding <div id="main_search">
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<input type="checkbox" name="product" class="prod_checkbx" value="IDD Buffet Base">IDD Buffet Base
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A: Add a form tag in your code like
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|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,407
|
\section{Introduction}\label{sec:introduction
Since the seminal works by~\cite{aharonov1993quantum, meyer1996quantum, farhi1998quantum},
quantum walks have been the subject of research in two decades.
They were originally proposed as a quantum generalization of random walks~\cite{spitzer2013principles}.
Asymptotic properties such as mixing time, mixing rate and hitting time
of quantum walks on a line and on general graphs have been studied extensively
~\cite{ambainis2001one, aharonov2001quantum,
moore2002quantum, childs2004spatial, krovi2006hitting}.
Applications of quantum walks in quantum information processing have also been investigated.
Especially, quantum walks can solve the element distinctness problem
~\cite{aaronson2004quantum, ambainis2007quantum} and
perform the quantum searching~\cite{szegedy2004quantum}.
In some applications, quantum walks based algorithms can even gain
exponential speedup over all possible classical algorithms~\cite{childs2003exponential}.
The discovery of their capability for universal quantum
computations~\cite{childs2009universal, lovett2010universal}
indicates that understanding quantum walks is helpful for
better understanding quantum computing itself.
For a more comprehensive review, we refer the readers to
~\cite{kempe2003quantum, venegas2012quantum} and the references within.
Lackadaisical quantum walks (LQWs), first considered by Wong et al.~\cite{wong2015grover},
are quantum analogous of lazy random walks.
This model also generalizes three-state quantum walks on a line
~\cite{inui2005one, falkner2014weak, vstefavnak2014limit, wang2016grover},
which only have one self-loop at each vertex.
In~\cite{wong2015grover}, the authors mainly investigate the effect of extra
self-loops on Grover's algorithm when formulated as search for a marked vertex on complete graphs.
They find that adding self-loops can either slow down or boost the success probability
by choosing different coin operators.
On the other hand, three-state quantum walks on a line have been investigated exhaustively.
Most notably, if the walker of a three-state quantum walk is initialized at one site,
it will be trapped with large probability near the origin after walking enough steps~\cite{inui2005one, falkner2014weak}.
This phenomenon is previously found in quantum walks on
square lattices~\cite{inui2004localization} and is called localization.
Researches show that the localization effect happens with a broad family of coin
operators in three-state quantum walks
~\cite{vstefavnak2012continuous,vstefavnak2014stability,vstefavnak2014limit}.
Moreover, a weak limit theorem is recently derived in~\cite{falkner2014weak, machida2015limit}
for arbitrary coin initial state and coin operator.
However, the properties of LQWs, such as localization and spread behavior,
are still open. In this paper, we give a in-depth study the LQWs on a line.
Since the lackadaisical model is more complicated than the standard one,
we could expect more intrinsic characteristics.
The rest of this paper is organized as follows.
In Sec.~\ref{sec:definition}, we give formal definitions of LQWs
and describe the Fourier transformation method which is often used in analyzing quantum walks.
In Sec.~\ref{sec:localization}, we provide a mathematical framework for the walker's localization probability
on the time limit.
In Sec.~\ref{sec:velocity}, we find the explicit forms to compute the velocities of
the left- and right-travelling peaks appeared in the walker's probability distribution.
And in Sec.~\ref{sec:weak-limit}, we obtain a long time approximation for the entire probability density function
and prove that all LQWs spread ballistically.
Finally, we conclude in Sec.~\ref{sec:conclusion}.
\section{Definitions}\label{sec:definition
\subsection{Lackadaisical quantum walks}
In this paper, a LQW is defined to be a quantum walk
on an infinite line with $\tau$ self-loops attached to each vertex.
An illustrative example is given in Fig.~\ref{fig:self-loops}, in which each vertex has 2 additional self-nodes.
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{fig_loops.png}
\caption{An illustrative example of an infinite line with $2$ self-loops attached to each vertex.}\label{fig:self-loops}
\end{figure}
We term the number of self-loops $\tau$ as the \textit{laziness factor}.
If $\tau=0$, it is the standard quantum walk (also called the Hadamard walk).
In this paper we consider $\tau>0$.
It's obvious that in lazy random walks, the greater the $\tau$ is, the more the walker prefers to stay.
The total system of a LQW with laziness factor $\tau$
is given by $ \mathcal{H} = \mathcal{H}_P \otimes \mathcal{H}_C$,
where $\mathcal{H}_P$ is the position space defined as
$$ \mathcal{H}_P = \textrm{Span}\{\ket{n}, n \in \mathbb{Z}\},$$
and $\mathcal{H}_C$ is the coin space.
In each step, the walker has $\Delta$ ($\Delta = \tau + 2$) choices -
it can move to the left, or the right,
or just stay in current position via a self-loop.
For each of these options, we assign a standard basis of the coin space $\mathcal{H}_C$.
Thus $\mathcal{H}_C$ is defined as
$$\mathcal{H}_C = \mathbb{C}^\Delta = \textrm{Span}\{\ket{1}, \;\ket{2}, \;\cdots, \;\ket{\Delta}\}. $$
A single step of quantum walk is given by
$U = S \cdot (\mathbb{I}_P \otimes C)$
where $S$ is the position shift operator, $\mathbb{I}_P$ is the identity of
$\mathcal{H}_P$ and $C$ is the coin flip operator.
For LQWs, the position shift operator $S$ is
\begin{eqnarray*}
S &=& \sum_{n \in \mathbb{Z}}\Big\{
\ket{n-1}\bra{n} \otimes \ket{1}\bra{1} + \ket{n+1}\bra{n} \otimes \ket{2}\bra{2} \\
~ &~& \quad\quad + \sum_{j=3}^{\Delta} \ket{n}\bra{n} \otimes \ket{j}\bra{j} \Big\}.
\end{eqnarray*}
For the coin operator $C$, a common choice is the Grover operator $G$, which is defined as
\begin{equation}\label{eq:coin-operator}
\operator{G} = \frac{1}{\Delta}
\begin{pmatrix}
-\tau & 2 & 2 & \cdots & 2 \\
2 & -\tau & 2 & \cdots & 2 \\
2 & 2 & -\tau & \cdots & 2 \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
2 & 2 & \cdots & 2 & -\tau
\end{pmatrix}.
\end{equation}
Let
$\ket{\Psi(t, n)} = [\psi_1(t, n), \psi_2(t, n), \cdots, \psi_\Delta(t, n)]^\dag \in \mathcal{H}_C$
be the probability amplitude of the walker at position $n$ at time $t$,
then the system state can be expressed by
$$
\ket{\Psi(t)} = \sum_{n \in \mathbb{Z}} \ket{n}\otimes \ket{\Psi(t, n)}.
$$
$\ket{\Psi(t)}$ can be obtained by applying $U$ to the initial state $\ket{\Psi(0)}$ for $t$ times,
i.e. $\ket{\Psi(t)} = U^t \ket{\Psi(0)}$.
The walker $X_t$ can be found at position $n$ at time $t$ with probability
\begin{equation}\label{eq:calculate-prob}
\mathbb{P}(X_t = n) = \langle\Psi(t,n) \vert \Psi(t,n)\rangle = \sum_{j=1}^{\Delta}\vert \psi_j(t,n) \vert^2.
\end{equation}
Expanding $\ket{\Psi(t+1)} = U \ket{\Psi(t)}$ in terms of $\ket{\Psi(t, n)}$,
we obtain the master equation for the walker at position $n$
\begin{eqnarray}\label{eq:recurrence}
\ket{\Psi(t+1, n)} &=& \quad G_1\ket{\Psi(t, n+1)} + G_2\ket{\Psi(t, n-1)} \nonumber\\
~ &~& + \sum_{j=3}^{\Delta}G_j\ket{\Psi(t, n)},
\end{eqnarray}
where $G_j = \sum_{k=1}^{\Delta}G_{j,k}\ket{j}\bra{k}, j = 1, 2, \cdots, \Delta$, and
$G_{j,k}$ are the elements of $G$ defined in Eq.~\ref{eq:coin-operator}.
\subsection{Fourier analysis}
Eq.~\ref{eq:recurrence} can be solved by Fourier transformation on the system state
\begin{equation}\label{eq:fourier}
\ket{\tilde{\Psi}(t, k)} = \sum_{n \in \mathbb{Z}} e^{-ikn} \ket{\Psi(t, n)},\;\; k \in (-\pi, \pi].
\end{equation}
From now on, a tilde indicates quantities with a $k$ dependence.
The inverse Fourier transform is
\begin{equation}\label{eq:inverse-fourier}
\ket{\Psi(t, n)} = \int_{-\pi}^{\pi}\frac{dk}{2\pi} e^{ink} \ket{\tilde{\Psi}(t, k)}.
\end{equation}
Substituting Eq.~\ref{eq:fourier} to Eq.~\ref{eq:recurrence} yields the master equation in the Fourier space
\begin{equation}\label{eq:fourier-recurrence}
\tilde{\Psi}(t+1, k) =
\underbrace{ \big[ G_1e^{ik} + G_2e^{-ik} + \sum_{j=3}^{\Delta}G_j\big]}_{\tilde{U}_k} \tilde{\Psi}(t, k).
\end{equation}
Let $\kappa = e^{ik}$, $\tilde{U}_k$ has the form of
\begin{equation*}
\tilde{U}_k = \frac{1}{\Delta}
\begin{pmatrix}
-\tau\kappa & 2\kappa & 2\kappa & \cdots & 2\kappa \\
2/\kappa & -\tau/\kappa & 2/\kappa & \cdots & 2/\kappa \\
2 & 2 & -\tau & \cdots & 2 \\
\cdots & \cdots & \cdots & \cdots & \cdots \\
2 & 2 & \cdots & 2 & -\tau
\end{pmatrix}.
\end{equation*}
Since $\tilde{U}_k$ is unitary, its eigenvalues have the forms of $\lambda_j = e^{i\omega_j}$.
We denote $\ket{\lambda_j}$ as the corresponding eigenvectors.
After some calculation, we get explicit forms of the eigenvalues
\begin{equation*}
\omega_j =
\left\{ \begin{array}{ll}
\theta, & j=1,\\
-\theta, & j=2,\\
0, & j=3, \\
\pi, & j \geq 4, \\
\end{array} \right.
\end{equation*}
where $\theta$ satisfies
\begin{eqnarray*}
\cos\theta &=& -\frac{\tau\cos k + 2}{\tau + 2}, \\
\sin\theta &=& \frac{\sqrt{\tau(1 - \cos k)(\tau + 4 + \tau\cos k)}}{\tau + 2}.
\end{eqnarray*}
The corresponding eigenvectors are
\begin{equation*}
\ket{\lambda_j} = \sqrt{N_j}
\begin{pmatrix}
\frac{1}{1 + e^{i(\omega_j - k)}} \vspace{0.1in}\\
\frac{1}{1 + e^{i(\omega_j + k)}} \vspace{0.1in}\\
\frac{1}{1 + e^{i\omega_j}} \vspace{0.1in}\\
\vdots \\
\vdots \vspace{0.1in} \\
\frac{1}{1 + e^{i\omega_j}}
\end{pmatrix}, \;\mbox{for}\; j = 1, 2, 3;
\end{equation*}
\begin{equation}\label{eq:eigenvectors}
\ket{\lambda_j} = \frac{1}{\sqrt{2}}
\begin{pmatrix}
0 \\
0 \\
-1 \\
0 \\
\vdots \\
0 \\
1 \\
0 \\
\vdots
\end{pmatrix}
\begin{matrix}
~ \\
~ \\
~ \\
~ \\
~ \\
~ \\
\leftarrow \mbox{$j$-th row} \\
~ \\
~
\end{matrix}, \;\forall j \geq 4,
\end{equation}
where $N_j$ is the corresponding normalization factor.
Putting $\tilde{U}_k$ in its eigenbasis, we can rewrite Eq.~\ref{eq:fourier-recurrence} as
\begin{eqnarray}\label{eq:evolution-equation}
\tilde{\Psi}(t, k) &=& \tilde{U}_k \tilde{\Psi}(t, k) = \tilde{U}_k^t \tilde{\Psi}(0, k) \nonumber\\
~ &=& \sum_{j=1}^{\Delta}\lambda_j^t\ketbra{\lambda_j} \tilde{\Psi}(0, k),
\end{eqnarray}
where $\tilde{\Psi}(0, k)$ is the Fourier transformed initial state.
In this paper, we assume the walker always starts at position $0$ and the initial coin state satisfies
$\ket{\Psi(0, 0)} = \alpha\ket{1} + \beta\ket{2}$, where $\alpha, \beta \in \mathbb{C}$, $\alpha^2 + \beta^2 = 1$.
This assumption is quite reasonable if
we want to have a same initial state for walks with different $\tau$.
We guarantee the quantum walker starts with its coin state in superposition of only left and right bases.
Therefore, the system's initial state can be formulated as
\begin{equation}\label{eq:init-state}
\Psi(0) = \sum_{n \in \mathbb{Z}} \delta_{n,0} \ket{0} \otimes \big[\alpha\ket{1} + \beta\ket{2}\big],
\end{equation}
where $\delta_{n,0}$ is the Kronecker function.
By Eq.~\ref{eq:fourier} the Fourier transformed system's initial state becomes
\begin{equation}\label{eq:dfted-init-state}
\tilde{\Psi}(0, k) = [\alpha, \beta, 0, \cdots, 0]^\dag, \quad \forall k \in (-\pi, \pi].
\end{equation}
\section{Probability at Origin}\label{sec:localization
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{fig_origin.png}
\caption{Numerical probabilities of finding the walker at the origin as a function of walking steps
for LQWs with various laziness factors $\tau=1$ (blue dots), $\tau=6$ (red dots),
and $\tau=20$ (green dots). These factors are carefully chosen to show different oscillating behaviors.
The initial coin state is $\alpha=\frac{1}{2}$, $\beta=\frac{i}{2}$ for all walks.
The horizontal lines are the corresponding theoretical localization probabilities obtained
by Eq.~\ref{eq:localization-prob}.
}\label{fig:origin-probability}
\end{figure}
In this section, we focus on the localization phenomenon on LQWs.
To determine whether the localization will occur at the origin, we need to calculate
$\lim_{t \rightarrow \infty} \mathbb{P}(X_t = 0)$.
Let the probability be $\hat{P}_0$, where we use a caret ($\hat{~}$) to indicate
the asymptotic limit of $t$.
\begin{theorem}\label{thm:localization-prob}
For a LQW with laziness factor $\tau$,
if the walker starts with the state given by Eq.~\ref{eq:init-state},
the asymptotic limit of the probability of the walker at origin is
\begin{equation}\label{eq:localization-prob}
\hat{P}_0 = 2\cdot\frac{\tau + 4 - 2\sqrt{2\tau+4}}{\tau^2}.
\end{equation}
\end{theorem}
It's obvious from the theorem that, if the walker starts on a superposition of only left and right directions,
the localization probability of the walker is independent on the coin initial state, and is
totally dominated by the laziness factor $\tau$.
When $\tau=1$, we get $\hat{P}_0 = 2(5-2\sqrt{6})$.
This result coincides with Eq.~15 derived in~\cite{falkner2014weak} when $\beta=0$.
We perform numerical simulations and the conclusions are summarized in Fig.~\ref{fig:origin-probability}.
The figure manifests the probabilities $\mathbb{P}(X_t = 0)$ oscillate around their corresponding
theoretical limiting values $\hat{P}_0$
for $\tau=1$, $\tau=6$, and $\tau=20$.
It is clearly that, for different laziness factors,
the probability at the origin oscillates periodically with different patterns.
These oscillations clearly exhibit tendencies to converge, indicating that
the walker does have a non-zero probability to be localized.
Furthermore, we observe that the larger the $\tau$ is, the faster the probabilities converge.
\begin{proof}
By Eq.~\ref{eq:calculate-prob}, we have
\begin{equation}\label{eq:origin-prob-expression}
\hat{P}_0 \equiv \lim_{t \rightarrow \infty} \mathbb{P}(X_t = n)
= \lim_{t \rightarrow \infty} \langle\Psi(t,0) \vert \Psi(t,0)\rangle.
\end{equation}
To obtain $\hat{P}_0$, we have to calculate
$\lim_{t \rightarrow \infty} \ket{\Psi(t,0)}$.
Substitute Eq.~\ref{eq:evolution-equation} into Eq.~\ref{eq:inverse-fourier} and let $n = 0$,
we derive the explicit form for $\ket{\Psi(t,0)}$
\begin{equation*}
\ket{\Psi(t, 0)} = \int_{-\pi}^{\pi}\frac{dk}{2\pi} \tilde{\Psi}(t, k)
= \int_{-\pi}^{\pi}\frac{dk}{2\pi} \sum_{j=1}^{\Delta}
\lambda_j^t\ket{\lambda_j}\langle\lambda_j\vert\tilde{\Psi}(0, k)\rangle.
\end{equation*}
From \textbf{Lemma 1} in~\cite{lyu2015localization}, we know that
the contributions to $\ket{\Psi(t, 0)}$ from items with $j=1, 2$ in the above equation
are negligible when $t$ approaches infinity.
As a result, $\ket{\Psi(t, 0)}$ is totally determined by the integrals with $j\geq 3$.
Since $\lambda_3 = 1$, and $\forall j \geq 4,\lambda_j = -1$,
we can further simplify the equation above by substituting into these constant eigenvalues.
The final expression is shown in Eq.~\ref{eq:origin-amplitude-2}.
In this equation, only $\ket{\lambda_3}$ is a function of $k$,
while for all $j \geq 4, \ket{\lambda_j}$ and
$\tilde{\Psi}(0, k)$ are independent on $k$ according to Eq.~\ref{eq:eigenvectors} and~\ref{eq:dfted-init-state} respectively.
Actually, Eq.~\ref{eq:origin-amplitude-2} can be understood as a series of linear maps from the
initial state $\tilde{\Psi}(0, k)$ to $\Psi(t, 0)$.
The linear maps are represented by a set of transformation matrices $F_j$ defined as
\begin{widetext}
\begin{eqnarray}\label{eq:origin-amplitude-2}
\lim_{t \rightarrow \infty} \ket{\Psi(t, 0)} &\sim& \int_{-\pi}^{\pi}\frac{dk}{2\pi}
\Big[ \ketbra{\lambda_3} + \sum_{j=4}^{\Delta}(-1)^t\ketbra{\lambda_j} \Big] \ket{\tilde{\Psi}(0, k)} \nonumber \\
~ &=& \Bigg[ \int_{-\pi}^{\pi}\frac{dk}{2\pi}\ketbra{\lambda_3}
+ \sum_{j=4}^{\Delta}(-1)^t \int_{-\pi}^{\pi}\frac{dk}{2\pi}\ketbra{\lambda_j} \Bigg] \ket{\tilde{\Psi}(0, k)},
\end{eqnarray}
\end{widetext}
\begin{eqnarray}
F_3 &=& \int_{-\pi}^{\pi}\frac{dk}{2\pi} \ketbra{\lambda_3},\label{eq:f_3} \\
F_j &=& \int_{-\pi}^{\pi}\frac{dk}{2\pi} \ketbra{\lambda_j} = \ketbra{\lambda_j}, \; \forall j \geq 4.\label{eq:f_j}
\end{eqnarray}
The matrices $F_j$ capture all information about the walker's behavior at its original position
when $t \rightarrow \infty$. The existence of localization is directly related to the system's initial state
via the matrices $F_j$.
For all $j \geq 4$, $F_j$ is independent on $k$, so it is a constant matrix and can be easily calculated.
The exact form of $F_3$ can be obtained by exploiting the eigenvector $\ket{\lambda_3}$.
Let $\kappa_1 = \frac{2}{1 + e^{-ik}}$, $\kappa_2 = \frac{2}{1 + e^{ik}}$,
it's obvious that $\kappa_1^\dag = \kappa_2$ and $\kappa_2^\dag = \kappa_1$.
Substitute $\omega_3 = 0$ into Eq.~\ref{eq:eigenvectors} for $j=3$,
we get the explicit form of $\ket{\lambda_3}$
\begin{eqnarray*}
\ket{\lambda_3} &=& \sqrt{N_3}\big[\kappa_1,\; \kappa_2,\; 1,\; \cdots,\; 1\big]^\dag,
\end{eqnarray*}
where
$
N_3 = \frac{1}{\kappa_1\kappa_1^\dag + \kappa_2\kappa_2^\dag + \tau}
= \frac{\tau + 4 + \tau\cos k}{1 + \cos k}
$ is the normalization factor.
Then
\begin{equation*}
F_3 = \int_{-\pi}^{\pi}\frac{dk}{2\pi}N_3
\begin{pmatrix}
\kappa_1\kappa_2 & \kappa_1\kappa_1 & \kappa_1 & \cdots & \kappa_1 \\
\kappa_2\kappa_2 & \kappa_2\kappa_1 & \kappa_2 & \cdots & \kappa_2 \\
\kappa_2 & \kappa_1 & 1 & \cdots & 1 \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
\kappa_2 & \kappa_1 & 1 & \cdots & 1
\end{pmatrix}.
\end{equation*}
Define
$\Theta_1 = \frac{1}{\tau} - \frac{\sqrt{2\tau + 4}}{\tau(\tau + 2)}$,
$\Theta_2 = \frac{\sqrt{2\tau + 4}}{2\tau + 4}$, and
$\Theta_3 = \frac{2}{\tau} - \frac{(\tau+4)\sqrt{2\tau + 4}}{2\tau(\tau + 2)}$,
we can show after some tedious calculations
\begin{eqnarray*}
&~&\int_{-\pi}^{\pi}\frac{dk}{2\pi}N_3 =
\int_{-\pi}^{\pi}\frac{dk}{2\pi}N_3\kappa_1 =
\int_{-\pi}^{\pi}\frac{dk}{2\pi}N_3\kappa_2 = \Theta_1, \\
&~&\int_{-\pi}^{\pi}\frac{dk}{2\pi}N_3\kappa_1\kappa_2 =
\int_{-\pi}^{\pi}\frac{dk}{2\pi}N_3\kappa_2\kappa_1 = \Theta_2,\\
&~&\int_{-\pi}^{\pi}\frac{dk}{2\pi}N_3\kappa_1\kappa_1 =
\int_{-\pi}^{\pi}\frac{dk}{2\pi}N_3\kappa_2\kappa_2 = \Theta_3.
\end{eqnarray*}
Thus the explicit form of $F_3$ is
\begin{equation*}
F_3 = \begin{pmatrix}
\Theta_2 & \Theta_3 & \Theta_1 & \cdots & \Theta_1 \\
\Theta_3 & \Theta_2 & \Theta_1 & \cdots & \Theta_1 \\
\Theta_1 & \Theta_1 & \Theta_1 & \cdots & \Theta_1 \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
\Theta_1 & \Theta_1 & \Theta_1 & \cdots & \Theta_1
\end{pmatrix}.
\end{equation*}
Substitute $F_j$ into Eq.~\ref{eq:origin-amplitude-2}, we get
\begin{equation}\label{eq:final-state}
\lim_{t \rightarrow \infty} \ket{\Psi(t, 0)} = \Big[ F_3 + \sum_{j=4}^{\Delta}(-1)^t F_j \Big] \ket{\tilde{\Psi}(0, k)},
\end{equation}
where $\ket{\tilde{\Psi}(0, k)}$ is given in Eq.~\ref{eq:dfted-init-state}.
Let $\lim_{t \rightarrow \infty} \ket{\Psi(t, 0)} = [\phi_1, \cdots, \phi_\Delta]^\dag$,
as $F_j$ has no impact on the first two components of $\ket{\tilde{\Psi}(0, k)}$ for all $j \geq 4$,
Eq.~\ref{eq:final-state} can be easily solved:
\begin{eqnarray*}
\phi_1 &=& \Theta_2\alpha + \Theta_3\beta, \\
\phi_2 &=& \Theta_3\alpha + \Theta_2\beta, \\
\phi_j &=& \Theta_1(\alpha + \beta), \; \forall j \geq 3.
\end{eqnarray*}
The asymptotic limit of the probability at origin $\hat{P}_0$ can be obtained now
\begin{equation*}
\hat{P}_0 = \vert\phi_1\vert^2 + \vert\phi_2\vert^2 + \tau\vert\phi_3\vert^2
= 2\cdot\frac{\tau + 4 - 2\sqrt{2\tau+4}}{\tau^2}.
\end{equation*}
\end{proof}
Though Theorem~\ref{thm:localization-prob} only considers localization probabilities
for a special class of initial states (given in Eq.~\ref{eq:init-state}),
we should point out that we are able to calculate the localization probability
by Eq.~\ref{eq:origin-prob-expression} and~\ref{eq:final-state} for arbitrary system's initial state
that satisfies
\begin{equation*}
\Psi(0) = \sum_{n \in \mathbb{Z}} \delta_{n,0} \ket{0} \otimes \sum_{j=1}^{\Delta} \alpha_j\ket{j},
\end{equation*}
where $\alpha_j \in \mathbb{C}$, and $\sum_{j=1}^{\Delta} \vert\alpha_j\vert^2 = 1$.
\section{Peak Velocity}\label{sec:velocity
In this section, we determine the peak velocity at which LQWs spread on the line.
The analytical method we use here is first described in~\cite{vstefavnak2012continuous}.
From their arguments, we know that the peak velocity is given by the first order of the stationary
points of the phase
\begin{equation*}
\tilde{\omega}_j \equiv \omega_j - \frac{n}{t}k.
\end{equation*}
The stationary point of the second order of $\tilde{\omega}_j$ corresponds to the solution of $k$.
We should notice that both the first and second derivatives of $\tilde{\omega}_j$ with respect to $k$ vanish.
Therefore in order to obtain the peak velocity, we need to solve equations
\begin{eqnarray}\label{eq:first-second-order}
\frac{d\tilde{\omega}_j}{dk} &=& \frac{d\omega_j}{dk} - \frac{n}{t} = 0, \nonumber\\
\frac{d^2\tilde{\omega}_j}{dk^2} &=& \frac{d^2\omega_j}{dk^2} = 0.
\end{eqnarray}
Assume $k_0$ is the solution of the second equation in Eq.~\ref{eq:first-second-order}, then
by the first equation we obtain the position of the peak after $t$ steps
$$
n = \frac{d\omega_j}{dk}\Big\vert_{k_0} t.
$$
The peak propagates with a constant velocity $\frac{d\omega_j}{dk}\big\vert_{k_0}$.
Now we show the peak velocities for LQWs on a line.
As the phases $\omega_j$ are constant for all $j \geq 3$, we immediately know that their corresponding
peak velocities are $v_S = 0$. This is easy to understand as the constant phases result in
the central peak of the probability distribution staying .
Thus the velocities of left and right travelling peaks are dominated by $\omega_{1,2}$.
We find the equations in Eq.~\ref{eq:first-second-order} can be solved
by investigating $\omega_1$ and $\omega_2$
\begin{eqnarray}
\frac{d\omega_{1,2}}{dk} &=& \pm \frac{\tau\sin k}{\sqrt{\tau(1-\cos k)(\tau\cos k + \tau + 4)}},\label{eq:first-derivate} \\
\frac{d^2\omega_{1,2}}{dk^2} &=& \pm 2\sqrt{\frac{\tau(1 - \cos k)}{ (\tau\cos k + \tau + 4)^3}}. \nonumber
\end{eqnarray}
In $k \in (-\pi, \pi]$, $d^2\omega_{1,2}/dk^2=0$ has a solution when $k_0 = 0$.
Evaluating $\omega_{1,2}/dk$ at $k_0$, we get the peak velocities of the left and right traveling probabilities
\begin{eqnarray}
v_R &=& \lim_{k \rightarrow 0^{+}} \frac{d \omega_2}{dk} = \sqrt{\frac{\tau}{\tau + 2}}, \label{eq:right-velocity} \\
v_L &=& \lim_{k \rightarrow 0^{+}} \frac{d \omega_1}{dk} = - \sqrt{\frac{\tau}{\tau + 2}}.\label{eq:left-velocity}
\end{eqnarray}
When the laziness factor satisfies $\tau=1$,
we recover the results presented in~\cite{vstefavnak2012continuous}.
As an illustrative example, we plot the walker's probability distribution of the LQW
whose laziness factor is $10$ in Fig.~\ref{fig:peak-velocity}.
The probability distribution contains three dominant peaks,
the left and right travelling peaks are given by the peak velocities $v_L$ and $v_R$ respectively.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{fig_peak.png}
\caption{ Probability distributions of LQWs after $T = 50$ steps,
for various laziness factors $\tau$.
The coin initial state is $\alpha=1/\sqrt{2}$, $\beta=i/\sqrt{2}$,
This state will give a symmetric walk.
We can easily identify three dominant peaks in each probability distribution.
We visualize the right peaks $\sqrt{1/3}T \approx 29$, $\sqrt{5/6}T \approx 46$ in grid lines
for $\tau=1$ and $\tau=10$. These theoretical peaks coincide with the positions of peaks
obtained from numerical simulations.
}\label{fig:peak-velocity}
\end{figure}
From Eq.~\ref{eq:right-velocity} (Eq.~\ref{eq:left-velocity}) we can see that as the laziness factor increases,
the right (left) peak velocity also becomes larger.
In this sense, we can control the spread behavior the quantum walker
and achieve faster spreading than the standard quantum walks.
In~\cite{vstefavnak2012continuous}, the authors offer a different way to
control the spread behavior of the walker by tuning the parameter $\rho$ of
the generalized Grover coin operator (see Eq.~14 in their paper), the underlying
quantum walks are still three-state quantum walks.
While in our paper we actually propose a \textit{multi-}state quantum walk scheme by
introducing different number of self-loops to each vertex,
the spread behavior of the walker can be controlled by tuning the laziness factor $\tau$,
the underlying coin operator is always Grover operator.
In the extreme case we have
$
\tau \rightarrow \infty \Rightarrow \vert v_{R,L} \vert = 1.
$
This indicates that if there is infinite self-loops in each vertex,
the quantum walker will propagate on the line with constant speed $1$.
This can be explained by investigating the coin operator $G$ defined in Eq.~\ref{eq:coin-operator}.
When $\tau \rightarrow \infty$, $G$ satisfies
\begin{eqnarray*}
\lim_{\tau \rightarrow \infty} G &\equiv& \lim_{\tau \rightarrow \infty} 2\ket{\psi}\bra{\psi} - \mathbb{I}_C \\
~ &=& \lim_{\tau \rightarrow \infty} \frac{2}{\tau+2}\sum_{j=1}\sum_{k=1}\ket{j}\bra{k} - \mathbb{I}_C \sim -\mathbb{I}_C,
\end{eqnarray*}
where $\mathbb{I}_C$ is the identity of $\mathcal{H}_C$.
That is, when $\tau \rightarrow \infty$, the coin operator $G$ approximates to $-\mathbb{I}_C$,
which results in a trivial quantum walk.
This fast spread behavior of LQWs is striking different from lazy random walks,
in which the additional self-loops will slow down the spread speed.
In the extreme case where $\tau \rightarrow \infty$, the classical walker will localize
in the origin and never spread.
\section{Weak Limit}\label{sec:weak-limit
In this section, we present a weak limit distribution for the rescaled LQW $X_t/t$ as $t \rightarrow \infty$.
It expresses an asymptotic behavior of the walk after long enough time.
The limit distribution is composed of a Dirac $\delta$-function
related to the localization probability calculated in Sec.~\ref{sec:localization} and
a continuous function with a compact support whose domain is given by
the peak velocities given in Sec.~\ref{sec:velocity}.
We also prove that LQWs spread ballistically.
The analytical method we use in this section is first proposed in~\cite{grimmett2004weak}
and we mainly follow the proof procedure outlined in~\cite{machida2015limit}.
What's more, one should keep in mind that in this paper we only consider a special
class of system's initial states defined in Eq.~\ref{eq:init-state}.
\begin{theorem}\label{thm:weak-limit}
For any real number $x$, we have
\begin{equation*}
\lim_{t \rightarrow \infty} \mathbb{P}(\frac{X_t}{t} \geq x) =
\int_{-\infty}^{x}dy \Big\{ \delta_{0,y}\hat{P} + f(y)I_{(-\Omega, \Omega)}(y) \Big\},
\end{equation*}
where
\begin{itemize}
\item $\delta_{0,y}$ is the Dirac $\delta$-function at the origin,
\item $\hat{P}$ is the sum of localization probabilities in all positions and satisfies
\begin{equation*}
\hat{P} = \frac{\sqrt{2\tau + 4}}{2\tau + 4}
+ 2\Big\{\frac{2}{\tau} - \frac{(\tau+4)\sqrt{2\tau + 4}}{2\tau(\tau + 2)}\Big\}\mathfrak{R}(\alpha^\dag\beta),
\end{equation*}
\item $f(y)$ is the weak limit density function defined in Eq.~\ref{eq:f-x},
\item $\Omega = \sqrt{\frac{\tau}{\tau+2}}$ is the bound of the compact support domain, and
\item $I_{\Gamma}(y)$ is the compact support function whose domain is $\Gamma$ and defined as
\begin{equation*}
I_{\Gamma}(y) = \left\{ \begin{array}{ll}
1, & y \in \Gamma, \\
0, & y \notin \Gamma.
\end{array} \right.
\end{equation*}
\end{itemize}
\end{theorem}
From the theorem we can see that the limit density function rescaled by time has a compact support
and its domain $(-\Omega, \Omega)$ is totally determined by the walker's travelling peak velocities.
A weak limit theorem of three-state walks is presented in~\cite{machida2015limit}.
Our results are the same as theirs when we let $\tau=1$ and set the parameters $c=-1/3$, $s=2\sqrt{2}/3$, $\beta=0$
in Theorem 2 of their paper.
One should note the difference between $\hat{P}$ and $\hat{P}_0$ (the localization probability at the origin)
studied in Sec.~\ref{sec:localization}.
Actually, $\hat{P}$ is the sum of localization probabilities in all positions,
i.e., $\hat{P} = \sum_{n\in\mathbb{Z}} \hat{P}_n$.
We are unable to derive an analytic expression for $\hat{P}_n$ for $n \neq 0$,
but luckily we can still calculate $\hat{P}$.
\begin{proof}
The $r$-th moment of the quantum walker's probability distribution can be calculated as
\begin{eqnarray*}
\mathbb{E}(X_t^r) &=& \sum_{n \in \mathbb{Z}} n^r \mathbb{P}(X_t = n) \\
~ &=& \int_{-\pi}^{\pi}\frac{dk}{2\pi}
\langle \tilde{\Psi}(t, k)\vert\Big( D^r\ket{\tilde{\Psi}(t, k)} \Big) \\
~ &=& (t)_r \int_{-\pi}^{\pi}\frac{dk}{2\pi}
\sum_{j=1}^{\Delta} \Big( i\frac{\lambda_j^{\prime}}{\lambda_j}\Big)^r
\Big\vert \langle \lambda_j \vert \tilde{\Psi}(0, k) \rangle \Big\vert^2 + O(t^{r-1}),
\end{eqnarray*}
where $D = i(d/dk)$ and $(t)_r = t(t-1)\cdots(t-r+1)$.
To have $X_t$ spatially rescaled by time, we divide both sides of the above equation by $t^r$ and
take a limit on $t$
\begin{eqnarray}\label{eq:moment-limit}
\lim_{t \rightarrow \infty} \mathbb{E}\Big[\Big(\frac{X_t}{t}\Big)^r \Big] &=&
\quad\sum_{j=1}^{2}\int_{-\pi}^{\pi}\frac{dk}{2\pi}
\Big( i\frac{\lambda_j^{\prime}}{\lambda_j}\Big)^r
\Big\vert \langle \lambda_j \vert \tilde{\Psi}(0, k) \rangle \Big\vert^2 \nonumber\\
~ &~& + \sum_{j=3}^{\Delta}\int_{-\pi}^{\pi}\frac{dk}{2\pi}
\Big\vert \langle \lambda_j \vert \tilde{\Psi}(0, k) \rangle \Big\vert^2.
\end{eqnarray}
As $F_j$ has no impact on the first two components of $\ket{\tilde{\Psi}(0, k)}$ for all $j \geq 4$,
the second term in Eq.~\ref{eq:moment-limit} can be easily calculated by making use of
the transformation matrix $F_3$ defined in Eq.~\ref{eq:f_3}
\begin{eqnarray}\label{eq:part1}
\sum_{j=3}^{\Delta}\int_{-\pi}^{\pi}\frac{dk}{2\pi}
\Big\vert \langle \lambda_j \vert \tilde{\Psi}(0, k) \rangle \Big\vert^2
&=&
\int_{-\pi}^{\pi}\frac{dk}{2\pi}\Big\vert \langle \lambda_3 \vert \tilde{\Psi}(0, k) \rangle \Big\vert^2\nonumber\\
~ &=&
\bra{\tilde{\Psi}(0, k)}\cdot F_3\cdot \ket{\tilde{\Psi}(0, k)} \nonumber\\
~ &=&
\Theta_2 + 2\Theta_3\mathfrak{R}(\alpha^\dag\beta).
\end{eqnarray}
Then we calculate the first term.
As $\lambda_j = e^{i\omega_j}$, we can get the derivation of $\lambda_j$
using the expressions for $d\omega_j/dk$ obtained in Eq.~\ref{eq:first-derivate}
for $j = 1, 2$
$$
i\frac{\lambda_j^\prime}{\lambda_j}
= -\frac{d\omega_j}{dk}
= (-1)^{j-1}\frac{\tau\sin k}{\sqrt{\tau(1-\cos k)(\tau\cos k + \tau + 4)}}.
$$
Putting $i\lambda_j^\prime/\lambda_j = x$ in the integrals of Eq.~\ref{eq:moment-limit} and after
some tedious calculations, we are able to show that
\begin{equation}\label{eq:part2}
\sum_{j=1}^{2}\int_{-\pi}^{\pi}\frac{dk}{2\pi}
\Big( i\frac{\lambda_j^{\prime}}{\lambda_j}\Big)^r
\Big\vert \langle \lambda_j \vert \tilde{\Psi}(0, k) \rangle \Big\vert^2
=
\int_{-\sqrt{\frac{\tau}{\tau+2}}}^{\sqrt{\frac{\tau}{\tau+2}}} x^rf(x) dx,
\end{equation}
where $f(x)$ satisfies
\begin{widetext}
\begin{equation}\label{eq:f-x}
f(x) = \frac{1}{\pi(1 - x^2)\sqrt{2\tau - 2(\tau+2)x^2}}
\Big\{ 1 + 2\mathfrak{R}(\alpha^\dag\beta)
+ 2(\vert\beta\vert^2-\vert\alpha\vert^2)x
+ \Big(1 - 2\mathfrak{R}(\alpha^\dag\beta)\frac{\tau+4}{\tau}\Big)x^2 \Big\}.
\end{equation}
\end{widetext}
Substitute Eq.~\ref{eq:part1} and \ref{eq:part2} into Eq.~\ref{eq:moment-limit},
we obtain the $r$-th moment of the quantum walker's probability distribution
\begin{eqnarray*}
\lim_{t \rightarrow \infty} \mathbb{E}\Big[\Big(\frac{X_t}{t}\Big)^r \Big] &=&
\hat{P} + \int_{-\sqrt{\frac{\tau}{\tau+2}}}^{\sqrt{\frac{\tau}{\tau+2}}} x^rf(x) dx \nonumber\\
~ &=& \int_{-\infty}^{\infty}
x^r\Big\{ \delta_{0,x}\hat{P} + f(x)I_{(-\Omega, \Omega)}(x) \Big\}dx.
\end{eqnarray*}
\end{proof}
As a corollary of the weak limit theorem, we prove all LQWs
with system initial states defined in Eq.~\ref{eq:init-state} spread ballistically
and obtain an analytical expression for the variance of the walker's probability distribution.
The variance of a walker's probability distribution is defined as
$$
\sigma^2 = E(X_t^2) - E^2(X_t) \sim ct^\alpha,
$$
where $c$ is the spread coefficient, and $\alpha$ is the spread exponent.
The spread behavior of a quantum walk is determined by the spread exponent of the variance.
If $\alpha = 2$, we say that the walk spreads ballistically;
If $\alpha = 1$, we say that the walk spreads diffusively.
It has been proved that for standard quantum walks $\alpha = 2$~\cite{chandrashekar2008optimizing},
while for random walks $\alpha = 1$~\cite{venegas2012quantum}.
\begin{corollary}\label{thm:coe}
For a LQW whose laziness factor is $\tau$,
if the walker starts with the system initial state given in Eq.~\ref{eq:init-state},
the variance of the walker's probability distribution satisfies
\begin{equation*}
\sigma^2 = c(\tau, \alpha, \beta) t^2,
\end{equation*}
where $c(\tau, \alpha, \beta)$ is the spread coefficient defined in Eq.~\ref{eq:coe-solution}.
\end{corollary}
We can see easily from Corollary~\ref{thm:coe} that all lackadaisical quantum walks spread ballistically
for system initial states defined in Eq.~\ref{eq:init-state} as the spread exponent is $2$.
Moreover, we obtain an analytical solution for the spread coefficient $c(\tau, \alpha, \beta)$ of the variance
in Eq.~\ref{eq:coe-solution}, from which we find that it is dependent on both $\tau$ and coin initial state $\alpha$, $\beta$
and the laziness factor $\tau$.
By tuning the parameters $\tau$, $\alpha$ and $\beta$, we can achieve arbitrary spread coefficients
in the range $(0, 1)$.
We conduct numerical simulations to calculate the spread coefficients for different laziness factors and
the comparison between numerical and theoretical results are illustrated in Fig.~\ref{fig:coe}.
\begin{proof}
By Theorem~\ref{thm:weak-limit} it is easy to see that
\begin{eqnarray*}
\sigma^2 &=& E(X_t^2) - E^2(X_t) \\
~ &\sim& t^2\int_{-\Omega}^{\Omega} x^2f(x) dx - t^2 \Big\{\int_{-\Omega}^{\Omega} xf(x) dx\Big\}^2 \\
~ &=& c(\tau, \alpha, \beta) \; t^2,
\end{eqnarray*}
where the coefficient function $c(\tau, \alpha, \beta)$ is defined as
\begin{equation}\label{eq:coe-expression}
c(\tau, \alpha, \beta) = \int_{-\Omega}^{\Omega} x^2f(x) dx - \Big\{\int_{-\Omega}^{\Omega} xf(x) dx\Big\}^2.
\end{equation}
Solving Eq.~\ref{eq:coe-expression}, we obtain the analytical solution for $c(\tau, \alpha, \beta)$ which has the form of
\begin{widetext}
\begin{equation}\label{eq:coe-solution}
c(\tau, \alpha, \beta) =
1 - \frac{(5\tau+8)\sqrt{2\tau+4}}{(2\tau+4)^2}
+ \Bigg\{ \frac{2(\tau^2+12\tau+16)\sqrt{2\tau+4}}{\tau(2\tau+4)^2} - \frac{4}{\tau} \Bigg\}\mathfrak{R}(\alpha^\dag\beta)
- \Bigg\{\Big(1 - \frac{\sqrt{2\tau+4}}{\tau+2}\Big)(\vert\beta\vert^2-\vert\alpha\vert^2) \Bigg\}^2.
\end{equation}
\end{widetext}
\end{proof}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{fig_coe.png}
\caption{Numerical and theoretical spread coefficients for different laziness factors.
The system initial state is $\alpha=\frac{1 + \sqrt{2}i}{2}$, $\beta=\frac{\sqrt{2}(1 + i)}{4}$.
This state is designated to guarantee that
both $\mathfrak{R}(\alpha^\dag\beta)$ and $\vert\beta\vert^2-\vert\alpha\vert^2$ do not equal to $0$.
The theoretical results are obtained by Eq.~\ref{eq:coe-solution}.
The numerical results are got by fitting the numerical data to function $ct^\alpha$ in Matlab.
The slight difference between two curves is due to that
the numerical data is obtained by running the walks for only 1000 steps.}\label{fig:coe}
\end{figure}
\section{Conclusion}\label{sec:conclusion
In this paper, we analyze in detail the properties of LQWs on a line
for arbitrary laziness factor $\tau$.
First, we study the localization phenomenon shown in the walks.
With the discrete Fourier transformation method,
we are able to present an explicit form for the localization probability of the walker
in the limit of $t \rightarrow \infty$.
The limiting coin state is obtained by a set of linear maps $F_j$ on the initial coin state.
This set of $F_j$ contain all information required to depict the walker's behavior at the origin.
The localization probability is the inner product of the limiting coin state,
which is shown independent on the initial coin state, and totally determined by $\tau$.
We also calculate the velocities of the left and right-travelling probability peaks
appeared in the walker's probability distribution.
We can control the spread behavior the quantum walks and achieve faster spreading than
the standard quantum walks by tuning the laziness factor.
Furthermore, we show that when $\tau$ approaches infinity, the LQW degenerates to a trivial walk.
At last, we calculate rigorously the system state
and get a long time approximation for the entire probability density function.
The density function has both the Dirac $\delta$-function and a continuous function
with a compact support whose domain is determined by the peak velocities.
As an application of the density function,
we prove that all LQWs spread ballistically,
and give an analytic solution for the variance of the walker's probability distribution.
The analytical results we obtain illustrate interesting characteristics of
LQWs compared to standard quantum walks and the corresponding lazy random walks.
For example, it is obvious that the greater the $\tau$ is, the more the walker prefers to stay in lazy random walks.
However, a lackadaisical quantum walker spread even faster with the increment of $\tau$.
That's why we say the lackadaisical quantum walker is \textbf{not} lazy at all.
\section*{Acknowledgement
The authors want to thank Haixing Hu, Qunyong Zhang, Xiaohui Tian and Huaying Liu for
the insightful discussions. K. W. wants to thank Takuya Machida for his kind help.
This work is supported by the National Natural Science Foundation of China
(Grant Nos. 61300050, 91321312, 61321491) and
the Chinese National Natural Science Foundation of Innovation Team (Grant No. 61321491).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,184
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I want to display:none if user hovers my banner for 500ms, but the following JQuery code is not working. Where is mistake?
If you want it to hide the banner when you've hovered over it for 500ms, then you need to save a reference to the DOM element being hidden. You probably also want to clear the timer if you've stopped hovering before the timeout fires. You'll need the signature that takes an in AND out handler separately. Store the timer handle and clear it when you stop hovering if it hasn't already expired.
The this variable changes meaning based on scope. Once inside the the function in the setTimeout() call this no longer refers to the .banner element.
So you need to "save" that reference so you can use in the function in the setTimeout() call.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 3,207
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Q: Is there a trick to invert a $3\times 3$ matrix? Consider any matrix which is $3\times3$ or bigger, but always square.
Is there a trick to invert quickly a matrix of this type without computing all the minors?
Thanks in advance.
A: Yes, there is: Gauß reduction, downwards, then upwards, on your matrix. Simultaneously, perform the same row operations on the unit matrix. The transformed unit matrix is the inverse.
A: Here is a method based on the Cayley-Hamilton theorem and Bocher's formula. Write the characteristic polynomial of an $n \times n$ matrix $A$ as $$\det(\lambda I-A)=\lambda^n+\beta_1\lambda^{n-1}+...+\beta_n.$$ The Cayley-Hamilton theorem says that$$A^n+\beta_1A^{n-1}+...+\beta_nI=0,\text {so} $$ $$A(A^{n-1}+\beta_1A^{n-2}+...+\beta_{n-1}I)=-\beta_nI$$ Thus, if $\beta_n=0$, the inverse does not exist; otherwise, the inverse is given by $$A^{-1}=\frac {-1}{\beta_n}(A^{n-1}+\beta_1A^{n-2}+...+\beta_{n-1}I)$$ To calculate the $\beta_{\nu}$ efficiently, use Bocher's formula $$\beta_1=-\mu_1,\text { } \beta_{\nu}=\frac{-1}{\nu}(\beta_{\nu -1}\mu_1+...+\beta_1\mu_{\nu -1}+\mu_{\nu})$$ where $\mu_i=tr(A^i)$ The most effective way to use this formula is to evaluate the matrix polynomial by "synthetic division," so for $n=3$ we have $$A^{-1}=\frac {-1}{\beta_3}(A(A+\beta_1I)+\beta_2I)$$ Note that we do not have to store the powers of $A$. All we have to do is evaluate successive powers of $A$ and use them to calculate the traces and $\beta_i$as we go along. This method works over any field of characteristic 0 or of characteristic greater than $n.$ It also works over any commutative ring with identity in which all the elements 2,3, ... ,$n$ are invertible.
A: It is often much more work to invert the matrix and use the inverse than to just go ahead and solve the linear system of equations of interest in the first place. Sure, matrix inverses are of huge theoretical importance, but you know the saying: "In theory, theory and practice are the same; in practice, they are very different".
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
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package com.adaptris.fs;
/**
* <p>
* Subclass of <code>FsException</code> indicating that a file has not be found.
* </p>
*/
public class FsFileNotFoundException extends FsException {
/**
*
*/
private static final long serialVersionUID = 2009081801L;
/**
* <p>
* Creates a new instance.
* </p>
*/
public FsFileNotFoundException() { }
/**
* <p>
* Creates a new instance with a reference to a previous
* <code>Exception</code>.
* </p>
* @param cause a previous, causal <code>Exception</code>
*/
public FsFileNotFoundException(Exception cause) {
super(cause);
}
/**
* <p>
* Creates a new instance with a description of the <code>Exception</code>.
* </p>
* @param description of the <code>Exception</code>
*/
public FsFileNotFoundException(String description) {
super(description);
}
/**
* <p>
* Creates a new instance with a reference to a previous
* <code>Exception</code> and a description of the <code>Exception</code>.
* </p>
* @param description of the <code>Exception</code>
* @param cause previous <code>Exception</code>
*/
public FsFileNotFoundException(String description, Exception cause) {
super(description, cause);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,604
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Home Justice Officials Tout Intensive Crime Reduction Efforts in Baltimore
Officials Tout Intensive Crime Reduction Efforts in Baltimore
William F. Zorzi
Federal, state and local law enforcement officials had the chance to exhale Wednesday – albeit briefly – as their collaborative efforts of the last 60 days showed some success in slowing the crippling violent crime that has gripped Baltimore over the last year.
Maryland Gov. Lawrence J. Hogan Jr. (R) took the lead in presenting an update on the joint initiative to combat violent crime in the city at a news conference at the Edward A. Garmatz U.S. Courthouse in Baltimore.
Baltimore Mayor Catherine E. Pugh listens as Gov. Lawrence J. Hogan Jr. describes recent crime-fighting initiatives in the city. Photo by William F. Zorzi
"The goal of this aggressive joint federal, state and local surge operation was to apprehend as many violent criminals as possible from the streets of Baltimore City," Hogan said.
The result, he said, was more than 500 arrests in Baltimore, including 259 of the city's most violent criminals. Among that number are 10 individuals who have been charged with murder, 10 charged with attempted murder, 21 charged with serious gun or weapons offenses, and 76 charged with aggravated assault or assault.
"When it comes to these repeat violent offenders, and the people shooting and killing people on the streets of Baltimore City, I want to be absolutely clear: We have no tolerance whatsoever to those who are engaging in this violent activity and terrorizing the streets of Baltimore," Hogan said.
"Make no mistake about it," he added. "We will find you, we will arrest you, and we will prosecute you to the fullest extent of the law."
Hogan went on to say that as a result of what he called "joint enhanced visibility patrols," law enforcement officials executed 44,135 premises checks, responded to 5,665 calls for service, and issued 5,333 citations. He said they "closed an additional 261 high-priority warrants on the most violent criminals in the city, and we made 263 additional arrests."
During the 60-day sweep, he said, parole and probation officers conducted 2,463 site visits, and 379 additional probation warrants were served.
Hogan was flanked by Baltimore Mayor Catherine E. Pugh, a Democrat, and Col. William M. Pallozzi, superintendent of the Maryland State Police. Beside them were acting Baltimore Police Commissioner Darryl D. De Sousa, who is awaiting confirmation by the City Council, and Daniel L. Board Jr., special agent in charge of the Baltimore Field Division of the federal Bureau of Alcohol, Tobacco, Firearms and Explosives (ATF).
So far in 2018, homicides, shootings and violent crime generally are down sharply in Baltimore, where there have been more than 300 killings in each of the last three years. Last year, there were 342 homicides, a record number per capita.
"We're going in the right direction now. It's still early, but we're encouraged by what we see so far," said De Sousa, a 30-year veteran of the Baltimore department. "This initiative was on steroids. So, this is what we needed to give us that jolt."
U.S. Marshal Johnny L. Hughes, a former Maryland state trooper, hosted the update in the federal offices.
Hughes, who resides in Baltimore, praised Hogan and Pugh, as well as De Sousa, whom he said he has known for years – both as a police officer and someone who has tended to the homeless and less fortunate along Baltimore's Old Town Mall on N. Gay Street, as part of a church-related outreach program.
"I know you got heart for the city, and you have a heart for people, and you're a good man," Hughes said of the acting police commissioner.
Other federal, state and local law enforcement authorities filled the conference room, and Hogan seemed to thank them all: the U.S. Marshals Service, FBI, ATF, Drug Enforcement Administration, Department of Homeland Security and U.S. Secret Service, as well as all the Maryland state police agencies and Division of Parole and Probation.
Hogan began the news conference by referencing the death of off-duty Prince George's County police officer, Cpl. Mujahid A. Ramzziddin, 51, a 14-year veteran of the department who was shot and killed Wednesday while protecting a neighbor in need of help in a domestic dispute. He ordered state flags to be flown at half-staff.
De Sousa, who was appointed by Pugh last month after she fired Police Commissioner Kevin F. Davis, was confirmed unanimously Wednesday night by a City Council committee, which in turn sent his name to the full body for approval.
Before her brief remarks, Pugh thanked Hogan for his help, particularly in dispatching more than 200 state parole and probation officers to Baltimore and assigning two to each of the city's nine police districts – 18 total — as had been the case at one time.
"This police department, I believe, under the direction of our commissioner designee De Sousa will make strides in how we attack and reduce violence in our city," Pugh said. "These kinds of collaborative efforts only can mean one thing for Baltimore: That we will become a safer city. that is the goal of this administration, that is the goal of this collaborative.
"We can make Baltimore one of the safest cities in America," she said.
Darryl D. De Sousa
Bill Zorzi was a Baltimore Sun reporter and editor for nearly 20 years, focusing on government and politics. An Annapolis bureau veteran, he wrote a weekly column, "The Political Game" for the paper.Zorzi and another former Sun reporter, David Simon, are longtime collaborators on acclaimed television projects, including the HBO series, "The Wire," and the HBO miniseries "Show Me a Hero," which dealt with an explosive housing desegregation case in Yonkers, NY.
Trump Beats Frosh With Dismissal of Emoluments Claims in Appellate Court
Amid Public Outcry, Montgomery Lawmakers Look to Create Policing Commission
Karen Sutter February 22, 2018 at 7:37 am
Pugh's words did not reflect her body language, which I believe exposes her real attitude toward Hogan's remarks.
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Looking to fill the stocking of your favorite stoner this holiday season? Well, look no further. We've curated the perfect list of gadgets and gizmos that any pot lover would be thrilled to find on Christmas morning.
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{"url":"https:\/\/www.rdocumentation.org\/packages\/MBESS\/versions\/4.6.0","text":"# MBESS v4.6.0\n\n0\n\nMonthly downloads\n\n0th\n\nPercentile\n\n## The MBESS R Package\n\nImplements methods that useful in designing research studies and analyzing data, with particular emphasis on methods that are developed for or used within the behavioral, educational, and social sciences (broadly defined). That being said, many of the methods implemented within MBESS are applicable to a wide variety of disciplines. MBESS has a suite of functions for a variety of related topics, such as effect sizes, confidence intervals for effect sizes (including standardized effect sizes and noncentral effect sizes), sample size planning (from the accuracy in parameter estimation [AIPE], power analytic, equivalence, and minimum-risk point estimation perspectives), mediation analysis, various properties of distributions, and a variety of utility functions. MBESS (pronounced 'em-bes') was originally an acronym for 'Methods for the Behavioral, Educational, and Social Sciences,' but at this point MBESS contains methods applicable and used in a wide variety of fields and is an orphan acronym, in the sense that what was an acronym is now literally its name. MBESS has greatly benefited from others, see <http:\/\/nd.edu\/~kkelley\/site\/MBESS.html> for a detailed list of those that have contributed and other details.\n\n## Functions in MBESS\n\n Name Description aipe.smd Sample size planning for the standardized mean different from the accuracy in parameter estimation approach ci.cv Confidence interval for the coefficient of variation ci.src Confidence Interval for a Standardized Regression Coefficient ci.cc Confidence interval for the population correlation coefficient ci.snr Confidence Interval for the Signal-To-Noise Ratio ci.sc.ancova Confidence interval for a standardized contrast in ANCOVA with one covariate ancova.random.data Generate random data for an ANCOVA model ci.sm Confidence Interval for the Standardized Mean HS.data Complete Data Set of Holzinger and Swineford's (1939) Study ci.reg.coef Confidence interval for a regression coefficient MBESS MBESS conf.limits.nct Confidence limits for a noncentrality parameter from a t-distribution conf.limits.ncf Confidence limits for noncentral F parameters ci.c Confidence interval for a contrast in a fixed effects ANOVA ci.pvaf Confidence Interval for the Proportion of Variance Accounted for (in the dependent variable by knowing the levels of the factor) ci.c.ancova Confidence interval for an (unstandardized) contrast in ANCOVA with one covariate covmat.from.cfm Covariance matrix from confirmatory (single) factor model. smd.c Standardized mean difference using the control group as the basis of standardization signal.to.noise.R2 Signal to noise using squared multiple correlation coefficient cor2cov Correlation Matrix to Covariance Matrix Conversion s.u Unbiased estimate of the population standard deviation smd Standardized mean difference ci.rc Confidence Interval for a Regression Coefficient ci.smd.c Confidence limits for the standardized mean difference using the control group standard deviation as the divisor. ci.smd Confidence limits for the standardized mean difference. ci.reliability Confidence Interval for a Reliability Coefficient ss.aipe.smd.sensitivity Sensitivity analysis for sample size given the Accuracy in Parameter Estimation approach for the standardized mean difference. ci.srsnr Confidence Interval for the Square Root of the Signal-To-Noise Ratio mr.cv Minimum risk point estimation of the population coefficient of variation ss.aipe.R2.sensitivity Sensitivity analysis for sample size planning with the goal of Accuracy in Parameter Estimation (i.e., a narrow observed confidence interval) ss.aipe.rmsea Sample size planning for RMSEA in SEM mediation.effect.plot Visualizing mediation effects ss.aipe.reliability Sample Size Planning for Accuracy in Parameter Estimation for Reliability Coefficients. ss.aipe.R2 Sample Size Planning for Accuracy in Parameter Estimation for the multiple correlation coefficient. mediation.effect.bar.plot Bar plots of mediation effects ss.aipe.src sample size necessary for the accuracy in parameter estimation approach for a standardized regression coefficient of interest ss.aipe.crd Find target sample sizes for the accuracy in unstandardized conditions means estimation in CRD ss.aipe.crd.es Find target sample sizes for the accuracy in standardized conditions means estimation in CRD ci.rmsea Confidence interval for the population root mean square error of approximation ss.aipe.src.sensitivity Sensitivity analysis for sample size planing from the Accuracy in Parameter Estimation Perspective for the standardized regression coefficient ss.power.R2 Function to plan sample size so that the test of the squared multiple correlation coefficient is sufficiently powerful. ss.aipe.rmsea.sensitivity a priori Monte Carlo simulation for sample size planning for RMSEA in SEM power.density.equivalence.md Density for power of two one-sided tests procedure (TOST) for equivalence ss.aipe.c.ancova.sensitivity Sensitivity analysis for sample size planning for the (unstandardized) contrast in randomized ANCOVA from the Accuracy in Parameter Estimation (AIPE) Perspective conf.limits.nc.chisq Confidence limits for noncentral chi square parameters ss.aipe.cv Sample size planning for the coefficient of variation given the goal of Accuracy in Parameter Estimation approach to sample size planning ss.aipe.sc Sample size planning for Accuracy in Parameter Estimation (AIPE) of the standardized contrast in ANOVA power.equivalence.md Power of Two One-Sided Tests Procedure (TOST) for Equivalence mr.smd Minimum risk point estimation of the population standardized mean difference ss.aipe.cv.sensitivity Sensitivity analysis for sample size planning given the Accuracy in Parameter Estimation approach for the coefficient of variation. ci.sc Confidence Interval for a Standardized Contrast in a Fixed Effects ANOVA intr.plot.2d Plotting Conditional Regression Lines with Interactions in Two Dimensions cv Function to calculate the regular (which is also biased) estimate of the coefficient of variation or the unbiased estimate of the coefficient of variation. ss.aipe.pcm Sample size planning for polynomial change models in longitudinal study ss.aipe.sem.path.sensitiv a priori Monte Carlo simulation for sample size planning for SEM targeted effects ss.aipe.rc Sample size necessary for the accuracy in parameter estimation approach for an unstandardized regression coefficient of interest power.equivalence.md.plot Plot power of Two One-Sided Tests Procedure (TOST) for Equivalence prof.salary Cohen et. al. (2003)'s professor salary data set ss.aipe.reg.coef Sample size necessary for the accuracy in parameter estimation approach for a regression coefficient of interest intr.plot Regression Surface Containing Interaction ci.R2 Confidence interval for the population squared multiple correlation coefficient vit.fitted Visualize individual trajectories with fitted curve and quality of fit ss.aipe.sm.sensitivity Sensitivity analysis for sample size planning for the standardized mean from the Accuracy in Parameter Estimation (AIPE) Perspective transform_Z.r Transform Fischer's Z into the scale of a correlation coefficient ss.aipe.rc.sensitivity Sensitivity analysis for sample size planing from the Accuracy in Parameter Estimation Perspective for the unstandardized regression coefficient transform_r.Z Transform a correlation coefficient (r) into the scale of Fischer's Z ss.aipe.smd Sample size planning for the standardized mean difference from the Accuracy in Parameter Estimation (AIPE) perspective ss.aipe.reg.coef.sensitivity Sensitivity analysis for sample size planning from the Accuracy in Parameter Estimation Perspective for the (standardized and unstandardized) regression coefficient mediation Effect sizes and confidence intervals in a mediation model ss.aipe.sm Sample size planning for Accuracy in Parameter Estimation (AIPE) of the standardized mean ss.aipe.c.ancova Sample size planning for a contrast in randomized ANCOVA from the Accuracy in Parameter Estimation (AIPE) perspective ss.aipe.sem.path Sample size planning for SEM targeted effects ss.aipe.sc.ancova.sensitivity Sensitivity analysis for the sample size planning method for standardized ANCOVA contrast ss.power.reg.coef sample size for a targeted regression coefficient ss.aipe.sc.sensitivity Sensitivity analysis for sample size planning for the standardized ANOVA contrast from the Accuracy in Parameter Estimation (AIPE) Perspective ss.aipe.sc.ancova Sample size planning from the AIPE perspective for standardized ANCOVA contrasts ss.aipe.c Sample size planning for an ANOVA contrast from the Accuracy in Parameter Estimation (AIPE) perspective ss.power.sem Sample size planning for structural equation modeling from the power analysis perspective t.and.smd.conversion Conversion functions for noncentral t-distribution theta.2.Sigma.theta Compute the model-implied covariance matrix of an SEM model verify.ss.aipe.R2 Internal MBESS function for verifying the sample size in ss.aipe.R2 vit Visualize individual trajectories ss.power.pcm Sample size planning for power for polynomial change models ss.power.rc sample size for a targeted regression coefficient var.ete The Variance of the Estimated Treatment Effect at Selected Covariate Values in a Two-group ANCOVA. upsilon This function implements the upsilon effect size statistic as described in Lachowicz, Preacher, & Kelley (in press) for mediation. Sigma.2.SigmaStar Construct a covariance matrix with specified error of approximation ci.R Confidence interval for the multiple correlation coefficient Variance.R2 Variance of squared multiple correlation coefficient Cor.Mat.Lomax Correlation matrix for Lomax (1983) data set CFA.1 One-factor confirmatory factor analysis model Cor.Mat.MM Correlation matrix for Maruyama & McGarvey (1980) data set Expected.R2 Expected value of the squared multiple correlation coefficient Gardner.LD The Gardner learning data, which was used by L.R. Tucker F.and.R2.Noncentral.Conversion Conversion functions from noncentral noncentral values to their corresponding and vice versa, for those related to the F-test and R Square. No Results!\n\n## Details\n\n Type Package Date 2019-6-12 License GPL-2 | GPL-3 URL http:\/\/nd.edu\/~kkelley\/site\/MBESS.html RoxygenNote 6.0.0 NeedsCompilation no Packaged 2019-06-12 19:59:57 UTC; kkelley Repository CRAN Date\/Publication 2019-06-12 20:30:03 UTC\n imports boot , gsl , lavaan , MASS , methods , mnormt , nlme , OpenMx , parallel , sem , semTools depends R (>= 3.2.0) , stats Contributors","date":"2020-01-26 20:11:08","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4231332838535309, \"perplexity\": 3866.2032104717096}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-05\/segments\/1579251690379.95\/warc\/CC-MAIN-20200126195918-20200126225918-00305.warc.gz\"}"}
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NEFU National and International Recognition
University in brief
The Mammoth Museum
Institute of the East
NEFU Endowment Fund
NEFU in rankings
English Taught Courses and Programs
Diamond Studies
University of the Arctic
Learn Russian at NEFU
Short lab course in Japan
Inactive (completed) programs
Joint Research Projects
NEFU School of International Studies
Arctic Innovation Center (AIC)
Small Innovation Businesses
Vestnik of NEFU
NEFU grant
Research facts and figures
NEFU Laboratories
NEFU Scientific Schools
NEFU Educational Scientific Centres
Areas of Scientific Research
Life in Yakutsk
Visa support form for students
NEFU International Club
About the Republic of Sakha (Yakutia)
Students social life
Projects&activities
Center of Educational Materials (supported by Goethe-Institute)
French Resource Center
Deutsche Tage in Jakutsk 2019
Anatoly SHESTAKOV / head adminstration
Aisen Nikolaev: NEFU should become one of the flagships of higher education worldwide
On September 4, the head of Yakutia, Aisen Nikolaev, at a meeting of the Presidium of the State Council of the Russian Federation, emphasized that North-Eastern and Far Eastern Federal Universities should become flagships of higher education at the world level.
In his report, the head of the republic said that within the frameworks of the national program for the development of the Far East, the priority is to increase human capital and prevent the threat of urbanization from the regions of the Far Eastern Federal District.
The head emphasized that in addressing this issue, the main focus is on education - general and professional.
"The flagships of higher education — the Far Eastern and North-Eastern Federal Universities — should reach the world level. Special measures are also needed - for example, free training of young people at the best universities in Russia and worldwide who connect their future with the Far Eastern Federal District," he said.
Also, the head of Yakutia added in an interview that the republic is becoming increasingly interesting for business: "This is an incentive that would encourage young people to remain in the Far East. Injust a year, we rose 30 positions up in the national ranking of investment attractiveness of subjects - from the 52nd to 22nd place. Traditionally, there are companies from Asian countries among foreign investors in Yakutia - Japan and China."
The rate on these countries is explained by the geographical location and level of human capital. "Specialists in this country work are being trained on the basis of NEFU, and it is their self-realization that has led investors to come to Yakutia and have someone to work with and with whom to communicate," Aisen Nikolaev said.
Author: Kristina STEPANOVA, NEFU Nesroom
Copying of articles is allowed only if there is an active (clickable) link to the source page of North-Eastern Federal University. Link must be directly next to the material, must be visible and direct (without using the java-scripts).
Sector of Corporate Websites is not responsible for the information provided by the structural units
58 Belinsky str. Yakutsk, 677000
international@s-vfu.ru
2020 © M.K. Ammosov North-Eastern Federal University .
Sector of Corporate Websites
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{"url":"https:\/\/www.gamedev.net\/forums\/topic\/625642-c-question-about-variable-declarations\/","text":"# [C++] Question about Variable Declarations\n\nThis topic is 2090 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.\n\n## Recommended Posts\n\nThis is not really a code problem I am having more of a why isn't this allowed by the C++ standard, I googled a bit but couldn't find a solution and would like to have an answer to the question.\n\nSay we have this piece of sample code:\n class SimpleTestObject { public: SimpleTestObject() : a(0) {} SimpleTestObject(int a) : a(a) {} void print() { std::cout << a << std::endl; } private: int a; }; int _tmain(int argc, _TCHAR* argv[]) { SimpleTestObject obj(); obj.print(); return 0; } \n\nWe get a \"error C2228: left of '.print' must have class\/struct\/union\" and to remove this error I need to remove the \"()\" on the line above. However if I pass a number in this constructor it compiles without complaining.\nMy question is why am I not allowed to called the default constructor explicitly, and more over why does the compiler think I am defining a function in a function?\n\n##### Share on other sites\nIf it's allowed, how can you declare a function with no parameters and return an object of SimpleTestObject?\n\nWhen never you write,\nSimpleTestObject ThisIsAFunction();\n\nThe compiler has to treat it as an object.\nThen there is no way you can write a function like that. Edited by wqking\n\n##### Share on other sites\n\n... [color=#282828][font=helvetica, arial, verdana, tahoma, sans-serif]\n\n### [background=rgb(250, 251, 252)]why does the compiler think I am defining a function in a function?[\/background][\/font] [\/quote] Because according to the syntax rules of C++, that is (almost) what you are doing. To be accurate, what the compiler thinks is that you are declaring a function called \"obj\" which takes no parameters and returns a SimpleTestObject. See also C++'s most vexing parse.\n\n##### Share on other sites\n\n...\n\n[background=rgb(250, 251, 252)]why does the compiler think I am defining a function in a function?[\/background]\n\nBecause according to the syntax rules of C++, that is (almost) what you are doing. To be accurate, what the compiler thinks is that you are declaring a function called \"obj\" which takes no parameters and returns a SimpleTestObject.\n\n[\/quote]\n\nYeah I have seen that link before must have forgotten about it, cheers though","date":"2018-02-19 12:38:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.22072811424732208, \"perplexity\": 1986.865561510853}, \"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-2018-09\/segments\/1518891812584.40\/warc\/CC-MAIN-20180219111908-20180219131908-00521.warc.gz\"}"}
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The Next Generation StormPro2 weather storm tracker with Intelicharge. The future of storm tracking!
The new StormPro2 storm tracking system utilizes ground-breaking new technology and represents a quantum leap forward in storm tracking capabilities. Featuring next generation design, these lightning detectors for sale use advanced new charging circuitry to eliminate the need for the constant purchasing and disposing of environmentally unfriendly batteries .
With the new Intelicharge technology, the StormPro2 weather storm tracker can be used and charged anywhere in the world. Plus, storm distances are now automatically displayed in both Miles and Kilometers making this new system the most versatile portable storm tracking device ever made.
The StormPro2 lightning detectors for sale with Intelicharge are available in multiple housing colors including Safety Yellow, Carbon Tech and Camouflage.
Wall charger is included with these weather storm trackers.
Available accessories include: Wall mounting pod, Hard shell carry case and DC power adapter.
Tracks dangerous thunderstorm activity from over 130 miles.
Large backlit LCD display with multiple storm status updates including: Storm Distance, ETA and Time to Clear, etc.
Automatically displays in both Miles and Kilometers.
Visual and Audible storm warning alerts.
Rechargeable on any current and plug configuration in the world.
Automatic 5 hours shut-down function.
Low battery indicator. Spanish and other languages available upon request.
StormPro2 lightning detectors for sale are manufactured under license from Spectrum Electronics, Auburn, GA.
Great company to deal with. I will use them again.
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| 9,587
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Чемпионат мира по полумарафону 2010 года прошёл 16 октября в китайском городе Наньнин. Решение о проведение соревнований в городе Наньнин было принято 20 ноября 2009 года на заседании совета IAAF в Монако. Общий призовой фонд соревнований составил 245 000 долларов США. За первое место было денежное вознаграждение в размере 30 000 долларов. За победу в командном первенстве 15 000 долларов.
Дистанция полумарафона проходила по улицам города. Всего было проведено 2 забега — мужчины и женщины. Также разыгрывались победители в командном первенстве — складываются три лучших результата от страны и по сумме наименьшего времени определялись чемпионы.
Призовой фонд
Общий призовой фон соревнований составил 245 000 долларов США.
В личном первенстве
1-е место — 30 000
2-е место — 15 000
3-е место — 10 000
4-е место — 7000
5-е место — 5000
6-е место — 3000
В командном зачёте
1-е место — 15 000
2-е место — 12 000
3-е место — 9000
4-е место — 7500
5-е место — 6000
6-е место — 3000
Результаты
Мужчины
Женщины
Примечания
Ссылки
Результаты
Чемпионат мира по полумарафону
2010 год в лёгкой атлетике
События 16 октября
Октябрь 2010 года
Спорт в Китае в 2010 году
Наньнин
Чемпионаты мира в Китае
Международные соревнования по лёгкой атлетике в Китае
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{
"redpajama_set_name": "RedPajamaWikipedia"
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Q: How to use JvisualVM to profile Spark on Yarn cluster? I know how to do this work on single executor host, just export JMX port.
But some potential performance issue only can be happened in large cluster, if we add JMX options to "spark.executor.extraJavaOptions", listen address of multiple executors on a host will conflict.
I also tried to start a jstatd daemon, but the visualvm can't do cpu profiling in this way(it has message on UI).
Is there way to profiling Spark application in large scale cluster especially pinpoint what does each thread do in some time and cpu cost profiling? Not considering commercial tools.
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{
"redpajama_set_name": "RedPajamaStackExchange"
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| 2,119
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