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Writer, communications strategist, creative consultant. I work with mission-driven organisations — from development agencies and cultural institutions to companies — to build their brand, engage audiences and improve impact. I'm passionate about telling stories that reveal untold truths, highlight unsung heroes and celebrate hidden beauty. | {
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Q: Select based on timestamp and update timestamp with zero How do I select records from a date field which has a time (HH:MM:SS.Milisecond) value greater than zero from a MongoDB collection and update it with a time (HH:MM:SS) value as zero by keeping the date value the same as the existing value in a Python script?
The current data would look like as below -
1) "createdDate" : ISODate("2015-10-10T00:00:00Z")
2) "createdDate" : ISODate("2015-10-11T00:00:00Z")
3) "createdDate" : ISODate("2015-10-12T00:00:00Z")
4) "createdDate" : ISODate("2015-10-13T01:04:30.515Z")
5) "createdDate" : ISODate("2015-10-14T02:05:50.516Z")
6) "createdDate" : ISODate("2015-10-15T03:06:60.517Z")
7) "createdDate" : ISODate("2015-10-16T04:07:80.518Z")
How do I select only rows 4, 5, 6, and 7 using mongodbsql and update it with the timestamp as zero in a Python script?
After an update, the data would look like as below -
1) "createdDate" : ISODate("2015-10-10T00:00:00Z")
2) "createdDate" : ISODate("2015-10-11T00:00:00Z")
3) "createdDate" : ISODate("2015-10-12T00:00:00Z")
4) "createdDate" : ISODate("2015-10-13T00:00:00Z")
5) "createdDate" : ISODate("2015-10-14T00:00:00Z")
6) "createdDate" : ISODate("2015-10-15T00:00:00Z")
7) "createdDate" : ISODate("2015-10-16T00:00:00Z")
A: ISODate() is represented as a datetime object by PyMongo. MongoDB assumes that dates and times are in UTC. There are several ways to get midnight (start of a day) for a given UTC time d:
>>> from datetime import datetime, time, timedelta
>>> d = datetime(2015, 10, 13, 1, 4, 30, 515000)
>>> datetime(d.year, d.month, d.day) # @user3100115' answer
datetime.datetime(2015, 10, 13, 0, 0) # 369 ns
>>> datetime.fromordinal(d.toordinal()) # 451 ns
datetime.datetime(2015, 10, 13, 0, 0)
>>> datetime.combine(d, time.min) # 609 ns
datetime.datetime(2015, 10, 13, 0, 0)
>>> d - (d - d.min) % timedelta(days=1) # Python 3
datetime.datetime(2015, 10, 13, 0, 0) # 1.87 µs
>>> datetime(*d.timetuple()[:3])
datetime.datetime(2015, 10, 13, 0, 0) # 2.34 µs
>>> from calendar import timegm
>>> datetime.utcfromtimestamp((timegm(d.timetuple()) // 86400) * 86400) # POSIX
datetime.datetime(2015, 10, 13, 0, 0) # 4.72 µs
A: The best way to update your documents and set the time to 00:00:00 is using the datetime module, because createdDate is a datetime object in Python, so you can use the datetime instance attributes day, year, and month.
from datetime import datetime
from pymongo import MongoClient
client = MongoClient()
db = client.test
collection = db.collection
bulkOp = collection.initialize_ordered_bulk_op()
count = 0
for doc in collection.find():
year = doc['createdDate'].year
month = doc['createdDate'].month
day = doc['createdDate'].day
new_date = datetime(year, month, day)
bulkOp.find({'_id': doc['_id']}).update({'$set': {'createdDate': new_date}})
count = count + 1
if count == 125:
bulkOp.execute()
bulkOp = collection.initialize_ordered_bulk_op()
if count % 125 != 0:
bulkOp.execute()
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,060 |
Q: SQL Proper Case Function Query Refinement I have a table of names that I am trying to convert from UPPERCASE to Proper Case. And the below code does almost exactly what I am. When I was testing it out I noticed that I had folks who had Roman Numerals in their name, Mc* and O'* in the table. Naturally the query converted any multiple Roman Numeral into Iv like it was supposed to and any MCDONALDS or O'DANIEL were converted into Mcdonalds and O'daniel. I was trying to figure out how to make a clean change to this function so I can run my update query but I'm still peacemilling my SQL knowledge together in off hours. Any help/suggestions would be much appreciated. I did a google search and found several examples but the ones I tried didn't work. The amount of corrections I would have to do is relatively minor (17 corrections out of 1000 row table), but I'd like to try and tidy it up to limit as many human errors as possible.
Thank you in advance for your help.
CREATE FUNCTION [dbo].[f_ProperCase]
(@Text as varchar(80))
RETURNS varchar(80) as
BEGIN
DECLARE @Reset bit
DECLARE @Ret varchar(80)
DECLARE @i int
DECLARE @c char(1)
SELECT @Reset = 1, @i=1, @Ret = ''
WHILE @i <= LEN(@Text)
SELECT @c= SUBSTRING(@Text,@i,1),
@Ret = @Ret + CASE WHEN @Reset=1 THEN UPPER(@c) ELSE LOWER(@c) END,
@Reset= CASE WHEN
CASE WHEN SUBSTRING(@Text,@i-4,5) like '_[a-z] [DOL]''' THEN 1
WHEN SUBSTRING(@Text,@i-4,5) like '_[a-z] [D][I]' THEN 1
WHEN SUBSTRING(@Text,@i-4,5) like '_[a-z] [M][C]' THEN 1
WHEN SUBSTRING(@Text,@i-4,5) like '_[a-z] [M][c][*]' THEN 1 --Convert MCDONALD to McDonald
WHEN SUBSTRING(@Text,@I-4,5) like '_[a-z] [O][''][*]' THEN 1 --Convert O'DONNEL to O'Donnel
ELSE 0
END = 1
THEN 1
ELSE CASE WHEN @c like '[a-zA-Z]' or @c in ('''') THEN 0
ELSE 1
END
END,
@i = @i +1
RETURN @Ret
end
A: I would do it differently:
CREATE FUNCTION [dbo].[f_ProperCase]
(@Text as varchar(80))
RETURNS varchar(80) as
BEGIN
DECLARE @Reset bit
DECLARE @Ret varchar(80)
DECLARE @i int
DECLARE @c char(1)
DECLARE @Text1 varchar(81)
SELECT @Reset = 1, @i=1, @Ret = '', @Text1 = ' ' + @Text
WHILE @i <= LEN(@Text1)
SELECT @c= SUBSTRING(@Text1,@i,1),
@Ret = @Ret + CASE WHEN @Reset=1 THEN UPPER(@c) ELSE LOWER(@c) END,
@Reset= CASE WHEN
CASE WHEN SUBSTRING(@Text1,@i-2,3) like ' [DdOoLl]''' THEN 1
WHEN SUBSTRING(@Text1,@i-2,4) like ' [Mm][cC][a-zA-Z]' THEN 1
WHEN SUBSTRING(@Text1,@i-3,5) like ' [Mm][Aa][cC][a-zA-Z]' THEN 1
ELSE 0
END = 1
THEN 1
ELSE CASE WHEN @c like '[a-zA-Z]' or @c in ('''') THEN 0
ELSE 1
END
END,
@i = @i +1
RETURN stuff(@Ret, 1, 1, '')
end
This function supports O', L', D', as well as Mc, and Mac. The function is also converts from any case (not only the upper case) to the proper case
SQL Fiddle
select dbo.f_ProperCase('CORMACK')
,dbo.f_ProperCase('Mcdonald ronald')
,dbo.f_ProperCase('o''hara')
| | | |
|---------|-----------------|--------|
| Cormack | McDonald Ronald | O'Hara |
A: I tweaked some to handle suffixes being included in the name. This handled almost anything I threw at it. We sometimes had hyphenated names as well as periods after some suffixes. I know handling from I-IX was overkill, but was easy enough to add the X to the V check.
ALTER FUNCTION dbo.udf_ProperCase
(@Text as varchar(1024))
RETURNS varchar(1024)
AS
/*
Created to ProperCase most common LastName (Mc/Mac/O'/D'/L') and handle I(first)-IX(nineteenth) suffixes
Original Code: https://stackoverflow.com/questions/22923616/sql-proper-case-function-query-refinement
SELECT dbo.udf_ProperCase('iitest demetrii mcdonald o''neil victor second 2nd ii iii iv v vi vii viii ix x xi test')
*/
BEGIN
DECLARE @Reset bit
DECLARE @Ret varchar(1024)
DECLARE @i int
DECLARE @c char(1)
DECLARE @Text1 varchar(1025)
SELECT @Reset = 1, @i=1, @Ret = '', @Text1 = ' ' + @Text + ' ' --Ensure one space before to make first char upper and after to handle suffixes
--Loop through each character, comparing prior/next to determine if need to handle
WHILE @i <= LEN(@Text1)
BEGIN
SELECT
@c= SUBSTRING(@Text1,@i,1)
,@Ret = @Ret + CASE WHEN @Reset=1 THEN UPPER(@c) ELSE LOWER(@c) END
,@Reset= CASE
WHEN
CASE
WHEN SUBSTRING(@Text1,@i-2,3) like '[ -][DdOoLl][''`]' THEN 1 --Names (including hyphenated) beginning with D/O/L
WHEN SUBSTRING(@Text1,@i-2,4) like ' [Mm][cC][a-zA-Z]' THEN 1 --Names with Mc
WHEN SUBSTRING(@Text1,@i-3,5) like ' [Mm][Aa][cC][a-zA-Z]' THEN 1 --Names with Mac
WHEN SUBSTRING(@Text1,@i-1,4) like ' [Ii][Ii][ .]' THEN 1 --Handle II (include ending with Space or period)
WHEN SUBSTRING(@Text1,@i-1,5) like ' [Ii][Ii][Ii][ .]' THEN 1 --Handle III
WHEN SUBSTRING(@Text1,@i-2,4) like ' [Ii][Ii][ .]' THEN 1 --Handle II
WHEN SUBSTRING(@Text1,@i-2,5) like ' [Ii][Ii][Ii][ .]' THEN 1 --Handle III
WHEN SUBSTRING(@Text1,@i-3,4) like ' [Ii][Ii][ .]' THEN 1 --Handle II
WHEN SUBSTRING(@Text1,@i-3,5) like ' [Ii][Ii][Ii][ .]' THEN 1 --Handle III
WHEN SUBSTRING(@Text1,@i-1,4) like ' [Ii][VvXx][ .]' THEN 1 --Handle IV
WHEN SUBSTRING(@Text1,@i-1,4) like ' [VvXx][Ii][ .]' THEN 1 --Handle VI
WHEN SUBSTRING(@Text1,@i-1,5) like ' [VvXx][Ii][Ii][ .]' THEN 1 --Handle VII
WHEN SUBSTRING(@Text1,@i-1,6) like ' [VvXx][Ii][Ii][Ii][ .]' THEN 1 --Handle VIII
WHEN SUBSTRING(@Text1,@i-2,4) like ' [VvXx][Ii][ .]' THEN 1 --Handle VI
WHEN SUBSTRING(@Text1,@i-2,5) like ' [VvXx][Ii][Ii][ .]' THEN 1 --Handle VII
WHEN SUBSTRING(@Text1,@i-2,6) like ' [VvXx][Ii][Ii][Ii][ .]' THEN 1 --Handle VIII
WHEN SUBSTRING(@Text1,@i-3,4) like ' [VvXx][Ii][ .]' THEN 1 --Handle VI
WHEN SUBSTRING(@Text1,@i-3,5) like ' [VvXx][Ii][Ii][ .]' THEN 1 --Handle VII
WHEN SUBSTRING(@Text1,@i-3,6) like ' [VvXx][Ii][Ii][Ii][ .]' THEN 1 --Handle VIII
ELSE 0
END = 1 THEN 1
ELSE
CASE
WHEN @c like '[a-zA-Z`]' or @c in ('''') or @c like '[0-9]' THEN 0 --If any letter, single-quote or number, then keep next lowercase
ELSE 1 --Anything else (e.g. Space dash, And(&), etc. make next Upper-Case)
END
END
,@i = @i +1
END
RETURN stuff(@Ret, 1, 1, '')
END
GO
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,689 |
Q: React-native - Background bluetooth tracking possible? I want to develop an app, if the user get a bluetooth signal if the app is not opened, send a push notification. How is that possible? Should i save the bluetooth signal in a server database and send a notification over firebase?
I read, that iOS background tasks are a grey area.
I orient myself on the corona warn app. There this app is also in the background, tracks the bluetooth signals and gets a push (probably via a server then, right?).
| {
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} | 2,592 |
Q: Spark 2.0 with Zeppelin 0.6.1 - SQLContext not available I am running spark 2.0 and zeppelin-0.6.1-bin-all on a Linux server. The default spark notebook runs just fine, but when I try to create and run a new notebook in pyspark using sqlContext I get the error "py4j.Py4JException: Method createDataFrame([class java.util.ArrayList, class java.util.ArrayList, null]) does not exist"
I tried running a simple code,
%pyspark
wordsDF = sqlContext.createDataFrame([('cat',), ('elephant',), ('rat',), ('rat',), ('cat', )], ['word'])
wordsDF.show()
print type(wordsDF)
wordsDF.printSchema()
I get the error,
Traceback (most recent call last):
File "/tmp/zeppelin_pyspark-7635635698598314374.py", line 266, in
raise Exception(traceback.format_exc())
Exception: Traceback (most recent call last):
File "/tmp/zeppelin_pyspark-7635635698598314374.py", line 259, in
exec(code)
File "", line 1, in
File "/spark/spark-2.0.0-bin-hadoop2.7/python/pyspark/sql/context.py", line 299, in createDataFrame
return self.sparkSession.createDataFrame(data, schema, samplingRatio)
File "/spark/spark-2.0.0-bin-hadoop2.7/python/lib/py4j-0.10.1-src.zip/py4j/java_gateway.py", line 933, in __call__
answer, self.gateway_client, self.target_id, self.name)
File "/spark/spark-2.0.0-bin-hadoop2.7/python/pyspark/sql/utils.py", line 63, in deco
return f(*a, **kw)
File "/spark/spark-2.0.0-bin-hadoop2.7/python/lib/py4j-0.10.1-src.zip/py4j/protocol.py", line 316, in get_return_value
format(target_id, ".", name, value))
Py4JError: An error occurred while calling o48.createDataFrame. Trace:
py4j.Py4JException: Method createDataFrame([class java.util.ArrayList, class java.util.ArrayList, null]) does not exist
at py4j.reflection.ReflectionEngine.getMethod(ReflectionEngine.java:318)
at py4j.reflection.ReflectionEngine.getMethod(ReflectionEngine.java:326)
at py4j.Gateway.invoke(Gateway.java:272)
at py4j.commands.AbstractCommand.invokeMethod(AbstractCommand.java:128)
at py4j.commands.CallCommand.execute(CallCommand.java:79)
at py4j.GatewayConnection.run(GatewayConnection.java:211)
at java.lang.Thread.run(Thread.java:745)
When I try the same code with "sqlContext = SQLContext(sc)" it works just fine.
I have tried setting the interpreter "zeppelin.spark.useHiveContext false" configuration but it did not work.
I must obviously be missing something since this is such a simple operation. Please advice if there is any other configuration to be set or what I am missing.
I tested the same piece of code with Zeppelin 0.6.0 and it is working fine.
A: SparkSession is the default entry-point for Spark 2.0.0, which is mapped to spark in Zeppelin 0.6.1 (as it is in the Spark shell). Have you tried spark.createDataFrame(...)?
| {
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\section{Introduction}
The B method~\cite{b-book} is an outstanding formal method for safety critical systems, in particular, railway systems. Such systems have quite strong requirements such as real-time constraints and fault (in)tolerance due to the nature of their problem domain that may involve lives. Therefore, they are natural candidates for the application of formal methods for their specification and validation.
One of the main components of the B method are machine descriptions in the Abstract Machine Notation (AMN). They describe how a software component must behave in a safety-critical system.
Such descriptions are suitable for an automata-based interpretation and therefore can be naturally specified as transition systems~\cite{Clarke:2000:MC:332656} and validated by techniques such as model checking~\cite{Clarke:2000:MC:332656}.
This paper proposes B Maude, a tool for the validation of AMN descriptions. Our approach applies formal semantics of programming languages techniques. It is comprised by a set of basic programming languages constructs, inspired on Peter Mosses' Component-based Semantics~\cite{Mosses:2009:CS:1596486.1596489} whose semantics are given in terms of a generalization of Gordon Plotkin's Interpreting Automata~\cite{plotkin}. We named the resulting formalism $\uppi$ Framework~\cite{2018arXiv180504650B}. The semantics of AMN statements is given in terms of $\uppi$ constructions, or $\uppi$ Lib as we call it. B Maude is the implementation of the $\uppi$ Framework and the transformation from AMN to $\uppi$ in the Rewriting Logic language Maude~\cite{maude/2007}. It allows for execution by rewriting, symbolic search with narrowing and Linear Temporal Logic model checking of AMN descriptions. The B Maude prototype can be downloaded from \url{https://github.com/ChristianoBraga/BMaude}.
This paper is organized as follows. In Section~\ref{sec:preliminaries} we recall the syntax of AMN and the use of Maude as a formal meta-tool. Section~\ref{sec:modpi} outlines the $\uppi$ Framework. Section~\ref{sec:pi-den-for-amn} formalizes AMN in $\uppi$ by giving denotations of AMN statements as $\uppi$ Lib constructions. Section~\ref{sec:bmaude} talks about B Maude by showing the application of search and model checking to an AMN description and outlines their implementation in B Maude. Section~\ref{sec:related-work} discusses work related to the formal semantics of AMN. Section~\ref{sec:conclusion} wraps up this paper with a summary of its contents and points to future work.
\section{Preliminaries}\label{sec:preliminaries}
\subsection{Abstract Machine Notation grammar and example}\label{sec:amn-cfg-and-ex}
In B Maude, we organize the Abstract Machine Notation (AMN) syntax in two parts. The first one is a context-free grammar for AMN expressions and commands, called Generalized Substitution Language. The second part declares machine level statements such as machines, variables, constants, values and operations. The AMN grammar will be used in Section~\ref{sec:pi-den-for-amn} when we specify the semantics of AMN as denotations in the $\uppi$ Framework.
Only an excerpt is covered, due to space constraints, just enough to describe the example in Listing~\ref{amn:mutex}.
The Generalized Substitution Language, which essentially defines \emph{expressions} and \emph{commands}, is comprised by:
\begin{itemize}
\item identifiers, denoted by non-terminal $\langle\mathit{GSLIdentifiers}\rangle$\footnote{Non-terminals $\langle\mathit{ID}\rangle$, $\langle\mathit{RAT}\rangle$ and $\langle\mathit{BOOL}\rangle$ are left unspecified but simply denote what their names imply.},
\item arithmetic expressions, denoted by non-terminal $\langle\mathit{GSLExpression}\rangle$, predicates, denoted by non-terminal $\langle\mathit{GSLPredicate}\rangle$, and
\item substitutions, denoted by non-terminal $\langle\mathit{GSLSubstitution}\rangle$.
\end{itemize}
The notation for expressions and predicates is quite standard. Substitutions are essentially commands in AMN. Command `\_\texttt{:=}\_' denotes an assignment (an infix operator where `\_' denotes the positions of its operands), `\texttt{IF$\_$THEN$\_$END}' and `\texttt{IF\_THEN\_ELSE\_END}' denote conditionals, and `\texttt{WHILE\_DO}' denotes unbounded repetition.
Operator `\_\texttt{OR}\_' represents the bounded choice substitution. Keyword `\texttt{BEGIN\_END}' is a declaration and yields a scope where its substitutions must be evaluated within. The GSL grammar (excerpt) in BNF notation is as follows.
\begin{grammar}
<GSLIdentifiers> ::= `bid('<ID>`)' | `grat(' <RAT>`)' | `gboo(' <BOOL>`)'
<GSLExpression> ::= <GSLIdentifiers> | <GSLPredicate>
\alt <GSLExpression> `+' <GSLExpression> | $\ldots$
<GSLPredicate> ::= <GSLExpression> `==' <GSLExpression>
\alt <GSLExpression> `/\char`\\' <GSLExpression> | $\ldots$
<GSLSubstitution> ::=
<GSLIdentifiers> := <GSLExpression>
\alt `IF'~<GSLPredicate>~`THEN'~<GSLSubstitution>~\\`ELSE'~<GSLSubstitution>~`END'
\alt `WHILE' <GSLPredicate> `DO' <GSLSubstitution>
\alt <GSLSubstitution> `OR' <GSLSubstitution>
\alt `BEGIN' <GSLSubstitution> `END' <GSLSubstitution> | $\ldots$
\end{grammar}
The context-free grammar for AMN's machine-related statements
essentially defines \emph{declarations}:
\begin{itemize}
\item a `\texttt{MACHINE\_\_END}' declaration, denoted by non-terminal $\langle\mathit{AMNMachine}\rangle$, declares variables, constants, initializations for either or both, and operations,
\item a `\texttt{VARIABLES}' declaration (resp. `\texttt{CONSTANTS}'), in $\langle\mathit{AMNAbsVariables}\rangle$, is only a list of identifiers, and
\item a $\langle\mathit{AMNValuation}\rangle$ declaration associates a substitution to the initialization of a variable or constant. An operation declaration, denoted by non-terminal $\langle\mathit{AMNOperation}\rangle$, associates a substitution to an (operation) identifier and a list of (identifier) parameters.
\end{itemize}
The AMN grammar in BNF notation is as follows.
\begin{grammar}
<AMNMachine> ::= `MACHINE' <GSLIdentifiers> <AMNClauses> `END' | `MACHINE' <GSLIdentifiers> `END'
<AMNClauses> ::= <AMNAbsVariables> | <AMNAbsConstants> | <AMNOperations> | \\ <AMNValuesClause> | <AMNClauses> <AMNClauses>
<AMNAbsVariables> ::= `VARIABLES' <AMNIdList>
<AMNAbsConstants> ::= `CONSTANTS' <AMNIdList>
<AMNIdList> ::= <AMNIdList> `,' <AMNIdList>
<AMNValuesClause> ::= `VALUES' <AMNValSet>
<AMNValuation> ::= <GSLIdentifiers> `=' <GSLExpression>
<AMNValSet> ::= <AMNValSet> `;' <AMNValSet>
<AMNOperations> ::= `OPERATIONS' <AMNOpSet>
<AMNOperation> ::= <GSLIdentifiers> `=' <GSLSubstitution>
<AMNOpSet> ::= <AMNOpSet> `;' <AMNOpSet>
<AMNOperation> ::= <GSLIdentifiers> `(' <AMNIdList> `)' <GSLSubstitution>
\end{grammar}
\subsubsection{The \texttt{MUTEX} machine.}
As an illustrative example, Listing~\ref{amn:mutex} declares the \texttt{MUTEX} machine (which is actually executable in B Maude). It will be our running example in this paper. In Section~\ref{sec:bmaude}, we will use it to explain how B Maude is implemented. The \texttt{MUTEX} machine specifies a simple mutual exclusion protocol with two processes competing to enter a critical section. Each process, represented by an abstract variable in the \texttt{MUTEX} machine, can be in one of two possible states: \texttt{idle}, \texttt{wait}, or \texttt{crit}, encoded as constants in the machine. The protocol is encoded as the (parameterless) operation \texttt{mutex} that simply runs forever and, depending on the state of the processes, may non-deterministically change one of the processes state. For example, substitution
{\scriptsize\begin{amn}
IF p1 == idle /\ p2 == idle THEN p1 := wait OR p2 := wait ELSE $\ldots$
\end{amn}}
\noindent declares that, in a situation where both \texttt{p1} and \texttt{p2} are in the \texttt{idle} state, either one or both of them may change to \texttt{wait}.
\begin{amn}[caption=\texttt{MUTEX} protocol in the Abstract Machine Notation, label=amn:mutex]
MACHINE MUTEX
VARIABLES p1 , p2
CONSTANTS idle , wait , crit
VALUES
p1 = 0 ; p2 = 0 ;
idle = 0 ; wait = 1 ; crit = 2
OPERATIONS
mutex =
WHILE true DO
BEGIN
IF p1 == idle /\ p2 == idle
THEN (p1 := wait OR p2 := wait)
ELSE IF p1 == idle /\ p2 == wait
THEN p1 := wait OR p2 := crit
ELSE IF p1 == idle /\ p2 == crit
THEN p1 := wait OR p2 := idle
ELSE IF p1 == wait /\ p2 == idle
THEN p1 := crit OR p2 := wait
ELSE IF p1 == wait /\ p2 == wait
THEN p1 := crit OR p2 := crit
ELSE IF p1 == wait /\ p2 == crit
THEN p2 := idle
ELSE IF p1 == crit /\ p2 == idle
THEN p1 := idle OR p2 := wait
ELSE IF p1 == crit /\ p2 == wait
THEN p1 := idle
END END END END END END END END
END
END
\end{amn}
\subsection{Maude as a meta-tool}\label{sec:pre-maude}
The Maude system and language~\cite{maude/2007} is a high-performance implementation of Rewriting Logic~\cite{meseguer92}, a formalism for the specification of concurrent systems that has been shown to be able to represent quite naturally many logical and semantic frameworks~\cite{MartiOliet2002}.
A (concurrent) system is specified by a rewrite system $\mathcal{R} = (\Sigma, E \cup A, R)$ where $\Sigma$ denotes its signature, $E$ the set of equations, $A$ a set of axioms, and $R$ the set of rules. The equational theory $(\Sigma, E \cup A)$ specifies the \emph{states} of the system, which are terms in the $\Sigma$-algebra modulo the set of $E$ equations and $A$ axioms, such as associativity, commutativity and identity. Combinations of such axioms give rise to different rewrite theories such that rewriting takes place \emph{modulo} such axioms. Rules $R$ specify the (possibly) non-terminating behavior, that takes place modulo the equational theory $(\Sigma, E \cup A)$.
Computations in Maude are represented by rewrites according to either equations, rules or both in a given module. Functional modules may only declare equations while system modules may declare both equations and rules. Equations are assumed (that is, yield proof-obligations) to be Church-Rosser and terminating~\cite{Baader:1998:TR:280474}. Rules have to be coherent: no rewrite should be missed by alternating between the application of rules and equations.
An equational theory $\mathcal{E} = (\Sigma, E \cup A)$ is declared in Maude as a \emph{functional} module with concrete syntax `\texttt{fmod $\mathcal{E}$ is $I$ $\Sigma$ $E$ endfm}', where $I$ is the list of modules to be included by $\mathcal{E}$. A sort $s$ in $\Sigma$ is declared with syntax `\texttt{sort} $s$'. A subsort inclusion between $s$ and $s'$ is declared with syntax `\texttt{subsort $s$ < $s'$}'. An operation $o :\ \stackrel{\to}{s_i} \ \to s'$ in $\Sigma$ is declared with syntax `\texttt{op $o$ : $\stackrel{\to}{s_i}$ -> $s'$ [$A^*$] .}', where $\stackrel{\to}{s_i}$ is the domain of $o$ denoted by the product of $s_i$ sorts, $1 \le i \le n$, $n \in \mathbbm{N}$, and $A^*$ is a list of $A$ attributes including `\texttt{assoc}', `\texttt{comm}', `\texttt{id: $c$}', `\texttt{idem}', that denote what their names imply, and $c$ is a constructor operator. Equations are declared with syntax `$\mathtt{eq} ~L~ \mathtt{=} ~R~.$' where $L$ and $R$ are terms in $\mathcal{T}_{\Sigma/A}(X)$, the $\Sigma$-algebra with variables in $X$, modulo axioms $A$. Equations can be conditional, declared, essentially, with syntax `\texttt{ceq $L$ = $R$ if $\bigwedge^n_i$ $L_i$ = $R_i$ .}', where $n \in \mathbbm{N}$, and $L_i$, $R_i \in \mathcal{T}_{\Sigma/A}(X)$. A rewrite theory $\mathcal{R} = (\Sigma, E \cup A, R)$ is declared in Maude as a \emph{system} module with concrete syntax `\texttt{mod $\mathcal{R}$ is $I$ $\Sigma$ $E$ $R$ endm}' where $\Sigma$ and $E$ are declared with the same syntax as for functional modules and rules in $R$ are declared with syntax `\texttt{rl [l] : $L$ => $R$ .}' where $l$ is the rule label. As equations, rules can also be conditional, declared with syntax `\texttt{crl $L$ => $R$ if $\bigwedge^n_i$ $L_i$ = $R_i$ $\land$ $\bigwedge^m_j$ $L_j$ => $R_j$ .}', where $L_j$, $R_j \in \mathcal{T}_{\Sigma/A}(X)$ and $m \in \mathbbm{N}$.
One of the distinctive characteristics of Maude is the support to meta-programming. Meta-level applications in Maude, applications that use meta-programming, apply the so called \emph{descent functions}~\cite[Ch.14]{maude/2007}. Such functions use a representation of modules as terms in a \emph{universal} theory, implemented in Maude as a system module called META-LEVEL in its prelude. Some of the descent functions are metaParse, metaReduce, metaRewrite and metaSearch.
\begin{itemize}
\item Function metaParse receives a (meta-represented) module denoting a grammar, a set of quoted identifiers representing the (user) input and a quoted identifier representing the rule that should be applied to the given input qids, and returns a term in the signature of the given module.
\item Descent function metaReduce receives a (meta-represented) module and a (meta-represented) term and returns the (meta-represented) canonical form of the given term by the exhaustive application of the (Church-Rosser and terminating) \emph{equations}, only, of the given module. An interesting example of metaReduce is the invocation of the model checker at the meta-level: (i) first, module MODEL-CHECKER must be included in a module that also includes the Maude description of the system to be analyzed, and (ii) one may invoke metaReduce of a meta-representation of a term that is a call to function modelCheck, with appropriate parameters, defined in module MODEL-CHECKER.
\item Function metaRewrite simplifies, in a certain number of steps, a given term according to both equations and rules (assumed coherent, that is, no term is missed by the alternate application of equations and rules) of the given module.
\item The descent function metaSearch looks for terms that match a given \emph{pattern}, from a given term, according to a choice of rewrite relation from $\Rightarrow^*$, $\Rightarrow^+$, $\Rightarrow^!$, denoting the reflexive-transitive closure of the rewrite relation, the transitive closure of the rewrite relation or the rewrite relation that produces only canonical forms.
\end{itemize}
\subsection{Writing a compiler in Maude}\label{sec:comp-in-maude}
A compiler for programs in a (source) language $L$ into programs in a (target) language $L'$ can be written in Maude as a meta-level application $\mathcal{C}$. The main components of $\mathcal{C}$ are: (i) a context-free grammar for $L$, (ii) the abstract syntax of $L$, (iii) a parser for programs in $L$ that generates abstract syntax trees for $L$, (iv) a transformer from the abstract syntax of a program in $L$ into the abstract syntax of a program in $L'$, (v) a pretty-printer for the abstract syntax of programs in $L'$, and (vi) a command-line user-interface for interacting with the compiler .
The context-free grammar of a language $G = (V, T, P, S)$, where $V$ is the set of variables or non-terminals, $T$ is the set of terminals, $P$ is the set of productions of the form $V \to \alpha$, with $\alpha \in (V \cup T)^*$, and $S \not\in (V \cup T)$ the initial symbol, is represented, in Maude, as a functional module $\mathcal{G} = (\Sigma, \emptyset \cup A)$, that is, $E = \emptyset$, where, essentially, non-terminals are captured as sorts in $\Sigma$, non-terminal relations are captured by subsort inclusion, also in $\Sigma$, and terminals are represented as operations with appropriate signature in $\Sigma$, possibly with properties, such as associativity, declared in $A$.
The parser for $L$ is a meta-function in a functional module $\mathcal{P} = (\Sigma, E)$ in Maude that includes, at least, the (i) functional module denoting the grammar of $L$, (ii) the functional module denoting the abstract syntax of $L$ and (iii) the functional module META-LEVEL. The set $E$ of equations in $\mathcal{P}$ are defined by structural induction on the syntax of $L$ \emph{encoded as meta-terms in Maude}, that is, they are such that the left-hand side of an equation is a meta-term denoting a statement in $L$ and its right-hand side is a term in the initial algebra of the functional module denoting the abstract syntax of $L$.
A transformer from the AST of $L$ to the AST of $L'$ is a meta-function in a functional module $\mathcal{T} = (\Sigma, E)$ including, at least, the functional modules for the AST for $L$ and $L'$ and the META-LEVEL. Each equation in $E$ is such that: (i) its left-hand side is given by a term with variables in the initial algebra of the functional module representing the AST of $L$, and, similarly, (ii) its right-hand side denotes a term with variables in the initial algebra of the functional module representing the AST of $L'$. When $L'$ is Maude itself, that is, the compiler generates \emph{a rewrite theory representing the rewriting logic semantics of} $L$, the many tools available in Maude, such as the rewrite module axioms engine, narrowing and Linear Temporal Logic model checker, are ``lifted'' to programs in $L$ through its rewriting logic semantics.
A pretty-printer for the AST of $L'$ is a meta-function in a functional module $\mathcal{PP} = (\Sigma, E)$ such that the equations in $E$ produce a list of quoted identifiers from a term in the initial algebra of the functional module denoting the AST of $L'$.
\section{$\uppi$ Framework}\label{sec:modpi}
In~\cite{2018arXiv180504650B}, the first author outlines the $\uppi$ Framework, a simple semantic framework for compiler construction based on Peter Mosses' Component-based Semantics (CBS)~\cite{Mosses:2009:CS:1596486.1596489} and Gordon Plotkin's Interpreting Automata~\cite{plotkin}. The idea is to implement a formal core language that can be reused in the implementation of different compilers. The framework has two components: (i) $\uppi$ Lib, a library of basic programming language constructs inspired by Mosses' CBS and based on previous work of the first author with others, and (ii) $\uppi$ automata, an automata-based formalism for the specification of the semantics of programming language constructs that generalizes Plotkin's Interpreting Automata. Each component is discussed next, in Sections~\ref{sec:uppi-lib} and \ref{sec:uppi-automata}, respectively.
\subsection{$\uppi$ Lib signature}\label{sec:uppi-lib}
$\uppi$ Lib is a subset of Constructive MSOS~\cite{Mosses:2004:FCF}, as implemented in~\cite[Ch. 6]{msc-chalub}.
The signature of $\uppi$ Lib is organized in five parts, and implemented in four different modules in Maude: (i) expressions, that include basic values (such as rational numbers and Boolean values), identifiers, arithmetic and Boolean operations, (ii) commands, statements that produce side effects to the memory store, (iii) declarations, which are statements that construct the constant environment, (iv) output and (v) abnormal termination.
Grammar~\ref{grm:uppi-lib} declares the CFG for $\uppi$ Lib identifiers, arithmetic expressions and commands. Given their simplicity, they should be self-explanatory. Due to space constraints, we limit our exposition to these two syntactic classes only.
\begin{Grammar}
\begin{grammar}\scriptsize
<Control> ::= <Exp> | `ADD' | `SUB' | `MUL' | `DIV'
<Exp> ::= <Id> | <AExp> | <BExp>
<Id> ::= `idn' <ID>
<AExp> ::= <RAT> | <AOp> <AExp> <AExp>
<AOp> ::= `add' | `sub' | `mul' | `div'
<Control> ::= <Cmd> | `ASSIGN' | `LOOP' | `IF'
<Cmd> ::= `nop' | `choice' <Cmd> <Cmd> | \\
`assign' <Id> <Exp> | `loop' <BExp> <Cmd>
\end{grammar}
\caption{$\uppi$ Lib excerpt}
\label{grm:uppi-lib}
\end{Grammar}
\subsection{$\uppi$ automata}\label{sec:uppi-automata}
$\uppi$ automata are a generalization of Plotkin's Interpreting Automata as defined in~\cite{plotkin}.
Let $\mathcal{L}$ be a programming language generated by a Context Free Grammar (CFG) $G = (V, T, P, S)$ defined in the standard way where $V$ is the finite set of variables (or non-terminals), $T$ is the set of terminals, $P \subseteq V \times (V \cup T)^*$ and $S \not\in V$ is the start symbol of $G$.
An interpreting automaton for $\mathcal{L}$ is a tuple $\mathcal{I} = (\Sigma, \Gamma, \rightarrow, \gamma_0, F)$ where $\Sigma = T$, $\Gamma$ is the set of configurations, $\rightarrow \subseteq \Gamma \times \Gamma$ is the transition relation, $\gamma_0 \in \Gamma$ is initial configuration, and $F$ the unitary set of final configurations with the single element $\langle \emptyset, \emptyset, \emptyset \rangle$. Configurations in \(\Gamma\) are triples of the general form
\(
\Gamma = \mathit{Value~Stack} \times \mathit{Memory} \times \mathit{Control~Stack},
\)
where $\mathit{Value~Stack} = (L(G))^*$ with $L(G)$ the language accepted by $G$,
the set \(\mathit{Memory}\) is a finite map \(\mathit{Var}
\to_{\mathit{fin}} \mathit{Storable}\) with $\mathit{Var} \in V$ and $\mathit{Storable} \subseteq T^*$, and the
$\mathit{Control~Stack} = (L(G) \cup \mathit{KW})^*$, where $\mathit{KW}$ is the set of keywords of $\mathcal{L}$.
A computation in $\mathcal{I}$ is defined as $\rightarrow^*$, the reflexive-transitive closure of the transition relation.
$\uppi$ automata are interpreting automata
whose \emph{configurations are sets of semantic components} that include, at least,
a $\mathit{Value~Stack}$, a $\mathit{Memory}$ and a $\mathit{Control~Stack}$.
Plotkin's stacks and memory in
Interpreting Automata (or environment and stores of Structural
Operational Semantics) are generalized to the concept of \emph{semantic
component}, as proposed by Peter Mosses in the Modular
SOS~\cite{Mosses:2004:MSOS} approach to the formal semantics of
programming languages.
Formally, a $\uppi$ automaton is an interpreting automaton $\mathcal{I}$ where, given an abstract preorder \emph{Sem}, for semantic components, the configurations of $\mathcal{I}$ are $\Gamma = \uplus^n_{i=1} \mathit{Sem}$, with $n
\in \mathbbm{N}$, $\uplus$ denoting the disjoint union operation of
$n$ semantic components, with $\mathit{Value~Stack}$, $\mathit{Memory}$ and $\mathit{Control~Stack}$ subsets of $\mathit{Sem}$.
Let us look now at the $\uppi$ automaton for the fragments of $\uppi$ Lib CFG in Grammar~\ref{grm:uppi-lib}, for arithmetic expressions and commands.
The values in the $\mathit{Value~Stack}$ are
elements of the set
$\mathbbm{B} \cup \mathbbm{R} \cup \langle\mathit{Id}\rangle \cup \langle\mathit{BExp}\rangle \cup \langle\mathit{Cmd}\rangle,$
where \(\mathbbm{B}\) is the
set of Boolean values, \(\mathbbm{R}\) is the set of rational numbers. The $\mathit{Control~Stack}\) is defined as the set
$(\langle\mathit{Cmd}\rangle \cup \langle\mathit{BExp}\rangle \cup \langle\mathit{AExp}\rangle \cup \mathit{KW})^*$, with
$\mathit{KW}= \{\mathtt{ADD}, \mathtt{SUB}, \mathtt{MUL}, \mathtt{DIV}, \mathtt{ASSIGN},\mathtt{IF}, \mathtt{LOOP}\}$.
Informally, the computations of a $\uppi$ automaton mimic the behavior of a calculator
in Lukasiewicz postfix notation, also known as reverse Polish notation. A typical computation of a $\uppi$ automaton
\emph{interprets} a statement $c(p_1, p_2, \ldots, p_n) \in L(G)$ on the top of $\mathit{Control~Stack}$ \(C\) of a
configuration \(\gamma = (S, M, C) \cup \gamma'\), where $\gamma' \in \Gamma$, by unfolding its subtrees $p_i \in L(G)$ and $c \in \mathit{KW}$ that are then pushed back into $C$, possibly updating the $\mathit{Value~Stack}$
\(S\) with intermediary results of the interpretation of the $c(p_1, p_2, \ldots, p_n)$, and the $\mathit{Memory}$, should $c(p_1, p_2, \ldots, p_n) \in L(\langle\mathit{Cmd}\rangle)$.
For the transition relation of $\mathcal{I}$, let us consider the rules for
arithmetic sum expressions,
\begin{eqnarray}
\label{eq:val}\langle S, M, n~ C \rangle \cup \gamma & \Rightarrow & \langle n ~ S, M, C \rangle \cup \gamma \\
\label{eq:add1}\langle S, M, \mathtt{add}(e_1, e_2) ~ C \rangle \cup \gamma & \Rightarrow &
\label{eq:add2} \langle S, M, e_1 ~ e_2 ~ \mathtt{ADD}~ C \rangle \cup \gamma \\
\langle n ~ m ~ S, M, \mathtt{ADD}~ C \rangle \cup \gamma & \Rightarrow & \langle (n + m)~ S, M, C \rangle \cup \gamma
\end{eqnarray}
where \(e_i\) are meta-variables for arithmetic expressions, and \(n, m
\in \mathbbm{R}\). Rule~\ref{eq:add1} specifies that when the
arithmetic expression \(\mathtt{add(e_1, e_2)}\) is on top of the control
stack \(C\), then operator \texttt{ADD} should be pushed to \(C\) and then expression's operands \(e_1\) and \(e_2\) will be recursively
evaluated, as a computation is the reflexive-transitive closure of
relation \(\rightarrow\), leading to a configuration with an element in $\mathbbm{B} \cup \mathbbm{R}$ left on
top of the value stack $S$, as specified by
Rule~\ref{eq:val}. Finally, when $\mathtt{ADD}$ is on top of the control
stack $C$, and there are two natural numbers on top of $S$, they are
popped, added and pushed back to the top of $S$.
The rules for unbounded loop have standard meaning, as specified by Rules~\ref{eq:loop-unfold} to~\ref{eq:false-loop}. When a \texttt{loop} instruction is on top of the control stack, (i) the loop is pushed to the top of the value stack, the keyword \texttt{LOOP} and the loop's predicate expression (denoted by variable $E$ in Rule~\ref{eq:loop-unfold}) are pushed to the top of the control stack, (ii) if the recursive evaluation of $E$ produces \emph{true} on top of the value stack, the loop instruction and the body of the loop instruction are pushed to the top of the control stack, and (iii) if the recursive evaluation of $E$ produces \emph{false} on top of the control stack, the program evaluation continues with the remainder of the control stack $C$.
\begin{eqnarray}
\label{eq:loop-unfold}
\langle S, \mathtt{loop(E, K)}~ C \rangle \cup \gamma & \Rightarrow &
\langle (\mathtt{loop(E, K)}~ S), E ~\mathtt{LOOP}~ C \rangle \cup \gamma \\
\label{eq:true-loop}
\langle \mathit{true} ~\mathtt{loop(E, K)}~ S, \mathtt{LOOP}~ C \rangle \cup \gamma & \Rightarrow &
\langle S, K ~\mathtt{loop(E, K)}~ C \rangle \cup \gamma \\
\label{eq:false-loop}
\langle \mathit{false} ~\mathtt{loop(E, K)}~ S, \mathtt{LOOP}~ C \rangle \cup \gamma & \Rightarrow &
\langle S, C \rangle \cup \gamma
\end{eqnarray}
\subsection{Model checking $\uppi$ automata}\label{sec:mc-pia}
Model checking (e.g.~\cite{Clarke:2000:MC:332656}) is perhaps the most popular formal method for the validation of concurrent systems. The fact that it is an \emph{automata-based automated validation technique} makes it a nice candidate to join a simple framework for language construction that also aims at validation, such as the one used in this paper.
This section recalls the syntax and semantics for (a subset of) Linear Temporal Logic, one of the modal logics used in model checking, and discusses how to use this technique to validate $\uppi$ automata, only the necessary to follow Section~\ref{sec:amn-mc}.
The syntax of Linear Temporal Logic is given by the following grammar
\[\begin{array}{c}\scriptsize
\phi ::= \top ~|~ \bot ~|~ p ~|~ \neg(\phi) ~|~ (\phi \land \phi) ~|~ (\phi \lor \phi) ~|~ (\phi \to \phi) ~|~
(\Diamond \phi) ~|~ (\Box \phi)
\end{array}\]
\noindent where connectives $\Diamond$, $\Box$
are called \emph{temporal modalities}. They denote ``Future state'' and ``Globally (all future states)''. There is a precedence among them given by: first unary modalities, in the following order $\neg$, $\Diamond$ and $\Box$, then binary modalities, in the following order, $\land, \lor$ and $\to$.
The standard models for Modal Logics (e.g.~\cite{goldblatt}) are Kripke structures, triples $\mathcal{K} = (W, R, L)$ where $W$ is a set of worlds, $R \subseteq W \times W$ is the world accessibility relation and $L : W \to 2^{\mathit{AP}}$ is the labeling function that associates to a world a set of atomic propositions that hold in the given world. Depending on the modalities (or operators in the logic) and the properties of $R$, different Modal Logics arise such as Linear Temporal Logic.
A \emph{path} in a Kripke structure $\mathcal{K}$ represents a possible (infinite) scenario (or computation) of a system in terms of its states. The path $\tau = s_1 \to s_2 \to \ldots$ is an example. A \emph{suffix} of $\tau$ denoted $\tau^i$ is a sequence of states starting in $i$-th state.
Let $\mathcal{K} = (W, R, L)$ be a Kripke structure and $\tau = s_1 \to \ldots$ a path in $\mathcal{K}$. Satisfaction of an LTL formula $\phi$ in a path $\tau$, denoted $\tau \models \varphi$ is defined as follows,
\[\begin{array}{l}
\tau \models \top, \qquad \tau \not\models \bot, \qquad \tau \models p ~\mathit{iff}~ p \in L(s_1), \quad \tau \models \neg\phi ~\mathit{iff}~ \tau \not\models \phi, \\
\tau \models \phi_1 \land \phi_2 ~\mbox{iff}~ \tau \models \phi_1 ~\mbox{and}~ \tau \models \phi_2, \\
\tau \models \phi_1 \lor \phi_2 ~\mbox{iff}~ \tau \models \phi_1 ~\mbox{or}~ \tau \models \phi_2, \\
\tau \models \phi_1 \to \phi_2 ~\mbox{iff}~ \tau \models \phi_2 ~\mbox{whenever}~ \tau \models \phi_1, \\
\tau \models \Box \phi ~\mbox{iff}~ \mbox{for all}~ i \ge 1, \tau^i \models \phi, \\
\tau \models \Diamond \phi ~\mbox{iff}~ \mbox{there is some}~ i \ge 1, \tau^i \models \phi. \\
\end{array}\]
A $\uppi$ automata, when understood as a transition system, is also a \emph{frame}, that is, $\mathcal{F} = (W, R)$, where $W$ is the set of worlds and $R$ the accessibility relation. A Kripke structure is defined from a frame representing a $\uppi$ automata by declaring the labeling function with the following state proposition scheme:
\begin{eqnarray}
\label{eq:state-prop}\forall \sigma \in \mathit{Memory}, v \in \mathit{Index}(\sigma), r \in \mathit{Storable}, \nonumber\\
\langle \sigma, \ldots \rangle \models p_v(r) =_{\mathit{def}} (\sigma(v) = r),
\end{eqnarray}
meaning that for every variable $v$ in the index of the memory store component (which is a necessary semantic component) there exists a unary proposition $p_v$ that holds in every state where $v$ is bound to $p_v$'s parameter in the memory store. A \emph{poetic license} is taken here and $\uppi$ automata, from now on, refers to the pair composed by a $\uppi$ automata and its state propositions.
As an illustrative specification, used in Section~\ref{sec:amn-mc}, the LTL formula $\Box \neg[p_1(\mathit{crit}) \land p_2(\mathit{crit})]$ specifies safety (``nothing bad happens''), in this case both $p_1$ and $p_2$ in the critical section, when $p_i$ are state proposition formulae denoting the states of two processes and $\mathit{crit}$ is a constant denoting that a given process is in the critical section, and formula $\Box[p_1(\mathit{try}) \to \Diamond (p_1(\mathit{crit}))]$ specifies liveness (``something good eventually happens''), by stating that if a process, $p_1$ in this case, tries to enter the critical section it will eventually do so.
\section{$\uppi$ denotations for the Abstract Machine Notation}\label{sec:pi-den-for-amn}
This section formalizes the meaning of B's Abstract Machine Notation (AMN) in terms of $\uppi$ denotations. The subset of the AMN considered in this section is (almost) in bijection with $\uppi$ Lib, so the formalization is quite easy to follow.
\newcommand{\PIDEN}[1]{\llbracket #1 \rrbracket_\uppi}
The $\uppi$ denotational semantics of AMN, denoted by function $\llbracket\cdot\rrbracket_\uppi$, simply maps AMN statements, that is, arithmetic expressions, predicates, substitutions, machine declarations, variable declarations, constant declarations, initialization declarations, and operation declarations, into $\uppi$ denotations in a quite direct way.
\begin{PiDen}
Let \texttt{F}, \texttt{I}, \texttt{P}, \texttt{V} in $\langle\mathit{GSLIdentifiers}\rangle$, \texttt{R} in $\langle\mathit{RAT}\rangle$, \texttt{B} in $\langle\mathit{BOOL}\rangle$, and \texttt{E}, $\mathtt{E_{i}}$ in $\langle\mathit{GSLExpression}\rangle$, $\mathtt{P}$ in $\langle\mathit{GSLPredicate}\rangle$, $\mathtt{S}$, $\mathtt{S_i}$ in $\langle\mathit{GSLSubstitution}\rangle$, $\mathtt{A}, \mathtt{A_i}$ in $\langle\mathit{GSLActuals}\rangle$, $\mathtt{FS}$ in $\langle\mathit{AMNIdList}\rangle$, $\mathtt{O}$ in $\langle\mathit{AMNOperation}\rangle$, $\mathtt{OS}$ in $\langle\mathit{AMNOpSet}\rangle$, $\mathtt{VS}$ in $\langle\mathit{AMNValSet}\rangle$, $\mathtt{VC}$ in $\langle\mathit{AMNValuesClause}\rangle$, $\mathtt{AV}$ in $\langle\mathit{AMNAbsVariables}\rangle$, in
{\scriptsize \begin{align}
\llbracket \mathtt{I} \rrbracket_\uppi & = \mathit{id}(\mathtt{I}), \\
\llbracket \mathtt{grat}(\mathtt{R}) \rrbracket_\uppi & = \mathit{rat}(\mathtt{R}), \\
\llbracket \mathtt{gboo}(\mathtt{B}) \rrbracket_\uppi & = \mathit{boo}(\mathtt{B}), \\
\label{eq:sum}\llbracket \mathtt{E_{\mathtt{1}} ~\mbox{\tt+}~ E_{\mathtt{2}}}\rrbracket_\uppi & =
\mathit{add(\llbracket \mathtt{E_{\mathtt{1}}}\rrbracket_\uppi, \llbracket \mathtt{E}_{\mathtt{2}} \rrbracket_\uppi)}, \\
\llbracket \mathtt{E_{\mathtt{1}} ~\mbox{\tt==}~ E_{\mathtt{2}}}\rrbracket_\uppi & =
\mathit{eq(\llbracket \mathtt{E_{\mathtt{1}}}\rrbracket_\uppi, \llbracket \mathtt{E}_{\mathtt{2}} \rrbracket_\uppi)}, \\
\label{eq:and}\llbracket \mathtt{E_{\mathtt{1}} ~\mbox{\tt/\char`\\}~ E_{\mathtt{2}}} \rrbracket_\uppi & =
\mathit{and(\llbracket \mathtt{E_{\mathtt{1}}}\rrbracket_\uppi, \llbracket \mathtt{E}_{\mathtt{2}} \rrbracket_\uppi)}, \\
\label{eq:assign}\llbracket \mathtt{I ~\mbox{\tt :=}~ E}\rrbracket_\uppi & =
\mathit{assign(\llbracket \mathtt{I} \rrbracket_\uppi, \llbracket \mathtt{E} \rrbracket_\uppi)}, \\
\llbracket \mathtt{S_{\mathtt{1}} ~\mbox{\tt OR}~ S_{\mathtt{2}}} \rrbracket_\uppi & =
\mathit{choice(\llbracket \mathtt{S_{\mathtt{1}}} \rrbracket_\uppi, \llbracket \mathtt{S_{\mathtt{2}}} \rrbracket_\uppi)} , \\
\llbracket \mathtt{\mbox{\tt IF}~ P ~\mbox{\tt THEN}~ S_{\mathtt{1}} ~\mbox{\tt ELSE}~ S_{\mathtt{2}}} \rrbracket_\uppi & =
\mathit{if(\llbracket \mathtt{P} \rrbracket_\uppi, \llbracket \mathtt{S_{\mathtt{1}}} \rrbracket_\uppi, \llbracket \mathtt{S_{\mathtt{2}}} \rrbracket_\uppi)}, \\
\llbracket \mathtt{\mbox{\tt WHILE}~ P ~\mbox{\tt DO}~ S} \rrbracket_\uppi & =
\mathit{loop(\llbracket \mathtt{P} \rrbracket_\uppi, \llbracket \mathtt{S} \rrbracket_\uppi)} \\
\PIDEN{\mbox{\tt MACHINE} ~\mathtt{I} ~\mathtt{AV}~ \mathtt{C} ~\mathtt{VC}~ \mathtt{OP}~ \mbox{\tt END}} & =
\mathit{dec}(\PIDEN{\mathtt{AV}, \mathtt{VC}}, \mathit{dec}(\PIDEN{\mathtt{C}, \mathtt{VC}}, \PIDEN{\mathtt{OP}})), \\
\PIDEN{\mbox{\tt MACHINE} ~\mathtt{I} ~\mathtt{AV}~\mathtt{VC}~ \mathtt{OP}~ \mbox{\tt END}} & =
\mathit{dec}(\PIDEN{\mathtt{AV}, \mathtt{VC}}, \PIDEN{\mathtt{OP}}), \\
\PIDEN{\mbox{\tt MACHINE} ~\mathtt{I} ~ \mathtt{C} ~\mathtt{VC}~ \mathtt{OP}~ \mbox{\tt END}} & =
\mathit{dec}(\PIDEN{\mathtt{C}, \mathtt{VC}}, \PIDEN{\mathtt{OP}}), \\
\PIDEN{\mbox{\tt MACHINE} ~\mathtt{I} ~ \mathtt{OP}~ \mbox{\tt END}} & = \PIDEN{\mathtt{OP}}, \\
\PIDEN{\mbox{\tt VARIABLES} ~\mathtt{IS}, \mbox{\tt VALUES} ~\mathtt{VS}} & = \PIDEN{\mathtt{IS}, \mathtt{VS}} , \\
\PIDEN{\mathtt{I}, \mathtt{I} \mbox{\tt =} \mathtt{E}} & = \mathit{ref}(\PIDEN{\mathtt{I}}, \PIDEN{\mathtt{E}}), \\
\PIDEN{\mathtt{I}, (\mathtt{I} \mbox{\tt =} \mathtt{E} \mbox{\tt ;} \mathtt{VS})} & = \mathit{ref}(\PIDEN{\mathtt{I}}, \PIDEN{\mathtt{E}}), \\
\PIDEN{(\mathtt{I} \mbox{\tt ,} \mathtt{IS}), (\mathtt{I} \mbox{\tt =} \mathtt{E} \mbox{\tt ;} \mathtt{VS})} & = \mathit{dec}(\mathit{ref}(\PIDEN{\mathtt{I}}, \PIDEN{\mathtt{E}}), \PIDEN{\mathtt{IS}, \mathtt{VS}}), \\
\PIDEN{\mathit{I}, (\mathtt{I'} \mbox{\tt =} \mathtt{E} \mbox{\tt ;} \mathtt{VS})} & = \PIDEN{\mathtt{I}, \mathtt{VS}}, \mathit{if} \mathtt{I} \not=\mathtt{I'} \\
\PIDEN{(\mathtt{I} \mbox{\tt ,} \mathtt{IS}), (\mathtt{I'} \mbox{\tt =} \mathtt{E} \mbox{\tt ;} \mathtt{VS})} & = \PIDEN{(\mathtt{I} \mbox{\tt ,} \mathtt{IS}), \mathtt{VS}}, \mathit{if} \mathtt{I} \not=\mathtt{I'} \\
\PIDEN{\mbox{\tt OPERATIONS} ~\mathtt{O}} & = \PIDEN{\mathtt{O}}, \\
\PIDEN{\mbox{\tt OPERATIONS} ~\mathtt{OS}} & = \PIDEN{\mathtt{OS}}, \\
\PIDEN{\mathtt{O} \mbox{\tt ;} \mathtt{OS}} & = \mathit{dec}(\PIDEN{\mathtt{O}}, \PIDEN{\mathtt{OS}}), \\
\PIDEN{\mathtt{P} \mbox{\tt=} \mathtt{S}} & = \mathit{prc}(\PIDEN{\mathtt{P}}, \mathit{blk}(\PIDEN{\mathtt{S}})), \\
\PIDEN{\mathtt{P} \mbox{\tt(} \mathtt{FS} \mbox{\tt)} \mbox{\tt=} \mathtt{S}} & = \mathit{prc}(\PIDEN{\mathtt{P}}, \PIDEN{\mathtt{FS}}^\mathit{for}, \mathit{blk}(\PIDEN{\mathtt{S}})), \\
\PIDEN{\mathtt{F}}^\mathit{for} & = \mathit{par}(\PIDEN{\mathtt{F}}^\mathit{for}), \\
\PIDEN{\mathtt{F}\mbox{\tt ,} \mathtt{FS}}^\mathit{for} & = \mathit{for}(\PIDEN{\mathtt{F}}^\mathit{for}, \PIDEN{\mathtt{FS}}^\mathit{for}), \\
\llbracket \mathtt{I} \mbox{\tt(}\mbox{\tt)} \rrbracket_\uppi & = \mathit{cal}(\llbracket \mathtt{I}\rrbracket_\uppi), \\
\llbracket \mathtt{I} \mbox{\tt(}\mathtt{E} \mbox{\tt)} \rrbracket_\uppi & = \mathit{cal}(\llbracket \mathtt{I} \rrbracket_\uppi, \llbracket \mathtt{E}\rrbracket_\uppi ), \\
\llbracket \mathtt{I} \mbox{\tt(}\mathtt{A} \mbox{\tt)} \rrbracket_\uppi & = \mathit{cal}(\llbracket \mathtt{I}\rrbracket_\uppi, \llbracket \mathtt{A}\rrbracket^\mathit{act}_\uppi ), \\
\llbracket \mathtt{E} \rrbracket^\mathit{act}_\uppi & = \llbracket \mathtt{E} \rrbracket_\uppi, \\
\llbracket \mathtt{E} \mbox{\tt,} \mathtt{A} \rrbracket^\mathit{act}_\uppi & = \mathit{act}(\llbracket \mathtt{E}\rrbracket_\uppi, \llbracket \mathtt{A}\rrbracket^\mathit{act}_\uppi ).
\end{align}}
\caption{Abstract Machine Notation}
\label{piden:amn}
\end{PiDen}
GSL expressions, such as sum and conjunction, are mapped to $\uppi$ Lib expressions, \emph{add} and \emph{and} in Equations~\ref{eq:sum} and~\ref{eq:and}. GSL substitutions are mapped to commands, such as \texttt{`\_:=\_} being denoted by \texttt{assign} in Equation~\ref{eq:assign}.
A `\texttt{MACHINE\_\_END}' declaration is associated with a \emph{dec} construction in $\uppi$ Lib. Such a declaration is a composition of further declarations for variables and constants, initialized according with the expressions in the `\texttt{VALUES}' clause, and operations. Variable declarations give rise to \emph{ref} declarations, that associate a location in the memory with a given identifier in the environment. (Constants are denoted by \emph{cns} $\uppi$ Lib declarations.) An operation declaration gives rise to a \emph{prc} declaration that binds an abstraction (a list of formal parameters together with a block of commands) to an identifier in the environment. An operation call substitution is translated into a \emph{cal} $\uppi$ Lib command, parametrized by the expressions resulting from the translations of the GSLExpressions given as actual parameters in the given operation call.
\section{B Maude}\label{sec:bmaude}
B Maude is a formal executable environment for the Abstract Machine Notation that implements, in the Maude language, the $\uppi$ denotational semantics described in Section~\ref{sec:pi-den-for-amn}.
The main objective of B Maude is to endow the B method with the validation techniques available in the Maude system, both the buit-in ones, such as rewriting, narrowing, rewriting modulo SMT and model checking, and user-defined ones
The current version of \href{https://github.com/ChristianoBraga/BMaude}{B Maude} is called \textsc{uru\c{c}\'u amarela} (a bee common in the Rio de Janeiro area), it requires Maude Alpha 115 or later to run due to some of the narrowing-based techniques available in this version. (Even though it is not directly available from Maude's web site, Alpha 115 can be obtained free of charge by joining the Maude (Ab)users group by sending an email to {\small\email{maude-users@cs.uiuc.edu}}.)
Figure~\ref{fig:bmaude-splash-screen} displays B Maude banner and the output of correctly loading the \texttt{MUTEX} machine. (The logo makes a ``visual pun'' with hexagons from a bee hive and components, the sound of the B method's name and its objective of giving support to component-based development.)
\begin{figure}[ht]\centering
\includegraphics[width=0.5\textwidth]{bmaude-splash-screen2}
\caption{B Maude banner and loading a machine}\label{fig:bmaude-splash-screen}
\end{figure}
B Maude's architecture is comprised essentially by the following components (recall the discussion in Section~\ref{sec:comp-in-maude}), together with the Maude implementation of the $\uppi$ Framework (see Section~\ref{sec:modpi}): (i) B Maude \emph{parser} that, given a list of (quoted) identifiers, produces a term in the initial algebra of the module \texttt{AMN-SYNTAX}, representing B Maude's grammar;
(ii) B Maude to $\uppi$ Lib \emph{compiler}, that is exactly the implementation of the $\uppi$ denotations of Section~\ref{sec:pi-den-for-amn} in Maude; (iii) B Maude \emph{read-eval-loop}, a user interface that processes commands, given in terms of quoted identifiers, calls the appropriate meta-function; (iv) B Maude \emph{pretty-printer}, that translates the results coming from the meta-functions to user level representation, such as the output of the model checker in terms of $\uppi$ Lib constructions, into B Maude syntax. In the following sections we describe the implementation of some (due to space constraints) of the functionality implemented in the tool:
(i) searching the state space, in Section~\ref{sec:search}, and (ii) model checking machines, in Section~\ref{sec:amn-mc}.
\subsection{Searching the computation graph of a GSL substitution} \label{sec:search}
One may animate a B Maude description or check for invariants using the \texttt{search} command. Figure~\ref{fig:mutex-simulation} displays the execution of a \texttt{search} command querying for the first solution that satisfies the constraint \texttt{p1 = 2}, denoting the situation where process $1$ is in the critical section.
The \texttt{search} command is handled in two levels. First, the syntax of the command is checked at module \texttt{BMAUDE-INTERFACE}, essentially to make sure that the conditions are well-formed. Then, if that is the case, rule \texttt{search}, in module \texttt{BMAUDE-COM\-MANDS}, is applied. First the descent function \texttt{metaSearch} is invoked by \texttt{bMaudeSearch}, with appropriate parameters. If \texttt{metaSearch} succeeds, then \texttt{bMaudePrintTrace} pret\-ty-prints the trace, from the \emph{initial state}, given by the $\uppi$ automaton state
\begin{maude}[caption=Initial state for \texttt{metaSearch}, label=lst:initial-state]
< cnt : blk(compile(M:AMNMachine), compile(S:GSLSubstitution)) ecs,
env : noEnv, sto : noStore , val : evs ,
locs : noLocs , out : evs , exc : CNT >,
\end{maude}
to the \texttt{N:Nat}-th solution, by invoking the descent function \texttt{metaSearchPath}. The initial state is such that the control stack \texttt{cnt} has a block resulting from the commands yielded by compilation (or $\uppi$ denotation) of the given GSL substitution \texttt{S:GSL\-Subs\-ti\-tu\-ti\-on} (a call to operation \texttt{mutex}, in Figure~\ref{fig:mutex-simulation}) together with the declarations resulting from the compilation of the machine \texttt{M:AMNMachine} (machine \texttt{MUTEX}, in this example) and remaining semantic components initialized to their default initial values. (For instance, \texttt{evs} is short for empty-value-stack and \texttt{CNT}, short for continue, means ``normal execution'', as opposed to an \texttt{EXT}, short for exit, meaning abnormal termination, that $\uppi$ Lib construction \emph{exit} may raise.)
\begin{maude}
crl [search] :
< search N:Nat S:GSLSubstitution C:Condition ; M:AMNMachine ; QIL > =>
< idle ; M:AMNMachine ;
if T:ResultTriple? :: ResultTriple
then
bMaudePrintTrace(S:GSLSubstitution, M:AMNMachine, N:Nat)
else
printNoSolutionOrError(S:GSLSubstitution)
fi >
if T:ResultTriple? := bMaudeSearch(S:GSLSubstitution, M:AMNMachine, N:Nat)
\end{maude}
\begin{figure}[ht]\centering
\includegraphics[width=.5\textwidth]{mutex-simulation2}
\caption{Mutex simulation in B Maude}\label{fig:mutex-simulation}
\end{figure}
Figure~\ref{fig:mutex-safety} is an example of the use of the \texttt{search} command to check for an \emph{invariant} property. In this example, a \emph{safety} property is checked by querying for a state such that both processes are in the critical section (that is, \texttt{p1 = 2 and p2 = 2}). Since no solution is found, then the protocol implemented by operation \texttt{mutex} is safe.
\begin{figure}[ht]\centering
\includegraphics[width=.5\textwidth]{mutex-safety2}
\caption{Checking mutex safety invariant with search in B Maude}\label{fig:mutex-safety}
\end{figure}
\subsection{Linear Temporal Logic model checking AMN Machines}\label{sec:amn-mc}
In Section~\ref{sec:mc-pia} we discussed how to model check $\uppi$ automata and in Section~\ref{sec:pi-den-for-amn} we gave denotations for Abstract Machine Notation (AMN) statements as $\uppi$ Lib constructions. Their combination yields the foundation to model check AMN descriptions for Linear Temporal Logic (LTL) properties. In B Maude, this is encoded as a meta-function that invokes the Maude LTL model checker to validate
\[
\mathcal{K}(\llbracket \mathit{A} \rrbracket_\uppi), s_0 \models \varphi,
\]
where $\mathcal{K}(\llbracket \mathit{A} \rrbracket_\uppi)$ is the Kripke structure associated with the $\uppi$ denotations for AMN description $A$, $s_0$ is defined as in Listing~\ref{lst:initial-state}, and $\varphi$ is an LTL formula such that the atomic propositions are instances of the Equation Schema~\ref{eq:state-prop}, as defined in Section~\ref{sec:mc-pia}.
As with the \texttt{search} command, the model check command is handled in two levels. The first level makes sure that the command is well-formed. Then, Rule \texttt{mc} is applied and actually invokes the model checker by calling \texttt{metaReduce} with: (i) the module resulting from the application of operation \texttt{makeMCModule}, that creates a meta-module that includes Maude's MODEL-CHECKER module and declares state propositions resulting from the variable declarations in $A$ and, (ii) a meta-term denoting an invocation of the \texttt{modelCheck} operation (in module MODEL-CHECKER) that model checks $\mathcal{K}(\llbracket \mathit{A} \rrbracket_\uppi)$, starting in the $\uppi$ state that has the $\uppi$ denotation of \texttt{S:GSLSubstitution} on top of the control stack, as in Listing~\ref{lst:initial-state}, for the LTL properties encoded in \texttt{T:Term}.
\begin{maude}[caption=]
crl [mc] :
< mc S:GSLSubstitution T:Term ; M:AMNMachine ; QIL > =>
< idle ; M:AMNMachine ;
if (properResultPair?(T:ResultPair?))
then printModelCheckResult(T:ResultPair?))
else printModelCheckError(S:GSLSubstitution, T:Term)
fi >
if T:ResultPair? :=
metaReduce(makeMCModule(M:AMNMachine),
'_`,_|=?_[upTerm(M:AMNMachine), upTerm(S:GSLSubstitution), T:Term]) .
\end{maude}
In Figure~\ref{fig:mutex-liveness} we see that machine \texttt{MUTEX} does not have the liveness property, that specifies that if a process tries to enter the critical section it will eventually do so, by showing a counter-example where the system enters a loop with only one of the processes, $\mathtt{p_2}$ in this scenario, accessing the critical section.
\begin{figure}[ht]\centering
\includegraphics[width=.5\textwidth]{mutex-liveness2}
\caption{Checking mutex liveness in B Maude}\label{fig:mutex-liveness}
\end{figure}
\section{Related work}\label{sec:related-work}
In~\cite{etmf:2016} the present authors together with David Deharbe and Anamaria Moreira proposed a Structural Operational Semantics for the Generalized Substitution Language (GSL). The SOS semantics for GSL had a straight forward representation in the Maude language by encoding SOS transition rules as conditional rules in Maude. Moreover, the Conditional Rewriting Logic Semantics of GSL in Maude was shown to have an equivalent Unconditional Rewriting Logic Semantics by considering conditional rewrites in the main rewrite ``thread''. The inspiration for the approach used there is the same here: Plotkin's Interpreting Automata. We have further developed this approach for operational semantics that is now called $\uppi$ Framework.
The semantics of the Abstract Machine Notation is then described as denotations in the $\uppi$ Framework, as described in Section~\ref{sec:pi-den-for-amn}.
GSL is presented in~\cite{b-book} with a weakest precondition (WP) semantics. Most of the work on B and GSL relies on this semantics. An interesting example is the work of Dunne in \cite{Dunne2002}, which includes additional information concerning the variables in scope at a substitution to solve some delicate issues that restrict what can be stated in B due to limitations of the pure WP semantics. On the other hand, some related work proposing the embedding of B into other formalisms with strong tool support, such as Isabelle/HOL, can be found in the literature~\cite{Chartier1998,Deharbe2016}. As in the current research, the purpose of those works is combining strengths of both worlds, to achieve further proof or animation goals. On a more theoretical line, but also with similar goals, the work in~\cite{Zeyda2005} proposes a prospective value semantics to define the effect of GSL substitutions on values and expressions. The meaning of a computation, specified in GSL, is given in terms of the values of an expression, if the computation is carried out. In the current paper we contribute to this line of work by discussing GSL operational semantics, its rewriting logic semantics, using Maude as the specification language for Rewriting Logic theories, and a prototype execution environment in the Maude system.
\section{Conclusion}\label{sec:conclusion}
In this paper we have presented B Maude, an executable environment for the Abstract Machine Notation (AMN) language of the B method. Descriptions in AMN may be interpreted as automata and analyzed with automata-based methods such as model checking. We denote AMN statements as constructions in the $\uppi$ Framework, called $\uppi$ Lib constructions, which have an automata-based semantics given in terms of $\uppi$ automata. B Maude is then a compiler from AMN descriptions into $\uppi$ Lib constructions in Maude together with an implementation of the $\uppi$ Framework in Maude. B Maude allows for the execution by rewriting, symbolic search with narrowing and Linear Temporal Logic model checking of AMN descriptions by applying these techniques to the Maude representation of the given AMN description.
B Maude does not yet cover the complete AMN language and does not give support to the specification and reasoning on refinements. This is left to future work together with the inclusion of new validation techniques.
\paragraph*{Acknowledgements.} The $\uppi$ Framework is the result of a long term collaboration among the first author and Fabricio Chalub, Edward Hermann Haeusler, Jos\'e Meseguer and Peter D. Mosses, to whom he is deeply grateful.
\def\sortunder#1{}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,429 |
Many skillfully orchestrated events came to pass which brought the disciples of Jesus to this most astounding question. These events happened in their lives as testimony to the gospel of Jesus Christ, which is the power of God unto salvation, beginning with Peter, James and John accompanying Jesus to the top of a mountain. The scriptural account indicates that there was a six day period in which nothing which Jesus said or did was recorded. There are no conflicting events which could have persuaded the minds of these three men nor are there any distractions, such as a multitude of people or belligerent Pharisees accosting the assembly, which could have occupied their hearts and minds. These men were chosen of God to observe His glory and power while at the same time they were not given to understand the meaning of what they witnessed.
These few days of inactivity would seem to indicate that the disciples were well rested as they ascended the mountain, yet "Peter and they that were with him were heavy with sleep" (Luke 9:32). There is no indication of the time of day that this trip took place, the height of the ascent or the duration of the walk but the power of Almighty God enabled them to walk the walk and then to fall asleep as His glory was being manifested. God did not need their assistance as His appearance in the likeness of sinful flesh was changed into the brilliance of His glory. Nor did He require their cooperation or recognition as Moses and Elijah appeared and spoke with Jesus. They were brought along to bear witness to this wondrous manifestation but when it was over, "they kept it close and told no man in those days and of those things which they had seen" (9:36).
How did these men identify Moses and Elijah? It could be concluded that the Anointed One was the only one shinning and therefore it was obvious which one was Jesus but there were no visual records of Moses or Elijah and certainly no one living who could identify either of them. So, when these three men awoke from their sleep, how ever long it was, the raiment of Jesus was so bright that "no fuller on earth can white them, and there appeared unto them Elias, Moses and they were talking with Jesus" (Mk. 9:3b-4). The men who represented the Law and the prophets were speaking with the Elect Servant of Yehovah God who had come in the fullness of time to manifest, "that all things must be fulfilled which were written in the Law of Moses and the prophets and the psalms concerning Me" (Lk. 24:44). They were not briefing Him as to what came next or preparing Him for the work that lay before Him. He came to make an open demonstration of the work of God which was finished from the foundation of the world and so the Spirit revealed to Peter, James and John the identity of these men. This understanding did not come by the power of man, nor of the wisdom of the flesh "but My Father which is in heaven" gave them this spiritual understanding.
This event was taking place just a few short days after Jesus asked the question, "Who do men say that I am?" Peter was given of the Holy Spirit to answer that He is Messiah, the Son of the living God. Jesus assures Peter that this revelation was not of Adam or of anything to do with this world, yet a few days later, when He was beginning to tell the disciples of the upcoming events, Peter rebukes Him saying; "Be it far from you Lord, this shall not be unto you". Jesus rebuked Peter for his inability in the flesh to know or understand the things of God and now this same man was standing amazed and frightened at the sight of the glorified Messiah with Moses and Elijah. The same Spirit which revealed the truth about Messiah to him had also revealed the true identity of these other men and the significance of their presence.
The Holy Spirit gave these men this understanding, contrary to any natural ability, and He caused them to be exceedingly afraid. Along with the knowledge of the identity of these men came the understanding of what they represented and a most amazing representation. When Moses went into the mountain and met with Yehovah God, the glory of God caused his face to glow. This was so frightening to the people and Aaron to the point that they could not steadfastly look upon his face and were afraid to come near him (Ex. 34:29ff). Jesus, however, was not reflecting the light of the glory of God, for He was the fullness of the Godhead, bodily. He was changed in form and appeared altered to manifest the authority of God in, through and over all that He did and to verify the truth. The Law demands that at the minimum two witnesses were required for a matter to be established (Deut. 19:15). Jesus bore witness to the truth in that He came as the chosen servant of God (Is. 42:1). He fulfilled the sign of the prophesy as the Son being given and, even though He is the King of Peace (Solomon), He was manifested as the Prince of Peace (Is. 9:6). He humbled Himself and took on the form of the servant to do the work assigned unto Him and He came as that prophet to speak the Word of Yehovah. Moses, the prophets and the Psalms all speak of the coming of the Anointed One and Jesus was revealed unto these three men, in His glory, speaking of all things which must needs come to pass.
The excitement of this moment enticed Peter to venerate this momentous occasion. He immediately wanted to raise a monument to the three figures which he was given to see according to all that he was taught of men was about to come to pass. The coming of Elijah signified the coming of the end and the disciples knew it (Mal. 4:5). When they descended off the mount they asked Jesus concerning this but His answer was not according to the dogma they had been taught. The natural man had seen a visible event and had carnally concluded that the earthly kingdom was to be revealed. The answer that Jesus gave them seemed to support their anticipation as He said that Elijah must indeed come and He added to "restore all things" (Matt.17:11). The prophet said that Elijah must come before 'the great and dreadful Day of Yehovah' which was in the end of the world and even today at the 'Sedar' meal in celebration of the Passover, there is an empty chair for the immanent return of Elijah. Peter believed that this day was come, Messiah was to set up His kingdom on earth, Israel was to once again be the most powerful nation and he was to sit upon one of the twelve thrones and rule over Israel. This is what they had been taught, this is what they believed and so when the Lord ascended back into heaven they asked Him; "Will you now at this time restore the kingdom to Israel" (Acts. 1:6).
Then came a cloud which overshadowed them and a voice proclaimed Jesus as the Son of God and commanded that these men "hear Him". As if there was not enough fear, wonder and amazement happening God spoke in the thickness of this cloud, just as He did at Mt. Sinai when He gave Moses the Law (Lk. 9:36). When the voice had ceased and the cloud was withdrawn, they saw only Jesus and as they descended the mount they asked Him why people say Elijah must come first. He confirmed that the prophesy stated that Elijah must come and added that Elijah had already come, in the person of John the baptizer; they logically expected the kingdom to follow. The holy Spirit gave them knowledge in the recognition of the men, the voice of God declared the identity of Jesus and His authority and Jesus told them not to speak of what they saw until after He was delivered up, crucified and risen from the dead; "And they kept that saying with themselves, questioning one another what the rising from the dead should mean" (Mk. 9:10). Neither Peter, James nor John were given to write about this encounter as it stands as a testimony to the fact that God does not need Adam to help Him with His work; Adam is His work.
"And when He came to disciples, He saw a great multitude about them and the scribes, questioning with them" (Mk. 9:14).
Moses was ordered to descend off the mount to find that the children of Israel had risen up to play and there was a noise of war in the camp (Ex. 32:17) so when Jesus came off the mount He found the adversary had surrounded the camp. There was one who had brought his infirmed son to the disciples for them to cast out the demon within while the scribes were accosting them because they could not do it. Jesus rebukes them all as being an 'incredulously corrupt' generation because they did not believe the truth nor seek after Him. They sought a quick fix of all their problems and the disciples could not help for they too were of this generation. They did not have faith, for Adam is devoid of this gift, they sought to ingratiate themselves as they attempted to cast out without authority and they demonstrated the pride of Edom as they thought to be in the place of God to this man with the possessed child.
This was not a small gathering as the decree of Yehovah God demanded that this testimony be given in open session with many witnesses. It was full of emotion on all sides as it demonstrated that without faith it is impossible to please God or to come to the knowledge of the truth. The disciples could not extricate the spirit within because they had not the faith in Adam to do such things. They acted according to the flesh in their attempt to 'witness for God' and they manifested the impotence of man. Jesus said; "if you had faith the size of a grain of mustard seed" (Matt. 17:20), which signifies that the natural man is lacking even this diminutive quantity of that which is the fruit of the Spirit.
Prayer and supplications are not catalysts which prompt God into action or grant Him the liberty to perform His Holy will, they are the evidence of the Spirit within. The Spirit of Almighty God knows the mind of Almighty God and thusly the prayers of the saints ascend before the throne of the Father continually "with groanings which cannot be uttered, and He that searches the hearts knows what the mind of the Spirit is, because He makes intercession for the saints according to God" (Rms. 8:26f).
Jesus casts out the devil and restores the child to his father. "They were all amazed at the mighty power of God, but while they wondered every one at all things which Jesus did" (Lk 9:43) He continued to teach the disciples how that He must go to Jerusalem and be betrayed by the hands of man "and they should kill Him and the third day He shall be raised again; and they were exceedingly sorrowful" (Matt. 17:23). The concept of death was not foreign to them nor was the resurrection from the dead. They knew the story of the man who when he touched the bones of Elisha rose from the dead (II Kings 13:21), they had recently witnessed the resurrection of the widows son at Nain and the daughter of Jairus in Galilee but the concept of the Messiah dying was not in their Sabbath School materials. They expected a powerful military leader who would come and deliver them from the tyranny of man, restore the kingdom of Israel here on earth and reward them as rulers and leaders in this newly established government.
The power and majesty of Almighty God was being openly manifested in the sight of these great multitudes and the men whom Jesus called to be His disciples were arguing which one of them was greater than the others. He openly demonstrated that they are weak and confused in the flesh but their minds were filled with the idea of the kingdom. They could not keep their eyes open when He was transformed, they could not cast out the demon in the child, they could not answer the scribes and instead of being humbled for their incompetence, they were jousting for prominence and power. Jesus, perceiving the thoughts of their hearts, took a child and set him before Him. He was not impressed with the perfection or innocence of the small child who had not yet arrived at the age of accountability nor was He blessing all little children who are caught up into heaven if they die at a young age; He was setting before His disciple an example. The child had not yet learned the customs, traditions and corruption of the flesh. He carried no baggage of doctrines or creeds and he did not care about his worth before God.
The Jews claimed their heritage in Abraham as their right of inheritance into the kingdom of God. The work mongers cry out, 'Lord, Lord' as they recite the list of accomplishments they have performed and the religionists lift up their water baptisms, church member ships and, of course, their prayer of faith as reasons for God to be impressed with man. God is not impressed. He is not a respecter of persons nor does He react to the actions of His creations. His ways are beyond finding out and His wonders are too spectacular for the naked eyes of man to behold. The arrogance of the disciples, as to how great they were, was further demonstrated when they saw one casting out demons and they rebuked him. Jesus said; "Do not forbid him for he that is not against us is for us" (Lk. 9:49). When the question of sin came up they were not abased as to their own inability but rather asked how many times they should forgive when their brother sins against them (Matt. 18:15ff). Jesus again rebukes them by telling them that they should forgive as they have been forgiven. This concept will prompt the child of grace to consider his own sin against God and his brother before he demands contrition of another, for if Yehovah should mark iniquity, "who shall stand?" (Ps. 130:3).
This arrogance was demonstrated again when the time was come that He should be delivered up, He turned His face as though He would go to Jerusalem and sent messengers before Him into a village of the Samaritans to make ready for Him. Yet they would not receive Him and the disciples, in light of all that was happening, asked if they should call down fire from heaven, as Elijah did, to consume these 'infidels' from off the earth. Jesus again rebukes them and tells them that they do not know "what manner of spirit you are, for the Son of man is not come to destroy men's lives, but to save them" (Lk. 9:55f). This again was contrary to the commonly taught and widely accepted doctrine of the earthly kingdom of God.
Then came the questions about divorce and remarriage to which Jesus replied that Moses gave these provisions to men because of the "hardness of your hearts he wrote you this precept but from the beginning of the creation, God made them male and female". Another inquired as to how he could become a child of the kingdom and inherit eternal life. Jesus gave no altar call or told him to recite a prayer or two at the anxious bench; rather He asked him what the Law said. The man was so sure of himself that he affirmed that he had kept all these commandments from his youth upward but when Jesus told him the one thing that he lacked, he went away sorrowful for he had much wealth.
If Jesus was referring to a gate in the city which was so narrow and low that it was both impractical and of great difficulty for a camel to pass through, then His statement about salvation must bear the same comparative nature. He emphatically states that it is impossible for man to save himself or another; therefore there can be no possibility of a man stooping down, turning sideways or even crawling to curry the favour of the Lord God Almighty to deliver him. Salvation is of Yehovah and not of man.
Peter immediately begins to elaborate upon all that he and the others had done for the Lord but Jesus is not impressed. When in the final analysis it is disclosed that the sons of Adam have done all that they were bid to do by the Master, they will all at best be unprofitable servants of God and every child of the kingdom shall have the added blessing of afflictions. A person who abandoned everything for family, cause or beliefs may indeed have a feeling of nobility for their denying of the flesh and commitment to the matter but a child of grace shall receive all this in persecution. This is the identification of the stranger in the land. The children of the king of the land do not pay tribute to the king but strangers do. Jesus instructed Peter to go down to the water and cast in a hook. The first fish that he brought forth would have a coin in the mouth which Peter was to take and pay his tribute to the ruler of the land. This testifies to the power of God and His constant provisions in all things as well as the fact that His kingdom and His children are strangers in this world. The modern teaching of the earthly kingdom was shattered by the teaching of the King of the kingdom of our God. The importance and worth of the children of the dust was manifested in a series of event that happened in a short period of time and Jesus concludes this session with; "But many first shall be last and the last first" (10:31).
It is not by works of righteousness which a person may do or because of their zeal or heritage that God shows forth His grace upon them. His eternal love demands His attention and care upon those for whom He bled and died. These live by His faith and walk in the light of the Lord. They grow in the grace and knowledge of His will as they see clearer and clearer the wretchedness of this flesh and the vanity of this world. Jesus systematically dismantled every confidence that man can bring forth to show that the only hope that the Beloved has his in the Lord God Omnipotent. The proclamation of the gospel unto the nations is that Yehovah reigns, and the comfort of His people is that He has shown forth His salvation and all the world has seen it and that all flesh is grass. There can be no comfort in Adam who is deceived; his heart is desperately wicked, his mind is corrupted with the things of this world and his passions are after the needs and wants of the body of this death.
"Wherefore let him that thinks he stands take heed lest he fall" (I Cor. 10:12).
To be carnally minded is death and to have hope in this world is most miserable. The testimony of the Lord to His people is found in His chastening hand and the furnace of affliction. The greatest joy in this life is wormwood to the joy of the witness of the Spirit of God to the Spirit of His dear children that they are His sons, through the trying of their faith. The elite of this world are accepted of man and they have their reward. The children of the Kingdom of the Israel of our God are hated of all men for His name sake; they are despised, rejected and cast aside as fools but the Father will never leave them nor forsake them. He loves them with an eternal love and disciplines them according to the greatness of that love; there fore they love one another. "By this all men know that you are my disciples, because you have love one for another" (John 13:35). | {
"redpajama_set_name": "RedPajamaC4"
} | 1,897 |
International Medical University (IMU) är ett privat engelskspråkigt universitet i Kuala Lumpur och Malaysias ledande privata medicinska universitet. Det grundades 1992 och är verksamt inom forskning och utbildning inom medicin och sjukvård med en stark internationell inriktning. Den största aktieägaren är den statsägda kommersiella investeringsfonden Khazanah Nasional. Khazanah Nasional är även den största aktieägaren i det Singapore-baserade företaget Parkway Holdings, Sydostasiens största privata sjukhus- och vårdföretag.
Universitetet erbjuder ett antal kurser inom medicin, odontologi, farmaci, farmaceutisk kemi, omvårdnad, vårdvetenskap, näringslära, psykologi, biomedicin, medicinsk bioteknologi, kiropraktik och kinesisk medicin.
Källor
Externa länkar
International Medical University
Universitet och högskolor i Malaysia
Medicinska universitet och högskolor
Utbildningsinstitutioner bildade 1992
Kuala Lumpur | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 321 |
Q: Comparing elements in a list in Python's for -loop What is wrong in the method end in the code?
The method end returns always 1 although it should return 0 with the current data.
# return 1 if the sum of four consecutive elements equal the sum over other sum of the other three sums
# else return 0
# Eg the current sums "35 34 34 34" should return 0
data = "2|15|14|4|12|6|7|9|8|10|11|5|13|3|2|16"
arra = data.split("|");
def do_row ( arra, n ):
return arra[4*n:4 + 4*n]
def row_summa (row):
return sum(map(int,row))
def end ( summat ): # problem here!
equality = 1
for i in summat[2:5]:
print "Comparing: ", summat[1], " and ", i, ".\n"
if summat[1] != i:
equality = 0
print equality
for i in range(0,4):
summat = []
summat.append( row_summa( do_row(arra,i) ) )
print row_summa ( do_row(arra,i) )
summa = 0
end(summat)
A: I can't really tell what you're trying to do here, but I can certainly say why end() returns 1 instead of 0. In your last for loop, you reset summat to [] at the start of the loop, so at the end, summat only contains a single value (the one you most recently appended on). So when you ask for summat[2:5] on a list of a single item, Python returns an empty list (as there are no values in that range) - in which case there are no chances for equality to be set to zero because the loop in end never runs.
A: I think you may have an off-by-one error. Remember that array indexes in Python start at 0, not 1. So where you do this:
for i in summat[2:5]:
print "Comparing: ", summat[1], " and ", i, ".\n"
if summat[1] != i:
equality = 0
you are not looking at summat[0] at all. Try perhaps:
for i in summat[1:4]:
print "Comparing: ", summat[0], " and ", i, ".\n"
if summat[0] != i:
equality = 0
A: First off, end doesn't return 1. It returns None. It prints 1. Kind of deceptive if you're running it from the command line.
Second, when you call end, summat is equal to [34]. So this:
for i in summat[2:5]:
never even executes. It won't do anything unless summat contains at least 3 elements.
A: You have two problems. Initialising summat to [] inside the loop, also the off by one error Greg mentioned
data = "2|15|14|4|12|6|7|9|8|10|11|5|13|3|2|16"
arra = data.split("|");
def do_row ( arra, n ):
return arra[4*n:4 + 4*n]
def row_summa (row):
return sum(map(int,row))
def end ( summat ): # problem here!
equality = 1
for i in summat[1:]: # 1 <=== IS THE SECOND ELEMENT
print "Comparing: ", summat[0], " and ", i, ".\n"
if summat[0] != i:
equality = 0
print equality
summat = [] # <=== DO THIS BEFORE THE LOOP
for i in range(0,4):
summat.append( row_summa( do_row(arra,i) ) )
print row_summa ( do_row(arra,i) )
summa = 0
end(summat)
A: You should also study this piece of code
data = "2|15|14|4|12|6|7|9|8|10|11|5|13|3|2|16"
arra = map(int,data.split("|"))
summat = [sum(arra[i:i+4]) for i in range(0,len(arra),4)]
print summat
print len(set(summat))==1
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,818 |
STORIES THAT SHOCKED THE WORLD
INFORMATION-EDUCATION The Kiss of Life
The Kiss of Life
by Rahul Pandey on 10:21 AM in INFORMATION-EDUCATION
"The Kiss of Life": A utility worker, J.D. Thompson, giving mouth-to-mouth to co-worker Randall G. Champion after he contacted a high voltage wire -July 17, 1967.
Details : This photo shows two power linemen, Randall Champion and J. D. Thompson, at the top of a utility pole. They had been performing routine maintenance when Champion brushed one of the high voltage lines at the very top. These are the lines that can be heard "singing" with electricity. Over 4000 volts entered Champion's body and instantly stopped his heart (an electric chair uses about 2000 volts).
His safety harness prevented a fall, and Thompson, who had been ascending below him, quickly reached him and performed mouth-to-mouth resuscitation. He was unable to perform CPR given the circumstances, but continued breathing into Champion's lungs until he felt a slight pulse, then unbuckled his harness and descended with him on his shoulder. Thompson and another worker administered CPR on the ground, and Champion was moderately revived by the time paramedics arrived, eventually making a full recovery
INFORMATION-EDUCATION
By Rahul Pandey at 10:21 AM
Labels: INFORMATION-EDUCATION
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Violette was born into one of the wealthiest families in Old New Orleans to Robért and Yvette. They once lived in a beautiful home ...
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Architect of Cellular Jail The Cellular Jail , which is also known as Kala Paani (Black water), is the most prominent landmark o...
King Luther Assassination
The American black civil rights leader , Dr Martin Luther King, has been assassinated. Dr King was shot dead in the southern US city of ...
"World's youngest serial killer from India" - Armadeep Sada
"India's youngest serial killer" - Armadeep Sada An eight-year-old boy was being described as "India's youngest ...
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It is a legend's birthday today and we can't keep calm. MS Dhoni is celebrating his 38th birthday and to be said out loud he is as han...
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Failures are part of our life. All the successful people that we see around us were not get success easily. They had to go through a lot t...
How To Get Girlfriend On Facebook or other Social Networking Sites
Are You Looking For Girlfriend ? How to find Girlfriend ? How to get a girl friend online? Dating, nothing in this world is as difficult ...
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The assassination of Rajiv Gandhi, the ex-Prime Minister of India , occurred as a result of a suicide bombing in Sriperumbudur, near Chen...
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Designed with by Shockedyou | Distributed by Blogger Themes | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,960 |
describe('searchController', function() {
var searchFactoryMock, sentimentTrendsFactoryMock, ctrl, searchTerm, $q, rootScope, scope, httpBackend, storageFactoryMock, presentationFactoryMock, searchResult;
beforeEach(module('LoveIt'));
beforeEach(inject(function($rootScope, _$q_, $controller, $httpBackend){
scope = $rootScope.$new();
$q = _$q_;
httpBackend = $httpBackend;
searchFactoryMock = {query: function(){} };
sentimentTrendsFactoryMock = {setSearchTerm: function(){},setSearchResult: function(){} };
storageFactoryMock = {getHistory: function(){}, setHistory: function (){} };
searchResult = [];
spyOn(storageFactoryMock,'getHistory').and.returnValue(searchResult);
searchTerm = {search_term: 'Test searchTerm'};
presentationFactoryMock = {evaluateSearch: function(){}, getFullColorScheme: function(){}};
ctrl = $controller('searchController', {
$scope: scope,
searchFactory: searchFactoryMock,
sentimentTrendsFactory: sentimentTrendsFactoryMock,
storageFactory: storageFactoryMock,
presentationFactory: presentationFactoryMock
});
}));
describe('#getHistory', function(){
it('is called when ctrl is loaded', function(){
expect(ctrl.searches).toEqual(searchResult);
});
it('local storage is called to retrieve previous results', function(){
expect(storageFactoryMock.getHistory).toHaveBeenCalled();
});
it('results from local storage is entered into searches', function(){
expect(ctrl.searches).toEqual(searchResult);
});
});
describe('#setHistory', function(){
var key;
beforeEach(function(){
spyOn(storageFactoryMock,'setHistory');
key = 'resultHistory';
});
// it('local storage is called to retrieve previous results', function(){
// storageFactoryMock.setHistory(key,searchResult);
// expect(storageFactoryMock.setHistory).toHaveBeenCalledWith(key, searchResult);
// });
it('is called when a user moves from the page', function(){
scope.$broadcast("$routeChangeStart");
expect(storageFactoryMock.setHistory).toHaveBeenCalledWith(key,searchResult);
});
});
describe('#deleteSearch', function(){
beforeEach(function(){
spyOn(sentimentTrendsFactoryMock,'setSearchTerm');
spyOn(storageFactoryMock,'setHistory');
var deferred = $q.defer();
deferred.resolve({data:'some value'});
spyOn(searchFactoryMock,'query').and.returnValue( deferred.promise );
httpBackend.whenGET('partials/main-search.html').respond({data: 'Success'});
ctrl.makeSearch(searchTerm);
});
it('a user can delete a search item', function(){
ctrl.deleteResult('some value');
expect(ctrl.searches.length).toEqual(0);
});
it ('updates the local storage', function(){
ctrl.deleteResult('some value');
expect(storageFactoryMock.setHistory).toHaveBeenCalled();
});
});
describe('#makeSearch', function(){
beforeEach(function(){
var deferred = $q.defer();
deferred.resolve({data:'some value'});
spyOn(searchFactoryMock,'query').and.returnValue( deferred.promise );
httpBackend.whenGET('partials/main-search.html').respond({data: 'Success'});
});
it('searchFactory is called', function(){
ctrl.makeSearch(searchTerm);
expect(searchFactoryMock.query).toHaveBeenCalled();
});
it('returns the response data', function(){
ctrl.makeSearch(searchTerm);
scope.$apply();
expect(ctrl.searches.length).toEqual(1);
});
});
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,733 |
import os
import sys
os.environ['PYTHONUNBUFFERED'] = '1'
os.environ['MXNET_CUDNN_AUTOTUNE_DEFAULT'] = '0'
os.environ['MXNET_ENABLE_GPU_P2P'] = '0'
#os.environ['MXNET_ENGINE_TYPE'] = 'NaiveEngine'
this_dir = os.path.dirname(__file__)
sys.path.insert(0, os.path.join(this_dir, '..', '..', 'faster_rcnn'))
import train_end2end
import test
if __name__ == "__main__":
train_end2end.main()
test.main()
| {
"redpajama_set_name": "RedPajamaGithub"
} | 623 |
\section{Training Details}\label{appendix:training_details}
Unless stated otherwise, all experiments were conducted with the following training configuration. As a default, we used the TResNet-M model \cite{ridnik2021tresnet}, pre-trained on ImageNet-21k dataset \cite{ridnik2021imagenet21k}. The model was fine-tuned using Adam optimizer \cite{kingma2017adam} and 1-cycle cosine annealing policy
\cite{smith2018disciplined} with a maximal learning rate of 2e-4 for training OpenImages and MS-COCO, and 6e-4 for training LVIS.
We used true-weight-decay \cite{loshchilov2017decoupled} of 3e-4 and
standard ImageNet augmentations. For fair comparison to previously published results on OpenImages V3, we also trained a ResNet-101 model, pre-trained on ImageNet.
In the OpenImages experiments we used the following hyper-parameters: $\eta=0.05$, $K=200$, $\gamma_u=7$, $\gamma_-=2$ and $\gamma_+=1$. In LVIS we used: $\gamma_u=1$, $\gamma_-=0$ and $\gamma_+=0$.
\section{Soft Label Prior}\label{appendix:a}
Herein, we will explore a soft alternative for integrating the label prior in the loss.
We follow equation (\ref{eq:pasl}) and define the un-annotaetd weights by
\begin{equation}
\omega_c = \exp(-\alpha \hat{P}_r(c))
\end{equation}
where $\alpha$ is the decay factor. In Table \ref{table: opim_v6_soft_labels} we compare the soft label prior to the configuration used in section \ref{sec:selective_loss}.
\begin{table}[hbt!]
\centering
\begin{tabular}{ |M{3.6cm}||M{1.3cm}|M{1.3cm}|}
\hline
Method & mAP(C) & mAP(O)\\
\hline
P-ASL, \textit{Selective} & \textbf{86.72} & 93.57 \\
P-ASL, \textit{Selective} (soft) & 86.62 & \textbf{93.59} \\
\hline
\end{tabular}
\caption{\textbf{OpenImages (V6) results using soft label prior.} We used $\alpha=10$.}
\label{table: opim_v6_soft_labels}
\vspace{-0.2cm}
\end{table}
As the soft label prior provided with lower mAP(C) results, we did not use it in our experiments.
\section{Results on MS-COCO}\label{appendix:b}
In this section, we will present the results obtained on a partially annotated version of MS-COCO, based on the fixed per class (FPC) simulation scheme. Note that in this experiment, the class distribution measured by the number of annotations is no longer meaningful, as all classes have the same number of annotations. The mAP results, as well as the average precision (AP) scores for the class "Person", are presented in Figure \ref{fig:coco_selective}. The \textit{Negative} mode produces higher mAP (computed over all the classes) compared to the \textit{Ignore} mode. However, as the frequent class "Person" is present in most of the images, the \textit{Negative} mode is inferior, especially in the cases of a small number of annotations. Using the \textit{Selective} approach, top results can be achieved for both mAP and the person AP. In Figure \ref{fig:coco_top_10}, we show the top 10 frequent classes obtained using our procedure for estimating the class distribution as described in section \ref{sec: estimate_class_distribution}, and compared them to those obtained using the original class frequencies in MS-COCO. As can be seen, most of the frequent classes measured by the original distribution are also highly ranked by our estimator.
\begin{figure}[t!]
\begin{subfigure}[a]{.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{Figures/coco_all_ap.eps}
\caption{}
\end{subfigure}%
\begin{subfigure}[h]{.23\textwidth }
\centering
\includegraphics[width=1.0\linewidth]{Figures/coco_person_ap.eps}
\caption{}
\end{subfigure}
\caption{\textbf{Results on MS-COCO (FPC).} .}
\label{fig:coco_selective}
\vspace{-3mm}
\end{figure}
\begin{figure}[t!]
\begin{subfigure}[a]{.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{Figures/coco_orig_tag_freq_0.eps}
\caption{Original top frequencies}
\end{subfigure}%
\begin{subfigure}[h]{.23\textwidth }
\centering
\includegraphics[width=1.0\linewidth]{Figures/coco_ignore_freq_0.eps}
\caption{Estimated top frequencies}
\end{subfigure}
\caption{\textbf{Class frequency estimation in MS-COCO.} Top frequent classes measured by (a) original class distribution and (b) estimated class distribution. The estimated top 10 frequent classes are included in the original top classes.}
\label{fig:coco_top_10}
\vspace{-0.3mm}
\end{figure}
\section{Frequent classes in OpenImages}\label{appendix:c}
We add more results of the class distribution estimated by our approach (detailed in \ref{sec: estimate_class_distribution}) for OpenImages dataset. See Figure \ref{fig:opim_freq_appendix}.
\begin{figure*}[t!]
\centering
\begin{subfigure}[a]{.30\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{Figures/opim_tag_freq_1.eps}
\caption{}
\end{subfigure}%
\begin{subfigure}[h]{.30\textwidth }
\centering
\includegraphics[width=1.0\linewidth]{Figures/opim_tag_freq_2.eps}
\caption{}
\end{subfigure}
\begin{subfigure}[h]{.30\textwidth }
\centering
\includegraphics[width=1.0\linewidth]{Figures/opim_tag_freq_3.eps}
\caption{}
\end{subfigure}
\caption{\textbf{Estimating the calss distribution in OpenImages.} Additional top 60 frequent classes as estimated by our approach.}
\label{fig:opim_freq_appendix}
\vspace{-3mm}
\end{figure*}
\section{Frequent classes in LVIS}\label{appendix:d}
In Figure \ref{fig:estimate_classs_lvis} we plot the top frequent classes in LVIS, obtained by our estimator detailed in section \ref{sec: estimate_class_distribution}.
Also in LVIS, it can be seen that the most estimated frequent classes are related to common objects as "Person", "Shirt", "Trousers", "Shoe", etc.
\begin{figure} [t!]
\vspace{-0.1cm}
\centering
\includegraphics[scale=.42]{Figures/lvis_tag_freq_0.eps}
\vspace{-0.1cm}
\caption{\textbf{Estimating the calss distribution in LVIS.}
Top 20 frequent classes estimated by the \textit{Ignore} model.}
\label{fig:estimate_classs_lvis}
\vspace{-0.1cm}
\end{figure}
\end{appendices}
\subsection{Impact of Annotation Schemes}
\label{sec:annotation_schemes}
As aforementioned in section \ref{sec:how_to_treat}, the scheme used for annotating the dataset can substantially induce the learning process. Specifically, the choice of how to treat the un-annotated labels is highly influenced by the annotation scheme. To demonstrate that, we simulate two partial annotation schemes on the original fully annotated MS-COCO dataset \cite{lin2014microsoft}.
MS-COCO includes 80 classes, 82,081 training samples, and 40,137 validation samples, following the 2014 split. The two simulated annotation schemes are detailed as follows:
\noindent\textbf{Fixed per class (FPC).}
For each class, we randomly sample a fixed number of positive annotations, denoted by $N_s$, and the same number of negative annotations. The rest of the annotations are dropped.
\noindent\textbf{Random per annotation (RPA).}
We omit each annotation with probability $p$. Note that this simulation preserves the true class distribution of the data.
In Figure \ref{fig:impact_modes}, we show results obtained using each one of the simulation schemes for each primary mode (\textit{Ignore} and \textit{Negative}) while varying $N_s$ and $p$ values. As can be seen, while in RPA (Figure \ref{fig:impact_modes}(a)), the \textit{Ignore} mode consistently shows better results, in FPC (Figure \ref{fig:impact_modes}(b)), the \textit{Negative} mode is superior.
Note that as we keep more of the annotated labels (by either increasing $N_s$ or decreasing $p$), the gap between the two training modes is reduced, catching the maximal result.
The phenomenons observed in the two case studies we simulated are also related in real practical procedures for partially annotating multi-label datasets. While in the FPC simulation, the class distribution is completely vanished and cannot be inferred by the number of positive annotations ($N_s$ for $c=1,..., C$), the RPA scheme preserves the class distribution.
\begin{figure}[t!]
\begin{subfigure}[a]{.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{Figures/coco_rps.eps}
\caption{}
\end{subfigure}%
\begin{subfigure}[h]{.23\textwidth }
\centering
\includegraphics[width=1.0\linewidth]{Figures/coco_fpc.eps}
\caption{}
\end{subfigure}
\caption{\textbf{Impact of annotation schemes.} mAP results obtained using the RPA (a) and the FPC (b) simulation schemes for each primary mode. While in RPA, \textit{Ignore} mode consistently shows better results, in FPC, the \textit{Negative} mode is superior.}
\label{fig:impact_modes}
\vspace{-3mm}
\end{figure}
\begin{figure} [t!]
\vspace{-0.2cm}
\centering
\includegraphics[scale=.38]{Figures/coco_spearman.eps}
\vspace{-0.1cm}
\caption{\textbf{Spearman correlation between the true class distribution and the estimated distribution.}
Unlike the \textit{Negative} mode, training a model using \textit{Ignore} mode is well suited for estimating the class distribution.}
\label{fig:coco_spearman}
\vspace{-0.1cm}
\end{figure}
\subsection{Estimating the Label Prior}
\label{sec:results_estimate_class_prior}
To demonstrate the estimation quality of the class distribution obtained by the approach proposed in section \ref{sec: estimate_class_distribution}, we follow the FPC simulation scheme applied on the MS-COCO dataset (as described in section \ref{sec:annotation_schemes}), where a constant number of 1,000 annotations remained for each class. Because MS-COCO is a fully annotated dataset, we can compare the estimated class distribution (i.e. the label prior) to the true class distribution inferred by the original number of annotations. In particular, we measure the similarity between the original class frequencies and the estimated ones using the Spearman correlation test. In figure \ref{fig:coco_spearman}, we show the Spearman correlation scores while varying the number of top-ranked classes. We also show the results obtained with \textit{Negative} mode as a reference. Specifically, the Spearman correlation computed over all the 80 classes, with the estimator obtained using the \textit{Ignore} mode is $0.81$, demonstrating the estimator's effectiveness. In the next section, we will show how it benefits the overall classification results. Also, in appendix \ref{appendix:b} we present the top frequent classes measured by our estimator and compare them to those obtained by the original class frequencies in MS-COCO.
\section{Related Work}
Several methods had been proposed to tackle the partial labeling challenge. \cite{durand2019learning} offered a partial binary cross-entropy (CE) loss to weigh each sample according to the proportion of known labels, where the un-annotated labels are simply ignored in the loss computation.
In \cite{DBLP:conf/nips/KunduT20} they proposed to involve also the un-annotated labels in the loss, treating them as negative while smoothing their contribution by incorporating a temperature parameter in their sigmoid function. An interactive approach was presented in \cite{Huynh-mll:CVPR20} whose loss is composed of cross-entropy for the annotated labels and a smoothness term as a regularization. A curriculum learning strategy was also used in \cite{durand2019learning} to complete the missing labels.
Instead of using the same training mode for all classes, in this paper we propose adjusting the training mode, either as \textit{Ignore} or \textit{Negative} for each class individually, relying on probabilistic based conditions. Also, we introduce a key challenge in partial labeling, concerning the inability to infer the class distribution directly from the number of annotations, and suggest an estimation procedure to handle this.
Other methods were proposed in \cite{yu2013largescale, yang2016improving, wu2018multilabel} to cope with partial labels, for example by a low-rank
empirical risk minimization \cite{yu2013largescale} or by learning structured semantic correlations \cite{yang2016improving}. However, they are not scalable to large datasets and their optimization procedures are not well adapted to deep neural networks.
\begin{figure*} [t!]
\centering
\includegraphics[scale=.36]{Figures/approach_scheme_v5.eps}
\vspace{-0.3cm}
\caption{\textbf{Proposed approach.} First, a class distribution estimation phase is performed to obtain a reliable label prior using a temporary network trained with the \textit{Ignore} mode. Then, the target model is trained using a \textit{Selective} approach which assigns a \textit{Negative} or \textit{Ignore} mode for each label based on its estimated prior and likelihood.}
\label{fig:approach_scheme}
\vspace{-0.0cm}
\end{figure*}
Positive Unlabeled (PU) is also related to partial labeling \cite{DBLP:journals/corr/abs-1811-04820,jiang2020improving,hammoudeh2020learning}. The difference is that PU learning approaches use only positive and un-annotated labels without any negative annotations.
\section{Introduction}
\input{introduction}
\section{Learning from Partial Annotations}
\input{section_2}
\section{Proposed Approach}
\input{section_3}
\section{Experimental Study }
\input{experiments}
\section{Benchmark Results}
\input{results}
\section{Conclusion}
\input{conclusion}
{\small
\bibliographystyle{ieee_fullname.bst}
\subsection{OpenImages V6}
\vspace{-0.1cm}
Openimages V6 is a large-scale multi-label dataset \cite{Kuznetsova_2020}, consists of 9 million training images, 41,620 validation samples, and 125,456 test samples. It is a partially annotated dataset, with 9,600 trainable classes.
In Table \ref{table: opim_v6}, we present the mAP results obtained by our proposed \textit{Selective} method and compare them to other approaches. Interestingly, \textit{Ignore} mode produces better results than \textit{Negative} mode, as OpenImages contains many under-annotated frequent classes such as colors and other general classes (see Figure \ref{fig:estimate_classs_opim}). Using \textit{Negative} mode adds a massive label noise and harms the learning of many common classes. In Table \ref{table: opim_v6_backbones}, we present results for different network architectures. Specifically, using TResNet-L \cite{ridnik2021tresnet}, we achieve state-of-the-art result of $87.34$ mAP score.
To show the impact of decoupling the focusing parameters of the annotated and un-annotated loss terms in P-ASL as proposed in equation (\ref{eq:pasl}), we varied the negative focusing parameter $\gamma_-$, while fixing $\gamma_u=7$. The results are presented in Figure \ref{fig:pasl_opim}. The case of $\gamma_-=7$ represents the standard ASL \cite{ben2020asymmetric}. As can be seen, the mAP score increases as we lower $\gamma_-$, up to 2. It indicates that lowering the decay rate for the annotated negative term boosts their contribution to the loss.
\begin{figure} [t!]
\vspace{-0cm}
\centering
\includegraphics[scale=.36]{Figures/PASL_opim.eps}
\vspace{-0.1cm}
\caption{\textbf{Impact of decoupling the focusing parameters.}
We set the un-annotated focusing to $\gamma_u=7$ and varied the annotated negative focusing $\gamma_-$.}
\label{fig:pasl_opim}
\vspace{-0.5cm}
\end{figure}
\begin{figure} [t!]
\vspace{-0.3cm}
\centering
\includegraphics[scale=.36]{Figures/selective_ablation_opim.eps}
\vspace{-0.1cm}
\caption{\textbf{Ablation study of the \textit{Selective} approach components.}
mAP results are shown for different numbers of top likelihood labels, $K$. We show results for the case of using only the likelihood condition $\Omega_L$, and with both conditions $\Omega_L \cup \Omega_P$.}
\label{fig:selective_ablation_opim}
\vspace{-0.2cm}
\end{figure}
In Figure \ref{fig:selective_ablation_opim}, we show the mAP scores while varying the number of top likelihood classes, $K$ as defined in equation (\ref{eq:likelihood}). Note that setting $K=0$ is equivalent to use $\textit{Negative}$ mode. Training with high enough $K$ becomes similar to training using $\textit{Ignore}$ mode.
The highest mAP results are obtained with both the likelihood and prior conditions.
\subsection{OpenImages V3}
\vspace{-0.1cm}
To be compatible with previously published results, we used the OpenImages V3 which contains 5,000 trainable classes. We follow the comparison setting described in \cite{Huynh-mll:CVPR20}. Also, for a fair comparison we used the ResNet-101 \cite{he2016deep} backbone, pre-trained on the ImageNet dataset.
In Table \ref{table: opim_v3}, we show the mAP score results obtained using previous approaches and compared them to our $\textit{Selective}$ method. As shown, our method significantly outperforms previous approaches that deal with partial annotation in a multi-label setting.
\vspace{-0.1cm}
\subsection{LVIS}
\vspace{-0.1cm}
LVIS is a partially labeled dataset originally annotated for object detection and image segmentation, that was adopted as a multi-label classification benchmark. It consists of 100,170 images for training and 19,822 images for testing. It contains 1,203 classes. In Table \ref{table:lvis}, we present a comparison between different approaches on the LVIS dataset. As can be seen, in this case, the \textit{Negative} mode is better, compared to the \textit{Ignore} mode. This can be related to the fact that most of the labels are related to specific objects which do no appear frequently in the images. The most frequent class is ``Person". Therefore we also added its average precision to Table \ref{table:lvis}. Note that the \textit{Ignore} model better learns the class ``Person" compared to the one trained with \textit{Negative} mode. Using the P-ASL with the \textit{Selective} mode, we were able to obtain superior mAP results as well as top average precision even for the most frequent class, ``Person".
\begin{table}[t!]
\centering
\begin{tabular}{ |p{2.8cm}||M{1.0cm}|M{1.0cm}| M{1.0cm}|}
\hline
Method & mAP(C) & mAP(O) & Person-AP\\
\hline
CE, \textit{Ignore} & 74.49 & 95.70 & 99.81 \\
wCE \cite{durand2019learning}, \textit{Ignore} & 74.15 & 95.20 & 99.80\\
\hline
CE, \textit{Negative} & 77.82 & 96.66 & 97.20 \\
SE \cite{DBLP:conf/nips/KunduT20}, \textit{Negative} & 77.81 & 96.60 & 97.28 \\
ASL \cite{ben2020asymmetric}, \textit{Negative} & 78.32 & 96.77 & 98.43\\
\hline
P-ASL,\textit{Selective} & \textbf{78.57} & \textbf{96.80} & \textbf{99.81}\\
\hline
\end{tabular}
\caption{\textbf{Results for LVIS.} The \textit{Selective} approach with P-ASL improves both mAP(C) and mAP(O) scores. Also, it provides top result for the frequent class "Person".}
\label{table:lvis}
\end{table}
\subsection{Problem Formulation}
\label{section_2_modes}
Given a partially annotated multi-label dataset with $C$ classes, each sample $\mathbf{x} \in \mathcal{X}$, corresponding to a specific image, and is annotated by a label vector $\mathbf{y}=\{y_c\}_{c=1}^{C}$, where $y_c \in \{-1, 0, 1\}$ denotes whether the class $c$ is present in the image (`$1$'), absent (`$-1$') or unknown (`$0$').
For a given image, we denote the sets of positive and negative labels as $\mathcal{P}_{\mathbf{x}}=\{c|y_c=1\}$, and $\mathcal{N}_{\mathbf{x}}=\{c|y_c=-1\}$, respectively. The set of un-annotated labels is denoted by $\mathcal{U}_{\mathbf{x}}=\{c|y_c=0\}$. Note that typically, $| \mathcal{P}_{\mathbf{x}} \cup \mathcal{N}_{\mathbf{x}} | \ll |\mathcal{U}_{\mathbf{x}}|$.
A general form of the partially annotated multi-label classification loss can be defined as follows,
\begin{equation}
\mathcal{L}(\mathbf{x})=\sum_{c \in \mathcal{P}_{\mathbf{x}}}\mathcal{L}_{c}^{+}(\mathbf{x})
+\sum_{c \in \mathcal{N}_{\mathbf{x}}}\mathcal{L}_{c}^{-}(\mathbf{x})
+\sum_{c \in \mathcal{U}_{\mathbf{x}}}\mathcal{L}_{c}^{u}(\mathbf{x})
\end{equation}
where $\mathcal{L}_{c}^{+}(\mathbf{x})$, $\mathcal{L}_{c}^{-}(\mathbf{x})$ and $\mathcal{L}_{c}^{u}(\mathbf{x})$ are the loss terms of the positive, negative and un-annotated labels for sample $\mathbf{x}$, respectively.
Given a set of $N$ labeled samples $\{\mathbf{x}_i, \mathbf{y}_i\}_{i=1}^{N}$, our goal is to train a neural-network model $f(\mathbf{x}; \boldsymbol{\theta})$, parametrized by $\boldsymbol{\theta}$, to predict the presence or absence of each class given an input image.
We denote by $\mathbf{p}=\{p_c\}_{c=1}^{C}$ the class prediction vector, computed by the model: $p_c=\sigma(z_c)$ where $\sigma(\cdot)$ is the sigmoid function, and $z_c$ is the output logit corresponding to class $c$.
For example, applying the binary CE loss while considering only the annotated labels is defined by setting the loss terms as $\mathcal{L}_{c}^{+}(\mathbf{x}) = \log(p_c)$, $\mathcal{L}_{c}^{-}(\mathbf{x})=\log(1-p_c)$ and $\mathcal{L}_{c}^{u}(\mathbf{x})=0$.
\subsection{How to Treat the Un-annotated Labels?}
\label{sec:how_to_treat}
Typically, the number of un-annotated labels is much higher than the annotated ones.
Therefore, the question of how to treat the un-annotated labels may have a considerable impact on the learning process. Herein, we will first define the two primary training modes and detail their strengths and limitations. Then, in light of these insights, we will propose a class aware mechanism which may better handle the un-annotated labels.
\textbf{Mode \textit{Ignore}}. The basic scheme for handling the un-annotated labels is simply to ignore them, as suggested in \cite{durand2019learning}. In this mode we set $\mathcal{L}_{c}^{u}(\mathbf{x})=0$.
This way the training data is not contaminated with wrong annotations. However, its drawback is that it enables to use only a subset of the data. For example, in OpenImages dataset, the number of samples with either positive or negative annotations for the class ``Cat" is only $\sim0.9\%$ of the training data. This may lead to a sub-optimal classification boundary when the annotated negative labels do not sufficiently cover the space of the negative class. See illustration in Figure \ref{fig:illustration}(b).
\textbf{Mode \textit{Negative}}. In typical multi-label datasets, the chance of a specific label to appear in an image is very low. For example, in the fully-annotated MS-COCO dataset \cite{lin2014microsoft}, a label is annotated as negative with a probability of $ \sim 0.96$. Based on this prior assumption, a reasonable choice would be to treat the un-annotated labels as negative, i.e. setting $\mathcal{L}_{c}^{u}(\mathbf{x})=\mathcal{L}_{c}^{-}(\mathbf{x})$.
This working mode was also suggested in \cite{DBLP:conf/nips/KunduT20}. While this mode enables the utilization of the entire dataset, it suffers from two main limitations. First, it may wrongly annotate positive labels as negative annotations, adding label noise to the training.
Secondly, this mode inherently triggers a high imbalance between negative and positive samples.
Balancing them, for example by down-weighting the contribution of the negative samples, may diminish the impact of the valuable annotated negative samples. These weaknesses are illustrated in Figure \ref{fig:illustration}(c).
The question of which mode to choose has no unequivocal answer. It depends on various conditions and may have its origin in the annotation scheme used. In section \ref{sec:annotation_schemes}, we will show that different partial annotation procedures can lead to favor different loss modes (See Figure \ref{fig:impact_modes}).
Moreover, as discussed in the next section, the used mode can influence each class differently, depends upon the class presence frequency in the data and the number of available annotations.
\subsection{Class Distribution in Partial Annotation}
\label{sec:class_distribution}
As aforementioned, in multi-label datasets the majority of labels are present in only a small fraction of the data. For example, in MS-COCO, $89\%$ of the classes appear in less than $5\%$ of the data.
Thus, treating all un-annotated labels as negative may improve the discriminative power for many classes, as more real negative samples are involved in the training, while the added label noise is negligible. However, this may significantly harm the learning of classes whose number of positive annotations in the dataset is much lower than the actual number of samples they appear in. Consider the case of the class ``Person" in MS-COCO. It is present in $55\%$ of the data (45,200 samples). Now, suppose that only a subset of 1,000 positive annotations are available, and the rest are switched to negative.
It means that during the training, most of the prediction errors are due to wrong annotations. In this case, the optimization will be degraded and the network confidence will be decayed considerably. Hence, it will be beneficial to first identify the frequent labels and handle them differently in the loss.
\subsubsection{Positive Annotations Deficiency}
To identify the frequent labels, we need to reliably acquire their distribution in the data.
While in fully annotated datasets it can be easily obtained by counting the number of annotations per class and normalizing by the total number of samples, in partially annotated datasets it is not straightforward. While one may suggest counting the number of positive annotations for each class, the resulted numbers are misleading and are usually not proportional to the true class frequencies. For example, in OpenImages (V6), we found that many common and general classes which are frequently present in images are labeled with very few positive annotations. For example, general classes such as ``Daytime", ``Event" or ``Design" are labeled in only 1,709, 1,517 and 1,394 images (out of 9M), respectively. Color classes which massively appear in images are also rarely annotated. ``Black" and ``White" classes are labeled in only 1,688 and 1,497 images, respectively. We may assume that classes such as ``Daytime" or ``White" are present in much more than $0.02\%$ of the samples. Similarly, in LVIS dataset, the classes ``Person" and ``Shirt" are annotated in only 1,928 and 1,942 samples, respectively, while they practically appear in much more images (note that in MS-COCO, which shares the same images with LVIS, the class ``Person" appears in $55 \% $ of the samples).
Note that the labels are not necessarily annotated according to their dominance in the image. In Figure \ref{fig:lip_yellow}, we show examples of three images and corresponding annotations of the classes ``Lip" and ``Yellow". As can be seen, the left image was not annotated with neither ``Lip" nor ``Yellow" although these labels are present and dominant in it. Also, ``Lip" is annotated in only 1,121 images which is highly deficient in view of the fact that the class ``Human face" is annotated in 327,899 images.
According to the above-mentioned observations, the number of positive annotations cannot be used to measure the class frequencies in partially labeled datasets. In section \ref{sec: estimate_class_distribution}, we will propose a simple yet effective approach for estimating the class distribution from the data.
\subsection{Asymmetric Loss for Partial Labeling}
To mitigate the high negative-positive imbalance problem, we adopt the asymmetric loss (ASL) proposed in \cite{ben2020asymmetric} as the base loss for the multi-label classification task. It enables to dynamically focus on the hard samples while at the same time controlling the contribution propagated from the positive and negative samples.
First, let us denote the basic term of the focal loss \cite{tsung2017focal} for a given class $c$, by:
\begin{equation}
\mathcal{L}_{\scaleto{F}{3.5pt}}(p_c,\gamma) = (1-p_c)^{\gamma}\log p_c
\end{equation}
where $\gamma$ is the focusing parameter, which adjusts the decay rate of the easy samples. Then, we define the partially annotated loss as follows,
\begin{equation}
\label{eq:pasl}
\begin{split}
\mathcal{L}(\mathbf{x}) & = \smashoperator[lr]{\sum_{c \in \mathcal{P}_{\mathbf{x}}}}\mathcal{L}_{\scaleto{F}{3.5pt}}(p_c, \gamma^{+})
\\ &+\smashoperator[lr]{\sum_{c \in \mathcal{N}_{\mathbf{x}}}}\mathcal{L}_{\scaleto{F}{3.5pt}}(1-p_c, \gamma^{-})
+\smashoperator[lr]{\sum_{c \in \mathcal{U}_{\mathbf{x}}}}\omega_c\mathcal{L}_{\scaleto{F}{3.5pt}}(1-p_c, \gamma^{u})
\end{split}
\end{equation}
where $\gamma^+$, $\gamma^-$ and $\gamma^u$ are the focusing parameters for the positive, negative and un-annotated labels, respectively. $\omega_c$ is the \textit{selectivity} parameter and it is introduced in section \ref{sec:selective_loss}.
We usually set $\gamma^{+} < \gamma^{-}$ to decay the positive term with a lower rate than the negative one because the positive samples are infrequent compared to the negative samples. In addition, as for a given class, the negative annotated samples are \textit{verified} ground-truth we are interested in preserving their contribution to the loss. Therefore, we suggest decoupling of the focusing parameter of the annotated negative labels from the un-annotated one, allowing us to set a lower decay rate for the annotated negative labels: $\gamma^{-} < \gamma^{u}$. This way, the impact of the annotated negative samples on establishing the classification boundary for each class is higher (see Figure \ref{fig:illustration}(d)).
We term this form of asymmetric loss as \textit{Partial}-ASL (P-ASL).
\subsection{Class-aware Selective Loss}\label{sec:selective_loss}
As described in section \ref{section_2_modes}, both \textit{Ignore} and \textit{Negative} modes are supported by inadequate assumptions for the partial annotation problem.
In this section, we propose a selective approach for adjusting the mode per individual class. The core idea is to examine the probability of each un-annotated label being present in a given sample $\mathbf{x}$. Un-annotated labels that are suspected as positive will be ignored. The others will be treated as negative.
For that purpose, we define two probabilistic values: \textit{label likelihood} and \textit{label prior}, and detail their usage in the following section.
These two quantities are complementary to each other. The label likelihood enables to dynamically ignore the loss contribution of a label in a given image by inspecting its visual content. The label prior extracts useful information of the estimated class frequencies in the data and uses it regardless of the specific image content.
\textbf{Label likelihood.}
Defined by the conditional probability of an un-annotated label $c$ of being positive given the image and the model parameters. i.e.
\begin{equation}
P(y_c = 1 | \mathbf{x}; \boldsymbol{\theta}) ; \quad \forall c \in \mathcal{U}_{\mathbf{x}}
\end{equation}
It can be simply estimated by the network prediction $\{p_c\}_{c \in \mathcal{U}_{\mathbf{x}}}$ throughout the training. A high $p_c$ may imply that the un-annotated label $c$ appears in the image, and treating it as negative may lead to an error. Accordingly, the label $c$ should be ignored. In practice, we allow for $K$ un-annotated labels with top prediction values to be ignored. i.e.
\begin{equation}\label{eq:likelihood}
\Omega_{L} = \big\{c \in \mathcal{U}_{\mathbf{x}} \mid c \in \text{TopK}(\{p_c\})\big\}
\end{equation}
where the $\text{TopK}(\cdot)$ operator returns the indices of the top $K$ elements of the input vector. The algorithm scheme is illustrated in Figure \ref{fig:likelihood_scheme}. Note that this implementation enables us to ``walk" on a continuous scale between the \textit{Negative} and \textit{Ignore} modes. Setting $K=0$ corresponds to \textit{Negative} mode, as no un-annotated label is ignored. $K=C$ equivalents to the \textit{Ignore} mode, as all un-annotated labels are ignored.
\textbf{Label prior.} Defined by the probability of a label $c$ being present in an image. It can also be viewed as the actual label presence frequency in the data. We are interested in the label prior for the un-annotated labels,
\begin{equation}
P(y_c = 1) ; \quad \forall c \in \mathcal{U}_{\mathbf{x}}.
\end{equation}
According to section \ref{sec:class_distribution}, the label prior should be estimated from the data, as the class distribution is hidden in partially annotated datasets. In the next section (\ref{sec: estimate_class_distribution}), we will introduce the scheme for estimating the label prior. Meanwhile, let us denote by $\hat{P}_r(c)$ the label prior estimator for class $c$. We are interested in disabling the loss contribution of labels with high prior values. These labels are formally defined by the following set,
\begin{equation}
\Omega_{P} = \big\{c \in \mathcal{U}_{\mathbf{x}} \mid \hat{P}_r(c) > \eta \big\}
\end{equation}
where $\eta \in [0, 1]$ represents the minimum fraction of the data determining a label to be ignored.
\begin{figure} [t!]
\centering
\includegraphics[scale=.35]{Figures/likelihood_scheme_v2.eps}
\vspace{-0.3cm}
\caption{\textbf{Using the label likelihood.} Un-annotated labels with highest network confidence may be related to positive items in the image. Thus, we ignore them in the loss computation.}
\label{fig:likelihood_scheme}
\vspace{-0.1cm}
\end{figure}
Finally, we denote the set of labels whose loss contribution are ignored, as the union of the two previously computed sets,
\begin{equation}
\Omega_{\text{Ignore}} = \Omega_{L} \cup \Omega_{P}.
\end{equation}
Accordingly, we set the parameter $\omega_c$ in equation (\ref{eq:pasl}) as follows,
\begin{equation}\label{eq:omega_c}
\omega_c =
\begin{cases}
0 \quad c \in \Omega_{\text{Ignore}}\\
1 \quad c \notin \Omega_{\text{Ignore}}\\
\end{cases}
\end{equation}
Note that we have explored other alternatives for implementing the label prior in the loss function. In particular, in appendix \ref{appendix:a} we compare a soft method that integrates the label prior by setting $\omega_c=\exp(-\alpha\hat{P}_r(c)); \forall c \notin \Omega_{L}$, and show that using a hard decision mechanism, as proposed in equation (\ref{eq:omega_c}), produces better results.
\subsection{Estimating the Class Distribution}
\label{sec: estimate_class_distribution}
We aim at estimating the class distribution in a representative dataset $\mathcal{X}$. For that, we first need to assess the presence of each class in every image in the data, i.e. we would like to first approximate the probability of a class $c$ being present in an image $\mathbf{x} \in \mathcal{X}$: $P(y_c= 1|\mathbf{x})$.
To that end, we propose training a model parametrized by $\boldsymbol{\theta}$, for predicting each class in a given image, i.e. $P(y_c= 1|\mathbf{x}; \boldsymbol{\theta})$.
Afterwards, the model is applied on the sample set $\mathcal{X}$ (e.g. the training data). The label prior can then be estimated by calculating the expectation,
\begin{equation}
\vspace{-0.15cm}
P(y_c=1; \mathbf{\theta}) = \frac{1}{|\mathcal{X}|}\sum_{\mathbf{x} \in \mathcal{X}}P(y_c= 1|\mathbf{x}; \boldsymbol{\theta}).
\vspace{-0.1cm}
\end{equation}
For estimating the label priors, we train the model in \textit{Ignore} mode. While the discriminative power of the \textit{Negative} mode may be stronger for majority of the labels, it may fail to provide a reliable prediction values for frequent classes with small number of positive annotations. Propagating abundance of gradient errors from wrong negative annotations will decay the expected returned prediction for those classes and will fail to approximate $P(y_c= 1|\mathbf{x})$.
Consequently, our suggested estimation for the class distribution is given by,
\begin{equation} \label{eq:prior}
\hat{P}_r(c) = P(y_c=1; \mathbf{\boldsymbol{\theta}_{\text{Ignore}}}),
\end{equation}
where $\boldsymbol{\theta}_{\text{Ignore}}$ denotes the model parameters trained in \textit{Ignore} mode.
In section \ref{sec:results_estimate_class_prior}, we will empirically show the effectiveness of the \textit{Ignore} mode in ranking the class frequencies and the inapplicability of the \textit{Negative} mode to do that.
To qualitatively show the estimation effectiveness, we present in Figure \ref{fig:estimate_classs_opim} the top 20 frequent classes in OpenImages (V6) as estimated by our proposed procedure. Note that all the top classes are commonly present in images such as colors (``White", ``Black", ``Blue" etc.) or general classes such as ``Photograph", ``Light", ``Daytime" or ``Line". In appendix \ref{appendix:c}, we show the next top 60 estimated classes. Also, in appendix \ref{appendix:d}, we provides the top 20 estimated frequent classes for LVIS dataset.
\begin{figure} [t!]
\vspace{-0.1cm}
\centering
\includegraphics[scale=.38]{Figures/opim_tag_freq_0.eps}
\vspace{-0.1cm}
\caption{\textbf{Estimating the class distribution in OpenImages.}
Top 20 frequent classes estimated by the \textit{Ignore} model. Top classes are all related to common labels such as colors or general concepts.}
\label{fig:estimate_classs_opim}
\vspace{-0.3cm}
\end{figure}
| {
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-Stories of encouters in other places inhabited by "unbelievers" and "evildoers".
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Special Report: Fighting Breaks Out Again In Zaire's Southern Province, Shaba. 1978
VLVA4Y8ARUY4LRHGMGZDW1FRXCNU6-SPECIAL-REPORT-FIGHTING-BREAKS-OUT-AGAIN-IN-ZAIRES-SOUTHERN
Fighting has broken out in Shaba, the southern province of Zaire, for the second time in 14 months.
GV: President Mobuto Sese Seko talking to troops. (2 SHOTS)
SCU: President Mobutu talking to officers walking along road. (2 SHOTS)
GV PAN: Helicopter flies overhead, PAN DOWN TO troops repairing damaged bridge. (3 SHOTS)
SV: Moroccan troops on vehicle.
CU: Zaire troops in machine gun post. (2 SHOTS)
1960 CU: Moise Tshombe looks on as Katanga flag raised. (2 SHOTS) (B&W)
GV: Armoured car along road firing gun, troops taking cover as bomb explodes, troops firing at roadside, running into bush. (4 SHOTS)
CU: Wrecked vehicles and civilians at roadside. (2 SHOTS)
1964 CU: Tshombe leaves aircraft, greeted by General Mobutu, takes salute. (4 SHOTS)
1965 CU: Mobutu drinking with ministers, surrounded by crowd; Mobutu salutes and walks away. (3 SHOTS)
GV: Copper mines, Shaba. (3 SHOTS)
1976 GV: Cuban soldiers waling in streets of Luanda. (4 SHOTS) (Col)
GV: Cuban soldiers on truck with Angolans waving as they drive along road.
GV: Cuban on trucks towing guns through streets of Luanda. (2 SHOTS)
Initials JS/2350
Background: Fighting has broken out in Shaba, the southern province of Zaire, for the second time in 14 months. About 4,000 men of the National Liberation Front of the Congo -- a movement that opposes the government of President Mobutu Sese Seko -- are reported to have entered the province last Thursday night (11 May). Zaire's Ambassador in Paris admitted today (Monday) that they have captured the town of Mutshatsha, but he denied claims that the copper mining centre of Kolwezi has also fallen.
SYNOPSIS: Mutshatsha was captured by the rebels in last year's invasion -- but recaptured by government forces after about a month. President Mobutu went there immediately afterwards to see the situation at first hand, and encourage his troops. The attacks are a direct challenge to his government in a province that has a history of opposition to central authority in Zaire.
The Liberation Front are claiming to have inflicted heavy losses on the government forces. President Mobutu had asked the United States, Britain, France, Belgium and China for help. Last year, a thousand Moroccans, airlifted in by the French, took part in the drive against the rebels.
The Front's origins go back to the civil war in the Congo, after Katanga province seceded. Moise Tshombe declared it a separate state in 1960, two weeks after Belgium had granted the Congo independence. The United Nations intervened, at the request of the central government.
Several time, their forces were involved in major clashes with the Katangese. Eventually, the Katangan forces were disbanded. Some joined the regular Congolese army. Others disappeared into the bush, or fled into exile. It is these who have reappeared in Shaba, as katanga is now called.
Mr. Tshombe himself went to become Prime Minister of the Congo for more than a year -- till he himself was deposed and banished to exile in Europe. The man waiting to greet him was the Commander in Chief, General Mobutu-who took power a month after Mr. Tshombe's fall.
General Mobutu became President in 1965, when the country was seething in the aftermath of rebellions in the north and east. Ever since, ha has maintained strict control, which his opponents call dictatorship.
Kolwezi, the main target of the Liberation Front, is in the heart of Zaire' copper belt. Depression of the copper market has brought hardship to Shaba, which is adding to the unrest.
Last year, Zaire blamed Angola and Cuba -- alleging that Cuban soldiers in Angola had trained the rebel troops, and that the Soviet Union had armed them. This time, the Zaire news agency says the rebels came across the Zambian border, not the Angolan. But it says they again had the support of the Soviet Union and Cuba, and also of Algeria and the Libyan Jamahiriyah.
Plans for the invasion, it claims, were worked out in Havana and completed in Algiers -- and it says Cubans have been identified in the fighting at Mutshatsha.
ZAIRE'S
PROVINCE,
SHABA.
VLVA4Y8ARUY4LRHGMGZDW1FRXCNU6
Japanese War Criminals Identified
Chinese victims identify Japanese prisoners in Hong Kong who are responsible for the murders of their relatives and friends
Malaysia: Captured Indonesians Identify Dead Comrades
MALAYSIAN TROOPS TODAY (SUNDAY) KILLED FOUR INDONESIANS AND CAPTURED FOUR MORE DURING A CLASH IN JOHORE STATE, IN SOUTHERN MALAYA.
Gi Identifies Captured Koreans
American soldier identifies Korean prisoners responsible for wounding him and killing others
Greece: Bodies Of Boeing Crash Victims Identified At Athens Morgue.
The possibility of sabotage in Sunday's (8 September) air crash off the Greek coast was ruled out by the Athens Coroner on Tuesday (10 September).
South Korea: Captured North Korea Infiltrator, Arrested During Seoul Raid, Identifies Bodies Of Comrades
SECOND LIEUTENANT SHIN-JO KIM, THE ONLY NORTH KOREAN INFILTRATOR ARRESTED DURING LAST SUNDAY'S RAID (21 JANUARY) ON SEOUL, SOUTH KOREA, WAS ON THURSDAY (25 JANUARY) CALLED UPON TO IDENTIFY THE CORPSES OF 13 OF HIS COMRADES KILLED DURING AND AFTER THE ATTACK ON THE PRESIDENTIAL PALACE.
Syria: Russian Consul Identifies Body Of Russian Woman Teacher Killed In Israeli Air Raid On Damascus
The body of a Russian woman teacher was recovered in Damascus on Saturday (October 13th) from the wreckage of the Soviet Cultural Centre which was severely damaged in an Israeli air raid on the Syrian capital.
U.S.A.: Rabbi Ringer Identifies Eichman As Killer
Visnews went to AN apartment in Brooklyn February 10 and Marty Wheldon asked Rabbi Aaron Ringer to recall the events in Auschwits when Rabbi Ringer claims he saw Eichmann commit a murder of a small child.
Indonesia: Japanese Relatives Of Plane Crash Passengers Identify Bodies.
As the grim task of trying to identify the remains of those killed in the Pan American airliner crash in Northern Bali continued, on Thursday (April 25) a party of relatives of Japanese people killed in the crash travelled to a camp set up by the Indonesian Army 5 miles (8kms) from the crash site. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,562 |
Le musée d'histoire de la médecine et de la pharmacie de Lyon est un musée d'histoire de la médecine et de pharmacie créé en 1914 par Alexandre Lacassagne et situé à Lyon, dans le bâtiment de la faculté de médecine de l'université Claude-Bernard Lyon 1.
Description
Le musée est le fruit d'une collection d'Alexandre Lacassagne depuis 1896 sur les instruments et pratiques médicales ainsi que leur évolution.
Notes et références
Voir aussi
Article connexe
Liste des musées de Lyon
Musée d'histoire de la médecine
Musée à Lyon
8e arrondissement de Lyon
Université Claude-Bernard-Lyon-I
Grange Blanche
Lyon | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 318 |
Q: How to enable/disable a button within a DataGridPro hopefully someone can help here. I have a table with a save and cancel button at the end of each row. I'm attempting to disable the save button until a cell within the row has been edited. The code here is:
const columns = [
{
field: "name",
headerName: "Name",
},
{
field: "age",
headerName: "Age",
},
{
field: "actions",
type: "actions",
headerName: "Actions"
cellClassName: 'actions'
getActions: ({id}) => {
if(isInEditMode) {
return [
<GridActionsCellItem
icon={Tooltip title="Save Changes" arrow><SaveIcon/></Tooltip>
label="Save"
onClick={handleSave(id)}
color="primary"
/>,
<GridActionsCellItem
icon={Tooltip title="Cancel Changes" arrow><CancelIcon/></Tooltip>
label="Cancel"
className="textPrimary"
onClick={handleCancel(id)}
color="inherit"
/>
];
},
},
},
];
So the idea is as soon as the value within the name or age cells change the Save button would become enabled, otherwise remain disabled.
Hope this makes sense, and many thanks in advance.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 582 |
Bucket Detective is a video game released in 2018 by The Whale Husband, a collaboration between American independent developer Jesse Barksdale and Finnish developer Samu Kovanen. Described as a "dark comedy horror game", Bucket Detective is an adventure game in which an illiterate and perverse author endures personal and moral sacrifices made to a cult in order to receive the literary inspiration to publish a novel. Bucket Detective, in line with Barksdale's previous game, The Static Speaks My Name, was strongly influenced by personal themes of "depression, anxiety and obsession" and designed in ways to disorient and confuse the player. The game, which was the first commercial project from Barksdale, received a positive reception from critics, with particular attention directed at the game's unusual and grotesque premise and disturbing content.
Plot
Bucket Detective is played from the perspective of David Davids, a narcissistic, sexually neurotic, misogynistic man of middle age. An illiterate narration describes Davidson's desire to write a best-selling novel, Bucket Detective, because "famous book make it impossible for girls to resist sex ". A friend of Davids directs to a mysterious building under the promise that if they complete all that is asked of them, they will get what they desire. At various stages of navigating the building, Davids is able to dismember parts of his body, to receive passages from Bucket Detective, ultimately leading him to sever his finger, hand, and feet in pursuit of finishing the book.
The player's journey through the building is guided by found diary entries and an audio tour narrated by Gwen Sleeveless, a man raised as a child in service to the occupants of the building. As the player explores the building, Sleeveless' recordings explains that it is occupied by a cult who worship the Dark Lord Mishreal, founded by the Two Fathers: Dr. Z.W. Francis and Jedidah Holcombe. The player inadvertently becomes initiated to the cult and pass through a series of tests. Their final task is to bring about the rebirth of the Dark Lord through receiving the Genderless Child, a baby borne from Beth, a woman imprisoned and impregnated against her will by the Two Fathers. To complete the ritual, the player is tasked with delivering the Genderless Child to an elevator, upon which its closure causes the elevator to spew blood, implying the child has been sacrificed to the Dark Lord.
The game features five different endings depending on the choices made by the player during gameplay, with each providing an epilogue on the success or failure of Davidson's book, Bucket Detective. The player is presented with the choice at the end of the game to either leave, leave with Beth's child, or complete the sacrifice to bring about the rebirth of the Dark Lord and the end of the world. In most cases, Davids gains little insight or benefit from his experience, using the publishing of Bucket Detective to pick up women and ignore the consequences of his actions. In another instance, if the player leaves the baby behind and fails to complete the ritual, Davids realises he is a masochist and indulges in his self-destructive tendencies by removing all of his other body parts. However, the player is also able to exit the building upon entry, ending the game immediately.
Gameplay
Bucket Detective is a narrative-based adventure game in which players progress the game through exploration of the cult's building, with limited interaction with objects including to read notes providing background context, use items to complete puzzles, and press buttons to play Sleeveless' audio tour in various areas of the building. In some stages, progression to new areas is undertaken by the mechanic of interacting with machines to sacrifice parts of the player's body in order to contribute one of four sentences to 'the book Bucket Detective, selected through dialog options. As the game's story progresses, the player sacrifices larger and more impairing parts of their body to continue, such as their legs, ultimately affecting player movement from a walk to a crawl, or moving in a wheelchair. Barksdale stated these mechanics were based in a desire for the player to be "constantly aware of their body in a way the players normally are not".
The game also features several puzzles designed with the purpose of serving as steps in the ritual to bring about Mishreal's rebirth. Puzzle mechanics include the use of items, including finding stamps from separate offices of the Fathers to process a document, and using knowledge of the game's story to resolve the puzzles: for instance, players must complete a puzzle by positioning the urns of the Fathers, solvable by concealing the urns from one another in reflection of the narrative that the Fathers could not stand to be within sight of one another. Barksdale stated that the design of these puzzles intended to "teach the player more about the game world" by "re-enacting events which occurred (in) the game's lore." However, some puzzles, including a pattern-based "meditation game", were designed to be intentionally incoherent to induce a sense of disorientation and confusion.
Development
Following the release of his previous game in 2015, The Static Speaks My Name, Barksdale entered a period of creative frustration marked by a period of unemployment and illness. Barksdale began development of Bucket Detective in August 2015, immediately prior to commencing study for a Master of Arts at Aalto University School of Arts, Design and Architecture. The game was developed collaboratively by Barksdale over a seventeen month period between 2015 to 2017 with his Aalto University colleagues Samu Kovanen as a co-developer and programmer, and input from Mikael Immonen as a 3D artist, and Denis Zlobin as a composer. Co-designer Samu Kovanen stated that the game was developed as a "portfolio piece" that would represent "the best we could create by ourselves", with Kovanen minimising the use of third-party tools and dedicating time to implementing custom physics and character control systems to showcase the skills gained in the development process. Both Barksdale and Kovanen would use their development of Bucket Development as a case study for a thesis written in completion of their study at Aalto University. The development team secured a grant from the Finnish organisation AVEK (the Promotion Centre for Audiovisual Culture) to complete the project.
The writing and themes of Bucket Detective were strongly influenced by Barksdale's personal experiences. Barksdale stated that the overarching theme of the game is that "each of us has the ability to destroy ourselves in the pursuit of things we think will make us happy," in line with the his personal experiences of "depression, anxiety and obsession" that run through his works, including The Static Speaks My Name. Barksdale stated that the game's body horror elements had been inspired by his repeated hospitalisation with appendicitis during this time. Furthermore, he stated his experiences with religious education in childhood also influenced the themes around cults in the game, using religious archetypes of "a charismatic leader, salvation comes to those who sacrifice, (and) a day of rebirth (and) judgement which is coming soon." Barksdale stated the game's visual style was drawn heavily from the Gothic Revival aesthetic shown in films such as Rosemary's Baby and The Grand Budapest Hotel, and the design of the New York apartments The Dakota. Further, Barksdale attempted for the game to be "funnier" and possess more "overt humor" than its predecessor, The Static Speaks My Name, using cutscenes and dialogue that balance a "funny" tone with "weird and dark" content.
Bucket Detective was showcased in April 2017 at A MAZE, an international festival showcasing art and experimental games. The game was selected as one of 25 nominees at the festival, with the game contesting the narrative catergory. Barksdale personally attended the event to exhibit the game on April 26-29 2017.
Reception
Bucket Detective received generally positive reviews from critics, with praise directed for its unorthodox and compelling tone. Writing for PC Gamer, Tom Sykes stated the game was a "disturbing, gross, and occasionally pretty funny," stating "there's some truly horrible subject matter, and a bit of gore, but also an assortment of wonderful illustrations, a great script and a memorable soundtrack." Wiehahn Diederichs of Gearburn described the game as "incomprehensibly bizarre" and "one of those games that, upon completion, will leave you speechless and confused as your brain helplessly attempts to process what you've just experienced," praising the game for "(challenging) established notions of gameplay." Save or Quit praised the game's design as a "dark, twisted and cynical piece of work" that "captures a lust for detail" through its visual and level design.
Several critics expressed mixed views on the role of the characterisation in the game. Writing for Unwinnable, Khee Hoon Chan noted that "there hasn't been a lot of video games that feature protagonists as repugnant as Davids," stating "by featuring an irksome, self-absorbed man as its lead, Bucket Detective has shown itself to be hilarious, yet horrifying in its stark, detached presentation of abuse and violence," praising the characterisation for avoiding an "overbearing narrative" and "emotional posturing". Adam Smith of Rock Paper Shotgun stated "David is such a ludicrous idiot that stepping into his shoes is just an excuse to use him as a punching bag, something that the game acknowledges, and when other characters elicit more sympathy with a few seconds of screen-time, there's no space to step outside David and away from his grim, grubby little path." Save or Quit found the character of Sleeveless to be a "a character consistent with the overall tone and mood (of the game), and within that, a piercing social commentary on the dangers of misogyny, violence, and cult mentality."
Some critics expressed mixed to negative reception of the game's execution of its themes and narrative. Writing for Game Critics, Rebekah Ocker stated that the game's "story left something to be desired," finding that the game had a "persistent element of disjointedness" and implying that the treatment of themes of "pedophilia, rape and murder" in the game "felt wrong". Similarly, Rachael Brearton of Indie Hive described the game as "uncomfortable", dismissing the game as "an awkward combination of sensitive and potentially triggering subject matter and sporadic attempts at humour which is often difficult to laugh at and even feels distasteful at times." Pavan Shamdasani of the South China Morning Post gave a mixed review of the game, stating the game was "hard to recommend: it certainly achieves its goal, depicting a greedy, shameless human being and all the things he'd do to become a paperback writer. Barksdale has a warped sensibility, and it's certainly original – I'm just wondering if another medium would suit it better."
Legacy
Barksdale stated that Bucket Detective was a success, selling "several thousand copies" as of eight months of release, and attributed its success along with The Static Speaks My Name to open avenues into professional work in the video game industry, such as writing for the 2018 title Overkill's The Walking Dead. In a postmortem of the game, although Barksdale expressed disappointment that Bucket Detective had been overshadowed by attention towards his previous work, The Static Speaks My Name, he stated the game helped him overcome the "pressure of making games" as a "false expectation", and gave him "freedom to have my next project be whatever the fuck I wanted it to be." Barksdale cited the significant planning and work in the creation of Bucket Detective as prompting a shift away from video game development, stating "there are still games I want to make, and I'd be surprised if I never make another game, but my focus has shifted."
References
2017 video games
Adventure games
Windows games
MacOS games
Single-player video games
Indie video games
Video games developed in Finland
Video games about cults | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,600 |
Q: Suppressing log4net startup messages Is there a proper way to suppress these Log4Net startup messages?
log4net: Configuration update mode [Merge].
log4net: Logger [root] Level string is [ALL].
log4net: Logger [root] level set to [name="ALL",value=-2147483648].
log4net: Loading Appender [Console] type: [log4net.Appender.ConsoleAppender]
log4net: Converter [message] Option [] Format [min=-1,max=2147483647,leftAlign=False]
log4net: Converter [newline] Option [] Format [min=-1,max=2147483647,leftAlign=False]
log4net: Setting Property [ConversionPattern] to String value [%5level - %message%newline]
log4net: Converter [level] Option [] Format [min=5,max=2147483647,leftAlign=False]
log4net: Converter [literal] Option [ - ] Format [min=-1,max=2147483647,leftAlign=False]
log4net: Converter [message] Option [] Format [min=-1,max=2147483647,leftAlign=False]
log4net: Converter [newline] Option [] Format [min=-1,max=2147483647,leftAlign=False]
log4net: Setting Property [Layout] to object [log4net.Layout.PatternLayout]
log4net: Created Appender [Console]
log4net: Adding appender named [Console] to logger [root].
log4net: Loading Appender [RollingFile] type: [log4net.Appender.RollingFileAppender]
log4net: Setting Property [File] to String value [migrator.log]
log4net: Setting Property [AppendToFile] to Boolean value [True]
log4net: Setting Property [RollingStyle] to RollingMode value [Size]
log4net: Setting Property [MaxSizeRollBackups] to Int32 value [10]
log4net: Setting Property [MaximumFileSize] to String value [10MB]
log4net: Setting Property [StaticLogFileName] to Boolean value [True]
log4net: Converter [message] Option [] Format [min=-1,max=2147483647,leftAlign=False]
log4net: Converter [newline] Option [] Format [min=-1,max=2147483647,leftAlign=False]
log4net: Setting Property [ConversionPattern] to String value [%-5p %d %5rms %-22.22c{1} %-18.18M - %m%n]
log4net: Setting Property [ConversionPattern] to String value [%date [%thread] %-5level %logger - %message%newline]
log4net: Converter [date] Option [] Format [min=-1,max=2147483647,leftAlign=False]
log4net: Converter [literal] Option [ [] Format [min=-1,max=2147483647,leftAlign=False]
log4net: Converter [thread] Option [] Format [min=-1,max=2147483647,leftAlign=False]
log4net: Converter [literal] Option [] ] Format [min=-1,max=2147483647,leftAlign=False]
log4net: Converter [level] Option [] Format [min=5,max=2147483647,leftAlign=True]
log4net: Converter [literal] Option [ ] Format [min=-1,max=2147483647,leftAlign=False]
log4net: Converter [logger] Option [] Format [min=-1,max=2147483647,leftAlign=False]
log4net: Converter [literal] Option [ - ] Format [min=-1,max=2147483647,leftAlign=False]
log4net: Converter [message] Option [] Format [min=-1,max=2147483647,leftAlign=False]
log4net: Converter [newline] Option [] Format [min=-1,max=2147483647,leftAlign=False]
log4net: Setting Property [Layout] to object [log4net.Layout.PatternLayout]
log4net: Searched for existing files in [C:\dev\src\SAS-3.4\code\importers\Migrator\Migrator\bin\Release]
log4net: curSizeRollBackups starts at [0]
log4net: Opening file for writing [C:\dev\src\SAS-3.4\code\importers\Migrator\Migrator\bin\Release\migrator.log] append [True]
log4net: Created Appender [RollingFile]
log4net: Adding appender named [RollingFile] to logger [root].
We used this trick:
private static void TemporarilyHideLog4NetOnConsoleMessagesOnStartup()
{
var consoleOut = Console.Out;
var memoryStream = new MemoryStream();
var redirection = new StreamWriter(memoryStream );
Console.SetOut(redirection );
XmlConfigurator.Configure();
Console.SetOut(consoleOut );
}
Which seems to work but appears just wondering if there is an official API for this.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,783 |
\section{Axioptomechanical coupling}
\label{sec:app1}
The axioptomechanical coupling $g_a^{(0)}$ (see Eq.~\eqref{eq:Heffq}) parametrizes the effective single photon-photon-phonon coupling in the presence of an axion field. We formally derive such an effective coupling and show that
it can be reduced to the usual optomechanical coupling in the absence of the axion, assuming specific polarizations of the photon fields.
In position space, the number density operator in the effective Hamiltonian described in Eq.~$\eqref{eq:Heff}$ reads $n({\bf r}) = \sum_i \delta^3 \left({\bf r} -{\bf r}_i \right) $. In momentum space, a single phonon excitation state is given by
\begin{equation} \label{eq:}
\begin{split}
| {\bf k} \rangle = \frac{1}{\sqrt{n_0 S({\bf k})}} n_{{\bf k}} | 0 \rangle,
\end{split}
\end{equation}
where $|0\rangle $ is the ground state, $S({\bf k})$ is the static form factor of the target material, and $n_{{\bf k}}$
can be written in terms of phonon creation and annihilation operators as
\begin{equation} \label{eq:}
\begin{split}
n_{{\bf k}} = \sqrt{n_0 S ({\bf k})} (b_{-{\bf k}} - b^{\dagger}_{{\bf k}}),
\end{split}
\end{equation}
which is the Fourier transform of the number density operator in position space
\begin{equation} \label{eq:Mq1}
\begin{split}
n({\bf r}) = n_0 + V^{-1/2} \sum_{{\bf k}} e^{i {\bf k} \cdot {\bf r} } n_{{\bf k}},
\end{split}
\end{equation}
where $n_0$ is the average number density of the material.
Optical modes in the cavity correspond to electromagnetic fields with quantized momenta ${\bf p_i}$:
\begin{equation} \label{eq:quantizationp}
\begin{split}
{\bf p_i} = \frac{\pi}{L} \left(i_{x}, i_{y}, i_{z}\right),
\end{split}
\end{equation}
where $i_{x,y,z}$ are integer numbers, and $L$ is the cavity length assumed to be identical in the three dimensions. In our set-up, the lasers drive the optical states to be
one-directional along the cavity axis, defined as~the $z-$direction, {\it i.e.} $i_x, i_y = 0$. Then, optical modes can be labeled by a single integer number $i \equiv i_{z}$, and the momentum $|{\bf p}_i| = p_{i,z} = |i_z| \, \pi/L$.
Following \cite{scully1999quantum}, the electromagnetic fields can be decomposed as:
\begin{equation} \label{eq:Eq2}
\begin{split}
& {\bf E}({\bf r}, t)
=
\sum_{i}
\hat{\epsilon}_{i} \mathcal{E}_{{i}} a_{i}(t) e^{ - i {\bf p}_{i} \cdot {\bf r}} + {\rm h.c.},\\
& {\bf B} ({\bf r},t) = \sum_{i} \frac{ {\bf p}_{i} \times\hat{\epsilon}_{i} }{\omega_{i}} \mathcal{E}_{{i}} a_{i}(t) e^{ - i {\bf p}_{i} \cdot {\bf r}} + {\rm h.c.} , \\
\end{split}
\end{equation}
where $\hat{\epsilon}_{i}$ is the polarization vector of the mode, and the effective amplitude reads $\mathcal{E}_{{i}} = \sqrt{\frac{ \omega_{i} }{2 \varepsilon_0 V}} $ with $\omega_{i} = |{\bf p}_{i}| $. $a_{i}^{\dagger}$ and $a_{i}$ are the creation and annihilation operators for the corresponding eigenmode, respectively.
The acoustic modes are also quantized with the same boundary condition. Similar to the optical modes, in one dimension, the acoustic momentum is labeled by a mode number $k$, $|{\bf k}_{{\rm {\bf m}},k}| = k_{{\rm m},k,z} =k \pi/L$, where the subscript `m' stands for {\it mechanical}. Accordingly, we use the mode number $k$ to label the creation and annihilation operator $b_{{\bf k}_{{\rm {\bf m}},k}}^{\dagger} ,b_{\bf -{\bf k}_{{\rm {\bf m}},k}} =b^{\dagger}_k, b_k $ corresponding to such mode.
Following~Eqs.~\eqref{eq:Mq1}
and \eqref{eq:Eq2}, the interacting Hamiltonian in~Eq.~\eqref{eq:Heff} can be written in momentum space as
\begin{eqnarray} \label{eq:Heffall}
H_{\rm eff}
& \supset &
\frac{\alpha}{2} g_{a\gamma\gamma} a_0
\sum_{i,j,k}
\sqrt{\frac{n_0 S(-{\bf k}_{{\rm {\bf m}},k})}{V}} \mathcal{E}_{i} \mathcal{E}_{j}
\left( \hat{{\bf p}}_{j} - \hat{{\bf p}}_{i} \right) \cdot (\hat{\mathbf{\epsilon}}_{j} \times \hat{{\bf \epsilon}}^*_{i} )
\int d^3 {\bf r}
e^{i ( {\bf k}_{{\bf a}} + {\bf p}_{i} - {\bf p}_{j}- {\bf k}_{{\rm {\bf m}},k} ) \cdot {\bf r} }
a^{\dagger}_{j} a^{}_{i} b^\dagger_{k} \nonumber\\
& =&
\frac{\alpha}{2} g_{a\gamma\gamma} a_0
\sum_{i,j,k}
\sqrt{\frac{n_0 S(-{\bf k}_{{\rm {\bf m}},k})}{V}} \sqrt{\omega_{i} \omega_{j}} \frac{1}{2 \varepsilon_0 V}
\left( \hat{p}_{j} - \hat{p}_{i} \right) \cdot (\hat{\epsilon}_{j} \times \hat{\epsilon}^*_{i} )\,
a^{\dagger}_{j} a^{}_{i} b^\dagger_{k} \nonumber \\
&&\quad \times
\begin{cases}
\quad (2\pi)^3\delta^3 ({\bf k}_{{\bf a}} + {\bf p}_{i} - {\bf p}_{j}- {\bf k}_{{\rm {\bf m}},k} ) ,\quad & {\rm full~space}, \\
\displaystyle \prod_{d = x,y,z} \!\!\!L_d \, {\rm sinc} \left(\frac{L_d}{2} \left( k_{a,d} + p_{{i},d} - p_{{j},d} - k_{{\rm m},k,d} \right) \right), \quad & {\rm cavity~}(L_x,L_y,L_z),
\end{cases}
\end{eqnarray}
where we have rewritten the axion field $a({\bf r}, t) = a(t) e^{i {\bf k_{\rm a}} \cdot {\bf r} } + $h.c.~with the simplification $\omega_a \sim m_a$. Kinematics in the regime where $m_a \ll \omega_{i,j}$ requires the initial and final photons to be back-to-back (see discussion in the following section~\ref{sec:appkin}), {\it i.e.}~$\hat{{\bf p}}_{j} = - \hat{{\bf p}}_{i} $, which is further confirmed by the above relation as the coefficient is proportional to $\hat{{\bf p}}_{j} - \hat{{\bf p}}_{i} $. From the polarization vector factor, we observe that for circular polarized photons, the initial and final optical modes should have opposite handedness for the polarization cross product not to vanish, such that
\begin{equation}
\begin{split}
\left( \hat{{\bf p}}_{j} - \hat{{\bf p}}_{i} \right) \cdot (\hat{{\bf \epsilon}}_{j} \times \hat{\epsilon}^*_{i} ) = \frac{2}{\varepsilon_{r}},
\end{split}
\end{equation}
where the relative permittivity comes from the normalization of the polarization vector in the medium.
Matching into the effective Hamiltonian in Eq.~\eqref{eq:Heffq}, the momentum dependent axioptomechanical coupling reads
\begin{equation} \label{eq:g0D1}
\begin{split}
g_a^{(0)} ({\bf p}_{i},{\bf p}_{j},{\bf k}_{{\rm {\bf m}},k} ,{\bf k_{\rm {\bf a}}}) =
\sqrt{\frac{ S(-{\bf k}_{{\rm {\bf m}},k})}{n_0 V}} \sqrt{\omega_{i} \omega_{j}} \frac{n_0 \alpha }{ 2\varepsilon_0 \varepsilon_r }
\begin{cases}
\displaystyle \frac{(2\pi)^3}{ V} \delta^3 ({\bf k}_{{\bf a}} +{\bf p}_{i} - {\bf p}_{j} - {\bf k}_{{\rm {\bf m}},k} ) ,\quad & {\rm full~space}, \\
\displaystyle \frac{1}{ V} \!\!\! \prod_{d = x,y,z} \!\!\! L_d \, {\rm sinc} \phi_d , \quad &{\rm cavity~}(L_x,L_y,L_z),
\end{cases}
\\
\end{split}
\end{equation}
where the phases $\phi_d \equiv \frac{L_d}{2} \, \left( k_{a,d} + p_{{i},d} - p_{{j},d} - k_{{\rm m},k,d} \right)$ for $d = x,y,z$ in the cavity setup. Note that momentum conservation in the cavity is represented by a mode overlap factor, $a_\text{ovl}$,
\begin{equation}
\begin{split}
\frac{(2\pi)^3}{ V} \delta^3 ({\bf k}_{{\bf a}} + {\bf p}_{i} - {\bf p}_{j}- {\bf k}_{{\rm {\bf m}},k} ) \quad \rightarrow \quad a_{\rm ovl} \equiv
\frac{1}{ V} \!\!\! \prod_{d = x,y,z} \!\!\! L_d \,{\rm sinc} \phi_d,
\end{split}
\label{eq:Vdelta}
\end{equation}
which will define the kinematic/phase matching conditions, discussed in the next section.
In the linear dispersion regime, the static structure function reads $S({\bf k}_{{\rm {\bf m}},k}) = |{\bf k}_{{\rm {\bf m}},k}|/(2 m_{0} c_s)$, where $m_0$ is the nucleon mass of the target. According to the Clausius-Mossotti relation, the polarizability of the medium is related to the relative permittivity as
\begin{equation}
\begin{split}
\frac{n_0 \alpha}{2 \varepsilon_0 \varepsilon_r } = \frac{3}{2} \frac{\varepsilon_r-1}{\varepsilon_r+2}\frac{1}{\varepsilon_r}.
\end{split}
\label{eq:}
\end{equation}
Incorporating the overlap factor into the axioptomechanical coupling, we finally have
\begin{equation} \label{eq:g0D2}
\begin{split}
g_a^{(0)}
& = \sqrt{ \omega_{i} \omega_{j} } \sqrt{\frac{|{\bf k}_{{\rm {\bf m}},k}| }{2 c_s\rho V_{\rm mode} }}
\frac{3}{2} \frac{\varepsilon_r-1}{\varepsilon_r+2}\frac{1}{\varepsilon_r}
a_{\rm ovl}, \\
\end{split}
\end{equation}
where $\rho_0 = m_0 n_0$ is the mass density of the target. Note that the volume factor should be the mode volume as the electric and magnetic fields in~Eq.~\eqref{eq:Eq2} are only non-zero within the volume of the optical mode, $V_\text{mode} = L^2 \lambda_\text{opt} / \sqrt{\epsilon_r}$.
\section{Phase matching condition and the kinematics}
\label{sec:appkin}
In this section, we derive the kinematics and the phase matching condition of the axion absorption process, {\it i.e.}~
\begin{equation} \label{eq:}
\begin{split}
\gamma_{1}\, ({\bf p_{\bf 1}})\, + a \,({\bf k_{\rm {\bf a}}} )\, \rightarrow \gamma_{2} \,({\bf p_{\bf 2}})\, + \phi\, ({\bf k_{\rm {\bf m}}}) ,
\end{split}
\end{equation}
and specify the axion mass regime for the conditions to be valid.
The momentum and energy conservation read
\begin{equation} \label{eq:cons-all}
\begin{split}
{\bf p_{\bf 1}} + {\bf k_{\rm {\bf a}}} &= {\bf p_{\bf 2}} + {\bf k_{\rm {\bf m}}} \quad \rightarrow \quad
\omega_1 \, \hat{{\bf n}}_{\bf 1} + m_a v \, \hat{{\bf n}}_{\bf a} = \omega_2\, \hat{{\bf n}}_{\bf 2} + \frac{\Omega_{\rm m}}{c_s}\, \hat{{\bf n}}_{\rm {\bf m}},\\
\omega_1 + m_a &= \omega_2 + \Omega_{\rm m},
\end{split}
\end{equation}
where we have considered generic directions of the initial and final states, characterized by the unit vectors $\hat{{\bf n}}_{\bf i}$ with $i = 1,2$, a, and m. The axion energy, $\omega_a$, follows a Boltzmann distribution as defined in~Eq.~\eqref{eq:Boltzmann}. For simplicity, we have again used $\omega_a \sim m_a$. Defining $\cos\theta \equiv \hat{{\bf n}}_{\bf 1} \cdot \hat{{\bf n}}_{\bf 2} $ and $\cos \theta' \equiv \hat{{\bf n}}_{\bf a} \cdot \hat{{\bf n}}_{\rm {\bf m}}$, energy and momentum conservation, solving for $\omega_1$, give
\begin{equation} \label{eq:omega1}
\begin{split}
\omega_1
&= \frac{\Omega_{\rm m}}{2} \left(1 - \frac{m_a}{\Omega_{\rm m}} \pm \frac{1}{|\sin \frac{\theta}{2}|}
\sqrt{ \frac{1}{c_s^2} - \cos^2 \frac{\theta}{2} + \left( v^2 - \cos^2 \frac{\theta}{2} \right) \frac{m_a^2}{\Omega_{\rm m}^2}+ 2 \left(\cos^2 \frac{\theta}{2} - \frac{v}{c_s} \cos\theta' \right) \frac{m_a}{\Omega_{\rm m}} } \,
\right).
\end{split}
\end{equation}
As the lasers are applied along the cavity axis perpendicular to the end mirrors, the photon states are driven to be one-directional.
In such a case, the $\theta$ angle can only take two values: $\theta =0$ or $\theta = \pi$. The photon momentum is unphysical, given a non-zero $\Omega_{\rm m}$, when $\theta = 0$. The only physical solution is $\theta = \pi$, {\it i.e.}~the incoming and outgoing photons are back-to-back. In this case, Eq.~\eqref{eq:omega1} simplifies to
\begin{equation} \label{eq:w11}
\begin{split}
\omega_1
&= \frac{\Omega_{\rm m}}{2} \left(1 - \frac{m_a}{\Omega_{\rm m}} \pm
\sqrt{ \frac{1}{c_s^2} +v^2 \frac{m_a^2}{\Omega_{\rm m}^2}-2 \frac{v}{c_s} \cos\theta' \frac{m_a}{\Omega_{\rm m}} } \,
\right).
\end{split}
\end{equation}
For the range of axion masses we are interested in, $m_a v \ll m_a \ll \Omega_{\rm m}\,c_s^{-1} \approx \omega_1 \sim $ eV, the term dependent on the angle $\theta'$ is always negligible. Therefore, we will assume $\theta' = 0$ in the following. Note that, with $c_s \ll 1$,~Eq.~\eqref{eq:w11} further simplifies to take the approximate form $\omega_1 \approx \Omega_{\rm m}/(2 c_s)$.
Fixing the angles as argued above, the energy and momentum conservation relations in~Eq.~\eqref{eq:cons-all} are satisfied exactly with
\begin{equation} \label{eq:cons-exact}
\begin{split}
\Omega_{\rm m} = c_s \frac{2 \omega_1 + (1+v)m_a}{1+c_s},\quad \text{ and } \quad \omega_2 = \frac{ (1-c_s ) \omega_1 + (1-v\,c_s)m_a}{1+c_s}.
\end{split}
\end{equation}
Note that we have used the continuous momentum conservation above. However, in a cavity, where the optical and acoustic modes are quantized, the momentum matching is parametrized by the mode overlap factor $a_{\rm ovl}$ given in~Eq.~\eqref{eq:Vdelta} (see also Eq.~\eqref{eq:aovl} in the main text).
To respect the kinematic matching condition, with fixed $\omega_1$ for a given $m_a$, $\omega_2$ and $\Omega_{\rm m}$ should be chosen to satisfy $a_{\rm ovl} \sim 1$. Hence, assuming the pump laser being on-resonance with the cavity,
\begin{equation}\label{eq:nm}
n_{\rm m} = n_1 + n_2 + \left[ \frac{v\, m_a}{\omega_{\rm FSR}} \right] = 2 n_1 + \left[\frac{m_a-\Omega_{\rm m}}{\omega_{\rm FSR}}\right] + \left [ \frac{v\, m_a}{\omega_{\rm FSR}} \right ] ,
\end{equation}
which is the cavity realization of Eq.~\eqref{eq:cons-exact} and perfectly agrees with it when the lasers and the mechanical drive are on-resonance. Above and hereafter, the square brackets denote the integer part of their argument. $\omega_{\rm FSR} \equiv \pi/L$ (free spectral range) defines the frequency separation between consecutive optical modes. Note that the last term in Eq.~\eqref{eq:nm}, coming from the axion momentum, is zero for $m_a < \text{meV}$ in a meter-long cavity. In the above equation and hereafter, we take $n_i$ to be the natural number labelling the mode $i$, {\it i.e.} $n_i = |i_z|$ where $i_z$ is defined in Eq.~\eqref{eq:quantizationp}. Eq.~\eqref{eq:sincmatch} in the main text is recovered taking into account that $\Omega_{\rm m} \sim \omega_{\rm FSR} \ll n_1 \, \omega_{\rm FSR}$.
Requiring $\omega_2$ to be a resonance mode frequency, Eq.~\eqref{eq:cons-exact} specifies how the cavity length must be tuned,
\begin{equation} \label{eq:modematch}
\begin{split}
n_1 - n_2 = \frac{L}{\pi} \frac{2 c_s \omega_{{\rm pump}} - (1- v\, c_s) m_a}{1+c_s},
\end{split}
\end{equation}
where we have assumed the pump laser to be on-resonance, {\it i.e.} $\omega_1 = \omega_{\rm pump}$.
Note that, in the absence of an axion, the Stokes and anti-Stokes sidebands reside at $n_{\rm S,aS} \approx n_1 \approx n_{\rm m}/2$. For example, for a meter-long cavity, $n_{\rm S,aS} = n_1 \pm 2$ with $n_1 \sim \mathcal{O}(10^6) $. From~Eq.~\eqref{eq:modematch}, one observes that in the light axion regime ($m_a \lesssim \omega_{\rm FSR}$), the axion sideband resides at the same mode as the Stokes sideband, while in the heavy regime ($m_a \gtrsim \omega_{\rm FSR}$,) the axion sideband resides at a different mode. We note that, depending on the experimental strategy, $\omega_2$ can be slightly off-resonance, and the phase matching can be mildly violated with a suppressed $a_{\rm ovl}$.
\section{Derivation for rates of axion and optomechanical processes}
\label{sec:Srate}
We now derive rates for the relevant processes with the axioptomechanical and optomechanical couplings.
The optical and acoustic modes in the cavity can be driven by external photon lasers and a mechanical drive. The Hamiltonian terms describing the driven system read \cite{PhysRevA.30.1386,PhysRevA.31.3761,Aspelmeyer:2013lha,10.1093/oso/9780198828143.003.0005,RevModPhys.82.1155}
\begin{equation} \label{eq:}
\begin{split}
H_{\rm opt}
\supset \sum_n
\omega_{n} a_n(t) a_n^{\dagger} (t) + \sum_n \sqrt{\kappa} \left( a_{\rm in} (t) a_n^{\dagger} (t) + {\rm h.c.} \right) + H_{\rm laser},
\end{split}
\end{equation}
for the optical modes, and similarly for the acoustic modes
\begin{equation} \label{eq:}
\begin{split}
H_{\rm ac} \supset \sum_k \Omega_{{\rm m},k} b_k(t) b_k^{\dagger} (t) + \sum_k \sqrt{\Gamma_{\rm m}} \left( b_{\rm in} (t) b_k^{\dagger} (t) + {\rm h.c.} \right)+ H_{\rm mechanical~drive}.
\end{split}
\end{equation}
The sums run over integer numbers $n,k = 1,2,\cdots$, corresponding to the optical and acoustic cavity modes $a_n$ and $b_k$, respectively, with frequency
\begin{equation} \label{eq:}
\begin{split}
\omega_n = |{\bf p}_n|= n \frac{\pi}{L},\quad \text{ and } \quad \Omega_{{\rm m},k} = c_s|{\bf k}_{{\rm {\bf m}},k}| = k \frac{\pi c_s}{L}.
\end{split}
\end{equation}
Above, $\kappa$ and $\Gamma_{\rm m}$ are the optical and mechanical decay rates, respectively, given by the properties of the cavity and the filling material,
\begin{equation} \label{eq:}
\begin{split}
\kappa = \frac{ \pi}{\mathcal{F}_{\rm opt} L},\quad \text{ and } \quad
\Gamma_{\rm m} = \frac{c_s \pi}{\mathcal{F}_{\rm ac} L},
\end{split}
\end{equation}
where the optical and acoustic finesse of the cavity are determined by the reflectivity of the end mirrors, $r_{1,2}^{\rm (ac),(opt)}$, as follows
\begin{equation} \label{eq:}
\begin{split}
\mathcal{F}_{\rm ac,opt} = \frac{1}{\pi} \, \frac{1-r^{\rm (ac),(opt)}_1r^{\rm (ac),(opt)}_2}{r^{\rm (ac),(opt)}_1 r^{\rm (ac),(opt)}_2}.
\end{split}
\end{equation}
The optical and acoustic modes in the cavity are coherent states, denoted as $| \alpha_i \rangle$ and $|\beta_i \rangle$, respectively, where the $i$ labels the cavity mode. These are eigenstates of the creation and annihilation operators,
\begin{equation}
\begin{split}
& a_n | \alpha_n \rangle = \alpha_n | \alpha_n \rangle ,\qquad \text{ where } \quad |\alpha_n|^2 = \langle \alpha_n | a_n^{\dagger} a_n | \alpha_n \rangle = N_{\gamma,n}^{\rm circ} , \\
& b_k | \beta_k \rangle = \beta_k | \beta_k \rangle ,\, \, \qquad \,\, \! \text{ where } \quad |\beta_k|^2 = \langle \beta_k | b_k^{\dagger} b_k | \beta_{k} \rangle = N_{\phi,k}^{\rm circ} .
\end{split}
\end{equation}
The density of states of the driven optical and acoustic modes is determined by the photon lasers and the mechanical drive. For finite width laser input described by
\begin{equation} \label{eq:}
\begin{split}
a_{\rm in}(t) = \alpha_{\rm L} e^{-i\omega_{\rm L}t - \kappa_{\rm L} |t|/2},
\end{split}
\end{equation}
where $\omega_{\rm L}$ is the laser frequency and $\kappa_{\rm L}$ is the laser width, according to the input-output theorem, the optical density of states reads
\begin{equation} \label{eq:}
\begin{split}
\rho(\omega) = \frac{1}{\pi} \frac{\kappa_{\rm L}/2}{(\omega - \omega_{\rm L})^2 + (\kappa_{\rm L}/2)^2},
\end{split}
\end{equation}
assuming $\kappa_{\rm L} \ll \kappa$. The total photon population number in the mode $n$ is related to the laser power $P_{\rm L}$ as
\begin{equation} \label{eq:ng}
\begin{split}
N_{\gamma,n}^{\rm circ} = |\alpha_n|^2 = \frac{4 \, P_{\rm L}}{\omega_{\rm L} \kappa} \frac{(\kappa/2)^2}{(\omega_{\rm L} - \omega_{n})^2 + (\kappa/2)^2} \equiv N_{\gamma,{\rm L}} \frac{(\kappa/2)^2}{(\omega_{\rm L} - \omega_{n})^2 + (\kappa/2)^2}.
\end{split}
\end{equation}
The cavity mode with the frequency closest to the laser frequency has the largest population. Thus, we consider processes with only the closest resonant mode to the optical laser frequency. In the following, $N_{\gamma,{\rm pump}}^{\rm circ}$ ($N_{\gamma,{\rm probe}}^{\rm circ}$) will refer to the optical mode $n$ excited by the pump (probe) laser, {\it i.e.} the closest mode to the laser frequency. For the mechanical mode, similarly, the density of states reads
\begin{equation} \label{eq:}
\begin{split}
\rho_{\rm m}(\Omega) = \frac{1}{\pi} \frac{\Gamma_{\rm L}/2}{(\Omega - \Omega_{\rm m})^2 + (\Gamma_{\rm L}/2)^2},
\end{split}
\end{equation}
with $\Omega_{\rm m}$ and $\Gamma_{\rm L}$ being the frequency and the width of the mechanical drive, respectively. The phonon population in the driven mode is also reduced from the total input if $\Omega_{\rm m} \ne \Omega_{{\rm m},k}$ according to
\begin{equation} \label{eq:nphi}
\begin{split}
N_{\phi,k}^{\rm circ} = |\beta_k|^2 = N_{\phi} \frac{(\Gamma_{\rm m}/2)^2}{(\Omega_{\rm m} - \Omega_{{\rm m},k})^2 + (\Gamma_{\rm m}/2)^2}.
\end{split}
\end{equation}
We do not specify how the input phonon population $N_{\phi}$ is related to the mechanical drive, instead, we take fixed values limited by various considerations to be discussed in the next section. As in the previous case, we only consider the acoustic resonant mode closest to the frequency of the coherent phonons and drop the index $k$ to simplify notation.
In the absence of an optical laser or mechanical drive, the optical and acoustic modes have density of states determined by the cavity decay rates
\begin{equation} \label{eq:}
\begin{split}
\rho_{n} (\omega) =
\frac{1}{\pi}\frac{\kappa/2}{(\omega - \omega_{n})^2 + (\kappa/2)^2},\quad \text{ and } \quad
\rho_{{\rm m}, k} (\Omega) = \frac{1}{\pi}
\frac{\Gamma_{\rm m}/2}{(\Omega - \Omega_{{\rm m}, k})^2 + (\Gamma_{\rm m}/2)^2}.
\end{split}
\end{equation}
These are useful when deriving un-stimulated processes in the absence of one or more lasers.
We use Fermi's golden rule to derive the interaction rate. Firstly, we consider the process where all the incoming and outgoing photon and phonon modes are driven, {\it i.e.}~populated, with the photon lasers and the mechanical drive. Note that, as reasoned above, we only calculate the interaction rate for the initial optical mode with the closest resonance frequency $\omega_{n}$ to the pump laser frequency, {\it i.e.}~the mode number $n_\text{pump} = [ \omega_{\rm pump}/ \omega_{\rm FSR} ]$, and similarly for the final optical and acoustic modes, taken to be the closest to $\omega_{\rm probe}$ and $\Omega_{\rm m}$, respectively; $n_\text{probe} = [ \omega_{\rm probe}/ \omega_{\rm FSR} ]$, and $n_{\Omega_{\rm m}} = [\Omega_{\rm m}/(c_s \omega_{\rm FSR})]$. The corresponding detuning for each mode is defined as in the main text,
\begin{equation}
\Delta_{\rm pump,probe} \equiv \omega_{\rm pump,probe} - n_\text{pump,probe}\, \omega_{\rm FSR}, \quad \text{ and } \quad \Delta_{\rm m} \equiv \Omega_{\rm m} - n_{\Omega_{\rm m}} \, \omega_{\rm FSR} \,c_s.
\end{equation}
Treating the axion as a classical field, according to Fermi's golden rule, the rate reads
\begin{equation}
\begin{split}
\Gamma
&=\sum_{I,F} | \langle \alpha_{{\rm pump}}, \alpha_{{\rm probe}}, \beta_{\Omega_{\rm m}} | H_{\rm eff} | \alpha_{{\rm pump}}, \alpha_{{\rm probe}}, \beta_{\Omega_{\rm m}} \rangle |^2 (2\pi) \delta ( \omega_{I} - \omega_F ) \\
&= (2\pi) \sum_{I,F} | \langle \alpha_{{\rm pump}}, \alpha_{{\rm probe}}, \beta_{\Omega_{\rm m}} | \sum_{i, j, k} g_a^{(0)} \big({\bf p}_{i},{\bf p}_{j},{\bf k}_{{\rm {\bf m}},k} ,{\bf k_{\rm {\bf a}}} \big) g_{a\gamma\gamma} a(t) a_{i} a^{\dagger}_{j} b_k^\dagger |\alpha_{{\rm pump}}, \alpha_{{\rm probe}}, \beta_{\Omega_{\rm m}} \rangle |^2 \delta ( \omega_{I} - \omega_F ) \\
&= (2\pi) \sum_{I,F} \left| g_a^{(0)} \right|^2 g_{a\gamma\gamma}^2 |a(t)|^2
|\alpha_{{\rm pump}}|^2\, |\alpha_{{\rm probe}}|^2\, |\beta_{\Omega_{\rm m}}|^2
(2\pi) \delta (\omega_{I} - \omega_F) \\
&= (2\pi) g_{a\gamma\gamma}^2 \left| g_a^{(0)} \right|^2 \frac{2\rho_a}{m_a^2} N_{\gamma,{\rm pump}}^{\rm circ} N_{\gamma,{\rm probe}}^{\rm circ} N_{\phi}^{\rm circ}
\\
&\qquad \times \int d \omega_a \, d \omega_1 \, d \omega_2 \, d\Omega \, B_{m_a}(\omega_a) L(\omega_1-\omega_{\rm pump},\kappa_{\rm L}) L(\omega_2-\omega_{\rm probe},\kappa_{\rm L}) L(\Omega - \Omega_{\rm m}, \Gamma_{\rm L}) \delta (\omega_1 + \omega_a - \omega_2 - \Omega ) \\
& \simeq (2\pi) g_{a\gamma\gamma}^2 \left| g_a^{(0)} \right|^2 \frac{2\rho_a}{m_a^2} N_{\gamma,{\rm pump}}^{\rm circ} N_{\gamma,{\rm probe}}^{\rm circ} N_{\phi}^{\rm circ} \\
& \qquad \qquad \qquad \quad \ \ \qquad \times
\begin{cases}
\int d \omega_2 \, B_{m_a}(\omega_2 + \Omega_{\rm m} - \omega_{\rm pump})L(\omega_2-\omega_{\rm probe},\kappa_{\rm L}) ,\quad &\Delta f_a \gg \kappa_{\rm L} \\
\int d \omega_2 \, L(\omega_2 + \Omega_{\rm m} - m_a - \Delta f_a/2 -\omega_{\rm pump},\kappa_{\rm L}) L(\omega_2-\omega_{\rm probe},\kappa_{\rm L}) , \quad &\Delta f_a \ll \kappa_{\rm L}\\
\end{cases} \\
& \simeq (2\pi) g_{a\gamma\gamma}^2 \left| g_a^{(0)} \right|^2 \frac{2\rho_a}{m_a^2} N_{\gamma, {\rm pump}}^{\rm circ} N_{\gamma, {\rm probe}}^{\rm circ} N_{\phi}^{\rm circ}
\begin{cases}
\displaystyle B_{m_a}(\omega_{\rm probe} + \Omega_{\rm m} - \omega_{\rm pump}),\quad &\Delta f_a \gg \kappa_{\rm L}, \\
\displaystyle L(\omega_{\rm probe} + \Omega_{\rm m} - \omega_{\rm pump}-m_a - \Delta f_a/2,2 \kappa_{\rm L})
,\quad &\Delta f_a \ll \kappa_{\rm L}. \\
\end{cases}
\end{split}
\end{equation}
where we abbreviate a Lorentzian distribution with a full-width-at-half-maximum $\delta x$ and a central value $x_0$ as $L(x - x_0,\delta x)$, as displayed in Eq.~\eqref{eq:LorentzianStandard}. The Boltzmann distribution, $B_{m_a}(\omega_a)$, is given in Eq.~\eqref{eq:Boltzmann}. $\omega_1$ and $\omega_2$ are the frequencies of the initial and final photons, respectively, as defined in the main text.
Note that the sum over initial and final states, $\sum_{I,F}$, is non-trivial because of the finite width of the optical and acoustic modes.
For reference, we derive the rate for a single photon counting search scheme, where there is no probe laser and thus the final state photons are spontaneously emitted. The rate in this case reads
\begin{equation} \label{eq:sb}
\begin{split}
\Gamma
&=\sum_{I,F} | \langle \alpha_{{\rm pump}}, 0 , \beta_{\Omega_{\rm m}} | H_{\rm eff} |\langle \alpha_{{\rm pump}}, 1 , \beta_{\Omega_{\rm m}} \rangle |^2 (2\pi) \delta ( \omega_{I} - \omega_F ) \\
&= (2\pi) g_{a\gamma\gamma}^2 \left| g_a^{(0)} \right|^2 \frac{2\rho_a}{m_a^2} N_{\gamma, {\rm pump}}^{\rm circ} N_{\phi}^{\rm circ} \\
&\qquad \times \int d \omega_a \, d \omega_1 \, d \omega_2 \, d\Omega \, B_{m_a}(\omega_a) L(\omega_1-\omega_{\rm pump},\kappa_{\rm L})
L(\omega_2 - \omega_{n_2},\kappa)
L(\Omega - \Omega_{\rm m}, \Gamma_{\rm L}) \delta (\omega_1 + \omega_a - \omega_2 - \Omega ) \\
&\simeq (2\pi) g_{a\gamma\gamma}^2 \left| g_a^{(0)} \right|^2 \frac{2\rho_a}{m_a^2} N_{\gamma,{\rm pump}}^{\rm circ} N_{\phi}^{\rm circ} \int d \omega_2 \, B_{m_a}( \omega_2 + \Omega_{\rm m} - \omega_{\rm pump})
L(\omega_2 - \omega_{n_2},\kappa)
\\
& \simeq (2\pi) g_{a\gamma\gamma}^2 \left| g_a^{(0)} \right|^2 \frac{2\rho_a}{m_a^2} N_{\gamma,{\rm pump}}^{\rm circ} N_{\phi}^{\rm circ}
\begin{cases}
\displaystyle L(\omega_{\rm pump} + m_a + \Delta f_a/2 - \Omega_{\rm m} - \omega_{n_2}, \kappa),\quad &\kappa \gg \Delta f_a, \\
\displaystyle B_{m_a}( \Omega_{\rm m} + \omega_{n_2}- \omega_{\rm pump} ), \quad & \kappa \ll \Delta f_a.
\end{cases}
\end{split}
\end{equation}
Above, $\omega_{n_2}$ is the closest resonance frequency to the outgoing photon.
Note that we have assumed the laser and mechanical drive widths $\kappa_{\rm L}, \Gamma_{\rm L} \ll \kappa$. Such spontaneously emitted photons from the axion sideband could be detected with a single photon detector. Note that, because the frequency of the phonon population is chosen to kinematically match the axion mass ($a_{\rm ovl} \sim 1$), the search is narrow-band in its nature up to $\omega_{\rm FSR}$, within which the momentum conservation can be violated in the cavity setup.
A band-pass filter can be then employed to forbid most of the background photons from the transmitted pump laser and the Stokes, anti-Stokes sidebands. Considering background photons within a pass band of width $\kappa_{F}$, the signal-to-background ratio for the single photon detection
reads
\begin{equation}
\frac{S}{B} \approx 8 \sqrt{\frac{2\pi}{e}} \left(\frac{2\rho_a}{m_a^2} g_{a\gamma \gamma}^2\right) \frac{|g_a^{(0)}|^2 m_a^2}{\kappa^2\kappa_L\Delta f_a} \frac{N_{\gamma,\text{pump}}^{\rm circ} N_\phi^{\rm circ} }{ N_{\rm bkg} }.
\label{eq:SB}
\end{equation}
We have assumed $\Delta f_a \gg \kappa > \kappa_{\rm F}$ as such a scheme is optimal in the large axion mass regime, where the signal window is located far away from where the background photons peak, as can be seen in~Eq.~\eqref{eq:sb} and~Fig.~\eqref{fig:scheme}.
$N_{\rm bkg}$ is the total number of background photons populating the cavity, that can be approximated by $N_{\rm bkg} \sim N_{\gamma,\text{pump}}^{\rm circ}$. Due to the different polarizations between the background photons and axion-induced photons, the background can be further reduced by polarization filters.
Note that such a background typically dominates over the dark count of the detector. With a meter-long cavity, $\mathcal{F}_{\rm opt} /\pi= 10^6$, $\kappa_L = 1 \,{\rm Hz}$, and $N_{\phi} = 10^{16}$, such a scheme is only sensitive to $(g_{a\gamma\gamma}/{\rm GeV}^{-1}) \sim 4 \times 10^{-2}\, (m_a / {\rm eV})^{1/2}$.
\section{Stokes and anti-Stokes processes and the phonon number limit}
\label{sec:SMphononN}
While the initial photon and the phonon states are populated by driving the cavity, the Stokes/anti-Stokes sidebands are populated spontaneously. Note that, because in principle the photons from the probe laser have a different polarization than the photons from the pump laser, the probe laser does not populate the final states participating in the Stokes and anti-Stokes processes. However, it may be experimentally challenging to control the polarization of the total photons in the probe laser. Even if a given percent of the probe population has the necessary polarization to contribute, in the regime of axion masses $m_a, |m_a - 2 \, \Omega_{\rm m}| \gg \kappa_{\rm L}$, such contribution is negligible.
Assuming a non-populated photon final state, the rate for the Stokes background reads
\begin{equation}
\begin{split}
\Gamma_{\rm S}
&=\sum_{I,F} | \langle \alpha_{\text{pump}}, 0, \beta_{\Omega_{\rm m}} | H_{\rm eff} | \alpha_\text{pump}, 1, \beta_{\Omega_{\rm m}} \rangle |^2 (2\pi) \delta ( \omega_{I} - \omega_F ) \\
&= (2\pi) |g_0|^2 N_{\gamma,{\rm pump}}^{\rm circ} N_{\phi}^{\rm circ} \int d \omega_1 \, d \omega_2 \, d\Omega \, L(\omega_1 - \omega_{\rm pump},\kappa_{\rm L}) L(\omega_2 - \omega_{n_{\rm S}},\kappa) L(\Omega - \Omega_{\rm m},\Gamma_{\rm L}) \delta (\omega_1 - \omega_2 - \Omega ) \\
&\simeq (2\pi) |g_0|^2 N_{\gamma,{\rm pump}}^{\rm circ} N_{\phi}^{\rm circ} \int d \omega_2 \, L(\omega_2 + \Omega_{\rm m} - \omega_{\rm pump},\kappa_{\rm L}) L(\omega_2 - \omega_{n_{\rm S}},\kappa) \\
&\simeq (2\pi) |g_0|^2 N_{\gamma,{\rm pump}}^{\rm circ} N_{\phi}^{\rm circ} \, L(\omega_{\rm pump} - \Omega_{\rm m} - \omega_{n_{\rm S}},\kappa), \\
\end{split}
\end{equation}
where $\omega_{n_{\rm S}}$ is the closest optical mode to the frequency of the Stokes photon.
Similarly, for the anti-Stokes background,
\begin{equation} \label{eq:}
\begin{split}
\Gamma_{\rm aS}
&= (2\pi) |g_0|^2 N_{\gamma,{\rm pump}}^{\rm circ} N_{\phi}^{\rm circ} L(\omega_{\rm pump} + \Omega_{\rm m} - \omega_{n_{\rm aS}},\kappa), \\
\end{split}
\end{equation}
where $\omega_{n_{\rm aS}}$ is now the closest optical mode to the frequency of the anti-Stokes photon. Such rates should not be too large to deplete the pump laser. As a crude estimate, we require that
\begin{equation} \label{eq:DepletionCondition}
\begin{split}
\Gamma_{\rm S,aS} \lesssim 50\% \, \Gamma_{\rm pump},
\end{split}
\end{equation}
where $\Gamma_{\rm pump} = P_{\rm pump}/\omega_{\rm opt} = N_{\gamma,{\rm pump}} \,\kappa / 4$. In this case, assuming the pump laser being on-resonance with the cavity, Eq.~\eqref{eq:DepletionCondition} requires that
\begin{equation} \label{eq:}
\begin{split}
N_{\phi}^{\rm circ} \lesssim \frac{\kappa}{16 \pi\, |g_0|^2\, L(\omega_{\rm pump} \pm \Omega_{\rm m} - \omega_{n_{\rm S,aS}},\kappa) } .
\end{split}
\end{equation}
Note that, given that the pump laser is on-resonance, the argument in the Lorentzian is sensitive to $(\Omega_{\rm m} \ {\rm mod} \ \omega_{\rm FSR})$. As for long-meter cavities $\Omega_{\rm m} \sim \omega_{\rm FSR} \gg \kappa$, one generically expects $N_\phi^{\rm circ} < {\cal O}(\Omega_{\rm m}^2 / (8 \pi |g_0|^2 )) \simeq {\cal O}(10^{16}) (L/{\rm m})^2$ for superfluid Helium and $\lambda_{\rm pump} = 1.5 \, \mu$m.
However, assuming also the probe laser to be on-resonance with the cavity, energy conservation demands that $(\Omega_{\rm m} \ \text{mod} \ \omega_{\rm FSR}) = m_a$. Hence, an axion mass heavier than $m_a \gtrsim 10^8 |g_0|$ is required to host $N_\phi^{\rm circ} \sim 10^{16}$ without depleting the pump laser. For lighter axions, the Stokes and anti-Stokes sidebands are closer to the axion sideband and the depletion is more drastic.
In such a case, one could dial the cavity such that the axion sideband ({\it i.e.} the probe laser) is slightly off-resonance to suppress the depletion. The loss of circulating probe photons can in principle be compensated by a larger laser power.
Such a limit is derived assuming $a_{\rm ovl} = 1$ for $g_0$. However, in the heavy axion mass regime, as $\Omega_{\rm m}$ is chosen such that the phase matching condition is exact for the axion process, the phase matching condition cannot be satisfied exactly for the Stokes and anti-Stokes processes. Thus a suppression factor from the mode mismatch should apply as
\begin{equation} \label{eq:}
\begin{split}
& \Gamma_{\rm S,aS} \rightarrow
\text{sinc}^2 \left( \frac{\pi}{2} \big (n_{\text{pump}} +n_{\rm S,aS} - n_{\rm m}) \right)\, \Gamma_{\rm S,aS} = \text{sinc}^2 \left( \frac{\pi}{2} \left[ \frac{m_a}{\omega_{\rm FSR}} \right] \right) \, \Gamma_{\rm S,aS} ,
\end{split}
\end{equation}
where $n_{\rm S,aS} = \omega_{n_{\rm S,aS}} / \omega_{\rm FSR}$.
Such a suppression can be significant for axions with $m_a > \omega_{\rm FSR}$, what allows for more phonons to be driven into the cavity without depleting the pump further. Note that similar processes stimulated by the phonon and the probe laser also occur, which, however, are much more suppressed due to the less favorable kinematic matching ($a_{\rm ovl} \ll 1$).
Ultimately, $N_\phi^{\rm circ}$ is limited by the amount of acoustic energy $U_{\rm m}$ that the cavity can host,
\begin{equation} \label{eq:numberphononsmax}
\begin{split}
N_{\phi}^{\rm circ}= \frac{U_{\rm m}}{\Omega_{\rm m}},\quad \text{ and }\quad
U_{\rm m} \lesssim \frac{1}{2} \, \rho_{\rm He} \, c_s^2 \left(\frac{\delta \rho_{\rm He}}{\rho_{\rm He}}\right)^2 V_{\rm mode}.
\end{split}
\end{equation}
Imposing $\delta \rho_{\rm He} / \rho_{\rm He} < 10^{-3}$, the number of phonons should not surpass $N_{\phi}^{\rm circ} \lesssim 10^{19} (L/{\rm m})^2$, assuming $\lambda_{\rm pump} = 1.5 \, \mu\text{m}$.
The non-depletion limit also characterizes the so-called weak coupling regime, {\it i.e.}~$g \lesssim \kappa$, where $g \sim g_0 \sqrt{N^{\rm circ}_{\phi} /[1+(2\, \Delta_{\rm S}/\kappa)^2]}$~\cite{Aspelmeyer:2013lha}, where $\Delta_{\rm S}$ is the detuning between the Stokes photon frequency and the closest optical mode. Close to the strong coupling limit, {\it i.e.}~$g \gtrsim \kappa $, non-linear optomechanical couplings of the order of $\alpha^2\,n({\bf r})^2$ start to contribute.
On the other hand, the leading-order Fermi's golden rule we have used is only valid in the weak coupling regime, beyond which, a second-order Fermi's golden rule should be employed to derive multi-phonon emission processes, that is on the same order as the non-linear optomechanical interaction. Thus, in the light axion regime ($N_{\phi} \sim 10^{16}$), non-linear and second-order processes, leading to secondary axion-less sidebands, should also be calculated to determine the total depletion rate and the background, which are however generically weaker than the single emission Stokes and anti-Stokes processes as higher order corrections.
We postpone such calculations to a later paper and use the limit estimated above in this work.
\section{Scalings of the sensitivity curves with the input parameters}
\label{sec:e}
In this section we display the explicit scaling of the $g_{a\gamma \gamma}$ vs $m_a$ sensitivity curves as a function of the experimental input factors. These factors are of four kinds: parameters related with the geometry of the cavity (length $L$ and optical finesse ${\cal F}_\text{opt}$), the material used to fill it (density $\rho$, speed of sound $c_s$, relative permittivity $\epsilon_r$), the photon lasers and the mechanical drive ($N_\phi$, $\kappa_L$, $P_\text{pump}$ and $P_\text{probe}$), and the axion parameters ($\rho_a$ and $m_a$).
First, we discuss specifics of the scanning strategy to cover an extended axion mass range, that defines the reach to $g_{a\gamma\gamma}$ over a given integration time per each scan, $t_{\rm int}$, and the coverage in axion masses for a total exposure time of the experiment, $T_{\rm exp}$. As has been discussed in the main text, the reach curve is determined by two subsequent measurements,
separated by a probe frequency window $\delta$.
The generic scanning strategy consists of matching $\delta$ to the relevant width of $P_\text{sig}$, up to an order one parameter $\epsilon$, in order not to lose on sensitivity:
\begin{enumerate}[(a)]
\item For $\Delta f_a <2\, \kappa_L$, {\it i.e.} $m_a \lesssim 5 \text{ neV} (\kappa_L/\text{Hz})$, the scanning step is set by $\delta = \epsilon \kappa_L$.
Two successive measurements overlap at $\delta/2 \approx \Delta + \Omega_{\rm m} - m_a $, where
\begin{equation}
{\rm SNR} \propto L(\delta/2 , 2 \kappa_{\rm L}) = \frac{1}{\pi \kappa_{\rm L}}\frac{1}{\epsilon^2/4+1}.
\end{equation}
The coverage in axion masses grows linearly with the exposure time of the experiment as
\begin{equation}
\text{coverage} = \epsilon \, \kappa_{\rm L} \, \frac{T_{\rm exp}}{t_{\rm int}},
\end{equation}
which for $\epsilon = 1$ allows one to scan $\sim 20 \text{ neV/year}$ ($\sim 0.2 \, \mu{\rm eV/year}$) for $\kappa_L = {\rm 1 Hz}$ (10 Hz) and $t_\text{int} = 1\, \text{s}$ .
\item If $\Delta f_a > 2\,\kappa_L$, {\it i.e.} $m_a \gtrsim 5 \text{ neV}(\kappa_L/\text{Hz})$, the width of the axion Boltzmann distribution sets the scanning step, $\delta
= \epsilon \, m_a v^2 / (4\pi)$. Separating consecutive scans by a frequency $\delta$, two successive measurements overlap at the condition $ \delta/(e^{2 \delta / \Delta f_a } - 1) \approx \Delta + \Omega_{\rm m} - m_a$.
Hence, at the intersection it follows
\begin{equation}
{\rm SNR}\propto B_{m_a}( \Delta + \Omega_{\rm m} ) = \frac{1}{\Delta f_a} \left(\frac{2}{\pi} \left(\frac{\epsilon}{e^{\epsilon}-1} \right) e^{\displaystyle \epsilon/(e^{\epsilon}-1)}\right)^{\frac{1}{2}}.
\end{equation}
The coverage in axion masses over an experimental time $T_{\rm exp}$ is given by
\begin{equation}
\text{coverage} = m_a^{(0)} \left(
\big(\epsilon
\frac{ v^2}{4\pi}
+1 \big)^{\frac{T_\text{exp}}{t_\text{int}}-1}-1\right),
\end{equation}
where $m_a^{(0)}$ is the axion mass that peaks the sensitivity of the initial measurement of the scan. Taking $\epsilon = 1$, one can cover $ \sim 70 \, m_a^{(0)}$, which is almost two orders of magnitude in axion mass per year for $t_\text{int} = 1 \, \text{s}$.
\end{enumerate}
Having specified the choice of the scanning spacing $\delta$, and assuming that $t_\text{int} \geq \kappa_L^{-1}$,
\begin{equation}
g_{a\gamma \gamma} \! \propto \! \left(\rho^{1/2} c_s^{1/2} \frac{\epsilon_r +2}{\epsilon_r-1}\epsilon_r^{3/4}\right) \!\! \left(\frac{1}{{\cal F}_\text{opt}^{5/4}} \frac{1}{L^{1/4}}\right) \!\!\left( \frac{1}{\omega_\text{opt}^{5/4}} \frac{1}{P_\text{pump}^{1/2}}\frac{1}{P_\text{probe}^{1/4}} \frac{1}{N_\phi^{1/2}} \right) \!\! \frac{m_a }{\rho_a^{1/2}} \times
\begin{cases}
m_a^{\tfrac{1}{2}} \left(\frac{e^\epsilon-1}{\displaystyle e^{\frac{\epsilon}{1-\epsilon}}\epsilon}\right)^{\tfrac{1}{4}} \left(\frac{1}{t_\text{int} \kappa_L}\right)^{\tfrac{1}{4}}\!\!\!\!\!, & t_\text{int} > (2\kappa_L)^{-1}\! > \tau_a,\\
\kappa_L^{\tfrac{1}{2}}(1+\epsilon^2/4)^{\tfrac{1}{2}} \left(\frac{1}{t_\text{int} m_a}\right)^{\tfrac{1}{4}} \!\!\!\!\!,& t_\text{int} > \tau_a > (2\kappa_L)^{-1}\!\!\!\!\! ,\\
\kappa_L^{\tfrac{1}{2}}(1+\epsilon^2/4)^{\tfrac{1}{2}} \left(\frac{1}{t_\text{int} m_a}\right)^{\tfrac{1}{2}} \!\!\!\!\!, & t_\text{int} < \tau_a .\\
\end{cases}
\end{equation}
Therefore, the sensitivity in the reach plots will scale with the axion mass, depending on how the integration time compares to the axion and laser coherence times, as follows
\begin{equation}
g_{a\gamma \gamma} \propto \begin{cases} m_a^{3/2}, &t_\text{int} > (2\kappa_L)^{-1}>\tau_a ,\\
m_a^{3/4}, & t_\text{int} > \tau_a > (2\kappa_L)^{-1} ,\\
m_a^{1/2}, &t_\text{int} < \tau_a .\\
\end{cases}
\end{equation}
Note that, in the heavy axion regime, if one saturates the number of phonons that the cavity can host, limited by the density perturbations in the material as stated in Eq.~\eqref{eq:numberphononsmax},
\begin{equation}
N_\phi^{\rm max} \propto \left(\frac{\rho}{\sqrt{\epsilon_r}} c_s \left( \frac{\delta \rho}{\rho}\right)^2\right) \left(\frac{L^2}{\omega_{\rm opt}^2}\right),
\end{equation}
the sensitivity scales with $L$ as $g_{a\gamma \gamma} \propto L^{-5/4}$, while the sensitivity becomes independent on the speed of sound and density of the filling material. Therefore, in such regime, the cavity length, $L$, and the optical finesse, ${\cal F}_{\rm opt}$, are the parameters with highest impact on the sensitivity.
\end{document}
| {
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Chamaecyparis formosensis é uma espécie de conífera da família Cupressaceae.
Apenas pode ser encontrada em Taiwan.
Esta espécie está ameaçada por perda de habitat.
Referências
Conifer Specialist Group 2000. Chamaecyparis formosensis. 2006 IUCN Red List of Threatened Species. Dados de 10 de Julho de 2007.
Cupressaceae | {
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{"url":"https:\/\/math.stackexchange.com\/questions\/4006890\/disproving-asymptotic-lower-bound-of-central-binomial-coefficient","text":"# Disproving asymptotic lower bound of central binomial coefficient\n\nThis is sort of a cross-post from here, since I'm looking for a more mathematical proof.\n\nI was recently learning about binomial coefficients and was wondering about how to disprove 2nCn (or the central binomial coefficient) not being lower-bounded by 4^n; in other words:\n\n$$\\binom{2n}{n} \\notin \\Omega (4^n)$$\n\nSome extremely generous bounds can be easily constructed, such as the following:\n\n$$\\frac{4^n}{2n+1} \\leq \\binom{2n}{n} \\leq 4^n \\text{ for all } n \\geq 1$$\n\nI sought to prove by contradiction, so to assume:\n\n$$\\exists c_1 \\text{ or } c_2 : 0 \\lt c_14^n \\leq \\frac{4^n}{2n+1} \\leq c_24^n \\leq \\binom{2n}{n} \\leq 4^n$$\n\nClearly, $$c_1$$ cannot exist, since $$\\lim_{n \\rightarrow \\infty}{\\frac{1}{2n + 1}} = 0$$. It can also be seen that $$c_2 \\in (0,1]$$. And... I'm stuck. Intuitively, it seems rather obvious that $$c_2$$ cannot exist.\n\nI am aware a similar question has been asked here, but there wasn't really a proof provided. I'm also aware that you could prove $$\\lim_{n \\rightarrow \\infty}\\frac{\\binom{2n}{n}}{4^n} = 0$$, but I was wondering if there was another way to do so - particularly, by proving that $$c_2$$ cannot exist.\n\nYou can prove by induction on $$n\\geq 1$$ that $$\\binom{2n}{n} \\leq \\frac{4^n}{\\sqrt{n}}.$$","date":"2021-06-15 01:14:53","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 11, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7989049553871155, \"perplexity\": 103.54775565414919}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623487614006.8\/warc\/CC-MAIN-20210614232115-20210615022115-00253.warc.gz\"}"} | null | null |
Die Ordensgemeinschaft der Alexianerbrüder, kurz Alexianer (, Ordenskürzel: CFA; historisch auch Celliten und anders), ist eine römisch-katholische Brüdergemeinschaft, die vor allem in der Krankenpflege tätig ist.
Strukturen
Gegenwart
Im jahr 2016 gab es etwa 70 Ordensbrüder in sieben Ländern, im Juni 2020 waren es noch 53 Brüder, davon 10 in Deutschland.
Die Kongregation besteht aus zwei Provinzen und zwei Regionen
Provinz der Unbefleckten Empfängnis in den USA, den Philippinen und Ungarn, gegründet 1866
St. Alexius Provinz Deutschland, mit drei Konventen in Aachen, Neuss und Münster. Das Provinzialat befindet sich in Münster. Die Stiftung der deutschen Ordensprovinz betreibt die Alexianer GmbH mit zahlreichen Krankenhäusern und Pflegeeinrichtungen in Deutschland und in Kerala (Indien).
Region Herz Jesu in Großbritannien und Irland mit Häusern in Manchester, London, Dublin und Limerick, gegründet 1875 in Manchester
Region in Belgien, mit Häusern in Boechout, Tirlemont, Grimbergen und Henri-Chapelle, diese werden wahrscheinlich nur noch von Laien geführt
Das Generalat befindet sich in Signal Mountain in Tennessee in den USA. Der Generalprior ist seit 2016 Lawrence Krueger.
Die Alexianer sind nach ihrem Schutzheiligen Alexius von Edessa benannt.
Historische Entwicklung der Strukturen
Im Mittelalter gab es fünf Provinzen in Brabant, Flandern, dem Rheinland und Sachsen (= Niedersachsen und Provinz Sachsen). Letztere löste sich im 16. Jahrhundert auf.
1717 führten die schon lange bestehenden Spannungen zwischen den Konventen der rheinischen Provinz Overland zur Trennung der Aachener und Neusser Alexianer vom Kölner Konvent.
In der zweiten Hälfte des 19. Jahrhunderts gründeten Aachener Alexianer Niederlassungen in England, Irland und den USA. Damit entwickelte sich die Aachener Ordensgemeinschaft 1866 zur weltweiten Kongregation der Alexianer mit vier Provinzen, davon zwei in Deutschland sowie jeweils eine in Großbritannien und den USA.
Die Provinzen unterstanden, wie schon in den Anfängen der Ordensgemeinschaft, der päpstlichen Jurisdiktion.
1968 schlossen sich die Neusser und Kölner Alexianer zu einer eigenständigen Kongregationen bischöflichen Rechts zusammen, 1990 traten sie dann der weltweiten Kongregation bei. Im Jahr 2008 fusionierten die beiden deutschen Provinzen mit den Sitzen in Aachen und Neuss zur St. Alexius Provinz Deutschland.
Geschichte
Mittelalter
Die Anfänge der Bewegung sind nicht eindeutig festzustellen.
In Flandern (Mechelen 1306, Antwerpen 1342) und am Niederrhein (Köln 1306) gab es seit dem frühen 14. Jahrhundert Gemeinschaften von Laienbrüdern, die Kranke pflegten. Diese wurden als Matemans (Flandern), Lollarden oder Begharden (Köln) bezeichnet.
Besonders während der großen Pestepidemie 1348/49 leisteten sie wichtige Tätigkeiten auch bei der Totenbestattung.
Die einzelnen städtischen Konvente gaben sich jeweils eigene Satzungen und unterstellten sich den regionalen Bischöfen.
1431 bezeichnete sie Papst Eugen IV. als Celliten. Diese Bezeichnung wurde seitdem häufiger für sie verwendet.
1468 nahmen sie auf ihrem ersten Generalkapitel in Lüttich die Augustinerregel an, worauf sie Papst Sixtus IV. 1472 als Orden anerkannte.
Von 1480 ist die erstmalige Bezeichnung Alexianer bekannt, benannt nach dem heilige Alexius von Edessa, der sein Leben in Armut und mit dem Dienst an Bedürftigen und Kranken verbrachte.
Unter Papst Julius II. wurden sie 1502 als exemter Orden bestätigt, mit dem Wahlspruch caritas Christi urget nos – "Uns treibt die Liebe Christi" .
Neuzeit
Nach dem Dreißigjährigen Krieg und dem Rückgang der Pest übernahmen die Alexianer verstärkt mit der Pflege psychisch Kranker neue Aufgaben. So entstanden die ersten psychiatrischen Kliniken.
Um 1805 blieben einige Alexianerniederlassungen von der Säkularisation verschont, da sie vor allem Krankenpflege betrieben.
20. und 21. Jahrhundert
In den Jahren der NS-Diktatur gerieten durch das Gesetz zur Verhütung erbkranken Nachwuchses vom 14. Juli 1933 auch die Alexianer in Bedrängnis. Sie versuchten, die ihnen anvertrauten Bewohner ihrer Häuser vor den "Euthanasie"-Morden zu schützen, u. a. indem sie ihre Schutzbefohlenen in die belgischen Häuser des Ordens verlegten – doch oft vergebens. Bruder Gereon Wittkamp, der Rektor von Haus Kannen bei Münster, berichtete dem Bischof von Münster, Clemens August Graf von Galen, von einer Mordaktion an 106 Bewohnern von Haus Kannen. Dessen berühmte Predigt am 3. August 1941 und seine Strafanzeige gegen die Mörder trugen entscheidend dazu bei, dass die Mordaktion ausgesetzt wurde.
Im Januar 2013 gründete die Ordensgemeinschaft der Alexianerbrüder eine rechtsfähige kirchliche Stiftung des bürgerlichen Rechts. Diese ist seitdem Gesellschafter der Alexianer GmbH, unter deren Dach sämtliche Einrichtungen und Dienste der Alexianer in Deutschland zusammengefasst sind. Gegenwärtig betreibt die Alexianer GmbH Krankenhäuser, medizinische Versorgungseinrichtungen sowie Einrichtungen der Senioren-, Eingliederungs- und Jugendhilfe in fünf Bundesländern.
Historische Alexianerkonvente
Historische Bezeichnungen
Die Konvente wurden in historischen Erwähnungen in den ersten Jahrzehnten verschieden bezeichnet
Lollarden
Begarden, z. B. Köln 1306
Willige Arme
Trullebrodern (= Trullbrüder, Trollbrüder), von Trull = Diener?
Celliten
Cellenbrüder
Cellenbroeders, niederländisch
Matemans, niederländisch im 14. Jahrhundert
Verbreitung
Alexianerkonvente gab es vor allem in Flandern, Brabant und den Niederlanden, sowie am Rhein und im mitteldeutsch-niedersächsischen Raum.
Alexianerkonvente in deutschen Territorien
Mittelalterliche Gründungen
Alexianerkloster Aachen, seit 1334
Alexianer Braunschweig, um 1472 erhielten ein Haus, 1677 letzter Bruder gestorben, danach Armenhaus, Waisenhaus, Lazarett, nicht erhalten
Alexianer Frankfurt am Main
Trollmönch Goslar
Trillkloster Halberstadt, 1375–um 1540, um 1701–1810 wieder genutzt, 1860/75 Armenhaus, nicht erhalten
Alexianer Hamburg
Alexianer Helmstedt, 1503–1526 in der Stolzengasse, dann an Kloster Hamersleben gegeben
Lüllekehaus Hildesheim
Alexianer Koblenz, seit 1354
Alexianer Köln
Alexianer Neuss
Alexianer Straßburg, Elsass
Alexianer oder Engelbrüder Trier, seit 1434
Alexianer Worms
Neuzeitliche Gründungen
Elsdorf (?)
Krefeld
Mönchengladbach, St. Josef, 1859–1956
Münster, Haus Kannen
Alexianerkonvente in Belgien und den Niederlanden
In Flandern, Brabant und den Niederlanden gab es einige Alexianerniederlassungen, von denen einige relativ große Anlagen waren (Amsterdam). Sie werden heute meist als Cellenbroeders (Cellenbrüder) bzw. Cellebroederkloosters (Cellenbrüderkloster) bezeichnet.
Belgien
Antwerpen, seit 1342
Brügge
Gent
Löwen (Leuven), seit 1345
Mechelen, seit 1305, ältestes bekanntes belgisches Alexianerkloster
Niederlande
Amsterdam
Kampen
Maastricht
Nijmegen, jetzt Musezum
Literatur
Gesamtdarstellungen
Margery Frisbie: Die Geschichte der Alexianerbrüder. Editions Sadifa Media, Kehl 1984. ISBN 3-88786-008-X.
Christopher J. Kauffman: Geschichte der Alexianerbrüder:
Bd. 1: Von 1300 bis 1789: Sie haben den Tod vertraut gemacht. Gemeinschaft der Alexianerbrüder, Aachen 1980. (englischsprachige Ausgabe: Tamers of death. The history of the Alexian Brothers from 1300 to 1789. Seabury Press, New York 1976. ISBN 0-8164-0314-7)
Bd. 2: Von 1789 bis zur Gegenwart: Dienst am Kranken. Gemeinschaft der Alexianerbrüder, Aachen 1980. (englischsprachige Ausgabe: The Ministry of healing. The history of the Alexian brothers from 1789 to the present. Seabury Press, New York 1978. ISBN 0-8164-0387-2)
Lexikonartikel
Kurzdarstellungen
Max Heimbucher: Die Congregationen und Orden der katholischen Kirche. 2. Auflage, Zweiter Band. Paderborn, 1907, S. 233, mit guter kurzer Darstellung
Weblinks
Alexianerkloster Offizielle Website der Ordensgemeinschaft
Stiftung der AlexianerbrüderOffizielle Website
The Alexian Brothers International Offizielle Website (englisch)
Alexians Catholic Encyclopedia, 1903 (deutsch)
Einzelnachweise
Krankenpflegeorden
Wohlfahrtsorganisation
Organisation (Tennessee)
Alexius von Edessa | {
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package org.asteriskjava.manager.action;
/**
* The ZapHangupAction hangs up a zap channel.
*
* @author srt
* @version $Id$
*/
public class ZapHangupAction extends AbstractManagerAction {
/**
* Serializable version identifier
*/
private static final long serialVersionUID = -4064616334146097495L;
private Integer zapChannel;
/**
* Creates a new empty ZapHangupAction.
*/
public ZapHangupAction() {
}
/**
* Creates a new ZapHangupAction that hangs up the given zap channel.
*
* @param zapChannel the number of the zap channel to hang up
* @since 0.2
*/
public ZapHangupAction(Integer zapChannel) {
this.zapChannel = zapChannel;
}
/**
* Returns the name of this action, i.e. "ZapHangup".
*/
@Override
public String getAction() {
return "ZapHangup";
}
/**
* Returns the number of the zap channel to hangup.
*/
public Integer getZapChannel() {
return zapChannel;
}
/**
* Sets the number of the zap channel to hangup.<p>
* This property is mandatory.
*/
public void setZapChannel(Integer channel) {
this.zapChannel = channel;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
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Property: Judge withdraws from developer's trial amid pressure
Justice Mohammed Idris of the FCT High Court has withdrawn from the alleged fraud brought against a property developer, Cecil Osakwe, citing external pressure. Justice…
By ....
Mon, 19 Sep 2022 15:35:26 WAT
Justice Mohammed Idris of the FCT High Court has withdrawn from the alleged fraud brought against a property developer, Cecil Osakwe, citing external pressure.
Justice Idris, on Monday, remitted the case file, which was brought by the office of the Attorney General of the Federation, to the Chief Judge of the FCT High Court for advice.
Supreme Court judges drop to 13 as Justice Aboki retires
Judges' welfare top on Buhari's agenda —Malami
The Attorney General of the Federation, Abubakar Malami (SAN), had filed the two-count charges against Osakwe following a petition by Asabe Waziri claiming that his company, Abeh Signature Apartments collected the sum of N130 million without delivering possession to her.
But in a letter to the Malami by counsel to Osakwe, Victor Giwa, the company advised him not to allow his "highly regarded" office to be used to defame his client who is an honest businessman.
Giwa explained that the repossession of the two units of the 2-bedroom apartment at Mekong Close, Maitama, Abuja, priced at N130 million each, totalling N260 million, for which the staff of the Nigerian National Petroleum Corporation (NNPC) was granted possession after a part payment of N140 million, was due to her "violation of the covenants of the apartment."
He said, "Based on various breaches and her mode of payment, our client approached Asabe Waziri to terminate the transaction and offered to refund her money, an offer Asabe bluntly refused, prompting our client to approach the FCT High Court."
A copy of the enrolled order of the FCT High Court presided by Justice Musa Othman showed that the court had on February 17, 2022, upheld Osakwe's application terminating the contract, refunding and repossessing the apartments in suit no: FCT/BW/CV/2435/2021 that the mode of payments employed for the purchase violated the money laundering laws.
"That in view of the termination of the contract for the purchase of the two flats of Abeh Court by the claimant, the defendant (Waziri) can no longer claim or exercise ownership over the said two flats," the judge ruled.
"Consequently, the court orders the claimant (Abeh Signatures) to immediately refund the entire monies paid to it by the defendant (Waziri), (including the legal and agency fees) and further orders the defendant to immediately hand over possession of the two flats, being flats 3C and 38 of Abeh Court to the claimant," he added.
Court Tells Civil Servant To Vacate Property, Orders Refund
EFCC auctions Diezani, others' property
Court quashes EFCC's bid to takeover Kano businessman's property
Court to rule on Ekweremadu's plea against seized properties Jan 25 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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<?php
class Crop {
// Загрузка списка культур
public static function getListCrop(){
$db = Db::getConnection();
$result = $db->query("SELECT * FROM crop_head_veg WHERE NOT status_delete = 1");
$result->setFetchMode(PDO::FETCH_ASSOC);
$listCrop = $result->fetchAll();
return $listCrop;
}
//Добавление новой культуры в БД
public static function createCrop($name_crop_ua, $yield_min, $yield_max, $cleaning_qty, $drying_qty, $cleaning_price, $drying_price, $storing_price, $selling_price, $other_operating_1, $other_operating_2, $other_operating_3){
$db = Db::getConnection();
$db->query("INSERT INTO crop_head_veg ( name_crop_ua, yield_min, yield_max, cleaning_qty, drying_qty, cleaning_price, drying_price, storing_price, selling_price, other_operating_1, other_operating_2, other_operating_3)
VALUE ('$name_crop_ua','$yield_min','$yield_max','$cleaning_qty','$drying_qty','$cleaning_price','$drying_price','$storing_price','$selling_price', '$other_operating_1','$other_operating_2' ,'$other_operating_3' )");
return true;
}
//Загрузка одной культуры
public static function getCropId($id){
$db = Db::getConnection();
$result = $db->query("SELECT * FROM crop_head_veg WHERE id_crop=$id");
$result->setFetchMode(PDO::FETCH_ASSOC);
$idCrop = $result->fetchAll();
return $idCrop;
}
//Редактирование культуры
public static function editSaveCrop($id, $name_crop_ua, $yield_min, $yield_max, $cleaning_qty, $drying_qty, $cleaning_price, $drying_price, $storing_price, $selling_price, $other_operating_1, $other_operating_2, $other_operating_3){
$db = Db::getConnection();
$db->query("UPDATE crop_head_veg SET name_crop_ua = '$name_crop_ua', yield_min = '$yield_min', yield_max = '$yield_max', cleaning_qty = '$cleaning_qty', drying_qty = '$drying_qty', cleaning_price = '$cleaning_price', drying_price = '$drying_price', storing_price = '$storing_price', selling_price = '$selling_price', other_operating_1 = '$other_operating_1', other_operating_2 = '$other_operating_2', other_operating_3 = '$other_operating_3'
WHERE id_crop = '$id'");
return true;
}
//Удаление культуры
public static function removeCrop($id){
$db = Db::getConnection();
$db->query("UPDATE crop_head_veg SET status_delete = '1' WHERE id_crop = $id");
return true;
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 5,945 |
Don Giovanni in Sicilia è uno sceneggiato televisivo del 1977 diretto da Guglielmo Morandi, tratto dal romanzo omonimo di Vitaliano Brancati, adattato da Giuseppe Cassieri e trasmesso in tre puntate.
Lo sceneggiato è basato sul romanzo breve, scritto a Zafferana Etnea nel 1940, pubblicato successivamente nel 1941. È il primo romanzo del cd. gallismo teorizzato da Brancati. Il protagonista è impersonato dal cantante-attore Domenico Modugno. Fra gli altri interpreti figurano anche l'attrice Rosanna Schiaffino e l'attore Leopoldo Trieste.
Trama
In una città come Catania, appena uscita dalla guerra, vive una società variegata. Accanto alla decaduta e spiantata aristocrazia di origine feudale si muovono i soliti personaggi delle piazze e delle vie principali: il farmacista, il commendatore, il barone, l'avvocato, il cavaliere, i giovani rampanti spinti dal «germe dei viaggi», tutti riuniti nel solito rinomato bar, fulcro e centro vitale di una espansione urbana inaspettata. Un bar che si trasforma in salotto improvvisato dove osservare le donne più belle, uomini innegabilmente attratti da «I piaceri del discorrere della donna».
All'appassionata ricerca di donne e di avventure fugaci pensa sempre Giovanni Percolla, il quarantenne commerciante che vive in una casa grande, arredata con mobili antichi, circondato dall'affetto fin troppo premuroso e castrante di tre sorelle non sposate e dell'anziana domestica Agatina.
Giovanni passeggia con il gruppo di amici, Ciccio e Saretto Muscarà, incantati nel ricordare le descrizioni poetiche di don Procopio, attorniato sempre da giovani, che all'arrivo in città delle nuove "donnine", animava le serate altrimenti noiose. Inoltre, Ciccio ha il problema di dover nascondere ai genitori, di ritorno da Taormina, l'ultima invenzione della scienza in campo sessuale: una bambola-statua di ceralacca importata da Parigi che commuove perfino le generazioni più anziane, tale è la precisione dei dettagli e le movenze sensuali.
Ninetta dei Marconella, la ragazza più ambita di Catania, non perde occasione di mandare sguardi "pubblici" a Giovanni, suscitando invidie e perplessità tra amici e conoscenti.
Critica
La prima [puntata] è apparsa ben costruita dalla sceneggiatura di Giuseppe Cassieri ed egregiamente diretta da Guglielmo Morandi. La Sicilia descritta nel libro ha avuto una sua accurata e puntuale estrinsecazione visiva; i campioni di « gallismo » che nella narrativa di Brancati posseggono vivo risalto, sono guardati con il medesimo acume che li caratterizza nell'ironia della pagina scritta.
Distribuzione
Andato in onda sulla Rete 1 dal 2 al 16 gennaio 1977, mentre la replica è stata trasmessa di pomeriggio sulla Rete 2 dal 24 febbraio al 2 marzo 1982. Ancora una volta venne trasmessa su Rete 3 dal 23 al 29 settembre 1983.
Note
Collegamenti esterni
Miniserie televisive di Rai 1
Fiction televisive della Rai | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,299 |
{"url":"https:\/\/de.maplesoft.com\/support\/help\/maple\/view.aspx?path=examples%2Flinearode","text":"Closed-form Solutions of Linear Differential Equations - Maple Programming Help\n\nHome : Support : Online Help : Applications and Example Worksheets : Differential Equations : examples\/linearode\n\nClosed-form Solutions of Linear Differential Equations\n\nThe Maple dsolve\u00a0command allows determination of closed-form solutions for linear differential equations.","date":"2021-01-17 21:46:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 53, \"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\": 0.8517008423805237, \"perplexity\": 1706.5508849506462}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-04\/segments\/1610703513194.17\/warc\/CC-MAIN-20210117205246-20210117235246-00521.warc.gz\"}"} | null | null |
Words Move is a 4-track EP by Get Smart!, released on Syntax Records on December 8, 1981. It was recorded at Ramona Studios in Lawrence, KS and produced by Dave Stuckey, of the Dave and Deke Combo. The track "Eat, Sleep A Go-Go" from this EP was included on the Sub Pop #5 cassette compilation.
Track listing
Side 1:
"Disillusion" - 2:03
"Where Did This Week Go?" - 1:55
Side 2:
"Eat, Sleep A Go-Go" - 2:17
"This Is Style" - 2:48
Reception
"Get Smart! is a trio of Kansans with brains and a beat. You may prefer bassist Lisa Wertman's vocals over guitarist Marc Koch's but eventually every stark melody on this EP will wheedle its way into your head (and then move onto your feet). This seven incher also features the same version of GS! smash hit "Eat, Sleep A Go-Go" that you heard on Sub Pop 5" (New York Rocker, June 1982)
"We're particularly impressed with Words Move, the new four-song EP debut from Get Smart!. Highlights are "Where Did This Week Go?" a song about being dog-tired to a slowed down B-52's beat (or Pylon at its natural speed) and "Eat, Sleep A Go-Go," which is minimal to Cramps proportions with its own unique set of dissonances." (Cary Baker, Illinois Entertainer, 1982)
"Four-song offering of sparse brains-and-beat music from a Lawrence, KS new-rock trio that already has reaped some praise with independent releases." (Bart Becker, Lincoln Evening Journal, 1982)
"Twangy, big-beat minimalism with a Dave Byrne-ish singer" (Trouser Press, 1982)
"Get Smart sports a catchy, beaty minimalism and a modern "lost in a supermarket" viewpoint. A strong debut, more encouraging new music from the Midwest, I'm looking forward to hearing more. Sharp packaging too." (Jim Manion, The Bloomington Free Rider, 1982)
Personnel
Marc Koch - vocals, guitar
Lisa Wertman Crowe - vocals, bass
Frank Loose - drums
References
External links
Get Smart! discography
1981 debut EPs
Get Smart! (band) albums | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,418 |
package org.apache.beam.runners.dataflow.util;
import org.apache.beam.runners.dataflow.options.DataflowPipelineOptions;
import org.apache.beam.sdk.options.PipelineOptions;
import org.apache.beam.sdk.util.PathValidator;
import org.apache.beam.sdk.util.gcsfs.GcsPath;
import com.google.common.base.Preconditions;
import java.io.IOException;
/**
* GCP implementation of {@link PathValidator}. Only GCS paths are allowed.
*/
public class DataflowPathValidator implements PathValidator {
private DataflowPipelineOptions dataflowOptions;
DataflowPathValidator(DataflowPipelineOptions options) {
this.dataflowOptions = options;
}
public static DataflowPathValidator fromOptions(PipelineOptions options) {
return new DataflowPathValidator(options.as(DataflowPipelineOptions.class));
}
/**
* Validates the the input GCS path is accessible and that the path
* is well formed.
*/
@Override
public String validateInputFilePatternSupported(String filepattern) {
GcsPath gcsPath = getGcsPath(filepattern);
Preconditions.checkArgument(
dataflowOptions.getGcsUtil().isGcsPatternSupported(gcsPath.getObject()));
String returnValue = verifyPath(filepattern);
verifyPathIsAccessible(filepattern, "Could not find file %s");
return returnValue;
}
/**
* Validates the the output GCS path is accessible and that the path
* is well formed.
*/
@Override
public String validateOutputFilePrefixSupported(String filePrefix) {
String returnValue = verifyPath(filePrefix);
verifyPathIsAccessible(filePrefix, "Output path does not exist or is not writeable: %s");
return returnValue;
}
@Override
public String verifyPath(String path) {
GcsPath gcsPath = getGcsPath(path);
Preconditions.checkArgument(gcsPath.isAbsolute(),
"Must provide absolute paths for Dataflow");
Preconditions.checkArgument(!gcsPath.getObject().contains("//"),
"Dataflow Service does not allow objects with consecutive slashes");
return gcsPath.toResourceName();
}
private void verifyPathIsAccessible(String path, String errorMessage) {
GcsPath gcsPath = getGcsPath(path);
try {
Preconditions.checkArgument(dataflowOptions.getGcsUtil().bucketExists(gcsPath),
errorMessage, path);
} catch (IOException e) {
throw new RuntimeException(
String.format("Unable to verify that GCS bucket gs://%s exists.", gcsPath.getBucket()),
e);
}
}
private GcsPath getGcsPath(String path) {
try {
return GcsPath.fromUri(path);
} catch (IllegalArgumentException e) {
throw new IllegalArgumentException(String.format(
"%s expected a valid 'gs://' path but was given '%s'",
dataflowOptions.getRunner().getSimpleName(), path), e);
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,115 |
.class final Lo/aj$Aux;
.super Lo/aj$If;
# annotations
.annotation system Ldalvik/annotation/EnclosingClass;
value = Lo/aj;
.end annotation
.annotation system Ldalvik/annotation/InnerClass;
accessFlags = 0x18
name = "Aux"
.end annotation
# direct methods
.method constructor <init>(Lcom/google/android/gms/common/data/DataHolder;)V
.locals 0
invoke-direct {p0, p1}, Lo/aj$If;-><init>(Lcom/google/android/gms/common/data/DataHolder;)V
return-void
.end method
# virtual methods
.method public ˊ(Lo/fx;Lcom/google/android/gms/games/multiplayer/realtime/Room;)V
.locals 0
invoke-interface {p1, p2}, Lo/fx;->ˎ(Lcom/google/android/gms/games/multiplayer/realtime/Room;)V
return-void
.end method
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,963 |
<?xml version="1.0" encoding="utf-8"?>
<LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"
android:layout_width="fill_parent"
android:layout_height="fill_parent"
android:orientation="vertical" >
<ScrollView
android:id="@+id/scrollViewFullList"
android:layout_width="fill_parent"
android:layout_height="fill_parent" >
<LinearLayout
android:id="@+id/linearLayoutFullList"
android:layout_width="match_parent"
android:layout_height="match_parent"
android:orientation="vertical">
</LinearLayout>
</ScrollView>
</LinearLayout>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,614 |
{"url":"https:\/\/atdotde.blogspot.com\/","text":"Tuesday, February 12, 2019\n\nVisits to a Bohmian village\n\nOver all of my physics life, I have been under the local influence of some Gaul villages that have ideas about physics that are not 100% aligned with the main stream views: When I was a student in Hamburg, I was good friends with people working on algebraic quantum field theory. Of course there were opinions that they were the only people seriously working on QFT as they were proving theorems while others dealt with perturbative series only that are known to diverge and are thus obviously worthless. Funnily enough they were literally sitting above the HERA tunnel where electron proton collisions took place that were very well described by exactly those divergent series. Still, I learned a lot from these people and would say there are few that have thought more deeply about structural properties of quantum physics. These days, I use more and more of these things in my own teaching (in particular in our Mathematical Quantum Mechanics and Mathematical Statistical Physics classes as well as when thinking about foundations, see below) and even some other physicists start using their language.\n\nLater, as a PhD student at the Albert Einstein Institute in Potsdam, there was an accumulation point of people from the Loop Quantum Gravity community with Thomas Thiemann and Renate Loll having long term positions and many others frequently visiting. As you probably know, a bit later, I decided (together with Giuseppe Policastro) to look into this more deeply resulting in a series of papers there were well received at least amongst our peers and about which I am still a bit proud.\n\nNow, I have been in Munich for over ten years. And here at the LMU math department there is a group calling themselves the\u00a0Workgroup Mathematical Foundations of Physics. And let's be honest, I call them the Bohmians (and sometimes the Bohemians). And once more, most people believe that the Bohmian interpretation of quantum mechanics is just a fringe approach that is not worth wasting any time on. You will have already guessed it: I did so none the less. So here is a condensed report of what I learned and what I think should be the official opinion on this approach. This is an informal write up of a notes paper that I put on the arXiv today.\n\nBohmians don't like about the usual (termed Copenhagen lacking a better word) approach to quantum mechanics that you are not allowed to talk about so many things and that the observer plays such a prominent role by determining via a measurement what aspect is real an what is not. They think this is far too subjective. So rather, they want quantum mechanics to be about\u00a0particles that then are allowed to follow trajectories.\n\n\"But we know this is impossible!\" I hear you cry. So, let's see how this works. The key observation is that the Schr\u00f6dinger equation for a Hamilton operator of the form kinetic term (possibly with magnetic field) plus potential term, has \u00a0a conserved current\n\n$$j = \\bar\\psi\\nabla\\psi - (\\nabla\\bar\\psi)\\psi.$$\n\nSo as your probability density is $\\rho=\\bar\\psi\\psi$, you can think of that being made up of particles moving with a velocity field\n\n$$v = j\/\\rho = 2\\Im(\\nabla \\psi\/\\psi).$$\n\nWhat this buys you is that if you have a bunch of particles that is initially distributed like the probability density and follows the flow of the velocity field it will also later be distributed like $|\\psi |^2$.\n\nWhat is important is that they keep the Schr\u00f6dinger equation in tact. So everything that you can do with the original Schr\u00f6dinger equation (i.e. everything) can be done in the Bohmian approach as well. \u00a0If you set up your Hamiltonian to describe a double slit experiment, the Bohmian particles will flow nicely to the screen and arrange themselves in interference fringes (as the probability density does). So you will never come to a situation where any experimental outcome will differ \u00a0from what the Copenhagen prescription predicts.\n\nThe price you have to pay, however, is that you end up with a very non-local theory: The velocity field lives in configuration space, so the velocity of every particle depends on the position of all other particles in the universe. I would say, this is already a show stopper (given what we know about quantum field theory whose raison d'\u00eatre is locality) but let's ignore this aesthetic concern.\n\nWhat got me into this business was the attempt to understand how the set-ups like Bell's inequality and GHZ and the like work out that are supposed to show that quantum mechanics cannot be classical (technically that the state space cannot be described as local probability densities). The problem with those is that they are often phrased in terms of spin degrees of freedom which have Hamiltonians that are not directly of the form above. You can use a Stern-Gerlach-type apparatus to translate the spin degree of freedom to a positional but at the price of a Hamiltonian that is not explicitly know let alone for which you can analytically solve the Schr\u00f6dinger equation. So you don't see much.\n\nBut from Reinhard Werner and collaborators I learned how to set up qubit-like algebras from positional observables of free particles (at different times, so get something non-commuting which you need to make use of entanglement as a specific quantum resource). So here is my favourite example:\n\nYou start with two particles each following a free time evolution but confined to an interval. You set those up in a particular entangled state (stationary as it is an eigenstate of the Hamiltonian) built from the two lowest levels of the particle in the box. And then you observe for each particle if it is in the left or the right half of the interval.\n\nFrom symmetry considerations (details in my paper) you can see that each particle is with the same probability on the left and the right. But they are anti-correlated when measured at the same time. But when measured at different times, the correlation oscillates like the cosine of the time difference.\n\nFrom the Bohmian perspective, for the static initial state, the velocity field vanishes everywhere, nothing moves. But in order to capture the time dependent correlations, as soon as one particle has been measured, the position of the second particle has to oscillate in the box (how the measurement works in detail is not specified in the Bohmian approach since it involves other degrees of freedom and remember, everything depends on everything\u00a0but somehow it has to work since you want to produce the correlations that are predicted by the Copenhagen approach).\n\n The trajectory of the second particle depending on its initial position\n\nThis is somehow the Bohmian version of the collapse of the wave function but they would never phrase it that way.\n\nAnd here is where it becomes problematic: If you could see the Bohmian particle moving you could decide if the other particle has been measured (it would oscillate) or not (it would stand still). No matter where the other particle is located. With this observation you could build a telephone that transmits information instantaneously, something that should not exist. So you have to conclude you must not be able to look at the second particle and see if it oscillates or not.\n\nBohmians \u00a0tell you you cannot because all you are supposed to observer about the particles are their positions (and not their velocity). And if you try to measure the velocity by measuring the position at two instants in time you don't because the first observation disturbs the particle so much that it invalidates the original state.\n\nAs it turns out, you are not allowed to observe anything else about the particles than that they are distributed like $|\\psi |^2$ because if you could, you could build a similar telephone (at least statistically) as I explain the in the paper (this fact is known in the Bohm literature but I found it nowhere so clearly demonstrated as in this two particle system).\n\nMy conclusion is that the Bohm approach adds something (the particle positions) to the wave function but then in the end tells you you are not allowed to observe this or have any knowledge of this beyond what is already encoded in the wave function. It's like making up an invisible friend.\n\nPS: If you haven't seen \"Bohemian Rhapsody\", yet, you should, even if there are good reasons to criticise the dramatisation of real events.\n\nThursday, January 17, 2019\n\nToday, there was news about a huge database containing 773 million email address \/ password pairs became public. On Have I Been Pawned\u00a0you can check if any of your email addresses is in this database (or any similar one). I bet it is (mine are).\n\nThese lists are very probably the source for the spam emails that have been around for a number of months where the spammer claims they broke into your account and tries to prove it by telling you your password. Hopefully, this is only a years old LinkedIn password that you have changed aeons ago.\n\nTo make sure, you actually want to search not for your email but for your password. But of course, you don't want to tell anybody your password. To this end, I have written a small perl script that checks for your password without telling anybody by doing a calculation locally on your computer. You can find it on GitHub.\n\nFriday, October 26, 2018\n\nInterfere and it didn't happen\n\nI am a bit late for the party, but also wanted to share my two cents on the paper \"Quantum theory cannot consistently describe the use of itself\"\u00a0by Frauchiger and Renner. After sitting down and working out the math for myself, I found that the analysis in this paper\u00a0and the blogpost by Scot (including many of the the 160+ comments, some by Renner) share a lot with what I am about to say but maybe I can still contribute a slight twist.\n\nColeman on GHZS\n\nMy background is the talk \"Quantum Mechanics In Your Face\" by Sidney Coleman which I consider as the best argument why quantum mechanics cannot be described by a local and realistic theory (from which I would conclude it is not realistic). In a nutshell, the argument goes like this: Consider the three qubit state state\n\n$$\\Psi=\\frac 1{\\sqrt 2}(\\uparrow\\uparrow\\uparrow-\\downarrow\\downarrow\\downarrow)$$\n\nwhich is both an eigenstate of eigenvalue -1 for $\\sigma_z\\otimes\\sigma_z\\otimes\\sigma_z$ and an eigenstate of eigenvalue +1 for $\\sigma_x\\otimes\\sigma_x\\otimes\\sigma_z$ or any permutation. This means that, given that the individual outcomes of measuring a $\\sigma$-matrix on a qubit is $\\pm 1$, when measuring all in the z-direction there will be an odd number of -1 results but if two spins are measured in x-direction and one in z-direction there is an even number of -1's.\n\nThe latter tells us that the outcome of one z-measurement is the product of the two x-measurements on the other two spins. But multiplying this for all three spins we get that in shorthand $ZZZ=(XXX)^2=+1$ in contradiction to the -1 eigenvalue for all z-measurments.\n\nThe conclusion is (unless you assume some non-local conspiracy between the spins) that one has to take serious the fact that on a given spin I cannot measure both $\\sigma_x$ and $\\sigma_z$ and thus when actually measuring the latter I must not even assume that $X$ has some (although unknown) value $\\pm 1$ as it leads to the contradiction. Stuff that I cannot measure does not have a value (that is also my understanding of what \"not realistic\" means).\n\nFruchtiger and Renner\n\nNow to the recent Nature paper. In short, they are dealing with two qubits (by which I only mean two state systems). The first is in a box L' (I will try to use the somewhat unfortunate nomenclature from the paper) and the second in in a box L (L stands for lab). For L, we use the usual z-basis of $\\uparrow$ and $\\downarrow$ as well as the x-basis $\\leftarrow = \\frac 1{\\sqrt 2}(\\downarrow - \\uparrow)$ \u00a0and $\\rightarrow = \\frac 1{\\sqrt 2}(\\downarrow + \\uparrow)$ . Similarly, for L' we use the basis $h$ and $t$ (heads and tails as it refers to a coin) as well as $o = \\frac 1{\\sqrt 2}(h - t)$ and $f = \\frac 1{\\sqrt 2}(h+f)$. \u00a0The two qubits are prepared in the state\n\n$$\\Phi = \\frac{h\\otimes\\downarrow + \\sqrt 2 t\\otimes \\rightarrow}{\\sqrt 3}$$.\n\nClearly, a measurement of $t$ in box L' implies that box L has to contain the state $\\rightarrow$. Call this observation A.\n\nLet's re-express $\\rightarrow$ in the x-basis:\n\n$$\\Phi =\\frac {h\\otimes \\downarrow + t\\otimes \\downarrow + t\\otimes\\uparrow}{\\sqrt 3}$$\n\nFrom which one concludes that an observer inside box L that measures $\\uparrow$ concludes that the qubit in box L' is in state $t$. Call this observation B.\n\nSimilarly, we can express the same state in the x-basis for L':\n\n$$\\Phi = \\frac{4 f\\otimes \\downarrow+ f\\otimes \\uparrow - o\\otimes \\uparrow}{\\sqrt 3}$$\n\nFrom this once can conclude that measuring $o$ for the state of L' one can conclude that L is in the state $\\uparrow$. Call this observation C.\n\nUsing now C, B and A one is tempted to conclude that observing L' to be in state $o$ implies that L is in state $\\rightarrow$. When we express the state in the $ht\\leftarrow\\rightarrow$-basis, however, we get\n\n$$\\Phi = \\frac{f\\otimes\\leftarrow+ 3f\\otimes \\rightarrow + o\\otimes\\leftarrow - o\\otimes \\rightarrow}{\\sqrt{12}}.$$\n\nso with probability 1\/12 we find both $o$ \u00a0and $\\leftarrow$. Again, we hit a contradiction.\n\nOne is tempted to use the same way out as above in the three qubit case and say one should not argue about contrafactual measurements that are incompatible with measurements that were actually performed. But Frauchiger and Renner found a set-up which seems to avoid that.\n\nThey have observers F and F' (\"friends\") inside the boxes that do the measurements in the $ht$ and $\\uparrow\\downarrow$ basis whereas later observers W and W' measure the state of the boxes including the observer F and F' in the $of$ and $\\leftarrow\\rightarrow$ basis. \u00a0So, at each stage of A,B,C the corresponding measurement has actually taken place and is not contrafactual!\n\nInterference and it did not happen\n\nI believe the way out is to realise that at least from a retrospective perspective, this analysis stretches the language and in particular the word \"measurement\" to the extreme. In order for W' to measure the state of L' in the $of$-basis, he has to interfere the contents including F' coherently such that there is no leftover of information from F''s measurement of $ht$ remaining. Thus, when W''s measurement is performed one should not really say that F''s measurement has in any real sense happened as no possible information is left over. So it is in any practical sense contrafactual.\n\nTo see the alternative, consider a variant of the experiment where a tiny bit of information (maybe the position of one air molecule or the excitation of one of F''s neutrons) escapes the interference. Let's call the two possible states of that qubit of information $H$ and $T$ (not necessarily orthogonal) and consider instead the state where that neutron is also entangled with the first qubit:\n\n$$\\tilde \\Phi = \\frac{h\\otimes\\downarrow\\otimes H + \\sqrt 2 t\\otimes \\rightarrow\\otimes T}{\\sqrt 3}$$.\n\nThen, the result of step C becomes\n\n$$\\tilde\\Phi = \\frac{f\\otimes \\downarrow\\otimes H+ o\\otimes \\downarrow\\otimes H+f\\otimes \\downarrow\\otimes T-o\\otimes\\downarrow\\otimes T + f\\otimes \\uparrow\\otimes T-o \\otimes\\uparrow\\times T}{\\sqrt 6}.$$\n\nWe see that now there is a term containing $o\\otimes\\downarrow\\otimes(H-T)$. Thus, as long as the two possible states of the air molecule\/neuron are actually different, observation C is no longer valid and the whole contradiction goes away.\n\nThis makes it clear that the whole argument relies of the fact that when W' is doing his measurement any remnant of the measurement by his friend F' is eliminated and thus one should view the measurement of F' as if it never happened. Measuring L' in the $of$-basis really erases the measurement of F' in the complementary $ht$-basis.\n\nWednesday, October 17, 2018\n\nBavarian electoral system\n\nLast Sunday, we had the election for the federal state of Bavaria. Since the electoral system is kind of odd (but not as odd as first past the post), I would like to analyse how some variations (assuming the actual distribution of votes) in the rule would have worked out. So, first, here is how actually, the seats are distributed: Each voter gets two ballots: On the first ballot, each party lists one candidate from the local constituency and you can select one. On the second ballot, you can vote for a party list (it's even more complicated because also there, you can select individual candidates to determine the position on the list but let's ignore that for today).\n\nThen in each constituency, the votes on ballot one are counted. The candidate with the most votes (like in first past the pole) gets elected for parliament directly (and is called a \"direct candidate\"). Then over all, the votes for each party on both ballots\u00a0(this is where the system differs from the federal elections) are summed up. All votes for parties with less then 5% of the grand total of all votes are discarded (actually including their direct candidates but this is not of a partial concern). Let's call the rest the \"reduced total\". According to the fraction of each party in this reduced total the seats are distributed.\n\nOf course the first problem is that you can only distribute seats in integer multiples of 1. This is solved using the Hare-Niemeyer-method: You first distribute the integer parts. This clearly leaves fewer seats open than the number of parties. Those you then give to the parties where the rounding error to the integer below was greatest. Check out the wikipedia page explaining how this can lead to a party losing seats when the total number of seats available is increased.\n\nBecause this is what happens in the next step: Remember that we already allocated a number of seats to constituency winners in the first round. Those count towards the number of seats that each party is supposed to get in step two according to the fraction of votes. Now, it can happen, that a party has won more direct candidates than seats allocated in step two. If that happens, more seats are added to the total number of seats and distributed according to the rules of step two until each party has been allocated at least the number of seats as direct candidates. This happens in particular if one party is stronger than all the other ones leading to that party winning almost all direct candidates (as in Bavaria this happened to the CSU which won all direct candidates except five in Munich and one in W\u00fcrzburg which were won by the Greens).\n\nA final complication is that Bavaria is split into seven electoral districts and the above procedure is for each district separately. So there are seven times rounding and adding seats procedures.\n\nSunday's election resulted in the following distribution of seats:\n\nAfter the whole procedure, there are 205 seats distributed as follows\n\n\u2022 CSU 85 (41.5% of seats)\n\u2022 SPD 22 (10.7% of seats)\n\u2022 FW 27 (13.2% of seats)\n\u2022 GREENS 38 (18.5% of seats)\n\u2022 FDP 11 (5.4% of seats)\n\u2022 AFD 22 (10.7% of seats)\n\nNow, for example one can calculate the distribution without districts throwing just everything in a single super-district. Then there are 208 seats distributed as\n\n\u2022 CSU 85 (40.8%)\n\u2022 SPD 22 (10.6%)\n\u2022 FW 26 (12.5%)\n\u2022 GREENS 40 (19.2%)\n\u2022 FDP 12 (5.8%)\n\u2022 AFD 23 (11.1%)\nYou can see that in particular the CSU, the party with the biggest number of votes profits from doing the rounding 7 times rather than just once and the last three parties would benefit from giving up districts.\n\nBut then there is actually an issue of negative weight of votes: The greens are particularly strong in Munich where they managed to win 5 direct seats. If instead those seats would have gone to the CSU (as elsewhere), the number of seats for Oberbayern, the district Munich belongs to would have had to be increased to accommodate those addition direct candidates for the CSU increasing the weight of Oberbayern compared to the other districts which would then be beneficial for the greens as they are particularly strong in Oberbayern: So if I give all the direct candidates to the CSU (without modifying the numbers of total votes), I get the follwing distribution:\n221 seats\n\u2022 CSU 91 (41.2%)\n\u2022 SPD 24 (10.9%)\n\u2022 FW 28 (12,6%)\n\u2022 GREENS 42 (19.0%)\n\u2022 FDP 12 (5.4%)\n\u2022 AFD 24 (10.9%)\nThat is, there greens would have gotten a higher fraction of seats if they had won less constituencies. Voting for green candidates in Munich actually hurt the party as a whole!\n\nThe effect is not so big that it actually changes majorities (CSU and FW are likely to form a coalition) but still, the constitutional court does not like (predictable) negative weight of votes. Let's see if somebody challenges this election and what that would lead to.\n\nThe perl script I used to do this analysis is here.\n\nPostscript:\nThe above analysis in the last point is not entirely fair as not to win a constituency means getting fewer votes which then are missing from the grand total. Taking this into account makes the effect smaller. In fact, subtracting the votes from the greens that they were leading by in the constituencies they won leads to an almost zero effect:\n\nSeats: 220\n\u2022 CSU\u00a0 91 41.4%\n\u2022 SPD\u00a0 24 10.9%\n\u2022 FW\u00a0 28 12.7%\n\u2022 GREENS\u00a0 41 18.6%\n\u2022 FDP\u00a0 12 5.4%\n\u2022 AFD\u00a0 24 10.9%\nLetting the greens win M\u00fcnchen Mitte (a newly created constituency that was supposed to act like a bad bank for the CSU taking up all central Munich more left leaning voters, do I hear somebody say \"Gerrymandering\"?) yields\n\nSeats: 217\n\u2022 CSU\u00a0 90 41.5%\n\u2022 SPD\u00a0 23 10.6%\n\u2022 FW\u00a0 28 12.9%\n\u2022 GREENS\u00a0 41 18.9%\n\u2022 FDP\u00a0 12 5.5%\n\u2022 AFD\u00a0 23 10.6%\nOr letting them win all but Moosach and W\u00fcrzbug-Stadt where the lead was the smallest:\n\nSeats: 210\n\n\u2022 CSU\u00a0 87 41.4%\n\u2022 SPD\u00a0 22 10.5%\n\u2022 FW\u00a0 27 12.9%\n\u2022 GREENS\u00a0 40 19.0%\n\u2022 FDP\u00a0 11 5.2%\n\u2022 AFD\u00a0 23 11.0%\n\nThursday, March 29, 2018\n\nMachine Learning for Physics?!?\n\nToday was the last day of a nice workshop here at the Arnold Sommerfeld Center organised by Thomas Grimm and Sven Krippendorf on the use of Big Data and Machine Learning in string theory. While the former (at this workshop mainly in the form of developments following Kreuzer\/Skarke\u00a0and taking it further for F-theory constructions, orbifolds and the like) appears to be quite advanced as of today, the latter is still in its very early days. At best.\n\nI got the impression that for many physicists that have not yet spent too much time with this, deep learning and in particular deep neural networks are expected to be some kind of silver bullet that can answer all kinds of questions that humans have not been able to answer despite some effort. I think this hope is at best premature and looking at the (admittedly impressive) examples where it works (playing Go, classifying images, speech recognition, event filtering at LHC) these seem to be more like those problems where humans have at least a rough idea how to solve them (if it is not something that humans do everyday like understanding text) and also roughly how one would code it but that are too messy or vague to be treated by a traditional program.\n\nSo, during some of the less entertaining talks I sat down and thought about problems where I would expect neural networks to perform badly. And then, if this approach fails even in simpler cases that are fully under control one should maybe curb the expectations for the more complex cases that one would love to have the answer for. In the case of the workshop that would be guessing some topological (discrete) data (that depends very discontinuously on the model parameters). Here a simple problem would be a 2-torus wrapped by two 1-branes. And the computer is supposed to compute the number of matter generations arising from open strings at the intersections, i.e. given two branes (in terms of their slope w.r.t. the cycles of the torus) how often do they intersect? Of course these numbers depend sensitively on the slope (as a real number) as for rational slopes $p\/q$ and $m\/n$ the intersection number is the absolute value of $pn-qm$. My guess would be that this is almost impossible to get right for a neural network, let alone the much more complicated variants of this simple problem.\n\nRelated but with the possibility for nicer pictures is the following: Can a neural network learn the shape of the Mandelbrot set? Let me remind those of you who cannot remember the 80ies anymore, for a complex number c you recursively apply the function\n$f_c(z)= z^2 +c$\nstarting from 0 and ask if this stays bounded (a quick check shows that once you are outside\u00a0$|z| < 2$ you cannot avoid running to infinity). You color the point c in the complex plane according to the number of times you have to apply f_c to 0 to leave this circle. I decided to do this for complex numbers x+iy in the rectangle -0.74\nI have written a small mathematica program to compute this image. Built into mathematica is also a neural network: You can feed training data to the function Predict[], for me these were 1,000,000 points in this rectangle and the number of steps it takes to leave the 2-ball. Then mathematica thinks for about 24 hours and spits out a predictor function. Then you can plot this as well:\n\nThere is some similarity but clearly it has no idea about the fractal nature of the Mandelbrot set. If you really believe in magic powers of neural networks, you might even hope that once it learned the function for this rectangle one could extrapolate to outside this rectangle. Well, at least in this case, this hope is not justified: The neural network thinks the correct continuation looks like this:\nEhm. No.\n\nAll this of course with the caveat that I am no expert on neural networks and I did not attempt anything to tune the result. I only took the neural network function built into mathematica. Maybe, with a bit of coding and TensorFlow one can do much better. But on the other hand, this is a simple two dimensional problem. At least for traditional approaches this should be much simpler than the other much higher dimensional problems the physicists are really interested in.\n\nThursday, December 14, 2017\n\nWhat are the odds?\n\nIt's the time of year, you give out special problems in your classes. So this is mine for the blog. It is motivated by this picture of the home secretaries of the German federal states after their annual meeting as well as some recent discussions on Facebook:\nI would like to call it Summers' problem:\n\nLet's have two real random variables $M$ and $F$ that are drawn according to two probability distributions $\\rho_{M\/F}(x)$ (for starters you may both assume to be Gaussians but possibly with different mean and variance). Take $N$ draws from each and order the $2N$ results. What is the probability that the $k$ largest ones are all from $M$ rather than $F$? Express your results in terms of the $\\rho_{M\/F}(x)$. We are also interested in asymptotic results for $N$ large and $k$ fixed as well as $N$ and $k$ large but $k\/N$ fixed.\n\nLast bonus question: How many of the people that say that they hire only based on merit and end up with an all male board realise that by this they say that women are not as good by quite a margin?\n\nThursday, November 09, 2017\n\nWhy is there a supercontinent cycle?\n\nOne of the most influential books of my early childhood was my \"Kinderatlas\"\nThere were many things to learn about the world (maps were actually only the last third of the book) and for example I blame my fascination for scuba diving on this book. Also last year, when we visited the Mont-Dor\u00e9 in Auvergne and I had to explain how volcanos are formed to my kids to make them forget how many stairs were still ahead of them to the summit, I did that while mentally picturing the pages in that book about plate tectonics.\n\nBut there is one thing I about tectonics that has been bothering me for a long time and I still haven't found a good explanation for (or at least an acknowledgement that there is something to explain): Since the days of Alfred Wegener we know that the jigsaw puzzle pieces of the continents fit in a way that geologists believe that some hundred million years ago they were all connected as a supercontinent Pangea.\n\nBy Original upload by en:User:Tbower - USGS animation A08, Public Domain, Link\n\nIn fact, that was only the last in a series of supercontinents, that keep forming and breaking up in the \"supercontinent cycle\".\n\nBy SimplisticReps - Own work, CC BY-SA 4.0, Link\n\nSo here is the question: I am happy with the idea of several (say $N$) plates roughly containing a continent each that a floating around on the magma driven by all kinds of convection processes in the liquid part of the earth. They are moving around in a pattern that looks to me to be pretty chaotic (in the non-technical sense) and of course for random motion you would expect that from time to time two of those collide and then maybe stick for a while.\n\nThen it would be possible that also a third plate collides with the two but that would be a coincidence (like two random lines typically intersect but if you have three lines they would typically intersect in pairs but typically not in a triple intersection). But to form a supercontinent, you need all $N$ plates to miraculously collide at the same time. This order-$N$ process seems to be highly unlikely when random let alone the fact that it seems to repeat. So this motion cannot be random (yes, Sabine, this is a naturalness argument). This needs an explanation.\n\nSo, why, every few hundred million years, do all the land masses of the earth assemble on side of the earth?\n\nOne explanation could for example be that during those tines, the center of mass of the earth is not in the symmetry center so the water of the oceans flow to one side of the earth and reveals the seabed on the opposite side of the earth. Then you would have essentially one big island. But this seems not to be the case as the continents (those parts that are above sea-level) appear to be stable on much longer time scales. It is not that the seabed comes up on one side and the land on the other goes under water but the land masses actually move around to meet on one side.\n\nI have already asked this question whenever I ran into people with a geosciences education but it is still open (and I have to admit that in a non-zero number of cases I failed to even make the question clear that an $N$-body collision needs an explanation). 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\section*{Introduction}
Let $G$ be a Poisson Lie group and $\mathfrak g$ the Lie algebra of $G$. Let
$\Ugh$ be the quantization of the universal enveloping algebra ${\U}(\g)$
along the corresponding Lie bialgebra structure on $\mathfrak g$. Consider
two Poisson homogeneous $G$-manifolds, $M:=G/G_M$ and $N:=G/G_N$
such that the stabilizer $G_M$ is a subgroup in $G_N$. Then there
exists a natural projection $M \to N$ of $G$-spaces, and it makes
$M$ into a $G$-bundle over $N$ with the fiber $G_N/G_M$. Suppose
this projection is a Poisson map. It is natural to consider the
problem of equivariant quantization of such a bundle. By this we
understand a $\Ugh$-equivariant quantization of the function algebras
together with the co-projection $F(M)\hookleftarrow F(N)$, to a
morphism $F_\ensuremath{\hbar} (M)\hookleftarrow F_\ensuremath{\hbar} (N)$ of $\Ugh$-algebras.
In the present paper, we quantize orbit bundles for the case when
$G=GL_n(\mathbb{C})$, and $\mathfrak g$ is equipped with a factorizable Lie
bialgebra structure. We assume the stabilizers $G_M$
and $G_N$ to be Levi subgroups of $G$.
Specifically for the $GL_n(\mathbb{C})$-case, those are precisely
reductive subgroups of maximal rank.
Then the $G$-varieties $M$ and $N$ can be
realized as semisimple coadjoint orbits $O_1,O_2\subset{\mathfrak{g}}^*$.
The Poisson structure on $\mathcal{O}_i$ is obtained by restriction
from a Poisson structure on ${\mathfrak{g}}^*$. The latter is a linear combination of
the $G$-invariant
Kostant-Kirillov-Lie-Souriau bracket (KKLS) and the
Semenov-Tian-Shansky bracket (STS).
In our case, $G=GL_n(\mathbb{C})$, they
are compatible, i.e. the Schouten bracket
between the two is equal to zero.
The Poisson bracket on $\mathcal{O}_i$ is not $G$-invariant, however, it makes
$\mathcal{O}_i$ Poisson-Lie manifolds over the Poisson-Lie group $G$.
Moreover, it is the only such bracket on $\mathcal{O}_i$ obtained by restriction from ${\mathfrak{g}}^*$.
Explicit quantization of semisimple orbits has been constructed in
\cite{DoninJMudrovAI:ExplicitQuantiz} for the special case of the
standard, or Drinfeld-Jimbo, quantum group $\Ugh$.
In the present paper, we extend that quantization to the case of any
factorizable Lie bialgebra structure on $\mathfrak g$ and the
corresponding quantum group.
We also describe all semisimple Poisson-Lie orbit bundles
$O_1\toO_2$ in ${\mathfrak{g}}^*$.
In particualar, we show that $O_2$ is necessarily
symmetric.
We explicitly construct a $\Ugh$-equivariant quantization
of the projection map $\prj$ for all orbit bundles.
The paper is organized as follows.
\noindent
In Section~1 we recall some
definitions concerning equivariant quantization.
\noindent
In Section~2 we establish necessary and sufficient conditions for
an orbit bundle to be Poisson.
\noindent
In Section~\ref{sect:REAlgebras} we study the behavior of algebras
defined by a modified (quadratic-linear) Reflection Equation
under twist of quantum groups.
\noindent
In Section~\ref{sect:DMQuantization} we use results of the previous
section to extend the double quantization of orbits
\cite{DoninJMudrovAI:ExplicitQuantiz} to the case of the quantum
group defined by an arbitrary factorizable classical r-matrix.
\noindent
In Section~\ref{sect:QnOfBundles} we prove that any Poisson orbit
bundle admits a $\Ugh$-equivariant quantization, so the conditions
of Section \ref{sect:Poisson-LieOrB} are also sufficient.
We give an explicit formula for the quantized bundle map.
\noindent
There is an Appendix at
the end of the paper where we study certain properties of the q-trace functions.
\subsubsection*{Acknowledgements}
We thank Steven Shnider whose valuable remarks led to a
substantial improvement of the text.
This work was supported by the EPSRC grant C511166
and by the RFBR grant 06-01-00451.
\section{Generalities on equivariant quantization}
\subsection{Deformation quantization of Poisson varieties}
Let $M$ be a variety with a Poisson bracket $\pb$ and $A=\fun[M]$ be the
algebra of polynomial functions $M\to\mathbb{C}$.
Recall the following definition (see e.g. \cite{DrinfeldW:Doklad}):
\begin{defin}
\label{def:QnOfPoissManifs} An algebra $(A_\ensuremath{\hbar},\star)$ over the ring
${\mathbb{C}[\![\ensuremath{\hbar}]\!]}$ of formal power series is called {\em quantization} of
$(A,\pb)$ if:
\begin{enumerate}
\item[ (i)]
$A_\ensuremath{\hbar}$ is a free ${\mathbb{C}[\![\ensuremath{\hbar}]\!]}$-module;
\item[ (ii)]
As a $\mathbb{C}$-algebra, the quotient $A_\ensuremath{\hbar}/\ensuremath{\hbar} A_\ensuremath{\hbar}$ is isomorphic to
$A$;
\item[ (iii)]
If $a,b\in A$ then $\displaystyle\frac{a\star b-b\star
a}{\ensuremath{\hbar}}\equiv\pb(a,b)$ modulo $\ensuremath{\hbar}$.
\end{enumerate}
The Poisson bracket $\pb$ is called {\em the infinitesimal} of
$(A_\ensuremath{\hbar},\star)$.
\end{defin}
\begin{rem}\label{nb:Pb}
The deformed multiplication is expanded as an $\hbar$-series:
$\displaystyle a\star b=\sum_{k=0}^\infty m_k(a\otimes b)\ensuremath{\hbar}^k$
for $a,b\in A\subset A_\hbar$.
Therefore one has $\pb(a,b)=m_1(a\otimes b) - m_1(b\otimes a)$.
\end{rem}
Let $(M,\pb_M)$ and $(N,\pb_N)$ be two Poisson varieties, $A_\ensuremath{\hbar}$
and $B_\ensuremath{\hbar}$ some quantizations of the function algebras $A=\fun[M]$
and $B=\fun[N]$ respectively, $f\colon B\to A$ a morphism of Poisson
algebras.
\begin{defin}
\label{def:QnOfPoissMorphs}
A homomorphism $f_\ensuremath{\hbar}\colon B_\ensuremath{\hbar}\to A_\ensuremath{\hbar}$ of
${\mathbb{C}[\![\ensuremath{\hbar}]\!]}$-algebras is called a {\em quantization of the map} $f$ if the
induced morphism $f_0\colon B_\ensuremath{\hbar}/\ensuremath{\hbar} B_\ensuremath{\hbar}\to A_\ensuremath{\hbar}/\ensuremath{\hbar}
A_\ensuremath{\hbar}$ of $\mathbb{C}$-algebras coincides with $f$.
\end{defin}
\begin{prop}
Suppose there exists a quantization of $f\colon B\to A$.
Then $f$ is a Poisson map.
\end{prop}
\begin{proof}
Denote by $m_{A_\ensuremath{\hbar}}$ the multiplication in $A_\ensuremath{\hbar}$ and by $m_A$ the
undeformed multiplication in $A$. The map $f_\ensuremath{\hbar}$ is supposed to
be an algebra homomorphism, i.e. $f_\ensuremath{\hbar}\comp
m_{B_\ensuremath{\hbar}}=m_{A_\ensuremath{\hbar}}\comp\left(f_\ensuremath{\hbar}\otimes f_\ensuremath{\hbar}\right)$. Consider the
infinitesimal part of this equality:
\begin{equation*}
f\comp m_{B_\ensuremath{\hbar},1}+f_1\comp m_B= m_A\comp\left(f_1\otimes f\right)+
m_A\comp\left(f\otimes f_1\right)+ m_{A_\ensuremath{\hbar},1}\comp\left(f\otimes f\right).
\end{equation*}
Applying this equality first to $a\otimes b\in A\otimes A$, then to $b\otimes a$
and taking the difference, one obtains $f\bigl(\pb_M(a,b) \bigr) =
\pb_N\bigl(f(a),f(b)\bigr)$ where $\pb_M(a,b)=m_{A_\ensuremath{\hbar},1}(a,b)-m_{A_\ensuremath{\hbar},1}(b,a)$ and
$\pb_N(a,b)=m_{B_\ensuremath{\hbar},1}(a,b)-m_{B_\ensuremath{\hbar},1}(b,a)$.
\end{proof}
\subsection{Quantization of $G$-varieties}
Consider a simple complex algebraic group $G$ and its Lie algebra $\mathfrak g$.
Suppose $G$ is a Poisson group; then $\mathfrak g$ is equipped with a quasitriangular
Lie bialgebra structure. Denote by $r\in \wedge^2\mathfrak g$
the corresponding classical r-matrix.
We consider only the factorizable case, when $r$ satisfies the {\em modified} classical Yang-Baxter equation.
By $\Ughr$ we denote the quantization of the universal enveloping algebra ${\U}(\g)$ along
$r$.
Consider the variety $M=K\backslash G$ where $K\subset G$ is a reductive subgroup in $G$ of maximal rank;
let $\mathfrak{k}$ denote its Lie algebra (we prefer to work with right coset spaces, so that the right $G$-action induces
a left action on functions).
Suppose $M$ is a Poisson $G$-variety, that is to say,
the action $G\times M\to M$ is Poisson. Set $A=\mathbb{C}[M]$ and let $A_\ensuremath{\hbar}$ be its quantization.
We expect the deformed
multiplication in $A_\ensuremath{\hbar}$ to be equivariant with respect to an
action of $\Ughr$. In other words, this multiplication should obey the
"Leibniz rule"
$$
x.(a\star b)=(x^{(1)}.a)\star (x^{(2)}.b)
$$
for all $a,b\in A$ and $x\in\Ughr$.
We use the standard Sweedler notation $x^{(1)}\otimes x^{(2)}$
for the coproduct $\Delta(x)$.
The infinitesimal of a
$\Ughr$-equivariant quantization $A_\ensuremath{\hbar}$
is always of the form (see \cite{DoninGurevicShnider:DoubleQuantiz})
\begin{equation}\label{eq:UhEquivariantPb}
\pb=\overleftarrow{r}+\overrightarrow{s}.
\end{equation}
Here $\overleftarrow{r}$ denotes the bivector field on
$M$ generated by $r\in \wedge^2\mathfrak g$ via the action of $G$,
and $\overrightarrow{s}$ denotes the invariant bivector field on
$M$ generated by an element $s\in\left(\wedge^2\mathfrak g/\mathfrak{k}\right)^\mathfrak{k}$.
The latter
should satisfy certain conditions, and the bivector field
$\overrightarrow{s}$ is called {\em quasi-Poisson structure} on $M$.
For simplicity, we call the generator $s$ a quasi-Poisson structure as well.
Recall the constructions of the bivector fields $\overleftarrow{r}$
and $\overrightarrow{s}$.
For $r\in\wedge^2\mathfrak g$ and $g\in G$, consider the bivector field $(R_g)_*r$
on $G$,
where $R_g:G\to G$ is the right translation $x\mapsto xg$.
This bivector field is left $G$-invariant, so it is left $K$-invariant.
Hence it is projectable to $M=K\backslash G$, and we denote the projection by
$\overleftarrow{r}$.
Note that $\overleftarrow{r}$ is not $G$-invariant.
To describe the bivector field $\overrightarrow{s}$,
lift $s$ from $\wedge^2\left(\mathfrak g/\mathfrak{k}\right)^\mathfrak{k}$
to $\left(\wedge^2\mathfrak g\right)^\mathfrak{k}$ and consider
a left $G$ invariant bivector field $(L_g)_*s$,
where $L_g:G\to G$ is the left translation $x\mapsto gx$.
It is also left $K$-invariant, hence it is projectable to $M=K\backslash G$.
We denote the projection by $\overrightarrow{s}$.
Any $G$ invariant bivector field on $M$ is obtained in this way.
Left and right invariant vector
fields on $G$ commute with each other, hence the Schouten bracket
$[\overleftarrow{r},\overrightarrow{s}]$ vanishes.
This implies that $p=\overleftarrow{r}+\overrightarrow{s}$ is
a Poisson bracket if and only if $[\overleftarrow{r},\overleftarrow{r}]=-[\overrightarrow{s},\overrightarrow{s}]$.
Recall that two Poisson brackets are called compatible if their any linear combination
is again a Poisson bracket. Suppose that the new bracket makes the variety $M$ Poisson over the Poisson
group $G$. Then the formula (\ref{eq:UhEquivariantPb}) suggests that
a Poisson bracket $\ensuremath{\varkappa}$ on $M$ is compatible with $p$ if and only if it is
$G$-invariant and $[\overrightarrow{s},\ensuremath{\varkappa}]=0$.
Next we recall the notion of 2-parameter, or double quantization (see,
for example, \cite{DoninJ:DoubleQuantizationOfLieGroups}).
\begin{defin}
\label{def:DoubleQnOfPoissManifs} Suppose that the commutative
algebra $A$ is endowed with two compatible Poisson brackets, $\pb$
and $\ensuremath{\varkappa}$, such that $\ensuremath{\varkappa}$ is $G$-invariant. An algebra
$(A_{\ensuremath{\hbar},t},\star)$ over the ring $\mathbb{C}{[\![}\ensuremath{\hbar},t{]\!]}$ of formal power
series in two variables is called {\em equivariant quantization} of
$(A,\pb,\ensuremath{\varkappa})$ if
\begin{enumerate}
\item[ (i)]
$A_{\ensuremath{\hbar},t}$ is a free module over $\mathbb{C}{[\![}\ensuremath{\hbar},t{]\!]}$;
\item[ (ii)]
The first order term of the deformed multiplication $\star$ is
$\ensuremath{\hbar}\pb+t\ensuremath{\varkappa}$;
\item[ (iii)]
The algebra $A_{\ensuremath{\hbar},t}$ is $\Ughr$-equivariant;
\item[ (iv)]
The quotient $A_{\ensuremath{\hbar},t}/\ensuremath{\hbar} A_{\ensuremath{\hbar},t}$ is a $G$-equivariant
(one-parameter) quantization of $(A,\ensuremath{\varkappa})$.
\end{enumerate}
\end{defin}
\section{Poisson-Lie orbit bundles}
\label{sect:Poisson-LieOrB}
\subsection{General remarks on coadjoint orbits in $\mathfrak{gl}_n^*(\mathbb{C})$}
\label{sect:GenRemarks}
Fix $G=GL_n(\mathbb{C})$ and put $\mathfrak g=\mathfrak{gl}_n(\mathbb{C})$. Choose the
algebra of diagonal matrices as a Cartan subalgebra
$\Cart\subset \mathfrak g$.
The non-degenerate $G$-invariant bilinear
form $\tr(XY)$ on $\mathfrak g$ allows us to think of $\Cart^*$
as a subset in $\mathfrak g^*$, the dual vector space for $\mathfrak g$.
Denote by $\ola$ the coadjoint orbit of a
semisimple element $\ensuremath{\lambda}\in\Cart^*\subset{\mathfrak{g}}^*$.
As a $G$-variety,
$\ola$ is isomorphic to $G^\ensuremath{\lambda}\backslash G$ where $G^\ensuremath{\lambda}$ is a Levi
subgroup of $G$. That is, the Lie algebra $\mathfrak g^\ensuremath{\lambda}$ of $G^\ensuremath{\lambda}$
is a Levi subalgebra of $\mathfrak g$ containing
$\Cart$.
The trace bilinear form on $\mathfrak g$ establishes a canonical
isomorphism between coadjoint orbits in ${\mathfrak{g}}^*$ and adjoint orbits
in $\mathfrak g$.
We will use this isomorphism without further noticing.
In the same way, we identify an element $\ensuremath{\lambda}\in\Cart^*$
with the corresponding diagonal matrix in $\Cart$.
Two diagonal matrices with different order of the entries belong
to the same $G$-orbit.
Hence we can
choose a representative of the orbit in which all equal entries are
grouped up together. In other words, we can think that
$\ensuremath{\lambda}=\diag(\ensuremath{\Lambda}_1,\ldots,\ensuremath{\Lambda}_l)$ where $\ensuremath{\Lambda}_i$ is the scalar
$n_i\times n_i$ matrix with $\ensuremath{\lambda}_i$ on its diagonal.
In particular, the orbit $\ola$ is determined by a pair
$(\mbox{\boldmath$\lambda$},\mathbf{n})$ where $\mbox{\boldmath$\lambda$}=(\ensuremath{\lambda}_1,\ldots,\ensuremath{\lambda}_l)$ is the row of pairwise
distinct eigenvalues of $\ensuremath{\lambda}$, and
$\mathbf{n}=(n_1,\ldots,n_l)$ and $\ensuremath{\lambda}_1,\dots,\ensuremath{\lambda}_l$ is the row of their
multiplicities.
Note that the orbit $\ola$ does not depend on
simultaneous permutations of the entries of $\mbox{\boldmath$\lambda$}$ and $\mathbf{n}$.
\subsection{The related Poisson structures on $\ola$}
The admissible quasi-Poisson bracket on a semisimple orbit has the form
(see \cite{DoninJ:QuanGMans},
\cite{DoninJOstapenkoV:EquivQn},
\cite{DoninGurevicShnider:DoubleQuantiz} and
\cite{KarolinskijE:ClassPoissHomSpace} for details):
$$
\sum_{1\leqslant i<j\leqslant l}c_{ij}\xi_{ij},
$$
where $c_{ij}$ are some coefficients depending on the eigenvalues of
$\ensuremath{\lambda}$ (see below for explicit formulas),
and $\xi_{ij}$ are defined as follows:
\begin{equation}\label{eq:BasisBivector}
\xi_{ij}=\sum_{s,t} E_{st}\wedge E_{ts} \mod \mathfrak g\wedge\mathfrak g^\ensuremath{\lambda}.
\end{equation}
Here $E_{st}$ is the $(s,t)$-th matrix unit, $l$ is the number of
different eigenvalues of $\ensuremath{\lambda}$, and the sum is taken over
$n_1+n_2+\dots+n_{i-1}< s \leqslant n_1+n_2+\dots+n_i$,
$n_1+n_2+\dots+n_{j-1}< t \leqslant n_1+n_2+\dots+n_j$, where $n_i$
denotes the multiplicity of the eigenvalue $\ensuremath{\lambda}_i$.
\subsubsection*{The KKSL Poisson bracket $\ensuremath{\varkappa}_\ensuremath{\lambda}$}
The Kirillov-Kostant-Lie-Souriau bra\-cket $\ensuremath{\varkappa}_\ensuremath{\lambda}$ is induced on
$\ola$ from the Lie structure on $\mathfrak g$.
For a semisimple (co)adjoint $GL_n(\mathbb{C})$-orbit $\ola$ it is given
by the following expression:
\begin{equation*}
\ensuremath{\varkappa}_\ensuremath{\lambda}=\sum_{1\leqslant i<j\leqslant l}\frac{1}{\ensuremath{\lambda}_i-\ensuremath{\lambda}_j}\ \xi_{ij}.
\end{equation*}
This is a $G$-invariant non-degenerate Poisson bracket.
\subsubsection*{The quasi-Poisson bracket $s^0_\ensuremath{\lambda}$.}
Consider the bivector field on $\ola$ restricted
from the Semenov-Tian-Shansky (STS)
bracket on $\End(\mathbb{C}^n)$, see \cite{STS:QDualityQDouble}.
Its quasi-Poisson part is generated by an element $s_\ensuremath{\lambda}^0\in\wedge^2(\mathfrak g/\mathfrak g^\ensuremath{\lambda})^{\mathfrak g^\ensuremath{\lambda}}$.
Specifically for the
case $G=GL_n(\mathbb{C})$, it takes the form:
$$
s_\ensuremath{\lambda}^0=\sum_{1\leqslant i<j\leqslant l} \frac{\ensuremath{\lambda}_i+\ensuremath{\lambda}_j}{\ensuremath{\lambda}_i-\ensuremath{\lambda}_j}\
\xi_{ij},
$$
see \cite{DoninJMudrovAI:MethodQuantChar}.
\subsubsection*{The brackets admitting $\Ughr$-equivariant
quantization}
In our case $G=GL_n(\mathbb{C})$, the Schouten bracket between $s_\ensuremath{\lambda}^0$
and $\ensuremath{\varkappa}_\ensuremath{\lambda}$ vanishes.
Thus, by adding a multiple of the KKLS
bracket to that generated by $s_\ensuremath{\lambda}^0$, one obtains the general form
for a quasi-Poisson bracket on $\ola$ admitting a $\Ughr$-equivariant
quantization:
\begin{equation}\label{eq:sla}
s_\ensuremath{\lambda}=s_\ensuremath{\lambda}^0+a \ensuremath{\varkappa}_\ensuremath{\lambda}= \sum_{1\leqslant i<j\leqslant
l}\frac{\ensuremath{\lambda}_i+\ensuremath{\lambda}_j+a}{\ensuremath{\lambda}_i-\ensuremath{\lambda}_j}\ \xi_{ij}, \mbox{ where }
a\in\mathbb{C}.
\end{equation}
Recall once again that we consider only those Poisson structures
that are restricted from ${\mathfrak{g}}^*$.
\subsection{The structure of Poisson orbit bundles}
Here we give the necessary and sufficient conditions for
an orbit map $\prj\colon \left(\ola,s_\ensuremath{\lambda}\right)\to\left(\omu,s_\mu\right)$
to be Poisson.
Recall that we do not distinguish between an element $\ensuremath{\lambda}\in{\mathfrak{g}}^*$ and
the corresponding diagonal matrix $\diag(\ensuremath{\Lambda}_1,\ldots,\ensuremath{\Lambda}_l)$.
\begin{prop}\label{prop:P}
Any semisimple orbit bundle $\ola\to\omu$ is of the form
$X\mapsto\Prj(X)$ where
$\Prj$ is a polynomial in one variable with complex coefficients.
\end{prop}
\begin{proof}
If an equivariant map $\left(\ola,s_\ensuremath{\lambda}\right)\to\left(\omu,s_\mu\right)$
brings $\ensuremath{\lambda}$ to $\mu$, then the isotropy group $G^\ensuremath{\lambda}$ is
a subgroup in $G^\mu$; this gives an inclusion $\mathfrak g^\ensuremath{\lambda}\subset \mathfrak g^\mu$ of their Lie algebras.
Consider $\mathfrak g=\mathfrak{gl}_n(\mathbb{C})$ as an associative algebra, $\mathfrak g\cong\End(\mathbb{C}^n)$,
and denote by $\Z(\mathfrak g^\ensuremath{\lambda})\subset\mathfrak g$ the centralizer of
$\mathfrak g^\ensuremath{\lambda}$.
Since $\mathfrak g^\ensuremath{\lambda}$ is a Levi subalgebra,
$\Z(\mathfrak g^\ensuremath{\lambda})$ is a semisimple commutative associative
algebra generated by $\ensuremath{\lambda}$ and by the unit matrix.
For any polynomial
$P$ in one variable the mapping $X\mapsto P(X)$, $X\in \mathfrak g$, is
$G$-equivariant.
Hence it suffices to check that $\mu = \Prj(\ensuremath{\lambda})$
for some polynomial $\Prj$.
The inclusion $\mathfrak g^\ensuremath{\lambda}\subset \mathfrak g^\mu$ implies the inclusion $Z(\mathfrak g^\mu)\subset Z(\mathfrak g^\ensuremath{\lambda})$.
Therefore the matrix $\mu$ is a polynomial in
$\ensuremath{\lambda}$.
\end{proof}
\begin{rem}
We will use the same symbol $\Prj$ for both the orbit map
$\ola\to\omu$ and for the corresponding polynomial.
\end{rem}
\subsubsection*{The graph of an orbit bundle}
It is convenient to use
the following graphical presentation of an orbit bundle $\Prj\colon\omu\to\ola$.
The orbit $\ola$ has a representative in the form of diagonal matrix
$\ensuremath{\lambda}=\diag(\ensuremath{\Lambda}_1,\ldots,\ensuremath{\Lambda}_l)$.
Set $\mu=\Prj(\ensuremath{\lambda})$, then $\mu$ also has the form
$\mu=\diag(\mathrm{M}_1,\ldots,\mathrm{M}_m)$, where $\mathrm{M}_j$ denotes the scalar
block corresponding to the eigenvalue $\mu_j$ of $\mu$.
Using this, denote by $\Gamma_\Prj$ the bipart type
graph whose upper nodes have labels $1,\dots,l$ corresponding to the
blocks $\ensuremath{\Lambda}_1,\dots,\ensuremath{\Lambda}_l$, and the lower nodes are labeled by
$1,\dots,m$ corresponding to the blocks $\mathrm{M}_1,\dots,\mathrm{M}_m$.
The $i$-th node of the upper part of $\Gamma_\Prj$ is connected to the
$\alpha$-th node of the lower part if and only if
$\Prj(\ensuremath{\lambda}_i)=\mu_\alpha$.
Note that each upper node has exactly one edge.
Since the map $\Prj$ is surjective, each
lower node is connected to some upper node.
Therefore the graph $\Gamma_\Prj$ is a disjoint union of trees of
the form:
\begin{center}\setlength{\unitlength}{.7in}
\begin{picture}(0,-1)
\multiput(-.9,0)(.3,0){3}{\circle{.08}}
\put(.45,0){\circle{.08}}
\put(0,-.9){\circle{.08}}
\put(-.88,-.03){\line(1,-1){.863}}
\put(-.59,-.03){\line(2,-3){.565}}
\put(-.3,-.03){\line(1,-3){.279}}
\put(.445,-.03){\line(-1,-2){.423}}
\put(-.03,-1.2){$\alpha$}
\multiput(-.06,-.05)(.12,0){3}{$\cdot$}
\end{picture}
\end{center}\vspace{1.0in}
Each tree is a connected component of $\Gamma_\Prj$
and it is labeled by the blocks of $\mu$.
The graph $\Gamma_\Prj$ gives a complete description of
the map $\Prj$.
\subsubsection*{The graph $\Gamma_\Prj$ for $\Prj$ Poisson}
Both $\ola$ and $\omu$ are endowed with Poisson
structures (\ref{eq:sla}).
The tangent space of $\ola$ at the point $\ensuremath{\lambda}$ is isomorphic as
a vector space to the quotient $\mathfrak g/\mathfrak g^\ensuremath{\lambda}$.
Recall that $\mathfrak g^\ensuremath{\lambda}\subset \mathfrak g^\mu$.
The tangent map $\prj_*:\mathfrak g/\mathfrak g^\ensuremath{\lambda}\to\mathfrak g/\mathfrak g^\mu$ of $\prj$ is given by
the formula:
$$
\prj_*(X)=\left\{
\begin{array}{l}
0, \mathrm{\ if\ } X\in\mathfrak g^\mu/\mathfrak g^\ensuremath{\lambda},\\
\ \\
X \mathrm{\ otherwise.}
\end{array}
\right.
$$
An element $X\in\mathfrak g\setminus\Cart$ cannot generate a vector field on
$\omu$ since it is not even $\Cart$-invariant.
However, the element like $\xi_{ij}$ (see formula
(\ref{eq:BasisBivector})) does generate a bivector field on $\omu$.
The map $\prj$ is Poisson if and only if
$\prj_*(s_\ensuremath{\lambda})=s_\mu$.
The tangent map $\prj_*$ is determined by
its values $\prj_*(\xi_{ij})=\xi_{\alpha\beta}$ where $i,j$ run over
the upper nodes of the graph $\Gamma_\Prj$ while $\alpha,\beta$ run
over the lower nodes of $\Gamma_\Prj$.
\begin{lm}\label{lm:GraphOfPoissBundle}
Let $\Prj\colon \ola\to\omu$ be a Poisson orbit bundle w.r.t.
the Poisson structure on $\ola$ determined by $s_\ensuremath{\lambda}=\sum
c_{ij}\xi_{ij}$ and some Poisson structure on $\omu$.
Suppose that $i\leqslant j<p\leqslant s$ are such {\em upper} nodes that $i$
and $j$ belong to the same connected component $\alpha$, and $p$, $s$ also
belong to the same connected component $\beta$ of $\Gamma_\Prj$.
Then $c_{ip}=c_{js}$.
\end{lm}
\begin{proof}
If $\Prj$ is Poisson then $s_\ensuremath{\lambda}$ is projectable under $\prj_*$.
Thus if $\prj_*(\xi_{ip})$ and $\prj_*(\xi_{js})$
enter the same basis bivector $\xi_{\alpha\beta}$, then $c_{ip}=c_{js}$.
\end{proof}
\subsubsection*{Classification of Poisson bundles}
We call a coadjoint orbit $\omu$ {\em symmetric} if the
corresponding matrix $\mathrm{M}$ has exactly two different eigenvalues.
\begin{thm}\label{th:MainPoiss}
A $GL_n(\mathbb{C})$-equivariant map $\prj\colon O_1\toO_2$ is Poisson if and
only if the following three conditions are satisfied:
\begin{enumerate}
\item[{\em (a)}]
The orbit $O_2$ is symmetric;
\item[{\em (b)}]
There exist $\ensuremath{\lambda}\inO_1$ and $\mu\inO_2$ such that
$\prj(\ensuremath{\lambda})=\mu$ and the multiplicity $n_1$ of the eigenvalue $\ensuremath{\lambda}_1$
is equal to the multiplicity $m_1$ of the eigenvalue $\mu_1$;
\item[{\em (c)}]
The Poisson structures on $O_1$ and $O_2$ are defined by $s_\ensuremath{\lambda}
= s_\ensuremath{\lambda}^0 - 2\ensuremath{\lambda}_1 \ensuremath{\varkappa}_\ensuremath{\lambda}$ and $s_\mu = s_\mu^0 - 2\mu_1 \ensuremath{\varkappa}_\mu$
respectively.
\end{enumerate}
\end{thm}
\begin{proof}
Show first that if $\Prj\colon O_1\toO_2$ is a Poisson map,
then the orbit $O_2$ is symmetric, i.e. the graph $\Gamma_\Prj$
consists of exactly two connected components.
Indeed, suppose that $\Gamma_\Prj$ has the
form \vspace{.3in}
\begin{center}\setlength{\unitlength}{.7in}
\begin{picture}(0,-2)
\put(-1.6,0){\circle{.08}}
\put(-1.2,0){\circle{.08}}
\put(-.4,0){\circle{.08}}
\put(0,0){\circle{.08}}
\put(-2.0,-1){\circle{.08}}
\put(-.8,-1){\circle{.08}}
\put(.4,-1){\circle{.08}}
\put(-1.6,-.04){\line(-2,-5){.37}}
\put(-1.2,-.04){\line(2,-5){.37}}
\put(-.4,-.04){\line(-2,-5){.37}}
\put(0,-.04){\line(2,-5){.37}}
\put(-1.7,.2){$i$}
\put(-1.4,.2){$i+1$}
\put(-.5,.2){$j$}
\put(-.2,.2){$j+1$}
\put(-2.05,-1.4){$1$}
\put(-0.85,-1.4){$2$}
\put(.35,-1.4){$3$}
\multiput(-2.05,-.05)(.12,0){3}{$\cdot$}
\multiput(-.99,-.05)(.12,0){4}{$\cdot$}
\multiput(.15,-.05)(.12,0){3}{$\cdot$}
\put(1.5,-.1){$\ola$}
\put(1.5,-1.1){$\omu$}
\put(1.6,-.3){\vector(0,-1){.5}}
\put(1.72,-.6){$\Prj$}
\end{picture}
\end{center}\vspace{.3in}\vs\vspace{.3in}
Since $\Prj$ is Poisson, Lemma~\ref{th:MainPoiss} leads to the
following system of conditions:
$$
\left\{
\begin{array}{l}
c_{i,i+1}=c_{i,j}\\
c_{i+1,j+1}=c_{j,j+1},
\end{array}
\right.
$$
where $\displaystyle c_{ij}=\frac{\ensuremath{\lambda}_i+\ensuremath{\lambda}_j+a}{\ensuremath{\lambda}_i-\ensuremath{\lambda}_j}$, see formula
(\ref{eq:sla}). Thus we should solve the following overdefined
system of linear equations in $a$:
$$
\left\{
\begin{array}{l}
\displaystyle \frac{\ensuremath{\lambda}_i+\ensuremath{\lambda}_{i+1}+a}{\ensuremath{\lambda}_i-\ensuremath{\lambda}_{i+1}}=
\frac{\ensuremath{\lambda}_{i}+\ensuremath{\lambda}_{j}+a}{\ensuremath{\lambda}_{i}-\ensuremath{\lambda}_{j}} \\
\ \\
\displaystyle \frac{\ensuremath{\lambda}_{i+1}+\ensuremath{\lambda}_{j+1}+a}{\ensuremath{\lambda}_{i+1}-\ensuremath{\lambda}_{j+1}}=
\frac{\ensuremath{\lambda}_{j}+\ensuremath{\lambda}_{j+1}+a}{\ensuremath{\lambda}_{j}-\ensuremath{\lambda}_{j+1}}
\end{array}
\right.
$$
This system is inconsistent, thus our hypothesis is wrong, and
$\Gamma_\Prj$ consists of exactly two connected trees.
This means that
$\mu$ has exactly two different eigenvalues:
$\mu=\diag(\mathrm{M}_1,\mathrm{M}_2)$, i.e. it is symmetric.
We now show that if $\Prj$ is Poisson, then either
$\mathrm{mult}\>\ensuremath{\lambda}_1=\mathrm{mult}\>\mu_1$ or $\mathrm{mult}\>\ensuremath{\lambda}_l=\mathrm{mult}\>\mu_2$,
where $\mathrm{mult}$ means the multiplicity of the corresponding eigenvalue.
In other words, we need to prove that
if $\Prj$ is Poisson, then one of the two connected components of
the graph $\Gamma_\Prj$ contains precisely one edge.
Suppose, to the contrary, that $\Gamma_\Prj$ is of the form:
\vspace{.3in}
\begin{center}\setlength{\unitlength}{.7in}
\begin{picture}(0,-1)
\put(-1.2,0){\circle{.08}}
\put(-.6,0){\circle{.08}}
\put(0,0){\circle{.08}}
\put(.6,0){\circle{.08}}
\put(-.6,-1){\circle{.08}}
\put(0,-1){\circle{.08}}
\put(-1.2,-.04){\line(3,-5){.57}}
\put(-.6,-.04){\line(0,-1){.93}}
\put(0,-.04){\line(0,-1){.93}}
\put(.6,-.04){\line(-3,-5){.57}}
\put(-1.3,.2){$1$}
\put(-.65,.2){$i$}
\put(-.25,.2){$i+1$}
\put(.6,.2){$l$}
\put(-.65,-1.4){$1$}
\put(-.03,-1.4){$2$}
\multiput(-1.05,-.05)(.12,0){3}{$\cdot$}
\multiput(.15,-.05)(.12,0){3}{$\cdot$}
\put(1.5,-.1){$\ola$}
\put(1.5,-1.1){$\omu$}
\put(1.6,-.3){\vector(0,-1){.5}}
\put(1.72,-.6){$\Prj$}
\end{picture}
\end{center}\vspace{.3in}\vs\vspace{.3in}
Again, by Lemma~\ref{th:MainPoiss} the following system of
conditions should be satisfied: $c_{1,i+1}=c_{i,i+1}=c_{i,l}$.
This is a system of equations in $a$:
$$
\frac{\ensuremath{\lambda}_{1}+\ensuremath{\lambda}_{i+1}+a}{\ensuremath{\lambda}_{1}-\ensuremath{\lambda}_{i+1}}=
\frac{\ensuremath{\lambda}_i+\ensuremath{\lambda}_{i+1}+a}{\ensuremath{\lambda}_i-\ensuremath{\lambda}_{i+1}}=
\frac{\ensuremath{\lambda}_{i}+\ensuremath{\lambda}_{l}+a}{\ensuremath{\lambda}_{i}-\ensuremath{\lambda}_{l}},
$$
which obviously has no solution.
Thus one of the two connected
components of $\Gamma_\Prj$ contains one edge.
By appropriate choice of
the representatives $\ensuremath{\lambda}\in\ola$ and $\mu\in\omu$, one can assume
that to be the left component:
\vspace{.3in}
\begin{center}\setlength{\unitlength}{.7in}
\begin{picture}(0,-1)
\put(-.6,0){\circle{.08}}
\put(0,0){\circle{.08}}
\put(.6,0){\circle{.08}}
\put(-.6,-1){\circle{.08}}
\put(0,-1){\circle{.08}}
\put(-.6,-.04){\line(0,-1){.93}}
\put(0,-.04){\line(0,-1){.93}}
\put(.6,-.04){\line(-3,-5){.57}}
\put(-.7,.2){$1$}
\put(-.1,.2){$2$}
\put(.6,.2){$l$}
\put(-.65,-1.4){$1$}
\put(-.03,-1.4){$2$}
\multiput(.15,-.05)(.12,0){3}{$\cdot$}
\put(1.5,-.1){$\ola$}
\put(1.5,-1.1){$\omu$}
\put(1.6,-.3){\vector(0,-1){.5}}
\put(1.72,-.6){$\Prj$}
\end{picture}
\end{center}\vspace{.3in}\vs\vspace{.3in}
i.e. that the multiplicities of $\ensuremath{\lambda}_1$ and $\mu_1$
coincide.
Lemma \ref{lm:GraphOfPoissBundle} applied to the above graph gives rise
to the overdefined system of $l-1$ linear equations
$c_{12}=c_{13}=\ldots=c_{1l}$. This system is
consistent, and the solution is $a=-2\ensuremath{\lambda}_1$. Hence the
quasi-Poisson brackets on $O_i$ are as in (c).
Now we prove the sufficiency of the conditions (a)--(c).
Indeed, consider an invariant mapping $\Prj\colon O_1\toO_2$
corresponding to the above graph.
Suppose that $\Prj(\ensuremath{\lambda})=\mu$. Then for $s_\ensuremath{\lambda} =
s_\ensuremath{\lambda}^0 - 2\ensuremath{\lambda}_1 \ensuremath{\varkappa}_\ensuremath{\lambda}$ and $s_\mu = s_\mu^0 - 2\mu_1 \ensuremath{\varkappa}_\mu$ one
has $\Prj_*(s_\ensuremath{\lambda})=s_\mu$, i.e. $\Prj$ is a Poisson map with respect
to those brackets.
\end{proof}
\subsection{Explicit formula for the map $\prj$}
Denote by $\ensuremath{\lambda}$ and $\mu$ elements of
$\Cart$ satisfying the conditions of Theorem~\ref{th:MainPoiss} and by
$\ola$ and $\omu$ their adjoint orbits.
For the purpose of quantization, we need an explicit expression for
$\prj$.
Recall that $\ensuremath{\lambda}_1,\ensuremath{\lambda}_2,\dots,\ensuremath{\lambda}_l$ denote all the distinct
eigenvalues of $\ensuremath{\lambda}\in\mathfrak g$, so we need to find a polynomial
$\Prj(x)$ of degree $l-1$ in one variable with complex coefficients
such that
$\Prj(\ensuremath{\lambda}_1)=\mu_1$ and $\Prj(\ensuremath{\lambda}_2)=\ldots=\Prj(\ensuremath{\lambda}_l)=\mu_2$.
These $l$ equations determine $\Prj$ uniquely:
\begin{equation}\label{eq:DefOfP}
\Prj(x)=(\mu_1-\mu_2)\prod_{i=2}^l\frac{x-\ensuremath{\lambda}_i}{\ensuremath{\lambda}_1-\ensuremath{\lambda}_i} +
\mu_2.
\end{equation}
\section{Reflection equation algebras}
\label{sect:REAlgebras}
An equivariant two-parameter quantization of coadjoint orbits is
constructed in \cite{DoninJMudrovAI:ExplicitQuantiz} for the
special case of the standard, or Drinfeld-Jimbo, quantum group
$\Ugh$.
In this section we extend the quantization of \cite{DoninJMudrovAI:ExplicitQuantiz} for an arbitrary
quantum group $\Ughr$, not necessarily the standard.
Note that
possible factorizable Lie bialgebra structures on $\mathfrak g$ are
parameterized by Belavin-Drinfeld triples and a subspace in $\mathfrak h$,
\cite{BelawinDrinfeld:CYBE}.
Quantization of the universal enveloping algebra along any Lie
bialgebra structure has been constructed in
\cite{EtinhoffKazdan:QuantBialg}.
To describe quantized coadjoint orbits explicitly, we need some
facts about the so called {\em modified reflection equation} (mRE)
algebra.
It is a two parameter quantization of the polynomial ring
on the vector space of $n\times n$-matrices.
The quantized orbits will be
presented as quotients of the mRE algebra by certain ideals which
are deformations of the classical ideals of the orbits.
In the present section we study how mRE algebras transform under
twist of quantum groups.
\subsection{The definition and basic facts}
\label{subsection:mREA}
Let $\Ughr$ be a quantization of the universal enveloping algebra of
$\mathfrak g=\mathfrak{gl}_n(\mathbb{C})$ along a classical r-matrix $r$.
Let $R\in \End(\mathbb{C}^n\otimes
\mathbb{C}^n){[\![}\ensuremath{\hbar}{]\!]}$ be the image of its universal R-matrix in the basic
representation in $\mathbb{C}^n{[\![}\ensuremath{\hbar}{]\!]}$.
Denote by $S:=\sigma R\in
\End(\mathbb{C}^n\otimes\mathbb{C}^n){[\![}\hbar{]\!]}$ the quantum permutation, where $\sigma$
designates the usual flip $\mathbb{C}^n\otimes\mathbb{C}^n\to \mathbb{C}^n\otimes\mathbb{C}^n$, $u\otimes
v\mapsto v\otimes u$.
\begin{defin}
The mRE algebra $\mathcal{L}$ is an associative unital
algebra over the ring $\mathbb{C}{[\![}\hbar{]\!]}[t]$ generated by the entries of
the matrix $L=\left(L_{ij}\right)_{i,j=1}^n$
modulo the relations
\begin{equation}\label{eq:mRE}
[SL_2S,L_2]=-qt[S,L_2] \in \End(\mathbb{C}^n\otimes
\mathbb{C}^n)\otimes\mathcal{L},
\end{equation}
where $L_2:=1\otimes L$ and $q:=e^\hbar$.
\end{defin}
The action of $\Ughr$ on the algebra $\mathcal{L}$ is given by the formula
\begin{equation}\label{eq:action}
x\triangleright L= \rho\bigl(\gamma(x^{(1)})\bigr)L\rho(x^{(2)}),
\end{equation}
where $\gamma$ is the antipode of $\Ughr$ and $\rho$ is the representation
$\Ughr\to\End(\mathbb{C}^n){[\![}\ensuremath{\hbar}{]\!]}$.
The algebra $\mathcal{L}$ is a $\Ughr$-equivariant
quantization of the polynomial ring $\mathbb{C}[\End(\mathbb{C}^n)]$
with $\ensuremath{\hbar} s^0_\ensuremath{\lambda}-t\kappa_\ensuremath{\lambda}$
being the linear term of the deformed multiplication,
\cite{DoninJMudrovAI:ExplicitQuantiz}.
\begin{rem}
The relations (\ref{eq:mRE}) are called the
{\em modified reflection equation} (mRE).
These relations become quadratic when $t=0$.
The corresponding quotient $\mathcal{L}/t\mathcal{L}$ is called
{\em quadratic} or simply {\em reflection equation} (RE) algebra.
This algebra can be defined
for any quasitriangular Hopf algebra $\mathcal{H}$ and its
representation.
When $\mathcal{H}$ is a quantized
universal enveloping algebra of an algebraic matrix group $G$,
then a certain quotient of the quadratic RE algebra yields a
(one-parameter) quantization of $\mathbb{C}[G]$.
The mRE algebra $\mathcal{L}$ as a two-parameter quantization of the
coordinate ring on the matrix space is special for the case $\mathfrak g=\mathfrak{gl}_n(\mathbb{C})$.
\end{rem}
Let us describe the center $Z(\mathcal{L})$ of the algebra $\mathcal{L}$.
First of all, $Z(\mathcal{L})$ coincides with
the subalgebra of $\Ughr$-invariants and it is isomorphic to
$Z_0\otimes \mathbb{C}{[\![}\ensuremath{\hbar}{]\!]}[t]$, where $Z_0\subset\mathbb{C}[\End(\mathbb{C}^n)]$ is the
subalgebra of classical invariants.
To describe $Z(\mathcal{L})$, consider the
matrix $R^*:=\bigl((R^{t_1})^{-1}\bigr)^{t_1}=R^*_1\otimes R^*_2$,
where $t_1$ means transposition of the first tensor
component. This matrix is equal to $\mathcal{R}_1\otimes\gamma(\mathcal{R}_2)$ evaluated
in the basic representation.
Define the matrix
$D:=\nu(R_1^*R_2^*)\in \End(\mathbb{C}^n){[\![}\ensuremath{\hbar}{]\!]}$,
where $\nu$ is a scalar.
It is convenient to choose $\nu$ such that
$\displaystyle\tr(D)=\frac{1-q^{-2n}}{1-q^{-2}}$.
Put $\tau_m=\tr_q(L^m):=\tr(DL^m)\in Z(\mathcal{L})$ for $m=1,2,\dots$.
Then the $Z(\mathcal{L})$ is a polynomial ${\mathbb{C}[\![\ensuremath{\hbar}]\!]}$-algebra
generated by $\tau_1,\dots,\tau_{n-1}$.
\subsection{mRE algebras and twist}
The quantum group $\Ughr$ is a twist of the standard quantization
$\Ugh$.
Let $\mathcal{F}\in \Ugh^{\otimes 2}$ be the corresponding twisting cocycle.
It is an invertible element satisfying the identities
\begin{equation}\label{eq:cocycle}
(\Delta\otimes \id)(\mathcal{F})\mathcal{F}_{12}=(\id \otimes \Delta)(\mathcal{F})\mathcal{F}_{23},
\end{equation}
\begin{equation}
\label{eq:cocycle_norm}
(\varepsilon\otimes \id)(\mathcal{F})=1\otimes 1=(\id \otimes \varepsilon)(\mathcal{F}),
\end{equation}
where $\varepsilon$ is the counit in $\Ugh$.
As an associative algebra, $\Ughr$ coincides with $\Ugh$
but has a different comultiplication, $x\mapsto\mathcal{F}^{-1} \Delta(x)\mathcal{F}$.
The antipode is transformed accordingly, see \cite{DrinfeldW:AlmostCocomHopfAlgs}.
Recall that
a twist of Hopf algebras induces a transformation of module
algebras, which we also call twist.
Given a $\Ugh$-algebra $\mathcal{A}$
one gets an algebra over $\Ughr$ with the new multiplication
$a\otimes b\mapsto(\mathcal{F}_{1} a)(\mathcal{F}_{2} b)$ for $a,b\in \mathcal{A}$.
Applying $\mathcal{F}$ to the mRE
algebra, one apparently destroys the form of relations (\ref{eq:mRE}).
Nevertheless, the mRE algebras corresponding to
$\Ughr$ and $\Ugh$
are still related by $\mathcal{F}$, as we now demonstrate.
Let $\mathcal{A}$ and $\widetilde \mathcal{A}$ denote the quadratic RE algebras
corresponding to quantum groups $\Ugh$ and $\Ughr$ respectively.
We assume that they are extended trivially to $\mathbb{C}[t]$-algebras.
Let $\widetilde \mathcal{A}'$ be the twist of the algebra $\mathcal{A}$ by
the cocycle $\mathcal{F}$.
Denote by $\{Q_{ij}\}\subset \mathcal{A}$ and $\{\widetilde Q_{ij}\}\subset \widetilde\mathcal{A}$
the generators satisfying the quadratic RE, i.e. the equation similar to
(\ref{eq:mRE}) but with zero in the r.h.s. These algebras
are quantizations of $\mathbb{C}[\End(\mathbb{C}^n)]$ along the STS brackets. As was shown in
\cite{KulishPMudrovA:dRE}, $\widetilde\mathcal{A}'$ is isomorphic to $\widetilde\mathcal{A}$
as $\Ughr$-module algebras,
and the isomorphism $\phi\colon \widetilde\mathcal{A}\to\widetilde\mathcal{A}'$ is given by the
formula
\begin{equation}\label{eq:RETwist}
\widetilde{Q}\mapsto (\rho\circ\gamma)(\mathcal{F}_{1} \zeta)Q\rho(\mathcal{F}_{2}).
\end{equation}
Here $\zeta:=\mathcal{F}^{-1}_{2}\gamma^{-1}(\mathcal{F}^{-1}_{1})\in \Ugh$ is the
element which participates in definition of the antipode $\tilde{\gamma}$ of $\Ughr$,
namely $\tilde \gamma(x)=\gamma(\zeta^{-1} x\zeta)$
for all $x\in \Ughr$.
Choose the new generators $\{K_{ij}\}\subset \widetilde\mathcal{A}'$ by setting
$K_{ij}:=Q_{ij}-t\ensuremath{\delta}_{ij}$ and similarly for $\{\widetilde K_{ij}\}\subset \widetilde\mathcal{A}$.
Note that $K_{ij}$ are also generators of $\mathcal{A}$.
\begin{lm}\label{lm:LinGen}
The isomorphism $\phi$ given by the formula (\ref{eq:RETwist}) defines a linear map
$\Span\bigl(\widetilde K_{ij}\bigr)\to \Span(K_{ij})$ through the formula
\begin{equation}
\label{Phi}
(\id\otimes \phi)(\widetilde K)= (\Phi\otimes \id)(K),
\end{equation}
where $\Phi$ is an invertible linear operator $\End(V)\to \End(V)$ acting by the rule
$\Phi(X)=(\rho\circ\gamma)(\mathcal{F}_{1}\zeta)X\rho(\mathcal{F}_{2})$
\end{lm}
\begin{proof}
Evaluating $\phi$ on the generators we find
$$
\phi(\widetilde K)=\rho\big(\gamma(\mathcal{F}_{1}\zeta)\big)Q\rho(\mathcal{F}_{2})-t=
\rho\big(\gamma(\mathcal{F}_{1}\zeta)\big)K\rho\big(\mathcal{F}_{2}\big)
+t\rho\big(\gamma(\mathcal{F}_{1}\zeta)\mathcal{F}_{2}\big)-t.
$$
The assertion will be proved if we show that $\gamma(\zeta)\gamma(\mathcal{F}_1)\mathcal{F}_2=1$.
But this is a well known fact from the twist theory, see \cite{DrinfeldW:AlmostCocomHopfAlgs}.
\end{proof}
Denote by $\mathcal{L}$ and by $\widetilde \mathcal{L}$ the mRE algebras corresponding to
$\Ugh$ and $\Ughr$, respectively.
\begin{prop}\label{prop:REtwisted}
The algebra $\widetilde \mathcal{L}$ is isomorphic to the twist of
$\mathcal{L}$ by the cocycle $\mathcal{F}$.
\end{prop}
\begin{proof}
Let $\widetilde\mathcal{A}'$ and $\widetilde \mathcal{L}'$ be respectively the twists of the
algebras $\mathcal{A}$ and $\mathcal{L}$ by the cocycle $\mathcal{F}$.
The algebra $\mathcal{A}$
admits an embedding in $\mathcal{L}$ through the assignment
\begin{equation}
\label{embed}
K\mapsto (1-q^{-2}) L.
\end{equation}
This embedding induces an embedding $\widetilde\mathcal{A}'\hookrightarrow
\widetilde\mathcal{L}'$ of the twisted algebras. Let us prove that the isomorphism
(\ref{eq:RETwist}) extends to an isomorphism $\widetilde \mathcal{L}\to\widetilde \mathcal{L}'$.
Denote by $\mathbb{C}(\!(\hbar)\!)$ the field of Laurent formal series in
$\hbar$. First of all notice that the mapping (\ref{embed}) is
invertible over $\mathbb{C}(\!(\hbar)\!)$.
Further, the mapping (\ref{eq:RETwist}) induces the isomorphism
$$
\widetilde \mathcal{L}\otimes_{{\mathbb{C}[\![\ensuremath{\hbar}]\!]}}\mathbb{C}(\!(\hbar)\!)\simeq
\widetilde\mathcal{A}\otimes_{{\mathbb{C}[\![\ensuremath{\hbar}]\!]}}\mathbb{C}(\!(\hbar)\!) \longrightarrow
\widetilde\mathcal{A}'\otimes_{{\mathbb{C}[\![\ensuremath{\hbar}]\!]}}\mathbb{C}(\!(\hbar)\!)\simeq
\widetilde\mathcal{L}'\otimes_{{\mathbb{C}[\![\ensuremath{\hbar}]\!]}}\mathbb{C}(\!(\hbar)\!).$$ of $\mathbb{C}(\!(\hbar)\!)$-algebras,
which we denote by $\hat \phi$.
Since $\widetilde\mathcal{L}$ and $\widetilde\mathcal{L}'$ are
free over $\mathbb{C}{[\![}\ensuremath{\hbar}{]\!]}$, we have the inclusions
$\widetilde \mathcal{L}\subset \widetilde \mathcal{L}\otimes_{{\mathbb{C}[\![\ensuremath{\hbar}]\!]}}\mathbb{C}(\!(\hbar)\!)$
and $ \widetilde\mathcal{L}'\subset \widetilde\mathcal{L}'\otimes_{{\mathbb{C}[\![\ensuremath{\hbar}]\!]}}\mathbb{C}(\!(\hbar)\!)$.
It is therefore sufficient to check that the
image of $\widetilde \mathcal{L}$ under
$\hat \phi$ lies in $ \widetilde\mathcal{L}'$ and similarly for the inverse of $\hat \phi$.
Introduce the linear operator $\Phi^*\colon\Span(K_{ij})\to \Span(K_{ij})$
through the equality $(\Phi\otimes \id)(K)=(\id\otimes \Phi^*)(K)$
(the dual conjugate of $\Phi$).
Evaluate $\hat \phi$ on a monomial in the generators $\widetilde L_{ij}$:
$$
\hat\phi (\widetilde L_{i_1 j_1}\ldots \widetilde L_{i_k j_k})
=
\hat\phi \left(\frac{1}{\omega}\widetilde K_{i_1 j_1}\ldots \frac{1}{\omega}\widetilde K_{i_k j_k}\right)
=
\frac{1}{\omega}\Phi^*(K_{i_1 j_1})\ldots \frac{1}{\omega}\Phi^*(K_{i_k j_k}),
$$
where $\omega=1-q^{-2}$.
The last equality is obtained using Lemma~\ref{lm:LinGen}.
But the rightmost expression is
$\Phi^*(L_{i_1 j_1})\ldots \Phi^*(L_{i_k j_k})\in \widetilde \mathcal{L}'$.
In the same fashion, one can check that ${\hat \phi}^{-1}(\widetilde \mathcal{L}') \subset \widetilde \mathcal{L}$.
\end{proof}
\begin{cor}\label{cor:twist}
Let $\ola$ be a semisimple coadjoint orbit. For any quantum group $\Ughr$ there
exists a $\Ughr$-equivariant quantization of $\fun$ which is a
quotient of the mRE algebra associated with $\Ughr$.
\end{cor}
\begin{proof}
Let $\mathcal{B}$ be the quantization of $\fun$ corresponding to the
standard quantum group $\Ugh$.
It is a quotient of the mRE algebra
$\mathcal{L}$. The twisted module algebra $\widetilde{\mathcal{B}'}$ is a
$\Ughr$-quantization of $\fun$.
It is a quotient of the algebra
$\widetilde\mathcal{L}'$, which is isomorphic to $\widetilde\mathcal{L}$ by
Proposition~\ref{prop:REtwisted}.
\end{proof}
\subsection{More on RE algebras and twists}
\label{subsect:Twist&REA}
We are going to derive a description of quantum orbits for
an arbitrary quantum group from that corresponding to
the Drinfeld-Jimbo quantum group.
To this end, we need
some facts about Hopf algebras.
As we argued in the previous section (see Proposition~\ref{prop:REtwisted}),
the twist of the (modified) reflection
equation algebra associated with a quantum group
is isomorphic to the (modified) reflection equation algebra
associated with the twisted quantum group.
In this section we obtain a more detailed information about
that isomorphism.
We start with the following auxiliary algebraic assertion.
\begin{lm}\label{4-3}
Let $\mathcal{H}$ be a Hopf algebra with multiplication $m$, comultiplication $\Delta$,
and invertible antipode $\gamma$. Suppose $\mathcal{F}\in \mathcal{H}\otimes \mathcal{H}$ is a twisting cocycle.
Then
$$
m_{23}\circ\gamma_3\Bigl((\Delta\otimes \Delta)(\mathcal{F})(\mathcal{F}\otimes\mathcal{F})(\zeta\otimes 1\otimes \zeta\otimes 1)\Bigr)=
\mathcal{F}_{1}\zeta\otimes 1\otimes \mathcal{F}_{2},
$$
where the argument in the left-hand-side belongs to $\mathcal{H}^{\otimes 4}$.
\end{lm}
\begin{proof}
Applying the cocycle equation (\ref{eq:cocycle}) to
$(\Delta\otimes \Delta)(\mathcal{F})\mathcal{F}_{34}$, we obtain for the left-hand side
the expression
$$
\mathcal{F}_1^{(1)}\mathcal{F}_{1'}^{(1)}\mathcal{F}_{1''}\zeta\otimes \mathcal{F}_1^{(2)}\mathcal{F}_{1'}^{(2)}\mathcal{F}_{2''}\gamma(\mathcal{F}_1^{(3)}\mathcal{F}_{2'}\zeta)\otimes \mathcal{F}_2.
$$
In order to distinguish between different copies of $\mathcal{F}$, the subscripts are marked with dashes.
We apply the cocycle equation to $\mathcal{F}_{1'}^{(1)}\mathcal{F}_{1''}\otimes \mathcal{F}_{1'}^{(2)}\mathcal{F}_{2''}\otimes \mathcal{F}_{2'}$
and obtain
$$
\mathcal{F}_1^{(1)}\mathcal{F}_{1'}\zeta\otimes \mathcal{F}_1^{(2)}\mathcal{F}_{2'}^{(1)}\mathcal{F}_{1''}\gamma(\mathcal{F}_1^{(3)}\mathcal{F}_{2'}^{(2)}\mathcal{F}_{2''}\zeta)\otimes \mathcal{F}_2.
$$
Now the statement immediately follows from the equalities
$
\mathcal{F}_{1}\gamma(\zeta)\gamma(\mathcal{F}_{2})=1
$
and (\ref{eq:cocycle_norm}).
\end{proof}
Suppose that $\mathcal{H}$ is a quasitriangular Hopf algebra and
let $(V,\rho)$ be a finite dimensional representation of $\mathcal{H}$.
We say that a matrix $A\in \End(V)\otimes \mathcal{A}$ is {\em invariant}, if
$h\triangleright A=\rho\bigl(\gamma(h^{(1)})\bigr)A \rho(h^{(2)})$ for
all $h\in \mathcal{H}$, where $h\triangleright A$ denotes the action (\ref{eq:action}).
Let $\mathcal{A}$ and $\widetilde \mathcal{A}$ be the (quadratic) RE algebras
corresponding to
the Hopf algebras $\mathcal{H}$ and $\widetilde \mathcal{H}$, where $\widetilde \mathcal{H}$
is the twist of $\mathcal{H}$ by the cocycle $\mathcal{F}$.
The map (\ref{eq:RETwist}) implements an equivariant isomorphism
of $\widetilde\mathcal{H}$-module algebras $\widetilde \mathcal{A}\to \widetilde \mathcal{A}'$ where $\widetilde \mathcal{A}'$
is the twist of $\mathcal{A}$ by $\mathcal{F}$.
We can also consider $\phi$ as an isomorphism $\widetilde \mathcal{A}\to \mathcal{A}$ of $\mathcal{H}$-modules.
For an invariant matrix $\widetilde A\in \End(V)\otimes\widetilde \mathcal{A}$
we have
\begin{equation}\label{eq:AtA}
(\id \otimes \phi)(\widetilde A)=(\Phi\otimes \id)(A),
\end{equation}
where
$A$ is an
invariant matrix in $\End(V)\otimes\mathcal{A}$.
(For the definition of the operator $\Phi$, see Lemma \ref{lm:LinGen}).
\begin{prop}\label{A*B}
Suppose that $\widetilde A$ and $\widetilde B$ are invariant matrices from $\End(V)\otimes\widetilde \mathcal{A}$.
Then
$(\id\otimes \phi)(\widetilde A\widetilde B)=(\Phi\otimes\id) (AB)$,
where $A$ and $B$ are invariant matrices from $\End(V)\otimes\mathcal{A}$ defined by
(\ref{eq:AtA}).
\end {prop}
\begin{proof}
Follows from Lemma~\ref{4-3}.
\end{proof}
For any invariant matrix $A\in \End(V)\otimes\mathcal{A}$
we define an invariant (hence central) element
$\tr_q(A):=\tr_V\bigl(\mathcal{R}_1\gamma(\mathcal{R}_2)A\bigr)\in \mathcal{A}$.
Note that here we suppress
the representation symbol and we do not care
about the normalizing scalar, contrary to Section~\ref{subsection:mREA}.
\begin{prop}\label{tr-tr0}
Suppose that $\widetilde A$ and $A$ are invariant matrices with coefficients
in $\widetilde\mathcal{A}$ and $\mathcal{A}$, respectively, related by (\ref{eq:AtA}). Then
$$
(\tr_q\otimes \phi)(\widetilde A)=\tr_q(A).$$
\end{prop}
\begin{proof}
Suppressing the representation symbol $\rho$, we find
$$
(\tr_q\otimes \phi)(\widetilde A)=\tr\Bigl(\widetilde \mathcal{R}_1\widetilde\gamma(\widetilde \mathcal{R}_2)\gamma(\mathcal{F}_1\zeta)A\mathcal{F}_2\Bigr),
$$
where $\widetilde \gamma$ is the antipode in $\widetilde \mathcal{H}$, $\widetilde \gamma(x)=\gamma(\zeta^{-1}x\zeta)$.
But
$$
\mathcal{F}_2 \widetilde\mathcal{R}_1\widetilde \gamma(\widetilde \mathcal{R}_2)\gamma(\mathcal{F}_1\zeta)=
\mathcal{F}_2\widetilde\mathcal{R}_1\gamma(\mathcal{F}_1\widetilde \mathcal{R}_2\zeta)
=\mathcal{R}_1\mathcal{F}_1\gamma(\mathcal{R}_2\mathcal{F}_2\zeta)
=\mathcal{R}_1\gamma(\mathcal{R}_2)
$$
because of the equality $\mathcal{F}_1\gamma(\zeta)\gamma(\mathcal{F}_2)=1$.
This proves the assertion.
\end{proof}
Denote by $\{Q_{ij}\}\subset \mathcal{A}$ the RE generators
considered simultaneously as generators for $\widetilde \mathcal{A}'$
(the latter coincides with $\mathcal{A}$ as an $\mathcal{H}$-module
and has the same system of generators as an algebra).
Let $\{\widetilde Q_{ij}\}$ denote the RE generators of $ \widetilde \mathcal{A}$.
The matrices $Q$ and $\widetilde Q$ are invariant and so are
their powers relative to the multiplications in
$\mathcal{A}$ and $\widetilde \mathcal{A}$, respectively.
The isomorphism $\phi$ relates $\widetilde Q$ and $Q$
by the formula (\ref{eq:AtA}).
The following result is an immediate corollary of Propositions~\ref{A*B} and \ref{tr-tr0}.
\begin{prop}\label{tr-tr}
Regard the algebra isomorphism $\phi\colon \widetilde\mathcal{A}\to \widetilde\mathcal{A}'$ as an isomorphism $\widetilde \mathcal{A}\to \mathcal{A}$
of vector spaces. Then
$
(\tr_q\otimes \phi)(\widetilde Q^m)=\tr_q(Q^m).$
\end{prop}
Now let $\widetilde \mathcal{L}$ and $\mathcal{L}$ be the mRE algebras corresponding
to $\widetilde \mathcal{H}$ and $\mathcal{H}$.
Let $\{\widetilde L_{ij}\}\subset \widetilde \mathcal{L}$ and $\{L_{ij}\}\subset \mathcal{L}$
be their mRE generators. Put $\widetilde \mathcal{L}'$ to be the twist of $\mathcal{L}$ by $\mathcal{F}$.
Regard the algebra isomorphism $\phi\colon\widetilde \mathcal{L}\to \widetilde \mathcal{L}'$
extending the isomorphism $\phi\colon\widetilde \mathcal{A}\to \widetilde \mathcal{A}'$
as an isomorphism $\widetilde \mathcal{L}\to \mathcal{L}$ of vector spaces.
\begin{prop}\label{phi_preserves_submodules}
\begin{enumerate}
\item[{\em (a)}]
The map $\phi$ preserves q-traces:
$
(\tr_q\otimes \phi)(\widetilde L^m)=\tr_q(L^m).
$
\item[{\em (b)}]
For any polynomial $P$ in one variable,
$(\id \otimes \phi)\bigl(P(\widetilde L)\bigr)=(\Phi \otimes \id)\bigl(P(L)\bigr)$.
\end{enumerate}
\end{prop}
\begin{proof}
The proof readily follows from
Propositions \ref{A*B} and \ref{tr-tr} and
the fact that the twist extends from the quadratic RE algebras to
the modified RE algebras, by Proposition \ref{prop:REtwisted}.
\end{proof}
\section{$\Ughr$-equivariant quantization of orbits}
\label{sect:DMQuantization}
In this section we give a description of a 2-parameter
quantization of the function algebra $\fun$ starting from an
arbitrary (factorizable) classical r-matrix.
This generalizes the construction
given in \cite{DoninJMudrovAI:ExplicitQuantiz}.
The linear term of
this quantization (or, more precisely, the "quasi-Poisson part" of
it (see formula (\ref{eq:UhEquivariantPb})) is
$\ensuremath{\hbar} s^0_\ensuremath{\lambda}+t\ensuremath{\varkappa}_\ensuremath{\lambda}$ where $\ensuremath{\hbar}$ and $t$ are formal parameters.
Reducing this to a one-parameter quantization corresponding to the
curve $t=\ensuremath{\lambda}_1\left(e^{-2\ensuremath{\hbar}}-1\right)$ on the plane $(\ensuremath{\hbar},t)$, we
get a quantization $\funh$ with the linear term
$\ensuremath{\hbar}\left(s^0_\ensuremath{\lambda}-2\ensuremath{\lambda}_1\ensuremath{\varkappa}_\ensuremath{\lambda}\right)$.
\subsection{Algebraic description of coadjoint orbits}
Organize the generators of the symmetric algebra $\Sg$ in an
$n\times n$ matrix $L=(L_{ij})$, then $\Sg=\mathbb{C}[L_{ij}]$.
The algebra $\fun$ of polynomial functions on $\ola$ is a quotient
of $\mathbb{C}[L_{ij}]$ by two sets of relations. The first set of $n^2$
relations can be written in the matrix form as
\begin{equation}\label{eq:CH}
\left(L-\ensuremath{\lambda}_1\right)\dots\left(L-\ensuremath{\lambda}_l\right)=0,
\end{equation}
where $(x-\ensuremath{\lambda}_1)\dots(x-\ensuremath{\lambda}_l)$ is the minimal polynomial for $\ensuremath{\lambda}$.
To distinguish the orbits corresponding to the same eigenvalues with
different multiplicities, one should impose the following {\em trace
conditions}:
\begin{equation}\label{eq:mults}
\tr\left(L^r\right)=\sum_{j=1}^l n_j\ensuremath{\lambda}_j^r,\quad r=1,\dots,l-1,
\end{equation}
where $\displaystyle\sum_{j=1}^l n_j=n$. It is known that the ideal generated
by (\ref{eq:CH}) and (\ref{eq:mults}) is radical, hence it is
precisely the ideal of functions vanishing on $\ola$.
\subsection{On central characters of the mRE algebra}
\label{sec:DMquantization}
To describe quantum orbits explicitly, we need $q$-analogs of the
polynomials in the right-hand side of (\ref{eq:mults}), i.e. quantum
trace functions. For every $m\in \mathbb{N}$ put
$\displaystyle\hat m:=\frac{1-q^{-2n}}{1-q^{-2}}$.
Fix
$\mbox{\boldmath$\lambda$}:=(\ensuremath{\lambda}_1,\ldots,\ensuremath{\lambda}_l)$ and $\hat{\mathbf{n}}:= (\hat n_1,\ldots,\hat n_l)$
assuming $\ensuremath{\lambda}_i$ pairwise distinct; put also
$\tilde\mbox{\boldmath$\lambda$}=(\tilde\ensuremath{\lambda}_1,\ldots,\tilde\ensuremath{\lambda}_l)$, where
$\displaystyle\widetilde\ensuremath{\lambda}_i=\ensuremath{\lambda}_i-\frac{t}{\omega}$.
Consider the family of
functions $\vartheta_r(\mbox{\boldmath$\lambda$},\hat{\mathbf{n}},q^{-2},t)$, $r=0,\ldots,\infty$, defined
by
\begin{equation}\label{eq:qmults}
\vartheta_r(\mbox{\boldmath$\lambda$}, \hat{\mathbf{n}},q^{-2},t):=\sum_{i=1}^l C_i(\widetilde\mbox{\boldmath$\lambda$} ,
\hat{\mathbf{n}},\omega)\ensuremath{\lambda}_i^r,
\end{equation}
where
\begin{equation}
C_i(\mbox{\boldmath$\lambda$} , \hat{\mathbf{n}},\omega):=\hat n_i\prod_{j\not =i}
\Bigl(1+\omega\frac{\hat n_j \ensuremath{\lambda}_j}{\ensuremath{\lambda}_i-\ensuremath{\lambda}_j}\Bigr).
\end{equation}
(Recall that we use the notation $\omega=1-q^{-2}$).
Although manifestly rational, the functions $\vartheta_r$ are in fact
polynomials in all arguments, see
\cite{DoninJMudrovAI:ExplicitQuantiz}.
In the classical limit $\omega\to 0$, the
function $\vartheta_r$ turns into the classical trace function
$\displaystyle\sum_{i=1}^l n_i\ensuremath{\lambda}_i^r$.
Fix a polynomial $\Prj$ in one variable with coefficients in $\mathbb{C}$.
Consider the quotient of $\mathcal{L}$ by the
$\Ughr$-invariant ideal of relations $\Prj(L)=0$.
Denote by $Z_\Prj$ its subalgebra of invariants.
We call a homomorphism $Z_\Prj\to\mathbb{C}{[\![}\ensuremath{\hbar}{]\!]}[t]$
{\em a character} of $Z_\Prj$.
The meaning of the functions $\vartheta_r(\mbox{\boldmath$\lambda$}, \hat{\mathbf{n}},q^{-2},t)$
is explained by the following proposition.
\begin{prop}\label{center_in_M_p}
The algebra $Z_P$ is a free module over $\mathbb{C}{[\![}\ensuremath{\hbar}{]\!]}[t]$.
The characters of $Z_\Prj$ are given by the formulas
\begin{equation}\label{eq:char}
\chi_{\hat \mathbf{n}}\colon\tau_r\mapsto\vartheta_r(\mbox{\boldmath$\lambda$}, \hat{\mathbf{n}},q^{-2},t),
\quad r=1,\ldots, \infty,
\end{equation}
and define an embedding of $Z_\Prj$ in the direct sum
$\bigoplus_{\hat{\mathbf{n}}}\mathbb{C}{[\![}\ensuremath{\hbar}{]\!]}[t]$.
This embedding becomes an isomorphism
over $\mathbb{C}{[\![} \hbar,t{]\!]}$.
\end{prop}
This proposition is proved in \cite{DoninJMudrovAI:ExplicitQuantiz}
for the case of standard quantum group.
One can prove it for
the general quantum group $\Ughr$ using similar arguments.
\subsection{The DM quantization of coadjoint orbits}
Now we are in possession of all ingredients for construction of
quantum orbits. We will work over the ring of scalars being $\mathbb{C}{[\![} \hbar,t{]\!]}$.
\begin{thm}\label{th:DMQuantization}
Let $\Ughr$ be {\em any} quasitriangular quantization of ${\U}(\g)$ along
a factorizable Lie bialgebra $\mathfrak g$. Let
let $\mathbb{C}_{\hbar,t}[{\mathfrak{g}}^*]$ be the corresponding mRE algebra generated
by $n^2$ entries of the matrix $L$. Then the quotient of
$\mathbb{C}_{\hbar,t}[{\mathfrak{g}}^*]$ by the ideal of relations
\begin{equation}\label{eq:MinPolynomCondition}
(L-\ensuremath{\lambda}_1)\ldots(L-\ensuremath{\lambda}_l)=0,
\end{equation}
\begin{equation}\label{eq:qTraceCondition}
\tr_q(L^r)=\vartheta_r(\mbox{\boldmath$\lambda$}, \hat{\mathbf{n}}, q^{-2},t,), \quad
m=1,\ldots,l-1,
\end{equation}
is a $\Ughr$-equivariant quantization of the orbit of matrices with
eigenvalues $\mbox{\boldmath$\lambda$}$ of multiplicities $\mathbf{n}$.
\end{thm}
\begin{proof}
The description of the quantized ideal of the orbit can be deduced
from Corollary \ref{cor:twist} and Proposition \ref{center_in_M_p} using
deformation arguments.
We will give an alternative proof based on the results
of Section~\ref{subsect:Twist&REA}, deriving the quantized ideal
of the orbit from the Drinfeld-Jimbo case.
Let $\widetilde \mathcal{L}$ and $\mathcal{L}$ denote the mRE algebras corresponding
to $\Ughr$ and $\Ugh$, respectively. The quantum group
$\Ughr$ is the twist of $\Ugh$ by a cocycle $\mathcal{F}$. Denote by $\widetilde \mathcal{L}'$
the corresponding twist of $\mathcal{L}$; that is a module algebra over $\Ughr$.
By Proposition \ref{prop:REtwisted}, there is an equivariant
isomorphism of algebras $\phi\colon \widetilde \mathcal{L}\to \widetilde \mathcal{L}'$.
The map $\phi$ is determined by formula (\ref{Phi}), where
the matrices $\widetilde K$ and $K$ should be replaced by
$\widetilde L$ and $L$, respectively.
Denote by $\mathcal{B}$ the quantization of the orbit $O_\ensuremath{\lambda}$ which is
equivariant under $\Ugh$.
It is a quotient of $\mathcal{L}$ by the ideal $\mathcal{J}$ of relations
(\ref{eq:MinPolynomCondition})
and (\ref{eq:qTraceCondition}).
The twist $\widetilde \mathcal{B}'$ of the algebra $\mathcal{B}$ by $\mathcal{F}$ is a quantization
of $O_\ensuremath{\lambda}$ which is equivariant under $\Ughr$.
It is a quotient of $\widetilde \mathcal{L}'$ by the ideal $\widetilde \mathcal{J}'$ which
coincides with $\mathcal{J}$ as a vector space.
Moreover, $\widetilde \mathcal{J}'$ is generated by the same submodule as $\mathcal{J}$ in $\mathcal{L}$.
In our case that submodule is spanned by the elements
of the matrix $P(L)$ and the kernel of the central character of $\mathcal{L}$. Consider the
equivariant isomorphism $\phi^{-1}\colon \widetilde \mathcal{L}'\to \widetilde \mathcal{L}$.
By Proposition \ref{phi_preserves_submodules}, it sends $\Span \bigl(P(L)_{ij}\bigr)$
to $\Span \bigl(P(\widetilde L)_{ij}\bigr)$ and preserves the q-traces.
This proves the theorem.
\end{proof}
\begin{rem}
In \cite{MudrovA:QConjClasses}, a description similar to
Theorem \ref{th:DMQuantization} of semisimple quantum conjugacy classes
of the Drinfeld-Jimbo matrix quantum groups is given.
Using the same arguments as in the proof of Theorem \ref{th:DMQuantization}
and the results of Section \ref{subsect:Twist&REA},
the quantization of \cite{MudrovA:QConjClasses} extends to arbitrary
quantum groups of the classical series.
\end{rem}
\section{Quantization of orbit bundles in $\mathfrak{gl}_n^*(\mathbb{C})$}
\label{sect:QnOfBundles}
In this section we prove that all orbit bundles admit
$\Ughr$-equivariant quantization and give the explicit construction.
We start with the following
algebraic lemma \cite{DoninJMudrovAI:MethodQuantChar} which we prove
here for the sake of completeness.
\begin{lm}\label{lm:AlgLemma}
Let $Q(x)$ be a polynomial over a field $F$ of zero characteristic,
$\alpha, \beta$ some elements of $F$, and $L,S$ elements of an
associative algebra with unit over $F$ satisfying the following
conditions:
\begin{enumerate}
\item[\em (a)]
$[SLS,L]=0$,
\item[\em (b)]
$S^2=\alpha S+1$,
\item[\em (c)]
$LQ(L)=\beta L$.
\end{enumerate}
Then one has $[SQ(L)S,Q(L)]=0$.
\end{lm}
\begin{rem}
The algebra generated by $S$ and $L$ subject to conditions (a)--(c) is a special case of cyclotomic affine Hecke algebra of rank 1.
\end{rem}
\begin{proof}
Prove, using the induction on $m\geq 1$, that $[SL^mS,Q(L)]=0$. The
induction base, $m=1$, holds true for one checks readily that (a)
implies $[SLS,L^k]=0$ for any $k$. Now, suppose $[SL^mS,Q(L)]=0$,
then one has using (b):
\begin{gather}
[SL^{m+1}S,Q(L)]=[SL1L^mS,Q(L)]=[SL(S^2-\alpha S)L^mS,Q(L)]=\nonumber \\
=[SLS^2L^m S,Q(L)]-\alpha[SLSL^mS,Q(L)].\label{eq:One}
\end{gather}
According to the induction assumption, both $SLS$ and $SL^mS$ commute
with $Q(L)$, thus
$$
[SLS^2L^mS,Q(L)]=[(SLS)(SL^mS),Q(L)]=0.
$$
The last term in (\ref{eq:One}) is treated as follows:
\begin{eqnarray*}
[SLSL^mS,Q(L)]&=&
SL\bigl(SL^mSQ(L)\bigr)-\bigl(Q(L)SLS\bigr)L^mS\\
&=&SLQ(L)SL^mS-SLSL^mQ(L)S \\
&=&\beta SLSL^mS-\beta SLSL^mS=0,
\end{eqnarray*}
where the induction assumption and (c) were used.
\end{proof}
Recall from see Proposition~\ref{prop:P} that any orbit map is determined
by a polynomial $\Prj$ in one variable.
\begin{thm}\label{th:Main}
Fix a factorizable quantum group $\Ughr$, where $\mathfrak g=\mathfrak{gl}_n(\mathbb{C})$.
Let $\ola$
and $\omu$ be two orbits in $\mathfrak g$ satisfying the condintions of
Theorem~\ref{th:MainPoiss}, and denote by
$\mathbb{C}_\ensuremath{\hbar}[\ola]=\mathbb{C}_\ensuremath{\hbar}[L_{\ensuremath{\lambda},ij}]$ and
$\funh[\omu]=\mathbb{C}_\ensuremath{\hbar}[L_{\mu,ij}]$ their quantizations from
Theorem~\ref{th:DMQuantization} with
$t=\ensuremath{\lambda}_1\left(e^{-2\ensuremath{\hbar}}-1\right)$.
Then the assignment
$L_\mu\mapsto \Prj(L_\ensuremath{\lambda})$, where the polynomial $\Prj$ is
given by (\ref{eq:DefOfP}), is a $\Ughr$-equivariant quantization of
the orbit bundle $\ola\to\omu$ determined by $P$.
\end{thm}
\begin{proof}
Denote by $\prj^*$ the algebra monomorphism $\fun[\omu]\to\fun$
corresponding to the map $\prj$.
Both $L_\ensuremath{\lambda}$ and $L_\mu$ are
subject to the relations (\ref{eq:CH}) and (\ref{eq:mults}).
The algebra homomorphism $\prj^*\colon \fun[\omu]\to\fun$ is determined
by the correspondence $L_\mu\mapsto P(L_\ensuremath{\lambda})$.
We need to prove that
the same correspondence defines a $\mathbb{C}{[\![}\ensuremath{\hbar}{]\!]}$-algebra monomorphism
$\funh[\omu]\to\funh$, i.e. that the matrix $\prj(L_\ensuremath{\lambda})$
satisfies the same relations as the matrix $L_\mu$.
1. Check the relation: $[SP(L_\ensuremath{\lambda})
S,P(L_\ensuremath{\lambda})]=\mu_1(q-q^{-1})[S,P(L_\ensuremath{\lambda})]$.
It can be written in the form:
\begin{equation}\label{eq:REforP}
[S\left(P(L_\ensuremath{\lambda})-\mu_1\right) S,P(L_\ensuremath{\lambda})-\mu_1]=0,
\end{equation}
as $S$ is a Hecke matrix.
It is easy to check that $(L_\ensuremath{\lambda}-\ensuremath{\lambda}_1)\left( P(L_\ensuremath{\lambda})-\mu_1
\right)=(\mu_2-\mu_1)(L_\ensuremath{\lambda}-\ensuremath{\lambda}_1)$. Now set $\beta:=\mu_2-\mu_1$,
$L:=L_\ensuremath{\lambda}-\ensuremath{\lambda}_1$, $Q(x):=P(x+\ensuremath{\lambda}_1)-\mu_1$, then
(\ref{eq:REforP}) follows from Lemma~\ref{lm:AlgLemma}.
2. Check the relation:
\begin{equation}\label{eq:CHforP}
(P(L_\ensuremath{\lambda})-\mu_1)(P(L_\ensuremath{\lambda})-\mu_2)=0.
\end{equation}
Substituting (\ref{eq:DefOfP}) into the l.h.s. of (\ref{eq:CHforP}),
one gets
\begin{equation}\label{eq:CheckingHC}
\prod_{i=2}^l(L_\ensuremath{\lambda}-\ensuremath{\lambda}_i) \left(
\prod_{i=2}^l(L_\ensuremath{\lambda}-\ensuremath{\lambda}_i)-\prod_{i=2}^l(\ensuremath{\lambda}_1-\ensuremath{\lambda}_i) \right)
\end{equation}
up to a constant multiple. The expression in the big brackets is
divisible by $L_\ensuremath{\lambda}-\ensuremath{\lambda}_1$.
Indeed, for any polynomial $f(x)$,
the polynomial in two variables $F(x,y):=f(x)-f(y)$ is divisible by
$x-y$.
This implies that (\ref{eq:CheckingHC}) is divisible by the
minimal polynomial of $\ensuremath{\lambda}$, so it is equal to zero.
3. In order to check the q-Trace Condition,
\begin{equation}\label{eq:TraceCondForP}
\tr_q P(L_\ensuremath{\lambda})=\tr_q(L_\mu),
\end{equation}
we put $\mbox{\boldmath$\nu$}=\hat{\mathbf{n}}=(\hat n_1,\ldots,\hat n_l)$ and $\omega:=1-q^{-2}$
in the functions $C_i(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega)$, see Appendix, formula
(\ref{eq:Cs}). (As above, $\hat n_i =
\frac{1-q^{-2n_i}}{1-q^{-2}}$).
Replacing $\mbox{\boldmath$\lambda$}$, $\mbox{\boldmath$\mu$}$, $L_\ensuremath{\lambda}$, $L_\mu$ and $\Prj(x)$
by $(0, \ensuremath{\lambda}_2-\ensuremath{\lambda}_1,\ldots,\ensuremath{\lambda}_l-\ensuremath{\lambda}_1)$, $(0,\mu_2-\mu_1)$,
$L_\ensuremath{\lambda}-\ensuremath{\lambda}_1$, $L_\mu-\mu_1$ and $\Prj(x+\ensuremath{\lambda}_1)-\mu_1$
respectively, one reduces the problem to the case $\ensuremath{\lambda}_1=\mu_1=0$.
So, it suffices to prove that the condition (\ref{eq:TraceCondForP})
is satisfied when $\ensuremath{\lambda}_1=0$ and $\Prj(0)=0$.
By assumption, $\mu_1=\Prj(0)=0$, therefore one has:
\begin{equation}\label{eq:TrMu}
\qtr L_\mu=C_2(\mbox{\boldmath$\mu$},\hat{\mathbf{m}},q)\mu_2=\hat n'\mu_2,
\end{equation}
where $\mathbf{m}=(n_1,n')$ and $\displaystyle n':=\sum_{i=2}^l n_i$. On the other
hand,
\begin{equation}\label{eq:TrLa}
\qtr\bigl(P(L_\ensuremath{\lambda})\bigr)= \sum_{i=1}^l P(\ensuremath{\lambda}_i)C_i(\mbox{\boldmath$\lambda$},\hat{\mathbf{n}},q)=
\mu_2\sum_{i=2}^l C_j(\mbox{\boldmath$\lambda$},\hat{\mathbf{n}},q),
\end{equation}
because $\Prj(\ensuremath{\lambda}_1)=\Prj(0)=0$. By Corollary~\ref{prop:Reductio}
(see Appendix),
\begin{equation}\label{eq:SumOfCs}
\displaystyle\sum_{i=2}^l C_i(\mbox{\boldmath$\lambda$}, \hat{\mathbf{n}}, \omega)=\hat n',
\end{equation}
since $1-\omega\hat n_i=q^{-2n_i}$. Substituting (\ref{eq:SumOfCs})
into (\ref{eq:TrLa}), one concludes that the latter is equal to
(\ref{eq:TrMu}).
\end{proof}
\section*{Appendix}
In this section, we study some properties of the coefficients $C_i$
in (\ref{eq:qmults}), which were announced without proof in
\cite{DoninJMudrovAI:GeneralizedVermaMod}.
For $1\leqslant i\leqslant l$,
define a function of $2l+1$ variables $\mbox{\boldmath$\lambda$}=(\ensuremath{\lambda}_1,\ldots,\ensuremath{\lambda}_l)$,
$\mbox{\boldmath$\nu$}=(\nu_1,\ldots,\nu_l)$ and $\omega$:
\begin{equation}\label{eq:Cs}
C_i(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega):= \nu_i\prod_{\substack{1\leqslant j\leqslant l\\j\not
=i}} \left(1+\omega\frac{\nu_{j}\ensuremath{\lambda}_{j}}{\ensuremath{\lambda}_i-\ensuremath{\lambda}_j}\right),
\end{equation}
and also another function of the same variables:
\begin{equation*}
S(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega)=\sum_{i=1}^l C_i(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega).
\end{equation*}
These functions were introduced in
\cite{DoninJMudrovAI:ExplicitQuantiz} and
\cite{DoninJMudrovAI:GeneralizedVermaMod}. Our goal is to prove
Proposition~\ref{prop:MainLemmaOnC} below.
Obviously, $S(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega)$ is stable under simultaneous
permutations of the entries of $\mbox{\boldmath$\lambda$}$ and the entries of $\mbox{\boldmath$\nu$}$.
In fact, a stronger statement is true:
\begin{lm}
$S(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega)$ is a symmetric function of $\mbox{\boldmath$\lambda$}$.
\end{lm}
\begin{proof}
It suffices to show that $S(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega)$ is stable under the
transposition $\ensuremath{\lambda}_1\leftrightarrow \ensuremath{\lambda}_2$.
First, opening the
brackets in (\ref{eq:Cs}) one gets
\begin{equation}\label{eq:CsExpanded}
C_i(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega)= \nu_i+\nu_i\sum_{k=1}^{l-1}\omega^ k
\sum_{j_1<\ldots<j_ k} \frac{\nu_{j_1}\ensuremath{\lambda}_{j_1}}{\ensuremath{\lambda}_i-\ensuremath{\lambda}_{j_1}}
\ldots \frac{\nu_{j_ k}\ensuremath{\lambda}_{j_ k}}{\ensuremath{\lambda}_i-\ensuremath{\lambda}_{j_ k}}.
\end{equation}
In this form, the functions $C_i$ were introduced in \cite{DoninJMudrovAI:ExplicitQuantiz}.
The multiplicative form (\ref{eq:Cs}) appeared in
\cite{GurewitzSaponov:GeomNCOrbits}.
All the terms in (\ref{eq:CsExpanded}) containing $\ensuremath{\lambda}_1$ and
$\ensuremath{\lambda}_2$ can be arranged into sums of the following three forms:
\begin{gather*}
\nu_j\frac{\nu_1\ensuremath{\lambda}_1}{\ensuremath{\lambda}_j-\ensuremath{\lambda}_1} \frac{\nu_2\ensuremath{\lambda}_2}{\ensuremath{\lambda}_j-\ensuremath{\lambda}_2}f
=\nu_j\nu_1\nu_2\frac{\ensuremath{\lambda}_1}{\ensuremath{\lambda}_j-\ensuremath{\lambda}_1}
\frac{\ensuremath{\lambda}_2}{\ensuremath{\lambda}_j-\ensuremath{\lambda}_2}f, \\
\ \\
\nu_1\frac{\nu_2\ensuremath{\lambda}_2}{\ensuremath{\lambda}_2-\ensuremath{\lambda}_1}f+
\nu_2\frac{\nu_1\ensuremath{\lambda}_1}{\ensuremath{\lambda}_1-\ensuremath{\lambda}_2}f=
\nu_1\nu_2f, \\
\ \\
\nu_j\frac{\nu_1\ensuremath{\lambda}_1}{\ensuremath{\lambda}_j-\ensuremath{\lambda}_1}f+
\nu_j\frac{\nu_2\ensuremath{\lambda}_2}{\ensuremath{\lambda}_j-\ensuremath{\lambda}_2}f+
\nu_1\frac{\nu_j\ensuremath{\lambda}_j}{\ensuremath{\lambda}_1-\ensuremath{\lambda}_j}f+
\nu_2\frac{\nu_j\ensuremath{\lambda}_j}{\ensuremath{\lambda}_2-\ensuremath{\lambda}_j}f
=-(\nu_j\nu_1+\nu_j\nu_2)f,
\end{gather*}
with $j\not=1,2$, and $f$ being independent on $\ensuremath{\lambda}_1$ and $\ensuremath{\lambda}_2$.
It is seen that the expressions in the right hand sides are stable
under the transposition $\ensuremath{\lambda}_1\leftrightarrow \ensuremath{\lambda}_2$.
\end{proof}
\begin{prop}\label{prop:MainLemmaOnC}
$\displaystyle\omega S(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega)=1-\prod_{i=1}^l(1-\omega\nu_i)$.
\end{prop}
\begin{proof}
Prove first that $S(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega)$ does not actually depend on
$\mbox{\boldmath$\lambda$}$. Fix $\mbox{\boldmath$\nu$}$ and $\omega$, and consider $S(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega)$ as a
rational function of $\mbox{\boldmath$\lambda$}$ only. This function is homogeneous of
degree zero.
Reducing $S(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega)$ to the common denominator
$\prod_{i<j}(\ensuremath{\lambda}_i-\ensuremath{\lambda}_j)$ we obtain a ratio of two homogeneous
polynomials of the same degree.
Since $S(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega)$ is a
symmetric function of $\mbox{\boldmath$\lambda$}$, the numerator is divisible by
$\prod_{i<j}(\ensuremath{\lambda}_i-\ensuremath{\lambda}_j)$ because the ring of polynomials is a
unique factorization domain.
Since the numerator of
$S(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega)$ has the same degree as the denominator,
$S(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega)$ is independent on $\mbox{\boldmath$\lambda$}$.
Now put $\ensuremath{\lambda}_l=0$, then it follows from (\ref{eq:Cs}) that
$S(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega)=S(\mbox{\boldmath$\lambda$}',\mbox{\boldmath$\nu$}',\omega)+
\nu_l\prod_{i=1}^{l-1}(1-\omega\nu_i)$, where
$\mbox{\boldmath$\lambda$}'=(\ensuremath{\lambda}_1,\ldots,\ensuremath{\lambda}_{l-1})$ and
$\mbox{\boldmath$\nu$}'=(\nu_1,\ldots,\nu_{l-1})$. Finally, one applies the induction
on $l$.
\end{proof}
\begin{cor}\label{prop:Reductio}
\begin{enumerate}
\item[{\em (a)}]
If $\ensuremath{\lambda}_1=0$ then $\displaystyle\omega\sum_{i=2}^l C_i(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega)
=1-\prod_{i=2}^l (1-\omega\nu_i)$.
\item[{\em (b)}]
Denote $\mbox{\boldmath$\lambda$}'=(\ensuremath{\lambda}_2,\ldots,\ensuremath{\lambda}_l)$,
$\mbox{\boldmath$\nu$}'=(\nu_2,\ldots,\nu_l)\in\mathbb{C}^{l-1}$, and suppose that
$\ensuremath{\lambda}_1=0$.
Then $\displaystyle\sum_{i=2}^l C_i(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega)
=\sum_{i=2}^l C_i(\mbox{\boldmath$\lambda$}',\mbox{\boldmath$\nu$}',\omega)$.
\item[{\em (c)}]
One has $\displaystyle\sum_{i=1}^l C_i(\mbox{\boldmath$\lambda$}, \hat{\mathbf{n}}, \omega)=\hat n$.
\end{enumerate}
\end{cor}
\begin{proof}
(a) Denote $S'(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega)=\displaystyle\omega\sum_{i=2}^l
C_i(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega)$.
Then
\begin{gather*}
\omega S'(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega)=
\omega\bigl(S(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega)-C_1(\mbox{\boldmath$\lambda$},\mbox{\boldmath$\nu$},\omega)\bigr)=\\
=1-\prod_{i=1}^l(1-\omega\nu_i)-\omega\nu_1\prod_{i=2}^l
(1-\omega\nu_i)=1-\prod_{i=2}^l (1-\omega\nu_i).
\end{gather*}
(b) Obvious.
(c) Note that $1-\omega\hat n_i=q^{-2n_i}=e^{-2n_i\ensuremath{\hbar}}$, recall that
$\displaystyle n=\sum_{i=1}^l n_i$ and use
Proposition~\ref{prop:MainLemmaOnC}.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,588 |
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> [ DEFERRABLE | NOT DEFERRABLE ] [ INITIALLY DEFERRED | INITIALLY IMMEDIATE ]
VALIDATE CONSTRAINT <TT
CLASS="REPLACEABLE"
><I
>constraint_name</I
></TT
>
DROP CONSTRAINT [ IF EXISTS ] <TT
CLASS="REPLACEABLE"
><I
>constraint_name</I
></TT
> [ RESTRICT | CASCADE ]
DISABLE TRIGGER [ <TT
CLASS="REPLACEABLE"
><I
>trigger_name</I
></TT
> | ALL | USER ]
ENABLE TRIGGER [ <TT
CLASS="REPLACEABLE"
><I
>trigger_name</I
></TT
> | ALL | USER ]
ENABLE REPLICA TRIGGER <TT
CLASS="REPLACEABLE"
><I
>trigger_name</I
></TT
>
ENABLE ALWAYS TRIGGER <TT
CLASS="REPLACEABLE"
><I
>trigger_name</I
></TT
>
DISABLE RULE <TT
CLASS="REPLACEABLE"
><I
>rewrite_rule_name</I
></TT
>
ENABLE RULE <TT
CLASS="REPLACEABLE"
><I
>rewrite_rule_name</I
></TT
>
ENABLE REPLICA RULE <TT
CLASS="REPLACEABLE"
><I
>rewrite_rule_name</I
></TT
>
ENABLE ALWAYS RULE <TT
CLASS="REPLACEABLE"
><I
>rewrite_rule_name</I
></TT
>
DISABLE ROW LEVEL SECURITY
ENABLE ROW LEVEL SECURITY
FORCE ROW LEVEL SECURITY
NO FORCE ROW LEVEL SECURITY
CLUSTER ON <TT
CLASS="REPLACEABLE"
><I
>index_name</I
></TT
>
SET WITHOUT CLUSTER
SET WITH OIDS
SET WITHOUT OIDS
SET TABLESPACE <TT
CLASS="REPLACEABLE"
><I
>new_tablespace</I
></TT
>
SET { LOGGED | UNLOGGED }
SET ( <TT
CLASS="REPLACEABLE"
><I
>storage_parameter</I
></TT
> = <TT
CLASS="REPLACEABLE"
><I
>value</I
></TT
> [, ... ] )
RESET ( <TT
CLASS="REPLACEABLE"
><I
>storage_parameter</I
></TT
> [, ... ] )
INHERIT <TT
CLASS="REPLACEABLE"
><I
>parent_table</I
></TT
>
NO INHERIT <TT
CLASS="REPLACEABLE"
><I
>parent_table</I
></TT
>
OF <TT
CLASS="REPLACEABLE"
><I
>type_name</I
></TT
>
NOT OF
OWNER TO { <TT
CLASS="REPLACEABLE"
><I
>new_owner</I
></TT
> | CURRENT_USER | SESSION_USER }
REPLICA IDENTITY { DEFAULT | USING INDEX <TT
CLASS="REPLACEABLE"
><I
>index_name</I
></TT
> | FULL | NOTHING }
<SPAN
CLASS="phrase"
><SPAN
CLASS="PHRASE"
>and <TT
CLASS="REPLACEABLE"
><I
>table_constraint_using_index</I
></TT
> is:</SPAN
></SPAN
>
[ CONSTRAINT <TT
CLASS="REPLACEABLE"
><I
>constraint_name</I
></TT
> ]
{ UNIQUE | PRIMARY KEY } USING INDEX <TT
CLASS="REPLACEABLE"
><I
>index_name</I
></TT
>
[ DEFERRABLE | NOT DEFERRABLE ] [ INITIALLY DEFERRED | INITIALLY IMMEDIATE ]</PRE
></DIV
><DIV
CLASS="REFSECT1"
><A
NAME="AEN75341"
></A
><H2
>Description</H2
><P
> <TT
CLASS="COMMAND"
>ALTER TABLE</TT
> changes the definition of an existing table.
There are several subforms described below. Note that the lock level required
may differ for each subform. An <TT
CLASS="LITERAL"
>ACCESS EXCLUSIVE</TT
> lock is held
unless explicitly noted. When multiple subcommands are listed, the lock
held will be the strictest one required from any subcommand.
<P
></P
></P><DIV
CLASS="VARIABLELIST"
><DL
><DT
><TT
CLASS="LITERAL"
>ADD COLUMN [ IF NOT EXISTS ]</TT
></DT
><DD
><P
> This form adds a new column to the table, using the same syntax as
<A
HREF="sql-createtable.html"
>CREATE TABLE</A
>. If <TT
CLASS="LITERAL"
>IF NOT EXISTS</TT
>
is specified and a column already exists with this name,
no error is thrown.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>DROP COLUMN [ IF EXISTS ]</TT
></DT
><DD
><P
> This form drops a column from a table. Indexes and
table constraints involving the column will be automatically
dropped as well. You will need to say <TT
CLASS="LITERAL"
>CASCADE</TT
> if
anything outside the table depends on the column, for example,
foreign key references or views.
If <TT
CLASS="LITERAL"
>IF EXISTS</TT
> is specified and the column
does not exist, no error is thrown. In this case a notice
is issued instead.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>SET DATA TYPE</TT
></DT
><DD
><P
> This form changes the type of a column of a table. Indexes and
simple table constraints involving the column will be automatically
converted to use the new column type by reparsing the originally
supplied expression.
The optional <TT
CLASS="LITERAL"
>COLLATE</TT
> clause specifies a collation
for the new column; if omitted, the collation is the default for the
new column type.
The optional <TT
CLASS="LITERAL"
>USING</TT
>
clause specifies how to compute the new column value from the old;
if omitted, the default conversion is the same as an assignment
cast from old data type to new. A <TT
CLASS="LITERAL"
>USING</TT
>
clause must be provided if there is no implicit or assignment
cast from old to new type.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>SET</TT
>/<TT
CLASS="LITERAL"
>DROP DEFAULT</TT
></DT
><DD
><P
> These forms set or remove the default value for a column.
Default values only apply in subsequent <TT
CLASS="COMMAND"
>INSERT</TT
>
or <TT
CLASS="COMMAND"
>UPDATE</TT
> commands; they do not cause rows already in the
table to change.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>SET</TT
>/<TT
CLASS="LITERAL"
>DROP NOT NULL</TT
></DT
><DD
><P
> These forms change whether a column is marked to allow null
values or to reject null values. You can only use <TT
CLASS="LITERAL"
>SET
NOT NULL</TT
> when the column contains no null values.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>SET STATISTICS</TT
></DT
><DD
><P
> This form
sets the per-column statistics-gathering target for subsequent
<A
HREF="sql-analyze.html"
>ANALYZE</A
> operations.
The target can be set in the range 0 to 10000; alternatively, set it
to -1 to revert to using the system default statistics
target (<A
HREF="runtime-config-query.html#GUC-DEFAULT-STATISTICS-TARGET"
>default_statistics_target</A
>).
For more information on the use of statistics by the
<SPAN
CLASS="PRODUCTNAME"
>PostgreSQL</SPAN
> query planner, refer to
<A
HREF="planner-stats.html"
>Section 14.2</A
>.
</P
><P
> <TT
CLASS="LITERAL"
>SET STATISTICS</TT
> acquires a
<TT
CLASS="LITERAL"
>SHARE UPDATE EXCLUSIVE</TT
> lock.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>SET ( <TT
CLASS="REPLACEABLE"
><I
>attribute_option</I
></TT
> = <TT
CLASS="REPLACEABLE"
><I
>value</I
></TT
> [, ... ] )</TT
><BR><TT
CLASS="LITERAL"
>RESET ( <TT
CLASS="REPLACEABLE"
><I
>attribute_option</I
></TT
> [, ... ] )</TT
></DT
><DD
><P
> This form sets or resets per-attribute options. Currently, the only
defined per-attribute options are <TT
CLASS="LITERAL"
>n_distinct</TT
> and
<TT
CLASS="LITERAL"
>n_distinct_inherited</TT
>, which override the
number-of-distinct-values estimates made by subsequent
<A
HREF="sql-analyze.html"
>ANALYZE</A
>
operations. <TT
CLASS="LITERAL"
>n_distinct</TT
> affects the statistics for the table
itself, while <TT
CLASS="LITERAL"
>n_distinct_inherited</TT
> affects the statistics
gathered for the table plus its inheritance children. When set to a
positive value, <TT
CLASS="COMMAND"
>ANALYZE</TT
> will assume that the column contains
exactly the specified number of distinct nonnull values. When set to a
negative value, which must be greater
than or equal to -1, <TT
CLASS="COMMAND"
>ANALYZE</TT
> will assume that the number of
distinct nonnull values in the column is linear in the size of the
table; the exact count is to be computed by multiplying the estimated
table size by the absolute value of the given number. For example,
a value of -1 implies that all values in the column are distinct, while
a value of -0.5 implies that each value appears twice on the average.
This can be useful when the size of the table changes over time, since
the multiplication by the number of rows in the table is not performed
until query planning time. Specify a value of 0 to revert to estimating
the number of distinct values normally. For more information on the use
of statistics by the <SPAN
CLASS="PRODUCTNAME"
>PostgreSQL</SPAN
> query
planner, refer to <A
HREF="planner-stats.html"
>Section 14.2</A
>.
</P
><P
> Changing per-attribute options acquires a
<TT
CLASS="LITERAL"
>SHARE UPDATE EXCLUSIVE</TT
> lock.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>SET STORAGE</TT
>
</DT
><DD
><P
> This form sets the storage mode for a column. This controls whether this
column is held inline or in a secondary <ACRONYM
CLASS="ACRONYM"
>TOAST</ACRONYM
> table, and
whether the data
should be compressed or not. <TT
CLASS="LITERAL"
>PLAIN</TT
> must be used
for fixed-length values such as <TT
CLASS="TYPE"
>integer</TT
> and is
inline, uncompressed. <TT
CLASS="LITERAL"
>MAIN</TT
> is for inline,
compressible data. <TT
CLASS="LITERAL"
>EXTERNAL</TT
> is for external,
uncompressed data, and <TT
CLASS="LITERAL"
>EXTENDED</TT
> is for external,
compressed data. <TT
CLASS="LITERAL"
>EXTENDED</TT
> is the default for most
data types that support non-<TT
CLASS="LITERAL"
>PLAIN</TT
> storage.
Use of <TT
CLASS="LITERAL"
>EXTERNAL</TT
> will make substring operations on
very large <TT
CLASS="TYPE"
>text</TT
> and <TT
CLASS="TYPE"
>bytea</TT
> values run faster,
at the penalty of increased storage space. Note that
<TT
CLASS="LITERAL"
>SET STORAGE</TT
> doesn't itself change anything in the table,
it just sets the strategy to be pursued during future table updates.
See <A
HREF="storage-toast.html"
>Section 65.2</A
> for more information.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>ADD <TT
CLASS="REPLACEABLE"
><I
>table_constraint</I
></TT
> [ NOT VALID ]</TT
></DT
><DD
><P
> This form adds a new constraint to a table using the same syntax as
<A
HREF="sql-createtable.html"
>CREATE TABLE</A
>, plus the option <TT
CLASS="LITERAL"
>NOT
VALID</TT
>, which is currently only allowed for foreign key
and CHECK constraints.
If the constraint is marked <TT
CLASS="LITERAL"
>NOT VALID</TT
>, the
potentially-lengthy initial check to verify that all rows in the table
satisfy the constraint is skipped. The constraint will still be
enforced against subsequent inserts or updates (that is, they'll fail
unless there is a matching row in the referenced table, in the case
of foreign keys; and they'll fail unless the new row matches the
specified check constraints). But the
database will not assume that the constraint holds for all rows in
the table, until it is validated by using the <TT
CLASS="LITERAL"
>VALIDATE
CONSTRAINT</TT
> option.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>ADD <TT
CLASS="REPLACEABLE"
><I
>table_constraint_using_index</I
></TT
></TT
></DT
><DD
><P
> This form adds a new <TT
CLASS="LITERAL"
>PRIMARY KEY</TT
> or <TT
CLASS="LITERAL"
>UNIQUE</TT
>
constraint to a table based on an existing unique index. All the
columns of the index will be included in the constraint.
</P
><P
> The index cannot have expression columns nor be a partial index.
Also, it must be a b-tree index with default sort ordering. These
restrictions ensure that the index is equivalent to one that would be
built by a regular <TT
CLASS="LITERAL"
>ADD PRIMARY KEY</TT
> or <TT
CLASS="LITERAL"
>ADD UNIQUE</TT
>
command.
</P
><P
> If <TT
CLASS="LITERAL"
>PRIMARY KEY</TT
> is specified, and the index's columns are not
already marked <TT
CLASS="LITERAL"
>NOT NULL</TT
>, then this command will attempt to
do <TT
CLASS="LITERAL"
>ALTER COLUMN SET NOT NULL</TT
> against each such column.
That requires a full table scan to verify the column(s) contain no
nulls. In all other cases, this is a fast operation.
</P
><P
> If a constraint name is provided then the index will be renamed to match
the constraint name. Otherwise the constraint will be named the same as
the index.
</P
><P
> After this command is executed, the index is <SPAN
CLASS="QUOTE"
>"owned"</SPAN
> by the
constraint, in the same way as if the index had been built by
a regular <TT
CLASS="LITERAL"
>ADD PRIMARY KEY</TT
> or <TT
CLASS="LITERAL"
>ADD UNIQUE</TT
>
command. In particular, dropping the constraint will make the index
disappear too.
</P
><DIV
CLASS="NOTE"
><BLOCKQUOTE
CLASS="NOTE"
><P
><B
>Note: </B
> Adding a constraint using an existing index can be helpful in
situations where a new constraint needs to be added without blocking
table updates for a long time. To do that, create the index using
<TT
CLASS="COMMAND"
>CREATE INDEX CONCURRENTLY</TT
>, and then install it as an
official constraint using this syntax. See the example below.
</P
></BLOCKQUOTE
></DIV
></DD
><DT
><TT
CLASS="LITERAL"
>ALTER CONSTRAINT</TT
></DT
><DD
><P
> This form alters the attributes of a constraint that was previously
created. Currently only foreign key constraints may be altered.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>VALIDATE CONSTRAINT</TT
></DT
><DD
><P
> This form validates a foreign key or check constraint that was previously created
as <TT
CLASS="LITERAL"
>NOT VALID</TT
>, by scanning the table to ensure there
are no rows for which the constraint is not satisfied.
Nothing happens if the constraint is already marked valid.
</P
><P
> Validation can be a long process on larger tables. The value of separating
validation from initial creation is that you can defer validation to less
busy times, or can be used to give additional time to correct pre-existing
errors while preventing new errors. Note also that validation on its own
does not prevent normal write commands against the table while it runs.
</P
><P
> Validation acquires only a <TT
CLASS="LITERAL"
>SHARE UPDATE EXCLUSIVE</TT
> lock
on the table being altered. If the constraint is a foreign key then
a <TT
CLASS="LITERAL"
>ROW SHARE</TT
> lock is also required on
the table referenced by the constraint.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>DROP CONSTRAINT [ IF EXISTS ]</TT
></DT
><DD
><P
> This form drops the specified constraint on a table.
If <TT
CLASS="LITERAL"
>IF EXISTS</TT
> is specified and the constraint
does not exist, no error is thrown. In this case a notice is issued instead.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>DISABLE</TT
>/<TT
CLASS="LITERAL"
>ENABLE [ REPLICA | ALWAYS ] TRIGGER</TT
></DT
><DD
><P
> These forms configure the firing of trigger(s) belonging to the table.
A disabled trigger is still known to the system, but is not executed
when its triggering event occurs. For a deferred trigger, the enable
status is checked when the event occurs, not when the trigger function
is actually executed. One can disable or enable a single
trigger specified by name, or all triggers on the table, or only
user triggers (this option excludes internally generated constraint
triggers such as those that are used to implement foreign key
constraints or deferrable uniqueness and exclusion constraints).
Disabling or enabling internally generated constraint triggers
requires superuser privileges; it should be done with caution since
of course the integrity of the constraint cannot be guaranteed if the
triggers are not executed.
The trigger firing mechanism is also affected by the configuration
variable <A
HREF="runtime-config-client.html#GUC-SESSION-REPLICATION-ROLE"
>session_replication_role</A
>. Simply enabled
triggers will fire when the replication role is <SPAN
CLASS="QUOTE"
>"origin"</SPAN
>
(the default) or <SPAN
CLASS="QUOTE"
>"local"</SPAN
>. Triggers configured as <TT
CLASS="LITERAL"
>ENABLE
REPLICA</TT
> will only fire if the session is in <SPAN
CLASS="QUOTE"
>"replica"</SPAN
>
mode, and triggers configured as <TT
CLASS="LITERAL"
>ENABLE ALWAYS</TT
> will
fire regardless of the current replication mode.
</P
><P
> This command acquires a <TT
CLASS="LITERAL"
>SHARE ROW EXCLUSIVE</TT
> lock.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>DISABLE</TT
>/<TT
CLASS="LITERAL"
>ENABLE [ REPLICA | ALWAYS ] RULE</TT
></DT
><DD
><P
> These forms configure the firing of rewrite rules belonging to the table.
A disabled rule is still known to the system, but is not applied
during query rewriting. The semantics are as for disabled/enabled
triggers. This configuration is ignored for <TT
CLASS="LITERAL"
>ON SELECT</TT
> rules, which
are always applied in order to keep views working even if the current
session is in a non-default replication role.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>DISABLE</TT
>/<TT
CLASS="LITERAL"
>ENABLE ROW LEVEL SECURITY</TT
></DT
><DD
><P
> These forms control the application of row security policies belonging
to the table. If enabled and no policies exist for the table, then a
default-deny policy is applied. Note that policies can exist for a table
even if row level security is disabled - in this case, the policies will
NOT be applied and the policies will be ignored.
See also
<A
HREF="sql-createpolicy.html"
>CREATE POLICY</A
>.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>NO FORCE</TT
>/<TT
CLASS="LITERAL"
>FORCE ROW LEVEL SECURITY</TT
></DT
><DD
><P
> These forms control the application of row security policies belonging
to the table when the user is the table owner. If enabled, row level
security policies will be applied when the user is the table owner. If
disabled (the default) then row level security will not be applied when
the user is the table owner.
See also
<A
HREF="sql-createpolicy.html"
>CREATE POLICY</A
>.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>CLUSTER ON</TT
></DT
><DD
><P
> This form selects the default index for future
<A
HREF="sql-cluster.html"
>CLUSTER</A
>
operations. It does not actually re-cluster the table.
</P
><P
> Changing cluster options acquires a <TT
CLASS="LITERAL"
>SHARE UPDATE EXCLUSIVE</TT
> lock.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>SET WITHOUT CLUSTER</TT
></DT
><DD
><P
> This form removes the most recently used
<A
HREF="sql-cluster.html"
>CLUSTER</A
>
index specification from the table. This affects
future cluster operations that don't specify an index.
</P
><P
> Changing cluster options acquires a <TT
CLASS="LITERAL"
>SHARE UPDATE EXCLUSIVE</TT
> lock.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>SET WITH OIDS</TT
></DT
><DD
><P
> This form adds an <TT
CLASS="LITERAL"
>oid</TT
> system column to the
table (see <A
HREF="ddl-system-columns.html"
>Section 5.4</A
>).
It does nothing if the table already has OIDs.
</P
><P
> Note that this is not equivalent to <TT
CLASS="LITERAL"
>ADD COLUMN oid oid</TT
>;
that would add a normal column that happened to be named
<TT
CLASS="LITERAL"
>oid</TT
>, not a system column.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>SET WITHOUT OIDS</TT
></DT
><DD
><P
> This form removes the <TT
CLASS="LITERAL"
>oid</TT
> system column from the
table. This is exactly equivalent to
<TT
CLASS="LITERAL"
>DROP COLUMN oid RESTRICT</TT
>,
except that it will not complain if there is already no
<TT
CLASS="LITERAL"
>oid</TT
> column.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>SET TABLESPACE</TT
></DT
><DD
><P
> This form changes the table's tablespace to the specified tablespace and
moves the data file(s) associated with the table to the new tablespace.
Indexes on the table, if any, are not moved; but they can be moved
separately with additional <TT
CLASS="LITERAL"
>SET TABLESPACE</TT
> commands.
All tables in the current database in a tablespace can be moved by using
the <TT
CLASS="LITERAL"
>ALL IN TABLESPACE</TT
> form, which will lock all tables
to be moved first and then move each one. This form also supports
<TT
CLASS="LITERAL"
>OWNED BY</TT
>, which will only move tables owned by the
roles specified. If the <TT
CLASS="LITERAL"
>NOWAIT</TT
> option is specified
then the command will fail if it is unable to acquire all of the locks
required immediately. Note that system catalogs are not moved by this
command, use <TT
CLASS="COMMAND"
>ALTER DATABASE</TT
> or explicit
<TT
CLASS="COMMAND"
>ALTER TABLE</TT
> invocations instead if desired. The
<TT
CLASS="LITERAL"
>information_schema</TT
> relations are not considered part
of the system catalogs and will be moved.
See also
<A
HREF="sql-createtablespace.html"
>CREATE TABLESPACE</A
>.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>SET { LOGGED | UNLOGGED }</TT
></DT
><DD
><P
> This form changes the table from unlogged to logged or vice-versa
(see <A
HREF="sql-createtable.html#SQL-CREATETABLE-UNLOGGED"
><I
CLASS="TERM"
><TT
CLASS="LITERAL"
>UNLOGGED</TT
></I
></A
>). It cannot be applied
to a temporary table.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>SET ( <TT
CLASS="REPLACEABLE"
><I
>storage_parameter</I
></TT
> = <TT
CLASS="REPLACEABLE"
><I
>value</I
></TT
> [, ... ] )</TT
></DT
><DD
><P
> This form changes one or more storage parameters for the table. See
<A
HREF="sql-createtable.html#SQL-CREATETABLE-STORAGE-PARAMETERS"
><I
>Storage Parameters</I
></A
>
for details on the available parameters. Note that the table contents
will not be modified immediately by this command; depending on the
parameter you might need to rewrite the table to get the desired effects.
That can be done with <A
HREF="sql-vacuum.html"
>VACUUM
FULL</A
>, <A
HREF="sql-cluster.html"
>CLUSTER</A
> or one of the forms
of <TT
CLASS="COMMAND"
>ALTER TABLE</TT
> that forces a table rewrite.
</P
><P
> Changing fillfactor and autovacuum storage parameters acquires a <TT
CLASS="LITERAL"
>SHARE UPDATE EXCLUSIVE</TT
> lock.
</P
><DIV
CLASS="NOTE"
><BLOCKQUOTE
CLASS="NOTE"
><P
><B
>Note: </B
> While <TT
CLASS="COMMAND"
>CREATE TABLE</TT
> allows <TT
CLASS="LITERAL"
>OIDS</TT
> to be specified
in the <TT
CLASS="LITERAL"
>WITH (<TT
CLASS="REPLACEABLE"
><I
>storage_parameter</I
></TT
>)</TT
> syntax,
<TT
CLASS="COMMAND"
>ALTER TABLE</TT
> does not treat <TT
CLASS="LITERAL"
>OIDS</TT
> as a
storage parameter. Instead use the <TT
CLASS="LITERAL"
>SET WITH OIDS</TT
>
and <TT
CLASS="LITERAL"
>SET WITHOUT OIDS</TT
> forms to change OID status.
</P
></BLOCKQUOTE
></DIV
></DD
><DT
><TT
CLASS="LITERAL"
>RESET ( <TT
CLASS="REPLACEABLE"
><I
>storage_parameter</I
></TT
> [, ... ] )</TT
></DT
><DD
><P
> This form resets one or more storage parameters to their
defaults. As with <TT
CLASS="LITERAL"
>SET</TT
>, a table rewrite might be
needed to update the table entirely.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>INHERIT <TT
CLASS="REPLACEABLE"
><I
>parent_table</I
></TT
></TT
></DT
><DD
><P
> This form adds the target table as a new child of the specified parent
table. Subsequently, queries against the parent will include records
of the target table. To be added as a child, the target table must
already contain all the same columns as the parent (it could have
additional columns, too). The columns must have matching data types,
and if they have <TT
CLASS="LITERAL"
>NOT NULL</TT
> constraints in the parent
then they must also have <TT
CLASS="LITERAL"
>NOT NULL</TT
> constraints in the
child.
</P
><P
> There must also be matching child-table constraints for all
<TT
CLASS="LITERAL"
>CHECK</TT
> constraints of the parent, except those
marked non-inheritable (that is, created with <TT
CLASS="LITERAL"
>ALTER TABLE ... ADD CONSTRAINT ... NO INHERIT</TT
>)
in the parent, which are ignored; all child-table constraints matched
must not be marked non-inheritable.
Currently
<TT
CLASS="LITERAL"
>UNIQUE</TT
>, <TT
CLASS="LITERAL"
>PRIMARY KEY</TT
>, and
<TT
CLASS="LITERAL"
>FOREIGN KEY</TT
> constraints are not considered, but
this might change in the future.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>NO INHERIT <TT
CLASS="REPLACEABLE"
><I
>parent_table</I
></TT
></TT
></DT
><DD
><P
> This form removes the target table from the list of children of the
specified parent table.
Queries against the parent table will no longer include records drawn
from the target table.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>OF <TT
CLASS="REPLACEABLE"
><I
>type_name</I
></TT
></TT
></DT
><DD
><P
> This form links the table to a composite type as though <TT
CLASS="COMMAND"
>CREATE
TABLE OF</TT
> had formed it. The table's list of column names and types
must precisely match that of the composite type; the presence of
an <TT
CLASS="LITERAL"
>oid</TT
> system column is permitted to differ. The table must
not inherit from any other table. These restrictions ensure
that <TT
CLASS="COMMAND"
>CREATE TABLE OF</TT
> would permit an equivalent table
definition.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>NOT OF</TT
></DT
><DD
><P
> This form dissociates a typed table from its type.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>OWNER</TT
></DT
><DD
><P
> This form changes the owner of the table, sequence, view, materialized view,
or foreign table to the specified user.
</P
></DD
><DT
><A
NAME="SQL-CREATETABLE-REPLICA-IDENTITY"
></A
><TT
CLASS="LITERAL"
>REPLICA IDENTITY</TT
></DT
><DD
><P
> This form changes the information which is written to the write-ahead log
to identify rows which are updated or deleted. This option has no effect
except when logical replication is in use. <TT
CLASS="LITERAL"
>DEFAULT</TT
>
(the default for non-system tables) records the
old values of the columns of the primary key, if any. <TT
CLASS="LITERAL"
>USING INDEX</TT
>
records the old values of the columns covered by the named index, which
must be unique, not partial, not deferrable, and include only columns marked
<TT
CLASS="LITERAL"
>NOT NULL</TT
>. <TT
CLASS="LITERAL"
>FULL</TT
> records the old values of all columns
in the row. <TT
CLASS="LITERAL"
>NOTHING</TT
> records no information about the old row.
(This is the default for system tables.)
In all cases, no old values are logged unless at least one of the columns
that would be logged differs between the old and new versions of the row.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>RENAME</TT
></DT
><DD
><P
> The <TT
CLASS="LITERAL"
>RENAME</TT
> forms change the name of a table
(or an index, sequence, view, materialized view, or foreign table), the name
of an individual column in a table, or the name of a constraint of the table.
There is no effect on the stored data.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>SET SCHEMA</TT
></DT
><DD
><P
> This form moves the table into another schema. Associated indexes,
constraints, and sequences owned by table columns are moved as well.
</P
></DD
></DL
></DIV
><P>
</P
><P
> All the forms of ALTER TABLE that act on a single table, except
<TT
CLASS="LITERAL"
>RENAME</TT
>, and <TT
CLASS="LITERAL"
>SET SCHEMA</TT
>
can be combined into a list of multiple alterations to applied together.
For example, it
is possible to add several columns and/or alter the type of several
columns in a single command. This is particularly useful with large
tables, since only one pass over the table need be made.
</P
><P
> You must own the table to use <TT
CLASS="COMMAND"
>ALTER TABLE</TT
>.
To change the schema or tablespace of a table, you must also have
<TT
CLASS="LITERAL"
>CREATE</TT
> privilege on the new schema or tablespace.
To add the table as a new child of a parent table, you must own the
parent table as well.
To alter the owner, you must also be a direct or indirect member of the new
owning role, and that role must have <TT
CLASS="LITERAL"
>CREATE</TT
> privilege on
the table's schema. (These restrictions enforce that altering the owner
doesn't do anything you couldn't do by dropping and recreating the table.
However, a superuser can alter ownership of any table anyway.)
To add a column or alter a column type or use the <TT
CLASS="LITERAL"
>OF</TT
>
clause, you must also have <TT
CLASS="LITERAL"
>USAGE</TT
> privilege on the data
type.
</P
></DIV
><DIV
CLASS="REFSECT1"
><A
NAME="AEN75679"
></A
><H2
>Parameters</H2
><P
></P
><DIV
CLASS="VARIABLELIST"
><DL
><DT
><TT
CLASS="LITERAL"
>IF EXISTS</TT
></DT
><DD
><P
> Do not throw an error if the table does not exist. A notice is issued
in this case.
</P
></DD
><DT
><TT
CLASS="REPLACEABLE"
><I
>name</I
></TT
></DT
><DD
><P
> The name (optionally schema-qualified) of an existing table to
alter. If <TT
CLASS="LITERAL"
>ONLY</TT
> is specified before the table name, only
that table is altered. If <TT
CLASS="LITERAL"
>ONLY</TT
> is not specified, the table
and all its descendant tables (if any) are altered. Optionally,
<TT
CLASS="LITERAL"
>*</TT
> can be specified after the table name to explicitly
indicate that descendant tables are included.
</P
></DD
><DT
><TT
CLASS="REPLACEABLE"
><I
>column_name</I
></TT
></DT
><DD
><P
> Name of a new or existing column.
</P
></DD
><DT
><TT
CLASS="REPLACEABLE"
><I
>new_column_name</I
></TT
></DT
><DD
><P
> New name for an existing column.
</P
></DD
><DT
><TT
CLASS="REPLACEABLE"
><I
>new_name</I
></TT
></DT
><DD
><P
> New name for the table.
</P
></DD
><DT
><TT
CLASS="REPLACEABLE"
><I
>data_type</I
></TT
></DT
><DD
><P
> Data type of the new column, or new data type for an existing
column.
</P
></DD
><DT
><TT
CLASS="REPLACEABLE"
><I
>table_constraint</I
></TT
></DT
><DD
><P
> New table constraint for the table.
</P
></DD
><DT
><TT
CLASS="REPLACEABLE"
><I
>constraint_name</I
></TT
></DT
><DD
><P
> Name of a new or existing constraint.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>CASCADE</TT
></DT
><DD
><P
> Automatically drop objects that depend on the dropped column
or constraint (for example, views referencing the column),
and in turn all objects that depend on those objects
(see <A
HREF="ddl-depend.html"
>Section 5.13</A
>).
</P
></DD
><DT
><TT
CLASS="LITERAL"
>RESTRICT</TT
></DT
><DD
><P
> Refuse to drop the column or constraint if there are any dependent
objects. This is the default behavior.
</P
></DD
><DT
><TT
CLASS="REPLACEABLE"
><I
>trigger_name</I
></TT
></DT
><DD
><P
> Name of a single trigger to disable or enable.
</P
></DD
><DT
><TT
CLASS="LITERAL"
>ALL</TT
></DT
><DD
><P
> Disable or enable all triggers belonging to the table.
(This requires superuser privilege if any of the triggers are
internally generated constraint triggers such as those that are used
to implement foreign key constraints or deferrable uniqueness and
exclusion constraints.)
</P
></DD
><DT
><TT
CLASS="LITERAL"
>USER</TT
></DT
><DD
><P
> Disable or enable all triggers belonging to the table except for
internally generated constraint triggers such as those that are used
to implement foreign key constraints or deferrable uniqueness and
exclusion constraints.
</P
></DD
><DT
><TT
CLASS="REPLACEABLE"
><I
>index_name</I
></TT
></DT
><DD
><P
> The name of an existing index.
</P
></DD
><DT
><TT
CLASS="REPLACEABLE"
><I
>storage_parameter</I
></TT
></DT
><DD
><P
> The name of a table storage parameter.
</P
></DD
><DT
><TT
CLASS="REPLACEABLE"
><I
>value</I
></TT
></DT
><DD
><P
> The new value for a table storage parameter.
This might be a number or a word depending on the parameter.
</P
></DD
><DT
><TT
CLASS="REPLACEABLE"
><I
>parent_table</I
></TT
></DT
><DD
><P
> A parent table to associate or de-associate with this table.
</P
></DD
><DT
><TT
CLASS="REPLACEABLE"
><I
>new_owner</I
></TT
></DT
><DD
><P
> The user name of the new owner of the table.
</P
></DD
><DT
><TT
CLASS="REPLACEABLE"
><I
>new_tablespace</I
></TT
></DT
><DD
><P
> The name of the tablespace to which the table will be moved.
</P
></DD
><DT
><TT
CLASS="REPLACEABLE"
><I
>new_schema</I
></TT
></DT
><DD
><P
> The name of the schema to which the table will be moved.
</P
></DD
></DL
></DIV
></DIV
><DIV
CLASS="REFSECT1"
><A
NAME="AEN75786"
></A
><H2
>Notes</H2
><P
> The key word <TT
CLASS="LITERAL"
>COLUMN</TT
> is noise and can be omitted.
</P
><P
> When a column is added with <TT
CLASS="LITERAL"
>ADD COLUMN</TT
>, all existing
rows in the table are initialized with the column's default value
(NULL if no <TT
CLASS="LITERAL"
>DEFAULT</TT
> clause is specified).
If there is no <TT
CLASS="LITERAL"
>DEFAULT</TT
> clause, this is merely a metadata
change and does not require any immediate update of the table's data;
the added NULL values are supplied on readout, instead.
</P
><P
> Adding a column with a <TT
CLASS="LITERAL"
>DEFAULT</TT
> clause or changing the type of
an existing column will require the entire table and its indexes to be
rewritten. As an exception when changing the type of an existing column,
if the <TT
CLASS="LITERAL"
>USING</TT
> clause does not change the column
contents and the old type is either binary coercible to the new type or
an unconstrained domain over the new type, a table rewrite is not needed;
but any indexes on the affected columns must still be rebuilt. Adding or
removing a system <TT
CLASS="LITERAL"
>oid</TT
> column also requires rewriting the entire
table. Table and/or index rebuilds may take a significant amount of time
for a large table; and will temporarily require as much as double the disk
space.
</P
><P
> Adding a <TT
CLASS="LITERAL"
>CHECK</TT
> or <TT
CLASS="LITERAL"
>NOT NULL</TT
> constraint requires
scanning the table to verify that existing rows meet the constraint,
but does not require a table rewrite.
</P
><P
> The main reason for providing the option to specify multiple changes
in a single <TT
CLASS="COMMAND"
>ALTER TABLE</TT
> is that multiple table scans or
rewrites can thereby be combined into a single pass over the table.
</P
><P
> The <TT
CLASS="LITERAL"
>DROP COLUMN</TT
> form does not physically remove
the column, but simply makes it invisible to SQL operations. Subsequent
insert and update operations in the table will store a null value for the
column. Thus, dropping a column is quick but it will not immediately
reduce the on-disk size of your table, as the space occupied
by the dropped column is not reclaimed. The space will be
reclaimed over time as existing rows are updated. (These statements do
not apply when dropping the system <TT
CLASS="LITERAL"
>oid</TT
> column; that is done
with an immediate rewrite.)
</P
><P
> To force immediate reclamation of space occupied by a dropped column,
you can execute one of the forms of <TT
CLASS="COMMAND"
>ALTER TABLE</TT
> that
performs a rewrite of the whole table. This results in reconstructing
each row with the dropped column replaced by a null value.
</P
><P
> The rewriting forms of <TT
CLASS="COMMAND"
>ALTER TABLE</TT
> are not MVCC-safe.
After a table rewrite, the table will appear empty to concurrent
transactions, if they are using a snapshot taken before the rewrite
occurred. See <A
HREF="mvcc-caveats.html"
>Section 13.5</A
> for more details.
</P
><P
> The <TT
CLASS="LITERAL"
>USING</TT
> option of <TT
CLASS="LITERAL"
>SET DATA TYPE</TT
> can actually
specify any expression involving the old values of the row; that is, it
can refer to other columns as well as the one being converted. This allows
very general conversions to be done with the <TT
CLASS="LITERAL"
>SET DATA TYPE</TT
>
syntax. Because of this flexibility, the <TT
CLASS="LITERAL"
>USING</TT
>
expression is not applied to the column's default value (if any); the
result might not be a constant expression as required for a default.
This means that when there is no implicit or assignment cast from old to
new type, <TT
CLASS="LITERAL"
>SET DATA TYPE</TT
> might fail to convert the default even
though a <TT
CLASS="LITERAL"
>USING</TT
> clause is supplied. In such cases,
drop the default with <TT
CLASS="LITERAL"
>DROP DEFAULT</TT
>, perform the <TT
CLASS="LITERAL"
>ALTER
TYPE</TT
>, and then use <TT
CLASS="LITERAL"
>SET DEFAULT</TT
> to add a suitable new
default. Similar considerations apply to indexes and constraints involving
the column.
</P
><P
> If a table has any descendant tables, it is not permitted to add,
rename, or change the type of a column, or rename an inherited constraint
in the parent table without doing
the same to the descendants. That is, <TT
CLASS="COMMAND"
>ALTER TABLE ONLY</TT
>
will be rejected. This ensures that the descendants always have
columns matching the parent.
</P
><P
> A recursive <TT
CLASS="LITERAL"
>DROP COLUMN</TT
> operation will remove a
descendant table's column only if the descendant does not inherit
that column from any other parents and never had an independent
definition of the column. A nonrecursive <TT
CLASS="LITERAL"
>DROP
COLUMN</TT
> (i.e., <TT
CLASS="COMMAND"
>ALTER TABLE ONLY ... DROP
COLUMN</TT
>) never removes any descendant columns, but
instead marks them as independently defined rather than inherited.
</P
><P
> The <TT
CLASS="LITERAL"
>TRIGGER</TT
>, <TT
CLASS="LITERAL"
>CLUSTER</TT
>, <TT
CLASS="LITERAL"
>OWNER</TT
>,
and <TT
CLASS="LITERAL"
>TABLESPACE</TT
> actions never recurse to descendant tables;
that is, they always act as though <TT
CLASS="LITERAL"
>ONLY</TT
> were specified.
Adding a constraint recurses only for <TT
CLASS="LITERAL"
>CHECK</TT
> constraints
that are not marked <TT
CLASS="LITERAL"
>NO INHERIT</TT
>.
</P
><P
> Changing any part of a system catalog table is not permitted.
</P
><P
> Refer to <A
HREF="sql-createtable.html"
>CREATE TABLE</A
> for a further description of valid
parameters. <A
HREF="ddl.html"
>Chapter 5</A
> has further information on
inheritance.
</P
></DIV
><DIV
CLASS="REFSECT1"
><A
NAME="AEN75839"
></A
><H2
>Examples</H2
><P
> To add a column of type <TT
CLASS="TYPE"
>varchar</TT
> to a table:
</P><PRE
CLASS="PROGRAMLISTING"
>ALTER TABLE distributors ADD COLUMN address varchar(30);</PRE
><P>
</P
><P
> To drop a column from a table:
</P><PRE
CLASS="PROGRAMLISTING"
>ALTER TABLE distributors DROP COLUMN address RESTRICT;</PRE
><P>
</P
><P
> To change the types of two existing columns in one operation:
</P><PRE
CLASS="PROGRAMLISTING"
>ALTER TABLE distributors
ALTER COLUMN address TYPE varchar(80),
ALTER COLUMN name TYPE varchar(100);</PRE
><P>
</P
><P
> To change an integer column containing Unix timestamps to <TT
CLASS="TYPE"
>timestamp
with time zone</TT
> via a <TT
CLASS="LITERAL"
>USING</TT
> clause:
</P><PRE
CLASS="PROGRAMLISTING"
>ALTER TABLE foo
ALTER COLUMN foo_timestamp SET DATA TYPE timestamp with time zone
USING
timestamp with time zone 'epoch' + foo_timestamp * interval '1 second';</PRE
><P>
</P
><P
> The same, when the column has a default expression that won't automatically
cast to the new data type:
</P><PRE
CLASS="PROGRAMLISTING"
>ALTER TABLE foo
ALTER COLUMN foo_timestamp DROP DEFAULT,
ALTER COLUMN foo_timestamp TYPE timestamp with time zone
USING
timestamp with time zone 'epoch' + foo_timestamp * interval '1 second',
ALTER COLUMN foo_timestamp SET DEFAULT now();</PRE
><P>
</P
><P
> To rename an existing column:
</P><PRE
CLASS="PROGRAMLISTING"
>ALTER TABLE distributors RENAME COLUMN address TO city;</PRE
><P>
</P
><P
> To rename an existing table:
</P><PRE
CLASS="PROGRAMLISTING"
>ALTER TABLE distributors RENAME TO suppliers;</PRE
><P>
</P
><P
> To rename an existing constraint:
</P><PRE
CLASS="PROGRAMLISTING"
>ALTER TABLE distributors RENAME CONSTRAINT zipchk TO zip_check;</PRE
><P>
</P
><P
> To add a not-null constraint to a column:
</P><PRE
CLASS="PROGRAMLISTING"
>ALTER TABLE distributors ALTER COLUMN street SET NOT NULL;</PRE
><P>
To remove a not-null constraint from a column:
</P><PRE
CLASS="PROGRAMLISTING"
>ALTER TABLE distributors ALTER COLUMN street DROP NOT NULL;</PRE
><P>
</P
><P
> To add a check constraint to a table and all its children:
</P><PRE
CLASS="PROGRAMLISTING"
>ALTER TABLE distributors ADD CONSTRAINT zipchk CHECK (char_length(zipcode) = 5);</PRE
><P>
</P
><P
> To add a check constraint only to a table and not to its children:
</P><PRE
CLASS="PROGRAMLISTING"
>ALTER TABLE distributors ADD CONSTRAINT zipchk CHECK (char_length(zipcode) = 5) NO INHERIT;</PRE
><P>
(The check constraint will not be inherited by future children, either.)
</P
><P
> To remove a check constraint from a table and all its children:
</P><PRE
CLASS="PROGRAMLISTING"
>ALTER TABLE distributors DROP CONSTRAINT zipchk;</PRE
><P>
</P
><P
> To remove a check constraint from one table only:
</P><PRE
CLASS="PROGRAMLISTING"
>ALTER TABLE ONLY distributors DROP CONSTRAINT zipchk;</PRE
><P>
(The check constraint remains in place for any child tables.)
</P
><P
> To add a foreign key constraint to a table:
</P><PRE
CLASS="PROGRAMLISTING"
>ALTER TABLE distributors ADD CONSTRAINT distfk FOREIGN KEY (address) REFERENCES addresses (address);</PRE
><P>
</P
><P
> To add a foreign key constraint to a table with the least impact on other work:
</P><PRE
CLASS="PROGRAMLISTING"
>ALTER TABLE distributors ADD CONSTRAINT distfk FOREIGN KEY (address) REFERENCES addresses (address) NOT VALID;
ALTER TABLE distributors VALIDATE CONSTRAINT distfk;</PRE
><P>
</P
><P
> To add a (multicolumn) unique constraint to a table:
</P><PRE
CLASS="PROGRAMLISTING"
>ALTER TABLE distributors ADD CONSTRAINT dist_id_zipcode_key UNIQUE (dist_id, zipcode);</PRE
><P>
</P
><P
> To add an automatically named primary key constraint to a table, noting
that a table can only ever have one primary key:
</P><PRE
CLASS="PROGRAMLISTING"
>ALTER TABLE distributors ADD PRIMARY KEY (dist_id);</PRE
><P>
</P
><P
> To move a table to a different tablespace:
</P><PRE
CLASS="PROGRAMLISTING"
>ALTER TABLE distributors SET TABLESPACE fasttablespace;</PRE
><P>
</P
><P
> To move a table to a different schema:
</P><PRE
CLASS="PROGRAMLISTING"
>ALTER TABLE myschema.distributors SET SCHEMA yourschema;</PRE
><P>
</P
><P
> To recreate a primary key constraint, without blocking updates while the
index is rebuilt:
</P><PRE
CLASS="PROGRAMLISTING"
>CREATE UNIQUE INDEX CONCURRENTLY dist_id_temp_idx ON distributors (dist_id);
ALTER TABLE distributors DROP CONSTRAINT distributors_pkey,
ADD CONSTRAINT distributors_pkey PRIMARY KEY USING INDEX dist_id_temp_idx;</PRE
><P></P
></DIV
><DIV
CLASS="REFSECT1"
><A
NAME="AEN75885"
></A
><H2
>Compatibility</H2
><P
> The forms <TT
CLASS="LITERAL"
>ADD</TT
> (without <TT
CLASS="LITERAL"
>USING INDEX</TT
>),
<TT
CLASS="LITERAL"
>DROP</TT
>, <TT
CLASS="LITERAL"
>SET DEFAULT</TT
>,
and <TT
CLASS="LITERAL"
>SET DATA TYPE</TT
> (without <TT
CLASS="LITERAL"
>USING</TT
>)
conform with the SQL standard. The other forms are
<SPAN
CLASS="PRODUCTNAME"
>PostgreSQL</SPAN
> extensions of the SQL standard.
Also, the ability to specify more than one manipulation in a single
<TT
CLASS="COMMAND"
>ALTER TABLE</TT
> command is an extension.
</P
><P
> <TT
CLASS="COMMAND"
>ALTER TABLE DROP COLUMN</TT
> can be used to drop the only
column of a table, leaving a zero-column table. This is an
extension of SQL, which disallows zero-column tables.
</P
></DIV
><DIV
CLASS="REFSECT1"
><A
NAME="AEN75898"
></A
><H2
>See Also</H2
><A
HREF="sql-createtable.html"
>CREATE TABLE</A
></DIV
><DIV
CLASS="NAVFOOTER"
><HR
ALIGN="LEFT"
WIDTH="100%"><TABLE
SUMMARY="Footer navigation table"
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BORDER="0"
CELLPADDING="0"
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><TR
><TD
WIDTH="33%"
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VALIGN="top"
><A
HREF="sql-altersystem.html"
ACCESSKEY="P"
>Prev</A
></TD
><TD
WIDTH="34%"
ALIGN="center"
VALIGN="top"
><A
HREF="index.html"
ACCESSKEY="H"
>Home</A
></TD
><TD
WIDTH="33%"
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VALIGN="top"
><A
HREF="sql-altertablespace.html"
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> | {
"redpajama_set_name": "RedPajamaGithub"
} | 6,859 |
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"redpajama_set_name": "RedPajamaC4"
} | 8,001 |
Thanks for checking up on us, we arrived safely but was stuck with unpacking and cleaning up the mess here, hence the delay. Just want to thank you for an amazing holiday. Jason looooved the beach, he is already making a huge drama wanting to go back ;) say hi to Ranjith for us. He was a wonderful guide and was patient enough with Jason's huge list of questions, which says a lot about his character.
Overall we really loved the time spent in Sri Lanka and hope to visit again very soon.
We had a wonderful stay! Thanks to you all. Raj was a star!!
Thank you very much for an excellent holiday!!
Thank you for a fantastic tour in wonderful Sri Lanka!
Ceylon Tours and Mr.Kannangara together make us falling in love with Sri Lanka. We'll recommend both of you to our friends that wish to visit Sri Lanka! Ayubowan!!
Our recent tour to Sri Lanka was outstanding.We saw leopard, 86 species of birds, multiple UNESCO world heritage sights and significant historical and religious sites.
Our guide was very competent and an excellent, patient driver.
We particularly appreciated the complementary dinner at the Cinnamon Grand Hotel where we could sit out under the Banyon tree and enjoy delicacies from Sri Lanka served by people who could patiently explain any dish available.
We look forward to returning.Do you have a trip to the North??? If so please furnish details.
We would highly recommend Ceylon Tours and Pradeep is one of the main reason for doing so. | {
"redpajama_set_name": "RedPajamaC4"
} | 487 |
\section{RADON BACKGROUNDS FOR SUPERCDMS SNOLAB}
A potential source of dominant backgrounds for many rare-event searches or screening detectors is from radon daughter plate-out~\cite{Simgen:2013dlh,Schnee:2014eea}. Backgrounds from $^{210}$Pb and the recoiling $^{206}$Pb nucleus from the $\alpha$ decay of $^{210}$Po were the dominant low-energy backgrounds for XMASS~\cite{Kobayashi:2015lla,Hiraide:2015cba}, SuperCDMS Soudan~\cite{Agnese:2014aa}, and EDELWEISS~\cite{Navick:2013hgx}. These backgrounds remain the dominant surface backgrounds for EDELWEISS-III~\cite{Armengaud:2017rzu}. Mitigation of radon daughters on surfaces has also been critical for SuperNEMO~\cite{Mott:2013nzg} and CUORE~\cite{wangSUthesis}. Both neutrons from ($\alpha,n$) reactions and $^{206}$Pb recoils are important for LZ~\cite{lux2014backgrounds}, XENON1T~\cite{Aprile:2017ilq}, and DarkSide~\cite{Agnes:2014bvk}.
Radon-daughter backgrounds are important to the expected low-mass sensitivity of the SuperCDMS SNOLAB experiment, which will use detectors of germanium and silicon to search for dark matter interactions~\cite{Agnese:2016cpb}. Although the SuperCDMS Interleaved Z-sensitive Ionization and Phonon (iZIP) detectors provide excellent rejection of surface events above 8\,keV~\cite{Agnese:2013ixa}, at lower energies the rejection is expected to worsen to the point that radon-daughter backgrounds may dominate~\cite{Agnese:2016cpb}. The situation is similar for the SuperCDMS high-voltage (HV) detectors, which provide the experiment's lowest energy threshold by amplifying the ionization signal~\cite{Agnese:2014aa,Agnese:2017jvy}. Below energies of $\sim$0.5\,keV, rejection of events on the detector sidewall surface, based on the relative sizes of signals in the detectors' phonon sensors, becomes ineffective, so radon-daughter surface backgrounds are expected to dominate over bulk backgrounds for $^{210}$Pb concentrations $\gtrsim$50\,nBq\,cm$^{-2}$. Figure~\ref{sens_plot} shows how the expected sensitivity of SuperCDMS SNOLAB varies for different surface concentrations of $^{210}$Pb.
Without mitigation, radon daughter plate-out during assembly underground at SNOLAB could dominate the expected background. The duration of the SNOLAB assembly is estimated at about 150 hours. At the average SNOLAB radon concentration of 130\,Bq\,m$^{-3}$, the expected plate-out would be about 70\,nBq\,cm$^{-2}$ of $^{210}$Pb. To make this plate-out rate negligible, SuperCDMS SNOLAB has a goal of achieving a radon concentration underground of $<$0.1\,Bq\,m$^{-3}$ in its assembly cleanroom underground.
\begin{figure}[h]
\includegraphics[width=0.80\textwidth]{SensitivityPlot_170716}
\caption{Projected sensitivity to the spin-independent dark matter-nucleon scattering cross-section as a function of dark matter mass for Si and Ge iZIP and Si and Ge HV detectors of the SuperCDMS SNOLAB experiment~\cite{Agnese:2017jvy,Kurinsky:2016fvj}. Each set of four dashed lines represents projected sensitivities for (from bottom to top) 0, 100, 300, and 600\,nBq\,cm$^{-2}$ $^{210}$Pb surface contamination on the detector sidewalls (i.e., a sum across the detector and copper surfaces). At low masses, the sensitivity degrades significantly with increasing $^{210}$Pb contamination above 50\,nBq\,cm$^{-2}$.}
\label{sens_plot}
\end{figure}
\section{THE DEMONSTRATION RADON-MITIGATION SYSTEM AT SD MINES}
Two systems of classes may create a radon-mitigated, breathable-air environment. One type flows continuously through a filtration column, while the second swings flow back and forth using two or more filtration columns. The filtration columns are usually filled with activated carbon. The continuous flow system (e.g.~\cite{Mount:2017aa,Nachab:2007zz}) operates on the basis that a considerable fraction of radon decays before exiting the column. For an ideal column, the final radon concentration $C_\text{final}=C_\text{initial}\exp{\left(-t_\mathrm{BT}/\tau_\mathrm{Rn}\right)}$, where $C_\text{initial}$ is the radon concentration of the input air, $t_\mathrm{BT}$ is the characteristic break-through time of the filter, and $\tau_\mathrm{Rn}=5.516$\,days is the mean lifetime of radon. The break-through time is the mean-time it takes for radon to pass through the carbon filter. To increase the break-through time, and therefore make a continuous flow system practical, one must cool the carbon to reduce desorption of radon. Continuous systems are commercially available (e.g.~\cite{Mount:2017aa,Nachab:2007zz}) and typically achieve reduction factors of $\sim$1000$\times$.
In a swing flow system, two or more filtration columns are used. While air is filtered through one column, the other is regenerated using either low pressure or high temperatures to allow radon to desorb efficiently and be exhausted. For a vacuum-swing-adsorption (VSA) system (e.g.~\cite{Pocar:2005kp,Pocar:2003dw,Street:2015nwa}), high-radon input air is filtered through the first column while the second column is pumped down to $\sim$20\,Torr. Well before the break-through time, the path of the high-radon input air is switched so that it flows through the second column instead, allowing the first to regenerate. For an ideal column, no radon reaches the output. Swing flow systems are more complicated both in operation and analysis. A VSA system (e.g. Fig.~\ref{VSA_Diagram_image}) can potentially outperform a continuous flow system even at a lower cost. Temperature-swing systems (e.g.~\cite{Hallin:LRT2010}) should provide best performance but at the highest cost and complexity.
The VSA system built at SD Mines (Fig.~\ref{VSA_Diagram_image}) was based closely on the Princeton design~\cite{Pocar:2003dw,Pocar:2005kp}. An air blower provides as much as $\sim$100 cubic feet per minute (cfm) to the VSA system input. The air is then chilled, dehumidified, and chilled again, to prevent moisture in the air from collecting on the carbon surfaces, and without heating the air (since radon desorbs from carbon more at lower temperatures). The high-radon air then passes through one of the two carbon columns. Radon adsorbs preferentially to carbon relative to nitrogen and oxygen because both radon and carbon are nonpolar and therefore experience induced dipole interactions described by the Van der Waals force. Because radon spends a higher fraction of its time adsorbed on the carbon, it moves with a net speed much slower than that of air. Before any radon has made it all the way through the column, the air flowing out is low-radon air. During this time (as much as hours), the second column is pumped down to $\sim$20\,Torr by a high-speed vacuum system and $\sim$10\% of the low-radon air flowing from the first column (called the purge flow) is introduced to the second column to help remove its radon. After less than an hour, the second column is regenerated (i.e., the radon is removed) and it may be used as a filter just as the first was. Before radon travels to the end of the first column, the system ``swings" such that high-radon air flows through the second column and the first is regenerated. In this way, low radon air is continuously supplied to the low-radon clean room without interruption.
\begin{figure}[!tbp]
\begin{tabular}{c c}
\includegraphics[width=0.50\textwidth]{VSA_Diagram_1} &
\includegraphics[width=0.46\textwidth]{VSA_Image}
\end{tabular}
\caption{\textit{Left}: Diagram of the vacuum-swing-adsorption radon mitigation system, showing the path of airflow (arrows) through open valves (solid circles), through the carbon tank at atmospheric pressure and to the system output, while ~10\% of the output air (small arrows) is diverted through a butterfly valve, backwards through the low-pressure carbon tank and exhausted at the vacuum pump. By opening the closed valves (crossed circles) and closing the opened ones, the roles of the tanks may be reversed. \textit{Right}: Image of the VSA system at SD Mines. Labels show the air blower, air chillers, dehumidifier, VSA system controller in front of valve cluster, carbon columns, roughing pump and booster.}
\label{VSA_Diagram_image}
\end{figure}
An improvement to the original design of this system (e.g.~\cite{Pocar:2003dw,Street:2015nwa}) is in the way the regenerated column is brought back up to atmosphere. Previously~\cite{Pocar:2003dw,Street:2015nwa}, once a column was regenerated it was brought up to atmosphere by opening a second channel connected directly to the air blower, which bypassed the air chillers and dehumidifier. This choice maximized the amount of air supplied to the filling column in order to prevent air from being pulled from the clean room (which when under-pressured would pull high-radon air in from the lab environment). Our improved system instead closes the valve to the roughing pump after regeneration and then slowly fills the column up to atmosphere through the purge flow line; at 7\,cfm, it takes about 15 minutes to approach atmosphere (from $\sim$18\,Torr). This technique, termed the ``slow-fill method," reduces the number of valves needed by the VSA system, prevents the introduction of radon to the newly regenerated column before use, and maintains mechanical stability (because raising these columns to atmosphere quickly can produce sudden stresses on components).
\section{The Radon-Mitigated Clean Room and Results}
The VSA system supplies low-radon air for a radon-mitigated clean room has been commissioned at SD Mines (Fig.~\ref{CR}, \textit{Left}). The clean room was constructed primarily from aluminum, because radon does not easily diffuse through metals. The clean room also has polycarbonate windows that are sufficiently thick (6.4\,mm) such that radon decays before diffusing completely through. Make-up air is supplied at $10-90\,$cfm and replaces the clean room air volume quickly enough to make negligible radon emanating from materials inside the clean room. The achieved room radon concentration described below limits the activity from materials inside to be $<$1000\,Bq.
As shown in~\ref{CR} (\textit{Right}), the VSA system has demonstrated radon reduction meeting the goal of the SuperCDMS SNOLAB experiment. The upper limit on the radon concentration of the radon-mitigated clean room in February 2016 was $<$0.067\,Bq$\,$m$^{-3}$ at 90\% confidence. This shows a $>$1000$\times$ radon reduction between the VSA system input and the radon-mitigated clean room. Note that the VSA system output must have an even lower radon concentration than the cleanroom. This upper limit on radon concentration in the clean room is currently limited by the sensitivity of our Durridge Rad7 radon monitor used to make the measurement.
\begin{figure}[!tbp]
\begin{tabular}{c c}
\includegraphics[width=0.45\textwidth]{RnMitigatedCR} &
\includegraphics[width=0.52\textwidth]{LabAir_vs_CR_reduction_170314_5}
\end{tabular}
\caption{\textit{Left}: The radon-mitigated clean room at SD Mines with interior view inset. Labels show an overpressure gauge, control panel for an internal heating, ventilation and air conditioning (HVAC) unit, the HVAC and steel ducting from the VSA system. The clean room is made from aluminum and has an HVAC that recirculates air ($\sim$1000\,cfm) through four HEPA filters. \textit{Left inset}: The HVAC inside avoids the difficulty of preventing high-radon air from the lab from being introduced to the clean room through the HVAC's recirculation blower region. \textit{Right}: Results of the SD Mines VSA system and clean room. The input air provided to the VSA system (upper error bars) had an average radon concentration (upper solid line) of $79.6\pm0.7\,$Bq$\,$m$^{-3}$, while the air inside the radon-mitigated clean room (dotted line) was measured to be $<$0.067\,Bq$\,$m$^{-3}$ at 90\% confidence. This shows a reduction of $>$1000$\times$. The solid curve with error bars represents the measured concentration in the clean room binned over 24 hours.}
\label{CR}
\end{figure}
\section{SIMULATION AND EXPECTED PERFORMANCE OF THE VSA SYSTEM}
The simulation of the VSA system provides predictions of performance outside of what we can directly measure. It aids in understanding our current VSA system and improvements of future systems in terms of cost and performance. The simulation models radon diffusion and translation through an activated carbon column, while accounting for column temperature, column pressure (as a function of the longitudinal distance in a column and time), radon emanation from the carbon (with activity of 0.01\,Bq\,kg$^{-1}$~\cite{Pocar:2003dw}) and radon decay. The simulation calculates the radon concentration as a function of position and time inside both of the VSA system's carbon columns during operation.
In each carbon column, the radon concentration (having units of Bq\,m$^{-3}$) may be denoted as the \textit{radon vector} $\boldsymbol{c}(x,t)$. Each element of the radon vector is the radon concentration in a position $x$ along the length of the column. Operators evolve these radon vectors in ways that represent the VSA system operation. There are three operators: one for filtering (flowing air forward through a column), one for regenerating a carbon column by flowing air backward through it, and one for slow-filling (bringing a column up to atmosphere). Each operator is a matrix in which the $j^\mathrm{th}$ row evolves the $j^\mathrm{th}$ position of $\boldsymbol{c}(x,t)$ by $\Delta t=1\,$min, which is the temporal resolution chosen for the simulation. The linear velocity of radon moving through a column is
\begin{equation}
v_\mathrm{Rn}(x,t) = v_\mathrm{air}(x,t) A_\circ \exp{\left(-T_\circ/T\right)},
\end{equation}
where $v_\mathrm{air}(x,t)$ is the linear air flow at a position $x$ and time $t$ through the column, $A_\circ$ is the radon adsorption factor, and $T_\circ \approx 3500\,$K is the critical temperature of radon adsorbing to carbon~\cite{Strong:1979aa}. The operators are built upon a normal distribution that has been modified to account for a nonzero spatial binning and depends on the linear radon velocity $v_\mathrm{Rn}$ and the diffusion coefficient $D$ (which we assume is constant) of radon passing through carbon:
\begin{equation}
\mathrm{Normal\_Dist}(x; v_\mathrm{Rn},D,j,\Delta t) = \frac{1}{2}\left[\mathrm{erf}\left(\dfrac{(x+\frac{1}{2}) -v_\mathrm{Rn}\Delta t - j}{\sqrt{4D\Delta t}}\right) - \mathrm{erf}\left(\dfrac{(x-\frac{1}{2}) -v_\mathrm{Rn}\Delta t - j}{\sqrt{4D\Delta t}}\right)\right] e^{-\Delta t/\tau},
\label{dist}
\end{equation}
where $e^{-\Delta t/\tau}$ accounts for the decay of radon after one minutes and $\tau$ is the mean-lifetime of radon. This normal distribution model gives similar results to the n-theoretical stages framework (e.g.,~\cite{Pocar:2003dw,Strong:1979aa}). Each operator is built upon Eq.~\ref{dist}, but with different radon velocities to account for filtering, regeneration or slow-fill.
\subsection{Characterization and Expected Radon Concentration at the VSA System Output}
\begin{figure}[t]
\begin{tabular}{c c}
\includegraphics[width=0.5\textwidth]{170727_ref_of_data_170212_5_RedChiSqrd0_98_withInput} &
\includegraphics[width=0.5\textwidth]{RegenCurves_170726}
\end{tabular}
\caption{\textit{Left}: A measured break-through curve (vertical error bars) at 90\,cfm behind a simulated break-through curve (solid) with a reduced $\chi^2$ of 0.98 and p-value of 0.22. This fit suggests $D=32\,$cm$^2$\,hour$^{-1}$ and $A_\circ=1.17$ for a breakthrough time $t_\mathrm{BT}=9.26\,$hours. The dashed line represents the radon concentration at the input, shifted by the break-through time, accounting for radon decay. \textit{Right}: Predicted radon concentration at the VSA system output versus clean room make-up air flow for various purge flows (curves), with a 100\,Bq\,m$^{-3}$ input radon concentration, compared to the measured upper limit of 0.067\,Bq\,m$^{-3}$ shown in Fig.~\ref{CR}. The star-curve represents the system as configured in February 2016 while the other curves represent the current system. The simulation shows that purge flow above 5 cfm does not further reduce the radon concentration. Radon emanating from the activated carbon dominates the radon concentration at the output for flows $<$200\,cfm. at low make-up air flow ($<$10\,cfm) and determines the lower limit on radon concentration. The contribution of radon emanating from the carbon decreases with increasing make-up air flow, until, at higher flows, radon begins breaking-through the columns due to only partial regeneration.}
\label{fig:BT_Curve}
\end{figure}
For the simulation to make accurate predictions, internal parameters must be inferred from measurements. These parameters are the diffusion coefficient $D$ and the radon adsorption factor $A_\circ$, both of which depend on details of the carbon and its packing.
The internal parameters are determined with so-called break-through curves (Fig.~\ref{fig:BT_Curve}, \textit{Left}). A break-through curve is produced when high radon air flows through a single, regenerated carbon column. At first, the air exiting the column was low radon. Radon adsorbs and desorbs from the carbon surfaces as it moves through a carbon column and diffuses over time according to Eq.~\ref{dist}. Eventually, radon begins ``breaking through" the column until the output of the column is similar to the input delayed by the breakthrough time but slightly decayed and diffused. The diffusion coefficient determines the spread of the radon distribution and $A_\circ$ determines its translation (modifying the radon velocity). Figure~\ref{fig:BT_Curve} (\textit{Left}) shows that the simulation agrees well with the data, although small systematic shifts suggest an even more accurate model may be possible.
The second characterization is from the equilibrium radon concentration at the VSA system output. This comparison is important because it examines not only the forward flow of the simulation but also regeneration and slow fill. Regeneration and slow-fill components of the simulation may be examined closely. This form of characterization is currently ongoing.
Once the simulation has been characterized, many VSA system operating parameters may be explored. Figure~\ref{fig:BT_Curve} (\textit{Right}) shows the simulated radon concentration at the VSA system output versus clean room make-up air flow for several values of the purge flow. The star-curve represents the VSA system configuration during February 2016. The other curves represent radon concentration at output for the current system configured for 40\,min of regeneration, 15\,min of slow-fill, and purge flows of 2, 3, or 5\,cfm. The simulation indicates that the VSA system should achieve radon reduction $>$100$\times$ larger than is typically achieved with (more expensive) continuous flow systems.
\section{ACKNOWLEDGMENTS}
This work was supported in part by the National Science Foundation (Grant No. PHY-1506033 and EEC-1461190) and the Pacific Northwest National Laboratory (PNNL), operated by Battelle for the U.S. Department of Energy (DOE) under Contract Nos. DE-AC05-76RL01830.
\bibliographystyle{aipnum-cp}
| {
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{"url":"https:\/\/mathematica.stackexchange.com\/questions\/105926\/fitting-bivariate-data-using-findfit","text":"# Fitting bivariate data using FindFit [closed]\n\nI am trying to fit the following bivariate data set using the FindFit function:\n\ndata = {{0.017, 1.091}, {0.034, 1.054}, {0.051, 1.130}, {0.068,\n1.226}, {0.085, 1.184}, {0.102, 1.307}, {0.119, 1.250}, {0.136,\n1.326}, {0.153, 1.324}, {0.17, 1.336}, {0.187, 1.314}, {0.204,\n1.382}, {0.221, 1.333}, {0.238, 1.402}, {0.255, 1.316}, {0.272,\n1.474}, {0.289, 1.382}, {0.306, 1.308}}\n\nHowever, when I use the function I get an error:\n\nFindFit[data, {a + b*Exp[-k*x]}, {a, b, k}, {x, y}]\n\nFindFit::cvmit: Failed to converge to the requested accuracy or precision within 100 iterations. >>\n\nAny help on how to fix this expression would be appreciated.\n\n\u2022 Check this prior thread or this other for simultaneous fitting. \u2013\u00a0Daniel Lichtblau Feb 8 '16 at 18:21\n\u2022 Are you sure about the {x, y} argument? You get the above warning only when you use {x}. Using the function as above yields another message: FindFit::fitc: Number of coordinates (1) is not equal to the number of variables (2). >> \u2013\u00a0Sjoerd C. de Vries Feb 8 '16 at 18:44\n\nI wonder if I misunderstand your problem, but it seems to me that perhaps your exponential decay model is inappropriate to your data:\n\ndata = {{0.017, 1.091}, {0.034, 1.054}, {0.051, 1.130}, {0.068, 1.226}, {0.085, 1.184},\n{0.102, 1.307}, {0.119, 1.250}, {0.136, 1.326}, {0.153, 1.324}, {0.17, 1.336},\n{0.187, 1.314}, {0.204, 1.382}, {0.221, 1.333}, {0.238, 1.402}, {0.255, 1.316},\n{0.272, 1.474}, {0.289, 1.382}, {0.306, 1.308}};\n\nListPlot[data, PlotRange -> All]\n\nPerhaps an exponential rise to max model such as $a\\ (1-b\\ e^{-kx})$ would be more appropriate:\n\nfit = FindFit[data, a (1 - b Exp[-k*x]), {a, b, k}, x]\n\n(* Out: {a -> 1.3988, b -> 0.301228, k -> 10.6606} *)\n\nThis seems to reproduce your data better:\n\nShow[{\nListPlot[data, PlotStyle -> {PointSize[0.015], Red}],\nPlot[a (1 - b Exp[-k*x]) \/. fit, {x, 0, 0.35}]},\nPlotRange -> All\n]\n\n\u2022 note the original expression (after fixing syntax issues) gives this same result if you supply a good initial guess for k : FindFit[data, a + b*Exp[-k*x], {a, b, {k, 20}}, x] \u2013\u00a0george2079 Feb 9 '16 at 15:57","date":"2020-02-17 17:22:39","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\": 0.1911621242761612, \"perplexity\": 11940.147825318096}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875142603.80\/warc\/CC-MAIN-20200217145609-20200217175609-00345.warc.gz\"}"} | null | null |
Don't miss this great CADILLAC!
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"redpajama_set_name": "RedPajamaC4"
} | 5,837 |
"""LibSVM"""
from tensorflow import sparse
from tensorflow_io.python.ops import core_ops
def decode_libsvm(content, num_features, dtype=None, label_dtype=None):
"""Convert Libsvm records to a tensor of label and a tensor of feature.
Args:
content: A `Tensor` of type `string`. Each string is a record/row in
the Libsvm format.
num_features: The number of features.
dtype: The type of the output feature tensor. Default to tf.float32.
label_dtype: The type of the output label tensor. Default to tf.int64.
Returns:
features: A `SparseTensor` of the shape `[input_shape, num_features]`.
labels: A `Tensor` of the same shape as content.
"""
labels, indices, values, shape = core_ops.io_decode_libsvm(
content, num_features, dtype=dtype, label_dtype=label_dtype
)
return sparse.SparseTensor(indices, values, shape), labels
def re2_full_match(input, pattern): # pylint: disable=redefined-builtin
"""Extract regex groups
Args:
input: A `tf.string` tensor
pattern: A pattern string.
"""
return core_ops.io_re2_full_match(input, pattern)
def read_text(filename, **kwargs):
"""read_text"""
memory = kwargs.get("memory", "")
offset = kwargs.get("offset", 0)
length = kwargs.get("length", -1)
return core_ops.io_read_text(filename, offset=offset, length=length, memory=memory)
class TextOutputSequence:
"""TextOutputSequence"""
def __init__(self, filenames):
"""Create a `TextOutputSequence`."""
self._filenames = filenames
self._resource = core_ops.io_text_output_sequence(destination=filenames)
def setitem(self, index, item):
core_ops.io_text_output_sequence_set_item(self._resource, index, item)
| {
"redpajama_set_name": "RedPajamaGithub"
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Hanna Dorota Szwed (ur. 1950) – polska kardiolog, prof. dr hab. nauk medycznych.
Życiorys
Hanna Szwed urodziła się w 1950. W 1990 uzyskała stopień doktora habilitowanego na podstawie oceny dorobku naukowego i pracy zatytułowanej Dynamika zaburzeń czynności u chorych z zawałem serca leczonych fibrynolitycznie i koronaroplastyką, a 14 grudnia 1999 otrzymała tytuł naukowy profesora nauk medycznych. Pełni funkcję profesora zwyczajnego w Instytucie Kardiologii im. Prymasa Tysiąclecia Kardynała Stefana Wyszyńskiego w Warszawie.
Została członkiem władz Polskiego Towarzystwa Kardiologicznego.
Publikacje
1998: Utility of transesophageal atrial pacing to estimate the results of catheter ablation of supraventricular tachycardias
2005: Porównawcza ocena skuteczności klinicznej dwóch dawek czteroazotanu pentaerytrytylu (100 mg i 30 mg) u pacjentów ze stabilną chorobą wieńcową
2005: Ocena żywotności mięśnia sercowego po zawale
2006: Wieloośrodkowy Ogólnopolski System Monitorowania Standardu Podstawowej Opieki Kardiologicznej w Warunkach Podstawowej Opieki zdrowotnej (POLKARD-SPOK) – metody przyjęte podczas realizacji programu
2010: High concentrations of B-type natriuretic peptide and left ventricular diastolic dysfunction in patients with paroxysmal/persistent atrial fibrillation as possible markers of conversion into permanent form of arrhythmia: 1-year prospective evaluation
2014: Translation and cultural adaptation of a Patient Perception of Arrhythmia Questionnaire in Poland
Przypisy
Członkowie Polskiego Towarzystwa Kardiologicznego
Polscy kardiolodzy
Pracownicy Instytutu Kardiologii w Warszawie
Urodzeni w 1950 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,332 |
Q: Replacing with backreference in TextMate issue I am using TextMate to replace expression [my_expression] consisting in characters between open and closed brackets by {my_expression}; so I tried to replace
\[[^]]*\]
by
{$1}
The regex matches the correct expression, but the replacement gives {$1}, so that the variable is not recognised. Can someone has an idea ?
A: You forgot to escape a character, [^]] should be [^\]].
You also need a capture group. $1 is back-referencing the 1st Capture Group, and you had no capture groups, so use the following Regex:
\[([^\]]*)\]
This adds () around [^\]]*, so the data inside the [] is captured. For more info, see this page on Capture Groups
However, this RegEx is shorter:
\[(.*?)\]
Also substituting with {$1}
Live Demo on Regex101
A: Use a capturing group (...):
\[([^\]]*)\]
The $1 is a backreference to the text enclosed with [...].
Here is the regex demo and also Numbered Backreferences.
Also, the TextMate docs:
1. Syntax elements
(...) group
20.4.1 Captures
To reference a capture, use $n where n is the capture register number. Using $0 means the entire match.
And also:
*
*If you want to use [, -, ] as a normal character in a character class, you should escape these characters by \.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,807 |
The Dobrodošao u Klub Tour (also styled as DUK Tour) was the second headlining concert tour by Croatian pop-folk singer Severina. It was launched to support of her eleventh studio album Dobrodošao u klub (2012). It was officially announced in February 2013, with dates for Balkan venues revealed. The tour began on 23 March 2013 in Rijeka, Croatia at Dvorana Mladosti, and concluded on 6 December of the same year in Split, Croatia at the Spaladium Arena.
The tour was also included some festival concerts. On 29 June 2013, Her tour was also a part of Celebration of Croatia's accession to the European Union. Severina also performed at the Strumica Open Festival, Macedonia on 18 July 2013.
Set list
The concert set list consists of 29 songs. As part of the concert, a part of the foreign song "Harlem Shake" was included.
"Italiana"
"Uzbuna"
"Lola"
"Harlem Shake"
"Tarapana"
"Mili Moj"
"Gade"
"Daj da biram"
"Tango" (Maestro dance Crew)"
"Dobrodošao u klub"
"Kamen Oko Vrata"
"Ostavljena"
"Prijateljice"
"Kradeš sve"
"Ko Me Tjero"
"Tuge od sna"
"Grad bez ljudi"
"Molitva: Gardelin"
"Pogled ispod obrva"
"Tridesete"
"Šta me sad pitaš šta mi je"
"Virujen u te"
"Maestro dance Crew/2CELLOS"
"Krivi spoj"
"Ajde, ajde zlato moje"
"Gas Gas"
"Ja samo pjevam"
"Brad Pitt"
Tour dates
Box office score data
References
2013 concert tours | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,886 |
Owen Keir Davidson (født 4. oktober 1943 i Melbourne, Australien) er en tennisspiller fra Australien. Han var en af verdens bedste doublespillere i 1960'erne og 1970'erne og vandt i løbet af sin karriere 13 grand slam-titler: 2 i herredouble og 11 i mixed double, heraf 8 titler med Billie Jean King som makker. Hans bedste grand slam-resultat i single var semifinalepladsen ved WImbledon-mesterskaberne 1966.
Han vandt 29 WTA-turneringer i single, heraf seks Tier I-titler, og 69 WTA-doubletitler, heraf to WTA Tour Championships og 16 titler på Tier I-niveau.
Han blev i 2010 valgt ind i International Tennis Hall of Fame.
Eksterne kilder/henvisninger
Tennisspillere fra Australien
Personer fra Melbourne | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,160 |
Sir Wilfrid Edward Le Gros Clark (5 June 1895 – 28 June 1971) was a British anatomist, surgeon, primatologist and palaeoanthropologist, today best remembered for his contribution to the study of human evolution. He was Dr Lee's Professor of Anatomy at the University of Oxford.
Education
Le Gros Clark was educated at Blundell's School and subsequently admitted as a medical student to St Thomas' Hospital Medical School in Lambeth.
Career
After qualification he immediately joined the Royal Army Medical Corps as a medical officer and was sent to France early in 1918. He caught diphtheria and was sent back to England to recover, following which he spent the remainder of the war as a medical officer at ''No. 8 Stationary Hospital'' at Wimereux in northern France.
Following a period in the Department of Anatomy at St Thomas' Hospital Medical School he was appointed as Principal Medical Officer to the Sarawak Government. He was subsequently appointed as Professor of Anatomy at St Bartholomew's Hospital Medical School, followed by a period as Professor of Anatomy at St Thomas' Hospital Medical School and finally, in 1934, he was invited to take over as the Dr. Lee's Professor of Anatomy (and effectively the Chair of the Anatomy Department) at the University of Oxford. The following year he was elected a Fellow of the Royal Society. Le Gros Clark was also editor of the Journal of Anatomy between 1938 and 1945.
In 1953, Le Gros Clark was one of three men (the others being Joseph Weiner and Kenneth Oakley) who proved that the Piltdown Man was a forgery.
In 1960, Le Gros Clark was elected to the American Philosophical Society. He was awarded the Royal Society's Royal Medal in 1961 and delivered their Ferrier Lecture in 1956. He was elected President of the Anatomical Society of Great Britain and Ireland for 1951 to 1953.
Papers relating to Le Gros Clark, his grandfather the surgeon Frederick Le Gros Clark and his brother Cyril Le Gros Clark (former Chief Secretary of Sarawak, who was murdered by the Japanese in 1945 after a period of detention at Batu Lintang camp in Borneo) are held at the Bodleian Library (Special Collections and Western Manuscripts) at Oxford University. During his career Le Gros Clark published numerous papers on human evolution and palaeontology, and an autobiography.
References
External links
1895 births
1971 deaths
People educated at Blundell's School
British surgeons
Fellows of the Royal College of Surgeons
British paleoanthropologists
Human evolution theorists
Fellows of the Royal Society
Foreign associates of the National Academy of Sciences
Royal Medal winners
20th-century British writers
Dr Lee's Professors of Anatomy
20th-century surgeons
British Army personnel of World War I
Royal Army Medical Corps officers
Journal of Anatomy editors
Members of the American Philosophical Society | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,448 |
Q: Pass value by reference to a thread and modify I want to modify a Qlist by reference in a thread because the sorting could take a few seconds.
But it seems like that I can't modify the List. Outside of the thread the List has not beend changed.
In QMainwindow:
QThread *thread = new QThread();
FarmSortWorker *farmsort_worker = new FarmSortWorker();
farmsort_worker->setFarmData(farm_list);
farmsort_worker->moveToThread(thread);
connect(farmsort_worker, &FarmSortWorker::error, [=](QString error_msg){
qDebug() << error_msg;
logger->logEvent(error_msg, Logger::ERROR);
});
connect(thread, &QThread::started, farmsort_worker, &FarmSortWorker::processSort);
connect(farmsort_worker, &FarmSortWorker::finished, thread, &QThread::quit);
connect(farmsort_worker, &FarmSortWorker::finished, farmsort_worker, &QThread::deleteLater);
connect(thread, &QThread::finished, thread, &QThread::deleteLater);
thread->start();
My Thread (.h)
#ifndef FARMSORTWORKER_H
#define FARMSORTWORKER_H
#include <QObject>
#include "../framcontainer.h"
#include <QList>
#include <qDebug>
#include <QString>
class FarmSortWorker : public QObject
{
Q_OBJECT
public:
explicit FarmSortWorker(QObject *parent = 0);
~FarmSortWorker();
void setFarmData(QList<FramContainer> &farm_container);
private:
QList<FramContainer> farm_container;
signals:
void error(QString error);
void finished();
public slots:
void processSort();
};
#endif // FARMSORTWORKER_H
My Thread (.cpp)
#include "farmsortworker.h"
FarmSortWorker::FarmSortWorker(QObject *parent) :
QObject(parent)
{
}
FarmSortWorker::~FarmSortWorker()
{
}
void FarmSortWorker::setFarmData(QList<FramContainer> &farm_container)
{
this->farm_container = farm_container;
}
void FarmSortWorker::processSort()
{
qDebug() << "count:" << farm_container.size();
for(int i = 0; i < farm_container.size(); i++)
{
FramContainer park = farm_container.at(i);
qDebug() << "original:" << park.getFarmName();
}
for(int i = 0; i < farm_container.size(); i++)
{
FramContainer *park =& farm_container[i];
park->setFarmName("test");
}
for(int i = 0; i < farm_container.size(); i++)
{
FramContainer park = farm_container.at(i);
qDebug() << "neu:" << park.getFarmName();
}
emit finished();
}
Thank you all in advance.
A: It's because you save a copy of the list in your thread class, not a reference.
Instead of having a separate function to set the list, pass it (by reference) to the constructor, and have it set the reference variable:
class FarmSortWorker : public QObject
{
Q_OBJECT
public:
explicit FarmSortWorker(QList<FramContainer>& list, QObject *parent = 0);
...
private:
QList<FramContainer>& farm_container; // Store as a reference
...
};
...
FarmSortWorker::FarmSortWorker(QList<FramContainer>& list, QObject *parent) :
QObject(parent), farm_container(list)
{
}
...
QThread *thread = new QThread();
FarmSortWorker *farmsort_worker = new FarmSortWorker(farm_list);
...
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,194 |
Q: Is there an alternative for the soon-to-be-deprecated pandas.util.testing.assert_produces_warning? I don't see one in the API, though the warning suggests it should be available:
In [1]: import pandas
In [2]: from pandas.util.testing import assert_produces_warning
/home/mghenis/anaconda3/bin/ipython:1: FutureWarning: pandas.util.testing is deprecated. Use the functions in the public API at pandas.testing instead.
#!/home/mghenis/anaconda3/bin/python
In [3]: from pandas.testing import assert_produces_warning
---------------------------------------------------------------------------
ImportError Traceback (most recent call last)
<ipython-input-3-38c6b9f78373> in <module>
----> 1 from pandas.testing import assert_produces_warning
ImportError: cannot import name 'assert_produces_warning' from 'pandas.testing' (/home/mghenis/anaconda3/lib/python3.7/site-packages/pandas/testing.py)
A: They've moved this into the "private" API in _testing.py.
Really you should probably be using pytest's with pytest.warns() methods directly for this and I suspect that's why they moved it, though I can't find the discussion around that choice.
A: *
*The source code for assert_produces_warning states that it is a wrapper for warnings.catch_warnings.
*
*It seems using warnings.catch_warnings is the best option.
*Examples
*From pandas 1.0.0 Release Notes:
*
*The pandas.util.testing module has been deprecated. Use the public API in pandas.testing documented at Testing functions (GH16232).
*As per pandas 0.25 Test Warnings for pandas.util.testing.assert_produces_warning
*
*We prefer this to the pytest.warns context manager because ours checks that the warning's stacklevel is set correctly.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,113 |
/*
* Zed Attack Proxy (ZAP) and its related class files.
*
* ZAP is an HTTP/HTTPS proxy for assessing web application security.
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.zaproxy.zap.extension.httppanel.view.syntaxhighlight;
import java.awt.Color;
import java.awt.Component;
import java.util.LinkedList;
import java.util.List;
import java.util.Observable;
import java.util.Observer;
import java.util.Vector;
import java.util.regex.Pattern;
import javax.swing.Action;
import javax.swing.JMenuItem;
import javax.swing.JPopupMenu;
import javax.swing.JViewport;
import javax.swing.text.BadLocationException;
import javax.swing.text.DefaultHighlighter;
import javax.swing.text.Highlighter;
import javax.swing.text.Highlighter.HighlightPainter;
import org.apache.commons.configuration.FileConfiguration;
import org.apache.log4j.Logger;
import org.fife.ui.rsyntaxtextarea.AbstractTokenMakerFactory;
import org.fife.ui.rsyntaxtextarea.RSyntaxDocument;
import org.fife.ui.rsyntaxtextarea.RSyntaxTextArea;
import org.fife.ui.rsyntaxtextarea.SyntaxConstants;
import org.fife.ui.rtextarea.RTextArea;
import org.fife.ui.rtextarea.RTextScrollPane;
import org.parosproxy.paros.Constant;
import org.parosproxy.paros.extension.ExtensionPopupMenuItem;
import org.parosproxy.paros.view.View;
import org.zaproxy.zap.extension.httppanel.Message;
import org.zaproxy.zap.extension.httppanel.view.syntaxhighlight.menus.SyntaxMenu;
import org.zaproxy.zap.extension.httppanel.view.syntaxhighlight.menus.ViewMenu;
import org.zaproxy.zap.extension.search.SearchMatch;
import org.zaproxy.zap.view.HighlightSearchEntry;
import org.zaproxy.zap.view.HighlighterManager;
public abstract class HttpPanelSyntaxHighlightTextArea extends RSyntaxTextArea implements Observer {
private static final long serialVersionUID = -9082089105656842054L;
private static Logger log = Logger.getLogger(HttpPanelSyntaxHighlightTextArea.class);
public static final String PLAIN_SYNTAX_LABEL = Constant.messages.getString("http.panel.view.syntaxtext.syntax.plain");
private static final String ANTI_ALIASING = "aa";
private static final String SHOW_LINE_NUMBERS = "linenumbers";
private static final String WORD_WRAP = "wordwrap";
private static final String HIGHLIGHT_CURRENT_LINE = "highlightline";
private static final String FADE_CURRENT_HIGHLIGHT_LINE = "fadehighlightline";
private static final String SHOW_WHITESPACE_CHARACTERS = "whitespaces";
private static final String SHOW_NEWLINE_CHARACTERS = "newlines";
private static final String MARK_OCCURRENCES = "markocurrences";
private static final String ROUNDED_SELECTION_EDGES = "roundedselection";
private static final String BRACKET_MATCHING = "bracketmatch";
private static final String ANIMATED_BRACKET_MATCHING = "animatedbracketmatch";
private Message message;
private Vector<SyntaxStyle> syntaxStyles;
private static SyntaxMenu syntaxMenu = null;
private static ViewMenu viewMenu = null;
private static TextAreaMenuItem cutAction = null;
private static TextAreaMenuItem copyAction = null;
private static TextAreaMenuItem pasteAction = null;
private static TextAreaMenuItem deleteAction = null;
private static TextAreaMenuItem undoAction = null;
private static TextAreaMenuItem redoAction = null;
private static TextAreaMenuItem selectAllAction = null;
public HttpPanelSyntaxHighlightTextArea() {
((RSyntaxDocument)getDocument()).setTokenMakerFactory(getTokenMakerFactory());
setSyntaxEditingStyle(SyntaxConstants.SYNTAX_STYLE_NONE);
syntaxStyles = new Vector<>();
addSyntaxStyle(PLAIN_SYNTAX_LABEL, SyntaxConstants.SYNTAX_STYLE_NONE);
if (syntaxMenu == null) {
initActions();
}
setPopupMenu(null);
this.message = null;
setHyperlinksEnabled(false);
setAntiAliasingEnabled(true);
setLineWrap(true);
setHighlightCurrentLine(false);
setFadeCurrentLineHighlight(false);
setWhitespaceVisible(false);
setEOLMarkersVisible(false);
setMarkOccurrences(false);
setBracketMatchingEnabled(false);
setAnimateBracketMatching(false);
setAutoIndentEnabled(false);
setCloseCurlyBraces(false);
setCloseMarkupTags(false);
setClearWhitespaceLinesEnabled(false);
initHighlighter();
}
@Override
protected JPopupMenu createPopupMenu() {
return null;
}
private void initHighlighter() {
HighlighterManager highlighter = HighlighterManager.getInstance();
highlighter.addObserver(this);
if (message != null) {
highlightAll();
}
}
// Highlight all search strings from HighlightManager
private void highlightAll() {
HighlighterManager highlighter = HighlighterManager.getInstance();
LinkedList<HighlightSearchEntry> highlights = highlighter.getHighlights();
for (HighlightSearchEntry entry: highlights) {
highlightEntryParser(entry);
}
}
// Parse the TextArea data and search the HighlightEntry strings
// Highlight all found strings
private void highlightEntryParser(HighlightSearchEntry entry) {
String text;
int lastPos = 0;
text = this.getText();
Highlighter hilite = this.getHighlighter();
HighlightPainter painter = new DefaultHighlighter.DefaultHighlightPainter(entry.getColor());
while ( (lastPos = text.indexOf(entry.getToken(), lastPos)) > -1) {
try {
hilite.addHighlight(lastPos, lastPos + entry.getToken().length(), painter);
lastPos += entry.getToken().length();
} catch (BadLocationException e) {
log.warn("Could not highlight entry", e);
}
}
}
@Override
// Apply highlights after a setText()
public void setText(String s) {
super.setText(s);
highlightAll();
}
public abstract void search(Pattern p, List<SearchMatch> matches);
// highlight a specific SearchMatch in the editor
public abstract void highlight(SearchMatch sm);
protected void highlight(int start, int end) {
Highlighter hilite = this.getHighlighter();
HighlightPainter painter = new DefaultHighlighter.DefaultHighlightPainter(Color.LIGHT_GRAY);
try {
// DOBIN
removeAllHighlights();
hilite.addHighlight(start, end, painter);
this.setCaretPosition(start);
} catch (BadLocationException e) {
log.error(e.getMessage(), e);
}
}
public Object highlight(int start, int end, HighlightPainter painter) {
try {
Object highlightReference = getHighlighter().addHighlight(start, end, painter);
this.setCaretPosition(start);
return highlightReference;
} catch (BadLocationException e) {
log.error(e.getMessage(), e);
}
return null;
}
public void removeHighlight(Object highlightReference) {
getHighlighter().removeHighlight(highlightReference);
}
private void removeAllHighlights() {
Highlighter hilite = this.getHighlighter();
hilite.removeAllHighlights();
}
@Override
// HighlighterManager called us
// there is either
// - a new highlight
// - something other (added several, deleted, ...).
public void update(Observable arg0, Object arg1) {
if (arg1 == null) {
// Re-highlight everything
removeAllHighlights();
highlightAll();
} else {
// Add specific highlight
HighlightSearchEntry token = (HighlightSearchEntry) arg1;
highlightEntryParser(token);
}
this.invalidate();
}
public void setMessage(Message aMessage) {
this.message = aMessage;
}
public Message getMessage() {
return message;
}
public void loadConfiguration(String key, FileConfiguration fileConfiguration) {
setAntiAliasingEnabled(fileConfiguration.getBoolean(key + ANTI_ALIASING, this.getAntiAliasingEnabled()));
Component c = getParent();
if (c instanceof JViewport) {
c = c.getParent();
if (c instanceof RTextScrollPane) {
final RTextScrollPane scrollPane = (RTextScrollPane)c;
scrollPane.setLineNumbersEnabled(fileConfiguration.getBoolean(key + SHOW_LINE_NUMBERS, scrollPane.getLineNumbersEnabled()));
}
}
setLineWrap(fileConfiguration.getBoolean(key + WORD_WRAP, this.getLineWrap()));
setHighlightCurrentLine(fileConfiguration.getBoolean(key + HIGHLIGHT_CURRENT_LINE, this.getHighlightCurrentLine()));
setFadeCurrentLineHighlight(fileConfiguration.getBoolean(key + FADE_CURRENT_HIGHLIGHT_LINE, this.getFadeCurrentLineHighlight()));
setWhitespaceVisible(fileConfiguration.getBoolean(key + SHOW_WHITESPACE_CHARACTERS, this.isWhitespaceVisible()));
setEOLMarkersVisible(fileConfiguration.getBoolean(key + SHOW_NEWLINE_CHARACTERS, this.getEOLMarkersVisible()));
setMarkOccurrences(fileConfiguration.getBoolean(key + MARK_OCCURRENCES, this.getMarkOccurrences()));
setRoundedSelectionEdges(fileConfiguration.getBoolean(key + ROUNDED_SELECTION_EDGES, this.getRoundedSelectionEdges()));
setBracketMatchingEnabled(fileConfiguration.getBoolean(key + BRACKET_MATCHING, this.isBracketMatchingEnabled()));
setAnimateBracketMatching(fileConfiguration.getBoolean(key + ANIMATED_BRACKET_MATCHING, this.getAnimateBracketMatching()));
}
public void saveConfiguration(String key, FileConfiguration fileConfiguration) {
fileConfiguration.setProperty(key + ANTI_ALIASING, Boolean.valueOf(this.getAntiAliasingEnabled()));
Component c = getParent();
if (c instanceof JViewport) {
c = c.getParent();
if (c instanceof RTextScrollPane) {
final RTextScrollPane scrollPane = (RTextScrollPane)c;
fileConfiguration.setProperty(key + SHOW_LINE_NUMBERS, Boolean.valueOf(scrollPane.getLineNumbersEnabled()));
}
}
fileConfiguration.setProperty(key + WORD_WRAP, Boolean.valueOf(this.getLineWrap()));
fileConfiguration.setProperty(key + HIGHLIGHT_CURRENT_LINE, Boolean.valueOf(this.getHighlightCurrentLine()));
fileConfiguration.setProperty(key + FADE_CURRENT_HIGHLIGHT_LINE, Boolean.valueOf(this.getFadeCurrentLineHighlight()));
fileConfiguration.setProperty(key + SHOW_WHITESPACE_CHARACTERS, Boolean.valueOf(this.isWhitespaceVisible()));
fileConfiguration.setProperty(key + SHOW_NEWLINE_CHARACTERS, Boolean.valueOf(this.getEOLMarkersVisible()));
fileConfiguration.setProperty(key + MARK_OCCURRENCES, Boolean.valueOf(this.getMarkOccurrences()));
fileConfiguration.setProperty(key + ROUNDED_SELECTION_EDGES, Boolean.valueOf(this.getRoundedSelectionEdges()));
fileConfiguration.setProperty(key + BRACKET_MATCHING, Boolean.valueOf(this.isBracketMatchingEnabled()));
fileConfiguration.setProperty(key + ANIMATED_BRACKET_MATCHING, Boolean.valueOf(this.getAnimateBracketMatching()));
}
public Vector<SyntaxStyle> getSyntaxStyles() {
return syntaxStyles;
}
protected void addSyntaxStyle(String label, String styleKey) {
syntaxStyles.add(new SyntaxStyle(label, styleKey));
}
protected abstract CustomTokenMakerFactory getTokenMakerFactory();
private static synchronized void initActions() {
if (syntaxMenu == null) {
syntaxMenu = new SyntaxMenu();
viewMenu = new ViewMenu();
undoAction = new TextAreaMenuItem(RTextArea.UNDO_ACTION, true, false);
redoAction = new TextAreaMenuItem(RTextArea.REDO_ACTION, false, true);
cutAction = new TextAreaMenuItem(RTextArea.CUT_ACTION, false, false);
copyAction = new TextAreaMenuItem(RTextArea.COPY_ACTION, false, false);
pasteAction = new TextAreaMenuItem(RTextArea.PASTE_ACTION, false, false);
deleteAction = new TextAreaMenuItem(RTextArea.DELETE_ACTION, false, true);
selectAllAction = new TextAreaMenuItem(RTextArea.SELECT_ALL_ACTION, false, false);
final List<JMenuItem> mainPopupMenuItems = View.getSingleton().getPopupList();
mainPopupMenuItems.add(syntaxMenu);
mainPopupMenuItems.add(viewMenu);
mainPopupMenuItems.add(undoAction);
mainPopupMenuItems.add(redoAction);
mainPopupMenuItems.add(cutAction);
mainPopupMenuItems.add(copyAction);
mainPopupMenuItems.add(pasteAction);
mainPopupMenuItems.add(deleteAction);
mainPopupMenuItems.add(selectAllAction);
}
}
public static class SyntaxStyle {
private String label;
private String styleKey;
public SyntaxStyle(String label, String styleKey) {
this.label = label;
this.styleKey = styleKey;
}
public String getLabel() {
return label;
}
public String getStyleKey() {
return styleKey;
}
}
protected static class CustomTokenMakerFactory extends AbstractTokenMakerFactory {
@Override
protected void initTokenMakerMap() {
String pkg = "org.fife.ui.rsyntaxtextarea.modes.";
putMapping(SYNTAX_STYLE_NONE, pkg + "PlainTextTokenMaker");
}
}
private static class TextAreaMenuItem extends ExtensionPopupMenuItem {
private static final long serialVersionUID = -8369459846515841057L;
private int actionId;
private boolean precedeWithSeparator;
private boolean succeedWithSeparator;
public TextAreaMenuItem(int actionId, boolean precedeWithSeparator, boolean succeedWithSeparator) throws IllegalArgumentException {
this.actionId = actionId;
this.precedeWithSeparator = precedeWithSeparator;
this.succeedWithSeparator = succeedWithSeparator;
Action action = RTextArea.getAction(actionId);
if(action == null) {
throw new IllegalArgumentException("Action not found with id: " + actionId);
}
setAction(action);
}
@Override
public boolean isEnableForComponent(Component invoker) {
if (invoker instanceof HttpPanelSyntaxHighlightTextArea) {
HttpPanelSyntaxHighlightTextArea httpPanelTextArea = (HttpPanelSyntaxHighlightTextArea)invoker;
switch(actionId) {
case RTextArea.CUT_ACTION:
if (!httpPanelTextArea.isEditable()) {
this.setEnabled(false);
}
break;
case RTextArea.DELETE_ACTION:
case RTextArea.PASTE_ACTION:
this.setEnabled(httpPanelTextArea.isEditable());
break;
case RTextArea.SELECT_ALL_ACTION:
this.setEnabled(httpPanelTextArea.getDocument().getLength() != 0);
break;
}
return true;
}
return false;
}
@Override
public boolean precedeWithSeparator() {
return precedeWithSeparator;
}
@Override
public boolean succeedWithSeparator() {
return succeedWithSeparator;
}
@Override
public boolean isSafe() {
return true;
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,720 |
Tsugaike's dual terrain park setup not only gives riders the chance to fight boredom by experiencing two different settings, but also helps ensure freestyle enthusiasts have uninterrupted park access despite changing conditions during the course of the long snow season.
* The exact dates of operation & number and variety of items are subject to weather and snowpack conditions.
*Shown above are the previous year's items. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,947 |
{"url":"https:\/\/physics.stackexchange.com\/questions\/590797\/what-is-the-relationship-between-torque-and-angular-momentum-for-a-non-point-mas","text":"# What is the relationship between torque and angular momentum for a non-point mass?\n\nFor a point mass, $$\\tau=I\\alpha$$ and $$L=I \\omega$$.\n\nWhat if we apply a force to the edge of an ellipse, if the torque is $$rF\\sin\\theta$$, where $$r$$ is the distance between the point where the force is applied to the centre of mass of the ellipse, then the equation relating angular momentum and this force will then be $$rFt\\sin\\theta=I\\omega$$ or $$mvr\\sin\\theta=I\\omega$$, where \"I\" will be the moment of inertia of ellipse about its centre of mass. However, $$mvr\\sin\\theta$$ is the angular momentum of a point mass instead of an ellipse.\n\nWhat is the mistake? Thank you.","date":"2021-10-18 10:00:07","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 7, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.850793182849884, \"perplexity\": 65.55892459802986}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323585201.94\/warc\/CC-MAIN-20211018093606-20211018123606-00521.warc.gz\"}"} | null | null |
/*
* ArrayBuffer.prototype.slice()
*/
/*@include util-buffer.js@*/
/*---
custom: true
---*/
/*===
false 4 41424344
false 3 424344
false 1 44
false 0
false 4 41424344
false 2 4243
false 2 4243
false 3 424344
false 0
false 0
===*/
function arrayBufferSliceBasicTest() {
var b;
var buf = new ArrayBuffer(4);
var u8 = new Uint8Array(buf);
u8[0] = 0x41; u8[1] = 0x42; u8[2] = 0x43; u8[3] = 0x44; // ABCD
// No arguments, return copy.
b = buf.slice();
print(b === buf, b.byteLength, printableBuffer(b));
// Single argument, offset.
b = buf.slice(1);
print(b === buf, b.byteLength, printableBuffer(b));
// Negative argument is interpreted from end of buffer. This is not
// clearly specified in Khronos specification (it talks about clamping)
// but ES2015 clarifies handling for negative indices:
// http://www.ecma-international.org/ecma-262/6.0/index.html#sec-%typedarray%.prototype.slice
b = buf.slice(-1);
print(b === buf, b.byteLength, printableBuffer(b));
// Indices are clamped, after taking account the negative number handling.
b = buf.slice(10);
print(b === buf, b.byteLength, printableBuffer(b));
b = buf.slice(-100);
print(b === buf, b.byteLength, printableBuffer(b));
// End argument behaves similarly.
b = buf.slice(1, 3);
print(b === buf, b.byteLength, printableBuffer(b));
b = buf.slice(1, -1);
print(b === buf, b.byteLength, printableBuffer(b));
b = buf.slice(1, 10);
print(b === buf, b.byteLength, printableBuffer(b));
b = buf.slice(1, -100);
print(b === buf, b.byteLength, printableBuffer(b));
// Crossed indices result in a zero size buffer.
b = buf.slice(3, 2);
print(b === buf, b.byteLength, printableBuffer(b));
}
try {
arrayBufferSliceBasicTest();
} catch (e) {
print(e.stack || e);
}
/*===
ArrayBuffer slice() bruteforce test
0 0 0 TypeError
0 0 1 TypeError
0 0 2 TypeError
0 0 3 TypeError
0 0 4 TypeError
0 0 5 TypeError
0 1 0 TypeError
0 1 1 TypeError
0 1 2 TypeError
0 1 3 TypeError
0 1 4 TypeError
0 1 5 TypeError
0 2 0 TypeError
0 2 1 TypeError
0 2 2 TypeError
0 2 3 TypeError
0 2 4 TypeError
0 2 5 TypeError
0 3 0 TypeError
0 3 1 TypeError
0 3 2 TypeError
0 3 3 TypeError
0 3 4 TypeError
0 3 5 TypeError
0 4 0 TypeError
0 4 1 TypeError
0 4 2 TypeError
0 4 3 TypeError
0 4 4 TypeError
0 4 5 TypeError
0 5 0 TypeError
0 5 1 TypeError
0 5 2 TypeError
0 5 3 TypeError
0 5 4 TypeError
0 5 5 TypeError
0 6 0 TypeError
0 6 1 TypeError
0 6 2 TypeError
0 6 3 TypeError
0 6 4 TypeError
0 6 5 TypeError
0 7 0 TypeError
0 7 1 TypeError
0 7 2 TypeError
0 7 3 TypeError
0 7 4 TypeError
0 7 5 TypeError
1 0 0 object undefined 0 undefined undefined
1 0 1 object undefined 0 undefined undefined
1 0 2 object undefined 0 undefined undefined
1 0 3 object undefined 0 undefined undefined
1 0 4 object undefined 0 undefined undefined
1 0 5 object undefined 0 undefined undefined
1 1 0 object undefined 0 undefined undefined
1 1 1 object undefined 0 undefined undefined
1 1 2 object undefined 0 undefined undefined
1 1 3 object undefined 0 undefined undefined
1 1 4 object undefined 0 undefined undefined
1 1 5 object undefined 0 undefined undefined
1 2 0 object undefined 0 undefined undefined
1 2 1 object undefined 0 undefined undefined
1 2 2 object undefined 0 undefined undefined
1 2 3 object undefined 0 undefined undefined
1 2 4 object undefined 0 undefined undefined
1 2 5 object undefined 0 undefined undefined
1 3 0 object undefined 0 undefined undefined
1 3 1 object undefined 0 undefined undefined
1 3 2 object undefined 0 undefined undefined
1 3 3 object undefined 0 undefined undefined
1 3 4 object undefined 0 undefined undefined
1 3 5 object undefined 0 undefined undefined
1 4 0 object undefined 0 undefined undefined
1 4 1 object undefined 0 undefined undefined
1 4 2 object undefined 0 undefined undefined
1 4 3 object undefined 0 undefined undefined
1 4 4 object undefined 0 undefined undefined
1 4 5 object undefined 0 undefined undefined
1 5 0 object undefined 0 undefined undefined
1 5 1 object undefined 0 undefined undefined
1 5 2 object undefined 0 undefined undefined
1 5 3 object undefined 0 undefined undefined
1 5 4 object undefined 0 undefined undefined
1 5 5 object undefined 0 undefined undefined
1 6 0 object undefined 0 undefined undefined
1 6 1 object undefined 0 undefined undefined
1 6 2 object undefined 0 undefined undefined
1 6 3 object undefined 0 undefined undefined
1 6 4 object undefined 0 undefined undefined
1 6 5 object undefined 0 undefined undefined
1 7 0 object undefined 0 undefined undefined
1 7 1 object undefined 0 undefined undefined
1 7 2 object undefined 0 undefined undefined
1 7 3 object undefined 0 undefined undefined
1 7 4 object undefined 0 undefined undefined
1 7 5 object undefined 0 undefined undefined
2 0 0 object undefined 4 undefined undefined 41424344
2 0 1 object undefined 4 undefined undefined 41424344
2 0 2 object undefined 4 undefined undefined 41424344
2 0 3 object undefined 4 undefined undefined 41424344
2 0 4 object undefined 4 undefined undefined 41424344
2 0 5 object undefined 4 undefined undefined 41424344
2 1 0 object undefined 4 undefined undefined 41424344
2 1 1 object undefined 3 undefined undefined 414243
2 1 2 object undefined 0 undefined undefined
2 1 3 object undefined 1 undefined undefined 41
2 1 4 object undefined 4 undefined undefined 41424344
2 1 5 object undefined 4 undefined undefined 41424344
2 2 0 object undefined 1 undefined undefined 44
2 2 1 object undefined 0 undefined undefined
2 2 2 object undefined 0 undefined undefined
2 2 3 object undefined 0 undefined undefined
2 2 4 object undefined 1 undefined undefined 44
2 2 5 object undefined 1 undefined undefined 44
2 3 0 object undefined 4 undefined undefined 41424344
2 3 1 object undefined 3 undefined undefined 414243
2 3 2 object undefined 0 undefined undefined
2 3 3 object undefined 1 undefined undefined 41
2 3 4 object undefined 4 undefined undefined 41424344
2 3 5 object undefined 4 undefined undefined 41424344
2 4 0 object undefined 3 undefined undefined 424344
2 4 1 object undefined 2 undefined undefined 4243
2 4 2 object undefined 0 undefined undefined
2 4 3 object undefined 0 undefined undefined
2 4 4 object undefined 3 undefined undefined 424344
2 4 5 object undefined 3 undefined undefined 424344
2 5 0 object undefined 1 undefined undefined 44
2 5 1 object undefined 0 undefined undefined
2 5 2 object undefined 0 undefined undefined
2 5 3 object undefined 0 undefined undefined
2 5 4 object undefined 1 undefined undefined 44
2 5 5 object undefined 1 undefined undefined 44
2 6 0 object undefined 0 undefined undefined
2 6 1 object undefined 0 undefined undefined
2 6 2 object undefined 0 undefined undefined
2 6 3 object undefined 0 undefined undefined
2 6 4 object undefined 0 undefined undefined
2 6 5 object undefined 0 undefined undefined
2 7 0 object undefined 0 undefined undefined
2 7 1 object undefined 0 undefined undefined
2 7 2 object undefined 0 undefined undefined
2 7 3 object undefined 0 undefined undefined
2 7 4 object undefined 0 undefined undefined
2 7 5 object undefined 0 undefined undefined
3 0 0 object undefined 8 undefined undefined 6162636465666768
3 0 1 object undefined 8 undefined undefined 6162636465666768
3 0 2 object undefined 8 undefined undefined 6162636465666768
3 0 3 object undefined 8 undefined undefined 6162636465666768
3 0 4 object undefined 8 undefined undefined 6162636465666768
3 0 5 object undefined 8 undefined undefined 6162636465666768
3 1 0 object undefined 8 undefined undefined 6162636465666768
3 1 1 object undefined 7 undefined undefined 61626364656667
3 1 2 object undefined 0 undefined undefined
3 1 3 object undefined 1 undefined undefined 61
3 1 4 object undefined 5 undefined undefined 6162636465
3 1 5 object undefined 8 undefined undefined 6162636465666768
3 2 0 object undefined 1 undefined undefined 68
3 2 1 object undefined 0 undefined undefined
3 2 2 object undefined 0 undefined undefined
3 2 3 object undefined 0 undefined undefined
3 2 4 object undefined 0 undefined undefined
3 2 5 object undefined 1 undefined undefined 68
3 3 0 object undefined 8 undefined undefined 6162636465666768
3 3 1 object undefined 7 undefined undefined 61626364656667
3 3 2 object undefined 0 undefined undefined
3 3 3 object undefined 1 undefined undefined 61
3 3 4 object undefined 5 undefined undefined 6162636465
3 3 5 object undefined 8 undefined undefined 6162636465666768
3 4 0 object undefined 7 undefined undefined 62636465666768
3 4 1 object undefined 6 undefined undefined 626364656667
3 4 2 object undefined 0 undefined undefined
3 4 3 object undefined 0 undefined undefined
3 4 4 object undefined 4 undefined undefined 62636465
3 4 5 object undefined 7 undefined undefined 62636465666768
3 5 0 object undefined 5 undefined undefined 6465666768
3 5 1 object undefined 4 undefined undefined 64656667
3 5 2 object undefined 0 undefined undefined
3 5 3 object undefined 0 undefined undefined
3 5 4 object undefined 2 undefined undefined 6465
3 5 5 object undefined 5 undefined undefined 6465666768
3 6 0 object undefined 1 undefined undefined 68
3 6 1 object undefined 0 undefined undefined
3 6 2 object undefined 0 undefined undefined
3 6 3 object undefined 0 undefined undefined
3 6 4 object undefined 0 undefined undefined
3 6 5 object undefined 1 undefined undefined 68
3 7 0 object undefined 0 undefined undefined
3 7 1 object undefined 0 undefined undefined
3 7 2 object undefined 0 undefined undefined
3 7 3 object undefined 0 undefined undefined
3 7 4 object undefined 0 undefined undefined
3 7 5 object undefined 0 undefined undefined
===*/
function arrayBufferSliceBruteForceTest() {
/*
* Some differences to Khronos spec:
*
* - First argument defaults to zero if missing (mandatory in spec)
*/
var i;
var b0 = new ArrayBuffer(0);
b0.name = 'b0';
var b1 = new ArrayBuffer(4);
b1.name = 'b1';
var u8 = new Uint8Array(b1);
for (i = 0; i < 4; i++) { u8[i] = 0x41 + i; } // ABCD
var b2 = new ArrayBuffer(8);
b2.name = 'b2';
var u8 = new Uint8Array(b2);
for (i = 0; i < 8; i++) { u8[i] = 0x61 + i; } // abcdefgh
[ 123, b0, b1, b2 ].forEach(function (thisValue, idx1) {
[ 'NONE', -10, -1, 0, 1, 3.9, 7.1, 15 ].forEach(function (begin, idx2) {
[ 'NONE', -1, 0, 1.9, 5, 9 ].forEach(function (end, idx3) {
var b;
try {
if (begin === 'NONE') {
b = thisValue.slice();
} else if (end === 'NONE') {
b = thisValue.slice(begin);
} else {
b = thisValue.slice(begin, end);
}
print(idx1, idx2, idx3, typeof b, b.length, b.byteLength,
b.byteOffset, b.BYTES_PER_ELEMENT,
printableBuffer(b));
} catch (e) {
print(idx1, idx2, idx3, e.name);
}
});
});
});
}
try {
print('ArrayBuffer slice() bruteforce test');
arrayBufferSliceBruteForceTest();
} catch (e) {
print(e.stack || e);
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,423 |
namespace('App.Bundle.AddressBook.Entity');
App.Bundle.AddressBook.Entity.Profile = function () {
Sy.Entity.call(this);
this.firstname = null;
this.lastname = null;
this.fullname = null;
this.address = null;
this.zipcode = null;
this.city = null;
this.phone = null;
this.picture = null;
};
App.Bundle.AddressBook.Entity.Profile.prototype = Object.create(Sy.Entity.prototype, {
INDEXES: {
value: ['fullname']
},
/**
* Return the picture
*
* @return {File}
*/
getPicture: {
value: function () {
return this.picture;
}
},
/**
* Picture setter
* Used to prevent the form from setting empty value
*/
setPicture: {
value: function (picture) {
if (picture) {
this.picture = picture;
}
return this;
}
},
/**
* Check if the picture is set
*
* @return {Boolean}
*/
hasPicture: {
value: function () {
return typeof this.picture === 'string';
}
},
/**
* Replace the fake picture path by the image file
*
* @param {File} picture
*
* @return {App.Bundle.AddressBook.Entity.Profile} self
*/
setPictureFile: {
value: function (picture) {
this.picture = picture;
return this;
}
},
/**
* Return the picture blob url
*
* @return {String}
*/
getPictureUrl: {
value: function () {
if (this.picture instanceof Blob) {
return URL.createObjectURL(this.picture);
} else {
return 'http://www.gravatar.com/avatar/00000000000000000000000000000000?s=125';
}
}
},
/**
* Return the firstname + the lastname
*
* @return {String}
*/
getFullname: {
value: function () {
return this.firstname + ' ' + this.lastname;
}
},
/**
* Prevent to actually set the fullname
*
* @param {String} fullname
*
* @return {App.Bundle.AddressBook.Entity.Profile} self
*/
setFullname: {
value: function (fullname) {
return this;
}
}
}); | {
"redpajama_set_name": "RedPajamaGithub"
} | 2,358 |
One of the most unique features of the Ackland Snap-Cap is our "Hands-Free" perimeter. This adhesive strip around the edge of the frame allows you to display both rigid and flex-face media. That's a huge advantage over typical flip frames that expands your display options. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,522 |
//CurrentPageController implements IPageController
var CurrentPageController = function () {
//constructor
};
CurrentPageController.prototype = (function () {
//private
var childObj = {
//staff begin
//staff end
//pageEvent begin
//pageEvent end
//controlEvent begin
//controlEvent end
//private action begin
//private action end
//other method begin
PageLoaded : function (sender, args) {
var oThis = this;
},
//other method end
};
return childObj;
})();
CurrentPageController.Implement(IPageController);
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,441 |
Erland Johnsen, né le à Moss (Norvège), est un footballeur norvégien, qui évoluait au poste de défenseur central au Chelsea Football Club et en équipe de Norvège.
Johnsen a marqué deux buts lors de ses dix-neuf sélections avec l'équipe de Norvège entre 1988 et 1995.
Carrière
1983-1988 : Moss FK
1988-1989 : Bayern Munich
1989-1997 : Chelsea Football Club
1997-1998 : Rosenborg BK
1998-1999 : Strømsgodset IF
Palmarès
En équipe nationale
19 sélections et 2 buts avec l'équipe de Norvège entre 1988 et 1995.
Participation à la coupe du monde 1994.
Avec Moss FK
Vainqueur du Championnat de Norvège de football en 1987.
Vainqueur de la Coupe de Norvège de football en 1983.
Avec le Bayern de Munich
Vainqueur du Championnat d'Allemagne de football en 1989.
Avec Chelsea
Vainqueur de la Coupe d'Angleterre de football en 1997.
Avec Rosenborg BK
Vainqueur du Championnat de Norvège de football en 1998.
Liens externes
Footballeur international norvégien
Naissance en avril 1967
Naissance à Moss
Joueur du Rosenborg BK
Joueur du Chelsea FC
Joueur du Bayern Munich
Joueur du Strømsgodset IF
Joueur du Moss FK | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,056 |
{"url":"https:\/\/math.stackexchange.com\/questions\/3506442\/combinatorics-problem-with-connecting-computers","text":"# Combinatorics problem with connecting computers\n\nA system administrator has to connect 16 computers to 4 network switches. Exactly 3 computers must be connected to the first network switch.\n\n1. Show that there must exist a switch that has at least 5 connected computers.\n2. In how many ways can we connect the computers if we know that they are all different?\n3. In how many ways can we connect the computers if we know that they are all the same?\n\nI tried solving the third question with first finding the $$3$$-combinations for all $$16$$ computers: $$\\binom{16+3-1}{16} = 153$$ ways and then for the remaining $$3$$ switches I used the equation: $$x_1+x_2+x_3=13$$, with $$x_1$$ having at least $$5$$ connected computers, meaning $$x_1>4$$. The solution I got was $$84$$ ways. $$153+84$$ ways $$= 237$$ ways.\n\nI don't know if this is the right approach. I would like to know if I am on the right path and what am I doing wrong.\n\n\u2022 Please edit your question to show what you have attempted and explain where you are stuck so that you receive responses that address the specific difficulties you are encountering. This tutorial explains how to typeset mathematics on this site. \u2013\u00a0N. F. Taussig Jan 12 '20 at 17:45\n\u2022 Welcome to MSE. You'll get a lot more help, and fewer votes to close, if you show that you have made a real effort to solve the problem yourself. What are your thoughts? What have you tried? How far did you get? Where are you stuck? This question is likely to be closed if you don't add more context. Please respond by editing the question body. Many people browsing questions will vote to close without reading the comments. \u2013\u00a0saulspatz Jan 12 '20 at 17:47\n\u2022 Did you mean all $16$ computers? Also, can a computer be connected to more than one switch? \u2013\u00a0N. F. Taussig Jan 12 '20 at 21:17\n\u2022 Yes I mean 16, not 24. I edited my mistake. Also, a computer can be connected to only one switch. \u2013\u00a0lina Jan 12 '20 at 21:42\n\nWe are told that exactly three computers are connected to the first network switch. That means $$16 - 3 = 13$$ computers are connected to the remaining three network switches. If we let $$x_i$$ be the number of computers connected to the $$i$$th network switch, then $$x_2 + x_3 + x_4 = 13$$ which is an equation in the nonnegative integers. A particular solution of the equation corresponds to the placement of $$3 - 1 = 2$$ addition signs in a row of $$13$$ ones. For instance, $$1 1 + 1 1 1 1 1 1 + 1 1 1 1 1$$ corresponds to the solution $$x_2 = 2$$, $$x_3 = 6$$, $$x_4 = 5$$. The number of such solutions is $$\\binom{13 + 3 - 1}{3 - 1} = \\binom{15}{2} = 105$$ since we must choose which two of the $$15$$ positions required for $$13$$ ones and two addition signs will be filled with addition signs. Thus, there are $$105$$ ways to connect $$16$$ identical computers to the four network switches if exactly three of those computers are connected to the first network switch.\nIt looks like you were misled by the wording of the first part of the question. If exactly three computers are connected to the first switch, then $$13$$ computers must be connected to the remaining network switches. At least one of those three switches must be connected to at least five computers, otherwise there would only be at most $$3 + 4 + 4 + 4 = 15$$ computers connected to the network switches, a contradiction. Since exactly three computers must be connected to the first network switch, the case in which $$x_1 > 4$$ is ruled out.\n\u2022 If the computers are all different, choose which three of them are connected to the first network switch. Each of the remaining $13$ computers must be connected to one of the other three switches. Thus, there are $\\binom{16}{3}3^{13}$ ways to connect the computers to the network switches if all the computers are different and exactly three of them must be connected to the first network switch. \u2013\u00a0N. F. Taussig Jan 13 '20 at 17:51","date":"2021-03-08 13:06:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 29, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8912765979766846, \"perplexity\": 180.12586215588973}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178375439.77\/warc\/CC-MAIN-20210308112849-20210308142849-00062.warc.gz\"}"} | null | null |
The long history of US intervention in other countries' elections has been omitted from discussions of Russia's alleged meddling.
The collective hysteria over fake news, Russia's alleged role in the DNC hack, and the unsubstantiated kompramat that supposedly links Donald Trump to Vladimir Putin has reached a fever pitch. But mainstream cable news and the Washington intelligentsia have somehow neglected to connect it to a crucial piece of the US history: its long-standing tradition, euphemistically known as the Truman Doctrine, of intervening in democratic elections abroad to promote its commercial and ideological interests.
Truman's doctrine would "support free peoples," he proclaimed in March 1947, "who are resisting attempted subjugation by armed minorities or by outside pressures." Indeed, he and his successors would go to great lengths to keep this promise during the Cold War. American presidents repeatedly directed the CIA to overthrow freely elected leaders in Iran, Guatemala, the Congo, and Chile because they nationalized industries, threatened corporate interests, and obstructed the United States' imperial ambitions. American officials falsely branded these leaders as Communists, framed them as threats to national security, and authorized covert operations to replace them with dictators who would serve US interests.
Omission of this history from today's discourse on Russia and our adversaries prevents our leaders, and especially the American public, from realizing the same tools the United States used to interfere in others' affairs are now being used against us.
Shielded by this ignorance, it is easy for US officials to portray us as the victims of attacks rather than the inventors of the weapons. When Senator John McCain, for example, says, "If you're able to change the results of an election, then you have undermined the very fundamentals of democracy," he forgets to mention this is precisely what the United States did in Iran, Guatemala, the Congo, and Chile when they were just beginning to experience democracy.
In 1953, a US-backed military coup overthrew Iran's first democratically elected leader, Prime Minister Mohammad Mossadegh, in response to his decision to nationalize the highly lucrative oil industry, cutting off the gravy train the Anglo-Iranian Oil Company had been riding since 1909.
Time had honored the Western-educated leader the year before the coup as its man of the year, hailing him as "the most world-renowned man his ancient race had produced for centuries." Suddenly, because he wanted to use Iran's oil wealth to benefit his country, he was deemed a pinko.
Using American tax dollars to develop a network of Iranian agents and to bribe the regime's opponents, the CIA launched political warfare against Mossadegh. It distributed fake news via posters and newspapers that called him corrupt, anti-Islam, and the Soviet Union's ally, it encouraged religious leaders to criticize the prime minister from inside their mosques, and it enlisted street mobs to incite riots across Tehran.
Success finally came on August 19. Paid infiltrators played both sides: some posed as Tudeh party members attempting to foment revolution while others convinced the citizens to rise up against this threat. Eventually, amid growing anarchy, General Fazlollah Zahedi, paid off by the CIA, ordered his bribed military units to seize government facilities and Radio Tehran. He proclaimed himself "the lawful prime minister by the Shah's orders" and collected $1 million in cash from the CIA.
Soon after, the Shah — Washington's chosen dictator — assumed the throne, American oil companies moved in, and US-Iranian relations quickly warmed as the new regime squashed dissent, imprisoned opponents, and received unprecedented US arms shipments, American assistance to create the Monarchy's secret police, and US support to develop Iran's civilian nuclear program.
The American recipe for overthrow continued to evolve. Guatemala's freely elected president Jacobo Arbenz became the next target when his New Deal-style programs threatened the interests of American corporations. The powerful United Fruit Company, whose executives were in bed with a number of influential American officials — some of whom were former employees and some of whom had financial interests in the corporation — found Arbenz's policies especially worrying.
The Agrarian Reform Law of 1952 authorized the Guatemalan government to seize vast tracts of United Fruit's uncultivated acres. The next December, the CIA's Operation SUCCESS set in motion a six-month coup. CIA agents used Voice of Liberation radio to broadcast fake news describing an impending communist takeover and civilian uprisings, recruited a rebel army to sow unrest, distributed religious leaflets calling Catholics to revolt, and coordinated air raids that dropped bombs on military installations and other targets across Guatemala City.
These efforts turned popular and military support against Arbenz, forcing his resignation and paving the way for Castillo Armas — the United States' handpicked dictator — to become president. This established a long line of dictators, death squads, oppression, and near-genocide that wreaked havoc across Guatemala for the next four decades.
In 1960, American meddling escalated. President Eisenhower decided to skip the coup and go straight to assassination. He ordered — once again using the false pretense of an impending communist threat — the CIA to assassinate Congo's democratically elected prime minister Patrice Lumumba, the young nationalist who ended seven decades of brutal Belgian rule and promised Congolese citizens a better future with greater control over the country's natural resources.
When the plot to poison Lumumba failed, the CIA outsourced the job to Congolese accomplices and Belgian officers. With the help of Joseph Mobutu, the repressive military dictator installed by the United States who would rule for three decades, they eventually captured him and handed him over to his enemies. They tortured Lumumba, then murdered him by firing squad on January 17, 1961 — just three days before John F. Kennedy was inaugurated as the freely elected president of the United States.
During the early 1970s, commercial interests once again required CIA assistance. When Chile elected Salvador Allende, the anti-imperialist, working-class champion, he nationalized profitable American-dominated industries such as communications and copper, the resource gem for which Chile was the world's leading supplier.
President Nixon, who believed the false reports of Soviet influence he received from American and Chilean industrial titans, authorized the CIA to overthrow Allende, setting in motion a ferocious propaganda campaign that included distributing fake news, strangling economic development, conspiring with disgruntled Chilean officers, indirectly assassinating a senior military leader, and staging anti-government protests.
Our electoral integrity is a legitimate concern, and American officials should express outrage at Russia's alleged actions (should they turn out to be true). But every American should feel equally chilled by our history of undemocratic electoral interference around the world. Millions of Iranians, Guatemalans, Congolese, and Chileans suffered under the iron grips of unelected dictators the United States installed.
Hundreds of thousands died in the aftermath of these coups, countries split into civil wars, untold natural riches were stolen, and countless others endured unspeakable trauma and loss. Unforeseen blowback in response to these actions will likely continue in the years ahead.
Whether it was Russian spies or American couch potatoes who convinced Podesta to give up his password, the United States has not suffered the way other countries have — at least not yet. As the tools of American power proliferate and fall into the hands of our supposed adversaries — arms, nuclear weapons, coups, drones, and cyber-warfare — we must confront the reality that as long as the United States continues its habit of meddling abroad, other countries will be tempted to use these and other forms of covert action against us. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,844 |
require_relative 'movement'
# Base class for an empty, stupid entity. Will eventually be inherited
# by NPCs and monsters
# Difference between NPCs and Monsters is primarily in rendering and AI or interactibility
module Entities
class Base
extend Environment::BaseObject
include Entities::Movement
include Engines::Collidable
include Engines::Gravity
extend Utils::ActionHelpers
action_flag :jump
attr_entity [:updateable, :drawable, :gravitizable, :collidee]
attr_accessor :window, :x, :y, :vel_x, :vel_y, :w, :h, :pos, :prev_x, :prev_y, :prev_pos
def initialize(window, start_x, start_y, drawable)
@window = window
@x, @y = start_x, start_y
@w, @h = 32, 32
@vel_x, @vel_y = 0, 0
@pos = :rs
@drawable = drawable
end
def update
update_movement
end
def draw
@drawable.draw(x, y, pos)
end
def collide!(collision)
send collision.action, collision
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 248 |
Q: What is the derivative of $f (\mathrm X) = \mathrm X^3$? Let $E = M_{3}(\mathbb{R})$ and $f: E \rightarrow E$ which $f(X) = X^3$ is $f^{'}(X)H = 3X^2 H$ the derivative for this function?
I tried to prove that $r(H) \rightarrow 0$ where $r(H) = -X^3 + (X+H)^3 - 3X^2 H$ but i dont get in anywhere.
A: The directional derivative of $f (\mathrm X) = \mathrm X^3$ in the direction of $\mathrm V$ is
$$\lim_{t \to 0} \frac{1}{t} \left( f (\mathrm X + t \mathrm V) - f (\mathrm X) \right) = \lim_{t \to 0} \frac{1}{t}\left( (\mathrm X + t \mathrm V)^3 - \mathrm X^3 \right)$$
Since
$$(\mathrm X+ t \mathrm V)^3 = \mathrm X^3 + t (\mathrm X^2 \mathrm V + \mathrm X \mathrm V \mathrm X + \mathrm V \mathrm X^2) + t^2 (\mathrm X \mathrm V^2 + \mathrm V \mathrm X \mathrm V + \mathrm V^2 \mathrm X) + t^3 \mathrm V^3$$
we obtain
$$\lim_{t \to 0} \frac{1}{t} \left( f (\mathrm X + t \mathrm V) - f (\mathrm X) \right) = \mathrm X^2 \mathrm V + \mathrm X \mathrm V \mathrm X + \mathrm V \mathrm X^2$$
If $\mathrm X$ and $\mathrm V$ commute, then $\mathrm X^2 \mathrm V + \mathrm X \mathrm V \mathrm X + \mathrm V \mathrm X^2 = 3 \mathrm X^2 \mathrm V$.
A: If you're looking for a Fréchet derivative approach, we're looking for a bounded, linear map $A_X : E \to E$ that satisfies the following limit:
$$ \lim_{ V \to 0\text{ in } M_{3}(\Bbb{R})} \dfrac{\|f(X+V)-f(X)-A_{X}V \|_E}{\|V\|_E} = 0 $$
The argument of the limit can be re-written as: $$ \dfrac{\| X^2\color{red}{V} + X\color{red}{V}X + XV^2 + \color{red}{V}X^2 + VXV + V^3 - A_{X}V\|_{E}}{\|V\|_E} $$
We're looking for a map $A_X$ that's $\color{red}{ \text{linear in $V$} }$ so we can hazard a guess that: $$ A_{X}V = X^2V + XVX + VX^2 $$ By successive applications of the triangle inequality (in the numerator) and using the operator norm for a matrix that represents the linear map $\| \cdot \|_E = \| \cdot \|_{\text{op}}$ the result should follow.
It'll help with cancellation where you want terms going to $0$ that for linear maps $A$ & $B$: $$ \| AB \|_{\text{op}} \leqslant \|A\|_{\text{op}} \|B\|_{\text{op}} $$
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,277 |
\section{Introduction}\label{sec:intro}
\begin{figure}[t!]
\centering
\includegraphics[width=\linewidth]{intro2.pdf}
\caption{A semantic parser proactively interacts with the user in a friendly way to resolve its uncertainties.\nop{ In doing so it also accumulates targeted training data and continues improving itself autonomously without annotators or developers.} In doing so it also gets to imitate the user behavior and continue improving itself autonomously with the hope that eventually it may become as good as the user in interpreting their questions.}
\label{fig:intro}
\vspace{-10pt}
\end{figure}
Semantic parsing has found tremendous applications in building natural language interfaces that allow users to query data and invoke services without programming~\cite{woods1973progress,zettlemoyer2005learning,berant2013semantic, yih2015semantic, su2017building,yu2018spider}. The life cycle of a semantic parser typically consists of two stages: (1) \emph{bootstrapping}, where we keep collecting labeled data via trained annotators and/or crowdsourcing for model training until it reaches commercial-grade performance (e.g., 95\% accuracy on a surrogate test set), and (2) \emph{fine-tuning}, where we deploy the system, analyze the usage, and collect and annotate new data to address the identified problems or emerging needs.
However, it poses several challenges for scaling up or building semantic parsers for new domains:
(1) \emph{high bootstrapping cost} because mainstream neural parsing models are data-hungry and the annotation cost of semantic parsing data is relatively high,
(2) \emph{high fine-tuning cost} from continuously analyzing usage and annotating new data, and
(3) \emph{privacy risks} arising from exposing private user data to annotators and developers~\cite{techcrunch2019privacy}.
In this paper, we suggest an alternative methodology for building semantic parsers that could potentially address all the aforementioned problems. The key is to involve human users in the learning loop.
A semantic parser should be introspective of its uncertainties~\cite{dong2018confidence} and proactively prompt for demonstrations from the user, who knows the question best, to resolve them. In doing so, the semantic parser \nop{would be able to}can accumulate targeted training data and continue improving itself autonomously without involving any annotators or developers, hence also minimizing privacy risks. The bootstrapping cost could also be significantly reduced because an interactive system needs not to be almost perfectly accurate to be deployed. On the other hand, such interaction opens up the black box and allows users to know more about the reasoning underneath the system and better interpret the final results~\cite{su2018natural}. A human-in-the-loop methodology like this also opens the door for domain adaptation and personalization.
This work builds on the recent line of research on interactive semantic parsing~\cite{li2014constructing, chaurasia2017dialog, gur2018dialsql, yao2019model}. Specifically, \citet{yao2019model} provide a general framework, MISP (Model-based Interactive Semantic Parsing), which handles uncertainty modeling and natural language generation. We will leverage MISP for user interaction to prove the feasibility of the envisioned methodology. However, existing studies only focus on interacting with users to resolve uncertainties. None of them has fully addressed the crucial problem of \emph{how to continually learn from user interaction}, which is the technical focus of this study.
\nop{None of them has answered the crucial question of \emph{how to continually learn from user interaction}, which is the technical focus of this study.}
One form of user interaction explored for learning semantic parsers is asking users to validate the execution results~\cite{clarke2010driving,artzi2013weakly,iyer2017learning}. While appealing, in practice it may be a difficult task for real users because they would not need to ask the question if they knew the answer in the first place. We instead aim to learn semantic parsers from fine-grained interaction where users only need to answer simple questions covered by their background knowledge (Figure~\ref{fig:intro}). However, learning signals from such fine-grained interactions are bound to be sparse because the system needs to avoid asking too many questions and overwhelming the user, which poses a challenge for learning.
To tackle the problem, we propose \textsc{Neil}\xspace, a novel \emph{a\underline{N}notation-\underline{E}fficient \underline{I}mitation \underline{L}earning} algorithm for learning semantic parsers from such sparse, fine-grained demonstrations: The agent (semantic parser) only requests for demonstrations when it is uncertain about a state (parsing step).
For certain/confident states, actions chosen by the current policy are deemed correct and are executed to continue parsing. The policy is updated iteratively in a Dataset Aggregation fashion~\cite{ross2011reduction}. In each iteration, all the state-action pairs, demonstrated or confident, are included to form a new training set and train a new policy in a supervised way. Intuitively, using confident state-action pairs for training mitigates the sparsity issue, but it may also introduce training bias. We provide a theoretical analysis and show that, under mild assumptions, the impact of the bias and the quality of the \textsc{Neil}\xspace policy can be controlled by tuning the policy initialization and confidence estimation accuracy.
We also empirically compare \textsc{Neil}\xspace with a number of baselines on the text-to-SQL parsing task. On the WikiSQL~\cite{zhong2017seq2sql} dataset, we show that, \emph{when bootstrapped using only 10\% of the training data, \textsc{Neil}\xspace can achieve almost the same test accuracy (2\% absolute loss) as the full expert annotation baseline, while requiring less than 10\% of the annotations that the latter needs}, without even taking into account the different unit cost of annotations from users vs.\ domain experts. We also show that the quality of the final policy is largely determined by the quality of the initial policy, which provides empirical support for the theoretical analysis. Finally, we demonstrate that \textsc{Neil}\xspace can generalize to more complex semantic parsing tasks such as Spider~\cite{yu2018spider}.
\section{Related Work}\label{sec:related_work}
\begin{remark}[Interactive Semantic Parsing.]
Our work extends the recent idea of leveraging system-user interaction to improve semantic parsing on the fly~\cite{li2014constructing,he2016human,chaurasia2017dialog,su2018natural, gur2018dialsql,yao2018interactive,yao2019model,elgohary2020speak, zeng2020photon, SMDataflow2020}. \citet{gur2018dialsql} built a neural model to identify and correct error spans in generated queries via dialogues. \citet{yao2019model} formalized a model-based intelligent agent MISP, which enables user interaction via a policy probability-based uncertainty estimator, a grammar-based natural language generator, and a multi-choice question-answer interaction design. More recently, \citet{elgohary2020speak} crowdsourced a dataset for fixing incorrect SQL queries using free-form natural language feedback. {\citet{SMDataflow2020} constructed a contextual semantic parsing dataset where agents could trigger conversations to handle exceptions such as ambiguous or incomplete user commands.}
In this work, we seek to continually improve semantic parsers from such user interaction, a topic that is not carefully studied by the aforementioned work.
\end{remark}
\begin{remark}[Interactive Learning from Feedback.]
Learning interactively from user feedback has been studied in many NLP tasks \cite{sokolov2016learning, wang2016learning, wang2017naturalizing, nguyen2017reinforcement, gao2018april, abujabal2018never, hancock2019learning, kreutzer2019self}. Most relevant to us, \citet{hancock2019learning} constructed a self-feeding chatbot that improves itself from user satisfied responses and their feedback on unsatisfied ones.
In the field of semantic parsing, \citet{clarke2010driving, artzi2013weakly, iyer2017learning} learned semantic parsers from binary user feedback on whether executing the generated query yields correct results. However, often times {(especially in information-seeking scenarios)} it may not be very practical to expect end users able to validate the denotation correctness (e.g., consider validating an execution result ``\emph{103}'' for the question ``\emph{how many students have a GPA higher than 3.5}'' from a massive table). Active learning is also leveraged to save human annotations \cite{duong2018active,ni20aaai}.
Our work is complementary to this line of research as we focus on learning interactively from end users (not ``teachers'').
\end{remark}
\begin{remark}[Imitation Learning.]
Traditional imitation learning algorithms \cite{daume2009search,ross2010efficient,ross2011reduction,Ross2014ReinforcementAI} iteratively execute and train a policy by collecting expert demonstrations for every policy decision. Despite its efficacy, the learning demands costly annotations from experts. In contrast, we save expert effort by selectively requesting demonstrations. This idea is related to active imitation learning~\cite{chernova2009interactive,kimmaximum,judah2014active,zhang2017query}. For example, \citet{judah2014active} assumed a ``teacher'' and actively requested demonstrations for most informative trajectories in the unlabeled data pool.
Similar to us, \citet{chernova2009interactive} solicited demonstrations only for uncertain states. However, their algorithm simply abandons policy actions that are confident, leading to sparse training data. Instead, our algorithm utilizes confident policy actions to combat the sparsity issue and is additionally provided with a theoretical analysis.
{Concurrent with our work, \citet{daume20active} studied active imitation learning for structured prediction tasks such as named entity recognition. Our work instead focuses on semantic parsing, which presents a unique challenge of \emph{integrality}, i.e., the output sequence (a semantic parse) could only be correct \emph{as a whole} (as opposed to \emph{partially} correct) in order to yield the correct denotation. We therefore propose a new cost function (Section~\ref{sec:theoretical_analysis}) to theoretically analyze the factors that affect the efficacy of learning semantic parsers via imitation.}
\end{remark}
\section{Preliminaries} \label{sec:overview}
Formally, we assume the semantic parsing model generates a semantic parse by executing a sequence of actions $a_t$ (parsing decisions) at each time step $t$. In practice, the definition of an action depends on the specific semantic parsing model, as we will illustrate shortly. A state $s_t$ is then defined as a tuple of $(q, a_{1:t-1})$, where $q$ is the initial natural language question and $a_{1:t-1}=(a_1, ...,a_{t-1})$ is the current partial parse. In particular, the initial state $s_1=(q, \phi)$ contains only the question.
Denote a semantic parser as $\hat{\pi}$, which is a \emph{policy function} \cite{sutton2018reinforcement} that takes a state $s_t$ as input and outputs a probability distribution over the action space. The semantic parsing process can be formulated as sampling a \emph{trajectory} $\tau$ by alternately observing a state and sampling an action from the policy, i.e., $\tau=(s_1$, $a_1 \sim \hat{\pi}(s_1)$, ..., $s_T$, $a_T \sim \hat{\pi}(s_T))$, assuming a trajectory length $T$. The probability of the generated semantic parse becomes: $p_{\hat{\pi}}(a_{1:T}|s_1) = \prod_{t=1}^T p_{\hat{\pi}}(a_t|s_t)$.
An interactive semantic parser typically follows the aforementioned definition and requests the user's validation of a specific action $a_t$. Based on the feedback, a correct action $a_t^*$ can be inferred to replace the original $a_t$. The parsing process continues with $a_t^*$ afterwards.
{In this work, we adopt MISP \cite{yao2019model} as the back-end interactive semantic parsing framework, given that it is a principled framework for this purpose and can generalize to various kinds of semantic parsers and logical forms. However, we note that our proposed algorithm is not limited to MISP; it instead depicts a general algorithm for learning semantic parsers from user interaction. We illustrate the application of MISP to {a sketch-based parser,} SQLova \cite{hwang2019comprehensive}, as follows. More details and another example of how it applies to a non-sketch-based parser EditSQL \cite{zhang2019editing} can be found in Appendix~\ref{app:misp}.
}
\begin{remark}[Example.]
Consider the SQLova parser, which generates a query by filling ``slots'' in a pre-defined SQL sketch ``\texttt{SELECT Agg SCol WHERE WCol OP VAL}''.\nop{In practice, SQLova takes an additional step to predict the number of \texttt{WHERE} conditions. Since it is vague to request user feedback on such a prediction, we consider it as an ``implicit'' decision that controls the explicit SQL generation steps.} To complete the SQL query in Figure~\ref{fig:intro}, it first takes three steps: \texttt{SCol}=``\texttt{School/Club Team}'' ($a_1$), \texttt{Agg}=``\texttt{COUNT}'' ($a_2$) and \texttt{WCol}=``\texttt{School/Club Team}'' ($a_3$). MISP detects that $a_3$ is uncertain because its probability is lower than a pre-specified threshold. It validates $a_3$ with the user and corrects it with \texttt{WCol}=``\texttt{Player}'' ($a_3^*$). The parsing continues with
\texttt{OP}=``\texttt{=}'' ($a_4$) and \texttt{VAL}=``\texttt{jalen rose}'' ($a_5$). Here, the trajectory length $T=5$.
\end{remark}
\section{Learning Semantic Parsers from User Interaction} \label{sec:model}
In this section, we present \textsc{Neil}\xspace, an \textit{a\underline{N}notation-\underline{E}fficient \underline{I}mitation \underline{L}earning} algorithm that trains a parser from user interaction, without requiring a large amount of user feedback (or ``annotations''). This property is particularly important for end user-facing systems in practical use.
Note that while we apply \textsc{Neil}\xspace to semantic parsing in this work, in principle it can also be applied to other structured prediction tasks (e.g., machine translation).
\subsection{An Imitation Learning Formulation}
Under the interactive semantic parsing framework, a learning algorithm intuitively can aggregate $(s_t, a_t^*)$ pairs collected from user interactions and trains the parser to enforce $a_t^*$ under the state $s_t=(q, a_{1:t-1})$. However, this is not achievable by conventional supervised learning since the training needs to be conducted in an \emph{interactive} environment, where the partial parse $a_{1:t-1}$ is generated by the parser itself.
Instead, we formulate it as an imitation learning problem \cite{daume2009search, ross2010efficient}.
Consider the user as a \emph{demonstrator}, then the derived action $a_t^*$ can be viewed as an \emph{expert demonstration} which is interactively sampled from the demonstrator's policy (or \emph{expert policy}) $\pi^*$,\footnote{We follow the imitation learning literature and use ``expert'' to refer to the imitation target, but the user in our setting by no means needs to be a ``domain (SQL) expert''.} i.e., $a_t^* \sim \pi^*(s_t)$. The goal of our algorithm is thus to train policy $\hat{\pi}$ to imitate the expert policy $\pi^*$. A general procedure is described in Algorithm~\ref{alg:mispl_general} (Line 1--9), where $\hat{\pi}$ is learned iteratively for every $m$ user questions. In each iteration, the policy is retrained on an aggregated training data over the past iterations, following the Dataset Aggregation fashion in \cite{ross2011reduction}.
\begin{algorithm}[t!]
\begin{algorithmic}[1]
\Require Initial training data $D_0$, policy confidence threshold $\mu$.
\Ensure A trained policy $\hat{\pi}$.
\State Initialize $D \leftarrow D_0$.
\State Initialize $\hat{\pi}_1$ by training it on $D_0$.
\For{$i=1$ to $N$}
\State Observe $m$ user questions $q_j, j \in [1,m]$;
\State $D_i \leftarrow \bigcup_{j=1}^m \textsc{Parse\&Collect}(\mu, q_j, \hat{\pi}_i, \pi^*)$;
\State Aggregate dataset $D \leftarrow D \bigcup D_i$;
\State Train policy $\hat{\pi}_{i+1}$ on $D$ using Eq.~\eqref{eq:general}.
\EndFor
\State \textbf{return} best $\hat{\pi}_i$ on validation.
\vspace{2mm}
\Function{Parse\&Collect}{$\mu, q, \hat{\pi}_i, \pi^*$}
\State Initialize $D_i^\prime \leftarrow \emptyset$, $s_1=(q, \phi)$.
\For{$t=1$ to $T$}
\State Preview action $a_t = \argmax_{a} \hat{\pi}_i(s_t)$;
\If{$p_{\hat{\pi}_i}(a_t|s_t) \geq \mu$}
\State $w_t \leftarrow 1$;
\State Collect $D_i^\prime \leftarrow D_i^\prime \bigcup \{(s_t, a_t, w_t)\}$;
\State Execute $a_t$;
\Else
\State Trigger user interaction and derive expert demonstration $a_t^* \sim \pi^*(s_t)$;
\State $w_t \leftarrow 1$ if $a_t^*$ is valid; $0$ otherwise;
\State Collect $D_i^\prime \leftarrow D_i^\prime \bigcup \{(s_t, a_t^*, w_t)\}$;
\State Execute $a_t^*$.
\EndIf
\EndFor
\State \textbf{return} $D_i^\prime$.
\EndFunction
\end{algorithmic}
\caption{The \textsc{Neil}\xspace Algorithm}
\label{alg:mispl_general}
\end{algorithm}
\subsection{Annotation-efficient Imitation Learning}
Consider parsing a user question and collecting training data using the parser $\hat{\pi}_i$ in the $i$-th iteration (Line 5). A standard imitation learning algorithm such as \textsc{DAgger} \cite{ross2011reduction} usually requests expert demonstration $a_t^*$ for every state $s_t$ in the sampled trajectory. However, it requires a considerable amount of user annotations, which may not be practical when interacting with end users.
Instead, we propose to adopt an \emph{annotation-efficient} learning strategy in \textsc{Neil}\xspace, which saves user annotations by \emph{selectively} requesting user interactions, as indicated in function \textsc{Parse\&Collect}. In each parsing step, the system first previews whether it is confident about its own decision $a_t$ (Line 13--14), which is determined when its probability is no less than a threshold, i.e., $p_{\hat{\pi}_i}(a_t|s_t) \geq \mu$.\footnote{{The metric is shown effective for interactive semantic parsing in \citet{yao2019model}. Other confidence measures can also be explored, as we will discuss in Section~\ref{sec:future_work}.}}
In this case, the algorithm executes and collects \nop{the policy action}{its own action} $a_t$ (Line 15--17); otherwise, a system-user interaction will be triggered and the derived demonstration $a_t^* \sim \pi^*(s_t)$ will be collected and executed to continue parsing (Line 19--22).
Denote a collected state-action pair as $(s_t, \Tilde{a}_t)$, where $\Tilde{a}_t$ could be $a_t$ or $a_t^*$ depending on whether an interaction is requested. To train $\hat{\pi}_{i+1}$ (Line 7), our algorithm adopts a \emph{reduction-based} approach similar to \textsc{DAgger} and reduces imitation learning to iterative supervised learning. Formally, we define our training loss function as a \emph{weighted} negative log-likelihood:
\vspace{-2mm}
\begin{equation}\label{eq:general}
\small
\mathcal{L}(\hat{\pi}_{i+1}) = - \frac{1}{|D|} \sum_{(s_t, \Tilde{a}_t, w_t) \in D} w_t \log p_{\hat{\pi}_i}(\Tilde{a}_t|s_t),
\vspace{-2mm}
\end{equation}
where $D$ is the aggregated training data over $i$ iterations and $w_t$ denotes the weight of $(s_t, \Tilde{a}_t)$.
We consider assigning weight $w_t$ in three cases:
(1) For confident actions $a_t$, we set $w_t=1$. This essentially treats \nop{confident actions}{the system's own confident actions} as gold decisions, which resembles self-training \cite{scudder1965probability,nigam2000analyzing, mcclosky2006effective}.
(2) For user-confirmed decisions (\emph{valid demonstrations} $a_t^*$), such as enforcing a \texttt{WHERE} condition on ``\texttt{Player}'' in Figure~\ref{fig:intro}, $w_t$ is also set to 1 to encourage the parser to imitate the correct decisions from users.
(3) For uncertain actions that cannot be addressed via human interactions (\emph{invalid demonstrations} $a_t^*$, which are identified when the user selects ``None of the above options'' in Figure~\ref{fig:intro}), we assign $w_t=0$. This could happen when some of the incorrect precedent actions are not fixed. For example, in Figure~\ref{fig:intro}, if the system missed correcting the \texttt{WHERE} condition on ``\texttt{School/Club Team}'', then whatever value it generates after ``\texttt{WHERE School/Club Team=}'' is wrong, and thus any action $a_t^*$ derived from human feedback would be invalid. {In this case, the system selects the next available option without further validation and continues parsing.}
A possible training strategy to handle case (3)\nop{in this case} may set $w_t$ to be negative, similar to \citet{Welleck2020Neural}. However, empirically we find this strategy fails to train the parser to correct its mistake in generating ``\texttt{School/Club Team}'' but rather disturbs the model training.
By setting $w_t=0$, the impact of unaddressed actions is removed from training.
A similar solution is also adopted in \citet{petrushkov2018learning, kreutzer2019self}. As shown in Section~\ref{sec:experiments}, this way of training weight assignment enables stable improvement in iterative model learning while requiring fewer user annotations.
\section{Theoretical Analysis} \label{sec:theoretical_analysis}
While \textsc{Neil}\xspace enjoys the benefit of learning from a small amount of user feedback, one crucial question is whether it can still achieve the same level of performance as the traditional supervised approach (which trains a policy on full expert annotations, if one could afford that and manage the privacy risk). In this section, we prove that the performance gap between the two approaches is mainly determined by the learning policy's probability of trusting a confident action that turns out to be wrong, which can be controlled in practice.
Our analysis follows prior work ~\cite{ross2010efficient,ross2011reduction} to assume a unified trajectory length $T$ and an infinite number of training samples in each iteration (i.e., $m=\infty$ in Algorithm~\ref{alg:mispl_general}), such that the state space can be full explored by the learning policy. An analysis under the ``finite sample'' case can be found in Appendix~\ref{app:subsec:finite}.
\subsection{Cost Function for Analysis}
Unlike typical imitation learning tasks (e.g., Super Tux Kart \cite{ross2011reduction}), in semantic parsing, there exists only one gold trajectory semantically identical to the question.\footnote{We assume a canonical order for swappable components in a parse. In practice, it may be possible, though rare, for one question to have multiple gold parses.} Whenever a policy action is different from the gold one, the whole trajectory will not yield the correct semantic meaning and the parsing is deemed failed. {In other words, a well-performing semantic parser should be able to keep staying in the correct trajectory during the parsing.} Therefore, for theoretical analysis, we only analyze a policy's performance when it is conditioned on a \emph{gold partial parse}, i.e., $s_t \in d_{\pi^*}^t$, where $d_{\pi^*}^t$ is the state distribution in step $t$ when executing the expert policy $\pi^*$ for first $t$-1 steps.
Let $\ell(s, \hat{\pi}) = 1 - p_{\hat{\pi}}(a = a^*|s)$ be the loss of $\hat{\pi}$ making a mistake at state $s$.
We define the \emph{cost} (i.e., the inverse \emph{test-time} quality) of a policy as:
\begin{equation}
J(\hat{\pi}) = T \mathbb{E}_{s \sim d_{\pi^*}}\big[\ell(s,\hat{\pi}) \big],\label{eqn:3}
\vspace{-2mm}
\end{equation}
where $d_{\pi^*} = \frac{1}{T}\sum_{t=1}^T d_{\pi^*}^t$ denotes the average expert state distribution (assuming time step $t$ is a random variable uniformly sampled from $1 \sim T$). A detailed derivation is shown in Appendix~\ref{app:cost_function}.
{The \emph{better} a policy $\hat{\pi}$ is, the \emph{smaller} this cost becomes. Note that, by defining Eq.~\eqref{eqn:3}, we simplify the analysis from evaluating \emph{the whole trajectory sampled from $\hat{\pi}$} (as we do in experiments) to evaluating \emph{the expected single-step loss of $\hat{\pi}$ conditioned on a gold partial parse}. This cost function makes the analysis easier and meanwhile reflects a consistent relative performance among algorithms for comparison.
Next, we compare our \textsc{Neil}\xspace algorithm with the supervised approach by analyzing the upper bounds of their costs.
}
\subsection{Cost Bound of Supervised Approach}
A fully supervised system trains a parser on expert-annotated $(q, a_{1:T}^*)$ pairs, where the gold semantic parse $a_{1:T}^*$ can be viewed as generated by executing the expert policy $\pi^*$. This gives the policy $\hat{\pi}_{sup}$:
\begin{equation*}
\hat{\pi}_{sup} = \argmin_{\pi \in \Pi} \mathbb{E}_{s \sim d_{\pi^*}}[l(s, \pi)],
\vspace{-2mm}
\end{equation*}
where $\Pi$ is the policy space induced by the model architecture. A detailed derivation in Appendix~\ref{app:sup_cost_bound} shows the cost bound of the supervised approach:
\begin{theorem}\label{theorem:sup}
For supervised approach, let $\epsilon_N = \min_{\pi \in \Pi} \mathbb{E}_{s \sim d_{\pi^*}}[l(s, \pi)]$, then $J(\hat{\pi}_{sup}) = T \epsilon_N$.
\end{theorem}
The theorem gives an exact bound (as shown by the equality) since the supervised approach, given the ``infinite sample'' assumption, trains a policy under the same state distribution $d_{\pi^*}$ as the one being evaluated in the cost function (Eq.~\eqref{eqn:3}).
\subsection{Cost Bound of \textsc{Neil}\xspace Algorithm}
\label{subsec:obj_bound}
Recall that, in each training iteration, \textsc{Neil}\xspace samples trajectories by executing actions from both the previously learned policy $\hat{\pi}_i$ and the expert policy $\pi^*$ (when an interaction is requested). Let $\pi_i$ denote such a ``mixture'' policy. We derive the following cost bound of a \textsc{Neil}\xspace policy $\hat{\pi}$:
\begin{equation}
J(\hat{\pi}) \leq \frac{T}{N} \sum_{i=1}^N \big[ \mathbb{E}_{s \sim d_{\pi_i}}[\ell(s, \hat{\pi}_i)] + \ell_{max} ||d_{\pi_i} - d_{\pi^*}||_1 \big].\nonumber
\end{equation}
The bound is determined by two terms. The first term $\mathbb{E}_{s \sim d_{\pi_i}}[\ell(s, \hat{\pi}_i)]$ calculates the expected training loss of $\hat{\pi}_i$. Notice that, while the policy is \emph{trained} on states induced by the mixture policy ($s \sim d_{\pi_i}$), what matters to its \emph{test-time} quality is the policy's performance conditioned on a gold partial parse ($s \sim d_{\pi^*}$ in Eq.~\eqref{eqn:3}).
This state discrepancy, which does not exist in the supervised approach, explains the performance loss of \textsc{Neil}\xspace, and is bounded by the second term $\ell_{max} ||d_{\pi_i} - d_{\pi^*}||_1$, the weighted $L_1$ distance between $d_{\pi_i}$ and $d_{\pi^*}$.
{To bound the two terms, we employ a ``no-regret'' assumption \cite[see Appendix~\ref{app:no-regret}--\ref{app:mispl_cost_bound} for details]{kakade2009generalization, ross2011reduction}, which gives the theorem:}
\begin{theorem}\label{theorem:mispl}
For the proposed \textsc{Neil}\xspace algorithm, if $N$ is $\Tilde{O}(T)$, there exists a policy $\hat{\pi} \in \hat{\pi}_{1:N}$ s.t. $J(\hat{\pi}) \leq T \big[ \epsilon_N + \frac{2T \ell_{max}}{N} \sum_{i=1}^N e_i \big] + O(1)$.
\end{theorem}
Here, $\epsilon_N = \min_{\pi \in \Pi} \frac{1}{N} \sum_{i=1}^N \mathbb{E}_{s \sim d_{\pi_i}}[\ell(s, \pi)]$ denotes the best expected policy loss in \emph{hindsight}, and $e_i$ denotes the probability that $\hat{\pi}_i$ does not query the expert policy (i.e., being confident) but its own action is wrong under $d_{\pi^*}$.
{We note that a no-regret algorithm requires convexity of the loss function \cite{hazan2007logarithmic, kakade2009generalization}, which is not satisfied by neural network-based semantic parsers.
In general, proving theorems under a non-convex case is not trivial. Therefore, we follow the common {practice}\nop{routine} (e.g., \citet{DBLP:journals/corr/KingmaB14, j.2018on}) to theoretically analyze the convex case while empirically demonstrating the performance of our \textsc{Neil}\xspace algorithm with non-convex loss functions (i.e., when it applies to neural semantic parsers). More accurate regret bound for non-convex cases will be studied in the future.}
\begin{remark}[Remarks.]
{Compared with the supervised approach (Theorem~\ref{theorem:sup}), \textsc{Neil}\xspace's cost bound additionally contains a term of $\frac{1}{N}\sum_{i=1}^N e_i$, which, as we expect, comes from the aforementioned state discrepancy. Intuitively, if a learning policy frequently executes its own but wrong actions in training, the resulting training states $d_{\pi_i}$ will greatly deviate from the gold ones $d_{\pi^*}$. }
This finding inspires us to restrict the performance gap by reducing the learning policy's error rate \emph{when it does not query the expert}. Empirically this can be achieved by:
(1) \emph{Accurate confidence estimation}, so that actions deemed confident are generally correct, and (2) \emph{Moderate policy initialization}, such that in general the policy is less likely to make wrong actions throughout the iterative training.
For (1), we set a high confidence threshold $\mu$=0.95, which is demonstrated reliable for MISP \cite{yao2019model}. We then empirically validate (2) in experiments.
\end{remark}
\section{Experiments}\label{sec:experiments}
In this section, we conduct experiments to demonstrate the annotation efficiency of our \textsc{Neil}\xspace algorithm and that it can train semantic parsers for high performance when the parsers are reasonably initialized, which verifies our theoretical analysis.
\subsection{Experimental Setup}
We compare various systems on the WikiSQL dataset \cite{zhong2017seq2sql}. The dataset contains large-scale annotated question-SQL pairs (56,355 pairs for training) and thus serves as a good resource for experimenting iterative learning. For the base semantic parser, we choose SQLova \cite{hwang2019comprehensive}, one of the top-performing models on WikiSQL, to ensure a reasonable model capacity in terms of data utility along iterative training.
We experiment each system with three parser initialization settings, using 10\%, 5\% and 1\% of the total training data. During iterative learning, questions from the remaining training data arrive in a random order to simulate user questions and we simulate user feedback by directly comparing the synthesized query with the gold one.
{In each iteration, all systems access exactly the same user questions. Depending on how they solicit feedback, each system collects a different number of annotations. At the end of each iteration, we update each system by retraining its parser on its accumulated annotations and the initial training data, and report its \emph{(exact) query match accuracy} on the test set. We also report the \emph{accumulated number of annotations} that each system has requested after each training iteration, in order to compare their annotation efficiency.}
In experiments, we consider every 1,000 user questions as one training iteration (i.e., $m$=1,000 in Algorithm~\ref{alg:mispl_general}). We repeat the whole iterative training for three runs and report average results.
\emph{Reproducible details} are included in Appendix~\ref{app:implementation_details}.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.95\linewidth]{exp_figures/joint_test.png}
\caption{Parsing accuracy on WikiSQL test set when systems are trained with various numbers of user/expert annotations (top) and for different iterations (bottom). We experiment with three initialization settings, using 10\%, 5\% and 1\% of the training data respectively. Results on validation set can be found in Appendix~\ref{app:detailed_exp}.}
\label{fig:exp}
\end{figure*}
\subsection{System Comparison}
We denote our system as \textbf{MISP-\textsc{Neil}\xspace} since it leverages MISP in the back end of \textsc{Neil}\xspace.
We compare it with the traditional supervised approach (denoted as \textbf{Full Expert}). To investigate the skyline capability of our system, we also present a variant called \textbf{MISP-\textsc{Neil}*\xspace}, which is \emph{assumed} with perfect confidence measurement and interaction design, so that it can precisely identify and correct its mistakes during parsing. This is implemented by allowing the system to compare its synthesized query with the gold one. Note that this is not a realized automatic system; we show its performance as an upper bound of MISP-\textsc{Neil}\xspace.
On the other hand, although execution feedback-based learning systems \cite{clarke2010driving, artzi2013weakly, iyer2017learning} may not be very practical for end users, we include them nonetheless in the interest of a comprehensive comparison. This leads to two baselines. The \textbf{Binary User} system requests binary user feedback on whether \emph{executing} the generated SQL query returns correct database results and collects only queries with correct results to further improve the parser. The \textbf{Binary User+Expert} system additionally collects full expert SQL annotations when the generated SQL queries do not yield correct answers.
{Given the completely different nature of annotations from Binary User (which validate the \emph{denotation}) and those from Full Expert and MISP-\textsc{Neil}\xspace (which validate a semantic parse's \emph{constituents}), there may not exist a universally fair way to convert one's annotation consumption into the other's. Therefore, in the following sections, we only present and discuss Binary User(+Expert) in terms of their parsing accuracy under different training iterations.
To give an estimation of their annotation efficiency for reference, we design a compromised annotation calculation metric for Binary User(+Expert) and include their results on WikiSQL validation set in Appendix~\ref{app:detailed_exp}.
}
{Finally, while our MISP-\textsc{Neil}\xspace and the aforementioned baselines all leverage feedback from users or domain experts, an interesting question is how much gain they could obtain compared with using no annotation or feedback at all. To this end, we compare the systems with a \textbf{Self Train} baseline \cite{scudder1965probability, nigam2000analyzing, mcclosky2006effective}.
In each iteration, this baseline collects SQL queries generated \emph{by itself} as the new gold annotations for further training. We additionally apply a confidence threshold to improve the collection quality, i.e., only SQL queries with probability $p_{\hat{\pi}}(a_{1:T}|s_1)$ greater than 0.5 are included. This strategy empirically leads to better performance. Intuitively, we expect Self Train to perform no better than any other systems in our experiments, since no human feedback is provided to correct mistakes in its collection.}
\subsection{Experimental Results}\label{subsec:exp_results}
We evaluate each system by answering two research questions (RQs):
\begin{itemize}[noitemsep,topsep=1pt]
\item \emph{RQ1: Can the system improve a parser without requiring a large amount of annotations?}
\item \emph{RQ2: For interactive systems, while requiring weaker supervision, can they train the parser to reach a performance comparable to the traditional supervised system?}
\end{itemize}
For \textit{RQ1}, we measure the number of \emph{user/expert} annotations that a system requires to train a parser. For Full Expert, this number equals the trajectory length of the gold query (e.g., 5 for the query in Figure~\ref{fig:intro}); for MISP-\textsc{Neil}\xspace and MISP-\textsc{Neil}*\xspace, it is the number of user interactions during training.
Note that while we do not differentiate the actual (e.g., time/financial) cost of users from that of experts in this aspect, we emphasize that our system enjoys an additional benefit of collecting training examples from a much cheaper and more abundant source. {For Self Train, the number of annotations is always zero since it does not request any human feedback for the online user questions.}
Our results in Figure~\ref{fig:exp} (top) demonstrate that MISP-\textsc{Neil}\xspace consistently consumes a comparable or smaller amount of annotations to train the parser to reach the same parsing accuracy.
Figure~\ref{fig:q_curve} in Appendix further shows that, on average, it requires no more than one interaction for each user question along the training. Particularly in the 10\% initialization setting, MISP-\textsc{Neil}\xspace uses less than 10\% of the total annotations that Full Expert needs in the end. Given the limited size of WikiSQL training set, the simulation experiments currently can only show MISP-\textsc{Neil}\xspace's performance under a small number of annotations. However, we expect this gain to continue as it receives more user questions in the long-term deployment.
To answer \textit{RQ2}, Figure~\ref{fig:exp} (bottom) compares each system's accuracy after they have been trained for the same number of iterations. The results demonstrate that when a semantic parser is moderately initialized (10\%/5\% initialization setting), MISP-\textsc{Neil}\xspace can further improve it to reach a comparable accuracy as Full Expert (0.776/0.761 vs. 0.794 in the last iteration).
In the extremely weak 1\% initialization setting (using only around 500 initial training examples), all interactive learning systems suffer from a huge performance loss. This is consistent with our finding in theoretical analysis (Section~\ref{sec:theoretical_analysis}). In Appendix~\ref{app:subsec:theoretical_exp}, we plot the value of $e_i$, the probability that $\hat{\pi}_i$ makes a confident but wrong decision given a gold partial parse, showing that a better initialized policy generally obtains a smaller $e_i$ throughout the training and thus a tighter cost bound.
Our system also surpasses Binary User. We find that the inferior performance of Binary User is mainly due to the ``spurious program'' issue \cite{guu2017language}, i.e., a SQL query having correct execution results can still be incorrect in terms of semantics.\nop{For example, ``\texttt{WHERE C1=A}'' and ``\texttt{WHERE C1=A and C2=B}'' may give the same execution results when all database records satisfying ``\texttt{C1=A}'' also meet ``\texttt{C2=B}'' by accident. However, semantically the latter includes an extra condition which may not be specified by the question.} MISP-\textsc{Neil}\xspace circumvents this issue by directly validating the \emph{semantic meaning} of intermediate parsing decisions. {The performance of Binary User+Expert is close to Full Expert as it has additionally involved expert annotations on a considerable number of user questions, which on the other hand also leads to extra annotation overhead.}
When it is assumed with perfect interaction design and confidence estimator, MISP-\textsc{Neil}*\xspace shows striking superiority in both aspects. Since it always corrects wrong decisions immediately, MISP-\textsc{Neil}*\xspace can collect and derive the same training examples as Full Expert, and thus trains the parser to Full Expert's performance level in Figure~\ref{fig:exp} (bottom). However, it requires only 6\% of the annotations that Full Expert needs (Figure~\ref{fig:exp}, top). These observations imply large room for MISP-\textsc{Neil}\xspace to be improved in the future.
{Finally, we observe that all feedback-based learning systems outperform Self Train dramatically (Figure~\ref{fig:exp}, bottom). This verifies the benefit of {learning from human feedback}.}
\subsection{Generalize to Complex SQL Queries}\label{subsec:spider}
We next investigate whether MISP-\textsc{Neil}\xspace can generalize to the complex SQL queries in the Spider dataset \cite{yu2018spider}, which can contain complicated keywords like \texttt{GROUP BY}.
For the base semantic parser, we choose EditSQL \cite{zhang2019editing}, one of the open-sourced top models on Spider. Given the small size of Spider (7,377 question-SQL pairs for training after data cleaning; see Appendix~\ref{app:subsec:editsql} for details), we only experiment with one initialization setting, using 10\% of the training set.
Since \nop{all Spider models}EditSQL does not predict the specific values in a SQL query (e.g., ``\texttt{jalen rose}'' in Figure~\ref{fig:intro}), we cannot execute the generated query to simulate the binary execution feedback. Therefore, we only compare our system with Full Expert and Self Train.
Parsers are evaluated on Spider Dev set since its test set is not publicly available.
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{exp_figures/editsql_joint_10p_dev_avg.png}
\caption{Parsing accuracy on Spider Dev set when systems are trained with various numbers of user/expert annotations and for different iterations.}
\label{fig:editsql}
\vspace{-4mm}
\end{figure}
Figure~\ref{fig:editsql} (top) shows that MISP-\textsc{Neil}\xspace and MISP-\textsc{Neil}*\xspace consistently achieve comparable or better annotation efficiency while enjoying the advantage of learning from end user interaction. We expect this superiority to continue as the systems receive more user questions beyond Spider. Meanwhile, we also notice that the gain is smaller and MISP-\textsc{Neil}\xspace suffers from a large performance loss compared with Full Expert (Figure~\ref{fig:editsql}, bottom), due to the poor parser initialization and the SQL query complexity. This can be addressed via adopting better interaction designs and more accurate confidence estimation, as shown by MISP-\textsc{Neil}*\xspace. {Similarly as in WikiSQL experiments, Self Train performs worse than human-in-the-loop learning systems, as there is no means to correct wrong predictions in its collected annotations.}
\section{Conclusion and Future Work}\label{sec:future_work}
Our work shows the possibility of continually learning semantic parsers from fine-grained end user interaction. As a pilot study, we experiment systems with simulated user interaction. One important future work is thus to conduct large-scale user studies and train parsers from real user interaction. This is not trivial and has to account for uncertainties such as noisy user feedback. We also plan to derive a more realistic formulation of user/expert annotation costs by analyzing real user statistics (e.g., average time spent on each question).
In experiments, we observe that neural semantic parsers tend to be overconfident and training them with more data does not mitigate this issue. In the future, we will look into more accurate confidence measure via neural network calibration \cite{guo2017calibration} or using machine learning components (e.g., answer triggering \cite{zhao2017end} or a reinforced active selector \cite{fang2017learning}).
Finally, we believe our algorithm can be applied to save annotation effort for other NLP tasks, especially the low-resource ones \cite{mayhew2019named}.
\section*{Acknowledgments}
We would like to thank Khanh Nguyen and the anonymous reviewers for their helpful discussions or comments.
{The research conducted by the Ohio State University authors was sponsored in part by the Army Research Office under cooperative agreements W911NF-17-1-0412, NSF Grant IIS1815674, NSF CAREER \#1942980, Fujitsu gift grant, and Ohio Supercomputer Center \cite{OhioSupercomputerCenter1987}. The views and conclusions contained herein are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notice herein.}
The Spider dataset~\cite{yu2018spider} is distributed under the CC BY-SA 4.0 license.
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Q: Wrong number of rows when reading a CSV file I have a CSV file that I wrote with a Perl script. Files opens fine in Excel or in Simple Text and looks fine. It has 9 rows. However, when I count it with nrow() or dim(), I get 8 rows. This is causing downstream problems. Headers are 'a' to 'j'. Thanks.
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g 0.368286 0.133209 0.189854 0.329541 0.312233 0.371553 0.55966 0.663372 0.678283 0.811317 0.896647 0.887872 0.919798 0.945003 0.895565 0.968837 0.991214 0.987583 1.00316 1.020707 1.015521 0.985502 0.998961 0.976184 0.98338 0.973889 0.9674 0.968549 0.966232 0.966105 0.966297 0.965883 0.976568 0.999395 1.011924 1.03344 1.051626 1.059014 1.063355 1.068936 1.053488 1.045903 1.031994 1.008174 0.991796 0.972343 0.973369 0.969431 0.967085 0.963154 0.966865 0.962234 0.95759 0.96642 0.966713 0.974709 0.973966 0.981547 0.984093 0.991954 0.985628 0.996822 0.991295 0.98659 0.989936 0.978239 0.977446 1.025974 1.042636 1.040808 0.982495 0.991225 1.015466 1.008242 1.030642 1.004306 1.086892 1.097275 1.120253 1.138095 1.135337 1.209962 1.225443 1.224011 1.338381 1.450842 1.727673 1.719172 1.82727 2.074713 1.709345 2.290568 2.321692 1.542005 0.798422 8.181066 2.759658 5.513723 22.122134 13.63921 2.675538
h 0.30497 0.32974 0.424478 0.455078 0.523571 0.606559 0.660406 0.703971 0.729999 0.915297 0.981265 0.96674 0.925204 1.036524 0.953261 0.978409 0.987847 1.01834 1.019895 1.038827 1.024035 1.012836 0.994345 0.994459 0.972257 0.97309 0.978206 0.968312 0.964948 0.962225 0.96529 0.974432 0.975632 0.994073 1.010742 1.030179 1.042012 1.056884 1.058926 1.060281 1.059781 1.038339 1.029365 1.009022 0.986078 0.978875 0.9716 0.969215 0.958117 0.972496 0.968037 0.97107 0.958519 0.970863 0.969962 0.975005 0.978711 0.984085 0.984683 0.984162 0.996244 0.997889 0.994661 1.001441 0.985552 1.021569 1.000549 1.002552 0.997683 1.033186 1.013344 1.019947 1.057587 1.033291 1.069199 1.036226 1.168877 1.175308 1.22975 1.11576 1.122753 1.106146 1.224346 1.258167 1.290459 1.477277 1.427201 1.742816 1.558004 1.386269 1.910887 1.920479 1.08143 1.206672 2.082642 3.048554 3.085038 18.491473 12.365233 0 0
j 0.354463 0.327358 0.398802 0.473 0.602168 0.764142 0.819362 0.914898 0.823412 1.010715 0.854421 0.892255 0.981967 0.966507 1.021983 0.975027 0.961088 0.960516 0.971975 1.01222 0.979767 0.987716 0.98707 0.970561 0.963265 0.962699 0.958097 0.961291 0.952577 0.958112 0.9596 0.967041 0.97311 0.991703 1.006086 1.025563 1.040921 1.055595 1.05902 1.068992 1.060288 1.046614 1.037744 1.019254 0.996458 0.984552 0.986514 0.984358 0.977773 0.980087 0.972711 0.969122 0.975125 0.968046 0.968058 0.979439 0.97843 0.978518 0.98551 0.979352 0.983617 0.984822 0.986629 0.986932 0.991861 1.002382 0.999269 0.99465 0.994519 0.987402 1.000541 0.977929 0.976282 0.964102 1.032155 1.04334 1.063832 1.096302 1.105991 1.065358 1.106644 1.068104 1.064264 1.167453 1.278531 1.383359 1.417057 1.672739 1.39427 1.396529 1.94346 1.50906 1.274638 1.467406 2.337833 6.387922 0 11.099379 16.193772 4.992065 0
A: Functions in R can have default values for some parameters. For read.csv:
read.csv(file, header = TRUE, sep = ",", quote = "\"",
dec = ".", fill = TRUE, comment.char = "", ...)
header=TRUE means que first row is assume to be the header of the file. This means, it won't be recognized as data by R. If you read your file with header=FALSE
read.csv(file, header=FALSE)
you will get the 9 rows.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,050 |
Сулейма́н Бердука́ев () — российский боксёр чеченского происхождения, бронзовый призёр чемпионата России 2008 года, мастер спорта России.
Родился 19 ноября 1987 года в Веденском районе. В 2001 году стал заниматься боксом. Выпускник Грозненского государственного нефтяного института. Представляет спортивный клуб «Динамо» (Грозный). Тренер Беслан Кадыров.
Спортивные результаты
Победитель Кубка мира 2005 года среди нефтедобывающих стран;
Чемпионат России по боксу 2008 года — ;
Чемпион Южного Федерального округа 2008 года.
Литература
Ссылки
Боксёры Чечни
Выпускники Грозненского нефтяного университета
Боксёры России | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,044 |
class FormatPauper < FormatVintage
def format_pretty_name
"Pauper"
end
def in_format?(card)
return false if card.funny
card.printings.each do |printing|
next if @time and printing.release_date > @time
next if @excluded_sets.include?(printing.set_code)
return true if printing.rarity == "common" or printing.rarity == "basic"
end
false
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,466 |
Blumenkrantz is a German and Ashkenazi Jewish surname meaning "flower-wreath". Notable people with the surname include:
Avrohom Blumenkrantz, American rabbi
Jeff Blumenkrantz, American composer
Jewish surnames
German-language surnames
Yiddish-language surnames
Surnames from ornamental names | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,727 |
In 1979, Jerry Whitlock founded Whitlock Packaging after acquiring a closed and insolvent juice manufacturer. Through creative strategic partnering, Jerry Whitlock become the largest contract manufacturer of branded non-carbonated beverages in North America, Whitlock Packaging Corporation. Whitlock Packaging Corporation was twice the size of its nearest competitor. Under Mr. Whitlock's leadership, the company achieved significant growth through acquisitions, expansion of production facilities, product innovation and increased volume from major clients such as PepsiCo, Coke, Nestle, Arizona Tea, Campbell Soup, Ocean Spray, Lipton, Jumex, Goya, Apple & Eve, Monster, Rockstar, Gold Peak Tea , Pure Leaf Tea and Vitamin Water. Whitlock Packaging Corporation received a coveted Oklahoma "Industrial Development Award" along with other distinguished awards such as "The Minute Maid Company Award of Excellence" and the "PepsiCo Strategic Partner Award".
In 1987, Jerry Whitlock served as ambassador to China, Japan, Hong Kong and Korea. Again, in 1991 he served as ambassador to Russia, India, Nepal, Tibet and Vietnam. Here he was a member of the American People Ambassador Program, which was founded by former President Dwight D. Eisenhower. Whitlock represented the Delegation in the category of Food and Packaging Technology.
In 2004 Jerry Whitlock acquired Juice Bowl, a trusted 50-year-old juice brand serving schools and government feeding programs .In 2007 Cutting Edge Beverages was created by Jerry Whitlock. In 2016, Jerry Whitlock sold Whitlock Packaging to Refresco, the leading European bottler of soft drinks and fruit juices for A-brands and retailers.
Being a true believer in "giving back to the community", Mr. Whitlock donated the Whitlock Sports Complex in Fort Gibson. Twelve hundred children took the field on the first day of the grand opening. Other land donations were made to three Oklahoma towns, including monetary donations to numerous organizations such as civic clubs, churches, school educational programs and disaster relief assistance throughout the U.S. Jerry Whitlock also donated the Whitlock Guest House in Muskogee, OK to Women in Safe Homes (WISH), a non-profit organization. Whitlock Wishouse today is used as a fundraising vehicle. All money collected goes to support the WISH shelters to house and support battered women, children and their recovery.
He has travelled to and explored 117 countries to date with the desire to visit all 197 undisputed countries in the world. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,686 |
Vivienne Kelly was born and educated in Melbourne, where she now lives. She has worked in university administration, and has a PhD from Monash University on myth and history in Australia. _Cooee_ , her first novel, was shortlisted for the _Age_ Book of the Year in 2008. 'Passion Fruit' was included in _Best Australian Short Stories_ , and 'The Third Child' won the _Australian Women's Weekly_ short-story competition. _The Starlings_ is Vivienne Kelly's latest novel.
textpublishing.com.au
The Text Publishing Company
Swann House
22 William Street
Melbourne Victoria 3000
Australia
Copyright © 2008 Vivienne Kelly
The moral right of Vivienne Kelly to be identified as the author of this work has been asserted.
All rights reserved. Without limiting the rights under copyright above, no part of this publication shall be reproduced, stored in or introduced into a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photocopying, recording or otherwise), without the prior permission of both the copyright owner and the publisher of this book.
First published in 2008 by Scribe Publications
This edition published in 2017 by The Text Publishing Company
Cover design by Sandy Cull, gogoGingko
Cover images from Shutterstock
Page design by Jessica Horrocks
Typeset by J&M Typesetting
National Library of Australia Cataloguing-in-Publication entry
Creator: Kelly, Vivienne, author.
Title: Cooee/by Vivienne Kelly.
ISBN: 9781925498066 (paperback)
ISBN: 9781925410426 (ebook)
Subjects: Interpersonal relations—Fiction. Families—Fiction. Domestic fiction, Australian.
_For RJWS_
Contents
Part One
Part Two
Part Three
Part Four
Part Five
Who am I? It's a good question, when all's said and done. It's something I often ponder. To Sophie I am Gandie. To Kate I am Mum—a nice, normal name, which absolutely fits, given that Kate is such a nice, normal person. To Dominic (who repudiates relationships filial or maternal, doubtless for his own good reasons) I am Isabel. His partner Paula doesn't call me anything at all. To my sister Zoë, for reasons relating to a childhood joke long since forgotten, Minky. (Interestingly, I never had a name for Zoë. She was just Zoë.) My oldest friend and professional partner Beatrice calls me Izzie. I detest the name, but she's always done it.
To Max I was Belle. Belle, Bella, ma belle.
The hapless Gavin calls me Millie. This is because he has constructed and sportively elaborated an acronym for Mother-In-Law. When I say that this is one of Gavin's better jokes you will have some idea of his general standard. And Liam of course knows me as Gandie, because that's what his sister calls me.
What's a name, anyway? (Others have said this before me, I know.) It's a handle, for our acquaintances and relatives to use in their dealings with us. It's a handle that frequently tells us more about the people using it than ourselves.
I don't have a name for myself. Do any of us? Do we create names for ourselves, to add to our comfort or security or confidence or simply for a welcome clarification of one's role in a complex and bewildering world? I don't. I change, as chameleons are supposed to do (although I never really believe that's true: it seems one of the unlikelier myths, like pelicans piercing their breasts for blood—so implausible when you think of it, given the shape of a pelican's beak—or phoenixes being reborn from their own ashes); I change depending on who I am with, who is conversing with me, seducing me, launching the current demand at me. Sophie draws my sweetness, Kate my anger, Dominic my yearning.
Yet it seems to me I must somehow—mustn't I?—be more than the sum of the parts I play. I don't believe anybody profoundly knows me—least of all my granddaughter Sophie, who loves me best, and to whom in curious ways I feel closest.
But there's nobody now who takes in the whole inchoate lump of me, who understands the messy ins and outs, who gets past the particular stereotype he or she has constructed to deal with the tricky thing—the problematic and demanding thing—that the relationship with me seems always to turn into.
The acerbic mother, the resentful daughter, the thorny little sister, the distant wife, the passionate lover, the indulgent grandmother. How we all rely on these paradigms to navigate ourselves through the maze of communicating with people, of relating to our relations!
I've created stereotypes of my own, of course, for them. Kate is the dutiful daughter; Dominic the edgy, prickly, whimsical son; Zoë the irritating authoritative big sister. It's a matter of expectation—especially, I suspect, where one's children are concerned.
When those small explosive bundles of babies burst into our lives, each is freighted with its burden of parental expectations. I expected my children to be good children; I expected to love them; I expected them to love me. I wouldn't have been surprised, oddly, by major reversals: I wouldn't have been amazed to find a bit of hatred in the mix, because hatred is passion, almost a part of fierce loving. What I didn't expect were the minor blemishes, the trivial flies in the ointment. I didn't expect to be annoyed by them, to find that they produced inconsequential but perpetually discordant notes in the harmonies I was trying blindly to build into my life, the music I was trying to live.
You would die for your children. You would kill for your children. I thought it would be like that, and it was. Even when you don't like them much, you'd still die for them. It's such a big, sweeping emotion, you expect it to carry everything before it like a vast thundering avalanche, and instead it gets tripped up and subverted by pebbles, branches, bits and pieces aggravatingly strewn across the course of every day.
Children are the centre of one's emotional existence, the fuel for all one's hopes, the source of one's most profound joys; they are also bothersome and inconvenient and exasperating. This is the ambiguity no one is prepared for before actually becoming a parent; it is also the tiresome truth not enough people admit to after becoming one.
But my grandchild slid into my world without beforehand arousing my expectations, and I was therefore unprepared for her. When Kate put her into my arms with that fey and timorous smile of hers that always infuriates me even as it moves me, I had not anticipated the great surge of passion that flooded my body.
I gazed into Sophie's pale squashed face with its eyes closed into slits and its black eyelash silk forming a short fine crescent on either cheek. I started to tremble and I was afraid I would drop her. I sat down, abruptly, and admired this tiny, new person. I tried to stop the tears from rising. I glanced up, and saw Kate's face, which was now apprehensive, fearful.
Why must she always want my approval? I wondered, vexed with her for not looking proud and triumphant, as a woman should look who had produced such a miracle.
'Do you like her?' asked Kate.
'Don't be silly,' I said. 'How could I not like her? I adore her.'
'She is gorgeous, isn't she?' As if Kate needed to be reassured, I thought; as if she had to be told her own daughter was beautiful. Where were her wits?
'She's very gorgeous,' I said, steadily, still trying to deal with all this starry blood that was flushing through me, scouring my veins, thumping behind my eyes. 'Very beautiful.'
Kate leant back on the pillow and closed her eyes, as if vindicated.
It is an interesting question, this one of grandchildren. I think one of the reasons they tug at us so is that we have lost the bounce we had as parents: we are older and tireder and less elastic; we sag more, emotionally. We have lost our brightness. We have disappointed the people we love; they have disappointed us. The new child is the opportunity for a new relationship; it offers the chance of being a new person, one who is still bright, who does not disappoint.
I have tried never to disappoint Sophie. As she nibbled her way into my heart with her pearly little teeth, I did not need to try to love. But I have tried to be loveable. I have consciously tried to attach her to me, tried to beckon and fascinate and hold her, in ways I never tried with my own children—or perhaps was not capable of, with them.
Perhaps I give in to her too much. There was a curious incident when she was five or so. We had been shopping, and it turned out that what she wanted most in the world was a string of lilac beads. It was a small-enough thing, and I had no idea Kate would disapprove. Truly, I did not.
When we returned, the child with her sparkling trophy (I have to admit, they were dreadful, tawdry things, but she was only young after all) strung around her neck, Kate pulled me up sharp.
'I don't allow her to have jewellery.'
'For heaven's sake, Kate. It's a string of beads.'
'I don't think little girls should wear such things. You never let me wear them.' Suddenly Kate's eyes are brimming with tears; she is screaming at me. 'Why didn't you let _me_ have them? If it's only a string of beads, why wasn't _I_ allowed to have them?'
This is a grown woman, weeping for something that happened—or didn't—twenty years ago. This is typical of Kate: the big emotional storm over sheer trivia, the accusatory tears, the sudden tornado that blows up in an instant out of nowhere and disappears as rapidly.
'Kate,' I snap. Sophie is looking on, curious, interested, ready to be upset because her mother is upset. 'Kate, get hold of yourself, for goodness' sake.' I indicate with a jerk of my head what she already knows, that the child is there, that the child will be distressed at any moment. How ungovernable she is, with these swift storms! Has she really never grown up past this stage of simmering indignant adolescence?
I never did learn how to manage Kate. Zoë used to take me to task over her. Once, I remember, when Kate was I suppose fourteen or so and had done something silly, I spoke sharply to her and she fled.
'Why do you treat her like that?' says Zoë, speaking sharply herself, as she usually did. To me, anyway.
'Like what?'
'Why do you flick at her like that, all sarcastic?' 'Because I love her.' Isn't it obvious?
'It's cruel.'
I roll my eyes. _Cruel?_ I think. _Moi?_ This is Zoë speaking, my sister, she who bit painfully at my heels all through my childhood and adolescence, she who constantly berated and criticised and attacked and punished. And, suddenly, _I_ am the cruel one?
'Why don't you do it to Dominic, then?' she asks, accusingly.
Because I love him. What other answer is there? Because I love him, my dark, remote Dominic, who refuses ever to permit a diminutive form of his name. He has always scowled when called Dom, Dommy, Nic, Nicky; once we even tried Domenico: to himself he is always and only Dominic. _He_ knows what his name is and who he is.
My Dominic, my changeling son, who as a baby used wide-eyed to pump my breasts, gazing at me distrustfully. 'Where did you come from?' I'd ask him. 'Not from me, I'm certain. Someone swapped you, didn't they?' His steady stare agreed.
We love in different ways. Different people demand from us different love, different hopes, different agonies.
One of the surprising things about Dominic was that in some ways he was so much what I had hoped for, what I had even expected, albeit nervously. As dark as Kate was fair, as mercurial as she was placid, elfin where she was bovine. Physically and temperamentally, he was all I'd imagined; and as he grew older it became clear that his intelligence was as lucid and ranging as I'd assumed it would be. But the closeness I had also imagined, the intimacy and transparency that was to have existed between us: this never eventuated.
I suppose there were occasions on which Dominic as a toddler raised his chubby arms to me and did or said something that made me feel he loved and needed me. Offhand, I can't bring such occasions to mind. Always, he regarded me dispassionately and from a distance: across a playground, a table, a breast. Never a loving child, he has grown into an unloving adult.
It is astonishing, in fact, that I care for him as I do: so much of love is a response, an answering involuntary warmth to the warmth projected. It mirrors that warmth: no, not just mirrors it, reflects and expands it. Dominic has never tried to attract me, never tried to radiate this warmth, this tenderness, kindliness, that ought to exist between us, not only on my side.
But without the most minimal offering of affection from Dominic, finding itself treated with this entire absence of tenderness, my love simply becomes more stubborn: it digs itself in like some spiky leather-leaved plant whose only object is survival. It feeds off itself, since there is nothing else to sustain it.
Sophie is a loving child. Sweet-natured and gentle. It's a problem, really: how will she ever defend herself in a hostile world? So far she has met kindness at every turn—but, as we all know, that won't last.
Sometimes I think Kate ought to try to toughen her up a bit—so she'll be a more competent person than her mother, apart from anything else. But she says she hasn't the heart, and I understand that. I say _Humph_ , and cast my eyes upward, and drum my fingers, and tell Kate she's a fool; but I see what she means.
When the child comes to me (which she does, after school, Monday to Thursday, four o'clock prompt); when she slopes in the door with her wide, shy smile, her clear green eyes and her ready hug and kiss for Gandie, I melt, too. I look forward to seeing her, and I'm sorry when she leaves. Even when I love people dearly (and I am a loving person; I am not a cold person), I am usually at least a little relieved when they leave, when we part. Such a strain, sometimes, supporting relationships, maintaining conversations. Sophie's not a strain.
She's taken to asking me about days past, about when I was a young woman, about when her own mother was a child. I found this hard, at first. Kate was an unremarkable child by any standards, after all, and her childhood was undistinguished: I found it difficult to reach back through the tangle of memory, to pluck bright stories from the long-ago. But once I started, I found I could manage. For a twelve-year-old she is quite mature, I think. Kate would not at that age have been able to grasp some of the concepts her daughter easily handles. She startled me when she first began to ask questions.
'Gandie, you used to be married to Grandpa Steve, didn't you?'
I can see her, lolling against the kitchen table, her slim, tanned legs inelegantly splayed, white socks rucked about her ankles. It disconcerts me, this question: it comes like a mild bomb out of the blue, all kinds of straying tracer bullets to follow in its wake. I've never spoken to Sophie about Steve—why would I?—but somehow I've always assumed she understood the relationship.
We both turn up regularly for the big occasions—birthdays, school concerts and the like. It was difficult at first: we would eye each other suspiciously across the room, speak together only stiffly, draw a sigh of relief when we could withdraw from each other. I did, anyway, and I assume it was the same for him. As these things do, it has become easier over time.
We even sometimes go to each other's houses for a family event, although I think that is stretching the tape: how friendly can divorced couples ever be, after all? How can we step smoothly to meet each other over the mess of the past, the mangled trust, the dead dreams?
We have surfed together on a surging sea of bile and bitterness, leaving in our single wake a foul detritus, a jetsam of anger and shame and guilt and betrayal. It is obscene, really obscene, to expect us to maintain civilities. Still, we do it, for Sophie's sake. At least, that's why I do it, and I assume it's the same for him. And for Liam, of course: we do it for him, too.
'You don't mind me asking, Gandie?' says Sophie, taking a bite of her apple and regarding me with slight anxiety.
'No, no,' I say, heartily, lying through my teeth. 'Of course I don't mind, sweetheart.' I do, of course. I mind terribly. She's fond of Steve: I don't want anyone filling her head with a lot of rubbish about the divorce, about how I behaved. 'Yes, we were married.'
'And then you stopped being married,' observes Sophie, cheerfully.
'Yes. Yes, we did.'
'Why, Gandie?' Ah, why?
'Well, honeybunch,' I say. 'It was all a long time ago, you know.'
'How long?'
I pause to think. I make more of a meal of it than I need to, as I count over my fingers, to try to give myself a bit of a buffer, a few moments to think, to make sure I don't blurt out something wrong. Steve and I separated when Kate was fifteen. Three years older than Sophie is now. Kate had Sophie when she was twenty. What an easy sum it is.
'Seventeen years,' I say, brightly. 'Good heavens, Sophie, seventeen years! What a long time that is. More than your life.'
Sophie absorbs this.
'But why?' she says. 'Whose fault was it?'
'Sophie, honey, people sometimes decide not to keep on living together. There's nothing wrong about it; it needn't be anyone's fault. Grandpa Steve and I just decided not to keep on living together. It was all perfectly friendly.'
I mouth these barefaced lies in a kindly voice, watching my granddaughter's face. I don't blame her for looking dissatisfied. Why do we try to protect our children in this hopeless way?
When they are little, we put them to bed with the hot, rank breath of the wolf, slavering down the front of Granny's nightgown; we torment them with stories of children whose parents desert them in thorny, hostile woods, of children whose parents force them into the world on murderous quests, of children who do battle with ogres, who are imprisoned alone forever in dank stone towers or pierced by glass shards or abused by vindictive stepmothers, children who go to bed in their flowerlike innocence and unexpectedly fall asleep for a hundred years, waking to find the world still cruel but differently so.
And we expect them to sleep soundly. _Don't let the bedbugs bite_ , we say, tapping their downy plump cheeks lightly as we leave them to the dark solitude of the bedroom. And then, when they are only a little older, we cannot find it in ourselves to tell them the plain truth, to admit that a marriage went badly wrong, that people lied and shouted and hated.
'And then you married Max,' she states, bending to pat the old dog Borrow, who has a penchant for apples and licks the sweet juice from her fingers.
I rearrange my features to try to produce an impression of offhandedness, of undisturbed good humour.
'Yes,' I say. 'I married Max.' Well, near enough. 'Gandie, you look so cross.'
'No, no, sweetheart.' I stretch my lips to show how unconcerned I am.
'You sure you don't mind me asking you about it?' 'Heavens, no. It's just—sweetheart, Sophie love—it's such a long time ago, and I don't know—I mean—why are you suddenly talking about these things?'
'Mum told me about it,' says Sophie, apparently losing interest and showing me what genuine unconcern looks like as she mooches off to glance at the newspaper, which is lying on the kitchen bench.
I will give Kate hell the next time I see her, I think, the old fury my daughter can always generate stirring deep and cold and snakelike inside me.
But then I think, later, she is twelve, after all, my Sophie. She will be adolescent, soon: to all intents and purposes, is already. She is of an age to take an interest in these things.
I suddenly wonder whether her periods have begun. Surely Kate would have told me? But I have noticed her little breast-buds under her white school T-shirts. A sliver of fear pricks my heart. It is at this point that your children start to change, to regard you with alien and uncommunicative eyes. It is when puberty kicks in, when hormones perk up, when the blistering new juices of life start to sizzle. _Hello_ , you say. _Hello? It's me here. It is I, your loving mother. I haven't changed_. But they have.
I dread the day this happens to my Sophie, the day when she ceases to run to me with the fierce pressure of her kisses and her sharp, squeezing hugs. I dread her being caught up in the pitiless impassive tides of womanhood, the impersonal coercive rhythms, the bleeding and the desire and the pain.
We ought to have some kind of ritual to welcome our daughters into their maturity, into the sisterhood. There ought to be a ceremony, a thumbed cross of blood on the forehead, the musty perfume of incense, prophecies from the crones. Picking up a box of tampons from the supermarket shelf doesn't rate.
Not that the sisterhood is up to much, though. On the whole, taken by and large, really one would sooner not be part of it.
The next time I was alone with Kate I mentioned Sophie's new interest in Max. I didn't shout; I wasn't heated. I was quite mild about it, and there was no need for her to look sideways at me in that infuriating kicked-puppy way she has.
'Sophie talked to me about Max,' I say.
She glances at me, drops her eyes. 'Yes?' she says, politely.
'You've been talking to her about Max.'
'She asked me.'
'She wouldn't have asked if she hadn't had some encouragement.'
'Sophie's quick to pick things up. She doesn't need encouragement.'
'She must have had some.'
'Mum,' says Kate, with a conspicuous effort at patience that irritates me. 'Mum, Max will come back one day. What are we meant to say to Sophie? Here's Max: he used to be married to Gandie, but we've never talked about him; we've never told you about him. What will she think then? Max was a fact in our lives, Mum. Is a fact. I know you and I don't speak of him, but there he is, just the same. And a fact in Sophie's life, too. We can't hide him. We can't pretend he never existed.'
'He won't come back.'
'I think he will,' says Kate, still not looking at me. Why will she never look at me when we speak? She never meets my eyes.
'Have her periods started?' I ask, abruptly.
Kate is startled. 'Sophie?'
'Who else?'
'Not yet.'
'She's twelve.'
'I know how old my daughter is, Mum.'
'Well, then. I only wondered.'
'Any day, I'd say. She's growing up fast.'
'I can see that,' I snap. Why does she always state the obvious in those dulcet explanatory tones? She drops her eyes again, ducks her head in that exasperating way she has.
'Mum, she is growing up. What I mean is, she's started to be interested in different things.'
'Do you mean sex?'
'Not really sex, no. Well, not that she's let on to me. But weddings, and babies, and that sort of thing. She's asked me lots about when I was little. When was I born, what happened, what sorts of toys did I have. That sort of thing.'
It's true. It isn't long before Sophie nobbles me about Kate as a baby.
'Mum was your first baby, wasn't she, Gandie?'
'Yes, honey.'
'Were you very excited?' 'Well, of course.'
'Did you have a nursery all ready for her?'
'I certainly did,' I say, brightly.
'And toys and things?'
'Yep.'
'Cuddly toys?'
'You bet.'
'Mum says her favourite toy was a blue bunny called Thumper.'
I think I do remember a moth-eaten ugly old rabbit. Kate used to take it everywhere with her and fall into despair when she lost it, which happened frequently. It drove me mad.
'Mum says Thumper was blue because you really wanted a boy and you got all blue things. Mum says you didn't really want a daughter.'
Sophie is munching a walnut brownie (I make them especially for her) and her enunciation is a little unclear, but there is no doubt she has just said this. She has presented it to me without animus, without any accusatory quality to the statement, in a matter-of-fact tone. It distresses me: it isn't true; and I've certainly never given Kate any reason to think it is.
'That's not the way it was,' I say. 'Of course I wanted your mum.'
'She says you didn't. Mum says you only ever wanted boys. Mum says you weren't happy till you had Dominic.'
'Well, darling, Mum's wrong.'
'Okay. But why did you have all blue things?'
'I didn't have all blue things. I had lots of different coloured things. In those days you didn't know what sex your baby was going to be, so I had some blue things and some pink.'
'And some yellow?'
'Probably.'
'And some white?'
'I suppose so. Sophie, it was a long time ago and I can't possibly remember what colour things I bought for your mum.'
'Okay. That's cool, Gandie.' She is pacific as ever once her stubbornness has borne fruit.
Why does Kate tell her these things? Why does she do it? Is it to get back at me, to awaken old guilts, old sadnesses? It wasn't my fault, anyway. And I never talked about it with her, so how can she be so sure she knows anyway?
Why is it a woman's fault, if she expects one sort of baby and another eventuates? Disappointment isn't a feeling you ask for, and it isn't something you can control. And I got over it pretty damn quick. I couldn't help it; I couldn't help feeling the thud when the bundle of hopes was dropped down the chute, the loss when someone who was confidently expected to turn up decided not to. I still remember it so well.
I am lying there in a sweat of pure relief, trying to understand that the pain, the racking pain, has gone, that my flesh is no longer bursting and I am not split in two up the middle after all, that I am still alive, that it is over, over, over. I am drifting, rather: I am just conscious enough for a sleepy voice in the back of my mind to be wondering if they gave me only pethidine—and, if they did, wow, what great stuff pethidine turns out to be.
I am aware that Steve is somewhat in awe of me, of what I have just accomplished, more or less single-handed. He is sitting beside me with his head down, overcome with emotion, stroking my hand. Normally I don't like this sort of contact from him, but right at the moment the stroking, which it seems to me betokens respect and admiration and wonderment, is perfectly appropriate and in fact rather soothing.
And I hear a voice—a woman's voice—saying: 'A lovely little girl!' And this is odd, because the voice sounds like the doctor's voice, and here is the doctor's head swimming moonlike in the air above me, and she is smiling and nodding. 'A girl,' she repeats. But surely I am the only woman in the room to have just given birth, and therefore surely the only baby in the room is mine? I glance sideways, to see if perhaps there is another bed beside me. There is not.
And a baby is being placed on me, on my breasts, and the doctor is still smiling and nodding and repeating: 'A girl, a lovely little girl.'
A girl? A girl? How can it be a girl?
The baby makes an angry, snuffling sound and moves its head. It is almost the colour of beetroot—a dark, bright, pink-purple colour, and it is slick with some sort of grease. What have they smeared on it, for Christ's sake?
'Hold your baby, dear,' says the midwife, bustling around and helping me move one of my limp arms up around the creature. 'It's a dear little girl. You put your arms around her and give her a cuddle, now.'
A _girl_.
Where is my patrician Leo, my cub, my small king, he of Steve's blue eyes and my crisp, dark curls (so unnecessary on me, so attractive on him)? Where is the QC with a taste for Mozart and Tolstoy, the neurosurgeon with the exquisite collection of minor Heidelberg artists? What happened to the first Australian Prime Minister to be also a poet, a musician, a philosopher? Where is he, this son of mine with his rueful smile and his charming manner and his sensitive fingers? My cellist, my Nobel Prize winner, my Test cricket captain? My Wimbledon champion, my Astronomer Royal? My dark chevalier? Where is my son, who over the last few months has become realer than real to me, brighter than bright, my companion and saviour and pride and triumph? Goddamn it, where has Leo _gone_?
Wherever it is, it's not here. He's padded off on his ghostly paws to some celestial jungle, some vine-draped lonely clearing where he lifts up his head and roars in the moonlight, piteous and small and solitary. He's there for good: he's never coming back. And here's this peculiar, misshapen creature, this girl, glistening purple-crimson under the labour-ward lights, making off-putting nos while it tries to crawl beetlelike up my body.
It wasn't my fault. Emotions are not amenable to schooling. We cannot pretend to feel something we do not feel. I mean, to ourselves. Of course we can pretend whatever we like to the world, and if we're clever enough the world is deceived. I deceived the world, I think.
Once, early on, I started to explain to my mother how I felt about my first-born, but her clear gaze was so horrified that I stopped. She told me I was tired, I didn't know what I was saying, everything would be all right. And I suppose it was, in a manner of speaking.
I've read a lot about that kind of thing since then. These days, people would be sympathetic and call me a victim of post-parturition stress disorder or some such thing, and I'd get lots of counselling and support. But it wasn't like that, then. You loved your child; you struggled on. Count your blessings, people said.
And of course I did love Kate, and so forth. I do love her. But it was hard. I've never been able to forget how hard it was.
I waited five years before deciding to have Dominic: I was so terrified he would be another girl, another blond, chubby, docile little girl who played with dolls and sat around amiably and enjoyed fiddling with frilly doll-sized clothes. The son I had expected Kate to be still dawdled in the shadows of my imagination, witty and satirical, elegant and sleek and quick, my dark prince, my soul's companion. So long as I delayed his birth, the safer he was from the ravages of reality.
Well, I got him. As soon as I saw him, I knew I'd got him. Dominic slid out of me with a minimum of fuss: he glanced around him, took stock of his new world, and determined his course. A baby of remarkable self-possession and clarity of vision; a child who knew what he wanted.
'He's not a cuddly baby, is he?' said the infant-welfare sister one day, thoughtfully. She picked Dominic off the scales, gave him a hug. He resisted, pushing against her. Dominic always resisted.
'I thought all babies were supposed to be cuddly,' I said.
'Good heavens, no.'
'My first was cuddly. My first wanted to be cuddled all the damn time.'
'Mmmm.' She glanced at me sidelong.
'I thought I was doing something wrong. With this one, I mean.'
'No,' she said, decisively. I still bless that woman when I think of her. I can't remember her name, but I loved her: I loved her for telling me Dominic's hostility wasn't my fault, that his unlovingness was not to be laid at my door.
She had a square jaw, steady eyes, large hands, a no-nonsense look about her. 'No, see how he pushes away from me. He doesn't like it: he doesn't want to be held. All babies are different. They're just people, you know. We forget that. It's so obvious, but we forget it. Babies are just small people. Not all people are cuddly. I shouldn't think you're a cuddly person, are you?'
'No,' I said, reluctantly, wondering why she asserted this with such confidence.
'Well, then. It doesn't mean he's not affectionate. He just doesn't like close physical contact. Not at this stage of his life, anyway.'
Such a relief! I had spent the first few months of Dominic's life being pushed away from him. I spent the next several years of Dominic's life in the same manner, but at least I knew now it was nothing to worry about. Satirical, composed, distant, critical, even disdainful, he resisted intimacy from infancy.
He used to remind me of that bit in _Twelfth Night_ where Olivia is rhapsodising about Cesario/Viola: ' _O, what a deal of scorn looks beautiful / In the contempt and anger of his lip!_ ' (I know that bit because I once played Olivia in a school production. I wanted to be Viola, and wear emerald satin capri pants, but it was Olivia or nothing.)
From the age of approximately three (which I believe was when he decided he was too old to hold my hand), Dominic did a great curled lip. It's the look aspired to by those surly male models in fashion magazines: it came naturally to him. And she was right, that large-handed infant-welfare lady: it didn't mean he was unaffectionate. He was a fervent child, but selectively fervent: he didn't bestow his love on just anyone, in the indiscriminate way Kate had.
Sophie is a cross between them, I suppose. Less obsessively fastidious than Dominic, more judicious than Kate. Readier with her embraces than her uncle, not as clingy as her mother. 'Love you, Gandie', Sophie will say, hugging me goodbye with her rapid firm touch. 'I love you too, sweetness,' I will say, but she's so quick, I'm often saying it to an empty room, a closing door.
The next time Sophie asked me about what she so insouciantly calls the olden days, I was ready for her. She wanted to know about our wedding, mine and Steve's. She'd want to know about my second wedding, too, one of these days, I knew, but sufficient unto the day and so on.
I'd gone through the old case under the bed and I'd pulled out the wedding album. I don't know why I have it and Steve doesn't, but that's how it is. I'd even spent a bit of time going through it on my own, turning the pages with amazement, with sorrow.
There Steve and I are, captured forever with imprudent smiles on our faces, foolish hopes in our hearts. We stand awkwardly, trying to follow the photographer's directions, both too self-conscious, too young, lacking in composure or the sense of the theatrical that permeates good photographs of big occasions, landmarks, milestones in one's life. I remember him telling Steve to stand on an angle, his feet just so: Steve simply couldn't manage it. His feet were like clown's feet, flopping over the ground in awkward dispositions. He turned too far, anxiously, then not far enough, shuffling, his face splitting into his daft grin.
I remember it as a black day, a disastrous day, the day of my greatest mistake. Yet we all look as happy-go-lucky as kids on a picnic. No prevision dulled our spirits; no disquiet fingered us with its bleak chill touch.
We look so innocent: that's what I can't quite fathom. I don't mean sexually innocent: Steve and I had slept together—not often, but a few times. I suppose there was a political and moral dimension to our innocence. We simply had no idea that such good intentions might take one on so shadowy and truncated a route, no idea of the sharp, mashing teeth of the traps that would jump from the dark corners of the path to cripple our feet and confuse our confident, stupid steps.
But Sophie is pleased with the photographs, and with the album itself, still in reasonably good condition, its ivory satiny cover not much discoloured, the photographs themselves still miraculously crisp and glossy behind their protective tissue. I suspect she is amused by certain details—my hair, for instance, starched and curled like some extravagant origami project, and the bridesmaids' dresses. These things date, after all. But she is satisfied with my dress, which was classic and elegant. Stylish, that was the word my mother used of it. At least I avoided the meringue syndrome.
'You were so slim, Gandie,' she says, reverently. 'And so beautiful.'
What she actually means is, I was so young. I am not plump even now, when slender has metamorphosed into bony and slight given way to lean; but I was svelte then, and lissom, and young. And, yes, beautiful, damn it. I am nearly as impressed as Sophie is by my willowy waist, my high breasts, my graceful arms.
She wants to know who everybody in the photos is: she recognises Zoë as bridesmaid, and my mother, and Steve's parents. It takes ages, as we go through it all: she wants to know what the cake was made of, and what sort of shoes I wore, and what the menu was. The menu? No, I can't for the life of me remember the menu. Didn't I keep a copy? No, I didn't. Or, if I did, it's long since gone, vanished down the obliterating slippery slide of discarded keepsakes.
The honeymoon? Well, yes, a week in Tasmania: not an inspired choice in June (so cold, the air so perishing cold in your lungs), but cheap and I suppose picturesque. We were keen on bushwalking (or at any rate Steve was and I pretended to be so), and went tramping around Cradle Mountain. I guess we enjoyed it.
The first night together? Yes, a city hotel. That part is fairly clear in my mind. I had not stayed before in such five- star splendour: the heart-shaped pink soaps, the flowers and chilled champagne thoughtfully ordered by Steve. It was all ravishing, and I was brimful of the novel delectable nature of my world as I undressed and showered and arranged the new white lacy nightie over my shoulders, tweaking it down to make it more revealing.
How strange it is to remember happiness when one has come since to realise its cause was so superficial, so mistaken. Memory is like layers, needing to be peeled off: finally you have done with the endless stripping, the painstaking unwrapping and reconstructing; finally you arrive at the kernel and see how pathetic and wizened it was, really, after all.
Not that making love was a source of great contentment, even in those days. I was already familiar with Steve's style in bed: the dog, the baker, the jackhammer. First he pawed me, hesitantly, like a dog who's desperate for approval. Then he became excited and started to knead my sparse flesh, as if he thought I was made of dough and would eventually rise if he expended enough energy. And then he lost all control: I'm sure I don't need to describe the jackhammer phase to you. But I knew nothing else, at that stage. And it's like poverty: if it's all you've known, you don't miss wealth.
I had hazy notions of climax and pleasure, but I thought these would occur at some distance into the marriage, when we knew more about each other's bodies, when we had fairly launched ourselves on the broad surge of marriage and togetherness, when presumably a more sophisticated lovemaking would supersede these first crude experiments, when...when...when.
But the worst came later. Although we had made love, we had never actually spent a night together. We had never, literally, slept together. I had envisaged a period of lying in each other's arms, whispering, murmuring, touching, sharing secrets, exchanging our sleepy, contented impressions of the wedding, me falling asleep on Steve's wide warm chest, within his strong embrace. But after Steve had penetrated me and obtained his staccato climactic peak, he patted my stomach benignly, muttered something bleary, and hunched himself over, falling asleep within a minute.
I lay there, disconcerted and trying not to feel deserted. It had been a big day, of course; and when he had got his speech over and done with he'd knocked back a few wines, and I was sure there was every excuse for fatigue; but I had expected something a little more conversational, a little more...well, attentive. Courteous. Some demonstration of consideration, of husbandly care.
And then, just as I started to succumb to the warmth and the wine and the tiredness, and drift into sleep myself, a great sound exhaled itself into the room. I jumped in fright.
It was a bubbling, cascading, raw sound: a sound of turgid, rumbling rhythms and insistent plosive beats, a gaseous, volcanic, assertive, cavernous noise. It terrified me. There was something subterranean about it, yet we were on the twentieth floor of a city hotel. It rolled and roiled around the room: it echoed into the corners and bounced aggressively off the walls and the ceiling. I sat upright and stared around the bedroom in consternation and fear: Steve was snoring.
I slept very little that night. I shook his shoulder, but he only grunted. I lay next to him, gazing into the dark, absorbing the blubbering, grunting tide as it rose and fell, rhythmically, without cessation, without interruption. I suppose I dropped off at around five or six in the morning, and small wonder if I was hazy at breakfast.
We had it delivered to the room: all part of the Lovebird Special Steve had paid for. A smart, diminutive waiter with a bald head and a long, black apron set up the table with a damask cloth and lots of really quite good silver. I had taken some toast and was helping myself to marmalade when Steve leant over the table and clasped my hand, immobilising it just as I was about to tip the spoon onto the side of my plate.
'I can't believe how lucky I am,' he whispered.
'Um,' I said, watching the marmalade drip onto the white damask. 'Honey, just let me—'
'I can't believe you chose me. I can't believe you said yes. You're my darling, my sweetheart, my lovely one. I'm the proudest man in the world, Isabel.'
Don't misunderstand me: it's pleasant to be adored. I'm no different from anybody: I can take a lot of adoration. But I'd had maybe two hours of sleep and I wasn't at my best and I didn't want to drip marmalade on the crisp white tablecloth. I tried to keep the twang out of my voice, but it was hard.
'Steve, love, just let go of my hand, would you.'
He looked absurdly stricken. He dropped my wrist as if it had burnt him. He goggled, slightly.
'You don't regret it?' he asked, tense, desperate. 'You don't regret marrying me?'
'No, of course not. I haven't had much sleep, that's all.'
'You do look tired,' he said, fondly. I had seen myself in the mirror earlier that morning and I could attest to this. It was an understatement and I should have been more grateful than I was. 'Maybe you were wound up, nervous? It was a big day, wasn't it, lovely? I was tired; I bet you were, too.'
'Well, yes, I was tired. But I couldn't sleep. I couldn't sleep at all.'
'Were you worried?' he asked, taking my hand again and pawing it.
'Steve, are you aware that you snore?'
'Well, yes, of course. All men snore, lovely. Women get used to it.'
I didn't like being called lovely all the time. I was starting to hope he wasn't going to make a habit of it. And all men didn't snore: I was sure of that.
'Well, isn't there anything you can do about it?'
'Do?'
'I don't know. A mask, or a clothes peg, or an operation, something. There must be something you can do. Steve, I can't sleep through that noise. I can't sleep. I haven't slept. I won't sleep. What can you do?' I could hear my voice, thin and high and rising to hysteria.
He looked concerned, but not enough. 'Lovely, all men snore. You talk to my mum. My dad snores. All men snore. Didn't you realise that?'
I didn't see that it followed, quite like that. But he was right. Not about all men snoring (Max didn't), but about getting used to it. I got used to it. I did learn to sleep through it—for the most part, anyway. Steve was so convinced that it was his bounden right to snore that he managed to convince me, not through argument but simply by smiling fondly and ignoring anything I said to the contrary.
'It's so sad, Gandie,' says Sophie, eventually, closing the album and solemnly regarding it lying in her lap.
I am a little nonplussed. Sad? She's been twittering on about how lovely it all is. Bridesmaids, and cake, and champagne, and all the associated fairy floss: I'm quite disconcerted by the rosy glow with which she surrounds it all. And suddenly we're sad? 'Sad that you're not still together, I mean,' she explains.
Well, I don't know. It seems to me an extremely good thing that we're not still together. Wasn't seventeen years of snoring enough?
And then again, I think, perhaps it's not fair to brush off seventeen years of marriage so abruptly, so flintily. It wasn't all bad. There were good times, especially at the start. Sex improved. Well, a bit, anyway. It's too easy, after a divorce, to slam into the ex: I've heard otherwise apparently nice women say the most appalling things about their exes, whipping themselves up into frenzies of hatred.
It's such a temptation, to make out it was always bad, that none of it was your fault, that you were always trapped somewhere you didn't want to be. Something in me always wants, meanly, to play devil's advocate. _Yes,_ I want to say, _but you didn't think that at the time, did you? I saw you, giggling and holding hands. I was at your wedding. I saw how it was between you, then. So don't pretend, now._
No, it wasn't all bad, though sometimes it suits me to pretend it was. I remember happy times. The wedding night was a disaster, but at least it had the advantage of making what followed seem a distinct improvement. For at least a year or two we travelled a mostly enjoyable learning curve, discovering curiously and, on the whole, happily how to live together.
I look back on the early photographs. We're picnicking; we're at the beach; we're at the playground or blowing out birthday candles or having a barbecue or fooling around in the backyard. Some of it was good. I was a sweeter, richer, funnier person then. I could be charming, and perhaps even loveable: my soul had more cushiony places, fewer prickles and spikes. Steve adored me, anyway, and I enjoyed being adored.
And I was fond of him, I admit, in the way that one becomes fond of a large and good-tempered dog—maybe a St Bernard, now that I think of it. If I hadn't fallen in love with Max, I probably would have finished up in the same crematorium niche as Steve, and not often thought of myself as desperately unhappy on the way through life. It's just that once you've discovered paradise you don't want to stay married to a St Bernard.
I don't know how to explain any of this to Sophie. Still, I suppose it's good that she will talk to me with this kind of frankness, this utterly translucent degree of openness. I try to be even-handed, as a grandmother, of course I do; but it's hard, sometimes. It doesn't always work, trying to spread yourself evenly.
And it makes a difference that Sophie is a girl, that she will be a woman, like me. It's much harder to talk to a small boy: he doesn't want to be like his grandmother when he grows up. Nor should he, of course: I'm not suggesting that Liam has to make a role model of me in order to communicate with me. But it's different, with women: we have common ground on which to pitch our tent and chat. I've explained this to Kate, to make sure she doesn't think I always favour Sophie.
Thinking about these things puts me in mind of Liam's fourth birthday. I remember it well. If Liam was turning four, Sophie must have been nine.
I was careful to turn up on time and with a good present: Kate had started to hint to me once or twice, in that tentative oblique way she has, that I appeared at times to be in danger of neglecting Liam. What she means, of course, is that I indulge Sophie. I'm aware that it's a danger, but, as I say, I _am_ aware of the dangers of partiality, and take care not to be trapped by it.
Liam was a pudgy dumpling of a child: he did a lot of sitting and smiling—radiantly, I grant you, but not very intelligently. He's very like his mother used to be. He is a sweet-natured child, I suppose, but has none of his sister's swift, dark charm, nor her quick, astute glance. He is all Kate: the sandy curls, the pink cheeks, the blue eyes, the slightly glassy stare.
Insipid. It has to be said. Sweet but insipid.
We divide comically into the dark and the fair, Steve and me and our offspring. We used to joke about it: he would refer to my genes as the devil's side of the family, and to his as the angels'. He is stocky and fair and blonde: so are Kate and Liam. I am slight and olive-skinned and dark, and so is Dominic. And Sophie, of course.
The luckless Gavin is a kind of pale fudgey brown, and long and gangling: his genes don't appear to be particularly insistent.
So I was careful, as I say. On time, a big present. I felt in my bones it was going to be a bad day when Gavin opened the door with a glass of beer in his hand, a cardboard pirate hat on his head, and a silly grin on his face.
'Millie!' he guffaws. 'My favourite mother-in-law.'
It puts my teeth on edge. But I see Kate and Liam hovering in the background, so I kneel and stretch out my arms to the child.
'Come to Gandie, sweet,' I cry. 'Come give Gandie a hug.' But he clings to Kate's leg. She nudges him gently forward. 'Give Gandie a hug, Liam. Gandie's come for your birthday party.'
'Look!' I call, holding up the brightly wrapped present, covered all over with balloons and Sesame Street characters. Inside there is a toy fire station with lots of fire engines and hoses and so forth. I don't know what boys like, really; but this seems to fit the bill. 'Something for a little boy! Something for a little boy who's turning four!'
But Liam is not persuadable. I get up, feeling a bit of a fool, dusting my knees down and straightening my skirt.
'Sorry, Mum. He's very shy, sometimes.'
'I'm his grandmother,' I say, perfectly reasonably. 'What is there to be shy about?'
'Well, Mum, just because you're his grandmother it doesn't make you less...er...'
'Less what?'
'Well, daunting, I suppose.'
'It might indeed make you more so,' says Dominic, glinting darkly from a corner of the room.
I catch a supercilious expression—quickly veiled—from his girlfriend of the time. Dominic was never without a girlfriend, even as a teenager; and by now he would have been in his mid-twenties. What was her name? Bec. I never liked her.
Steve is in the background, as usual, looking at the floor, as usual. The face of the hapless Gavin is over Steve's shoulder, bobbing away in the kitchen, looking worried. Zoë has on her disapproving expression and is bringing out a tray of something to eat; and Zoë's awful husband, Henry, is sitting next to Dominic and staring at the ceiling.
Hey, hey, the gang's all here.
Dominic and I glare at each other. I pick up the present from the floor and deposit it on the table. Liam starts to cry. I pick up the present from the table and hand it to Liam. He runs away, sobbing.
'You need to tone it down a bit, Isabel,' says Dominic, evilly. 'The poor child's confused: he's not used to such human warmth from you. It's like being exposed to a blast furnace when you've always had a two-bar radiator.'
Dominic has missed his vocation. He's a lawyer. He should be writing one of those dreadful bitchy gossip columns in a women's magazine.
I glance around, half waiting for someone to remonstrate. A light reprimand would have been acceptable. Just for someone to say: _Oh, come on, Dominic_. But nobody says anything.
From these not very auspicious beginnings you would say the day has to improve. This is not the case. We sit around and manufacture noises of the kind we imagine a non-dysfunctional family might generate under similar circumstances. I sip tea and think longingly of a stiff brandy and a cigarette. The brandy I can manage later on, when I blessedly return home; but I haven't smoked for years and seldom miss it, these days.
A chocolate birthday cake is produced, iced in Kate's habitual lopsided inexpert manner. The song is sung. Liam tries to blow out his four candles but succeeds only in showering spit over the bedaubed icing.
Sophie, always such a good big sister, helps him with the candles and then with taking around plates of crumbly slices. I pick at mine, leaving the icing. Liam eventually plucks up enough courage to open my present and it transpires that the identical object is already in his possession.
Well. How was I to know?
We limp along. Bec looks patronising; Dominic loses no chance to launch his poisonous little arrows into my quivering flesh. Kate gets pink and flustered. Steve sits silently and looks as if he wishes he were a million miles away. I'm sure I don't blame him: I do, too.
The hapless Gavin drinks slightly too much beer (Gavin is one of those men who assumes that at these occasions men require alcohol where women are permitted only non-spirituous beverages) and becomes hearty and talkative. Not that I can remember anything he says. At one stage Sophie whispers in Liam's ear and they leave the room together.
When they return, Kate, who has been watching for this, says anxiously: 'Liam has a surprise for you, Gandie. Sophie's been coaching him. Now, Liam!'
Liam stands foolishly in the centre of the room and regards the carpet with fascination. After some minutes of this he is persuaded to say, in a half-whisper: 'Gandie, Gandie. Sugar, sugar.'
I realise what it is. Sophie has taught him—well, tried to teach him—the little nonsense rhyme she and I made up, long ago, when she was around Liam's age.
_Gandie, Gandie, sugar candy_
_Legs all twisted, bent and bandy_
_Make sure she's all fine and dandy_
_Put her to bed with Andy Pandy_
_Tuck her in with a swig of brandy._
We used to chortle it together with disproportionate pleasure. It was one of the lovely things about Sophie, from when she was very tiny: the huge joy she was able to derive and communicate. She adored this silly rhyme.
Liam cannot see the point of it, and looks around in patent anguish when he has twice failed to get through the first line correctly. I try to be helpful and prompt him gently: he starts to cry again.
When I leave, Kate says to me, worriedly: 'I'm so sorry about Liam, Mum. He is very shy. It's just a phase.'
I'm tired, and my head has started to throb with that particular kind of vibration behind the eyes that so often presages a migraine. And I'm hurt by the whole thing anyway. I'm really hurt. I'm the child's grandmother: I'm there to be loved, aren't I?
So I probably snap at her more than I mean to. I say something about manners, I think, and four-year-olds not being too young to acquire such things. And then I'm surprised when, for once, she snaps back at me.
'If only you showed him a little warmth, Mum. If only you gave him half the warmth you give Sophie.'
I'm rather pleased. For once, she has the gumption to bite back. I scowl at her, just to show she hasn't got me bluffed, and I leave. I blunder home to all the large silent confusions of my life. I'll ring her in a day or two, and make peace. Probably she'll ring first, I think.
I can't remember, now, who rang whom. We always do make peace, though.
But she's wrong. I'm not a cold person.
The business of the rhymes is interesting. Sophie was herself not much older than Liam when she made up that rhyme. With a little help, I admit: still, it was her impetus.
The children in our family have always done this: made up rhymes. Zoë and I used to hurl them at each other as abuse. _Dizzy Izzy, in a tizzy! Doughy Zoë, gone to Moe! Stinky Minky!_ Mainly pretty meaningless, I suppose, but they were a game we played—in many different versions, in many different places—on and off right through our childhood. I'm not saying they were great poetry, or even minor poetry, but it can't have done our verbal skills any harm.
I still remember how stunned I was when Dominic, aged three (am I exaggerating?), cocked his head at me over his messy breakfast toast and chortled: ' _Funny honey, gone all runny!_ ' It was as if he knew he was participating in an old family tradition. And there was no way he could have known about it.
I'd never done it with Kate: she was a stolid and unimaginative child and had no interest in wordplay. I rang Zoë to tell her about Dominic and his honey, and we dissolved together in a rare sisterly exhibition of tears and laughter.
I still, sometimes, find myself haunted by doggerel. The words slide sideways into my mind and shift themselves into pulse and cadence and won't go away.
I'm not a cold person.
Sophie's interests are expanding. The other day she said: 'You used to be an architect, didn't, you, Gandie?'
'Used to be?' I say, meaning to make a joke of it but perhaps letting it come out more sharply than it should. 'What is this "used to be"? Are you putting your old Gandie on the scrapheap already?'
'Well, but I mean you used to do it all the time, didn't you? Not just a couple of days a week.'
'Yes, honey.'
'Why?'
'Why? A girl has to do something.'
'Yes, but I mean why did you choose to do that? Why did you decide to be an architect?'
'I thought I'd like it, I suppose.'
'And did you?'
'Yes. Very much.'
'Then why? What did you like so much about it?'
She is persistent, Sophie, in this new phase of hers: I am learning this. She will not be shrugged off. I try to explain it to her. It's all ancient history: it's hard to bring it back. It had something to do with the constructing of new things, the excitement of generation, the buzz of creativity. Something, too (let's be honest), to do with sheer randomness, a friend's father, an architect himself, suggesting it to me only half seriously. I might never have thought of it for myself, but his suggestion somehow took root and prospered.
And the drawing, the discipline of it, the precision. I had always wanted to be an artist. I try to describe it to her, this urge to produce wonderful oil paintings, watercolours, even sculptures, to see things nobody else could see, to _show_ that you could see things nobody else could see. I didn't have the talent. It always seemed to me that I had the first part of it: I could see. But I couldn't translate the seeing.
As an architect, however, I could learn to draw: I could do the technical side of things. I might not be able to paint glorious portraits or landscapes, but I could learn perspective, and precision. I could draw little, round shrubs, symmetrical obedient shapes, tiny detail penned in with meticulous care.
Even now, I don't use computers much for my sketches. I can do, of course, if I have to. But I choose not to, most of the time; I choose to draw. And I always try to do a watercolour for each project. I love doing the watercolours.
'That's one of yours, on the wall, isn't it?'
It is.
'That was your house, wasn't it, Gandie? You built it, didn't you?'
'I designed it, honey, and I lived in it.'
'With Max?'
'Yes, with Max.'
'What was the house called? It had a funny name, didn't it?'
'Rain. It was called Rain.'
'Why did you call it that?'
'I can't remember now.'
Sophie goes over and inspects the sketch. It's still there, framed, on the wall. God knows I never want to see Rain again as long as I live, but I can't bear to throw out my painting of it, when it had no history. It was so new, so fresh, it hadn't even been built, didn't exist except in my plans and my dreams and my mind's eye. It shimmered for me then.
'It's pretty,' she says, thoughtfully, as if she isn't quite sure. 'There's people in it, Gandie. Look, there's these two people, holding hands, looking at the house, over at the side. Are they you and Max?'
'No,' I lie. 'They're just two people. I always put people in my sketches. It makes it more fun.'
'This one looks like you. Look, she's short and thin, and she's got dark curly hair.'
I baulk, but silently, at 'short and thin': I had petite and lissom in mind.
'It's a coincidence, honey.'
'The man's tall. Was Max tall?'
'I suppose so. But those aren't us; they're not Max and me.'
'Who are they then?'
She doesn't believe me, of course. I'm not sure myself why I'm lying.
She examines the picture minutely, running her finger over it. It'll leave a mark on the glass, but I don't say anything. She's thinking.
'You always have funny names for things, Gandie. A house called Rain. A dog called Borrow.'
'Well, you know about Borrow.'
'I know. You used to share him with Mum. You used to borrow him, and she'd borrow him back.'
The dog raises his grizzled old Labrador head and looks at us. He knows he's being discussed.
'And it was Max who gave him to you? To you and Mum?'
Dear Christ, the child is relentless!
'Yes, honey, it was Max.'
She thinks, and then flashes one of her blinding smiles at me. ' _Gandie wears the cloak of Zorro, in a house called Rain with a dog named Borrow_.'
I smile. It's an effort, but I do smile. But she's not focusing on me: she's working on another little flash of creativity. It comes and she smiles again. ' _Lives in a house, for nothing it lacks, with a dog called Borrow and a man called Max_.'
For the rest of the day, into the evening, into bed, the knocking rhythms pursue me, but they're not Sophie's rhymes. They're my rhymes. I don't know where they come from but I know they're mine. Doleful and thudding, like a muffled hammer, they hum in my head: _Lived and loved in pain and sorrow, in a house named Rain with a dog named Borrow. Lived and loved, pain and sorrow, house named Rain, dog named Borrow._
Perhaps I am going mad.
But madness takes its time, evidently, to sneak up on one. I know this from the past. There are the dark times, the times of spiralling down inside the vast slow cyclone, its walls black and unyielding, slipping, slipping all the time, knowing you're slipping, not able to do anything about it. I've got a bit of slipping still to do, I think, before I reach the stage of irretrievable lunacy.
The endless questioning about the past didn't stop there.
Kate swore to me that she wasn't encouraging it. 'She's twelve,' Kate kept saying, placidly. 'She's growing up. She's taking an interest in all these sorts of things. If you try to shut her up she'll just think you're keeping things from her, Mum.'
Well, aren't I?
The next assault concerns not my wedding but Kate's.
'Well, honey,' I say. 'You should ask your mum about that. She's got a whole album: you must have seen it.'
Sophie knows this, but it turns out that what she's after isn't the formal statement, the embossed frontispiece, the endorsed public announcement.
'I like your photos of Mum's wedding better, Gandie. They're not so stiff and posed. Everyone looks happier in them.'
There's an irony. The photos in my album were taken by Max, who loved taking photos and had a gift for it. He was good at so many things. This was in pre-digital days, of course; and Max's camera (like everything else Max owned) was absolute top quality, state of the art. I can't remember what it was—German, I think. Leica? Whatever. All sorts of special lenses, but he never made a fuss of it: he would sneak up on you and take it before you realised, and then he'd grin at you, and maybe wink, and move on to his next victim.
The photographs Max took at Kate's wedding were terrific: I remember when they came back from the developers and I leant over his shoulder as he sorted through them. He was pleased with them; Kate and I were delighted. Sophie has good taste, as always. They have a quickness and vibrancy—and, sometimes, a wit—quite lacking in the official version, in which Kate and Gavin, both self-conscious and gauche anyway, look even more so as the professional carefully poses them, tries to make them appear happy, as they stare at their feet or glance wild-eyed at each other, as if to say: _My God, what have I done?_ Which, to be fair, is not what either of them was thinking. At least, I don't believe so.
So we get out the album and flip through. At least, since Max took most of them, I am spared having to look at him for the most part. There are a few of him. There's one I took, of him with his arm around Kate, who is looking up at him adoringly. There are a couple of him and me together, me in the fine silky little number I bought specially for the occasion, mother of the bride, floaty soft pink and dark sepia, high strappy heels; Max divinely handsome in his monkey suit. Most men these days have to go and hire a dinner suit for these occasions: Steve certainly did. Max owned his own, beautifully cut. I can't remember now who took these of us together, who grasped Max's camera and lined us both up through the delicate hairline sights. We look awfully happy.
Sophie lingers on them. 'It's a pretty dress,' she says.
'Yes.'
'It looks soft as.'
'It was.'
'Have you still got it?'
'I don't think so. It was twelve years ago, darling.'
She draws a breath and I can see she is working up to something.
'Do you miss Max a lot, Gandie?'
'Yes, honey. Of course I miss him.'
'When will he come back, do you think?'
'I don't know that he will.'
'Did you ever report him?'
'Report him?' I repeat, confused. What would I have reported him for, I am wondering.
'As a missing person.'
'Oh. No. No, I never did that.'
'But he was missing, wasn't he? Is missing.'
'I suppose so. In a way.'
'You don't mind me asking?'
Why can't I just say: _Sophie, Sophie darling, don't talk to me about Max. I do mind, I can't bear it, for Christ's sake stop it._
'No, of course not.'
Sophie looks at the photo again, traces around the edge of it with her slender, honey-brown finger. Sophie's like me: fine-boned, olive-skinned; she's got a tan all year round. She doesn't appreciate it yet, but she will, when she finds what a saving it is on pantyhose.
'He must come back one day,' she says. 'He must come back to you, Gandie.'
To my appalled horror I feel tears rising to my eyes. I turn the page and see a spread of Kate and Gavin fooling around with the cake, Gavin brandishing the knife in mock threat.
Sophie giggles. 'Dad's such a clown.' Well, that's certainly true.
When the children were small I would go in from time to time, in the evenings, to check on them—as all parents do, I suppose. Kate would always be asleep, curled up, rosy and boringly angelic. Dominic was frequently awake, ramrod-straight in the bed, his wide eyes glinting darkly.
'Why aren't you asleep, honey?' I'd sit down on the side of the bed. Sometimes, if I dared (as I've said, Dominic never liked being touched), I would run my hands through his thick dark hair.
'Can't.'
'Are you trying?'
'The more you try, the less it happens.'
That's true. Insomniac from way back, I know this.
'Have you been playing word games?' I know this is one of the things he does, when he can't sleep.
'Boring.'
'Are you worried about anything?'
'No.'
Eventually I would give up and leave, patting him in what was meant to be a consoling, maternal way. I'd peek in later, on my way to bed. Sometimes, even then, he'd be awake.
I never knew what he was thinking, though. Not then; not now. I talked to Zoë about it; I asked her advice. This was a rare step, for me: the trouble with asking advice from Zoë was that she gave it. Then she followed it up, to see if you'd done what she'd told you. It wasn't worth the trouble, asking Zoë what she thought about anything: you knew you would be liable to her swift persecutions, her rampant assaults.
But I suppose I must have approached her when I was feeling yielding or unconfident, forgetting that the aura of sisterhood was more attractive than the actual experience. Anyway, she was a teacher: she was supposed to understand child psychology. I remember she raised her eyebrows at me.
' _You_ worry about things, don't you, Minky?'
'Some things.'
'Some things, yes: that's what I mean.'
'Of course I do. Everybody worries about some things.'
'Well. Children worry, too.'
'He's three, Zoë.'
'The world is a worrying place, when you're three.'
'How do you know?'
'Use your brains.'
Well, that was Zoë all over. Use your brains, little sister.
Don't get me to work it out for you.
It's always been edgy, my relationship with Zoë. I don't know that I would call it dysfunctional, entirely. I don't know enough about what sisterly relationships are usually like to be able to make that judgement. I am five years younger than Zoë, and she's never let me forget it. I think when I was born she regarded me as a little-sister present, straight from the stork to her, my parents incidental to the transaction.
My character, my clothes, my moral development, my diet: Zoë studied them all earnestly, and never fell short of her duty when she felt good advice was required. Her shadow, murky and persistent and I suppose well meaning, leans over my childhood: wherever I was, whatever I did, Zoë was there to help things along. There were good parts to this, of course: when I started school, for instance, it was comfortable to know Zoë was around to give Chinese burns to anyone who teased me; Zoë was there to look after me if I was ill, if I fell over in the playground, if I forgot my lunch.
It was less comfortable, as we grew up, to find Zoë had read my diary (she gave me marks for it), vetoed potential boyfriends (when they rang, she told them I wasn't home), and insisted on monitoring the sorts of bras I bought. Whenever I complained she always said she did it for my own good, because she loved me.
She said the same thing, many years later, when I complained about her girlhood interventions. She did them because she loved me. I didn't disbelieve her. That was precisely the trouble with Zoë: she did things because she loved you. They were awful things, but she did love you, loved you according to her lights and with a total absence of imagination.
It would have been far easier to cope with her if she had acted out of malice. Contrary to what many people think, it isn't malice or evil that causes most of the world's problems: it's stupidity—it's people thinking they know best and not seeing how disastrously wrong they are.
As I've said, we used to make up rhymes when were young, but Zoë's frequently had an admonitory flavour. I've never used rhyming as a means of instruction or inculcation with Sophie: it's a matter of having fun, of having a giggle and encouraging her to play games with language. It's a skill she already has and it'll never do her any harm. Kate is perpetually astounded by it and shakes her head. 'She's so _clever_ ,' she says to me, in wonderment. 'Just like Dominic. He could always do that, too, couldn't he?'
Once, when Sophie and I were doing some cooking together, she chanted this to me:
_Great chefs are contumacious,_
_Disputatious and pugnacious_
_In kitchens that are spacious_
_They make soups that are audacious,_
_And salads all herbaceous._
_Their desserts are ostentatious_
_And their pancakes farinaceous._
'Did she truly make it up on her own?' Kate asked later.
'Yes, of course.'
'You didn't help with any of it? Even the farinaceous bit?'
'It started with the farinaceous bit. She discovered that it was a word that had to do with flour, and she just took it from there.'
'Amazing,' murmured Kate. 'Amazing.'
One of the things Sophie has achieved by the very fact of her existence is to inject a little normality into her extended family. Split as we are by old histories, replete with enmities matured like ripened cheese, we gather at family occasions with an inbuilt propensity to eyeball each other across the room as if battlelines were about to be declared.
Sophie overcomes these divisions; she traverses the sharp and jagged crevices in our family landscape; she presents to us an image of ourselves as harmonious and ordinary. A regular family, a commonplace and unremarkable group of related people who don't bicker and simmer, who don't harbour fetid suspicions or implacable hostilities about each other. It isn't a bit true, of course: we're so lost in our dysfunction, so crabbed and twisted by it, that we'll never emerge from it. It's like an indelible dye: it's imbued the fibre and substance of our relationships with its telltale stain.
So we're a dysfunctional family: but then again, what's dysfunction? Show me a functional family and I'll show you a pack of poseurs and fibbers. Our dysfunction is at any rate remedied, if not completely repaired, by Sophie: it's partly why she's so important, so necessary.
When she was eight or so, Dominic's passion for old cars was just starting: Sophie was scornful. He came around to Kate's one time to show off his latest purchase when I was there (clearly, he didn't know I was or he would have chosen another time), and Sophie poked fun at him. How did it go again, the nonsense rhyme she made up on the spot?
_Dominic, bominic, tinsel and stars,_
_Spends all his money on silly old cars,_
_Drives down the street in his burgundy Jag,_
_Bominic, Dominic, Oh! what a dag!_
And Dominic—proud, prickly, difficult Dominic—laughed and hugged her. Somehow Sophie could get away with all kinds of things I never could, where Dominic was concerned. I don't know how seriously close he's ever allowed her to get; but he certainly lowers the barricades for her.
I presume he lowers them for Paula, too. I wonder what he's told her—what he tells her—about me, I mean. I see her eyes resting on me sometimes with that kind of prissy speculative look she gets, and I wonder what he's shared with her, what she knows, what she guesses.
I wonder what he knows, too.
'What about your wedding, Gandie?' said Sophie to me one day, as I knew she would. 'I mean, your other wedding. Your wedding to Max.'
Ah. What about it? Well, it's a good question. I won't tell her how good. I'd anticipated the approach and I'm ready for her this time: I bring out the measly handful of photographs (they're not in an album: I never got around to doing that) and she studies them earnestly.
'Why did you get married on a beach?'
'Max decided to,' I say.
'Didn't you want to?'
'I didn't mind.'
'Was it sunset?'
'Yes.'
'Was it very romantic?'
'Yes.' I realise this must sound bald, but it's hard to enthuse.
'Where are all your guests, Gandie? Didn't they come to the beach?'
'We didn't have guests. Only witnesses. Two witnesses.'
'A wedding with no guests?' She is incredulous.
'We sort of eloped, honey.' She isn't impressed, I can tell.
'But didn't your mum come? Or Mum, or Dominic? Or Aunt Zoë?'
Hard to explain, especially from this distance: it's like something that happened to me in another life. Well, it is, really.
I could always tell her the truth, that Aunt Zoë (Zoë's her great-aunt, of course, but she always calls her Aunt) and my mother (if she'd still been alive, which she wasn't) would have stoned Max to death and eaten his liver sooner than come to our wedding. It doesn't seem worth it, though.
'Who took the photos, then?'
'One of the witnesses.'
'But why?'
'It's just the way it was, honey.'
Sophie studies one of the photographs closely, wrinkling the smooth skin around her eyes as she strains to see more clearly.
'Max is awfully handsome, isn't he, Gandie?'
Oh, yes. Max was handsome. Max was only _the_ most attractive man I'd ever laid eyes on. How many women get to make their marriage vows to the most attractive man they've ever seen, I wonder, bleakly. Cross Cary Grant with Sean Connery; add a dash of Robert Redford. (I'm aware that these names betray my age.) Shake together.
People turned to look at Max. They took one look at him and they were ready to be seduced by him. Except for my family, of course. Other people fell about for Max: they lay on their back and waggled their paws in the air. It wasn't just that he was so good-looking. He had slathers of charm. Pots of it. Vats of it.
'Yes, honey,' I say. 'Max was very handsome.'
'But he's probably still alive, isn't he? He wasn't very old, was he, when he went away? Won't he come back?'
'He won't come back, honey.'
Sophie looks as if she's going to cry.
'But, Gandie, he can't be dead. He's not dead. You don't think he's dead, do you?'
I find I can't answer, and turn to the photographs instead. But it doesn't work. Sophie can't understand why I haven't mounted these shots in a satin-bound album, why I keep them in an old envelope. She wants them carefully arranged and lovingly displayed, and even offers to do it all for me. But I can't bear it, suddenly; I can't bear the way she's pawing the photos, the questions she's asking, the way she assumes I'm going to want to talk about all of this.
I cut her off sharply, and I go into the kitchen to make myself a cup of coffee with my trembling hands and my loneliness and my desolation. Sophie doesn't follow me. She's a tactful child: she can tell she's overstepped some line she didn't know existed. When she says goodbye to me she gives me a fiercer hug than usual.
'Poor Gandie,' she mumbles into my neck. 'Poor Gandie.' I am poor Gandie, too. The episode upsets me and I spend the next day or so dangerously close to tears. It doesn't do, to stir all this up. Borrow senses my distress and shoves his damp muzzle into my hand, making those high-pitched staccato noises that are his way of saying: _Never mind_.
Strangely enough, however, the memories Sophie's interest awakens aren't so much of Max as of my mother's opposition to him. She was dying at the time: it gave her pronouncements an unfair and entirely spurious moral authority. Dying but not yet bedridden, as she was by the end: I remember her, engulfed by my father's old recliner chair, clasping and unclasping her tiny hands that always reminded me of the paws of some small and ineffectual animal: a rabbit, perhaps, or a guinea pig.
I hadn't left Steve yet, but I knew I was going to. I'd brought Max to meet her the previous day, and now I'd dropped in to gloat, to savour her capitulation to his glamour.
To my astonishment, she had failed to succumb. At once suave and earnest, unfailingly courteous, sober and gentle and intelligent, he had sat with her for over an hour, but she was unmoved by him.
'I don't like him,' she says, her hands starting their nervous wringing. She is looking down: she won't meet my eyes.
'Sorry?' I am completely taken aback.
I knew Zoë didn't like Max, but I put that down to jealousy and to the puritanical elder-sister streak that had so frequently during our lives snuffed out the small flickering flames that had given me pleasure.
'I don't like him. He's a con man, Isabel. I think he's a con man.'
'You've been listening to Zoë,' I say, contemptuously.
'What if I have? She's got sensible things to say. Why shouldn't I listen to her?'
'She doesn't see straight on this.' _As on so much else_ , I feel like adding.
'It's not Zoë that isn't seeing straight.'
'Mum,' I say. 'Mum, I can't believe this. He's a gorgeous man, a lovely man. Can't you see how kind and thoughtful he is?' And how sexy, I wanted to say—how ravishingly, gloriously sexy. 'Can't you see how we love each other?'
She shakes her head. 'Zoë says he's very rich,' she offers, peeking at me suspiciously.
'What's that got to do with it? It's not his money I'm in love with.'
'No, Isabel: I know that, darling. I'm not accusing you of being mercenary. But where did he get his money from?'
'He's an investor,' I say. 'He's an investor, an entrepreneur.' I don't at this stage have a clue where Max's money comes from, but I don't want to admit it to her. He could have blown up the Mint, for all I know or care. It doesn't matter.
The recliner creaks slightly as my mother rocks. As she becomes more agitated, she rocks faster. 'I don't know what's got into you, Isabel. You've got a lovely husband, lovely children. You've got everything a woman could possibly want. Career, money, security. And you're throwing it all away on this—this—mountebank.'
She releases the archaic word with a mouselike squeak. I say nothing. I am ridiculously disappointed.
'I'm glad your father isn't still alive,' she says, almost vengefully. 'He would be so upset. He loved you so much, Isabel. He was so proud of you.'
I think this is below the belt, and say so. My mother shrugs, weakly, peevishly. I know the cancer is causing her excruciating pain: the oncologist has told Zoë and me that she can't survive more than a year. I know this fact on its own should lead me to treat her kindly, but I am filled with anger at the unfairness of it all, and I say, more shortly than I mean to, that her opinion of Max is in any case quite immaterial, since he is in effect the summation and pinnacle of my heart's desire, all I have ever wanted in life, and nothing will change this.
I probably don't put this argument as well as I could; and inside me some small voice to which I am not listening tells me it isn't a convincing argument anyway. I am, however, by now irretrievably committed and, no matter how little I intend to wound my mother, I find myself launched on a tirade about the general unreasonableness of my family and the degree to which it has (especially Zoë) always conspired to ruin my life.
My mother may be a shrunken and vulnerable old woman, but she is stung by my remarks and retaliates with remarks of her own. From this point the outcome is predictable, and five minutes later I am slamming out of the house, leaving my dying mother weeping.
Zoë rings me that night (when I am in the middle of a stand-up battle with Steve) and bitterly rebukes me for my selfishness and cruelty.
'I couldn't help it,' I say. 'I can't help it if nobody likes what I'm doing, if nobody likes the new direction my life is taking. I can't help it if you don't try to enter into my feelings, if you don't try to sympathise with me and support me.'
'I've never done anything but support you,' snaps Zoë.
This is so manifestly and thoroughly untrue that I cannot think of any way to respond. Zoë misinterprets my silence as shamed acquiescence and, ill-advisedly, continues, in what she considers a softened tone.
'Minky, dear, you must think harder about all this. You're throwing your life away for nothing more than an infatuation. Think of Steve and the children. Think of all you've achieved together. You're not thinking of other people's lives, Minky.'
'Nobody's thinking of my life,' I reply, right on the cusp between screaming and sobbing. 'You're all talking as if I've engineered this whole thing. It's not my fault. I can't help it, that this has happened to me.'
'That's what you've always said,' says Zoë. ' _I can't help it. It's not my fault._ ' She modifies her voice to produce a self-pitying whimper with which she intends to parody me. 'It's what you've always said. Nothing's ever your fault. You know what I think? I think it's time you grew up.'
I slam the phone down.
None of it is edifying. Even I can tell that. But to me my fate seems so inescapable. I cannot begin to understand why nobody likes this glorious man who has entered my life and so thoroughly transformed it; but, since they don't like him and are therefore unable to sympathise with my feelings, they leave me no choice. I was prepared to meet them halfway, to explain and elaborate and perhaps even (to some minuscule extent) compromise. But it's their fault: they won't meet me, so I'm left with no option.
Why can't they see it? It's as if they view the world—or me, anyway—from behind the wobbling curtain of a giant heat-haze, and the vibrations through which they gaze, discontentedly squinting and peering, distort their vision. And the more they gaze, the worse it gets, and the more mistaken they are. I'm the only one with clear vision.
The next time I see my mother, we more or less make up. But she's still brimming with disapproval, and I'm still full of love (though not for her) and despair, and we are each profoundly dissatisfied with the other.
And, suddenly, everyone's on _Steve's_ side. Nobody, so far as I can tell, cares about _me_ , about what I might have had to put up with in this marriage, about the blanknesses and gaps that have grown and stretched into cavernous black holes during the seventeen years of our alleged connubial bliss. Nobody is even interested in hearing my point of view.
Dominic, certainly, has never been interested. Locked in his dark, elegant shell, double-insulated against the insistent disharmonies of life, as remote from me as if he lived in the Himalayas, Dominic doesn't care. He never has.
With every meeting a skirmish, each of us caught in increasingly plastic rules of engagement where nothing, really, is off-limits any more, we meet in a kind of no-man's-land from which all mother–son tetherings have been ruthlessly stripped. Not by me, I might add. I don't know how it came to this. We meet; he has a go at me; I have a go back. What am I meant to do when he's so consistently aggressive? Ignore him? Curl up and whimper?
I certainly won't give up on him. One day, I'm sure he'll return to me. Well, approach me might be more accurate: he hasn't ever been near enough to me to return, precisely. But I really tried with Dominic. We had lunch one day. I suggested that he meet me at Southbank. He works in the city, so it wasn't far to ask him to come. When I rang to invite him, he hesitated. I suppose he was simply surprised, which was not an unreasonable response. I hadn't ever made a request like this before; but it was shortly after I'd had a brief exchange with him—at some family occasion, I suppose—during which I had thought I'd detected in his eyes, in his voice, in his expression, some sentiment not totally alien to, let us say, mild affection. It seemed worth a try.
'It's all right,' I said, trying to make it sound bantering rather than testy. 'I don't have nefarious designs.'
He gave a small cough that might have started as a laugh. Or might not. 'No, I'm sure you don't. It's only that it's a busy week, and I'm just not sure...'
'Make it next week, then,' I offered. 'It's not urgent.'
He seemed to debate with himself over this. I pictured him, pencil poised over diary. Except, of course, that he probably does all this kind of thing on computer, or palm pilot.
'No,' he said, finally. 'No, let's make it this week, then.'
I'm always very proud of Dominic, when I'm with him publicly. He's so spruce, so damn good-looking. I always think, well, even if I was a bad wife to Steve (not that I admit that, not on balance anyway), I gave him this glorious son.
It's interesting, the categories we invent for ourselves. Bad wife, good wife. Good mother, bad mother. When I married Steve, I intended to be a good wife. It was an explicit resolution I made. When I had the children, I promised myself quite passionately that I would be a good mother. I was determined in this endeavour. I'm still not sure where it all went wrong, at what point these naïve aspirations started to trip each other up.
Good mother, bad mother.
Dominic is six minutes late, but I forgive him as I see him striding across the restaurant. I see women watching him, as he approaches me. One woman—just a girl, really, with blonde shoulder-length hair, in a rather pretty purple shirt—turns to look at the person with whom Dominic is lunching.
I've gone to trouble, of course. I'm wearing a black pant suit with a biscuity-coloured shirt and an expensive silk scarf with a kind of butterscotch and cream pattern of leaves on it. Whatever Dominic thinks of me, I don't want him to think I'm daggy. It's important to me that he doesn't mind being seen with me, even if he minds being _with_ me. The purple- shirted blonde decides I'm hardly competition and transfers her attention back to Dominic.
We don't kiss, of course. We never do. He nods, warily. I try to beam. He looks alarmed. I modify my expression.
I suppose I have imagined this as a leisurely and perhaps anecdotal lunch, both of us relaxing over a good wine, the river sparkling outside the restaurant's windows on this cool, sunny afternoon. It has seemed to me entirely possible, in this atmosphere, that all kinds of boundaries might be crossed, all kinds of fences knocked flying. In fact, as soon as Dominic sits down he glances at his watch.
'Sorry I'm late,' he says, but cursorily, as though he doesn't really mean it. 'It's a hell of a week. There's a lot happening.'
'We could have made it some other time,' I say, trying not to sound vexed. 'I told you, it wasn't urgent. We could have made it next week.'
He runs his hand through his hair. 'The thing is, Isabel, there's always a lot happening. It's quite hard to plan ahead. It seemed better to do it straightaway.'
The waiter looms and distributes menus. I study mine, and pass the wine list to him.
'Oh no,' says Dominic, as though shocked by the mere suggestion that he might imbibe an alcoholic beverage. 'I'll just have a mineral water, thanks. You have a wine, if you'd like: they do it by the glass here.'
'You don't want even a glass?'
'No, honestly. I've got to work this afternoon, you know.'
'Well, yes, but just one glass...'
Dominic is shaking his head, and I break off. If he doesn't want a drink, he doesn't want a drink, and naturally I'm not going to heckle him about it. But it's a bit of a shame, because it makes me feel that if I have a glass of wine I'm somehow dissipated (bad mother), and, because I'm nervous, I want a glass of wine. It'll settle me down.
I try not to look disappointed, and study the list of trendy little entrées and mains. Dominic pushes the menu away almost immediately. The waiter materialises at his side. What's his secret? I can't usually get hold of a waiter within fifteen minutes minimum.
'I'll go for the pasta of the day,' says Dominic. 'Isabel?'
I haven't actually made up my mind, but I say I'll have the same, entrée size.
'Have a glass of the house white,' says Dominic. 'It's good.' He doesn't even give his order for a drink, but a mineral water turns up almost instantly, along with my wine. He notices my slight surprise.
'They know me here,' he explains. 'They know what I always have.'
When I asked him to come to lunch, I also asked him for the name of a restaurant he'd like to go to. I meant it more as a special occasion, a treat, if you like, rather than just turning up somewhere he usually goes anyway. But he clearly hasn't seen it like this, and I can't help feeling hurt.
'Well,' he says, sitting back. 'Hello, Isabel.'
'Hello.'
'Anything in particular on your mind?'
'Should there be?'
'It's not usual, is it, our getting together as cosily as this?'
'No. No, it's not, but perhaps we ought to do it more often.'
'Ought we?'
'Dominic, what's happened to us?' I ask, a bit desperately.
I hadn't meant to do this, to jump into deep waters. All I'd meant to do was build a little bridge, maybe, the beginnings of a little bridge. But he's given me an opening: that cool 'ought we?' is a genuine opportunity, despite its chilly tone, and I'm not likely to get a better one.
And I'm so nervous: perhaps my judgement's off-kilter. Maybe this chink is something he's deliberately giving me; maybe it represents a sliver of opportunity that he positively wants me to grasp.
'What's happened to us, when we can't even be civil to each other any more? It usen't to be like this.'
He sips his mineral water, studies it, says nothing.
'Please. Couldn't we just try? Really, I don't want that much. Only...only to be able to be friends again.'
'Again?' says Dominic, with just that whippy little inflection at which he is so expert.
'Ah, Dominic,' I say. 'We were friends, weren't we?'
'I don't recall. Were we?'
'I don't understand. Dominic, I know I left home when you were still quite young...'
'I was ten,' says Dominic, flatly. 'I'd say ten was very young, not just quite young. Wouldn't you?'
Oh, Christ, I'm right in it now. I plough on.
'Yes, all right, ten is very young. I know it is. Dominic, things don't always work out the way we want them. I didn't mean it to happen the way it did. I didn't mean to leave you. I wanted to take you with me, remember?'
'Yes, but then you left.'
'Well, you wouldn't come.'
'You wouldn't stay.'
'Why should I have stayed?' I'm starting to get angry, and desperate.
'I was ten. That's why you should have stayed. I was ten. We go around in circles, don't we?'
'But that was years and years ago. You've grown up. I'm on my own now. Am I always and eternally to be defined by you as the mother who left you? Is that going to be my permanent status, forever? I mean, can't we move on from that?'
'Well. It isn't a definition I came up with, you know. It was what happened, not what I invented. To borrow that phrase you're so fond of, it wasn't my fault. You did leave me. You self-selected, as it were. I had nothing to do with it, except to have to cope with it.'
'Why do you hate me so?' I cry at Dominic, alarmed even as I speak by the keening note of my voice, the wretched and unpleasant nasal sob that catches at it. He gives me that narrow-eyed look I know and abhor.
'I don't hate you, Isabel. I just don't like you very much.'
The people at the tables near us aren't staring at us; they're averting their eyes, which is somehow worse. The waiter magically appears between us with two plates of pasta, and it occurs to me in a strange disjointed way that this is a strategy I've never before cottoned on to—if you want immediate service in a restaurant, cause an uncomfortable scene. The waiter makes much show of grinding black pepper and parmesan; we preserve a sullen silence until he retreats.
'Can't we start again?' I ask, trying to sound humble, not whiney.
'I don't think so,' says Dominic, stabbing at his pasta.
'Why not?' I'm speaking more quietly now, anxious not to embarrass him.
'I don't know how.'
And there we leave it. I am too shaken to try any further. I can see I'm going to get nowhere. We finish our pasta in silence, to the clear relief of the neighbouring tables. It's a farce. He glances again at his watch, mutters something, gets to his feet. I offer to pay. He shakes his head, settles the bill. Off he goes, back to the demanding world where there's so much to do and no mother to make a nuisance of herself.
This wasn't how I'd imagined it. When he was a child, I mean, or for that matter before he was born. I'd always thought of our being friends, easy with each other, casual and witty, undemonstrative perhaps but affectionate, sharing an understated empathy, an unarticulated but deep mutual trust. I'd sketched out a number of scenarios, in the torpid moments of pregnancy, before I even knew him; and then again later, when he was a baby, mistrustful and opaque. I'd bowl to him in the backyard. I'd take him to galleries, to Mozart concerts, to poetry readings. We'd play chess together, and when he grew up we'd sip chilled pale wine on summer evenings. We'd discuss philosophies, scientific theories, whatever. Our relationship would rest on shared pleasures, shared opinions, mutual appreciation. And love, of course.
Kate and Steve were, needless to say, absent from these touching scenes.
But, the relationship never worked, never meshed. It was worse, far worse, after the marriage broke up, but it was never the way I'd planned it, never the harmonious rapport I'd envisaged. Zoë would say this was my fault, of course. Zoë would say I got it wrong with both children.
Once, she visited when Dominic was little. He was still crawling: he must have been around twelve months, I suppose. Kate was six: that I do remember. She took a doll from him and his roar was passionate. I rebuked her. I wasn't savage; I wasn't mean: there was no need for her to burst into tears.
'She's very good with him, isn't she?' Zoë said, her voice just lifting towards that spiky tone it has so often when she addresses me.
I disapprove of discussing children in their presence. It leads to unpleasant precocity. I shrugged and nodded as noncommittally as I could. At this stage Kate was whining loudly and showing little sign of being good, so far as I could see.
'It's my special doll, Mummy,' she wailed.
'It's her special doll,' said Zoë, regarding me meaningfully.
'She shouldn't have left it where he could reach it.'
'Oh, have a heart, Isabel.'
'For God's sake,' I snapped. Kate ran howling into her bedroom, clutching the damn doll. Dominic, sensing conflict, sobbed more loudly.
'You're hopeless,' snarled Zoë. This is her idea of sisterly support. 'You've got no idea, have you?'
'What's that supposed to mean?'
'You favour him. All the time. It's always: "Kate, don't do this. Kate, don't do that."'
'Zoë, she's six. He's a baby.'
'You need to watch it. You're making some big mistakes.'
'Oh, come on.'
'I mean it, Minky. You're so hard on her, and of course she gets upset. She is good with him. There aren't many six-year- olds who'd be as careful, as gentle. She's a dear little girl, and you're going to cause huge problems for her if you keep on going this way. You'll cause problems for him, too, you mark my words.'
'You've got so much experience, I suppose,' I said, meanly.
'I may not have kids of my own, but I do have experience with kids in the classroom. I spend every day with kids. Anyway, you don't need experience to see the mistakes you're making. It's just a matter of common sense.'
This is a good example of Zoë's tact and sensitivity. In any case, she was demonstrably wrong. Kate has no problems that I'm aware of—or none, anyway, that are caused by me. She had a perfectly happy childhood. She appears to me, in spite of everything, to be a perfectly happy adult. And if Dominic hates me because I was too kind to him when he was a baby—well, what kind of sense does that make?
I kept thinking the relationship would improve. I gave up work for Dominic, so I could spend more time with him. It was harder, anyway, to work when I had two small children at home. I've heard other women say this, too: it's three times as hard with two children as with one. But he always fought me, even as a baby, even as a toddler. He fell over once and I cuddled him, crooned to him, savouring the moment, because he so seldom let me do this. 'Poor Dominic,' I murmured, rocking him. He exploded in fury and struck my face. 'I'm _not_ poor,' he screamed. 'I'm _not poor_.'
Well. What can you do?
This is one reason I so much appreciate Sophie's company, Sophie's unconditional love. I sometimes think Sophie and Max are the only two people who have ever really cared about me, who have truly loved me.
Yet I always feel, with pricking resentment, that I'm worthier of love than people seem to think. If only they could see me as I really am, if only Dominic could get past the protective spines I raise against him, if he could only lower the guns he mounts against me, if only we could meet in some pure space unhindered by our family, our history, our expectations of ourselves and each other, I'm sure he'd like me.
I'm a likeable person: I know I am. I'm a warm and loving person. _I love you, Dominic!_ Sometimes I think I ought to scream it at him, every time I see him, just to drive the point home, to make him notice me, listen to me, listen to what I want to force into that sleek, obdurate head of his.
I know women who are entirely unremarkable but who nevertheless manage to retain cordial relations with their children, their sons. Of course we all adopt party manners for the world's scrutiny, and I imagine there are rough patches in such relationships, dropped stitches, torn pages, dark, vicious corners of psyches that never see the light of day or face exposure to outside observation. Still, such people manage to maintain a reasonable façade.
I'm quite positive most sons aren't as savage to their mothers as Dominic is to me; most sons don't chip away at their mothers' fragile veneers with anything matching Dominic's deadly ruthlessness. I can't, however, adequately measure or comprehend their success against my failure. I can't see _why_ they've managed what should surely be a relatively simple manoeuvre, when I so conspicuously haven't. What is it about bringing up children, after all? What makes it so difficult?
I started off with intentions and resolutions as good as the next person's. I taught them manners, correct behaviour, consideration for others. I talked to them; I played with them; I didn't abuse them; I didn't take drugs; I didn't hit them; I tried not to shout at them. I know I split up the family, but it wasn't them I had to get away from: it was Steve, and I've always made that distinction clear.
I don't understand where it went wrong, at what particular point Dominic decided to limit his communications to sneer mode, why he woke up one day and perceived my maternal limitations with such devastating clarity.
These women I know who are successful mothers—who at any rate seem to be successful mothers—there's nothing so special about any of them. Intelligence doesn't appear to be a prerequisite, nor does talent, nor (let's face it) good looks. They've just bumbled along, the same as all of us. I once said to a woman I knew how lucky she was to have such happy and successful children, and she fixed me with a dirty look and said: 'Luck, Isabel, had nothing to do with it.'
Well, luck always has something to do with it, in my book. If you can put in so much effort and still fail, luck has a lot to do with it.
Sometimes I fantasise about the kinds of circumstances that might enlighten Dominic about the value of a mother's love. Well, not any mother. Me. Some rescue I could effect, some unfortunate condition he finds himself trapped by (imprisonment, perhaps, for some shameful crime?) in which I waft back into the mainstream of Dominic's life trailing forgiveness and compassion. _Gosh, Mum, where would I be without you?_ whispers a white-faced Dominic, as I gently raise him on my arm and spoon homemade beef tea between his unresisting lips.
Or perhaps we're hostages together (unlikely, this, I do recognise that element of the hypothesis), and I intercept the shot that's meant for him. Or then again, perhaps I'm simply dying, slowly and rather picturesquely (and let us hope painlessly), and Dominic in imagining his future without me is brought to his senses. _Gosh, Mum, what will I do without you?_ whispers a white-faced Dominic, reaching to grasp my fragile hand as he sits beside the bed repenting of his former brutality.
And then there's the final scenario, the funeral one. That's where I'm in my coffin (adorned simply, due to my distaste for ostentation—perhaps only with a single crimson long- stemmed rose) and Dominic, still white-faced, is at the microphone, or in the pulpit, or wherever, choking over the revelation of his long-hidden adoration of me. Ah, the poignancy!
For obvious reasons, this particular scenario isn't one I treat myself to except when I'm truly depressed, but on certain occasions it does afford a kind of odd, luxurious enjoyment. In many ways it's a scene I find more immediately attractive than contemplation of either of my weddings, or for that matter anyone else's.
Strange, how we circle around these milestone occasions. We eventually decide they were of significantly less importance than we'd thought at the time (what did it matter whether I married Steve or not, after all, in retrospect?), but still we open the albums, still we dream over the pictures, wondering. Mind you, I've done it only because Sophie has egged me on, but once she's awakened my interest I find I'll spend longer than is sensible poring, and remembering trivia the detail of which had lain slumbering in the farthest recesses of my mind for (for instance) thirty years.
Sophie in her recent pursuit of family weddings has shown little interest in Zoë's, but I suppose that's natural: I'm her grandmother, after all. Zoë's more distant, just a great-aunt. In those days it mattered more, getting married: it was the thing young girls had rammed down their throats, the thing they were taught to regard as the very apex of aspiration, the miraculous consummation of our lives. Zoë and I were lucky that our parents judged we ought to have careers, too, or at least the option of careers: many parents didn't think like that.
And Zoë's five years older than I am (the same distance, coincidentally, as there is between Kate and Dominic, and between Sophie and Liam: the women of our family all seem to wait five years before the progenitive urge returns). But it meant a lot, then, that Zoë was five years older and yet I was married first.
I was aware of the superiority it conferred on me, and I was not above taking advantage of it, either, after all the years of bullying and condescension. She did not marry until two years after I did, at the end of a protracted courtship: she was twenty-seven by the time she stalked down the aisle on Dad's arm. I believe I felt her mortification more keenly than she did herself, to have reached this ripe old age before entering into the holy state of matrimony. Strange, how things change.
At the time of Zoë's wedding I was heavily pregnant with Kate, who spent most of the ceremony alternating between kicking viciously at my bladder and bearing down on it with agonising determination. Zoë had worn an ivory silk thing, which to my mind somehow contrived to shout its understated elegance to the world (with her figure, she certainly couldn't have got away with frills and froth), and Henry had simpered above a tie that didn't suit him.
I found the whole thing fatiguing and profoundly uncomfortable and exceptionally boring; I remembered also feeling disproportionate relief upon reaching the end of the service without Kate's having caused a disaster.
I'd been relieved, though, that I didn't have to play bridesmaid. Nowadays you see young women tittupping down the aisle eight months gone—and they're the brides, let alone the bridesmaids—but in those times it was more rigid, decorous—call it what you will. It wasn't entirely a bad thing, having notions of decorum.
At any rate, I wasn't expected to participate. I sat through the ceremony, breathing heavily and wishing passionately, desperately, that the baby would stop its demoniac activity. Steve beamed beside me. He was so proud: it was hilarious yet strangely moving. I couldn't have had more fuss made of me, more attention drawn to me, if I'd been anticipating a virgin birth. Which I certainly wasn't.
Zoë, entering connubial bliss, seemed largely indifferent. She was an extremely competent bride, avoiding most of the available nuptial excesses, the flimflam and the fairy floss.
I had actually pretended, at my wedding, to be a little more flustered than I really was, in order to present an image of coy feminine confusion calculated (if I had only realised it) to confirm unenlightened prejudices still clung to by the congregation. My blushing uncertainty enabled Steve to assume the manly protective profile for which he yearned. He actually referred to me in his speech as 'the little woman'. I don't think any element of self-parody entered into this nomenclature.
My sister, by contrast, adopted no girlish strategies, no circumlocutory fluffiness. She was wise enough, I suppose, to know she couldn't get away with it. Businesslike, sturdy, methodical, she superintended such challenges as table arrangements and flowers with military precision. A wedding was like a lesson: it had to be planned and programmed and rehearsed and, finally, got through with minimal disruption, discipline being maintained and the whole proving an edifying experience for all concerned. Henry came along for the ride: that was as much as he had to do with any of it. Zoë steered her way through it all with, so far as I could see, minimal consultation with him. A likely prevision of their marriage, I thought.
Henry was such a puzzle to me. He taught science at the school where Zoë taught history. He still does. When Zoë first brought him home, I couldn't fathom him at all. Even in those days, when (I freely admit) many strange customs obtained, most men of Henry's age, which I suppose was his late twenties, didn't customarily wear three-piece suits.
I'd known Henry for over a year before I saw him in anything else. It made him look like an undertaker. He had a disconcerting habit of tilting his head back, appraising you through the lower half of his spectacles. Balding and bony, slightly stooped even then, a gold filling conspicuous in a front tooth, meticulous in his speech and careful in his movements, he seemed to me desiccated, ponderous and boring.
I recall teasing Zoë about him.
_Blind as a bat, bald as an eagle_
_Henry isn't sexy; Henry isn't regal_
_Caught like flies in amber in his sober three-piece suit_
_Henry isn't clever and nor is Henry cute._
I know it wasn't nice of me. Zoë wasn't nice to me, either.
Anyway, she didn't see the joke.
My parents liked Henry. Well, they liked Steve, too, so it shows what their judgement in these matters was like. My mother thought Steve's blond, square good looks and his doglike adoration of me made him a delightful and endearing young man, and a suitor of eligibility _par excellence_. The irremediably comic element I detected in Henry was not visible to her.
'She can't want to _live_ with him,' I remember saying, one evening after dinner. I think I was drying the dishes at the time.
'Why not?' My mother would have been wearing the bright pink rubber gloves that she always donned for this task, drawing them with elaborate care over her tiny hands. She was a slow and careful washer of dishes. It drove me mad, having to hang around while she scrubbed at specks minute as atoms and warily held spotless glasses up to the light.
'Oh, Lord, Mum. He's so frantically boring. He's such an old man. How could she even think of marrying him?' What I really meant was: _How could she think of going to bed with him?_
'He's a sensible man,' commented my mother. 'Have you heard the way he tries to tell jokes?'
'Delivering a good punchline isn't a mandatory skill for a husband.'
'No, but...'
'Actually,' said my mother, 'he has a good sense of humour. Very dry.'
I snorted and my mother frowned. 'He'll make her very happy.'
'They're not even engaged yet. Are they?'
'They will be,' said my mother with quiet certitude. 'They suit each other. You might find him unglamorous, Isabel, but he's a reliable, good-tempered man with a great capacity for loyalty. And he loves Zoë.'
I think I didn't care enough to argue the point. If Zoë wanted to wreck her future by attaching herself permanently to a tedious freak with bad teeth and bifocals, that was her funeral. I was getting on with my life—or that was how it seemed to me at the time. Amazing though it seems to me now, I was busy planning to marry Steve.
I'm not sure, now, why. It wasn't as if I was completely bluffed by the whole marriage scenario. I wanted my degree; I wanted my career. I wanted it all, I suppose. Well, why not? It was becoming evident that having it all might be possible: young women of my time, graduating in the early 1970s, were just starting to regard themselves as forerunners to a brave new world where injustice would never again be based on gender and where nothing would dare to intervene between oneself and the star one decided intrepidly to follow. Some of it has come true; a lot hasn't. In those days, there were no templates to follow, no blueprints to study. Not for careers, not for parenting, not for anything much. Mind you, I don't think you learn much about being a parent from anything but workplace training.
Being a grandparent, however, is different. I can't believe how dead easy it is, especially after all the times my own children ate me up alive and smacked their lips over me. It's so effortless, when Sophie wanders into my house in the late afternoon. 'Hi, Gandie,' she calls, and comes to give me a big hug and kiss. Never anything forced, never anything other than natural and affectionate and loving. It makes it so easy, to love back.
I must say, however, that the minuteness of her attention to the detail of my life is starting to wear thin. _No more wedding snaps_ , I pray, silently. I've had my fill of staring down old aisles and those who people them.
But she takes a different tack, the next time. I'd almost prefer the wedding snaps.
'Look what I found, Gandie,' she says, rummaging in her bag. 'You'll never guess.'
I wouldn't have, either. It's a photograph, again; but this time it's not of a person but a portrait. It's a photograph of Max's portrait. The original—the portrait, I mean, not its subject—stands face to the wall, covered in brown paper, in my garage. It's a long time since I've taken it out and checked it for mould or rat nibbles or anything else.
The photograph is rather dog-eared and, since it is taken from a lower angle, presents an odd perspective on the painting. Max's head seems, somehow, further away than it ought to be, and his stance appears awkward. Max was never awkward.
'It's Max, isn't it?'
'Yes, honey. It's Max.'
'Did you paint it, Gandie?'
'Me? Goodness no, darling. I just mess around in watercolours, and all I do are sketches. This is a photo of a proper oil portrait.'
'Where is it now?'
'I don't know,' I lie.
'Did Max take it with him when he went away?'
'Yes,' I say, relieved to have so neatly packaged a solution ready to hand. 'Yes, he took it with him.'
'Who painted it?'
'Honey, where did you get this photograph?' I've never seen it before. I certainly didn't take it.
Sophie looks equivocal.
'I just found it,' she says, airily. She sees that I am looking hard at her and shifts slightly.
'Sophie?'
'I found it in one of Mum's drawers. I wasn't looking for it, or anything. Mum knew I was looking in her drawer. She was going to lend me some perfume and she told me to go and look for it in her drawer. And this was stuck at the bottom of it.'
'Does she know you've got it, Sophie?'
'Well. Not really, I guess. I thought I'd just put it back and, you know, everything would be cool. She wouldn't mind, I'm sure. I just wanted to show it to you, Gandie. Tell me about it, go on.'
I know I should be cross with her, but she can always wheedle me. And I love it when we share a secret from which Kate is excluded. I know this is mean and unworthy of me; I know Kate is her mother and she ought to be more open with her mother. But the fact is, I love it when Sophie says: 'This is just for us, Gandie, okay?'
But I don't want to talk to her about the portrait. I fudge it, the way I am fudging so many things with Sophie, these days.
'It was a present,' I say. 'It was for his birthday.' 'Which birthday? A special birthday?'
'Yes, it was his fiftieth birthday.' I am aware that to Sophie fifty must seem astonishingly old, momentously old. But that's what it was for: Max's fiftieth birthday.
We'd grown fond of a gallery in Richmond: one of the artists who exhibited there regularly had caught our eye. She did portraits in oils—rather fluid, elongated things, almost El Greco-ish or even Dobellian, their exaggerated perspectives delicate and witty, their hues gentle, the likenesses in the best of them magical, wondrous things that caught essential quirks and personalities as well as simply registering identities.
Her name was Hilary Jacoby and she was a funny, fey wisp of a woman, with pale puffball hair and hard green eyes that to your immense surprise you found looking right through you, into your innermost soul, just when you'd decided she was right off the planet.
I commissioned a portrait of Max for his fiftieth birthday; he promised to commission one of me for mine. We were going to hang them side by side in the lounge. 'There'll be a hundred years of life up there on our wall,' he said.
I thought of this when I turned fifty, but wasn't tempted to do anything about it. Why commission a portrait of yourself for yourself? I haven't heard of Hilary for years: she may not even be alive any more. She was only a little older then than I am now, but who knows what's happened to her.
There, at any rate, in this old and rather creased photograph of his portrait, is Max, elegant as always in a crisp, white, open-neck shirt, tailored grey pants, black shiny shoes. He's sitting back in a dark leather armchair; his long and beautiful hands lie, relaxed, on its armrests. Beside him is a small table and on the table is a photograph of me, sitting at my desk at Rain. On my desk is a photograph of Max's portrait, with its photograph of me inside it, and so on. It was something Hilary wasn't keen on, but Max persuaded her: he loved tricksy little things, puzzles that made you look twice and three times, and charmed you. Sophie doesn't fail to notice it, of course.
'Look, Gandie!' she exclaims. 'It's a trick, isn't it? It goes back, and back, and back. It's so clever!'
'Yes. It is clever.'
'Who thought of doing that? Was that you?'
'No, honey, that was Max. Max liked puzzles. Conundrums, riddles—anything like that.'
'He's got such a _nice_ face,' says Sophie, studying the nice face closely.
'Yes.'
Sophie twines her arms about my neck. 'He just went away, didn't he, Gandie?' 'Yes, he did, honey.'
'Wouldn't you like to have him back?' she says, coaxingly, as if she were offering me an especially delicious chocolate.
'Oh, honey, it's all water under the bridge. When you're older you'll understand that. There's no point looking back, saying I wish, I wish.'
'Where's Max now, Gandie?'
'It sounds like one of those books, doesn't it?' I say with an effort, giving her a quick hug. 'Where's Wally? Where's Max now?'
'Yes, but don't put me off, Gandie.'
'I don't know, honey.'
'But he might come back?'
'I don't think so.'
'But he might, Gandie?'
'Sophie, darling, leave it. Max is dead, that's the truth of it. He's dead: he's not coming back. Let's not talk about it any more. And you make sure you pop that picture back in your mum's drawer, okay?'
'Yes, I will. Of course I will, Gandie. But listen, Gandie darling, you don't _know_ he's dead, do you? Mum says he's still alive. Mum says he'll come back, one day.'
Bloody Kate, interfering again, I think in fury. And, suddenly, something snaps. She's twisted the terrible knife enough. I can't take this any more.
'I do know he's dead,' I say, tightly. 'I do know it.'
She cocks her head, sceptical, a little dismayed. 'Do you know it for absolute certain sure? I mean, have you visited his grave?'
'Yes.' I reach out and hold her hand. 'I have visited his grave. I do know he's dead. Sophie, you mustn't tell this to anyone. It's our secret, okay? You mustn't tell your mum, or your dad, or anyone. Can you keep an important secret, like this?'
Sophie's eyes grow wide as she absorbs this. 'But, Gandie, Mum would want to know, wouldn't she? Mum really liked Max; she says he was the best friend she ever had. Why don't you tell Mum?'
'I can't explain,' I say, aware that I am gripping her hand too hard and loosening my grasp. 'When you're older, I'll explain. I can't talk about it, Sophie. Not at present. When you're older, I'll tell Mum. I'll explain everything. But I can't do it now. Believe me, darling, you mustn't say anything to your mother.'
At last, puzzled and chastened by my disclosure, she settles down to her milk and apple and homework. She looks serene, but what is she thinking? What does she make of what I've told her? What will she do about it?
She sprawls at the kitchen table to work; I sit on the other side of the bench, in the living room, which opens out between the kitchen and the small, pretty back garden, with its ferns and roses and two young birch trees. This is what we usually do when Sophie does her homework. She's at the kitchen table; I'm in the living room drinking coffee and reading the newspaper, available to answer questions.
I often sneak a look at her around the side of the paper when she doesn't know I'm doing it. I like to watch her sweet face creased with concentration; I like to watch her chewing her hair and mumbling figures to herself and doodling on the side of the page.
Today, however, I use the paper to hide my face from her perceptive eyes. My hands, I notice, are trembling. Does she believe me? I wonder. I think she does. Will she tell her mother? I don't think she will: Sophie and I have shared secrets before, and she's been reliable. But this is different. It matters.
When she goes, I say, quite lightly: 'You won't forget, Sophie? It's our secret, darling. Just ours.' She nods solemnly, and leaves me on my own, with nothing to do but fret, and gnaw my fingernails.
I don't want to think about Max. I don't want to think about anything, really: I just want to calm myself down and return to a state of dreary neutrality, so that Sophie's interest in these matters loses its edge, so that I escape this awful sense of staring down the crevasse.
I know that the more emotional I get, the more her ready sympathies are stirred, and the more she wants to know and understand. I should be grateful, I suppose: how many children are obsessed with their grandmother's past?
In my determination not to think about Max or his fiftieth birthday, I find myself thinking about mine instead. It wasn't as we'd envisaged it when Max spoke of hanging our portraits together. I would have let it pass, frankly. The celebration of having spent half a century battling away at life is bound to be an ambiguous kind of occasion, after all; although I suppose survival pure and simple is probably something to celebrate.
Anyway, Kate and Zoë between them were determined not to let the day pass without some kind of family get-together, God save me. They even had a mild disagreement about where it should be held. Kate wanted it at her place so that, she said, the children could feel part of the festivities; Zoë thought it more appropriate (she actually put it like that) that she and Henry host the event. Ever since Max's departure from my life Zoë has treated me with a kind of scrupulously considerate and self-conscious benevolence that drives me stark raving mad.
'I don't see why we have to do anything at all,' I say. 'I've never been a party person: you both know that. It's just another birthday. Let's let it go.'
'Rubbish, Minky,' says Zoë. 'It's not just another birthday. I had a big fiftieth and I want you to have the same.'
'We're not kids, Zoë. I don't have to have a party because you had a party.'
'Of course you must have a party, Mum,' says Kate, beaming, misinterpreting my genuine pleading for a variety of specious attention-getting. She thinks I'm merely making a fuss so that I can then gracefully give in. Zoë knows better: I can tell that from the look in her eye, but all her native persistence is to the fore and, terrier-like, she's not going to let go till she gets her way and forces it on me as well.
'I don't like fuss,' I say.
'You used to like it,' says Zoë. 'Remember your sixteenth birthday? You had a huge party. I helped with that one, too.'
'Zoë, you mightn't have noticed this, but I'm not sixteen any more.'
Zoë laughs immoderately, as if I'd cracked rather a funny joke, and continues to discuss menus and methodologies with Kate.
The event was held at Zoë and Henry's. I could have told Kate this would happen: someone with her pliancy is at a disadvantage when dealing with resolve of Zoë's calibre. They actually went to a lot of trouble, or Zoë did, anyway: I don't suppose Henry had much to do with it.
It's a big house, in which they rattle around like a pair of dried peas. Sometimes they have students staying in one of the three spare bedrooms: they both involve themselves in their school's international program and a succession of Chinese and Malaysian and Vietnamese and Thai adolescents have come and gone through Zoë's militant domesticity and Henry's fussy over-organisation. They are always suckers for good causes, Zoë and Henry.
Sometimes, as I've said, I think our family is as dysfunctional as families get; other times I marvel, startled into unwilling admiration, at how well we manage, all of us, how we scrub up for the big occasion. We've improved, no doubt about it. This is partly because of Sophie, as I've said, but it's also because we ourselves try so damned hard. Assiduously we paper over the cracks, snip off the dangling telltale threads; and we front up, bright-eyed and bushy-tailed, to the big occasion.
We carry out all this feverish remedial activity only partly for the outsiders who will attend and for whom some sort of fiction needs to be maintained; principally it is all for our own benefit, to fool ourselves into a temporary acceptance of family unity and bonhomie.
Oh, the poses we adopt, the grins we artlessly flash at each other, the greetings and wishes and presents, the outward registering of all levels of pleasure between satisfaction and delight, the chattering and gossip and kindnesses with which we spuriously bless each other on such occasions! All of these are part of the same deluded plan we have concocted towards an ephemeral new reality that depends on our cooperation and whose success is measured by the extent to which we are able temporarily to forget old divisions, old quarrels, old hatreds.
There's a lot of energy involved, when you think about it. Duplicity always does eat up energy, I suppose, even when it's half-instinctive, as this is. Why do we put ourselves to the trouble? Is it a tribal, collective impulse, designed to ensure our survival by protecting us all from the awful truth about ourselves and each other? Or is it a more individual response, a deliberate and explicit resolution from each of us, grounded in pure self-trickery? Whichever, in my family we're all past masters, and our mastery came to the fore for my birthday.
All the family were there, of course, even Liam, who can't have been more than three at the time. Kate insisted on his presence. Sophie's excitement was touching: Zoë arranged for her to propose a toast and she made up a little nonsense rhyme for it. How did it go?
_Swiftly Gandie's turning fifty_
_Gandie's always very thrifty_
_Will she be niftier, will she be thriftier_
_As she gets fitter and swiftly more fiftier?_
Something like that, anyway.
Steve kissed my cheek in an unforeseen access of affection (unforeseen by him as well as by me, I imagine—or perhaps it was premeditated; perhaps he had to steel himself to it). Gavin told several Millie jokes. Kate beamed. Liam played with small cars in a corner. Henry fretted over the drinks. Dominic brooded darkly. Zoë bustled around capably and presided triumphant over all. I tolerated it as best I could. That is, we all behaved true to type. I trust some of us enjoyed it: I know I didn't, much, but then I don't suppose my enjoyment was the principal aim of the undertaking.
How fatiguing these celebrations are! How many of them one has to attend! At what point in my life will I be permitted to say: _No. No more._ How old will I have to be before my children and grandchildren will simply say that Gandie is a cantankerous old so-and-so and there's no point inviting her along?
One of the surreal things about resting in the bosom of my family on such occasions is that I can look around me and it is almost as if Max never happened. There is Steve. There is Dominic, with Paula or whoever. There is Kate, with her husband and children. There are Zoë and Henry. It is all too normal for words.
Ah, Max, Max. Where did you appear from? Why did you come into my life at all? It wasn't so bad, as it was. Unremarkable, I grant you, but there are worse things than being unremarkable. Why did you storm my threshold, lay waste to my life, come prancing in with your glitter and your glamour? How did it all happen?
Max entered my life as a hot knife enters butter. Without warning, without resistance. Suddenly, I was transformed, penetrated. Perhaps impaled would be a better word. There was no prevision, no sly astrologer's forecast. One moment I knew nothing of his existence; the next, there he was, in my studio at the office, saying, charmingly: 'Hello. My name's Max Knight. You're Isabel Weaving? You designed a house for a friend of mine.'
I had been working, as I always do, in the square patch of sun directly beneath the skylight, puzzling over the doors to a courtyard. As I blinked and looked up it seemed to me that the bright flicker surrounding him was no trick of the eye but a sexual dazzle, an aura of danger and appeal, a cross-hatching of faint glitter festooning him like a profane halo.
'Hello,' I said, squinting. 'Which house? Which friend?'
I don't do a lot of work on houses. Beatrice and I tend to concentrate on commercial and sometimes government developments—buildings for small businesses, alterations to municipal offices, that sort of thing. Sometimes we'll do sets of villa units, if an up-market developer will give us enough land and not demand too much crowding on it: we're successful enough to be picky. But we don't do many houses. It's not that I don't enjoy them: it's just the way it's worked out. They don't on the whole come my way.
'Bianca Crawford,' he said. 'Ah, yes.'
The Crawfords were old friends of Steve's and they'd wanted an ultra-special beach house. They'd bought a block near Point Leo and driven us down to show it to me.
It was a great block: bush and ocean together. It whetted my appetite and I designed something for them at greatly reduced rates, partly because Steve wanted me to and I didn't have any objection, partly because the land set my blood racing: it was exposed on one side, sheltered on the other, on a rise, facing south-west into the bay. It offered fabulous sunsets and lots of challenges to do with wind and light and blending.
I finished up designing something flat and rambling, full of light and space, with small windows where you didn't expect them, vast sweeps of glass where you did. There were two courtyards where Bianca wrought wonders with native plants. She did the same in the main garden so that the house, in spite of being elevated, settled gently into its surroundings. Almost immediately, the whole thing looked as if it was meant to be there, had always been there.
It was more complex, and better, than your average beach house: they intended to retire there eventually and well before it was finished Bianca was starting to say she couldn't wait to get there permanently.
They called it Granada, because of some holiday they'd once spent in Spain. I have to say none of it looked very Spanish to me, but it was their house and they could have called it Patagonia or Upper Slaughter so far as I was concerned.
It had been a lot of fun and a great success, one of those projects where everything fell into place: the builder was snappy but meticulous and the Crawfords hadn't done any of the annoying things people who commission houses usually do, like endlessly changing their minds, or being cowardly about the drama the site offered. It had won one award and had been nominated for a couple more, and I'd been told it hadn't missed out by much. I'd thought at the time I could bear to do more of that sort of work, but it had been five or six years earlier and the business had hotted up and taken us in different directions.
And here was this stunning man, with his divine crinkly grin, wanting me to design another one like it.
It wasn't until much, much later that Bianca told me he wasn't really a friend at all. His silver Audi had pulled into the driveway one day when they were out on the terrace. He'd been all diffident charm: such a beautiful house; had admired it many times; could he possibly see inside; how imaginative; how striking; how understatedly gorgeous; who on earth had dreamt it up? They gave him a cup of tea and they chatted amiably and then he drove out of their lives and into mine.
It didn't matter: I took no notice. It had been cheeky of him, to claim acquaintance where none existed, but not a mortal sin after all. I taxed him with it and we laughed together about his nerve.
'I've bought a block,' he said, sitting down, all eagerness, passion reined in but peeking through, charisma blazing. 'I want a house like Granada on it. Can I show it to you? Will you do it for me?'
I always notice a man's eyes first. His eyes and his hands. Max had very black lashes and his eyes were grey and clear and smiling. His fingers were long and strong; his hands tanned and sinewy. He had beautiful oval fingernails, very clean.
Silly, to remember a man's fingernails.
His voice was wonderful. It was a cultured voice but not affected. Deep, with a burr to it that might have been Irish once, a long time ago. Sometimes I would listen to Max without hearing him properly because it was the tone and lilt and timbre of his voice that entranced me, not what the voice was actually saying.
When he had gone I waited five minutes or so, and then strolled into the little reception area off which our offices, Beatrice's and mine, gave. Dawn was on the telephone. Bea was there, rifling through a filing cabinet.
'Who was that?' she asked.
'A new client.'
'Where did he pop up from then?'
'Nowhere, really. He's a friend of the Crawfords. He wants me to do something like Granada for him.'
'Where?'
'Somewhere in Hawthorn.'
'He's got the land? In Hawthorn? Enough land for a place like Granada in Hawthorn?'
'I believe so. Roughly, anyway. He said it was a big block.'
Bea continued to flick through the files, her disapproval patent in the sharp jabbing of her fingers and the stiffness of her back.
'Did you meet him through the Crawfords?'
'I've only just met him.'
'You never knew him before he walked through the door then?'
'Never.'
'Izzie, why have you got that stupid smile on your face?' said Bea.
I hadn't known of it, but I got rid of it as fast as I could.
Dawn put the telephone down. 'You don't know him?' she asked.
'No. Never laid eyes on him till then. Why?'
'Well, I guess it was the way he walked in. He kind of looked at your door and said "Isabel in, is she?" and headed straight for it. Normally I would have given you warning, shown him in. But he seemed so confident, I thought he must be a friend.'
Max was always like that. He had an extraordinary capacity to get his bearings in a place, in a situation: he could walk into a room and know instantly what its layout was, who stood where, what strategic advantages it offered him, what traps lay in wait. He had a preternaturally keen nose for opportunity and a talent for judging when and how to sidle towards it.
'As if he owned the place,' snapped Bea. 'He looks like a con man, Izzie. Take care. Don't do anything till you get a hefty deposit.'
I thought she was jealous. I still think she was jealous.
They talk about love at first sight. Who can say? I don't suppose I really fell in love with Max during that very first meeting: it couldn't have lasted more than ten minutes, anyway. Who can say what it was or how it happened? Maybe I was having an active-pheromone day.
But the attraction was instant and I felt I'd never understood the word 'magnetic' before. It was as if my entire body were composed of iron filings that had snapped to attention: every skerrick of me was drawn to him, yearning at him, screaming for him.
Later that week I drove to meet him at the site. It was winter: one of those cold, still mornings with wisps of mist still hanging furtively, swollen black clouds above. It was indeed a large block—almost a double block—in what the real-estate agents are fond of calling a quiet leafy street in the heart of Hawthorn. You wouldn't have thought such a site could still exist in quite so built-up an area: it must have been worth a mint.
I wore my black leather boots and my scarlet coat. Normally, then as now, I wore quieter colours—olive, biscuit, tan, cream, and always lots of black; but I saw this coat and fell in love with it, as they say. You fall in love with coats, with men, with babies. I was very conscious, on this pewter morning, of how colourful I was, how I moved through the dulled landscape, a splash of geranium-red against the muted misty background.
It was the first time I noticed this very particular effect Max had on me, that he forced me to watch myself so that I was constantly aware of how I appeared to him: it was like having an internal video of me running in my brain. It was to be the same when we were in bed. I was always conscious of how I looked to him, especially in my nakedness. It was disconcerting at first, but then I grew accustomed to it: when Max was no longer in bed with me, it was not only him I missed but the curious new perspective I had acquired of myself: grainy slow-moving star of my own private cult movies.
But that day we were not naked, not yet. He was in grey: charcoal Italian jumper, grey tailored pants, good black shoes. His hair was silvering: it must have once—not so very long ago—have been jet black, as black as his eyelashes, as black as my hair.
I walked around the block, marvelling. There were two grand lemon-scented gums at the north-west corner, great white shafts thrusting through the ashen air, dissolving into their pale plumes of delicate long leaves at their extremities. They were as mighty as cathedral columns, as white as flour, as bones. You could smell their crisp fresh fragrance in the damp still air. I could see how the sun would glitter off them, as it glittered off the slanting windows I was already imagining.
'You'll keep these?' he asked, a shade anxiously.
I nodded. It would have been murderous to do anything else. Already I could see how I might use them: a half-frame to the entrance, great flagless masts off-centre from the main action. Granada had depended partly for its effect on asymmetrical angles and unpredictable planes, on nooks and sweeps and perspectives of sudden enchantment. I could do something very like that with this site, dissimilar though it was to Granada's ground. My fingers tingled with it.
The site was flat and scraped, unnaturally tidy in the way sites look when wreckers and backhoes have been through. Not much grass; no bush or shrubs. Only the two giant eucalypts. It had been prepared: it was ready for population.
'How did you come by this?' I asked. 'There must have been something here. You must have wrecked it?'
'It was a cottage. Not very big. I moved it and sold it. They sliced it down the middle and took it away on two of those huge trailer things, with _Wide Load_ all over them. Quite a spectacle, in fact. Then I had people in to clean up. It's all ready to go. A blank page, for you to draw on.'
Judging by the houses in the rest of the street, it couldn't have been too small a cottage. If he had the resources to do that, I thought, to dispense with what must have been a pretty good house in so summary a way, he had the resources to build a fabulous home.
'You could have a pool,' I said. 'It's big enough. Do you want a pool?'
'Love a pool.'
'Or a tennis court?' But I wanted him to have a pool: I wanted the house to have water near it, water as part of its thought patterns, just as Granada had.
'Don't play tennis.'
'Do you swim?'
'All the time.'
'That's settled, then.'
'Not bright blue,' he said.
'Sorry?'
'Not bright blue. The pool. Something subtler.'
In the event the pool had latte-coloured tiles, shot through with a darker shimmer, like raw silk. It was a gorgeous pool: it twinkled in the sun with a pale, exotic gleam. Inviting, coolly seductive.
The house sprang around me. Stairs flowed in shallow curves; doorways unfolded; space crystallised into shapes: my head hummed with the rhythms of it. Its colours danced: colours of salt and bark, leaf and foam. Rainwater and sunlight. Gravel and moss. The texture and grain of it all curtained me. I was practically orgasmic with it.
He was watching me. 'You can do something with it?'
'I can do something good with it. I can do something very good. What sort of a budget are we thinking?'
He shrugged. 'It's my dream house. The house of a lifetime. Of what's left of a lifetime, at any rate. No holds barred.' 'I have _carte blanche_ , in effect?' I could hardly believe it.
'Absolutely.'
'No limits? You're seriously telling me I have no limits?'
'Not unreasonable ones.' He smiled. He had a crooked smile, a way of slightly crinkling his nose.
'It's incredible. It's what all architects long for.'
'Well, you've got it. I want to be in on the planning, mind you.'
'You will be. Oh, you will be.'
'And I don't want a mansion,' he said, glancing around at the surroundings, the old solid wealth on both sides. 'I don't want something that shouts or sticks out like a sore thumb. I want something like Granada: I want something that nestles in, that belongs, and settles itself into its surroundings, something that you have to look and look at to see how beautiful it is, something you keep discovering.'
I nodded. It was amazing, the feel the man had for all I held most dear. I would give him a house that sang, a symphony of a house, a house like no other in the world.
And then, without warning, the sky burst and it poured. It rained as if it hadn't rained for a hundred years and the sky had been saving it up all that time. Max had parked his car, his silver Audi, down the road a little way. We sprinted for it. We couldn't have been more than thirty metres away but we were pretty much soaked by the time we scrambled in, tumbling over each other in the back seat, laughing, breathless, wet hair in our faces, rivulets dribbling down our necks.
Of course he kissed me. Of course I kissed him back.
I can't even remember in what order things happened then. I'd fallen into the maelstrom, and there was no way out.
It didn't all happen suddenly, not all of it. I know that. I did try. I tried to withstand it, to pretend it hadn't happened, but finally that was impossible. It was six months from that day, that first visit to the site of Rain, before I slept with Max; it was a year before I left Steve. A delay like that shows something, doesn't it? I did try: for a while I tried hard. But there was no point. There was nothing I could do.
How can I describe that year? What can I say to give some idea of the change that came over me? It was not a change that I willed or welcomed. It was beyond my control. It was as if the molecules in my body quivered and melted and recast themselves in a new physical pattern.
I was a different person: that was what none of them could see. It was as if I had lived all my life in an invisible eggshell: my birth had been just one hatching; another was required. And as I broke through the eggshell I smelt different smells; I heard different sounds. I swear I saw different colours. It was like discovering that all your life you had lived in a vast deep shadow that was now withdrawn: suddenly you stepped into the sunlight. What were you meant to do? Creep back into the shadow again? Exchange the sweet fizz in your veins for the old sludge? Refuse to see the new colours? Turn your face away from it all?
My sister Zoë told me I had caused our mother's death by leaving Steve, by splitting the family.
'That would be in addition to the cancer, Zoë?' I said.
I felt guilty enough: why should I be made to feel worse? She understood no more than anyone: I was helpless. What was I meant to do? Look my only happiness, all my happiness, straight in the eye and refuse it? It was as if I could see my old life only through the wrong end of a telescope—as a series of tiny black figures on a horizon. Steve was the tiniest, and his distant cavortings interested me the least. I could not focus on him. I almost could not see him at all.
Zoë pursued me with biblical enthusiasm, squealing sin and retribution and hellfire, screaming wreckage and mayhem. I hadn't ever realised she felt so strongly about the religious side of things, nor that she was so fond of Steve, about whom she had always been somewhat condescending. 'Steevil the weevil', she'd called him, all those years ago when I became engaged to him and she had no fiancé. Now I reminded her of that and she told me I was being childish.
'Excuse me?' I said. 'I'm the childish one? You called him Steevil the weevil and I'm the childish one?'
'Don't trivialise this issue!' she yelled at me: this is one of the admonitions people grab at when they are at a loss. 'I don't know what's happened to you! You're infatuated, blinded! This man has changed you into somebody I don't know!'
And much more of the same, all emphatically punctuated. 'It's the sex,' she said to me one day, with a ferocity that approached real viciousness. 'I understand that, of course. It's just the sex. That'll fade away, sooner rather than later, and you'll look back and wonder what you ever saw in him.'
I didn't bother to reply, because it was such a foolish accusation. Sex on its own is terrific, but it doesn't finally matter as much as other things. Making love with Max was wonderful, but it was not for sex, nor for the glamour, that I fled to Max, curled in his embrace, hid under his protecting wings. It was for his kindness and gentleness, his ready responsiveness to me, his love and care. It was for his thoughtfulness and sympathy, his loyalty, his unending, unwavering support through everything we ever did together. I'd never had anything like it.
My family and Steve had always carped at me ( _The trouble with you, Minky..._ ), criticised me ( _Lovely, why can't you just..._ ), found fault ( _Isabel, dear, couldn't you try a little harder..._ ). Nobody had ever simply loved and sustained me as Max did. This was what I found I couldn't adequately explain, what nobody would believe or comprehend.
Imagine you had lived your life within curtains, in silence, and one day someone had lifted the curtains and immediately shown you, let us say, a Rembrandt painting, played you a Mozart symphony. Imagine you had lived your life in snow and then discovered fire. Could you glimpse it all past the curtain's quiver, hear its echoes dying in the distance, and turn your back on it?
When we revisit the past, we are told, we inadvertently change it; we crystallise it anew, altering the chemical structure of our memories subtly but inevitably, definitively. Just by selecting words, casting our net into retrospective waters, we catch different fish. We try to reconstitute the flavour of the original experience and in that effort inevitably mislay something. I am making it sound as if it was all instant and perfect. It wasn't: I've said, I didn't leave Steve for over a year. And it wasn't all ecstatic either.
As I try now to connect these days, to construct the narrative faithfully, it is like trying to make a necklace out of different beads. It shouldn't look like a string of pearls when I've finished. Some of the beads do indeed have a fine soft bloom to them, like a pearl; others are jagged and crooked; some are delicate and flowery; others glisten with a dangerous obsidian polish.
Years later, when I had sold the house called Rain and was living on my own again—or, rather, with Borrow—Zoë asked me if I regretted any of it. She still couldn't understand that it was an irrelevant question. I had no choice in the matter. I was out of control.
It was like asking me if I regretted World War II, or the French Revolution. It wasn't up to me to regret, because it wasn't possible for me at the time to make choices other than those that imposed themselves upon me, those that became historical truth entirely without my connivance or intention. A storm can't choose to do other than it does. I was caught in the storm, at one with the hurricane.
I'd gone feral.
And it went sharply against the grain for me: that was the other thing everybody failed to appreciate. I'd never been a rebel. I'd had my faint resentments about life, my lingering suspicions about unfair treatment and inherent worth. I was utterly happy when I married Steve. I was unawakened and ignorant; but I was pretty happy, too.
I came out of an uneventful childhood, from loving parents, from the unknowing shadows of middle-class endeavour and the circumscriptions of middle-class satisfaction. I wanted a career, certainly; but I wanted a husband, too, and children, and all the domesticity that went with those aspirations. It never crossed my mind, when I married Steve, that I would want one day to unmarry him.
Divorced people were other people. People who had affairs were other people. People who fell newly in love, people for whom the world glittered like bushland freshly washed after rain, people who did wild and unpredictable things were other people.
Yet I had always sensed that there was something extra and dangerous in me, a capacity to do something radically unstable, threatening. Zoë didn't have it: Zoë marched militarily down the road of her life, not consenting to be distracted or delayed or to whiffle off on a sortie to the left or right. Steve didn't have it: Steve was wedded to comfort and habit and regularity as well as to me.
But I'd always feared and cherished the knowledge that there was inside me something bright and resilient and exotic—a great vivid tropical tree, perhaps, with glossy leaves and gaudy parrots flashing and gabbling in the thick branches—something that could explode and change the course of my life.
Sometimes I had felt it within me, this thing, pushing upwards like the blind nubby eyes of potatoes; but always I had firmly squashed it, because I had been frightened of what would happen if I let it go.
And now it had sprouted and was growing wild. It was still frightening, but it was something also to grin about, to relish. I had been so docile, always. All my life, even with Zoë, except when her demands became truly intolerable, I had been compliant, submissive—a good girl. I had always acted within the parameters of people's expectations of me. Pliant, tractable, biddable, docile, meek: I had been a veritable thesaurus of obedience. But now I was supine no longer. Now I was living my life for me.
Steve and I were savage with each other, over that year. At first he was deliberately obtuse: he refused to understand what I was saying. Before I said anything at all, I tried to let him find out. I left a trail of hints, a long string of unanswered questions that he had only to ask himself, or me, in order to discover how his life trembled on a precipitous brink. I left a packet of condoms in the bathroom. (Steve had had a vasectomy.) He didn't advert to them. I left a compromising note from Max on the floor of the bedroom: he obstinately refused to pick it up. Even when I wasn't seeing Max I was late home from work, with specious excuses. I never explained incoming telephone calls; I did my best to make the most innocent conversation sound mysterious, dangerous. Nothing worked. He had to have his nose rubbed in it before he could catch any scent, pick up any trail.
And when I eventually hurled it at him so he couldn't dodge it any more, I wasn't interested in rational discussion. I tried to provoke him, I think. I didn't know that was what I was doing, but I think that's how it worked. I was guilty, and I was infuriated by my guilt, and I wanted him to behave badly in order to justify my own behaviour. I wanted him to hit me, to bellow at me, to cast me out.
At the beginning, he would simply sit at the kitchen table and stare at me with bloodhound eyes. It drove me mad. It took him at least a month to work out that I really meant what I was telling him; and even after that it was a while before he lathered up a true rage.
But then we were off. It was hammer and tongs. We'd wait till the children were out of the way, but only barely, sometimes. I didn't care. The children were as distant and tiny to me as everyone else. I couldn't even look at Kate. I persuaded myself Dominic was too young to understand, that when he was older he would forgive me, see my side of it.
Dominic was ten. Plenty old enough to understand, as it turned out; and not of a forgiving disposition or an elastic perception.
'What's _wrong_ with it?' Steve shouted bitterly at me one day. 'What's wrong with the bloody marriage? We've been married more than fifteen years, for Christ's sake, and you never complained before. What the fuck's wrong with it?'
In a way I could see what he meant. I hadn't complained; there'd been nothing, in a way, to complain about. And yet, when I looked back, I thought I had been endlessly forbearing. About the snoring, about lots of things.
The main thing, it seemed to me, that I had been long-suffering about, was the boredom. It had come home to me quite rapidly, after marrying Steve, that there were to be no fireworks in life. I'm not quite sure, now, what I expected of marriage, although I wanted it so badly. I do recall thinking that Steve was a comforting, steady presence—in a teddy- bearish kind of way—and I think I believed he would provide stability. I suppose I was right about that.
The thing is, you might want stability but kick against sheer immovability. Steve was great on routine. All-bran and sultanas for breakfast. The 8.10 from the local station, Monday to Friday. If he missed it, he still got to work on time; but missing it up-ended the day into Steve's version of really significant cataclysm. Saturday afternoon for gardening and listening to footy on the radio. The news at seven. Ringing his mum on Thursday nights. Sex on Friday nights, Sunday mornings.
It wasn't that I minded any of it in particular. It was fine, really. It was irritating, a lot of the time, but it was safe. He was so safe. He looked after me so much: he took such good care of me it drove me mad. He discovered I had a fear of spiders: he was all concern. He would hear me screech or see me go rigid and he would come running. 'I'll take care of it, lovely,' he would say. 'Just a little old spider, is it? I'll take care of that. Don't you worry about a thing.'
After dinner, at night, sometimes he would sit and beam at me foolishly. If I asked him why he was looking at me like that, he'd say: 'I'm just so happy to be with you, lovely. I'm such a lucky guy.' And that was comforting, and pleasant; and I became used to it, even if I did resist being called lovely the whole time.
Habit ensnared me, too. And it was better in the beginning, when we both had a routine: he'd go off to catch the 8.10 to the city; I'd hop into the Cortina and do the ten-minute drive to Bea's office. The business belonged entirely to Bea, in those days: I'd only just graduated when I started there. It was before she took me on as a partner; that came later.
*
And for two years or so, that was perfectly acceptable. But after I'd had Kate, it all changed. I'd decided to spend Kate's first two years at home. Steve and I had grappled earnestly with questions about motherhood and childcare, and we'd agreed I'd be a full-time mother for the first little while at least. We could afford it; and he'd be happier, he said, to think that his lovely was at home caring for the kids.
But then I found out about looking after children. I found out about the loneliness and the insecurity and the boredom and the worry, about the prison a tiny baby could build for you. I found out about being left alone in the house in the morning with a small child and the indescribable smell of damp toast crumbs.
It wasn't that Kate was a difficult baby. She was dead easy. Everyone kept telling me how lucky I was, pointing out how placid she was. She was sleeping through at nights within a couple of months. She was a cherub, a prodigy. Everybody said so, especially my mother. I asked her one day if she'd ever felt bored when she had Zoë and she looked at me as if I were a child molester, a monster.
Within six weeks I'd rung Bea. 'I can't stand it,' I said.
'What on earth do you mean?' asked Bea, in shocked tones.
I'd driven to the office the previous week, Kate in tow, and she and Dawn had carried on endlessly about Kate and what an unparalleled model of a baby she was.
'I mean, it's gorgeous, she's gorgeous, she's adorable, motherhood's terrific, I've never been so fulfilled. That's what everyone expects me to say and that's what I say; I say it to everybody. It's even true, in a way. Bea, don't be like everybody else, please don't. I can't stand it. Can I come back? You've got enough work, haven't you? You miss me, don't you? You haven't got anyone else yet, have you?'
Steve wasn't happy, but there wasn't a thing he could do about it and he was sensible enough to realise that. 'I just want you to be happy, lovely,' he said. 'You know that. So long as you're happy. It's just that we talked it through and I didn't think we'd be doing it this way; I thought we'd be doing it the way we talked about. You're not worried about the money, are you, Isabel? There's plenty of money. We don't need for you to work: you know that, don't you.'
He hated it, that I wasn't maternal enough to want to be with Kate every moment of every day. It embarrassed him in front of his mother, our friends. I pointed out to him that Kate didn't suffer from it: Kate was happy to be with anyone at all. The girls at the childcare centre adored her because she never gave them a moment's bother. He disliked that: he disliked thinking that Kate did perfectly well without me, but it was true.
During the terrible year of our gradual, creaking separation he recalled this, and accused me of never having really loved our children, never having really loved him. It wasn't true. I did love my children. I do love my children. I'd die for them. I had loved him—in so far as I had understood what love was all about, then. But it was over. Over, over, over. Max had flared into my life like a comet, beautiful, dangerous, fascinating. Nothing could ever be the same. Not love, not anything.
And in the meantime, while everyone shouted and harangued, we were planning the house, Max and I. Soon the permits had been obtained and we could start building. I've never known a project like it: everything went with miraculous smoothness. I thought it was a signifier of some divine approbation of our union, our love.
All the bits fell into place: there was none of the usual nonsense of trying to get the plumber to talk to the plasterer, or the bricklayer to tell the carpenter what he was going to do and when he was going to do it.
Then I found Max was liberally greasing the palms of almost everyone involved. At first I was annoyed: I confronted him over it. He was astonished. He thought I was ridiculously naïve, not to understand that this was how things were done; this was how you got things done. Then I thought it was funny. If he had so much money—which seemed to be the case—that he could do this without noticing it, so much that it simply didn't matter, why should I stop him? He enjoyed it, anyway: he enjoyed the comfort of largesse, of distributing ridiculous bribes.
There was a kind of wildness about it all, an intoxication. I was conscious that Dawn and Bea were watching me with harassed eyes as I came and went in the office. I didn't care.
'What does he do?' Bea asked me one day.
_What does he do?_ I felt like replying. _He turns my blood to honey; he transforms me; he comes to me in a shimmering shower of nine-carat gold_.
I looked at her. 'Lots of things. Rather well, in fact.'
'What does he _do_?' Bea said, crossly. 'You know what I mean. Where does all the money come from? Tax evasion? Prostitution? Drugs? White slave traffic? Izzie, you don't know this man. You don't know where he comes from, who he is. You know nothing, nothing about him. What in Christ's name are you doing?'
I shrugged. Max was a businessman: that was all he had ever told me. I was quite incurious about his profession. I hadn't ever properly understood what Steve did in the bank, beyond putting on a suit and turning up to work every day. Why did it matter?
Still, after Bea started pestering me it did occur to me to ask Max where his apparently endless supply of cash actually came from.
'What do you do?' I said.
He burst out laughing. 'I wondered when you'd ask. They've been at you, haven't they? "What does he _do_?" they're saying to you. "Why don't you find out what he _does_?"'
Nobody could have been less troubled, more open, readier to find amusement in the interest people might express in his occupation.
'I consult,' he said. 'I develop. I invest. I keep fingers in lots of pies.'
'You're telling me not to bother my pretty head about it?'
'I'll tell you anything you want,' he replied, breezily. 'There's no single easy answer, that's all. I've got dozens of business interests, dozens of investments. I do all sorts of things, my darling Belle. I buy land and build things and then sell them as big packages. Then I buy more land and build more things. I own multi-storey car parks; I own rubbish tips. I own cranes.'
'Cranes?' I said, bizarre images of stalky long-beaked birds flapping through my mind. 'You run a rent-a-flock?'
'Industrial cranes,' he said, laughing. 'I lease them to builders, to large-scale developers. You've no idea how boring it all is,' he continued, diffident, modest, humorous, spreading his hands in that self-deprecatory, slightly puzzled gesture with which I was to grow so familiar, which indeed was to haunt me, years later. 'Lots of people pay me lots of money for all kinds of different things. Business advice, marketing advice, planning advice. It's all to do with bean-counting, really.'
I didn't actually care. It was just as well I didn't, for nothing much was clarified during our marriage. I was to discover, after we started to live together, that letters rarely came for Max.
'I use a post-office box,' he told me. 'More convenient, easier to keep tabs on.'
Much of his business seemed to be conducted in other people's offices or else in expensive restaurants. I met no colleagues, no associates. None of this bothered me in the slightest. This was all peripheral stuff, cloudy trivial filaments hanging inconsequentially around the solid gorgeous fabric of our relationship.
One day, when we were in my office, he used the telephone.
Halfway through the conversation—it was obviously about money, about shares, I recall, but I didn't try to follow it: I didn't want him to think I was inquisitive, and in any case it was impenetrable to me—he laughed, and started speaking in Italian. It seemed to me that he spoke with extraordinary speed and fluency: he sounded like a native.
'I didn't know you spoke Italian,' I said, later.
He glanced at me. 'I speak a few languages. Some better than others, some worse. It's helpful, for business.'
'Do you travel to those countries?'
'A bit.'
He did travel during the years we lived together, but not frequently. Once or twice to Bangkok; two or three times to Singapore. Indonesia, a couple of times. Maybe half-a-dozen trips to Sydney, the same to Perth.
He did spend a good deal of time on the telephone: his sleek, little study in Rain (mahogany desk, burgundy leather chairs) had separate telephone and fax lines; and that was when it was quite unusual for people to have such an arrangement at home. He did send and receive faxes, but unobtrusively, without fuss, and I think infrequently, too.
He didn't welcome anyone fiddling with his things, and I soon learnt to leave his study alone, not to search through the papers on the desk, not to open the drawers. It didn't bother me. He was a reserved and private man: he had a right to keep his papers to himself if he wanted to.
Nor, I realise now, did I ever find out much about his life—his life, that is, before meeting me. He wasn't evasive; it just wasn't something we talked about. He had grown up in Sydney, the only child of patrician and distant parents. There had been a brief marriage, when he was still a very young man: his wife had died of a catastrophic and rapacious cancer within three years. Her name had been Caroline, and there had been no children. He looked faintly baffled when he spoke of this first marriage: when I said this to him he laughed, wryly.
'It's so long ago, Belle, that's all. It's as if it happened to somebody else. I swear I loved her: I adored her. But now I can't really recall what she looked like, to tell you the brutal truth.'
After her death he had travelled, lived for a while in Canada, for a while in London. It had been a rootless kind of existence, he said, dismissively. Comfortable enough, but finally without direction, without substance.
On his return to Australia he had made Sydney his headquarters, but he had always had a yen to live in Melbourne; and, after a windfall from some unusually successful enterprise, he had bought the land on which Rain was built with the intention of settling down for at least four or five years.
'And now I'm here for life,' he said, grinning at me. 'Well, for as long as you're here, anyway, my darling.'
He never avoided a question I put to him about his life. He never volunteered information, either. I didn't mind. As far as I was concerned, I'd emigrated to a brilliant new country: I didn't talk much about the old one, either. I suppose I assumed that, as we continued to live together, as we became more and more accustomed to each other, we would gradually learn more of each other's lives; our pasts would roll out slowly for each other's minute inspection, just as the years ahead would unfurl, like long, lush carpets, intricately patterned and richly interwoven, waiting to be trodden on.
But while Max and I worked on our house, our home, I was still living with Steve, and he and I were engaged in an intense and concentrated war. It was only by small degrees that he accepted the inevitability of the split. Towards the end there were awful nights when he wept—noisily, unattractively—and begged me to stay, begged me to give the marriage another go.
He offered me different domestic arrangements as a compromise; he said he would welcome Max at our house, that I could go and spend nights with him sometimes, that he would promise not to have sex with me for a year or more, that he would do anything, anything at all, if only I would stay with him.
All of it was impossible. He was desperate, mad. I'm not sure that it was me he cared about, so much as the life he had made, the life in which he was comfortable. I was up-ending his life, and he would never forgive me.
Eventually, however, he accepted it. And we started to talk about what would happen with the children.
'I'll take Dominic,' I say.
'Will you now?' replies Steve, rather unpleasantly.
'Yes. I will.'
'Not Kate?'
'It would be better if we split them,' I say. 'You know we talked about this: we don't want to have custody battles, that sort of thing. It's silly. The only people who profit are the lawyers: we both know that. I thought we'd agreed, it would be better to have one each. It isn't as if they're going to be broken up for good: they'll have plenty of time together. It isn't as if they're inseparable anyway, either: most of their time they spend apart.'
'I realise this. Why don't you take Kate? That's what I'm saying, Isabel. I'm not suggesting you take them both. I'll have them both, gladly, but I don't think that's what Kate wants. I'm suggesting you take Kate instead of Dominic.'
I stare at him. This is not an alternative I have considered. Why would I take Kate? Doesn't he understand? It's Dominic I want, my prince, my dark angel.
'But Dominic's mine,' I say, stupidly. 'We've always said that. Dominic's mine; Kate's yours.'
'Don't be so idiotic. We said that as a joke, a parody on ourselves if you like. The dark and the light. It doesn't work like that. You know it doesn't.'
'I can't take Kate.'
'Isabel, you can be so blind. Kate adores you. Dominic doesn't.'
'Dominic does love me.' With terror I feel the prickle behind my eyelids, tears approaching. It will never do, to start crying at this point. I have to be strong, to demonstrate that I can be strong.
'Of course he loves you,' says Steve, with that exaggeratedly patient manner I so resent. 'That's not what I said. You don't listen to what I say, Isabel: sometimes I think you've never listened to what I say. You'll break Kate's heart if you leave her.'
But I insist. Dominic will come with me. Kate will stay with her father, in whose mould she is cast and whose habits and predilections she understands and to a large extent shares.
I do not take Dominic's feelings into account. Why should I? He is only little: he is ten. He is too young to know what he really wants. Of course he will do as he is told.
Dominic, of course, has never done as he is told. I find a quiet time to speak with him. I try to cuddle him (an ill-advised strategy), tell him how I love him, how happy we will be together. He refuses to listen.
'If you're leaving, well, okay then,' he says. 'Cool. You leave. I'm staying with Dad.'
I reason. I flatter. I schmooze.
'No,' says Dominic, regarding me with unfriendly eyes. Eyes like pebbles.
I represent to him how splendid life will be with me, with godlike Max, in our beautiful house, sharing our bliss.
Dominic measures me with his stern and pebbly eyes, and finds my dimensions lacking. 'No,' he says. And he will not be moved. Not then, not ever.
'I told you,' says Steve when I tell him. 'I told you, but you wouldn't listen. You never listen. What are you going to do about Kate then?'
'I'll think about it.'
'Well, don't think too long. She knows it's happening; she's worried sick. She's waiting for you to talk to her.'
But I talk to Max, first. I find this embarrassing, as in our previous discussions I have always assumed Dominic will be the child who joins us; I have assured Max of Dominic's virtues and painted vivid pictures of how he loves me and how he will love Max, how well we will all get on together, how beautifully our lives will mesh.
Max has met Dominic and has (I suspect) found him sulky and distant, but is too well-mannered to say so. Kate has played little part in our discussions. I find it hard to explain to Max that my beloved son doesn't wish to join us.
'He'll come for weekends and so on, I imagine?' says Max, consolingly.
'Well, yes, I suppose so,' I say, already hearing Dominic's flinty refusal.
'Well, then. There's no worry, is there?'
'No.'
And Max enchants Kate by bearing her off to various exotic shops where together they handle fabrics and inspect colours for the purpose of furnishing her bedroom: he even manages to educate her a little towards his own severe style of elegance, distracting her from the twee white four-poster and the ruffled pink curtains she at first covets. Kate has always had an unfortunate penchant for tinsel and frilly knickers.
It's rapidly clear that she adores him; at fifteen, ripe for crushes, deep in adolescence, she hatches a fully fledged worship of him. He's faintly amused, but flattered. I'm pleased, of course; and ironically it makes for greater harmony between us than she and I have ever enjoyed. But I miss Dominic's dark sardonic presence in my life.
I had little to do with Kate's bedroom, but I revelled in furnishing the rest of the house. For now we knew that we would live in this house together, that it would be not solely his but ours. We would sit on the patio during summer twilights; we would knock together dinners in the gleaming kitchen at the end of the day's work, we would dive into the pale beautiful pool. We were building not just a house but lives, furnishing not just rooms but years.
We would sleep together and make love in the glorious main bedroom, so spacious and light; and waking, would together enjoy the view, framed by the lemon-scented gums we had so carefully preserved. Together we would gaze up through the vast clear shaft of the skylight Max had insisted upon, directly over the bed, inspecting the weather in the mornings, the moonlight at night.
Normally I had little to do with soft furnishings: Bea used an interior decorator on a casual basis and we called her in when clients didn't have the time or the nous to construct a heart for the body we had made them. I was asked for advice sometimes: Bianca had got me to talk her through Granada, and she had done a lot of what I'd suggested. But here I could run riot.
Not that Max was the sort of man to run riot. Nothing but the best, nothing over the top, was his rule. In one sense he was reckless: he loved lavish gestures and he could spend more extravagantly than anyone I'd ever come across, but he wasn't stupid and he didn't buy rubbish. His taste was immaculate.
We fleshed out the house's skeleton together, choosing the carpets and timbers, the tiles and the paint. We spent hours there, experimenting with colour and texture, holding up drapes, comparing ceramics and woodgrains. When we had these right, we moved to curtains and furniture, cushions and pictures, lampshades and ornaments and lights. It was one long, incredible spendfest.
I'd never known anyone who spent money like this. Nothing was too expensive for him. In the living room we chose lamps that spilt pools of brandy and whisky; cushions the colour of burnt toffee and a dark honey, like honey mixed with rust, to scatter over the fine milky leather of the acres of sofa stretching in a soft crescent around the fireplace. I felt drunk every time I walked in there, intoxicated by the room's simple magnificence, the beauty of colour and sweep, stone and glass.
One evening I went there after work (it was before we were living together, and we used to meet there for an hour or so at the end of the day, for our daily charge, our shot of ecstasy). Max was standing before the fireplace (its façade a streaked amber stone) rolling something over in his hands. As I entered, he turned and held it out to me.
It was a sculpture, about the size of a grapefruit. It was round and smooth and startlingly heavy, a deep polished cream, marbled with the amber of the hearthstone. One's hands curved around it with gratitude for its cold smoothness, its satisfying weight and density. Its centre was carved into a hollow, crossed over with swathes of the stone, passing and weaving in heavy ribbons that intersected and dissolved back into the substance of the stone.
It was like a highly stylised version of one of those Chinese puzzle spheres that nest within each other, except that there was only one of it. I weighed it and turned it and stroked it: I didn't want to put it down. I cradled it in my hands.
'Where did you get it?'
'That gallery in Richmond,' he said. 'The one on the corner, you know. It had those woodcuts you liked. Isn't it exquisite?'
'It's perfect. What sort of stone is it?'
'Don't know. I thought it might be onyx or something of the kind, but I think it's too grainy, too heavy. The gallery bloke didn't know.'
'It's meteor stone,' I said. 'It came flashing down to earth, just for us. It's our meteor.'
'It is, too,' he said, hugging me.
*
On the day I finally left Steve, he wept inconsolably. I took little—only clothes, and a few books. He ought to have been pleased, that I left him with everything, that I didn't force him to a division of property. I wanted nothing. He could have the house; he could have the car, the shares. None of it meant anything to me.
Max bought me a new car. He bought me a Fred Williams painting and a crisp, glittering diamond ring.
Zoë spat bitter things into my ear. Bea wrung her hands.
My mother cried.
'You're just being bought,' she sobbed.
I didn't say that if Max had wanted to buy me he could have managed the transaction a lot more cheaply.
I did keep on trying to explain to them, to my family and my friends. But they none of them wanted to know. Even my mother, who was at the time so ill, even she—knowing death approached, knowing that within the year she would leave us—was harsh to me after I'd moved in with Max.
'Thank God your father's not alive, Isabel,' she said. 'He was so fond of Steve. We all are.'
'Well, I'm not. It seems to me I'm the one who counts, here.'
She shook her head. 'Of course you're still fond of him,' she said, impatiently, just as if I were still ten, a child who needed reminding of common sense. 'This is some kind of mad aberration. You'll grow out of it. You'll regret this, Isabel: you mark my words. You'll regret it.'
She astounded me. I had thought Max would fascinate away her prejudices in five minutes. She was usually susceptible to personable men, and I'd expected Max's sheer good looks and fluent charm to knock her over, if not immediately, at least in time. She remained conspicuously unaffected; and her obduracy was reflected by almost everyone I knew. Kate was the only person to respond with warmth to him.
Everyone seemed to assume that I had been bewitched by his glamour, by the sheen and gloss of him. Of course he was glamorous; of course he was sexy; but his glamour and his sexual charge weren't all there was to love about him, and it enraged me that nobody seemed able to see this.
Everyone apparently assumed the man I had fallen for was nothing more than a silver-haired debonair stud, who had seduced me and bemused me with wealth and sophistication and flair. Was I so shallow? Did they think me so easily enamoured, so ripe for cheap enchantment?
His gifts were amazing. No less amazing was the manner in which he delivered them. If Steve gave a present, it was heralded and trumpeted from afar. It seemed Max's gifts slid sideways from his pocket, like those of some captivating and talented conjuror.
'The start of your investment portfolio,' he said, kissing me as he gave me the ring. It had not just one diamond but three, set elegantly on a simple band of white gold.
I'd never known generosity like it, nor extravagance. It was dizzying, intoxicating. He would bring things home for us, for Kate and me—a bikini for me, a pair of sandals for her, a silk wall-hanging, a Mixmaster, a silver cocktail shaker, theatre tickets, a Swedish glass vase, pewter candelabra. I had to tell him to stop.
'You'll be coming home with frankincense and myrrh next,' I said. 'It's all getting ridiculous.'
He laughed at me, but did as I asked. He could see my worry about Kate: her world had been turned on its head and this giddy shower of presents was doing nothing to restore a sense of reality to her life.
*
So we all settled down, more or less. I continued to work in the partnership with Bea; Kate continued to go to school; Max continued to pursue his mysterious financial occupations.
Little by little, we made Rain the house we wanted it to be. We added pictures, ornaments, rugs. The pool was constructed and we admired its pale brilliance; Max employed the most expensive landscape gardener in Melbourne to encircle it in bright gardens and a tempting, shaded patio.
Summer came and we swam daily. I swam, too, though I'd never been much of a swimmer or a sun-worshipper; but I joined Max in relishing the whole experience: the clean cool splash of the water as we dived in, the ice-clinking drinks he made for us as we lounged on the long cedar chairs, the slow twilights.
It was rather like entering the heady ambience of an exotic travelogue, like seeing ourselves up on a wide and colourful screen across which privileged people, glistening like olives, gambolled expensively.
And during the short, warm summer nights we tumbled into bed and made love, Max's long lean fingers travelling across my body sensitively, imaginatively, always with a planned and steady vision of my pleasure, of our pleasure. Sometimes it was intense and direct and shattering; at other times, a slow journey towards the concentrated passion of our delight. He laughed at me; he said I had never been awakened before.
It was true. I came to Max a virgin in everything relating to sex except its actual prosaic progenitive happening. I didn't know about foreplay or afterplay. I didn't know about play. I didn't know about teasing, or timing, or waiting, or holding out, or submitting, or frenzy or peace or anything.
I had never understood the theatre of sex, its drama, the frisson preceding it or the harmony concluding it. Nobody had ever done these things with me before, and I was left astonished and breathless and disbelieving and grateful. All my life I'd been frigid, but nobody had told me; nobody had explained it to me. I'd mistaken the tepid enthusiasm Steve had sometimes managed to arouse in me for the real thing. And suddenly I had arrived in a new life, a life in which I was as new, as freshly awakened and untouched, as everything around me.
For a little while at least, life proceeded in this unbroken golden string of days, this peaceful meandering, isolated from the rest of the world. I remember we moved into Rain in December, just before Christmas. I remember it so clearly because Christmas was our goal, and we made it with only a few days to spare.
After Christmas, I recall it as if the months stretched out endlessly in the hedonistic vein I have described. It can only have been a matter of three or four weeks, however: we were all on holiday during January and it was January that gave us this magical spell. Then the spell was broken: there came a telephone call from Zoë, who had assiduously ignored me since I had left Steve.
'It's Mum,' she said, without preamble. 'Yes?'
'She's dying.'
'We've been told that many times these last two years.'
I was suspicious. Zoë had already made a number of claims on me by invoking our dying mother. I loved her, too; I cared that she was dying, of course I did. But I was resentful of all the emotional blackmail Zoë had already tried, and I was wary of sudden claims of impending death. It was also true that we had been on a more-or-less permanent alert, she and I, like people on a platform expecting a train that never comes.
Five years earlier, the doctors had told us our mother had a maximum of three years to go; two years ago, that death was imminent. Yet she continued. My father, a big strong man who looked as if he would last forever, had died many years before with no warning and astonishing speed. Frail and small as she was, the more murderously the cancer raced through her body, the more stubbornly my mother clung to life. When this happens, people say admiringly of the ill person that he or she is a real fighter.
I'd never thought of my mother as a fighter. Where I was concerned, her mode of fighting seemed to be to burst into tears and tell me I was wrecking her life, which I didn't really count as fighting so much as underhanded and manipulative tactics designed to wear me down through the unwelcome accumulation of guilt. They were successful tactics, too, in so far as guilt was certainly generated. But I had vowed that I wouldn't change my life because of them. And I dreaded deathbed demands, weighted with all the unfair burdens of guilt and love and expectation.
'I don't know what's wrong with you,' said Zoë, crossly. 'Of course she's dying. We know that. We know that, yet you haven't visited her since before Christmas.'
'Firstly, we know that she's dying, but not that it's about to happen. Secondly, when my family's treating me like a bloody leper, why should I feel as if I have to visit them daily?'
'She's in hospital.'
'She's been in hospital before.'
'This time it's different.'
'You said that last time.'
'Well,' said Zoë. 'Ignore me, then. Take no notice. Go ahead and do exactly as you please. Just don't blame me when I ring you to tell you she's dead. When I ring and tell you our mother's passed away, don't tell me you couldn't help it. Okay?'
That is the kind of vicious stab at which Zoë has always excelled.
When I visit the hospital (which I do the following day), I can see that Zoë is in fact right. There is a different look about my mother, a new look of being dragged down, rubbed away. Somehow her features seem blurrier, her face sunken. She lies in the narrow metal bed, apparatus snaking around her, up her nose, down her side. She looks tiny and apathetic. I bend over to kiss her and she turns her cheek, with slight impatience, as if to repel the caress. I feel hurt.
'How are you?' I say, searching around for a chair that isn't piled high with books and Kleenex boxes and bedjackets. (Does anyone in the world apart from my mother still wear bedjackets? I wonder.)
'How do I look?'
I know as well as the next person that to ask a dying person how she is mightn't sound smart. But you have to say something, don't you? It's a formal expression designed to communicate interest and concern. It's a way of saying: _Here I am, and I care about you, and I'm not sure how to negotiate all of this_. _So I'm asking you how you are. I know it's a dumb question, but you're meant to answer it politely._ If I know this, why doesn't she?
I've brought some red roses, which I lay on the tray straddling her bed. I know that she loves red roses, and I've gone to some trouble to get these.
'You'll need to get a vase,' she says, petulantly. 'The nurses say they haven't got time to be arranging flowers.'
'Thank you for the beautiful flowers, Isabel,' I murmur to myself, looking about for a vase. I know it's rude of me, bad-tempered, but the provocation is great. A spasm passes over my mother's face; I'm not sure whether she's in sudden pain or whether she's heard my resentful muttering. Heaven knows, when I actually want her to hear things, she's always too deaf.
Why are there never any empty vases in hospital rooms?
'There's a room down the corridor,' says my mother.
I glance out the door. The corridor stretches endlessly on both sides.
'Which side?'
'How would I know which side? Have I ever been there? Do I look as if I can jump out of bed and go wandering down the corridor?'
I grab the flowers, stomp out of the room and mooch down the corridor. I can't find anything that looks remotely like a room where vases might live. Nobody shows any sign of wanting to help me.
God, I hate hospitals.
Predictably, the vase room when I eventually discover it is down the other end from the direction I've been travelling. As is always the case, none of the vases is the right size for my roses.
I take my time arranging them, in a vast receptacle clearly meant to accommodate small trees, and hope that by the time I get back someone else will be visiting my mother, so that I won't have to talk to her and can leave early. This makes me feel ignoble and selfish and therefore even guiltier.
When I return, I'm still the only visitor. I plonk the vase on the shelf, displace some jetsam from a chair, and sit down.
'It's too _big_ ,' she whispers, in evident anguish. I decide to ignore this.
'It's not such a bad room,' I say. 'You've got quite a good view.'
'Have I? It's a pity I can't sit high enough to see it, then.'
'At least you've got a room to yourself.'
'I get lonely.'
She and I both know perfectly well that, if some poor unfortunate were to be sharing the room with her, her complaints would lift the roof. So I disregard this play for sympathy and try another tack.
'Is the food good?'
'I can't taste it.'
'Why's that?'
'It's the medication. It makes everything taste like cardboard.'
'That's a pity,' I say, vaguely. Conversation lapses. I gaze out the window (she's right, in fact; she can't see it, and that's a shame: it's a pretty view, of big, old oaks and smart, cosy terrace houses) and wonder if grumpiness is an automatic by- product of the dying process.
'So you're living with him, then?' says my mother.
I'm startled. 'Sorry?'
'With him. With that man. You're living with him.'
'Max. His name is Max. Yes. I'm living with Max.'
She folds her lips and another spasm jumps over her face. 'Steve visited me this morning,' she says.
'Why tell me?'
'He's a good man, Isabel. I hope you don't live to regret what you've done to him. I hope you don't live to regret what you've done to your little family.'
'What you mean is you're damn sure I will and you'll be pleased if I do.'
My mother closes her eyes and turns her head. 'You never would be told. You never would be told anything, would you? You always had to do it all your own way, didn't you? You always were a wilful, artful child. A cold, uncaring child, you were.'
For someone who's dying, she spits the words out with extraordinary clarity and energy. Clearly, she doesn't mind that they hurt me, that they're unfair.
It's not long before I leave, feeling under all the circumstances that my visit hasn't been a great success. As I bend to kiss her I see a tear slipping down her papery old skin. This is clearly not something I can do anything about and I beat a rapid retreat, sighing in relief as I escape the hospital and find safety in the sunny day outside.
Zoë rings me that night.
'Did you have to upset her like that?' she asks, furiously.
'Excuse me?'
'Can't you ever just be nice to her? She's dying. How many times do I have to tell you that?'
'I didn't do a thing to her.'
'She was crying when I came in.'
'So it has to be my fault?'
'It was your fault. She told me it was your fault.'
'What did I do?'
'You snapped at her. She said you bit her head off.'
' _I_ bit _her_ head off? She didn't happen to mention some of the remarks she made to me?'
'She's dying, Isabel.'
'I have that message, actually. Loud and clear. I don't see that dying gives you the right to say whatever you feel like saying. She always was a tactless cow and dying isn't improving her.'
I don't really mean to say anything like this: of course I don't; but by this stage I'm seriously upset and I simply don't see why everything is always and inevitably my fault.
'What a bitch you are,' says Zoë. 'She's in the most terrible pain. Why should she be thinking about your feelings?'
'It'd certainly be a change for anyone to do that.'
'Isabel, don't you ever think of anyone but yourself?'
I slam the phone down. Most conversations with my family are finishing with the slammed phone these days.
The next few weeks are not much fun. Our holiday mood, abruptly shattered, returns only to mock us with its memory. Kate goes with Steve to visit her grandmother and creeps around the house with a stricken look for days afterward, glancing at me reproachfully every so often when she thinks I'm not looking.
Everyone in the world is against me. Except for Max. Max is delicate and considerate; he brings me bunches of flowers and small gifts; he drops gentle kisses on my head; he makes tender love to me until I tremble with the tense joy of it. He takes me out to dinner. He obviously thinks I endure a lot from my family, and puts himself out to compensate for this. We all go back to work. Well, I go back to work. Max says he goes back to work, but it's hard to tell the difference, so far as I can see. Kate spends hours in her bedroom and says she's reading texts in preparation for the new school year. Maybe she is.
I go twice more to visit my mother. The first time Zoë is there, too, and monopolises the conversation to such an extent that I might just as well not be there.
The second time, she's asleep. My relief is overpowering. I sit quietly, so if Zoë pesters me I can say I stayed for a decent time; and, after a little while, I find myself observing my mother's face.
I hadn't realised how old she has become—how old the cancer has made her. I do a couple of sums in my head and confirm that in fact she's only sixty-three. It's terrible, that she should look so old, that her face should have collapsed and that her bare arms are so skinny and crêpelike.
I remember when my father died, but that was a heart attack and it came out of the blue. Better to do that, I think: better to take everyone by surprise; far better than this apparently endless degeneration, this cruel, slow, deteriorative process that eats one up so. I've never watched anyone dying like this, bit by bit. Little by little.
Will I die like this? Will a cancer eat me like this, and will Max loyally visit me when I lie in hospital? Will he bring me roses? Will I snap at him and tell him to find a proper vase? Will Dominic visit me? Will Dominic sit by my bed and express penitence?
Perhaps Steve will come, and tell me he now appreciates the depth of his deficiencies, his perfidy, and comprehends finally why I had to leave him, why I had to escape from the hell our marriage had turned into.
I'm tired and depressed, and these are affecting pictures. I become aware that I'm crying and rummage in my bag for a tissue.
It's at this point that the door opens and Zoë enters. Her belligerence on seeing me immediately dissolves when she realises that I am weeping: she drops to her knees beside me and envelops me in a hug, a massive hug, a benevolent, loving, forgiving, big-sisterly hug.
'Oh, Minky!' she says. 'Oh, Minky, dearest Minky!'
So I am reinstated—temporarily, anyway—in Zoë's good books, and we have a nice cry together. It doesn't seem necessary to explain to her exactly why I'm crying; and I'm not sure that I could do this in any case. Perhaps it really is my mother I'm grieving for. In fact, I'm sure it is. Sorrow works on us in complex ways, after all; and my mother's weakness, the corrosion of her body, its gradual decline, the knowledge that the end is on its inexorable way—these have all powerfully distressed me. Possibly I've even been rather brave, to withstand my grief so steadfastly thus far.
In any case, Zoë's mollification is for the moment complete. We mop up our tears and sit silently for another ten minutes. My mother half wakes then, but she's blurry and confused, and seems not to know with any certainty who we are, and soon drops off again. Softly we tiptoe out, and part in mutual sisterly amicability.
Zoë rings me the next day, and tells me our mother is dead. I know it sounds absurd, but it comes as a shock. I hadn't expected it so soon. For so long we had been reminded—frequently by my mother herself—that her end was approaching, momentous and inescapable, that it was hard to realise that the cataclysm was no longer imminent. Suddenly, swiftly, it had happened, and she had gone.
Zoë made most of the funeral arrangements: she consulted me about various details and indeed I accompanied her to the undertaker, but it was clear that she was in her organisational element and I was definitely superfluous. I didn't mind.
I couldn't think of worse things to do than choosing a coffin, deciding what sorts of flowers to decorate the coffin with, working out what music to play. Zoë enjoyed all of it and it obviously gave her solace. I let her fly solo. It was an efficient funeral and I was grateful to her for managing it so competently.
The goodwill between Zoë and me did not last, and it was not long before she returned to the same accusations and the same blistering diatribes. I figured in these alternately as Jezebel and Jill the Ripper, and either way they became tedious.
I was having enough difficulty in coping with my mother's death. It wasn't that I missed her. It was that I didn't. I was depressed and upset, but not because I had lost a loved and loving mother. I suppose she did love me, and I suppose I did love her, but it was hard somehow to recover that love, to recall it fully and experience it again as part of what people like to call the grieving process.
I kept pushing away the knowledge of her death; I kept thinking that I would deal with it later, when I could accustom myself to it and think about it properly. But when I tried to think about it properly, I found I couldn't.
Questions of death, questions of life. How they torment us. Dominic as a child was fascinated by the bad banksia men and adopted the term 'deadybones' until it drove us mad. He made it the subject of one of his rhymes, along with his old teddy bear Fred.
_Ned the Red shot Fred the Ted_
_in the head_
_as he lay in bed._
_Beddy-byes, Teddy dies;_
_Teddy bones all deadybones._
_Ready, steady, Teddy, go!_
He would chant it again and again, capering and spinning like a dervish, shouting the last line in explosive glee, grinning evilly.
'Do you ever worry about Dominic?' Steve asked me once, tentatively, after one of these performances.
'No,' I lied.
'You don't find that teddy rhyme of his...well, a little disquieting? He's only six, after all.' Steve scratched his head, regarding me with ill-concealed anxiety.
'No,' I said, calmly, though I agreed with Steve, deep down.
It _was_ disquieting. But, now, as I look back, I think it might have taught him to understand death, not to fear it, to live with it. I couldn't live with it, not at any rate with my mother's death. Perhaps one needs to be a touch disquieting to survive. Perhaps being disquieting is something to be fostered.
I actually tried to make myself cry, by hauling out all of the best memories I had of my mother. I remembered, for instance, the time Zoë had bossed me about the garden. I suppose I was about five, which would have made her ten. I think we had been given a children's gardening set; or perhaps it was only Zoë's. There was a little, wooden-handled rake, I recall, its prongs painted red, and a blue spade. Somewhere there is a photograph of Zoë holding these items, smirking.
Our father had dug us a space in the backyard so that we could plant our own flowerbed. He went to a good deal of trouble: the bed was kidney-shaped and beautifully edged. I can't remember now what the cause of the quarrel was, but perhaps we wanted different flowers, or perhaps Zoë regarded me as slave labour and forced me into too much digging and raking.
At any rate, it all ended in tears and I recall my mother sweeping me, sobbing, into her arms and bearing me off. It is this feeling of being suddenly swooped upon and borne off, a shred of dandelion upon a great maternal gust, rescued, safe, that principally stays with me.
As I clung to my mother, startled by her action but relishing it, I can remember a gush of emotion for her, a kind of grateful daughterly access of affection, which I now tried unsuccessfully to re-create.
Recollection of other, similar incidents was equally unavailing: I didn't understand why. One loves one's parents: of course one does. One is sad when they die. Isn't one?
I wanted to grieve; I wanted to mourn. I wanted to deserve the kindness and mute concern, the slightly deferential consideration people like Bea and Dawn were extending to me. The discovery that I couldn't plunge myself into these processes made me abstracted and bad-tempered. I tried to explain this to Max.
'I don't feel it as I think I ought to.'
'Is there an "ought" about how you feel grief?' he asked.
'Possibly not, but even I feel there's something unsatisfactory about my response.'
'Why?'
'Well.' My brain was sluggish.
It was late, time to go to bed. We were sitting in the living room, on the milky leather lounge suite with its long curves that were somehow both sumptuous and pristine. I picked up the meteor stone from the coffee table and rolled it from one palm to another, absent-mindedly relishing its coolness and density, its clean spherical perfection.
'It's so hard to know how to put it, Max. I expect to feel sorrow, and I look inside myself for the place where the sorrow ought to be, where it's kept, so to speak, and I can't find it there.'
'What do you find instead?'
'A kind of blankness, to tell you the truth. Almost an emptiness.'
'Mightn't that be an aspect of sorrow?'
'I don't know? Might it?'
'Tell me, my beautiful Belle, why does it worry you so?'
'Because I worry that I'm a monster,' I said, hearing my voice break. 'Because I worry that if I can't properly register grief, I can't properly register love, I can't love. She was my mother, for God's sake: why can't I feel her death more, like Zoë does, like Kate does? Even Dominic is more upset than I am, Max. I ought to be inconsolable; I ought to be bursting into tears; I ought to be devastated, and I'm just not.'
He came and sat beside me, and held my face in his hands. 'Do you love me?'
'You know I do.'
He kissed me, warmly, tenderly. 'You are the most loving, most caring, kindest, dearest woman in the world. I adore you. You mustn't do this to yourself.'
That seemed to terminate the conversation. We went to bed. We didn't make love that night, but we lay like spoons, me fitting into the contours of his lean body, him holding me. I fell asleep like this.
He brought me such peace, such confidence.
He brought more than that, too. He came home one day, perhaps a month after my mother's death, with a large white box. He set it on the kitchen bench and looked at me with his beguiling crooked smile.
It had been a rough day at work: I was peeling potatoes, I think, and drinking wine, and trying to overcome the dragging depression that continued to tug me downwards, some days.
'Where's Katie?' he asked. They often attached the suffix to each other's names. She was Katie; he was Maxie. I don't know why.
'In her room,' I said, eyeing the box and feeling a faint thud of disappointment that it evidently wasn't for me.
He called Kate and she came running down the stairs.
'Now,' he said, in a considering way. 'What are we going to do, here?'
A faint scuffle came from the box and Kate's face lit up. 'A pet!' she cried. 'Maxie, have you bought me a pet?'
'Well, now,' he said, stroking his chin.
Max loved these situations. He was never happier than when he was opening his wallet and throwing hundred-dollar bills around. Partly this was a matter of pure generosity; partly it was a case of an innate love of splendour, of luxury, of abundant surprise and lavish gesture.
'Well, now,' he said, pulling a comical face. 'There's a bit of a problem.'
Kate and I laughed and he grinned back at us.
'The thing is, you see, I have a present and I'm not sure who it's for.'
'Show it to us,' crowed Kate. 'If you show it to us we'll work out who it's for.'
'Ah, no,' he said. 'It wouldn't work, you see. You'll both want it.'
He was teasing Kate, but his eyes sought mine and smiled in the manner of someone who was sharing a superb and immense secret with the only person who mattered. He could always do this to me; he could always make me feel that everyone else in the world was peripheral: I was the core and the heart and the centre of everything for him.
Kate's excitement was becoming uncontrollable, and he started to lift the box's lid, only to pause and say anxiously: 'You'll have to share it, mind. You'll both have to share it. You'll have to borrow it from each other.'
And there he was, Borrow, a golden Labrador puppy with impeccable lineage and huge floppy paws and a heart-melting smile. He scrambled out of the box and into Kate's plump and eager arms, wagging his tail and whimpering with enthusiasm and wriggling and licking her face while she issued rapturous little screams.
Borrow was to become so integral a part of our lives that it was hard to believe we had not always had him. Noble and stupid, he treated Max and Kate with all the adulation his giant heart could muster, which was considerable. They were his favourites; me he tolerated.
And yet he has finished up, happily enough, with me and with neither of his heroes; it is with me that his days will amble to what I hope will be a benign and pain-free end. Old and venerable now, with white whiskers, increasingly cloudy eyes and a gammy hip, he limps after me and good-naturedly pretends he doesn't remember the dark side of me. Do dogs remember? I am sure Borrow does.
It wasn't only the dog that arrived in my life with such panache, such bold munificence. Addicted as he was to the grand entrance, the dispensation of gifts, Max kept on spoiling himself and me. He walked in the door one evening with a large, flat, silver box, an extravagant crimson ribbon jauntily tied on top of it. I eyed it inquisitively.
'I bought you a wedding dress,' he said, all crinkly grin, presenting it to me.
It was so like Max. No preliminaries, no questions. There was no question, of course. No proposal. We were head over heels, still; then and always. We both understood that we were partners forever, that we couldn't live without each other.
'Who am I marrying?' I asked, undoing the box. 'When?' He laughed.
It was a floaty, fine thing: layers of shimmer, lilac and aquamarine, crystal and snow. It had drifted right out of the silky expensive-smelling pages of _Vogue_ into my delighted arms. The material was so delicate you could have passed the entire dress through a wedding ring. He had bought shoes, too: fragile jewelled sandals, with soft leather soles and sleek gleaming straps.
He was extraordinary like that: how many men are there in the world who could walk into a dress shop or a shoe shop and choose so unerringly, with such imagination and confidence? He always knew sizes and colours; he understood my preferences and my style and everything about me.
'They're not my colours,' I said, marvelling, letting the fineness of it slip through my fingers, like bright shadows of silk, almost no substance to it.
'They're not the colours you usually wear,' said Max, dropping a kiss on the back of my neck. 'It doesn't mean they're not your colours.'
'It'll make me look sallow.'
He snorted. 'Try it on and see.' I did. I didn't look sallow.
'I thought a beach wedding,' he said, while I twirled in front of the mirror, craning to see different views, liking them all. 'Twilight.'
'When?'
'Whenever you say.'
I tried to describe all this to Zoë. I made a real effort.
'I couldn't bear it,' she said.
'Couldn't bear what?'
'Such an overbearing attitude. I don't know what's happened to you, Minky: once upon a time you wouldn't have put up with being pushed around like this.'
'He's not pushing me around,' I snapped. 'He doesn't push me around.'
'I couldn't bear it. It's not natural, for a man to act that way. How is it he knows so much about women's clothes? You ask him that.'
'You're just jealous.'
She laughed, nastily. 'Jealous? Of you and your con man? Don't be silly.'
'I'd be jealous,' I said, spite getting the better of me. 'If I was married to Henry, I'd be jealous of just about every other woman on the face of the earth.'
She gave me a wounded look. 'Anyway, those colours don't suit you.'
Well, it's no wonder I didn't invite her to the wedding, which happened about four weeks afterwards, just as he'd said, on a fine, still evening, on a beach down on the Peninsula, near Point Leo, at twilight.
I got Bianca Crawford and her husband to be witnesses; no one else came. I didn't want anybody else. Bianca was about the only one of my acquaintances not to resist Max. We had coffee once, and I told her about how unfair everybody was, and she laughed and leant forward and tapped my cheek. Bianca was half-Italian and when she felt like it she would become very European.
'Don't you worry about it, _cara_. You go and live your own life. Don't you give a damn about any of them, old killjoys.'
'You like him, don't you, Bianca?'
She rolled her eyes. 'Like? Like? I fancy him, my dear, and so do they, if they were only honest. Oh, but he is a dangerous man, you mark my words, Isabel. _E uomo molto pericoloso_ , _cara_. Believe me. You take care with your Max.'
I laughed, knowing how safe Max in fact was. I could see why she called him _pericoloso_ : it had to do with the sharp edge of his sexuality, the vertiginous quality of it, the physical frisson he generated, the quiet sizzle of his mere presence. But he wasn't really dangerous at all. I'd never felt safer with anyone in my life.
So Bianca was delighted to be invited. She used up a roll of film with languishing shots of us standing in our finery on the flat white sand, gazing together at the sunset, walking hand in hand into the golden beyond, kissing against a tangerine sky. She sent us copies later and Max said they looked like ads for Jim Beam and coke.
Max had a friend who was a celebrant. She was a plump, motherly lady who wore a rather lovely, black tailored pant suit and read a beautiful service; and she kissed me afterwards with, it seemed to me, real feeling and affection.
We drove to a swanky hotel in Portsea and got tipsy over dinner and made love several times in a gorgeous suite with a four-poster bed. It was so romantic I almost didn't feel married. That's a laugh, now.
I kept my name. Weaving is my own name, not Steve's. At the time of our marriage, it was unusual for a woman to retain her maiden name, but Steve's surname is Bell. Even so rigid a conservative as my mother could see that I didn't want to be called Isabel Bell. Steve's mother wasn't so sympathetic, but her name was Marjorie, so the problem hadn't presented itself for her. And by now, of course, I had a professional name—not a huge one, but in certain circles well known enough for me to want to keep it.
Bea was as difficult about it as everybody else. I told her I'd be away on the Monday (married on Saturday: two days' honeymoon didn't seem unreasonable to me). She asked why and I told her. She sniffed disapprovingly.
'Oh, Christ, Bea!' I said. 'You're my oldest friend as well as my partner. Don't you go and put me in the doghouse, too.'
'It's not a matter of putting you in the doghouse. It's a matter of seeing you stuff up your life, your husband's life, your kids' lives.'
'Steve's not my husband any more. The divorce is through. I haven't stuffed up anyone's life. I'm just living my own.'
'Izzie,' she said, stretching out her hand. 'Izzie, just living your own life isn't possible. You know that. Everything we do touches on other people, on other people's lives. You _know_ that. It's so awful to watch. I just find it infinitely depressing to see you permanently infatuated with that—that— _gigolo._ '
I shrugged and turned away. I was so wounded and so angry that I couldn't trust myself to say anything.
It hurt a lot, of course, that everyone was so harsh to me, so determined to judge me, not to understand. Sometimes I felt oppressed by the malice they all seemed suddenly to be sprouting, by the burble and hiss of spite. At first, as I've said, I tried to explain. To Bea, to Zoë. I could no longer explain to my mother, since she was dead, but she hadn't wanted to understand, any more than they did.
But I was happy; I was incredibly, wildly happy: that was what they couldn't understand, how essential that happiness had become to me. Happiness bubbled through my blood: I felt I had little neon glows flickering like Christmas lights through my veins. When I walked down the street, when I went to the shops, it seemed to me everyone must notice and admire my incandescence.
*
Yet, although I was so happy, it was at this time that the Lost Dream started. This is not something I can explain. I woke one night next to Max, perhaps two hours after making love, in a cold sweat, weeping.
I had never before woken from a dream, from sleep, with actual tears sliding down my face. The dream had begun as it always has done, subsequently, with a stroll through bushland, walking through the dull silver gleam and tangy air of eucalypts. I am content: it is warm; I am secure; the bush is beautiful, restful. Magpies chortle; rosellas dart and chatter.
But the air turns colder and the path more winding and the trees more swarming, pushing their way over the path. I walk on, knowing I have to get to the end of the path, knowing I have to reach a certain point. But it is colder and darker all the time. I panic; I try to turn back, but the seething trees block my way. And I cry _cooee_.
It is what we were taught to do as children, if we walked or played in the bush—at regular intervals, to stop and shout _cooee._ It is what Australian children have always been taught to do. Well, that is the myth, anyway. Most Australian children these days are brought up in the synthetic environment of the suburbs: they may see a gum tree in a park occasionally but they wouldn't know real bush if they fell over it.
But there are old legends of the bush, old legends of the children who were lost in it—some whose bodies were never found. The bush scoops a wandering child into its heart quicker than you would believe. So, if you are lost, you cry _cooee_. And back, strong and sweet, comes the answering _cooee_ of father or brother or friend. If you remain within cooee, we are told, you will never be lost.
And now I call _cooee, cooee—_ but I have gone beyond cooee; there is nobody to hear me; nothing around me but the teeming, muttering bush that could contain anything, anything at all.
I feel rather than hear a low vibration emanating from the crowding leaves, which quiver above me, sage-green, pale silver, thin as paper, thin as skin. I am so cold I can hardly breathe. The wild cackle of kookaburras explodes in my ears.
My heart's in a flurry; it's like a lunatic, plump pigeon hurling itself against the cage of my ribs, heavy, panic- stricken.
_Cooee_ , I cry, and again, in accelerating desperation. _Cooee_. No one answers; no one is there to answer; I am alone in the world. Fear possesses me utterly. I wake, weeping.
Max was not generally speaking a patient man, not perhaps somebody who readily sympathised with psychic ailments and traumas of the imagination; but he was always tender with me after this dream. He used to call it the Lost Dream, capitalising it somehow, as if to acknowledge its searing power over me, its horror. I still have the dream sometimes; and now it doubly sears me: when I wake from it, sweating and shaking, I miss his hard arms around me, his murmur in my ear, his long, kind fingers stroking my neck, soothing me.
I miss him so. I miss his kind-heartedness, his gentleness, his generosity.
How they disparaged him, my friends and relations! They insisted—in direct contradiction to all my evidence—that he was dubious and underhanded. Crooked, grubby.
'What have you put your name to?' Bea asked me one day at work.
'What do you mean?'
'Have you signed papers for him? Have you stood surety for him, or anything like that?'
'You're referring to Max?'
'Come on, Izzie: you know I am.'
'I haven't, but there's no reason why I shouldn't, and if he asks me to I certainly will.'
'Listen, love,' said Bea, sitting down on the chair in front of my desk where Max had sat when I had first met him. 'Izzie, just don't sign anything, okay?'
'You all drive me mad,' I said, furiously. 'Completely mad. He's a lovely, lovely man, and I'm the luckiest woman in the world to be married to him, and you all carry on as if he were a serial killer.'
'It's up to you, Isabel. I'm telling you. Don't sign anything.'
'Why shouldn't I? Why do you all behave as if there must be some deep, shameful secret attached to him? I'm telling you, he's the most straightforward, transparent, honest man I've ever known. Leave me alone, Bea.'
Bea did her heavy sigh and trudged out.
'Sign these, ma Belle, my sweetheart,' said Max one night not long after this conversation, brandishing a sheaf of papers and dropping a kiss on my forehead.
'What are they?'
'A surprise, my love. A small surprise for the belle of my heart.'
I unfolded the papers. 'Can I look at them?'
Max burst out laughing. 'Is it likely I'd ask you to sign anything without looking at it?'
I leafed slowly through them. They were legal papers, title papers.
'Max,' I said. 'Max, what are you doing?'
'I'm giving you Rain.'
I couldn't believe it. 'What do you mean?'
'Look, we're married now, aren't we? We should share things. All my worldly goods, and all that.'
'But this isn't sharing. This is giving.'
'I love you, Bella. I love giving things to you.'
'Max, this is ridiculous.'
He spread his hands. 'You're going to beat the truth out of me,' he said, shaking his head mournfully. 'The fact is, I'm only doing it as a tax break.'
'Max. Darling Max. Please be serious.'
'Well, it seems a shame to be serious when I don't have to, but I'll do my best for a little while. Bella, my love, this is our house; we live in it together, but it's all in my name, and you had so much input into it that it's absurd for you not to own it. Rain is your invention, my darling. Your creation.'
'We did it together.'
'Yes, but you were the guiding spirit, the inspiration, the genius of it all. You know you were. This is only a logical extension of that fact. The other thing, my sweetheart, if I'm going to be perfectly honest, is that it actually is a good tax move for me. My accountant tells me I ought to cut down on my assets. He's advised me on this one. I always take my accountant's advice.'
'But you can't just give something away like that, can you? Aren't there tax penalties to gifts, too?'
'Arthur says to trust him,' said Max, grinning. 'As I always do. He'll manage it all for me. But in the meantime we need you to sign these. Don't do it now, but read through them so you're sure you understand what's going on, and tomorrow, if you can take an hour or so off, we'll wander down to Arthur's solicitor and talk about it so you understand it all and we'll make it all legal.'
He was set on this, fixed on the course he had described. I tried to dissuade him, at first. I was uncomfortable about accepting so enormous a present. But Max could be stubborn, when he wanted something badly enough, and it became apparent that this was something he wanted. We signed the papers; the title was handed to me. Rain became mine.
I told Bea, after it had happened, of this transaction. I must say I told her with triumph, with a strong flavour of _I told you so_ mixed with _How could you say those things_. She was nonplussed, though she tried not to show it.
'Well, good,' she said, staring at me across her desk. 'Good. I'm pleased for you, Izzie.'
'So you were wrong. Aren't you going to admit you were wrong, Bea?'
'Nothing's proven about my rightness or wrongness. Not yet.'
She was unrelenting. I didn't care—or not much, anyway. I had Max to comfort me. Max, and Rain. I'd never owned property before. When Steve and I married the house had been in his name. I'd never questioned that, though now I find it hard to believe that I could have been quite so pliant, so yielding to the master's voice.
I wandered through Rain now, familiarising myself to the curiousness of possessing it. I ran my hand along the jarrah banister and gazed out the study window, feeling a kind of new richness—a material, selfish richness, which complemented my inner and less selfish spiritual richness. _All this is mine_ , I said to myself, in active and questioning wonderment at the strangeness of it.
Max never mentioned it. It had been done; the gift had been given, and that was the end of the matter so far as he was concerned. We lived in the house together as if it belonged to us together, without reference to who its actual owner was.
Sometimes I would wake before Max did, and I would lie there studying him, not just his face, but the contours of his body. He was a lean man, tanned and sinewy, strong and incisive in his movements. His presence dominated: it was impossible to be in a room where Max was and not know he was there; and somehow this innate and effortless habit of authority made him seem all the more vulnerable when he was asleep. Sometimes he would lie on his back, one arm flung up so that the inner arm was exposed. I found its whiteness, its susceptibility, intensely moving. He had curiously pointed elbows. Is it ridiculous to swoon at a man's elbows? I dare say.
I was obsessed by him, infatuated with him. I freely admit it. But it wasn't merely obsession or infatuation. It was a dense, profound and lasting love. Luminous and expansive, it had invaded every particle of me. He was the love of my life.
The supposed love of Kate's life, Gavin, hove onto the horizon when Kate was, I suppose, eighteen and studying not very hard at a fine arts course. Fine arts was the kind of woolly thing she _would_ be interested in. He was a postgraduate, studying for his doctorate, which in Kate's eyes, I suppose, invested him with some _cachet_. He was unimpressive from the start: tall, gangly, he would come to pick Kate up and (because she was never ready) would be trapped in conversation with Max and me.
Max was especially good at this kind of encounter: easy and fluent in his behaviour and his language, he was adept at the kind of genial bonhomie that comes in so useful in potentially strained situations. But Max never really succeeded with Gavin.
He had a quick stuttery laugh, did Gavin; and nervous feet and hands; and he was too long; there was too much of him to occupy any given space, and he was uncomfortably aware of this. He was like a giant fifteen-year-old: he'd never properly matured, never got past the gawky self-conscious stage. If he'd had any physical coordination worth talking about he would have been great at basketball; but he hadn't. His head seemed to bob around in a manner disconcertingly separate from his angular shoulders. He had no social grace whatsoever. I found him painful to be with.
I couldn't understand what the attraction was. Kate had had boyfriends before, and to a man they had manifested more appeal than Gavin. But the liaison persisted and she appeared to enjoy it. Summer was coming on, when Gavin first appeared, and the pair of them spent much time in the pool at Rain.
Kate had always been shy about being seen in bathers, but for some reason (God only knows what) the company of Gavin seemed to confer on her a new aura of physical and sexual confidence: to my astonishment, she bought an expensive turquoise bikini, and freely let herself be observed in it.
Kate has always been plump, and pink and white, too, which are not promising coordinates for bikini wearers. But somehow, on her, it worked. Her body was firm and well- proportioned; she moved gracefully; the bikini was well-cut; and, although she did not tan beyond the palest café-au-lait, her fairness lent her a nymphlike aspect that was quite fetching.
A new Kate emerged, languorous and quietly confident, one whom I shouldn't have been surprised to find swimming naked in the pool on moonlit evenings, pearly and dripping in the silver air, the silver water. Gavin frolicked awkwardly around her, lanky and adoring. Max liked him more than I did.
'He's not stupid,' he insisted.
I raised my eyebrows.
'He's not. Look, you can't be stupid and be writing a PhD on Rousseau.'
'I've known a fair few stupid people who had PhDs.'
'Believe me, he's not one. And he's very sincere.'
'I'd prefer another quality, I think. Sincerity is such a dispensable virtue.'
'He's genuinely in love with her.'
I shrugged. But I didn't spend much time thinking about Gavin, because I didn't expect him to become a fixture in my life. I realised, of course, that they were sleeping together: when he started sheepishly to appear for breakfast, it was not a difficult conclusion to reach. I didn't think much about it, except to register a vague incredulity: it seemed to me going to bed with Gavin would be like going to bed with an extremely large daddy-long-legs.
Around this time, the business—mine and Bea's—started to expand. Bea and I had always managed quite happily, just the two of us; after struggling for a while, we made a small name for ourselves and hummed along very nicely for several years. Suddenly, it seemed, we were making more of a name, and people wanted more of us.
We had seen a gap in the market and had developed a good line in adapting old houses—especially terrace houses—for offices: streamlined, modern offices that nevertheless retained the spirit (and what people like to call the ambience) of the original houses, with their cool, high ceilings and narrow stairwells and elaborate cornices. It seemed our fame had spread. Bea one day had an invitation to quote on a row of old terraces in Sydney.
'It's worth a fortune to us,' she said, leafing through the photographs the developer had sent her. 'Look at this, Izzie—and this. They're great little houses, and it's the Rocks; it's very central. Such good publicity.'
I picked up the photos and agreed. The houses looked as if they were run down, but so far as you could tell they were still solid and well worth the effort and expense required for the conversion.
It was a question of who would go. We hadn't had much interstate work, but when we did it was usually Bea who went: she liked the buzz of it; and she had a brother in Sydney she liked to catch up with. This time, however, it didn't suit her.
She looked hopefully at me. 'I don't suppose...?'
'I don't mind,' I said, musing. 'It does look interesting, and it would be a shame to miss out. A shame not to even try, at any rate.'
In fact I was tempted by the prospect. Steve had always disliked it when I travelled on business: arrangements had to be made for the children and, furthermore, my absence reminded him that he was not the sole provider, the great white hunter, to which condition he aspired.
By agreeing to undertake the task, I was able to demonstrate to Bea that I was now in a more independent and mature relationship, that I could carry my professional share of responsibilities, that Max was not going to jump up and down in a petulant fit because I was away for a few days. Also, to tell the truth, I thought that I should like to show Max that I, too, had pressing business interests; I, too, had a professional life whose demands I was required to meet, able to meet. I hadn't been in Sydney for years and the opportunity was beguiling.
*
So I spent five days in Sydney—and, apart from missing Max, I enjoyed myself thoroughly. Even missing Max had its advantages: I knew that the absence was short, with a defined conclusion and an inevitable joyful reunion, which would be made the more rapturous because of the sting caused by the separation. I anticipated the reunion with such sharpness, such keen-edged imagination, that I could almost relish the period preceding it in the thought of what was to follow.
Bea found other things for me to do—contacts to follow up, an exhibition to attend—and I worked hard for three days and turned into a tourist for the last two days of the week. I travelled on ferries and enjoyed the breezes and the views from the deck; I wandered around the Opera House; I shopped in Double Bay; I had cocktails at night before dinner. It was all consistent with the kind of easy, well-heeled, prosperous existence to which Max had introduced me; and, charmed, I found that I was able to create and savour it myself even when he was not there to share it with me.
I told him this on the phone on the last night of my visit, and heard his deep delighted chuckle.
'What you've taught me!' I said. 'I've turned into a regular lotus-eater.'
'Do you good. I've never known anyone who deserved to eat lotuses more than you do. I'm pleased I've shown you how to have a good time, even if you're having it without me.'
'It's a talent to be fostered, I'm finding.'
'And you're fostering it.'
'I certainly am.'
'Good for you, my darling Bella.'
That night I was unusually wakeful. I switched impatiently from one channel to another on the TV in the opulent hotel to which I had treated myself. (Bea had said the practice could sustain three stars; I stretched it to five.) I raided the minibar. I ate peanuts and tried a can of rum and coke, which I didn't usually drink and didn't much enjoy.
I went through the papers I had collected over the last few days and sorted them into order for Bea. I lingered over some of my sketches. The terraces had been less enticing than I'd expected: something could be done with them, certainly, but their fundamental dinginess was a matter of construction and siting rather than neglect, and possible remedies brought their own difficulties. I didn't think we should take it on, even apart from the travel that was going to be involved in the project.
Although I knew I was going to see him the next day, I missed Max with a nearly unbearable pang. I could not calm my restlessness and the night stretched ahead of me black and bleak. I stared out the window and saw that Darling Harbour, several floors below, was bright with lights and busy with people—people strolling, chatting, dining at the pavement cafés. Small brilliant boats, which I presumed were water taxis, bobbed and slid across the dark water.
As is the way with these modern glossy hotels, I couldn't open the window, which meant I couldn't hear anything. It gave the scene a bizarrely unreal and distant quality, as if I were watching it on television with the mute button on. The lights strung along the waterside were like jewels: it sounds clichéd to say so, but they had the prismatic glint, the twinkle and the lure of jewels.
On impulse I grabbed a jacket and went on down. It felt as if I were doing something immensely daring, something slightly dangerous and even risqué. Steve would certainly have been horrified ( _No, no, lovely_ , I could hear him saying. _You can't go down there on your own. You just hang on a tick and I'll pop on_ _some shoes and come with you_ ); Max would have been puzzled that I even thought twice about it.
It was a mild night, only a flick of cool breeze coming off the water. People were everywhere. It had looked crowded and jolly from my bedroom window: being in the middle of it was at once more and less cheerful than I'd imagined: the buzz of conversation and laughter was intoxicating; but I was alone inside all the merriment, and felt conspicuous and isolated. I sat at a waterside café and ordered a brandy.
The alcohol bit into my veins and I started to relax. The café had a guitarist, a dark young man strumming a guitar. Something about him reminded me of Dominic: not so much his dark good looks, I think, as an expression in his eye, a tilt to his head. I found myself staring at him, pondering almost abstractly the set of his shoulders, the concentration of his closed lips. I was looking at him but I was thinking of Dominic, and was embarrassed to register a few seconds late that he was gazing back at me. I felt colour rush up my throat to my face, smiled apologetically, and transferred my attention to the pavement, the water, the boats.
Five minutes later I became aware that the young guitarist was standing by my table.
'You permit?' he asked. His voice was soft and rich, accented. Spanish? I wasn't sure.
Confused, I nodded, and he sat next to me, carefully propping his guitar on the ground beside him.
'I think I have not seen you before,' he commented, with one of the most rare and charming smiles I have ever seen. 'You are tourist? Not local?'
'I'm from Melbourne,' I said.
I could be your mother, I thought.
I was flattered by his attentiveness, his concentration on me. Something like this hadn't happened to me before. When I'd been young and a likely candidate, I'd been too carefully sheltered. And marriage to Steve had done nothing to loosen the shackles.
I was flustered, but there was something exhilarating, something fascinating, about the situation: here I was, with a son almost as old as this beautiful young man with the warm adoring eyes; and here was he, with his appreciative gaze and his ravishing smile, paying court to me.
A waiter brought him coffee.
'Holiday?'
'Work, mainly,' I said, wondering if this luscious creature was indeed picking me up and concluding that he probably was.
'You are here for much longer?'
'I go home tomorrow.'
He nodded, sipped his coffee. 'You liked my music?'
It was all feeling more and more surreal.
'You play superbly,' I said, politely.
He sipped his coffee again and turned his glorious brown eyes on me.
'I can play for you. Just for you. You will enjoy.'
I started to demur.
He flashed his excellent teeth again.
'I am not so expensive,' he said.
Suddenly I understood what was happening, saw how naïve I'd been.
'No,' I said. 'No. Sorry, but no.'
He lifted his gracefully arched brows, looking more than ever like Dominic. His eyes widened. Bedroom eyes, gigolo's eyes.
'Please go,' I said.
He shrugged and went.
Two minutes later, I left, too. They charged me for his coffee, but I didn't have the confidence or presence of mind to argue the toss. I felt a fool, an absolute fool. Here I had been, preening myself on how attractive I was, how desirable I must be. Nothing but a fool, a middle-aged old chook deluding herself.
I strode back to the hotel (how different from the way I had sauntered out), let myself into my room with shaking hands. The telephone was ringing.
'My beautiful Bella,' said Max's voice in my ear. 'Just ringing to say goodnight, my darling. Together again tomorrow. I didn't wake you, did I?'
'I was in the shower,' I said.
'Sleep well, Bella, my love. I love you.'
'I love you, too. I really do, Max. I really do love you. You know that, don't you?'
He sounded slightly surprised. 'You know it, too, my love.'
'Yes,' I said. 'Yes. You sleep well, too.'
But it was a long time before I slept. The incident had deeply upset me, and I wasn't sure quite why. Partly, of course, I simply felt foolish. There I was, forty-one, silly enough to think that a boy making calf 's eyes at me was smitten. I'd been a different sort of target, and it had taken me too long to realise it.
I think it was also a question of the things Bea and Zoë had said about Max, the way they had both so contemptuously called him a gigolo, the way their contempt had washed not only over Max for being a gigolo but also over me for my evident readiness to succumb to a gigolo. _Gigolo_ , I kept repeating to myself. What an awful word it was, redolent of sleaze and self-interest and corruption.
Max was at the airport to pick me up the next morning. I'd never been more pleased to see him. I felt as if I'd sailed back into harbour, as if I was rocking safely again, lapped, unharmed after a close shave out on the open waters. I didn't tell him what had happened with the young guitarist. There was no reason why I should, of course; still, it felt odd to keep something from Max, to whom I told everything, in whom I confided as I had never confided in anyone before.
Doubly glad to be back home, I nestled into Rain, consciously luxuriating in its beauty and comfort, letting its surfaces and textures, its angles of light and its slanting shadows, soothe me. It was easy to be soothed.
During the day I worked with Bea: the business was booming and, although we didn't pick up on the Sydney terraces, more and more work was coming our way, and we were doing it well and enjoying it. Bea started to speak about employing a third architect. I didn't mind much one way or the other, so long as it was somebody who could be trusted with the name we were building for attention to detail, smooth, unfussy plans, flowing space, subtle planes and clever use of light.
There was plenty of work and I worked hard, but at the end of the day I went home to Max and enjoyed the life I had created for myself. I rang Dominic regularly: he didn't want to talk to me, but that would come in time.
When I look back on this period, I think it was the happiest time of my life. I had descended a little from the dizzy heights of my first wild infatuation with Max: that had been intensely happy, it is true; but it had also been accompanied by much trauma. Now, as we headed into our fourth year together, our life had settled and established itself: we were comfortable together in ways that were essentially domestic, undramatic, snug.
Our snugness was shared by Gavin, who turned up more and more frequently. We got to a stage in which we no longer noticed his presence at breakfast: it seemed as regular and predictable as Kate's own and I poured his orange juice without thinking about it.
It was no surprise when they appeared one night, silly and giggly and high as kites, interrupting our pre-dinner Scotch and announcing their engagement with all the erupting excitement of five-year-olds on their way to a circus. They declared that they intended to marry soon, soon!
I was conscious of a vague, undefined disappointment, but I tried not to show it and I certainly didn't make any attempt to dissuade them. Gavin didn't light up any candles for me, but I wasn't Kate, and in fact when I thought about it I found it impossible to imagine Kate marrying someone I might have liked.
Gavin was at least relatively inoffensive: he didn't put my teeth on edge more than three or four times a day. He did seem genuinely in love with Kate; he was working as a part- time tutor and seemed quite likely to turn into an academic (heaven knew there wasn't much else I could imagine him doing), so there was some prospect of his being able to support a family.
We all drank champagne and hugged and kissed and said how wonderful, and part of me half meant it. I knew Kate was very young, but so had I been when I'd married for the first time. If mothers chose their daughters' partners, after all, the world wouldn't work any better than it does now.
I suppose it was two or three weeks later—and a week or two before the wedding—that I came across Kate throwing up in the toilet.
'Is there anything you think you ought to tell me?' I asked, when she'd got herself cleaned up. We were in the kitchen and she was sipping the peppermint tea I'd made her. Peppermint tea had helped me through the early stages of both pregnancies, and it seemed to be doing the trick for Kate, too: she still looked peaky, but her colour was starting to return, and so was her composure.
She smiled her fey smile and cradled her mug in her hands.
'Well. Yes, I guess so.'
'You're pregnant?'
'I suppose so, Mum. Are you cross?'
'I'm not cross, no. I didn't think you and Gavin had been playing Scrabble up there in your room all night. But do you _know_ , Kate? Have you done the tests?'
'Oh, yes,' said Kate, smiling again. 'It all seems pretty clear. The right colour and everything, you know. The blue line.'
This was incomprehensible to me. There hadn't been a blue line in my day.
'But have you seen a doctor?'
'Not yet. I will do. I've got an appointment.'
'Is that why you wanted to marry so quickly, Kate?'
'Not really. I mean, it isn't like it used to be, is it? Nobody really minds these days, do they? I guess we just couldn't see any point in waiting.'
'And Gavin knows?'
'Well, yes, Mum,' said Kate, looking slightly hurt. 'Of course he does. He's terribly thrilled.'
'Kate,' I said, very earnestly. 'Kate, you know you don't have to get married, if you don't want to. I don't want you to feel trapped.'
She looked astonished. 'Why would I feel trapped?'
'Well, only that if you thought—because of this, I mean because of being pregnant—that you had to get married, I'm only saying, you don't have to.'
'But, Mum. That's what I've been saying. I don't feel I have to get married. I _want_ to get married.'
'To Gavin?'
'Of course to Gavin. There's no other candidate, is there?' said Kate, laughing.
'If you're sure,' I said.
'Of course I'm sure.'
'I just want you to be happy, Kate.'
'I'm going to be happy.'
'Okay. That's fine, then.' I stood. I was running late for work.
She put up her arms to me. 'Aren't you pleased, Mum?'
I embraced her, somewhat awkwardly, murmuring reassurance.
Was I pleased? I wasn't sure. I thought about it as I drove to work. I should have guessed, of course: Kate and Gavin were no more competent to supervise their own contraception than the guinea pigs Kate had kept when she was eight. I wasn't sure if I was pleased, to tell the truth. I was only forty-one. I wasn't at all sure that I was ready to be a grandmother.
To my relief, Kate didn't want a big wedding. I'd had some misgivings about this: two years before, she'd been bridesmaid for one of her cousins on Steve's side of the family. Sarah had gone right over the top on the tulle and the glitter. The bridesmaids had frothed down the aisle like a trio of big white cappuccinos, preceding the large and twinkling pavlova of Sarah in her wedding dress. Kate had been very taken by it all, and I confidently expected a replay in her decisions about her own marriage. It didn't eventuate, however. Whether it was the lingering morning sickness, or consideration for her father's budget, she requested and got something a lot smaller, a lot quieter.
After the wedding, she settled into domestic contentment with the dreadful Gavin, rather as if all her dreams had been fulfilled and this was all she wanted of life. To be pregnant and not quite barefoot.
Well, I suppose I wasn't too far from that, when I married Steve.
They settled into a nasty little house with cheap secondhand furniture Steve helped them to buy; Steve also gave them some old things, things he and I had bought when we were first married. I contributed the nursery furniture, which was neither second-hand nor cheap. I must say, Kate was most appreciative.
She painted the nursery herself, and got me to help her put up a wallpaper frieze with elves and hedgehogs and whatnot scampering over it. We bought a walnut cradle with white lacy sheets, and a gleaming white cot, and a sumptuous navy pram with a suspension that would lull any baby to sleep. I started to dawdle in babywear shops, feeling a maternalism to which I was unaccustomed, and bought the odd growsuit and fluffy toy. It was all rather pleasant.
Towards the end of Kate's pregnancy, late on a Sunday afternoon during which she and Gavin had visited us, I came across Max standing absorbed in thought, chin in hand, in the back garden.
'A penny for them,' I said, snuggling into his side and kissing his cheek.
'Do you still want the pool?'
I was taken aback. 'It's you who wants the pool, not me. I don't mind one way or the other.'
'It's a beautiful pool,' he said, thoughtfully.
'You've always said so.'
'I'm just thinking. There's not much of a garden, is there? Not around this side. The pool takes it all up. We're going to have a grandchild. Well, I know it'll be your grandchild, but I think of it as ours. I'm just thinking, wouldn't we rather have a garden for the little tyke to run around in?'
'I don't know. Wouldn't you miss it?'
'I'm not sure.' He rubbed his chin. 'I'm not sure I wouldn't rather have a bigger garden, you know. Couple of rose beds, perhaps? A herb patch? Maybe a winding path, a big tree or two? Bit of a lawn?'
'I don't mind,' I said.
And it was true: I didn't. Max could do this sometimes: we'd been through it with Rain's sunroom, which at first he'd loved. After a year, he found there was too much lemon in it, and eventually, once he'd worked up steam, the room was completely repainted and refurnished. Suddenly, what had been perfectly adequate turned out to be misguided, a foolish error, unsatisfactory from the start.
So I wasn't surprised that he wanted a garden, and I wasn't surprised when, in the next couple of days, he started to contact landscapers and pool demolishers. Max had never been a procrastinator.
He devoted immense effort to the planning, and, instead of returning to the swanky landscape gardener he'd originally employed for Rain, held endless discussions with Jack, of Jack's Landscape Solutions, recommended by some business acquaintance. Jack was a fatherly man with a ginger beard, which he had the habit of twirling, deep in thought.
Together they pored over diagrams and garden books. Max went out to nurseries and examined their stock, engaging in intense conversations with the nursery staff about anticipated heights and growth habits and conditions and colours and heaven only knows what else. What Max did, he did thoroughly.
And then Sophie was born.
It is impossible adequately to describe how Sophie's new presence, her existence as a new person, affected me. Everything was more fun. I fizzed more; I laughed more. Making love was better than ever; I embarked on an extravagant series of kitchen explorations, buying new equipment, cooking new things, trying new flavours. I attacked the front garden, which it seemed to me needed a lift, and planted dozens of seedlings, most of which died within a week because I forgot to water them. Max came across me humming one day: I was ironing, and I hate ironing. He laughed at me.
'You're worse than Pollyanna. It's like living with Julie Andrews: suddenly the hills are alive.'
I was startled. I knew I felt different, but I hadn't realised I'd shown it so clearly.
'You are utterly transparent, my love,' he said, kissing the back of my neck. 'That tiny little mite has transformed your life completely, and there's no point in denying it.'
Well, I didn't want to deny it. Something melted inside me every time I picked her up, every time I saw her. And relations between Kate and me were so good, too: after a lifetime of tension, we were finally doing the mother–daughter thing.
I started to work part-time, turning up maybe three days a week if they were lucky. Bea complained loudly, and I laughed and said I was learning how to be a granny and she'd have to get used to it.
I drove over frequently and helped Kate to bathe Sophie: peacefully we marvelled together at her perfection, stroking her cheeks, kissing her tummy, cuddling her, giggling like a couple of schoolgirls.
It seemed to me that Kate liked me more than she ever had before. I certainly liked her. She was deft and gentle with Sophie, and sweet. She was willing to defer to me, to ask my advice. 'What should I do here?' she would ask. 'How should I handle this, or that? Did I do this, when I was a baby?' And so on.
New harmonies established themselves, crept into our conversations, our modes of discussion. We conferred over what name I could be called. Nanna? Grandma? Gran? We settled on Gandie because it was a bit different, and I liked its rhythm, its slow movement.
We would sit together quietly as Kate breastfed. I took to knitting small things in fleecy white or soft pink wool, on fragile metal needles that clicked companionably through our amicable chat, our compatible silence. We were there together, hanging over her cradle, when she smiled her first smile. 'You can't say _that's_ wind!' Kate cried in triumph, and we hugged. We actually hugged.
Picture postcard, we were. No, better: high art. I see us in retrospect, heads bent, sitting in the slanting sunshine of a rustic Dutch interior. Tendrils of hair, shimmering in the faint and dusty light, escape from our starched white caps. Blue hyacinths make a statement in an earthenware vase; a solid curvaceous milk jug squats on the wooden table. The baby kicks and coos. There's probably a damsel playing a dulcimer in a courtyard out the back.
Well, it happened. It was brief, I grant you; but it happened.
*
When Sophie was about three months old and we were engaged in one of these idyllic scenes, Kate said one day: 'Families are so important, aren't they, Mum.'
'Mmmm,' I say, with some wariness. I am conscious of being vulnerable in this kind of discussion.
'And ritual's important, too, I think. Don't you?'
'Possibly,' I say. I have no idea where this is heading.
'Dad asked me the other day if we were going to have her christened.'
'Ah,' I say. Religion has never meant much to me. It seems clear to me that if there is a God up there he's just as baffled by everything as I am, and he's therefore not much use to me. I was faintly surprised, after we were married, to discover that Steve had C-of-E leanings. Not that they ever made much difference to anything.
'And are you?' I ask.
'No, I don't think so,' says Kate, reflectively, detaching Sophie from her nipple and lifting her to her shoulder. 'There you go, my precious, see how you feel after that little lot. I talked about it with Gavin. He doesn't really mind: he'll do whatever I want, he says.'
That'd be right, I think. Hopeless.
'And I don't really want the full christening deal. You know? It seems hypocritical, somehow. I mean, we never go to church, and I really don't want all the trappings and so forth. But we agreed we wanted something. Just something to sort of celebrate Sophie, and her arrival into the world, and how perfect she is. To welcome her formally, I suppose. She's had enough, I think. Can you take her for a moment, Mum?'
I receive Sophie into my arms. She gives me a bleary grin and my insides liquefy into warm honey. It keeps happening, the miracle of holding her and feeling myself melt with the wondrous beauty of it, the beauty of her.
Kate is doing up her bra. 'So I thought, perhaps, let's have a kind of reception.'
'A reception?'
'Well, yes. I mean literally. A receiving of her, into the family. You know? Just like a party, really.'
I don't know, at all, but I'm feeling soppy these days and I'm prepared to agree to pretty much anything Kate suggests. Within reason.
'Of course,' she goes on, thoughtfully, 'it would mean you and Dad would be here together. Would that be uncomfortable for you, do you think?'
It will be exceedingly uncomfortable, of course. As I've explained, Steve and I didn't split up with the kind of cold-fish amity other couples seem to achieve on occasion. We were bitter and hot and hurt; we shouted and screamed; we wounded each other with express calculation. We never actually hit each other, but we came close. Well, I did, anyway. And now we avoid each other with resolute circumspection.
But I have thought about all of this. Steve and I are grandparents, now, and perhaps it is time to face up to our joint obligations. Grandchildren are different. There will be birthdays, Christmases, school sports, school concerts. There will be ballet performances and netball matches and tennis finals and maybe eisteddfods and piano competitions. Trophy presentations, prize-givings, all the trappings that accompany clear public evidence of excellence. (Not that I'm necessarily expecting these, you understand; but they might happen all the same.) There will be graduations and debuts. And then there will be more weddings. So if I can manage to be grown up about all of this, Steve can, too.
'That's fine,' I say, casually. 'Max will have to be invited, of course.'
Kate nods. 'Max is part of the family, too,' she says, seriously.
I am pleased she says this. I tell him about it, later, that evening, when we're having a whisky in the light stone courtyard leading in from Rain's well-equipped kitchen—silvery granite-topped benches, best-quality Swedish appliances, capacious and ingenious cupboards in pale mountain ash.
He tinkles the ice in his glass, gazing down at it and saying nothing. I think I can tell, however, from his faint crinkly smile, that he's glad Kate has said this, that he interprets it as her cementing his relationship with Sophie, with her family.
So Kate's reception proceeds. She chats about it a lot: it fills her mind. Invitations are sent out for a Sunday afternoon. We are each to bring a gift: not a physical, wrapped gift (though I intend to bring one of those, too: I have bought a glorious snowy crochet wrap which will, I realise, be of strictly limited use but drapes superbly and seems to me to establish my gravitas as a grandmother), but a metaphoric gift, a spiritual gift, which requires each of us to nominate the essential quality that in an ideal world we would pass on to a baby aspiring to the best of everything. In short, we are to be fairy godmothers.
I give much thought to what I will propose as my gift to Sophie. I am not blind to the possibilities this occasion offers for Steve and me to insert vengeful blades sleekly between each other's ribs, but I am determined to rise above this, to demonstrate my large-mindedness, my benign serenity. Should Steve meanly come up with some quality such as fidelity, I am prepared to look mildly hurt and to forgo my corresponding opportunity. Forewarned, I smugly believe myself forearmed.
Max is interested in what he sees as Kate's ingenuity and engages to think seriously about his own gift, but for some reason we do not discuss our intentions.
Before we leave for the reception, I glance out the back door and see that the wattle is in vibrant bloom. Yellow has always seemed to me to be Sophie's natural colour, and even at that stage of her life its radiance carried its own appropriateness for her special glow, her sunlit essence, her goldenness. I rush out with secateurs and mutilate the young trees, carving out great swathes of vivid flowers, which deposit themselves all over the fine leather of Max's back seat on the way over. Ever since then, whenever I see the electric lemon fuzz of early cootamundra blossom, I remember Sophie's fairy godmother party.
I realise when we get there that the wattle has been a mistake: Kate exclaims over its beauty but is clearly at a loss when trying to find a suitable vase, and it makes the devil of a mess of her carpet while she struggles with it. Never mind: it gives us a way to get through the awkwardness of Steve, who has already arrived and has obviously decided to behave well. He nods at me, shakes Max's hand mournfully, and mutters something which may not be gracious but at least is not provocative.
So we're all sitting around and dandling Sophie and covering awkward gaps with exclamations over her beauty and intelligence. Then, at a sign from Kate, the hapless Gavin shambles over to the fireplace and looks about him with myopic satisfaction.
'Thank you all for coming,' he says, and meanders on for some time about Sophie, and Kate, and how happy he is to be married to Kate, and the importance of families.
We sit on the cheap, velveteen lounge suite and listen to him with more or less pleased looks on our faces. Kate is on my right, I recall; holding Sophie and watching Gavin with approval. Dominic is beyond her, tight, tense, closed in. (Dominic was about fifteen. He still had the slim fawnlike look of prepubescence; when he wore bathers you could see the delicate angular projections of his shoulder blades, jutting out as if his skeletal structure hid wings that were trying to bud through the bones. His chest and arms hadn't filled out. His voice was just breaking and his skin still had a childish sheen to it. He was so beautiful he made you weep.) Steve is beyond Dominic; his shoulders are hunched and he stares at the carpet.
And on the other side of the room—in the old armchairs Steve has given Kate, the armchairs he and I bought when we were first married—sit Zoë and Henry. I hadn't expected Zoë and Henry to be here and I'm not sure why they are: I'd thought it was only the immediate family; but I'm determined to be good-tempered about everything today. Max is next to me (I've made sure of that), on the other side of me from Kate. I see Steve glancing at him, every now and then. Sophie is making gurgling noises, bless her.
Gavin comes, with slow humour and elephantine tread, to the end of his speech. He beams at us all and then says, with no particular emphasis: 'And my gift to my daughter is charm.' And he goes and sits down.
Kate breaks in to protest. 'You have to say why,' she cries, dissolving into laughter.
We have all been instructed thus. It is not enough simply to give: we must speak to our gifts, describing the manner in which we wish them to manifest themselves and the reasons for our choice. Gavin has, of course, forgotten this. Smiling ineffectually, he lumbers to his feet again and considers.
'I haven't got it,' he says. 'Charm, I mean. I've never had it.' Some people make ineffectual demurring noises. (I don't.) 'No, I haven't,' he continues. 'I know I haven't. I've always wanted it. I've always been—how would you put it— _beguiled_ by charming people. They make you want to be with them; they make you want their good opinion; they make you want to do things for them. People who have that—they've got the battle won. That's what I want for my little girl. If she's got that, I reckon everything else will follow.'
Kate applauds and the rest of us, unsure about whether we are meant to join in, clap in a piecemeal kind of way as Gavin, blushing and ungainly, sits down again and tries to look as if he has some marginal capacity for self-possession. Later on, as we have a quiet drink in the courtyard at Rain, Max will say to me that he found this speech touching.
'Touching?' I say, mystified.
Max regards me in what seems a slightly odd way. 'You really don't like Gavin, do you?'
Briefly, I investigate my emotions.
'I don't like him or dislike him. I don't know what she sees in him. But I don't specially dislike him. It's just that he irritates me. How can she go to bed with him, Max? He's like a long, thin puppet, a clown puppet; his head bobs around as if his neck's on a spring; he's so gangly and bumbly and he always looks so damn worried.'
'But you don't dislike him,' says Max, laughing.
'No. He drives me mad, but I don't dislike him.'
'It was a very honest speech,' says Max.
Anyway, whether Gavin was touching or not, we now move on to our next contributor, skulking palely in the corner.
'Dad!' calls Kate, who is the self-elected emcee for this occasion. 'Your turn.'
Steve has actually written out a speech, which he drags from a back pocket and self-consciously unfolds. 'I have thought a good deal about this,' he reads, flat and hurried. 'I have found it a difficult task.'
I school my features to look earnest: it will not do to snigger.
'It is a new thing for me to have a granddaughter,' continues Steve. 'I find I am quite surprised by it. It seems to me only the other day that Kate was still a little child herself and I do not know where the years have gone. But here we are, and suddenly here is Sophie, who is now a part of our family.'
Steve pauses and takes a deep breath; a light sweat gleams on his upper lip. I am puzzled. Grace and sophistication are not attributes of Steve's: I realise this; indeed, no one knows it better than I, but he can normally exhibit a kind of bluff self-possession that serves him better than this awful rehearsed floundering. Suddenly I wonder if he is as disconcerted by my presence—and, of course, by Max's—as I am by his. I suppose it's possible.
'I have thought a lot about Kate's request and I have tried to come up with something that is appropriate for me to give Sophie as her gift from me,' Steve goes on, so clearly uncomfortable and embarrassed that these sensations are communicated to us all like a rabid infection.
'Of course I understand that whatever I come up with it will only be my wish for Sophie and not actually a real gift, no matter how much I would like it to be. Still, it is important to wish for the right thing. And so I have decided to wish for Sophie not beauty or wealth or talent but something else that I think is more important. I am wishing her resilience. You don't ever know what life will bring you, and whatever it does bring you, you have to try to bounce back from. This is something all of us know from different experiences, in different ways.'
He pauses and glances around, in a dissatisfied, unfocused way. 'So,' he finishes, lamely. 'So. That is my wish for Sophie.'
Again, as he sits down, the patchy applause dribbles around, Kate clapping over-enthusiastically and the rest of us still not sure about joining in. It's a bit like a school prize-giving where none of the students like each other very much. I'm relieved. Resilience is okay, as Steve's wish. Fidelity I would have had trouble with.
'Max!' Kate declares. 'Your turn.'
Max stands, with the graceful diffidence he's so good at. He's not at all nervous, unlike his predecessors, but he's not going to err on the side of over-confidence either. Oh God, he speaks so well, and I am so proud of him! He says, with his own gorgeous quiet understated charm, how glad he is to be here, how much he appreciates everyone's forbearance and warmth and how good it is to feel part of a family.
He goes on: 'I've never been part of a family—or at least, not for so long that I almost can't remember it. I'd been on my own for a long time, before I met Isabel. Frankly, I'd expected always to be on my own. And it's such a privilege, such an honour, to be accepted among you all.'
He pauses, and glances around with that candid and somehow vulnerable look of his, the one that always slays me.
'Like all of you, I've thought a lot about what I'd like to bequeath to this bewitching scrap of humanity here. And I thought perhaps at first I'd focus on this element of her life: her centredness in a family, the way she nestles in the heart of her family. It's an extended family, I grant you, and my membership of it is a surprising part of it, I know. But it's a group of related people who all—whatever else they do or don't share—do share a profound and confident wish that Sophie's life will be the best life that it can possibly be, that we can possibly help to make it.
'And so it seemed to me that it would be redundant to wish Sophie a strong and embracing family. She already has that. It encircles her and protects her; it provides her with all the support and love and sustenance she needs. And so the best gift I can give Sophie is a spirit of adventure, a spirit that will tempt her away from the safety and comfort and shelter of her family into the excitement and challenge of the unknown. A spirit that will lead her to test herself, to find how limitless her limits are, to leap into space and fly.'
Max actually goes over to Kate, and leans over her to kiss the top of Sophie's head. It's quite a theatrical thing to do, and I think a brave one. We all clap him (I am a little too enthusiastic, but really, I am so proud of him), and Kate turns to Dominic.
Dominic is unfazed, of course. He has notes—three or four small squares of paper—and holds them carefully, periodically shuffling the front one to the back as he speaks. He has prepared so well that he knows which notes are on which piece of paper.
'So much,' he says, glancing around in a rather peremptory way, as if to make sure he has our attention. 'So much one would like to offer to Sophie. So much one would like to have oneself, that somehow one doesn't have. I suspect we will all offer Sophie something that shines the more brightly for us because it isn't in our own possession.'
_Shines the more brightly!_ Ah, what an orator. What a future this kid has! How articulate, for fifteen! I recall Steve telling me (in one of our stilted antagonistic sessions during which we consult about the children) that Dominic is in the debating club at school. Even the president of it, perhaps. I must find out.
'I am no exception,' continues Dominic. 'I'd like Sophie to have something I've always wanted. Try as I might, it's always eluded me.' He pauses.
There is something I can't quite read about him. He's speaking so eloquently, so lucidly: he's mixing it so impressively with the adults; you'd expect him to be enjoying himself. Dominic's used to occupying the limelight in a variety of contexts and normally he adopts its sheen with aplomb. Not so today. There's something missing. He's not relishing any of this.
'What I wish for Sophie...'
There is an awful second or two during which he seems to look into the middle distance and gulp slightly, and forget what he's saying. Just as I'm becoming nervous, however, he gains control and returns to us.
'I'd like Sophie to believe in herself,' says Dominic, to my utter astonishment.
Was there ever a child, I think, born with greater self-belief than Dominic?
'I'd like her to be deeply confident within herself. Not stupidly arrogant, not up herself. I'm not talking ego; I'm not talking conceit. Just a realistic and abiding self-belief. I reckon that'll see her through more than anything else.'
I am so thunderstruck by this assertion and its implications that I miss a beat or two of Kate's mild and inconsequential connecting remarks and the next thing I know Zoë has bounced up.
'It's been a very interesting exercise,' says Zoë. 'I'd like to thank Kate for the opportunity to think through some of the questions that her request inevitably generated. In thinking about what we would wish to give Sophie, as Dominic has so intelligently pointed out, we have all thought about what we ourselves would like to possess. And that naturally leads to a reappraisal of our own lives. Henry and I'—she ducks her head towards Henry in an acknowledgement—'have spent many hours discussing the matter. It's been most instructive.'
She smiles at us all, in a pleased way, as if we should be especially grateful for the chance at self-instruction.
'What to wish for?' proceeds Zoë, clearly enjoying herself. 'Kate didn't make it clear whether she wanted our gifts to be _qualities_ , that is, an integral part of Sophie, or _conditions_ , things like money, extrinsic to Sophie but available to her. So it was, for instance, open to any of us to choose great wealth, or immunity from disease. These would be good gifts. Instead, all of us so far have concentrated on qualities. This is what Henry and I decided, too, though of course I mustn't pre-empt Henry's choices in this matter.'
I suppose this is how Zoë talks when she gives a class. It's rapid, crisp, articulate, assertive. She feels no doubt, no diffidence. The outlines are clear; the colours are bright. We sit there, listening like obedient schoolchildren.
'I think the reason we have all instinctively focused on qualities is that we implicitly acknowledge our own roles in our destinies. It would be possible to wish for Sophie, say, happiness, but happiness is something we achieve ourselves, not something that can be handed around on a plate. If Sophie is to be happy—and I certainly hope she will be—she must win that happiness herself. Nobody can give it to her.'
I'm trying not to look bored, but it's getting harder by the moment. It's just like Zoë, to turn all of this from a family celebration into a purely didactic occasion in which she can herself play a leading part and make a lot of unfounded assumptions about life and how other people live it. I recognise in her comments a barbed reference to my own circumstances, and hear her voice as it has sounded forth on other occasions, earnestly telling me that I am the master of my fate, the captain of my soul, and so on.
Zoë smiles, holding her hands clasped in front of her. She looks as if she is about to announce the winner of the under-twelve hurdles.
'I wish Sophie a love of learning,' she says. 'Through learning Sophie can become whatever she wishes. She can discover great opportunities and she can make the most of those opportunities. In some ways I feel this was an entirely predictable choice for me to make—exactly what you would expect of a tunnel-visioned, obstinate, old-fashioned, chalk-'n'-talk teacher.'
She laughs, slightly, as if she thinks this description of herself will seem inadequate to her audience. 'But there it is. Sophie, I wish you a fine education, and I wish you the capacity to make the most of it.'
Zoë looks as if she will go on for a long time in exhortation and explication, but fortunately Kate breaks in with delighted exclamations. She turns then to Henry.
I am to be last, then. How am I to read this, I wonder? Does the last person speak more weightily, or am I something of a postscript to the proceedings?
All I can think of, for a few moments, as I try to concentrate on Henry, is how very much I have always disliked him. Then his words arrest this train of thought. He's been saying something about the specialness of the occasion, the bright promise of a new life, and so forth.
Then he says, in his finicky way, pronouncing all his words carefully in that over-particular manner he appears for some reason to cultivate: 'None of this would matter, of course'—(I'm not sure what it is that wouldn't matter)—'if we only focus on those attributes, those characteristics, that in fact we do see as important, as significant.'
That's Henry all over. Never settle for one word if two will do the trick. Zoë always maintains he is a wonderful teacher. 'Inspirational,' says Zoë, with bull-like force, so you don't feel like contradicting her. 'Simply inspirational. He has them hanging on his every word.' I don't believe it, though.
'Indeed,' says Henry, beaming around, 'as _necessary_. However, let us not forget, in the daily striving, the competition, the conflict that constitute our lives, that other elements, other characteristics, characteristics that have nothing to do with competition, are still an integral part of our beings. In the great competitions that life sets us, enters us for, if I may put it that way, there are still other aspects of our personalities that are just as important, although they are not attuned to winning, to victory. One of these is the capacity to love.'
_Shit_ , I think.
'It is this capacity to love—to love and to accept love—that stands us in such good stead in the significant relationships of our lives. Without it, we are unable to give or to receive the affection that in old-fashioned terms used to be called loving-kindness, that provides the links for all our friendships, the cement for our families, the single thing that holds us all together.'
The fact is that Henry has stolen my thunder. My gift to Sophie was going to be the capacity to love. And, in fact, the capacity to be loved, which he seems to have cottoned on to also. In the midst of my irritation and confusion, I am astounded that Henry's desiccated soul is capable of formulating these concepts. Goodness knows where he's got them from.
But what am I going to say? It's nearly my turn. While Henry rabbits on about love and its importance or significance or whatever, I'm thinking as hard as I can. I've got my little speech all prepared. I've known what I'm going to say: I haven't got notes because I don't need them; unlike Steve, I'm perfectly capable of making a simple statement without rustling my way through several sheets of A4. But the simple statement I was going to make suddenly isn't available to me any more. So far as I can tell, Henry's taken pretty much an identical line, and left me high and dry.
And how am I going to look? Everyone else has gone to a lot of trouble; everyone's thought about Kate's request and made an effort to meet it. So have I, of course, but it's not going to look like that. I, who love Sophie best of all: I'm going to be the one who looks inadequate, ill-prepared, uncaring. It's not fair.
I'm thinking so hard that I don't realise Henry has come to an end and everybody is looking at me.
'Earth to Isabel,' says Gavin. 'Earth to Isabel.'
I can never see why this is supposed to be such a witty thing to say, nor why people say it to me. Steve used it all the time and it drove me insane. _I am listening_ , I felt like saying to him. _I'm listening, but you're not saying anything_.
'Sorry,' I say. 'The thing is...' The thing is, of course, that I don't want to say what the thing is. I don't want to admit that Henry and I jumped on the same tram. I'm not quite sure why this is. Well, perhaps I am. I can't stand Henry and don't wish to be bracketed with him. What I wanted for Sophie was special, and Henry's gone and spoilt it. I decide to improvise.
'I've been sitting here, listening to you all,' I start. I look around and smile winningly. Well, I try to. I don't want any of them thinking I'm a bad grandmother. So many of them think I'm a bad mother. Wrongly, wrongly; but it's what they think.
'And it's been fascinating, seeing what you've all chosen.'
By mistake I look at Dominic, and find his gaze on me, stern and brimming with judgement and perhaps slightly baffled, too, as if he's wondering what I've done to deserve being there at all. He's wearing that awful, fastidious, unforgiving look he gets. I turn my eyes from him.
I look around this group of people most of whom I do not love, although they are through genetic and legal ties the closest people to me in all the world. I see Max's concern, Kate's puzzlement, Zoë's irritation, Dominic's antagonism. I see Gavin's embarrassment, Steve's unease and Henry's disdain. I see Sophie, burbling away, quietly sociable, on her mother's knee.
And suddenly it comes to me. So many of these people have combined to thwart and frustrate me over the years; so many of them have militated against my happiness in different ways. I know what I want for my granddaughter, growing up amongst them.
'I wish this for Sophie,' I say, firmly and steadily. 'I want her to have a mind of her own. I want her to think for herself, to grow in strength and independence of character, not to be frightened to trust her own judgement, and to act as she sees fit, as she thinks best.'
Everybody looks relieved that I've actually thought of something to say and have said it, even if they don't much like the thing itself. Kate gives me one of her big gooey smiles.
'What about you, Kate?' asks Dominic. 'Don't you have a gift for Sophie?'
Kate gives him a gooey smile, too.
'I do,' she says, as if this were a matter for particular congratulation.
She pauses, apparently gathering her thoughts. God knows, it shouldn't take her long.
'I want to thank you all. Each and every one of you has given Sophie something truly worth having, something she'll be grateful to you for to the end of her days. I'm going to make a little book for her, with pictures in it, and I'm going to include all her gifts in it, all the wishes you've made for her and the presents you've given her. It will mean she can look back to this day and see how lucky she is to be at the centre of such a loving family, such a good and thoughtful and generous family.'
It seems as if she really thinks we have verifiably given Sophie the things of which we spoke, rather than simply speaking of them. Kate has always dwelt in her own special version of la-la land. Would that it were so easy! Would that I could have organised for Dominic to have a loving heart, for Kate to have some brains!
'My own gift might seem an odd one,' says Kate. 'I guess it isn't obvious, straightaway, as something you'd choose to give a child. But it just seems to me so important, I can't tell you. I'm not even sure what to call it. It's got to do, I suppose, with catching the wave, knowing the right time. That makes it sound like opportunism, jumping on the bandwagon sort of thing, but I don't mean it as purely an opportunistic thing. It's more like—well, catching the wave's the best way I can put it, I think. Seeing when something is the right time for something and trusting yourself to act on it.'
'Don't die wondering,' says Gavin, helpfully. 'That's it,' she agrees. 'Don't die wondering.'
And on this note we disperse, drink, eat, chat, more or less awkwardly. I've been quite nervous about this occasion, as it represents the first time, really, that Steve and I have consented to meet each other as part of the family. It doesn't go too badly. We manage to have a conversation, or at least to be in the same conversing group, and neither of us positively scratches the other's eyes out. It has to be a good sign.
Max isn't a chatty man, but he's a bit quiet even for him in the car on the way home. It isn't until later, when we're sipping whisky in Rain's peaceful courtyard, that he refers to the afternoon, making his comment about Gavin. Then, to my surprise, he congratulates me on my family.
'You have to be joking,' I say.
'No, why?'
'My family! Actually, love, I don't usually regard them as grounds for congratulation.'
'I know. But you'd have to admit they all came up looking pretty good today.'
'Did they?' I consider it. 'In what way, would you say?'
'Well, they'd all thought about it, hadn't they? They'd all tried hard.'
'I did, too. I'd practised endlessly, what I was going to say. But then Henry pretty much said it for me, so I had to think about something else.'
'Ah. I wondered what prompted the stage fright. You did well, Bella: that was a good present you gave her, especially if you thought of it at short notice.'
Max stretches out and contemplatively swirls the liquor and ice around in his glass.
'Yours was pretty good, too.'
'I hope it will stand her in good stead,' he says, seriously. 'It's a dangerous thing to wish on someone, a sense of adventure.'
'It hasn't done you any harm.'
'No. Maybe not.' He smiles and leans over to me, strokes my arm, makes me tingle. 'It didn't stop me from meeting you, anyway. That's been my biggest adventure.'
There's a brief pause, spent in mutual smug contemplation.
It's one of the nice things about being one half of a happy couple, that you can do this sort of thing.
After a little while, I say: 'I was surprised by what Dominic said, though.'
'What bit of it?'
'The bit about lacking self-confidence. I would have said it was one of the principal things Dominic absolutely doesn't lack.'
'Truly?'
'You surely don't think Dominic an unconfident person?'
'Well,' says Max, 'I haven't been given the opportunity to get to know Dominic very much at all. But I've never thought of him as particularly confident.'
I am baffled by this, and say so.
'He didn't seem very confident today, though. Did he?'
'Perhaps not, but I've seen Dominic in that sort of situation before—one, I mean, where he has to speak more or less publicly—and he always carries it off just fine.'
'I thought he was feeling very exposed,' says Max. 'I thought he was being forced to reveal more of himself than he wanted to. I suppose all of us were, but I think Dominic in particular found it uncomfortable, an invasion of privacy. He had to draw the curtain aside, didn't he? He had to let people in. He didn't like it.'
I mull this over.
'But he's not unconfident,' I say, eventually. 'He's not an unconfident person.'
Max shrugs. 'You know him far better than I do, Bella, obviously. But I've never thought of Dominic as very confident. Confident people don't need to put up the barriers Dominic puts up.'
I'm genuinely puzzled by this. I see what Max means, but it doesn't square with how I think of Dominic. To me, he always seems as swift and strong and confident as an otter, a tiger, a hawk. How can Max think he isn't?
'Don't worry about it,' says Max with a slight smile. 'You look so anxious, Bella. There's nothing to worry about.'
I let it go, but it returns to niggle at me. Dominic: unconfident?
He certainly doesn't seem unconfident the next time I see him, which happens to be at his school's speech night. He picks up several prizes, makes a cogent contribution to a kind of mini-debate they offer as part of the evening's entertainment, and generally seems as untroubled as ever by attention. I mention this to Max (who doesn't come, as he thinks it tactless to inflict himself on Dominic at these occasions) when I return home. He nods, cheerily.
'It's only that you said he was unconfident,' I say.
'I'm probably wrong.'
'But he seemed so much on top of everything, Max. I wish you could have seen him.'
I am bubbling over with undisguised pride: it's pleasant to have produced a child who so conspicuously achieves high standards in everything he attempts. A number of Dominic's teachers have made admiring remarks about him during the evening. Dominic himself, doubtless borne along and uplifted by his success, has even attained a degree of civility to me.
Max pours my drink and hands it to me.
'I believe you, my darling Bella,' he says, with his charming grin. 'No question.'
I lean back, kick off my shoes, relax. 'You know what I was thinking? I was thinking, heavens, if Kate can have a child like Sophie, imagine what Dominic's children will be like.'
Max looks puzzled.
'Well,' I say, suddenly realising that I'm not sounding kind. I like to sound kind, for Max. 'What I mean is, I know Kate's lovely, and of course we love her very much, but in terms of beauty and brain power...well. You know what I mean.'
'Kate's intelligent,' says Max, with a hint of shortness. 'Kate's got a lot of perception, a lot of insight.'
'Yes. Of course I agree.' (I don't.) 'But...'
'Look,' says Max, 'I know Sophie's the most brilliant child ever to exist, but she isn't very old yet, Bella darling. Don't count too much on her intelligence and beauty, will you?'
I shrug. Somewhere, dimly, I intuit a degree of what one might almost call displeasure in Max's responses. But it's been a good night, and I don't want to dig too deep, or to enmesh myself in a tricky argument. Happily, I concede to his caution.
Soon, we go to bed and make love; and making love is as glorious and radiant as it always is. There seems no limit to our luck, our felicity, the extraordinary generosity of the providence that watches silently over us.
So I moved into a new phase. If not a genuine fairy godmother, certainly a devoted grandmother.
Not long after Kate's little celebration for Sophie, Max and I went over to their house one afternoon. Max didn't usually come with me on these occasions. He derived great enjoyment, he said, from my new status, and he delighted in my pleasure with a tender and attentive mirth that moved me deeply; but he didn't find it necessary always to be there, hanging onto my sleeve. He was a very sensitive man, Max, and highly attuned to female responsiveness, feminine priorities.
He handled Sophie deftly enough himself—surprising, in a man who had never had children—but he preferred to see me holding her, to chuckle at my raptness and to encourage my captivation. He said that he thought it good for me to spend time with Kate, to strengthen my relationship with her at what he called a very special time—which it was—rather than insisting on being there all the time himself.
But on this day he drove me over, and stayed twenty minutes or so before going on to a business appointment—one of those mysterious business appointments into the mists of which he disappeared, debonair and unassuming, one of the appointments about which information was never forthcoming. Not that I sought such information, then or ever.
The previous day the demolition of the pool had commenced. A small swarm of men had arrived at an early hour and made themselves at home in our back garden. We'd already drained the pool. Jackhammers had noisily shattered the pale tiles and trucks had deposited neat slagheaps of filler earth around its perimeter. Before we left for Kate's, I remember, we conducted a brief inspection of the back. The cavity in the ground gaped roughly as if a bomb had dropped in it: it was hard to believe we had lounged around it, dived so hedonistically into it. It all looked awful, but Max was pleased that the work had finally started.
'You won't know it,' he said. 'Going to be brilliant. Instant garden. Nothing like it.'
Kate's cleaning lady was there, when we arrived. We had given her twelve months' worth of cleaning lady, Max and I, when Sophie was born: I remembered so clearly Kate's own infancy and the difficulty I had found in ordering the most trivial components of one's life in a halfway efficient manner. (Not that Kate hadn't managed pretty well. She took to motherhood, I must say, better than I had.) She was a nice lady, Charmian: she had badly dyed, tight auburn curls and a hard-bitten, scrappy, sinewy look about her, but she was soft as butter inside, and _gooed_ and _gaaed_ around Sophie as much as any of us did. She cleaned the bathroom a treat: I remember that.
Max had gone. Kate was on the phone to some friend who had rung. I was playing with Sophie. Charmian had just finished vacuuming and came into the living room. She tickled Sophie under the chin.
'Goodbye, my precious,' she said. 'I'll see you again next week, I expect.'
Sophie was lying on my lap. She cooed and waved her fists about.
'You can certainly see the likeness to her grandfather,' observed Charmian.
'Sorry?' I said. I didn't think Sophie resembled Steve in the slightest.
'Even at this age, you can see the nose, I reckon. And the way her eyes are set. But specially the mouth. Your husband's a very good-looking man, isn't he? Striking. She'll be the same, I shouldn't wonder.'
I went to explain, laughingly, that she had it wrong, that Max was my husband but not Kate's father, and as I did so I glanced down at Sophie on my knees and the words—as the phrase has it—died on my lips. Died quite dead, all quite deadybones, as Dominic would once have said.
Sophie lay on her back, beaming up at us, and as I looked at her she gave a delighted little clucking noise and crinkled her nose.
I had seen that nose-crinkle before.
'You hadn't seen it?' asked Charmian, misunderstanding my silence. 'Likenesses are strange, I always think. Families often don't notice them: it takes an outsider to see what's right before your eyes. Funny things, genes. See you later.'
I said something. Charmian left. I continued to sit, Sophie beaming up at me.
I suppose I was slow. It didn't all fall into place immediately. Right in front of me it had certainly been, but that didn't mean I comprehended it rapidly. Comprehension was a tall order, at that moment.
But as I sat there, some of the pieces started to fall into place. I recalled Kate's new maturity, her unselfconscious indolence and her adolescent glamour, the new charge of sexuality flickering through her. I recalled the bikini, the hours around the pool. I recalled, quite specifically, the satiny way her skin shone when she climbed out of the pool. I recalled my week in Sydney.
It's always the oldest stories in the world that make least sense when they happen to you.
And this one made no sense at all. For against my memories of these things, against my enforced new interpretation of absences and silences, of laughter and glances, of intonation and gesture, stood all my experience of Max, all the experience of our love and my trust. My absolute trust.
It was not within the range of likelihood that Max had cold-bloodedly, deliberately deceived me. The understanding that we shared simply transcended that possibility. Max's kindness, his loyalty and generosity and openness and sweet nature—all his qualities came crowding into my head. It was not possible that this man had betrayed me. Not with the icy intention of betrayal. Not knowingly, not as a plan.
Yet betrayed I was. I gave my little finger to Sophie and she grabbed it, chuckling with that uncomplicated, entirely unmanipulative delight that makes babies such a joy. I drew it away gently and she pulled it back and stuck it in her mouth. She chuckled again, crinkling her nose.
At this point, Kate entered. She took one look at me and nodded, quite calmly.
'You've seen it, haven't you?' she said, rather as if we were discussing a television program.
I couldn't say anything.
She bent over and took Sophie. 'I knew you'd see it soon,' she said, matter-of-factly. 'It's time for her sleep. I'll put her down. I'll be back in a moment.'
When she returned she sat opposite me and regarded me levelly.
'Well,' she said. 'I understand it's a shock for you. Do you want to talk about it?'
I tried to speak and couldn't. I tried again. I stood up. I didn't want to be sitting down. She stood, too, and put her hand out as if she meant to comfort me. I winced away.
'Talk. What good will talking do?' In my own ears my voice sounded vicious, its bitterness spraying uncontrollably.
She said nothing. She interlaced her fingers and looked down at them, as if she were considering some conundrum they posed.
'Why?' I asked. 'Why did you do it?'
'You're blaming me?'
'You can ask that?'
'Why assume it's my fault?'
I stared at her. 'You're telling me he raped you?'
'Good God, no.'
'Then you must have seduced him.'
She actually laughed. 'Mum, can't you imagine making love outside rape or seduction? I was in love with him. I am in love with him.'
'What the fuck do you mean?'
She flinched, at the word, I suppose. I thought it was rich of her to take exception to my language when she'd just seduced my husband.
'I've been in love with Max for a long time. Since I was sixteen or so, I think. You're not the only person who's allowed to fall in love, Mum.'
I raised my hand to hit her, but she stepped back, alarm on her face.
'I only slept with him once,' she said. 'You saw Max and you fell hopelessly in love with him and you decided you had to have him. I saw Max and I fell hopelessly in love with him and I slept with him once. It's all I asked. I knew I couldn't have him, not permanently. But why shouldn't I make love with him, just once?'
'Why shouldn't you? My God, you can say to me, why shouldn't you?'
'Well, why would it matter, just once? The opportunity came. I didn't think it would matter, just once.'
'You caught the wave. I wondered what you'd meant, at your precious reception.'
She nodded. 'I caught the wave,' she said, simply.
'You're obscene. What about Gavin?'
'What about him?'
'You've cheated on him.'
'I don't think so,' said Kate, with maddening placidity. 'I don't think it's cheating on him. I wasn't married to him then, anyway. Such concern for Gavin, Mum. You surprise me.'
'You knew you were pregnant to Max when you married Gavin?'
'I knew I was pregnant.'
'Does Gavin know?'
'I'm not sure. Sometimes I think he does; sometimes I think he doesn't.'
'Aren't you afraid I'll tell him?'
'It's hardly in your best interests,' said Kate, composedly.
'You're pretty bloody cool about it all.'
'I've had time to think about it. You'll be cooler, too, when you've had a bit of time.'
'I don't think so.'
'Mum, let's leave it for the moment. It's been a big shock for you. In a little while, I'll talk about it as much as you want.'
'What makes you think I want to talk about it at all?'
'It has to be dealt with,' said my daughter. 'It's happened, that's all. It's something that's happened, and we can't make it unhappen. We just have to deal with it; we have to find the best way to deal with it.'
I started towards the door. Something was thudding in my head, behind my eyes. I seemed to have lost peripheral vision, but I could see where the door was and I headed for it.
'Don't forget your bag,' said Kate, holding it out to me.
I snatched it, but something else occurred to me. 'Who knows?'
'I don't know.'
'Who have you told?'
'Nobody.'
'Have you and Max discussed it?'
'Not a word.'
'That's absurd. You can't expect me to believe it.'
'It's true, though.'
'You expect me to believe Max doesn't know?'
'Of course he knows,' said Kate, patiently. 'He can't help knowing. But so far we haven't talked about it. I don't think anyone else has noticed. I knew you would. I'm not at all sure Dad will notice, or Dominic. I'm not sure about Gavin. I don't think it's the sort of thing men necessarily do notice. Especially if they're not expecting it.'
I shook my head and fled, helpless in the fury and shock and disbelief that throbbed through me, possessing me outright.
I caught the bus home. It was just as well Max had taken the car: I was in no fit state to drive. As it was I missed my stop and had to walk back to Rain. Images played themselves before my eyes: the young satiny sheen of Kate's skin; Kate's hair, its brightness and softness; Kate's clear blue eyes, her pretty hands, the new confidence displaying itself in her stance, her languor, her suave new sexuality. Max's hands, Max's lean body, Max's hands, his hands, his hands. His hands that caressed me; his hands that brought me such delight; his long, fine, strong, beautiful hands.
He had not arrived home. I went into the lounge. I couldn't sit down. I walked up and down, up and down. I poured myself a brandy. And another.
When he came home, he looked at me just once and a kind of defeat came over his face, a resignation.
'You know. You've seen it, haven't you?'
'How could you?' I said. 'How could you?'
He shrugged, wearily, and went to pour himself a drink. 'I don't know how it happened, Bella. It was when you were in Sydney.'
'I'd worked that out, thank you.'
'We missed you, Kate and I.'
'Indeed?'
'Well, I know it sounds fatuous. But we did. We missed you. Kate said, let's do something special. So I went and got a crayfish. We had crayfish and a bottle of wine for dinner. Then we jumped into the pool.'
'And?'
'I can't explain it, Bella. These things happen. We'd had a glass too many. It was a hot night.'
'I bet.'
'It's partly why I wanted to get rid of the pool. It kept reminding me.'
'I'm sure.'
Max took a swig of his whisky and placed his glass on the shelf above the fireplace, next to the photograph of us holding hands, standing on the white sands at Point Leo, half silhouetted against the darkly golden sunset. He looked down, and then up. He partially spread his hands and then let them fall, in that puzzled, self-deprecatory movement so characteristic of him. His face was sad, drawn.
He stood there, by the fireplace, one foot on its chunky hearth of slate with its streaks of amber and gold. I was by then sitting on the creamy leather sofa, leaning forward towards the coffee table, where we kept the beautiful object we had christened the meteor stone. I think I was clenching and unclenching my fists. I remember looking down and seeing my hands resembling claws—slight, powerless, little claws. They talk about seeing things through a red haze. I don't recall the redness, but I certainly had a haze hanging thick over my eyes.
I picked up the meteor stone and threw it at him. It struck him on the left temple.
Max looked at me, for a moment, through clear and perplexed eyes. His right hand jerked up, slightly, towards me, and then fell, loose—as if it no longer belonged to him—to his side.
Without a sound, without even swaying, without trying to save himself or break his fall, he toppled. He fell the way a tree falls, heavy and straight. As he crashed onto the floor his head struck the hearth's rich slate, which we had chosen together.
I don't remember what I did next. I think I just sat there. Next thing I found myself crouched by him. I called his name. I shook him. I tried to feel his pulse, to detect his breath; I put my head to his chest to try to discern his heartbeat; I screamed at him.
But there was no doubt about it. He was dead. All quite deadybones.
Borrow came loping over and set up a shrill keening that vibrated through my skull. I hit him hard over his aristocratic nose and he stopped abruptly, with a look of astonishment. I'd never hit him before.
Sometimes even now Borrow looks at me slantwise, his expression revealing his belief that I am someone not to be trusted, someone who at any moment might run amok, throwing, striking, killing. Perhaps I am. Well, obviously I am.
I felt guiltier, at that moment, about hitting Borrow than about killing Max. Borrow's pain was real and graspable; Max's death wasn't.
I sat on the floor, next to my love, the love of my life, he who had fathered my daughter's daughter, my granddaughter, my stepdaughter. Borrow lay on the other side of him, head between paws, his eyes dolefully fixed on me.
I don't know how long I sat there. Max's eyes were open. I thought they might hold an expression of bewilderment, or recrimination, or hurt; but they just looked dead. Fishy, really.
Finally I leant over and closed the lids, as gently as I could, and was startled by the slight but palpable fleshy resistance they offered. At some stage I went and poured myself another brandy. My hands were shaking. I remember watching my hand shake as it tipped the bottle over the glass. The brandy went all over the benchtop. Scrupulously I wiped it up. Max hated mess.
I went back into the lounge. I sat on one of the deep leather armchairs and let my body sink into its resilient softness and rested my head against one of the honey-rust silk cushions and sipped my brandy while Max's portrait, so lovingly commissioned, so triumphantly hung, gazed enigmatically down at me.
I suppose there were all sorts of options open to me, but I didn't consider many of them. It never occurred to me, for instance, to ring the police, or an ambulance, or to say there had been an accident. Of course I would have rung an ambulance if there had been the slightest chance of his still being alive. (It was amazing, how dead he was: I didn't know you could die as fast, as decisively, as that.)
It never occurred to me to ring either of my children, or my sister. Funnily enough, I did think of ringing Steve. There was something ineffably consoling at that moment about the thought of Steve—stocky, slow, practical, careful. _I'll take care of it, lovely_ , he would say. _Just a little old corpse, is it? I'll take care of that. Don't you worry about a thing_. A pity, really, that a phone call to Steve was out of the question.
It didn't occur to me either to go out, to come back a few hours later, to find the body and raise the alarm. Well, that would have been a daft thing to do, anyway. Someone would have cottoned on. Some balding, unnecessary neighbour, behind his curtains, straight out of a 1940s Hollywood crime movie, would have noticed, would have blabbed to the good- looking detective in his soigné hat and knee-length coat.
It was clear to me that I would be seen as culpable—that indeed in some sort of technical sense I was culpable—and that I had to dissemble in order to save my own skin. It was a question of how best this deception could be managed, not of whether I would indulge in it. I could not bear it, that I might have murdered Max. I could not bear to have people saying I had murdered Max. I had to slip through the sudden giant padlock I had fastened on my life.
I did think of disappearing. I had enough money in the bank. I had a passport; I had credit cards. I could lock the house and go and catch a plane somewhere. Anywhere. But it seemed too difficult, too unwieldy, too outlandish somehow. And it required decision and purpose, and a capacity to deal with complex activity, and these were things of which I was then incapable.
I watched Max. I didn't know much about rigor mortis. I supposed it would set in soon. I looked at the indentation in his fragile skull, at the darkness beneath it, the livid pool that had formed under the skin of his temple, near the roots of his thick and silvery hair. No blood had been shed. It was astonishing, that there was no blood. I thought it was good, a bloodless death: it would be easier to clean up. The pale oatmeal carpet was unblemished: it bore no trace of death.
I contemplated the fact that Max would never again make love with me—nor, for that matter, with anybody else; not my daughter, not anybody. I thought of the cruel extraordinariness of time: approximately twelve hours ago I had arisen, a wife, a grandmother, a fulfilled and happy woman, singing in the shower, eating muesli for breakfast, brushing my teeth carefully afterwards to extract the little pieces of nuts and grains. And now, I was a widow, a murderess; my granddaughter was also my stepdaughter. The transformation had been breathtaking. I was astray in the awful bafflement of it all.
I can see now, of course, that I was in shock. But at the time it seemed to me I was functioning with commendable lucidity, that the cogs were all spinning and connecting to each other with at least usual efficiency, that indeed I had somehow achieved a singular purity of vision.
I breathed silently while death and guilt and anger and fear and other great nameless things swirled around me, while possible new, murky futures unfolded like dark, narrow paths on all sides, while Borrow uneasily snuffled and my lover gradually cooled and stiffened.
After a while—it could have been twenty minutes; it could have been hours—I set down my glass of brandy, which under the circumstances I had managed to stretch out pretty well. I walked over to Max's corpse. I squatted beside him and I stroked his hair.
His head was very cold.
I didn't know what to do. I felt in his jacket pocket and took out his wallet. It was an old wallet, but good quality: it was a soft, dark leather, with claret-coloured silk lining. I'd never looked at it closely before: Max didn't appreciate people messing with his belongings; and I wouldn't normally have dreamt of examining his wallet or his desk or his papers or anything else belonging to him.
I opened it and found my picture inside. It was a nasty shock, considering everything. There I was, grinning insouciantly up at me. There were a number of credit cards and what appeared to be a huge amount of cash. I counted it, quietly and methodically, pleased in a way to have something to do. There were thirty-five crisp, neat hundred-dollar notes, and a medley of smaller denominations as well. It was a very orderly wallet.
Steve used to stuff everything in his wallet: bits of paper, addresses, cards, advertising blurb, receipts. Max had been so meticulous about everything. I was amazed that he carried so much cash around. He was never short of cash, of course, but three thousand dollars plus seemed excessive. I was cross, in fact: it wasn't only excessive but careless. Anything might have happened to him, carrying as much as that around with him.
But I had no time or energy to think about it: I replaced the notes carefully and took the wallet upstairs, and put it on the table beside our bed, on Max's side, next to the novel he had been reading, a murder mystery by P.D. James.
Only a night or two ago he had asked lazily, as he closed the book and turned the bedside lamp off, why women wrote the best murder mysteries. I'd told him it was because we had the most devious minds, and he'd laughed and pulled me towards him and...Well, there was no point in thinking about that.
I went back downstairs, stepping heavily on the carpeted treads. Max's body was still there, exactly where I had left it. It seemed odd, somehow: I had so competently extinguished his spirit, it seemed as if his body should also have evaporated, vanishing into the air with a wreath of white smoke. Yet here it lay, stubbornly corporeal, dead but otherwise fit and svelte, as Max always was, requiring disposal.
And suddenly it was so easy, to see how the disposal was going to work. It was as if there was a divine dispensation, looking down benignly, knowing all along this was going to happen, having arranged for me in advance the perfect—the brilliant—solution. Its timeliness was breathtaking. How frequently during the course of my life, after all, would I need to dispose of a body? It was nothing short of phenomenal.
I was going to have to move him. I tugged at his ankle, experimentally. Nothing happened except that Borrow started whimpering again. I tugged again. Nothing.
He was a dead weight, of course.
'Shut up,' I said, fiercely, to the dog. He shot me a resentful leer and dropped his head between his paws again. I stood and pondered.
We had a trolley in the garage. We'd got it for moving furniture around when we were setting up in the house. Feeling gratitude for my forethought in providing direct access between house and garage, I padded into the garage with its chilly concrete floor and disentangled the trolley from the garden tools. I knocked a spade and it clanged against the Audi's silver bonnet as it fell. No dent. Not that it mattered, of course: Max was past fretting over the duco. Carefully I steered the trolley into the lounge.
It was a good trolley, with triple wheels and strong webbing that you could use to stabilise large items such as refrigerators. We'd paid a bit extra for a reliable article: always thoughtful consumers, we had eschewed the cheaper varieties, which looked as if they'd buckle under anything with real bulk or weight. I laid the trolley on its back and, after a number of ham-handed attempts, rolled Max onto it.
I deduced from the chill inflexibility of his limbs that rigor mortis was setting in; but if anything this made the process easier. I buckled the webbing straps tightly and experimentally raised the trolley handles. It was surprisingly easy to manoeuvre. Borrow looked on, lifting his lip slightly as if to demonstrate his contempt for the proceedings. I didn't blame him.
I went out to the switches by the back door and made sure that all the outside sensor lights were turned off. I was proud of my composure: I was thinking of everything. Softly I opened the door and wheeled Max out into the night. The neighbours' lights were off. I stood a moment, to allow my eyes to accustom themselves to the silver-shot darkness of the moonlight. Borrow loped to one of the wattles and absent-mindedly urinated against its trunk.
I rolled Max over to the gaping mass of clay and broken tile where the glossy pale pool had sparkled in sunlight, its water transparent against the latte ceramic sheen of its tiles. I undid the webbing buckles and tipped him in. It was hard: he was very heavy and for a moment I thought I wasn't going to be able to gain sufficient leverage. It was a relief when he toppled.
My eyes were more used to the moonlight now: I could see the dim outline of his body, sprawled where once he had dived with easy grace, his olive skin glistening in the water, his silver hair plastered to his skull.
I rolled the trolley back to the shed and brought out the spade. I noticed the wheelbarrow was full of weeds and grass clippings, so I also wheeled that out and tipped it into the great cavity. There were prunings, too, from the jasmine Max had trimmed back at the weekend. It all went in there. I slipped and slid down the edge of the slope, cutting my hand on a bit of tile.
Borrow lay in his sphinx position at the edge of the hole, concealing his confusion by asserting disengagement from my disreputable actions. I scrabbled and scraped, doglike, at the bottom of the hole, under the naked, callous moon, covering up the dreadful thing I had created. Please, God, I thought, don't let the neighbours wake up and look out. Please, God, make the neighbours keep on sleeping.
It was hard work, and I had to try to be quiet, too, which made it harder. When I thought I had done enough, I climbed up. I cleaned the spade on the grass, replaced it, and went back inside. Blood dripped on the kitchen floor from the wound to my hand. I wiped it up very carefully and wrapped a tea towel around the gash. I would bandage it properly later, I thought. I went back into the lounge and inspected it. So odd, I thought again, that there was no blood—or, rather, that the only blood was mine. The room was blank and peaceful, volunteering no information about the violence that had so recently ruptured it.
I examined the hearthstone, to see whether there were telltale hairs, whether some vestigial evidence clung to it. I could see nothing, but I wiped it down anyway, using the clean part of the tea towel. I took Max's half-empty whisky glass and rinsed it in the kitchen.
I returned to the lounge and gazed around. I noticed the meteor stone, where it had rolled away into the corner of the room. I picked it up and scrutinised it closely. Even on this, the murder weapon, I could see no incriminating evidence—no smear of congealing blood, no tuft of hair, no scrape or abrasion on its smooth surface to betray what it had done, what I had done. I dabbed at it with the tea towel, but it didn't need cleaning. It was clean as a whistle already.
I replaced it on the table, where it belonged. But it vexed me there. Its character had changed: it now looked squat and evil, as if it harboured of its own volition an unpleasant secret that it might speak aloud bluntly to some stranger entering the room. Formerly it had symbolised the happiness and harmony of our union; now its presence mocked and corrupted that memory. Also, it was evidence.
I took it outside and chucked it, too, into the large cubic hole in the garden where our beautiful pool had nestled. I heard the crack and thud of its landing.
I showered. I bandaged my hand. I rinsed the blood- stained tea towel. I went to bed. I lay awake in bed, staring at the ceiling, at the walls, at the blankness all around me. After a while I started to shake uncontrollably. I remember I was relieved by this: it demonstrated to me that I had retained some shred of humanity, that I was not a monster.
And then the tears came. I wept for hours. Borrow crawled up from his mat onto the bed, which he was not allowed to do, and I grabbed him very tight and shook and sobbed until the morning.
Then I slept, for an hour or two. I had a mad painful dream in which Max was forcing me to think of a rhyme for the word _corpse_ , and I woke up mumbling _borps, morps, porps, dorps._ _Forceps. Forceps is the closest I can get, Max._ And Max loomed impossibly high and dark above me, clenching his fists and threatening me, shouting at me (as he had never shouted in life) that _forceps_ wasn't good enough, that I had to think of a proper rhyme.
And then I got up and I showered again, for a long time, and I started to face up to a new kind of life.
I didn't see anyone for a couple of days, except for the contractor who turned up to topple the slagheaps of filler earth into the fractured shell of the pool. This did not happen the day immediately after I killed Max, but the following morning. He came very early, when it was still half-dark, and buzzed around in the grey air in his bright little bobcat, trundling like a bulbous orange beetle across the garden.
I glanced out at him from time to time, my chest and throat tight, a hammering inside my head, wondering how he could possibly cross and recross the garden so frequently, so blithely unaware of what lay beneath him. The beetle's throbbing rumble stopped at one point, and I peered out to see what was happening.
The contractor was standing by the void of the pool, looking meditatively across it. He was having a smoko. I could see the bluish trail of the cigarette fading into the morning air. Inhale, exhale; inhale, exhale. He looked down at the corner where Max lay, and my heart thudded harder. He threw in his stub. He sighed and stretched and flexed his arms, and ambled back into his machine. He continued to empty quantities of dirt into the space where the pool had been, and departed.
The next day Jack, of Jack's Landscape Solutions, arrived to check out the site, twirling his beard and promising graders and rollers and ready-made turf and rose bushes and instant flowerbeds, all of which duly eventuated over the following couple of weeks without my doing anything to prompt it. Max had organised it all, of course, with his usual precision, his reliable attention to detail.
It looked terrific when it was all complete. Jack was disappointed that Max wasn't around to see the finished product; I apologised and said he was on an unexpected business trip.
I rang work and said I had a stomach bug and wouldn't be there for a few days. I didn't answer the telephone: I didn't even listen to the messages on the answering machine. I didn't go out, except to empty the letterbox after dark. I didn't bother to open the letters: I just didn't want to leave the letterbox looking untended.
Borrow was cross, because he wanted a walk. It wasn't until the fourth day that he got one. I slunk out in my sunglasses and ran immediately into one of the neighbours, out doing the edges of his lawn. He waved jauntily and I waved back.
I can't remember much about those days. I was in shock, or denial, or limbo. Wherever. Certainly, I was somewhere I'd never been before. Max's brutal quenching had engulfed me with such thoroughness, such rapidity, that I could do nothing but crouch and hide. I didn't even cry very much, after that first night. The tears welled up sometimes, but as the days passed it was as if the reservoir within was drying up, its bed hard and eroded and cracked. A drought inside.
I thought a lot—but to no great purpose—about death. I found it hard to credit that Max was really dead; I found it even harder to believe I was responsible. I dreamt the Lost Dream again and again at night, waking in sweat, trembling, turning to the embrace that no longer held me, the arms that were no longer there.
I thought about murder. Surely it hadn't really been murder? I hadn't meant it, after all. _Let's call it manslaughter_ , I said to myself. It sounded so much better. And I thought about punishments. The state no longer sanctioned capital punishment. If I was discovered, exposed, I wouldn't actually die. I wouldn't be hanged. This was quite a consoling thought. But what would happen? Would I be jailed? And for how long?
What did you wear in jail? I wondered. Some sort of uniform, I supposed. I hoped it wouldn't be yellow. Or pink.
Then again, perhaps it was murder. Perhaps I was a murderer. A murderess. Perhaps intention didn't have anything to do with it. In any case, who could say what I had intended? It was only my word to say I hadn't meant it. People might not believe me. People probably wouldn't.
I went around in circles, dreary and lonely and frightened.
On perhaps the fourth evening the doorbell rang. Dominic stood there.
'Honey! Dominic!' I said, brightly.
He stared at me. I opened the security door to let him in. His gait was curious, not his usual light, elastic tread, rather shambling, rather as if he were sleepwalking.
'What is it?'
I was alarmed. I thought perhaps something had happened to Steve. Had Steve died? Was Dominic coming to tell me that Steve had died? Had both my husbands died together? At a level that I can't possibly explain I thought this sublimely amusing, if it were the case. I prepared myself to restrain laughter.
'Dominic,' I repeated. 'What is it? How did you get here?' And then I saw the bicycle, leaning against the wall behind him. 'You rode here?' I asked, stupidly. 'It's miles, Dominic. It's miles from Dad's place to here. How long did it take you?'
I noticed then that he didn't look well. His skin had a chalky colour and his eyes seemed huge charcoal blobs, Boyd eyes. He reminded me of the black-crayon stick people he used to draw as a child: dangly, puppetlike, white triangular faces with vast eyes like lumps of coal. A different Dominic from the suave young man at Kate's party for Sophie, from the prizewinner at speech night.
'You cow,' he said. 'You bitch.'
I was dumbfounded. How did he know? How could he possibly know what I had done? He didn't even like Max, I thought. Why is he so angry, when he didn't even like Max? I couldn't think of anything to say.
'You just don't care, do you? You couldn't give a fuck about me.'
Him? Care about him? What the hell did _he_ have to do with it?
He laughed. 'You don't even know, do you? You don't even remember.'
I did remember, then. It trickled back into my consciousness. I'd been supposed to meet him at an appointment and take him home. I'd forgotten. Could that possibly be what this was about? _Dominic_ , I felt like saying, _Dominic, get your priorities straight. A man has died here, and you're fussing about a dental appointment?_
'You're so fucking self-absorbed.'
'Dominic,' I said. _Dominic, apple of my eye, light of my life, stay and support of my feeble old age_. At least, I didn't say all of that, but I thought it, and I hoped it was strongly implied in my gaze.
'You were supposed to be there. You were supposed to take me home. I waited for an _hour_ after, and you didn't show up. How could you forget me like that?'
'I'm sorry. I know, I know, you had a dentist's appointment, didn't you?'
'I had a _wisdom tooth_ pulled out,' screamed Dominic. 'It hurt like hell. It went wrong. He couldn't get at it. You were supposed to be there. There was no one to take me home. It was bloody dark by the time I realised you weren't ever coming. I was bleeding. I was bleeding and it was dark and you didn't _come_.'
'I'm sorry. I haven't been well, honey. I'm sorry. Things haven't been good.'
'I'm supposed to feel sorry for you? Things haven't been good? Things haven't been fucking good for me, either, but you haven't bothered to enquire, have you?'
I was aghast but fascinated. I'd never seen Dominic like this. I'd seen him sarcastic and scornful and bitter and wry, but I'd never seen him flamingly angry, like this. To my astonishment I saw that he was weeping. Tears seeped from his large, black eyes and made sparkling snail-tracks down his cheeks. I didn't think I'd ever seen Dominic weep before—at any rate, not since he was a toddler.
He was fifteen. He was behaving like Kate, who sobbed over trivia and came out in awful pink stains and blotches. I'd forgotten about picking him up from the dentist, for Christ's sake, and he was crying. I couldn't understand it. _Pull yourself togethe_ r, I felt like saying. _Dominic, if you want something to cry about, contemplate the fact that your mother's a murderer. You want to weep, boy? I'll give you something to weep about._
'Honey, I'm really sorry, truly I am, but I don't see why it's so important. I overlooked it. I'm sorry if it was uncomfortable for you. I'm sorry, but I didn't mean to do it. Next time I'll be there.'
'Next time isn't the _point_ ,' he said, suddenly sounding immensely tired, even bored. 'Whether you meant to do it isn't the point. Being sorry isn't the point. The point is that you didn't come when you were supposed to. The point is that you forgot.'
'It's not my fault. It's truly not my fault, Dominic. I couldn't help it.'
'That's what you always say,' he threw at me, belligerence returning.
'You've got no idea how much I've had on my mind. I've been having a really bad time, Dominic.' I heard my voice waver. I wasn't putting it on to impress him: it was genuinely wavering, all on its own. 'Max has left me, Dominic.'
And as I said it, I nearly believed it. It was what I'd decided on. Just to say, well, we quarrelled. He left. I'd tried to think my way through what happened after that, but it was hard. I couldn't quite work it out. But it would be enough, as a first step. And here I was, taking the first step. The lie slipped from my lips smoothly and easily.
For a moment he didn't say anything. He just looked at me, his eyes wet and burning. And then he made a funny, small sound, something like a cough, half sob, half laugh. 'You know what it feels like, then, don't you?'
I found I couldn't say anything. The manifest unfairness, the sheer cruelty of it, kicked me too sharply. And then he turned and left, wheeling the bike down the front steps, mounting it in a swift, fluid movement, riding off with decision and crispness.
Absalom, Absalom.
Oh, my son, Dominic. My son, my son, Dominic.
But it forced me back into the land of the living, Dominic's visit. I started to pick up pieces. I had to. I listened to the answering machine. I opened the letters, and paid the bills. I turned up to work again. I cleaned the house, top to bottom, scrubbing obsessively; I hired a gardener; I took Borrow to the vet to get his shots. I took him for a few extra walks, too. I thought about going to see Kate, who had left two messages on the answering machine, the second one gratifyingly desperate. I thought about Sophie. I thought a lot about Sophie.
I gathered a number of Max's clothes, laying them out carefully on the bed and working out what he would need if he were to go away for an extended holiday. I packed his elegant, charcoal suitcase with changes of underwear and hankies and socks, and spare shoes and an umbrella and a jacket in case it rained.
I checked his overnight bag and cast an eye over his deodorant and toothpaste and shampoo. Max was fussy about his shampoo: he didn't like using the sort hotels give you. His gleaming black wetpack was kept fully packed: it was his habit to keep it always ready for use, so everything was in pretty good order. I replaced the toothpaste with a fresh tube, just to be sure; and I included his blood-pressure medication and his vitamin tablets.
I put the cases in the boot of my car and drove into Spencer Street Station one Friday afternoon. It was busy; there were people streaming everywhere. I felt extraordinarily conspicuous, but calm common sense (which I could dimly perceive somewhere in the distance near the horizon) told me nobody was paying me any attention. I hired a luggage locker and left the cases in it. The ticket they gave me said that if I didn't claim my property within two weeks it would be confiscated and eventually disposed of. I tore up the ticket and the printout of the lock combination and threw them out in a street bin. Then I drove home.
He'd gone away. We'd quarrelled and he'd gone away. He'd walked out of the house carrying his suitcase, all packed, all ready to go. He'd gone away and I didn't know where he'd gone. He must have deposited the case in the locker himself: it was nothing to do with me. He'd put the case in the locker and gone off to do something else and never came back. He'd met with an accident. Or perhaps there had been foul play. Perhaps he had been murdered. Why would anyone murder Max? I don't know, officer.
I looked around our bathroom and threw out Max's toothbrush and his comb, and his aftershave and a few nondescript tubes and suchlike. A minimalist in these matters, he had left little for me to do. I'd packed his favourite razor.
Max's wallet was lying by the side of the bed, where I had put it after he died. I hadn't known what to do with it. I'd taken and used some of the cash, but most of it was still there. I put the wallet back, inside his bedside drawer. I saw that there were three or four letters in there—opened envelopes, anyway, with papers in them, in a careful pile at the back corner of the drawer. I picked them up and turned them over. I noticed they had someone else's name on them, not Max's. I didn't care. I didn't want to know. I felt like an insect, splattered against the windscreen of the great truck Life. Later, I thought, I'd look at them later. Right then I didn't have the strength.
'When is later?' Dominic used to ask, when he was a child, waiting for some treat. 'When is later, Mummy?' And we made up a nonsense rhyme about it, which he would chant endlessly.
_Later, later, alligator,_
_Put him on a merchant freighter,_
_Feed him salt and fried potater, Shoot him 'cos he is a traitor._
Dominic would come back, I thought. He would be sorry, and he would come back. I would be calm and peaceful, when he did: I would utter no recriminations, no blame.
Max wouldn't come back, but I was the only person who knew that.
I went to see Kate. By then it had been, I think, six days since I'd seen her. It was the longest I'd been away from them since Sophie's birth. As I rang her doorbell I remembered that I had meant to telephone first. It had slipped my mind. I noted that, and thought that I'd have to be careful: it didn't matter so much this time, but now I couldn't afford for things to slip my mind.
I'd thought so much about this. Not so much about the visit, as what it betokened. I'd had to try to work through all of it, to work through what I was prepared to do, what I wasn't prepared to give up.
The single thing, the factor that mattered, the consideration that towered above all other considerations, was Sophie. I was worried that I wouldn't feel the same way about Sophie: I'd been wondering how it would be, now that I knew her genesis, now that I knew she was no longer just my granddaughter, no longer the daughter of Gavin and Kate. But I'd felt something new and precious enter my life when I held Sophie for the first time, and I wasn't willing to relinquish that, not without at least trying.
So far as Kate was concerned, I'd had to think my way through the morass I was trapped in. To my surprise, I didn't hate her. At first I had; at first, my hatred had swamped me. Now, however, I'd calmed down. I won't say I was looking forward to seeing her, but I was concentrating on thinking of her as Sophie's mother. Not my daughter, not Max's lover. Just the mother of this child who mattered so much.
Kate answered the door holding Sophie. Her shirt was crumpled: it had been hastily pulled down and I could tell she had been feeding the baby. She didn't look surprised to see me. She just held the door open and stepped back.
'Hello,' I said.
'Hello.'
'You're feeding? I'm sorry to interrupt,' I said, formally.
'I've just finished. Do you want to hold her?'
She held Sophie towards me and I took her. She was sleepy and cuddled into me. All the rich warmth flooded back into my veins; I felt the same mesmeric joy, the same startling physical bliss at the contact with her. I was enormously relieved. It was just the same: an entirely involuntary, entirely inevitable response. I held her and the miracle of her impressed itself on me, just as it had the first time I'd held her. I was simply filled with love for her.
I knew I was doing the right thing; I knew that no matter how this hurt me, it was something I had to do, for my own survival. The connection with Sophie was something I couldn't jeopardise, couldn't bear even to think of losing.
Kate was adjusting her bra strap, watching me. 'Coffee?'
'I'd like a coffee,' I said. 'Is she ready for a sleep?'
'She probably needs a burp. Do you want to hold her for a while?'
I followed Kate into the kitchen and put Sophie up on my shoulder, rubbing her small solid back, smelling the clean, milky smell of her. Kate put the kettle on and got together mugs and sugar and so forth.
We sat down at the kitchen table, facing each other. 'What are you going to do?' asked Kate, stirring her coffee.
'I'm not going to do anything. I'm never going to talk about this again, Kate. It's all over, it's all in the past. It didn't happen.'
'It did happen, though,' said Kate, with a hardihood I hadn't expected. 'It did happen, and we both know it did happen. How are you going to cope with that?'
'That's my business.'
'As she grows older, as likenesses develop—as likenesses _may_ develop—how are you going to feel?'
'I'll be all right. It's not her fault.'
Sophie gave a contented burp.
'Of course it isn't her _fault_ ,' observed Kate, with the shadow of a smile. 'We know that. But it's not a matter of fault, is it? Not, I mean, so far as she's concerned. She is who she is, and it may not be her fault, but she still is that person.'
'I don't want to talk about it any more. I've said that. That's it.'
Kate sipped her coffee, rubbed her chin. 'I'd like to explain. I'd like to explain how it happened.'
'I don't want to listen.' I heard my voice rising and tried to take the edge off it. 'I don't want to know anything. I never want to talk about it again.'
She chewed her lip.
'There's one thing I have to tell you,' I said. 'Max has left me.'
She looked at me narrowly and nodded, almost as if she had expected this. 'Will he come back?'
'No, I don't think so.'
'You quarrelled?'
'It's not your business what happened.'
'I suppose you're right.'
I turned slightly so that she could see Sophie's face over my shoulder. 'Is she asleep?'
Kate nodded again. 'Fast asleep. I'll put her down.'
When she came back, she sat opposite me.
'There's one thing I have to say,' she said. 'I want you to know it. I heard what you said: I understand you don't want to talk about it. But this is something I have to tell you. I have to. You always said Max was the love of your life. Well, he was the love of my life, too. I adored him. I'll never love anyone else, not like that.' 'You've got a husband,' I said, furiously. 'What about your husband?'
'What about yours?' snapped Kate. I wasn't used to Kate biting back. 'What about yours? You had a husband, too, when you met Max. Gavin is a darling. I love Gavin dearly. I'm going to be a good wife to Gavin. But there'll never be anyone else but Max, for me. I've given him up. I know he's yours.
'I'm just saying, so you know—all right?—it wasn't a whim; it wasn't something I did idly, without thinking about it. It mattered to me more than anything else in the world has ever mattered. It was that important. I didn't mean to hurt you. I didn't mean you ever to know. It was the most important thing in the world, for me. I just want you to know that.'
Her voice was starting to shake. I couldn't bear any of this. Why was this necessary? Why did I have to know anything? For Sophie's sake, I was going to try to forgive her, or at least to pretend that none of it had happened: wasn't that enough? Wasn't that more than she had any right to expect?
She stopped, and we sat there in silence.
'I'm not patching this up for you,' I said. I knew it would hurt her. I wanted to hurt her. 'I'm patching it up for her. For Sophie. You don't matter.'
Kate winced slightly. 'Oh,' she said, with a dry bitterness I'd never heard in her voice before. 'You didn't have to say that, Mum. I know I don't matter. I've never mattered, have I?'
I was enraged by this idiotic grandstanding. I was worked very thin by this stage, like pastry that you roll, and roll again, trying to make it stretch, and it starts to tatter at the edges and to spread into faint withering patches that eventually split and gape. I was stretched so far I couldn't possibly stretch any further; I couldn't cover any more space. She had no right to say such things. What right had _she_ to be bitter?
'That's a stupid and hurtful thing to say,' I shouted. 'Haven't you hurt me enough without attacking me like that?' I wanted to hit her; I wanted to hit her very hard. 'It's not fair,' I cried into her startled face. 'I haven't deserved this from you.'
I felt the surge of anger within me as a surge of blood; my head was full of it, dizzy with it. If I had had a knife, I think at that moment I would have lurched at her and plunged it into her.
Well, perhaps I would; perhaps I wouldn't. There's something about the imagined meeting between flesh and blade that's very disconcerting. But if I'd had a gun, I would have shot her. I would have shot her as much for the shocked and wounded look she turned on me, as for what she'd said, what she'd done.
'Never mind, Mum,' she said, quietly. 'So long as we do it, patch it up, I mean, so long as that happens, it doesn't matter why we're doing it, does it? If it's for Sophie, okay, it's for Sophie.'
I didn't stay long. I was shaking. I was afraid of what I might say, what I might do, if I stayed. I drank my coffee (it was awful coffee: Kate's always made bad coffee) and left.
I paused at the door, I remember, and looked at her. 'Tell the others, will you? I don't want to have to talk about it to anyone.'
'Okay,' she said.
I took another step and then I paused again and turned. 'I mean, tell them that Max and I have split up. I don't mean tell anybody anything else.'
'Am I likely to?'
'I don't know if you're likely to or not. But I'm telling you, nobody's to know.'
Kate swallowed. 'Mum, believe me, I've told nobody. I'm not going to tell anybody.'
'Promise?'
She nodded and then said in a dry, croaky voice: 'I promise.'
And then I left.
I don't recollect a lot of that time very clearly. Perhaps my subconscious has ensured that I don't. I got up; I went to work; I came home from work. Borrow stopped regarding me reproachfully: at least, I thought he did.
I made sure I saw Sophie two or three times a week. I babysat whenever I could. I must say, Gavin and Kate had it made with me: I was every young parent's dream grandmother, even if I had killed the baby's father. Sometimes, when Kate was tired, I even spent the night with them and walked around with Sophie through the wee small hours. She wasn't a good sleeper, in her first year. My infatuation with her grew and prospered. I outgrew an obsessive early propensity to trace Max's lean elegance in her tiny, chiselled features.
Fortunately, the likeness was intermittent, fleeting: it depended more on expression and angle than anything else, and seemed to me moreover to diminish with time, although that was a hard thing to judge. It did not take long for Sophie to acquire a presence and personality that were all hers and had nothing to do with her progenitors—although people often said (they say it still) that in her dark, quick slenderness, her neat limbs and fine bones, she took after me. These days, I look at Sophie and think about her and speak with her and hug her and love her without even thinking of Max.
Well, perhaps not entirely. But almost.
One evening, quite early in the piece, Zoë visited. Kate had told her Max and I had split up. I knew I'd told her to tell everyone, but then I wished she hadn't. There she was, my sister, when I opened the door, arms extended, replete with coo and drivel.
'Minky!' she cried. 'Minky, darling!' She enfolded me in her capable embrace. I stepped backwards, but it was impossible to evade. Zoë in full flight is simply something that can't be evaded. In she came, glancing around her, absorbing every detail even as she surrounded me with her crisp perfume and her unctuous readiness to comfort and support. She wanted me to throw myself upon her breast. _Come home, Minky,_ she wanted to say. _All is forgiven._
Well, I wasn't going to do that, and if she had learnt anything at all about me during the forty-two-odd years we'd known each other by then, she wouldn't have expected it. She wanted me to weep on her bosom because my love had left me; she wanted me to admit my sins and tell her how right she'd been and how sorry I was. She wanted me to pour it all out to her, open my heart, weep. She wanted to know every last, sordid detail. She was disappointed. _He's gone for good?_ she wanted to ask me. _Never to return? Never mind, darling, you're better off without him._
I wasn't better off, and I wasn't going to pretend I was. Not then, not ever.
In fact, one of the major irritations of this time and the following few months was the way in which people treated me: full of sober consolation, a reverential and pitying circumspection as if Max had died. Which he had, of course, but they didn't know that. What had happened to me was much worse, much more traumatic, than a mere death: it seemed to me I deserved far more sympathy than anybody was prepared to give me; and it exasperated me that I could explain this to nobody, could weep truthfully on nobody's shoulder.
When I told Bea, she was just as pleased as Zoë was, but she handled it with more tact. (Zoë and tact have always been strangers to each other.) I told Bea that Max and I had split. She nodded non-committally, her eyes on my face.
'Are you all right, Izzie?'
'Yes.'
'Do you want to talk about it?'
'No.'
'Okay,' she said. 'Thanks for telling me. Any time you want a cup of tea, or something stronger for that matter, let me know.'
And that was that. I dare say she jumped up and did cartwheels when I'd left her office, but at least she didn't make her delight evident. Nor did she display Zoë's itch to find out exactly what had transpired. Zoë never did like it when I had secrets from her.
When you have secrets in a family, after a while you start to wonder who knows them. Who knows, who guesses, who hasn't a clue? Sometimes, when our family gets together on one of those appalling occasions—birthdays, Christmas, whatever—I glance around the room; I briefly study their faces. Who knows the identity of Sophie's father? Does Zoë? Does Gavin speculate? Dominic? Steve? No, surely not Steve. Has anyone else detected Max's finely drawn mouth in Sophie's exquisite lips, his long, slender hands in her small, neat fingers? Who knows?
I found I became edgy on my own in Rain. I developed the habit of keeping the radio on at all times, so I never walked into silence. I still do that, even now. I can't bear a silent house. I hated sleeping in the capacious bed, in the vast acreage of the main bedroom, under the skylight, the heavy square sky always falling into my eyes, into my head. I depended on Borrow a lot. Once, I remember, he had to spend a night at the vet's for some minor operation. With the comfort of his presence withdrawn, I hardly slept.
I'd never been a nervous person, and I didn't believe in ghosts. Well, I didn't think I did. But solitude didn't suit me any more; and Rain, which had been a comfortable house for two or three of us, became ridiculously large. I thought of selling: this was one step I could take without legal complications, since Max had given the place to me. But it was all too hard and required too much energy, too much thought and action. In any case, I loved Rain; it had housed my happiest days. I tried not to think of what lay in the garden, under the smart, new flowerbeds. I procrastinated.
One night, about four months after Max's death, a light knock came at the front door. It was slightly later than I would normally have expected Foxtel or Jehovah's Witnesses, and I opened up cautiously, wishing I'd locked the security door. A small and unremarkable man was standing there. He was balding and his hands were in his pockets. He wore a striped shirt and a dingy bomber jacket; his face was lined, ratlike and attentive.
'Can I have a word with Matty?' he said. His voice had the hint of a soft lilt to it. Irish?
'Matty?' I said. 'I'm afraid you have the wrong house.'
He shook his head cheerfully. 'The fool I am. It's Max I'm after.'
'Max?' My mouth was dry. Silly of me, not to be more prepared, when it happened. I had known moments like this would come; I'd rehearsed them. 'I'm sorry,' I said. 'Max isn't here.'
'When will he be back?'
'No. What I mean is, Max doesn't live here any more.' It sounded absurd, when I said it, like a pop song.
His eyes widened slightly. 'Then it's a forwarding address I'm after.'
'He didn't give me one.'
'I find that hard to believe.'
'Sorry,' I said, starting to shut the door.
In a quick and supple movement the little man prevented me from doing so, and suddenly, instead of being outside the door looking in at me, he was inside the door, next to me, his hand closed hard on my wrist.
Borrow was just behind me, having ambled after me to see who was at the door. He barked once, uncertainly.
'Good dog,' said the bald-headed man, leaning over and scratching him behind the ear. Borrow wagged his tail and slobbered cheerfully over the man's hand, glancing at me smugly. 'You'll forgive me. I'll take a quick look around.'
My heart was pounding so hard I could hear it in my ears. 'You can't do that. Get out.'
He was already halfway up the stairs. Borrow watched him amiably.
'I'll call the police.' My voice sounded quavery and foolish. 'You can if you like,' he said, laughing. 'I'll be gone in two minutes, and it'll take them forty to come, minimum. So no harm done. But Matty mightn't like it.'
'There is no Matty,' I called after him. I followed him up the stairs.
He was a quick mover: he'd already looked in the ensuite. He had returned to the bedroom and was inspecting Max's wardrobe, which was pretty well non-existent: most of what I hadn't packed in the suitcase I'd taken by then to the Salvos.
He turned to me. He looked oddly at a loss. 'What happened? He's gone. I can tell he's gone. Where did he go?'
'I don't know,' I said.
We stood there and eyed each other.
'I'll be on my way,' he said, eventually. 'But I'll be back. When you hear from Matty, you tell him Colin wants to speak with him. Okay?'
And he was down the stairs and out the front door. I followed him to the landing and called after him, but he'd gone. I looked out the window and saw him get into a nondescript dark blue hatchback and drive off.
I sat down on the bed. My heart's hammering had taken over my body, and I concentrated unsuccessfully on diminishing its force. My legs felt weak. It had happened so quickly. Well, nothing had happened, really, but what might happen? What might not happen, next time? Was I to be perpetually constrained by my guilt from taking action whenever I was threatened? I seemed to visualise a future as in a series of oblique mirrors, a conga line of endless bald-headed little men bursting into my house whenever they felt like it, charging up my stairs, searching my bathroom cupboard and my wardrobe, peering under my bed, through my drawers.
And what did he mean by _Matty_?
I stood up and my legs felt so wobbly that I sat down again, speedily. Eventually I lurched downstairs, gripping the rail, settling both feet firmly on each tread before attempting the next. I reached the kitchen safely, poured myself a slug of brandy in one of Max's elegant Swedish tumblers.
I sat in the kitchen and sipped it until I felt its fire enter me and strengthen me. I'm turning into an old soak, I thought. Off to the brandy bottle every time something goes wrong. It was true: I had been drinking too much. It always seems more, and worse, when you do it on your own; but it does help so.
Then I grabbed the keys and locked the security doors, front and back. Borrow accompanied me, full of benign importance.
'You weren't any bloody use,' I said to him, bitterly. 'Call yourself a watchdog?'
He wagged his tail happily and licked my hand.
I went back upstairs. I sat on the bed and opened the drawer in Max's bedside table, where I had kept his wallet, where I had noticed the envelopes. They were still there, four of them, the addresses all handwritten but all in different writing. The address was the same for all: a post-office box in North Ringwood.
Why North Ringwood, God help us? We lived in leafy Hawthorn: why drive out to the godforsaken edge of the city to collect your mail?
Two envelopes were addressed to Max Knight. One was addressed to Martin Ritter. The last was addressed to Matthew Templar. They were all neatly slit open at the top. Max always used a rather evil-looking letter-opener with a sharp curved blade, like a miniature silver scimitar. It was still in the drawer.
I sat and looked at them. Some ancient schoolgirl memory awoke in my brain. I went downstairs, carrying the envelopes gingerly, and found the old German dictionary.
It was as I had remembered, or half-remembered. 'Ritter' was German for 'knight'.
Hadn't the Templars been crusaders? Knights? Max Knight. Martin Ritter. Matthew Templar. Max Knight. Martin Knight. Matthew Knight.
Oh, Max, Max. If that was your name, which it seems it wasn't. What a wag you were. It would almost have been funny, if it hadn't been so frightening.
I turned the envelopes over in my hands. There was little point now in being prissy about Max's privacy. I opened them one at a time, taking care to keep the contents of each next to its envelope.
Inside the two envelopes addressed to Max Knight there were two cheques, one for twenty thousand dollars and one for thirty-two thousand dollars. They were made out to Max and stamped non-negotiable. The signatures were the same, but illegible.
Inside the envelope addressed to Martin Ritter there were four photographs—black and white, roughly cut at the edges—of the kind you take in passport photo booths. Each was of an Asian girl, head and shoulders. None looked older than fifteen or so. A name was written in black ink, in delicate careful handwriting, on the back of each. There was May, Susie, Kylie and Lindy Lou. All were pretty and smiling. Susie's expression was a trifle forced, but the rest of them beamed at the camera like true professionals. A folded piece of flimsy paper—like old-fashioned airmail paper—was also in the envelope. On it, in Max's unmistakeable spiky writing, was written: _SQ 277, 14.50, 15/7._
In the envelope addressed to Matthew Templar there was some stiff yellow paper with a number typed on it, apparently by an old-fashioned typewriter. At first I thought it might be a telephone number, but it had too many digits. Fourteen digits. It could have been the number of a bank account.
I sat and regarded these things and heard Bea's voice in my head. _What does he do? Where does all the money come from? Tax evasion? Prostitution? Drugs? White slave traffic?_ I heard the despairing pitch of her voice. _You know nothing, nothing about him. What in Christ's name are you doing?—_ I looked at the paper with Max's writing on it. _SQ_? Flight prefix, Singapore Airlines? Yes, I was sure of it. I'd picked Max up at the airport from Singapore Airlines flights. _14.50_ that would be the time. _15/7—_ date.
I looked at Lindy Lou and Kylie. I found I didn't want to know anything about them. I didn't like the expression in Kylie's eyes. I fingered the cheques. One of them, I noticed, was from a German bank, although the envelope had been locally posted. I realised I didn't even know what bank Max was with. I'd seen him use ATMs, from time to time, but I'd never seen him walk into a bank.
I went back upstairs and replaced the envelopes in the drawer. I took out the wallet. I'd looked at it briefly, before, but it was time for an exhaustive search.
There was still a thousand dollars, more or less, inside, in crisp hundred-dollar notes. I'd used a lot of the money by this stage: I'd regarded it as a kind of informal banking arrangement, one that hadn't required me to find an ATM or a friendly teller. (People still used tellers, sometimes, in those days; and some of them were friendly.) I hadn't used the credit cards, nor examined the rest of the wallet's compartments. Now, I turned it inside out.
There were five credit cards, one of which was AMEX and the others different versions of Visa and MasterCard. All were gold, or platinum, or whatever signified a VIP cardholder in some way. All bore Max's name except the Visa. That belonged to Matthew Templar.
There was a driving licence. I examined this carefully. It was made out in the name of Maximilian Knight. His date of birth was 4 August, which was when we'd always celebrated his birthday. His age here was consonant with what he'd told me. He'd been fifty-two when he died. I turned it over in my hands, feeling it, weighing it, squinting at it.
I retrieved my own driving licence from my purse and compared the two. If Max's licence had been forged, the forger had been talented. Reliable. Well, Max liked having reliable people working for him. He'd often told me that. When Rain was being built, he'd been angered by unreliability and tardiness. If somebody told him something was going to happen, it had to happen.
It was ridiculous, I thought crossly to myself. Why should I expect Max's driving licence to be forged?
Max Knight. Martin Ritter. Matthew Templar.
I went back to the wallet. His Medicare card. His private health insurance card. Both apparently in order, the name of Maximilian Knight embossed on the plastic.
There was the photograph of me, of course.
There were two receipts. One was for petrol, the other for the dinner we'd eaten at a local restaurant the week before he'd died.
What would happen to the credit cards, I wondered. To the accounts, I mean. Did he use them? Where would the bills be sent? To North Ringwood? They didn't come to Rain, I was sure of that. All the household bills—electricity, telephone, and so forth—came to the house, as well as the MasterCard we shared, but none of these. What would happen when they were not paid? How much was owing on them? Did the banks have Rain's address?
There was nothing else in the wallet. Bare and neat, it gave nothing away.
I wondered if I should go out to the post-office box at North Ringwood and investigate it. What illicit and unwelcome mail might be accumulating behind its small, bland exterior? Would I need to provide identification? I'd never used a post-office box.
What had the little man meant when he said Matty wouldn't like it if I were to call the police? Who had I been living with? Who was this man I had married?
It came to me that post-office boxes were small and black and mounted in rows on walls outside post offices. I'd seen people unlocking them and walking away. You walked up; you unlocked your box; you walked away. Nobody stopped you; nobody asked for ID or anything else. You just turned up.
With a key. You turned up with a key.
I'd put Max's keys in the suitcase in the station locker. I hadn't anticipated needing them. I'd glanced quickly at them: I'd seen the house keys and the car keys, of which naturally I had duplicates. There must have been other keys: the post- office-box key, for example; perhaps the key to a place of business; perhaps the key to a house where Kylie and her friends lived.
I'd never seen Max with other keys, only the bunch I now regretted disposing of. I looked quickly through the other drawers in the bedside table, but there was nothing there apart from a couple of prescriptions, a nail file, a calculator, the vicious little curving letter-opener, a packet of mints: the sorts of innocent items any man might have in his bedside table.
Max had a study, too, of course. I had taken some delight in creating it. It had smoke-grey carpet, burgundy leather chairs, a magnificent mahogany desk, framed Chinese prints on the walls. The furnishings were all dark and sumptuous, but the room itself was bright and light-filled because I had designed it with two huge north-facing windows that looked down over the pool. Where the pool had once been.
Max had always joked that it was the only design fault the house had, that his study window lured him away from his study.
He hadn't spent a great deal of time here. Maybe two or three hours a week. He had always kept it fastidiously tidy—never with piles of papers lying around, never any mess. I'd hardly been in it, since he'd died.
I came in now, looked on the desk, in the desk drawers. Bills, a birthday card I'd given him. Pens, paper. His Mont Blanc fountain pen. The kind of bits and pieces you expect to find in a desk. A folder with some papers in it. House insurance. Car registration. Max had remarkably few papers, all things considered. I was amazed by how little there was.
What on earth had he done with all the paraphernalia of his life, the endless forms and identification papers, the letters and bills and receipts, the shifty bits of paper that flutter through life around us, mapping our lives, exposing our histories, defining our limits, recording our tiniest transactions? If he owned shares or stocks, if he conducted mysterious business through consultancies, if he made money, surely there must be paper trails somewhere, trails that I or the tax man or anyone else could follow?
My own desk, downstairs in a study-cum-sitting room where I very occasionally worked and which we called the studio, was crammed with such extraneous material.
No cheque book, no bank book. No passport. And no keys.
There was a filing cabinet, though. A handsome, shining, black two-drawer filing cabinet. Of course. That was where everything was. But it, too, was locked.
So many keys, I suddenly needed! So many keys, where I had thought none existed. So many locked doors dividing me from the things I needed to know.
There was nothing I could do about the box at North Ringwood—or, at any rate, nothing I could see at that point. But the filing cabinet belonged to me; it was in my house and I had some kind of power over it, control in a zone where control was spinning away from me.
I rang a locksmith the next morning and it took him approximately two and a half minutes to open the filing cabinet. He was a middle-aged chap with grey stubble over his chin, a bag full of tools and a mouth full of chat. I said as languidly as I could manage that I'd lost the keys, such a bore, just couldn't find them.
'Ah, it's a bugger, isn't it?' he said, settling down to work. 'You lock something up all safe and sound and you pop away the key in a nice safe place and the next moment you can't remember what you've done with it. A real bugger. And there'll be something in here you need as a matter of urgency, I suppose? There always is.'
'Just some papers. They're not really urgent, only that I do need them, sooner or later, and I thought I might as well get it all done before there's any urgency.'
I don't know why it is that whenever I lie I rattle on compulsively, adding trivial details which seem to me to lend conviction to an increasingly threadbare story.
'Well, you know,' he said, 'now that you've called me and tackled the problem head on, you'll probably find those keys tomorrow.' He seemed to take a lot of pleasure in his forecast.
'It always happens,' he said, grinning. 'And you'll be saying to yourself, my goodness me, why did I bother to go and get that bugger of a locksmith, when all along the keys were right here? That's what you'll be saying to yourself.'
All the while he was working, it seemed to me, his beady eyes were glancing around the room, noting its contents, memorising the layout. I wondered if I were becoming clinically paranoiac.
'Here we go,' he said, easing the lock out. 'Not a hard one, this one. Here we go, nice and easy. See, I haven't had to do much damage at all: I can pop in a new lock for you and you'll never know anything happened. Good as new.'
He rolled the top drawer out. Apart from a half-dozen or so suspension files, it was completely empty. He glanced at me. 'Nothing in this one.'
'They're all in the bottom one,' I said, off-handedly.
He paused. 'Will I pull it out for you, then?'
'Yes, of course,' I said, because it seemed foolish to say anything else; I thought it would seem suspicious to prevaricate, to demand privacy. The worst that could happen was that the lower drawer, too, would be empty.
He pushed the top drawer back in and opened the lower one. I heard the intake of his breath. I glanced down and saw the revolver lying at the bottom of the drawer.
I couldn't think of anything to say. Nor could he.
'That's what you expected?' he asked, finally.
'It's my husband's gun,' I said, reflecting that this at least was certainly true. 'I didn't realise he'd put it in here. He's away at the present. He's on a business trip.'
'He's got a licence?'
'Well, of course. He shoots with a local club.'
Amazing, how rapidly one could invent things.
Both of us were staring at the evil-looking thing.
'It's a Smith and Wesson,' he observed.
'Yes.'
'The sort James Bond uses. Isn't it?'
'I have no idea,' I said. In fact I didn't think it was. Steve was crazy about James Bond and some of it had rubbed off on me: I seemed to remember that James Bond customarily carried a Walther. But I wasn't about to engage in a debate on such matters.
'Is it loaded?'
'I don't know. No. No, I shouldn't think so.'
'Well then. I'll put this new lock in for you, will I?'
'Thank you,' I said, coolly.
When he left, he asked me for an amount I found surprising. It's not unusual, of course, for tradesmen to do this: what's unusual is for them to charge you something you think a fair amount; but it really did seem exorbitant.
It occurred to me I was being blackmailed, that he felt he could freely ask a ridiculous sum because he sensed something dodgy going on. He was right, of course. I paid him without query. I suppose it was cowardly of me, and perhaps stupid, but I so overwhelmingly wanted him out of the house, I probably would have paid him three times as much just to get rid of him.
When he had gone, I returned to the study and took out the gun. I weighed it in my hand. It wasn't bulky, but was still surprisingly heavy. I had always thought revolvers like that tucked neatly in a pocket or holster, that you wouldn't really be aware of their presence about your person until you were in need of them. This was so heavy, you couldn't possibly forget it. I examined it carefully, and wondered suddenly if it had Max's fingerprints on it, if my own fingerprints had just obliterated his. I knew so little about guns that I did not even know whether it was loaded or not.
It is a strange and frightening experience to handle a gun. The sight of a gun is so recognisable: it is familiar to us from movies and television, from gangster fiction and war documentaries; it is a part of our culture, a common pattern on the wallpaper of our lives.
Yet actually to hold one, to consider it and balance it, to stretch out your arm and point it as if you were truly threatening someone—these are foreign and bizarre actions, as alien to the comfort of our souls as anything can be. The object exerts an uncanny fascination, so that touching or stroking or gripping it forces you to recognise that it is so much more than a mere object, so replete with its potential and with the menace deriving from its function. It's unlike anything else. Its function is unmistakable and singular, and it cannot be contemplated without reference to that function.
I could not accommodate the gun to the texture of ordinary life. I felt like a character who had landed in the wrong movie. I'd been looking all my life for something along the lines of _The Princess Bride_ , and instead I'd found myself deep in the latest Tarantino release. I returned the revolver to its drawer and, a deep and unfamiliar disquiet in my heart, locked the cabinet with the shiny, new key the locksmith had provided.
That evening I sat down with Borrow and a brandy and reviewed my situation.
It was plain enough, I suppose, if unpalatable. I was a widow. I was a widow of means, but it appeared to me that any of my means might be stripped from me at any point. My own income was adequate but not magnificent. Max's bequest was huge, but Max's bequest appeared to be not just a house but a house on sand. Pretty wobbly sand, too.
My marriage had been to a man I had never known. I had loved this man more dearly than I had ever loved anyone in my life, more (I was convinced) than I would ever love anyone again. Yet he had betrayed me monstrously and it seemed reasonable to assume or at least to speculate that his wealth had been accumulated by unorthodox and possibly illegal methods. If this were so, I supposed I might, as the inheritor of his possessions, face criminal charges. I might be convicted of aiding and abetting. I might in fact have aided and abetted, without realising I had done so.
And he was dead. Nobody knew this but I. Nobody else knew I was a widow. And, given the circumstances of his death, that gave me some protection.
Where did this leave me? Was I in danger? It seemed preposterous to regard myself as in danger of any kind, but I had lost my toehold on the tectonic plates of my life: they were slipping and sliding and could finish up anywhere. The gaps they left in the wake of their eruptions were alarming already, and seemed to be widening speedily. I was going to fall down one of those crevices any day now.
It was like inventing a whole new landscape and inserting myself in it. It was something of a desert, this landscape, with many thorn trees and a good deal of hot sand and not much in the way of oases.
So I had to find my way out of it. But it was hard to make my mind move. Normally I am a quick thinker, I believe: but now, stymied and shocked as I was, trying to devise strategies was slow going. I tried to force myself into a logical train of thought, imagining outcome, calculating consequence and risk.
All I could really think of was that I had known one fine, glimmering light, one unrepeatable moment of apparent perfection, one dazzling flame of passion and—yes, purity—and that I had permanently quenched it. Yet the darkness I inhabited was not my fault. It was the fault of those who had betrayed me. The dazzling flame had been treacherous.
Well, then, it must be my responsibility to discover the sunlight again. It wouldn't ever again have the brilliance that had drenched me with Max's coming, but surely I could ferret my way through the gloom that had descended; surely I could find a point from which I could start to try to refashion my life.
When Steve was trying to work something out, he used a thing he proudly called his Thought Strategy: he had been taught it at some staff development seminar or other.
SWAT. Strengths, weaknesses...
I ran into a brick wall here. Strengths, I muttered to myself. Strengths, weaknesses. Borrow whimpered quietly in his sleep. I stared at the wall.
I had it. Not SWAT but SWOT. Strengths, weaknesses, opportunities, threats.
What were my strengths? Off-hand, I could bring none to mind. Weaknesses there were aplenty. Ditto threats. Opportunities? Concentrate, concentrate. I felt my mind slipping off track. I had another brandy.
Well, if I had a strength it had to be financial stability. I had enough money. On the other hand, of course, I had now to suppose that Max's ways of making money might have been at least open to question, in which case I might have my money taken off me. Since I had been married to Max, I wondered, might I be liable for his debts? It was surely likely. What had Max done, exactly? Photographs of young women were not in themselves incriminating items. They might have been models, actresses, applicants for respectable jobs. Because a mysterious man called Colin had burst into the house, it didn't follow that Max had been involved in illegal activities. Was I being ridiculous?
But there was the question of the name—the three names. And now there was the revolver, black and deadly in its dark drawer. And, the more I thought, the more it seemed to me I had been unforgivably stupid.
Max had never answered a straight question on the origins of his improbable wealth. I'd never asked him one. I couldn't get Bea's voice out of my head. _Tax evasion? Prostitution? Drugs? White slave traffic?_ I recalled his fabulous largesse, his evidently limitless funds, his casual teasing whose function was finally always evasion. I recalled the business trips whose purpose and outcome remained cloudy. Well, murky rather than cloudy, one had to say.
But we had known each other intimately, hadn't we? We had been transparent to each other, hadn't we? There was no point in dwelling on any of that. What I had to do was focus on what I had to do.
It seemed to me that the first thing was to sell Rain. Some little time previously, the thought would have dismayed me, struck a dead chill into the roots of my soul. But I couldn't bear Rain any longer. That man Colin had invaded with such ease, such neat, irritating panache; and there would be more invaders, I was sure. I could lock them out only for a brief spell: they would find their way in and would torment me. It was up to me to evade them, to make it impossible for them to up-end my life.
And Max lay outside, under the smooth turf and flowerbeds of Rain's brand-new back garden, under the rubble of the cool and gleaming pool I had designed for him. No, I didn't want to stay.
Well, Max had given me Rain. The title was in my bank: nothing mysterious or evasive about it, thank Christ. It belonged to me outright, and there should be no difficulty in selling it. I would call an agent the next day. Max had invested enough in the house: property was booming and probably the investment would prove to have ripened speedily.
What next?
I hadn't changed my name, of course. If people wanted to find me, it would be Isabel Weaving they would have to find, not Isabel Knight. Nor Isabel Ritter, Templar, whatever. At first this seemed rather a strength, but, as I thought my way through it further, I didn't think it mattered much. If Max had mixed with the sort of pals who were likely to seek me out and (for instance) gun me down, they presumably had kept tabs on me enough to know my name and my whereabouts.
Yet why would anyone want to harm me? If they thought Max had absconded with funds; if they thought I knew Max's whereabouts...Or, to put a different spin on it, if they thought Max had absconded but that I had access to the funds. Then I might well be a person of considerable interest to them.
Surely, however, nobody would guess the truth? Nobody would guess what had really happened to Max. Would they? The truth was so unlikely, so extraordinary and implausible, nobody could possibly guess it.
This was a strength, I thought, not a threat. If it were an opportunity of some kind, I couldn't see how.
Then I thought of the revolver. My first instinct had been to creep out at dead of night (an evocative phrase, that) and hurl it into the nearest deep-flowing river. Perhaps, however, I should keep it as protection? Yet surely that was laughable: I, who had never fired a gun in my life, to use the thing for safety. I'd be more likely to shoot myself in the foot. Or the neck. And if I didn't manage to kill myself, I'd surely kill someone else. That would make me a double murderer. On the way to serial status, in fact.
So. I'd sell Rain. I'd get rid of the gun.
I was starting to feel mildly better. Until now I'd been like a one-finned fish, swimming in circles and sinking to boot. If I could regain control—or at least the illusion of control—over my life, I might regain something more than control: I might feel that sanity and peace were possible again. For some time, now, they hadn't looked like accessible options.
The sooner I acted, the sooner I would be able to disentangle myself from a scenario whose ramifications were becoming alarming.
I rang three agents the next morning and chose the one I thought the sharpest and least scrupulous. You could see from the glitter in his eyes how much he wanted the property, how far he'd go to get it and to sell it for a top price. He wouldn't even have to lie. He didn't know about the back garden and what lay in it.
'I want to move fast,' I said, and he nodded ecstatically.
The house was clean already, but I got cleaners in. I got gardeners in, too, and I rented a huge skip and spent two or three days manically tossing out as much as I could.
I spent five solid days driving around looking at units and houses nearer to Kate and Gavin and eventually found one I liked enough to imagine myself living there. I put a deposit on it and told my agent that I wanted Rain sold within a month. As it turned out he managed it in ten days.
Then there was the gun to dispose of. It was easy, in fact. I took Borrow for a long walk, one night. It was dark and chilly; I wore a parka. In its deep left pocket was the gun, with a heavy axehead (I'd found it in the garage; I don't know where it came from), swathed tightly together in an old rag. It knocked against my thigh as I walked.
Borrow was excited: for some reason he loves walking at night and at first he tended to caper slightly, puppylike, displaying his pleasure in the unexpected treat. It was a long walk, though—nearly two miles, down to the river—and he settled eventually, padding beside me contentedly.
When we arrived at the bridge, I threw the awful package over. It splashed, but nobody was there to hear it. Cars spurted back and forth; a tram rumbled past. No other pedestrians were around. It was so dark that I couldn't see down into the water. I turned and headed home, faintly surprised that it had been so easy. A criminal life appeared less demanding than I had always believed.
Then there was the car. The silver Audi, his pride and joy, which had stayed in the garage since his death. It was the third silver Audi he'd had since I'd known him, his third pride and joy.
For Max, a new car had been like a new suit—perhaps even a new shirt. He had liked the luxury of new cars but the reliability of what he knew, the comfort of what he was accustomed to. He'd been happy with a silver Audi (well, why wouldn't he have been?) and that was what he'd continued to buy. I imagine the salesmen at the Audi dealer knew him well.
I couldn't throw the car in the river. The registration papers were in the folder in Max's study. It was Max's, not mine. I couldn't sell it. I didn't want to drive it.
If Colin came back, if anyone else came, could I give away the car, use it as a bribe? No: that wasn't a good idea. It was too much: it would make them suspicious. More suspicious. I imagined the scene: Colin sliding into the house again, refusing to go until I told him where Max was. Or Matty. Or Martin. Whoever. _I don't know_ , I'd say. _I don't know, but here: take his very expensive car. Will that do?_
No. That wouldn't look good.
I had keys, of course: each of us had owned a set of keys, although I had my own car and had rarely driven the Audi. I should have got rid of it somehow at the very beginning, I thought. I hadn't been thinking properly, not to do that. But then, of course, nobody knew that I hadn't. Nobody had seen inside the garage. The natural thing for Max to do would be to drive away in his car, and I could still make it look as if that's what had happened.
So I drove the car out the next Friday evening, when the shops were open late, and headed for somewhere a long way away. I'd decided a big shopping centre would be best.
I'd gone through the glove box and removed anything that would provide easy identification. Not that it would matter in the long run, of course; the registration would identify the car and its owner. But I figured the more time I gave myself the better it would be.
I parked the beautiful silver thing in a multi-storey car park in a shopping centre on the city fringe. I'd never been there before, and I didn't plan to return. I lowered a back window a tiny bit, placed the key in the glove box, patted the Audi on its sleek and bulbous bonnet and caught a bus back into the city.
So that was the end of the car, I thought. For the time being, at least.
These decisions, these actions, gave me a new sense of control, a sense that I had some sort of power over what was happening to my life. Until now I had felt perilously like one of the crude cartoon characters in the computer games of the time—dashing hither and yon, avoiding random assaults from traffic, buckets of water, arrows and boulders. Dominic had a game like that: he was remarkably good at it. With Dominic at the joystick the round-headed man had a genuine chance. I didn't know how much chance I had: I seemed to live with a permanently breathless feeling that a boulder was about to drop on me and shatter me.
And the boulder did drop, one day. The sharp-eyed real-estate agent had sold Rain, but I was still living there. I was about to leave: I think the move was scheduled for two or three days distant, and I had started to congratulate myself on evading the buckets of water and the arrows. Colin had never reappeared. Nobody else had descended on me. It seemed I was safe.
I was drinking black coffee before going to work, glancing around me at the denuded, too-clean kitchen and wondering how I would adapt again to an ordinary, little kitchen—without a long sweep of polished granite bench, without dozens of smooth capacious cupboards, without a vast walk-in pantry—when the knock rattled the front door. A purposeful, deliberate knock, a knock that meant business.
This time, it was a policeman. I knew he was a policeman because he was in uniform. I thought I would have preferred a gangster.
He was a middle-aged, thickset man with short spatulate fingers and pale eyes. He didn't look friendly. He flashed his ID at me. I tried to look nonchalant and slightly surprised. I raised my eyebrows in what I hoped was an approximation of a kind of lazy detachment, and unlocked and opened the security door. I always kept it locked, now.
'Frank Pritchard,' he said. 'Inspector Frank Pritchard. Mrs Knight? Mrs Isabel Knight?'
'I don't use that name,' I said.
'You're married to Maximilian Knight?'
'Yes,' I said, with a _so-what-if-I-am?_ inflection. 'But my name is Weaving.'
'Could I have ten minutes of your time, Ms Weaving?'
There wasn't much point in saying no. And I was impressed by the carefully enunciated 'Ms'. Most policemen, I felt, wouldn't have been up to it: they would have messed around with Mrs and Miss and finished up mumbling something that could have been anything.
I led him into the lounge and saw him glancing around with what struck me as disproportionate interest. I gestured vaguely at the sweep of cream leather and he sat down. Some people felt overpowered by all the leather and perched on the edge of the seats. Inspector Pritchard wasn't. He sat firmly foursquare, and gave me his full and undivided attention. I reciprocated.
'Mr Knight's not in?' he said.
'Mr Knight hasn't been in for some time.'
'He no longer lives here, ma'am?'
'That's correct.'
'Might I ask on what date he moved from this residence?'
He really did talk like that. And he really did have a notebook, too, and a pencil.
'I don't remember the exact date,' I lied. 'It must be six or seven months ago.'
'You're not expecting him back?'
'I'm not.'
'Do you have a forwarding address?'
'I don't.'
'A contact number?'
'No, I don't. Look, what's this all about?' I injected mild irritation into my voice.
'We're conducting an investigation,' said Inspector Pritchard, examining the end of his pencil.
'Into what?'
'Into the whereabouts and activities of Mr Knight.'
'I know nothing about either.'
'Forgive me, ma'am, but if you're married to Mr Knight I think you must have some knowledge of Mr Knight's activities.'
'I don't regard myself as married to him any more.'
'Do you mind explaining that to me, ma'am? Are you divorced?'
'Separated.'
'Since what date, may I ask?'
'I'm not sure that it's any of your business.'
'You'll let me be the judge of that, perhaps.'
'Well, perhaps I won't,' I said, thinking that perhaps it was time I started to show a bit of temper. He wasn't being rude or unpleasant, but he had control and he wasn't about to relinquish it. But if I was too subservient, if I tried to mollify him, wouldn't that make him think I was hiding something? On the other hand, I didn't want to get him offside.
'You'd be well advised to cooperate, ma'am.'
Nowadays it's quite common to call women _ma'am_. Those infuriating people who ring you up and want your money call you _ma'am_ ; so do people behind counters, and ticket inspectors, parking officials, people like that. In those days it wasn't so usual. It made me feel uncomfortable, as if he was jeering at me up his navy serge sleeve.
'I don't understand,' I said, assuming the tone of someone who was trying to be reasonable against mounting odds. 'I don't see why you're conducting an investigation. Why won't you tell me? If my husband had done something wrong, something criminal, don't you think I'd know about it?'
'Yes. Yes, I'm thinking you would know about it. So why don't you tell me what you do know, Ms Weaving?'
'Like what?'
'Like what does your husband do?'
I stared at him. Anxiety sprang a level or two and bloomed into panic.
'What is it, ma'am?'
'I'm wondering if I should get a lawyer.'
'If you want to get in a lawyer, Ms Weaving, you're entitled to. At any point. But I'm only asking about Mr Knight's occupation, after all. I'm laying no charges. I'm not thinking I'm going to arrest you. I'm not even necessarily thinking I'm going to arrest Mr Knight, when I find him. I'm not thinking of laying charges, at present. I'm gathering information, you see. It's a friendly chat. You'd be ill-advised, I'm thinking, to call in a lawyer on the basis of me asking you for a friendly chat.'
'Well,' I said, uncertainly.
He waited.
'What do you want to know?'
'Principally, I want to know how I can contact Mr Knight.'
'I can't tell you that.'
'No idea at all, ma'am?'
'Honestly. None.'
'No leads, no possibilities, no friends he might be staying with?'
I shook my head. 'I just don't know. I swear I'm telling you the truth. My husband—my ex-husband I think of him as—didn't have many friends. I didn't know them, anyway.'
'How long were you married, ma'am?'
'Nearly five years.'
'You were married for five years and you don't know any of his friends?'
'Well, I didn't. It may sound odd. But he was a solitary sort of man.'
'Business associates?'
'I never met them.'
'What does he do, ma'am?'
'He had...business interests,' I said, desperately. 'He owned things and he rented them out and he bought things and sold them. He was a consultant. He advised people about investments and...and things like that.'
'Ma'am, are you aware that you're speaking in the past tense?'
'It _is_ past, for me,' I said, kicking myself. 'I suppose he still does all of those things, but I don't have anything to do with him any longer, so I think of it all in the past tense.'
'When did you divorce Mr Knight, ma'am?'
'I'm not divorced. We're separated. I told you that.'
'Are you expecting a divorce?'
It was a funny question, when you thought about it. Not that I was grinning.
'Yes. I suppose so. I don't know.'
'So how did he come to leave?'
'We had a quarrel.'
'A quarrel.'
'That's what I said, yes. It was a very significant quarrel. We'd been unhappy for some time,' I said, improvising frenetically. 'It was just...the last straw.'
'And he left?'
'Yes, he left.'
'He just walked out the door?'
'No. He packed. He packed lots of things. And he left.'
'And you've heard nothing since?'
'Nothing at all.'
'Forgive me, ma'am,' he said, with a sudden unexpected gentleness, 'but that's hardly likely, is it? I'm thinking, a man isn't going to walk out of your life as quickly and easily as that, is he? There must be things you have in common, things that needed to be worked out between you both. Money, financial arrangements, so forth.'
'We had separate financial arrangements.' I could feel sweat condensing and crawling down the back of my neck, down my spine. 'There was nothing to work out.'
'Your house?'
'The house is mine.'
'You don't find it odd, that you haven't heard from him?'
'I didn't expect to hear from him.'
'Such a final break,' he said, marvelling. 'Such a quick, final break.'
'I told you, the marriage had been going wrong for some time. It was just the final thing. The final disaster.'
'And what caused the final...er...disaster?'
'I don't have to tell you that.'
'You don't,' he agreed, almost paternally. 'So, when you didn't hear from him, you weren't surprised?'
'Not at all.'
'You hadn't thought there might be a reconciliation?'
'It was clear that there wasn't going to be a reconciliation.'
'And he didn't forget anything?'
'Forget?'
'When he packed. He didn't forget anything, and come back for it later.'
'No.'
'An expert packer,' said Inspector Pritchard, admiringly. 'Me, now, I always forget something when I pack. Specially if I'm in a hurry.'
I said nothing.
'And none of your family's heard from him?'
'No. So far as I know.'
'Mail?'
'I believe he had a post-office box for business mail.'
'So he did.' The inspector nodded, as if he approved of this arrangement. 'And it didn't occur to you to worry about him? When he walked so completely out of your life, I mean?'
'Why should I worry?'
'Why, indeed?' he agreed. 'You wouldn't have thought, for instance, of reporting him as a missing person?'
'Why should I do that?'
'Such an absolutely total disappearance? And you never thought he might have met with foul play?'
'No. Why should he?'
The inspector shook his head in apparent perplexity and looked down at his notebook, in which he had, from time to time, made tidy and apparently leisurely notes.
'You've sold the house, Ms Weaving?' Of course, he would have seen the real-estate board out the front, with its flamboyant _SOLD_ sign.
'Yes. It was mine. I told you. The house was mine to sell.' I knew I was sounding defensive.
'Ah. And you're moving elsewhere?'
'I am, yes.'
'May I have that address, ma'am?'
I told him. He gave me his card and suggested I might like to ring him if I remembered anything else.
Not bloody likely, I thought.
'Lovely place,' he said, looking around as I showed him out.
'Yes.'
'Cost you a pang or two to move, I should think?'
'I guess.'
'Built it yourself, I understand?'
'Yes,' I said, wondering how he knew that. He smiled at me as if he knew what I was thinking. He paused at the door.
Go, I was thinking. I was suffused by fear and fury. Just go.
'You'll be in touch, then. You'll be in touch if you think of anything. Or if you hear from Mr Knight, of course?'
'Yes,' I said, stopping myself from saying: _But I won't hear from him._
After he had gone, I tried to stop the sweat and the trembling. I sat down and closed my eyes and breathed deeply.
I should have been better prepared. What a fool I was, not to have thought my way through it all. Of course, people didn't separate like that, not with that kind of suddenness, that abrupt and volcanic rupture. People didn't just walk out of a five-year marriage and not return, not like that. Both the inspector and I knew that. I should have thought my way through the story more carefully, should have negotiated its twists and turns and sharp treacherous corners with more prudence, more care.
But he couldn't prove anything. Nobody could prove anything. Not unless they dug up the back garden. Max had packed and left. He'd walked out of the house. He'd driven his car away. Maybe he'd met with foul play after that. How would I know? I'd have no way of knowing. Anything could have happened. All I had to do was remember that I didn't know anything, that I had no way of knowing anything.
Only the day after I'd moved, there he was at my new front door. 'Nice little place, ma'am,' he said.
'Yes.'
'A bit different from your last home, though.'
'Yes.'
He stood there, gazing at me. He wasn't going away.
'Would you like to come in?' I said.
'I'm thinking I'd like five minutes of your time, ma'am. If that's convenient.'
'It's not.'
He grinned. Ponderously he came in. We sat on the new furniture (flowered Sanderson linen) in the living room. The cream leather suite had no chance of fitting into this house, and I'd sold it to the new owners of Rain, so it would stay where it belonged. I'd chosen the Sanderson because it was so unlike anything at Rain: time for something completely different, I'd thought.
'A different style of place altogether,' he observed.
'Yes.'
He shifted, looked at his fingers. 'The fact is, ma'am, I don't know how to raise this matter with you.'
I tried to give him a look that was stony without being overtly hostile. Neutral, I thought. Be neutral.
'What's it connected with?' I asked.
'Well. It's connected with your marriage.'
'My marriage?'
'Your marriage to Mr Knight.'
'In what way, connected?'
'Well, when did you marry Mr Knight, ma'am?'
I gave him the date.
'And where?'
On the beach, I told him. 'Down Point Leo way.'
'I see. And you'd have the certificate?'
'Of course I do.'
'Would you be able to lay your hand on it?'
I wondered where the hell I'd put it.
'If I was still in Rain—in my old house—I'd know where it was. I'm not sure, here. I've only just moved; you know that.' I gestured around the room at the cardboard cartons, the piles of books, the mess. 'Honestly, I don't know if I can find it. It's here somewhere. It'll take me a while to get everything sorted out.'
'Ah.'
'Why do you want to see it? What's it going to prove?'
'Ma'am,' said Inspector Pritchard, 'since you've broken up and all, and since you don't expect to hear from Mr Knight again, perhaps this won't matter to you. But it does seem to us as if no legal wedding ceremony took place.'
'No legal ceremony? But I've got the proof. I've got the photographs; I've got the certificate.'
'It seems as if the ceremony may not have been all you thought it.'
I couldn't understand what he was talking about. 'Look, I was there; I was the damn bride. What on earth do you mean, it may not have been all I thought it? I thought it was a wedding. It _was_ a wedding.'
So he explained it to me.
Maximilian Knight, he said, had a wife in the US and another in Perth. He had married the wife in the US under another name (Pritchard didn't say what) some twenty years ago. He hadn't divorced her. He'd come back to Australia, about twelve years ago, and he'd married the wife in Perth. He'd been Martin Ritter then. She'd been Mrs Martin Ritter. (For reasons I would find it impossible to explain, I was obscurely grateful that she hadn't been Mrs Max Knight.) He hadn't divorced her, either. Not that you could divorce someone when you weren't legally married to her, of course.
'You'll have to tell me again,' I said. 'I'm not taking any of this in.'
'You don't look well, ma'am,' he said. I think he was really concerned. It was small wonder if I'd gone grey. I felt nauseous. I needed a drink, but I didn't want to have one while he was there.
He told me again. It didn't sound any better the second time around.
'It can't be true,' I said. 'It can't be true. Why should I believe you? How do you know all of this?'
He explained that, too. They'd been looking into Mr Knight. They hadn't been entirely satisfied about some of the activities Mr Knight had engaged in. There were questions, for instance, about Mr Knight's bank accounts, about his travel. There were questions about Mr Knight's name, about his history, about his intentions. Questions hung thickly all around Mr Knight.
'Two wives. Two previous wives. You must be mistaken. You _must_ be mistaken. He told me he was married once. But she died. She died of cancer.'
He wasn't willing to admit any mistake. 'Were there children?'
He wasn't sure.
I went to the carton that contained photographs. I was after the photograph of our wedding, the sunset, the beach, the deep mellow light, the photograph beside which Max had stood when I killed him. I found it at the bottom and drew it out. I hadn't been going to display it.
'Look,' I said. 'Look. This is when we got married.'
Seriously, soberly, he examined it.
'It's a wedding, isn't it? See?'
'I can see that it looks like a wedding, ma'am. I don't dispute that. I don't dispute that the event took place. I'm only saying, it wasn't legal.'
'But I have the certificate. There was a celebrant.'
'What was his name?'
'It was a her, a woman. I've got a photo of her somewhere, too. Everything was signed, witnessed.'
'I'm sure it was. But what I'm saying, ma'am, is that the ceremony wasn't legal. It was a con job, I'm afraid.'
'A con job?'
'You don't have to take my word for it. Go to the registry office. Births, deaths and marriages. It's in Collins Street, in the city. You go down there and you ask for a copy of the marriage certificate. They won't have it.'
'But why?' I asked, stricken. 'Why? It makes no sense. Why should he go through a charade like that?'
I think I'd forgotten at this stage that Pritchard was a policeman, that he was lined up on the enemy front.
'He couldn't marry you. He already had a wife. He already had two wives.'
'But he could have divorced them. He _must_ have divorced them.'
'I imagine the difficulties related to his name.'
'His name?'
'Ma'am,' said Pritchard in a burst of what seemed very like true candour. 'You have to understand, that wasn't Mr Knight's real name, Maximilian Knight.'
'It was on his driver's licence,' I said, as if that proved something.
'Forged.'
'It didn't look forged.'
'Good forgeries don't look forged, that's the thing about them. Or maybe the papers he used for it were forged. See, for a long time we were interested in the activities of Martin Ritter. We were asked by our colleagues in the US to look into Mr Ritter. Mr Ritter was hard to track. And then, when we did find him, suddenly he wasn't around. Gone. Disappeared. Phut. So we had to start all over again. It took us quite some time to work out that Mr Ritter and Mr Knight were the same person. Then there was Malcolm Baron.'
'Baron?' I was reeling.
'Yes. Malcolm Baron.' He eyed me. 'So you knew about Martin Ritter, but not about Malcolm Baron?'
'I found an envelope,' I said, slowly. 'I found an envelope addressed to Martin Ritter. It didn't make sense. But, no, I've never heard of Malcolm Baron.'
I wondered briefly if I should mention Matthew Templar, but decided against it.
'Do you still have that envelope, ma'am?'
'No.' That was true, in fact. I'd burnt it. I'd burnt the pictures of May and Susie, Kylie and Lindy Lou.
'What was in the envelope?' he asked.
They talk about people watching like a hawk. Inspector Pritchard's eye's were entirely hawklike. Hooded, sharp, bright. Quite at odds with his fubsy and unremarkable face.
'Nothing,' I said. And then, seized with a curiosity so sharp it bit into me, I asked: 'What was his real name? Do you know?' Pritchard shrugged. 'We don't know. I'm almost inclined to say Mr Knight himself didn't know. There have been so many, you see.'
'What did he do?'
'You mean, what crimes did he commit? You don't know?'
'How could I know? There was always plenty of money. Heaps of money.'
'I imagine there was,' he said, dryly.
'He told me—well, I told you what he told me. Consultancies, investment, properties. But you're saying there were illegal activities.'
'We don't know the full extent. We suspect him of involvement in a number of illegal activities.'
'Yes, but what _sort_ of illegal activities? Are you saying he was a criminal? Is a criminal?'
'I was hoping you could tell me some of those activities yourself, ma'am.'
'Did he do drugs?' I ask, baldly.
Pritchard's eyes never left my face. 'We don't think he did drugs himself, ma'am. We think he may have sold them, though: yes. Not a user. A dealer.'
He would give me no more information. Well, I had enough to cope with, I suppose. Before he left, looking at me narrowly, he said: 'I didn't expect you to be so distressed, Ms Weaving.'
'Didn't expect me to be distressed that I wasn't really married to a man I regarded as my husband for five years?'
'No. By your own account, you didn't much care for him, by the end. You weren't much distressed, when he left. You were getting on with your life quite nicely, ma'am, according to you. Marriage had been going downhill for some time, you said. Inevitable outcome. So no, I didn't expect you to care so much.'
'You overestimated me, I'm afraid. What I told you was true. It wasn't good. It had all been going downhill. But...but...'
To my immense relief, I felt the sobs coming. I let my voice shake; let the tears roll down. I'd felt frozen, until then. It was starting to come home to me, now. I didn't have to act at all.
'It's been such a shock,' I managed to say. Well, that was certainly true.
After he left (he couldn't cope with tears, which was a useful thing to know, or might have been, if I'd been able to summon them at will), I felt as if the shock would never dissipate, that it would continue to reverberate through me all my days, a little continuous tremor that would never die down. _Two wives_ , I kept saying to myself. _Two wives_. _At_ least _two wives._ Bea's voice returned again and again. _What does he do? Where does all the money come from? Tax evasion? Prostitution? Drugs? White slave traffic? You know nothing, nothing about him. What in Christ's name are you doing?_
What in Christ's name had I done?
I continued to unpack, stacking, tidying, folding, ordering, emptying, filling, smoothing, dusting, arranging, making hundreds of trivial decisions. Methodically I flattened the cardboard cartons as I took out all the household things, all the pots and pans and glasses and plates, the photographs and vases and books and pictures.
I'd thrown a lot out, but a lot remained. Kate offered to help, but I didn't accept: I wanted to sift through everything myself, to know exactly where everything was. It was part of the business of wresting back control over my life.
Now, as I sorted through all the flotsam, all the bits and pieces that turned out to be so much less important than I'd once thought, I kept thinking: _He'd been married twice before. He'd been married twice before._
What were they like, these wives of Max? Had they been like me? Physical attractions, sexual attractions, are supposed to remain consistent to type. Had they been dark and quick and slight? For some reason I found myself imagining them as statuesque blondes. I didn't want them to be like me. I understood none of it. I knew he had loved me; I knew this just as surely as I knew I had loved him. _What were you doing?_ I screamed at him inside my head. _What were you thinking? How could you let this happen to me?_
The fact that he had engaged in criminal pursuits (as Frank Pritchard would have put it) troubled me far less than did the previous marriages. I felt a little ashamed when I realised this was the case; still, it was so. Lindy Lou and Kylie and the rest of them had given me a nasty shock, but their presence in Max's life seemed to me a professional matter, as it were. I had absorbed the shock of their appearance. They were extraneous; they had not invaded his emotional life—or even, I thought, his sexual life. They posed no threat; they remained at a psychic distance from him and from me. I supposed he had made money out of them. Well, that was different from falling in love with them. It was the wives who bothered me.
Over the next six months or so, Frank Pritchard and I got to be on first-name terms. He dropped in rather a lot, after the visit in which he first told me about Max's wives.
Sometimes he had a cup of coffee. Sometimes he asked me questions: during the first couple of visits, he pressed me quite hard. I thought he was trying to exploit my new knowledge of Max's previous marriages: he was hoping my hurt and resentment would lead me to betray Max. I continued steadfast in my denial of knowledge of Max's activities or whereabouts, and gradually he came to believe me, or so I started tentatively to believe.
He let information slip, from time to time. Heroin, he said. Prostitution of minors. He said a number of pretty unpleasant things, which included embezzlement and fraud, but those two were probably the worst.
I thought of us as locked in deadly (albeit extremely civilised) battle, but sometimes we had quite pleasant chats. I didn't exactly relax with him, but I had to concentrate on not letting my guard down, which I was sure was his aim. I grew to know his knock—that businesslike knock on my door, which never failed to jolt my heart. The hardest thing was to open the door to him looking untroubled.
Once, Kate dropped by with Sophie while he was there. Unfortunately, he was in uniform at the time (sometimes he was, sometimes not). Her eyes popped when she saw him. I had to introduce them, to explain that Kate was my daughter. Frank's eyes flickered over her in the manner that had become so familiar to me. Taking everything in. He did the same to Sophie, but in a cursory way. I had one moment of sharp anxiety, in case he saw the likeness; but then I realised that Frank had never met Max, knew him only through photographs.
Sophie was just starting to walk then, I recall, and when Kate put her down her attempts to wobble around took centre stage in the way that young children can instantly manage.
Frank made a couple of admiring comments; there was some desultory small talk about toddlers. He started to leave, but stopped at the door, in that infuriating way he had of pretending (I was sure it was a pretence) that something had just occurred to him.
'I don't suppose you've seen your stepfather recently?' he asked.
Flummoxed, Kate shook her head.
'Or heard from him?'
'No,' she said, all at sea. 'Absolutely not.'
He fumbled in his inner pocket, brought out a card, gave it to her. 'In case you do.'
Kate rounded on me when he had left. 'What the hell was all that about?'
I was silently cursing Frank. I'd really wanted to get through all of this without having the rest of the family know anything about it, and I'd thought I was going to succeed.
'He sometimes comes by,' I said, tiredly. 'It seems there are things about Max's business affairs that they think aren't quite right.'
'Why don't they ask Max, then?'
'They can't find him.'
'Well, you know where he is, don't you?'
'No.'
She looked at me sideways. She knew she was on forbidden territory, that I didn't want to talk about Max, that she wasn't allowed to. But she didn't stop.
'What sort of things aren't in order?'
'I don't know. I think there's a couple of tax queries, things like that,' I lied.
Kate lunged at Sophie, who was heading for a low table with breakables on it. 'You wouldn't have a policeman asking about tax stuff, would you?'
'I don't know.'
'Well, but you wouldn't,' she persisted. 'You'd have someone from the tax office, not a policeman. Sophie, love, come here to Mummy and leave that alone. You wouldn't, Mum.'
'I don't know about that,' I said, heading Sophie off at another pass. Sophie was very enterprising and surprisingly quick on her feet for one so unstable.
'But listen—'
'I'm not talking about it.'
'For God's sake!' she said, in exasperation, whether aimed at me or at Sophie it was hard to say.
Sophie fell over and started to cry. I picked her up and comforted her, but she wanted her mother. I passed her over, with a prick of envy.
Kate cuddled her. 'Just tell me this,' she said. 'Do you honestly not know where he is?'
'I don't have to tell you anything,' I said, just in case she hadn't got the point. 'But no, I don't. And I don't want to. And that's all I'm saying.'
She wasn't satisfied; she wanted to go on talking. But I wouldn't, and she knew it. She gave in. Kate always gives in, eventually. She's got no spine.
Quite soon after that—a week or so, I think, Frank contacted me and asked me to go down to the local station with him. Some property had been recovered, he said. Property of Mr Knight's. He'd like it if I could identify it for him.
I thought it must be the car, and it was.
It was sad. Somebody had beaten the heart out of Max's beautiful Audi. It was on the back of a tow truck in the car park behind the station: it wasn't even driveable. The luxurious soft leather seats had been slashed; the windows were broken; the boot lid had been wrenched off; someone had lit a fire in the back seat.
I'd thought, after I'd driven it that last time, perhaps I ought to have worn gloves; perhaps they'd powder it or swab it or do whatever they do to get fingerprints off it and they'd be able to tell I'd been the last one to drive it. I needn't have worried. It would have been as difficult as trying to get fingerprints off a prickly pear.
Poor battered, blackened, ravaged Audi. Max would have wept. The registration plates were gone, but the police had traced it through the engine number, Frank said. Was it Max's, did I think?
I shrugged. 'How can I tell?' I asked, walking around the poor devastated thing. 'If the engine number is right, I guess it must be. But there's no way I can tell. There was a scratch on the duco at the back, I remember, that he was going to have fixed. But there's not much point in looking for a scratch on that now, is there?'
'And Mr Knight drove this when he left?'
'Yes.'
'And you've not seen it since?'
'No.'
'Well,' he said, looking dissatisfied.
I didn't think he'd wanted me to identify it. What was the point? They had the engine number. That was enough. The car had been in Max's name: I'd seen the papers. What use was there in asking me to identify something that was nine-tenths destroyed and unrecognisable anyway? He wanted me to be upset; he wanted me to show emotion. I wasn't going to.
'Does it concern you?' he asked, abruptly.
'Seeing the car like this? Of course.'
'But don't you worry, now, about what's happened to Mr Knight? He was fond of this car, wasn't he?'
'Yes, he was.'
'So he wouldn't have been responsible for this damage himself?'
'Good heavens, no.'
'Then what do you think has happened to him, ma'am? Aren't you concerned about that? Where do you think he is?'
'Frank, we've been through this so many times. Given what you've told me, I imagine he's in Rio de Janeiro chatting up the local talent.'
'He would have just ditched the car?'
'No, he wouldn't. He would have been careful with it. He would have left it somewhere it would have been safe. It must have been stolen.'
'He didn't report it.'
'Well, you're always telling me he had good reason not to go anywhere near the police. Anyway, he probably doesn't know the car's been stolen. He's probably left the country.'
'We have no record of his doing so,' said Frank.
'How many thousands of people fly in and out of Australia every week?'
'Not so many that we can't keep track of some of them,' he insisted, stubbornly. 'Mr Knight hasn't left the country.'
'You can't be sure,' I said. And, of course, he couldn't.
After this episode, Frank stopped coming for a while. It's as if it was his last fling, his last attempt to prise out of me all he can. I think he thought seeing the devastated Audi might make me crack. Having failed, I think he's given up. I even think he might finally accept that I know nothing more than I say I do. I almost stop listening for the knock on the door.
But it comes again. It comes one night, quite late. I've decided to have a drink. I've actually decided this quite early in the evening, and by the time Frank knocks at the door I'm right into it. Perhaps I'm on the fourth brandy? And the brandies have got bigger and less diluted on the way through the evening.
'Hello, Frank,' I say, opening the door. Then I hiccup. The hiccup wafts in the air between us before drifting, gently and featherlike, to the floor, where it rests. Frank's eyes—grave, even troubled—regard it there before he looks straight at me.
'Hello, Isabel,' he says. 'I hope it's not too late. I was driving by and saw your light on.'
'No, not a bit,' I say, foggily, waving him in. I'm almost pleased to see him. It gets lonely, drinking on your own. I tell him this, and press a brandy on him, fortuitously refilling my own glass in the process. I'm a little surprised that he accepts, as I had a notion that cops are not meant to drink on duty, but I guess Frank's on top of what he's doing. Perhaps he's not on duty? It's all a bit fuzzy and I can't quite focus on anything as I know I should be doing.
We sit at the kitchen table and observe each other silently for a period. I know what he's thinking. He's thinking he's found me sozzled. He's thinking I'm cornered; he's thinking he can extract from me anything he likes. His eyes are bright with anticipation of my errors, my contradictions, my admissions.
Fat chance, Frank, I think. But I think it a bit blearily. My mind isn't crisp enough; I know this. It might not be equal to my stubborn determination to give nothing away. Silently I double my resolve and, with an immense effort, concentrate.
Well, sozzled I may be, and cornered, too; but I am buoyed by the competition of the situation (he's a very competitive person, Frank, but so am I) and by the knowledge that a ratlike cunning is budding deep within me. I can keep Frank out there on the perimeter, where he belongs. I can stop Frank from hopping the fence, stalking closer to me, closer to the truth. I have to stop Frank.
The thing is, he can see I'm a little under the weather. It's written all over his face, his excitement that he's finally caught me with my guard well down. And if I can keep one step ahead of him, if I can manage not to give anything away this time of all times, perhaps I'll convince him at last.
I smile at Frank. I try to make it a winning smile, and hope I don't look merely predatory. Or drunk.
'So,' I say, trying to cross my legs and almost getting there. It's a good thing Frank can't see under the table, I think. Or perhaps he can?
Frank's equal to this ploy. 'So,' he says, smiling.
There's a pause.
'Don't you ever go home?' I try.
'On my way.'
'Doesn't your wife get cross with you when you're so late home?'
'No wife.'
This is something I've often wondered about Frank. Somehow, to me he has always looked married, and I'm mildly surprised that he isn't. I'm sure he's not gay.
'Can we talk about Max?' he says.
I'm even more surprised by this. Frank is usually rather more subtle in his approach: he'll stalk around for a while before he pounces. Furthermore, he's always referred to Max as Mr Knight, a careful formality I've occasionally found bizarre. It turns out Frank wants me to tell him about Max and me, about how we met, about what he was like.
'I've never met him, remember,' he says. 'I'd like to get a feel for the man.'
So I tell him. It's an expurgated version, certainly; but I answer all his questions ( _yes, we met at my office: he wanted an architect to design his house; no, I didn't meet many of his friends; no, we didn't travel together much; yes, he did give me my dog_ ). And so on.
While I am steadily answering his cruel questions, managing what I think is a halfway decent impersonation of a woman who doesn't give much of a damn any more, Frank leans over and scratches Borrow's chin on the exact place he loves it. I am really irritated by this, and try not to show it.
All in all, it's my finest hour, I think. It is so hard, presenting these details of everything that has mattered most, dazzled brightest, in my life, without breaking down; but I do.
Once or twice, affected I suppose by the alcohol, I am almost tempted to say to hell with it all and tell Frank everything. I'm so sick of him persecuting me in his courteous, obstinate way; and he's a nice man; surely he'd understand. It'd be such a relief, to tell somebody all about it.
But I manage to retrieve myself from this idiocy. I remain polite, slightly distant, slightly puzzled by his interest, by the way it's all turned out. I don't think I give anything away at all.
When he leaves, after half an hour or so, I sit quietly and gaze into space. I haven't heard his car drive off, and he's tried this trick on me once before. And, sure enough, there he is, at the door again, having (he says) forgotten one small question, his eyes positively ablaze with pale fire at the possibility of catching me out somehow. But lo! I am calm, unflustered. I answer the inconsequential remaining question; he goes; I hear the car leave.
In fact the strain of it all has pretty much exhausted me. But it turns out to be worth it. Frank doesn't come again. He's given up. It takes me a long time to be sure of this, and even when I am practically sure I still continue to listen for that knock, the resolute knock that signifies Frank's return.
Sometimes I wake during the night, suddenly persuaded I have heard it, the knock that presages investigation, conviction, punishment. My heart thumps as I lie in bed, listening as if my life depended on it, which perhaps it does.
Sometimes I get up and pad barefoot out to the slanted window through which I can see the front porch. There's never anybody there. Gradually, slowly, I start to believe I might have satisfied Frank; he might not return. I can get on with my life.
Well, I can almost get on with my life. Before I can properly do so, there's something I have to sort out. I do some homework. I look up addresses. I fly to Perth and hire a car.
It's a fine sunny day, and the ritzier bits of Perth unfold before me, wide streets and bright gardens. I find the address I'm looking for, the only M. Ritter. Immediately I know I've hit paydirt. There's an old silver Audi in the drive.
The woman who opens the door isn't statuesque, but she is blonde. That ash-streaked, expensive blonde that goes so well with a lustrous golden tan, which she has. She's on the plump side, but she's pretty. She's in jeans and a T-shirt, but they're classy jeans and a designer T-shirt, and her thin, gold leather sandals are just gorgeous: I feel like asking her where she bought them. Her expression is unfriendly.
After a second or two I realise I am staring at her mutely, drinking in every detail of her appearance.
'I'm sorry,' I say. 'I'm really sorry. Please, can I talk to you. Just for a few minutes?'
I'm stupid, of course: such an important meeting, and I've rehearsed nothing, prepared nothing. It's not like me. I've been so consumed with it all, so consumed with the thought that I'll perhaps actually be meeting one of the wives, that I'm not ready: I haven't got my act together.
'My name's Weaving. But that won't mean anything to you. I was married to Max Knight.'
Her eyes widen and harden. She starts to close the door.
'Please. Please. I'm from Melbourne. I won't ever bother you again. I'm going home tomorrow. I came over here just to meet you. Please. I only want ten minutes. Then I'll go and you'll never see me again.'
'What do you want to ask me?' Her voice is cold.
'I only want to ask you some questions. Please. I lived with him for five years. There are things I don't understand. I thought if we talked we might both understand more about it, about him.'
She hesitates. Then she steps back, nods. 'Ten minutes.' We sit opposite each other in a remarkably pleasant room, all sun and chintzy wicker chairs and big extravagant pot plants and glass-topped tables. I don't like wicker furniture very much, and I don't like pot plants very much, but she's put it all together (at least I suppose it was she who put it together) so that it looks summery and airy and fresh.
'My name's Isabel,' I say.
'I'm Meryl.'
'Thank you for seeing me.'
A long pause settles between us. We examine each other. I dare say it's an unusually frank appraisal between two women who have never before met each other, but we need it. I certainly do, anyway.
My initial impression is reinforced: this is a classy lady. She has that costly sheen of preservation about her that only really chic, really wealthy people get. She's chubby but it's fit chub, firm and gym-toned flesh. Her eyes are bright. She has terrific teeth.
'How did you find me?' she asks.
'A policeman.'
'Cops are pigs,' she says, with a vigour that surprises me.
'At least, he didn't tell me how to find you, but he told me Max had married in Perth when he was calling himself Martin Ritter. So I looked up Ritters in the phone book. There aren't many. It wasn't hard.'
'He left you, too?' she asks, abruptly.
'Yes.'
'How long ago?'
'Nearly a year.'
'Did you know it was going to happen? Did he give you any warning?'
'No,' I say, reflecting that this would have been a difficult thing for Max to have managed.
'Me neither. One day he just went away. He left a letter. Did he leave you a letter?'
'No.'
She raises her eyebrows. 'Unusual. He usually leaves a letter.'
'Usually?'
'Well, he did with me, and apparently he did with my predecessors.'
'Your predecessors?'
'Didn't you know? He's a serial bigamist, dear.'
'I knew there was one other wife.'
'A whole trail of them, probably. I know of three.' She looks at me. 'Four, now.'
'Are the others in America?'
'Yes. Well, two of them are. God knows how many others exist, or where they all are.'
_More than two other wives_ , I think.
'Meryl,' I say, experimenting with her name. 'How did he make his money?'
She smiles, thinly. 'You don't know?'
I shake my head. I've got a pretty good idea, or at least I think I have, but I want to hear it from her.
'Three ways, mainly. He imports prostitutes from Asian countries, and he launders money. But principally it's drugs.'
_Tax evasion? Prostitution? Drugs? White slave traffic?_ What an irony, I think. I must make sure Bea never finds out she was right after all.
'Drugs?'
'As I understand it, heroin. I believe he dabbles in boutique stuff, too, from time to time. But the basis is heroin—at least, it was then, and I assume it's the same now. That's where most of the money came from. Honestly, didn't you know?'
'He told me he was a consultant of sorts.'
'So he is. A consultant on whores and heroin. He has other interests, too, of course—legit ones, to give him a cover. I believe he does a nice line in hiring out heavy industrial machinery. But most of it's prostitution and drugs. You must have suspected something? Did he throw money around?'
'Yes. But I didn't want to think about it. I didn't want to think anything was wrong.'
'I was a bit like that, too,' she says. 'Common sense was telling me something was wrong. Nobody has as much money as that: I knew that. But I didn't want to face up to it, either.'
I realise her eyes are on my ring.
'That's a fabulous diamond,' she says. She holds up her hand and waves it at me. The diamond ring on her middle finger glitters. Its design is similar to mine.
'Tell me,' she says. 'Did he give you a Fred Williams?'
I can only nod.
'I sold mine. I didn't like it. And I thought a painting might be repossessed. I wanted the hard cash. I advise you to sell yours, too.' This is an even more surreal conversation than I expected it to be.
'What did he say in his letter? If you don't mind my asking.'
She sighs. 'He said he loved me, we'd had a great time, he was an adventurer at heart, sorry not to live up to what I'd have liked him to be, ta-ta. Classic stuff. The cops turned up about a week after he'd left. He always manages to keep one step ahead of them, apparently. They weren't very nice about it.'
I stare at her. I don't know what I expected, but it wasn't this. Probably it should have been, but it wasn't.
'Look,' she says. 'I know it's hard. Try to see the bright side. He's probably left you pretty well off?'
I nod.
'Well, take my advice, get rid of stuff like paintings. I did; I converted practically everything into cash. Except the diamond ring and the car. I liked the car, and I kept it. It still runs okay, though it's really old now. But that's a precaution: you probably don't need to worry. Martin has excellent accountants. The cops made lots of threats, but they never actually took anything away. Martin never seems to leave a paper trail. Except when everything's kosher, of course, and then he leaves a really clear one. He enjoys that.'
Every time she calls Max _Martin_ it's a surprisingly intimate irritation, as if someone were stroking a feather across my face. I try to ignore it. 'How long were you married?' I ask.
'Well, of course, we weren't married. You weren't either, I take it?'
'No. Not as it turns out.'
'We were together around four years.'
I sit there, running through it all in my mind. It's so much to take in. Meryl looks at me with something rather like compassion. She actually leans over and pats my hand.
'Listen to me,' she says. 'It gets better. Really, it's only just happened to you, hasn't it? He's a bastard, and I know he's a bastard, but at least he leaves people better off than he finds them, in a material sense, anyway. I mean, I'll never have to work again. I bet you won't either.'
I stare at her. I'm overcome—well, nearly overcome—by a desire to sink my face into that plump, fit bosom and tell her everything. Everything. I come very close. She misinterprets my expression (which must be an odd one) and pats my hand again, almost in a motherly way.
Then she starts blurting things out. 'I've thought an awful lot about Martin,' she says. 'I think he just has different needs from most people. He _is_ an adventurer. I mean, I know it's no excuse. I'm not trying to make excuses for him. But I've tried to understand what happened to us. I just think he can't help it. Honestly, I thought we were so happy. I believed we were so happy. I certainly was. I've never been so happy before, and I do believe he was, too, for a while.
'And he's very generous—he wants to give, all the time; he wants to shower you with stuff: you must have had that, too. Jewellery, clothes, all kinds of things. He loves doing that. He loves coming in and being Father Christmas, being a good fellow, and spreading good cheer and all the rest of it.
'Mind you, it's partly because he's guilty, because he's totally unable to be faithful. So he cheats, then he heaps presents on you because he feels guilty. But finally he gets bored. I mean, it's partly keeping ahead of the cops, and he has to do that, because I reckon they've got enough to put him away for a thousand years if they ever caught him. The drugs alone—I mean, hell.
'But it isn't only that: he just wants to keep moving on. The woman isn't born who can keep on satisfying him. He needs to be an impostor; he needs to have mystery and adventure and all the rest of it. You'll never find him, you know. He sheds everything—name, identity, everything. He's probably not even in Australia by now. He's got friends who make him a whole new set of papers and off he goes again.
'And listen, love, you're well rid of him. I know you love him, and I know you miss him, but you think about how he makes his money, and you ask yourself if you really want to profit from that. Do you know any heroin addicts? Well, I do, and let me tell you, it's not good, to be making your money out of poor lost souls like that. I've sometimes thought, if I had any principles, any real principles, I'd give it all away to a refuge and go and do work for addicts. But I guess I haven't got principles. Sometimes I think it's soiled money, I don't want any of it. But look, it's money, and it's there, and all the harm's been done already. So I keep it, and I live a comfortable life. And, after all, why shouldn't I? I suffered enough, when he left me. I deserve something.' She pauses. 'Heavens, I don't usually let go like that. I wasn't going to talk to you at all.'
'Why not?' I ask, almost humbly.
'I thought you might be a cop.'
'Me?' I start to laugh. It's funny: you can't get away from it.
Imagine thinking I'm a cop.
She smiles. She really has got excellent teeth.
'That's better,' she says, encouragingly. 'Look at me. I've lived through it. I've got my life together. You can do it, too. It's amazing: at the time you think you'll never recover, but you do: believe me. You do. Are the cops pestering you?'
'Yes. A bit. There's one cop. He comes around a lot.'
She nods, recognising the scenario. 'That'll keep going for a while, probably. But look, Martin really is clever. They tried so hard to pin stuff onto me, but they never managed to. He didn't leave anything that connected me with anything he'd done. He will have done the same with you. He tied up all the ends: really, he was so careful. They made lots of threats; they hung around. Then they got bored and they moved on, went to nobble someone who was easier game. These days, I never hear from them. That'll happen to you, too, mark my words.'
I suppose it will; but of course I didn't give Max time to tie up all the ends. The impulse to confide has evaporated, now, and I'm in no danger of telling her anything else. There's one thing I want to ask, though.
'Was he unfaithful to you?' I enquire, trying to make it sound as if this is a normal question to ask someone you've just met. Heaven knows why I'm worried: the rest of the conversation hasn't exactly been normal.
'Oh, God, yes. All the time. He can't help it, you see. I just didn't realise.' She looks at me narrowly. 'You poor darling. You're still in love with him, aren't you?'
I nod. There doesn't seem a lot left to say. We actually kiss goodbye, after a brief and embarrassed hesitation.
'Did you have children?' I ask her at the door.
She shakes her head; her peachy face suddenly sags, slightly. 'I'm infertile,' she says, simply. 'And you? Did you have children?'
'Not with Max. From my first marriage, yes.'
'Hang onto them. Kids are worth more than men, any day.'
I suppose she's right. I think about this, on the way home, on the plane. I think about Meryl, and about Max, and about Martin. I see what she means, about his being an adventurer, about his liking to move on, to shed old identities and don new ones like coats.
It's like an old fairytale, in a way: the swineherd who's really a prince (or vice versa, come to that); the impostor who competes in arcane games and wins them, who appears from nowhere, whose real identity isn't known till the end of the story, who changes his shape, pushes his luck, inveigles his way into bed with the princess.
Well, Max was a good inveigler. And adventuring was the gift he gave Sophie: that makes more sense, now. I think, too, of what Meryl has told me about the drugs, the heroin. I still have such trouble thinking of this.
All the evidence to the contrary notwithstanding, I cannot think of Max as anything but kind. I remember the photographs of Kylie and her friends; still I cannot imagine Max as malevolent or even uncaring. I know he betrayed me and lied to me, but in fact he didn't leave me; I don't believe he would have left me. Really, my only reservation, the only big black cross against his name, is his sin with Kate.
And I blame Kate for that.
I stare out the window at the apparently endless red desert beneath, and wonder what will come next.
*
As it happens, to my surprise and relief, nothing much does come next.
Gradually, very gradually, I start to relax. There's a long period during which nothing really happens to disturb the even tenor of the way I've grittily carved out for myself. I don't mind this: a patch of boredom is hardly unwelcome. I go to work. I cut back to part-time.
Sophie grows; Liam is born and grows. We all get older. Dominic goes to university, leaves university, becomes a solicitor, makes a lot of money, has a succession of sophisticated untalkative girlfriends who all look down their noses at me.
Steve, rather to my surprise, fails to find a new partner. Even after Dominic leaves home to set up the first in a succession of glossy bachelor pads, Steve remains in the old family home. It's far too big for him on his own, but he says he likes it, likes pottering around.
He and I mellow, a little, towards each other, or at any rate we pretend we do. Zoë and Henry continue to live energetic lives devoted to teaching, good causes and annoying as many of the family as they can. We all continue to turn up at the obligatory family gatherings.
I turn fifty. I notice silver strands in my hair and an unattractively crêpey look to my neck. The years roll past. Suddenly, Sophie is twelve. _Twelve!_ we all say to each other. _Fancy our Sophie being twelve!_ as if we hadn't all known that this would happen, as if she hadn't had an eleventh birthday the previous year.
The Lost Dream continues to return from time to time, always painful, always dislocating, always forcing home Max's physical absence. Slowly it worsens. The bush is darker, the muttering louder, the path longer. The leaves still quiver above me, thin and sharp, but now they transform themselves into a thousand pale, silver knives: I know that if they drop they will split my flesh.
I drink, perhaps, a little more than I should, from time to time. At first, I listen always for the knock that will herald my exposure and the retribution that it seems to me, in my more depressed phases, might in the end be inevitable. But curiously, miraculously, he doesn't come. Gradually, I forget about Frank—or, at least, let him sink to the back of my mind.
It's hard to distinguish between days, weeks, months. At the beginning of these grey, flat years—in which Sophie is the only bright, distinctive feature, Sophie's voice the only voice to touch a responsive chord in me, Sophie's milestones the only events that matter—I still think about Max a lot.
I try not to think about him, of course, and by dint of keeping busy, and still working most days, and making monstrous efforts, I do in fact manage to disengage my life from Max's overwhelming absence, during the daytime at any rate. It's in the long, lonely evenings, during the sleepless nights, that Max's ghost and I chase each other wearily around the bleak arena of my memories. By the time Sophie is twelve, this happens less frequently.
The drinking, on the other hand, probably happens more frequently. I can't help it. It makes such a difference, to sit down with a brandy at the end of the day, to feel its warmth and vitality seeping through your tired, stringy veins. I always remember the night I nearly came to grief with Frank, when I nearly told him too much, because of that same inner glow; and I'm usually careful not to risk that. If I know people are coming, I don't touch it.
But people don't come very often.
*
And then, one night, Kate rings.
'Mum,' she says. 'Sophie and I thought we might come and see you. Now. Is that all right?'
It is around eight o'clock and there is a television program I was planning to watch. I pause. Something in Kate's voice—some tight, stretched quality that wasn't at all characteristic of her—alerts me to an incipient drama, a crisis of some kind.
'Right now?'
'Is that okay? You're not going out?'
'No. Kate, what—?'
'We'll be there soon,' she says, and hangs up.
They come in, hand in hand. Kate looks tense. Sophie has a tiny, nervous smile playing around her lips as if she deserves praise but doesn't expect it.
'Mum, Sophie's done something,' says Kate. 'It may not matter and it may matter: I don't know. But I thought she ought to come and tell you what she's done.'
'It can't be so bad,' I say, consolingly, not having the faintest clue what she's talking about. We all sit down.
'I had a visitor yesterday,' says Kate. 'His name is Frank Pritchard. I think I met him once, ages and ages ago, here. He was visiting you and he asked me to get in touch with him if I knew anything about Max. I've never forgotten it because I thought it was so weird. He asked me a number of questions. I think you need to know about this, Mum.'
If they wanted my attention, they've got it.
'Tell Gandie what you've been doing,' says Kate, turning to Sophie.
'I went to the police,' says Sophie.
It's so astonishing, so out of the blue, I can't begin to understand. At first I think I've misheard. It takes a while to fall into place.
'What for, honey?'
'I was trying to get Max back. I was trying to get him back for you, Gandie, because I knew you wanted him back.'
I can't say anything. All I can do is stare at Sophie in dismay. God only knows what I look like.
'Tell her everything,' says Kate. 'Tell it to her, the way you told it to me.'
Uncertainly, Sophie unveils the horror. It appears that she has been deeply moved by my suffering. For some time it has been apparent to her that I am, God help me, still _very much in love_ with the absent Max. It turns out that Max has exercised more fascination over Sophie than either Kate or I had supposed.
She has always thought he has a lovely face. She has always thought how cool it would be if he were to return from whatever mysterious corner of the earth he inhabits, solitary and depressed, longing to return to Gandie. She has always thought he must be still _very much in love_ with me, and that our reconciliation would be only a matter of his realising that, yes, in fact, I still care for him. This realisation would inevitably then lead to a reunion and to subsequent endless felicity for us both.
By the time we get to this point Sophie is starting to understand that this initiative of hers isn't going down well. Her smile is fading. She pauses and looks at Kate, who tells her to go on.
So. For some time Sophie has harboured this dream that it will be she—and only she—who plays the leading part in reuniting the riven pair, but she hasn't in fact seen how this can be accomplished.
And then she happens to watch a program on television about missing persons. This program makes it blessedly clear that the police are the right people to approach. For how can the police do anything constructive about tracing a missing person if they do not know that the said person is indeed missing? And I have told Sophie quite plainly—haven't I?—that Max is to all intents and purposes a missing person and that I have never instigated a search for him.
So she sallies forth, our Sophie, to the local cop shop. At first, nobody will take much notice: they treat her as if she's a little girl. Not wise, to put Sophie off like this. So she insists, and she gets a toe in the door, and eventually a junior cop—a constable, she thinks—takes her statement.
And, says Sophie, with a faint intonation of triumph, when she says the name of Max Knight, the constable sits up and starts to take an interest.
And, quite quickly really, only a week or so later, she's talking to this very nice man whose name is Mr Pritchard. He's very interested in what she has to say. Very sympathetic. At this point Sophie halts her narrative and shoots a scared glance at me. She's mostly been looking at the floor, until now.
What she sees in my face obviously doesn't reassure her. Her voice trails off and her gaze reverts to the floor.
'What in Christ's name did you tell Pritchard?' I ask. To my own ears my voice sounds harsh, croaky. Kate glances at me, startled.
It's not hard, to see how it's gone. Frank has clearly been on the ball, has enticed Sophie to chatter away artlessly about her beloved grandmother and her grandmother's perplexing missing spouse. I can imagine it so clearly: Sophie burbling away about the love story she has imagined with such enthusiasm, Frank nodding, smiling, encouraging. A query here, a prompt there. He's a personable-enough man, Frank: he wouldn't have had trouble enlisting Sophie's confidence.
Does her dear old grandmother think Max is dead? he asks. Well, says Sophie; she's so sad about him: she thinks he'll never come back. Of course, he might be dead. But what if he isn't? she asks. What if the cops could find him, and present him to Gandie. Wouldn't she be surprised?
Well, she certainly would.
Christ, Christ, what have I told Sophie? I pull my brains apart trying to reconstruct our conversations.
And Sophie proceeds to remind me, falteringly: she visited me again, and we talked about Max again, the time she showed me the photo she found in her mother's drawer, and this time— _this_ time—I told her—didn't I?—that Max was dead. Truly dead. In fact, so dead is Max that he lies in a grave I've visited.
Kate's eyes are on me.
And by this time, of course, Sophie has got the cops looking for a missing person, so she feels a bit guilty to have put them to all this trouble. She thinks she'd better head them off. So she goes back and she tells Pritchard. She tells Pritchard that Max is dead. She tells Pritchard that Gandie knows where Max is buried.
She is starting to snivel.
'Sophie,' I say, and I still don't recognise my voice. 'That was a secret. That was our secret. I told you not to tell anyone.'
Sophie responds that I really only told her not to tell her mother. And anyway, she figured I wouldn't want the police wasting their time.
Kate's eyes are wide and hard, and full of consternation and judgement.
'You're not cross, are you, Gandie?' says Sophie in a small voice.
'Jesus Christ,' I say, in this voice that isn't mine, that is coming from somewhere outside me. 'Jesus Christ, Sophie. You little cow. What have you done?'
Sophie flinches, shrinks into herself.
And suddenly Kate is up and at me. 'She meant well,' she cries, flailing her arms around ineffectually. 'She meant nothing but good. She did it out of love for you.'
I'm so livid, I can't think of anything to say.
'Don't be angry with her. She didn't mean to do anything wrong.'
'Sophie,' I say. 'Sophie, next time just...just...'
But of course this is an idiotic thing to say, since there won't be a next time. The damage is all done. Sophie glances up at me, her lips quivering. I stare at her and for the first time in my life I find myself regarding my granddaughter without love.
All I can think of is that she has stirred up old muck, long-ago sludge, from the bottom of my pond, that she has muddied the water that I had come to think of as clear and safe. I am no longer safe.
For so many years, I have worked patiently and carefully, constructing protection, devising masks and invisible cloaks and hideaways and shields and God knows what, all designed to maintain my security, and now, on a childish whim, on an ignorant and foolish caprice, she has jeopardised all my travail.
I can't bear it.
Sophie has now gone perfectly white and is staring at the floor. Kate is starting to cry. Trust Kate. She stands, wiping her nose.
'We'll go now, Mum,' she says. 'I wanted Sophie to tell you herself what she'd done and she's done that. I know this has been a shock to you, but I think she's been very brave, don't you?'
'Brave,' I say, hoarse with the absolute horror of it all. 'Brave. My God.'
Kate yanks Sophie to her feet.
'I'm sorry, Gandie.'
'Get out,' I shout. 'Just get out.'
They do. Kate pauses in the doorway.
'Are you at home tomorrow morning?' she asks.
'Why?'
I'm not trying to sound encouraging.
'We need to talk. I'll be over.'
And she is. We sit in my little kitchen and look at each other over the teapot.
'Mum,' says Kate. 'Please. You must tell me what happened to Max.'
'I don't know what you mean.'
'That man Sophie was put onto, that Pritchard man who visited me: he's the one I once found at your place, isn't he?'
I try to put her off, but she cuts over me.
'I know he's the same man. Don't snow me. It's important. You know it's important. Mum, we have to talk about this. Has that man been in touch with you? Since Sophie spoke to him, I mean.'
'No.'
'When did you last see him?'
'Years ago.'
'Do you think he'll come again?'
I try to laugh dryly, but it comes out as an odd rasping sort of noise.
'I should think so. After what Sophie's told him.'
'I don't understand any of this, Mum. Tell me about Max. Tell me what happened to him.'
'Nothing happened to him. We split.'
'But something did happen,' says Kate. 'You told Sophie he was dead. You told Sophie you'd seen his grave. You have to tell me. Pritchard's questioned me once already, and I'm sure he'll try again.'
'What did he ask you?'
'When I stopped living with you and Max. What my relationship was with him. Things like that.'
'What did you say?'
'What could I say? I told him. That is, I told him I stopped living with you when I got married; I told him Max and I got on fine. But it seems to me, Mum, there are things I don't know, and I need to know them if I'm to give the right answers.'
I say nothing.
'Anyway,' says Kate, 'I know something's happened to Max.'
'How can you be so sure? You don't know anything. You can't.'
'Yes, I can. I do know.'
'Why?'
'He's never contacted me,' says Kate, quietly. 'It's been twelve years since Sophie was born and he's never contacted me. I know he would have.'
'You know no such thing.'
'Yes, I do.'
We eye each other.
'Listen,' says Kate. 'The way I see it is this. You and Max quarrelled about me, and Sophie, and you split up. Then Max disappeared, and someone reported it, all those years ago, and that man Pritchard was looking for him. And now he's looking for him again.
'Aren't you surprised by how much interest they're taking in Sophie's story, Mum? She's just a kid. They wouldn't be taking this sort of interest if they didn't think something was seriously wrong. And that wasn't true, what you told me, was it? About Max's business affairs, I mean? They weren't looking for him because of tax stuff, were they?'
'No. No, it wasn't tax. They were very interested in finding Max, and not because they thought he was just missing. There were big problems, Kate. Big problems.'
'Tell me,' she says.
Well. She's supposed to be an adult. She might as well know some of it, anyway. I tell her about the photos of Kylie and Lindy Lou, and about their obvious interpretation.
'I don't know what you're talking about,' says Kate, tears springing to her eyes. 'Max was a good man. He was a good man, a kind man. He wouldn't do something like that.'
I shrug.
'Perhaps they were children he was sponsoring. Third-world children, through one of those charity programs,' says Kate, desperately.
I tell her about the gun. I have to say I'm getting a perverse pleasure out of all this. My illusions were splintered to smithereens twelve years ago: why should hers be intact?
'A gun?' she whispers, her eyes wide.
'And then there was the heroin trading,' I say, brutally.
Kate can't say anything. She's looking at me in total horror. 'A drug dealer,' she murmurs to herself, as if I'm not there. 'Max is a _drug dealer_?'
I consider telling her about the wives, about Meryl, about the serial bigamy, but I don't. Is this due to some vestigial pride, I wonder? If I tell her Max was a born philanderer, it'll make her feel worse, but it'll also diminish my own relationship with him. I don't want to do that. I don't want to tell her that the soppy photographs taken after the ceremony on the beach signify nothing, nothing at all, that we were never married.
I can't bear to tell her that.
No, I can't bear that. But an odd thing is happening to me nevertheless. After all these years of silence, all these years when the only person I've ever discussed Max with has been Frank (if you don't count Meryl), it's as if there's a melt within me. I can practically feel the ice crack, the snow soften, the chill streams start to flow. And perhaps Kate knows already. Perhaps she's guessed what happened anyway.
But she is in deep shock. She certainly hadn't guessed anything of what had generated Max's copious funds.
'How long have you known all this?' she asks.
'All the time. Since then. Well, some of it only appeared gradually. Say, ten years.'
She shakes her head. 'Are you _sure_ it's all true? Are you quite, quite sure there hasn't been some awful mistake?'
'Quite sure.'
Kate mulls it over.
'So where is he?' she asks, suddenly. 'Tell me, Mum.'
And the melt takes over. I tell her about that evening. I tell her about the meteor stone.
'It wasn't my fault,' I say. 'I didn't mean it to happen. It was an accident.'
Kate's eyes are fixed on my face as if there's some invisible mechanism hooking them there.
'You killed him?' Her voice is so faint, I can hardly hear it.
'I didn't mean to. It just happened. It wasn't my fault.'
'You threw the stone at him.'
There is that, of course. In simple terms, in terms of cause and effect, I suppose this is true.
'But I didn't mean to. I didn't mean it to happen.'
Kate's body contracts into itself, winces away from me.
Her eyes have doubled in size.
'Why are you looking at me like that?' I ask her.
'You can ask me that?'
'Kate, I've told you, it was a terrible accident. It must have been one chance in a thousand, one chance in a million even, that the stone would hit him like that, that he would fall like that. It wasn't something I meant to happen.'
'I left you with my baby. I let you look after my baby, and you had Max's blood on your hands.'
This is a foolishly melodramatic and unhelpful way of looking at things, besides being extremely hurtful, and I tell her just that. She doesn't seem to be listening.
'I knew something happened. I knew something must have happened. But this—but this! How can I bear it?'
Her hands are twisting, each kneading the other. Tears run down her cheeks. She makes a little moaning sound.
'Did you not guess at all?' I ask, curious. I've always wondered if someone had guessed. Someone in the family, I mean.
She shakes her head. She's still moaning in that slight, ineffectual way. 'How could I?' she asks. 'How could anyone imagine such a thing?'
'Kate,' I say, urgently, speared by a sudden and dreadful anxiety. 'You mustn't breathe a word.'
'Am I likely to? Am I likely to tell anyone this? Oh, my God.'
She's shaking, and rocking backwards and forwards.
I find myself wondering whether it's really so very bad. Of course (well, of _course_ ) I know murder's a bad thing. And I can see it must be a jolt for her. But it really was more a case of manslaughter than murder (this argument has over the years acquired amplified persuasive value, for me). And it isn't as if I meant to do it. It isn't as if it wasn't an accident. I suppose I've become over time so used to having killed Max, so used to the knowledge of my having killed him, that the act's moral edge has blunted somewhat for me. I can't help feeling Kate is slightly overreacting.
'What did you do with him?'
'What did I do?' I don't immediately understand what she means. I've just told her, haven't I? I killed him.
'With the...with the body?'
'It was when the swimming pool was being filled in,' I say, watching her to see how she'll take it. 'Remember? It was all dug out.'
'So...?'
'So I took the body out to the garden, at the back, and I hid it at the bottom of the hole where the pool had been. Among the rubble. Before they filled it in.'
'You carried it out? The body? You were able to carry it?'
'I used the furniture trolley.'
This circumstantial anecdote is almost too much for her.
'So Max—Max is buried in the back garden at Rain?'
'Yes.'
There is a long, long pause while she absorbs this.
'And now?' she says, finally. 'Will anything happen because of what Sophie did?'
'I would imagine so. I would imagine Pritchard will turn up again. At first he was just pestering me because he thought I knew where Max was; he thought I'd lead him to Max. But now he'll be smelling a rat. Oh, he'll be back: don't you worry. He'll be back. Everything was all right, Kate. I had everything all sorted, until Sophie went and wrecked it all.'
'Everything was all right? You murdered Max, and you're telling me everything was all right?'
'I didn't murder him,' I hiss at her. 'I keep telling you: it was an accident.'
She shakes her head again. Her eyes are fixed on some point on the floor, and she's hugging herself in a strange way. She doesn't stay for long, which I suppose is not surprising.
'I need to get used to this,' she says, her voice breaking. 'You've got to give me time.' She scurries out the door, dabbing her eyes, giving me a nervous sidelong glance as she goes, as if she thinks I'll take a swipe at her on the way. It's not out of the question.
Well, she was the one who wanted it. She was the one who wanted to come around and have a comprehensive talk about everything. It was her fault, anyway. Well, you could argue that it was her fault. None of it would have happened if she'd stayed out of my husband's bed.
And she's offered no sympathy at all. Nothing like: _Oh, Mum, how awful it must have been for you. Living with that, all these years._ Nothing like that.
I pull myself back, at that point. It might be a little much, to expect Kate to be sympathetic right away. It's been a shock for her, I can see that.
Still. Haven't I been brave, and lonely, too? Aren't I to be pitied?
For a couple of days, nothing happens. I worry about Sophie, but I'm still so angry with her that I think it's better to let the whole thing die down, to wait until I can be calmer about it, to wait until...Until what?
I realise that I've frightened her; I realise, too, that she did, as Kate says, mean well. I resolve to fix things up between us. But not yet. I can't do it yet.
All I can think of is the knock on the door.
It takes three days. And then, in the evening, it comes, and there he is.
'My God,' I say, trying to sound spontaneous and cheerful and innocent. 'Inspector Pritchard, as I live and breathe. Where have you sprung from, Frank?'
I've been thinking, thinking, thinking about this.
He grins, a bit foolishly. I've always thought Frank's foolish grin is a deliberate ploy. It's a good one. He indicates his shoulder and the stuff on it in a gesture which at first I don't understand.
'Superintendent, actually.'
'Wow,' I say, hoping it doesn't sound perfunctory.
And then, when we've got past the civilities and I'm pouring a cup of tea, he says: 'You're probably surprised to see me again?'
'Well. Not entirely. I think you've been talking to my daughter and my granddaughter, haven't you?'
I say this with the utmost composure. The stream of tea from the teapot's spout doesn't so much as quiver.
'Well,' says Frank. 'Your granddaughter's been talking to me.'
'She's such a romantic,' I say, brightly, offering Scotch fingers.
Frank is watching me. He's not going to make it easy. 'A romantic?'
'Goodness, yes. Couldn't you tell?'
'I'm not sure. She seemed a truthful child to me.'
He still has that rather ponderous way of speaking; still surveys his flat fingertips while he chooses his words. But then he looks up, and his eyes search me.
I try to look intelligent and unconcernedly interested.
Alert, not anxious.
'It seems Mr Knight is dead?'
'Ah,' I say, shaking my head in a manner that implies a world of unspoken stories, unexplored explanations.
'So Mr Knight's not dead?' asks Frank, politely.
'You're going to think this is a bit weird, Frank. I'm almost embarrassed to tell you.'
He chuckles, reassuringly. I almost expect him to lean over and pat my knee, so fatherly is he.
'Try me,' he says. 'You wouldn't believe the number of weird things I've heard in my time.'
'Well, Frank. Sophie's very romantic.'
'You said that, yes.'
'Kate and I—Kate's Sophie's mother, remember: you once met her here—Kate and I have been a bit worried, you see. Sophie has this tendency to fantasise, and lately she's been getting a bit fixated on Max.'
'Fixated?' says Frank, interested.
'Yes. Well, that might not be quite the right word. But for some reason she's been looking at old photographs, and poring over them, and it seems she's created quite a little story for herself, out of Max.'
'A story. Well, well. Kids!'
I take heart from this. 'Of course, Max was a handsome man.'
'A very handsome man,' says Frank, generously.
'And a bit of an adventurer, I suppose.'
'Indeed.' Frank's tone is dry.
'Well, from her point of view, I mean. Of course, she doesn't know the things you've told me. I can't tell her those. But it does seem as if she's been—well, romanticising him. Daydreaming. And Kate and I were both a bit worried about it.'
'Kate knows about her stepfather's—er—adventuring tendencies?'
Oh, nice one, Frank, I think.
'Oh, yes. Of course, I've told Kate what he was really like. What he was really doing.'
'Of course.'
'Anyway. So I thought, perhaps best to give the whole thing a smart knock on the head. Sophie had been carrying on, saying how he might come back, saying how she was sure we still loved each other. You know what teenage girls are like?'
'My word, yes,' says Frank.
'So—I suppose it sounds a bit silly, but honestly we were getting quite concerned—I told her he was dead. Just to put it all to rest, get it all out of her mind. And of course, it might well be true by now anyway.'
'And the grave...?'
'The grave?'
'Your granddaughter—nice little girl, isn't she?—she said you'd visited the grave.'
'Well. That was a bit of extra information designed to... um...make it all a bit more convincing.'
'Corroborative detail, intended to give artistic verisimilitude to an otherwise bald and unconvincing narrative?' suggests Frank, helpfully.
I'm not sure where Frank is coming from, with this. It rings a bell, but for the moment I can't quite place it and decide to ignore it.
'I guess so. The point is, we needed to find some strategy to divert her, to stop her carrying on. So we thought the best thing to do was just to say, well, he's gone.'
'Good thinking.' Then Frank pauses, apparently puzzled. 'Then why would you have told Sophie not to tell her mother? If you and she had cooked this up between you?'
It's my turn to look puzzled, concealing my anger and panic as best I can. Why in God's name did Sophie have to pass on this sort of minutiae?
'I told her it was our secret, certainly. But not to tell Kate?
No. Sophie's obviously got confused on that one.'
'Ah,' says Frank, pensive now. 'Such a smart-seeming kid, too.'
I try not to glare at him. 'Well, that's how it happened.' It does sound a bit lame.
Frank nods affably. 'And what do you think?'
I'm at a loss. Think? What am I supposed to be thinking?
'What I mean is, do you think Mr Knight really is dead?'
'Well, I guess we have to accept that it's certainly possible.'
'When we last spoke, you didn't think he was dead. In fact, you were sure he wasn't.'
'Years have passed, Frank.'
'During which something has happened to change your mind? I can accept that, but I'd like to know what.' His tone alters. 'Or perhaps you weren't telling the truth earlier? Perhaps you knew then that he was dead? Perhaps that's something you've known all along?'
This is suddenly more confrontational than I'm expecting. I keep my voice low, reasonable.
'Frank, it was years ago. I knew nothing then and I know nothing now. But, as time goes by—well, we have to think it's more likely, don't we?'
He is silent, and I think perhaps it's time to bring the battle right up to him.
'Frank,' I say, hurt. 'All those years ago, when you came to see me, when I kept telling you I didn't know where Max was and you wouldn't believe me—did you think then I'd had something to do with his disappearing like that?' I pause, then I think it's safe, perhaps advisable, to say the unsayable, to approach the unapproachable. 'Did you think I'd _killed_ Max?'
'Well, no, actually. We didn't think he was dead, you see. We were thinking, he's gone to ground, and sooner or later he'll pop his head up above the parapets, and then we'll nab him. But he never has, you see. And there was a rather large amount of money he could have got his hands on, without too much trouble at all, and he didn't. And there hasn't been one reliable sighting, here or overseas. Also, some items came to light.'
'Items?'
'Items from a suitcase.'
I'd always wondered about that suitcase. I do the baffled look.
'By the time we got to it, it had been gone through pretty thoroughly and I imagine some of the contents had gone missing. But a few remained, and one of them had Mr Knight's name on it. A medication, it was. But what it looked like to us, you see, was that Mr Knight was still manufacturing his own disappearance, perhaps with a little bit of help from you, not that he _actually_ had disappeared.
'But it didn't make sense, you see. We had it from you that Mr Knight had taken his car, so we couldn't see what he'd be doing leaving a suitcase in a railway station locker. It didn't add up, not really. So at that stage, no, all I thought was that you knew where he was and you weren't talking. I wasn't even sure of that. We had nothing to go on. But I did notice, and I did make a note at the time, that you tended to talk of Mr Knight as if he were in the past. He _was_ , not he _is_. He _did_ , not he _does_. And I'm puzzled, Mrs Knight. Ms Weaving.'
'Isabel. You used to call me Isabel, Frank.'
'Isabel. I'm puzzled, Isabel. You see, I tend to believe that Mr Knight is dead. I tend now to believe that he was dead even when we were looking for him, twelve or so years ago. And now—yes, now I think you knew he was dead. I'm not sure what was happening to you back then. I don't know whether you'd been blackmailed or bribed or bullied. Or none of the above. But I'd certainly like you to tell me.'
I stare at him with absolute blankness. It seems to me my responses here are really important. He's flying a kite, I can see that. He's doing his old thing of trying to unsettle me, catching me off guard. I don't want to give him even the faintest sniff of anything untoward.
It doesn't matter what he thinks or suspects or wonders about. What matters is that I don't open the door to him, that I don't open it even a crack, even a skerrick, that he remains entirely ignorant of all the goodies I'm hiding, the veritable treasure chest of what he would call relevant material. He can suspect and wonder all he likes, so long as he isn't certain, so long as he doesn't _know_.
On the whole, I don't think I handle it badly. I'm shocked and astonished and, yes, hurt; but mainly I'm just tired. It all happened so long ago.
'For Christ's sake, Frank. You're really scraping the bottom of the barrel, aren't you?' I say this wearily, heavily aware of his eyes resting on my face (they're small eyes, rather too close together: my mother would have said he wasn't to be trusted), affability forgotten, the bloodhound gene taking over, his body fairly trembling with alertness and suspicion and mistrust. I maintain my fatigue, my incredulity.
I've never been so pleased, to see the back of him. When he finally goes I'm limp with exhaustion and damp with sweat. I'd thought it was all over. I'd thought he'd lost interest, that I'd fought him off, that I could relax. But I'm right back in it again now. Thanks to Sophie, I can't stop worrying, not yet.
The knock—the final knock, I mean, the knock that tells me it's all over, red rover—is still out there somewhere, waiting to happen. It's like the iceberg, bobbing around in the Atlantic (well, really big icebergs probably don't bob; they probably just mooch along, heavy and full of destiny), patiently expecting the Titanic to happen. And then: _Wham!_
The next time I see Kate, I draw her aside.
'Pritchard's been,' I say.
'And?'
'And what?'
'Does he know?'
'He doesn't know. Not yet.'
'But he suspects? Because of Sophie, he suspects?'
'God knows what he suspects,' I say. 'He certainly thinks something odd has been happening.'
Kate looks unhappy, as well she might. I don't suppose anyone really wants their mother convicted of murder.
I couldn't bear to stay at home: I was too edgy. I went to a couple of movies, but that was hopeless as a strategy for self-distraction. I couldn't have told you what they were about, not even two minutes after I'd walked out of the cinema.
So I went to work. The arrangement I had with Bea was that I would work two days a week. She gave me a narrow look, one morning.
'This is the fourth time you've turned up this week, Izzie.'
'So?'
'The practice can't afford to pay you extra, you know that.'
This was manifestly untrue, in fact. Bea was making so much money she didn't know what to do with it all. I didn't care, though.
'I'm restless,' I said. 'You don't have to pay me. I'm happy to put in a few extra hours. That new job's got a couple of tricksy bits. I'm just ironing them out.'
She raised her eyebrows.
I wandered into my office and worked, sitting under the same skylight where I had sat when I first saw Max—how long ago? Nearly twenty years.
I doodled at my sketches for our latest project, a vast old mansion in the hills that someone with more money than sense wanted to convert into a health resort. I wanted to tell him to knock it all down and start from scratch. He was set on old-world charm and what he called art-deco flair. The place wasn't remotely influenced by art deco, and I'd tried to explain that to him, too, but eventually I'd given up: when a client doesn't want to hear something, it's no use telling them.
Inspiration wouldn't come. It was a huge, leaden manor with cold corridors, small windows, tiny attics and antiquated plumbing. Nothing would save it. I was running behind schedule on it, and starting to loathe the whole project. Bea agreed with me that it was hopeless, but pointed out that it was worth a heap to us. The incentive didn't seem enough, somehow.
There's a knock on my door. _Not here_ , I think, suddenly panicking.
But it's only Kate.
'Hi,' she says, cheerily, presumably for Dawn's benefit. 'I was passing.'
'Hi,' I say.
She comes in, closes the door behind her, sits down. 'I was going to ring. Then I thought, might they be tapping your line?'
'I can't imagine they are,' I say, although in fact I have privately canvassed this possibility and concluded it is highly likely. 'Anyway, you could have rung my mobile.' I'd often thought how Max would have loved mobile phones—their convenience, their glitziness, their whizzbangery.
'They tap mobiles, too.'
'How do you know?'
'They do,' said Kate, definitively. 'Anyway, it doesn't matter if they are or they aren't: I just thought, we have to talk.'
'I don't see why.'
'Mum, I've got an idea.'
These are not words to fill me with confidence, not from Kate, but it's clear I'm not going to be able to stop her.
'I've been thinking,' she says. She has an odd look on her face. 'I've got a plan.'
And she tells me what it is.
'What I thought is this. I'll go to see Superintendent Pritchard.'
I stare at her. She's leaping right into the lion's mouth?
'It's a natural thing to do. I'm Sophie's mother, after all. I'm the mother of the child who has been telling him things, asking him to do things. We've talked already. I'll ring him up. I'll say I want to see him, want to talk some more.'
'But why?' I ask, dazed.
'You'll see.' She pauses, thinks. 'You may not like what I'm going to do. But it's the only thing I can see to do. I think it might work, so please, Mum, just consider it. Okay?'
I look at her, mutely. My mistrust is profound. First Sophie meddles in my affairs; now Kate. What in God's name is she hatching?
'So,' she says. 'I go to see Pritchard. I'll be rather cool, you know. I'll say, he should have contacted me earlier about Sophie. I'll say, I could have saved him a lot of trouble. Sophie is an emotional child and she's got a bee in her bonnet. I'll say, she's got the wrong end of the stick, and if he acts on anything she said, he'll find himself in trouble. Up the creek without a paddle. Chasing mares' teeth.'
'Mares' nests,' I say, automatically. 'Hens' teeth; mares' nests.' The thought of Kate thinking she can threaten Pritchard is making me physically ill, the bile rising in my throat.
'Whatever. So then he'll ask why. So I'll say, I don't really understand what his interest in my stepfather was, and you won't tell me. And he'll say, why not? And I'll say, there were reasons why you won't talk to me about Max. And he'll say, what are those reasons.' She pauses.
'Yes,' I say, swallowing. 'Go on.'
'So I'll tell him. I'll say, the reason my mother and Max split up was that Max was having an affair with me. And I'll say, he kept on having it. I'll say Max went to Sydney after you and he split up—I thought Sydney would be a good place to have him live, because it's so big, so crowded, you couldn't possibly prove someone hadn't been in Sydney—but I'll say he came down, every so often, to see me. I'll say this went on for around two years.
'And then I'll say he went overseas. I'll say, I've had postcards from Prague and Mexico. I thought Prague and Mexico would be good because they're a long way away and a long way apart. I'll say I last heard from Max maybe two, three years ago. And I'll say I don't have the postcards any more, because I didn't want my husband to see them.'
Kate pauses. She's twisting her fingers, knitting them unconsciously into the old game of church and steeple that she and I used to play together when she was tiny. _Open the doors and let out the people_ , I nearly say.
'Do you see what I'm doing, Mum? It'll all work, you see. It'll explain for him why you never told him why Max left, because of course you didn't want to admit that he'd slept with your daughter. And it explains why Max left, too. And it makes him think Max kept on being alive long after he left you.'
She muses for a moment. 'And, the thing is, I think I can make him believe me. There'll be enough truth in the lies I'll be telling for me to pretend to myself it is real, that everything I'm saying has really happened. In point of fact, and you might as well know this, all I'll be doing is telling him a story I've told myself. When Max and you split up, I thought Max would come to see me. I was sure he would. I didn't think in a million years he'd go away without somehow coming to see me. Because he knew about Sophie, of course he did. Max wasn't stupid: Max would have taken one look at Sophie and he would have known. And we needed to talk about it, and I know we would have talked about it. But he never came.
'Now, of course, I know why not. But then I didn't know, I just thought, where's he gone? I thought, sooner or later, he's bound to come. And I kept imagining I'd open the door and there he'd be, and then, because, you know, you and he had broken up, we could...well. We could get together again. Just for a little while. And I kept imagining that he'd gone somewhere not very far away, and that he'd turn up. And everything would be the way it had been, could be. Well, that was the story I told myself. So if I tell this to Pritchard, I'll only be telling him what I wanted to happen, what I'd talked myself into believing was going to happen.' She breaks off, tiredly.
I sit in a trance and watch her hands. _Here's the church; here's the steeple._
'I don't know what to say,' I eventually manage. But the rage flickering within me is going to tell me what to say, very soon. I feel it grow. I let it grow.
'I think it might work, Mum. It might make you safe. I want it to make you safe.'
'You think he'll believe you?' I snap.
'I don't know. Yes, I think so. I think I can be very believable.'
'Kate,' I say, the tsunami inside me swelling, overflowing, rushing out of control. 'Kate, this would be the most...the most half-baked, idiotic, perverse, _stupid_ thing to do. Can you not see that?'
She looks at me, sideways, clearly surprised. 'It'll prove Max was alive, after he was really dead. Why will that be stupid?'
'There is no reason in the world for Frank to believe any of this half-cocked crap. Can't you see? All he will think is that you and I are conspiring. All he will think is that I've been foolish enough to set you onto him, to spin him a story to make him believe Max is still alive. Can't you see how transparent it is, how ridiculous?'
She simply stares at me.
'You're a fool,' I say. 'Thank Christ you haven't done anything.' A dreadful suspicion smites me. 'You haven't, have you? You haven't done anything? You haven't spoken to him again?'
'No.' She has a blank look.
Can she really not comprehend? What right does she have to break the secret, the secret of her and Max, the secret she told me she'd never tell? Can she really think Frank will be such an idiot as to fall for a story like this, that she can maintain a defence against all his art and skill, his little levers and screws, his delicate chiselling and his slow, relentless assaults? And how dare she lay claim to Max? How dare she imply that his interest in her could have been anything other than a passing aberration?
'Thank God for that, anyway,' I snarl.
'But I want to save you, Mum. You're in danger, and I want to save you.'
'If I want to be saved, I'll let you know, thank you very much.'
'Is that all you can say?' Kate's voice breaks. 'Is that all you can say to me? I've planned this all for you. You mightn't like what I've thought of, but it might make you safe. I'm telling you, I'll perjure myself for you. I'll do anything for you. I only want to try to help you.'
'For God's sake, Kate,' I say. 'Fuck off, will you? Just fuck off. And mind your own business, you and Sophie both. Just stop meddling. Leave me alone.'
Tears stream down her cheeks. 'Ever since I can remember,' she sobs. 'Ever since—oh, God, what does it matter? I've always wanted...'
She can't go on, and it looks as if I won't ever know what it is she always wanted. My anger against her balloons. I cannot bear the thought that she could even consider telling Frank about her and Max, about her and Max in bed together. I can't bear to imagine Frank sitting at my kitchen table, his eyes meeting mine, the knowledge of Kate and Max, of what Kate and Max did, shared between us. The thought of Frank's sympathy, his genial commiseration, is intolerable.
Kate's sobs increase, multiply. People will hear. Dawn will hear, and anyone else who's around. I remember that Bea has an important meeting with clients in her office this afternoon. I look at Kate helplessly. I go over to her, pat her shoulder. I find my hand grasped with a ferocious strength. She pulls me down to her and hugs me wildly, hysterically. It's very uncomfortable, and I try gently to disengage myself. Irritated beyond enduring, I make soothing noises, glancing at the door and hoping Bea can't hear. Dawn probably can, out in reception.
'I'm so sorry.' She weeps into my shoulder. 'I'm so sorry, to get it all wrong.'
Gradually, she quietens. She looks terrible. I'm still livid with her, but I suppose I can try to get over it. I pat her some more, trying to quash my fury and despair. What more can she expect of me?
She grabs my box of tissues and blows her nose noisily, several times. She stands, regards me with an unreadable look on her face.
'I love you so much, Mum,' she says. And is gone.
The telephone rings just after three o'clock in the morning. My heart leaps from its normal position and rattles against my throat as I reach out of bed for the handset. I expect Frank, Frank ringing me in the middle of the night, ready to catch me adrift, to trap me. Instead, I hear Henry. 'Isabel?'
'Henry,' I mumble, blurrily cross as I surface from sleep. Here I am, badly frightened, thinking everything is over, that Frank is ringing me to say he knows it all, no use going on, and it's only Henry. What the hell is he ringing for?
His voice shakes. 'I've got bad news, dear.'
Henry's never called me dear in his life. Suddenly I'm bolt awake.
'It's Zoë.' He stops, choking.
'Yes, Henry,' I say, keeping my voice steady. An accident, I think. She's hurt, that's all. A car accident, maybe. She can't be dead. Not Zoë, not my sturdy, vigorous sister. 'Tell me, Henry. Tell me.'
'Dead.'
'How?' I ask, transfixed.
'Heart attack. Acute myocardial infarction.' It's so Henry, to add that.
'But _how_? I don't understand. Was she ill? Was there no warning? What happened? What in God's name happened?'
'It was out of the blue.' He sounds apologetic, as if he has been uncharacteristically careless. 'No warning. None at all.'
'When, Henry? Where? Where are you ringing from?'
'I'm at home.'
I picture him, alone in his study, in his big, faded, brown corduroy chair.
'What happened?'
'I don't know,' Henry says, suddenly sounding old and exhausted and confused, and with odd pauses. 'I don't know, Isabel. We went to...bed as usual. She got up, went to...the bathroom. I heard her cry out. She was...on the floor.'
And that was it. She had lost consciousness even before he'd reached her. He rang the ambulance. He said they had arrived rapidly—not more than five minutes. But she was dead.
It is strange that Henry and I should have this bond, this powerful link of finding our spouses dead on the floor. It's different for him, of course: he didn't put her there. Well, I presume he didn't. I briefly contemplate the possibility of explaining the irony to him, but decide against it. He's not going to appreciate this.
My mind's in overdrive. I don't know why this happens to me, why this feverishness grips me in a crisis, why it crowds out my grief, my shock. Already I'm thinking, when will the funeral be? Has he told anyone else yet? Ought I to go over? I'm even wondering what I'll wear to the funeral. My brain's manufacturing thoughts to prevent it from thinking the things it doesn't want to confront. That must be what's happening. I try to concentrate.
'Would you like me to come over, Henry?'
'No,' he says, to my huge relief. 'No, there's no point.'
'Are you all right?'
He says he is. We exchange a couple of bleak remarks, promise to be in touch the next day, hang up.
It's so hard to imagine a world without Zoë. I lie in bed and think about it. I can't sleep, so I get up and make myself a cup of tea. For once, I don't feel like a drink: I'm not sure why not. Borrow pads after me and regards me sorrowfully.
I try to cry. I actually do squeeze out a tear or two. What am I feeling? It's hard to say. I'm surprised, certainly. I recall Zoë's stocky figure, her positive stride, her firm, strong voice, her annoying way of articulating words very clearly in case the person to whom she was speaking was mildly retarded. Well, that's how it sounded. She didn't look like a candidate for a heart attack. What has Henry been thinking of? How has he allowed this to happen?
I manage not to ask him these questions when he rings me the following day. I still feel stunned, trapped in the ice in some frozen hinterland where the full realisation of Zoë's death is withheld from me. Soon I will comprehend it and cope with it.
'We're all getting together to talk about the funeral, Isabel,' says Henry, sounding weary. 'Would you like to come over?'
Who's _we_ , I wonder? Henry's an only child; he and Zoë are childless; all our parents are dead. I suppose Zoë had friends who want a say in it all. I don't like Zoë's friends, most of whom are authoritarian women who have spent their lives bellowing in classrooms, and Henry doesn't sound as if he especially wants me there anyway. I excuse myself and ask him to keep me in touch.
'You don't want to speak?' he asks, surprise inflecting his voice.
'At the funeral? God, no.'
'Are you sure, Isabel?'
I tell him I am sure.
I hate funerals. In fact I remember once saying this to Zoë, who replied sarcastically that there was nothing special about me; nobody actually enjoyed funerals.
I think she was wrong. I think some people do enjoy funerals. They remind us, after all, that we're still alive, a condition not applicable to the person at the centre of the occasion. I've been to lots of funerals now, and I've often observed a kind of avid relief rampant in the congregation, an ungodly solace perhaps derived from outliving the deceased and perhaps from finally being able to speak frankly of him or her with impunity.
When we marry we take on other people's families, their births and deaths and traumas, their scandals and skeletons. This is a fact that certainly hadn't revealed itself to me when I skipped so smugly down the aisle to Steve, awaiting me with a goofy look on his square face.
It started to become clear when we'd been married only a year or so. An elderly aunt of his died and I discovered to my alarm that Steve and his entire family confidently expected my attendance at her farewell. I'd never been to a funeral. It sounds silly: I was twenty-one, after all. I suppose I had remained unusually untouched by death. Two grandparents had died during my childhood, but nobody had expected me to go to their funerals. I didn't think funerals were my business. I panicked at the thought of this one.
'I didn't even know her,' I said to Steve. 'I only ever met her once. Why do I have to go?'
He had that pained expression I was starting even then to resent so sharply.
'She was my aunt, lovely. Of course we have to go. We both have to go.'
'But I didn't _know_ her,' I wailed, despairing of making him see reason. 'Can't you say I've got morning sickness?' Already I was pregnant with Kate. 'It wouldn't be a lie.'
I was not certain why my response was so negative. Normally I did what Steve asked me to do: I regarded docility as one of the wifely panoply of virtues to which I aspired. But this time my alarm was disproportionate: I passionately did not want to be a strand of the tapestry of this dead woman's life; I did not want to be caught up in a grief that was none of my business, a loss that I had never felt.
'It wouldn't be true either,' said Steve. 'You're over the morning sickness, lovely: you know you are.'
I could not withstand his obduracy. So he hauled me off to his Aunt Bessie's funeral, where, ironically, I did indeed feel nauseous, although this was (as he pointed out) probably because of the heat of the day and the closeness inside the church, rather than because of my pregnancy. I hated it all but I was morbidly fascinated by it, too.
And I understand now what I didn't then, that in marrying him I assumed for him (as he did for me) afflictions and encumbrances such as illnesses and deaths and funerals, that marriages entail families and joining in what families do. When we marry, we share lives; we learn customs; we adopt relations; we inherit deaths.
Except marriage to Max, of course—or at least what passed for marriage to Max. Max had no family: no mother to terrorise me (as Steve's had), no father to glance appreciatively up my legs and down my cleavage (as Steve's had), no sister to patronise me (as Steve's had). No family at all.
*
And so I go to my sister's funeral. It is a typical Melbourne day—cool, bleak, grey, with the occasional perverse flash of sunshine disrupting the dismal threat of drizzle.
I sit in the funeral chapel, front pew (membership of the deceased's family offers me these privileges), and gaze at the coffin, which undertakers always insist in calling a casket. (I note that these undertakers wear brass badges on their lapels that identify them as _bereavement consultants_ , whatever that might mean.)
The coffin is bright and smart and snazzy and rather too ornate, its decorated brass handles gleaming and its dark mahogany veneer polished to the clean, brilliant gloss of a new mirror. She would have thought it tacky: Zoë had taste, after all, and she wouldn't have liked the tawdry dazzle of the brass, the fussy curlicues on the handles, the insufficiently solid look the whole thing somehow has. It is so narrow, it is hard to believe Zoë's robust figure is contained within.
The taped music is Vivaldi, and seems too sprightly for the occasion. I don't think Zoë had any particular affection for (or indeed knowledge of) Vivaldi, and presume this choice is Henry's flight of fancy. Zoë was a Mahler person, a Wagner person. The Valkyries might have farewelled her, or the Meistersingers, not Vivaldi.
Henry is out in the foyer, looking solemn and burdened, thanking people for coming. Who knows? Probably he does indeed feel burdened; perhaps he is genuinely grateful.
I remember the last time I sat in a church for a ceremony at which Zoë was a central participant. It was her wedding, some thirty years ago. A little more than thirty years. Well, I hadn't much enjoyed that, either.
I find myself glancing around, every so often, to the back of the church. I pretend to myself that I'm doing this out of interest in who will attend. But it's Frank who's on my mind, Frank's shadow I expect to see lurking at the back, sliding behind the door, treading up the aisle in that measured, weighty way he has.
The Vivaldi finishes and the service begins. It is quiet and unexceptional. Henry (always technically proficient, as becomes a science teacher) speaks with the accompaniment of a powerpoint screening that shows photographs of Zoë at various stages of her life. Zoë as a baby, toddler, schoolgirl, teenager, debutante, graduate, bride, daughter, sister, wife, aunt, teacher. All the roles of her life, neatly clipped out and presented. I am in some of them, especially the early ones. I feel oddly uncomfortable about this, and find myself thinking that Henry should have asked me if I minded. I don't like it, being on show in this way: it's one thing in a family living room or a photograph album, quite another in a funeral chapel with a coffin next to one.
Here we are, Zoë and me, neatly attired in our school uniform, our hair freshly combed, standing awkwardly together, holding hands (a thing we never normally did, as I recall). Here we are sitting on our father's knee, too old to be doing so comfortably, displaying forced grins. Here is Zoë in a long dress, going to her first ball. Here she is marrying Henry in his silly tie, looking for once as if she mildly likes him. Here she is being auntly, holding Kate as a newborn. There she is, in her coffin; here I am, in my pew. It is bothersome.
My thoughts stray as Henry speaks. I cannot believe Zoë has died: I am cross with her for doing so. She is—was—only five years older than I am, and it seems to me she had no business relinquishing her spirit so easily, with so little warning to the rest of us, so little fight.
I know they say the grief issuing from bereavement takes you inexorably through several stages, of which anger is one. This isn't like that, though: it isn't profound, burning anger, but exasperation, much as I might feel if we were playing Scrabble and I knew she wasn't trying. It worries me that I cannot tap any deeper feeling than this for my sister. I have not grieved for Zoë, any more than I grieved for my mother. Can it be that I do not feel grief? Surely I loved her. She was crabby and annoying and difficult, but surely I did love her?
Why can't I _feel_?
Henry is saying something about Zoë always having wanted children of her own, about her being a fond aunt, about the fondness deriving partially from her own dashed hopes, her profound disappointment at her own barrenness.
I don't think this is true. I don't think Zoë wanted children at all; and Henry certainly didn't: it would have up-ended their neat lives, their organised lists, their careful careers, their complex diaries, their tidy house.
She never forgot the children's birthdays, I'll give her that. But she was perfectly happy, it seems to me, to remain an aunt, to forgo all the messiness and indignity and turmoil of motherhood, retaining an arm's-length intimacy, a burdenless responsibility, an affectionate distance.
I am annoyed by Henry's easy assumption that Zoë's childlessness was cause for sorrow or sympathy, and I stop listening for a while.
I try to put Frank out of my mind, but it's an effort. He's starting to hover in the pew directly behind me, his flat capable fingers about to descend vice-like on my shoulder.
Deliberately I turn to repel him and find myself gazing into the reproachful eyes of a mousy-looking woman I've never seen before in my life, who presumably is an ex-colleague, or an ex-student, or somehow a beneficiary of my sister's efficient and intrusive habits. I turn back again.
I can never go to a funeral without visualising my own. I don't know if everybody does this: I suppose it's egotistical and perhaps irreverent. But I can't help it. While Henry speaks, reading in his finicky tones from a piece of paper that he has obviously prepared with meticulous care on his computer, I can't help reflecting that there will be no husband to speak at my funeral.
Not that husbands or wives usually do speak, anyway; Henry has already made reference to this and said that he feels this is something he needs to do; he feels for some reason I cannot understand that he will not have bidden Zoë goodbye unless he does so formally, publicly. He looks tired, but his voice doesn't shake and he seems to have himself well under control.
I never did think they were very fond of each other. I wouldn't want a husband who could get up and say a few pernickety, dry-eyed words about me when I died.
But if Steve won't speak at my funeral—and Max certainly can't—who will? Who will be left? Will Kate say anything? Will Dominic? Will they care? What will they say? I steal a look at Dominic, sitting in the front pew on the other side of the chapel, next to his sister. He is on his own: Paula hasn't come. Steve has: he is sitting in the same pew as Kate, further down. Sophie is there, in her school uniform, on Kate's other side. Kate told me she wanted to come; so did Liam, but she sent Liam to school as usual, because she thought he wasn't old enough. She's probably right.
Well, Sophie's twelve. She's old enough to know people die; people go away for good. She is sniffling quietly into a tissue. She knows I'm here, but hasn't looked over at me. She's still upset with me after our last meeting.
Kate is upset, too, but she and I have achieved our precarious semi-reconciliation. I can tell nothing from Dominic's expression. Is this something he felt he had to come to? It meant time away from the office, after all, time stolen from the all-important things he has to do that crowd in on him and prevent him from having a proper lunch. He is soberly suited, eyes cast down. Was he impelled by a sense of propriety? Did he care much about Zoë? Does he care much about anybody?
Just as I look across, I see Kate lean towards him, whispering something; he leans back, nods, smiles. He puts his hand on Kate's for a second.
Let's replay that.
He puts his hand on Kate's hand. I feel the shock of it quiver through me. This is Dominic: Dominic the isolate, the ascetic, the austere loner; Dominic who hates to be touched, who doesn't kiss, doesn't hug, doesn't do any of those vaguely social things people expect of one. And here he is, touching his sister. Unbidden, evidently in compassion and—well, yes, love.
Why are they all sitting over there, anyway? Why are they sitting with their father? I'm more bereaved than he is: she was my sister, after all. Steve didn't even like Zoë very much.
I can't recall the last time Dominic willingly touched me, let me touch him. Probably during his toilet training.
I find myself thinking about the apparent closeness with his sister that Dominic has just manifested. Will she talk with him, I wonder, uneasily. Will she tell him any of the things she now knows? Dominic is a lawyer. Not a criminal lawyer, but a lawyer just the same. He probably has strong views about people who kill people, whether they do so inadvertently or not.
Suddenly I feel immensely vulnerable because of Kate's knowledge. I wish I'd never told her anything. I don't think she'd deliberately betray me: it's not that she's untrustworthy in that way. But she's such a fool, such a hopeless idiot, she might easily say something to incriminate me without meaning to.
There goes that shadow again, behind me, Frank's shadow, the quick blurred movement of Frank's hand about to grip my shoulder. For a moment, I tense.
Will Kate come and visit me if I go to prison? Probably, I think. Will Dominic? Much less certain, I imagine.
Perhaps, though, Dominic might be nicer to me. If I went to prison, I mean. Surely he'd be a bit sorry for me?
My attention is drawn momentarily by something Henry is saying about Zoë being warm and generous, somebody young people could easily relate to, somebody her pupils readily confided in. I'd as soon confide in a death adder, I think; and then I feel ashamed of myself.
Seriously, what will happen at my funeral? Will anybody even come? Will anybody bother? What music will they choose? I don't want Vivaldi. But what do I want? And how will they know what I want?
Now Henry is saying something about Kate.
And, suddenly, Kate is standing, going pink and blotchy around the cheeks, as might have been expected, holding a piece of paper in her hand. She is making her way to the microphone.
Nobody told me Kate was going to speak. Henry asked me if I wanted to; he didn't say a word about Kate doing so. I didn't want to. I don't know most of these people: they are Zoë's friends, Zoë's professional acquaintances, Zoë's ex-students.
I've met some of them, but they have nothing to do with me. Why should I open my heart to them and speak to them about my dead sister? I don't know what I'd say, in any case. I don't know what's in my heart: it's closed, even to me.
But Henry mentioned nothing about Kate speaking. What on earth will she say? Kate and Zoë weren't a bit close. Is she going to make a fool of herself? She hates doing things like this: Steve and I used to have terrible trouble with her whenever some kind of performance was demanded of her at school. There's nothing Kate hates more than standing up in front of a crowd of people.
And it's really very crowded, not just full but spilling over, and mainly with people I've never seen in my life. Ex-students? I can't think of a teacher whose funeral I would willingly have gone to, but I imagine they are ex-students: they're mainly pretty young. Why on earth is Kate doing this? Why is she exposing herself like this?
When Henry said _we_ like that, did he mean him and Kate and Steve and Dominic?
Well, Kate got on reasonably well with her, I know. Zoë made an effort: she was not acerbic with Kate as she was with me. She did not positively hound Kate, or criticise her, or flay her. But surely they weren't close.
Kate looks dreadful. The pink blotches make her resemble a plague carrier; she's been crying and the flesh around her eyes is soft and swollen; she's ill at ease, shifting from one foot to the other. I feel a trifle cross with Henry. Why is he forcing her to go through this?
Kate starts to speak. 'My aunt Zoë was a very special person,' she informs her audience, in faltering tones.
Well. Here's originality.
'Dominic and I always used to like it when we were little and Aunt Zoë visited,' Kate confides to the chapel at large.
It's another of the irritating things she does: _Aunt_ Zoë; _Uncle_ Henry. It infantilises her. They were always taught to call their relatives by their Christian names, none of this circuitous aunting and uncling. Sophie and Liam call their uncle plain _Dominic_. Well, Sophie does. I can't recall what Liam says, in fact. And I certainly don't recall my children jumping up and down for joy when Zoë happened to drop by, which she did infrequently.
Kate continues, feet shuffling, blotches flaring, looking as if she's about to fall over. 'Whenever Aunt Zoë came, she'd always talk to us as if we were real people.'
And when did anyone not, I wonder?
'What I mean is, she wouldn't talk to us as if we were kids. She wouldn't talk down to us. And she'd tell us what she thought. She wouldn't be soft on us. I remember once I didn't do well at school in something, and everyone else was inclined to say, oh, well, never mind, and Aunt Zoë said to me: "Kate, you can do better than that. What went wrong?" And I had to think about it, and I had to come up with what I thought went wrong, and she said: "Well, next time, you'll know what to do, won't you?" And I did.
'Dominic and I used to think she would be the most fabulous teacher, and I can see'—she looks up, smiling faintly—'I can see, from the numbers of you here today who must have been her students, you all must remember her as the most inspiring teacher imaginable. She was always very honest, that's what I valued most about Aunt Zoë.
'She was very direct, and she told you exactly what she thought, and you always knew you could trust her. And she knew we trusted her, and we knew she loved us. I hope she knew that we loved her. Goodbye, Aunty Zoë. We'll always remember you, and we'll always love you.'
Kate sniffles once more, casts a desperate apologetic look at Henry, and goes to sit down again while an approving kind of sugary murmur creeps across the chapel.
I remain still. I become aware that I am sitting stiffly, my muscles at full stretch, and it occurs to me that my face may look stiff, too. I arrange it in order to minimise its stiffness, its amazement, its pure steaming rage at my daughter's soppy sentimentalising, her distortions, her craven fibs.
Zoë may well have felt affection, to some extent, for my children, but she had no capacity to interact with them—or with other people's children—spontaneously. This is something I recognise since I, too, suffer from it; but I did better than she did, with my own children anyway. I suppose she may have been a spectacularly good teacher of adolescents (though I must say I doubt it); but she did not have the gift for speaking with small children; she was unable to address them save through a bracing interrogative style under which Kate (little as she seems to recall it) positively wilted. Her directness wasn't because she was honest; it was because she was undiplomatic and insensitive.
Steve and I used to joke about it. _Your bloody sergeant major of a sister_ , he used to call her. Involuntarily I glance over in Steve's direction and see that he is nodding and smiling encouragingly at Kate, who is still dabbing at her eyes.
Yes, indeed: I remember Stalinist cross-examinations like that which Kate has just described so winningly: _Well, Minky, why did you do that? You can do better than that, Minky. What went wrong? How can you do better next time?_ On it would roll, deadly as a puffball fish, inescapable as a guided missile. Caught in her sights, one could only shuffle in agony. And, invariably, to finish, in a despairing wail, as if you were the most moronic person on the face of the earth: _What were you_ thinking _, Minky?_
But none of this had anything to do with self-improvement, with doing better next time, with strengthening oneself or one's relationship with Zoë. It was only to pander to her ego, to foster in Zoë the entirely erroneous impression that she was necessary to one's development. It didn't issue from love or warmth. It issued from a paranoid need to dominate, to intimidate, to prove how indispensable she was, in short to bully.
I feel like jumping up and shouting all this to the assembled snivelling congregation. Then I see to my horror that Dominic is approaching the lectern. What the hell is he going to say?
He doesn't say much, as it turns out; but what he does say is to the point. Dominic has, as I know, a gift for this sort of thing. He doesn't allow himself to be overcome by Kate's brand of sentimentalism; he speaks crisply, without fuss.
How handsome he is!
He tells a couple of mildly funny stories; subtly he intimates that it would be foolish to deny that Zoë could be a difficult person. Well, a bit of a character, anyway. I start to relax.
Then he says: 'Zoë was a remarkable aunt, and a no less remarkable sister. It is not possible to conclude this tribute to her, to her energy and compassion and her sheer zest for life, without referring to her relationship with my mother, to whom she was devoted.'
My rage is swelling again. Why is Dominic doing this to me?
'For Zoë,' he continues, 'Isabel's welfare was always paramount. When Isabel was born, Zoë's delight knew no bounds. Zoë would have done anything at all for her little sister. She adored her as a baby, a toddler, a child and as a woman. Nowhere did Zoë's exceptional generosity of spirit show itself more clearly than in her love for my mother, who wishes to join with us in this tribute to her greatly beloved big sister.'
I glare at Dominic, who pretends that he does not notice, and will not meet my eyes. He says one or two more things, to which I do not listen, and steps down.
How dare he? I am thinking, the question pounding through my head with the force of an elephant stampede. I am so furious that I can sense the urgent particular throb of the blood in my temples. I am so furious that I could strangle Dominic.
Why should he have done this? Why impose upon me a wish I do not feel, a love I do not acknowledge, an obligation I deny? And, if he was going to say anything of the kind, why should he have made it sound as if the generosity was all on her side? It wasn't easy to be loved by Zoë, to be perpetually battered by her powerful zeal and her strident self-righteousness. What about _my_ generosity of spirit? Why doesn't that rate a mention?
I am only dimly aware of the rest of the service, but sooner than I expect I find to my relief that it is over and the front pews are dispersing.
I am surprised, when I turn around, to see how full the church now is. Again I think I see Frank at the far corner of my peripheral vision; it turns out to be some innocent person with no great resemblance to Frank on any count. We all file out into the uncertain weather, where a few cold drops are starting to fall, and two or three people whom I distantly remember approach me and say trite and hypocritical things about what a shock it is and how much we are all going to miss Zoë.
_Speak for yourself_ , I feel like saying, but do not. I excuse myself as soon as I decently can, and look for Dominic; but he is caught up with other groups and I can dimly sense by now that it's not going to be a good idea to stalk up to him and demand an immediate explanation.
A private cremation follows the service. I consider absenting myself, but in the end I decide to be generous and I go. Kate (who hasn't asked my advice on this matter) brings Sophie, who is too young for such a thing, and I am infuriated by her insensitivity.
I buttonhole Dominic at the end of it, as we trail towards our cars.
'Why did you say that?' I ask. 'About me, about me wanting to join in the tribute.'
He looks down at me. It's annoying, that he's so much taller than I am.
'I did ring you about it,' he says, sounding bored. 'You weren't home, or at any rate you didn't answer, and I thought I could surely assume you'd want one of us to say something. Henry said you didn't want to speak. I presumed you were too distraught to do so.' (He draws out the word 'distraught', accentuates it, makes of it an insult.)
It is probably true that he rang. I often ignore the telephone, especially if I've been drinking. But it is no excuse.
'I gave you no permission to say anything on my behalf.'
'Isabel, I'm trying to explain that it wasn't up to you. Henry wanted you to be included. He knew Zoë would have wanted you included, somehow, and so it was important that this was in fact achieved. Kate and I discussed it. The most significant thing appeared to us to be that Henry have the sort of ceremony he wanted, the sort he thought Zoë would want. That's all there is to it.'
'You should have asked me.'
He shrugs and starts to move away. 'I'm sorry you should feel like that. I must say, I don't see what the fuss is about.'
So cold, so formal! If only he had put his arm around my shoulder, given me a squeeze, said: _Hey, Mum, c'mon._ If only he would come halfway to meet me, only condescend to give the faintest impression—just sometimes—that he cares about me. I'd melt in a moment. I'd be so loving.
I turn on my heel and stride to my car. It's hard to stride in the black patent high heels I'm wearing, but I manage it pretty well. Out of the corner of my eye I catch the expression on his face as I move away. It's quite blank.
It's that blankness, I think, that finally pierces my armour, triggers the reaction I turn out (to my surprise) to have been suppressing all along.
As I drive home I think about Dominic's blank face, about Dominic's refusal ever to engage with me, to talk with me, to love me. It comes to me that the blankness will last forever, that it will never be replaced by concern or responsiveness or even friendliness. My only son will stare blankly at my face for the rest of his life. He will never love me. I discover that tears are slipping down my face, slowly at first and then faster and wetter and heavier until I can scarcely see the road, the traffic.
By the time I get home my face is smeared with mascara and snot and my sobs are out of control, causing me to shake violently. Somehow I park my car and I stumble inside my house where I weep and weep and weep. I weep for my son, for the relationship I will never have with him. I weep for my love, who lies dead because I killed him. I weep for my sister, whose absence I suddenly feel like a cavernous and painful rupture in my heart. I weep for my lostness, for my aloneness, for the final disappearance of hope from my wilted life.
Later that evening I sit alone, sipping my brandy. I wonder wearily what Frank is doing; in some sense I realise that I am waiting for Frank, that now I am always waiting for Frank. I feel exhausted, depressed, betrayed beyond all bearing.
Who has betrayed me? I don't know, but I am sure it is the sourness of betrayal on my tongue, the flat acrid taste of the disappointment betrayal brings, a taste like wine gone bad.
I also feel guilty. It is my fault that Max has never had a funeral, never been farewelled, celebrated, squabbled over. Max has had no decorous grave, no purifying flame, no sober headstone. Nobody has played Vivaldi for him; nobody has shown photographs of his childhood to an appreciative and tearful congregation; nobody has delivered a eulogy. There he lies, or what's left of him, in a hole of rubble. Does he know, I wonder? Does he care?
As always, the brandy helps. It dulls the edge of the pain. I know it dulls the edge of thought, too, but there is nobody now to care whether my thoughts are muddled or not. Its consoling warmth is miraculous; miraculous, too, the steadiness and ease with which it seeps through my body. It comforts and pillows me; it insulates me from my own loneliness.
And still I am listening for the knock on the door, the knock of a man who will tell me that he knows what I am guilty of, a man who will neither compromise nor negotiate, who will arrest me and deprive me of everything.
Perhaps Borrow will die before I have to desert him. He is an old dog. He creaks; he totters, a little, especially on the left rear leg, where he had the operation a few years ago. His eyes are clouded. He'll die, soon: he'll die, selfishly, and leave me all alone.
And Sophie will grow and become different, and—perhaps, probably, certainly—love me less. Already I have alienated her. For God's sake, I think, I've snapped at her _once—_ after years of love and devotion and gentleness and constancy. But that one lapse was enough: already, her disengagement has clearly started.
As we both grow older, I suppose I shall have to try to retrieve the relationship. I'm not certain that's possible. I'm still angry with her. I have the right to be angry with her. As she grows into adulthood, as she becomes a mother herself, it seems inevitable that we will drift from each other. It requires so much energy to remain close with people, to have quarrels with them and forgive them and reconcile with them.
Maintaining relationships is like maintaining cars: you have to keep servicing them, repairing parts, mending tyres, polishing and replacing and tightening. Not enough people care about their relationship with me. Not enough people love me, or love me enough. Nobody wants to polish up the screws, the nuts, the bolts. Nobody cares about making sure the components are tight enough, making sure everything's in working condition.
Maintaining bodies is problematic, too, of course. My own body is starting to give way. I'm only in my mid-fifties, but I can feel it through me, the grim loosening, the start of the slow vicious collapse. I grow old. Nothing is left for me but growing older. I'll mutter, when I'm old, like a witch, and my great-grandchildren will be frightened of me, especially if I'm in prison. And it's quite likely that that's where I'll be.
I'll grow hairs on my chin. My body will weaken and creak, its bones slowly corroding, disintegrating until they snap like celery, and my mind will fray and crumble like old lace. And then I will be dead, too. All quite deadybones.
Surely, surely, I have deserved better than this. Haven't I?
# TABLE OF CONTENTS
1. COVER PAGE
2. AUTHOR BIO
3. TITLE PAGE
4. CONTENTS
5. DEDICATION
6. PART ONE
7.
8. PART TWO
9.
10. PART THREE
11.
12. PART FOUR
13.
14. PART FIVE
15.
16. COPYRIGHT PAGE
## LIST OF PAGES
1. i
2. ii
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4. iv
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| {
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Kip Carpenter (Kalamazoo, 30 de abril de 1979) es un deportista estadounidense que compitió en patinaje de velocidad sobre hielo.
Participó en dos Juegos Olímpicos de Invierno, en los años 2002 y 2006, obteniendo una medalla de bronce en Salt Lake City 2002, en la prueba de 500 m.
Palmarés internacional
Referencias
Patinadores de velocidad de Estados Unidos
Patinadores de velocidad en los Juegos Olímpicos de Salt Lake City 2002
Patinadores de velocidad en los Juegos Olímpicos de Turín 2006
Medallistas olímpicos de bronce de patinaje de velocidad
Medallistas olímpicos de bronce de Salt Lake City 2002
Medallistas olímpicos de bronce de Estados Unidos
Deportistas de Estados Unidos en los Juegos Olímpicos de Salt Lake City 2002
Deportistas de Estados Unidos en los Juegos Olímpicos de Turín 2006 | {
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{"url":"https:\/\/zbmath.org\/?q=an:1097.03012&format=complete","text":"# zbMATH \u2014 the first resource for mathematics\n\nFunctional representation of preiterative\/combinatory formalism. (English) Zbl\u00a01097.03012\nSummary: Both formalisms model systems of multi-argument selfmaps closed under composition as well as argument permutation and identification by using abstract algebraic operations for these transformations and substitutions. Developed are additional requirements for each system to be representable as a system of concrete selfmaps on some set in which these operations act in the expected way.\n##### MSC:\n 03B40 Combinatory logic and lambda calculus 03C05 Equational classes, universal algebra in model theory 08A02 Relational systems, laws of composition\nFull Text:\n##### References:\n [1] CURRY H. B.-FEYS R.: Combinatory Logic. N. Holland, Amsterdam, 1968. \u00b7 Zbl\u00a00197.00601 [2] DICKER R. M.: The substitutive law. Proc. London Math. Soc. 13 (1963), 493-510. \u00b7 Zbl\u00a00122.25501 [3] FLEISCHER I.: Semigroup of not bijective finite selfmaps of an infinite set. Algebra Universalis 33 (1995), 186-190; Semigroup Forum 58 (1999), 468-470. \u00b7 Zbl\u00a00821.03030 [4] HALMOS P. R.: Algebraic Logic. Chelsea Publ. Comp., New York, 1962. \u00b7 Zbl\u00a00101.01101 [5] HINDLEY J. R.-SELDIN J.: Introduction to Combinators and X-Calculus. London Math. Soc. Stud. Texts 1, Cambridge Univ. Press, Cambridge, 1986. \u00b7 Zbl\u00a00614.03014 [6] HOWIE J. M.: An Introduction to Semigroup Theory. London Math. Soc. Monogr. 7, Academic Press, London-New York-San Francisco, 1976. \u00b7 Zbl\u00a00355.20056 [7] J\u00d3NSSON B.: Defining relations for full semigroups of finite transformations. Michigan Math. J. 9 (1962), 77-85. \u00b7 Zbl\u00a00111.03803 [8] LAUSCH H.-N\u00d6BAUER W.: Algebra of Polynomials. North-Holland Math. Library 5, North-Holland Publ. Comp.\/Amer. Elsevier Publ. Comp., Inc, Amsterdam-London\/New York, 1973. \u00b7 Zbl\u00a00283.12101 [9] MA\u013dCEV A. I.: Iterative Algebras and Pos\\?s Varieties (Russian). [ [10] MENGER K.: On substitutive algebra and its syntax. Z. Math. Logik Grundlag. Math. 10 (1964), 81-104. \u00b7 Zbl\u00a00132.24601 [11] ROSENBERG I. G.: Ma\u013ecev algebras for universal algebra terms. Algebraic Logic and Universal Algebra in Computer Science, Conference, Ames, Iowa, USA, June 1-4, 1988. Proceedings (C. H. Bergman et al., Lecture Notes in Comput. Sci. 425, Springer-Verlag, Berlin, 1990, pp. 195-208. [12] SCH\u00d6NFINKEL M.: Bausteine der Mathematischen Logik. Math. Ann. 92 (1924), 305-316. [ \u00b7 JFM\u00a050.0023.01 [13] STENLUND S.: Combinators, X-Terms and Proof Theory. D. Reidel, Dordrecht, 1972. \u00b7 Zbl\u00a00248.02032 [14] WHITLOCK H. I.: A composition algebra for multiplace functions. Math. Ann. 157 (1964), 167-178. \u00b7 Zbl\u00a00126.03501\nThis reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.","date":"2021-10-22 18:24:01","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\": 0.8229910731315613, \"perplexity\": 5417.132422859544}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323585518.54\/warc\/CC-MAIN-20211022181017-20211022211017-00049.warc.gz\"}"} | null | null |
Q: CamlQuery Lookup not correct I need to display the elements in Column "NetworkNodes" only SRV1 and SRV3 in Lookup, how to write correctly?
The remaining elements of the SRV are not needed.
Doesn't work like that:
Query = @"<Where><And><Eq><FieldRef Name='NetworkNodes' /><Value Type='Lookup'>" + "SRV1" + "</Value></Eq><Eq><FieldRef Name='NetworkNodes' /><Value Type='Lookup'>" + "SRV3" + "</Value></Eq></And></Where>"
A: If the lookup column does not allow to select multiple values then you should try your query like:
<Where>
<Or>
<Eq>
<FieldRef Name='NetworkNodes'/>
<Value Type='Lookup'>SRV1</Value>
</Eq>
<Eq>
<FieldRef Name='NetworkNodes'/>
<Value Type='Lookup'>SRV3</Value>
</Eq>
</Or>
</Where>
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Wymondham and Attleborough Mercury > News
Dad fears for family's future as he battles rare brain tumour
Gareth Stevens (left) with his fiancee Jennie Stevens, 39, and their three sons, Joseph, 13, Christopher, 10 and Louis, 4. - Credit: Gareth Stevens
It's hard to imagine the devastation which is felt when someone is told they have a brain tumour.
But for father-of-three Gareth Stevens, who is battling a rare condition, his worst fear is not being able to provide for his family.
Gareth Stevens (right) with his fiancee Jennie Stevens, 39, and their three sons, Joseph, 13, Christopher, 10 and Louis, 4. - Credit: Gareth Stevens
The ex-military man, from Attleborough, is preparing for a gruelling six weeks of radiotherapy in the biggest fight of his life.
But during what is already an extremely difficult time for his family, they have the added worry about how their bills will be paid while he is forced to stop work.
The 41-year-old, a HGV driver who transports tanks for the Ministry of Defence and remains a sponsored reservist in the Army, first detected something was wrong around 16 months ago.
As the coronavirus pandemic hit and face-to-face doctor appointments were reduced, he struggled to get a diagnosis for the headaches and severe pain he was experiencing.
Despite various appointments with his GP practice in Attleborough and the Norfolk and Norwich University Hospital, Mr Stevens was initially misdiagnosed with Trigeminal Neuralgia.
Gareth Stevens (right) is an ex-military man who has been diagnosed with rare brain tumour. Here he is pictured with his sons at a Remembrance Day stall. - Credit: Gareth Stevens
10 The fight is on to save a Norfolk village's Post Office
He said: "I started suffering with some pain, headaches, and jaw pain which, on a couple of occasions, they diagnosed as low mood and depression.
"The amount of pain that I was in and having to deal with that on a daily basis really made me quite depressed. I found it difficult to do most things.
"Throughout the 15 months I had my medication swapped numerous times. I almost felt like a guinea pig - they weren't really sure what was wrong with me.
"Also, during that time my cousin was suffering with a brain tumour and going through treatment.
"I mentioned this to doctors and said is there any chance this could be linked, but they dismissed that immediately and diagnosed me with Trigeminal Neuralgia."
Gareth Stevens with his fiancee, Jennie Stevens, after his operation to remove his brain tumour. - Credit: Gareth Stevens
The NHS Norfolk and Waveney Clinical Commissioning Group said they are unable to comment on individual cases but wished Mr Stevens the best with his treatment and thanked him for highlighting the issue.
A spokesman for the Norfolk and Norwich University Hospital also added: "We'd like to wish Mr Stevens all the best with his ongoing treatment and we would be happy to speak to him directly to answer any questions or concerns he has."
Eventually, Mr Stevens - who served in the Army from 1997 to 2010 - was booked in for a CT scan on November 7 and that was when doctors discovered a large mass around his right temporal lobe.
Gareth Stevens with his fiancee, Jennie Stevens. - Credit: Gareth Stevens
"They explained to me that they were extremely surprised that I was walking, talking and that I had driven myself to the hospital," he said. "I'm a heavy goods driver and work for the military transporting tanks.
"I had been doing that throughout the entire time I had been suffering from the pain. That's how dangerous things were."
Following the scan, Mr Stevens was correctly diagnosed with Chordoid Meningioma – which has a 100pc recurrence rate - and on Wednesday November 17 he underwent an operation at Addenbrooke's hospital in Cambridge to have the tumour removed.
After spending Christmas at home with his financee, Jennie Stevens, 39, and their three sons, Joseph, 13, Christopher, 10 and Louis, four, Mr Stevens is now waiting to start six weeks of intense radiotherapy.
He added: "My licence has been taken off me, so as a family we are scraping our way through on one wage. My fiancée also has to take time off work to get me back and forward to my appointments.
"As well as my job driving, I was working part time as a sponsored reservist back in the military. But even that has been taken away from me.
"I'm not sure whether I will be able to go back to it in the future. That would be a massive loss."
Gareth Stevens with his three sons, Joseph, 13 and Christopher, 10. Mr Stevens, from Attleborough, has been diagnosed with a rare brain tumour. - Credit: Gareth Stevens
To help support the family a gofundme page has now been set up with hopes of raising enough money to cover their rent while Mr Stevens goes through treatment.
To donate you can visit the donation page here, https://www.gofundme.com/f/gareth-brain-tumour-help-a-family.
Delays and road partially blocked after crash near Wymondham | {
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For a CV etc. check me out on LinkedIn or if you can stomach it, follow me on twitter. If this is all you're going to look at: My name is Ferry Biedermann, I'm a journalist, Middle East analyst, magazine maker and short fiction writer. My work appears in the Financial Times, Jane's, CNBC and elsewhere. It has also popped up in the Washington Post, Foreign Policy, the Chicago Tribune, IPS and more. | {
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Епитафи Скелићима на гробљу Буковац у Ртарима представљају епиграфска и фамилијарна сведочанства преписана са старих надгробних споменика у доњодрагачевском селу Ртари, Oпштина Лучани.
Скелићи из Ртара
Предак данашњих Скелића, познат само по надимку као "Скело" (био је скелеџија), доселио се у Ртаре у време Кочине Крајине, бежећи од Турака. Зна се да је Скело имао сина Обрада, а да је Обрад са супругом Иконијом имао сина Ђока. Родослов Скелића грана се од Аксентија, Ђоковог сина кога је добио са другом супругом Вукосавом.
Чланови ове фамилије дуго носили су презиме Скеловић (помињу се и као Ђоковићи), да би се од до 70-их година 19. века усталило презиме Скелић.
Скелићи данас живе у Ртарима, Чачку, Београду и Крагујевцу. Славе Стевањдан.
Епитафи на споменицима Скелићима
Споменик Миљку Скелићу (†1870)
Овде поч. раб. Божи
МИЉКО
син Ђока Скелића
поживи 18 го.
а умро 22 јануара 1870 го.
Споменик Ђоки Скелићу (†1872)
Овде почива раба божи
ЂОКО Скелић
пож. 62 г
а умре 25 маја 1872 год.
Споменик Аксентију Скелићу (†1890)
ИС ХР
Овде почива раба божи
АКСЕНТИЈЕ Скелић из Ртара.
Поживи 42 г.
а умро 9 марта 1890 г.
Спомен подиже му брат Антоније
и синови Ђорђе Божидар и Љубомир
Споменик Антонију Скелићу (†1900)
ИС ХР
Зде почива Раба божи
АНТОНИЈЕ Скелић из Ртара
поживи 60 г
умро 27 децембра 1900 г.
Спомен подижу му синовац Ђорђе
Божидар и Љубо
и његова жена Марија
Споменик Ђорђу Скелићу (†1906)
Овде поч. раб. Божи
покојни ЂОРЂЕ Скелић
који поживи 32 год.
а умре 20 године 1906 марта месеца
Спомен под. супр. Вида
и Ева кћи и зет Миљко
Споменик Марици Скелић (†1912)
Овде почива раба божија
МАРИЦА
супруга покој. Антонија Скелића из Ртара
која часно и поштено поживи 71 г.
а престави се 2.3.1912 г.
Бог да јој душу прости
Спомен подиже јој кћи Винка Петковић
из Горачића
Споменик сестрама Скелић: П (†1911) и Сибинки (†1913)
Овде почивају две сестре
П. 20 г.
и СИБИНКА пож. 13 год.
Умреше старија 1911 год.
и млађа 1913 г.
Спомен им подигоше мајка Вида
свак Миљко Раденковић и сестра Јева
Споменик Станојки Скелић (†1939)
СТАНОЈКА Скелић
оч. Раденковић
пож. 3 г.
Умре 4 авг. 1939. ГОД.
Спомен јој под. мајка Јева
и отац Миљко
Споменик Види Скелић (†1958)
Овде почива
ВИДА Скелић
поживи 82 г.
Умре 5.10.1958 г.
Спомен подиже
поштована ћерка Јева
и зет Миљко и унуци
Референце
Литература
Спољашње везе
Порекло
Надгробни споменици у Драгачеву
Драгачевски епитафи | {
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Figlia di Spyridōn Marinatos (con cui ha collaborato), ha effettuato studi classici ed archeologici negli Stati Uniti e si è laureata nel 1978 presso l'Università del Colorado a Boulder. Ha insegnato antichità classiche ed archeologia presso l'Oberline College in Ohio e alla stessa università del Colorado in Boulder.
I suoi interessi principali sono legati alla storia delle religioni antiche e alla società minoica negli ambiti relativi all'influenza della religione nei comportamenti della classe dominante, e dei culti religiosi in generale. Suoi contributi sono apparsi nelle principali riviste di filologia classica, antichistica e storia delle religioni (The American historical review, Numen, Classical Review, The Journal of Religion, Journal of Hellenic Studies, Classical Philology, Gnomon).
È inoltre autrice delle guide turistiche relative a Lindos (1983), Akrotiri (1983) e Creta (1984) e di numerosi testi ad uso scolastico per circolazione interna nelle università.
Opere principali
Thucydides and Religion, Konigstein, Hain 1981
Sanctuaries and Cults in the Aegean Bronze Age (con Robin Hägg) (Stockholm-Lund, Svenska institutet i Athen, 1981
The Minoan Thalassocracy: Myth and Reality (con Robin Hägg), Stockholm-Göteborg, Svenska institutet i Athen, 1984
Art and religion in Thera: reconstructing a Bronze Age society, Athens, D. & I. Mathioulakis, 1984
Minoan sacrificial ritual: cult practice and symbolism , Stockholm, P. Åström, 1986.
Minoan religion: ritual, image, and symbol, Columbia, University of South Carolina Press, 1993
Greek sanctuaries: new approaches (con Robin Hägg), London-New York, Routledge, 1993
The goddess and the warrior: the naked goddess and mistress of animals in early Greek religion, London-New York, Routledge, 2000
Minoan kingship and the solar goddess: a Near Eastern koine, Urbana, University of Illinois Press, 2009
Note
Collegamenti esterni
Nanno Marinatos, Goddess and monster in Ansichten griechischer Rituale: Geburtstags-Symposium für Walter Burkert Walter de Gruyter, 1998 | {
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0.14.0 2022-07-08
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- Add ocamlformat (#177)
- Handle missing variables as null (#184)
- Show default value in introspection query (#194)
- Support block strings in the parser (#198)
- Handle skip/include directives on fragment spreads (#200)
- Improved handling of recursive arguments and objects (#199)
- Fix websocket conflict (#206)
- Update deprecated Fmt functions (#206)
- Use Yojson `t` types instead of deprecated `json` type (#208)
- Raise minimum `rresult` version (#209)
0.13.0 2019-09-21
---------------------------------
- Support for custom errors (#166)
- Add `__type` introspection field (#163)
- Better parameter handling for `graphql-cohttp` (#156)
- Fix use of default variables (#158)
- Fix merging of field selections (#174)
0.12.2 2019-04-02
---------------------------------
- Remove open of Result (#154)
0.12.1 2019-03-29
---------------------------------
- Disable Yojson deprecation warnings (#148)
- Fix routing in graphql-cohttp (#150)
0.12.0 2019-03-22
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- Remove Str from cohttp-graphql (#146)
0.11.0 2019-03-03
---------------------------------
- Minor parser improvements (#139)
- Replace Str with Re (#144)
0.10.0 2019-02-14
---------------------------------
- Fix digestif dependency in graphql-cohttp (#141)
0.9.0 2019-02-08
---------------------------------
- Skip and include directives (#117)
- Expose more data to resolvers (#127)
- Add the `errors` key first in the response JSON (#131)
- Rewrite parser to Menhir and replace sexp with fmt (#132)
- Support for websockets as transport (#133)
0.8.0 2018-12-06
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- Subscription support (#101)
- Add path to errors in response (#112)
- Improved escaped character handling (#114)
- Improve error messages for invalid arguments (#128)
0.7.0 2018-08-10
---------------------------------
- Allow returning errors from resolve function of io_field (#69)
- Support for union and interface types (#70)
- Expose HTTP request to context construction function in Graphql_lwt.Server.start (#88)
- Fix error response from Graphql_lwt.Server (#96)
- Allow passing operation name to Graphql_lwt.Server (#103)
- Querying undefined fields gives validation error (#105)
- Fix parsing of enums with E (#109)
0.6.0 2018-03-31
---------------------------------
- Prevent fragment cycles (#78)
- graphql-lwt depends on cohttp 1.0.0 (#71)
0.5.0 2018-01-18
---------------------------------
- Depend on angstrom 0.7.0 (#64)
- Fix parsing of quoted strings (#64)
- Make custom argument types generalizable (#72)
- Deduplicate arg types in introspection result (#75)
- 4.06 compatibility (#76)
0.4.0 2017-09-17
---------------------------------
- Parse tabs as whitespace (#62)
- Move parser to separate package (#63)
0.3.0 2017-08-31
---------------------------------
- Built-in HTTP server for graphql-lwt (#60)
0.2.0 2017-07-02
---------------------------------
- Support deprecation of fields (#53)
- Support documentation and deprecation of enum values (#54)
0.1.0 2017-05-25
---------------------------------
Initial public release.
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,779 |
Start a New Discussion. It's not so hard to do rocketjump by yourself after a little practice: Rocket jumping off a wall will give you incredible speed, but a short jump. This item is incompatible with Team Fortress 2.
Sentry Gun rockets can also be used to jump; Sentry rockets will launch the Pyro higher into the air than the Soldier's rockets, but also inflict more damage. Learn more... Wings of Glory Kill an enemy soldier while both you and the target are airborne. Post as a guest Name.
With enough Medics using needle boosting , the player could be launched very high on a wave of needles. Fire a single rocket immediately after you've looked down. Showing 1 - 14 of 14 comments. Advanced sticky jumping techniques are similar to a Soldier's rocket jumping techniques, but the mechanics are somewhat different because of the sticky's arming period.
I've found that not crouching helps for more "lateral" rocket jumps. Current visibility: It can be used to clear gaps or jump to higher areas that are normally unreachable by the Pyro, which can prove useful for ambushing. By aiming the compression blast at his feet, the Pyro can redirect the Soldier 's or Sentry Gun 's rocket into a powerful rocket jump.
Created by. As Arda Xi mentioned, if you are crouching when the rocket explodes, you will go higher. Examples of potential locations for a Building Step jump include RED's advanced spawn on Dustbowl 's stage 3 which allows Engineers to return to spawn easily and RED's right side of Gold Rush 's stage 1 which allows Engineers to reach a common setup location from an unexpected direction.
Note that savvy enemies are capable of using a Dispenser for a jump, just as its Engineer can. Equipping the Atomizer allows the Scout to perform a triple jump while fully deployed. Once you start getting closer to the ground fire a rocket below you to keep momentum. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,047 |
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<p class="text-center"><a href="shear.html#The%20XShearImage">The XShearImage</a> • <a href="shear.html#DeskewImage">DeskewImage</a> • <a href="shear.html#IntegralRotateImage">IntegralRotateImage</a> • <a href="shear.html#ShearImage">ShearImage</a> • <a href="shear.html#ShearRotateImage">ShearRotateImage</a></p>
<h2><a href="http://www.imagemagick.org/api/MagickCore/shear_8c.html" id="The_XShearImage">The XShearImage</a></h2>
<p>The XShearImage() and YShearImage() methods are based on the paper "A Fast Algorithm for General Raster Rotatation" by Alan W. Paeth, Graphics Interface '86 (Vancouver). ShearRotateImage() is adapted from a similar method based on the Paeth paper written by Michael Halle of the Spatial Imaging Group, MIT Media Lab.</p>
<h2><a href="http://www.imagemagick.org/api/MagickCore/shear_8c.html" id="DeskewImage">DeskewImage</a></h2>
<p>DeskewImage() removes skew from the image. Skew is an artifact that occurs in scanned images because of the camera being misaligned, imperfections in the scanning or surface, or simply because the paper was not placed completely flat when scanned.</p>
<p>The amount of rotation calculated to deskew the image is saved in the artifact "deskew:angle".</p>
<p>If the artifact "deskew:auto-crop" is given the image will be automatically cropped of the excess background. The value is the border width of all pixels around the edge that will be used to determine an average border color for the automatic trim.</p>
<p>The format of the DeskewImage method is:</p>
<pre class="text">
Image *DeskewImage(const Image *image,const double threshold,
ExceptionInfo *exception)
</pre>
<p>A description of each parameter follows:</p>
<dd>
</dd>
<dd> </dd>
<dl class="dl-horizontal">
<dt>image</dt>
<dd>the image. </dd>
<dd> </dd>
<dt>threshold</dt>
<dd>separate background from foreground. </dd>
<dd> </dd>
<dt>exception</dt>
<dd>return any errors or warnings in this structure. </dd>
<dd> </dd>
</dl>
<h2><a href="http://www.imagemagick.org/api/MagickCore/shear_8c.html" id="IntegralRotateImage">IntegralRotateImage</a></h2>
<p>IntegralRotateImage() rotates the image an integral of 90 degrees. It allocates the memory necessary for the new Image structure and returns a pointer to the rotated image.</p>
<p>The format of the IntegralRotateImage method is:</p>
<pre class="text">
Image *IntegralRotateImage(const Image *image,size_t rotations,
ExceptionInfo *exception)
</pre>
<p>A description of each parameter follows.</p>
<dt>image</dt>
<p>the image.</p>
<dt>rotations</dt>
<p>Specifies the number of 90 degree rotations.</p>
<h2><a href="http://www.imagemagick.org/api/MagickCore/shear_8c.html" id="ShearImage">ShearImage</a></h2>
<p>ShearImage() creates a new image that is a shear_image copy of an existing one. Shearing slides one edge of an image along the X or Y axis, creating a parallelogram. An X direction shear slides an edge along the X axis, while a Y direction shear slides an edge along the Y axis. The amount of the shear is controlled by a shear angle. For X direction shears, x_shear is measured relative to the Y axis, and similarly, for Y direction shears y_shear is measured relative to the X axis. Empty triangles left over from shearing the image are filled with the background color defined by member 'background_color' of the image.. ShearImage() allocates the memory necessary for the new Image structure and returns a pointer to the new image.</p>
<p>ShearImage() is based on the paper "A Fast Algorithm for General Raster Rotatation" by Alan W. Paeth.</p>
<p>The format of the ShearImage method is:</p>
<pre class="text">
Image *ShearImage(const Image *image,const double x_shear,
const double y_shear,ExceptionInfo *exception)
</pre>
<p>A description of each parameter follows.</p>
<dt>image</dt>
<p>the image.</p>
<dt>x_shear, y_shear</dt>
<p>Specifies the number of degrees to shear the image.</p>
<dt>exception</dt>
<p>return any errors or warnings in this structure.</p>
<h2><a href="http://www.imagemagick.org/api/MagickCore/shear_8c.html" id="ShearRotateImage">ShearRotateImage</a></h2>
<p>ShearRotateImage() creates a new image that is a rotated copy of an existing one. Positive angles rotate counter-clockwise (right-hand rule), while negative angles rotate clockwise. Rotated images are usually larger than the originals and have 'empty' triangular corners. X axis. Empty triangles left over from shearing the image are filled with the background color defined by member 'background_color' of the image. ShearRotateImage allocates the memory necessary for the new Image structure and returns a pointer to the new image.</p>
<p>ShearRotateImage() is based on the paper "A Fast Algorithm for General Raster Rotatation" by Alan W. Paeth. ShearRotateImage is adapted from a similar method based on the Paeth paper written by Michael Halle of the Spatial Imaging Group, MIT Media Lab.</p>
<p>The format of the ShearRotateImage method is:</p>
<pre class="text">
Image *ShearRotateImage(const Image *image,const double degrees,
ExceptionInfo *exception)
</pre>
<p>A description of each parameter follows.</p>
<dt>image</dt>
<p>the image.</p>
<dt>degrees</dt>
<p>Specifies the number of degrees to rotate the image.</p>
<dt>exception</dt>
<p>return any errors or warnings in this structure.</p>
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,144 |
package apple.quartzcore;
import apple.NSObject;
import apple.coregraphics.struct.CGSize;
import apple.foundation.NSArray;
import apple.foundation.NSCoder;
import apple.foundation.NSMethodSignature;
import apple.foundation.NSSet;
import apple.quartzcore.protocol.CAAction;
import org.moe.natj.c.ann.FunctionPtr;
import org.moe.natj.general.NatJ;
import org.moe.natj.general.Pointer;
import org.moe.natj.general.ann.ByValue;
import org.moe.natj.general.ann.Generated;
import org.moe.natj.general.ann.Library;
import org.moe.natj.general.ann.Mapped;
import org.moe.natj.general.ann.MappedReturn;
import org.moe.natj.general.ann.NFloat;
import org.moe.natj.general.ann.NInt;
import org.moe.natj.general.ann.NUInt;
import org.moe.natj.general.ann.Owned;
import org.moe.natj.general.ann.Runtime;
import org.moe.natj.general.ptr.VoidPtr;
import org.moe.natj.objc.Class;
import org.moe.natj.objc.ObjCRuntime;
import org.moe.natj.objc.SEL;
import org.moe.natj.objc.ann.ObjCClassBinding;
import org.moe.natj.objc.ann.ProtocolClassMethod;
import org.moe.natj.objc.ann.Selector;
import org.moe.natj.objc.map.ObjCObjectMapper;
@Generated
@Library("QuartzCore")
@Runtime(ObjCRuntime.class)
@ObjCClassBinding
public class CATiledLayer extends CALayer {
static {
NatJ.register();
}
@Generated
protected CATiledLayer(Pointer peer) {
super(peer);
}
@Generated
@Selector("accessInstanceVariablesDirectly")
public static native boolean accessInstanceVariablesDirectly();
@Generated
@Owned
@Selector("alloc")
public static native CATiledLayer alloc();
@Owned
@Generated
@Selector("allocWithZone:")
public static native CATiledLayer allocWithZone(VoidPtr zone);
@Generated
@Selector("automaticallyNotifiesObserversForKey:")
public static native boolean automaticallyNotifiesObserversForKey(String key);
@Generated
@Selector("cancelPreviousPerformRequestsWithTarget:")
public static native void cancelPreviousPerformRequestsWithTarget(@Mapped(ObjCObjectMapper.class) Object aTarget);
@Generated
@Selector("cancelPreviousPerformRequestsWithTarget:selector:object:")
public static native void cancelPreviousPerformRequestsWithTargetSelectorObject(
@Mapped(ObjCObjectMapper.class) Object aTarget, SEL aSelector,
@Mapped(ObjCObjectMapper.class) Object anArgument);
@Generated
@Selector("classFallbacksForKeyedArchiver")
public static native NSArray<String> classFallbacksForKeyedArchiver();
@Generated
@Selector("classForKeyedUnarchiver")
public static native Class classForKeyedUnarchiver();
@Generated
@Selector("debugDescription")
public static native String debugDescription_static();
@Generated
@Selector("defaultActionForKey:")
@MappedReturn(ObjCObjectMapper.class)
public static native CAAction defaultActionForKey(String event);
@Generated
@Selector("defaultValueForKey:")
@MappedReturn(ObjCObjectMapper.class)
public static native Object defaultValueForKey(String key);
@Generated
@Selector("description")
public static native String description_static();
/**
* The time in seconds that newly added images take to "fade-in" to the
* rendered representation of the tiled layer. The default implementation
* returns 0.25 seconds.
*/
@Generated
@Selector("fadeDuration")
public static native double fadeDuration();
@Generated
@Selector("hash")
@NUInt
public static native long hash_static();
@Generated
@Selector("instanceMethodForSelector:")
@FunctionPtr(name = "call_instanceMethodForSelector_ret")
public static native NSObject.Function_instanceMethodForSelector_ret instanceMethodForSelector(SEL aSelector);
@Generated
@Selector("instanceMethodSignatureForSelector:")
public static native NSMethodSignature instanceMethodSignatureForSelector(SEL aSelector);
@Generated
@Selector("instancesRespondToSelector:")
public static native boolean instancesRespondToSelector(SEL aSelector);
@Generated
@Selector("isSubclassOfClass:")
public static native boolean isSubclassOfClass(Class aClass);
@Generated
@Selector("keyPathsForValuesAffectingValueForKey:")
public static native NSSet<String> keyPathsForValuesAffectingValueForKey(String key);
@Generated
@Selector("layer")
public static native CATiledLayer layer();
@Generated
@Selector("needsDisplayForKey:")
public static native boolean needsDisplayForKey(String key);
@Generated
@Owned
@Selector("new")
public static native CATiledLayer new_objc();
@Generated
@Selector("resolveClassMethod:")
public static native boolean resolveClassMethod(SEL sel);
@Generated
@Selector("resolveInstanceMethod:")
public static native boolean resolveInstanceMethod(SEL sel);
@Generated
@Selector("setVersion:")
public static native void setVersion_static(@NInt long aVersion);
@Generated
@Selector("superclass")
public static native Class superclass_static();
@Generated
@Selector("version")
@NInt
public static native long version_static();
@Generated
@Selector("init")
public native CATiledLayer init();
@Generated
@Selector("initWithCoder:")
public native CATiledLayer initWithCoder(NSCoder coder);
@Generated
@Selector("initWithLayer:")
public native CATiledLayer initWithLayer(@Mapped(ObjCObjectMapper.class) Object layer);
/**
* The number of levels of detail maintained by this layer. Defaults to
* one. Each LOD is half the resolution of the previous level. If too
* many levels are specified for the current size of the layer, then
* the number of levels is clamped to the maximum value (the bottom
* most LOD must contain at least a single pixel in each dimension).
*/
@Generated
@Selector("levelsOfDetail")
@NUInt
public native long levelsOfDetail();
/**
* The number of magnified levels of detail for this layer. Defaults to
* zero. Each previous level of detail is twice the resolution of the
* later. E.g. specifying 'levelsOfDetailBias' of two means that the
* layer devotes two of its specified levels of detail to
* magnification, i.e. 2x and 4x.
*/
@Generated
@Selector("levelsOfDetailBias")
@NUInt
public native long levelsOfDetailBias();
/**
* The number of levels of detail maintained by this layer. Defaults to
* one. Each LOD is half the resolution of the previous level. If too
* many levels are specified for the current size of the layer, then
* the number of levels is clamped to the maximum value (the bottom
* most LOD must contain at least a single pixel in each dimension).
*/
@Generated
@Selector("setLevelsOfDetail:")
public native void setLevelsOfDetail(@NUInt long value);
/**
* The number of magnified levels of detail for this layer. Defaults to
* zero. Each previous level of detail is twice the resolution of the
* later. E.g. specifying 'levelsOfDetailBias' of two means that the
* layer devotes two of its specified levels of detail to
* magnification, i.e. 2x and 4x.
*/
@Generated
@Selector("setLevelsOfDetailBias:")
public native void setLevelsOfDetailBias(@NUInt long value);
/**
* The maximum size of each tile used to create the layer's content.
* Defaults to (256, 256). Note that there is a maximum tile size, and
* requests for tiles larger than that limit will cause a suitable
* value to be substituted.
*/
@Generated
@Selector("setTileSize:")
public native void setTileSize(@ByValue CGSize value);
/**
* The maximum size of each tile used to create the layer's content.
* Defaults to (256, 256). Note that there is a maximum tile size, and
* requests for tiles larger than that limit will cause a suitable
* value to be substituted.
*/
@Generated
@Selector("tileSize")
@ByValue
public native CGSize tileSize();
@Generated
@Selector("supportsSecureCoding")
public static native boolean supportsSecureCoding();
@Generated
@ProtocolClassMethod("supportsSecureCoding")
public boolean _supportsSecureCoding() {
return supportsSecureCoding();
}
@Generated
@Selector("cornerCurveExpansionFactor:")
@NFloat
public static native double cornerCurveExpansionFactor(String curve);
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,698 |
Pinctada, perłopław perłorodny – rodzaj małży z rodziny perłopławowatych (Pteriidae). W języku polskim są nazywane, wraz z rodzajem Pteria, perłopławami.
Zasiedla wody ciepłe, głównie tropikalne i subtropikalne. Ma zasięg globalny. Największy gatunek Pinctada maxima
dorasta do 30 cm i 5,5 kg wagi. Długowieczne gatunki Pinctada maxima i Pinctada margaritifera żyją do 15 lat.
Rodzaj ten jest głównym producentem wysokiej jakości morskich pereł jubilerskich. Wytwarzana przez niego masa perłowa i perły są stosowane do wyrobu ozdób.
Systematyka
Do rodzaju Pinctada zaliczane są gatunki (Lista za: ):
Pinctada albina (Lamarck, 1819)
Pinctada capensis (G.B. Sowerby III, 1890)
Pinctada chemnitzii (Philippi, 1849)
Pinctada imbricata Röding, 1798
Pinctada longisquamosa (Dunker, 1852)
Pinctada maculata (Gould, 1850)
Pinctada margaritifera (Linnaeus, 1758)
Pinctada maxima (Jameson, 1901)
Pinctada mazatlanica (Hanley, 1855)
Pinctada nigra (Gould, 1850)
Pinctada colymbus Röding, 1798 uznany za Pteria colymbus (Röding, 1798)
Pinctada epitheca Iredale, 1939 uznany za Pinctada chemnitzii (Philippi, 1849)
Pinctada foliacea Röding, 1798 uznany za Pinctada margaritifera (Linnaeus, 1758)
Pinctada fucata (Gould, 1850) uznany za Pinctada imbricata fucata (Gould, 1850)
Pinctada galtsoffi Bartsch, 1931 uznany za Pinctada margaritifera (Linnaeus, 1758)
Pinctada penguin Röding, 1798 uznany za Pteria penguin (Röding, 1798)
Pinctada perrutila Iredale, 1939 uznany za Pinctada albina (Lamarck, 1819)
Pinctada radiata (Leach, 1814) uznany za Pinctada imbricata radiata (Leach, 1814)
Pinctada vulgaris Schumacher uznany za Pinctada imbricata Röding, 1798
Przypisy
Bibliografia
Nitkoskrzelne
Perły | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,181 |
Imagine your modern, boring car being transformed to sound like a powerful, expensive Super Sports Car!
Shift your car into first gear, put your foot down on the accelerator and listen to the exiciting, powerful driving experience. Every phrase of your driving will be covered: accelerating, decelerating, shifting, cruising and idling.
The convincing sound is designed to fool your senses, making you feel like the proud owner of an expensive luxury car.
Recorded from a real V12 "Ferrari", simply plug it into your cigarette lighter socker, select a frequency, turn up your volume and listen to the high-quality sound.
When you get bored of the sound, you can simply plug in your MP3 Player, iPod or any audio device with a 3.5mm audio socket and listen to your favourite songs using your car's speakers. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,198 |
{"url":"https:\/\/www.physicsforums.com\/threads\/simple-complex-numbers-integral.254682\/","text":"# Simple complex numbers integral\n\n1. Sep 8, 2008\n\n### elcotufa\n\n1. The problem statement, all variables and given\/known data\nIntegrate using complex numbers\n$$\\int\\limits_0^{2\\pi} cos^4(\\theta)$$\n\n2. Relevant equations\n$$cos^4(\\theta)= (\\frac{e^{j\\theta} + e^{-j\\theta}}2)^4$$\n3. The attempt at a solution\n$$\\frac 1{2^4} (e^{j\\theta} + e^{-j\\theta})^4$$\n\nI got\n$$\\frac 1{2^4} \\int^{2\\pi}_0 (e^{4j\\theta}+4e^{2j\\theta}+4e^{-2j\\theta}+e^{-4j\\theta}+6)$$\n\nAfter this I am not sure what to do\n\nThe integral of $$\\int e^{4j}$$ would be $$\\frac{e^{4j\\theta}}{4j}$$?\n\nHow do I cancel them?\n\nInput appreciated\n\n2. Sep 8, 2008\n\n### Dick\n\nSure, that's the integral of e^(4j*theta). You'll notice if you evaluate it from 0 to 2*pi the result is 0. The same for all the other exponentials. The only term that contributes is the 6.\n\n3. Sep 8, 2008\n\nthanks man","date":"2017-01-21 18:09:12","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\": 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\": 0.6774449348449707, \"perplexity\": 2024.1795824764345}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-04\/segments\/1484560281162.88\/warc\/CC-MAIN-20170116095121-00083-ip-10-171-10-70.ec2.internal.warc.gz\"}"} | null | null |
{"url":"http:\/\/www.math.uni-bonn.de\/ag\/topo\/abstracts\/2014-01-07-lustig","text":"# Bonn Topology Group - Abstracts\n\n### Talk\n\nMARTIN LUSTIG (Marseille): Fixed currents for hyperbolic automorphisms of free groups (07\/01\/2014)\n\n### Abstract\n\nIt is known that for an irreducible hyperbolic automorphisms \\phi of a finitely generated non-abelian free group F_n there are precisely two currents \\mu_+ and \\mu_- which are projectively fixed, i.e. \\phi(\\mu_+) = \\lambda_+ \\mu_+ and \\phi(\\mu_-) = \\lambda_- \\mu_-, for some positive \"stretching factors\" \\lambda_+ > 1 and \\lambda_- < 1.\n\nIn this talk we will explain how to generalize this result to general hyperbolic automorphisms \\psi of F_n, and exhibit a natural bijection from the set of projectively fixed currents to the set of non-negative (row) eigenvectors for the transition matrix of a train track representative of \\psi or of \\psi^{-1}.\n\nWe will also apply this result to exhibit the first known example of any R-tree T in compactified Outer space which is not \"diagonally equalizable\".\n\nBack to seminar page","date":"2018-10-23 07:09:36","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\": 0.9582949876785278, \"perplexity\": 2019.0686080583744}, \"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-2018-43\/segments\/1539583516117.80\/warc\/CC-MAIN-20181023065305-20181023090805-00204.warc.gz\"}"} | null | null |
Q: Better If statement with type checking C# I'm currently working on a .NET 4.7.1 application. I have an If-statement to check the data type and call an handler method accordingly.
My current If statement looks like this:
// object msg = MyHelper.Deserialize(xmlString);
if (msg is Tomato) Handle_Tomato((Tomato)msg);
if (msg is Apple) Handle_Apple((Apple)msg);
if (msg is Banana) Handle_Banana((Banana)msg);
if (msg is Orange) Handle_Orange((Orange)msg);
msg is basically an object deserialized from a string.
I was wondering, if there is a better way to write my if statements?
Thank you very much!
A: As Sweeper mentions in the comments, from C# 7.0 you can use the The is type pattern expression
if (msg is Tomato tomato) Handle_Tomato(tomato);
You could also use pattern matching with a switch statement (Type pattern) since C# 7.0
The type pattern enables concise type evaluation and conversion. When
used with the switch statement to perform pattern matching, it tests
whether an expression can be converted to a specified type and, if it
can be, casts it to a variable of that type.
switch(msg)
{
case Tomato tomato : Handle_Tomato(tomato); break;
case Apple apple : Handle_Apple(apple); break;
...
}
A: I'd strongly suggest not to do such checks. What if in the future there are dozens of different types? Your if statement will increase and be unmaintainable. What if the type changes? You'd have to change all the if statements as well.
You could solve this by using an interface. You already have the classes.
interface IHandler
{
void Execute();
}
class Orange : IHandler
{
public void Execute()
{
// do your orange stuff
}
}
class Tomato : IHandler
{
public void Execute()
{
// do your tomato stuff
}
}
It can be called like this.
if (msg is IHandler) ((IHandler)msg).Execute();
A: I think the easiest would be to use a switch/case
switch (msg)
{
case Tomato t:
Handle_Tomato(t);
break;
case Apple a:
Handle_Apple(a);
break;
case Banana b:
Handle_Banana(b);
break;
case Orange o:
Handle_Orange(o);
break;
}
A: Use a dictionary. I foresee your if will explode with new cases in future, generally speaking, walls of if and large switch statements are bad code. In a similar situation I created something like this:
private static readonly Dictionary<RuntimeTypeHandle, Action<object>> handleMsgImplementations = new Dictionary<RuntimeTypeHandle, Action<object>>
{
{ typeof(Tomato).TypeHandle, x => Handle_Tomato((Tomato)x) },
// etc...
};
// Then instead of if, use this (prepare a catch for Invalid Key or use a TryGetValue)
handleMsgImplementations[msg.GetType().TypeHandle](msg);
I get TypeHandle because I like to use a value type for the key.
EDIT: @TheGeneral answer is the best, also, the C# compiler creates a dictionary under the hood when the amount cases starts to damage performance. I keep my answer because I believe adds value.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,886 |
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1963 austin-healey sprite mk ii
2dr Convertible 4-cyl. 1098cc/56hp 2x1bbl
$8,000 Avg. Value*
Images are general in nature and may not reflect the specific vehicle selected.
History of the 1962-1969 Austin-Healey Sprite
Austin-Healey's famous "Bugeye" Sprite was a low-cost and tiny no-frills open sports car that was intended to be affordable fun. The British Motor Company (BMC) got the mix right as the model quickly became a sales and sporting success. Nearly 50,000 were sold from 1958-1961, with the overwhelming majority destined for export to North America. Building on this momentum, Austin-Healey updated the Sprite for the 1962 model year.
The Mk II Sprite kept the Bugeye's unit-body construction, but received a complete makeover. A more conventional grille and headlight setup replaced the Bugeye's cheerful visage, with the car taking on styling that would be seen on the MG Midget sister car a few months later, as well as the MGB. (Construction of the Sprite had always been contracted to MG since Jensen's body works—which constructed the big Healey sixes—could not cope with the anticipated demand.)
The Mk II Sprites initially ran with virtually the same 948-cc A-series OHV 4-cylinder engines as before, but by October 1962, the longer stroke 1098-cc version was adopted, allowing a power increase from 46 bhp to 56 bhp. The car's outright straight line speed was limited by this engine's displacement and power, but its light weight and low center of gravity made it a terrific handler, and the Sprite earned a reputation on the rally circuit.
By 1964, a series of revisions announced the Mk III Sprite. A new curved windshield was employed, as were roll-up side windows, exterior door handles, and a revamped engine that had 3 additional bhp and larger main bearings, thereby enhancing longevity. The rear suspension was updated and improved at the same time as well.
The Mk IV Sprite arrived in October 1966, most notably with the new (and detuned) 1275-cc engine that was used on the Mini Cooper S. Power output was raised to 65 bhp in the process. By 1969, big Healey cars had been out of production for two years, and the dealer network in the US had dwindled to the point where marketing the same car under two brands was a challenge. With MG's successful MGB still selling well, the Sprite disappeared while the Midget soldiered on.
Today, later Sprites are much cheaper than Bugeye Sprites and still deliver much of the same driving experience, which makes them good bargains. Purists tend to prefer the Mk II Sprites (1962-1964), while practical-minded owners prefer Mk IIIs and Mk IVs, mainly due to their enhanced weather protection and more comfortable suspension setup. Any of these choices are relatively easy to own and work on, with parts being easily available and mechanicals being straightforward. Performance enhancements are popular with these cars as well, which makes them interesting choices for rallying.
1963 austin-healey sprite mk ii Info
2dr Convertible
4-cyl. 1098cc/56hp 2x1bbl
*Please note: All prices shown here are based on various data sources, as detailed in About Our Prices. For all Hagerty Insurance clients: The values shown do not imply coverage in this amount. In the event of a claim, the guaranteed value(s) on your policy declarations page is the amount your vehicle(s) is covered for, even if the value displayed here is different. If you would like to discuss your Hagerty Insurance policy, please call us at 877-922-9701.
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,353 |
package com.wistronits.wh.annbot.ui;
import android.app.DialogFragment;
import android.os.Bundle;
import android.support.v7.app.AppCompatActivity;
import android.support.v7.widget.DefaultItemAnimator;
import android.support.v7.widget.GridLayoutManager;
import android.support.v7.widget.RecyclerView;
import android.text.TextUtils;
import android.view.View;
import android.widget.EditText;
import android.widget.ImageView;
import android.widget.LinearLayout;
import android.widget.RelativeLayout;
import android.widget.TextView;
import com.cyberon.event.CheckedHeadsetEvent;
import com.wistronits.wh.annbot.R;
import com.wistronits.wh.annbot.base.BaseApplication;
import com.wistronits.wh.annbot.business.biz.components.DaggerRecordComponent;
import com.wistronits.wh.annbot.business.biz.components.NetComponent;
import com.wistronits.wh.annbot.business.biz.modules.RecordModule;
import com.wistronits.wh.annbot.business.mvp.AddNuringPresenter;
import com.wistronits.wh.annbot.business.mvp.GetVoiceListPresenter;
import com.wistronits.wh.annbot.business.mvp.contract.AddNuringContract;
import com.wistronits.wh.annbot.business.mvp.contract.RecordDetailContract;
import com.wistronits.wh.annbot.data.Constants;
import com.wistronits.wh.annbot.data.SPManager;
import com.wistronits.wh.annbot.data.greendao.DBManager;
import com.wistronits.wh.annbot.models.RecordDetailModel;
import com.wistronits.wh.annbot.models.UserModel;
import com.wistronits.wh.annbot.net.SyncVoice;
import com.wistronits.wh.annbot.ui.adapter.AddImageAdapter;
import com.wistronits.wh.annbot.ui.adapter.NurseAllAdapter;
import com.wistronits.wh.annbot.ui.fragment.TimePickerFragment;
import com.wistronits.wh.annbot.ui.view.PlayVoiceListener;
import com.wistronits.wh.annbot.ui.view.SpacesItemDecoration;
import com.wistronits.wh.annbot.ui.view.timepicker.TimePickerDialog;
import com.wistronits.wh.annbot.utils.DataUtil;
import com.wistronits.wh.annbot.utils.DialogUtil;
import com.wistronits.wh.annbot.utils.SharePreferenceHelper;
import com.wistronits.wh.annbot.utils.TimeUtil;
import java.util.ArrayList;
import java.util.Calendar;
import java.util.List;
import javax.inject.Inject;
import butterknife.BindView;
import butterknife.ButterKnife;
import butterknife.OnClick;
import cn.finalteam.rxgalleryfinal.bean.MediaBean;
import de.greenrobot.event.EventBus;
import static com.wistronits.wh.annbot.utils.DataUtil.pad;
/**
* Created by WH1107012 on 2017/8/29.
*/
public class RecordEditActivity extends AppCompatActivity implements TimePickerFragment.CallBackValue, RecordDetailContract.View, AddNuringContract.view {
@BindView(R.id.ll_modify_back)
LinearLayout llBack;
@BindView(R.id.tv_back)
TextView tvBack;
@BindView(R.id.ll_modify_complete)
LinearLayout llComplete;
@BindView(R.id.et_modify_content)
EditText etContent;
@BindView(R.id.rv_modify_pic)
RecyclerView rvPic;
@BindView(R.id.layout_add_pic_empty)
RelativeLayout rlAddPic;
@BindView(R.id.tv_modify_time)
TextView tvTime;
@BindView(R.id.tv_voice_name)
TextView tvVoiceName;
@BindView(R.id.iv_voice_play)
ImageView ivVoicePlay;
@BindView(R.id.tv_modify_play)
TextView tvModifyPlay;
@BindView(R.id.tv_voice_length)
TextView tvVoiceLength;
@BindView(R.id.iv_annbot_status)
ImageView ivStatus;
private AddImageAdapter adapter;
private int imageSize = 0;
@Inject
GetVoiceListPresenter mPresenter;
private AddNuringPresenter picPresenter;
private RecordDetailModel recordDetailModel;
private List<RecordDetailModel.ImageModel> requestImageList;
private List<RecordDetailModel.ImageModel> imageModelList;
private boolean isPlay = false;
private String recordId;
private String recordTime;
private String modifyRecordTime;
private SyncVoice syncVoice;
private String voiceID;
private SharePreferenceHelper spHelper;
private final Calendar mCalendar = Calendar.getInstance();
private int hour;
private int minitue;
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.activity_record_edit);
ButterKnife.bind(this);
EventBus.getDefault().register(this);
initDagger();
initData();
initView();
}
/**
* 返回设备状态
*
* @param checkedHeadsetEvent
*/
public void onEventMainThread(CheckedHeadsetEvent checkedHeadsetEvent) {
int connState = checkedHeadsetEvent.getConnState();
if (connState == 1) {
//连接中 跳转主界面
} else if (connState == 2) {
//已连接 跳转主界面
if (spHelper != null) {
spHelper.put(Constants.DeviceKey.DEVICE_STATUS, 2);
}
ivStatus.setImageResource(R.drawable.ic_annbot_on);
} else if (connState == 0) {
//未连接 跳转连接设备界面
if (spHelper != null) {
spHelper.put(Constants.DeviceKey.DEVICE_STATUS, 0);
}
ivStatus.setImageResource(R.drawable.ic_annbot_off);
}
}
/**
* <p>initDagger</p>
*
* @Description 初始化dagger2
*/
private void initDagger() {
NetComponent netComponent = BaseApplication.get(this).getNetComponent();
DaggerRecordComponent.builder()
.netComponent(netComponent)
.recordModule(new RecordModule(this))
.build()
.inject(this);
picPresenter = new AddNuringPresenter();
syncVoice = SyncVoice.getInstance(RecordEditActivity.this);
}
/**
* <p>initData</p>
*
* @Description 初始化数据
*/
private void initData() {
spHelper = new SharePreferenceHelper();
if (spHelper != null) {
int status = (int) spHelper.get(Constants.DeviceKey.DEVICE_STATUS, -1);
ivStatus.setImageResource(status == 2 ? R.drawable.ic_annbot_on : R.drawable.ic_annbot_off);
}
requestImageList = new ArrayList<>();
imageModelList = new ArrayList<>();
recordDetailModel = (RecordDetailModel) getIntent().getSerializableExtra(Constants.ExtraKey.RECORD_DETAIL_MODEL);
if (recordDetailModel != null) {
recordId = recordDetailModel.getRecordId();
recordTime = recordDetailModel.getRecordTime();
modifyRecordTime = recordDetailModel.getModifiedRecordTime();
String[] times = modifyRecordTime.split(":");
hour = Integer.parseInt(times[0]);
minitue = Integer.parseInt(times[1]);
List<RecordDetailModel.ImageModel> imageModels = recordDetailModel.getImages();
if (imageModels.isEmpty()) {
rlAddPic.setVisibility(View.VISIBLE);
rvPic.setVisibility(View.GONE);
} else {
rlAddPic.setVisibility(View.GONE);
rvPic.setVisibility(View.VISIBLE);
}
etContent.setText(recordDetailModel.getContentText());
tvTime.setText(modifyRecordTime);
tvVoiceName.setText("錄音記錄-" + recordTime);
List<RecordDetailModel.Voice> voices = recordDetailModel.getVoice();
if (!voices.isEmpty()) {
String voiceLength = recordDetailModel.getVoice().get(0).getVoiceMinute();
voiceID = recordDetailModel.getVoice().get(0).getFileId();
tvVoiceLength.setText(voiceLength);
}
requestImageList = recordDetailModel.getImages();
imageSize = requestImageList.size();
}
}
/**
* <p>initView</p>
*
* @Description 初始化界面控件
*/
private void initView() {
tvBack.setText("上一頁");
rvPic.setLayoutManager(new GridLayoutManager(RecordEditActivity.this, 3));
adapter = new AddImageAdapter(RecordEditActivity.this, requestImageList);
//設置圖片間隔距離
rvPic.addItemDecoration(new SpacesItemDecoration(10));
rvPic.setAdapter(adapter);
rvPic.setItemAnimator(new DefaultItemAnimator());
adapter.setOnItemClickListener(new NurseAllAdapter.OnItemClickListener() {
@Override
public void onItemClick(View view, int position) {
switch (view.getId()) {
case R.id.iv_add_pic:
//图片选择数量
int addSize = 6 - imageSize;
int reSize = addSize % 3;
if (reSize == 0) {
reSize = 3;
}
picPresenter.addPhto(reSize, RecordEditActivity.this);
break;
case R.id.iv_add_delete:
requestImageList.remove(position);
imageSize = requestImageList.size();
adapter.setRefresh();
break;
}
}
});
}
@OnClick({R.id.tv_modify_time, R.id.layout_add_pic_empty, R.id.ll_modify_back, R.id.ll_modify_complete, R.id.iv_voice_play})
public void onViewClicked(View view) {
switch (view.getId()) {
case R.id.tv_modify_time:
/* DialogFragment dialogFragment = new TimePickerFragment();
dialogFragment.show(getFragmentManager(), "設定記錄時間");*/
showTimeDialog(hour, minitue);
break;
case R.id.layout_add_pic_empty:
picPresenter.addPhto(3, RecordEditActivity.this);
break;
case R.id.ll_modify_back:
setResult(RESULT_CANCELED);
RecordEditActivity.this.finish();
break;
case R.id.ll_modify_complete:
DialogUtil.showLoad(RecordEditActivity.this, "加載中...");
recordDetailModel.setContentText(etContent.getText().toString().trim());
recordDetailModel.setOpNo(1);
recordDetailModel.setRecordId(recordId);
recordDetailModel.setUpdTime(TimeUtil.getCurentTime());
recordDetailModel.setRecordTime(TimeUtil.getCurentTime());
if (modifyRecordTime.length() < 10) {
String[] times = modifyRecordTime.split(":");
modifyRecordTime = TimeUtil.setDefaultFormat(Integer.parseInt(times[0]), Integer.parseInt(times[1]));
}
recordDetailModel.setModifiedRecordTime(modifyRecordTime);
int dataSize = requestImageList.size();
// 清除三张空白图片
for (int i = 0; i < 3; i++) {
dataSize = requestImageList.size();
requestImageList.remove(dataSize - 1);
}
// 清除有imageId(此处考虑只有list前面的数据才有imageId)
dataSize = requestImageList.size();
imageModelList = requestImageList;
for (int i = 0; i < dataSize; i++) {
if (!TextUtils.isEmpty(imageModelList.get(0).getFileId())) {
imageModelList.remove(0);
}
}
recordDetailModel.setImages(imageModelList);
recordDetailModel.setRecordUserId(SPManager.getUserInfo().getUserName());
mPresenter.uploadPersonRecord(recordDetailModel);
break;
case R.id.iv_voice_play:
if (!isPlay) {
syncVoice.playVoice(0, voiceID, new PlayVoiceListener() {
@Override
public void onStartPlay() {
ivVoicePlay.setBackgroundResource(R.mipmap.btn_voice_stop);
tvModifyPlay.setVisibility(View.VISIBLE);
tvModifyPlay.setText("加载中");
}
@Override
public void onLoadComplete() {
tvModifyPlay.setText("播放中");
}
@Override
public void onComplete() {
ivVoicePlay.setBackgroundResource(R.mipmap.btn_voice_play);
tvModifyPlay.setVisibility(View.GONE);
isPlay = false;
}
@Override
public void onError() {
ivVoicePlay.setBackgroundResource(R.mipmap.btn_voice_play);
tvModifyPlay.setText("播放异常");
isPlay = false;
}
});
isPlay = true;
} else {
ivVoicePlay.setBackgroundResource(R.mipmap.btn_voice_play);
tvModifyPlay.setVisibility(View.GONE);
isPlay = false;
syncVoice.stop();
}
break;
}
}
@Override
public void SendMessageValue(int hour, int minute) {
if (hour < 10)
tvTime.setText("0" + hour + ":" + minute);
else
tvTime.setText(hour + ":" + minute);
modifyRecordTime = TimeUtil.setDefaultFormat(hour, minute);
}
/**
* <p>showTimeDialog</p>
* @param hour
* @param minitue
* @Description 顯示時間選擇框
*/
public void showTimeDialog(int hour, int minitue) {
TimePickerDialog.newInstance(new TimePickerDialog.OnTimeSetListener() {
@Override
public void onTimeSet(TimePickerDialog dialog, int hourOfDay, int minute) {
modifyRecordTime = new StringBuilder().append(pad(hourOfDay)).append(":").append(pad(minute)).toString();
tvTime.setText(modifyRecordTime);
}
@Override
public void onTimeCleared(TimePickerDialog timePickerDialog) {
}
}, hour, minitue, true, true)
.show(getFragmentManager(), "timePicker");
}
@Override
public void updateItemData(int postion, RecordDetailModel detailModel) {
}
@Override
public void getVioceListSuccess(List<RecordDetailModel> list) {
DialogUtil.unShowLoad();
if (!recordDetailModel.isSync()) {
DBManager.getInstance(RecordEditActivity.this).deleteNursingByID(recordDetailModel.getId());
}
RecordEditActivity.this.finish();
}
@Override
public void getVioceListFailure(String message) {
DialogUtil.unShowLoad();
DialogUtil.showShort(message);
requestImageList.clear();
requestImageList.addAll(imageModelList);
adapter.notifyDataSetChanged();
}
@Override
public void getVioceListFailure(String resultCode, String message) {
DialogUtil.unShowLoad();
DialogUtil.showShort(message);
requestImageList.clear();
requestImageList.addAll(imageModelList);
adapter.notifyDataSetChanged();
if (resultCode.equals(Constants.RESULT_FAILURE_TOKEN)) {
DialogUtil.tokenExpiredDialog(RecordEditActivity.this);
}
}
@Override
public void showLoading(String title) {
}
@Override
public void stopLoading() {
}
@Override
public void showErrorTip(String Text) {
}
@Override
public void returnSearchUser(List<UserModel> userModelList) {
}
@Override
public void returnPhto(List<MediaBean> mediaList) {
int size = mediaList.size();
if (size > 0) {
rlAddPic.setVisibility(View.GONE);
rvPic.setVisibility(View.VISIBLE);
}
for (int i = 0; i < size; i++) {
RecordDetailModel.ImageModel model = recordDetailModel.new ImageModel();
model.setSequenceNo((imageSize + i) % 3 + 1);
String[] types = mediaList.get(i).getMimeType().split("/");
model.setFileExtension(types[1]);
model.setImageLength((int) mediaList.get(i).getLength());
model.setImagePath(mediaList.get(i).getOriginalPath());
requestImageList.add(model);
}
imageSize = +size;
adapter.setRefresh();
}
@Override
public void returnUploadRespon(String uploadState) {
}
@Override
protected void onDestroy() {
super.onDestroy();
EventBus.getDefault().unregister(this);
syncVoice.stop();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,764 |
Officers Matt Schneider and Mark Lindsey asked Wheatcroft for his ID and he inquired why he had to show it, as he was not driving the vehicle. Police allege he went to stuff something in his backpack, and, after pressing the taser against his arm and telling him to "relax," Schneider twisted Wheatcroft's arm behind his back and pulled him from the car while still restrained by his seat belt.
I hope the tasered victim gets millions….
"police officers approached a vehicle for a signal turn violation"……..?
And, the passenger got tasered for THAT? What has happened to the USA? I wonder if these 'Police officers' had any training in Israel, which apparently a lot of American Policemen receive. It sounds like the sort of treatment that a Palestinian would receive from the Israeli police.
I agree with both Leland and Brian with an augmenting "WTF?"…but: why not a word about the '"family friend" driver?? (I'm submitting this without adding any independent/original thoughts for a "test" purpose: I've tried multiple times to comment on an Aletho News post regarding Syria, but each attempt is either "You've already submitted…" or the post reverts to its initial appearance [that displays one comment by another commenter]. Here, I'm going to submit for the prime purpose of ascertaining whether this comment will be accepted…or whether my Syria comment has been "censored" or otherwise been rejected [tech glitch by Akismet or … ?]. My firm belief is that my Syria comment had/has nothing controversial to warrant non-acceptance and is in fact greatly complimentary toward author Jeremy Salt and Aletho News….
This comment by me was just accepted/posted as is typical for my submissions. WTF?
I have no idea what is happening with your comments Robert. It does sound like an Akismet tech glitch. I never have time to peruse comments rejected by Akismet as spam.
Thanks for a great comment and evocative 3+-minute video clip. How can any human view the infinite universe and our nearer environs and become captive to arrogance and greed? Total-spectrum dominance, indeed…. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,299 |
title: Kristi Noem, South Dakota Congressperson
name: Kristi Noem
avatar: /ui/img/avatars/Kristi_Noem.jpg
party: republican
state: south-dakota
type: congressperson
lasthall: 2017-03-18
phone: 202-225-2801
never:
twitter: RepKristiNoem
---
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,805 |
Splitting Amazon's second headquarters between two locations would dilute the company's original promise of a megadeal, but it could also relieve Amazon of being blamed for worsening traffic and increased housing prices.
Amazon built The Spheres, which house indoor gardens, as an alternative working space for its employees in downtown Seattle. It's part of the campus of 44 Amazon buildings, with more under construction.
Amazon's second headquarters will be split between two locations, according to two people with knowledge of the discussions. The plan would bring up to 50,000 jobs, split between the two cities. The average salary has been promised to pay more than $100,000 annually over the next 10 to 15 years.
Amazon is still in the final stages of negotiations, the sources say, but Crystal City in Arlington, Va., is expected to pick up one-half of the deal, the people told NPR. Crystal City is a suburb of Washington, D.C.
New York City has been reported as a potential second location.
The surprise decision to divide the second headquarters between two cities that already have a considerable Amazon presence is an anticlimactic ending for the much-hyped, Olympic-style search. In all, 238 cities, counties and states had submitted bids hoping to woo Amazon during the fall of 2017. Amazon had narrowed the list to 20 finalists in January.
The choice of splitting the deal for so-called HQ2 was in part prompted by concerns that a single location may not be able to draw the needed 50,000 highly educated technical employees, said one person.
Dividing the new jobs between two cities could also relieve Amazon of being singularly blamed for a rapid influx of wealthy techies who could worsen traffic and increase housing prices. These urban problems have been controversial in Seattle, which is Amazon's original headquarters. The company has grown to occupy more than 40 office buildings in the heart of downtown Seattle.
Local authorities at the two locations are still sure to face questions about any taxpayer-funded perks promised to Amazon. Localities across the nation have been preparing financial subsidies worth millions of dollars.
The split of the second headquarters dilutes the company's original promise of a megadeal, which cities and counties had been chasing with stunts and gimmicks, investing thousands of dollars in websites and presentations. The contest prompted comparisons to a corporate beauty pageant or "the Olympics of the corporate world."
For Amazon, the HQ2 fanfare has added to a year of striking growth and significant milestones. In September, the company's stock value briefly topped $1 trillion. And before that, CEO Jeff Bezos — whose fortune is tied to his Amazon stake — became the world's wealthiest man in recent history.
Amazon representatives did not comment on reports of two cities splitting the second headquarters. | {
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} | 8,480 |
Q: how to access 6 different functions within for loop in C? I am writing a code in C. I have to calculate some coefficients named as:
k1, k2, k3, k4
I have 6 different functions named as:
func1, func2, func3, .....func6
Very inefficient way is to write a code like this:
/* find k1 for all 6 functions */
/* k[0][0] is k1 for func1 */
/* k[0][1] is k1 for func2 */
......
......
/* k[0][5] is k1 for func6 */
/* h is some constant */
k[0][0] = h*func1()
k[0][1] = h*func2()
k[0][2] = h*func3()
k[0][3] = h*func4()
k[0][4] = h*func5()
k[0][5] = h*func6()
Similarly I have to find out k2 for all 6 functions.
Again very inefficient way would be :
k[1][0] = h*func1()
k[1][1] = h*func2()
k[1][2] = h*func3()
k[1][3] = h*func4()
k[1][4] = h*func5()
k[1][5] = h*func6()
And similar things for remaining k3 & k4 would be:
/* find K3 for all 6 functions */
.............
.............
/* find K4 for all 6 functions */
.............
.............
I want to avoid all this.
I want a way so that I can call 6 functions for each coefficient k within a for loop. Something like this:
for(i=0; i<=3; i++)
{
for(j=0; j<=5; j++)
{
k[i][j] = h*func[...]
/* where func[...] means some way for calling 6 functions */
}
}
May be some array of functions ??
Any help will be highly appreciated.
Thanks.
A: If all the 6 functions have the same prototype, I assume they have - for example receives void and returns int, then you can use an array of function pointers:
typedef int (*FUNCTION_PTR)(void);
FUNCTION_PTR funcs[6] = { func1, func2, ... };
for(i = 0; i <= 3; i++)
{
for(j = 0; j <= 5; j++)
{
k[i][j] = h * func[j]();
}
}
A: You can declare an array of pointers to functions of the same type. Some people prefer to use typedefs for this, but you can also declare the array directly. If f is a pointer to a function, you can call the function with (*f)(...), where the ellipsis represents the arguments. But you can also call the function with f(...).
#include <stdio.h>
double f1(int);
double f2(int);
double f3(int);
double f4(int);
double f5(int);
double f6(int);
int main(void)
{
/* With typedef */
// typedef double (*Fn_ptr)(int);
// Fn_ptr funcs[6] = { f1, f2, f3, f4, f5, f6 };
/* Without typedef */
double (*funcs[6])(int) = { f1, f2, f3, f4, f5, f6 };
double k[4][6];
double h = 0.1;
for (size_t i = 0; i < 4; i++)
{
for (size_t j = 0; j < 6; j++)
{
k[i][j] = h*funcs[j](i);
}
}
for (size_t i = 0; i < 4; i++)
{
for (size_t j = 0; j < 6; j++)
{
printf("%10f", k[i][j]);
}
putchar('\n');
}
return 0;
}
double f1(int x)
{
return x + 1;
}
double f2(int x)
{
return x + 2;
}
double f3(int x)
{
return x + 3;
}
double f4(int x)
{
return x + 4;
}
double f5(int x)
{
return x + 5;
}
double f6(int x)
{
return x + 6;
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,271 |
{"url":"https:\/\/courses.energyexcursions.com\/courses\/in-pursuit-of-the-safe-well\/lessons\/primary-well-control\/topic\/hydrostatic-pressure\/","text":"Energy Excursions\n\n# Hydrostatic Pressure\n\nHydrostatic pressure represents the force per unit area exerted by a column of fluid at a particular depth due to the action of gravity. Hydrostatic calculations apply to a fluid at rest, meaning other pressure changes associated with flow are not accounted for, which simplifies the calculation tremendously and is usually sufficiently accurate for basic analysis.\n\nTo calculate the hydrostatic pressure in a well, engineers use an equation like the following:\n\n$\\require{textmacros}\\textit{Hydrostatic pressure (HSP)} = {\\textit{Density}} \\times {\\textit{Gravity}} \\times \\textit{Height}$\n\n### U.S. Petroleum Industry Uses Oilfield Units\n\nIn the U.S. petroleum industry, calculations are done in \u201coilfield units\u201d, and pressure is measured in units of pounds force (lbf) per square inch, abbreviated as psi, so the units for HSP will be psi. In oilfield units, fluid density is measured in pounds mass (lbm) per gallon, written as lbm\/gal or abbreviated as ppg. Freshwater has a fluid density of 8.34 ppg. Comparing these units to the SI system, lbm is comparable to kilograms (kg), and lbf is comparable to Newtons (N).\n\n### Engineers Regularly Have to Convert Units\n\nThe Gravity term in the equation above refers to the gravitational constant, $$g$$, divided by a unit conversion factor, $$g_c$$, or $$\\text{Gravity} = \\frac{g}{g_c}$$. In oilfield units, $$g$$ = 32.2 ft\/s2, and $$g_c$$ = 32.2 (lbm-ft\/s2)\/lbf. This sounds complicated, but if you multiply the gravity term out, the result is pretty simple.\n\n$\\textit{Gravity} =\\frac{g}{g_{c}} =\\frac{32.2\\frac{\\text{ft} }{\\text{s}^{2} } }{\\frac{32.2\\ \\text{lb}_{\\text{m} } \\cdot \\frac{\\text{ft} }{\\text{s}^{2} } }{\\text{lb}_{\\text{f} } } } =\\frac{1\\ \\text{lb}_{\\text{f} } }{1\\ \\text{lb}_{\\text{m} } }$\n\nOr in other words, the force caused by gravity, in oilfield units, is 1 pound force (lbf) for every 1 pound mass (lbm). Another way to say this is 1 lbm \u201cweighs\u201d 1 lbf. Using this conversion information and given that water has a density (mass\/volume) of 8.34 lbm\/gal, we can conclude that water weighs 8.34 lbf\/gal. For drilling operations, this normalized fluid weight (force\/volume) is called the mud weight. Given that in Oilfield Units the mass and weight have the same numerical value (because 1 lbm weights 1 lbf), the two terms are used synonymously, and has led to the practice where the density of the mud in drilling is called the mud weight.\n\n### An Aside Concerning SI Units\n\nA comparable representation of Gravity in SI units may look more familiar to you.\u00a0 The gravitational constant is $$g$$ = 9.8 m\/s2.\u00a0 The unit conversion factor is $$g_c$$ = 1 (kg-m\/s2)\/N.\u00a0 Multiplying as before,\n\n$\\textit{Gravity} =\\frac{g}{g_{c}} =\\frac{9.8\\frac{\\text{m} }{\\text{s}^{2} } }{\\frac{1\\ \\text{kg} \\cdot \\frac{\\text{m} }{\\text{s}^{2} } }{\\text{N} } } =\\frac{9.8\\ \\text{N} }{1\\ \\text{kg} }$\n\nIn other words, this equation says the force caused by gravity exerts 9.8 N for every kg, or a kg weighs 9.8 N. So unlike the Oilfield Unit system, the mass and weight of an object will have different numeric values, and fluid density and normalized weight cannot be used synonymously. In SI units, water has a density of 1,000 kg\/m3, but since a kg weighs 9.8 N, water \u201cweighs\u201d 9,800 N\/m3.\n\nThe Height in this equation refers to the fluid column and is described as the \u201ctrue vertical depth (TVD)\u201d expressed in feet for oilfield units. We use true vertical depth because sometimes a well deviates slightly from vertical, making the \u201cmeasured depth\u201d, or the distance traveled by the drill bit, greater than or equal to the TVD. But gravity acts vertically toward the center of the earth, so TVD is the value that is important.\n\nAn example calculation can help illustrate how hydrostatic calculations are made, and can guide you through the unit conversions required. Assume a vertical wellbore is 1,000 ft deep (TVD), and it is filled with fresh water, which has a density of 8.34 ppg.\u00a0 What is the hydrostatic pressure at the bottom of this well?\n\n$\\textit{HSP} =8.34\\frac{\\text{lb}_{\\text{m} } }{\\text{gal} } \\times 1\\frac{\\text{lb}_{\\text{f} } }{\\text{lb}_{\\text{m} } } \\times 1000\\ \\text{ft} =8340\\ \\frac{\\text{lb}_{\\text{f} } }{\\text{gal} }$\n\nNow you can see the units for this result seem odd, but let\u2019s analyze it. Lbf is a unit of force. Gallons (gal) is a unit of volume, which is length3. \u201cft\u201d is a unit of length. Multiplying the units gives force \u00d7 length\/length3 or force\/length2 which is force\/area, which is the definition of pressure. But in oilfield units, we express pressure in psi, not lbf-ft\/gal, so we need to convert \u201cft\u201d and \u201cgal\u201d into \u201cin\u201d and \u201cin3\u201d, respectively, as follows:\n\n$\\textit{HSP} =8340\\frac{\\text{lb}_{\\text{f} } \\cdot \\text{ft} }{\\text{gal} } \\times 12\\frac{\\text{in} }{\\text{ft} } \\times 1\\frac{\\text{gal} }{231\\ \\text{in}^{3} } =433\\ \\frac{\\text{lb}_{\\text{f} } }{\\text{in}^{2} } =\\ 433\\ \\text{psi}$\n\n### Conversions Result in a Handy Formula Format\n\nSo the fluid pressure at the bottom of a 1,000 ft tall column of fresh water in our well exerts 433 psi. Multiplying out this entire string of conversions every time you want to know the pressure in your well can be time consuming, so engineers develop shortcuts for common calculations to make things more efficient. Our answer above says that gravity exerts 433 psi of pressure for every 1,000 ft of freshwater column height (433 psi per 1,000 ft), which we can normalize to 0.433 psi per ft or 0.433 psi\/ft. We call this value of 0.433 psi\/ft the freshwater Pressure Gradient (PG),\u00a0 which is the rate at which pressure changes with increasing freshwater column height. Knowing this, we can modify our hydrostatic pressure equation to a more \u201chandy\u201d format, assuming freshwater, as\n\n$\\textit{HSP} =0.433\\frac{\\text{psi} }{\\text{ft} } \\times \\text{height}$\n\nWhat if we have something other than freshwater, or more specifically, something with a density different from freshwater? Let\u2019s rewrite our HSP equation to allow for a generic Pressure Gradient (PG), which is to relax the assumption that we have freshwater with a PG = 0.433 psi\/ft.\n\n$\\textit{HSP} =\\textit{PG} \\times \\textit{height}$\n\nComparing to our original HSP equation, we can see that the following must be true:\n\n$\\textit{PG}=\\textit{density} \\times \\textit{gravity}$\n\nThese two equations show that PG and consequently HSP are directly proportional to the density of the fluid. Being \u201cdirectly proportional\u201d means if the density changes by a certain ratio, the pressure or the pressure gradient will change by that exact same ratio. The ratio of the density of one substance compared to some other substance considered a standard is called the Specific Gravity (SG), and for fluids it is convenient to use freshwater as the standard.\u00a0 Drilling mud is usually more dense than freshwater, being made of saline water with some solids added (typically clay).\n\n## 1.32\n\n#### Correct.\n\n11 ppg is greater than 8.34 ppg by 1.32 times, which is the specific gravity of the drilling mud relative to water.\n\n## 0.572 psi\/ft\n\n#### Correct.\n\n(PG of drilling mud) = (SG of drilling mud) \u00d7 (PG of freshwater) = 1.32 \u00d7 (0.433 psi\/ft) = 0.572 psi\/ft\n\n## 572 psi\n\n#### Correct.\n\nHSP = (0.572 psi\/ft) \u00d7 (1,000 ft) = 572 psi\n\nIncreasing the SG of the drilling mud increases the hydrostatic pressure in the wellbore proportionally.\n\n## None of the above\n\n#### Incorrect.\n\nThis example calculation, examining the pressure of a fluid with a higher SG than water, demonstrates the predominant mechanism by which drillers exert primary well control: they increase the pressure at the bottom of the well by increasing the density of the wellbore fluid, or in driller\u2019s language, they \u201cincrease the mud weight.\u201d","date":"2022-11-26 18:43:40","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.7659851908683777, \"perplexity\": 1658.4630466028725}, \"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\/1669446708046.99\/warc\/CC-MAIN-20221126180719-20221126210719-00181.warc.gz\"}"} | null | null |
from __future__ import absolute_import
# Copyright (c) 2010-2019 openpyxl
from openpyxl.compat import unicode
from openpyxl.descriptors.serialisable import Serialisable
from openpyxl.descriptors import (
Typed,
String,
Bool,
Integer,
NoneSet,
Sequence,
)
from openpyxl.descriptors.excel import Relation, ExtensionList
from openpyxl.descriptors.nested import NestedText
from openpyxl.descriptors.sequence import NestedSequence, ValueSequence
from openpyxl.packaging.relationship import (
Relationship,
get_rels_path,
get_dependents
)
from openpyxl.xml.constants import SHEET_MAIN_NS
from openpyxl.xml.functions import fromstring
"""Manage links to external Workbooks"""
class ExternalCell(Serialisable):
r = String()
t = NoneSet(values=(['b', 'd', 'n', 'e', 's', 'str', 'inlineStr']))
vm = Integer(allow_none=True)
v = NestedText(allow_none=True, expected_type=unicode)
def __init__(self,
r=None,
t=None,
vm=None,
v=None,
):
self.r = r
self.t = t
self.vm = vm
self.v = v
class ExternalRow(Serialisable):
r = Integer()
cell = Sequence(expected_type=ExternalCell)
__elements__ = ('cell',)
def __init__(self,
r=(),
cell=None,
):
self.r = r
self.cell = cell
class ExternalSheetData(Serialisable):
sheetId = Integer()
refreshError = Bool(allow_none=True)
row = Sequence(expected_type=ExternalRow)
__elements__ = ('row',)
def __init__(self,
sheetId=None,
refreshError=None,
row=(),
):
self.sheetId = sheetId
self.refreshError = refreshError
self.row = row
class ExternalSheetDataSet(Serialisable):
sheetData = Sequence(expected_type=ExternalSheetData, )
__elements__ = ('sheetData',)
def __init__(self,
sheetData=None,
):
self.sheetData = sheetData
class ExternalSheetNames(Serialisable):
sheetName = ValueSequence(expected_type=unicode)
__elements__ = ('sheetName',)
def __init__(self,
sheetName=(),
):
self.sheetName = sheetName
class ExternalDefinedName(Serialisable):
tagname = "definedName"
name = String()
refersTo = String(allow_none=True)
sheetId = Integer(allow_none=True)
def __init__(self,
name=None,
refersTo=None,
sheetId=None,
):
self.name = name
self.refersTo = refersTo
self.sheetId = sheetId
class ExternalBook(Serialisable):
tagname = "externalBook"
sheetNames = Typed(expected_type=ExternalSheetNames, allow_none=True)
definedNames = NestedSequence(expected_type=ExternalDefinedName)
sheetDataSet = Typed(expected_type=ExternalSheetDataSet, allow_none=True)
id = Relation()
__elements__ = ('sheetNames', 'definedNames', 'sheetDataSet')
def __init__(self,
sheetNames=None,
definedNames=(),
sheetDataSet=None,
id=None,
):
self.sheetNames = sheetNames
self.definedNames = definedNames
self.sheetDataSet = sheetDataSet
self.id = id
class ExternalLink(Serialisable):
tagname = "externalLink"
_id = None
_path = "/xl/externalLinks/externalLink{0}.xml"
_rel_type = "externalLink"
mime_type = "application/vnd.openxmlformats-officedocument.spreadsheetml.externalLink+xml"
externalBook = Typed(expected_type=ExternalBook, allow_none=True)
file_link = Typed(expected_type=Relationship, allow_none=True) # link to external file
__elements__ = ('externalBook', )
def __init__(self,
externalBook=None,
ddeLink=None,
oleLink=None,
extLst=None,
):
self.externalBook = externalBook
# ignore other items for the moment.
def to_tree(self):
node = super(ExternalLink, self).to_tree()
node.set("xmlns", SHEET_MAIN_NS)
return node
@property
def path(self):
return self._path.format(self._id)
def read_external_link(archive, book_path):
src = archive.read(book_path)
node = fromstring(src)
book = ExternalLink.from_tree(node)
link_path = get_rels_path(book_path)
deps = get_dependents(archive, link_path)
book.file_link = deps.Relationship[0]
return book
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,730 |
{"url":"https:\/\/www.physicsforums.com\/threads\/what-would-you-write.221049\/","text":"# What would you write?\n\n1. Mar 10, 2008\n\n### ManDay\n\nI wonder how you'd handle it. For 2,345,000 would you rather write\n\n\u2022 2.234 x 10^6\n\u2022 2.234M\n\u2022 2.234e6\nExcept for exams I never use anything but the e...-Notation. Therefore: 2.234e6. Using the SI Prefixes I just feel not capable of because I hardly can remember the value of any besides the few standard ones like m,k,M,\u00b5..., no matter how many times I try to remember them. Writing ... x10^... however feels way to elaborate.\n\nWhat do you use? What is chiefly accepted by the physicists community?\n\n2. Mar 10, 2008\n\n### stewartcs\n\nIt is just a matter of preference. I prefer to use 1 x 10^6 simply because one cannot misinterpret it.\n\nCS\n\n3. Mar 10, 2008\n\n### dst\n\nStandard notation is the clearest of them all and it also makes calculations involving fractions easier. I've noticed that in exams they've started using ridiculously farfetched units just to make it slightly harder.\n\n4. Mar 10, 2008\n\n### ManDay\n\nIf you got used to it evaluating fractions becomes just as-easy with the exp-notation.\n\n<cut>Not right. I should think before I write<\/cut>\n\nLast edited: Mar 10, 2008\n5. Mar 10, 2008\n\n### Claude Bile\n\nUsing e notation is probably the least accepted in publications because of potential confusion with the constant e.\n\nWhere possible I use formal scientific notation, i.e. 2.234 x 10^6 (units).\n\nIn many instances though, it is actually less confusing to use an acceptable SI prefix - such as using the nano prefix when stating optical wavelengths, e.g. saying 633 nm instead of 6.33 x 10^-7 m.\n\nAt the end of the day, you use the notation that is the least confusing!\n\nClaude.\n\n6. Mar 10, 2008\n\n### Staff: Mentor\n\nAs far as I can remember, students started writing stuff in e-notation only after pocket calculators came along that use it because their small displays can't handle normal exponents. Also, computer programming languages use it for scientific notation because simple ASCII text can't display exponents. Properly-printed scientific notation is of the form $6.02 \\times 10^{23}$, and that's the way people always wrote it by hand when I was a student. Unless of course they chose to use the metric prefixes instead.\n\nI personally don't mind if students use e-notation in homework assignments and on tests, but in a formal report or paper I think it looks unprofessional. I would comment on it and ask students to change it in those situations.\n\n7. Mar 10, 2008\n\n### Danger\n\nSince I'm fairly informal, I generally use metric. For bigger numbers, I go with x10^n because I don't know all of the proper prefices.\nIn a school or employment situation, I'd ask the powers-that-be which they prefer and go with that.\nI've never even heard of e-notation, but thanks to jt for the explanation.\n\n8. Mar 11, 2008\n\n### ManDay\n\nThis is correct. Yet I think that this cannot nessesarily be considered \"adapting calculator notation\" (which you didn't, of course) but it's rather just a quicker and more convenient way to write it. As I said I prefer e.. over x10^... because it takes less than half the time to write.\n\nx 10^... is mathematical standard. But Physicist never really cared what mathematicans think is correct. So why would you stick to the ellaborate x 10^... if you have got a convenient shorthand at your disposal?\n\n9. Mar 11, 2008\n\n### pam\n\nAm I the only one who noticed you changed the number?\n\n10. Mar 11, 2008\n\n### stewartcs\n\nBecause Euler's number (e) has a very specific meaning in mathematics and may convolute things if you use that notation.\n\nI wouldn't consider it more convenient if it has the potential to confuse the reader.\n\nCS\n\n11. Mar 11, 2008\n\n### stewartcs\n\nI presumed this was a typo. Of course if it is not, then I wouldn't write it any of those ways!\n\nCS","date":"2018-06-17 22:51:20","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.709730863571167, \"perplexity\": 2169.3404690778616}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-26\/segments\/1529267859817.15\/warc\/CC-MAIN-20180617213237-20180617233237-00285.warc.gz\"}"} | null | null |
"School Prayer" is a poem written by American poet and naturalist Diane Ackerman; it is the first of 50 poems in Ackerman's book I Praise My Destroyer, which was published in 1998. "School Prayer" is a pledge to protect and revere nature, in every form it may appear. The poem was recited and discussed by Garrison Keillor on his daily podcast The Writer's Almanac.
Structure
"School Prayer" consists of 23 lines of varying length and is written in free verse, therefore lacking any rhyme scheme or regular meter. The natural speech pattern that School prayer obtains from its free verse prose allows the poem to read like a prayer or a pledge.
Subject
Joseph Kelly of The Seagull Book of Poems makes note of the pledge-like qualities of "School Prayer", and suggests that the poem is meant to be compared to ritual pledges such as the Pledge of Allegiance. This suggestion is bolstered by the title of the poem, as the Pledge of Allegiance is primarily said in American schools at the beginning of the day, making it a sort of school prayer.
The content of "School Prayer" consists primarily of Ackerman pledging to protect and revere nature in a secular, yet spiritual, way. Writer Maria Popova on her Brain Pickings blog, notes this notion of secular spirituality in "School Prayer", and concludes that School Prayer succeeds in drawing attention to the presence and "the world's fullness". Ackerman herself talks about the importance of developing spirituality, not in reference to a god, but in reference to personal values.
Ackerman has also said that her writing is about nature and human nature, lending more support to the concept that School Prayer is commenting on the idea of a secular spirituality. It also adds credence to the notion that "School Prayer" is melding together nature and human nature, and is setting up all forms of nature as something to be respected and worshiped.
Critical reception
There has been some critical interest in Ackerman's I Praise My Destroyer, and "School Prayer" in particular. John Taylor, writing for Poetry, finds that though "School Prayer" has honorable intention behind it, it falls a bit short, saying that the poem calls to attention a warning from André Gide: "it is with noble sentiments bad literature gets made".
However, not all view "School Prayer" in a negative light. Maria Popova declares the poem a "powerful invitation" to be aware of nature and participate in its beauty. "School Prayer" also found its way to Women's voices for Change, where it was described as a guiding source for school children.
References
American poems | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,026 |
Q: Threading a .Net web service and providing a repsonse before continuing processing Let me start with the fact that similar forms of this question have been asked here, yet there are no working examples or useable answers to the subject (that I have been able to find). This post will attempt to provide a more concise example and advance a code block to that end. With reference to the following links that are similar though not conclusive/working examples.
similar link #1 Background worker
similar link #2 Multi Threading in web service
similar link #3 threading in ASP.net
similar link #4 acknowledge quickly but continue processing
Other links:
#1 Async Web service Call
#2 Calling Web Service Async functions from web page
#3 MS ThreadPool.QueueUserWorkItem Method
#4 MS Multithreading Walk Through Forms based
The problem is returning a response to a client consuming a web service so that the client can continue on its way while the web service completes processing tasks. The web service (after authentication) receives a file name from the client that should be retrieved from a distant ftp server. Once the file has been downloaded to the web service hosting server, the client can be released, as the file name was valid and has been copied to the local server.
The web server, after returning a simple 'true' Boolean response to the client, continues to crunch the downloaded file into a database and happily waits for the name of the next file.
The current call is a SOAP message and this is a traditional asmx web service.
Web Service:
<WebService details .....
Public sourceFile As String
Public myData as Boolean = False
<WebMethod()> _
Public Function fileReady_Log2(ByVal user As String, ByVal PW As String, ByVal fileName As String) As String
' This function receives a web service call notifying this server to retrieve a file from a remote ftp server When the file is downloaded it is processed into this servers database.
'
' This routine attempts to process an ASYNC routine to respond to the call and release the caller as soon as possible while continuing processing requirements.
' Validation removed for simplicity
If authorized Then
' Need to retrieve ftp server information and pull down the file and load it to the database.
' Call Async function
sourceFile = filename
Dim myObject = New ftpThread()
myObject.SendDataAsync()
' Ok, now return the result to client.
Return myData
Dim ftpInfo As New ftpFileImport ' continue with processing the uploaded file.
results = ftpInfo.retrieveLogFile(fileName, deviceID, usr)
Return results
End If
Return results
End Function
Currently this is the structure of the calling functions. I have tried a number of variations and decided now is a good time to stop and ask for help.
ftpThread.vb
Imports Microsoft.VisualBasic
Imports System.Threading
Public Class ftpThread
Public Sub SendDataAsync()
ThreadPool.QueueUserWorkItem(Sub(getFtpFileExists(sourceFile )))
End Sub
End Class
I am also getting an Expression Expected exception on the word Sub above.
ftp functions This process works synchronously, need it to be asynchronous.
Public Class ftpFileExistsCheck
Public Sub getFtpFileExists()
' This all works without ASYNC
' Retrieve FTP credentials from DB
' Set up ftp request
' download the file.
' But continuing process to DB at this point causes client to wait.
myData = True
End Sub
End Class
Thought I could figure this out from all I have read, but I just can't get past setting up the correct coding for ASYNC processing.
A: In your ASMX service implementation, you should spin up a WCF client and invoke a one way service operation, passing the message straight across. This not only lets you solve your problem but when you can upgrade off of ASMX, you can go directly to the new but still familiar WCF endpoint.
*
*Implement a WCF service that mirrors your existing ASMX methods you need this kind of functionality in
*Make each service operation a one way operation
*Modify your existing ASMX service to consume the new WCF service and enjoy
Another alternative would be to use Windows Workflow Foundation (WF). WF lets you compensate an activity with the result of some service call. This means your service continues progressing until you need the value from the service, similar to how async and await works. You could create a WCF Workflow Service that represents your existing ASMX service and make the workflow a one way service operation, then fire off the entire workflow and dequeue work items in the workflow until completion or some expiration time or whatever criteria you need to indicate the workflow should stop expecting more work.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,578 |
Acanthocereus is een geslacht van cactussen. De soorten komen voor in het tropische deel van het Amerikaanse continent, van het uiterste zuiden van Florida, op de Caraïben, in Mexico tot in het zuidelijke deel van Centraal-Amerika en Colombia.
Soorten
Acanthocereus baxaniensis (Karw. ex Pfeiff.) Borg
Acanthocereus colombianus Britton & Rose
Acanthocereus horridus Britton & Rose
Acanthocereus occidentalis Britton & Rose
Acanthocereus subinermis Britton & Rose
Acanthocereus tetragonus (L.) Hummelinck
Cactaceae | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,588 |
What Is Codependency?
Recognizing the Signs
Wendy Rose Gould
Wendy Rose Gould is a lifestyle reporter with over a decade of experience covering health and wellness topics.
Verywell / Alison Czinkota
Codependency refers to a mental, emotional, physical, and/or spiritual reliance on a partner, friend, or family member.
"The term was originally coined in the 1950s in the context of Alcoholics Anonymous to support partners of individuals who abused substances, and who were entwined in the toxic lives of those they cared for," says Dr. Renee Exelbert, a licensed psychologist and author based in New York.
This is still true — but today, codependency covers a much broader spectrum.
Codependency is not a clinical diagnosis or a formally categorized personality disorder on its own. Generally speaking, codependency incorporates aspects of attachment style patterns developed in early childhood, and it can also overlap with other personality disorders, including dependent personality disorder.
The Link Between Borderline and Dependent Personality Disorders
The Varying Forms of Codependency
Codependency can come in all shapes and sizes and varying levels of severity. "Foundationally, it is due to poor concept of self and poor boundaries, including an inability to have an opinion or say no," says Dr. Mark Mayfield, a licensed professional counselor (LPC).
He adds that codependency can develop in all sorts of relationships, such as parent-child, partner-partner, spouse-spouse, and even coworker-boss.
Signs of Codependency
As outlined above, codependency refers to an imbalanced relationship pattern where one person assumes responsibility for meeting another person's needs to the exclusion of acknowledging their own needs or feelings.
Dr. Exelberg
"Codependency is a circular relationship in which one person needs the other person, who in turn, needs to be needed. The codependent person, known as 'the giver,' feels worthless unless they are needed by — and making sacrifices for — the enabler, otherwise known as 'the taker.'
— Dr. Exelberg
Codependent relationships are thus constructed around an inequity of power that promotes the needs of the taker, leaving the giver to keep on giving often at the sacrifice of themselves. According to Dr. Mayfield and Dr. Exelbert, signs of codependency might include some, but not necessarily all, the following:
A sense of "walking on eggshells" to avoid conflict with the other person.
Feeling the need to check in with the other person and/or ask permission to do daily tasks.
Often being the one who apologizes—even if you have done nothing wrong.
Feeling sorry for the other person even when they hurt you.
Regularly trying to change or rescue troubled, addicted, or under-functioning people whose problems go beyond one person's ability to fix them.
Doing anything for the other person, even if it makes you feel uncomfortable.
Putting the other person on a pedestal despite the fact that they don't merit this position.
A need for other people to like you in order to feel good about yourself.
Struggling to find any time for yourself, especially if your free time consistently goes to the other person.
Feeling as if you've lost a sense of yourself or within the relationship.
Boundaries in Relationships and Stress
Why Codependency Is an Unhealthy Dynamic
While everyone has loved ones and feels responsible for those loved ones, it can be unhealthy when someone's identity is contingent upon someone else.
"Codependency does not refer to all caring behavior or feelings — but only those that are excessive to an unhealthy degree. Responsibility for relationships with others needs to coexist with responsibility to self," says Dr. Exelbert.
"This dynamic has also been referred to as a 'relationship addiction' because people with codependency often form relationships that are one-sided, emotionally destructive, and/or abusive."
In that sense, the inherent problem of codependency is that the individual loses their true sense of self since they're pouring so much into someone else.
Even if "the giver" doesn't feel this way immediately — they likely enjoy giving their love and being relied upon — it can develop to very unhealthy degrees as the relationship progresses.
Another inherent issue is that it becomes difficult for "the giver" to extricate themselves from the relationship since they might feel the other person relies on them so much — even if they know in their gut it is the right thing to do. Conversely, "the taker" will feel so reliant on the other that they can have difficulty leaving a toxic relationship, as well.
How to Reduce Codependent Tendencies
The first step in reducing codependent tendencies is to focus on self-awareness. This can be done on your own, of course, but Dr. Mayfield also stresses the importance of therapy to help you really unravel your codependent tendencies.
He adds, "Many who struggle with codependency don't seek help until their life begins to fall apart. My advice is to be proactive and seek help."
Once you're on that journey, try your best to do the following:
Become president of your own fan club. "Learn to speak lovingly and positively to yourself, and resist the impulse to self-criticize," says. Dr. Exelbert.
Take small steps towards some separation in the relationship. Seek activities outside of the relationship and invest in new friendships. Focus on figuring out the things that make you who you are, and then expand upon them.
When tempted to think or worry about someone else, actively turn your attention inward. This takes practice, so be kind to yourself along the way.
"Stand up for yourself if someone criticizes, undermines, or tries to control you," says Dr. Exelbert. By working on building your own sense of self-esteem, you'll find more strength in yourself.
Don't be afraid to say "no" to someone when you don't really want to do something.
If one-on-one therapy doesn't appeal to you, consider trying a support group or group psychotherapy, suggests Dr. Exelbert.
There's even an organization called Codependents Anonymous (CoDa) that addresses "needing to be needed" and past relationship dynamics.
Press Play for Advice On Building Confidence
Hosted by Editor-in-Chief and therapist Amy Morin, LCSW, this episode of The Verywell Mind Podcast shares how to build your confidence and self-esteem. Click below to listen now.
Follow Now: Apple Podcasts / Spotify / Google Podcasts / RSS
Codependency is a nuanced behavior that comes in many forms and levels of intensity. It often leads to an unhealthy relationship dynamic that progressively gets worse over time as the codependent person loses a sense of themselves. Self-awareness and active redirection from the behavior is key in reducing codependent tendencies; be kind to yourself as you work through years of learned behavior.
How to Build a Relationship Based on Interdependence
6 Different Types of Relationships You May Find Yourself In
What to Know About Cuckold Relationships
Interdependence Can Build a Lasting and Safe Relationship
How to Recognize Someone With Covert Narcissism
How to Stop Being a People-Pleaser
How to Stop Being Codependent
10 Red Flags in Relationships
What to Do If Your Partner Doesn't Want to Get Married
Coping With Insecurity in a Relationship
Healing After Narcissistic Abuse: What Does Healing Look Like?
How to Handle Living With a Narcissist
What It Means to Walk on Eggshells in a Relationship
What to Do If You or a Loved One Lack Empathy
What Is Narcissistic Rage?
What Is Triangulation in Psychology?
What Is Quiet Borderline Personality Disorder? | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,724 |
If your organization wants to offer access to applications and data to users of smartphones, tablets, thin clients, laptops and desktop systems simply and easily, XenDesktop from Citrix has the tools you need.
I was scanning through my notes after reading through Kevin Strohmeyer, Citrix's Sr. Manager, Product Marketing Enterprise Apps and Desktops, presentation on XenDesktop 5 Service Pack 1.
Citrix has succeed in melding several different forms of virtualization together to present a seamless experience. The company has managed to tie access virtualization, application virtualization and processing virtualization together in a way that makes it possible for people to access applications and data from nearly every type of networkable system without having to know which form of virtualization they're using at the moment.
Citrix XenDesktop 5 makes it possible for organizations to create, provision and then deploy client software such that access virtualization, application virtualization or processing virtualization (or some combination of those types of technology) are made available to users.
Citrix has also improved how virtual machine memory is utilized when the workload is presented as a virtual client or desktop running back in the data center. Citrix has found a way to make the virtual machine memory allocation dynamic so that server memory can be used optimally without requiring companies have extensive expertise in virtual machine configuration and tuning.
Desktop virtualization seems to be a topic that confuses many folks. I believe this is because some suppliers would prefer people be confused so that they could sell a limited function solution rather than offering a complete offering.
When the phrase desktop virtualization is used to describe making it possible for people to access a physical or virtual system remotely, access virtualization technology is used to capture the user interface portion of an application. It is then converted to a neutral format and then projected across the network to a device that can display the user interface and allow the user to enter and access information. This means that just about any type of network enabled device could be used to access the application.
When the phrase desktop virtualization is used to describe encapsulating an application using client-side application virtualization technology and then projecting it in whole or piecemeal to a remote system for execution. The application could either remain on that client device or be deleted once the user completes the task depending upon the settings used by the IT administrator. This means, of course, that the client system has run the operating system needed by the application. So, Windows applications would need to run on Windows executing on a PC or Laptop.
One or more virtual client systems could execute on a single physical client system. This allows personal applications to run side by side with locked down, corporate applications.
Local execution. Virtual client systems, that is a complete desktop or laptop system image that has been encapsulated in a virtual machine, could run on a local blade server. The user interface is projected to physical PCs, Laptops or Thin client systems using access virtualization technology.
Remote execution. Virtual client systems could run on a server that resides in the organizations data center. The user interface is projected to physical PCs, Laptops or Thin client systems using access virtualization technology.
Since the industry is using the same phrase, desktop virtualization, to describe all of these different approaches, the concept can be quite confusing to those unfamiliar with all of the different types of technology that could be pressed into service.
If your organization wants to offer access to applications and data to users of smartphones, tablets, thin clients, laptops and desktop systems simply and easily, XenDesktop from Citrix has the tools you need. Citrix has made it possible for its customers to create and then provision solutions that dynamically chose the type of virtualization needed at the moment. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,459 |
{"url":"http:\/\/dev.theomader.com\/2013\/12\/","text":"# Linear Depth\n\nSomething that seems to come up again and again is the topic of linear vs. non-linear depth. If you take a look at the standard DirectX projection matrix and do the math for the $z$ component, you\u2019ll end up with something like this\n\n$z' = \\frac{z_f}{z_f - z_n} (1 - \\frac{z_n}{z})$\n\nwhere $z$ is the depth value before projection, $z'$ is the depth value after projection and $z_n$, $z_f$ correspond to the near and far planes. So projection actually transforms $z$ into some variation of $1\/z$. The reason for this is simple: GPUs rasterize primitives in screen space and interpolate attribute data linearly in screen space as well. Linear depth $z$ in view space, however, becomes non-linear after projection and thus cannot be correctly interpolated by simple linear interpolators. Conversely, it turns out that $1\/z$ is linear in screen space. This is actually quite easy to see: Assume a plane in view space\n\n$Ax + By + Cz = D$\n\nPerspective projection transforms view space x and y coordinates to\n\n$x' = \\frac{x}{z}, \\qquad y' = \\frac{y}{z}$\n\nInserting these equations into the original plane equation yields\n\n$A x' z + B y' z + C z = D$\n\nwhich gives us\n\n$\\frac{1}{z} = \\frac{A}{D} x' + \\frac{B}{D} y' + \\frac{C}{D}$\n\nclearly showing that $1\/z$ is a linear function of screen space $x'$ and $y'$. This is illustrated quite nicely in this blog post by rendering ddx(z') and ddy(z') as color to the screen. The same holds for other generic attributes like texture coordinates: The GPU cannot directly interpolate $u$ and $v$, but will interpolate $u\/z$ and $v\/z$ instead. The attribute value will then be reconstructed per pixel by multiplying by $z$.\n\n# Depth Precision\n\nNow that we have established that the value that ends up in the depth buffer is not the depth but rather something related to $1\/z$, one might ask what kind of effect this will have on depth precision. After all, $1\/z$ is a highly non-linear function that will significantly warp the original depth values. Check out the graph below: I plotted the resulting $z'$ for the view space depth range $z \\in \\{0,\\dots,100\\}$ for different near plane values $z_n$:Notice how steep the function is on the first couple of meters. Almost the entire interval $z'\\in\\{0,\\dots,0.99\\}$ is spent on the first couple of meters.\n\nIn order to test this result empirically I wrote a small program that will sample the range $z \\in \\{z_n,\\dots,z_f\\}$ in regular intervals on the GPU, calculate the depth value $z'$ after projection and write it to some depth buffer of choice. The buffer is then read back to the CPU and view space depth is reconstructed for each sample. This allows us to calculate the error of original depth value vs. reconstructed depth value. Here are the results for the formats DXGI_FORMAT_D16_UNORM and DXGI_FORMAT_D32_FLOAT with the following configuration: $z_n = 0.1$, $z_f = 10000$:\nNote how the error for DXGI_FORMAT_D16_UNORM quickly approaches ridiculous proportions; 16 bit integer depth in combination with a projective transform is definitely a no go! Here\u2019s another plot to illustrate the error of DXGI_FORMAT_D32_FLOAT in more detail:Much better, though at the extremes we still get an error of over 100 meters. With some care though, this can greatly reduced: The shape of the hyperbolic $z'$ curve is largely determined by the near plane distance $z_n$. Even a slight change from $z_n=0.1$ to $z_n=0.25$ reduces the maximal error from $1.4\\%$ down to $0.26\\%$.\n\nI also tested DXGI_FORMAT_D24_UNORM_S8_UINT but the results were so close to DXGI_FORMAT_D32_FLOAT that I can only conclude that the driver internally maps the depth format to 32 bit float. Not that much of a surprise, this is exactly what the the AMD GCN architecture does as well.\n\n# Practical Considerations\n\n\u2022 First of all: Make sure that your near plane is as far away from the camera as you can afford it. This will flatten the hyperbolic $1\/z$ curve and provide much better depth precision far away from the viewer.\n\u2022 Unless you are in some crazy setting with hundreds of kilometers view distance and you are going for sub centimeter depth resolution, DXGI_FORMAT_D32_FLOAT should be good enough and on modern GPUs should come at no additional cost compared to DXGI_FORMAT_D24_UNORM_S8_UINT.\n\u2022 DXGI_FORMAT_D16_UNORM isn\u2019t really a choice for projective transforms. It can be quite valuable for orthographic projections though (for example sun shadow maps), reducing bandwidth by half compared to a 32 bit format.\n\n# Linear Depth\n\nAnd if you really really need linear depth you can write it via the SV_DEPTH semantic in the pixel shader. Beware though, you\u2019ll loose the early Z unless you use the variant SV_DepthGreater, or SV_DepthLessEqual. Check out this blog post for more details. In most cases though I would argue that non linear depth is just fine.","date":"2018-12-19 03:49:17","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 39, \"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\": 0.5468435287475586, \"perplexity\": 663.17192110506}, \"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-2018-51\/segments\/1544376830479.82\/warc\/CC-MAIN-20181219025453-20181219051453-00614.warc.gz\"}"} | null | null |
package usrvtest
import (
"time"
"github.com/achilleasa/usrv"
)
type RecorderOption func(rec *ResponseRecorder)
// Bind a transport to the recorder.
func WithTransport(transport usrv.Transport) RecorderOption {
return func(rec *ResponseRecorder) {
rec.Transport = transport
}
}
// Bind a message to the recorder.
func WithMessage(msg *usrv.Message) RecorderOption {
return func(rec *ResponseRecorder) {
rec.Message = msg
}
}
// ResponseRecorder is an implementation of usrv.ResponseWriter that
// records its mutations for later inspection in tests.
type ResponseRecorder struct {
Message *usrv.Message
Transport usrv.Transport
Flushed bool
}
// Create a new recorder.
func NewRecorder(options ...RecorderOption) *ResponseRecorder {
rec := &ResponseRecorder{
Message: &usrv.Message{
Headers: make(usrv.Header),
Timestamp: time.Now(),
},
}
for _, opt := range options {
opt(rec)
}
return rec
}
// Get header map.
func (w *ResponseRecorder) Header() usrv.Header {
return w.Message.Headers
}
// Write an error.
func (w *ResponseRecorder) WriteError(err error) error {
if w.Flushed {
return usrv.ErrResponseSent
}
w.Message.Headers.Set("error", err.Error())
return nil
}
// Write data payload.
func (w *ResponseRecorder) Write(data []byte) (int, error) {
if w.Flushed {
return 0, usrv.ErrResponseSent
}
w.Message.Payload = data
return len(data), nil
}
// Flush the response.
func (w *ResponseRecorder) Close() error {
if w.Flushed {
return usrv.ErrResponseSent
}
w.Flushed = true
if w.Transport != nil {
w.Transport.Send(w.Message)
}
return nil
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,638 |
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