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Now that you know about (plain) tenses, you can learn how to use helping/auxiliary verbs with them. There are a few helping verbs in Sinhala. You place the helping verb after the normal present tense verb. The tense verb may be modified a little. Note that tense does not change in structure because in any tense you will find a "nava:" verb at the end (but the effect of tense is there in the meaning indeed).
You can use helping verbs with "be sentences" too. There is a common pattern here. When you use a helping verb with a "be sentence" having an adjective or a noun or a prepositional part, you should include "viya" or "venna/venda/vennata" just before the helping verb, and you must keep the adjective or noun as is (without any suffix). Optionally, when you use a helping verb with a "be sentence" having a prepositional part, you should include "inna/innata/inda" just before the helping verb. You will understand this more clearly with examples given below.
Let's learn helping verbs now one by one.
Here, the "Xnava:" part ("X represents any vowel sound) of a "nava:" verb is removed and "-iya" is appended. When that "nava:" verb's beginning syllable rhymes like "na" (that is, there is a vowel sound "a"), then that syllable is changed to "nae" too; if it rhymes like "no", then it is changed to "ne". After this you just put "yuthui".
You can put "o:na", but the verb is modified to infinitive form of the tense verb (that you learned before). Actually this form is better because it is easier to make, and some verbs can support this method only.
There are irregular forms too. Specially the verb "karanava:" must be remembered because you find thousands of verbs having it; so all such verbs ending with "karanava:" behave in the same manner. Again remember that the "o:na" helping verb with the infinitive is same everywhere in form; so it is common and easy.
Now let's see how to make sentences.
Eya: heta potha kiyaviya yuthui.
Eya: heta potha kiyavanna o:na.
Oya: adha bath uyanna o:na.
Oya adha bath iviya yuthui.
Eya: potha kiyavamin/kiyava kiyava sitiya yuthui.
Eya: potha kiyavamin/kiyava kiyava sitinna o:na.
In the perfect tenses, there is another popular method. Here, you put "thibuna:" at the end of the perfect tense verb, and the verb itself is changed to the infinitive. Optionally, you may append "-ta" to the doer.
Mama e:ka balala: thibiya yuthui.
Mama e:ka balala: thiyenna o:na.
Eya: liyumak liya: thibiya yuthui.
Eya: liyumak liya: thiyenna o:na.
Eya:/Eya:ta liyumak liyanna thibuna: .
Lamaya: sellam karamin/kara kara sitiya yuthui.
Lamaya: sellam karamin/kara kara sitinna/inna o:na.
Lamaya:/Lamaya:ta sellam karamin/kara kara inna/sitinna thibuna: .
Gaha kapamin/kapa kapa sitiya yuthui.
Gaha kapamin/kapa kapa inna/sitinna o:na.
Mama lassanai. -> mama lassana viya yuthui.
-> mama lassana venna o:na.
Eya: kreedakayek. -> eya: kreedakayek viya yuthui.
-> eya: kreedakayek venna o:na.
Api bus eke: . -> api bus eke: viya yuthui.
-> api bus eke: venna o:na.
You can easily form the other variants of this sentence pattern. To form the positive question, append "-dha" at the end of the helping verb ("yuthuidha" is often simplified to "yuthudha"). To make the negative statement, put "naehae" after the helping verb ("yuthui" becomes "yuthu" here). To make the negative question, put "naedhdha" after the helping verb. In fact, you already knew those methods. Right?
Eya: potha kiyaviya yuthu naehae.
Eya: potha kiyavanna o:na naehae.
Eya: potha kiyaviya yuthu naedhdha?
Eya: potha kiyavanna o:na naedhdha?
Eya: potha kiyaviya yuthui ne:dha?
Eya: potha kiyavanna o:na ne:dha?
Eya: potha kiyiya yuthu naehae ne:dha?
Eya: potha kiyavanna o:na naehae ne:dha?
Gaha kapamin/kapa kapa thibiya yuthudha?
Gaha kapamin/kapa kapa thiyenna o:nadha?
Gaha kapala: thibiya yuthu naehae.
Gaha kapala: thiyenna o:na naehae.
Here too you put "puluvan" after the tense verb. The tense verb takes the infinitive form. Moreover, you should append "-ta" to the doer.
To make the negative statement, just replace "puluvan" with "baehae". To make the positive question, append "-dha" to "puluvan". And to make the negative question, append "-dha" to "baehae" (and often "baehaedha" is simplified to "baerida").
You do the same changes to the tense verb as you did in "puluvan" (can).
You can make the future time sentence of "puluvan" (can/be able to) with "puluvan ve:vi" (will be able to). As you can see "ve:vi" is the future time form of "venava:". | {
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Q: Unhandled Exception System.Data.SqlClient.SqlException I am new to programming and I am working on a project using SQL and VB in Visual Studio 2013. One of the errors that has me stumped is as follows:
Imports System.Data
Imports System.Data.SqlClient
This error occurs at the following line:
da.Fill(ds)
Under additional information the following message is included:
Additional information: Invalid object name 'Complaints'.
I do have a Form named "Complaints" What am I missing? Thanks for any help you can provide! Please find the code below:
Public Class DButil
Public cs As String
Public Function GetDataView(ByVal sql As String) As DataView
Dim ds As New DataSet
Dim da As New SqlDataAdapter(sql, cs)
da.Fill(ds)
Dim dv As New DataView(ds.Tables(0))
Return dv
End Function
Public Sub New()
Dim strPath As String = Replace(System.AppDomain.CurrentDomain.BaseDirectory, "bin\debug", "cms.mdf")
cs = "Data Source=(LocalDB)\v11.0;"
cs += "AttachDbFilename='C:\Users\Sean\Documents\Visual Studio 2013\Projects\349591\349591\cms.mdf';Integrated Security=True"
cs += "Integrated Security=True;Connect Timeout=30"
End Sub
End Class
A: From your comment
Private Sub LoadGrid()
Dim sql As String sql = "SELECT Table.ComplaintID, Complaint.OpenDate, Status.StatusTitle, "
sql += "Table.Location, Table.Description FROM Table "
sql += " INNER JOIN Status ON Table.StatusID = Status.StatusID ORDER BY Table.OpenDate DESC"
grdComplaints.DataSource = DB.GetDataView(sql)
End Sub
You use a field named OpenDate from a table named Complaint, but this table is not listed in the JOIN clause of the sql statement. You need to specify this table in the JOIN clause (or remove the column from the SELECT columns list )
Given the actual field names probably you need to set the relationship between Table.OpenDate and Complaint.OpenDate
Dim sql As String sql = "SELECT Table.ComplaintID, Complaint.OpenDate, Status.StatusTitle, "
sql += "Table.Location, Table.Description FROM Table "
sql += "INNER JOIN Status ON Table.StatusID = Status.StatusID "
sql += "INNER JOIN Complaint ON Table.OpenDate = Complaint.OpenDate "
sql += "ORDER BY Table.OpenDate DESC"
| {
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Lincoln é uma cidade no Distrito de Selwyn, na Canterbury Ilha Sul da Nova Zelândia. A cidade é localizada nas Planícies de Canterbury a oeste da Península de Banks. Fica situada a 22 km a sudeste de Christchurch. É a segunda maior cidade do distrito, apenas ultrapassada em número de habitantes pela sua vizinha Rolleston.
Educação, instituições de pesquisa e amenidades
Lincoln tem duas escolas, sendo uma primária e outra secundária.
Lincoln Primary School é uma escola primária que foi fundada em 1866.
Lincoln High School é uma escola secundária e foi fundada em 1959.
Lincoln é a sede da Lincoln University. Assim com a universidade, existem vários outros centros de pesquisa em Lincoln, incluindo AgResearch, Institute for Plant and Food Research, FAR (Foundation for Arable Research), e Landcare Research.
Lincoln também tem o primeiro supermercado da Nova Zelândia a ter geração de energia eólica, gerando parte de sua energia: o Lincoln New World.
Ligações externas
Christchurch-Little River Railtrail
Cidades da Nova Zelândia | {
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ArsenalNews.org
Napoli 2 – Arsenal 0 : Not enough for Napoli
Napoli is the first team that fail to reach the knockout stage in the UEFA Champions League after gaining 12 points!
Arsenal did the bare minimum. They need to forget that match ever happened and prepare for a tough match in The Etihad this weekend.
Scored for Napoli: Higuain 74′ min, Callejon 92′ min.
RED CARD! Arteta
76′ min – Arteta gets his second yellow card.
Napoli vs. Arsenal Match Preview – Qualification up for Grabs
The Gunners, playing their best football in many years, need only a draw tonight when they play against Napoli in order to progress to the knock-out stages of the Champions League.
Can Olivier Giroud score tonight against Napoli and help Arsenal qualify in first place?
Some interesting facts on both teams:
Napoli must win by a 3 goal margin to secure a place in the top sixteen.
Napoli must do better than Dortmund against Marseille to reach the knockout stage of the Champions League for the second time in the history of the Club. If Napoli win by a one or two goal margin and Borussia Dortmund beat Marseille, Napoli will be the first side to finish at the 3rd place in a CL group picking up 12 points.
The Partenopei haven't drawn a game over their last 18 fixtures in European competition.
Arsenal are on course to reach the knockout stage for the 11th season in a row in the Champions League.
Arsene Wenger's side were defeated 4-0 in their last trip to Italy (vs AC Milan in February 2012).
Arsenal have won five of their last six Champions League matches between this and last season.
Mesut Özil has only scored once in an away game in the Champions League back in April 2012.
Gonzalo Higuaín has never scored against an English team in the Champions League.
Arsenal & Everton Line-up: Arsenal bench Flamini as Giroud, Ozil and Cazorla lead attack
ARSENAL have released the line up to face Everton today in a match which could see the Gunners pull away from the pack with seven points.
Arsene Wenger decided to bench Flamini and have Giroud, Ozil and Cazorla and lead the attack. Everton have no changes in line-ups.
Arsenal line-up against Everton: Szczesny, Jenkinson, Koscielny, Mertesacker, Gibbs, Arteta, Ramsey, Wilshere , Giroud, Ozil, Cazorla.
Arsenal v Everton Match Preview – Sunday Dec 8, 2013
The Gunners may want to make changes after maintaining their four-point lead at the top in the Premier League on Wednesday against Hull.
Wenger will be tempted to recall the likes of Jack Wilshere, Mikel Arteta and Olivier Giroud while Theo Walcott could start for the first time since September.
Stay tuned for more as the game starts tomorrow at the Emirates Stadium, kick-off 16.00 (GMT).
Mourinho: Arsenal are Better and Have Made Less Mistakes
According to Chelsea manager Jose Mourinho, Arsenal made less mistakes so far and deserve to lead the premier league.
"Arsenal Are Better", he said, and we want to hear the same words again in May.
Arsenal 2 Hull City 0
Beating Hull City at home is part of a routine – a trend of victories.
Most of the game resembled more to a training exercise. Aaron Ramsey and Mesut Özil were superb and both Arsenal's goals were of the highest class. In such a day even Nicklas Bendtner scored.
Wenger wants Dzeko in January
Wenger intends to purchase as early as January another striker. The name of Dzeko is a strong candidate together with Karim Benzema, another player in Arsenal wish list for January transfer window.
The Bosnian striker, Edin Dzeko, whose contract until June 2015 , is not happy with his present condition in Manchester City and looking for a new team.
Nigeria starlet Kelechi Iheanacho is Under Arsenal's Radar
Kelechi Iheanacho led Nigeria to the title in Under-17 World Cup. According to Iheanacho's agent, Henry Galeano, the 17 year old talent is being suited by several teams including the Gunners.
Will Kelechi join Arsenal?
Speciall Betting Offer
Claim your free bet today with William Hill, one of the leading bookmakers online.
Bet £10 on Arsenal & get £30 in free bets. | {
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package com.cloud.agent.api;
public class DeleteBcfSegmentCommand extends BcfCommand {
private final String _tenantUuid;
private final String _networkUuid;
public DeleteBcfSegmentCommand(final String tenantUuid, final String networkUuid) {
this._tenantUuid = tenantUuid;
this._networkUuid = networkUuid;
}
public String getTenantUuid() {
return _tenantUuid;
}
public String getNetworkUuid() {
return _networkUuid;
}
}
| {
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South African Labour Party var ett sydafrikanskt reformistiskt socialistiskt parti bildat i maj 1910 i och med Sydafrikanska unionens grundande, som representerade den vita arbetarklassens intressen. Partiet var representerat i Sydafrikas underhus 1910-1958 med 21 parlamentsledamoter som bästa resultat (1921 års val) och regerade 1924-1948 med Nationalistpartiet och United Party som koalitionspartners. Mot slutet av 1940-talet förlorade partiet väljare och parlamentsledamöter som föredrog en mer konservativ hållning i rasfrågan och förlorade sina sista mandat 1958, samma år som den siste partiordföranden avgick.
1961 ställde det Labourättade Conservative Workers' Party, med en tydligt apartheidvänlig linje upp i underhusvalet i syfte att utmana Nationalistpartiet men fick endast några tusen röster.
Politiska partier i Sydafrika
Politiska partier bildade 1910 | {
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I didn't expect to be as moved as I was by the Eiffel Tower. But quickly it became a sight that I wanted to see over and over.
Constructed in 1889 as the entrance arch to the World's Fair that year, the Eiffel Tower is still the tallest structure in Paris at 1, 063 feet. The same height as an 81- storey building. It is the most visited paid monument in the world. I didn't pay anything to admire its beauty but there are three levels for visitors. Levels are accessed by stairs and elevator both at different prices. Both the first and second levels feature restaurants.
Being near the tower made me feel so hip and modern. Even though the structure is over 100 years old it is the symbol of Paris and its modernity and fun. There are areas all around with the tower in view to sit and relax and even enjoy some street entertainment.
On our last night Cynthia and I got some drinks-yes adult drinks-and had them on the lawn in front of the tower. I could think of no better way to say goodbye to Paris.
At night the tower lights up beautifully. And every 20 minutes or so it sparkles. People sit around ready to see the lights like they are waiting for a fireworks display. Once the first sparkle goes everyone ohhhs and ahhhs and it's such a beautiful sight. | {
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Ipomoea descolei är en vindeväxtart som beskrevs av O'donell. Ipomoea descolei ingår i släktet praktvindor, och familjen vindeväxter. Inga underarter finns listade i Catalogue of Life.
Källor
Praktvindor
descolei | {
"redpajama_set_name": "RedPajamaWikipedia"
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{"url":"http:\/\/www.ams.org\/mathscinet-getitem?mr=2143000","text":"MathSciNet bibliographic data MR2143000 (2006d:70019) 70F05 (70G65) Cari\u00f1ena, Jos\u00e9 F.; Ra\u00f1ada, Manuel F.; Santander, Mariano Central potentials on spaces of constant curvature: the Kepler problem on the two-dimensional sphere \\$S\\sp 2\\$$S\\sp 2$ and the hyperbolic plane \\$H\\sp 2\\$$H\\sp 2$. J. Math. Phys. 46 (2005), no. 5, 052702, 25 pp. Article\n\nFor users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.","date":"2016-02-08 07:06:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 2, \"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\": 0.9896767139434814, \"perplexity\": 7666.647612000693}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-07\/segments\/1454701152959.66\/warc\/CC-MAIN-20160205193912-00024-ip-10-236-182-209.ec2.internal.warc.gz\"}"} | null | null |
\section{Introduction} \label{0}
Coronal mass ejections (CMEs) are spectacular eruptions of plasma often accompanied with rapid release of magnetic energy from solar atmosphere. When CMEs propagate away from the Sun to the interplanetary space, they are often called interplanetary CMEs (ICMEs) which exhibit a distinct set of observational signatures from in situ measurements. They are recognized as drivers of major space weather events which could severely impact human activities in modern society. Erupted flares have a close relationship with CMEs and strong flares are often observed along with CMEs \citep{2011Chen}. In the past few decades, a lot of efforts have been made on the development of the flare-CME model in order to explain the underlying physical mechanism(s). Among them, the standard two-dimensional (2D) flare and CME model, so-called CSHKP model \citep[developed by][]{1964Carmichael, 1966Sturrock, 1974Hirayama, 1976KoppPneuman}, has successfully explained the morphological evolution of eruptive two-ribbon flares. One of the essential components in the model is a CME magnetic flux rope (MFR). Upon its ejection, field lines below the erupting rope reconnect and form an arcade of flare loops observed in the X-ray and extreme ultraviolet (EUV) wavelengths, and two ribbons observed in optical and ultraviolet (UV) wavelengths demarcating the feet of the arcade. In the 2D model, the same amount of magnetic flux encompassed by the flare loops is also turned into the poloidal flux of the erupting MFR, and this amount of flux can be measured from flare ribbons sweeping through photospheric magnetic field \citep{1984Forbes, 2004Qiu, 2007Qiu}. Since MFRs are generally believed to be the core magnetic structure of CMEs and ICMEs~\citep[e.g.,][]{2006Gibson, 2014Hu, 2017Gopalswamy, 2018Gopalswamy, 2020Liurui}, studies of MFR properties related to their formation and evolution remain an important topic to be explored. We note that hereafter we use ``MFR" for a more generic reference to a magnetic flux rope at different stages of evolution, and ``CME-MFR" for the specific reference of a magnetic flux rope at the final stage of a CME eruption (i.e., when the MFR has well formed after the flare reconnection).
Nowadays, the study of evolution and propagation of CMEs/ICMEs between Sun and Earth is greatly advanced with the help of multiple measurements from multi-spacecraft missions. The large-scale magnetic clouds (MCs) in ICMEs which are commonly detected in situ provide direct evidence for the existence of erupted CME-MFRs that come from the Sun~\citep{1992burlaga,2007Qiu,2014Hu}. In addition, there are also remote-sensing observations available throughout the interplanetary space. For example, the twin spacecraft, Solar TErrestrial RElations Observatory (STEREO, \citealt{2008Kaiser}), can trace CMEs from the high corona to the inner heliosphere via coronagraphic observations. The STEREO mission provides two more viewpoints towards the Sun, in addition to the viewpoint from Earth provided by the Solar and Heliospheric Observatory (SOHO), in the past decades. There are also various signatures for MFRs on the Sun from remote-sensing observations, including filaments, coronal cavities, sigmoids, and hot channels \citep{2017Cheng}. These solar phenomena can be unified into one framework as distinct manifestations of MFRs \citep{2020Liurui}. Most of the latest recognized observational features are attributed to the observations from the Solar Dynamics Observatory (SDO, \citealt{2012Pesnell}). Recent development of large ground-based solar telescopes also becomes an indispensable way to reveal the fine-scale structures and dynamics of MFR formation in the low corona \citep[e.g.,][]{2015Wang}.
Compared to various studies based on in situ measurements of MFR structures after the eruption, the origination of CME-MFRs before and during eruptions still remains elusive due to the complex environment in the solar source region and limited observations. At the present time, there are certain hypotheses on the formation process of MFRs. Some studies indicate that MFRs could exist prior to the eruption. For example, both \citet{2001Fan} and \citet{2004Magara} reported magnetohydrodynamic (MHD) simulation results that a twisted MFR initially formed below the photosphere can partially emerge into the low corona by magnetic buoyancy. While some studies suggest that the presence of pre-eruptive MFRs is not necessary and MFRs could be built up in the corona via magnetic reconnection processes associated with flares \citep{2003Amari, 2001Moore, 1999Antiochos, 2021Jiang2, 2021Jiang}. To understand the physical processes more precisely for the flare-CME events, extensions of the standard 2D flare model have been proposed to account for much broader ranges of quantitative measurements with three-dimensional (3D) features intrinsic to realistic solar eruptions \citep{2007Longcope, 2012Aulanier, 2017Priest, 2019Aulanier}. For example, quasi-3D models have been developed with a non-vanishing magnetic field component along the axis of the MFR and to illustrate a scenario that sequential reconnection along magnetic polarity inversion line (PIL) forms the MFR in the first place \citep{1989vanBallegooijen, 2007Longcope,2015Schmieder}. This scenario has been widely applied to infer and interpret the magnetic reconnection properties based on the observed flare ribbon morphology \citep{2002Qiu, 2004Qiu, 2010Qiu, 2014Hu, 2017Kazachenko, 2020Zhu}. From such analyses, \citet{2004Qiu} illustrated that there is a temporal correlation between the magnetic reconnection rate and the acceleration of the CME (considered as the eruptive MFR) in the low corona. Such a correlation has been further established by \citet{2020Zhu} based on a statistical study of $\sim$ 60 events. In addition, \citet{2007Qiu} and \citet{2014Hu} showed the correlation between the magnetic reconnection flux and the flux contents of the corresponding ICME/MC flux ropes based on modeling results employing in situ spacecraft measurements. These results support the hypothesis, especially concerning CME-MFRs, that the CME-MFR can be formed by magnetic reconnection during the corresponding flare process. And recent simulation results also indicate clearly that the reconnection flux contributes to the axial (toroidal) flux of the CME-MFR in the early stage \citep{2021Jiang2, 2018Satoshi}.
For the quantitative MFR identification in the solar source region, numerical models can be applied to find MFRs in addition to observations. For the topological analysis of a solar MFR, the 3D magnetic field configuration is commonly obtained through coronal magnetic field extrapolation based on photospheric magnetograms. A number of high resolution extrapolation results employing different numerical schemes has been compared to observations to study the properties of MFRs in the magnetically dominant environment on the Sun \citep{2008Schrijver, 2008Thalmann, 2009Wheatland, 2009DeRosa, 2012Wiegelmannb, 2012Sun, 2014Jiang2, 2016Guob}. Among many extrapolation studies, the force-free approximation is commonly adopted for the case of low plasma $\beta$ (the ratio between the plasma pressure and magnetic pressure) over certain heights in the solar atmosphere above the photosphere \citep{2001Gary}. Under this assumption, the non-magnetic forces including the inertial force can be ignored for a static and time stationary equilibrium. Therefore, the Lorentz force should be self-balanced, and it should satisfy the equation, $\mathbf{J} \times \mathbf{B} = 0$, which means that the electric current density \textbf{J} is parallel to the magnetic field \textbf{B}, with $\mathbf{J} = \alpha \mathbf{B}$ ($\alpha$ is the so-called force-free parameter). The simplest case is the potential field when $\alpha \equiv 0$. If $\alpha$ is not zero, there are two situations depending on whether $\alpha$ is constant or varying in space. One is the linear force-free field (LFFF) for $\alpha \equiv const$ and the other is the nonlinear force-free field (NLFFF). Since our interest lies in the magnetic structure in the ARs on a local scale with high spatial resolution, the most common and practical way to reconstruct the coronal magnetic field is the NLFFF extrapolation method.
There are a variety of numerical methods proposed to reconstruct the NLFFF for an AR from boundary conditions and sometimes pseudo-initial conditions, including the upward integration, Grad-Rubin iteration, MHD relaxation, optimization approach, and so on \citep[see a review by][]{2021Wiegelmann}. The computation speed and accuracy of different numerical methods can vary significantly given the differences in algorithms and their specific realizations in many aspects. We apply a kind of MHD-relaxation method with a conservation-element/solution-element (CESE) solver \citep{2011Jiang}. The so-called CESE-MHD-NLFFF code \citep{2013Jiang} has been tested by different benchmark cases \citep{1990Low, 2007vanBallegooijen, 2008Metcalf}. And it has also been widely applied to the extrapolation of realistic solar magnetic field data \citep{2013Jiang, 2014Jiang2, 2017Duan, 2019Duan}.
The understanding towards the formation and evolution process of the CME-MFRs will ultimately help us make a definitive and physical connection between the origin of solar MFRs (including the MFR before and after the eruption) and their interplanetary counterparts. Such a connection can be pursued through a quantitative study of MFR's physical characteristics (e.g., magnetic flux, field-line twist, and electric current). Specifically, one critical step is the detailed analysis of available solar observations in order to determine whether an MFR exists or how one can form prior to and during the eruption. Characterization of MFR properties not only plays a major role in understanding the physical mechanisms underlying solar eruption and the subsequent evolution, but also contributes to the improvement of the forecast ability in space weather.
In this paper, we carry out the coronal magnetic field extrapolation based on the method developed by \citet{2013Jiang} for two events to obtain 3D magnetic field topology of the AR prior to eruption. In addition to the magnetic field extrapolation, we estimate the possible locations of the MFR's footpoints prior to eruption and measure a number of magnetic field parameters (including the magnetic reconnection flux derived from flare ribbons) in the corresponding AR during the eruption process from different observations. Then the results from extrapolation and observation are combined to identify whether there is a coherent pre-existing MFR before the eruption and to interpret how such a structure may evolve during and following the corresponding flare and CME eruption process. The magnetic properties of the CME-MFR will be further analyzed and compared with the in situ ICME/MC modeling results which are obtained separately.
This article is organized as follows. The two selected events and extrapolation method are introduced in Section \ref{1}. Then we describe the identification methods and analyze results associated with the MFRs on the Sun in Section \ref{2}. The magnetic properties of MFRs are estimated quantitatively and presented in Section \ref{3} based on results from both the solar source region and in situ modeling. The conclusion is given in Section \ref{4}.
\section{Events Selection and Extrapolation Method}\label{1}
\subsection{Events Overview}
For the purpose of performing a quantitative study of the CME-MFR, we search for appropriate event candidates from a list of reconstructed MFRs based on photospheric magnetograms before the eruption by \citet{2019Duan}. They extrapolated the 3D magnetic field in the active regions for 45 major flare eruption events employing the NLFFF extrapolation code by \citet{2013Jiang}. With a set of criteria similar to those by \citet{2018Jing}, all major flares that are above GOES-class M5 and occurred within $45\degr$ of the solar disk center from 2011 January to 2017 December are selected. Moreover, we also examine the associations between flare and CME, and between CME and ICME to ensure that there exists a well-established one-to-one connection among a flare, an associated CME and an ICME based on the work by \citet{2020Zhu}. Two events are selected for this study as shown in Table \ref{twoevent}. Both ARs related to the two events located near the disk center. The CMEs associated with the corresponding flares in these two events have been both observed by SOHO and STEREO spacecraft close to the peak times of the corresponding flares \citep{2014Vemareddy,2015Cheng,2016Cheng,2017Joshi}. And the associated ICME/MC events were also observed by the WIND and ACE spacecraft at 1 au, which have been reported by \citet{2021Hu} and \citet{2021Kilpua}, respectively.
In event 1, an M6.5 flare was produced at $\sim$ 6:55 UT on 2013 April 11 (N07E13). Then a halo CME appeared in the field of view of SOHO/LASCO after 07:24 UT, and the same CME was also observed simultaneously by both STEREO A and B spacecraft after $\sim$ 07:39 UT. The corresponding ICME/MC passing Earth was detected about three days later \citep{2021Hu}. Similar examination is conducted for the second event, which started with an X1.6 flare peaking at $\sim$ 17:45 UT on 2014 September 10. There was also a halo CME following the flare based on the observations from SOHO/LASCO and the coronagraph of STEREO B (data from STEREO A was unavailable during event 2). After two days, the WIND spacecraft encountered the subsequent ICME/MC structure at Earth \citep{2021Kilpua}. Therefore, the connections of CME-MFRs from the Sun to Earth are well established for these two events. We will mainly present the quantitative study of the CME-MFR before the eruption hereafter.
\subsection{CESE-NLFFF-MHD Extrapolation Method}
The CESE-MHD-NLFFF code solves the MHD momentum equation and the magnetic induction equation iteratively until a stationary magnetic field solution is reached, similar to a magnetofrictional approach. As a special case of the MHD relaxation method, the magnetofrictional method includes an artificial dissipative term $\textbf{D(v)}$ to balance the momentum equation with flow velocity \textbf{v}. Specifically, in the CESE-MHD-NLFFF code, $\textbf{D(v)}$ is written in a frictional form $\nu \rho \textbf{v}$ \citep[see below, and][]{2012Jiang} along with some modifications in order to utilize the existing CESE-MHD solver. The modified momentum equation and the induction equation are written as \citep{2012Jiang, 2013Jiang},
\begin{equation}\label{eq1}
\frac{\partial (\rho \textbf{v})}{\partial t} =(\nabla \times \textbf{B})\times \textbf{B} -\nu \rho \textbf{v}, \rho =|\textbf{B}|^{2} +\rho_{0}.
\end{equation}
\begin{equation}\label{eq2}
\frac{\partial \textbf{B}}{\partial t} = \nabla \times(\textbf{v} \times \textbf{B})
-\textbf{v} \nabla \cdot \textbf{B} + \nabla(\mu \nabla \cdot \textbf{B}).
\end{equation}
Here the (pseudo) mass density $\rho$ also contains the constant $\mu_{0}$ for simplicity and is assumed to be largely proportional to $|\textbf{B}|^{2}$ in order to roughly equalize the speed of the evolution of the entire field with nearly uniform Alfv\'en speed ($v_{A} =\frac{|\textbf{B}|}{\sqrt{\rho}} \approx const$). To enhance the ability of handling noisy data in realistic solar magnetograms, a small value $\rho_{0}$ is added, e.g., $\rho_{0}$ = 0.1 (in the same unit as $|\textbf{B}|^{2}$), to the original pseudo density $\rho$. Two extra terms are added to control the divergence of the magnetic field in the induction equation. More details can be found in \citet{2012Jiang, 2013Jiang}.
\subsection{Data Preprocessing and Grid Initialization}
Passing across the inhomogeneous plasma environment, the plasma $\beta$ could vary from $\beta >1$ in the photosphere to $\beta \ll 1$ in the low and middle corona, and to $\beta >1 $ again in the upper corona \citep*{1989Gary, 2001Gary}. So the force-free condition may not be always satisfied especially at the photosphere \citep{1995Metcalf}. One way to get a more consistent boundary condition for NLFFF extrapolation is to modify the raw photospheric magnetogram to mimic a force-free chromospheric magnetogram, by the so called preprocessing, which was first proposed by \citet{2006Wiegelmann}. We use the preprocessing code developed by \citet{2014Jiang} which is consistent with the CESE-MHD-NLFFF extrapolation code by adopting an optimal magnetic field splitting form. Such a procedure is designed to improve the quality of the raw magnetogram to make it closer to the force-free condition and smooth the raw data to help reduce the measurement uncertainties and numerical errors from the computation.
The high-resolution vector magnetograms are routinely observed by SDO/Helioseismic and Magnetic Imager (HMI; \citealt{2012Schou}). Specifically, the Space-weather HMI Active Region Patches (SHARPs, \citealt{2014Bobra}) vector magnetogram data product, \verb|hmi.sharp_cea_720s|, provided by SDO/HMI, is used as the input for the extrapolation. The SHARP data series provide maps following each patch of significant solar magnetic field for its entire lifetime and the data is also de-rotated to the disk center and remapped using the cylindrical equal area (CEA) Cartesian coordinates. Photospheric vector magnetograms are included with a cadence of 720 s and a spatial resolution of $0.5''$ ($\sim$ 0.36 Mm). For the two selected events, vector magnetograms from the ARs at least 10 minutes before the flare onset times (estimated from the soft X-ray measurement of the GOES satellite) are preprocessed to get necessary boundary conditions and derive the initial conditions from a potential field solver for the NLFFF extrapolations. The original magnetograms are rebinned from 0.5$''$ per pixel to 1$''$ per pixel for the preprocessing procedure and the subsequent extrapolations. Figures \ref{preprocess1} and \ref{preprocess2} show the overall smoothing effect as a result of preprocessing for the two events by comparing the raw and preprocessed maps of magnetograms and current density $J_{z}$ distributions. Random noise is obviously suppressed in the $J_{z}$ maps.
For the consideration of the speed and accuracy of the computation in terms of high-resolution and large field-of-view solar magnetograms, a non-uniform grid structure within a block-structured parallel computation framework is adopted with the help of PARAMESH software package \citep{2000MacNeice} for the CESE-MHD-NLFFF code. For the grid initialization, the whole computation domain includes the pre-set main computation region and the surrounding buffer zones to reduce the influence of the side boundaries \citep{2013Jiang} since the magnetic field at these numerical boundaries are simply fixed as their initial values (i.e., those of the potential field). Then the whole computational domain is divided into blocks with different spatial resolutions and all blocks have identical logical structures which are evenly distributed among processors. As we vary the grid size only in height (the $z$ dimension) for this study, the grid resolution matches the resolution of the magnetogram at the bottom boundary and decreases by four times at the top of the computational domain. After the grid initialization, the initial condition of the whole computation domain is assigned by a potential field solution derived from the input magnetogram by using the Green's function method \citep{1977Chiu, 2008Metcalf}.
\subsection{Convergence Study}
Before we start our analysis for the two events, we also examine the relaxation process by several metrics to obtain converged extrapolation results. These include the residual of the field between two successive iteration steps, the force-freeness of the numerical result \textit{CWsin}, the divergence-free condition $\langle|f_{i}|\rangle$ \citep{2000Wheatland, 2008Metcalf} and the total magnetic energy $E_{tot}$ (see their definitions in the Appendix).
For event 1, we carry out the extrapolation based on the whole SHARP vector magnetogram ($540'' \times 344''$) and then calculate the convergence metrics for every 200 iteration steps as shown in the first column of Figure \ref{converge_all} with the finest grid size $1''$. As shown in Figure \ref{converge_all}, the residual goes through a gradual increase before $\sim$ 6500 iteration steps because the bottom boundary condition drives the system away from the initial potential field \citep{2012Jiang}. Even though obvious fluctuations appear after the initial driving process, the overall trend of the residual toward the end is decreasing, accompanied by small oscillations. After $\sim$ 30,000 iteration steps, the residual is reduced to $\sim 10^{-5}$ and still maintains a declining trend. Other metrics also display a trend with little variation and both \textit{CWsin} and $\langle|f_{i}|\rangle$ reach relatively small values. Thus the extrapolation result can be considered as a converged solution. It is noticed that there are some oscillations in the convergence process, which may be caused by the broad distribution of the weak field and random noise from the input magnetogram. The total computation time (to converge until 40,000 iteration steps) took about 95 hours with 19 cores on a 24-core local desktop with 48 GB memory.
For event 2, the size of the SHARP magnetogram is $282''\times 266''$. One run is carried out with the smallest grid size $1''$ and the full size magnetogram. The second column of Figure \ref{converge_all} shows a smooth convergence process. The residual converges very fast after an initial rise exceeding $10^{-4}$ to an order of magnitude smaller, $<$ $10^{-5}$, within 11,000 iterations. All the other metrics show clear monotonic decreases and stabilize after $\sim$ 11,000 iterations, which is consistent with an optimal convergence pattern in the previous tests of this code \citep{2012Jiang, 2012Jiangb, 2013Jiang}. This convergence process is relatively smooth without any spurious oscillations, so a final solution with good indication of convergence is readily obtained for subsequent analysis. It took about 23 hours with 19 cores on the same local desktop for the extrapolation result to converge (after 20,000 iteration steps).
\section{Characterization of MFRs on the Sun}\label{2}
\subsection{MFR Identification Method}
Both extrapolation and observation results are critical for the MFR identification on the Sun. As for the observational analysis of MFRs, we analyze the data from Atmospheric Imaging Assembly (AIA, \citealt{2012Lemen}) and HMI on board the SDO spacecraft to study the evolution of the corresponding flares for the two events. SDO/AIA provides full-disk images in 7 extreme ultraviolet (EUV) and 2 ultraviolet (UV) wavelength channels with a high spatial resolution ($0.6''$ per pixel and a total of 4096 $\times$ 4096 pixels per image) and a moderate time cadence (12 s in EUV channels and 24 s in UV channels).
To provide additional support for the MFR identification and characterization of the corresponding CME-MFRs at a different stage besides the extrapolation, we also analyze the evolution of flare ribbons and the corresponding magnetic reconnection properties. Flare ribbons map the footpoints of reconnected field lines. Magnetic reconnection beneath the erupting MFR forms flare loops, and the same amount of reconnected magnetic flux is injected into the MFR. Reconnection may also take place between the erupting MFR and the ambient magnetic field, although this is not the main focus of this study. Therefore, magnetic reconnection flux associated with flare ribbons is useful to establish a quantitative connection between MFRs on the Sun (both before and after the flare eruption) and their interplanetary counterparts. The amount of accumulative magnetic reconnection flux can be measured by summing up the magnetic flux in newly brightened UV pixels within flare ribbons. In this study, we employ 1600 \r{A} data from SDO/AIA and vector magnetograms from SDO/HMI to measure the magnetic reconnection flux and magnetic reconnection rate from the brightening pixels, following the automated approach developed by \citet{2002Qiu,2004Qiu}. The brightening pixels are chosen when the intensity of a pixel is greater than $\sim$ 6 times the median intensity which is fixed and determined from the average of a 6-minute time period (for the region of interest before the eruption).
While an MFR is generally considered as a group of coherent winding field lines, it has not been quantitatively defined in a universal way. Identifying a coherent MFR based on the reconstructed coronal magnetic field derived from the real magnetogram can be difficult, given the complex magnetic topology. \citet{2016LiuR} suggested that the magnetic twist number $T_{w}$ can serve as a good proxy for finding the axis of an MFR. The twist number $T_{w}$ measures how many turns two infinitesimally close field lines wind about each other (see \citealt{2006Berger}), and is defined by
\begin{equation}
\begin{split}
T_{w} &= \int_{L} \frac{\mu_{0} J_{||}}{4\pi B} \,dl = \frac{1}{4\pi}\int_{L} \frac{(\nabla \times \textbf{B})\cdot \textbf{B} }{B^{2}} \,dl, \\
&=\frac{1}{4\pi}\int_{L} \alpha \,dl, \mbox{ if } \nabla \times \textbf{B} = \alpha \textbf{B}.
\end{split}
\end{equation}
Here $\alpha$ is the force-free parameter and the integral is taken along one magnetic field line with path length $L$, starting from one end point of the field line on the boundary to the other. For both events, extrapolation results are generated utilizing magnetograms that are chosen at least 10 minutes before the flare onset times. We calculate the twist number $T_{w}$ at each grid point in the whole volume with the same grid size as the resolution of the input magnetogram, i.e., 1$''$. Then we start the topology analysis with the definition of \citet{2016LiuR} that an MFR has a bundle of field lines spiraling around the same axis or each other by more than one turn ($|T_{w}| \geqslant 1$, see also \citealt{2019Duan}). Combined with the field-line topology, one may also require that such constrained MFR volume be a single tube without multiple bifurcations. In addition, the footpoints of MFRs should be restricted within or close to main flare ribbon areas identified from AIA observations, given the general relation between the magnetic reconnection process during flares and the formation of erupting CME-MFRs \citep{2001Moore, 2004Qiu, 2009Qiu,2020Zhu}.
\subsection{Results for AR 11719}
For event 1 in AR 11719, simultaneous observations of the flare's time evolution in SDO/AIA 94, 131 and 1600 \r{A} wavelength channels before and during the flare eruption are given in Figure \ref{f0}. From the EUV observations in 94 \r{A} and 131 \r{A}, some curved structures are present near the center before the flare eruption, which were recognized as hot channels in \citet{2016Cheng}. But such a sigmoid-like structure based on emission-line images does not necessarily yield a similarly continuous magnetic field-line configuration \citep{1999Titov, 2015Schmieder, 2016Cheng,2017Duan}, i.e., that of an MFR. Instead, these brightened features may correspond to groups of short sheared arcades which are discontinuous, based on the extrapolation result as to be demonstrated below. In the bottom panels of Figure \ref{f0} for the 1600 \r{A} UV observation, there is a typical flare morphology with two brightening ribbons lying nearly in parallel with each other, expanding and then drifting away from each other during the time evolution. The contours of flare ribbons coincide with the curved brightening structures in 131 \r{A} observation at the central region, especially towards the ``hooked'' ends, which gives us a rough estimation of possible positions for the MFR footpoints for this event.
The time evolution of the flare ribbons in the corresponding SDO/HMI magnetograms which are remapped to the sub areas in SDO/AIA's field of view is shown in Figure \ref{f2}, the left column. Besides, we add the X-ray flux measurement of the whole Sun provided by the GOES satellite for the wavelengths of soft X-ray (1 - 8 \r{A}) during the same time period in the right column, together with the concurrent measurements of accumulative magnetic reconnection flux and magnetic reconnection rate by the approach of \citet{2002Qiu}. This M6.5 flare eruption starts at $\sim$ 06:55 UT according to the rapid change of the soft X-ray flux curve, which is consistent with the onset of the magnetic reconnection flux increase shown in the second panel in the right column. Based on the average of the total unsigned magnetic flux in each enclosed ribbon area with one dominant polarity (either positive or negative, \citealt{2017Kazachenko}), the final accumulative magnetic reconnection flux reaches the magnitude of $17 \pm 2.8 \times 10^{20}$ Mx after the eruption. Given the association between the magnetic reconnection flux and the flux content of the corresponding ICME/MC flux ropes \citep[e.g.,][]{2007Qiu,2014Hu}, such flux measurement from flare ribbons can be helpful for making further connections of MFRs on the Sun and their in situ counterparts, as to be laid out in Section \ref{3}.
In Figure \ref{f11}a, some sample field lines are drawn over the corresponding AIA 94 \r{A} image to give a qualitative comparison between the extrapolation result and the observation. Several loop structures are recovered overlapping with selected field lines, and a set of twisted field lines lying around the PIL takes the shape resembling the middle of the inverse ``S'' sigmoid as seen in 94 \r{A} channel (see also Figure \ref{f11}c). Among the comparisons with the 1600 \r{A} observation in Figure \ref{f11}b, footpoints of the twisted field lines locating close to the flare ribbons are associated with grid points with negative $T_{w}$ values. On a plane near the bottom layer (at $z$ = 2$''$ above the photosphere), we pick all the points with $T_{w} \leqslant -1$ around the central sigmoidal structure, and plot field lines passing through this set of seed points. We eliminate open field lines which only have one end point attached to the bottom boundary thus not ``closed'', and also ill-defined $T_{w}$ values. As a result shown in Figure \ref{f11}b, four groups of field lines are distinguished starting with the selected seed points. Compared to the locations of the flare ribbons, three groups of field lines are excluded since a part of their footpoints extends out of the ribbon sites. Therefore the remaining bundle of field lines shown in Figure \ref{f11}c is identified to be the most likely candidate MFR for the 2013 April 11 event before the flare eruption. After determining the MFR, we can find the axial field line with the maximum $|T_{w}|$ of the MFR. The axis of the identified MFR in event 1 possesses $T_{w} = -1.5$ which lies close to the bottom boundary and reaches a maximum height at $z \sim$ 19$''$. The time sequence of flare ribbons after 6:42 UT is then co-aligned with the bottom boundary magnetogram and overplotted in the usual way, color coded by elapsed time in Figure \ref{f11}d. It shows that two groups of footpoints from the identified MFR locate on the opposite sides of the flare ribbons near the far ends, consistent with the scenario proposed by \citet{2001Moore}.
To further confirm the existence of the MFR, we also check different topological properties from a side view. In Figure \ref{f12}, the coherent MFR structure is still maintained with a different $|T_{w}|$ threshold as seen in Figure \ref{f12}a and \ref{f12}b. The distribution of the quantity $|\textbf{J}|/|\textbf{B}|$ as a proxy to current density is displayed in Figure \ref{f12}c and \ref{f12}d on a cross section plane nearly perpendicular to the MFR. The current density $|\mathbf{J}|$ itself also shows a similar distribution. Based on the current density distribution at the intersections between the identified MFR field lines and the vertical slice in Figure \ref{f12}c, the flux rope goes through a region with relatively high current density. The geometric boundary of an MFR can also be estimated by the location of a quasi-separatrix layer (QSL), a very thin layer where there is a strong gradient of the field line connectivity \citep{1996Demoulin}. Such a feature is typically defined mathematically by the high squashing factor Q \citep{2002Titov, 2021Vemareddy}. As shown in Figure \ref{f12}e and \ref{f12}f, the identified group of field lines with small Q values is surrounded indeed by a clear boundary with high squashing degree Q.
\subsection{Results for AR 12158}
Observations of the flare ribbon evolution before and during the flare eruption for event 2 in AR 12158 are shown in Figure \ref{f1}. Event 2 also exhibits a two-ribbon flare morphology, with two ribbon areas co-located near the two ends of an inverse ``S'' shape sigmoidal structure. The southward ribbon has a more dominant swept area in size than the other. Similarly, we use the overlapping regions between the flare ribbon areas and the curved brightening sigmoidal structures in 131 \r{A} observation to approximate the possible locations of MFR footpoints in this event.
In Figure \ref{f3}, we show the same set of panels for event 2, as in Figure \ref{f2}. We find that the initial enhancement of the X-ray flux is earlier than the significant increase of the magnetic reconnection flux. After a slow rise phase with small reconnection rate, a strong flare is produced quickly after $\sim$17:20 UT. The final accumulative magnetic reconnection flux reaches the magnitude of $47\pm 7.5 \times 10^{20}$ Mx for event 2. The flare ribbon morphology still exhibits general features for a ``two ribbon'' flare, albeit asymmetry of the spatial distributions is more pronounced, indicating perhaps more significant deviation from a ``standard'' 2D geometry.
For event 2, the configuration of magnetic field lines from the extrapolation result has a good visual correspondence with the AIA observations as shown in Figure \ref{f21}a and \ref{f21}b. There is a clear inverse ``S'' sigmoid structure near the center imaged by AIA 94 \r{A} passband. However, the core field in our extrapolation result mainly consists of several groups of sheared arcades structure over-arched by higher coronal loops, rather than one continuous inverse ``S'' structure (see also \citealt{2017Duan}). Based on the long-term evolution before the eruption, \citet{2015Cheng} found that there was a central sigmoid structure initially appearing in the AIA 94 \r{A} passband at $\sim$15:10 UT and then it had gone through repetitive disappearance and re-appearance processes. So they suggested that a nascent MFR was under formation prior to the major eruption by tether-cutting reconnection. After $\sim$16:55 UT, the sigmoid develops quickly and produces an X1.6 flare and a CME. In order to find a possible MFR structure prior to the flare, we take a look at the $T_{w}$ distribution and find that the majority of the core field region has a negative and relatively small twist number such that $|T_{w}| < 1$. This indicates the absence of a twisted coherent MFR according to the criterion we are using \citep{2016LiuR, 2019Duan}. In Figure \ref{f21}c and \ref{f21}d, we show the isosurfaces of $T_{w} = -1$ and $T_{w} = -0.8$ in the central volume. There is no coherent structure under the $T_{w} = -1$ criterion, though several coherent structures appear for a lower threshold in magnitude $T_{w} = -0.8$. Comparing these field line bundles with the locations of the flare ribbons, there are two coherent weakly twisted field line groups as presented in Figure \ref{f21}e. The left group has a maximum $|T_{w}| \sim 0.82$ but extends to a relatively far away location from the ribbon (also to a height of $z \sim$ 50$''$). It also appears to be nearly perpendicular to the local PIL. Another group of field lines lies close to the bottom boundary and has a maximum $|T_{w}| \sim 0.97$, which still, strictly speaking, fails to satisfy the MFR criterion. In addition, the current density distribution in Figure \ref{f21}f at the intersections of a vertical slice with two field line bundles also shows less clearly-defined concentrations in those field-line regions. So different from the NLFFF extrapolation result by \citet{2016Zhao} and the time-dependent magnetofrictional modeling result by \citet{2021Kilpua}, which demonstrated the existence of a strongly twisted MFR, we do not produce such a pre-existing MFR prior to the flare eruption for event 2.
\section{Estimation of magnetic properties of MFRs}\label{3}
After the identification of MFRs on the Sun for the two events, we look further into the magnetic properties of MFRs at different stages or locations and try to find a potential correlation among them. For example, the total magnetic flux (generally considered conserved) is one of the most important quantitative properties of MFRs that can be measured or derived to make a connection between CME-MFR and the corresponding ICME/MC \citep{2007Qiu, 2014Hu}.
Specifically, for the identified pre-existing MFR in event 1, we give below a quantitative description of its magnetic properties. The axial magnetic flux enclosed by the pre-existing MFR's footpoints is calculated for further analysis. Regions of footpoints are obtained by extracting the intersection points between field lines from the MFR and a slice parallel to the bottom boundary (photosphere). We choose a slice at a height of 1.0$''$ where two well-separated groups of footpoints are obtained. In general, more grid points are included under a smaller $|T_{w}|$ threshold value for the MFR criterion, i.e., more points with $|T_{w}|$ exceeding such a value. Here we denote the region dominated by the positive magnetic field in the MFR footpoints as `$FP_{+}$' and the region taken up mainly by the negative magnetic field in the MFR footpoints as `$FP_{-}$'. The total axial (or toroidal) magnetic flux for both regions is calculated based on $\Phi_{z} = \iint B_{z} dS$, where $B_{z}$ is the vertical magnetic field component. The integration is obtained by summing up the magnetic flux from all grid points (pixels) within the identified footpoints regions on the slice.
Table \ref{footpoints} shows the result of flux calculations of the identified MFR. The differences of $\Phi_{z}$ between $FP_{+}$ and $FP_{-}$ are within one order of magnitude for different $T_{w}$ criteria, though they get smaller for a larger $|T_{w}|$ threshold. Given that the largest $\Phi_{z}$ in magnitude is still significantly smaller than the reconnection flux measured from the flare ribbons after the eruption, especially for the typical criterion $|T_{w}| > 1.0$, we believe that the MFR found from the extrapolation is likely a seed MFR before the eruption. It should be noted that one group of footpoints ($FP_{-}$) locates closer to one main negative polarity of the magnetogram than the other group of footpoints ($FP_{+}$) to any main positive polarity. And $FP_{-}$ takes up a rather smaller area compared to $FP_{+}$, but the latter has much smaller average magnetic field $\langle B_{z}\rangle$, current density $\langle J_{z}\rangle$ and total current $I_{z}$.
A summary of quantitative results for MFRs in two events is given in Table \ref{summary}. The corresponding in situ modeling results for the two events are provided by \citet{2021Hu} by applying two magnetic cloud reconstruction methods. One of the modeling methods is the Grad-Shafranov reconstruction technique yielding a 2D configuration of the MFR \citep{2002Hu,Hu2017GSreview}. The other method is the optimization approach based on a more general linear force free formulation to obtain a more complex quasi-3D structure \citep{2021Hu,2021Hu2}. For event 1, the twist of the MFR identified in the source region is relatively consistent with the in situ modeling results of the MFR structure, considering the uncertainty of the total twist numbers. The axial magnetic flux calculated from the in situ modeling results is significantly larger than the seed MFR identified in the source region before the eruption, while the reconnection flux measured from the source region after the eruption is generally larger than the axial flux from the in situ modeling results. The poloidal magnetic flux, obtained from the in situ modeling results, appears to agree better with the reconnection flux, subject to the uncertainty in the axial length. Specifically if one assumes an axial length $\in [1, 2]$ au typically (see \citealt{2015Hu}) for an MC flux rope, this amounts to the total poloidal flux in the range 9.2 - 18$\times$10$^{20}$ Mx for event 1, based on the 2D MC modeling result.
For event 2, the axial (toroidal) flux content from the in situ 3D model agrees with the reconnection flux within their respective uncertainty ranges. Other parameters in the source region are not available (marked by ``...") since we did not find a pre-existing MFR structure before the eruption. The 2D MC model also failed to yield an acceptable solution. The twist of the MFR from the in situ modeling is generally larger than the twist we found in the groups of field lines in Figure \ref{f21} (the maximum $|T_{w}| \sim 0.97$). The CME-MFR containing significant amount of flux was likely formed during the eruption through dynamic evolution process in the solar atmosphere. Recently it was demonstrated by unique observational analysis and data-inspired numerical simulation \citep{2020Xing,2021Jiang2} the ``increase-to-decrease" behavior in the toroidal flux of the CME-MFR. However the applicability of such analysis to our events is beyond the scope of the current study. To study such a process usually requires discerning multiple flux systems with complex and constantly evolving topologies. And it remains a challenge to separate the toroidal and poloidal flux contents from the reconnection flux, although one pioneering approach developed by \citet{2009Qiu} for detailed analysis of the reconnection sequence can help in future studies.
\section{Conclusions}\label{4}
In this paper, we have identified the MFR structures in the solar source regions and established the connection between MFRs on the Sun and their in situ counterparts quantitatively for two selected events. One event began on 2013 April 11 (event 1, AR 11719) and the other on 2014 September 10 (event 2, AR 12158), respectively. Each event exhibits a sequence of flare, CME and the corresponding ICME observed by multiple space-borne instruments. We perform coronal magnetic field extrapolations for each AR by the CESE-NLFFF-MHD method, which utilizes the preprocessed photospheric magnetograms and the results are also examined through a set of convergence metrics. Remote-sensing observations from SDO are analyzed to find evidence of MFRs and trace the evolution of the associated flares. Specifically we measure the amount of magnetic reconnection flux by analyzing the temporal and spatial evolution of flare ribbons. We combine the extrapolation results with observations to identify MFRs on the Sun before the eruption and estimate their magnetic properties. The main results of our study are summarized as follows.
\begin{enumerate}
\item Observational evidence of MFR footpoints and associated magnetic reconnection flux during the flare eruption are inferred from multi-wavelength observations. From the comparison among EUV observations, there are signs of MFRs for the two events. Based on the flare ribbon measurements, the total magnetic reconnection flux reaches $17 \pm 2.8 \times 10^{20}$ Mx for event 1, and $47\pm 7.5 \times 10^{20}$ Mx for event 2, respectively, which corresponds to the amount of available flux to be injected into the final CME-MFRs.
\item From the combination of extrapolation and observation results, a coherent MFR structure before the flare eruption is identified for event 1. However, there is no pre-existing MFR found for event 2, based on the same set of MFR criteria, including the requirement for the field-line twist number $|T_w|>1.0$ and also both regions of the MFR field-line footpoints close to the main flare ribbons. For event 1, a coherent pre-eruption MFR is determined, which carries a maximum $T_{w} = -1.5$ and its two ends are located near the opposite ends of the respective flare ribbons across the PIL.
\item The magnetic properties of MFRs on the Sun are summarized and compared with their corresponding in situ modeling results from \citet{2021Hu} in Table~\ref{summary}. For event 1, the axial magnetic flux from in situ modeling results is in the order of $10^{20} \sim 10^{21}$ Mx, while the total magnetic reconnection flux after the eruption from the source region is in the order of $\sim 10^{21}$ Mx. Both are significantly larger than the flux in the identified pre-existing MFR's footpoints area which is in the order of $10^{19} \sim 10^{20}$ Mx (see Table~2).
\item For event 2, there is no pre-existing MFR identified. The amount of the magnetic reconnection flux, $47\pm 7.5 \times 10^{20}$ Mx, agrees with the corresponding ICME MFR toroidal flux, $\sim$ 16 - 81 $\times 10^{20}$ Mx, within the limits of the uncertainty ranges.
\end{enumerate}
These results for the two events indicate the dynamic and complex nature of the MFR formation during its evolution process while some properties (like magnetic flux and twist) are useful for making connections between the formation of MFRs on the Sun and their in situ characteristics in a quantitative manner. Based on these quantitative results, we conclude that the magnetic reconnection process, manifested during solar flares, injects significant amount of magnetic flux into the ensuing CME-MFR. For event 1, the identified pre-existing (or pre-eruption) MFR from the NLFFF extrapolation is likely a seed MFR before the eruption and additional flux is injected through the magnetic reconnection process associated with the flare.
Furthermore, based on the comparison among various inter-related magnetic flux contents and the corresponding flare ribbon morphology for each event, we conclude that for event 1, a quasi-2D configuration of the MFR is largely valid for which the poloidal flux is more meaningfully defined and compared more favorably with the corresponding reconnection flux than the axial flux. For event 2, however, we believe that the MFR topology deviates more from a 2D configuration, but is better described by a quasi-3D model for which the axial flux agrees with the reconnection flux. In this case, the poloidal flux is not readily defined geometrically because there does not exist a straight field line representing the central axis of a flux rope \citep[see][]{2021Hu}.
Therefore, for the 3D model, we choose to approximate the poloidal flux by the product of the average field-line twist and the axial flux \citep[see, e.g.,][]{2014Hu}.
This study represents an effort to make a physical connection between a solar MFR (including the MFR before and after the eruption) and the corresponding ICME/MC by the quantitative comparison of the magnetic properties under different scenarios through extrapolations and observations. It is usually not easy to envisage the existence of a coherent pre-existing MFR, reconstruct it before the eruption in the solar source region for a CME event and make one-to-one connection with its interplanetary counterpart. Efforts have been made continuously on the quantitative description of the MFR configuration with more advanced observations and improved numerical simulation techniques, which is helpful for further understanding the formation and evolution processes of the CME-MFRs. Future studies including more events will be carried out for a deeper understanding of CME-MFRs, where improvements to the extrapolation method and use of high-resolution ground-based data can be implemented.
\begin{acknowledgments}
W.H. and Q.H. acknowledge the support by NSF grants AGS-1954503 and AGS-1650854, and the NSO/DKIST Ambassador Program. C.W.J. acknowledges the support from National Natural Science Foundation of China (NSFC 42174200, 41822404), the Fundamental Research Funds for the Central Universities (Grant No. HIT.OCEF.2021033), and Shenzhen Technology Project JCYJ20190806142609035. A.P. would like to acknowledge the support by the Research Council of Norway through its Centres 458 of Excellence scheme, project number 262622, as well as through the Synergy Grant number 810218 459 (ERC-2018-SyG) of the European Research Council. Data for solar observations are provided by GOES, and the HMI and AIA instruments onboard SDO (\url{http://jsoc.stanford.edu/}). The Wind spacecraft measurements are accessed via the NASA CDAWeb: \url{https://cdaweb.gsfc.nasa.gov/index.html/}.
\end{acknowledgments}
\subsubsection*{#1}}
\pagestyle{headings}
\markright{Reference sheet: \texttt{natbib}}
\usepackage{shortvrb}
\MakeShortVerb{\|}
\begin{document}
\thispagestyle{plain}
\newcommand{\textsc{Bib}\TeX}{\textsc{Bib}\TeX}
\newcommand{\texttt{#1}\def\filedate{#2}\def\fileversion{#3}}}{\texttt{#1}\def\filedate{#2}\def\fileversion{#3}}}
\begin{center}{\bfseries\Large
Reference sheet for \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\ usage}\\
\large(Describing version \fileversion\ from \filedate)
\end{center}
\begin{quote}\slshape
For a more detailed description of the \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\ package, \LaTeX\ the
source file \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\texttt{.dtx}.
\end{quote}
\head{Overview}
The \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\ package is a reimplementation of the \LaTeX\ |\cite| command,
to work with both author--year and numerical citations. It is compatible with
the standard bibliographic style files, such as \texttt{plain.bst}, as well as
with those for \texttt{harvard}, \texttt{apalike}, \texttt{chicago},
\texttt{astron}, \texttt{authordate}, and of course \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}.
\head{Loading}
Load with |\usepackage[|\emph{options}|]{|\texttt{#1}\def\filedate{#2}\def\fileversion{#3}}|}|. See list of
\emph{options} at the end.
\head{Replacement bibliography styles}
I provide three new \texttt{.bst} files to replace the standard \LaTeX\
numerical ones:
\begin{quote}\ttfamily
plainnat.bst \qquad abbrvnat.bst \qquad unsrtnat.bst
\end{quote}
\head{Basic commands}
The \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\ package has two basic citation commands, |\citet| and
|\citep| for \emph{textual} and \emph{parenthetical} citations, respectively.
There also exist the starred versions |\citet*| and |\citep*| that print
the full author list, and not just the abbreviated one.
All of these may take one or two optional arguments to add some text before
and after the citation.
\begin{quote}
\begin{tabular}{l@{\quad$\Rightarrow$\quad}l}
|\citet{jon90}| & Jones et al. (1990)\\
|\citet[chap.~2]{jon90}| & Jones et al. (1990, chap.~2)\\[0.5ex]
|\citep{jon90}| & (Jones et al., 1990)\\
|\citep[chap.~2]{jon90}| & (Jones et al., 1990, chap.~2)\\
|\citep[see][]{jon90}| & (see Jones et al., 1990)\\
|\citep[see][chap.~2]{jon90}| & (see Jones et al., 1990, chap.~2)\\[0.5ex]
|\citet*{jon90}| & Jones, Baker, and Williams (1990)\\
|\citep*{jon90}| & (Jones, Baker, and Williams, 1990)
\end{tabular}
\end{quote}
\head{Multiple citations}
Multiple citations may be made by including more than one
citation key in the |\cite| command argument.
\begin{quote}
\begin{tabular}{l@{\quad$\Rightarrow$\quad}l}
|\citet{jon90,jam91}| & Jones et al. (1990); James et al. (1991)\\
|\citep{jon90,jam91}| & (Jones et al., 1990; James et al. 1991)\\
|\citep{jon90,jon91}| & (Jones et al., 1990, 1991)\\
|\citep{jon90a,jon90b}| & (Jones et al., 1990a,b)
\end{tabular}
\end{quote}
\head{Numerical mode}
These examples are for author--year citation mode. In numerical mode, the
results are different.
\begin{quote}
\begin{tabular}{l@{\quad$\Rightarrow$\quad}l}
|\citet{jon90}| & Jones et al. [21]\\
|\citet[chap.~2]{jon90}| & Jones et al. [21, chap.~2]\\[0.5ex]
|\citep{jon90}| & [21]\\
|\citep[chap.~2]{jon90}| & [21, chap.~2]\\
|\citep[see][]{jon90}| & [see 21]\\
|\citep[see][chap.~2]{jon90}| & [see 21, chap.~2]\\[0.5ex]
|\citep{jon90a,jon90b}| & [21, 32]
\end{tabular}
\end{quote}
\head{Suppressed parentheses}
As an alternative form of citation, |\citealt| is the same as |\citet| but
\emph{without parentheses}. Similarly, |\citealp| is |\citep| without
parentheses. Multiple references, notes, and the starred variants
also exist.
\begin{quote}
\begin{tabular}{l@{\quad$\Rightarrow$\quad}l}
|\citealt{jon90}| & Jones et al.\ 1990\\
|\citealt*{jon90}| & Jones, Baker, and Williams 1990\\
|\citealp{jon90}| & Jones et al., 1990\\
|\citealp*{jon90}| & Jones, Baker, and Williams, 1990\\
|\citealp{jon90,jam91}| & Jones et al., 1990; James et al., 1991\\
|\citealp[pg.~32]{jon90}| & Jones et al., 1990, pg.~32\\
|\citetext{priv.\ comm.}| & (priv.\ comm.)
\end{tabular}
\end{quote}
The |\citetext| command
allows arbitrary text to be placed in the current citation parentheses.
This may be used in combination with |\citealp|.
\head{Partial citations}
In author--year schemes, it is sometimes desirable to be able to refer to
the authors without the year, or vice versa. This is provided with the
extra commands
\begin{quote}
\begin{tabular}{l@{\quad$\Rightarrow$\quad}l}
|\citeauthor{jon90}| & Jones et al.\\
|\citeauthor*{jon90}| & Jones, Baker, and Williams\\
|\citeyear{jon90}| & 1990\\
|\citeyearpar{jon90}| & (1990)
\end{tabular}
\end{quote}
\head{Forcing upper cased names}
If the first author's name contains a \textsl{von} part, such as ``della
Robbia'', then |\citet{dRob98}| produces ``della Robbia (1998)'', even at the
beginning of a sentence. One can force the first letter to be in upper case
with the command |\Citet| instead. Other upper case commands also exist.
\begin{quote}
\begin{tabular}{rl@{\quad$\Rightarrow$\quad}l}
when & |\citet{dRob98}| & della Robbia (1998) \\
then & |\Citet{dRob98}| & Della Robbia (1998) \\
& |\Citep{dRob98}| & (Della Robbia, 1998) \\
& |\Citealt{dRob98}| & Della Robbia 1998 \\
& |\Citealp{dRob98}| & Della Robbia, 1998 \\
& |\Citeauthor{dRob98}| & Della Robbia
\end{tabular}
\end{quote}
These commands also exist in starred versions for full author names.
\head{Citation aliasing}
Sometimes one wants to refer to a reference with a special designation,
rather than by the authors, i.e. as Paper~I, Paper~II. Such aliases can be
defined and used, textual and/or parenthetical with:
\begin{quote}
\begin{tabular}{lcl}
|\defcitealias{jon90}{Paper~I}|\\
|\citetalias{jon90}| & $\Rightarrow$ & Paper~I\\
|\citepalias{jon90}| & $\Rightarrow$ & (Paper~I)
\end{tabular}
\end{quote}
These citation commands function much like |\citet| and |\citep|: they may
take multiple keys in the argument, may contain notes, and are marked as
hyperlinks.
\head{Selecting citation style and punctuation}
Use the command |\bibpunct| with one optional and 6 mandatory arguments:
\begin{enumerate}
\item the opening bracket symbol, default = (
\item the closing bracket symbol, default = )
\item the punctuation between multiple citations, default = ;
\item the letter `n' for numerical style, or `s' for numerical superscript
style, any other letter for
author--year, default = author--year;
\item the punctuation that comes between the author names and the year
\item the punctuation that comes between years or numbers when common author
lists are suppressed (default = ,);
\end{enumerate}
The optional argument is the character preceding a post-note, default is a
comma plus space. In redefining this character, one must include a space if
one is wanted.
Example~1, |\bibpunct{[}{]}{,}{a}{}{;}| changes the output of
\begin{quote}
|\citep{jon90,jon91,jam92}|
\end{quote}
into [Jones et al. 1990; 1991, James et al. 1992].
Example~2, |\bibpunct[; ]{(}{)}{,}{a}{}{;}| changes the output of
\begin{quote}
|\citep[and references therein]{jon90}|
\end{quote}
into (Jones et al. 1990; and references therein).
\head{Other formatting options}
Redefine |\bibsection| to the desired sectioning command for introducing
the list of references. This is normally |\section*| or |\chapter*|.
Define |\bibpreamble| to be any text that is to be printed after the heading but
before the actual list of references.
Define |\bibfont| to be a font declaration, e.g.\ |\small| to apply to
the list of references.
Define |\citenumfont| to be a font declaration or command like |\itshape|
or |\textit|.
Redefine |\bibnumfmt| as a command with an argument to format the numbers in
the list of references. The default definition is |[#1]|.
The indentation after the first line of each reference is given by
|\bibhang|; change this with the |\setlength| command.
The vertical spacing between references is set by |\bibsep|; change this with
the |\setlength| command.
\head{Automatic indexing of citations}
If one wishes to have the citations entered in the \texttt{.idx} indexing
file, it is only necessary to issue |\citeindextrue| at any point in the
document. All following |\cite| commands, of all variations, then insert
the corresponding entry to that file. With |\citeindexfalse|, these
entries will no longer be made.
\head{Use with \texttt{chapterbib} package}
The \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\ package is compatible with the \texttt{chapterbib} package
which makes it possible to have several bibliographies in one document.
The package makes use of the |\include| command, and each |\include|d file
has its own bibliography.
The order in which the \texttt{chapterbib} and \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\ packages are loaded
is unimportant.
The \texttt{chapterbib} package provides an option \texttt{sectionbib}
that puts the bibliography in a |\section*| instead of |\chapter*|,
something that makes sense if there is a bibliography in each chapter.
This option will not work when \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\ is also loaded; instead, add
the option to \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}.
Every |\include|d file must contain its own
|\bibliography| command where the bibliography is to appear. The database
files listed as arguments to this command can be different in each file,
of course. However, what is not so obvious, is that each file must also
contain a |\bibliographystyle| command, \emph{preferably with the same
style argument}.
\head{Sorting and compressing citations}
Do not use the \texttt{cite} package with \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}; rather use one of the
options \texttt{sort} or \texttt{sort\&compress}.
These also work with author--year citations, making multiple citations appear
in their order in the reference list.
\head{Long author list on first citation}
Use option \texttt{longnamesfirst} to have first citation automatically give
the full list of authors.
Suppress this for certain citations with |\shortcites{|\emph{key-list}|}|,
given before the first citation.
\head{Local configuration}
Any local recoding or definitions can be put in \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\texttt{.cfg} which
is read in after the main package file.
\head{Options that can be added to \texttt{\char`\\ usepackage}}
\begin{description}
\item[\ttfamily round] (default) for round parentheses;
\item[\ttfamily square] for square brackets;
\item[\ttfamily curly] for curly braces;
\item[\ttfamily angle] for angle brackets;
\item[\ttfamily colon] (default) to separate multiple citations with
colons;
\item[\ttfamily comma] to use commas as separaters;
\item[\ttfamily authoryear] (default) for author--year citations;
\item[\ttfamily numbers] for numerical citations;
\item[\ttfamily super] for superscripted numerical citations, as in
\textsl{Nature};
\item[\ttfamily sort] orders multiple citations into the sequence in
which they appear in the list of references;
\item[\ttfamily sort\&compress] as \texttt{sort} but in addition multiple
numerical citations are compressed if possible (as 3--6, 15);
\item[\ttfamily longnamesfirst] makes the first citation of any reference
the equivalent of the starred variant (full author list) and subsequent
citations normal (abbreviated list);
\item[\ttfamily sectionbib] redefines |\thebibliography| to issue
|\section*| instead of |\chapter*|; valid only for classes with a
|\chapter| command; to be used with the \texttt{chapterbib} package;
\item[\ttfamily nonamebreak] keeps all the authors' names in a citation on
one line; causes overfull hboxes but helps with some \texttt{hyperref}
problems.
\end{description}
\end{document}
| {
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{"url":"https:\/\/mturner.org\/devel\/playground\/entities","text":"# mturner.org\n\n### Entities with left-hand angle brackets\n\nDokuwiki converts certain combinations of characters to HTML entities; these entities are defined in conf\/entities.conf. Entities with left-hand angle brackets need special handling in ckgedit. Each example below shows such an entity in bold. Beneath each entity are two formats. The top format is the definition found in conf\/entities.conf. The second definition is what is needed by ckgedit in order to display the entity correctly and it should be added to conf\/entities.conf, or better yet to ''conf\/entities.local.conf 1), if you plan to use the entity in your documents. 2)\n\n# \u21d4\n\n<=>\n<=>\n\n# \u21d0\n\n<=\n<=;\n\n# \u2190\n\n<-\n<-\n2)\nPlease note: The original entity definition should not be deleted.\nNamespace Root","date":"2021-04-17 02:24:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6300469636917114, \"perplexity\": 2813.549993154149}, \"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-2021-17\/segments\/1618038098638.52\/warc\/CC-MAIN-20210417011815-20210417041815-00176.warc.gz\"}"} | null | null |
O Mercado de Jokkmokk (; PRONÚNCIA APROXIMADA ióque-móque) é um mercado de inverno na cidade de Jokkmokk, na província histórica da Lapónia, na Suécia.É realizado anualmente na primeira quinta-feira a sábado no mês de fevereiro, desde 1605, atraindo atualmente umas 50 000 pessoas. A temperatura da época ronda os 25 graus negativos, e o mercado proporciona corridas de renas e uns 500 pontos de venda.
Ligações externas
Página do Mercado de Jokkmokk
Atrações turísticas da Lapónia
Jokkmokk | {
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Copper Mine Point National Park
Virgin Gorda, British Virgin Islands
This prominent landmark dominates Mine Hill, on the cliffs of the southeastern tip of Virgin Gorda.
Surrounding the Copper Mine ruins there are many granite rock outcroppings, with additional deposits of quartz, feldspars, tin, copper and other clay minerals.
This abandoned copper mine played an important role in the history of Virgin Gorda. Spaniards passing through the BVI were the first Europeans to mine copper here in the early 18th century. Following a decline in mineral deposits in Cornwall, England in the 1800s, Cornish miners built the structure of which the ruins remain today. The mine closed in 1862, due to escalating expenses and low market prices.
There were 130 Cornish labourers and their families living on Virgin Gorda during this time. The ruins of their housing area and the operating center, containing the powerhouse, mine shafts, cistern, engine house and chimney are still visible, scattered across the slopes.
Long before the Cornish and Spanish miners arrived, Amerindians also used the area for copper. The copper was used to make tools and jewelry which were traded with other indigenous peoples from other islands.
Restoration works began in 1998 to stabilise the ruins, with the assistance of experts from Cornwall, England.
Mine Hill is also a habitat for the White-tailed tropicbirds (Phaethon lepturus) that nest in the rocky cliff crevices by the sea close to the southeastern corner of the Cornish Engine House. Departing from their seaside nests, they dive from incredible heights in order to feed on marine species, such as squid.
Area: 31.93 acres
Back to Parks
CONSERVING NATURE'S LITTLE SECRETS SINCE 1961
© 2017 NATIONAL PARKS TRUST OF THE VIRGIN ISLANDS
RESOURCES | FAQ | PRIVACY POLICY | CONTACT US | DONATE | {
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The Consequences of Intention, A History Boy, 18/07/2013
In writing news, I finished my edits of the first chapter of my ecophilosophy manuscript, and it didn't require quite the radical changes that I worried I'd have to make this summer. I still couldn't find a place to put that orphaned paragraph, but maybe I can find one for the idea elsewhere in this book, or in another project.
One of the typical divisions in contemporary moral philosophy is between deontological or duty-based and consequentialist moralities. These are often held (at least in mediocre undergraduate introductory courses) as the two pillars of moral thinking. Evaluating the goodness of an act will depend on what benefits or pains it produces as a consequence, or as well as it follows some moral duty arrived at through reasoning on moral intuitions or principles. In duty-based thinking, one's intentions play a key role in determining the rightness of an action. If you did a beneficial or morally right act, and you intended to do so or you intended to follow a moral duty, then that counts as more upstanding than if you'd done it by accident. You can see how our legal systems work as combinations of intentions and consequences. Accidental harms engender lesser punishments than intentional harms.
But I've always thought there was more to morality than this. It makes me something of a virtue ethicist, but I'm hesitant to identify as a virtue ethicist either. See, most of the discourse in virtue ethics today is on trying to figure out what the true virtues are. The discussions try to isolate the virtues in the same way, in deontological moral philosophy, thinkers try to isolate what the true moral duties are. And the expectation is that the virtues, like the duties, will be simple character traits that are unerringly and universally virtuous.
My own thinking about moral virtue is more complicated. (Do you see why I don't usually write straight articles about moral philosophy?) Because I don't think virtues are necessarily universal across all situations of action, or that the virtuousness of particular character features can be considered in isolation of the biological and psychological nature of humans. And when it comes to considering individual acts, I still think the psycho-biological context of human nature is important to consider ethically.
George Zimmerman, smiling
ambiguously
This murky ground between intention and consequent was floating in the back of many of my own reactions to the Zimmerman trial. Because ultimately, I can't judge how ridiculous Florida's self-defense and stand-your-ground laws are. Law isn't my area of expertise. The case was decided on the matter of who provoked the fight, and what constitutes provoking a fight. Throwing a punch? Or stalking someone down a dark, alienating suburban street? And some key jurors seemed to live in a state of reasonable doubt about reality, let alone the malicious intent of Zimmerman's actions.
What enrages me about Zimmerman is even more horrifying than the fact that he killed a 17-year-old young man, and that his racial prejudices probably motivated his suspicions of him. It's that he never seemed to feel bad about what he did. His brother's nonchalant dismissal of Trayvon's life in his Piers Morgan interview was a key sign. Killing someone shocks you; it's a destabilizing, terrifying, traumatizing event. It's a sign of your emotional health that if you kill someone, whether or not in self-defense, the event haunts you. Or at least upsets you.
Fred Allen, former supervisor of executions for the state of
Texas, and in many ways a broken person.
Zimmerman seemed calm. The act of killing never upset him or made him sad. In Werner Herzog's documentary Into the Abyss, one of the most powerful interviews is with Fred Allen, a former officer in charge of executions in Texas. He would strap down inmates and supervise the doctors administering lethal injections. He dealt every day with death that was entirely legitimated by his country's laws and his society's morality. One day, playing a part in so many deaths weighed too much, and he snapped. He had a nervous breakdown and had to leave his job, giving up his pension and financial security because he couldn't take part in killing anymore. I contrast Fred Allen with George Zimmerman, and I see something missing in Zimmerman. There's an obliviousness to the gravity of his actions. Leaving aside questions of legal accountability, killing a human is a weighty act.
And I think this aspect of human ethical life isn't considered by a lot of contemporary philosophy. The question always seems to be a matter of conformity to a moral principle to determine rightness, investigating the effects of an action to determine its benefits and harms, or whether it was motivated by virtuous or vicious character. The ethical weight of trauma does not seem to be widely considered, at least by the moral philosophy in which I've been educated over the last decade. Trauma ethics I think can be far more insightful than the tried-and-believed-to-be-true conceptual frameworks of the rest of moral philosophy. While I'm not yet sure where to start such an inquiry, I think it could be very intriguing for some philosopher out there to consider.
Zimmerman seems to have no personal remorse, no sense that he did anything wrong or even out of the ordinary. Philosophically and personally, that disturbs me.
Labels: Consequentialism, Deontology, George Zimmerman, History Boy, Moral Philosophy, Virtue Ethics, Werner Herzog
I once again feel the impulse to respond...
You wrote earlier about skepticism toward retributive or punitive justice, if memory serves. I'm curious how this links up with your discussion here, although I realize that your goal seems more to be to refine the concept than to connect this to the political realm.
What we seem to be confronted with here is a death caused by a man with minimal capacity for empathy and higher-order emotional experience (let us say for argument's sake). He killed a kid within the letter of the law (let's say) but wouldn't have done so had he had a normal or even minimally accepted level of empathy for the kid. This lack of empathy was likely the result of socialization processes (racism, a sort of generational gap re fashion, moral panic concerning crime) but may just be an internal condition of this guy.
I get the social side of the story, which is really civil-society work: legal sanctions need to be supplemented with interventions in social contexts where we see barriers to the sort of good will that is needed in this legal context.
So what are we to do with this guy in your perspective? I often detect what I think is a desire in your writing for higher standards of citizenship or humanhood or whatever, especially on this point of empathy. But how we cultivate this without creating a far vaster (and unrealistic for the US) system of punishment is unclear to me. Or in other words, help people see that kids in hoodies shouldn't be treated with hostility and suspicion.
What I see you saying here more specifically is that Zimmerman should be "punished" (however that gets defined, which you leave open) for failure of intention, failure of good will, failure of compassion and empathy.
This is a more pragmatic than conceptual approach and I don't know enough to know where trauma ethics would come in, but as always would be interested to hear your thoughts.
Adam Riggio July 18, 2013 at 6:37 AM
It's not that I ask the state to punish Zimmerman somehow as much as I think an act like killing someone should provoke a thought process in the actor that changes someone simply from recognizing the weight of the act. Trauma ethics is a vague gesture to what might be something like a new approach. That last sentence was the second time I'd used the phrase.
Moral philosophy today thinks largely in frameworks of duties, laws (whether moral/universal or temporal/partial/legal/customary), consequences, and virtues. This idea about trauma doesn't really fit into any of these frameworks, but I think it can have very interesting philosophical results.
I think my next History Boy post will be about Spinoza, and his distinguishing ethics from morality.
I wonder whether parole hearings include the type of moral conditioning that seems to be your concern. In making its decision, a parole board measures the probability of harm to society based on, among other things, indicators such as observed remorse. All things being equal, a prisoner who elicits credible remorse is more likely to be released on parole sooner than a prisoner who is judged to be feigning or not eliciting remorse. The ethical weight of, say, taking a life is, in this context, considered. "Because you do not feel remorse, and probably lack empathy, you present an unacceptable risk to society and cannot be released." The extent to which a person feels (and indicates the feeling of) the ethical weight of taking a life is a morally relevant fact in a consequentialist assessment.
Adam Riggio July 18, 2013 at 11:48 AM
Of course, in real life, all philosophical moral theories are true and overlap all the time. But no one to my knowledge has put remorse at the centre of a moral philosophy without reading it as a kind of conscience or intuition of an abstract moral principle. I'd be interested to see an immanentist version of a morality based on remorse. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,381 |
Q: How to stop busy javascript program? I am trying to write a nodejs express server. In that, I need to set time for execution. let say I made a request to /endpoint1 in express from the postman and that endpoint will trigger a function that contains an infinite for loop. My doubt is it possible to set the time limit for a single request?
....
app.get('/endpoint1',(req,res)=>{
while(true){}
});
....
A: You cannot do anything about that within the same execution context of your application.
What you could try to do is to have a controlling/observing parent process that spawns your application. And that application sends regularly "heartbeat" messages to the parent if those are missing the parent can restart the application.
In general, it is a good idea to write you express in such a way and in addition to this to use the cluster capability of node, so if one worker becomes unresponsive for whatever reason, that you only need to restart that one worker.
A: I guess this will work,
//type 1
var server = app.listen();
server.setTimeout(500000);
//type 2
app.post('/xxx', function (req, res) {
req.setTimeout(500000);
});
//type 3
app.post('/xxx', function (req, res) {
res.setTimeout(3000, function () {
console.log('here')
});
});
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,479 |
\section{}
\section{Introduction}
\par{Breast cancer is one of the most common malignant epithelial tumors in the world. According
to the latest statistics from the International Agency for Research on Cancer (IARC) of
the World Health Organization (WHO), there are 2.26 million new cases of breast cancer
worldwide, surpassing the 2.2 million cases of lung cancer. At the end of 2020, breast
cancer officially replaced lung cancer and became the world's largest
cancer~\cite{2020Breast}. Early treatment of breast cancer is vital, and doctors need to choose an
effective treatment plan based on its malignancy. Therefore, the detection of breast
cancer, the distinction between cancerous structure and the
identification of its malignant degree are valuable. Previously, there are
many techniques for detecting breast cancer, including Magnetic Resonance Imaging
(MRI)~\cite{murtaza2020deep}, Computed Tomography (CT), Positron Emission Tomography
(PET)~\cite{domingues2020using}, Ultrasound technology (US)~\cite{kozegar2020computer},
Mammograms (X-ray)~\cite{moghbel2020review} and Breast Temperature
Measurement~\cite{moghbel2013review}, etc.
\par{At present, histopathological diagnosis is
generally regarded as a ``gold standard''~\cite{de2019histopathologic}. Pathologists
need to observe the tissue lesions under the microscope to determine the cancerous area and the
degree of malignancy based on tissue structure, the nucleus and cytoplasm, and the growth pattern of the cells. To better analyze the different components of the tissue under the microscope, histopathologists usually stain the cut tissue.
In all
staining methods, Hematoxylin and Eosin staining is frequently used for more than a
hundred years, abbreviated as H\&E~\cite{gurcan2009histopathological}.
Fig.~\ref{FIG:BreastTissue} shows images of different types of breast tissues stained
with H\&E. Blue is the color of cell nuclei in excised tissues stained with hematoxylin,
and pink is the color of other structures (cytoplasm, matrix, etc.) stained with eosin. According to their biological behaviors, breast lesions can be divided into benign lesions (Fig.~\ref{FIG:BreastTissue} (b)) and malignant lesions (Fig.~\ref{FIG:BreastTissue} (c)(d)). Benign lesions generally grow slowly and do not exhibit an invasive growth pattern, while malignant tumor tends to metastasize to lymph nodes and distant organs.}
\begin{figure}[!h]
\centering
\includegraphics[width=1\textwidth]{Fig.1.pdf}
\caption{Breast tissue types based on H\&E staining.}
\label{FIG:BreastTissue}
\end{figure}
\par{However, it is very difficult for a histopathologist to observe the tissue with
the naked eyes and to manually analyze the visual information based on prior medical
knowledge. For one thing, manual analysis takes a lot of time. Since the histopathological
image itself will bring complexity and diversity due to subtle differences, cell overlap,
uneven color distribution and other reasons. For another, the objectivity of this kind
of analysis is unstable. The reason is that it depends to a large extent on the experience,
workload, and emotion of the histopathologist~\cite{li2019survey}. Authoritative
pathologists still need highly professional training and rich experience to make a
reliable diagnosis of patients, but the expertise and experience are also quite hard to
be inherited or innovated. Therefore, there is an urgent need for a system that can
realize \emph{Breast Histopathology Image Classification} (BHIC) to distinguish between
cancerous tissue (malignant tissue) and non-cancerous tissue (benign tissue) to help
pathologists make the diagnosis process more simple and efficient.}
\par{With the enhancement of computer calculating capacity and the continuous improvement
of storage performance, CAD technology has quickly become an indispensable technology in
the medical field, especially in the realm of histopathological image analysis. Currently,
the medical reliance on CAD systems has been increasing year by year. The histopathological
images can be quickly filtered and pre-classified by the CAD system, and then clinical
doctors can obtain the second idea of early diagnosis. This helps to reduce the burden
of pathologists and improve the efficiency of work. Meanwhile, it can also reduce the
influence of pathologists on subjectivity and personal experience differences. With the
assist of application technology in the CAD system, the difference between observers has
been further avoided. Above all, It is a work with research significance and broad
application prospects.}
\par{Faced with this situation, much research around the world is ongoing. However, most
of the existing methods are based on a single classifier. On the other hand, very few classification models based on image-level can achieve good classification results. This is because image-level classification is more complex:
Images under the same patient-level label are related, making it easier to obtain better
classification results; in contrast, image-level images only have relatively independent
labels, and there is no correlation between each label and less practical information
than patient-level images. While patient-level images are of great value in medical diagnosis, they sometimes add to the problem. For example, in the Medical Image Retrieval System~\cite{ghosh2011Review}, when the doctor encounters a difficult disease, a query image can be sent to the system for assisted diagnosis through the retrieval system. It is very inconvenient to upload full slices or a whole set of data, and image-level data can greatly simplify upload, so image-level analysis is also very valuable. Therefore, this paper proposes an image-level classification
method based on transfer learning and ensemble learning. It is believed that based on
the original research, ensemble learning and transfer learning can provide users with
easy-to-operate software and achieve better classification results. Since transfer
learning utilizes a pre-trained model, it can significantly adapt to a small dataset
in a short time~\cite{Rahaman-2020-IOCS}. Ensemble learning can complement the advantages
of multiple networks,
significantly improving the accuracy and generalization ability of the model. At the
same time, compared with the classification task under the patient-level label, the
classification of the lower image-level is more challenging. The workflow is shown in
Fig.~\ref{FIG:WorkflowWork}.}
\begin{figure}[!h]
\centering
\includegraphics[width=0.98\textwidth]{Fig.2.pdf}
\caption{Workflow of the proposed work.}
\label{FIG:WorkflowWork}
\end{figure}
As the workflow presented, the basic framework of this method is mainly composed
of four parts:
(a) Data augmentation. All of the images from the BreakHis dataset are divided into training, validation and test sets at the ratio of 7:2:1. Then the benign images are augmented through mirror flipping in order to balance the benign and malignant sample quantities.
(b) Transfer learning. The convolutional neural networks (CNNs), including VGG-16, VGG-19, Resnet-50, Inception-V3, Xception and DenseNet-201 are used for transfer learning to get applicable networks. According to the evaluation indicators, four out of six models are chosen as the base classifiers for the next process.
(c) Ensemble learning. This strategy based on the weighted voting method is used to further improve the classification performance. And the accuracy, one of the evaluation indicators, is chosen as the weight.
(d) Evaluation. The accuracy, precision, recall and F1-score are used as indicators to measure the classification ability of the whole algorithm.
\par{The main contributions of this study are summarized as follows:}
\par{(1). This subject has technical advantages. A new framework is proposed to solve
the classification problem of breast cancer histopathological images. In our previous
work~\cite{zhou2020comprehensive}, the application of classic neural networks and deep
learning in breast tissue pathological images from 2012 to early 2020 has been summarized
in detail. At the same time, combined with the analysis of the literature in recent
years, it is found that there are not many ideas for adopting the method of ensemble
learning, which is very promising for research.}
\par{(2). Sufficient experiments are guaranteed for results. In order to prevent the
singularity and limitation of a single classifier, six CNNs are used for training. Then,
the best four neural networks are ensembled, based on the weighted voting method. Through
multiple experiments, we find that when the accuracy is used as weight, the classification
result is the best.}
\par{(3). Excellent classification accuracy is achieved. The classification system
can effectively overcome the problem of small datasets, while greatly reducing training
time. On this basis, a classification accuracy of $98.90\%$ is obtained.}
\par{This paper is divided into the following chapters: Section~\ref{section:2}
introduces the related technologies of breast histopathological image classification.
Section~\ref{section:3} discusses the proposed image classification method based on
transfer learning and ensemble learning in detail. Section~\ref{section:4} introduces
the experimental settings, process, analysis and limitation of the results.
Section~\ref{section:5} summarizes this paper and puts forward the future plan under
this topic.}
\section{Method}
\label{section:2}
\subsection{Related work}
\par{This part mainly introduces the related works involved in the paper as well as
the research status in recent years.}
\subsubsection{Breast cancer and the application of CAD in breast histopathology}
\par{Among women, breast cancer accounts for one-quarter of cancer cases and one-sixth
of cancer deaths. It ranks first in most countries (159 out of 185 countries) in morbidity
and first in mortality among 110 countries~\cite{sung2021global}. In fact, women in all
regions of the world are at risk of breast cancer at any age after puberty, and the
incidence will increase in their later years.}
\par{The onset of breast cancer is caused by negative mutations in certain genes,
after which breast tissue cancer is evolved into malignant tumors. There are two major
types of breast cancer in medicine, namely carcinoma \emph{in situ} and invasive
carcinoma. Carcinoma \emph{in situ} refers to the early stage of breast cancer where
the growth of cancer cells is confined to ducts or lobules. Usually, the symptoms do
not manifest themselves, and the possibility of spreading or metastasis is small.
Over time, these cancer cells \emph{in situ} may gradually develop and invade the
surrounding breast tissue and then spread to nearby lymph nodes (specific regional
metastasis) or other organs in the body (distant metastasis). This is called invasive
carcinoma.}
\par{There are many pathogenic factors of breast cancer, mostly related to gender and age.
Other reasons mainly include obesity~\cite{renehan2008body}, lack of healthy
exercise~\cite{mctiernan2003recreational}, a high-protein diet, such as eating red meat
with exogenous hormones or carcinogenic by-products~\cite{levine2014low},
alcoholism~\cite{collaborative2002alcohol}, smoking~\cite{us2014health}, and using
oral contraceptives~\cite{hunter2010oral}. Most of these risk factors can be intervened
through health education in clinical practice and public health programs. Unfortunately,
even if all potential variables can be controlled, the risk of breast cancer can only
be reduced by up to $30\%$.}
\par{Numerous significant advances in computer science and medicine have created exciting
opportunities for medical experts and intelligent systems. CAD is the most direct and
obvious result of the organic integration of the two. CAD does not rely on the analysis
and skills of individual healthcare professionals and can make more objective and
fasterdiagnostic decisions. In addition, CAD can narrow the gap between experienced and
inexperienced medical personnel in diagnosis.}
\subsubsection{Deep learning model}
\par{Since 2012, the application of deep neural network (DNN) technology has become more
and more extensive, including image preprocessing, feature extraction, image postprocessing,
and classifier design. This development trend can be attributed to the rapid improvement
of hardware performance, which provides high feasibility for realizing DNN algorithms
with high computational complexity. In addition, an automatic feature extraction method
can be realized by DNN, through which it can be more robust to extract complex microscopic
breast tissue morphological features and structures. In the era of DNN, CNN is often preferred in image classification. Because of its superior performance in related fields, such as face recognition~\cite{co2017Face}, autonomous driving~\cite{gao2018Object}, COVID-19 image analysis~\cite{maram2021CovidXrayNet} and so on, CNN is also increasingly preferred in BHIC.}
\par{CNN is first proposed in~\cite{lecun1998gradient},
in which the backpropagation algorithm to the training of neural network structure is
applied. Compared with traditional image classification methods, CNN-based deep learning
can automatically learn features from massive amounts of data. Moreover, this method
effectively reduces the interference of subjective factors in the traditional method.
At the same time, CNN has been well applied in many fields, including object or human
body recognition, target tracking and detection, image segmentation and classification,
natural language processing and so forth. It has laid a good foundation for the application
of CNN in BHIA.}
\par{VGGNet~\cite{simonyan2014very} is the top two networks in the ImageNet Large-Scale
Visual Recognition Challenge (ILSVRC)~\cite{russakovsky2015imagenet} in 2014. The VGG-16 and
VGG-19 models are simple in structure, but consume many resources, training time, and
storage capacity. The Inception module is the core component of
GoogLeNet~\cite{szegedy2015going}. Based on the first two versions, Inception-V3 uses the
idea of factorization to split a two-dimensional convolution into two smaller modules:
$n \times n$ convolution into $1 \times n$ and $n \times 1$ convolution. It is beneficial
to reduce the number of parameters, speed up the calculation and further increase the
depth and nonlinearity of the network. The Xception network is an improved version of
Inception-V3 and is also regarded as an ``extreme'' version of the
Inception~\cite{chollet2017xception}. The network introduces a deep separable convolution
model, and an excellent effect can be obtained after comprehensive training data. One of
the basic modules that make up Resnet~\cite{he2016deep} is called the residual block,
which can effectively solve the problem of gradient disappearance in deep networks.
DenseNet~\cite{huang2017densely} is well-known for its particular structure. In forward
propagation, each layer is directly connected with all previous layers, and the input
of each layer comes from the output of all previous layers. Compared with the deep
residual network, the number of parameters that need to be trained is much lower than
that of Resnet to achieve the same accuracy.}
\par{With the proliferation of deep learning, people are increasingly pursuing stable models with good performance in all aspects, but the actual results are not so ideal. Therefore, the ensemble approach emerges as The Times require, which can complement each single classifier well. The work of \cite{Cao2020Ensemble} summarizes the recent applications in bioinformatics based on ensemble deep learning. Paper \cite{Yongquan2021Discussion} further discusses the development and limitations of ensemble learning in the era of deep learning, which is instructive for our research.}
\subsubsection{The development of breast histopathological image analysis}
\par{In the past ten years, CNN has had an outstanding performance in BHIC. In \cite{petushi2006large}, a third-party software (LNKNet package) containing a neural
network classifier is applied to evaluate two specific textures, namely the quantity
density of the two landmark substances, and a $90\%$ classification accuracy of
breast histopathology images is achieved.}
\par{In the paper~\cite{singh2010breast}, to classify breast histopathological images
stained by H\&E into four types, eight features and a three-layer forward/backward
artificial neural network (ANN) classifier are applied. Finally, a classification
accuracy of about 95$\%$ is obtained.}
\par{In~\cite{zhang2011breast,zhang2013breast1,zhang2013breast2}, an automatic
classification scheme is given. ``Random subspace ensemble'' is used to select and
aggregate the designed classifiers. Moreover, a classification accuracy of 95.22$\%$
is obtained on a public image dataset.}
\par{In the paper~\cite{shukla2017classification}, morphological features are extracted
to classify cancer cells and lung cancer cells in histopathological images. In the
experiment, the multi-layer perceptron based on the feedforward ANN model obtains
$80\%$ accuracy, 82.9$\%$ sensitivity, and 89.2$\%$ AUC successfully.}
\par{In~\cite{das2017classifying}, a CNN based classifier of full-slice histopathological
images is designed. First, the posterior estimate is obtained, which is derived from
the CNN with specific magnification. Then, the posterior estimates of these random
multi-views are vote-filtered to provide a slice-level diagnosis. Finally, 5-fold
cross-validation at the patient-level is used in the experiment, with an average
accuracy of 94.67$\%$, a sensitivity of 96$\%$, a specificity of 92$\%$, and an
F1-score of 96.24$\%$.}
\par{In the paper~\cite{gour2020residual}, a ReHist model is designed to classify benign
and malignant breast histopathological images, which is based on a 152-layer CNN with
residual learning. In the experiment, the histopathology images are first enhanced,
and then the ResHist model is trained end-to-end in the enhanced dataset by supervised
learning. After testing by ResHist model, the best accuracy of 92.52$\%$ and the F1-score
of 93.45$\%$ are achieved. In addition, to study the recognition ability of the ResHist
model for deep features, researchers input the extracted feature vectors into KNN,
Random Forest~\cite{breiman2001random}, secondary discriminant analysis, and SVM
classifiers. The best accuracy of 92.46$\%$ can be obtained when the deep functions
are fed back to the SVM classifier.}
\par{In~\cite{yan2020breast}, a new hybrid convolutional and cyclic DNN is created to
classify breast cancer histopathological images. The paper uses fine-tuned Inception-V3
to extract features for each image block. Then, the feature vectors are input to the
4-layer bidirectional long and short-term memory network~\cite{schuster1997bidirectional}
for feature fusion. In the experiment, the image is completely classified, and the
average accuracy is 91.3$\%$. It is worth noting that in this paper, a new dataset
containing 3,771 histopathological images of breast cancer is published, which covers
as many different sub-categories of different ages as possible.}
\par{In~\cite{kausar2019hwdcnn}, the deep CNN model based on Haar wavelet is introduced
to classify breast tissue pathological images, greatly reducing the calculation time
and improving the classification accuracy. In the experiment, the accuracy of 98.2$\%$
and 96.85$\%$ are achieved for the BACH dataset of four and two types and the BreakHis
dataset of multiple types, respectively.}
\par{In the classification work of~\cite{li2019classification}, an analysis technique
based on deep learning is proposed. The technique first uses a method based on CNN
and $k$-means for screening. Then Resnet-50 is used to extract features, and P-norm
pooling is used to get the final image features. In the end, SVM is applied for the
final image classification, with an accuracy of 95$\%$.}
\par{In~\cite{wang2018classification}, to distinguish four types of breast cancer in
histopathological images, a novel deep learning method is introduced. In the method,
hierarchical loss and global pooling are applied. VGG-16 and VGG-19 models are used as
the base deep learning network, and a dataset containing 400 images is used for testing.
Finally, an average accuracy of around 92$\%$ is obtained.}
\par{At present, there are a few types of researches on the classification of breast
histopathological images using ensemble learning methods. Among them, the classification
based on patient-level labels is commonly studied, such as researches in
Table.~\ref{TABLE:PatientLevelClassification}.}
\begin{table}[!h]
\centering
\caption{Researches on the patient-level classification of breast histopathological images.}
\begin{tabular}{llllr}
\toprule
Year & Method & Dataset & Transfer Learning CNNs & Accuracy \\ \midrule
2019 & Ensemble of DNNs \cite{kassani2019classification} & BreakHis & VGG-19, MobileNet-V2, DenseNet-201 & 98.13$\%$ \\
2019 & EMS-Net \cite{YANG2019EMSNet} & BreakHis(40 ×) & ResNet-101, ResNet-152, DenseNet-161 & 99.75$\%$ \\
2020 & Ensemble of DNNs \cite{Anda2020His} & Collected 544 Images & VGG-16, VGG-19 & 95.29$\%$ \\
2021 & MCUa \cite{Senousy2021MCUa} & BreakHis(40 ×) & ResNet-152, DenseNet-161 & 100.00$\%$ \\
2021 & 3E-Net \cite{Senousy2021ENet} & BreakHis(40 ×) & DenseNet-161, 6 image-wise CNNs & 99.95$\%$ \\ \bottomrule
\end{tabular}
\label{TABLE:PatientLevelClassification}
\end{table}
\par{From Table.~\ref{TABLE:PatientLevelClassification}, it can be seen that the existing
methods have achieved outstanding results in patient-level classification tasks. Although
patient-level classification is the most common clinical requirement, many studies
classify breast histopathological images at the image-level~\cite{Zhu-2021-HMEHE}. These
tasks are generally combined with patient-level classification to train and test the
proposed model, such as~\cite{Song2018Imagelevel,Li2020Imagelevel,hu2021Imagelevel}.
Nevertheless, since the image-level classification is more complex than the patient-level,
the classification accuracy is not relatively high in previous research. Therefore, this
paper focuses on the challenge of image-level classification. At the same time, the
characteristics of each transfer learning model are comprehensively considered, and
ensemble learning is carried out based on their complementarity.}
\subsection{Our method}
\subsubsection{Transfer learning}
\par{Transfer learning is a method of applying knowledge or patterns learned in a
specific field or task to different fields or problems~\cite{ribani2019survey}.
Traditionally, machine learning algorithms have strict training and test data
requirements, which require them to obey the same distribution. However, in reality,
this hypothesis cannot be fully established, resulting in many restrictions on the
practical application of machine learning. At the same time, the acquisition of
sufficient training data is another big problem for researchers. Therefore, scholars
have developed a special method to use a large amount of easy-collected data from
different fields to train learners and apply them to other similar scenarios. This
method is called transfer learning. In other words, if $X_{S}$ stands for the feature space and $X_{T}$ stands for the label space}, given a source domain
$D_{S}=\{ X_{S}, f_{S}(x)\} $ and learning task $T_{S}$, a target domain
$D_{T}=\{ X_{T}, f_{T}(x)\} $ and learning task $T_{T}$, transfer learning aims to
help improve the learning of the target predictive function $f_{T}(x)$ in $D_{T}$
using the knowledge in $D_{S}$ and $T_{S}$, where $D_{S}\neq D_{T}$, or $T_{S}\neq T_{T}$.
As for neural networks, researchers can directly use pre-trained models through a
large number of ready-made datasets. Then, reusable layers are selected, and the
output of these layers is used as input to train a network with fewer parameters
and a smaller scale. This small-scale network only needs to clarify the internal
relationships of specific problems and learn the implicit patterns through pre-trained
models~\cite{shoeleh2017graph}.
\par{At present, there are two main methods of applying transfer learning: one is to
fine-tune the parameters of the pre-trained network according to the required task;
the other is to use the pre-trained network as a feature extractor and then use these
features to train a new classifier. In this paper, we choose the former one.}
\par{The reasons why transfer learning is selected in this paper are as follows:}
\par{(1). Well-labeled histopathological images of breast cancer are relatively limited.
Due to the complexity of such images, it is costly for medical experts to annotate data.
Therefore, few publicly large-scale image datasets are available. Fortunately, transfer
learning can effectively overcome the problem of insufficient
data~\cite{hadad2017classification}.}
\par{(2). In the classification task of histopathology images, most of the preprocessing
models come from ILSVRC. Because of their stability in specific challenges, they can
be safely applied in breast cancer classification tasks.}
\par{(3). Transfer learning helps to improve accuracy or reduce the training
time~\cite{sarkar2018hands}, which is a crucial reason why transfer learning is
widely welcomed.}
\par{In this paper, the VGG series, Inception series, Resnet series, and DenseNet series
are carefully studied and compared, and six CNNs are selected to classify
breast histopathological images into benign and malignant tumors. They are: VGG-16,
VGG-19, Inception-V3, Xception, Resnet-50 and DenseNet-201. These classic networks are
selected because, on the one hand, these networks have passed numerous classification
tasks and can show high accuracy and stability in various datasets. In addition, the
advantages and shortcomings of these networks are systematically considered, and it
is highly possible to build a comprehensive ensemble network by using their complementarity.}
\par{For VGGNet, the VGG-16 and the VGG-19 network model are selected, because of their
simple network structure and excellent learning performance on many tasks. In terms of
the classification ability and parameter quantity, for the Inception series, Inception-V3
and Xception network models are adopted. In the Resnet series and the DenseNet series,
considering the calculation ability and other issues, we finally choose the Resnet-50
and the DenseNet-201 network model to conduct the research. In summary, a total of six
CNN models of VGG-16, VGG-19, Inception-V3, Xception, Resnet-50, and DenseNet-201 are
applied to the classification of breast histopathology images.}
\par{In the transfer learning based on these six CNNs, first of all, the massive labeled
images in the ImageNet dataset are applied separately, so these networks have good
classification capabilities. Then, the front ends of the trained model parameters in
these six CNNs are all frozen, so as not to destroy any information they contain in
future training. After that, the augmented breast histopathological images and their
labels are used to fine-tune the fully connected layers at the back end of the six
networks. Finally, the classification ability learned on ImageNet can be transferred
to the given pathological slices.}
\subsubsection{Ensemble learning}
\par{In the past few decades, ensemble learning has received more and more attention
in the field of computational intelligence and machine learning. Ensemble learning is
a process in which multiple models, such as classifiers, are combined according to a
certain method to solve a specific intelligent computing problem. It is mainly used
to improve the performance of the final model, such as classification, prediction,
and function estimation, or to reduce the impact of improper selection of the basic model.}
\par{The general framework of ensemble learning is summarized as follows: First, a
group of individual learners is generated; after that, the learners are effectively
combined through a specific strategy; finally, the expected experimental results can
be obtained~\cite{2009Ensemble}. Individual learners are usually generated from training
data by existing learning algorithms (such as decision tree, error propagation neural
network). Among them, using individual learners of the identical type (such as all
neural networks) is called homogeneous ensemble learning, and applying various
individual learners is called heterogeneous ensemble learning. In this paper,
homogeneous ensemble learning is used.}
\par{Recent fast ensemble deep learning techniques~\cite{Yang2020FTBME, Yang2021Local} can be applied in whole slide medical image analysis (f.e. predicting pCR from H\&E stained whole slide images~\cite{Yang2021Deep}) to reduce time and space overheads, at the expense of certain accuracy. However, we believe that the performance improvement is much more important in this task, so we choose some usual ensemble methods. There are three general combination strategies: voting,
averaging, and stacking. Nevertheless, for the task of this paper that aims to
achieve the binary classification of breast histopathological images, the voting
method is the most commonly used and easily manipulated one. This method refers to
the assumption that there are $T$ different classifiers ${h_{1}, h_{2},..., h_{T}}$,
and our goal is to predict the final category from the $l$ category markers
${c_{1}, c_{2},..., c_{l}}$ based on the output of the classifier. Typically, for
the sample $x$, the output result of the classifier $h_{i}$ is an $l$-dimensional
label vector $(h_{i}^{1}(x), h_{i}^{2}(x),..., h_{i}^{l}(x))^{T}$, where
$h_{i}^{j}(x)$ is the prediction output of the $h_{i}$ classifier on the class
$c_{j}$ label.}
\par{Voting can be divided into three types. Absolute majority voting: Different
categories will be marked by each classifier for voting. If the final number of
votes obtained by a certain type of mark exceeds $\frac{1}{2}$, the category is
regarded as the final output result; if the votes for all the categories do not
exceed $\frac{1}{2}$, the forecast is rejected. Relative majority voting: The side
with the most votes is deemed the winner; when there is a tie, one side is chosen
arbitrarily. Weighted voting: Different from the previous two, the corresponding
weights are assigned to different classifiers; if the classifier performs better,
a higher weight is assigned. Finally, the weighted votes of each category are summed,
and the one corresponding to the maximum value is regarded as the final result.}
\par{Although there are endless ensemble learning methods, considering their complexity
and ease of operation, the basic weighted voting strategy is applied in this paper,
and the formula is Eq.~(1). When given appropriate weights, weighted voting can be
both superior to the individual classifier with the best classification result, and
at the same time, superior to the absolute majority voting.}
\begin{equation}
H(x)=c_{arg_{j}max}\sum_{i=1}^{T}w_{i}h_{i}^{j}(x)
\label{Eq:1}
\end{equation}
\par{Among them, $w_{i}$ represents the weight of the classifier $h_{i}$. In practical
applications, similar to the weighted average method, the weight coefficients are
often normalized and are constrained to be $w_{i}\geq 0$ and $\sum_{i=1}^{T}w_{i}=1$.
Getting the right weight is very important. Suppose $l=(l_{1}, l_{2},...,l_{T})^{T}$
is the output of the individual classifier, where $l_{i}$ represents the prediction
result of the class label of the classifier $h_{i}$ on sample $x$. Let $p_{i}$ be
the precision of $h_{i}$, the combined output of the category label $c_{j}$ can be
expressed as Eq.~(2) using a Bayesian optimal discriminant function.}
\begin{equation}
H^{j}(x)=\log (P(c_{j})P(l|c_{j}))
\label{Eq:2}
\end{equation}
\par{When it is assumed that the output conditions of individual classifiers are
independent, Eq.~(2) can be reduced to Eq.~(3).}
\begin{equation}
H^{j}(x)=\log P(c_{j})+\sum_{i=1}^{T}h_{i}^{j}(x)\log \frac{p_{i}}{1-p_{i}}
\label{Eq:3}
\end{equation}
\par{When the first term of the above equation does not depend on the individual
classifier, the optimal weight of the weighted voting can be obtained from the
second term and satisfies Eq.~(4).}
\begin{equation}
w_{i}\propto\log \large\frac{p_{i}}{1-p_{i}}
\label{Eq:4}
\end{equation}
\par{Therefore, the optimal weight needs to be consistent with the performance of
the individual classifier, which is based on the case where the output of the
individual classifier is independent of each other. However, in the actual work of
this paper, the classifier is trained for the same problem. The output is usually
strongly correlated, and the independence assumption is not valid. So the weight is
set based on the evaluation indicators of the classifier, which is also a commonly
used method of weighting.}
\section{Result}
\label{section:3}
\subsection{Experimental settings}
\subsubsection{Image dataset}
\par{To implement the proposed model, a practical open source BreakHis dataset is used
in the research of this paper \cite{spanhol2015dataset}. BreakHis consists of 7,909
clinically representative microscopic images of breast tumor tissue, which are collected
from 82 patients using four magnifications.}
\par{Data source: The sample comes from a biopsy section of breast tissue, which is
marked by pathologists in Brazil's P\&D laboratory;}
\par{Staining method: H\&E staining;}
\par{Magnification: $50 \times$, $100 \times$, $200 \times$, $400 \times$;}
\par{Microscope: Olympus BX-50 system microscope, with 3.3 times magnification relay
lens, connected with Samsung digital color camera SCC-131AN;}
\par{Camera pixel size: 6.5 $\mu m$;}
\par{Image size: $700 \times 460$ pixels;}
\par{Image format: *.png;}
\par{Pixel bit depth: 24 bits (RGB three-channel image, 8 bits per channel,
$3 \times 8 = 24$);}
\par{Up to now, a total of 2,480 benign tumor samples and 5,429 malignant tumor
samples have been collected in the BreakHis dataset. Both of them are divided into
different subtypes, respectively. As shown in Fig.~\ref{FIG:SubTypeSamples}, the
first line is the four subtypes of benign tumors, including Adenosis (A),
Fibroadenoma (F), Tubular Adenoma (TA), and Phyllodes Tumor (PT). The second line
is the four subtypes of malignant tumors, including Ductal Carcinoma (DC),
Lobular Carcinoma (LC), Mucinous Carcinoma (MC), and Papillary Carcinoma (PC).}
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.98\textwidth]{Fig.3.pdf}
\caption{Benign/malignant tumor subtype samples}
\label{FIG:SubTypeSamples}
\end{figure}
\par{Therefore, the dataset can be used for both binary classification and
multi-classification tasks. In this paper, the binary classification of
benign/malignant tumors under four magnifications is studied. Table \ref{TABLE:ImageDistribution} shows the distribution of samples in this dataset.}
\begin{table}[!h]
\centering
\caption{Image distribution by magnification factor and class.}
\begin{tabular}{llll}
\toprule
Magnification & Benign \quad\quad\quad & Malignant \quad\quad & Total \quad \\
\midrule
$40 \times$ \quad\quad& 625 & 1,370 & 1,995 \\
$100 \times$ \quad\quad& 644 & 1,437 & 2,081 \\
$200 \times$ \quad\quad& 623 & 1,390 & 2,013 \\
$400 \times$ \quad\quad& 588 & 1,232 & 1,820 \\
Total \quad\quad& 2,480 & 5,429 & 7,909 \\
Number of patients \quad\quad& 24 & 58 & 82 \\
\bottomrule
\end{tabular}
\label{TABLE:ImageDistribution}
\end{table}
\subsubsection{Setting of experimental data}
\par{In clinical work, patient-level classification is essential. In the early
pre-experiment and review \cite{zhou2020comprehensive}, it has been found that many
methods can achieve excellent classification results under the label of the patient.
However, because the data information under the same patient label is highly
correlated, the patient-level classification is not challenging enough. Therefore,
the lower-level classification based on the image label is applied, and a better
result is achieved in the preliminary experiment.}
\par{This experiment uses a cross-validation method to separate the BreakHis dataset
into three mutually exclusive subsets. The subsets are randomly selected from the
dataset at a ratio of 7:1:2 and are separated into the training set, validation set
and test set. However, according to the introduction of the dataset above, it is
evident that the number of images under the two categories of benign (2,480) and
malignant (5,429) is seriously imbalanced. Approximately $69\%$ of images are samples
of malignant tumors. In order to tackle this problem, data augmentation strategies
are applied to the benign tumor samples in all of the subsets, namely horizontal
mirror flip and vertical mirror flip, with a total of 2,480 images generated.
Afterward, 469 images are randomly selected from the data obtained through the
vertical mirror flip through calculation. Finally, based on the original data
volume (2,480), adding the horizontal mirror flip (2,480) and the selected vertical
mirror flip dataset (469), the benign sample reaches 5,429 images. In this way, the
number of images in both two categories is equal, with 10,858 images. The amount of
samples in each subset is presented in Table~\ref{TABLE:Data}. In the early
research~\cite{Li2020GAN}, we have done data augmentation based on Generative Adversarial
Network (GAN) and achieved relatively good results. However, the focus of this paper
is not to generate a large number of new datasets, and only using mirror flipping
can satisfy this experiment.}
\begin{table}[!h]
\centering
\caption{Data setting.}
\begin{tabular}{llll}
\toprule
\quad\quad\quad & Benign \quad\quad\quad & Malignant \quad\quad & Total \quad \\
\midrule
Training set \quad\quad& 3,800 & 3,800 & 7,600 \\
Validation set \quad\quad& 543 & 543 & 1,086 \\
Test set \quad\quad& 1,086 & 1,086 & 2,172 \\
\bottomrule
\end{tabular}
\label{TABLE:Data}
\end{table}
\subsubsection{Experimental environment}
\par{The experiments involved in this paper are all developed based on the Python
language (version 3.6.8) under the Windows 10 operating system. All deep learning
models involved in the experiment are developed (training, test) entirely by the
Keras (version 2.24) framework. In addition, TensorFlow (version 1.12.0) is used
as the backend of Keras. The specific conditions of the workstation parameters
applied in the experiment are as follows: Intel(R) Core(TM) i7-8700 CPU (3.20 GHz),
32GB RAM, NVIDIA GEFORCE RTX 2080 8 GB.}
\subsection{Evaluation metrics}
\par{In the course of our experiments, the performance of the classifier needs to be
quantitatively evaluated. Appropriate evaluation indicators can reduce the deviation of
algorithm differentiation. In this paper, confusion matrixes are used to analyze various
evaluation measures. For the n-type classification problem, the confusion matrix is a
table of size $n \times n$. An accurate classifier represents most samples along the
diagonal of the matrix. In fact, the confusion matrix itself is not a performance
indicator, but almost all frequently-used indicators are based on it. Table~\ref{tbl3}
is a confusion matrix based on two classifications. Positive samples can be regarded
as samples of research interest. Thus, we consider samples of benign tumors as positive
and samples of malignant tumors as negative.}
\begin{table}[!h]
\centering
\caption{Confusion matrix based on binary classification.}
\begin{tabular}{lll}
\toprule
&\quad\quad Target class & \quad\quad \\
\multirow{-2}{*}{Output class} &\quad\quad Positive (P) &\quad\quad Negative (N) \\
\midrule
Positive (P) &\quad\quad TP & \quad\quad FP \\
Negative (N) &\quad\quad FN & \quad\quad TN \\
\bottomrule
\end{tabular}
\label{tbl3}
\end{table}
\par{TP (True Positive) is that an example is correctly identified as a positive;
TN (True Negative) is that an example is correctly identified as a negative;
FP (False Positive) is that negative cases are misclassified as positive ones;
FN (False negative) is that positive instances are incorrectly classified as negative.
This experiment evaluates the performance of six CNN classifiers mentioned above based
on these four values.}
\begin{table}[!h]
\centering
\caption{Evaluation metrics.}
\begin{tabular}{ll}
\toprule
Assessment & \quad\quad\quad Formula \\
\midrule
\quad Accuracy & \quad\quad\quad\large $\frac{TP+TN}{TP+FP+FN+TN}$ \\ [6pt]
\quad Precision & \quad\quad\quad\large $\frac{TP}{TP+FP}$ \\ [6pt]
\quad Recall & \quad\quad\quad\large $\frac{TP}{TP+FN}$ \\ [6pt]
\quad F1-score & \quad\quad\quad\large $\frac{2\times P\times R}{P+R}$ \\ [6pt]
\bottomrule
\end{tabular}
\label{TABLE:Evaluation}
\end{table}
\par{For classification tasks, accuracy, precision, recall and F1-score are considered
the most popular techniques for measuring the classification results. These evaluation
indicators are described in Table~\ref{TABLE:Evaluation}. The accuracy is the ratio of
the number of correctly classified samples to the total number of samples. The precision
reflects the proportion of true positive cases in the cases that are judged to be
positive, which represents how many benign tumors are predicted to be correct. The
recall reflects the proportion of cases judged to be positive in the total number of
positive cases. Low recall means that many patients with benign tumors are treated as
cancer, which is a major medical incident. Therefore, recall is of great significance
in medical image diagnosis. However, precision and recall are contradictory to a
certain extent. If the precision is intended to be improved, it can be achieved by
raising the criteria for positive classification. In other words, the number of samples
predicted to be positive is reduced. Although the precision has improved, the recall
will definitely decrease. On the contrary, if the recall needs to be improved, all
samples can be marked as positive. In this way, the recall can reach $100\%$, but the
precision will absolutely reduce. Therefore, F$\beta$-score is introduced, which is a
vital technique that balances precision and recall. When $\beta$ = 1, it is the
harmonic average of the two values, namely the F1-score.}
\subsection{Deep learning algorithm}
\subsubsection{CNN training process}
\par{The training process of CNN is divided into two stages: forward propagation
and backpropagation. The flow chart of its training is shown in
Fig.~\ref{FIG:WorkflowCNNs}. The whole training process includes (a) Forward Propagation and (b) Back Propagation. In forward propagation, six network models VGG-16, VGG-19, Inception-V3, Xception, Resnet-50 and DenseNet-201 are loaded first, and the weights of these neural networks are initialized. After that, the training data is input, and the output results are obtained through the convolution layer, the lower sampling layer and the full connection layer. Next, the error between output and target value is calculated. When the error meets the requirements, the weights and thresholds of each layer are saved. If not, the process of back propagation is entered, which intends to update the weight to reduce error. If the performance does not improve, the learning rate will be decayed. Then return to start the next iteration. The process for each CNN is summarized in Algorithm~\ref{alg1}.}
\begin{figure}[!h]
\centering
\includegraphics[width=0.7\textwidth]{Fig.4.pdf}
\caption{The workflow of neural networks.}
\label{FIG:WorkflowCNNs}
\end{figure}
\begin{algorithm}[!h]
\caption{Transfer learning algorithm in this paper}
\label{alg1}
\KwIn{The labeled dataset $S$, a base learning \textbf{Classifier} pre-trained on
ImageNet, and the maximum number of interations $N$.}
\textbf{Initialize} the initial weight vector, that $w_{jk}^{l}$ is the weight from the $k^{th}$ neuron in the $(l-1)^{th}$ layor to the $j^{th}$ neuron in the $l^{th}$ layer. $b^{l}_{j}$ and $a^{l}_{j}$ are the bias and the the activation (the output of the activation function) of the $j^{th}$ neuron in the $l^{th}$ layer, respectively.
\While{this model is improved}
{
\If{not first time executing this code}
{
Attenuating the learning rate
}
\For{$t=1, ..., N$}
{
Suppose the input is X and the output is Y. The value of $a^{l}_{j}$ depends on
the activation of the previous layer of neurons:
$a^{l}_{j}=\sigma(\sum_{k}w_{jk}^{l}a^{l-1}_{j}+b^{l}_{j})$\;
Rewrite the formular above into a matrix form:
$a^{l}=\sigma(w^{l}a^{l-1}+b^{l})$\;
Call \textbf{Classifier}, calculate the activation using
$a^{l}=\sigma(w^{l}a^{l-1}+b^{l})$ layer by layer, $X\rightarrow \hat{Y}$ \;
Calculate the loss function: $C=\frac{1}{2n}\sum_{x}\left \| y(x)-a^{L}(x)\right \|^{2}$.
Where $n$ is the total number of training samples $x$, $y=y(x)$ is the ground truth,
$L$ is the number of layers of the network, and $a^{L}(x)$ is the output vector of
the network\;
Calculate the value of $w$ when the error is the smallest by taking the partial d
erivative of $w_{jk}^{l}$\;
\textbf{Update} the weights in the direction of reducing errors\;
}
}
\KwOut{the CNN with trained weights and thresholds for each layer.}
\end{algorithm}
\par{Apart from transfer learning, six CNNs are also trained from scratch as a comparative
experiment. Table~\ref{TABLE:CNNsParameter} summarizes the parameters of these models.
Among them, both Top-1 Accuracy and Top-5 Accuracy refer to the accuracy of each neural
network on the ImageNet validation set. Depth refers to the topological depth of each
neural network, including the activation layer and batch normalization layer.}
\begin{table}[!h]
\centering
\caption{Overview of CNN parameters.}
\begin{tabular}{lrllrr }
\toprule
Model \quad\quad& Size & \quad\quad Top-1 Accuracy & Top-5 Accuracy & \quad Parameters & \quad\quad Depth \\
\midrule
VGG-16 \quad\quad& 528 MB &\quad\quad 71.3\% & 90.1\% &\quad 138,357,544 &\quad\quad 23 \\
VGG-19 \quad\quad& 549 MB &\quad\quad 71.3\% & 90.0\% &\quad 143,667,240 &\quad\quad 26 \\
Inception-V3 \quad\quad& 92 MB &\quad\quad 77.9\% & 93.7\% &\quad 23,851,784 &\quad\quad 159 \\
Xception \quad\quad& 88 MB &\quad\quad 79.0\% & 94.5\% &\quad 22,910,480 &\quad\quad 126 \\
Resnet-50 \quad\quad& 98 MB &\quad\quad 74.9\% & 92.1\% &\quad 25,636,712 &\quad\quad --- \\
DenseNet-201 \quad\quad& 80 MB &\quad\quad 77.3\% & 93.6\% &\quad 20,242,984 &\quad\quad 201 \\
\bottomrule
\end{tabular}
\label{TABLE:CNNsParameter}
\end{table}
\subsubsection{Classification performance evaluation}
\par{In the classification of breast histopathological images, some hyperparameters
are set. After many trials, these values are taken when the validation set obtains
the best result. The exponential decayed learning rate is adopted regardless of
training from scratch or training based on transfer learning. The decay steps are
set to 5, the decay rate is 0.1, the initial learning rate is $1 \times 10^{-4}$,
and the adaptive moment estimation (Adam) is selected as the optimizer. The other
hyperparameter settings in the experiment are as follows:}
\par{Training the network from scratch: The initial input size is $224 \times 224 \times 3$,
and the batch sizes of the VGG-16, VGG-19, Inception-V3, Xception, Resnet-50, and D
enseNet-201 networks are set to 32, 16, 26, 8, 26 and 16, respectively. The epoch is
specified as 60.}
\par{Transfer learning network: The initial input size is $460 \times 460 \times 3$,
the batch size is set to 16, and the epoch is 60. In Table \ref{TABLE:ScratchandTransfer},
the classification results obtained by the above six CNNs based on these two
training methods are respectively summarized.}
\begin{table}[!h]
\centering
\caption{The result and prediction time of CNN models on validation set (unit, $\%$).}
\begin{tabular}{llrrrrr}
\toprule
Model & Training Mode & \multicolumn{1}{l}{Accuracy} & \multicolumn{1}{l}{Precision} & \multicolumn{1}{l}{Recall} & \multicolumn{1}{l}{F1-score} & \multicolumn{1}{l}{Time (unit, $s$)} \\ \midrule
\multirow{2}{*}{VGG-16} & Training from scratch & 89.50 & 89.65 & 89.32 & 89.48 & 17.28 \\
& Transfer Learning & \textbf{95.49} & \textbf{95.40} & \textbf{95.58} & \textbf{95.49} & 32.50 \\
\multirow{2}{*}{VGG-19} & Training from scratch & 91.34 & 90.16 & 92.82 & 91.47 & 17.96 \\
& Transfer Learning & \textbf{95.03} & \textbf{93.27} & \textbf{97.05} & \textbf{95.13} & 38.61 \\
\multirow{2}{*}{Inception-V3} & Training from scratch & \textbf{94.48} & 90.59 & \textbf{99.26} & \textbf{94.73} & 42.75 \\
& Transfer Learning & 93.83 & \textbf{97.41} & 90.06 & 93.59 & 52.99 \\
\multirow{2}{*}{Xception} & Training from scratch & 89.96 & 86.29 & 95.03 & 90.45 & 31.04 \\
& Transfer Learning & \textbf{96.59} & \textbf{94.54} & \textbf{98.90} & \textbf{96.67} & 39.45 \\
\multirow{2}{*}{ResNet-50} & Training from scratch & 84.71 & 94.15 & 74.03 & 82.89 & 34.97 \\
& Transfer Learning & \textbf{98.90} & \textbf{98.72} & \textbf{99.08} & \textbf{98.90} & 42.72 \\
\multirow{2}{*}{DenseNet-201} & Training from scratch & 88.03 & 95.19 & 80.11 & 87.00 & 94.14 \\
& Transfer Learning & \textbf{98.25} & \textbf{96.95} & \textbf{99.63} & \textbf{98.27} & 103.06 \\ \bottomrule
\end{tabular}
\label{TABLE:ScratchandTransfer}
\end{table}
\par{From the statistical table of classification results, it can be concluded that
the effect of transfer learning is generally better than that of training from scratch.
Through transfer learning, the accuracy can be increased by around $3\%$ to $14\%$,
which is a very satisfactory result. In addition, it is also found that training the
Inception-V3 network from scratch is better than any other network using this method
for training. At the same time, the accuracy of Inception-V3 training from scratch
is $0.65\%$ higher than that of transfer learning. Both the recall and the F1-score
are also higher than using transfer learning, with the values of $99.26\%$ and
$94.73\%$, respectively. Nevertheless, this does not affect our subsequent choice
of Inception-V3 based on transfer learning for the next step. This is because the
training time of the network from scratch is generally more extended, and the
performance of the computer should also be taken into account. In the same environment,
training each network from scratch usually takes more than 2 hours. Therefore, transfer
learning is applied to all the networks for the following research.}
\subsubsection{Overall evaluation and visual analysis transfer learning networks}
\par{Fig.~\ref{FIG:TrainCurve} shows the transfer learning training curves of the
six CNNs: VGG-16, VGG-19, Inception-V3, Xception, Resnet-50 and DenseNet-201,
respectively, to observe the changes in the training process of these models.
It includes training set accuracy curve (blue curve), training set loss curve (green
curve), validation set accuracy curve (yellow curve), validation set loss curve (red
curve). As the number of training rounds increases, the accuracy curve shows an upward
trend, and the loss curve shows downward. This is what we expected, which means that
the performance of the models is gradually becoming stable.}
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.98\textwidth]{Fig.TrainCurve.pdf}
\caption{CNN training process curve.} %
\label{FIG:TrainCurve}
\end{figure}
Subsequently, these six networks are compared, and their confusion matrices are
shown in Fig.~\ref{FIG:CNNsMatrix}.
\begin{figure}[!h]
\centering
\includegraphics[width=0.98\textwidth]{Fig.CNNsMatrix.pdf}
\caption{Confusion matrix of CNN on validation set.}
\label{FIG:CNNsMatrix}
\end{figure}
\par{In the end, the Resnet-50 network performs best, with three indicators that top
all network models. To be precise, the accuracy is $98.90\%$, the precision reaches
$98.72\%$, and the F1-score reaches $98.90\%$. The second place is the DenseNet-201
network, which is comparable to the Resnet-50 network, at around $98\%$. Moreover,
the recall of the DenseNet-201 network is the highest among all models, reaching
$98.63\%$. In order to analyze the classification results more
intuitively, their evaluation indicators are drawn into a histogram, as shown in Fig.~\ref{FIG:BarChart}.}
\begin{figure}[!h]
\centering
\includegraphics[width=0.75\textwidth]{Fig.BarChart.pdf}
\caption{Comparison of evaluation indicators among six CNNs (unit, $\%$). The horizontal axis shows the value of the evaluation index, the vertical axis represents six different nodels, and the rectangular bars of different colors represent four evaluation indicators.}
\label{FIG:BarChart}
\end{figure}
\par{In the final evaluation, the results of the Inception-V3 network are relatively
the worst, in which the accuracy, recall and F1-score are the lowest among the six
models, with 9$4.48\%$, $90.06\%$ and $93.59\%$, respectively. The lowest value of
precision is obtained on VGG-19, which is $93.27\%$. For networks that belong to
the same series, such as the VGGNet series, the overall performance of the VGG-16 network
is better than that of the VGG-19 network. Three indicators calculated in the VGG-16
network are about $0.3\%$ to $2\%$ higher than those in the VGG-19 network. For the
Inception series, the performance of the Xception network is better than that of the
Inception-V3 network, and three of the evaluation indicators are about $2\%$ to $8\%$
higher than the Inception-V3. The performance of the DenseNet-201 network and the
Resnet-50 network are relatively good, with satisfactory classification results.
Finally, referring to the size, parameter amount, and network depth of the CNNs
in Table~\ref{TABLE:CNNsParameter}, four networks are chosen in the end, namely
VGG-16, Xception, Resnet-50 and DenseNet-201 for the follow-up ensemble experiment.}
\subsection{Ensemble learning}
\subsubsection{Ensemble pruning}
\par{Ensemble pruning refers to selecting a subset of the individual learners that
have been trained instead of all learners for the next step. It has two advantages:
First, the ensemble result can be obtained using a smaller network scale. This cannot
only reduce the storage overhead brought from storing model, but also reduce the
computing overhead corresponding to the output of the individual learner, thereby
improving the efficiency of the model. In addition, the generalization performance
of ensemble pruning is even better than the ensemble obtained by using all individual learners.}
\par{In this paper, two classifiers with relatively poor performance are pruned from
the six CNN models based on transfer learning, and four classifiers are selected as
individual learners, namely VGG-16, Xception, Resnet-50, and DenseNet-201 network
model. Since the accuracy is the main evaluation index of this experiment, $Accuracy$ of each network is used as the weight of our voting method. Firstly, each weight is quantized with the step length 0.01 into \{0, 0.01, 0.02, ..., 0.99, 1\}. Then, in a weigh vector $w_{k=1,2,...,6}$, the combination of $w_{(k,1)}$, $w_{(k,2)}$, $w_{(k,3)}$ leads to the highest matching accuracy $Accuracy_{k}$ is chosen. Finally, the selected $w_{k=1,2,...,6}$ are implemented the ensemble learning method. And finally, we got the best ensemble accuracy of 98.90$\%$ in this way.}
\par{Thus, the final experimental procedure is obtained. First, in order to achieve
data balance, data augmentation is applied to the BreakHis dataset. The method used
in this procedure is horizontal and vertical mirror flip. Then, the augmented data
is input into each transfer learning based CNN, and six individual classifiers are
obtained. After that, four classifiers with relatively good performance are selected and trained using the ensemble learning strategy of weighted
voting. Finally, the ensemble results are evaluated.}
\subsubsection{Evaluation of ensemble learning algorithms}
\par{After training, the four indicators of accuracy, precision, recall, and F1-score
are still used to evaluate the overall performance of the system.
Fig.~\ref{FIG:EnsembleMatrix} shows the confusion matrix of ensemble learning. It
can be found that the algorithm based on ensemble learning predicts 1 sample that
should be benign as malignant and predicts 11 samples that should be malignant
as benign.}
\begin{figure}[!h]
\centering
\includegraphics[width=0.40\textwidth]{Fig.EnsembleMatrix.pdf}
\caption{Confusion matrix of ensemble learning on validation set.}
\label{FIG:EnsembleMatrix}
\end{figure}
\par{Finally, the overall indicators of all single CNNs and our ensembled method
are evaluated on the test set and summarized in Table~\ref{TABLE:TestResult}. The
ensemble learning strategy finally achieves $98.90\%$ accuracy, $98.72\%$ precision,
$99.08\%$ recall, and $98.90\%$ F1-score. Obviously, the accuracy, recall, and F1-score
are higher than using any of the six CNNs based on transfer learning alone. After
comparison, this method effectively improves the accuracy by $0.1\%$ to $5.25\%$,
the recall by $0.09\%$ to $8.93\%$, and the F1-score by around $0.3\%$ to $5\%$. In
terms of precision, although it does not exceed the $99.07\%$ achieved by the
Resnet-50 network, its performance surpasses the other five models and wins second
place. In summary, the classification performance of the ensemble learning method
is generally better than that of the single transfer learning method.}
\begin{table}[!h]
\centering
\caption{Summary of classification results in testing process (unit, $\%$).}
\begin{tabular}{lllll}
\toprule
& \multicolumn{4}{l}{\quad\quad Evaluation metrics} \\
\multirow{-2}{*}{Model} &\quad\quad Accuracy &\quad\quad Precision &\quad\quad Recall &\quad\quad F1-score \\
\midrule
VGG-16 &\quad\quad 95.44 &\quad\quad 95.74 &\quad\quad 95.12 &\quad\quad 95.43 \\
VGG-19 &\quad\quad 94.66 &\quad\quad 92.47 &\quad\quad 97.24 &\quad\quad 94.79 \\
Inception-V3 &\quad\quad 93.65 &\quad\quad 96.93 &\quad\quad 90.15 &\quad\quad 93.42 \\
Xception &\quad\quad 96.04 &\quad\quad 94.25 &\quad\quad 98.07 &\quad\quad 96.12 \\
Resnet-50 &\quad\quad 98.80 &\quad\quad{\textbf{99.07}} &\quad\quad 98.53 &\quad\quad 98.53 \\
DenseNet-201 &\quad\quad 98.30 &\quad\quad 97.64 &\quad\quad 98.99 &\quad\quad 98.31 \\
Ensemble &\quad\quad{\textbf{98.90}} &\quad\quad 98.72 &\quad\quad{\textbf{99.08}} &\quad\quad{\textbf{99.90}} \\
\bottomrule
\end{tabular}
\label{TABLE:TestResult}
\end{table}
\section{Discussion}
\label{section:4}
\subsection{Comparitive experiment}
\par{To evaluate the ability of the breast histopathological image classification algorithm based on transfer learning and ensemble learning proposed in this paper, a comparative experiment on Transformer and MLP models is carried out using the dataset divided by this experiment. The latest methods are summarized in Table.~\ref{TABLE:Comparison}, which have shown great power in natural image classification. However, their generalization ability has not been developed well like CNN for the specific histopathological images, and the overall classification performance is not very good. Especially Transformer, because of its good ability to describe global information, there should be improved versions suitable for pathological images in the near future, making novel contributions to this field.}
\begin{table}[!h]
\centering
\caption{Comparison of accuracy between the existing method and our method on the same dataset.}
\begin{tabular}{llrr}
\toprule
\multicolumn{2}{l}{Model} & Accuracy (unit, $\%$) & Training time (unit, $s$) \\ \midrule
\multirow{7}{*}{Transformer} & BoTNet-50 \cite{srinivas2021BTFVR} & 90.75 & 4502 \\
& CaiT \cite{touvron2021going} & 96.70 & 5081 \\
& CoaT \cite{xu2021CCIT} & 92.91 & 420 \\
& DeiT \cite{touvron2021TDITD} & 90.51 & 2101 \\
& LeViT \cite{graham2021levit} & 93.42 & 13321 \\
& ViT \cite{dosovitskiy2020AIIWW} & 80.11 & 1242 \\
& T2T-ViT \cite{yuan2021tokens} & 80.02 & 2370 \\ \midrule
\multirow{3}{*}{MLP} & MLP-mixer \cite{tolstikhin2021mlp} & 83.84 & 5541 \\
& gMLP \cite{liu2021pay} & 93.88 & 8407 \\
& ResMLP \cite{touvron2021RFNFI} & 79.56 & 10341 \\ \bottomrule
\end{tabular}
\label{TABLE:Comparison}
\end{table}
\subsection{Comparison of previous reasearch}
\par{As can be seen from Table.~\ref{TABLE:TestResult}, although compared to some single classifiers, the accuracy of our ensemble network has not been significantly improved. However, the network has been enhanced in the evaluation of the four indicators. In particular, F1-score is even close to 100$\%$, reflecting that both precision and recall of the model are excellent.}
\par{Fig.~\ref{FIG:Evaluation} shows the comparison of some images that are not correctly classified and those that are correctly classified. It can be found that the correctly classified image contains almost all classification information of benign and malignant so that the proposed algorithm can easily identify it. There are two main reasons why the image is mispredicted. First, this type of image has a very high degree of similarity to another type, such as Fig.~\ref{FIG:Evaluation} (a) and Fig.~\ref{FIG:Evaluation} (b). When one type of image is similar to another type of texture, distribution and color, it will not be easy to identify correctly. In addition, the wrongly predicted images often contain very little information to distinguish between benign and malignant, such as Fig.~\ref{FIG:Evaluation} (c) and Fig. \ref{FIG:Evaluation} (d). Patch-based images are allowed to contain many blanks during segmentation. These reasons above can interfere with the model's correct classification of benign and malignant breast histopathological images.}
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.95\textwidth]{Fig.Evaluation.pdf}
\caption{An example of the classification result.
(a) and (c) are correctly classified images.
(b) and (d) are wrongly classified images.}
\label{FIG:Evaluation}
\end{figure}
\par{Additionally, we analyze the works based on deep learning on the BreakHis dataset. Especially for image-level classification in previous research, such as~\cite{Li2020Imagelevel} with less than $90\%$ and~\cite{hu2021Imagelevel} with around $91\%$, our method with 98.90$\%$ accuracy works more effective. Other ANNs and their results in different tasks are summarized in Table \ref{tbl8}. It should be noted that some studies may have done both binary and multiple classifications, but only the former results are considered. When compiling the tables, it is found that only a few studies used evaluation indicators other than accuracy. Even if the same indicator is employed (specifically, precision, recall and F1-score), the proposed method in this paper has a better performance. To facilitate statistics, only the accuracy in each study is discussed in detail. From the table, it can be concluded that in these algorithms, the accuracy of our method is effectively improved by 0.13$\%$ to 21.40$\%$.}
\begin{table}[!h]
\caption{Comparison of accuracy between the existing method and our method (unit, $\%$).}
\begin{tabular}{llllll}
\toprule
Year & Types of ANN that the research methods based on & Accuracy \\
\midrule
2016 & CNN \cite{bayramoglu2016deep} & 83.25 \\
2017 & LeNet and AlexNet \cite{spanhol2016breast,spanhol2017deep,spanhol2018automatic} & 85.00 \\
2017 & Transfer Learning based VGG-VD \cite{song2017adapting} & 87.00 \\
2017 & Transfer Learning based VGGNet \cite{zhi2017using} & 90.00 \\
2017 & CNN \cite{nejad2017classification} & 77.50 \\
2017 & CNN \cite{li2017using} & 90.00 \\
2017 & DCNN \cite{han2017breast} & 94.67 \\
2017 & DCNN \cite{wei2017deep} & 97.00 \\
2018 & Transfer Learning based VGG-16, VGG-19, and Resnet-50 \cite{mehra2018breast} & 92.60 \\
2018 & Inception-V1, Inception-V2 and ResNet-V1-50 \cite{2018InceptionResnet} & 95.00 \\
2018 & DCNN \cite{nahid2018histopathological1} & 87.75 \\
2018 & CNN and RNN \cite{nahid2018histopathological2} & 91.00 \\
2018 & DCNN \cite{nahid2018histopathological3} & 92.19 \\
2018 & CNN \cite{du2018breast} & 90.00 \\
2018 & DenseNet based CNN \cite{nawaz2018multi} & 95.40 \\
2018 & Resnet \cite{gandomkar2018framework} & 98.77 \\
2018 & DCNN \cite{cascianelli2018dimensionality} & 85.30 \\
2019 & SA-Net \cite{xu2019look} & 96.00 \\
2019 & Transfer Learning based Resnet-50, Inception-V2, Inception Resnet-V2 and Xception \cite{bhuiyan2019transfer} & 96.24 \\
2019 & DCNN \cite{xie2019deep} & 97.90 \\
2019 & Transfer Learning based VGG-16 and VGG-19 \cite{thuy2019fusing} & 98.10 \\
2020 & ResHist \cite{gour2020residual} & 92.52 \\
2021 & Our method & 98.90 \\
\bottomrule
\end{tabular}
\label{tbl8}
\end{table}
\section{Conclusion and future work}
\label{section:5}
\par{In this paper, a framework that combines transfer learning and ensemble learning
is proposed for the image-level classification of breast histopathological images. F
inally, an accuracy of $98.90\%$ is obtained. First of all, based on the pre-segmented
BreakHis dataset, training from scratch and transfer learning are applied separately
to train the six selected neural networks. Meanwhile, confusion matrices are used for
the comparison and analysis of algorithms. Considering the two perspectives of less
training time and high accuracy, we select the transfer learning method. After that,
based on the idea of ensemble pruning, four networks are selected as individual
classifiers. Through a large number of experiments, the weighted voting method with
accuracy as the weight is used to combine these classifiers. Finally, we use the ten
latest Transformer and MLP models to classify breast histopathological images on the
same image-level dataset. It turns out that our method is quite competitive and ranks
first in accuracy.}
\par{This research is dedicated to developing an algorithm to assist doctors in diagnosing
tumor types correctly. The goal is to promote the calculation speed and efficiency, as
well as the reliability of the final classification results. However, there is still
huge potential for further exploration and development. Firstly, classifiers with better
performance can be selected to further improve the final classification effect. Secondly,
in addition to the weighted voting method used in this paper, more ensemble strategies
and weights can be discussed in more depth. In addition, there are many types of breast
cancer, as well as many subtypes of tumors. This work is a two-class classification at
the image-level. In the future, a multi-classification system can be introduced on this
basis. Henceforth, we are committed to continuously developing more efficient and
high-accuracy classification models and will get involved in the image-level
multi-classification tasks.}
\section*{Author Contributions}
Yuchao Zheng: method, experiment, result analysis and paper writing.
Chen Li: method, result analysis, paper writing and proofreading.
Xiaomin Zhou: method, experiment.
Haoyuan Chen: comparitive experiment.
Hao Xu, Xinyu Huang and Marcin: data analysis, proofreading.
Yixin Li: comparitive experiment.
Haiqing Zhang: paper writing.
Xiaoyan Li and Hongzan Sun: medical knowledge.
All authors contributed to the article and approved the submitted version.
\section*{Acknowledgements}
This work is supported by National Natural Science Foundation of China (No. 61806047).
We thank Miss Zixian Li and Mr. Guoxian Li for their important discussion.
\section*{Conflict of Interest}
The authors declare that they have no conflict of interest in this paper.
\bibliographystyle{gbt7714-numerical}
| {
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} | 970 |
Franciscus Pahr var son till byggmästaren hos Georg II av Brieg i Schlesien, Jakob Pahr. Brodern Johan Baptista Pahr var också en känd byggmästare, i tjänst hos hertig Johan Albrecht I av Schwerin. Även de yngre bröderna Dominicus och Kristoffer Pahr var byggmästare.
Pahr arbetade under fadern under byggnationen av das Piastenschloss, därefter som byggmästare i Hainau och i Mecklenburg, inträdde 1558 i Hertig Ulrich av Güstrows tjänst och ledde för dennes räkning ombyggnaden av det då nyligen brandskadade schloss Güstrow. 1572 inkallades brodern Johan Baptista till Kalmar, och året därpå även Franciscus, och en yngre broder Dominicus. Fransiscus kom kort därefter att anställas vid byggnationen av Uppsala slott, och avled 1580 i Uppsala, varvid Antonius Wats efterträdde honom som byggmästare.
Källor
Vidare läsning
Män
Byggmästare
Personer från Schlesien | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,048 |
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|}
Het WTA-tennistoernooi van Quebec (officieel Bell Challenge) van 2007 vond plaats van 29 oktober tot en met 4 november 2007 in de Canadese stad Quebec. Het was de vijftiende editie van het toernooi. Er werd gespeeld op overdekte tapijtbanen.
Enkelspel
Titelhoudster Marion Bartoli was haar titel niet komen verdedigen.
De als eerste geplaatste Nicole Vaidišová sneuvelde al in de eerste ronde.
De ongeplaatste Lindsay Davenport uit de Verenigde Staten was met een wildcard tot het toernooi toegelaten. Zij versloeg op weg naar de finale twee reekshoofden: Angelique Kerber (7) en Vera Zvonarjova (2). In de eindstrijd zegevierde Davenport over de als derde geplaatste Oekraïense Joelija Vakoelenko in twee sets, waarmee zij voor het eerst in haar carrière het toernooi van Quebec op haar naam schreef. Het was haar 53e WTA-titel.
Geplaatste speelsters
Toernooischema
Finale
{{Wedstrijdschema laatste 2 zonder 3 met 3 sets
| RD1=Finale
| team-width=175
| RD1-seed1=3
| RD1-team1= Joelija Vakoelenko
| RD1-score1-1=4
| RD1-score1-2=1
| RD1-score1-3=
| RD1-seed2=WC
| RD1-team2= Lindsay Davenport
| RD1-score2-1=6
| RD1-score2-2=6
| RD1-score2-3=
}}
Bovenste helft
Onderste helft
Dubbelspel
Titelhoudsters Laura Granville en Carly Gullickson hadden zich niet voor deze editie van het toernooi ingeschreven.
Het eerste reekshoofd, Meilen Tu en Vera Zvonarjova, kwam niet verder dan de tweede ronde.
Het ongeplaatste Amerikaanse duo Christina Fusano / Raquel Kops-Jones''' won het toernooi. Zij versloegen in de finale het als derde geplaatste koppel Stéphanie Dubois en Renata Voráčová in twee sets, met een tiebreak om de tweede set te beslissen. Het was voor zowel Fusano als Kops-Jones haar eerste WTA-titel.
Geplaatste teams
Prijzengeld en WTA-punten
Toernooischema
Bron
Toernooischema WTA
Quebec 2007
Quebec | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,902 |
\section{Introduction}
Noncommutative Geometry can be described as a generalization of spin geometry based on the notion of \emph{spectral triple}. It has far-reaching applications in mathematics and particle physics. For the latter, one has to supplement the spectral triple with an action functional. Up to now two possibilites have emerged: the Connes-Lott action\cite{Connes-90}, which is a generalization of the Yang-Mills one, and the spectral action\cite{Chamseddine-96}, which can be seen as a generalization of the Einstein-Hilbert action. The spectral action has attracted more attention lately since it allows to unify gravity with the other forces. However, in order to do this, one has to switch from the formalism of spectral triples to another one in which the Dirac operator, which plays here the role of the metric, may vary. Such a formalism has been recently proposed\cite{Besnard-19-1} in the form of so-called \emph{algebraic backgrounds}. An unexpected consequence of this new point of view when applied to the Standard Model\footnote{It is important to note that what we call the Standard Model always include 3 generations of right-handed neutrinos.} is that the B-L symmetry has to be gauged. It has thus become necessary to study the predictions of the spectral action for the $U(1)_{B-L}$-extension of the Standard Model, and compare them to experiments. This can be done following the same steps as in Ref.~\onlinecite{Chamseddine-12}:
\begin{itemize}
\item Define an energy scale $\mu_{\rm unif}$ at which the spectral action can be applied (this will be a free parameter).
\item Since the spectral action depends only on two parameters, one obtains relations on coupling constants at the energy $\mu_{\rm unif}$: use them to derive the initial values of the coupling constants.
\item Run the renormalization group equations down to the accessible energies and compare to experiments.
\end{itemize}
The second point invariably necessitates some simplifying assumptions on the couplings. For instance in this paper we will assume that the coupling of the top dominates the quark sector. As for the Yukawa couplings of neutrinos, we will consider two opposite scenarios: A) only one Dirac and one Majorana coupling are non-zero, and B) the 3 Dirac couplings are equal, and so are the 3 Majorana couplings (universal couplings).
The third point boils down in checking that the masses of the top quark and Higgs boson at their own energy scale (pole masses) match the experimental values to an acceptable degree of precision. In this paper we will be happy with a precision of $5\%$, with 1-loop running and tree-level mass relations.
The main point of this paper is that the spectral action permits to derive constraints on the quartic couplings which must be satisfied in order for the minimum of the quartic potential to happen at a non-zero value for both Higgs fields of the model. What we will show is that in both scenarios A and B, these constraints are \emph{not} satisfied for the values of the parameters which yield acceptable results for the SM Higgs and top quark masses. More precisely, and in both cases, it is the B-L symmetry which is not broken. We also point out that even if we ignore this problem, or somehow cure it (for example with extra fields), another one will remain: if non-zero, the vevs of the two Higgses tend to be of similar order of magnitude, except if the quartic couplings are extremely fine-tuned. In fact, in scenarios A and B we show that this fine-tuning is not even possible since it contradicts the constraint derived from symmetry breaking. Note incidentally that none of these problems happen with the Connes-Lott action.
In section \ref{prelim} we introduce the general notations used in the rest of the paper as well as the mathematical definition of the B-L extended spectral standard model. We will assume a certain familiarity with Noncommutative Geometry in this section. Otherwise we refer to Ref.~\onlinecite{bes-20-a} for more detailed explanations. The reader may also directly skip to section \ref{fields} where we make the explicit connection between the fields as they appear in NCG and in physics. In section \ref{predictionSA} we derive the predictions of the spectral action for the B-L extended SM in general and the precise value of the couplings at unification scale, dependending on some parameters, in scenarios A and B. Section \ref{constraintsSB} contains the main point of the paper, namely the constraints on quartic couplings for the breaking of both the electroweak and the B-L symmetry. In section \ref{scenarA} we translate these constraints in the case of scenario A and show that they are not satisfied for the values of the parameters which are needed to obtain acceptable predictions for the pole masses of the top quark and Higgs boson. In section \ref{scenarB} we do the same in scenario B and obtain the same conclusion. Section \ref{conclusion} provides a brief discussion of the possible implications of the present work.
\section{Mathematical preliminaries}\label{prelim}
The spectral triple $\calS_F$ of the B-L extended Standard Model is the restriction to a particular subalgebra of the Pati-Salam spectral triple of Ref.~\onlinecite{Chamseddine-13-PS}. It is also the Euclidean version of the indefinite spectral triple presented in Refs.~\onlinecite{Besnard-19-1,bes-20-a}. We refer to these papers for details. The finite Hilbert space is the same as that of the Standard Model, namely
\begin{eqnarray}
\calH_F&=&\calH_R\oplus \calH_L \oplus \calH_{\overline{R}}\oplus\calH_{\overline{L}}\label{HF},
\end{eqnarray}
where $\calH_i$ is 24-dimensional and isomorphic to
\begin{eqnarray}
\calH_0 &=& (\CC^2\,\oplus\, \CC^2\otimes\CC_c^3)\otimes
\CC_g^N,\label{H0}
\end{eqnarray}
where
$N$ is the number of generations. In this paper we will only consider the case $N=3$. The finite algebra is
\begin{equation}
\calA_F=\CC\oplus \HH\oplus \CC \oplus M_3(\CC).
\end{equation}
To define the representation of $\calA_F$ on $\calH_F$, it is convenient to introduce, for any $a\in M_2(\CC)$ the notation
\begin{equation}
\tilde{a}:=a\oplus a\otimes 1_3
\end{equation}
acting on the first factor of the RHS of \eqref{H0}. The representation is then
\begin{equation}
\pi_F(\lambda,q,\mu,m)=\diag(\tilde q_\lambda,\tilde q,\mu\oplus 1_2\otimes m,\mu\oplus 1_2\otimes m)\otimes 1_N.
\end{equation}
where $q_\lambda=\begin{pmatrix}
\lambda&0\cr 0&\lambda^*
\end{pmatrix}$. The chirality, real structure and finite Dirac operator are the same as for the SM and given by
\begin{equation}
\chi_F={\rm diag}(1,-1,-1,1)\label{SMchirality}
\end{equation}
\begin{equation}
J_F=\begin{pmatrix}
0&0&1&0\cr
0&0&0&1\cr
1&0&0&0\cr
0&1&0&0
\end{pmatrix}\circ c.c.
\end{equation}
and
\begin{equation}
D_F=\begin{pmatrix}
0&\Upsilon^\dagger&M^\dagger&0\cr \Upsilon&0&0&0\cr M&0&0&\Upsilon^T\cr 0&0&\Upsilon^*&0
\end{pmatrix},\label{DF}
\end{equation}
where
\begin{equation}
\Upsilon=\begin{pmatrix}\Upsilon_\ell&0\cr 0&\Upsilon_q\otimes 1_3\end{pmatrix},\label{Y}
\end{equation}
with $\Upsilon_\ell,\Upsilon_q\in M_2(M_N(\CC))$ given by
\begin{equation}
\Upsilon_\ell=\begin{pmatrix}\Upsilon_\nu&0\cr 0&\Upsilon_e\end{pmatrix},\ \Upsilon_q=\begin{pmatrix}\Upsilon_u&0\cr 0&\Upsilon_d\end{pmatrix},\label{formeYell}
\end{equation}
where we have decomposed the $\CC^2$ factor using the $(u,d)$ basis, while
\begin{equation}
M= \begin{pmatrix}m&0\cr 0&0\end{pmatrix}\otimes\begin{pmatrix}1&0&0&0\cr 0&0&0&0\cr 0&0&0&0\cr 0&0&0&0\end{pmatrix}\label{formeM}
\end{equation}
where $m\in M_N(\CC)$ is a symmetric matrix (responsible for the type I see-saw mechanism). It is also important to describe the selfadjoint 1-forms of $\calS_F$. They come in two breeds: $\Phi(q)$ and $\sigma(z)$, where $q\in \HH$ and $z\in\CC$. They are defined by
\begin{eqnarray}
\Phi(q)&=&\begin{pmatrix}
0&\Upsilon^\dagger \tilde q^\dagger&0&0\cr
\tilde q \Upsilon&0&0&0\cr
0&0&0&0\cr
0&0&0&0
\end{pmatrix},\cr
\sigma(z)&=&\begin{pmatrix}
0&0&z^*M^\dagger&0\cr
0&0&0&0\cr
zM&0&0&0\cr
0&0&0&0
\end{pmatrix}.
\end{eqnarray}
Note that $J_F\sigma(z)J_F^{-1}=\sigma(z)$. A general selfadjoint 1-form is the sum of $\Phi(q)$ and $\sigma(z)$ for some $q$ and $z$.
\section{Field content}\label{fields}
The order 1 condition is not satisfied by $\calS_F$. However, weaker conditions hold\cite{bes-20-b} which suffice to obtain a well-defined and gauge-invariant bosonic configuration space consisting of fluctuated Dirac operators of the form
\begin{equation}
D_\omega=D+\omega+J\omega J^{-1}\label{fluct}
\end{equation}
where $\omega$ is a selfadjoint 1-form of the almost-commutative triple obtained by tensorizing $\calS_F$ with the (Euclidean) spacetime triple $\calS_M$ (which we do not recall here). There are two types of 1-forms: the 1-forms of $M$ with values in $\pi_F(\calA_F)$, which will yield the gauge fields, and the functions on $M$ with values in the finite 1-forms which correspond to the scalar fields.
\begin{rem}
It is important to
note that although the first-order condition is not satisfied, we did not include a correction to \eqref{fluct} as in Ref.~\onlinecite{Chamseddine-13}. The reason is that in this particular model it only amounts\cite{Besnard-19-3} to the redefinition $z\mapsto z^2$ of the complex field $z$.
\end{rem}
The explicit computation of \eqref{fluct} yields
\begin{align}
D+i\gamma^\mu\hat\otimes (X_\mu t_X+{1\over 2}gB_\mu t_Y+{1\over 2}g_wW^a_\mu t_W^a+{1\over 2}g_sG^a_\mu t_C^a&\cr
+g_{Z'}Z_\mu't_{B-L})+1\hat\otimes\Theta(q,z)&\label{extFluctuationdevelopee}
\end{align}
where $q$ and $z$ are respectively quaternion and complex fields and
\begin{equation}
\Theta(q,z)=\begin{pmatrix}
0&\Upsilon^\dagger \tilde q^\dagger&z^*M^\dagger&0\cr
\tilde q \Upsilon&0&0&0\cr
zM&0&0&\Upsilon^T\tilde q^T\cr
0&0&\tilde q^*\Upsilon^*&0
\end{pmatrix}.
\end{equation}
The SM Higgs doublet is, up to a normalization, the second column of the quaternion $q$ seen as a $2\times 2$ complex matrix. The fields $B_\mu$, $W_\mu^a$ and $G_\mu^a$ are the usual SM gauge fields, $Z_\mu'$ is the $Z'$-boson associated to B-L symmetry, and $X_\mu$ is an anomalous $U(1)$-field which is suppressed by the unimodularity condition\cite{Connes-Marcolli}. The Lie algebra generators are all of the form ${\rm diag}(\tau_R,\tau_L,\tau_R^*,\tau_L^*)\otimes 1_N$, where
\begin{eqnarray*}
\mbox{for }t_X:&&\tau_R=\begin{pmatrix}
0&0\cr 0&-2i \end{pmatrix}\oplus \begin{pmatrix} 0&0\cr 0&-2i\end{pmatrix}\otimes 1_3,\cr
&&\tau_L=-i1_2\oplus -i1_2\otimes 1_3,\cr
\mbox{ for }t_Y:&&\tau_R=\begin{pmatrix}0&0\cr 0&-2i\end{pmatrix}\oplus \begin{pmatrix}{4i\over 3}&0\cr 0&-{2i\over 3}\end{pmatrix}\otimes 1_3,\cr
&&\tau_L=-i1_2\oplus{i\over 3}1_2\otimes 1_3,\cr
\cr
\mbox{ for }t_{B-L}:&&\tau_R=\tau_L= -i1_2\oplus\frac{i}{3}1_2\otimes 1_3\cr
\mbox{for }t_{W}^a:&&\tau_R=0,\cr
&&\tau_L=i\sigma^a\oplus i\sigma^a\otimes 1_3 , a=1,2,3\cr
\mbox{ for }t_{C}^a:&&\tau_R=\tau_L=0\oplus 1_2\otimes i {\lambda^a}, a=1,\ldots,8
\end{eqnarray*}
The lack of factor $1/2$ in front of the $Z'$ coupling constant is explained by the Lie algebra generator which we really want to correspond to baryon minus lepton number instead of twice that value.
\section{The predictions of the Spectral Action}\label{predictionSA}
The bosonic part of the action is the so-called \emph{spectral action}
\begin{equation}
S_{b}(D_\omega)=\mathrm{Tr}(f(\frac{D_\omega^2}{\Lambda^2}))
\end{equation}
where $f$ is a cut-off function and $\Lambda$ an energy scale. This action has been computed for general almost-commutative manifolds\cite{Chamseddine-97,Dungen-12}. In particular, proposition 3.7 of Ref.~\onlinecite{Dungen-12}, which does not depend on the first-order condition\cite{Eckstein-18} yields the Lagrangian
\begin{eqnarray}
\calL&=&\frac{f_0}{2\pi^2}(\frac{5}{3}g^2B_{\mu\nu}B^{\mu\nu}+g_w^2W_{\mu\nu a}W^{\mu\nu a}+g_s^2G_{\mu\nu a}G^{\mu\nu a}\cr
&&+\frac{8}{3}g_{Z'}^2Z_{\mu\nu}'{Z'}^{\mu\nu}+\frac{8}{3}gg_{Z'}Z_{\mu\nu}'B^{\mu\nu})\cr
&&-\frac{f_2\Lambda^2}{2\pi^2}\mathrm{Tr}(\Theta^2)+\frac{f_0}{8\pi^2}\mathrm{Tr}(\Theta^4)+\frac{f_0}{8\pi^2}\mathrm{Tr}(D_\mu\Theta D^\mu\Theta)\cr
\end{eqnarray}
where $f_0,f_2$ are two constants (depending on the chosen function $f$). Recall that our normalizations for the fields are given in equation \eqref{extFluctuationdevelopee}.
The traces turn out to be
\begin{eqnarray}
\mathrm{Tr}(\Theta^2)&=&4A|H|^2+2B|z|^2\cr
\mathrm{Tr}(\Theta^4)&=&4\|\Upsilon^\dagger\Upsilon\|^2|H|^4+2\|M^\dagger M\|^2|z|^4\cr
&&+8\bra \Upsilon^\dagger\Upsilon,M^\dagger M\ket |z|^2|H|^2\cr
\mathrm{Tr}(D_\mu\Theta D^\mu\Theta)&=&4A|D_\mu H|^2+2B|D_\mu z|^2\label{3traces}
\end{eqnarray}
where the constants $A$ and $B$ are
\begin{eqnarray}
A&=&\Tr(\Upsilon_e\Upsilon_e^\dagger+\Upsilon_\nu \Upsilon_\nu^\dagger+3\Upsilon_u\Upsilon_u^\dagger+3\Upsilon_d\Upsilon_d^\dagger)\cr
B&=&\Tr(MM^\dagger)=\Tr(mm^\dagger),
\end{eqnarray}
and $H$ is the second column of the quaternion $q$. The first two traces of \eqref{3traces} are immediate. To obtain the third, we can observe that contribution of $q$ and $z$ are orthogonal to each other, so that we are reduced to deal with the cases $\Theta(q,0)=\Phi(q)+\Phi(q)^o$ and $\Theta(0,z)=\sigma(z)$. The first one belongs to the SM and can be found for instance in Ref.~\onlinecite{Dungen-12}, lemma 6.7. To compute the second one we recall that $D_\mu\sigma(z)=\partial_\mu \sigma(z)+[B_\mu,\sigma(z)]$, where $B_\mu$ are the anti-selfadjoint gauge fields. However the generators $t_Y,t_W^a$ and $t_C^a$ all commute with $\sigma(z)$ (which is a manifestation of the fact that $z$ is only charged under B-L). We thus have
\begin{eqnarray}
D_\mu\sigma(z)&=&\partial_\mu\sigma(z)+[Z_\mu't_{B-L},\sigma(z)]\cr
&=&\partial_\mu\sigma(z)+Z_\mu'\sigma(2i z)\cr
&=&\sigma(D_\mu z)
\end{eqnarray}
with $D_\mu z=\partial_\mu z+2iZ_\mu'z$ (from which we can read off that $z$ has B-L charge +2). The needed trace easily follows.
In order to get the gauge kinetic term in the form
\begin{equation}
\frac{1}{4}(B_{\mu \nu}B^{\mu \nu}+ W_{\mu \nu a}W^{\mu \nu a}+ G_{\mu \nu a}G^{\mu \nu a}+ Z_{\mu \nu}'{Z'}^{\mu \nu})+\frac{\kappa}{2}Z_{\mu \nu}'B^{\mu\nu}
\end{equation}
the gauge couplings have to satisfy:
\begin{eqnarray}
g_w^2=g_s^2={5\over 3}g^2={8\over 3}g_{Z'}^2=\frac{\pi^2}{2f_0},& \kappa=\sqrt{2\over 5}.
\end{eqnarray}
These values, including the kinetic mixing $\kappa$, are the same\cite{accomando} as in $SO(10)$ GUT. Similarly, the scalar kinetic term forces us to rescale the Higgses to
\begin{eqnarray}
H=\frac{\sqrt{2}\pi}{\sqrt{f_0A}}\phi,&&
z=\frac{2\pi}{\sqrt{f_0B}}\xi.\label{rescalehiggses}
\end{eqnarray}
When we rewrite the potential as
\begin{equation}
V(\phi,\xi)=\lambda_1|\phi|^4+\lambda_2|\xi|^4+\lambda_3|\phi|^2|\xi|^2+m_1^2|\phi|^2+m_2^2|\xi|^2
\end{equation}
we obtain
\begin{equation}
m_1^2=m_2^2=-\frac{4f_2\Lambda^2}{f_0}
\end{equation}
and the quartic couplings
\begin{eqnarray}
\lambda_1&=&\frac{2\pi^2\|\Upsilon^\dagger\Upsilon\|^2}{f_0A^2}\cr
\lambda_2&=&\frac{4\pi^2\|M^\dagger M\|^2}{f_0B^2}\cr
\lambda_3&=&\frac{8\pi^2\bra \Upsilon^\dagger \Upsilon,M^\dagger M\ket}{f_0AB}.\label{quartcoupSA}
\end{eqnarray}
These equations will be used below to obtain the values of the quartic couplings at the unification scale. But first we need to find the relation between the matrices $\Upsilon$ and $m$ and the mass matrices. For this we turn to the fermionic action. In Euclidean signature, the fermionic action is
\begin{equation}
S_f^{\rm Euc}(\omega,\Psi)=\frac{1}{2}(J\psi,D_\omega\Psi).
\end{equation}
This has the unfortunate consequence of requiring the matrices $\Upsilon$ and $M$ to be anti-hermitian\cite{Dungen-12}. This problem goes away when the correct physical signature is used\cite{bes-20-a}. They are related to the Dirac and Majorana mass matrices by
\begin{eqnarray}
\calM_D=i|H_{\rm min}|\Upsilon,\cr
\calM_R=i|z_{\rm min}|\frac{m}{2}
\end{eqnarray}
where $|H_{\rm min}|,|z_{\rm min}|$ realize the minimum of the potential. To see this we just isolate the Yukawa and Majorana terms in the fermionic action exactly as in Ref.~\onlinecite{bes-20-a} eq. (47) or Ref.~\onlinecite{Dungen-12} eq. (6.9). Reexpressing these relations in terms of fields $\phi,\xi$ thanks to \eqref{rescalehiggses} yields
\begin{eqnarray}
\Upsilon&=&\frac{-i\sqrt{f_0A}}{\pi v}\calM_D\cr
m&=&\frac{-i\sqrt{2f_0B}}{\pi v'}\calM_R
\end{eqnarray}
where $v$ and $v'$ are the vevs of $\phi$ and $\xi$, that is, the values of these fields corresponding to $|H|_{\rm min}$ and $|z|_{\rm min}$. In order to get rid of the vevs we define the coupling matrices $Y$ and $Y_N$ such that
\begin{eqnarray}
\calM_D&=&\frac{1}{\sqrt{2}} Yv,\cr
\calM_R&=&\sqrt{2}Y_Nv',
\end{eqnarray}
and we obtain
\begin{eqnarray}
\Upsilon&=&\frac{-i\sqrt{A}}{2g_w}Y,\cr
m&=&\frac{-i\sqrt{2B}}{g_w}Y_N.\label{decadix}
\end{eqnarray}
Inserting this in \eqref{quartcoupSA} we obtain
\begin{eqnarray}
\lambda_1&=&\frac{1}{4g_w^2}\|Y^\dagger Y\|^2,\cr
\lambda_2&=&\frac{32}{g_w^2}\|Y_N^\dagger Y_N\|^2,\cr
\lambda_3&=&\frac{8}{g_w^2}\bra Y_\nu^\dagger Y_\nu,Y_N^\dagger Y_N\ket.\label{quartcoupSA2}
\end{eqnarray}
From the definitions of $A,B$ and \eqref{decadix} we also obtain the relations :
\begin{eqnarray}
4g_w^2&=&\mathrm{Tr}(Y Y^\dagger),\cr
\frac{1}{2}g_w^2&=&\mathrm{Tr}(Y_NY_N^\dagger).\label{couprel}
\end{eqnarray}
In order to go further we have to make some assumptions on the hierarchy of right-handed neutrino masses. We will consider two different assumptions which are at the opposite extreme from one another and which we describe in the next two subsections.
\subsection{Scenario A: one dominant mass}
We first suppose that one Dirac mass $m_D$ (resp. one Majorana mass $m_R$) dominates the two others. In the appropriate basis we thus have $Y_\nu=y_\nu \Sigma$, where $\Sigma={\rm diag}(1,0,0)$. Similarly, $Y_N$ has also only one non-zero singular value. It can thus be written $Y_N=y_N U\Sigma U^T$ where $U$ is a unitary matrix whose first column only matters. Writing this column $[a,b,c]^T$, we obtain the parametrizations:
\begin{equation}
Y_\nu=\begin{pmatrix} y_\nu&0&0\cr 0&0&0\cr 0&0&0
\end{pmatrix},\quad
Y_N=y_N\begin{pmatrix}
a^2&ab&ac\cr ab&b^2&bc\cr ac&bc&c^2
\end{pmatrix},
\end{equation}
where
\begin{eqnarray}
a&=&\cos\theta e^{i\alpha},\cr
b&=&\sin\theta\cos\varphi e^{i\beta},\cr
c&=&\sin\theta\sin\varphi e^{i\gamma}
\end{eqnarray}
and $\alpha\in [0,\pi[$, $\beta,\gamma\in]-\pi,\pi]$, $\theta,\varphi\in[0,\pi/2[$ (for details see Ref.~\onlinecite{bes-20-a}). The complex phases turn out to have no consequence for the RG flow\cite{bes-20-a} so we can safely assume them to vanish, which makes $Y_N$ hermitian and $\Upsilon$ anti-hermitian, as required by the euclidean signature.
From \eqref{couprel} we then easily obtain
\begin{eqnarray}
y_t&=&\frac{2g_w}{\sqrt{3+\rho^2}},\cr
y_\nu&=&\rho y_t,\cr
y_N&=&\frac{g_w}{\sqrt{2}},\label{YukawaA}
\end{eqnarray}
where the second line is a definition of $\rho$. Now from \eqref{quartcoupSA2} and \eqref{YukawaA} we derive\footnote{It is useful to observe that $Y_N^\dagger Y_N=y_N^2U^* \Sigma U^T$.} the quartic couplings:
\begin{eqnarray}
\lambda_1&=&\frac{3+\rho^4}{(3+\rho^2)^2}4g_w^2\cr
\lambda_2&=&8g_w^2\cr
\lambda_3&=&\frac{16\rho^2g_w^2\cos^2\theta}{3+\rho^2}\label{quartinitSA}
\end{eqnarray}
One can check that the values of $\lambda_1,\lambda_2,\lambda_3$ for $\theta=0$ are consistent with Ref.~\onlinecite{Chamseddine-12}.
\subsection{Scenario B: universal coupling}
We now make the opposite assumption that $Y_\nu$ and $Y_N$ are both close to the identity: $Y_\nu=y_\nu I_3$ and $Y_N=y_N I_3$. We easily obtain by computations similar to those of the previous subsection the following initial conditions:
\begin{eqnarray}
y_t&=&\frac{2g_w}{\sqrt{3(1+\rho^2)}},\cr
y_\nu&=&\rho y_t,\cr
y_N&=&\frac{g_w}{\sqrt{6}},\cr
\lambda_1&=&\frac{1+\rho^4}{3(1+\rho^2)^2}4g_w^2,\cr
\lambda_2&=&\frac{8g_w^2}{3},\cr
\lambda_3&=&\frac{16\rho^2g_w^2}{3(1+\rho^2)}.\label{initSA}
\end{eqnarray}
\begin{rem}
Since $Y_N$ is required to be hermitian, having all its singular values equal forces it to be a multiple of the identity matrix. However one can wonder if it is an artifact coming from using the wrong metric signature. Maybe we are missing interesting cases where $Y_N$ has a single singular value and is not proportional to the identity. In that case we would have $Y_N=y_N UU^T$ with $U$ unitary as above. However equations \eqref{initSA} would remain unchanged. Indeed, the matrix $V=UU^T$ is unitary, and so is $M$, but $M$ enters \eqref{quartcoupSA} only through $M^\dagger M$. Moreover, one can see from the RGE (see appendix \ref{appendixRGE}) that $Y_\nu$ and $Y_N Y_N^*$ will stay proportional to the identity matrix. In these conditions, no running of any coupling will depend on $V$ and one can assume without loss of generality that $V=I_3$.
\end{rem}
\section{Constraints from symmetry breaking}\label{constraintsSB}
\subsection{Derivation in the general case}
In terms of the rescaled fields, the potential is $V(\phi,\xi)=q(|\phi|^2,|\xi|^2)$ where $q$ is the quadratic form
\begin{equation}
q(x,y)=\lambda_1x^2+\lambda_2y^2+\lambda_3xy+m_1^2(x+y).
\end{equation}
In order to have a minimum for the potential, the determinant $\lambda_1\lambda_2-\lambda_3^2/4$ of $q$ must be non-negative and this yields a first constraint on the quartic couplings at unification. Moreover, to have symmetry breaking, we need this minimum to be reached for non-vanishing $\phi$ and $\xi$. This means that the quadratic form $q$ must have a local minimum inside the positive quadrant. Leaving aside the non-realistic case of vanishing determinant, $q$ is positive definite and thus has a unique global minimum in $\RR^2$ at $(x_{\min},y_{\min})$ such that $\nabla q(x_{\min},y_{\min})=0$. Thus $x_{\min}$ and $y_{\min}$ are required to be positive and are the solutions of
\begin{eqnarray}
2\lambda_1 x+\lambda_3 y&=&-m_1^2\cr
\lambda_3 x+2\lambda_2 y&=&-m_1^2
\end{eqnarray}
which are
\begin{eqnarray}
x_{\min}&=&-m_1^2\frac{2\lambda_2-\lambda_3}{4\lambda_1\lambda_2-\lambda_3^2}\cr
y_{\min}&=&-m_1^2\frac{2\lambda_1-\lambda_3}{4\lambda_1\lambda_2-\lambda_3^2} \label{xminymin}
\end{eqnarray}
Since we assumed $4\lambda_1\lambda_2-\lambda_3^2> 0$ and we know $-m_1^2> 0$, this yields the constraints
\begin{equation}
\lambda_3< 2\min(\lambda_1,\lambda_2)\label{constraint}
\end{equation}
Conversely, observe from \eqref{quartcoupSA} that $\lambda_3>0$ (recall that the scalar product of two positive definite matrices is itself positive). It follows that \eqref{constraint} implies $4\lambda_1\lambda_2-\lambda_3^2> 0$. Hence \eqref{constraint} is a necessary and sufficient condition for both Higgses to have a non-zero vevs. If it is not met but $\det(q)$ is still $>0$, then the minimum of $q$ on the positive quadrant is reached on one of the axes. It is easy to know which one : if $\lambda_1<\lambda_2$ then $q(t,0)<q(0,t)$ for all $t>0$, so that the minimum is on the $x$-axis. The situation is symmetrical if $\lambda_2<\lambda_1$.
Let us summarize what we have shown:
\begin{itemize}
\item The electroweak and B-L symmetries are both broken iff $\lambda_3< 2\min(\lambda_1,\lambda_2)$.
\item If $4\lambda_1\lambda_2-\lambda_3^2>0$ and $2\lambda_1<\lambda_3<2\lambda_2$ the electroweak symmetry is broken, but not the B-L symmetry.
\item If $4\lambda_1\lambda_2-\lambda_3^2>0$ and $2\lambda_2<\lambda_3<2\lambda_1$ the B-L symmetry is broken, but not the electroweak symmetry.
\end{itemize}
Let us close this section by observing that even if the RHS of \eqref{xminymin} are both positive, they will be naturally of the same order of magnitude unless the quartic couplings are extremely fine-tuned. Indeed, calling $v$ and $v'$ the respective vevs of $\phi$ and $\xi$, one has $x_{\rm min}=v^2/2$ and $y_{\rm min}={v'}^2/2$ so that \eqref{xminymin} yields:
\begin{equation}
\frac{v^2}{{v'}^2}=\frac{2\lambda_2-\lambda_3}{2\lambda_1-\lambda_3}
\end{equation}
In order to have $v$ of the order $10^2$ GeV and $v'$ of the order $10^{14}$ GeV, and considering the denominator to be of order $1$, this means that we should have
\begin{equation}
\lambda_3=2\lambda_2-\epsilon, \label{finetuning}
\end{equation}
with $\epsilon\approx 10^{-24}$ ! This problem is very similar to the doublet-triplet splitting problem plaguing Grand Unified Theories\cite{Ilio}. It arises because the normalization of the kinetic term cancels the factors $A$ and $B$ in $\mathrm{Tr}(\Theta^2)$. This is a peculiarity of the spectral action: it is interesting to observe that this does not happen with the Connes-Lott action\cite{bes-20-a}. Note that in the above argument we have estimated the order of magnitude of the vevs using tree-level masses relations. To make it more precise we should take the running of the vevs into account and use renormalized mass formulas, which can be very involved\cite{sperling2013renormalization,irges2017renormalization,dudenas2020vacuum}. Anyway we cannot expect the renormalization effects to make the masses of the $W$ and $Z'$ bosons approximately equal at unification scale and separated by $12$ orders of magnitude at low energy in any natural way. At the very least, we can assume that the splitting of the mass scales require $\lambda_3$ to be much closer to $2\lambda_2$ than $2\lambda_1$. From \eqref{constraint} we then conclude that
\begin{equation}
\lambda_2<\lambda_1
\end{equation}
is an absolute necessity.
\section{The constraints in scenario A}\label{scenarA}
The first thing we can notice from \eqref{quartinitSA} is that $\lambda_1<\lambda_2$. It is thus impossible to simultaneously have $v$ and $v'$ both non-vanishing \emph{and} solve the ``mass-splitting problem''. We thus give up on the second issue and try to see if at least we can satisfy condition \eqref{constraint}. It is easy to see from \eqref{quartinitSA} that the stability condition $4\lambda_1\lambda_2-\lambda_3^2>0$ is satisfied so that \eqref{constraint} translates as
\begin{equation}
\cos^2\theta \le \frac{3+\rho^4}{2\rho^2(3+\rho^2)}\label{finalcond}
\end{equation}
Now we run down the RGE using the initial values given as functions of $g_w$ which we run up to the unification energy. We test values of $(\rho,\theta)$ in a lattice of spacing $10^{-2}$ in $\rho$ and $10^{-1}$ in $\theta$. For each point $(\rho,\theta)$ we compute the relative standard deviation $RSD_m$ of the top quark and Higgs bosons masses with respect to their experimental values. More precisely we compute
\begin{equation}
RSD_{m}=\frac{\sqrt{\scriptstyle{2((m_t(172)-172)^2+(m_h(125)-125)^2})}}{\scriptstyle{172+125}}.
\end{equation}
We see on this formula that the predicted pole masses are approximated by the running masses at the energy scale of the experimental pole masses. This simplification introduces only a very small error when the predicted pole masses are within a few percent of the experimental values. In table \ref{randomsearchSA} we display the interval to which $\rho$ must belong in order for $RSD_m$ to be less than $0.05$ for at least one value of $\theta$, for different values of $\mu_{\rm unif}$. We also displayed these results when a threshold correction is applied. Indeed, the $Z'$ boson and the complex scalar $\xi$ must have very high masses under which their couplings disappear from the RGE. We choose a threshold energy of $10^{14}$ GeV which is justified by the bounds on light neutrino masses. For a more detailed discussion, as well as the matching conditions, see Ref.~\onlinecite{bes-20-a}.
\begin{table}[hbtp]
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$\log_{10}(\mu/{\rm GeV})$ & 18 & 17 & 16 & 15 \\
\hline
$RSD_m<0.05$ (*) & $[1.05,1.45]$ & $[1.04,1.45]$ & $[1.02,1.40]$ & $[1.00,1.35]$ \\
\hline
\hline
$RSD_m<0.05$ (**) & $[1.33,1.76]$ & $[1.33;1.69]$ & $[1.32;1.66]$ & $[1.32,1.60]$ \\
\hline
\end{tabular}
\caption{Intervals of $\rho$ for which the standard deviation of the predicted masses of the top quark and Higgs boson becomes $<0.05$ for at lest one value of $\theta$. (*) no threshold correction. (**) a threshold is applied at $10^{14}$ GeV.}\label{randomsearchSA}
\end{table}
Thus, we see that agreement with expirements can be achieved in this model for some values of $(\rho,\theta)$. However, we find that for these values \eqref{finalcond} is \emph{never} satisfied. This can be seen in figure \ref{const1}, where we have plotted as functions of $\rho$ the minimal value of $\cos^2\theta$ among the values of $\theta$ which yield $RSD_m<0.05$ (in blue) and the RHS of \eqref{finalcond} (in red). We see that there exists a large gap between these curves, whatever the value of $\mu_{\rm unif}$. Applying a threshold correction only worsen the situation (figure \ref{const2}). We conclude from this study that in scenario A, and for the values of the parameter yielding an acceptable $RSD_m$, the B-L symmetry is not broken.
\begin{figure}[hbtp]
\includegraphics[scale=0.45]{minthetamu18et15nothresh.eps}
\caption{The blue curve is the minimum value of $\cos^2\theta$ among accepted $\theta$'s for a given $\rho$. If no $\theta$ is accepted the default value is $1$. The red curve represents the RHS of \eqref{finalcond}. Blue solid curve: $\mu_{\rm unif}=10^{18}$ GeV. Blue dashed curve: $\mu_{\rm unif}=10^{15}$ GeV. No threshold correction.}\label{const1}
\end{figure}
\begin{figure}[hbtp]
\includegraphics[scale=0.45]{const1.eps}
\caption{Same as figure \ref{const1} with a threshold at $10^{13}$ GeV.}\label{const2}
\end{figure}
\section{The constraint in scenario B}\label{scenarB}
In that case the stability condition is seen to be equivalent to $\rho<1$. We also have $\lambda_1<\lambda_2$, so that the mass-splitting problem is once again hopeless (in fact in this case \eqref{finetuning} is even impossible !) and we focus on \eqref{constraint}, which is easily translated thanks to \eqref{initSA} into
\begin{equation}
1-2\rho^2-\rho^4>0
\end{equation}
This means that $\rho$ has to satisfy
\begin{equation}
0\le\rho\le\sqrt{\sqrt{2}-1}\approx 0.64\label{constraintrho}
\end{equation}
In figure \ref{const3} we have drawn $RSD_m$ as a function of $\rho$ for different values of $\mu_{\rm unif}$. We see that in the region where \eqref{constraintrho} is satisfied (shaded in pink), the $RSD_m$ does not go below $0.06$, yielding only poor agreement with the experimental values of the top and Higgs masses. To make this more visible, we have displayed the predicted masses for the maximal value $\rho=0.64$ in table \ref{higgsculprit}, where we see that the Higgs mass is the culprit. Once again, the threshold corrections worsen the problem (figure \ref{const4}).
\begin{figure}[hbtp]
\includegraphics[scale=0.45]{rsdmrhoB.eps}
\caption{Plot of $RSD_m$ as a function of $\rho$ with values of $\mu_{\rm unif}$ ranging from $10^{15}$ to $10^{18}$ GeV. The region shaded in pink corresponds to values of $\rho$ satisfying constraint \eqref{finalcond}.}\label{const3}
\end{figure}
\begin{table}[hbtp]
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$\log_{10}(\mu/{\rm GeV})$ & 18 & 17 & 16 & 15 \\
\hline
$m_{\rm top}(172)$ & 171 & 170 & 169 & 168 \\
\hline
$m_{\rm Higgs}(125)$ & 147 & 146 & 144 & 142 \\
\hline
\end{tabular}
\caption{Predicted running top and Higgs masses in scenario B for $\rho=0.64$.}\label{higgsculprit}
\end{table}
\begin{figure}[hbtp]
\includegraphics[scale=0.45]{rsdmrhoBthresh14.eps}
\caption{Same as figure \ref{const3} with a threshold at energy $10^{14}$ GeV.}\label{const4}
\end{figure}
Thus in scenario B, we have seen that the B-L symmetry is not broken, unless we neglect the threshold corrections (which does not seem justified) and accept a prediction for the Higgs mass around $140$ GeV. As for the mass-splitting problem, the conclusion is the same as in scenario A: we cannot even solve it since $\lambda_1<\lambda_2$.
\section{Conclusion}\label{conclusion}
In this study we have shown that in the spectral B-L model, there exists an important tension between the experimental masses of the top and Higgs on the one hand and the necessity of breaking the B-L symmetry on the other. Are these results robust ? We have admitedly made several simplifications: we considered only the 1-loop RGE, ignored gravity, and studied only two scenarios for the neutrino couplings among an infinity of possibilities. However, the threshold corrections are dominant with respect to 2-loops effects \cite{bes-20-a} and worsen the problem, so we do not expect the 2-loop corrections to improve the situation. Gravity might have an important contribution when $\mu_{\rm unif}$ is taken to be close to the Planck scale, but is likely to be negligible when it is in the lower range of $10^{15}-10^{16}$ GeV, and as can be seen on figure \ref{const3} a higher unification energy makes the problem worse. Finally the two scenarios we considered for the neutrino couplings lie at opposite extremities of the spectrum. So we believe that our results are indeed robust. Moreover, we have seen that there exists a mass-splitting problem which seems to be even more serious, since it cannot be solved at all in the two scenarios we have considered, regardless of the experimental input.
What are the potential implications of these findings for the Noncommutative Geometry program ? It is known that in order to get the Higgs mass right, one needs to supplement the spectral SM by at least one scalar field. Since one does not want to ruin the beauty of the NCG approach by adding this field by hand, the most natural solution is to enlarge the spectral triple\footnote{For other approaches see Refs.~\onlinecite{Devastato-14,Farnsworth-15}.}, and the B-L extension presented here is arguably the simplest solution. As shown here, it probably does not work. The next obvious step is to go to the full Pati-Salam model, although this might raise other problems\cite{bes-20-b}. Moreover the RGE analysis of the Pati-Salam model depends on many assumptions so that the approach loses predictivity. Moreover, even if it turns out that there is possibility to break B-L symmetry in the spectral Pati-Salam model, something like equation \eqref{xminymin} probably remains true since it follows directly from the structure of the spectral action, so that it is not likely that the ``mass-splitting problem'' will go away, although this requires more investigations.
Other possible ways out could be found. First, the RGE used here might just not be adapted to the spectral action. What we have done, following Ref.~\onlinecite{Chamseddine-12} (among others), is to stop doing Noncommutative Geometry once the action is delivered, and proceed with the renormalization of the field theory with the appropriate Lagrangian, forgetting its geometrical origin. A recent work\cite{ChIlSu} might improve the situation in this respect, and change the conclusions of the present RGE analysis. We do not see, however, how it would affect the mass-splitting problem.
The issue could also be related to some conceptual problem in the theory itself. The euclidean nature of the spectral action comes to mind. Only time will tell if the issue raised here is a serious one, but even in that case that might not be necessarily a bad thing: healthy research program progress by overcoming both conceptual and experimental problems, and Noncommutative Geometry has already shown to be a healthy research programs several times in the past. Finally let us conclude by observing that neither of the two problems raised in this paper appears with the Connes-Lott action\cite{bes-20-a}, and this only adds to its merits.
\section{Acknowledgments}
I am indebted to Christian Brouder, Vytautas D{\=u}d{\.e}nas, Nikos Irges, Maximilian Löschner, Dominik Stöckinger, and Alexander Voigt, for very insightful discussions.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,485 |
taiga clematis adaptations
We hope you are enjoying Gardenerdy! It is mandatory to procure user consent prior to running these cookies on your website. Clematis 'Taiga' is a stunning Clematis that produces an abundance of unusual double flowers with striking violet-purple petals and contrasting vivid lime-green tips. Short-listed for Chelsea Plant of The Year 2017, Clematis Taiga has unusually large, purple and lime-colored flowers. Even shrubs and flowering plants are found in clearings. w�E�W���K����PB���d��HP�Z3"��J"%B:"ae���Ѡ��)��m�Y^��d��ce�9#?`�0 �6���Ky�I�:w1ǐ�B7_� ���n0N�*�$L ����"Ǿ҇N��KI{�d��a������~�>zs���]�9��v�6Y� d��H����y�g���W��. Its flowers are truly one of a kind. The southern regions are dense with trees, and are characterized by a closed canopy. In large amounts, these chemicals form a bluish haze in the atmosphere. Some of the large animals found in the taiga include moose, deer, and bears. Coniferous trees comprise a major part of the plant life in the taiga biome. Although the taiga biome does not offer favorable conditions for plant and animal life, these regions are not barren. As wildfires burn down the thick canopy, sunlight falls on the ground, thereby triggering germination of grasses. Clematis 'Taiga' Other names. Clematis 'Taiga' is a real show off climber, producing countless blooms throughout summer over a canvas of rich green foliage. endstream endobj 11 0 obj <> endobj 12 0 obj <> endobj 13 0 obj <>stream Clematis in pots should be given a liquid feed through the summer and early autumn. There are some plant and animal species that are adapted to the conditions of the taiga. The fallen pine needles on the ground do not decompose easily. We'll assume you're ok with this, but you can opt-out if you wish. 37 0 obj <>stream Take one look at Clematis 'Taiga' and you will see why. They become tolerant to very low temperatures. They grow in the dark understory of the forest, and are often found at the base of photosynthetic coniferous trees. � The taiga is a biome located south of the Arctic tundra and north of the temperate deciduous forests. The two tendrils at the base of the leaf stalk of Smilax probably represent stipules, and in clematis the leaf stalks themselves are able to coil round the stems of other plants, and so act as tendrils. Clematis 'Taiga' The double flowers of this late-blooming clematis are quite unusual. One such adaptation is the thick bark. Flowers through the summer. In return, they provide food to these fungi. Stunning double purple flowers with yellow-green tips are abundant in summer. ADD add to wishlist. The taiga biome amounts to around 30% of the world's forest cover. The taiga or boreal forests is a biome characterized by coniferous forests with pines, larches, and spruces as the dominant vegetation. This website uses cookies to improve your experience. The ghost plant is white, without any chlorophyll. 0 The plant can reach a width of 1 metre and height of 2.5 metres and should be grown in the sun. One species that does manage to survive in the cold north is the boreal chorus frog, an amphibian found in Canada and parts of the United States.. The roots spread wide so as to provide anchorage, and to absorb moisture and nutrients from a larger area. Botanical name. The snow that falls on the tree slides off easily, so that the branches do not break. Sometimes, the branches bend down due to snow buildup. It is characterized by long and cold winters and short summers. A unique new clematis, Taiga has the most extraordinary flowers, that change through the summer. Taiga Clematis Spacing. Clematis Florida Taiga found in: Clematis urophylla 'Winter Beauty', Clematis 'New Love', Clematis napaulensis, Clematis 'Miss Bateman', Clematis x.. across (12-18 cm) from early to late summer. %%EOF Taiga plant life is much less diverse, as compared to the rainforests. Purple Queen Of The Vines Plants For Sale | Clematis - Taiga. Take one look at Clematis 'Taiga' and you will see why. Coniferous trees shed their leaves on a regular basis, but they shed only a few leaves at a time, and the loss is unnoticeable. The habitat of a biome is determined by the climatic conditions of the place. La Clématite ou Clematis Taiga, obtenue tout récemment au Japon, est une véritable oeuvre d'art, unanimement primée au Plantarium 2016 dans la catégorie Jardin, balcon et plantes de patio. This is to conserve energy, which is required for growing new leaves after shedding. Clematis 'Taiga' (PBR) clematis (group 3) 5 5 1 star 1 star 1 star 1 star 1 star (4 reviews) Write review. 'Taiga' is a hardy climber that will trail up any fence, mailbox, arbor or wherever you want to enjoy these uniquely colorful blooms. × New and Unread Tree-Mails. Clematis in pots should be given a liquid feed through the summer and early autumn. This stunning new clematis was bred in Japan in 2007. Although the taiga biome does not offer favorable conditions for plant and animal life, these regions are not barren. Well, we're looking for good writers who want to spread the word. Trees like spruce may retain their leaves for around 15 years. Clematis 'Taiga' PBR . The Taiga Biome receives limited precipitation but it has many lakes and swamps that will attract birds. Clematis Clematis. These trees shed leaves during fall and regrow them during spring. Early in the season, deadheading is beneficial. 10 0 obj <> endobj Certain regions have a comparatively longer summer, and a warm and humid weather. This […] When it comes to the plants and trees in the taiga biome, you may identify two patterns of vegetation. Timing, Timing, Timing: When to Mulch Your Garden & Yard. It is difficult for the roots to grow deeper, as the soil beneath the surface is frozen. Forest fires are common in the taiga biome. In short, coniferous trees and mycorrhizal fungi share a mutually beneficial relationship. This prevents accumulation of snow on their branches. Throughout the season, the colours and shape of the flowers change. The taiga is a biome located south of the Arctic tundra and north of the temperate deciduous forests. By the end of summer, they have a pom-pom form. Layering is one of the reasons why these trees grow close to one another. Some of these adaptations include their shape, leaf type, root system, and color. With parentage from integrifolia and florida, it promises an ongoing show of … They release chemicals called terpenes, which have a pleasant smell. Taiga plants have to be hardy in order to survive not only the long, cold winter, but also the poor-quality soils typical of the biome. But opting out of some of these cookies may have an effect on your browsing experience. The taiga/boreal forest is the largest among the terrestrial biomes in the world, and stretches over North America and Eurasia. The habitat of a biome is determined by the climatic conditions of the place. Buy your Clematis Taiga direct online from Garden Centre Gardens4You.co.uk Money Back Guarantee If you are unhappy with the quality of the bulbs or plants at arrival or they fail to grow in their first growing season we will refund you or resend the products for free. The star-shaped blooms are blessed with 6 to 8 slightly overlapping tepals and a central tuft of contrasting red anthers. The fire provides suitable conditions for these cones to open and disperse seeds. This method is termed layering. The soil being thin and rocky, most of the coniferous trees have a shallow root system. Clematis 'Ramona' is a ravishing deciduous climber which produces an abundance of large, single, pale lavender-blue flowers, 5-7 in. Even the soil is thin, acidic, rocky, and infertile. They include birch, aspen, rowan, alder, balsam poplar, etc. Most of the taiga animals, like snowshoe rabbits and black bear, have a thick fur to protect themselves from the cold weather. But, there are certain plants that are parasitic on these fungi. Taiga: Plants Because the climate of the taiga is very cold, there is not a large variety of plant life. The purple petals have a greeninsh cream tip.The flowers produce layers of petals and appear in abundance in midsummer, and usually continue well into the autumn. Though coniferous trees are prominent in this biome, some types of deciduous trees, shrubs, flowering plants, grass, etc. The taiga biome is spread over continents and countries. More. endstream endobj startxref Clematis Taiga is a recently introduced variety, originally exclusive to Gardening Express, we have stocks of this amazing new variety available for immediate delivery, plus they're a decent size, ready to plant straight out in the garden, supplied in 2-3 litre containers. Some of them hibernate during winters. This is one of the best Waitrose Garden … 'Taiga' is an exotic looking clematis and is ideal as a feature plant in a patio container or against a south facing wall. The Picasso among plants! If the trees are exposed to such temperatures, before they harden, it may cause frostbite, which in turn can damage or kill them. There are some plant and animal species that are adapted to the conditions of the taiga. h��Vmo�8�+���D��i�Da�"�j��+!>����A!�ʿ��8��Z�U��xf�y2��! This species blooms from June to August. Adaptations shown by Climbing Plants. ... Plant adaptations Evergreens use a wide variety of physical adaptations. This is also said to be an adaptation that protects them from wind and cold. Although the taiga biome does not offer favorable conditions for plant and animal life, these regions are not barren. Except tamarack, other coniferous trees in the taiga biome are evergreen, and do not shed their leaves. Clematis Taiga. Taiga animals have thick furs and other special adaptations. Coniferous trees in the taiga biome are evergreen, except the tamarack that sheds leaves during fall, The fungus covers the roots, and forms a network of string-like structures, Forest fires trigger jack pine cones to open and disperse the seeds. are also seen. The subarctic is an area of the Northern Hemisphere that lies just south of the Arctic Circle.The taiga lies between the tundra to the north and temperate forests to the south. However, the tree species may vary from one region to another. Adaptations shown by Climbing Plants. All plants have features (adaptations) which help them to survive and reproduce in the places where they live (their habitat) Trees lose water through their leaves. Copyright © Gardenerdy & Buzzle.com, Inc. Clematis 'Taiga' Will grow in Full Sun Hardy Perennial Loved by Pollinators Needs support Will grow in Partial Shade Perfect for Pots Based on 2 reviews. Most of the coniferous trees grow in a conical shape, with drooping branches. The 3- to 6-inch (7.5–15 cm) flowers open semi-double, then fill in with extra petals until they're about as double as a flower can be. Apart from preventing water loss during winter, such shedding reduces the risk of snow buildup and breaking of branches. Out of these cookies, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. However, tamarack trees shed their needles during fall to avoid water loss during winter. You should dig the hole large enough to accommodate the plant, with most recommendations suggesting at least a two foot depth of soil amended with compost prior to planting. Pruning Your Clematis: 'Taiga' flowers in late summer on growth made in that season and should be pruned in late winter or early spring - simply cut back to just above a strong … This […] They are a breeze to grow and prune, simply cut back the stems to just above ground level at the seasons end. A major part of the taiga is located in Canada and Russia. A flower can be like a piece of art. The pinesap plant and the ghost plant belong to the same genus. Even the plants have some amazing adaptations. A flower can be like a piece of art. in stock (shipped in 3-5 working days) Quantity 1 Plus Minus. It is said that terpenes thicken clouds, which in turn results in cooling of the Earth! Some have attractive fluffy seedheads in autumn Sign up to receive the latest and greatest articles from our site automatically each week (give or take)...right to your inbox. During winters, coniferous trees undergo a process called hardening, to survive the freezing temperature. A true garden standout, Clematis Taiga works well combined with other plants in the landscape or solo in containers. An exciting new hybrid with unusually colored double flowers. Our site includes quite a bit of content, so if you're having an issue finding what you're looking for, go on ahead and use that search feature there! While it is difficult for most of the plant species to survive in the taiga biome, those which are found in the region are well-adapted to these climatic conditions. It's cold hardy to zone 6, and is a moderate grower with a climbing habit. across (12-18 cm) from early to late summer. The Picasso among plants! The white, thread-like mycorrhizal fungi that grow on the roots of coniferous trees help in decomposing pine needles. In summer and autumn, 'Taiga' blooms abundantly with bright blue-purple flowers. Taiga: Plants Because the climate of the taiga is very cold, there is not a large variety of plant life. Necessary cookies are absolutely essential for the website to function properly. A spectacular, showy clematis which looks like a passion flower, but is hardier and easier to grow. The most common type of tree found in the taiga is the conifer--trees that have cones. h�b``�d``Rc ��B���Y8��8���!���a�ä[]�*�[email protected]����n��� The Taiga Clematis is a continuous feed and fertilizer should be applied regularly. Size and nature of doubling will change through the seasons. We provide informative articles about gardening, lawn care and landscaping that you can come back to again and again when you have questions or want to learn more! However, these trees are adapted to the fire in different ways. 6789 Quail Hill Pkwy, Suite 211 Irvine CA 92603. Other common names clematis 'Taiga' Family Ranunculaceae Genus Clematis can be deciduous or evergreen shrubs or herbaceous perennials, mostly climbing by twining leaf-stalks, and often with showy flowers. The northern parts have very few trees, with drought-resistant lichens covering the ground. Clematis Taiga – the verdict. The soil is quite acidic and has few minerals. Some types of berries are also seen in the southern regions. Thus the trees get enough nutrients for photosynthesis. The taiga is a forest of the cold, subarctic region. Examples of smaller animals that live in the taiga are bobcats, squirrels, chipmunks, ermine, and moles. In these regions, the ground is covered with moss, especially peat moss. Clematis plants need plenty of space for adequate air flow as well as a rich, well-draining planting area. Would you like to write for us? Prune to around 50cm. It is covered by a deep layer of partially-decomposed conifer needles. Victory Gardening in 2020: Spring is Not Cancelled, Green Pavement Creates Beautiful Environmental Solutions, How to Use & Source Eco Mulch for Your Garden. The subarctic is an area of the Northern Hemisphere that lies just south of the Arctic Circle.The taiga lies between the tundra to the north and temperate forests to the south. If you're looking to use the Taiga Clematis for privacy purposes and looking for a seamless appearance, space it two to three feet, center on center. They include pines, spruces, larches, and firs. Unlike the broad leaves of deciduous trees, these needles do not lose much water through evaporation. Clematis Taiga is offered in a full gallon size with free shipping! The habitat of a biome is determined by the climatic conditions of the place. The two tendrils at the base of the leaf stalk of Smilax probably represent stipules, and in clematis the leaf stalks themselves are able to coil round the stems of other plants, and so act as tendrils. Cette variété de taille modeste est réellement unique en raison de ses grandes fleurs très doubles, évoquant un peu par leur architecture très singulière celles des dahlias cactus. The taiga biome has deciduous trees in some regions where the winter temperature is not very low. Cette variété de taille modeste est réellement unique en raison de ses grandes fleurs très doubles, évoquant un peu par leur architecture très singulière celles des dahlias cactus. Taiga: Animals The cold climate of the taiga prevents many animals from living there year-round. Though coniferous trees can make their own food through photosynthesis, they face a dearth of nutrients. They conserve heat during winter, and shed snow easily. Plant database entry for Clematis (Clematis florida 'Taiga') with 12 images and 34 data details. The most common type of tree found in the taiga is the conifer--trees that have cones. %PDF-1.5 %���� The star-shaped blooms are blessed with 6 to 8 slightly overlapping tepals and a central tuft of contrasting red anthers. They produce cones on the top branches that are located far from the ground. Clematis Taiga. It is also said that the dark green color of the leaves enables the trees to fasten the process of photosynthesis, by absorbing sunlight at a faster rate. The taiga is a forest of the cold, subarctic region. The pine sap too is dependent on the mycorrhizal fungi for food. These cookies do not store any personal information. Clematis are a garden favorite that are hardy bloomers that only get better as the years go by. The rich violet flowers, first bloom as singles, and become double blooms as the season progresses. Extraordinary news from our trial gardens: as a southern vine, the general assumption was that Taiga would perform best in zone 7 and south. It is covered by a deep layer of partially-decomposed conifer needles. All these plants and trees are adapted to the specific climatic conditions of the biome. The taiga is a home for those birds, which feed on the conifer seeds and berries. Though the climatic conditions of the taiga biome are not favorable for plant life, certain plants thrive well in these regions. These cookies will be stored in your browser only with your consent. Alaska, Canada, Scandinavia, and Siberia have taigas. Most of them migrate to nearby areas during snowfalls and food scarcity. Conifer trees are very common in the taiga biome. Animals found in Taiga Biome. Clematis 'Taiga', Clematis florida 'Taiga' Genus. Some of them, like the black spruce and jack pine have a special adaptation. Mycorrhizal fungi and coniferous trees share a mutually beneficial relationship. Is It Worth My Time to Use Natria Weed Killer? In short, these trees have leaves throughout the year, and they can start photosynthesis, as soon as they receive sunlight. This Wild Clematis is scrambling over other plants in the hedgerow using its leaf stalks to cling to them. With unmatched color, the glorious double purple and yellow-green flowers of Taiga Clematis add a wow factor to the garden. This category only includes cookies that ensures basic functionalities and security features of the website. We also use third-party cookies that help us analyze and understand how you use this website. They include the ghost plant (Indian pipe plant), pinedrops, and pinesap. 2 lt pot (60cm cane) £22.99. ��0 � While major fires destroy most of the coniferous trees, minor ones may not cause much damage. ... Plant adaptations Evergreens use a wide variety of physical adaptations. Pruning Your Clematis: 'Taiga' flowers in late summer on growth made in that season and should be pruned in late winter or early spring - simply cut back to just above a strong … Throughout the season, the colours and shape of the flowers change. Variety or Cultivar 'Taiga' _ 'Taiga' is a deciduous, perennial climber with pinnate leaves divided into ovate or lance-shaped, dark green leaflets and, from early summer to early autumn, large flowers with double, green-tipped, violet inner petals surrounded by lavender outer petals. h�bbd``b`[email protected]��H�$ ��@��$�2D c9 a� The roots of these plants extract nutrition from the mycorrhizal fungi. Adaptations of Plants in the Taiga Biome. Fore more new varieties check www.plantipp.eu or visit facebook.com/Plantipp Scientific name: Pseudacris maculata Conservation status: Least Concern The taiga, with its long winters and cold climate, is not an ideal habitat for amphibians. The population of insects is the highest, followed by that of birds, fish, and mammals. Adaptations of Plants in the Taiga Biome. If such branches come in contact with soil, they may develop roots and grow into new plants, which are totally identical to the parent plant. Clematis 'Kokonoe' A gorgeous clematis variety, Clematis 'Kokonoe' is a sister cultivar of our best selling Clematis 'Taiga'. It is a cold, inhospitable forest habitat in which winter can last for up to nine months. This website uses cookies to improve your experience while you navigate through the website. From summer through fall, this climber will delight you with loads of eye-catching blossoms.Flowers can grow to between 3 and 6 inches across, for a tropical look that thrives in the sun. Though the pinedrops plant is from a different genus, the method of deriving nutrition is the same as the ghost plant, by extracting nutrition from the mycorrhizal fungi. The soil is quite acidic and has few minerals. Its flowers are truly one of a kind. H ef�$)'�3��` �� Clematis 'Ramona' is a ravishing deciduous climber which produces an abundance of large, single, pale lavender-blue flowers, 5-7 in. The Taiga Biome receives limited precipitation but it has many lakes and swamps that will attract birds. Clematis,climbing plants and perennials specialist It is a cold, inhospitable forest habitat in which winter can last for up to nine months. You also have the option to opt-out of these cookies. 21 0 obj <>/Filter/FlateDecode/ID[<726D2028B6DEB54AA90274958DAC1B81>]/Index[10 28]/Info 9 0 R/Length 68/Prev 45096/Root 11 0 R/Size 38/Type/XRef/W[1 2 1]>>stream Coniferous trees are seen in large groups growing very tall and close. These trees share have some special features that help them survive in the taiga. Close × Share This Page. Height is 6 - 8ft. In summer and autumn, 'Taiga' blooms abundantly with bright blue-purple flowers. Some of these adaptations include their shape, leaf type, root system, and color. Alaska, Canada, Scandinavia, and Siberia have taigas. Even some types of fungi depend on wildfires for releasing their spores. Its compact habit makes it ideal for covering walls and fencing, or growing in a container as a spectacular patio feature. La Clématite ou Clematis Taiga, obtenue tout récemment au Japon, est une véritable oeuvre d'art, unanimement primée au Plantarium 2016 dans la catégorie Jardin, balcon et plantes de patio. The needle-like leaves of coniferous trees are long, thin, and waxy. Though the Taiga biome is characterized by coniferous forests, some deciduous trees are also found in certain regions. Clematis is a woodland vine renowned for its vigorous growth and stunning flower display. Walled Garden Nursery. Different species of the same genus may grow in different regions. We've now conducted over 3003 reviews in the Plants & Seeds category and have reached the conclusion based on a range of review criteria that Clematis Taiga is well worthy of its 9.1 score out of 10. Taiga plants have to be hardy in order to survive not only the long, cold winter, but also the poor-quality soils typical of the biome. Taiga Clematis - 2 Gallon Pot Every so often a plant comes along that just blows home gardeners and even professionals away...meet Taiga, a new Clematis that is sure to grab attention all summer long and into fall when it produces an abundance of exceptionally unique and … But our experts have been trialing Taiga for several years now in zone 6 with great success, and we are pleased to report that Taiga is reliably hardy to less than –10°F. There are some plant and animal species that are adapted to the conditions of the taiga. Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies.
Reindeer Vs Caribou, Pied Imperial Pigeon Call, Prix Vodka Poliakov, Sir Kensington's Special Mayonnaise Sauce, Cs6290 Final Exam, Msi Monitor Looks Pixelated, | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,084 |
Q: Static initialization of a map with a pair I'm trying to statically initialize a map that contains a pair:
typedef map<int, pair<int, int>> mytype;
static const mytype mymap = { 3, {3, 0} };
I'm using Visual Studio 2013, but I get the error:
error C2440: 'initializing' : cannot convert from 'initializer-list' to 'std::map<int,std::pair<int,int>,std::less<_Kty>,std::allocator<std::pair<const _Kty,_Ty>>>'
Any idea what would cause this? I thought VS2013 had this C++11 functionality.
A: You're missing one set of braces:
static const mytype mymap = { { 3, {3, 0} } };
^ ^ ^
| | pair<int,int> (value)
| pair<const key, value> (map element)
map<key, value>
A: The compiler thinks you want to initialize the map with two elements.
The correct syntax would be:
static const mytype mymap = { { 3, {3, 0} } };
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,034 |
\section{Introduction}
Due to the surge of interest in networks, such as the Internet
(e.g., The Internet Mapping Project by Hal Burch and Bill Cheswick),
resistor networks~\cite{newman2004finding}, the World Wide Web
(WWW)~\cite{barabasi1999emergence}, and social networks (e.g.,
friendship
network~\cite{moody2001race}), a plethora of network models have
been
proposed and studied in the last several decades. In this paper, we
investigate a network model that recently caught researchers'
attention---{\em Apollonian networks} (ANs). ANs arise from the
problem of space-filling packing of spheres, proposed by the ancient
Greek mathematician Apollonius of Perga. ANs possess a variety of
typical network characteristics,
which are summarized in the title of~\cite{andrade2005apollonian}:
scale free,
small world, Euclidean, space filling and matching graphs. Each of
these phrases is a significant area of modern network research
itself. In practice, ANs have found a lot of applications in
different scientific disciplines~\cite{almeida2013quantum,
huang2006walks,
lima2012nonequilibrium, pawela2015generalized, serva2013ising,
silva2013critical, souza2013discrete, wong2010partially,
xu2008coherent}. The wide application of this class of networks
motivates us to conduct the present research.
The counterpart of AN in the field of random network analysis is
called {\em Random Apollonian Network} (RAN). The study of RAN first
appeared in~\cite{zhou2005maximal}, where the power-law and the
clustering
coefficient were investigated. Since then, many more properties of
RANs have been uncovered by applied mathematicians and probablists:
The degree distribution was characterized by~\cite{frieze2014some};
the diameter was calculated by~\cite{ebrahimzadeh2014onlongest,
frieze2014some}; the length
of the longest path in RANs was determined
by~\cite{collevecchio2016longest, cooper2015long,
ebrahimzadeh2014onlongest}. All these research papers, however,
only
focused on planar RANs, the evolution of which is based on
continuing triangulation. Triangulated RANs are a special class of
(more general) RANs with network {\em index} taking value $3$. It
can be
shown that triangulated RANs are maximal planar graphs by the {\em
Kuratowski criterion}~\cite{kuratowski1930remarques}. It is
evident
that there is an underlying theory of {\em preferential attachment}
(PA)~\cite{massen2007preferential} in the evolution of RANs, where
PA is a critical manifestation in social sciences, rendering the
potential application of RANs in a wider range of fields (than what
has been found in the literature).
In this paper, we consider
a class of high-dimensional networks generalized from triangulated
RANs, i.e., high-dimensional random Apollonian Networks (HDRANs)
that refer to the RANs with a general network index $k \ge 3$.
HDRANs were first introduced by~\cite{zhang2006high}, where an
iterative
algorithm was designed to characterize several network properties
including degree distribution, clustering coefficient and diameter.
The exact
degree distribution of a vertex with a fixed label and the total
weight (a macro metric) were determined
by~\cite{zhang2016thedegree}. A follow-up study embedding RANs into
continuous time was given by~\cite{zhang2016distributions}. To the
best of our knowledge, there is almost no other work has been done
for
HDRANs in the literature.
The goal of this paper is to give a comprehensive study of HDRANs,
with specific focus on the investigation of several network
properties of common interest by utilizing some well-developed
methods; for instance, stochastic recurrences and \polya\ urns.
Some of the results, such as the degree distribution, are
directly extended from their counterparts for triangulated RANs. For
better readability, only the results and the underlying theory are
presented in the main body of the paper, but the associated
mathematical derivations are given in the appendix. A couple of
novel network properties, such as the sparsity and the
depth of HDRANs, are rigorously uncovered as well. Details will be
given in the
sequel.
The rest of the paper is
organized as follows. In Section~\ref{Sec:evolution}, we briefly
review the evolutionary process of HDRANs as well as some basic
graph invariants thereof. In the next five sections, the main
results of the analyzed properties are given; see a summary in
Table~\ref{Table:summary}.
\begin{table}[h!]
\begin{center}
\renewcommand*{\arraystretch}{1.2}
\begin{tabular}{|c|c|c|}
\hline
Section & Property & Method(s)
\\ \hline
$3$ & Degree profile {\rm I} & Two-dimensional induction
(extended from~\cite{frieze2014some})
\\ \hline
\multirow{2}{*}{$4$} & \multirow{2}{*}{Degree profile
{\rm II}} & Analytic
combinatorics~\cite{flajolet2006some}
\\ & & Triangular urns~\cite{zhang2016thedegree}
\\ \hline
$5$ & Small world & Local clustering coefficient
\\ \hline
$6$ & Sparsity & A proposed Gini index
\\ \hline
\multirow{3}{*}{$7$} & Total depth & Recurrence methods
\\ & Diameter & Results directly
from~\cite{cooper2014theheight}
\\ & The Wiener index & Numeric experiments
\\ \hline
\end{tabular}
\caption{Summary of the article}
\label{Table:summary}
\end{center}
\end{table}
In Section~\ref{Sec:concluding}, we give some concluding remarks and
propose some future work.
\section{Evolution of random Apollonian networks}
\label{Sec:evolution}
In this section, we review the evolution of a RAN of index $k \ge
3$. At time $n = 0$, we start with a {\em complete
graph}\footnote{In graph theory, a complete graph is a graph
such
that each pair of vertices therein is connected by an edge. A
complete graph on $k$ vertices is also called $k$-clique or
$k$-simplex. We shall interchangeably use these terms through
the
manuscript.} on $k$ vertices all of which are labeled with $0$.
At
each subsequent time point $n \ge 1$, a $k$-clique is chosen
uniformly at random among all active cliques in the network. A new
vertex labeled with $n$ is linked by $k$ edges to all the vertices
of the chosen clique. Then, the recruiting clique is deactivated. An
explanatory example of a RAN with index $k = 5$ is given in
Figure~\ref{Fig:evol}.
\begin{figure}[tbh]
\begin{center}
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(0.45, 0.52) node [circle=0.1,draw] {0}
(-0.133,0) -- (0.133,0)
(-0.33,0.52) -- (0.33,0.52)
(-0.158, 0.085) -- (-0.03, 0.755)
(0.158, 0.085) -- (0.03, 0.755)
(-0.41, 0.405) -- (-0.33, 0.09)
(0.41, 0.405) -- (0.33, 0.09)
(-0.357, 0.448) -- (0.141, 0.036)
(0.357, 0.448) -- (-0.141, 0.036)
(-0.357, 0.6) -- (-0.077, 0.78)
(0.357, 0.6) -- (0.077, 0.78);
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(1.857, 0.6) -- (1.577, 0.78);
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(1.658, 0.085) to[bend left=15] (1.8, 0.75)
(1.9, 0.63) to[bend right=30] (1.9, 0.75);
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\draw
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(3.45, 0.52) node [circle=0.1,draw] {0}
(3.35, 0.866) node [circle=0.1,draw] {1}
(3.55, 0.2) node [circle=0.1,draw] {2};
\draw[dashed]
(2.867,0) -- (3.133,0)
(2.67,0.52) -- (3.33,0.52)
(2.842,0.085) -- (2.97, 0.755)
(3.158, 0.085) -- (3.03, 0.755)
(2.59, 0.405) -- (2.67, 0.09)
(3.41, 0.405) -- (3.33, 0.09)
(2.643, 0.448) -- (3.141, 0.036)
(3.357, 0.448) -- (2.859, 0.036)
(2.643, 0.6) -- (2.923, 0.78)
(3.357, 0.6) -- (3.077, 0.78)
(3.125, 0.866) to[bend left=15] (3.235, 0.866)
(2.643, 0.6) to[bend left=75] (3.3, 0.97)
(2.842, 0.085) to[bend left=45] (3.25, 0.8)
(3.158, 0.085) to[bend left=15] (3.3, 0.75);
\draw
(3.4, 0.63) to[bend right=30] (3.4, 0.75)
(3.6, 0.32) to [bend right=45] (3.43, 0.78)
(3.52, 0.32) to [bend left=30] (3.05, 0.765)
(3.44, 0.25) to [bend left=15] (2.66,0.49)
(3.436, 0.195) to [bend right=10] (2.842, 0.085)
(3.47, 0.115) to [bend left=15] (3.37, 0.02);
\end{tikzpicture}
\caption{An example of the evolution of a HDRAN of index 5
in two steps; active cliques are those containing at
least
one solid edge.}
\label{Fig:evol}
\end{center}
\end{figure}
According to the evolutionary process described above, we obtain
some basic and deterministic graph invariants of a RAN with index
$k$ at time $n$: the number of vertices $V_{n}^{(k)} = k + n$, the
number of edges $E_{n}^{(k)} = k + nk$, and the number of active
cliques $\clique_n^{(k)} = 1 + (k - 1)n$. We note that RANs of
indices $1$ and $2$ are not considered in this paper, as their
structure lacks research interest. A RAN of index $1$ at time $n$ is
a single vertex labeled with $n$, while a RAN of index $2$ at time
$n$ is a path of length $n$.
\section{Degree profile {\rm I}}
\label{Sec:degree}
In this section, we investigate the degree profile of a RAN of index
$k \ge 3$. The random variable of prime interest is $X_{n,
j}^{(k)}$, the number of vertices of degree $j$ in a RAN of
index
$k$ at time $n$, for $j \ge k$, where the boundary condition arises
from the natural lower bound of the degree of vertices in
RANs\footnote{Upon joining into the network, every newcomer is
connected with $k$ existing vertices, leading to minimal
possible
degree $k$.}. It is also worthy of noting that the natural upper
bound for $j$ at time $n$ is $k + n - 1$.
The degree random variable that we consider in this section is
different from that investigated in~\cite{zhang2016thedegree}, and
the methods
developed in~\cite{zhang2016thedegree} are not amenable to this
study, which will
be explained in detail in the sequel. To distinguish the two kinds
of degree profiles, we call the one discussed in this section degree
profile {\rm I}. Specifically, we present two results of $X_{n,
j}^{(k)}$, which are respectively shown in
Theorems~\ref{Thm:L1bound} and~\ref{Thm:pbound}. In
Theorem~\ref{Thm:L1bound}, we prove that the difference between the
expectation of $X_{n, j}^{(k)}$ and a linear function of $n$ is
uniformly bounded, where the bound is determined. In
Theorem~\ref{Thm:pbound}, we show that $X_{n, j}^{(k)}$ concentrates
on its expectation with high probability, i.e., a focusing property.
\begin{theorem}
\label{Thm:L1bound}
Let $X_{n, j}^{(k)}$ be the number of vertices of degree $j$ in
a RAN of index $k$ at time $n$, for $j \ge k$. For each $n \in
\mathbb{N}$ and any $k \ge 3$, there exists a constant $b_{j,
k}$ such that
\begin{equation}
\label{Eq:degreeL1}
\left|\E \left[X_{n, j}^{(k)}\right] - b_{j, k} \, n\right|
\le \frac{2k^2}{2k - 1}.
\end{equation}
In particular, we have $b_{j, k} = \frac{\Gamma(j)\Gamma(2k -
1)}{\Gamma(j + k) \Gamma(k - 1)}$.
\end{theorem}
The proof of Theorem~\ref{Thm:L1bound} is based on an elementary
mathematical tool---induction. As suggested
in~\cite{frieze2014some}, we
split the cases of $j = k$ and $j > k$ in the proof. For the case of
$j = k$, we apply the traditional mathematical induction directly,
whereas we develop a two-dimensional induction based on an infinite
triangular array for the case of $j > k$. For the better readability
of the paper, we present the major steps of the proof in
Appendix~\ref{App:L1bound}.
In the proof of Theorem~\ref{Thm:L1bound}, we show that the mean of
$X_{n, j}^{(k)}$ scaled by $n$ converges to $b_{j, k}$ when $n$ is
large. When $j$ goes to infinity as well, we discover that $b_{j, k}
\sim j^{-k}$ according to the {\em Stirling's approximation}. This
implies that the degree distribution in HDRANs follows a {\em
power-law} property, where the exponent is the network index
$k$.
Consequently, HDRANs are {\em scale-free} networks. The power-law
property for planar RANs (i.e., $k = 3$) has been recovered
in~\cite{zhou2005maximal} numerically and in~\cite{frieze2014some}
analytically.
In addition, we are interested in the deviation of the random
variable $X_{n, j}$ from its expectation. In
Thoerem~\ref{Thm:pbound}, we develop a Chebyshev-type inequality.
\begin{theorem}
\label{Thm:pbound}
Let $X_{n, j}^{(k)}$ be the number of vertices of degree $j$ in
a RAN of index $k$ at time $n$, for $j \ge k$. For any $\lambda>
0$, we have
$$\Prob\left(\left|X_{n, j}^{(k)} - \E\left[X_{n, j}^{(k)}
\right]\right| \ge \lambda \right) \le e^{-\lambda^2/(8kn)}.$$
\end{theorem}
The proof of Theorem~\ref{Thm:pbound} is presented in
Appendix~\ref{App:pbound}. The main idea is to employ the {\em
Azuma-Hoeffding inequality}~\cite{azuma1967weighted} based on a
martingale
sequence. We remark that the exact same concentration result is
found for {\em random $k$-trees}~\cite{gao2009thedegree}. The author
of~\cite{gao2009thedegree} tackled the problem by using the methods
from tree
realization theory. The intrinsic reason of the identicality is
similarity in the evolutionary processes of HDRANs with index
$k$ and random $k$-trees.
Before ending this section, we would like to point out that the
methods in the proofs of Theorems~\ref{Thm:L1bound}
and~\ref{Thm:pbound} are extended from the ideas
in~\cite{frieze2014some}.
The results for planar RANs (a special case for $k = 3$) can be
found in~\cite[Theorem 1.1]{frieze2014some}.
\section{Degree profile {\rm II}}
\label{Sec:degree2}
Another type of degree profile that we look into is node-specified.
Let $D_{n, j}^{(k)}$ denote the degree of the node labeled with $j$
in a HDRAN of index $k$ at time $n$. This property was investigated
in~\cite{zhang2016thedegree}, where the growth of HDRANs was
represented by a
two-color \polya\ urn scheme~\cite{mahmoud2009polya}. \polya\ urn
appears to
be an appropriate model since it successfully captures the
evolutionary characteristics of highly dependent structures.
Noticing that the degree of a vertex is equal to the number of
cliques incident with it, the authors of~\cite{zhang2016thedegree}
introduced a
color code such that the active cliques incident with the node
labeled with $j$ were colored white, while all the rest were colored
blue. The associated urn scheme is governed by the {\em replacement
matrix}
$$\begin{pmatrix}
k - 2 & 1
\\ 0 & k - 1
\end{pmatrix}.$$
This replacement matrix is triangular, so the associated \polya\ urn
is called {\em triangular urn}. This class of urns has been
extensively studied in~\cite{flajolet2006some, janson2006limit,
zhang2015explicit}. The next
proposition specifies the exact distribution of $D_{j, n}^{(k)}$ as
well as its moments.
\begin{prop}
\label{Thm:degreedist}
Let $D_{n, j}^{(k)}$ be the degree of the node labeled with $j$
in a RAN of index $k$ at time $n$, for $n \ge j$. The
distribution of $D_{n, j}^{(k)}$ is given by
\begin{align*}
\Prob\left(D_{n, j}^{(k)} = k + \delta\right) &=
\frac{\Gamma(n - j + 1)\Gamma\left(j + \frac{1}{k -
1}\right)}{\Gamma\left(n + \frac{1}{k -
1}\right)}{{\delta +
\frac{2}{k - 2}} \choose \delta}
\\ &\qquad{}\times \sum_{r = 1}^{\delta} (-1)^{\delta}
{\delta \choose i} {{n - 2 - \frac{k - 2}{k - 1} r} \choose
{n - j}},
\end{align*}
for $\delta = 1, 2, \ldots, n - j$. The $s$-th moment of $D_{n,
j}^{(k)}$ is
\begin{align}
\E\left[\left(D_{n, j}^{(k)}\right)^{s\,}\right] &=
\frac{1}{(k - 2)^s} \left[(k(k - 3))^s + \sum_{r = 1}^{s}{s
\choose r} \frac{(k(k - 3))^{s - r}(k -
2)^r}{\left\langle j
+ 1/(k - 1)\right\rangle_{n - j}}\right. \nonumber
\\ &\qquad{}\left.\times \sum_{i = 1}^{r} (-1)^{r - i} {r
\brace i} \left\langle \frac{k}{k - 2} \right\rangle_i
\left\langle j + \frac{1}{k - 1} + \frac{k - 2}{k - 1}i
\right\rangle_{n - j}\right], \label{Eq:degreemoment}
\end{align}
where $\langle \cdot \rangle_{\cdot}$ represents the Pochhammer
symbol of rising factorial, and ${{\cdot} \brace {\cdot}}$
represents Stirling numbers of the second kind.
\end{prop}
The probability distribution function of $D_{n, j}^{(k)}$ is
obtained by exploiting the results in~\cite[Proposition
14]{flajolet2006some}, and the moments are recovered
from~\cite[Proposition
1]{zhang2016thedegree}. The asymptotic moments of $D_{n, j}^{(k)}$
are obtained
directly by applying the Stirling's approximation to
Equation~(\ref{Eq:degreemoment}); namely,
$$\E\left[\left(\frac{D_{n, j}^{(k)}}{n^{(k - 2)/(k - 1)}}\right)^{s
\,}\right] = \frac{\Gamma\left(j + \frac{1}{k - 1}\right)
\Gamma\left( s + \frac{k}{k - 2}\right)}{\Gamma\left(j + \frac{k
-
2}{k - 1}s + \frac{1}{k - 1}\right) \Gamma\left(\frac{k}{k -
2}\right)}.$$
In particular, the asymptotic mean of $D_{n, j}^{(k)}$ is given by
$$\E\left[D_{n, j}^{(k)}\right] \sim \frac{\frac{k}{k - 2}
\Gamma\left(j + \frac{1}{k - 1}\right)}{\Gamma(j + 1)} \, n^{(k
-
2)/(k - 1)},$$
implying a phase transition in $j = j(n)$:
$$\E\left[D_{n, j}^{(k)}\right] \sim
\begin{cases}
\frac{k}{k - 2} \left(\frac{n}{j(n)}\right)^{(k - 2)/(k - 1)},
\qquad &j = o(n),
\\ \frac{k}{k - 2} \left(k - 3 + \alpha^{-(k - 2)/(k - 1)}
\right), \qquad &j \sim \alpha n,
\end{cases}
$$
for some $\alpha > 0$.
\section{Small-world}
\label{Sec:smallworld}
In
this section, we look into the {\em small-world} property of HDRANs.
The term ``small-world'' was first coined
by~\cite{watts1998collective}. In the
paper, the authors suggested to use the average of {\em local
clustering coefficients} to assess the small-world effect of a
network; that is,
$$\hat{C}(n) = \frac{1}{V} \sum_{v} C_v(n),$$
where $V = V_n^{(k)}$ denotes the total number of vertices and
$C_v(n)$ is the
local clustering coefficient of vertex~$v$ at time $n$. The local
clustering coefficient of vertex $v$ is defined as the proportion of
the number of edges in the {\em open neighborhood} of $v$, i.e.,
$$C_v(n) = \frac{|\{e_{uw}, u, w \in
\mathcal{N}_v(n)\}|}{|\mathcal{N}_v(n)|(|\mathcal{N}_v(n) -
1|)/2},$$
where $\mathcal{N}_v(n)$ is the open neighborhood of $v$ at time
$n$, $e_{ij}$ denotes an edge between vertices $i$ and $j$, and
$|\cdot|$ represents the cardinality of a set.
For each newcomer, say $v^{*}$, to an HDRAN of index $k$, the open
neighborhood of $v^{*}$ is comprised by $k$ vertices of the simplex
chosen for recruiting $v^*$. Thus, the order of the open
neighborhood of $v^{*}$ is $k$, and the number of edges in the
neighborhood of $v^{*}$ is ${k \choose 2}$. Upon the first
appearance of $v^{*}$ in the network, the degree of $v^{*}$, denoted
$d_{v^{*}}(n)$, is $k$. As an active simplex containing $v^{*}$ is
selected for recruiting a newcomer in any subsequent time point,
$d_{v^{*}}(n)$ increases by $1$, and the number of edges of the
neighborhood of $v^{*}$ increases by $k - 1$. In general, for a
vertex $v$ of ${\rm deg}_v(n) = j$ at time $n$, the clustering
coefficient is given by
\begin{equation*}
C_v(n) = \frac{(k - 1)(j - k) + {k \choose 2}}{{j \choose 2}} =
\frac{(k - 1)(2j - k)}{j(j - 1)}.
\end{equation*}
Accordingly, the clustering coefficient of the entire network at
time $n$ is
\begin{equation*}
\hat{C}(n) = \frac{1}{n + k} \sum_{v}C_v(n) = \sum_{j = k}^{k +
n - 1} \frac{(k - 1)(2j - k)}{j(j - 1)} \times \frac{X_{n,
j}^{(k)}}{n + k},
\end{equation*}
where $X_{n, j}^{(k)}$ denotes the number of vertices of degree $j$
in the network at time $n$. When the network is large (i.e., $n \to
\infty$), the asymptotic clustering coefficient is given by
\begin{equation*}
\hat{C}(\infty) \approx \sum_{j = k}^{\infty} \frac{(k - 1)(2j
- k)}{j(j - 1)} \lim_{n \to \infty} \frac{\E \left[X_{n,
j}^{(k)}\right]}{n + k} = \sum_{j = k}^{\infty} \frac{(k
- 1)(2j
- k)}{j(j - 1)}
\frac{\Gamma(j) \Gamma(2k - 1)}{\Gamma(j + k) \Gamma(k - 1)},
\end{equation*}
where the second equality in the last display holds according to
Theorem~\ref{Thm:L1bound}. We simplify the expression of
$\hat{C}(\infty)$ by applying several algebraic results of gamma
function, and get
\begin{equation*}
\label{Eq:asymcc}
\hat{C}(\infty) \approx \frac{(k - 1)\Gamma(2k - 1)}{\Gamma(k -
1)} \sum_{j = k}^{\infty} \frac{(2j - k) \Gamma(j - 1)}{j \,
\Gamma(j + k)} = \frac{(k - 1)\Gamma(2k - 1)}{\Gamma(k - 1)}
\sum_{j =
k}^{\infty} \left(\frac{2 \, \Gamma(j - 1)}{\Gamma(j + k)} -
\frac{k \, \Gamma(j - 1)}{j \, \Gamma(j + k))} \right).
\end{equation*}
We evaluate the two terms in the summand one after another. The
first sum is given by
$$\sum_{j = k}^{\infty} \frac{2 \, \Gamma(j - 1)}{\Gamma(j + k)} =
\frac{2(2k - 1)\Gamma(k - 1)}{k \, \Gamma(2k)}.$$
The second sum is simplified to
$$\sum_{j = k}^{\infty} \frac{k \, \Gamma(j - 1)}{j \, \Gamma(j +
k))} = \frac{\Gamma(k - 1)\Hypergeometric{3}{2}{1, k - 1, k}{2k,
k +
1}{1}}{\Gamma(2k)}.$$
where $\Hypergeometric{3}{2}{\cdot}{\cdot}{\cdot}$ is a {\em
generalized hypergeometric function}. Putting them together, we
thus
have
$$\hat{C}(\infty) \approx \frac{k - 1}{2k - 1}\left(\frac{2(2k -
1)}{k} - \Hypergeometric{3}{2}{1, k - 1, k}{2k, k +
1}{1}\right).$$
Although hypergeometric functions cannot be written in closed forms
in general, we derive the analytical results of $\hat{C}(\infty)$
for several small values of $k$, and present them in
Table~\ref{Table:asymcc}. In particular, the estimated clustering
coefficient for triangulated RANs (i.e., $k = 3)$ based on our
calculation is $12 \pi^2 - 353/3 \approx 0.7686$, which is more
accurate than $46/3 - 36 \log(3/2) \approx 0.7366$~\cite[Equation
(6)]{zhou2005maximal}, according to a simulation
experiment ($0.7683$ based on the average of $50$ independent
samples, each of which is run over $10,000$ iterations).
\begin{table}[h]
\renewcommand{\arraystretch}{1.5}
\begin{center}
\begin{tabular}{|c|c|}
\hline
Network index ($k$) & $\hat{C}(\infty)$
\\ \hline
3 & $12 \pi^2 - \frac{353}{3}$
\\ \hline 4 & $120 \pi^2 - \frac{2367}{2}$
\\ \hline
5 & $\frac{2800}{3} \pi^2 - \frac{138161}{15}$
\\ \hline 6 & $6300 \pi^2 - \frac{746131}{12}$
\\ \hline
7 & $38808 \pi^2 - \frac{134056533}{350}$
\\ \hline 8 & $224224 \pi^2 - \frac{663900367}{300}$
\\ \hline
9 & $1235520 \pi^2 - \frac{26887974331}{2205}$
\\ \hline
10 & $6563700 \pi^2 - \frac{253941996039}{3920}$
\\ \hline
\end{tabular}
\end{center}
\caption{Asymptotic clustering coefficients of HDRANs with small
indicies $k$}
\label{Table:asymcc}
\end{table}
\section{Sparsity}
\label{Sec:sparsity}
{\em Sparsity} is a property of common interest in network
modeling~\cite{singh2015finding, verzelen2015community,
vinciotti2013robust}, as well as in
data
analytics~\cite{arnold2010specifying, buluc2011implementing}. As
opposed to
``dense,'' this
topology plays a key role when one defines sparse networks. Sparse
networks have fewer links than the maximum possible number of links
in the (complete) network of same order. In computer science, sparse
networks are considered to be somewhere dense or nowhere dense. The
investigation of sparsity of HDRANs is inspired by an article
recently published in the American Physics
Society~\cite{delgenio2011all}. It
was analytically and numerically proven in the article that the
probability of a scale-free network being dense is $0$, given that
the power-law coefficient falls between~$0$ and $2$.
One of the most commonly-used network topology to measure the
sparsity of a network $G(V, E)$ is the {\em link density} (also
known as {\em edge density} in the literature):
$${\rm density}(G) = \frac{|E|}{\binom{|V|}{2}}.$$
For a HDRAN of index $k$, denoted $\apol_{n}^{(k)}$, its link
density at time $n$ is a decreasing function of $n$, viz.,
$${\rm density}\left(\apol_{n}^{(k)}\right) =
\frac{E_n^{(k)}}{\binom{V_n^{(k)}}{2}} = \frac{k + nk}{\binom{k +
n}{2}} = \frac{2(k + nk)}{(k + n)(k + n - 1)}.$$
Observing that the link density of an HDRAN in any form is
deterministic given $k$ and $n$, we assert that this topology indeed
fails to expose the randomness or to capture the structure of
HDRANs. Other topologies that have been proposed to measure the
sparsity of both nonrandom and random networks include degeneracy,
arboricity, maximum average degree, etc. We refer the interested
readers to~\cite{nesetril2012sparsity} for textbook style
expositions of these
topologies and their properties.
In this section, we measure the sparsity of HDRANs via a classical
metric---the {\em Gini index}~\cite{gini1921measurement}. The Gini
index which
appears more often in economics is commonly used to measure the
inequality of income or wealth~\cite{dalton1920themeasurement,
gini1921measurement}. The utilization
of the Gini index as a sparsity measurement originates in electrical
engineering~\cite{hurley2009comparing}. More often, the Gini index
was used to
evaluate regularity of graphs~\cite{balaji2017thegini,
domicolo2020degree, zhang2019thedegree}. The Gini
index debuted as a sparsity measurement of networks
in~\cite{goswami2018sparsity}.
A graphical interpretation of the Gini index is the {\em Lorenz
curve}. As portrayed in Figure~\ref{Fig:exLorenz}, the Lorenz
curve
(thick black curve) splits the lower triangle of a unit square into
$A$ and $B$.
A well-established relationship between the Gini index and the
Lorenz curve is that the Gini index of the associated Lorenz curve
is equal to the ratio of ${\rm Area}(A)$ and ${\rm Area}(A + B)$,
equivalent to $1 - 2 \times {\rm Area}(B)$.
\begin{figure}[tbh]
\centering
\begin{tikzpicture}[scale=5]
\draw
(0, 0) -- (0, 1)
(0, 0) -- (1, 0)
(0, 1) -- (1, 1)
(1, 0) -- (1, 1)
(0, 0) -- (1, 1);
\draw[ultra thick]
(0, 0) to [bend right = 40] (1, 1);
\draw[fill=gray!20]
(0, 0) to [bend right = 40] (1, 1) -- (1, 0) -- (0, 0) --
cycle;
\node[text width = 0.1cm] at (0.57, 0.43) {$A$};
\node[text width = 0.1cm] at (0.8, 0.2) {$B$};
\end{tikzpicture}
\label{Fig:exLorenz}
\caption{An example of typical Lorenz curve}
\end{figure}
We construct the Gini index of HDRANs based on vertex degrees. At
time $n$, there is a total of $k + n$ vertices in $\apol_n^{(k)}$,
and the {\em admissible} degree set is $\mathcal{J} = \{k, k + 1,
\ldots, k + n\}$. According to Theorem~\ref{Thm:L1bound}, the mean
of the proportion of the number of vertices having degree $j \in
\mathcal{J}$ can be approximated by $\bigl(\Gamma(j) \Gamma(2k -
1)\bigr)/\bigl(\Gamma(j + k) \Gamma(k - 1)\bigr)$, when $n$ is
large. For simplicity, let us denote this mean proportion for each
pair of $j$ and $k$ by $\gamma(j, k)$. These $\gamma(j, k)$'s
altogether naturally form the Lorenz curve after being rearranged in
an ascending order. Note that
$$\frac{\partial}{\partial j} \, \gamma(j, k) = \frac{\bigl(
(\Psi(j) - \Psi(j + k) \bigr) 2^{2k - 2} (k - 1) \Gamma\left(k -
\frac{1}{2}\right) \Gamma(j)}{\Gamma\left(\frac{1}{2}\right)
\Gamma(j + k)} < 0,$$
where $\Psi(\cdot)$ is the {\em digamma function}, known to be
increasing on the positive real line. Hence, the function $\gamma(j,
k)$ is decreasing with respect to $j$.
Specifically, we build the Lorenz curve as follows. The bottom of
the unit square is equispaced into $(n + 1)$ segments. The bottom
left vertex is marked $0$ along with vertical value $0$. The
cumulative proportion value $\sum_{j = k + n - i + 1}^{k + n}
\gamma(j, k)$ is assigned to the $i$th segmentation from the left.
There is a total of $n$ segmentations between the bottom left and
bottom right vertices. Lastly, the vertical value for the bottom
right vertex is $\sum_{j = k}^{k + n} \gamma(j, k)$. The Lorenz
curve is comprised by smoothly connecting these assigned values in
order, from left to right.
In the next lemma, we show that the Lorenz curve that we established
in the last paragraph is well defined, i.e., the two ends of the
Lorenz curve respectively coincide with the bottom left and the top
right corners of the unit square.
\begin{lemma}
\label{Lem:lorenz}
We claim that
$$\lim_{n \to \infty} \sum_{j = k}^{k + n} \frac{\Gamma(j)
\Gamma(2k - 1)}{\Gamma(j + k) \Gamma(k - 1)} = 1 \quad
\mbox{\textit{and}} \quad
\lim_{n \to \infty} \frac{\sum_{j = k + n - i + 1}^{k + n}
\frac{\Gamma(j) \Gamma(2k - 1)}{\Gamma(j + k) \Gamma(k -
1)}}{i/n} = 0.$$
\end{lemma}
The proof of Lemma~\ref{Lem:lorenz} is presented in
Appendix~\ref{App:lorenz}. Next, we calculate ${\rm Area}(B)$,
equivalent to integrating the Lorenz curve from $0$ to $1$. For
large value of $n$, the integration can be approximated by applying
the {\em trapezoid rule}; that is,
\begingroup
\allowdisplaybreaks
\begin{align*}
{\rm Area}(B) &\approx \frac{1}{2 (n + 1)} \left[\sum_{j = k +
n}^{k + n} \frac{\Gamma(j) \Gamma(2k - 1)}{\Gamma(j + k)
\Gamma(k - 1)} \right.
\\ &\qquad{}+ \left(\sum_{j = k + n}^{k + n} \frac{\Gamma(j)
\Gamma(2k - 1)}{\Gamma(j + k) \Gamma(k - 1)} + \sum_{j = k +
n -
1}^{k + n} \frac{\Gamma(j) \Gamma(2k - 1)}{\Gamma(j + k)
\Gamma(k - 1)}\right)
\\ &\qquad{} + \cdots + \left.\left(\sum_{j = k + 1}^{k + n}
\frac{\Gamma(j) \Gamma(2k - 1)}{\Gamma(j + k) \Gamma(k - 1)} +
\sum_{j = k}^{k + n} \frac{\Gamma(j) \Gamma(2k - 1)}{\Gamma(j +
k) \Gamma(k - 1)} \right)\right]
\\&= \frac{1}{2(n + 1)}\left(\frac{3k - 2}{k - 2} - \frac{2^{2k
- 1}\bigl((k - 1)n + 2\bigr) \Gamma\left(k -
\frac{1}{2}\right)\Gamma(k + n + 1)}{(k -
2)\Gamma\left(\frac{1}{2}\right)\Gamma(2k + n)}\right).
\\&\sim n^{-1} - n^{1 - k},
\end{align*}
\endgroup
In what follows, the Gini index of an HDRAN of index $k$ at time $n$
is given by
\begin{align*}
&{\rm Gini}\left(\apol_{n}^{(k)}\right) = 1 - 2 \times {\rm
Area}(B)
\\ &\quad= 1 - \frac{1}{(n + 1)}\left(\frac{3k - 2}{k - 2} -
\frac{2^{2k - 1}\bigl((k - 1)n + 2\bigr) \Gamma\left(k -
\frac{1}{2}\right)\Gamma(k + n + 1)}{(k -
2)\Gamma\left(\frac{1}{2}\right)\Gamma(2k + n)}\right),
\end{align*}
the asymptotic equivalent of which is equal to $1$. A large value of
Gini index (ranging from $0$ to $1$) indicates an extremely
nonuniform distribution of vertex degrees, implying that all vertex
degrees are dominated by only a few classes, whereas a small value
of Gini index suggests vertex degrees are evenly distributed in
different degree classes. Thus, we conclude (asymptotically) high
sparseness of HDRANs.
We further verify our conclusion by conducting some simulation
experiments. In general, each network $G(V, E)$ is associated with a
unique $|V| \times |V|$ {\em adjacency matrix}, denoted $\matA =
(A_{ij})$, in which $A_{ij} = 1$ only when there is an edge linking
vertices $i$ and $j$, for $i, j \in V$; $0$, otherwise. If $G$ is
undirected, $\matA$ is symmetric. The degree of vertex $i$ thus can
be represented by the sum of $i$th row or the $i$th column in
$\matA$, allowing us to compute the Gini index of each simulated
network~$G$ through $\matA$ accordingly. For each $k = 3, 10, 30$,
we generate $100$ independent HDRANs at time $n = 5000$. The
comparison of Lorenz curves (based on the average of cumulative
degree proportion sequences) is given in Figure~\ref{Fig:Lorenz}.
\begin{figure}[tbh]
\centering
\includegraphics[width=\textwidth]{lorenzcurve-eps-converted-to}
\caption{Comparison of Lorenz Curves for simulated HDRANs of $k
= 3, 10, 30$ at time $n = 5000$}
\label{Fig:Lorenz}
\end{figure}
Besides, we calculate the Gini index of each of the $100$ simulated
HDRANs (of $k = 3, 10, 30$) at time $50,000$, and take the average;
The estimated Gini indices are $0.9970330$ (for $k = 3$),
$0.9990327$ (for $k = 10$), and $0.9997262$ (for $k = 30$). We do
not show the corresponding Lorenz curves as they are not visually
distinguishable.
\section{Depth, diameter and distance}
\label{Sec:diameter}
In this section, we investigate several distance-based properties of
HDRANs. The first measure that we look into is clique-based---{\em
depth}, which is defined (for HDRANs) recursively as follows. At
time $1$, the original $k$-clique is divided into $k$ simplexes, and
then is deactivated. The depth of each of the active $k$-cliques
equals $1$. At time $n > 1$, an existing active clique $\clique^{*}$
is chosen uniformly at random, and subdivided into $k$ new cliques
$\clique_1, \clique_2, \ldots, \clique_k$. Then, we have
$${\rm depth}(\clique_i) = {\rm depth}(\clique^{*}) + 1,$$
for all $i = 1, 2, \ldots, k$. An explanatory example of a RAN of
index $m = 5$ is shown in Figure~\ref{Fig:RANdepth}, where the
(active) cliques denoted by $\left(1, 0^{(1)}, 0^{(2)}, 0^{(3)},
0^{(5)}\right)$,
$\left(1, 0^{(1)}, 0^{(2)}, 0^{(4)}, 0^{(5)}\right)$, $\left(1,
0^{(1)}, 0^{(3)}, 0^{(4)}, 0^{(5)}\right)$, and $\left(1, 0^{(2)},
0^{(3)}, 0^{(4)}, 0^{(5)}\right)$ have depth $1$; all the rest have
depth $2$.
\begin{figure}[tbh]
\begin{center}
\begin{tikzpicture}[scale=3.77]
\draw
(2.75,0) node [circle,draw] {$0^{(2)}$}
(3.25,0) node [circle=0.1,draw] {$0^{(1)}$}
(3,0.866) node [circle=0.1,draw] {$0^{(4)}$}
(2.55, 0.52) node [circle=0.1,draw] {$0^{(3)}$}
(3.45, 0.52) node [circle=0.1,draw] {$0^{(5)}$}
(3.35, 0.866) node [circle,draw] {$\; \; 1 \; \;$}
(3.55, 0.2) node [circle=0.1,draw] {$\; \; 2 \; \;$};
\draw
(2.867,0) -- (3.133,0)
(2.67,0.52) -- (3.33,0.52)
(2.842,0.085) -- (2.97, 0.755)
(3.158, 0.085) -- (3.03, 0.755)
(2.59, 0.405) -- (2.67, 0.09)
(3.41, 0.405) -- (3.33, 0.09)
(2.643, 0.448) -- (3.141, 0.036)
(3.357, 0.448) -- (2.859, 0.036)
(2.643, 0.6) -- (2.923, 0.78)
(3.357, 0.6) -- (3.077, 0.78)
(3.125, 0.866) to[bend left=15] (3.235, 0.866)
(2.643, 0.6) to[bend left=75] (3.3, 0.97)
(2.842, 0.085) to[bend left=45] (3.25, 0.8)
(3.158, 0.085) to[bend left=15] (3.3, 0.75);
\draw
(3.4, 0.63) to[bend right=30] (3.4, 0.75)
(3.6, 0.32) to [bend right=45] (3.43, 0.78)
(3.52, 0.32) to [bend left=30] (3.05, 0.765)
(3.44, 0.25) to [bend left=15] (2.66,0.49)
(3.436, 0.195) to [bend right=10] (2.842, 0.085)
(3.47, 0.115) to [bend left=15] (3.37, 0.02);
\end{tikzpicture}
\caption{An example of a HDRAN of index 5 at step 2.}
\label{Fig:RANdepth}
\end{center}
\end{figure}
In contrast, {\em distance}, also known as {\em geodesic distance},
is a property based on pairwise vertices. In a given network $G(V,
E)$, the distance between a pair of arbitrary vertices $i, j \in V$,
denoted $d(i, j)$, is the number of edges in the shortest path (or
one of the shortest paths) connecting $i$ and $j$. A related
property, {\em diameter} of network $G$, denoted ${\rm
diameter}(G)$, is defined in a max-min manner: the greatest
length
of the shortest paths between every two verticies in $G$, i.e.,
$\max_{i, j \in V} \left\{d(i, j)\right\}$. see~\cite[page
82]{bondy2008graph} for fundamental properties of the diameter of a
graph.
For instance, the diameter of the HDRAN given in
Figure~\ref{Fig:RANdepth} is $2$, referring to the distance between
the vertices respectively labeled with $2$ and $0^{(5)}$.
It was introduced in~\cite{darrasse2007degree} that there exists an
one-to-one
relation between the evolution of HDRANs (of index $k$) and that of
$k$-ary trees\footnote{See~\cite[page 224]{storer2002anintroduction}
for the
definition of $k$-ary tree}. An illustrative example is
presented in
Figure~\ref{Fig:karytree}.
\begin{figure}[tbh]
\begin{center}
\centering
\begin{tikzpicture}[scale = 1.08]
\node[ellipse, draw] at (0, 0) (v0){$0^{(1)}, 0^{(2)},
0^{(3)}, 0^{(4)}, 0^{(5)}$};
\node[rectangle, draw] at (-5, -2) (v11){$1, 0^{(1)},
0^{(2)}, 0^{(3)}, 0^{(5)}$};
\node[rectangle, draw] at (-1.7, -2) (v12){$1, 0^{(1)},
0^{(2)}, 0^{(4)}, 0^{(5)}$};
\node[ellipse, draw] at (0, -3) (v13){$1, 0^{(1)},
0^{(2)}, 0^{(3)}, 0^{(4)}$};
\node[rectangle, draw] at (1.7, -2) (v14){$1, 0^{(1)},
0^{(3)}, 0^{(4)}, 0^{(5)}$};
\node[rectangle, draw] at (5, -2) (v15){$1, 0^{(2)},
0^{(3)}, 0^{(4)}, 0^{(5)}$};
\node[rectangle, draw] at (-5, -5) (v21){$1, 0^{(1)},
0^{(2)}, 0^{(3)}, 2$};
\node[rectangle, draw] at (-1.7, -5) (v22){$1, 0^{(1)},
0^{(2)}, 0^{(4)}, 2$};
\node[rectangle, draw] at (0, -6) (v23){$1, 0^{(1)},
0^{(3)}, 0^{(4)}, 2$};
\node[rectangle, draw] at (1.7, -5) (v24){$1, 0^{(2)},
0^{(3)}, 0^{(4)}, 2$};
\node[rectangle, draw] at (5, -5) (v25){$0^{(1)},
0^{(2)}, 0^{(3)}, 0^{(4)}, 2$};
\draw (v0)--(v11);
\draw (v0)--(v12);
\draw (v0)--(v13);
\draw (v0)--(v14);
\draw (v0)--(v15);
\draw (v13)--(v21);
\draw (v13)--(v22);
\draw (v13)--(v23);
\draw (v13)--(v24);
\draw (v13)--(v25);
\end{tikzpicture}
\caption{The evolution of the $5$-ary tree corresponding to
that of the HDRAN of index $5$ given in
Figure~\ref{Fig:RANdepth}. Elliptic (internal) nodes
refer
to inactive cliques, whereas rectangular (active) nodes
refer to active ones.}
\label{Fig:karytree}
\end{center}
\end{figure}
Active and inactive cliques in HDRANs (of index $k$) respectively
correspond to external and internal nodes in $k$-ary trees. Thus,
the total depth of active cliques in $\apol_{n}^{(k)}$ is equivalent
to the total depth\footnote{In tree structure, the depth of a node
is the number of links between the node and the root (of the
tree).}
of external nodes in the corresponding $k$-ary tree at time $n$,
denoted $\ktree{k}{n}$. In the literature, the total depth of
external nodes in $\ktree{k}{n}$ is also known as the {total
external path}, denoted by $\ext{k}{n}$ in our manuscript. For
uniformity, we use $\ext{k}{n}$ as the notation for the total depth
of active cliques in $\apol_{n}^{(k)}$ as well.
\begin{prop}
\label{Thm:ext}
Let $\ext{k}{n}$ be the total depth of active cliques in a HDRAN
of index $k$ at time $n$. The first two moments of $\ext{k}{n}$
are given by
\begin{align*}
\E\left[\ext{k}{n}\right] &= (kn - n + 1) \sum_{i = 0}^{n -
1} \frac{k}{k + (k - 1)i}.
\\ \E\left[\left(\ext{k}{n}\right)^2\right] &= \bigl((k -
1)n + m\bigr) \bigl((k - 1)n + 1\bigr) k E(k, n) +
O\left(n^2 \log{n}\right),
\end{align*}
where $E(k, n)$ is a function of $k$ and $n$, given in
Appendix~\ref{App:ext}.
\end{prop}
The proof of Proposition~\ref{Thm:ext} also can be found in
Appendix~\ref{App:ext}. As we know that
$$\sum_{i = 0}^{n - 1} \frac{k}{k + (k - 1)i} \sim \frac{k}{k - 1}
\log{n},$$
for large $n$, we hence conclude that the leading order of the
asymptotic expectation of $\ext{k}{n}$ is $kn\log{n}$.
The diameter of HDRANs is also considered. In~\cite{frieze2014some},
the
authors established an upper bound for the diameter of planar RANs
by utilizing a known result of the height of weighted $k$-ary
trees~\cite[Theorem 5]{broutin2006large}, i.e.,
$${\rm diameter} \left(\apol_{n}^{(3)}\right) \le \rho \log n,$$
where $\rho = 1/\eta$, and $\eta$ is the unique solution greater
than $1$ for $\eta - 1 - \log \eta = \log 3$. This upper bound can
be extended to $\apol_{n}^{(k)}$ effortlessly; that is,
$${\rm diameter} \left(\apol_{n}^{(k)}\right) \le \frac{2}{\rho^{*}
(k - 1)} \log n,$$
where $\rho^{*} = 1 / \eta^{*}$ is the unique solution greater than
$1$ for $\eta^{*} - 1 - \log \eta^{*} = \log k$. In addition, the
authors of~\cite{ebrahimzadeh2014onlongest} proved ${\rm diameter}
\left(\apol_{n}^{(3)}\right) \overset{a.s.}{\sim} c \log n$ by
estimating the height of a class of specifically-designed random
trees. The value of $c$ is approximately $1.668$. The asymptotic
expression of the diameter of more general $\apol_{n}^{(k)}$ was
developed by~\cite{cooper2014theheight} and
by~\cite{kolossvary2016degrees}. The
approach in~\cite{cooper2014theheight} was to utilize known results
of
continuous-time branching processes coupled with recurrence methods,
and the authors of~\cite{kolossvary2016degrees} coped with
difficulties by
characterizing vertex generations. We only state (without repeating
the proof) the weak law of the diameter of $\apol_{n}^{(k)}$
from~\cite{cooper2014theheight} (with a minor tweak) in the next
theorem.
\begin{theorem}[\mbox{~\cite[Theorem 2]{cooper2014theheight}}]
For $k \ge 3$, with high probability, we have
$${\rm diameter} \left(\apol_{n}^{(k)}\right) \sim c \log n,$$
where $c$ is the solution of
$$\frac{1}{c} = \sum_{\ell = 0}^{k - 1} \frac{k - 1}{\ell + a(k
- 1)},$$
in which the value of $a$ is given by
$$\frac{\Gamma(k + 1) \Gamma(ka)}{\Gamma\bigl((k - 1)a +
k\bigr)} \exp\left\{ \sum_{\ell = 0}^{k - 1} \frac{(k - 1)(a
+
1) - 1}{\ell + (k - 1)a} \right\} = 1.$$
Especially, as $k \to \infty$,
$$c \sim \frac{1}{k \log 2}.$$
\end{theorem}
A topological measure related to distance is the {\em Wiener index},
which was proposed by the chemist Harry
Wiener~\cite{wiener1947structural} to
study molecular branching of chemical compounds. For a network $G(V,
E)$, the Wiener index is defined as the sum of distances of all
paired vertices, i.e., $W(G) = \sum_{i, j \in V} d(i, j)$. The
Wiener index has been extensively studied for random
trees~\cite{dobrynin2001wiener, neininger2002thewiener}. For other
random structures, we
refer the readers to~\cite{bereg2007wiener, fuchs2015thewiener,
janson2003thewiener, wagner2006aclass, wagner2007ontheaverage,
wagner2012onthewiener}.
The methodologies for computing the Wiener index of random trees,
however, are
not adaptable to the study of RANs, as the bijection between RANs
and $k$-ary trees is based on a clique-to-node mapping. The high
dependency of active cliques (sharing vertices and edges)
substantially increases the challenge of formulating mathematical
relation between distance (vertex-based) and depth (clique-based).
There is only a few articles studying distance or related properties
in RANs. In~\cite{kolossvary2016degrees}, the authors proved that
the distance
of two arbitrary vertices in a HDRAN has both mean and variance of
order $\log n$, and that this distance follows a Gaussian law
asymptotically. However, it seems difficult to extend this result to
the Wiener index, as the covariance structure of the distances (of
all paired vertices) is unspecified. Planar RANs ($\apol_{n}^{(3)}$)
were considered in~\cite{bodini2008distances}. In this article, the
dominant term
of the total distance of all pairs of vertices was shown to be
$\sqrt{3 \pi} n^{5/2}/22$. The main idea was to consider an
enumerative generating function of the total distance, and then
decompose the total distance into interdistance and extradistance.
This approach can be extended to HDRANs of small network index~$k$,
but seemingly not applicable to HDRANs with general index $k$.
Therefore, the Wiener index of HDRANs remains an open problem.
We numerically explore the Wiener index of HDRANs via a series of
simulations. For $k = 3, 5, 8, 10$, we generate $500$ independent
HDRANs at time $2,000$, calculate the Wiener index for each
simulated
HDRAN, and use the kernel method to estimate the density. The plots
of the estimated densities are presented in
Figure~\ref{Fig:densityest}, where we find that they are
approximately bell-shaped, but not symmetric (positively skewed). By
observing these patterns, we conjecture that the limiting
distribution of the Wiener index of HDRANs does not follow a
Gaussian law.
\begin{figure}[tbh]
\centering
\begin{minipage}{0.48\textwidth}
\includegraphics[scale=0.23]{wiener3-eps-converted-to}
\end{minipage}
\begin{minipage}{0.48\textwidth}
\includegraphics[scale=0.23]{wiener5-eps-converted-to}
\end{minipage}
\\
\begin{minipage}{0.48\textwidth}
\includegraphics[scale=0.23]{wiener8-eps-converted-to}
\end{minipage}
\begin{minipage}{0.48\textwidth}
\includegraphics[scale=0.23]{wiener10-eps-converted-to}
\end{minipage}
\caption{Density estimation of the Wiener indices of HDRANs for
$k = 3, 5, 8, 10$}
\label{Fig:densityest}
\end{figure}
In addition, for each $k$, we apply the {\em Shapiro-Wilk} test to
the simulated data comprising $500$ Wiener indices, and receive the
following $p$-values: $0.0003$ for $k = 3$; $0.0024$ for $k = 5$;
$9.56 \times 10^{-8}$ for $k = 8$; and $0$ for $k = 10$. These
$p$-values are all statistically significant, in support of our
conjecture.
\section{Concluding remarks}
\label{Sec:concluding}
Finally, we address some concluding remarks and propose some
future work. We investigate several properties of high-dimensional
random Apollonian networks in this paper. Two types of degree
profiles are considered. For the first type, we show that the number
of vertices of a given degree concentrates on its expectation with
high probability. In the proof of Theorem~\ref{Thm:degreedist}, we
derive the $L_1$ limit of $X_{n, j}^{(k)}$, i.e.,
$$\lim_{n \to \infty} \E \left[X_{n, j}^{(k)} \right]=
\frac{\Gamma(j) \Gamma(2k - 1)}{\Gamma(j + k) \Gamma(k - 1)} \, n,$$
which suggests that the asymptotic expectation of $X_{n, j}^{(k)}$
experiences a phase transition. There are two regimes. According to
the Stirling's Approximation, we have
$$\E \left[X_{n, j}^{(k)} \right] \sim
\begin{cases}
\frac{\Gamma(j) \Gamma(2k - 1)}{\Gamma(j + k) \Gamma(k - 1)} \,
n, &\qquad \mbox{for fixed }j;
\\ \frac{\Gamma(2k - 1)}{\Gamma(k - 1)} \frac{n}{j^k}, &\qquad
\mbox{for }j \to \infty,
\end{cases}
$$
as $n \to \infty$.
For the second type of degree profile, the degree of a vertex of a
given label, we develop the probability mass function and the exact
moments by applying the analytic combinatorics methods and the
results in triangular \polya\ urns.
The next two properties that we investigate are the small world
property measured by the local clustering coefficient and the
sparsity measured by a proposed Gini index. We conclude that HDRANs
are highly clustered and sparse.
The last several properties that we look into are based on node
distance. According to an one-to-one relation between HDRANs and
$k$-ary trees, we compute the first two moments of the total depth
of active cliques in HDRANs. We also numerically study the Wiener
index, and conjecture that its limiting distribution is not normal
based on simulation results. The diameter of HDRANs is retrieved
from~\cite{cooper2015long}.
To end this section, we propose some future work. Our conjecture of
non-normality of the Wiener index is based on numerical experiments.
A more rigorous proof is needed. There remain many open problems for
HDRANs, such as the length of the longest path and the highest
vertex degree. One suggests carrying out some studies of the
stochastic
processes that
take place on HDRANs, especially those with applications to
mathematical physics, such as percolation and diffusion. We will
look into these open problems and report the results elsewhere.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,851 |
Today in history… the Battle of George Square
12:00am Jan 31, 2019 | Admin - Acorn Stairlifts UK & Lifestyle
ONE of the darkest days in the history of Glasgow took place 100 years ago today, when striking workers and police clashed violently in the "Battle of George Square".
When the First World War ended on November 11th, 1918, demobilised soldiers hoped to return to "a land fit for heroes", but what they actually came home to was widespread unemployment and grinding poverty.
The situation was at its worst in Britain's big cities, where major industries were immediately scaled down as soon as the 'war effort' was over. It meant there were few jobs for returning soldiers, while many existing workers were also laid off, raising the prospect of mass unemployment. In Glasgow the Scottish Trades Union Congress (TUC) and the Clyde Workers' Committee (CWC) – representing workers in the hard-hit shipbuilding trades – proposed a measure to ease the situation.
They wanted to reduce the newly-agreed 47-hour working week to 40 hours, reasoning that if everyone worked fewer hours, more jobs would be created. It made sense, but the employers, also facing tough times, would not budge and on January 27th around 3,000 workers went out on strike. After three days they had been joined by a further 40,000 workers in shipbuilding and engineering, with 'sympathy strikes' called at nearby coal mines and the local power station.
It quickly became Scotland's most widespread strike since a notorious week of unrest in 1820 which had similar origins, centred on unemployed soldiers back from the Napoleonic Wars. The growing Glasgow strike was discussed in Westminster, with fears it could spread and an instruction sent to Scottish Command (the Army HQ in Scotland) to be ready to deploy troops if necessary.
On Friday January 31st, 1919, up to 25,000 strikers congregated in George Square, the biggest of six civic squares in Glasgow city centre. They were awaiting an answer to their petition seeking a 40-hour week, which the CWC had delivered to the Lord Provost of Glasgow earlier in the week. The situation was tense, with strikers unhappy at receiving no answer, but it is unclear exactly how the gathering turned nasty.
What is known is that at 12-20pm, police launched a baton charge into a section of the striking workers, possibly to 'snatch' some of those identified as ringleaders. When the fighting started, CWC leaders David Kirkwood and Emanuel 'Manny' Shinwell were meeting with the Lord Provost in Glasgow City Chambers. On hearing the news, they immediately left for George Square to quell the trouble, but on arriving Kirkwood was knocked to the ground by a police baton and both he and Shinwell arrested, together with another union leader, William Gallacher.
All three were later charged with "instigating and inciting large crowds of persons to form part of a riotous mob", even though Kirkwood and Shinwell weren't even in the square when the fighting started. Kirkwood was acquitted at his trial when a photograph showed him lying unconscious on the ground before even reaching George Square.
After their baton charge, the outnumbered police retreated from George Square, but outbreaks of fighting between strikers and police, some on horseback, spread into nearby streets and continued into the night. With fears growing of "a Bolshevist uprising", the military were soon deployed, as previously planned. Troops from the nearest barracks at Maryhill were not used because it was feared they might side with the strikers, who were their neighbours and, in many cases, former soldiers themselves.
Instead they came from barracks in southern Scotland and northern England, with up to 12,000 troops made available. The emergency was deemed so serious that six medium-sized tanks were mobilised from the Royal Tank Regiment's base at Bovington Camp in Dorset, on England's south coast. They were loaded onto railway transporters and sent north on their 470-mile journey.
The first troops began arriving in Glasgow on the Friday evening, sending out armed patrols and setting up machine gun nests in George Square. Over the next few days, thousands more troops arrived, turning the quadrangle in front of the city chambers into a fortified encampment. It was all designed to send out a strong and clear message to the strikers, that any violent uprising would be swiftly and brutally suppressed – and it worked.
Remaining strikers quickly dispersed and by Sunday calm was restored. When the tanks arrived on Monday, their crews soon realised there would be no work for them. The strike was over, but many of its leaders were arrested and charged. Only two, William Gallacher and Manny Shinwell – were convicted, sentenced to five months and three months in prison respectively, and emerging as heroes of the workers' movement.
In the General Election of 1922, strike leaders Shinwell and Kirkwood were among 29 Labour MPs elected from Scotland, with Labour for the first time overtaking the Liberals as the main competition to the Conservatives. When the Tories called another General Election the following year to strengthen their numbers in Parliament, the gamble backfired and the Britain's first Labour government came to power under Scottish socialist Ramsay MacDonald. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,173 |
{"url":"https:\/\/repo.pw.edu.pl\/info\/phd\/WUTbe778e8435324d9c8d709615b8e79bba\/","text":"# Knowledge base: Warsaw University of Technology\n\nBack\n\n## Opracowanie metody badania b\u0142\u0119d\u00f3w dynamicznych w pomiarach skaningowych na maszynach wsp\u00f3\u0142rz\u0119dno\u015bciowych\n\n### Grzegorz Krajewski\n\n#### Abstract\n\nThrough the last decades, the coordinate metrology become the powerful and essential part inspection technique in a wide range of industries. The Coordinate Measuring Machine (CMM) is the main tool of this technique. It allows simple and easy measurement of even complicated three dimensional components, by gathering numbers of geometrical points. This results in the huge popularity of the CMM machine parts inspections. Scanning probes CMM are currently treated as a standard in coordinate metrology. Not only because of gathering high quantity of data in short time of probing but also significantly decreasing the inspection time. More efficiency in coordinate metrology equals faster measurement cycles with acceptable accuracy. But in fact, the scanning speed significantly influences the error budget. The hereby thesis is theoretical and experimental study of the CMM\u2019s dynamic properties. Based on the proposed model, the method of testing CMM\u2019s dynamic properties has been developed. The new method of investigating CMM\u2019s dynamic properties consist of the scanning measurement of the simple shape master artefact. The assembly of two flat reference plates has been used as a shape standard. The proposed method for testing the dynamic properties of a coordinate measuring machine is a reliable test for machine dynamics. Application of an angle artefact enables testing of the dynamic properties of the machine and checking it\u2019s ability to execute the measurement with a speed that guaranties the required accuracy. Based on this measurement, applying spectrum analysis CMM errors has been evaluated and assigned to the error sources. The experimental study has been extensively presented. Measurements have been carried out on two CMMs equipped with two types of probes (active and passive). The influence of a scanning speed, type of probe, artefact location and stylus length are taken into consideration. The effects of frictional interaction between the stylus tip and the part surface are also considered and mathematically explained. The whole thesis has been analyzed using statistical tests confirming dependence between compared factors. Developed method for testing dynamic errors of CMM have also been confirmed by another original measuring method. Detailed analysis of scanning process has been extensively presented and preceded by deep literature overview. This conforms original input into developing the knowledge of scanning measurements effects. The developed method is also a source of unique knowledge of optimizing scanning CMM measurements.\nRecord ID\nWUTbe778e8435324d9c8d709615b8e79bba\nDiploma type\nDoctor of Philosophy\nAuthor\nGrzegorz Krajewski Grzegorz Krajewski,, The Institute of Metrology and Biomedical Engineering (FM\/IMBE)Faculty of Mechatronics (FM)\nTitle in Polish\nOpracowanie metody badania b\u0142\u0119d\u00f3w dynamicznych w pomiarach skaningowych na maszynach wsp\u00f3\u0142rz\u0119dno\u015bciowych\nLanguage\n(pl) Polish\nCertifying Unit\nFaculty of Mechatronics (FM)\nDiscipline\nmechanical engineering \/ (technology domain) \/ (technological sciences)\nStatus\nFinished\nDefense Date\n03-12-2014\nTitle date\n17-12-2014\nSupervisor\nInternal reviewers\nEugeniusz Ratajczyk Eugeniusz Ratajczyk,, Faculty of Mechatronics (FM)\nExternal reviewers\nMa\u0142gorzata Poniatowska, prof. Politechnika Bia\u0142ostocka Ma\u0142gorzata Poniatowska, prof. Politechnika Bia\u0142ostocka,, Undefined Affiliation\nPages\n138\nKeywords in English\nThrough the last decades\nAbstract in English\nThrough the last decades, the coordinate metrology become the powerful and essential part inspection technique in a wide range of industries. The Coordinate Measuring Machine (CMM) is the main tool of this technique. It allows simple and easy measurement of even complicated three dimensional components, by gathering numbers of geometrical points. This results in the huge popularity of the CMM machine parts inspections. Scanning probes CMM are currently treated as a standard in coordinate metrology. Not only because of gathering high quantity of data in short time of probing but also significantly decreasing the inspection time. More efficiency in coordinate metrology equals faster measurement cycles with acceptable accuracy. But in fact, the scanning speed significantly influences the error budget. The hereby thesis is theoretical and experimental study of the CMM\u2019s dynamic properties. Based on the proposed model, the method of testing CMM\u2019s dynamic properties has been developed. The new method of investigating CMM\u2019s dynamic properties consist of the scanning measurement of the simple shape master artefact. The assembly of two flat reference plates has been used as a shape standard. The proposed method for testing the dynamic properties of a coordinate measuring machine is a reliable test for machine dynamics. Application of an angle artefact enables testing of the dynamic properties of the machine and checking it\u2019s ability to execute the measurement with a speed that guaranties the required accuracy. Based on this measurement, applying spectrum analysis CMM errors has been evaluated and assigned to the error sources. The experimental study has been extensively presented. Measurements have been carried out on two CMMs equipped with two types of probes (active and passive). The influence of a scanning speed, type of probe, artefact location and stylus length are taken into consideration. The effects of frictional interaction between the stylus tip and the part surface are also considered and mathematically explained. The whole thesis has been analyzed using statistical tests confirming dependence between compared factors. Developed method for testing dynamic errors of CMM have also been confirmed by another original measuring method. Detailed analysis of scanning process has been extensively presented and preceded by deep literature overview. This conforms original input into developing the knowledge of scanning measurements effects. The developed method is also a source of unique knowledge of optimizing scanning CMM measurements.\nThesis file\n\u2022 File: 1\nPraca Doktorska_Grzegorz Krajewski.pdf\nRequest a WCAG compliant version\nCitation count\n2\n\nUniform Resource Identifier\nhttps:\/\/repo.pw.edu.pl\/info\/phd\/WUTbe778e8435324d9c8d709615b8e79bba\/\nURN\nurn:pw-repo:WUTbe778e8435324d9c8d709615b8e79bba\n\nConfirmation\nAre you sure?","date":"2021-06-17 20:44:53","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\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3542560935020447, \"perplexity\": 3311.198499238051}, \"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\/1623487633444.37\/warc\/CC-MAIN-20210617192319-20210617222319-00192.warc.gz\"}"} | null | null |
Palestine Amiss in Priyamvada Gopal's Insurgent Empire
September 11, 2019 August 15, 2021 Churchill's KarmaLeave a comment
Britain has no anti-imperialist tradition. There may have been the occasional outburst from this or that literary, cultural or political figure but such outbursts have always had very limited public appeal. In recent years opinion polls have shown the British public has an overwhelming positive view of the days when the British imperialist writ ran supreme over a very good proportion of mankind. Tens of millions of souls may have perished in slavery and destitution; hundreds and thousands of millions of pounds may have been looted from what is now referred to as the Global South. But hey, let's look on the positive side of Empire: we did eventually abolish slavery and build the railroads in India.
The Cambridge University academic, Priyamvada Gopal over the last several years has become known as a critic of British imperialism, imperial nostalgia and also of contemporary British racism. Her latest book, Insurgent Empire: Anticolonial Resistance and British Dissent is apparently informed by her admiration of Edward Said. As she states, the "late Edward Said's work continues to nourish my mind and the impact of his thought will, hopefully, be evident throughout this book."[i] Professor Edward Said distinguished himself as one of the world's most preeminent intellectuals in the study of imperialism. Among his much esteemed books are Orientalism, The Question of Palestine and Culture and Imperialism. His writing was no doubt influenced by his status as a refugee as a result of British imperialism's policies in his native Palestine.
The hypothesis that drives Gopal's book is that anti-colonialist and anti-imperialist resistance in the British Empire influenced political dissent in Britain itself. In the introduction of the book there are numerous repetitions of this claim. For example, struggles in the Global South, "were not without impact on metropolitan ideologies and practices."[ii] And more forthrightly: Continue reading →
British Empire, Edward Said, Insurgent Empire, Palestine, Priyamvada Gopal
Unpacking the Zionism of ex-Revolutionary, Dr. James Heartfield
May 19, 2018 Churchill's KarmaLeave a comment
Between twenty-five to thirty years ago the only political groupings that guaranteed a fair hearing for the Palestinian cause are those that were and are commonly referred to as the "far left". The Labour Party had always been an overwhelmingly pro-Zionist organisation until very recently. For Muslim or Islamist groups, Palestine wasn't a major rallying issue for them back in say the late 1980s or early 1990s.
But even within the so-called "far-left" especially amongst the multitude of Trotskyist organisations one constantly encountered strong pro-Israeli sentiment which continues to this very day. Anyway, it was within this political environment that I came across a magazine called "Living Marxism" published by the Revolutionary Communist Party (RCP), a Trotskyist organisation whose leading guru was Frank Furedi. This organisation had ultimately split from the parent Trotskyist organisation that was headed by Tony Cliffe called, Socialist Workers Party (formerly International Socialists). Living Marxism had stood out from the other Trotsky magazines or journals because it took up more pro-Palestinian or pro-Arab positions than the others vis-à-vis the Zionist occupying entity and also during the first war on Iraq in 1991. Continue reading →
Ethnic Cleansing, Frank Furedi, James Heartfield, Living Marxism, Palestine, Spiked, Zionism
Is Minister Louis Farrakhan an anti-Semite?
March 12, 2018 July 2, 2020 Churchill's KarmaLeave a comment
Part of the course of any black, brown or other person of colour rising to any position of leadership is to weather accusations made by the inevitable Western establishment detractors. Usually, if they can't pin any financial discrepancy or moral impropriety then such leader will be accused to have a bigoted and irrational hostility to a demographic the West purports to be in love with, it's usually the Jewish people but could be others.
Contempt towards the Jewish people is anti-Semitism. Historically, this evil is associated with European political culture which ultimately manifested in the European holocaust in the early 1940s and took the lives of millions of European Jews. Some European Jews had seen a solution to European political cultural contempt for them by aligning with European imperialist elites with a view to transfer European Jews to an imperially administered colony in Africa or Asia. These latter Jews are referred to as Zionists. Obviously not all Jews were enamoured with this "solution" to a hitherto a European political-cultural condition.
In 1917, British imperialism found a role for European Jewish-Zionist and issued the 'Balfour Declaration' to colonise Palestine. It did so not because it had admiration for Jewish people, Continue reading →
anti-Semitism, Louis Farrakhan, Nation of Islam, Palestine, Zionism
Ken Livingstone and Zionism's only Truth.
May 1, 2016 Churchill's KarmaLeave a comment
The recent remarks of the first ever and former mayor of London, Ken Livingstone supposedly in support of another British Labour politician, Naz Shah, who had shared a social media post depicting a map of Israel transferred to the United States has ignited a debate on the extent of anti-Semitism in the British Labour Party.
In defence of Shah, Livingstone felt compelled to remind people that certain Zionists in 1930's Nazi Germany came into an agreement with elements in the Nazi regime to transfer German Jews to Palestine. And indeed there is nothing remotely mutually exclusive about being both anti-Semitic and pro-Zionist. But, why he needed to drag this minor episode of European Zionist history, the Haavara agreement, into the mix in a supposed defence of Shah is bewildering.
More bewildering when one considers the fact that British imperialism was the most consequential partner to the Zionist colonial settler project in Palestine in the inter-war period. In 1917 when the British government issued the Balfour Declaration there were between probably 70,000 Jews in Palestine as opposed to at least 700,000 Palestinians. The British Empire's policy was to establish a "national home for the Jewish people" and use its "best endeavours to facilitate" this achievement. Continue reading →
anti-Semitism, British Empire, Ken Livingstone, Labour Party, Palestine, Zionism
Britain's Colonial-Zionist Project in Palestine made the Creation of Saudi Arabia Necessary
November 20, 2015 October 8, 2020 Churchill's Karma1 Comment
The covert alliance between the Kingdom of Saudi Arabia (KSA) and the Zionist entity of Israel should be no surprise to any student of British imperialism. The problem is the study of British imperialism has very few students. Indeed, one can peruse any undergraduate or post-graduate university prospectus and rarely find a module in a Politics degree on the British Empire let alone a dedicated degree or Masters degree. Of course if the European led imperialist carnage in the four years between 1914 – 1918 tickles your cerebral cells then it's not too difficult to find an appropriate institution to teach this subject, but if you would like to delve into how and why the British Empire waged war on mankind for almost four hundred years you're practically on your own in this endeavour. One must admit, that from the British establishment's perspective, this is a remarkable achievement.
In late 2014, according to the American journal, "Foreign Affairs", the Saudi petroleum Minister Ali al-Naimi is reported to have said "His Majesty King Abdullah has always been a model for good relations between Saudi Arabia and other states and the Jewish state is no exception." Recently, Abdullah's successor King Salman expressed similar concerns to those of Israel's to the growing agreement between the United States and Iran over the latter's nuclear programme. This led some to report that Israel and KSA presented a "united front" in their opposition to the nuclear deal. This was not the first time the Zionists and Saudis have found themselves in the same corner in dealing with a common foe. In North Yemen in the 1960's, the Saudis were financing a British imperialist led mercenary army campaign against revolutionary republicans who had assumed authority after overthrowing the authoritarian, Imam. Gamal Abdul-Nasser's Egypt militarily backed the republicans, while the British induced the Saudis to support them in financing and arming remaining remnants of the Imam's supporters to stretch Nasser's forces. During this campaign, the British organised the Israelis to drop arms for the British proxies in North Yemen, 14 times. The British, in effect, militarily but covertly, brought the Zionists and Saudis together in 1960's North Yemen against their common foe.
However, one must go back to the 1920's to fully appreciate the origins of this informal and indirect alliance between KSA and the Zionist entity. Continue reading →
Arab World, Balfour Declaration, BBC, British Empire, Divide and Rule, Hijaz, Ibn Rashid, Ibn Saud, Palestine, Saudi Arabia, Sharif Hussain bin Ali, Syria, Wahhabis
RT @KawkabAlwaday: In2015, Saudi airstrike targeted residential neighbourhood near Sanaa airport. It destroyed 14 houses,including the hom… 2 hours ago | {
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} | 8,791 |
\section{Introduction}
\vspace{-0.05cm}
Fault Tree Analysis (FTA) constitutes a fundamental analytical tool aimed at modelling and evaluating how complex systems may fail \cite{FtaHandbook2002}. FTA is widely used in safety and reliability engineering as a risk assessment tool for a variety of industries such as aerospace, power plants, nuclear plants, and other high-hazard fields \cite{Ruijters2015}. Essentially, a fault tree (FT) involves a set of \textit{basic events} that are combined using logic operators (e.g. AND and OR gates) in order to model how these events may lead to an undesired system state represented at the root of the tree (top event). Basic events can be associated to hardware failures, human errors, and other cyber-physical conditions including cyber events such as software errors, communication failures, and cyber attacks. Let us consider a simple example.
\vspace{-0.05cm}
\subsection{Fault tree example}
\vspace{-0.05cm}
The fault tree shown in Fig. \ref{fig:simple-example} illustrates the different combinations of events that may lead to the failure of an hypothetical Fire Protection System (FPS) based on \cite{Kabir2017}. The FPS can fail if either the fire detection system or the fire suppression mechanism fails. In turn, the detection system can fail if both sensors fail simultaneously (events $x_1$ and $x_2$), while the suppression mechanism may fail if there is no water ($x3$), the sprinkler nozzles are blocked ($x_4$), or the triggering system does not work. The latter can fail if neither of its operation modes (automatic ($x_5$) or remotely operated) works properly. The remote control can fail if the communications channel fails ($x_6$) or the channel is not available due to a cyber attack, e.g. DDoS attack ($x_7$). Each basic event has an associated value that indicates its probability of occurrence~$p(x_i)$.
\begin{figure}[!t]
\centering
\includegraphics[width=0.46\textwidth]{images/fault-tree-v9.png}
\vspace{-0.2cm}
\caption{Fault tree of a cyber-physical fire protection system (simplified)}
\label{fig:simple-example}
\vspace{-0.6cm}
\end{figure}
\section{Problem description}
FTA comprises a broad family of methods and techniques used for qualitative and quantitative analysis. Qualitative techniques normally involve structural aspects of faults trees like single points of failure (SPOFs) and minimal cut sets (MCSs). MCSs are minimal combinations of events that together may lead to the failure of the top level event \cite{Ruijters2015}.
Quantitative analysis usually involves numerical outcomes such as failure probabilities. The present work lies in the intersection of these two families. On the one hand, we are interested in finding MCSs. On the other hand, we focus on the MCS whose probability is the highest among all possible MCSs.
We call this MCS the \textit{Maximum Probability Minimal Cut Set} (MPMCS).
Note that this optimisation problem not only relates to the structural minimal cut set in the fault tree but also to the probabilities assigned to the events in it.
A fault tree $F$ can be represented as a Boolean equation $f(t)$ that expresses the different ways in which the top event $t$ can be satisfied \cite{FtaHandbook2002}.
In our example, $f(t)$ is as follows:
\vspace{-0.15cm}
\fontsize{9}{9}\selectfont\sffamily{
\begin{equation}
\begin{array}{c}
f(t) = (x_1 \land x_2) \lor (x_3 \lor x_4 \lor (x_5 \land (x_6 \lor x_7))) \nonumber
\end{array}
\vspace{-0.15cm}
\end{equation}
}\normalsize \normalfont
The objective is to find the minimal set of logical variables that makes the equation $f(t)$ \textit{true} and whose joint probability is maximal among all minimal sets.
In our example, the MPMCS~is $\{x_1,x_2\}$ with a joint probability of $0.02$.
\section{Resolution method}
Our resolution method relies on Maximum Satisfiability (MaxSAT) techniques~\cite{Barrere-Arxiv2019}. A MaxSAT problem consists in finding a truth assignment that maximises the weight of the satisfied clauses. Equivalently, MaxSAT minimises the weight of the clauses it falsifies \cite{Davies2011}. A Weighted Partial MaxSAT problem involves \textit{soft clauses} with non-unit weights and it will try to minimise the penalty induced by falsified weighted variables. We use this last variant to solve our optimisation problem. The proposed resolution method involves six steps.
\textbf{Step 1 (Logical transformation).}
Since MaxSAT tries to maximise the number of satisfied clauses, and we are looking for minimal cut sets, we consider the complement of the equation $f(t)$ to represent the non-occurrence of the top event (system success): $X(t) = \neg f(t)$.
$X(t)$ models the \textit{Success Tree} and can be obtained directly from the original fault tree by complementing all the events and substituting OR gates by AND gates, and vice versa~\cite{FtaHandbook2002}:
\vspace{-0.4cm}
\fontsize{9}{9}\selectfont\sffamily{
\begin{equation}
\begin{array}{c}
X(t) = (\neg x_1 \lor \neg x_2) \land (\neg x_3 \land \neg x_4 \land (\neg x_5 \lor (\neg x_6 \land \neg x_7))) \nonumber
\end{array}
\vspace{-0.2cm}
\end{equation}
}\normalsize \normalfont
Since we are interested in minimising the number of satisfied clauses, which is opposed to what MaxSAT does (maximisation), we flip all logic gates but keep all events in their positive form. To explain why, let us reformulate $X(t)$ as $Y(t)$ where the logical variables are renamed as $y_i = \neg x_i$:
\vspace{-0.2cm}
\fontsize{9}{9}\selectfont\sffamily{
\begin{equation}
\begin{array}{c}
Y(t) = (y_1 \lor y_2) \land (y_3 \land y_4 \land (y_5 \lor (y_6 \land y_7))) \nonumber
\end{array}
\vspace{-0.2cm}
\end{equation}
}\normalsize \normalfont
We know that $\neg Y(t)=f(t)$ by definition. Therefore, we aim at maximising the number of satisfied variables $y_i$ to make $\neg Y(t) = true$.
But because the variables $y_i$ are the complement of the logical variables $x_i$, we are actually maximising the number of falsified variables $x_i$ and minimising the satisfied ones in $f(t)$. Such a minimal set in $f(t)$ constitutes an MCS in the fault tree.
\textbf{Step 2 (CNF conversion).}
SAT solvers normally consider input formulas in conjunctive normal form (CNF).
To avoid exponential computation times, we use the Tseitin transformation to produce, in polynomial time, a new formula in CNF that is not strictly equivalent to the original formula (because there are new variables) but is equisatisfiable~\cite{Barrere-Arxiv2019}.
This means that given an assignment of truth values, the new formula is satisfied if and only if the original formula is also satisfied.
\textbf{Step 3 (Probabilities transformation into log-space).}
In order to maximise the product of weighted decision variables in MaxSAT, we transform the weights $p(x_i)$ into $w_i = -\log(p(x_i))$ to produce positive values.
This means that the lower a probability $p(x_i)$, the higher its negative log value $w_i$.
Conversely, the higher the probability, the lower the $-log$ value, as shown in Table \ref{tab:example-probs} for our example fault tree.
\bgroup
{ \renewcommand{\arraystretch}{0.8}
\setlength{\tabcolsep}{0.4em}
\begin{table}[!h]
\vspace{-0.4cm}
\centering
\fontsize{7.4}{7.4}\selectfont
\caption{Fault tree probabilities and $-log$ values $w_i$}
\vspace{-0.3cm}
\begin{tabular}{c|c|c|c|c|c|c|c}
\toprule
\textbf{Probs.} & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $x_7$\\
\midrule
$p(x_i)$ & 0.2 & 0.1 & 0.001 & 0.002 & 0.05 & 0.1 & 0.05 \\
$w_i$ & 1.60944 & 2.30259 & 6.90776 & 6.21461 & 2.99573 & 2.30259 & 2.99573 \\
\bottomrule
\end{tabular}
\label{tab:example-probs}
\end{table}
}
\egroup
\vspace{-0.2cm}
\textbf{Step 4 (Weighted Partial MaxSAT instance).}
We define a soft clause for each decision variable in $\neg Y(t)$.
These soft clauses indicate the solver that each variable $y_i$ can be falsified with a certain penalty $w_i$, which corresponds to the transformed probability of event $x_i$ as shown in Table~\ref{tab:example-probs}.
The MaxSAT solver tries to minimise the total weight of falsified variables, and therefore, a solution to this problem yields a minimum vertex cut of the fault tree in logarithmic space.
Since the lowest logarithmic values correspond to the highest probabilities, the solution indicates the MCS with maximum joint probability, i.e. the MPMCS.
\textbf{Step 5 (Parallel MaxSAT resolution).}
We have experimentally observed that, quite often, SAT solvers are very good at some instances and not that good at others.
This is due to the different optimisation techniques used within solvers \cite{Davies2011}.
To address this issue, our tool executes multiple pre-configured solvers in parallel and picks up the solution of the solver that finishes first. This method provides a more stable behaviour in terms of performance and scalability.
\textbf{Step 6 (Reverse log-space transformation).}
The joint probability of the MPMCS~is computed by performing the reverse log-space transformation $P_F(t) = \exp(-1 \times \sum_{i} w_i )$, where $i$ indexes the events found in the MaxSAT solution.
\vspace{-0.05cm}
\section{Preliminary results and conclusion}
\vspace{-0.05cm}
We have developed an open source tool called MPMCS4FTA~that implements the proposed methodology and is publicly available at \cite{Barrere-MPMCS4FTA-Github}.
The tool runs in the command line and outputs the solution in a JSON file that is used to graphically display the fault tree and the MPMCS~in a web browser.
Fig.~\ref{fig:tool-example} shows the output of MPMCS4FTA~for our example fault tree.
The results of our analytical evaluation indicate that the method is able to scale to fault trees with thousands of nodes in seconds.
FTA is an essential technique to evaluate dependability in a wide range of systems.
The proposed MPMCS~is intended to extend the body of measures used in FTA and support fundamental activities such as decision making, risk assessment, and fault prioritisation.
As future work, we plan to evaluate different representation techniques (e.g. BDDs~\cite{Ruijters2015}) to address the MPMCS~problem and conduct a thorough comparison on performance and scalability. We also aim at extending our approach to include additional operators such as voting gates.
\begin{figure}[!t]
\centering
\includegraphics[width=0.41\textwidth]{images/snapshot3.png}
\vspace{-0.35cm}
\caption{Example scenario and MPMCS~with our tool MPMCS4FTA}
\label{fig:tool-example}
\vspace{-0.5cm}
\end{figure}
\vspace{-0.1cm}
\bibliographystyle{IEEEtran}
| {
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Quail Creek's seventh annual Health and Wellness Fair will be held on Friday, October 19, 2018 from 9:00 a.m.-1:00 p.m. Participating local businesses and health care professionals will be offering testing, samples and lots of information on a wide array of products and services that will help make you look and feel your best.
The Women of Quail Creek October luncheon – It's a fiesta! | {
"redpajama_set_name": "RedPajamaC4"
} | 5,638 |
TikkiBoxx | Hello Pretty. Buy design.
TikkiBoxx is a Proudly South African store stocking & manufacturing mostly handmade items and beautiful locally crafted wares.
We love to keep things fresh, unique and quirky.
Perfectly describing each and every product can be a hugely difficult task when dealing with unique, handmade items.
Please read our points below on how we try and provide the best possible way to exhibit our products on this store.
Producing intricate, written descriptions of each product can be tricky for some items and we therefore rely mainly on our images to represent these products.
In doing this we try our best to upload photos that are as accurate as possible to help describe each item, however there may be slight differences in colouring from time to time as perfectly capturing these images can give us a run for our money.
Most items uploaded onto our store have measurements available.
Due to the fact that the majority of our stock is handmade, there may be very few (and marginal) size discrepancies from item to item.
As per above, we try and capture each items image as best we can in our photos.
Sometimes our written descriptions may have some of our crazy vocabulary describing colour or patterns as we see them.
Please take special note that the majority of our items are HANDMADE.
By supplying these products, we offer you the chance to purchase something unique and specialised.
Please allow for minor imperfections (if any) when buying those items, as those make up the end product to be what it is.
WE also hope you see and feel the love that goes into those special items.
For a longer lifespan and more wearability, costume jewellery should always be cared for gently and generally be kept out of water eg. Taking showers, swimming etc.
Please note that all our childrens clothes are one-off garments.
They are made according to a sizing scheme which has proven to be somewhat petit.
We recommend that you should buy a size (maybe two sizes) bigger than your childs age just to be safe.
Unfortunately we don't offer custom made clothing (unless in much larger quantities).
We also only keep the stock that is uploaded onto this store and do not keep multiple sizes of any item unless stipulated in that products description.
We recommend that all items that are bought from our store that may need to be washed (clothing, bags, some dolls etc...) be done so by hand in cold water.
We try our best at giving you the best product possible. But should you be unsatisfied with your parcel when you receive it, give us a shout.
Items may only be refunded if they are returned in their original packaging, and in their original condition.
Unfortunately we will not offer any refunds for products purchased that are the incorrect size for what you need. Please be sure to read the product descriptions and the store policies to avoid this.
In the event of there being any refunds, shipping/posting costs may be at your own expense.
All items will be beautifully and securely packaged by us.
Items purchased may be collected from us (by appointment) so you will need to be in touch with us to arrange this.
We are situated in the EAST RAND, JHB.
Should you require your purchase to be deliciously TIKKIBOXX gift wrapped, please advise us. This service is free of charge. | {
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extern unsigned end;
void arch_init()
{
mem_init(&end, PHYS_RAM - (int) V2P(&end));
printk("Initialized memory allocator\n");
init_paging();
printk("Initialized kernel paging\n");
init_gdt();
init_idt();
printk("Initialized descriptors\n");
init_keyboard();
init_timer(100);
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,771 |
{"url":"https:\/\/math.stackexchange.com\/questions\/1982376\/prove-that-interval-0-1-is-not-compact","text":"# Prove that interval $(0,1]$ is not compact\n\nI'm trying to prove that interval $(0,1]$ is not compact by showing it doesn't have Heine-borel property.\n\nI know a set is compact if a set is closed and bounded or has BW property or has Heine-borel property. But I'm trying to use heine-borel property to prove that it is not compact. I know I have to use the definition of open cover to prove this, but I don't know how to begin.\n\nmy guess: in order to prove $(0,1]$ is not compact by showing it doesn't have heine-borel property, is to show that there exists open cover $(0,1]$ that cannot be reduced to a finite subcover. but then what would be $\\mathscr{U}$?\n\n\u2022 take the obvious open cover: the collection $(1\/n,2)$, say, for positive integers $n$. Oct 24, 2016 at 3:18\n\u2022 sorry i don't understand open cover too well. how is $(1\/n,2)$ an obvious open cover? @symplectomorphic Oct 24, 2016 at 3:19\n\u2022 sorry, what is the Heine Borel property? Oct 24, 2016 at 3:20\n\u2022 state the definition of an open cover and think about my example. Oct 24, 2016 at 3:21\n\u2022 @symplectomorphic ah i was dumb. i can even make $(1\/2^n,2)$ Oct 24, 2016 at 3:32\n\nYou want to construct an open cover for which no finite subcover can still cover all of the interval $(0,1]$. One way you might do this is to take a collection of covers $U_n = (a_n, 2)$, where $a_n \\to 0$.\nIf you create $a_n$ such that all $a_n > 0$, then clearly no finite subcover will still cover the interval.\nDefine $$I_{n} = (\\frac{1}{n}, 2)$$\nThen $$\\bigcup_{n \\in \\mathbb{N}} I_{n}$$ covers $$(0,1]$$ but it's easy to see that finitely many $$I_{n}$$ can't cover the interval. This is because if a finite cover $$\\{I_{n}\\}_{n=1}^{k}$$ existed, then this would not cover $$(0, \\frac{1}{k})$$ and thus wouldn't be a cover at all.","date":"2022-05-18 12:26:53","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\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 6, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7908146977424622, \"perplexity\": 154.14525568064767}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662522270.37\/warc\/CC-MAIN-20220518115411-20220518145411-00346.warc.gz\"}"} | null | null |
Text or Call: (575) 267-6564
RentTrack
A Brief History of New Mexico State University (NMSU)
by wpengine | Sep 16, 2020 | News & Events
New Mexico is known for many things — the Taos Pueblo community where people have continuously lived for more than one thousand years, Smokey Bear Historical Park, White Sands National Park, Carlsbad Caverns National Park, the Four Corners, the International UFO Museum and Research Center in Roswell, and much more. Throughout this great state, another noteworthy claim to fame includes universities with a rich background of academic excellence, especially when it comes to the history of New Mexico State University.
Founded in 1888 by Hiram Hadley, an Earlham College education teacher, NMSU was originally named Las Cruces College. The college began as a two-room building and included an elementary school, a university preparatory school, and a business school. The college later merged with the New Mexico College of Agriculture and Mechanic Arts, the state's land-grant college, and the new school opened in 1890.
It wasn't until 1960 when the New Mexico Constitution officially recognized the school as New Mexico State University. NMSU remains as the oldest public institution of higher education in the state of New Mexico.
Fast forward to the present day, and impressive numbers reflect how New Mexico State University has grown to become an outstanding educational beacon for students around the world:
More than 21,000 students are enrolled at the university.
Students have the option to choose from a plethora of degree programs with options for bachelor's, master's, and doctoral degrees, as well as certificates. The university offers 28 doctoral degree programs, 58 master's degree programs, and 96 baccalaureate majors.
The top five majors at the university include business, liberal arts and humanities, nursing, criminal justice and safety studies, and mechanical engineering.
NMSU enrolls a variety of students from 49 different states and 89 foreign countries.
The campus is a recognized NASA Space Grant College and the first honors college in New Mexico.
Alumni encompass more than 130,000 people worldwide.
The university is well known for being a Hispanic-serving institution and able to reach a multicultural population of students and communities across the state through its five campuses, a satellite learning center located in Albuquerque, cooperative extension offices in every county across the state, and 12 agriculture research and science centers.
And here's a fun fact about NMSU: The university has kept many traditions alive, including the "A". This is one of NMSU's most cherished traditions, bringing light to the Aggie pride that is surrounded on campus. The letter "A" can be seen on many of the buildings, items, and structures, but the most famous "A" stands 330 feet tall on the Tortugas Mountain east of campus.
NMSU also has a stand-out list of notable names who have called the campus home, including, musicians, actors, athletes, etc. — Joe Jackson, Roosevelt Skerrit, Sam Lacey, Cyrus Nowrasteh, Paul Wilbur Klipsch, Steve Pearce, and Tony Wragge, just to name a few.
Be sure to check out our other informative blogs here!
The Best Restaurants in Las Cruces, NM
How to Save Money as a College Student
Things to Do in Las Cruces, NM
The Flats at Ridgeview
2050 Wisconsin Avenue, Las Cruces, NM 88001
At the center of innovation in Las Cruces is a brand-new community. Although not owned or operated by BCOM, the developer's designed The Flats with BCOM's medical students in mind. Residents can enjoy a variety of exceptional amenities and an unparalleled location just minutes from the BCOM campus. Plus, our convenient by-the-bedroom leasing offers a hassle-free living experience. Relax after class by our resort-style pool, or deep dive into coursework and research in one of our individual study rooms with large wireless projection monitors and whiteboards — the choice is yours. The Flats at Ridgeview is owned by a limited liability corporation. BCOM has no ownership nor is it responsible for the operations. To ensure professional management, the ownership has contracted with Campus Advantage, a nationally recognized multifamily management company. As a result of being a proud donor to BCOM's student government association, Campus Advantage is allowed to advertise on this webpage. As well as to have internal promotions.
Become a vendor. | {
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Au bois est une nouvelle de Guy de Maupassant, parue en 1886.
Historique
Au bois est initialement publiée dans la revue Gil Blas du , puis dans le recueil Le Horla en 1887.
Résumé
Un garde champêtre, surprend, au bois Champioux, un couple de bourgeois mûrs s'abandonnant à ses instincts. Beaurain et son mari doivent s'expliquer devant monsieur le maire. Beaurain raconte comment elle et son mari se sont rencontrés.
Éditions
1886 - Au bois, dans Gil Blas
1887 - Au bois, dans Le Horla recueil paru chez l'éditeur Paul Ollendorff.
1979 - Au bois, dans Maupassant, Contes et Nouvelles, tome II, texte établi et annoté par Louis Forestier, éditions Gallimard, Bibliothèque de la Pléiade.
Lire
Lien vers la version de Au bois dans le recueil Le Horla
Notes et références
Nouvelle de Guy de Maupassant
Nouvelle française parue en 1886 | {
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{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/calculus\/thomas-calculus-13th-edition\/chapter-13-vector-valued-functions-and-motion-in-space-section-13-2-integrals-of-vector-functions-projectile-motion-exercises-13-2-page-753\/16","text":"# Chapter 13: Vector-Valued Functions and Motion in Space - Section 13.2 - Integrals of Vector Functions; Projectile Motion - Exercises 13.2 - Page 753: 16\n\n$$\\mathbf{r}=\\left(-\\frac{t^{2}}{2}+10\\right) \\mathbf{i}+\\left(-\\frac{t^{2}}{2}+10\\right) \\mathbf{j}+\\left(-\\frac{t^{2}}{2}+10\\right) \\mathbf{k}$$\n\n#### Work Step by Step\n\nSince \\begin{align*} \\frac{d \\mathbf{r}}{d t}&=\\int-(\\mathbf{i}+\\mathbf{j}+\\mathbf{k}) d t\\\\ &=-(t \\mathbf{i}+t \\mathbf{j}+t \\mathbf{k})+\\mathbf{C}_{1} \\end{align*} and $$\\frac{d \\mathbf{r}}{d t}(0)=\\mathbf{0} \\Rightarrow-(0 \\mathbf{i}+0 \\mathbf{j}+0 \\mathbf{k})+\\mathbf{C}_{1}=\\mathbf{0} \\Rightarrow \\mathbf{C}_{1}=\\mathbf{0}$$ Then $$\\frac{d \\mathbf{r}}{d t}=-(t \\mathbf{i}+t \\mathbf{j}+t \\mathbf{k})$$ and \\begin{align*} \\mathbf{r}&=\\int-(t \\mathbf{i}+t \\mathbf{j}+t \\mathbf{k}) d t\\\\ &=-\\left(\\frac{t^{2}}{2} \\mathbf{i}+\\frac{t}{2} \\mathbf{j}+\\frac{t^{2}}{2} \\mathbf{k}\\right)+\\mathbf{C}_{2} \\end{align*} $$\\mathbf{r}(0)=10 \\mathbf{i}+10 \\mathbf{j}+10 \\mathbf{k}$$ Then $$\\mathbf{C}_{2}=10 \\mathbf{i}+10 \\mathbf{j}+10 \\mathbf{k}$$ and $$\\mathbf{r}=\\left(-\\frac{t^{2}}{2}+10\\right) \\mathbf{i}+\\left(-\\frac{t^{2}}{2}+10\\right) \\mathbf{j}+\\left(-\\frac{t^{2}}{2}+10\\right) \\mathbf{k}$$\n\nAfter you claim an answer you\u2019ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide\u00a0feedback.","date":"2023-03-26 15:43:29","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\": 0.9548873901367188, \"perplexity\": 7334.829358787801}, \"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-2023-14\/segments\/1679296945473.69\/warc\/CC-MAIN-20230326142035-20230326172035-00658.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.qb365.in\/materials\/stateboard\/8th-maths-term-2-life-mathematics-one-mark-questions-with-answer-6593.html","text":"#### Term 2 Life Mathematics One Mark Questions with Answer\n\n8th Standard EM\n\nReg.No. :\n\u2022\n\u2022\n\u2022\n\u2022\n\u2022\n\u2022\n\nMaths\n\nTime : 00:45:00 Hrs\nTotal Marks : 35\n15 x 1 = 15\n1. 12% of 250 litres is the same as ________of 150 litres.\n\n(a)\n\n10%\n\n(b)\n\n15%\n\n(c)\n\n20%\n\n(d)\n\n30%\n\n2. If three candidates A, B and C is a 3 school election got 153, 245 and 102 votes respectively, the percentage of votes for the winner is___________.\n\n(a)\n\n48%\n\n(b)\n\n49%\n\n(c)\n\n50%\n\n(d)\n\n45%\n\n3. 15% of 25% of 10000 =___________.\n\n(a)\n\n375\n\n(b)\n\n400\n\n(c)\n\n425\n\n(d)\n\n475\n\n4. When 60 is subtracted from 60% of a number to give 60, the number is\n\n(a)\n\n60\n\n(b)\n\n100\n\n(c)\n\n150\n\n(d)\n\n200\n\n5. If 48%of 48 = 64% of x , then x =\n(a) (b) (c) (d)\n\n(a)\n\n64\n\n(b)\n\n56\n\n(c)\n\n42\n\n(d)\n\n36\n\n6. A fruit vendor sells fruits for Rs200 gaining Rs40. His gain percentage is\n\n(a)\n\n20%\n\n(b)\n\n22%\n\n(c)\n\n25%\n\n(d)\n\n16$\\frac { 2 }{ 3 }$%\n\n7. By selling a flower pot for Rs528, a woman gains 20%. At what price should she sell it to gain 25%?\n\n(a)\n\nRs500\n\n(b)\n\nRs.550\n\n(c)\n\nRs.553\n\n(d)\n\nRs.573\n\n8. A man buys an article for Rs150 and makes overhead expenses which are 12% of the cost price. At what price must he sell it to gain 5%?\n\n(a)\n\nRs.180\n\n(b)\n\nRs.168\n\n(c)\n\nRs.176.40\n\n(d)\n\nRs.85\n\n9. The price of a hat is Rs210. What is the marked price of the hat if it is bought at 16% discount?\n\n(a)\n\nRs.243\n\n(b)\n\nRs.176\n\n(c)\n\nRs.230\n\n(d)\n\nRs.250\n\n10. The single discount which is equivalent to two successive discount of 20%and 25% is\n\n(a)\n\n40%\n\n(b)\n\n45%\n\n(c)\n\n5%\n\n(d)\n\n22.5%\n\n11. Th e number of conversion periods, if the interest on a principal is compounded every two months is___________.\n\n(a)\n\n2\n\n(b)\n\n4\n\n(c)\n\n6\n\n(d)\n\n12\n\n12. The time taken for Rs.4400 to become Rs.4851 at 10%, compounded half yearly is _______.\n\n(a)\n\n6 months\n\n(b)\n\n1 year\n\n(c)\n\n1$\\frac { 1 }{ 2 }$years\n\n(d)\n\n2 years\n\n13. The cost of a machine is Rs.18000 and it depreciates at 16 \u00a0% annually. Its value after 2 years will be___________.\n\n(a)\n\nRs.12000\n\n(b)\n\nRs.12500\n\n(c)\n\nRs.15000\n\n(d)\n\nRs.16500\n\n14. The sum which amounts to Rs.2662 at 10% p.a in 3 years compounded yearly is_______.\n\n(a)\n\nRs.2000\n\n(b)\n\nRs.1800\n\n(c)\n\nRs.1500\n\n(d)\n\nRs.2500\n\n15. The cost of a machine is Rs.18000 and it depreciates at 16 % annually. Its value after 2 years will be___________.\n\n(a)\n\nRs.2000\n\n(b)\n\nRs.1500\n\n(c)\n\nRs.3000\n\n(d)\n\nRs.2500\n\n16. 15 x 1 = 15\n17. If 30%of x is 150, then x is _________.\n\n()\n\nx = 500\n\n18. 2 minutes is _________% to an hour.\n\n()\n\n$3\\frac { 1 }{ 3 } %$%\n\n19. If x % of x = 25, then x = ________.\n\n()\n\nx = 50\n\n20. In a school of 1400 students, there are 420 girls. The percentage of boys in the school is ________.\n\n()\n\n70 %\n\n21. 0.5252 is ________%.\n\n()\n\n52.52%\n\n22. Loss or gain percentage is always calculated on the__________.\n\n()\n\nCost price\n\n23. A mobile phone is sold for Rs8400 at a gain of 20%. The cost price of the mobile phone is________.\n\n()\n\nRs.7000\n\n24. An article is sold for Rs 555 at a loss of 7 $\\frac { 1 }{ 2 }$\u00a0%. The cost price of the article is ________.\n\n()\n\nRs.600\n\n25. The total bill amount of a shirt costing Rs575 and a T-shirt costing Rs325 with GST of 5 %is_______.\n\n()\n\nRs.945\n\n26. The compound interest on Rs.5000 at 12% p.a for 2 years compounded annually is ____________.\n\n()\n\n1272\n\n27. The compound interest on Rs.8000 at 10%p.a for 1 year, compounded half yearly is\n____________.\n\n()\n\nRs820\n\n28. The annual rate of growth in population of a town is 10%. If its present population is 26620, the population 3 years ago was_________.\n\n()\n\nRs.20,000\n\n29. The amount if the compound interest is calculated quarterly, is found using the formula __________.\n\n()\n\nA= P${ \\left( 1+\\frac { r }{ 100 } \\right) }^{ 4n }$\n\n30. The difference between the S.I and C.I for 2 years for a principal of Rs.5000 at the rate of interest 8%p.a is ___________.\n\n()\n\nRs.32\n\n31. Th e marked price of a mixer grinder is Rs.4500 is sold for Rs.4140 aft er discount. Th e rate of discount is __________.\n\n()\n\n11$\\frac { 1 }{ 9 }$%\n\n32. 5 x 1 = 5\n33. Depreciation value is calculated by the formula P${ \\left( 1+\\frac { r }{ 100 } \\right) }^{ n }$\n\n(a) True\n(b) False\n34. If the present population of a city is P and it increases at the rate of r% p.a, then the population n years ago would be P${ \\left( 1+\\frac { r }{ 100 } \\right) }^{ n }$\n\n(a) True\n(b) False\n35. The present value of a machine is Rs.16800. It depreciates at 25% p.a. It's worth after 2 years is Rs.9450.\n\n(a) True\n(b) False\n36. The time taken for Rs.1000 to become Rs.1331 at 20%p.a compounded annually is 3 years.\n\n(a) True\n(b) False\n37. The compound interest on Rs.16000 for 9 months at 20% p.a, compounded quarterly is Rs.2522.\n\n(a) True\n(b) False","date":"2019-12-16 08:34:17","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\": 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\": 0.5697930455207825, \"perplexity\": 2909.1630686365756}, \"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-2019-51\/segments\/1575541318556.99\/warc\/CC-MAIN-20191216065654-20191216093654-00207.warc.gz\"}"} | null | null |
package test.animation;
import com.test4android.R;
import android.animation.TypeEvaluator;
import android.animation.ValueAnimator;
import android.animation.ValueAnimator.AnimatorUpdateListener;
import android.app.Activity;
import android.graphics.PointF;
import android.os.Bundle;
import android.util.DisplayMetrics;
import android.view.View;
import android.view.Window;
import android.view.WindowManager;
import android.view.animation.BounceInterpolator;
import android.view.animation.LinearInterpolator;
import android.widget.ImageView;
public class AnimateFreeFall extends Activity {
private int screenHeight;
private int screenWidth;
private ImageView imageView;
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
requestWindowFeature(Window.FEATURE_NO_TITLE);
getWindow().setFlags(WindowManager.LayoutParams.FLAG_FULLSCREEN,
WindowManager.LayoutParams.FLAG_FULLSCREEN);
setContentView(R.layout.animate_free_fall);
DisplayMetrics metrics = new DisplayMetrics();
getWindowManager().getDefaultDisplay().getMetrics(metrics);
screenHeight = metrics.heightPixels;
screenWidth = metrics.widthPixels;
imageView = (ImageView) findViewById(R.id.im);
}
public void clean(View view) {
imageView.setTranslationX(0);
imageView.setTranslationY(0);
}
public void freefall(View view) {
final ValueAnimator animator = ValueAnimator.ofFloat(0, screenHeight
- imageView.getHeight());
animator.setTarget(view);
animator.setInterpolator(new BounceInterpolator());
animator.setDuration(1000).start();
animator.addUpdateListener(new AnimatorUpdateListener() {
@Override
public void onAnimationUpdate(ValueAnimator animation) {
Float value = (Float) animation.getAnimatedValue();
imageView.setTranslationY(value);
}
});
}
public void parabola(View view) {
ValueAnimator animator = ValueAnimator.ofObject(
new TypeEvaluator<PointF>() {
@Override
public PointF evaluate(float fraction, PointF arg1,
PointF arg2) {
PointF p = new PointF();
p.x = fraction * screenWidth;
p.y = fraction * fraction * 0.5f * screenHeight * 4f
* 0.5f;
return p;
}
}, new PointF(0, 0));
animator.setDuration(800);
animator.setInterpolator(new LinearInterpolator());
animator.start();
animator.addUpdateListener(new AnimatorUpdateListener() {
@Override
public void onAnimationUpdate(ValueAnimator animator) {
PointF p = (PointF) animator.getAnimatedValue();
imageView.setTranslationX(p.x);
imageView.setTranslationY(p.y);
}
});
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
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Q: VB.NET: Large Array - Faster Alternative? I am working with a very large csv file that keeps on growing. Basically, my program checks if a specific ID number is present. If it is present, is uses the data in the array. If it is not present, it downloads the data from the internet and adds it to the array.
Here is my array:
Public Shared ClientSideData(0 To 99999, 0 To 5) As String '"MLS Number","GLA","Latitude","Longitude","Year Built","Acreage","ClientSideNext"
Here is a sample line from the csv file:
"21633662","1631","40.298488","-74.052182","1950","0.1791"
And here is the very simple function I use to check if the MLS # is present:
Function ClientSideLoc(mls As String) As Integer
For i = 0 To GlobalVariables.ClientSideCount - 1
If GlobalVariables.ClientSideData(i, 0) = mls Then
Return i
Exit Function
End If
Next
Return -1
End Function
The CSV file adds upwards of 500 entries per day sometimes. It is growing fast. Once it surpasses 100,000 entries, my current plan is to go back and start overwriting the early entries with new ones. I have a feeling this ClientSideLoc function is going to end up slowing my program down more than if I just downloaded the missing info from the internet every time.
I am curious if there is a faster alternative? What would you recommend? Would a database be a better option? I am not familiar with them. Thank you
A: I suggest processing the csv file daily into a table in a database. Even something like Access can work for this. Then you can query the database table and not worry about memory consumption.
Also: this looks like about 76 bytes per record, if you have a CSV reader that lets you be efficient and parse columns to appropriate types:
MLS#: 8-char string * 4 bytes per unicode character
GLA: 4 byte int
Lat: 8 byte double
Long: 8 byte double
Year: 4 byte int
Acre: 8 byte double
Object reference: 12 byte object reference overhead
At 100,000 records, that's still less than 10MB. Memory use won't be a problem. Even the worst-case scenario where everything is a string works out to less than 20MB total.
Given the memory really won't be a problem, you should be fine to use a Dictionary(Of String, Listing), where Listing is a class you create to hold these records.
A: The Dictionary solution is what I was looking for. Doesn't help my memory issue, but that's okay. Answers this question for now. Thank you Craig!
| {
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Acknowledgements
Very special thanks to Helen Chamberlin, who edited the first edition of _Mad Meg_.
I made extensive use of the La Trobe University Library for texts on the rise of Mussolini and on the lives of the Italian émigrés in Paris, particularly Filippo Turati and Anna Kulischov. Without this library, which furnished the Italian Department at La Trobe, now sadly defunct, I would have remained ignorant of how Mussolini 'disappeared' the Risorgimento from Italy.
I travelled to France with the ingenious and resourceful Merlin Crossley at the time of the fiftieth anniversary of the round-up of the Paris Jews and the slaying of the leaders of the Italian resistance in Normandy.
I remain indebted to the work of Gianfranco Cresciani, whose book _Fascism, Anti-Fascism and Italians in Australia_ was a major source. I also read of the life of Omero Schiassi in books borrowed from the Baillieu Library at the University of Melbourne.
I would like to thank the following authors and publishers for permission to reprint material in this book: Ezra Pound _Perserae_ © 1926 New Directions Publishing Co; W.H. Auden _The Musee des Beaux Arts_ © Faber and Faber; 'Bee-bop-a-lula' © Davis and Vincent, reprinted with permission of Warner/Chapple Music (licence no L0432); 'Universal Soldier' © Buffy Sainte Marie, reprinted with permission of Southern Music Publishing Company Australasia, Pty Ltd; 'Raining in my Heart' © Boudleaux and Felice Bryant, Acuff Rose Music Inc.
This book was supported by the Literature Board of the Australia Council.
I dedicate this edition to all my friends and family who have shown me their good faith over the years.
Family Trees
Notes on Historical Characters
**Filippo Turati** | Leader of the moderate Italian socialists, co-led by Claudio Treves and Emanuele Modigliani (older brother of the painter) during Mussolini's rise to power.
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**'The Apparition'** | Anna Kulischov, Turati's lover and political mentor.
**Carlo and Nello Rosselli** | Carlo escaped from jail in Italy to become leader of the group _Giustizia e libertà_ and eventual leader of the Italian antifascists in Paris.
Nello, Carlo's younger brother, was a young liberal historian who tried to set up intelligent opposition to the fascist regime in Italy. He was assassinated with Carlo while visiting him in France.
Had they lived, the Rossellis would certainly have played a major role in the postwar reconstruction of Italy.
Contents
_Acknowledgements_
_Family Trees_
_Notes on Historical Characters_
PART ONE: Where the Nice Girls Live
**1** Christmas at 'Clare', 1956
**2** The Connotations of Sausages
**3** Perpetual Succour
**4** More about Mottoes
**5** Family Trees
**6** The Nice Girls
**7** Correspondence
**8** If
**9** Respect: the Odyssey of the Walnut Table
**10** Our Father's Daughters
PART TWO: Mad Meg
**11** The Art of Bamboozlement
**12** Encroachment
**13** The Politics of Kite-Flying and the Elusiveness of Cash
**14** Dadda's Beginnings
**15** Reason, Religion and Folly
**16** Mad Meg's Basket
**17** History and Moving into It
PART THREE: Good Intentions
**18** Christmas and an Explanation
**19** Uncle Nicola
**20** Displacement and Despoliation
**21** David
**22** A Table in the Presence of Enemies
**23** The Crushing
PART FOUR: The Promised Land
**24** Translating
**25** Modernity and Obsolescence
**26** Allegra
**27** Relations
**28** At Harry's Beach House
**29** Paris
**30** Faking It
**31** Justice and Liberty
**32** Homage to Bruegel
_About the author_
PART ONE
[WHERE THE
NICE
GIRLS LIVE](../Text/9781743583425_Split6.html#tch4)
ONE
Christmas at 'Clare', 1956
'AHEM, AHEM! WOULD you stop behaving like a hooligan?' Allegra bats her eyes and folds her arms across her chest.
'Hooligan, Madam? I don't know anyone called Hooligan. My name's Coretti.'
'Coretti? Who'd have a name like that? I don't believe you. You're a liar, little girl.'
'I'm not a liar. I'm thirty.'
'You are not thirty. You're ten.'
'I didn't say I was thirty. I said I was thirsty.'
'In that case, I'm thirsty, too. I'm going to have a gin,' and then she pours herself a drink of water from the round-bottomed jug in its cage on the wall. 'You can't have any. You're too young.'
There's a patterned metal contraption on the floor to rest your feet on. Allegra says it's a heater in the wintertime, but we've never been on the train in winter. Every year we come to 'Clare' in summer.
Allegra is moving along in front of me, her back swaying to her pony's gait. Over the soft, bruised paddocks, the sky trails a knife blade, raising red dust in the pasture.
Behind us the house lies low in its garden. We have left the trellis. From underneath it, in December, you can just see sky between the purple flowers and green leaves. The faint, persistent scent of wisteria threads the air. By January, the flowers that haven't fallen are transparent brown and the leaves form a withered grip around them.
Fly buzz. I am sitting badly on my pony and don't know how to handle it. Listening out from under the broad brim of my hat, I can hear the noxious bleating of sheep, and the carolling of magpies, clear and hard. I'm hot, my legs are short, my pony's fat, my boots are small.
Allegra has Granpa's old mare on the leading rein. Last year I was thrown by her. 'Pull on the bit when she does that, girlie!' Granpa yelled. 'Hurt her mouth!' As I was falling, Aunt Nina ran from the house, crying, 'Oh, that child!' So I was the one who was hurt, and not by the fall.
Allegra looks back, squinting. She can see I'm crying. She stops. She doesn't say a word. I slither down my pony's side and give the reins to her. I pick a big tree and squat down under it. The trees at Clare have coiled up fifty feet of drought and bumper season, sealing off the years. When you run your hands up the trunks, you can feel the years pulling you into them; rest an ear and you can hear them. The ring of our lives is being gathered in.
Before us were Granpa's parents, his wife Euphrosyne and his three sons. They're all dead now. How does it happen, what does it feel like, do you really wake up on the other side? I think I can see Euphrosyne sometimes, in her button boots and skirts. Why should she die? It doesn't make sense. She could stay here, we wouldn't mind that her clothes were out of date.
As I do up my jodhpurs, there's a tractor ticking round the paddocks in a cloud of dust. They are throwing out hay bales for the sheep. It's hot, though it's only eleven in the morning. In other years the pasture has been gold, and when you moved through it, looking down, there were ripples of mauve and turquoise, interrupted sometimes by a green stalk thrusting yellow buds against the breeze.
I mount my pony. Allegra takes up the slack of the old mare's lead and we move off slowly down the shadeless tracks towards 'Mountshannon'. Granpa lives there. In his father's day, Mountshannon was The House. Now there is Clare. It was built when Granpa married Euphrosyne, an upright lady with soft, dark eyes. They say it took him eight days to bring her here from Melbourne. His carriage had red-painted wheels and brass lanterns for travelling by night. It was beautifully sprung, our mother says. She is proud of that carriage and of the mother who was fit to ride in it.
Euphrosyne was descended from a lord. She left behind her silver-backed hairbrushes, her monogram on their handles, chiffoniers with velvet lining, damask tablecloths, a giant walnut table, rows of stately, matching chairs. Her father was a surgeon from Edinburgh who wrote an advice to his eleven daughters. _Though God has sent them among us for a reason,_ he wrote, _never marry a madman._ It's our mother's favourite saying.
Euphrosyne was a socialite. She threw balls in the hallway at Clare. Aunt Nina calls this place 'the vestibule'. Unlike other halls in grand houses in the district, this one runs transverse to the front verandah, so that, on fine nights, the verandah could also be used for dancing by opening all the French doors.
The light in the vestibule is yellowish and happy. The curtains and carpet runners are burgundy, edged with gold. Sometimes I see the ghosts of people dancing: our uncles, Hedley, Haydn and Shaver Motte, and their stately cousin, Vere de la Motte, whose eyebrows were looped and black to match her signature.
They believed in handwriting, these people: the letters from our uncles came in copperplate from the front in the Second World War. Uncle Hedley went to Jerusalem and Bethlehem, where he saw the manger. He believed in God. Uncle Haydn sent photos of camel races in Damascus and Uncle Shaver wrote home on Raffles Hotel paper from Singapore. _Dearest sisters,_ they wrote, and _Dear old Dad._
Euphrosyne had died before the war. A clothesline killed her. The crossbeam crowned her in a windstorm. Handwriting and noses she believed in. She pulled the children's noses at the dinner table so they wouldn't be snub. Euphrosyne Isabella Motte. A lady. I am named for her, but Isobel Euphrosyne, for how could I be as grand as the wife of Granpa?
But I am grand. Oh, I am very grand. I am extremely grand, and everything I see I own, those paddocks, those trees, those sheep, that tractor. And all the sky and all the earth are mine. I made them. I am God and none of the others know. Yes, I am God and they will have to watch themselves. 'Fear the name of the Lord,' I shall say when my day has come, 'the name of the Lord is Isobel Euphrosyne.'
We have come eastward through the paddocks for three miles. The land has begun to buckle and plunge. Gun-metal green marks the valley's edge and, rearing up out of it at the far side, a low, balding mountain is the boundary between Clare and Coolang. The trees stop abruptly at Coolang's fences. Down in the valley is Granpa's house, Mountshannon.
It gives me vertigo to ride down there. It's a steep tangle of native trees and the smell of dust and eucalyptus billows out of it. You could get lost. Allegra has lots of times – or so it seems on the thirsty waits crouched at the edge of the road when she goes in after wombats and possums. Once, a herd of wallabies nearly trampled me as I sat doodling in the dust, holding the pony reins, waiting, imagining her bitten by snakes as the ponies' velvet noses butted me and I ran my fingers over their quivering lips.
Mountshannon is peaceful. You can't hear woeful bleating way down here. Granpa doesn't like sheep; he used to run cattle. He likes the look of a 'decent beast' on pasture. Just as well he can't see any more... I close my eyes to feel what he must feel... the sway of the pony, the warming and cooling of the blood in my eyelids as I shift along under the shadows. Kookaburras are cackling, now soft, now loud. Insects buzz by my ears. I have to lie on my pony's neck to avoid the branches. I can feel his hot coat and coarse mane moving under my cheek. I can smell him. The smell of horse is kind of sweet, kind of sharp.
I sit up. And as my eyes come open, a flock of corellas busts out, white, from the treetops and startles my pony under me. I rein him back. Ahead of me, Allegra lets her pony have his head. 'Stop!' I yell out, 'It's dangerous!' But she doesn't listen, and perhaps she can't hear. Off she goes like The Abracadabra Lady – that's my name for her in this mood, The Abracadabra Lady – kicking up pillows of dust and making my pony sneeze. Her hair comes out from under her cap, a couple of long, rough clubs thrashing her back. Oh well, at least Allegra is not God.
I am. And if I were a jealous God, as it says in the Bible, I would give her sunstroke.
I remember sunstroke. From the Christmas before last I remember. You think your eyes are burnt out and you have to have ice cold bottles put behind your knees. It's a horrible sensation, as if your life were rushing for your hands and feet in a frantic effort to escape you.
I gave myself sunstroke as a trial punishment for Allegra if she should get out of hand. It is an excellent punishment and I devised it. She and Granpa have their secrets, but I have my weapons and one day I shall use them and show mercy. Then they will know what mercy is.
I suppose Granpa will die in Mountshannon one day. Where he was born. Eighty-seven years seems a very long time to have lived, but he is still alive as I slip my pony's reins through the disintegrating pickets of his front fence.
It's a timber house with a bull-nosed verandah running down the sides and across the front. The front door's open and I can see right down the hallway to the backyard, where the dogs are chained. They've set up a racket and are leaping forward as far as it's possible without choking; their taut legs hang quivering under their twisting, whining, prick-eared heads. Granpa is sitting on his chair on the verandah, belting divots out of the floor planks with his stick in an effort to shut them up.
'Is that you, girlie?' he calls. 'Come and give us a kiss.'
When you get off a horse, it's as if you have to learn to walk again. I'm a bit too big for last year's boots, and my feet are numb. I trudge over the hardened, scratched-up ground of the yard, where Allegra has cornered a white leghorn up the lemon tree and is poised, ready to grab it. I know what that means: Granpa's going to treat us to a slaughter. The yard stinks and the leghorns are racketing violently through the scanty bushes and blackberry tangle. The house hasn't been painted for years and the iron roof is rusty.
I grab Granpa's stick to stop it belting divots out of me as well as the planks and put my lips on the brown, leathery cheek, hold my breath, and kiss it. It's not that Granpa's dirty; in fact, he's very clean, he just stinks of tobacco. He cuddles me up so tightly, I nearly drop the thermos.
'Brought you some milk tea, Granpa. With a spot of whusky in it.'
'Whusky?'
'Whusky. Mum sent some up for Christmas. Real stuff. It's called Old Parr.'
'Scotch whusky?'
'Yep. She said it'd make the hairs on your chest curl.'
'Ain't got no hairs on my chest, daughty.'
'I knows that, Granpa. You're bald as a badger.'
'Badgers ain't bald.'
'C'n we have some?'
A great cacophony breaks from the lemon tree where Allegra has grabbed the chook by its scaly legs. She comes down, bottom-first, and bears it triumphantly back to the verandah. It's hanging upside down and flapping violently, its comb in the dust and its pink, reflective eye twisting in its socket so hard it could wrench out.
'Pour us a whusky tea, daughty,' Granpa says to me, 'while us bigguns get this chook here ready for Neeny.'
The axe is resting against a chopping block in the corner of the yard. Granpa keeps his axe sharp. He can sit for hours sometimes, running a whetstone over it. He's proud of his axe. I pull the tough old flywire door open on its rusty hinges as Allegra and Granpa struggle towards the chopping block, Granpa with his hand on Allegra's shoulder, Allegra stamping and dancing round the chook. The wire door slams behind me and I clop down the hallway to the kitchen at the back of the house. Where I'll stay till the killing's over, till they've had their fun and tried to set the headless thing running.
I take down three chipped, pale blue china cups and set them on the wooden running board of Granpa's sink. Hot light has baked dust onto the panes of the back window. There are dead flies in a layer on the sill and more flies stuck to the flypaper hanging from the cobwebbed, conical light shade where, obviously enough, Granpa has brushed his head, as there are fibres from his cloth hat on it and his hat is lying in the middle of the floor. I pick it up and sit, twirling it on my finger, on a wooden chair by the kitchen table. The oilcloth cover is sticky and smells rancid as I peel it back to rest my elbows on the table edge. I close my eyes.
Now God,
Please don't let Granpa chop his hand off or bring the axe down
on Allegra's head on Christmas Day. He is old, but let him die with both hands on of natural causes, and let my sister live. And please, God, don't let Uncle Garth get so drunk that he insists on doing the carving like last year, when he cut his thumb to the bone and pretended all Christmas Day that he hadn't and got blood poisoning on Boxing Day and we had to get his brother Tony over from Coolang to drive him to hospital because he wouldn't admit he was sick and had to be persuaded and Tony was so angry he drove over Aunt Nina's red standards and flattened them.
Amen.
There is a terrible squawking in the yard. The axe comes down and they start laughing.
I pour the whusky tea.
Poor Aunt Nina.
She pulls the disgusting heap of trampled chook from the sack and lets it drop on the kitchen bench. 'Blast him,' she mutters. Allegra let Granpa gallop across the home paddock. He forgot the gunny sack with the chook in it and it fell.
I put my hand on her wrist, 'It's all right, Neeny. I'll pluck it for you.' I hate plucking chooks, the feathers give me asthma. I've only ever plucked one and had bad dreams after. 'Bless you, darling,' she says, and kisses me. Allegra strides through the kitchen, slamming the back door. 'It hopped,' she says as she passes.
Uncle Garth is sitting under the clothesline with a tumblerful of Christmas cake rum, bellowing verses from the _Rubaiyat._ He always bellows, his voice is astonishing, his mouth a keen slice in his thick skin, the teeth strong, broad and discoloured, the tongue bright pink behind them. He has to suck in air as though through meshes. His mouth is always open, his head forced up; in his hands he holds a smudged glass, full or empty.
Aunt Nina grinds her teeth and begins reciting loudly, ' _I sometimes think that never blows so red the rose/As where some_ bally _Caesar bled._ ' 'Don't be upset, Neeny,' I whisper.
'I'm not upset, darling.' She clamps a wire hood down over the chook to keep the flies off. 'Did you have a lovely ride?'
'Oh yes,' I lie. 'I can understand why Granpa wants to stay at Mountshannon,' I say, though I cannot. Staying there means Aunt Nina has to ride out every day to feed him. She raises her eyebrows and wipes her chapped hands on a linen cloth. It is time to test her with my question, now. 'Granpa doesn't like foreign people, does he?'
'What makes you say that, darling?'
'I mean that Granpa doesn't like Dadda, does he?'
'Your father is a very talented man, and don't you forget it.'
'You don't like Dadda much, do you, Neeny?'
'He can be charming,' she says unconvincingly.
Although it's incredibly hot in the kitchen, the slowly roasting chicken adding to heat that's already in the nineties, I like sitting here. It smells terrific. Best of all, I like cutting the beans into long slivers with the beaner and eating a few of them while I sip from a glass of sherry I poured myself. In the back of my mind, I have a picture of my father feeding my mother sherry and beans. I'm sure that's the magic combination, whatever Allegra says about sausages and tutti frutti.
'Only,' Aunt Nina says absently, 'I don't know why he insists on painting naked women. I think women's bodies are so ugly, don't you?' I look at her, startled. I love the shape of Neeny: she's petite, with a grand bosom and what she calls 'dependable' legs – they're thick in the ankles. 'Not little girls, darling. Little girls' bodies are so pretty. Look at that child...'
Aunt Nina stares dreamily out the window to where Allegra is doing a handstand to amuse Uncle Garth. 'I'm a child, too,' I mumble.
'She's perfectly formed. Like a dancer. I wish you girls hadn't given away your ballet.'
'It was the teacher's fault. She called us spaghetti legs. I'm doing art on Saturdays now.'
'Are you just?'
'Yes. I am just. I painted you a picture for Christmas. Shall I go and get it?'
It's the moment I've been waiting for. I sock back the rest of my sherry, slip off the kitchen stool and race away towards the bedroom, where I've hidden my painting on a windowsill behind a blind. It's my first oil. Mr Millership, the teacher, wanted everyone to do still lives, but I could think of nothing worse than painting a bowl of apples and oranges – instead, I painted a picture of The Abracadabra Lady riding full tilt into Mountshannon. But she got too big for the paper and I had to stick on an extra bit for the horse's head. Mr Millership nearly had triplets. 'We'll make an artist of you yet, Miss Coretti,' he said, and, when I took the picture home, Dadda said, 'Well, well, well!' Three 'wells' from Dadda means you've hit the jackpot.
Allegra, who stays home on Saturdays now, for some reason suddenly fond of gardening, was very jealous. She even thought about art classes herself, but something about gardening holds her back. It isn't love of plants; she's already killed quite a few by forgetting they're there and treading on them. It's a band of high school boys who wander past our fence and push a certain person up our drive. The certain person's name is Jimmy Coote. Once he gave Allegra four lamb chops – his dad's a butcher. They nearly went bad before she'd let us eat them. She keeps the bones in a jewellery box.
So now, of course, she wants to go to high school and she even saved her pocket money and bought a state high uniform from the girl next door. Aunt Nina and our mother want us to go to their old school so we'll turn out _nice,_ but at their old school you have to wear hats and gloves and you're not allowed to eat your lunch in public. Allegra says that's ridiculous and I agree. But she still has to keep the high school uniform under her bed and only dresses up in it when Mum's not around.
My painting still hasn't quite dried. In the bedroom, Allegra has pulled up the blind and is prodding some of the wet parts with her finger. 'Isn't it funny,' she says, 'how it dries on top and stays wet underneath? I love this painting. It smells really terrific. You ought to be an artist when you grow up, Bel.'
I don't mind her touching it; it isn't sabotage. I like to touch the lumps on The Abracadabra Lady's chest myself, where the paint's thick. Her bosoms have become quite dull with fingering. I'm so proud of this painting, I can hardly believe it's my own work.
'It's re-al good,' drawls Allegra, 'but you haven't signed it. You ought to sign it, you know.'
'I have signed it. I signed it on the back.'
I turn it over. Dubiously. Because I'm not sure it's something I ought to let Allegra see. I've written in red ink: _A painting of The Abracadabra Lady by Scarlet Coretti, aged 10, December, 1956._
'Scarlet?' I knew this would happen. I should never have turned the damned thing over. I scuttle out of the room into the vestibule. The wiry head comes round the bedroom door, wrinkling its nose.
'Scar-let?' I scuttle into Aunt Nina's room, making straight for her dressing table where I seize a stick of carmine from her makeup tray and scribble wildly over the letters, deleting everything right back to the title. I can hear Aunt Nina's bed creaking behind me and the sound of Allegra pulling tiny feathers through the pink taffeta of the eiderdown and cackling, 'Scar-let?'
I am standing in the corner of Aunt Nina's room, fighting tears where the pink rosebud wallpaper meets in a splendidly concealed line. I have to keep the tears back for several reasons, not the least of which is that I'm holding _The Abracadabra Lady_ right side up. Even the ceiling's wallpapered, and it all matches, right down to the pillowslips where a froth of treacherous hair is quaking with silent laughter. I steady myself and breathe in deeply. 'Well, you called yourself Sapphire once.'
'Are you going to dress for lunch, Scarlet?'
'No, Sapphire, not unless you do.'
'I'm going to dress for lunch, Scarlet.'
'Are you, Sapphire? What are you going to wear?'
'Wouldn't _you_ like to know, Scarlet?'
'I know what you're going to wear, Sapphire. You're going to wear your confirmation hailspot muslin, because it's the only going-out dress you packed.'
'I'm not going to wear the dress I packed, Scarlet. You can wear it if you like.' Which comes as a surprise.
'Can I?' I spin round. 'Can I?'
'Only don't get it dirty.' She struts over to Aunt Nina's closet and opens the doors grandly.
'You'll get into serious trouble, Allegra...' but she is going for more than serious trouble, because she is pulling down Aunt Nina's 'hidden drawer', the one with our uncles' war letters in it. In there is a big box and in the big box is a lace wedding dress. Black lace. Aunt Nina dyed it black. When we told our mother about finding it, along with its dyed wedding veil and bouquet, she said, 'Oh, that! H'mph! It was from your aunt's first marriage.'
'She'll have a fit.'
'No, she won't.' She plucks the dress from its box, throws it on the bed, kicks off her riding boots and begins to strip. She is terribly proud of her bosom buds and won't let me see them. 'Skedaddle,' she says.
The hailspot muslin doesn't fit me perfectly. It's too long for a start, and in the second place, I popped two of the pearled buttons trying to do it up over my asthmatic rib cage. And of course, I got paint on it, little vermilion dabs wherever the index finger of my right hand touched. There's a streak of vermilion just under my chin. But I like myself in the mirror as I dash the two buttons round in my mouth, creating vacuums through the buttonholes so they stick to my tongue and I can click them on my teeth. The confirmation dress was bought by our mother off the hook at Georges because someone from the Sunday school offered her a cotton sheet and a pattern. Our mother never sews a stitch. Allegra was the envy of her grade this year, though she chickened out of confirmation in the end.
Back at Aunt Nina's bedroom, I am having great difficulty opening the door, as The Abracadabra Lady has stuffed a towel under it to stop intruders.
'Neeny's going to have a fit, you know,' I warn, as I finally enter the room and can see the nineteen-twenties bride shaping up before the mirror. She has rouged her cheeks and painted her lips.
'She won't,' says Allegra, primly.
'But she _will._ She had to get divorced.'
'Why'd she keep the dress, then?'
'Beats me.'
Outside, the garden has gone sour in the heat. Birds are panting round the lawns with their beaks wide open. Aunt Nina's standard reds are strewn on the ground as if some historical person like Cousin Vere had stripped her gloves off after a ride, made a pair of them and chucked them out the window into the Garden of Discarded Gloves. There's a story about Cousin Vere: a soldier came courting her here once and he gave her a medal with _'Honi soit qui mal y pense'_ written on it. She chucked it out this very window into this very garden and then, after he'd gone away disappointed, she and Aunt Nina and our mother got down on their knees and searched the roses for half a day looking for it. But all they found was Granpa's 'implement'. No one will tell us what the 'implement' is. Dadda says it's what he used to deknacker bulls, but when I called it the 'deknackerer', my mother, for some reason, was scandalised.
'What's a knacker?' I ask Allegra.
'It's not a knacker, for your information, Isobel. It's called an implement.'
'That's not what I meant.'
But Allegra has changed the subject. 'I wonder who her husband was,' she says.
'He ran away with a _South African!'_
'So?'
'You're going to get into serious trouble, Allegra.'
'I don't care.'
'Well, I do. I'm going. Goodbye.' I take my painting from the bed and, deciding that the muslin really doesn't fit me, I return to our room to change into my organdie, laying Allegra's dress on her bed and spitting the pearl buttons out on top of it.
But she hasn't chickened out, and I know that because when I am in the kitchen making my presentation to Aunt Nina, I can see Allegra's reflection in the side of the toaster. She is cuddling up to Granpa in the dining room. When Aunt Nina says my painting is very nice, but why don't I try painting ballet dancers, I am instructed to put it on the mantelpiece in the dining room so everyone can see it – except Granpa, to whom it has to be described. But later... because right now, Allegra is sitting on his knee, and even wearing the head-dress.
'What am I wearing, Granpa?'
'A... piece of toast.'
'No!' she laughs, 'be serious. What colour is it?'
'Puce!'
'Cold! What colour's the sky at night-time?'
'Ivory.'
It is not.'
'Mauve.'
'You're making it up.'
'Brown.'
'Freezing!'
'It is in a dust storm.'
'Granpa...' She nuzzles him. 'It's black, you old trout.'
'I'm not an old trout.'
'You are.'
'I'm a Murray cod.'
When Aunt Nina comes into the room at last, Allegra is rolling on the floor in the black wedding dress, laughing. She springs to her feet and races towards Neeny's arms, but Neeny is frozen stiff with rage, so stiff I doubt whether she can get her voice out. 'Take that off at once,' she says at last, and coldly. 'You children have been told not to meddle in that drawer. If ever you do again, I'll send you straight home.'
Both of us weep immediately. It is the severest thing Aunt Nina has ever said to us. She turns on her heel to go back into the kitchen, 'Go and find your uncle, Isobel. It's time he washed for lunch.'
Now Allegra is trailing away, crestfallen, down the vestibule and Uncle Garth is fast asleep on his chair under the clothesline, roasted hectic by the sun. His glass has fallen on the grass near his feet. I try to wake him. When talking won't do it, I try shaking. His voice is the first thing to surface. 'And when thyself with shining foot shall pass/ Among the gusts, star-splattered on the glass...' He takes a tumble from his seat, sits upright on the grass and pulls himself together, suddenly, 'Take no notice of the fellow, Isobel. He pees his pants at country dances. Isn't worth knowing. Take my word for it. A fellow like that reflects no credit on his maker. Damn poor design. Never mind the elegant timbre of his voice, his impressive memory, the times he was kind to you. He's a brute and he'll come to nothing.'
He hoists himself onto his feet and takes himself with military bearing to the outside bathroom. Aunt Nina insists that the men use the outside bathroom, because the cistern is sluggish inside the house and won't take a pelting.
We have been waiting quite a long time for Uncle at the table. Nina is still stern, and both of us still sorry. We've hugged her a couple of times, but she won't budge. This is very unlike Nina. Anyway, we've had to start without him, Granpa ceremoniously salting and peppering the entire table, making the tip of my nose so itchy, Aunt Nina keeps asking if I haven't a hanky because I keep smudging my nose up my face with a starched serviette.
The dining room is huge. It looks out over the long wisteria trellis shading the way to the paddocks. There is a full-grown jacaranda on a green lawn just outside the windows. The lawn stays green because it is in shade from midday onwards. Like the wisteria, the jacaranda is in flower, so at this moment, Spanish armadas of purple light are cruising through gold-green seas.
The walls of Clare are thick and it's really quite cool in here, though the day is so hot. I have goose pimples on my upper arms. Allegra and I are trying to stop our knives and forks from screeching. Our plates are rimmed in indigo and gold and in their centres there's a golden E for Euphrosyne on an indigo disc. I broke a sweet dish from this set when I was a baby. They're always telling me, there are only four sweet dishes left now, and it's supposed to be my fault, but I dare say someone else broke the other nineteen. Granpa's denture clicks. We're not supposed to notice, but how can we help it? Clicketty, clicketty, click! 'Isobel?' Aunt Nina wants me to go and find Uncle Garth.
I can hear the shower going full pelt as I walk down the path towards the outside bathroom. 'Uncle Garth!' I call, but there's no answer. Perhaps he can't hear me for the water. I push the bathroom door open a little way... he is lying on the floor of the shower recess, fully dressed. He may have slipped, because his head's jacked up in the tiled corner. He's very red in the face and his eyes are shut. Everything he's wearing is soaked through; I can see his underwear quite plainly. I don't know whether to be frightened or amused, but I'm too chicken to reach over his big slumped body to turn the water off.
Aunt Nina must suspect something's happened, because, as I'm leaving the bathroom, she's standing at the back door, clutching a serviette, uncertain.
'There seems to have been an accident,' I tell her, looking behind me to where Uncle Garth is still in the same position. Aunt Nina comes running. And I decide on fear. Aunt Nina turns the water off and asks me to help her drag him out of the recess onto the floor. With a strong feeling of revulsion, I am cradling his great heavy head in my organdie skirt to stop it bumping on the hard floor, as Aunt Nina drags him out by the feet. It is by no means easy and is going to take us a long time.
I wonder why Uncle Garth is like this... Aunt Nina says he can't help it. Mum says he does it because of the war. But Uncle Garth went to the first war and that was a long time ago. Granpa says Uncle Garth didn't even go to the second. He stopped in Brisbane with General MacArthur. Granpa says he was medically unfit, but Aunt Nina says that's disloyal and he was kept at home because all the educated men were. Uncle Garth studied as a doctor. But if he has an education, he doesn't use it – only recites poems all the time, but isn't reciting any now. And I lower his big red head and it goes 'honk!' slightly on the floor tiles. His hair is all sticky, it looks just like a run-over cat and he is dribbling. And making me cry just a bit, I feel so sorry for him. Aunt Nina keeps saying, 'It's only a man, Bel. No need to be afraid. It's only a poor man.'
She rolls up towels and places them under his neck, then frightens me more by feeling for his pulse.
I couldn't wait for her verdict. I came inside and sat bolt upright in my chair at the table, as if sitting up straight would restore tranquillity in a world stiff with doubts of the most nerve-racking kind. Allegra, in the white hailspot, went and lay down on the couch under the windows, face down. Over her head, the purple armada sails on through the coloured world. My shoes and socks, wet from the shower water, are lying next to each other neatly under the table and Granpa is prodding the floor rhythmically with his stick. 'She should never have married that man,' he says, over and over.
Aunt Nina is speaking to the doctor on the phone in the vestibule.
Sunstroke. Have you ever been sunstruck? It is like the bonging of a pendulum inside your head – right side, left side, right side, left side. Bong-bong, bong-bong, BONG-BONG. It is a horrible sensation, as if your eyes were fried and your skin were shrinking to tight elastic. Cold compresses and crushed mint under your nose... your fingers and toes blow up, your feet and hands, your ankles and wrists.
It is only a man. Not a run-over cat. So I shall take the bottles of lemonade from the fridge and hold them behind his knees. I shall roll up ice cubes in tea towels and hold them to his temples, his wrists, his ankles. I shall save him.
Aunt Nina is still on the phone, so I run to the kitchen in bare feet, fling open the fridge door and the bottle jumps from the shelf, comes down with a great explosion, peppering me with glass shards. I am bleeding. My blood fizzes through a pool of lemonade on the floor. I call out, 'Don't come! It's all right!' But it's not, quite.
As I begin to mop up, the phone rings sharply. It is not the doctor, from whom Aunt Nina is expecting a return call, but our mother, and Allegra keeps calling me to come and talk, but I can't very well as I am wielding a mop and bucket and quietly bleeding.
_Allegra:_ | You should see the lemon tree at Mountshannon, Mum. It's chock-a-block with lemons. And I caught a chook and Granpa chopped its head off and it hopped!
---|---
_Granpa:_ | She should never have married that man.
_Allegra:_ | We're having a lovely time. I can still gallop really well but Bel doesn't want to go fast.
_Granpa:_ | She should never have married that man.
Now I've finished mopping up, so I go outside and put my feet in the gully trap and turn the tap on them.
'Bel, what are you doing?'
'Nothing, Neeny.'
'You're covered in blood!'
'It was just a bottle. I thought I'd go and put it behind Uncle Garth's knees, because it's cool, you know, and good for sunstroke.' Blood keeps coming out the little wounds; I squash them flat on the couch grass by the tap. More blood spreads up between my toes.
'It isn't serious,' I say, looking up at her. 'I'm sorry, Neeny.'
She sighs, and brings me bandages. She sighs again as she binds my feet up.
'You'd better go and lie down,' she says.
So now I've been lying down for at least an hour, and I've just about matched up my hundred and thirtieth inverted lyrebird from the paisley pattern of our wallpaper when the door opens and chucks a shadow over the spot. It's Allegra. She says Granpa wants to go back to Mountshannon. She still has faded marks of make-up on her face.
'But I don't think you should come in that condition.'
Ordinarily it would be a cause for joy, but because I can't stop thinking about Uncle Garth unconscious on the bathroom floor I decide I'll stick my cuts into my riding boots and come along all the same. 'Someone'll have to carry his dinner,' I say.
On the ride back to Mountshannon with Granpa's dinner of leftover roast packed into a steamer to warm up for him at the other end, Allegra has been trying very hard to be merry about Christmas.
'I love Christmas,' she says. 'I love it how Uncle Garth puts the lights up in the elms every year and how you can sit out on the lawn at night and it's so cool and up above the coloured lights in the trees blinking on and off, you can count the stars in the black, black sky...'
But Granpa just plods along sullenly on Countess, his stubborn, blunt-cut profile outlined against a fierce sky. He wears his cloth hat pulled down to the bridge of his nose and holds his head high, so you wonder how much he can see behind the little round pair of dark glasses.
Granpa keeps a field of poppies on the go near Mountshannon. In spring they are red, but now we are rattling through the grey knobs on their spikes. Flies cluster in swarms on our backs and on the horses' flanks: there is the smell of dust and horse, and the last of the moisture in the paddock grasses is being drawn up through the shaky call of sheep.
What does Granpa think of, that every year he grows more bitter and bad-tempered?
Dear old Dad, we fought our way out of the mountains in the end. The Jerries gave us a terrific battering. It was bitterly cold. We were split up, and we lost our gear and our way. We found ourselves wandering in a poppy field. I picked one and pressed it in my notebook. I was going to send it on to the girls, but I lost my whole kit. Your loving son...
I don't understand why we die. I can never unscramble it. As if at some puzzle in the dark I try, but pieces are missing, hidden away.
At the top of the valley, Allegra's pony, used to having its head, grew skittish and difficult to control. I took Countess's leading rein from her and let her go on ahead.
Granpa and I continued silently down the rocky path, sometimes feeling the stab of hot light on our backs between the trees. Our shadows were growing long and eerie. When we reached the house, Allegra was feeding the dogs, so they made little fuss when Granpa swung himself out of the saddle, steadying his descent on my shoulders. To me he was heavy, so heavy, I thought I would buckle under him.
On the way back, the sun was still blazing angrily on the horizon, and we seemed to be galloping through a hell of light that robbed the land of its shapes and colours and dimension. Riding side by side for companionship, we barely spoke or altered pace until we reached the home paddock and reined the ponies back to a trot. We were silent as they searched out the oats from the creases in my fingers and Allegra lifted the sweaty saddles over the crossbeam in the stalls.
My feet swelled up so much from the ride that Allegra had to help me take my boots off in the bedroom. There was blood on my socks and I had to ease them away from the bandages where they stuck. It's hours now since Aunt Nina rang the doctor and she is sitting waiting for him in the lighted vestibule.
Allegra takes out the playing cards for a quiet game of poker. We talk in whispers. Uncle Garth is still not conscious. There is no help. The hands drove off to Scunthorpe for their Christmas party during the afternoon.
TWO
The Connotations of Sausages
FOR MANY YEARS, Clare existed in my mind as the glorious past devolved to the less than glorious present. It would be for Allegra and me to arrest the decline, even to restore, if we could, a past grandeur and a familial nobility. In better days, Granpa inherited Mountshannon from his father. It took in all of Clare and most of Coolang besides. Hot in summer and wet in winter, the country was suitable for either sheep or cattle.
Granpa loved cattle and he hated sheep. No one knew cattle the way he did. He had what was called 'an eye for a beast'. And an ear, too, it seems, because when his eyesight began to fail, twenty years before we were born, he was still asked for his opinion and taken to the saleyards where he listened to hearts, heard the rib vibrations set up by lowing and bellowing, and took note of hoof thud on dust and turf, its sonorousness and the propriety of its pace. He could tell if a cow had been hard-droved by whether her tread was sharp or dull. Yet he hadn't run cattle at Clare since 1910, the year our mother was born.
Sheep, said Granpa, had dull eyes and shiftless faces and no singing voices to speak of. A cow, on the other hand, bred, bore and provided – a versatile, comely beast. And it sang. He hoped he would die surrounded by cow song. In the days when he could see and there'd been cattle at Clare, he'd herded the home paddocks in the evenings, Countess's predecessor standing quietly in the gate. _There's heaven above, and night by night/I look right through its gorgeous roof,_ he'd sing in his tuneful baritone till the cattle, mesmerised, swaying, lowing harmoniously in cathedral lungs, would plod-dance to their resting place.
The Furlongers had always had sheep at Coolang. Furlonger men and hands whistled and spat and made their horses dash. It wasn't Granpa's style. He liked to gallop, but not around a herd. His herds moved as one beast, there was a heart to it, a head and a tail, a good man kept it that way. Country folk respected a good man. He had what they called 'instinct'.
But instinct didn't make a rich man. By Granpa's reckoning it should have; an instinctive cattle man ought to have been favoured over a sheep man who calculated. No Motte worth his salt – and Granpa loved his salt – ever calculated.
When Granpa met Euphrosyne, she was governessing at a station owned by her sister, Mrs Tennyson Taylor, née Gilchrist. Euphrosyne Isabella Gilchrist was a younger scion in a family of fourteen. She was slender, brown-eyed and was wearing her hair close-cropped after suffering scarlet fever. She was not built for robust country life, yet Granpa saw in the shape of her the shape of sons and daughters, and in her movements the dance of a joyful life. Sturdy in soul if not in body, Euphrosyne came to Clare and began the business of providing Granpa with his children, Nina first, then Haydn, Hedley, Shaver, whose real name was Robert, and last of all, our mother, Stella.
Until the time of Stella, the joyful life was actually lived at Clare. For fifteen years after the marriage, there were between a dozen and twenty people on the staff. Euphrosyne was known as a hostess. She could play, she could sing, she knew a lot of poetry. For Granpa she was life's adornment and its blessing. When heavy with Stella, however, she endured a disaster.
It began on a hot morning in February with the arrival at a gallop of Tom Furlonger, the father of Garth and Tony. He had just time to tell the Mottes that Coolang was on fire; then the wind changed and Clare began to go up. From the house they could see the fire scalp the eastern hill.
Clare homestead was three wings around a courtyard where the household, which in those days consisted of a cook, a housekeeper, a tutor, a governess, a manager, half-a-dozen maids and seven or eight hands, pushed furniture which had come Euphrosyne's way from Edinburgh and Granpa's way from County Clare, thinking to load the most valuable pieces ready for flight. But already it was too late. There was only time to run for the dam. Euphrosyne had the good sense to take with her her stallion and two mares, from which she bred remounts for the Indian Army.
While the fire raged and the sky rained flaming coals, Euphrosyne, big with Stella, baled water over her children, huddled under heavy damask tablecloths ripped from the walnut table, and over her mares and the terrified plunging stallion, and over the cook and the praying governess and over the trembling maids, while half a paddock away the men battled till the eyebrows and lashes were singed from their faces and, ultimately, they had fought off the flames from the house.
For those in the dam, there was squealing, grunting, roaring, neighing, shrieking, crackling, spitting, howling. There were heartbeats amid the sawing of maddened blowflies and the piteous shouts of men. There were cramp and smoke and the dam water growing hotter, and at one stage bizarre piano music, strident over the continuous splashing of water by a mother big with child whose prospects were being licked from earth by an infernal tongue.
Pigs burnt in the stye. You could smell them. Cattle perished at the fence rails. Birds dropped stone dead from the sky and most of the great trees, clothed in history, were turned to ashen fists. The grand piano played itself to death in the courtyard as the strings broke under a rain of coals, but Clare was saved.
And, miraculously, the valley of Mountshannon, thick with timber, was saved. Fire isn't as inexorable as flood; it only devours where the dragon turns its head.
Into this blighted world then came our mother, jolted into being two months early by catastrophe; she came at the nadir, but they called her Stella the Star.
No one in the district knew how to handle fire. It had never happened before. Insurance claims were underestimated: there was stock, of course, fencing, equipment and a house lost on one place, but what of the losses afterwards? Brucellosis got into the herds, footrot into the flocks. And a great wind blew away valuable topsoil before the regeneration got under way.
At first Granpa believed he could manage, but he soon felt that the place had stopped being right for cattle, he could not bear having to shoot the remnant of a burnt-out herd again. Half the hands would have to go, and if there couldn't be hands, there couldn't rightly be maids. The tutor was a luxury. For some time the only thing that kept the mortgagees from foreclosing was Euphrosyne's stallion, an excellent Waler called Oberon, who commanded a top fee.
The change in circumstances was hard on Euphrosyne. It meant her sons would never have the education she'd expected they'd have. She'd assumed they'd go to Edinburgh and Oxford as the men in her family had before them, but Haydn had to be content with two years at Melbourne Grammar and Hedley and Shaver had a year each at Sydney Grammar and Ballarat. Even then, these arrangements tied in with the presence of Motte and Gilchrist relations with contacts in these places, and there could be no thought of university. As for Aunt Nina, she was 'finished off ', rather late in the piece at nineteen, at an Anglican girls' school in Brighton, where she went from a house full of Motte cousins, including the romantic Cousin Vere, who was of her age.
At twenty-five Cousin Vere would take a trip to Hollywood, where she'd meet and marry a movie producer, but at nineteen, while Aunt Nina was 'finishing off ', she was dashing about Melbourne with a young man who made her momentarily famous by rolling his de Soto sports car, with her in it, on their way back to Brighton after the Flemington races. The introduction of traffic markings on the road meant little to well-to-do young men, who thought themselves above driving on one side. Markings were there for the riff-raff who really ought not to have afforded cars: at least, that was the opinion arrived at among Melbourne Mottes, who, for the next half century would be downing street lamps, neatening the corners of trams and ramming any other obstacles which were inconveniently situated between them and their destinations.
Aunt Nina was also on board the ill-fated de Soto, and she was in the company of the supposed son of the Bishop of Capetown, name of Weightly Lisle.
Weightly Lisle – naturally there aren't any photographs. Our mother would rue his absence of chin, his mouse-brown hair and decided lack of shoulder, but how would she have known? She was a child at the time. She probably never saw him. When Aunt Nina, no doubt feeling the weight of family poverty as well as the weight of Weightly, eloped with him, _Dad,_ wrote Haydn Motte to Cousin Vere, was _a broken man._ Vere felt she was to blame. Son of a bishop he might have been, but Weightly Lisle was also husband of a wife who ran tea rooms in Durban and father of her four children. The drama of Aunt Nina's elopement had taken on Gothic proportions even before this was appreciated, however. Granpa and the three Motte sons, eighteen, sixteen and fourteen, undertook the long journey to Melbourne, dug the cad from the nuptial townhouse in Jolimont and publicly horsewhipped him for not having been man enough to ask for a lady's hand. 1922. Granpa was fined £100.
Aunt Nina returned, disgraced, to Clare, where our mother's life was a toad in a kerosene tin, grass up to eye level and hat down to grass level – she was short. Euphrosyne was fifty-five and mortified. She resolved never to go out and show her face in public again. She, like her youngest child, became a hat on the move through herbage.
In the midst of this scandal was Garth Furlonger, twenty-six years old, a qualified doctor and veteran of the Somme. Suspected of being alcoholic, but no one would say for certain. He'd lain about in a stupor after the Somme, but the stupor in which he'd latterly been lying about was attributed to Aunt Nina's hasty actions, for he was known to love Aunt Nina; he'd loved her from childhood. The Mottes had thought it quite a joke. Until the fire, the Furlongers had been the people over the hill who ran sheep. Their saddles creaked and some of them talked through their noses. Anyway, as our mother put it, Garth Furlonger saw his opportunity, had his suit pressed and came and pressed his suit.
In those days, when there were three Motte sons, no one dreamt of Clare ending up in the hands of Garth Furlonger. He took Aunt Nina to live in Adelaide and proceeded to be sacked from practices the length and breadth of North Terrace while Aunt Nina wrote impassioned letters in his defence and on his behalf, pleading war neurosis and pain from a bayonet wound in the chest. At the same time, poor Aunt Nina lost babies at the rate of one per breeding season. The Mottes, with true medical acumen, put the losses down to Garth's drinking. It made the babies slippery.
So the Mottes lost first ground, and then ground and babies, to the Furlongers quite steadily from the fire of 1910 on until '42, when they lost three sons. At Clare the round-edged, Georgian drawers hid sharp things and all glass was a potential shattering. Soft though the footfall was on the heavy carpets, it bore the weight of misery.
Coolang began by leasing pasturage, then each year another paddock, a dam, a short stretch of the water course. Granpa started to call the Furlongers 'calculating people' whenever their name was mentioned. He lived for the day when history would right itself, but history didn't. Drought, begun in 1940, continued till 1945; from '45 to '51, there were shortages of labour; then migrants came and there was no money for making farm machinery. Following that, there was inflation.
By 1956 Uncle Garth was still alive, but unconscious in the shower recess at Clare where we had been forced to leave him after his fall. Nor did he die in Scunthorpe Hospital where he was taken by the Scunthorpe doctor at about half past nine on Christmas night, having become ill at approximately half past two in the afternoon. The journey from Scunthorpe to Clare was thirty minutes with an extravagant allowance for the opening and closing of gates.
Dear Sir,
Aunt Nina wrote to whomsoever might have seen fit to answer.
I write concerning the recent ill health of my husband, Garth Llewellyn Furlonger, who suffered a stroke on 25th December, 1956. My husband became ill at 2.30 p.m. and I sought help from the Scunthorpe Hospital by phone before 3.00 p.m. Despite my repeated telephoning and several requests for an ambulance, it was not until 9.30 p.m. that Dr Stimson, the local medical officer, arrived to attend my husband, who had been unconscious all that time.
I need not say that it does not take seven hours to travel fifteen miles, even on foot, and Dr Stimson came by car.
An ambulance was required, necessitating a further wait of twenty minutes.
I was a volunteer ambulance driver during the war in this district and a three and a half hour return journey was the longest I ever made in five years. Furthermore, whatever the day of the year, I always made sure I was contactable for emergencies.
I was unable to bring my husband into Scunthorpe myself as he is a sizeable man whom I cannot lift, and on Christmas Day I was alone except for my two young nieces and my aged father. I tried to contact some of our hands, but was unable to, and my brother-in-law at Coolang was unavailable.
As a result of his stroke, my husband is bedridden and has lost the sight in one eye. I feel sure that had he received attention sooner, the effects of his stroke would have been less severe.
I need not hesitate to say...
(And here Aunt Nina hesitated, as in her draft of the letter she has crossed out a long list of complaints, contenting herself finally with:)
I find the conduct of the local hospital staff reprehensible in the circumstances. I expect you will look into this matter.
Officially typed on the old Remington, single spaced with a carbon copy kept for reference, the letter went off to the Commonwealth department responsible for the health of war veterans.
Having enquired into the matter as asked, the department replied that on the 26th December 1955, the preceding Boxing Day, Dr Garth Furlonger had reportedly been brought into the casualty department of the Scunthorpe Hospital by his brother, Mr Antony Furlonger of Coolang, having sustained a self-inflicted wound to the left hand. As Dr Furlonger had been in an inebriated state and was suffering from an asthma attack, it had not been possible to administer anaesthetic in order to sew his wound. A wait of several hours ensued before Dr Furlonger's blood alcohol level was low enough for him to take an anaesthetic. During that time, Dr Furlonger failed to control his behaviour in a reasonable manner. His 'declaiming' could be heard throughout the hospital and was a disturbance to other patients.
Dr Stimson, who had been on duty on Boxing Day 1955, accordingly asked Mrs Furlonger on Christmas Day 1956 whether her husband wasn't simply inebriated. Asked how much liquor her husband had drunk in the morning before his fall, Mrs Furlonger had answered that to her knowledge the better part of a bottle of overproof rum had been consumed. This being the case, Dr Stimson suggested that Mrs Furlonger wait a couple of hours and then call again, as Dr Stimson had to attend a delivery. She was apprised of the correct procedure for drunkenness.
Aunt Nina persisted. Two hours was not seven hours. Her husband, far from raising his voice on this occasion, had been unconscious for most of Christmas Day 1956 and indeed for a whole week afterwards.
If there was a reply, Aunt Nina didn't pin this one to the back of her carbon copy. A letter to someone called Leon followed. It was dated Feb 1957 and begged him please, please to take her financial affairs in hand as she could no longer cope and was headed for the madhouse. The nieces, dear children, were now back in Melbourne having both sustained an outbreak of headlice in the disastrous period after Christmas when not only was Garth so ill, but her father moved from Mountshannon to Clare, no doubt in order to give her moral support, but oh, Leon, was it any wonder she was almost out of her mind?
The lice began as I sat cross-legged on my bed at Clare, bloody, rebandaged and playing some very poor poker hands.
Allegra was dealing herself royal flushes and doing me out of my supply of Christmas chocolate. When she went for my lime creams, I started to sob in a slow, hopeless fashion. And to scratch. To shut me up, Allegra opened the sock drawer and took out two of the little sherries she'd secreted in there before the day's drama had begun. Then she too started to scratch.
Consider the nit, _Pediculus humanus capitis._ It does not call itself that. We are not privy to its thoughts, though it probably chose us as its boon companions while we were metamorphosing from apes. The nit, though not literate, anticipated us in its flexible core and adapted its claws for grasping either the round follicle of European hair or the oval follicle of African hair. And as louse leapt from oval to round and back again, the cardinal types performed stud duties, producing crossbred and comeback lice capable of infesting every type of human head up to and including the Asian with its hair follicle in the shape of a bean. It stuck the eggs to the bases of hair shafts, whatever their racial origin, with insoluble glue.
Kero, metho and infusions of eucalyptus bark have been used to dislodge nit progeny. Aunt Nina, rather than compound her problems in Scunthorpe by going to the hospital dispensary for help, took to our heads, and later also to the heads of our grandfather and our prostrated uncle, with diluted sheep dip and curry combs which we were on no account to mix up or deliberately interchange with any article that might pass near the thick and springy crown of auburn hair that was louseless and hers.
Nits were a blow to Aunt Nina, whose class of people did not carry them. In spite of her, they appeared to have risen through the ranks, striving for cleaner and cleaner scalps, where subcutaneous blood vessels yielded more and more readily to the needle-like proboscis. Once on the better class of scalp, the nits, true to the ground plan shared by all living things, were most reluctant to leave.
Where had they come from? Aunt Nina was certain she knew. We had in our possession a photograph, taken in the month before Christmas, in which we were wearing funny clothes. Although the louse is unable to live without an ecology of human heads to cling to, Aunt Nina blamed its arrival among gentry on the dress Allegra was wearing. There were two reasons for this: firstly, it was a wedding dress and Aunt Nina saw her opportunity to let us know we were still in trouble over the Christmas Day episode, and secondly, we were foolhardy enough to tell her where the dress had come from.
Add an 'r' to our surname, and in Italian the _Corretti_ are the upright and well-bred ones. Notwithstanding, Aunt Nina didn't like our father.
Perhaps it was because he had said in his beautiful, lightly tinted accent that the word Motte was not, as our imaginative mother had claimed, a Huguenot corruption of the word _mot_ and therefore indicative of a learned tribe set upon by misfortune and driven to the Antipodes, but the French word for clod. Furthermore, the Mottes' knowledge of art could be described as a tendency to confuse Rubens with Botticelli, probably because the name Rubens was synonymous with bottoms in that quarter. As for Rembrandt, he drew sheds well, but warts were warts, and they'd been spotted on Rembrandt chins and noses, the general opinion being that he ought to have painted people reading books to cover them up.
Unbeknownst to the rest of the family, who found his artistic predilections and preoccupations bewildering, my father was a member of the postwar _avant garde._ He painted neither Great Men nor their wives. His work bore no relationship to the painting of the stud bull at Clare which was done in 1908 by an itinerant bushman called A. Twine.
A. Twine's father had been a dentist. I remember Aunt Nina telling my father so on one occasion at our home in Melbourne. He laughed in reply, a mellow, Latin laugh. Aunt Nina didn't say _How rude,_ but she'd worn her lips as though she was saying it. Desperate that she should see he wasn't laughing at her, just at the idea of what she'd said, I climbed up on her, saying, 'Neeny, Neeny, that isn't what he means.' I succeeded only in having her change the subject for my sake. Which put my father further on the outer.
My mother tried hard to take my father's part, but in doing so, she made bizarre mistakes. She would try, impossibly, to link him with Italian village painters who did altar pieces and frescoes for churches. But my father was not religious, not in the altar piece sense of the word, anyway.
I supposed the depth of Aunt Nina's problem with my father was really rubbish dumps, and it must be said that at ten I was beginning to feel there was more to life than traipsing after him through them. We did not go to the tip as it is customary for people to go, with trailers full of things to throw out. We went as connoisseurs: to view life's wrack, and to collect.
I can see my father sauntering out through the canyons of muck in his dump-visiting boots. I am sighing, but dramatic escapes of air from my person meet with no response. The shapely ears of Henry Coretti, with maps of Australia outlining the entrances to the auditory canals, are attuned to the dump sirens, not to me. These sirens, invisible to the innocent eye but all too evident to the initiated one, lie on their rotting chaises longues in their salons of squalor, calling throatily, Henry! Dadda has 'Englished' his name. Henry! the sirens call, longing to be ravished by his light blue eyes; Henry! and he tracks them down with relish.
He has disappeared, and I am at a place where a band of waste forms a delta, part-covered in dirt. The graders have been busy here forking the muck into a colossal pit. Two children are playing in the dank, shoulder-high declivity that looks from where I am standing like the margins of a woman's thighs. I suppose from their activities that the children are girls, sisters perhaps like Allegra and me, but younger. Their fair hair is cropped off, basin style. One wears overalls, the other, a green dress far too big for her: on her feet she has a battered pair of high-heeled shoes.
They have made themselves a house in the declivity with pieces of broken mirror; a buckled painting of a cow, marked off for cuts of meat; two rusted kitchen chairs; and part of a table on which they have placed jars and cans and bottles. Periodically, the child in the dress slummocks a circuit over the junk heaps, pushing a bashed pink baby's pram.
'Don't forget the provisions, Mum!' The child in the declivity has a surprisingly rasping voice. 'There's going to be a general alert today. Ya listenin'?' Her round head swivels above the mound. 'Kelly, I'm talkin' to ya!' Then she notices me. 'We're surrounded, Kelly. Ya better come 'ome. There's wogs up the hill!'
The child with the pram looks up at me on my mound of muck and grins. It is a pretty face, clear-skinned with bright blue eyes in it. The other child hurriedly scrambles out of her den. 'Give the kid 'ere!' she croaks and grabs the pram handle. Doncha realise the air's radioactive?'
'Aw, Maggie!'
'I'm not bloody Maggie, I'm yer bloody father. Ja want the baby to turn into a mutant?' She disappears down the ditch with the pram. 'Carmon, Kelly,' her voice rasps over the edge. 'We gotta go down in the fallout shelter. The Nazis've pushed the button!'
The child in the dress scrambles after her.
I am standing on top of a solid sea. Anchorless fridges loll off the ghastly bomboras into yawning cupboardfuls of jetsam. Mast-like standard lamps and broom handles mark the places where valiant articles have foundered. Inner tubes cruise through the mulch like stingrays.
Where is the safety of the modcon kitchen, purring away on the lino like a satisfied cat? There is the virtue in visiting other people. Why make friends with people worse off than ourselves when we might be in the company of cupboards that really close, seats with the Dunlopillo safely in their cushions, light globes like proud pears securely screwed in over brand-new, frosted-glass, dish-shaped shades?
As if some huge finger has trailed them out of the morass, five dead ibis lie one after another in a bedraggled line, their pathetic legs haywire. A dog is rolling in one of the corpses. A boy tramps by me, glaring. He has a bucket in his hand. He is blond, gap-toothed, probably the brother. His blue eyes squint. 'There's eels in there,' he hisses at me, pointing at the morass that borders the tip, 'eels. I'm gunna catch a bucketful of slippery, slimy eeeels. So waddaya starin' at?' He pulls a horrible face, reaches into his dilly bag and dangles a piece of putrid meat. 'Whassa matter, sister? Ya crazy or somefin? Scared?' He steps before me, brandishing his bait, and I take off like a rat through a living room.
I can see the top of my father's head moving through the muck-maze. I scale the foetid hills from the far sides and toboggan down them until I am a mere unsavoury hillock away, and I discover he is talking to someone. I wish for a pair of eyes I could hold up on a stick. Slowly I mount to put a face to the voice that is saying, 'They're me kids, little Ernie, 'n Maggie 'n Kelly. I'm the caretaker. Yeah. They give the job to a woman for a change.'
Woman? This is the most extreme example of the category I have yet witnessed. She is cube-shaped, her brick-red face round as a basketball. Her eyes, when they turn on me, still standing in amazement on my hilltop, are like a punch – bright blue. She is wearing men's castoffs. Over her shoulder hangs a long flaxen plait. She wears a bayonet through her belt. My father asks what her name is. 'Bridget Kelly.' She holds out a chock of a hand and pumps my father up and down until I expect water to spout out of his ears.
'Ya come here often, dontcha? I seen you before.' She puts her hands on her hips, squinting. 'Don't say much, do ya?' She cocks her head. 'Pickin's are better over there. Wanna come and see what I got?' She leads the way. 'Y'after hubcaps? I seen ya drive up. Got Holden hubcaps. Got a bumperbar'd probably fit your car. Found ten quid this mornin' in small change. People throw out anythin', anythin' at all. What's your name?'
'Henry,' my father says coyly.
'No kiddin'? You're that painter, aren't ya?'
'Ye-es,' sings my father, pensively. 'H-h-who told you that?' Dadda has a little way of shuddering his h's when he wants to know more but isn't sure it's polite to ask.
'Everyone knows it. You're part of the local colour, Henry. I got a friend, Gail. She's got one of your pitchas in a frame in her sittin' room. That one of the dame washing the wall. Gail's got another idea about it, Gail's _deep,_ ya see – she reckons it's a pitcha of a dame tryin' to wash out her shadow. Like the shadow's a kind of stain we put over everything. That right?'
'Could be,' Dadda purses his lips and nods seriously.
'Jew believe in ghosts, Henry? Believe in the Afterlife at all?'
He shakes his head. 'No-o. I can't say I do.'
'This dump's haunted, I'm tellin' you that much. Anyway, this is me haul for this mornin'.' She has filled several boxes with junk. 'It'd be better if I had me metal detector, but I lent it to Big Ernie and he never gave it back.'
'C-c-can you tell me, do you sell this stuff?' asks Dadda, peering into the boxes.
'Yeah. If you like. But you c'n 'ave whatever you fancy. I don't make money from friends. Just the bloody fascists, ha ha. You could fix that clock up, it'd go, and the table only wants a leg and a bit of paint. And look what I got here.' She holds up a white dress, stiff as a board, 'A weddin' dress, I reckon. Someone must've got sick of it hanging in the wardrobe takin' up room.'
My father takes hold of the dress and clouts it out a bit. Slaters run everywhere all over it. 'Ye-es,' he says, full of amusement, 'I will take that.'
As we are leaving, me, hauling my breaths up in asthmatic resignation, and him, craggy-faced and unimprovable, holding my hand, Bridget Kelly calls out, 'Bye Henry. I'll keep me eye out for yer next show.' I blush to the roots of my hair for the family escutcheon.
Later I painted eyes, out, on plates. Blue eyes. It was not the first time I strayed from the path towards sedate portraiture and mature renderings of the family bull.
It was the red of Bridget Kelly's face that made her eyes so blue. And the tan of Dadda's face that made his so luscious and so light. If I crawled through the little black hole in Dadda's eye I would get to Italy. There I would see lakes and ducks and speckled trout, mountains with snow on them and men in Tyrolean hats. There would be girls in blouses, and me, Isobel Euphrosyne, in a dirndl with a lace-up bodice; and the sky would be dad's-eye blue.
All that was long ago now; and far from being dad's-eye blue, the sky today can't even be discerned. A grey, invading rain has nudged up close to the house. I feel it has been advancing towards me for countless years and over unimaginable terrain, my image on the surface of its shield, till now, thrust up before my gaze, I see myself look over the windowsill as if over the edge of my grave.
The angles of middle age distort the clean line of my chin. The expression is arch, sardonic, not like me at all.
On the wall of the kitchen behind me is a large print of Bruegel's _Mad Meg._ She is on the rampage, like the past pressing forward, garnering the future into her basket. She is surrounded by catastrophe. Bruegelian eggs crack round her, bringing more catastrophe. She charges from right to left, but catastrophe rises up and falls in all directions. On she charges, on and on eternally, while you and I tumble into the picture plane and out again.
Somewhere else in the house is an old lady who knits, and if _Mad Meg_ can be likened to History, then she may be likened to Nature, a perpetual knitter. She doesn't know what she is knitting, she just knits, as long as there are needles, fingers and yarn.
I have been here sometime now. Other people would find it lonely, I suppose, but for me it is a visual and kinaesthetic paradise, richer far than life among people.
Until fairly recently, when my work began to absorb me completely, I could think of nothing but someone coming. I craved company. You, whoever you may be, were on the brink of arriving all the time, and in my dreams I was almost always finding you. Now it is different. When I feel the pain of absence weighing on me, like a diver's belt placed there to stop me from bursting to the surface, I put on music, the _Appassionata_ , Ravel's _Concerto for the Left Hand_ , _The Rite of Spring_ , fiery, frantic music that shakes my bones.
I have laid the books about my father on the walnut table. Whether you come or not, the books will be here waiting. They will tell you the so-called 'facts'. With facts, you can judge and condemn people, but you can't rebuild them. The table top is kinder on my face than glass is. Here I look warm and Byzantine, lightly, joyfully and quickly drawn. My father's eyes were these eyes, blue and slanted, and so were Allegra's.
Allegra wore the dress that Bridget Kelly gave my father, and I, a child's red waistcoat, said to have belonged to Uncle Shaver Motte. On my wrist was a man's watch given me by Uncle Garth the previous Christmas. The watch had been trodden on by Countess when she threw me. I was the groom in the photo and Allegra, the bride. We loved weddings for all their lascivious mystery.
We are standing on the front verandah of our house in suburban Melbourne, having made an altar of our chest of drawers by opening the drawers in a staggered tier and filling them to overflowing with our mother's best underwear and every piece of jewellery we could find.
Leone, our father's friend, is on the other side of the camera. She is backing into the garden, blushing pink beneath her ash blonde hair. She is plump and curvaceous. Her dress gapes a little bit around the waist and we can see where the top of her slip is dimpling the white flesh of her belly. She is lining us up, the box Brownie sitting between her long plump thumbs with painted nails that curve down right over the tops of them. Allegra grunts, 'There's a slater running round inside this dress.'
'Stand still and be serious,' I tell her. 'This is a serious occasion. You're my wife now and you have to do what I tell you.'
'What!'
'It's true. It says so. You have to.'
'Bull!'
'You do,' I say sedately and quietly add, 'and your name's Leone, for your information.'
A haze of spit sprays out from Allegra's face. 'What's yours?'
'Jimmy Coote.' Jimmy Coote is Allegra's boyfriend. She is beside herself with glee.
'Have you opened the wedding presents yet?' I ask, as the real Leone clicks away, the hem of her dress grazing the dandelions and the daisies. Dadda hasn't mown the grass for weeks because he's using the lawnmower for something else. In our garage he has made what is called an installation. The installation is to be painted for an exhibition entitled 'All Mod Cons'. So our lawnmower is hanging from a scaffold with its grass catcher gaping open. At one and the same time it is a lawnmower and a hanged man screaming. People get hanged for murder in this part of Australia, but, asks our mother, do they get hanged for murdering the grass?
The answer is no. And nor do they get hanged for murdering lawnmowers. We were once a two-lawnmower family, but now one of our lawnmowers has been capitally punished in the garage and the other is in pieces in the washing machine.
Allegra says, 'Stop it!' and this is because she can't stop _giggling._ She has opened the presents, I know, because it is the thought of one of them that is making her laugh.
'You'll have to put it back in the fridge,' she says and she means the sausage. But the sausage is rude and sly and as exciting as the chops Jimmy Coote gave Allegra. I have wrapped the sausage in cellophane so it goes _thwock_ in the palm of your hand and I know that we will not put it back in the fridge.
'Mum let me,' I say. And this is true. Mum doesn't like pork sausages, she says they're inferior meat. It's Dadda who buys them. We only eat rump or fillet or lamb, but he likes pork sausages.
Leone has stopped photographing and comes over to the verandah, picking her way through the shin-deep grass. She sits at our feet on the bottom step, winding the camera on and smiling up at us. 'You're both adorable,' she says. Leone loves us but we don't love her and so we don't tell her she is sitting right over the bull ants' nest.
There are bull ants everywhere in this neighbourhood. If you put a stick into one of their nests and dig it up, you find an underground tunnel, not like other ant nests with galleries and branches. There are only a few bull ants in one nest. They are nearly an inch long and have sickle jaws. Their bites hurt almost as much as a bee sting. A bull ant will be crawling up the inside of Leone's floral skirt right now, making for the juicy inside of her thigh.
Why don't we love Leone? She brings us presents and draws us in her children's books and praises our mother's cooking. And praises our mother's clothes and says all the time that our mother is beautiful. But sometimes she goes to Dadda's studio with him and the buttons on her dress don't do up properly, so we don't like her. Sometimes you can see the tips of her bosoms when she sits down. We wonder if she knows. We think she does.
Dadda's studio is in the backyard, right next to our bedroom. The door is opposite Allegra's window. When it is shut we don't know what's happening behind it because Dadda has put a blind on his window. It's like that song, _Green door, what's that secret you're keeping?_ Both the door and the blind are green.
We don't like the studio door to be shut. Leone is always touching her buttons when she comes out, and it's a habit that annoys our mother. Once our mother banged very loudly on the door before Leone's visit to Dadda had finished. When it was opened right away, Mum stopped being in a bad temper. We could see Leone sitting on a stool inside, fingering her buttons, her sketch pad on her lap. She was nervously moving her legs around so we could see Dadda's shoes in a jumble on the floor under the bench behind her. She had her shoes on. He had his shoes on. It was all right. But we still don't like Leone. She loves my paintings, but it makes no difference.
When I was smaller, Dadda gave me one of his shoes to make a doll's house in. It was smooth where the balls of his feet had rested and ridged where it had shaped itself around his toes: like the lino in our kitchen, tracked with our comings and goings. Leone had better leave my father's shoes alone.
When we know she's coming, we groan, but we don't go out. We try to stick with her, but sometimes she gets away. Our mother tells us not to badger her, that she's only come here to do her drawings and she has to get Dadda's opinion because Dadda is the expert. But our mother also has a saying and she says it sometimes when she isn't sure that her young friend Leone, whom she brought home to meet her beautiful, drawable daughters and to meet a real artist, hasn't come just to see the artist. The saying is, 'Beware of experts; an ex- is a has-been and a spurt is a drip under pressure.'
Allegra says she has bite lumps on her belly from the dump wedding dress. I look at Uncle Garth's broken watch and am about to tell Leone it's high time the guests went home, when she rises from the bottom step, uttering little shrieks. Just as I said, the bull ant went for the soft and fleshy inner part of her thigh. There are ants that milk caterpillars for the juices they exude. Her thigh may have seemed a magic caterpillar, capable of supplying juice for aeons.
Leone would be aging now, getting on for sixty, her firm flesh would have fallen away. Where once it was plump and smooth, now it would be a swaying bag of sinew and vein and no one would seek to bite her there. Not even a reprehensible man like Dadda.
Undoubtedly it was he who had given us lice; undoubtedly he was to blame for the sheep dip and curry comb summer of Uncle Garth's stroke. When, after a week, Uncle Garth came to and opened the single eye his mind now had access to, our grandfather, as has been said, had already moved up to Clare in order to be of comfort to Aunt Nina in her time of travail. Stark raving humanity and a sense of territorial integrity led Aunt Nina to have Uncle Garth brought home and ensconced distally in the second bedroom, which imposed a frontier on Granpa, his pipe and the various rubs he had for his rheumatism. Having heiresses with headlice in residence meant that the third bedroom, theirs, had to remain theirs. Rather than sleep like a guest in the fourth or fifth bedroom, Granpa and his appurtenances were housed, in the manner of a gentleman flung on hard times, in what had been the butler's quarters.
To stem the tide of her ill luck, Aunt Nina went to church each Sunday and belted out 'Oh God Our Help in Ages Past' in a stern, reproving contralto. Allegra stayed in bed while I spread lice to the congregation, and afterwards, as Aunt Nina crashed through the gears on the way home, I would prompt her to tell me why God, who had not been particularly helpful in her past, would suddenly offer his assistance now.
'He chasteneth those he loves,' she opined dolefully.
'Do you think you will go to Heaven, Neeny?'
'Good Heavens, child, it's not for me to say.'
'If you don't go to Heaven, no one will.'
She thought I was being truly kind. She couldn't have fathomed that I went along to church, not in a spirit of piety, but in one of amazed curiosity. The stone arches and stained glass at the Scunthorpe Anglicans bewitched me with their useless opulence. The church was nobody's home; God was not there. It was just a piece of Australian air parcelled up, albeit prettily, by the people who had built it. The pews, though they smelt nice, weren't even comfortable. There were no beds, and as far as I could tell, there wasn't any kitchen. Fervent prayer did nothing, I proved that time and time again.
Dear God,
I would pray,
If you're there, please send a ray of light through the memorial window to Mrs Jas Williams 1854–1900, even though there is a mirror bush on the other side of it and the light normally doesn't come through. You do not have to send the ray through any particular bit, you may choose any piece of the window you desire. You have ten seconds. If you do this I will believe in you forever and ever, Amen.
But it seemed God didn't fall for cheap tricks.
'I should get christened,' I announced one day on the journey home, thinking Aunt Nina would be delighted. But she was scandalised. I had already been christened. Uncle Garth was my godfather. Didn't I go to Sunday school and scripture?
The truth was that on scripture days at school I did drama with the non-sectarians and Sunday for me was dump day. I realised I couldn't tell Aunt Nina this and lied passionately to cover my tracks. To prove my religious bent, I took to reading the Bible as I deloused myself beside my prostrate uncle, who was pretty soon lousy too.
And, alas, he knew the Book of Job by heart. _'There was a man in the Land of Uz, whose name was Job; and that man was perfect and upright, and one that feared God and eschewed evil,'_ he would say in a melancholy slur. '... _My flesh is clothed with worms and clods of dust: my skin is broken and become loathsome.'_ I had done Aunt Nina a terrible disservice. ' _... My days are swifter than a weaver's shuttle and are spent without hope... O remember that my life is wind: mine eye shall no more see good... My breath is corrupt, my days are extinct, the graves are ready for me.'_
In the kitchen Nina grizzled to God under her breath, _'Hast Thou given the_ corpse _strength? Hast Thou clothed his neck with thunder?'_
We had to feed him sherry to shut him up. Granpa beat his stick up and down like a metronome. He could not openly castigate Uncle Garth because it was the Bible he lay quoting, and Granpa respected the word of God no matter whose mouth it issued from, even if that person should come from a family full of calculators.
THREE
Perpetual Succour
SOMETHING IS BEING worried by a heron across the courtyard outside, a frog maybe. The glistening tree litter from the stringybarks is humping up and throwing out javelins at the bird as it stalks, jabs, stalks, jabs: its sharp-beaked head making scissors with its neck. This house was built around the trees, following the land flow. In fine weather, forest robes the mountain range as far as the eye can see in one direction. Looking the other way, the view is of cleared hills, furred by blond grasses and cruised above by eagles and swifts; but now this heron, stitching the world in close.
The past is pressing forward, bringing its clatter of old fears. I catch my eye in the window glass again. Behind me _Mad Meg_ has armed herself with dishes, pans and carving knives and leads her band of hags against the Devil. Across the field of Hell she goes, sword on the rise, roars and exclamations mounting in her throat. What will she slay? Everything exists demonically and absurdly around her. The moment of her action tumbles onward and outward forever.
What is writ, is writ: our faces darting, hers, mine, through the landscapes of childhood, creeping up on the studio shed in the backyard.
Once there were chinks for our eyes all over it, but one by one they've been covered up. Once we played under the bench among his shoes, but then we were banished.
From the rear, the wobbling line where the wall meets the floor is marked by tufts of handleless bread-and-butter knives, a dangle of rusty toilet chain with a handsome wooden knob, a spray of eggbeater, bent bottle brush and scissor tip.
But we aren't allowed in. We aren't to worry him, our mother says. Our eyes pry past the door perpetually. We are very good at leaning, excellent at reading the insides of rooms from the undersides of tilted windows, fine at summing up from a doorway-full.
Babies can go in, little children with their mothers aren't unwelcome. Is it when you reach a certain age, is it ten, that entry is taboo?
In the garage, his arrangements: the wedding dress dancing a flamenco round a mop, a doorless refrigerator done out as a shrine: a plaster heart of Jesus mounted in the icebox. Transfixed, we stand there and wonder what he thinks of.
Dadda's arms: they are tanned and well formed. When he is among people he holds them just slightly bent at the elbows, and his fingers fidget against his thumbs, making his palms growl. The arm muscles twinge. He wants to get back to things. He closes one eye and laughs abstractedly, sending a double message. Lots of people choose not to understand and to think his laughter signifies their wit.
'Where will the money come from?' they ask. 'Who will buy a doorless refrigerator decorated with Roman Catholic icons?'
They were shrines, of course, shrines to the Mod Con, though we didn't know that then. Our mother would try to explain them away and get stuck on the Catholic aspect of things. Mottes looked upon Catholicism as a misfortune. 'The poor thing's a Catholic,' our mother would sometimes say, as if that explained away someone's need for spectacles or eye drops at the eye doctors' surgery where she worked. We didn't know what religion our father was, though our mother assumed he was Catholic because he was Italian, and once in a while she and Allegra and I traipsed up the only hill in our district to Our Lady of Perpetual Succour to stand at the back of a mass, just in case Dadda's soul should be salvaged by our doing so.
We fiddled with the holy water and went weak at the knees in an embarrassed kind of way. Once, when our mother went to the extreme of kissing her thumbnail, just to set herself apart as a religious curio, Allegra went right off the air and stalked out. Afterwards, she chucked herself downhill ahead of us in a burst of empassioned walking.
Our Lady of Perpetual Succour was a bluestone brontosaur of a building. There was a corpse behind the altar. Christ's knees, grazed, stuck out of a fireplace in the Lady Chapel. I was sure it was God's true home, because he had all his paintings there, money boxes to keep him going through hard times, Mary to do the cooking and cleaning, plenty of visitors, the bed with the corpse on it behind the altar, little brass doors and candles everywhere that looked promising in the kitchen line, and there was a toilet in the quadrangle. Pictures of Christ were freely available, with his name inscribed in the halo for those who couldn't afford aspirin to stare at during periods of nervous exhaustion, overwork and physical pain. I often picked one up for our mother on the way home from school.
The bluestone brontosaur was the only imposing building in the neighbourhood and it afforded the Catholics their only grandeur. Downhill from it, Catholic and Protestant alike lived in a state of mild decay. Heiresses we might have been, but we could hardly have been said to live the high life. When our mother was tired and allowed the fat from the chops to congeal on the griller, setting the blowflies in capsized attitudes, you could almost have called it genuine low. Except that the chops were lamb.
There were backyards in the vicinity of ours where kangaroos were gutted and their hides tanned. Dogs squabbled over the discarded tails in the street. In a certain bath there were sacks of pippies waiting to be scrubbed and bottled in brine for the fish shops. Most people kept chooks, so their backyards stank, were grassless and had rats. We just had the rats.
From the street, our house didn't appear perpendicular to the block. A 1920s house, the general trend was perpendicular, but there was a slight lean inwards, which made it look as if our garden were a witches' huddle over an unseen cauldron. Three old elms, ill planted round the front verandah, reached over their shoulders to pluck birds from the blue and stuff them in their pot. The rest of the front was tame enough, a sagging picket fence, a straggle of suburban flowers, the odd nest of sprouting bulbs, some of them lost in the grass, which was generally long and hid the path. It was a path up which you might have doubted many people would come, but you'd have been wrong, for our mother attracted people willy-nilly.
To visit our mother in the afternoons after school, in the surgery where she worked, was to have our blue eyes playfully examined by the partners, Rumpton and Rudge. Rumpton sometimes held Allegra by the chin and gazed at her, shaking his long-beaked head in admiration. Allegra curled her lip and knitted her brow when he did this, but 'Rumpty's' attentions made our mother fill with pride and deliver the fable of our coming inheritance. Clare, she would say, followed by a fictitious acreage, and you could tell she thought Rumpton and Rudge would rustle up a Medical Man for each of us one fine day.
We would climb the stairs to Rumpton and Rudge's and enter another age, and an ambience altogether different from the one at home, though home was not far away. Unlike our local GP's surgery, which you could identify from the smell created by stale cigarettes and linoleum, hot from the stream of sun through the smoke-hazed windows, Rumpton and Rudge's smelt opulent. Rumpton, the senior partner, believed that a decent, hard-wearing seat was made of leather – thick, brown leather – hide polished with hide, Rudge was inclined to call it. The leather chairs and couch were deep and hid children very satisfactorily. Behind our mother's desk of inlaid morocco, aspidistras flew in brass pots on the shady, colonnaded verandah, where there were rattan chairs from Singapore. From there you could see the pavilion of a British Colonial tennis club with its row of short, stout palm trees. Through French doors leading from the verandah and the waiting room, to the sound of tennis balls _pock-pocking_ at leisurely intervals, the examination room was a marvel of beautifully made little wooden drawers in handsome cabinets, comfy seats, ophthalmoscopes and eye charts.
Rumpton, tall and thin, wore three-piece suits and a pince-nez. Rudge, a late ape, wore clips on his shirtsleeves and called Rumpton 'Gloom Bottom'. Rumpton had a saturnine temperament while Rudge was jovial; the former called our mother Motto, the latter, O'Rourke. They loved her, and she was pretty satisfied with them, though their wives kept cropping up with little requests – would Motto mind minding the Pekinese while Mrs Rumpton went shopping? Would O'Rourkey be a dear and pick up some shirts from the cleaners in her lunch hour? Every now and then Stella Coretti née Motte wielded the family escutcheon – who did Marjorie Rudge think she was? Nothing in her family tree except phloem and xylem. And who did Marjorie Rumpton think she was addressing? And why were all eye doctors' wives called Marjorie, a name redolent of cold weather and cardigans?
No one could accuse Stella of lacking style. When in full throat over the Marjories, she could point to a thousand ways in which she was superior. Marjorie Rumpton, for instance, had had a little trip to a health farm after the birth of her second child, Denzil. A health farm was not a place where you'd find a Motte. How could you farm health, anyway? And _Denzil?_ Marjorie Rudge was worse; she had a fake fur coat.
And so Stella kept her head above her circumstances. And at Rumpton and Rudge's, between bouts with doctors' wives and _petit fours_ for morning tea, she picked up those many visitors given to treading the path to our front door.
Those who carried Aunt Nina's seal of approval called Stella Bunny. They knew her from her past and had rediscovered her 'doing' for Rumpton and Rudge. When they rapped on the front door, something out the back would invariably topple. They belonged to the valiant cake-bearing brigade known in our family as the Upright and Faithful. They asserted themselves over the cracked parlours of their poorer friends, their robust bottoms taking the sprung sofas in a patrician spirit, accompanied by the clumsy lunge of limbs. They were bossy, cheer-choked women and our mother went to great pains to please them. She was frequently and ostentatiously extravagant. She had been known to boast of owing a dressmaker two hundred pounds. That made them shriek, 'Oh, Bunny!' as she dolloped on the double cream to hide the cracks in plates. From her chest of treasures, she hauled the heavy damask cloths, the serviettes with silver rings, the English tea things. The house would smell of ironing. Glutinous puddles of starch would attract mosquitoes in the laundry. When she bothered, she had a way with flowers, so that any weed in her hands took on its own eccentric character. She mixed fennel with roses, bunched heads of lantana with belladonna lilies; in autumn she ripped trailers from the grapevine. Anything she dipped into a vase was immediate, flamboyant and arresting. While their eyes were on the 'dandelions, Bunny, how clever!' she'd zip the continental biscuits under their noses: and, lest they notice the pot holes in the floor, she'd turn a teaspoon over and ask if they could read the hallmark, she being too short-sighted.
The jolly, optimistic friends did not call often, however, their social calendars only offering a set amount of time for visiting the fallen.
Though afflicted with pathological goodness somewhat less seriously than Aunt Nina, Stella was also a sufferer. There weren't many eye doctors in Melbourne in the late fifties and early sixties, so Rumpton and Rudge saw a goodly swipe of Melbourne's population, from society queens to charity cases and everyone, irrespective of class, gravitated towards our mother, whose manner was irresistible, and whose solutions to their problems were unique.
One night a defecting diplomat turned up in my bed. Just as well I wasn't also in the bed when the press came, was the only opinion my mother cared to venture on the topic after the fuss died down. One morning we awoke to find all the chairs in the sitting room coralled around a hat and under the hat we recognised the lady who took the change at the ladies' room in Myer. When we turned up our noses we were told to be grateful for our lumpy kapok mattresses and other people's overcoats, which sometimes served as blankets on our beds.
She was kind to them irrespective of race, colour, creed, class or politics. Up the path they would come, waddling in gratitude, with enough _Women's Weeklies_ to keep hellfire stoked and glowing for the foreseeable future stuffed under their armpits. But although she was good at crisis solutions, she fell down on follow-up. Sometimes we had to hide below the windowsill, our mother saying 'Ssh, ssh, they'll hear you,' when we were being quiet as mice and she was making noises out of guiltiness. Up the path they'd tramp, just wanting to call in and say thanks, weighed down by ovaries swollen to the size of boulders with their troubles. She'd leave a glass of water casually on the verandah for them and pretend to be out, sitting in the kitchen with the blinds drawn. Up the path they would prance, when she'd successfully paired them off with the men of their choice, to skite and bellow in the sitting room, while she wielded damask cloths, Doulton china and sterling serviette rings just to let them know that she, Stella Motte, hadn't crawled out of the woodwork yesterday. She had a middle name – Clare. It stood for Castle.
One thing she was not without was relatives. These would brave any tangle and meet any circumstance at chin level to visit Bun. Many an attempt was made by the wife of a brigadeer or surgeon to appreciate the things 'that fellow Bun was married to' worried the art galleries with. Mottes, Gilchrists, Bloomfields, Beauchamps and Taylors, too, came calling, and not just on the distaff side. Men whose bellies dipped between their thighs had been known to teeter on the low mahogany chairs and allow their rumps to be tormented by a jiggered spring for hours, eating cake with Bun.
Then there were those guests you didn't really expect. The Indian with the copperhead snake in the Coke bottle, who tried to levitate the tantalus but couldn't even get the stoppers out of the decanters. There followed a night of poltergeist activity in which our mother kept appearing, probably by mistake but bathed in light, in the middle of our bedroom. The presence of a Yugoslavian boy, on parole from a boys' farm, who broke into the next-door neighbours' and stole their phone money, resulted in the appearance of a cassocked priest on our doorstep and a gentleman bringing up the rear with a pair of handcuffs. The gentleman sat in the parlour opening and closing the handcuffs in anticipation while the priest tried to jolly the boy's whereabouts out of Stella. But she would not budge. What's more, she served them lapsang souchong tea, which she knew they'd hate, till the gentleman had put the handcuffs away and promised on no account to use them. Then she called the boy to come down from the ceiling over our heads, where he'd been hiding among the rat traps, and give himself up.
In 1960 she bagged an Italian, a genuine Northerner from Ancona. And that was when we started to learn our father's language, because Aldo was an Italian tutor without a clientele. He set up shop in our parlour on Thursdays, and we weren't his only pupils. Other unfortunates, enlisted by treacherous mothers as they waited for eye drops, had to troop in, too, on Thursday afternoons to read the adventures of _Il piccolo Lord Fauntleroy_ for forty-five minutes straight.
' _Il piccolo lord va in vaccanza,'_ reads the new girl. Her name is Checkie Laurington and she has fuzzy blonde hair. She is two years older than Allegra and she takes these lessons seriously. Aldo prefers Allegra to her, which makes her say things like, 'If Allegra settled down, we could actually get Lord Fauntleroy onto _il treno.'_ Checkie Laurington's real name is Cecilia and she thinks that means she's Italian. She is almost the worst person we have ever met. But there is someone worse. Her mother.
Checkie goes to a grammar school several suburbs away from here, so why is it compulsory for her to learn Italian at our house? Could it be because it is compulsory for Mrs Laurington to scud up to the kerb outside in her Daimler in order to pick Checkie up and take her home? Mrs Laurington is an art dealer: Henri darling, which is what she calls our father, paints.
We don't learn much Italian on Thursday afternoons, because our mother doesn't come home from work till five-thirty; Aldo's lessons start at four and Mrs Laurington arrives to pick Checkie up at ten past. She streaks through the house saying she's early, but running like a woman who's late for a bus. Out the back, the studio door creaks open and we don't see any more of her till five twenty-nine, twenty-nine minutes after Aldo's gone and we've been sitting round while the other poor unfortunates go home and Checkie practises her phonetics. It's tiring. It means a lot of trips to the toilet for Allegra and me. But all that flushing doesn't yield results. All we know is that at five fifteen Mrs Laurington starts to shriek with laughter, great shuddering gales of it. Hee-hoo-hoo-hoo-hoo, Hee-hoo-hoo-hoo-hoo. At five twenty, she's going ha-ha, ha-ha and gasping for breath, and at five twenty-two we can hear the clasp on her handbag snap, which probably means she's putting on her lips.
Now Dadda is opening the door for her. Now she is singing, 'Byeee, Henri,' and Dadda is giving one of his double-meaning laughs. Now she is strolling through into the sitting room, saying, 'Come on, darling,' to neat and tidy fuzz-haired Checkie, and Dadda is looking away, back into his studio, his hand on the inside doorknob, his lovely profile framed in the door, the shapely nose, the lips with their extra upturning kink in either corner. He turns back, waves with two fingers, closes himself inside. 'Bye-ee, girls,' says awful Mrs Laurington and Checkie says exactly the same thing in the same way.
Our mother will be home in a few minutes' time. Mrs Laurington always drives the Daimler in the opposite direction from the one our mother takes. Checkie doesn't even look out the windows, just sits there poring over _Il piccolo Lord Fauntleroy._ She is learning to be her mother.
Anyway, we don't eat sausages. Only poor people eat ground-up meat. Proud people eat lamb chops. Allegra opens the fridge and slaps four chops on the griller and I start hoeing into the spuds with a knife. 'Bloody old brolga,' says Allegra as our mother is driving the key into the front door lock.
Brolga was a good description, but Viva Laurington was more in the plucked line than the feathered. Many people found in her something to adore. Allegra and I knew it couldn't possibly have been her charm; more than likely it was her husband Harry's cash.
Her appearances at our home began soon after our mother had removed a fly's wing from her right eye in the course of her professional duties. When she found out that Stella was married to Darling Henri, she still had the fly wing in her eye and it just about took flight. We were there at the surgery watching. It was Rudge's afternoon and he had been letting us try the big long lens glasses they use to make people's prescriptions with. Allegra was wearing them and I was being Rudge and putting the lenses in. 'Henri Coretti!' The Brolga trumpeted. We never knew our father's name could elicit such throes.
In our neighbourhood the father was not a figure to inspire respect and awe. There were many houses like ours where fatherly activity was at a low ebb. Not far from where we lived, however, just a matter of blocks, fathers wore shirts and caught buses to work. It seemed the right sort of thing to do. That type of father had desirable surroundings to go with him.
Heretofore we had spent our lives trying to bring Dadda into line. When I raised the topic of his failure to reform with Aunt Nina, she thought it might be because the full impact of the word _nice_ had not leapt the ethnic barrier to land with the right force in the right quarter of his mind. His foreignness was to be forgiven, tolerated and understood, by all means, but Aunt Nina considered it important that we should strive to acquaint him with the Right and the Proper, since he, through no fault of his own, might never have had these virtues revealed to him.
Allegra's eyes bulged to blimp size down the avenue of lenses. Here was our father receiving an honourable mention from a total stranger whose voice had plummeted an octave from the usual key in which conversations with our mother were held. 'And are these his children?' she demanded. The lenses rattled in the holders as Allegra tried to bring her into view.
At this stage Rudge put his head around the corner, from where he'd been talking to his stockbroker on the phone, and suggested that Allegra and I should pop out and buy some cakes. We were suddenly aware that this Mrs Laurington was someone important. At Rumpton and Rudge's, cakes were got up for bishops and millionaires. Within seconds the words 'art dealer' and 'gallery' were mentioned and an invitation to come and meet the shriek-inducer in person had been made and accepted.
Be it said that the name Laurington produced spasms of hastily prepared grace in several circles in Melbourne. Spontaneous ladders formed in shop girls' stockings when Rolls Royce Lauringtons nipped in to buy knick-knacks. When boards of directors were lacking a member, Lauringtons were rung. There were musical Lauringtons, medical Lauringtons, theological Lauringtons, legal and artistic Lauringtons.
To appease her threatened family gods, when Allegra and I arrived back in the surgery with beetroot and onion sandwiches dripping redly from white paper bags, our mother recited the diverting history of Euphrosyne's father, the surgeon from Edinburgh. His famous _Advice to Eleven Daughters_ served as a model for colonial upbringing for many a long and weary year. Our mother ended with her favourite quote from it: _'By all means show them Christian charity, for God has sent them among us for a reason, but never marry a madman.'_
The winged eyeball smiled mysteriously. 'You disobedient child, you,' she ventured snidely from the side of her mouth.
Why was Dadda such a difficulty? There was a drawer in my rib cage, beneath my heart, which contained my bits of Dadda in it. His dislike for new shoes, his smell, his shapeliness.
And then there was the photo of a bedroom he'd stuck behind our toilet door. It was bizarre; beside the single bed, resplendent under a sunburst cover, was a bombshell crowned with a laurel wreath. On the other side of the bed, arrowless, a statue of Saint Sebastian stood bleeding. Above, in an ornate golden frame, replete with curlicues and a white porcelain dove, was a painting of the bedroom's owner, depicted as a leper being kissed by Saint Francis of Assisi.
Whose bedroom was it, I wanted to know. And my father replied, 'D'Annunzio's.' And who was D'Annunzio, I asked. 'The so-called "Hero of Fiume".' And where was Fiume? 'Yugoslavia, where the boy who pinched the phone money came from.' And why was he a hero? 'Because after the First World War he invaded that city and made himself a kind of emperor there for a year or so. He was also a great Italian poet.'
'But you've put the photo in our dunny.'
'Where it belongs, Sibella.' Sibella was his special name for me. 'D'Annunzio ripped the trees out of his garden and planted bombs and flagpoles there instead.'
In my Dadda drawer was also _La Strada,_ my first Italian film. We sat together in a dank green cinema, Dadda and me, as if at the bottom of an old impressed tin tea caddy, two tokens from a Cornflakes packet, surrounded by dark Italian men. He guarded my chair with the curve of his arm. Dadda had seen the film before. 'Her name is Gelsomina,' he said. He used a baby voice when he spoke to me, as if he were reading me a children's story. 'Her mother sells her to a man called Zampanò because the family is very poor. Zampanò does a strongman act for a living. He and Gelsomina join up with a circus where there is a tightrope walker who is also a clown. Zampanò is very rough with Gelsomina, and the clown, who is gentle, wants her to run away with him. Zampanò and the clown fight over Gelsomina and get the sack from the circus. Gelsomina could have stayed at the circus if she wanted, but instead, she goes with Zampanò.'
'Why?'
'She is a simple girl, and poor, Sibella. It is hard for her to make a choice. Also, it's strange, but she loves Zampanò. She probably feels he needs her and maybe he does. Some of us don't choose the people we love, it just happens – we get hooked.'
'Is that why people tell fairy tales, Dadda? To cover up what really happens?'
'Well, I suppose it is, Sibella, when you think about it that way. Ye-es,' he said in a sing-song way he used for addressing children. Sometimes he would say it to Mrs Laurington and my heart would lurch.
I was surprised. _La Strada_ offered no snow-covered alps, no speckled trout, which was what my mother had told me when I asked about Italy. But there was something wonderful about it all the same: the incomparable Giulietta Masina. For weeks Giulietta/Gelsomina was my crush, my obsession. I wanted to change my name; I wanted to wear a bowler hat and take a trumpet to school, but alas, there was not a trumpet nor a bowler hat to be had in Melbourne. The best Bridget Kelly could do was an airforce cap and bugle. I couldn't get the bugle to play. No matter what I did, no music, not a parp let alone the wan and wonderful mystery of _La Strada,_ would flow from it. I was desolate. At school everyone laughed at me, yet _La Strada_ was my secret life; I wanted it to become real and thought if I willed it hard enough, it would. I wanted them to see it, just to see it, to see me be that astonishing urchin and to enter the story with me.
_La Strada_ made me hanker for more of Italy, but apart from Aldo and Dadda's Uncle Nicola we weren't to get much. Uncle Nicola only put in rare appearances at our house and, apart from him, Dadda only seemed to know one Italian. His name was Ben.
Ben was an old bloke who was supposed to do repairs for us every now and again, but he wasn't much good at repairing. Generally he just came and sat in the sun on Dadda's studio step and talked away, a little old saucepan on the boil, while Dadda made art inside. On one of his hands Ben had only three fingers. He'd smoke with the cigarette between his second and third fingers. The nicotine stains reached right up his arm. He'd smoke and prattle, smoke and prattle, until a semicircle of shoed-out butts surrounded his feet and he was ready to go home. We knew where he lived – down a skinny bitumen lane lined with palings behind a row of tomato plants in a dwelling with three walls, the fourth being a tarpaulin, like the curtain on a stage. He had a tortoiseshell cat, but we never knew her name. When he saw us, Ben would flick his three fingers in a little wave and nod.
Our mother was given tickets to the Dante Alighieri Society once, by someone visiting the surgery. Dadda wouldn't come. It seemed he hated the Dante Alighieris. 'But it's _Dante_ ,' our mother protested, wide-eyed, ' _Dante_.'
' _Nel mezzo del camin di nostra vita, mi ritrovai per una selva oscura, che la diritta via era smaritta_ ,' quoted Dadda, 'That's Dante, Stella. "In the middle of my life, I found myself in a dark forest where the straight way was lost".'
We didn't understand, but enjoyed the evening all the same. It was a recital by a Chinese girl who sang, among other things, 'Oh Waily, Waily, up the Bank'.
FOUR
More about Mottoes
A DOOR CREAKS. Through the lugubrious entrance hall, shuffling over the Mudgee slate and past the fireplace with its skirted copper hood, comes a little old lady, her white head bent, to visit me, the Midnight Knitter. Her hazel eyes are pin-pricky and sorrowful. Someone has broken into Fort Knox and stolen her bankbooks. Her hands are blue, her circulation's gone with her fortune. 'There, there,' I say, 'it isn't half as bad as you think it is.'
She has had an intruder, she tells me. Oh, for an intruder! Out here where no one comes. We go and look, down the passage and across the covered courtyard to the place where her bedroom is.
It is evening now. A mysterious light has come on outside her bedroom window, what could it be? 'I believe,' I say to her, 'it is the one the electricians came and put in so we could see our way across the courtyard at night.' They came in their overalls, doesn't she remember? One was called Chins and the other Jack Keogh. Chins is called Chins because he's part Chinese and grew three chins to keep up with his nickname. Jack Keogh is a left-side Keogh, left as you drive towards Canberra through the town, and not to be mistaken with the right-sided Keoghs, though one of them is also called Jack. The Keogh war goes back to the gold rush of 1860. So do the Chinese forebears of Chins. Irish and Chinese riddled the high, cold creeks for gold together in those days, just as now their descendants do the electrical work. Must be something about currents.
But who put the light on? Well, someone put it on. It didn't put itself on, did it? Where's the switch then? There isn't a switch. If I can find a switch she'll give me all the money she has in her purse. Except that her purse and bankbooks have gone, clean disappeared. Someone has broken in and taken them. But I can come into her place and choose a china ornament if I can find a switch.
I tell her I have to go into her place to find the switch, anyway. 'Oh no,' she says, 'this isn't my place. I live at Boris Pasternak and Jane Austen. You wouldn't find me living in a place like this.' She clambers up onto the railway sleeper that makes the doorstep to her bedroom and shuffles inside onto the higgledy-piggledy red brick floor. 'This is a rabbit warren.' She flings out a handful of arthritically noded fingers to the right of her. 'The walls are made of mud.' She flings out a handful of arthritically noded fingers to the left. 'It could be overrun with Italians any minute.'
'Don't you remember?' I ask. 'You moved from Boris Pasternak and Jane Austen years ago to marry my father.'
'Was he the Italian?'
'Yes,' I say.
'Why would I want to marry your father?'
'To have Allegra and me, I suppose.'
'You? I don't even know you.'
'I'm your daughter, Isobel.'
'Geoffrey was Isobel's father. Geoffrey Latimer. The Germans bayoneted him. He died in Syria with my brother, Shaver. The Red Cross sent us letters in French, and photos of the graves.'
'My father was called Henry,' I say.
'Well, I suppose he _would_ be.'
I find the light switch. 'See, this is the switch, this one here that looks different from the others. Remember, we asked the electrician to make it look different so you would know which one it was.'
She has never heard of such a thing. Next I'll be telling her the washing machine works.
'You'd be surprised,' I say.
Outside the rain is still falling. On and on and on. Little carts rolling down the iron roof, dragging the night along behind them.
I sit on the end of her bed, warming her poor old feet. She'd like to give me a china ornament but she hasn't any. Probably the intruder took them. I ask what he looked like. 'Ooh,' she says, 'he was that Italian. You know, they don't wear proper clothes and they never have clean fingernails. Or polished shoes, not like proper men.'
A proper man has suits for all occasions. A proper man washes his hands before he sits down to a meal and he endeavours to keep his fingernails clean even if he has been doing man's work. A proper man sends his daughters to grammar schools. His socks match. He takes pride in polished shoes. More than that, a proper man knows how to polish his own shoes.
The improper man wasn't keen on polish – I crack her old white toes above her bunion – high gloss on long boots was an especial phobia. When I played Captain Midnight at school wearing an old pair of Granpa's riding boots stuffed with newspaper, he said, 'Don't shine them. A bushranger wouldn't have.' But I was already out on the back steps with the polish. Granpa and Uncle Garth were proper men, gentlemen. They spat, rubbed, oiled and spoke of elbow grease. I, too, would be a gentleman. I thrust my arm down the cool length of the boot.
Granpa and Uncle Garth were still alive as I sat there polishing. The only person to die in January 1956 was Dadda's uncle, Nicola Coretti. Oddly enough, he was proper in his way and polished his shoes. The only relative of Dadda's we ever knew, Uncle Nicola, had a gold-tipped walking stick and wore a flower in the buttonhole of his frock coat. In spite of living in a single room in a boarding house in South Melbourne, he affected a large diamond ring which Dadda inherited when he died. His English wasn't good, but his voice in its heyday was supposed to have been impressive. We were told he had been given to public speaking, though quite where and what, it was difficult to ascertain. We understood he had lived in Melbourne for thirty years, but he was so different from anyone we'd met, we found that hard to believe. He took little notice of us or our mother, nor did we want him to take notice, as he was old and touchy with rheumatism.
When Uncle Nicola came to visit he sat in the studio with Dadda and quaffed vermouth. He died on an outing to Coleraine in the company of several university professors and lecturers, among whom must have been a botanist as it was said our great-uncle choked on a kex. We were a long time finding out that a kex was a dried-out stick of chervil, and having found out we were none the wiser as chervil was a plant unknown to us. We knew of fennel; it grew everywhere in the cracks and crannies of a neighbourhood rich in them. We knew of thistles, which abounded in the local parks, and of the blackberries choking Yarra bank and making it difficult for those who practised murder to dispose of the murderees. Allegra and I agreed that blackberries were probably the reason why murder was out of fashion in our region in the fifties, though a mere twenty years beforehand it had been quite a popular pastime. Imagining Uncle Nicola choking on a kex posed almost insurmountable difficulties. We thought of his sallow face turning a brilliant red above the white of his neat, clipped beard, then turning purple, and the cracking of his skeleton as academics slapped him on the back, then black and Uncle Nicola withdrawing entirely from ownership of such a face.
It was astonishing to think that his memorial service at the Unitarian Church in East Melbourne was packed out. Our mother told us some of the guests were very distinguished people.
Granpa and Uncle Garth, though probably made of no sterner stuff than Uncle Nicola, battled on through his choking and through the whole of the next year, by which time my days of bushranging were over. For three more years, during which the practice of polishing shoes among Corettis in Melbourne sank into oblivion as the tins of Nugget and Kiwi rusted round their slowly desiccating contents, Granpa and Uncle Garth wheezed and tapped, quoted and tapped their way into the future: the blind and the blind.
Aunt Nina ran from one to the other with draughts of sherry and pinches of salt. Then one day, unable to forestall the inevitable any longer, Granpa stopped tapping and bit off the air with his blunt old teeth – as if death were a matter of castrating life as he'd castrated lambs in his boyhood.
A fortnight later, victorious but weary, Uncle Garth stopped wheezing and the trap was sprung on his remaining eye.
Allegra was inconsolable when Granpa went, but I had to dredge tears up from the reservoir I kept for propriety's sake. She was appalled with me. By contrast, when our uncle died, it was I who found myself with a face wet out of pity. On both occasions Nina sang in a wobbling but strong contralto, her soft jowls quaking over her firm chin: ' _Rock of Ages, cleft for me/Let me hide myself in Thee_.' The stones of the Scunthorpe Anglicans would never forget her.
Our mother began to talk in the past tense of the fortune that was coming our way. Once or twice she toyed with the notion we'd been ruined. Allegra and I savoured this romantic state for quite a while, but among the people we knew, most had been born into ruin and hadn't had to wait to have it thrust upon them.
Aunt Nina continued on at Clare, though it was clear she would be lucky to salvage the house after the death duties. The Furlongers were supposed to be helping out by using the home paddocks to agist their sheep. In that way Aunt Nina could hold on to the land between the house and Mountshannon, but our mother was certain foul play was afoot. You couldn't tell her the lever wasn't being applied and Aunt Nina wouldn't end up out on her ear in an old people's home.
From the business at Clare our father stayed as aloof as he could. He was kind to Nina, however, wore a borrowed suit and helped with the funerals, only causing one bout of huffiness when his unmatching shoes were noticed as Granpa slid over the edge.
When Uncle Garth died, Dadda actually bought himself a pair of shoes to spare Aunt Nina hurt. They were polished black leather, but on the night of our arrival at Clare they mysteriously disappeared from the fourth bedroom where, in deference to the recently deceased, our parents had been installed.
Those of us in the know scoured the house, hoping to keep evidence of a thief to ourselves. Meanwhile, Dadda had to resort to some old slippers he found in Uncle Garth's den. On the morning of the funeral, seeing Dadda in her late husband's slippers, Aunt Nina marched off to her bedroom and produced a shoe box, saying, 'Well, I dare say Garth would prefer you wore his shoes to his funeral rather than his slippers.' Inside the box were Dadda's shoes. He lifted them out with a frown. Still with a frown, he put them on his feet. With half a frown, he drove us to the funeral, by the end of which the corners of his mouth were flickering up satanically, so that by the interment, the focus of attention was not on the flesh being clothed with its worms and clods of dust, but on the nitid toes of Dadda's shoes, which wriggled up and down, up and down, to the flickering of his smile. New they might have been, but they did not match.
'She really did believe they were Garth's,' whispered our mother. But if you thought about it that way, it made matters worse.
A series of paintings evolved from Uncle Garth's funeral. One was called _A Very Disreputable Man Wearing Highly Polished Shoes._ As for the shoes themselves, along with Granpa's riding boots and several pairs of Uncle Garth's shoes bequeathed to her shoeless brother-in-law by an Aunt Nina who said since one pair of Garth's shoes had fitted, there was no reason why the rest wouldn't also, they became part of a mobile that strutted and paraded arrogantly over our garage ceiling around a humble cluster of footwear Allegra and I had grown out of.
Some months later, Dadda and I drove to Clare, ostensibly so Dadda could help make an inventory and sort out what needed to be done. It was August. Scare-guns were going off at random all over Clare, where seed had been valiantly spread in the pastures. The white cockatoos who'd come to gobble it up rose in a motion like a sheet flapping on a clothesline, only to land again a little further off to peck between salvos.
Aunt Nina was thinking of roots and shoots, of soil binding and rainfall. The appearance on her premises of that fly in the ointment, my father, caused her spine to straighten and her lips to purse, even though she herself had invited him, as my mother should on no account feel left out of negotiations concerning the precarious family inheritance.
No doubt Dadda placed little importance on such things, but Clare had been the prime concern of three generations of Mottes, as he could see from all the paperwork she spread out for him daily on the walnut table in the vestibule. There were account books going back for years. The gorgeous copperplate of great-grandfather Augustus Motte gave way to the affable script of his son. In turn the affable script gave way to the gorgeous copperplate of Uncle Haydn and this yielded to Aunt Nina, whose hand got away from her so that words lounged across columns, squashing conclusions into tiny spaces that made the vowels pop out into thin air.
Dadda allowed himself to be shown Euphrosyne's family crest, a shattered oak, and to be read the motto _nec deficit alter,_ which Nina supposed meant 'Never fail another' but which some uncouth lout who thought he knew Latin had told her meant 'Never fail again'. Nina hadn't had the opportunity of learning Latin, but she was sure if she had learnt it _alter_ would have meant 'another', not 'again'. Dadda couldn't reassure her on that point. It all had to do with the case of _alter_ and whether another time was implied. She mustn't be so concerned; greater tragedies had occurred than misconstruing a family motto, and regarding Clare, all was not lost, not yet.
At this something was lost – Aunt Nina's temper. There was a standing up at the walnut table and a banging down of the account books. Mottes worth their salt bounded and plunged at the utterance of cold comforts. Had Dadda no pride? He struck Aunt Nina as a man without pride altogether. What had he been doing, for instance, when Mottes were being sunk off Crete, shot in Palestine and captured at the fall of Singapore? Why had he not been in his own country sinking, bombing and capturing?
It was an impasse. The antipathy between Dadda and Aunt Nina was insoluble. Dadda stood, his own lips pursed and his thumbs fidgeting wildly till his palms growled. For Aunt Nina's information, the sinking, bombing and capturing capacities of the Italian people had been in safe hands when he left them. And if she wanted a motto, why not _Me ne frego_ (Who cares) because that was the slogan with which his country had gone to war. And if she thought she could tell a man's worth by the boots he wore, why did she not lend her mind to the feet of Him in whom she had so much faith: that Anglican, Christ?
The light outside my mother's bedroom is making the spider webs in the courtyard shine. _Here –_ for the benefit of you who are not _here,_ I shall take this beautiful word as my motto. _Here_ means now and with me and it has come to Australia, to this mountainous, isolated place, via the English and, before them, the Frisians and, before them, the Saxons and the old High Germans. When the Romans were saying _adsum,_ the Germanic tribes described by Caesar in _De Bello Gallico_ were probably saying _Here!_ Why waste your time with _adsum_ when you could be _here?_ Here, it is the season of spiders, lace and grace, the culmination of all heres.
And it is where _Mad Meg_ on the kitchen wall is gathering the future into her basket, her sack, the Dutch oven slung over her gauntleted arm. There have been several heres. One was a room with a honey-coloured maple veneer suite in it. There was a steam train running down the grain of the wardrobe door. The door could only be shut by jamming it with a wedge of paper. Inside the clothes were packed so tight you could hardly get anything in or out. There were clothes and accoutrements everywhere; we even ran to a scotty-dog-shaped lump of coal, given to Allegra by the boyfriend after Jimmy Coote, whose chop bones she still kept in a satin-lined box. She could throw nothing out. I was a little less reverential, but hesitated guiltily at garbage tins and grieved just a bit when the lid clanged down.
From this chock-a-block room one day, when girls were changing into winter garb and boys were still vaulting the fences round the baths or being let down onto the concrete pool surrounds from roofs to avoid the turnstiles, we exited in furtive fashion via a window that deposited us in common suburban shrubbery to the side of our house. Through this tangle we scuttled and crouched the short distance to the street, snagging our black stockings and getting dust all over our black suede courtshoes. We were wearing straight skirts to the knee and floppy jumpers – hers powder blue, mine red. Beehive hairdos, French rolls. Not so conventional that everyone else was wearing them but, then, not so blatant that the daughters of public servants and kindergarten teachers were being summoned inside for fear they might set eyes on us and be transformed into rebels without a cause. Our suburb was being invaded. There was hardly a chook pen within crowing distance now, though there were still rats. The people who used to bottle the pippies had opened a fish and chip shop in the main street, and another shop sold kangaroo skin coats. The days when you'd find pippies in the bath and roo tails in the street were gone. Gone too was the phase of smoothing over 'ugly old Victorian facades' with rough-cast, as had happened in Melbourne for the '56 Olympics. The wrought-iron Mexicans siesta-ing over the house numbers were being clucked at and decried. Restorers had moved in. Quite a lot of iron lace was going up, a fair amount of it by mistake, on Art Deco.
We were on an important mission. It was a Saturday. We had left our house by the bedroom window so as not to be seen by the afternoon tea guests. These were The Brolga Laurington and her abominated daughter.
The Brolga had offered Allegra a job she didn't want to take. She'd been asked to the drive-in that night by her boyfriend, Macka, but there was no choice. To be offered a job by The Brolga was tantamount to a summons.
Terry McReady, known as Macka, long-faced, acned, with a footballer's waddle, had shot my sister a glance, all dark brown iris without white to it, since Terry McReady's sparsely lashed lids were worn closed along the greater part of their margins. Suddenly the blue and luxuriantly lashed eyes of my sister saw a boy transformed. His cap of ginger crinkle became beautifully sculpted old gold curls, his thick, punched lips, the broken mouth of Mars. Those bandy legs that jet-propelled the backward-leaning Macka through school cheers and a field of enemies towards the goal posts – Macka was a Rules man – and had Macka following after them, lurching stiffly now left, now right, had become for Allegra freckle-smattered God-limbs. A bond had formed across the playground, a bond as long as the distance between Allegra and Macka. The whole school disposed and disported itself like elves and fairies, dukes and amazons, around this pole of love: Macka was its ugliest, most decorated schoolboy and Allegra its most beautiful girl.
The public phone at the corner of our street is out of order. 'We'll just have to find another one,' I say. But I can tell Allegra is glad, ringing Macka would be traumatic. Though she's been going with him all year, his parents don't know he has a girlfriend. She can't bring herself to leave a message, so we have to go to the park over the road from his house and loiter until we can attract his attention. Otherwise he'll turn up tonight and rev the ute in the gutter and no one will come out. We start walking towards Macka's place, trying to look as though we aren't hurrying, though we are.
Boys in our neighbourhood are pretty weak at knocking on the front door. Furthermore, they are scared stiff of Darling Henry, who is not in the general run of fathers and keeps company with the dump woman, Bridget Kelly. They are scared even stiffer of Bridget, who comes calling at our house with specials. I suppose they think they might trip over her and be turned into toads.
Maggie Kelly started at high school this year and every Monday morning in first term she has had to stand on a bench in front of the assembled school because she hasn't paid the text book rental fee. Bridget says the fee is illegal because state education is meant to be free.
We decide to swing back and forth on the double swing opposite Macka's. Nonchalantly. It's hot in our beatnik clothes, and we're supposed to be at home finding something for our mother in our room, so time is limited. Allegra has chickened out at Macka's gate.
'Leave him a note,' I tell her, but there's the problem of no one looking in a letter box on a Saturday afternoon.
I'm really boiling in this jumper and want to go home, so in the end, I am the one to go up to the gate. In I walk while Allegra gasps, 'Don't!' behind me. Up the four green-painted concrete steps between the dahlias and the postal arrangements I go. From the doorstep I can see Macka's dad sitting up, polished and beery, in front of the telly. He is watching the tennis as a good dad should: Rod Laver, racquet sinister, droning adulatory commentary delivered by barely moving mouth off camera. I knock boldly three times and something raucous in the way of a dog goes crackers in the hall behind the door.
Macka's dad's beer glass goes _thunk_ on the table. Allegra is standing outside in the street, almost hidden by a tree, picking leaves off the front hedge.
'Gidday' says Macka's dad, very ruddy behind the flywire door.
'Gidday,' I say, 'Macka home?'
'Yer mean Billy or Terry, love?'
'Um... Terry.'
'No, love. Sorry. 'E's at the baths.'
'Is he? Um... I want to leave a message from my sister. Her name's Allegra. Could you tell him she's sick, but she'll see him at school on Monday? Okay?'
'Okay, love. Right you are. Nothin' serious, is it?'
'Um... no, she just fainted. Mum wants her to stay home tonight.'
'Okay, love, she'll be right. What was the name again?'
'Tell him Legs. Bye.'
'Bye,' he says, and closes the door.
I've done it. It's the approach that counts. Nobody puts two and two together if they find adding up difficult. I'm just glad it wasn't his mum who answered the door.
Allegra leaps out from behind a tree and grabs me like a maniac. 'What'd you say?' she asks. So I tell her. She smiles, a great burden has been lifted from her. She starts to skip, like a woman who has recently recovered from a fainting fit. And sing.
'It's like I always say,' I tell her. 'You gotta have the knack.'
On the way home we buy a packet of Peter Stuyvesant to practise being sophisticated with in the bedroom.
_The weatherman says clear today, He doesn't know you've gone away, And it's raining... aining... aining..._
'Bugger!' She kicks the radiogram with her foot. She has taken her shoes and stockings off, her toenails are painted purple. The needle skids onto the next track,
_... blue 'bout, Peggy, My Peggy-Soo-hoo-hoo, Hoo-hoo-hoo-hoo-hoo-hoo, Well I larv you, girl..._
Her foot disappears under a lump of junk on her unmade bed. We have found what we were supposed to be in here looking for half an hour ago. It is a piece of writing by me that our mother is very proud of. She needed it as Exhibit A in her campaign against The Brolga, but it's too late now, because The Brolga's gone out the back with Dadda.
' _When the archaeologists of the future unearth the 1960s_ ,' Allegra reads, ' _they will find vast rubbish tips where twentieth-century man sacrificed his chairs, tables and vacuum cleaners to appease the gods. They will think the gods of the twentieth century were middle-class deities who preferred lounge suites to virgins_.'
That's the bit our mother likes. Allegra lets go a smoke cube and looks at it cross-eyed, just in case. It dribbles down in front of her, holeless. She flaps it away with the pages of my essay and then starts reading out loud again.
_'The archaeologists will not realise that twentieth-century rubbish had a mind of its own. It was out to displace human beings._
_'In some homes, a great effort was being made to keep it in its place. An entire class of human being, the housewife, yearned for air- and water-tight plastic bags in which to imprison it. Little trucks with cardboard smiles attached to the radiator grilles prowled the suburbs dispensing detergents and toilet sterilisers to fend it off. And there were other trucks, pink- and yellow-striped like the Pied Piper, that sang and sprang around street corners to reward clean little children with cones full of ice-cream. Jawed trucks like dinosaurs called by twice weekly to collect the scourge. But in spite of all this, in one home in deepest darkest Melbourne, rubbish was taking a stand.'_
Despondently, Allegra hands me the weed, filter first. The trick is to draw in until the filter gets hot, then shape up your mouth and bop it out so your tongue makes the hole. I can get five rings off one puff, each new one passing through the one before. We're using my palette as an ashtray, but Allegra isn't even very good at ashing and so, when I lift my palette off the floor, it looks like a spatterwork and I have to thump it into the rug so it won't be noticed.
Allegra's bosom is a great disappointment to her. She is in her second-last year at school and will never be a lady now, all hope of grammar school has been abandoned. She is lying back on the bed, staring at the ceiling, tapping her ritzy toes to the music, singing softly and out of tune. She can't even click her fingers properly, they just make a sort of _plap_ and her dog collar bracelet – 'Legs and Macka', hearts entwined – slumps onto her hand.
_We-ell, bee-bop-a-loo-la,
She's my baby,
Bee-bop-a-loo-la
I don't mean maybe..._
Aunt Nina won't want us this Christmas. She didn't enjoy _Jailhouse Rock_ last year at the Scunthorpe Roxy, even though we explained the good bits to her and told her she didn't have to come if she didn't want to. She made us wear gloves. People must've thought we had something wrong with our hands. I expect this Christmas we'll be rearranging the weeds in our own front yard for the benefit of passing boys; that is, if the weeds are still standing. Like the rest of our house, they are in danger of collapse.
The knobs have fallen off the family stove. The knobs have also fallen off several of the inside doors, so when there's a draught, you can't get from one side of the house to the other without having to call someone on the far side to stick in the knob which you heard falling out of reach when the door slammed.
_'Ain nothin' but a houn' dawg,
Cryin' all the time.
'Ain never caught a rabbit,
An' you ain't no friend of mine._
And a healthy sprinkle of piano and drum makes the clothes that are sticking out shimmy in the open wardrobe, and sends a shower of lipsticks and false bosoms to the floor from the quaking dressing table. Allegra's swimsuit falls off her bed with a crash. This piece of apparatus, and another like it which belongs to me, is so stiffly under-girded that it stays facing forward whenever Allegra's skinny body turns sideways inside it. A hazardous garment, it cannot be lain down supine in for risk of baring what little there is to bare, or prone in for risk of cutting off the circulation.
The modern head of hair is a thing Aunt Nina desperately wants to put a comb through, but couldn't if she dared to try, since the top layer is bonded together with lacquer after the fashion of a shell, giving the lie to those foreign languages that insist on the hair being composed of separate entities. If you happen to be unfortunate enough to be caught in the rear by a gale you run the risk of being scalped.
The most prevalent cause of absenteeism from school is rock'n'roll injuries: feet speared by stiletto heels, backs twisted inside bustiers, rope burns to the knee region from stiffened petticoats, upper leg lacerations from snapped suspenders and permanent bruising and creasing of the waist from embedded hooks and belts.
Last week we decided to change our names. All the Janes at school are now called Jayne, the Sues are Peggy-Sue and the Margarets are Ann-Margaret, even Coral 'School' Mattress is called Cory now. I am known as Hell's Belle, but Allegra is, as she has always been at school, just Legs.
'Bugger,' she keeps on saying as gobful after gobful of Peter Stuyvesant fails to develop a central hole.
'You've got the wrong-shaped mouth.'
Her poor mouth is so strained with trying, it's sending her deaf. With one hand she is droopily netting scum off the goldfish bowl and flicking it out the window. 'You'd think he could buy her a new stove.'
'He' is up the back in his studio and so engrossed in his work and The Brolga Laurington he is most unlikely to notice the cigarette smoke billowing out our bedroom windows, which are stuck, through warping, neither up nor down.
'I mean, we must be the only family in Melbourne without a proper stove. And you'd think he could buy her a washing machine.'
I am tackling the slant on Buddy Holly's hornrims with my paintbrush. Our bedroom, though by no means as tidy as the Shrine of Remembrance, has become a place of pilgrimage. We entertain on a regular basis in here with ethereal rock'n'roll and contemplation of the Late Great, reconstructed on posters from record covers.
'He's class conscious,' I grunt. 'If you give in to middle-class urges like owning a washing machine that works and having somewhere to plug it in and a couple of taps to attach it to with actual water and electricity coming out the right holes, it's backsliding. And _I would not be a backslider/I'll tell you the reason why/Cause if I was a backslider, baby/ I wouldn't be ready to die._ And you gotta be ready to die, Legs. What would there be to sing about if everything ran like clockwork? He doesn't buy her a washing machine because he hasn't got any money.'
'He could borrow it.'
'There's his image to protect. You have to starve in a garret or at least stink in a shed for your art.'
'Why doesn't he get a job or something? I've got a job. If I can get a job, so can he.'
'What are you going to wear?'
'Oh God,' she sighs, smudging the remaining scum into the aquarium glass, 'she wants me to wear servants' clothes. I mean, servants' clothes at my own father's exhibition?'
The issue of servants' clothes is another reason Maggie Kelly has to stand on the bench at school – she hasn't paid for her sewing materials.
In First Form the girls are supposed to spend half the year sewing themselves a maid's uniform to wear in the second half, when they learn cooking. Bridget Kelly says it's a Hypocritical Practice left over from the old days when girls from our school and others like it were trained for domestic service. Bridget Kelly is proud that she is the only woman in the state of Victoria, probably in the whole of Australia, who is caretaker of a dump. Special regulations have had to be passed in the City Council just to describe her position. She is a Live-in Caretaker, Grade Two (female).
I join Allegra at the window, where she is using her reflection to metamorphose from a beatnik to a widgie by putting on white lipstick and teasing her hair. It being Saturday, the blind on the studio window is up and we can see inside. The Brolga Laurington is perched on one of Dadda's high stools, distributing a fortnight's gush and warble with her hands, even though it is only two days since she was last here. Mum was at work then.
'Do you reckon Dadda's a "sort"?' asks Allegra.
'Yeah.'
'She looks as though he's tickling her. Maybe he's got a feather tied to a piece of wire and he's running it up the insides of her legs. She's got a very stuck-up laugh. I don't think she's the sort who would climb down the social ladder gracefully, do you? I mean, she's probably "got an eye" for things that are stirring at the bottom of the social handbag and might keep her climbing upwards...'
'She'd better keep her claws off Dadda.'
'Duck! They're coming out!'
Under the windowsill we can hear the scraping of the studio door on the floor planks. I bop a ring through a ring through a ring.
I sit there with Allegra on the floor, the pair of us now dressed in black with black eyes and white lips, bopping smoke rings into our zoo of a room – not only do we have a moribund fish tank, but we also have a caged Peach Face called Leone in memory of ant-bitten Leone who has gone to London to seek her fortune, a white mouse endlessly fleeing in a rotating wheel, a grey cat called Silk and a pug called Botticelli.
During afternoon tea, our mother had dropped the creamcake she'd baked specially. It broke in two on the floor. She proceeded, while The Brolga Laurington trumpeted and trilled in controlled fashion, to pick it up again, put it back on the plate and dollop more cream goodnaturedly over the crack. Then The Brolga had repaired to the studio with Dadda and the removal of the last bits and pieces for the exhibition from the studio to the Daimler took place. And after that, the extended chat with the window wide open so our mother could see the entire action. Everything had an aura of Saturday innocence about it. Nasal phrases sailed through the air to the flicking of the silver bangles on the Laurington wrist. Dadda's voice was shady cool behind them.
'Old bitch,' says Legs from her possie in the past, 'ten bob for a whole night's work. It stinks.'
She has to hand out drinks and savouries.
'Girls...' Mum's head has appeared around the bedroom door. She sees us and gasps. Her eyes do a quick circuit of the room, her nostrils flare, she glares. 'Mrs Laurington is going now. Won't you come and say goodbye?'
Allegra sidles off her bed with a groan. Standing behind our mother in the next room, Checkie, who has grown tall and wears frocks, is trying to avoid touching anything in case something should come off and damage her.
The Brolga and Darling Henry are in the sitting room, all smiles.
'Such a talented family, darling,' The Brolga glubs and practically kisses our mother's face off. 'And Al-le-gra.' But Legs turns her face away just in time to deflect the kiss and leave me out of danger of copping it. I dash to the far side of the couch.
'Well, bye-bye, Allegra. See you tonight. Such shy little things, Stella, so pretty.'
'Say goodbye to Checkie, girls,' says our mother with narrowed eyes.
'Goodbye, Checkie,' we drawl in unison, so shy, so pretty, flicking bodgie chains around our fingers. Our mother can't wait to get her hands on us and wring our necks.
The poor old Midnight Knitter couldn't settle down because a man in shortie pyjamas came and stood outside her security door, and when she asked him, 'What are you doing there?' he said, 'What do you think?' The cheek! Proper men wear proper pyjamas, flannelette. And they put on a tie when they go visiting.
She is brewing coffee in the kitchen under the print of _Mad Meg._ Alongside her is a large, ornate Chinese lantern. Reg Sorby, the man who owns this house, made us a present of it months ago when we came here. One of his girlfriends left it behind. It's hard to imagine how you could 'leave behind' a lantern half the size of the average laundry, but there we are. Reg is not subtle, so there's no reason to suppose his girlfriends are.
He knows our story in the way a rich man knows anything, always requiring his opinion to prevail. For all that I like him and stand in his debt, he does not understand about Allegra or Mad Meg and I expect he never will. Mad Meg was our gallery. We bought it with our 'inheritance'. 'Why didn't you buy up art?' Reg wants to know. The fact is we had other ideas and other hopes, but men, and particularly rich men, need to have the final say; our ideas and our hopes are silverfish in the closets of their minds.
It would be comforting to be able to travel backwards through time and undo its kinks. But I wonder if an exposure of what was yet to come, issued by the fifteen-year-old Isobel in the sitting room where all the protagonists gathered after Dadda and Viva Laurington emerged from his studio, I wonder if the future had been foretold, even convincingly, whether postures would have changed. Would a condescending Brolga face have taken on a more sympathetic expression? Would a recognisably Mottean face, over which expressions traipsed in a continuous pantomime, buffoon to beauty queen, angel to arquebusier, have settled down? Would the dad's-eye blue eyes of the daughters have kept the glance arrows back from where they pinged off the bejewelled Brolga and pipped off Checkie Laurington, so that sight vibrations filled the room with opinion, and stances and attitudes were formed? Would the mystery man have stood forth and said enough is enough, instead of standing off to one side grinding his thumbs and letting his quiet gaze wander through the field of mothers and daughters all hostilely arrayed? I doubt it. There's a script written _sotto voce_ that we all obey.
FIVE
Family Trees
WHAT DID HE see in her? The closer we looked, the less we found. She seemed to us ugly, arrogant and condescending. 'She's intelligent,' our mother used to say. We wouldn't have granted her the grace of that adjective. To us, she was ruthless, hard, acquisitive, selfish. Necessary, some would have said, but not us.
The Harry Lauringtons were by no means the richest members of that august clan. By Laurington standards they were hardly rich at all. Harry's dead now, but there's a book about him. An only child, he was orphaned at fifteen when his parents drowned in a yachting tragedy. He was then brought up in his mother's sister's family. By the age of twenty-four, he had a few bits and pieces souvenired by his trustees from a heavily mortgaged estate. He had been to Oxford and had a degree in Art History. He was touring Europe when the war broke out, resulting in a swift departure for England from Trieste.
If his fellow Australians had a sinister impression of Harry because he was sophisticated, well educated, and dark-complexioned, this was intensified by an obvious physical handicap. At ten he had been the victim of infantile paralysis, resulting in a withered left arm. Perhaps to compensate for the defect, Harry dressed distinctively and smartly; there were such things in his wardrobe as three-piece suits. His casual clothing featured Scottish pullovers, hand-made shirts of Egyptian cotton, French and Italian shoes. There was plenty else he didn't wear until it became fashionable years after it had been bought: serapes from South America, ponchos from Mexico and caftans from India. Harry was always a man remarked upon.
When, in 1940 at the age of twenty-five, he was as a matter of course rejected for military service, Harry decided on a career no Laurington before him had taken. In among his fabulous selection of clothes were some black trousers with satin stripes down the sides, a blousey white shirt, black patent Italian pumps and a black water-weave satin cummerbund: the clothes of a _maître d'._ Harry Laurington became a restaurateur.
Siècle, as the restaurant was called, was in a _déclassé_ part of town. Arcaded windows gave out darkly onto a dual carriageway which was lower on the restaurant side than on the far side, the road being divided by a plantation that had seen better days. Root-bound palms laid their fronds over an unsightly mess of scraps blown into the plantation off the nearby beaches. Opposite, Harry Laurington had screwed coach lamps at intervals along his arcade of windows. From these, one assumed that the century referred to in the restaurant's name was the nineteenth.
Siècle scowled on the downhill side of a large car park, from which the patrons could enter beneath a covered way. Though the striped awning of the covered way was out of keeping with the brass coach lamps, it was in keeping with what Harry Laurington knew about Australians. The lure of a false front was everything. _Get them in with ersatz,_ could have been his motto, _then confront them with art._
In the early days of Siècle, it was cheap art he'd picked up on the quays of Paris on his trip to the continent before the war. But, slowly and shrewdly, Harry began to show Australians something that was happening in their midst. He began to hang the work of local painters.
Harry's father was a stockbroker. Typical of Melbourne gentry, he'd had a house in town and another at the beach. During the years of Harry's growing up, the town house had been lost, but the beach house remained as part of the inheritance. Though it was modest by family standards, and located on that drear bit of beach called Indented Head on the Bellarine Peninsula, the house itself was elegant. It was one of the earliest built on the coast. The high-pitched roof swept down over a verandah that ran along all sides, making it a great pity that the house, being on the flat land, had no view. Its garden, however, was planted with rare Australiana. _Eucalyptus rhodantha_ swung ungainly, carbuncled arms in a petrified dervish dance. Giant banksias, many-eyed, guarded the house from among serrated leaves. Grevilleas warded off evil by poking purple tongues through mouths of green.
The sea swelled in the shallows as if it were breathing in sleep. Low angles of light sought out the shapes in the interior; the hand of a woman had been there, a bride who had wanted everything in white. _Tatler_ and ballet programs lay on bedside tables in remembrance of a marriage; marriage had taken on the house with such conviction, Harry had had no wish to change it. He went there to be in the shrine of his parents' love.
But then there was the war, and the sea dreamt new realities. In February 1941 the HMAS _Sydney_ came home from the Mediterranean where she'd sunk an Italian cruiser and survived sixty bomb attacks. Harry felt restless and impotent. He knew several of the crew, but because of his handicap couldn't bring himself to join their celebrations. Then, at the end of November, all those young men were lost when the _Sydney_ was sunk by a German boat off the West Australian coast, and Harry was filled with a desperate malaise.
One morning in Swanston Street he teamed up with a young soldier who took him to an army canteen. The soldier, too, was full of unease. As they sat watching the Voluntary Aid Detachment girls trotting among the tables with trays and trolleys, living up to their nicknames, Virgins and Devils, the soldier began to draw them on the butcher's paper which served as a cloth. There was a lyrical compassion in his drawing. It was so unexpected, Harry wanted to know all about him and where his gift came from.
The soldier, Leslie Hallett, hailed from Yarra Valley, where his father was a timber feller. His mother had gone mad and thrown herself in front of a train when her only other child, a daughter, had run away from home. The father had put these calamities down to his own insufficiencies and began to drink heavily, with the result that Leslie left home at fifteen and came to Melbourne looking for his sister; that had been in 1935. He found where his sister had been living until a while before his arrival: it was a boarding house in St Kilda, run by a Jewish woman, a Mrs Hirsch.
Leslie liked Rose Hirsch immediately. She was a lively, bright-eyed little Frenchwoman in her twenties. While Australian snobs sailed on every available boat in order to kiss the hems of superior European traditions, Rose had sailed the other way, repelled by those same hems and nonplussed by the kissing of them.
She told Leslie his sister had gone to France. Leslie imagined she had gone with some man, but Rose assured him that wasn't the case. She had saved the money for her trip of her own accord, and was staying with Rose's relatives.
Rose, who had no children of her own, took Leslie under her wing and found him odd jobs among the Jews of Melbourne. When he wasn't working, she sat him down in her parlour and let him draw. The boy had a native talent with which Rose knew better than to interfere; she let it grow of its own accord, supplying only materials and the techniques for using them.
Leslie had stayed with Rose until he had been called up during 1941. Rose was broken-hearted and didn't want her 'so talented' boy to have to go and fight a war.
The evening of their encounter, Leslie took Harry Laurington to meet the inimitable Rose, whose boarding house was like no other. It was crammed with dolls and ancient toys, lace in the process of being tatted hung from pins on satin cushions. The tenants did their ablutions in a Bacchanalian mosaic cave and sidled their way to their rooms up steps and through corridors lined with wicker dolls' prams, miniature houses and rocking horses.
They found Rose dining on a clove of garlic and a tot of Pernod. She was weeping; down her sweet white porcelain face, generous blobs of water coursed a trail of mascara. She wore her thick black hair in Indian braids, and the tears splashed onto the white lace of her costume, which was more like a doll's dress than the dress of a modern woman. The Health Department had been round again and given her notice. Not enough toilets for the number of bottoms, it seemed. 'And zay say I am a fire hazard.' She contemplated this for a moment, biting her thumb and then flashed up, pummelled the table, and said, 'Zay're _right!'_
During the evening Rose impressed Harry with her indomitable vitality. On the surface gay and romantic, Rose had a deeper self that was shrewd and intelligent. Life for her was a system of transactions opening doors to richer, larger life. She was not rich in money and probably never would be, but her soul was luxuriant.
Until this meeting with Rose, Harry had thought Australia a confining place where the cultural milieu weighed down his hopes. The other people who had surrounded him took fright at the mention of a restaurant, but not so Rose. She put the blossoms on his dreams and made him decide to act. He would buy a suitable building, Rose's husband Laurent would be his cook. Rose, with her radiant manner, would help run the place.
Leslie Hallett found himself swept up in the idea of a place that would become the focus of new and fresh ideas, for Rose and Harry had talked of showing art, and having music and readings to liven the culturally undernourished palates of Australia. He abandoned himself to their dreaming and, though he was due back at his barracks by ten, stayed on and was still there talking in the early morning.
Before the sun was fully up, he found himself being taken on a car trip to Indented Head, where Harry had a mind to conceal him and foster the talent he had seen unfolding on butcher's paper in the canteen.
With Siècle Leslie Hallett's artistic career was born, and through Leslie Hallett, Harry's reputation as a dealer grew. By 1961, when Dadda first showed at Siècle, Leslie Hallett and his story were famous and Harry was known as the man who'd launched a great original talent.
Reg's roof leaks in several places so, in addition to the furious playing of rain, I can hear a smaller, more delicate polyphony of water dripping into buckets: each set of drips coming at regular intervals, but the intervals never coinciding. Reg says he can't be bothered to fix the flashing since he's always changing the roof line anyway.
There are photographs stuck on the noticeboard in the kitchen, newer ones on top of older ones, putting the assertive, jolly, latter-day Reg in the vanguard and leaving the radically slimmer youth to peek and peer through from the past. The kitchen is not closed off so much as partitioned by benches from the room with the walnut table and the log fire we need for warmth at this time of year. At the far end of the room there is a slow combustion stove, without which we would have to go about swathed in more clothing than would be comfortable. Reg looks out into the room, proud and amused. White curls scroll down over his jolly pink head. Embedded in the pinkness, a pair of sparkling blue eyes. He exudes well-being.
Round the corner, however, in a colder room where the fire is only lit for parties in winter – the room through which the Midnight Knitter has to come to brew her coffee – there is a self-portrait of a different Reg. This Reg is worried and downcast, his face in shadow.
Reg is a prolific painter. He might create twelve major works a year, thirty or forty smaller ones and innumerable drawings. The market for his work has always been strong and is firming. Painting is his life: all else is secondary. Or so he says.
After the war, he and his circle declared themselves against abstract art. To them it was iconoclastic, insidiously destroying the true language of art. This language resided in the image and the translation of the image into paint through the painter's imagination. They did not insist on an art derived from nature, but they did insist on images; dots and lines, however suave and elegant, were not enough, they could only ever be decoration.
Dadda was outside this circle. They dismissed him as a painter whose ideas were stuck in the past, allied to Dada and Surrealism. What they didn't see was the prophecy in his work, his discomfort with the prevailing world view, the products of which would come to threaten the earth with destruction. Dadda used to say that A + B might very well equal C, but C, once produced, could never again equal A + B.
Though Dadda painted images, he had nothing against abstraction. Objects of contemplation were all right by him, contemplation was never obsolete. He appreciated games and was given to them himself; there is optical and logical whimsy in many of his works.
While Reg and his circle stuck insistently with figuration, a younger generation, my contemporaries, rid itself of images, claiming that painting should not be interpreted by the artist for the viewer, but should respect the viewer's right to perceive and feel and be a part of the process of art making. If Reg regretted his anti-abstract stance, he ran in front of his regrets and staved them off in whatever way he could. He was never an intellectual. His work appeals to those who like intuitive, and, on the whole, narrative, painting. He is a sensualist; he doesn't like to have to stop and think. Thought might betray something in his art that has more to do with the nature of perception than the narrative power in what is perceived.
Though to all outward appearances Reg is sedate, I believe he is a driven man. No one around him goes wanting, and this is evidence of what drives him. He is a classic man, a provider, the quest of his life has been not so much to prove himself as a painter but to prove what a great provider he is. Providing is an obsession; he will take on any commission, no matter what, if he can see some way of making provision by it.
Reg, on the tail end of his middle years, has sat up covered in leeches in rainforests while bulldozers bore down on him; he has appeared as a tree at the Mardi Gras in Rio; he has cuddled wombats and prime ministers and painted tablecloths for charity; he has been up in a balloon and down in a diving bell. Pretty nearly all these things he has done with a glass of champagne in his hand. He is leaving his liver, one of nature's miracles, to a university medical school, along with the rest of him, ready pickled.
He has had five wives, but is without one at the moment. His ex-wives like him. Each year two or three of them get together and throw him a Christmas party. He supports the ex-wives in their ventures and is tolerant of the husbands, though they are not necessarily tolerant of him. He helps with the children whether they are his or someone else's. He helps with the grandchildren. If Reg had antlers, they would fill a banquet hall.
Reg paints large wild animals and naked women. He paints nature red in tooth and claw, and the pleasures of the flesh. Feminists attack him, critics leave him well alone. I have heard the word 'vulgar' spring to lips that shape his name, but he also paints images of war and violence, and there are times when he paints the hypocrisy, doubt, malice, greed or knowingness hidden in people's faces. His is a major talent which, in addition to his major works, has produced a plethora of minor ones.
Reg knows what it is to be afraid, but he does not so much race against time as roll against it. When his vision fails he takes another sip of champagne and plucks a woman from the tree of life. Yet when Reg looks back from his art into life, he knows he is alone, and that the curse of the artist is unique vision.
Reg is a collector as well as a painter, and among the paintings here are six by Leslie Hallett. Leslie Hallett died during the Second World War. Because he was undoubtedly a major talent, his work is as expensive as it is rare. The paintings here came up for auction during the incumbency of Reg's fourth wife, and she felt entitled to a share at the divorce. I understand there was quite a falling-out, but Reg, who is stubborn and lets go of nothing that matters, argued that the paintings were a set and had to be kept together. There was an out of court settlement and he bought the fourth wife off with a Jaguar.
I'd sooner have the Halletts than the Jag, and I'll bet the fourth wife would, too. It is not what they are worth; they are, quite simply, stunning paintings. They are not large, but fresh on the eye, bold and sweet in execution. They are paintings of the wartime VAD canteen. They date from 1941 to 1942, so, to paint them, he must have braved being caught by the military police. In one, a girl with smiling hazel eyes and dark foaming auburn hair is handing a soldier a plate of soup. There are not two faces like this one. It is a painting of my mother.
Think: the rumbling of my mother's tummy is an emanation that goes on forever through the universe as she swills the coffee flavour over her tastebuds. But the molecules are laying themselves down like logs across the access and passage to her memories. Not to remorse, though – the access and passage to remorse is not blocked up. Confession time. It's lucky I'm not far away from fifty, or these revelations would be as silver paper to a body made of tooth nerve.
_'That_ wide, Allegra,' she says to the absent daughter, chopping the air into a vicious three-foot width. She is snarling bitterly as she alludes to the windowsills at Clare, and their owners, who ought to have been rich if only history had done the proper thing by them. And now she is laying her story like a fat cat on the sill, perhaps in the hopes that some rich person will pick it up and stroke it. She is arch and soulful by turns.
Ideally, she says, she would have lived her life between the ages of just born and eleven, then none of this nonsense would have gone on: Italians and daughters and so forth. Princes would have given her sweets, duchesses would have cuddled her, Jane Austen would have written the house into immortality and set it in fields by Boris Pasternak. I ought to know what she means, whatever my name might happen to be, because she's sure I was there. She's seen my face before. It's got that chin.
I push the log further onto the fire and adjust her chair so she isn't exposed to draught. _Be kind to her,_ I tell myself; it's confession time and I suppose she thinks that all must out as she travels backward, backward, the memory doors thudding shut behind her as she goes.
Poor women, to have been so kind and yet so full of guilt! Euphrosyne, her mother, resented the timing of her birth. She has lived a contradiction ever since. Her name and face do not tell her story. Though her eyes danced her youth away, denying sadness, her face was a slope of snow before an avalanche. Sometimes the snow would shift, crack a little, an avalanche not allowed to happen. Smoothness covered the crevasses. The sweet exterior hid bitterness packing itself densely inside.
She had lost not only her brothers but also the man she was to have married, Geoffrey Latimer, her youngest brother's friend. Mottes worth their salt didn't cry. She didn't cry; instead, grief found its way into her movements, making her hopeless at getting tops off bottles or lids off cans. Keys wouldn't open doors for her, and all contraptions conspired to send her into frustrated rages.
From the roof of the bank where she'd been working in Sydney, as she'd watched one brother's regiment return, she'd allowed herself to look at the manager standing to attention by her side, flinchless. The soldiers marched by below, waving and smiling and she had felt herself at a cocktail party of the soul, safe in her dress and underclothes, her stocking ladderless on a roof in sunshine.
In her mind's eye there was Haydn, June 3rd, 1941, Crete, his lungs destroyed from the impact of a bomb going down the funnel of his ship; Shaver and Geoffrey last seen June 21st, 1941, Syria, dying of their wounds; and Hedley, February 14th, 1942, blown to pieces in a shell attack at the fall of Singapore. She knew about Hedley because someone had escaped and made his way home, helped for no other reason than kindness by ordinary people trying to live ordinary lives in the midst of other people's madness.
Sydney being the place where returning soldiers disembarked, it was also where Stella gazed out windows and saw phantom troop-carriers entering the Heads bringing phantom brothers: one tall and heroic, one strong and barrel-chested, the third a little fellow with a cowlick and a lopsided smile – Euphrosyne had always said that Shaver was born while he was looking the other way and had never caught up with his expression. And nothing was sadder to Stella than the arms of Sydney Harbour, yearning to gather in her lover: she used words like _yearn_ inside her head, but found she could not wrap four slaughters in such sentimental rugs. Thinking she would leave behind her torment, she came south in the May of '42 to the kinder city of Melbourne.
On the roof of the bank where she worked in Melbourne, she was taught how to fire a revolver, a skill she could put into practice in the event of a bank robbery. In the evenings and at weekends, she dispensed pathological goodness along with the soup at the Voluntary Aid Detachment canteen where Leslie Hallett must have seen and drawn her.
Wartime Melbourne lives on for her now in the chopping motion she makes with her hands. She has taken a flat in East Melbourne; the chop, with the hands parallel, is Clarendon Street, which lies between her chair and the bluestone chimney. She walks her fingers over it to the Fitzroy Gardens. Across the Fitzroy and Treasury gardens, indicated by two large, left-handed loops, while with her right hand, she holds Clarendon Street in place. It is a shortish walk into town to the bank each morning. In May 1943, when the fortunes of war have turned one way in Europe but the other in the Pacific, this is what the gardens look like: there has been rain and between the slim blades of well-mown grass, neatly illustrated in parallel lines where her fingernail greets the imagined picture plane, the ground is dark and wet. Underfoot the asphalt paths are deep bluey-grey and very cold.
Fog skulks in the tops of trees.
The puddles are still because nature isn't talking.
Green mottle on grey trunks; orange mottle on yellow.
To strike off to one side of the path would be to find it slippery and unsuitable for walking; water would certainly get into shoes and shoes would reach work with soak lines on them.
Work is situated in the city buildings, well hidden by trees. These, in the main, are elms. Elms turning yellow, guarding the paths, keeping back with strenuous arms the giant figs, the sorrowing willows, the globe-shaped Golden Elms, the shimmering poplars, the crisping oaks and plane trees, the pittosporum covered in variegated spear heads.
In the autumn light of 8 a.m., the trees look like assembled royalty. A spill of red down a golden gown. Filigree coronets on claret cushions. Parchment on plush.
Through the branches she glimpses the sky, bruised and broody. Even though the Japanese have been forced from the Solomon Islands, they have taken Timor and Java and are rumoured to be amassing a tremendous force to Australia's north. Nina's told her of women on the land who are keeping cyanide by in the event of an invasion. On Tuesday, Darwin was bombed and they won't say what the losses were, except that they were heavy. It is just three weeks since the papers finally admitted that the situation in Australia is desperate.
Stella feels tears in her eyes as she passes thick-waisted Diana with her swept-up bronze hairdo and hounds. She likes this statue for its bounding step, the lively contradiction between the form and the metal it is cast in. She has read somewhere that Diana is not just the Goddess of Hunting but also of Mood and Contemplation, and this too seems a contradiction. Diana will be lucky not to end up as a gun barrel.
Mist on the glass conservatory walls behind her, where there's an exhibition of tropical plants and, just quietly, it's hilarious how the anthurium lilies look like penises – all different sizes and colours. If Nina knew what went through one's head, she'd be scandalised.
The cold from the footpaths gets to her ankles, though Nina has supplied lisle stockings, several deniers of thickness. There's a sheen on her ankles seen to come and go under her nose and a coolness on her leg tops under her skirt, as well as a slight pinching sensation from the suspenders. Over her step-ins, her taffeta-lined skirt slips: left, right, left, creating a bit of electricity about her hips. Hat pinned to head: grey, with three magenta feathers sitting close. Leghorn feathers, dyed. Envelope handbag with two smart flaps, one grey, one white. Thin grey gloves. Left, right.
She is a smart, sweet, neat little woman thinking of her new fiancé. She doesn't know him very well, but what does that matter now? He is twenty-six. He has been in Australia about six years. The nationality stated on his passport is French, though he is Italian and spent the first nine years of his life in Milan. His father seems to have been a newspaper editor. He says he lived opposite a marvellous castle where he used to play in the moat. Poor people used to come to the moat to trap cats, then they seemed to kill and skin them and hang them in streams until they were ready to cook. Sounded a bit like what the shearers did with rabbits on Clare.
She is thinking also of her fiancé's uncle, Nicola Coretti, a charming, dapper old fellow, even if his name is that of a woman in English, a language he doesn't speak.
Before the war, her fiancé says, Nicola Coretti was the subject of an investigation by the Italian Consul-General in Australia. The Consul-General has now been detained, along with other consular staff, including the husband of Stella's friend Marietta, who she is sure has nothing to do with the investigation. After all, Marietta and Lamberto come from Parma, and that's in the north, just like Milan, so there's no reason Stella can think of why they wouldn't take to Zio Nicola. It's all just a little troubling, though, because the wedding's tomorrow and Marietta, who lives upstairs from Stella, is lending her flat for the reception. Marietta hasn't met Zio Nicola: Henry said it might be better not to mention him.
Marietta runs an Italian language school and is most amusing. She has said she will protect Stella when the Allies lose the war. Even though Stella has lost Geoffrey and all her brothers, she has not yet, in her heart, lost the war. After all, in February the Germans were being routed on the Russian front, and there were anti-German demonstrations going on all over Italy.
It was in February, as the Italian Consul-General was whiling away the hours 'until victory' in prison, that Stella encountered Marietta on the way to a pawn shop and bought from her a beautiful set of ceramic coffee cups, to spare her the embarrassment of hocking them. Stella had entered the bank at a trot, laughing at Marietta's pricelessness, the cups in a hatbox. It was her lunch break and she was a trifle late. The hatbox was decorated with travel stickers which said Milano, Venezia, Roma, Cremona, _eccetera_ , _eccetera_. She plunked it down on her desk, dashed over to the teller's window, whipped the Next Teller Please sign out from under the grille, where it had been keeping the hordes at bay on the counter, and there was a handsome man with light blue eyes in front of her, wanting to open a bank account.
'Coretti,' she repeated after hearing his name, and explained forthwith, as she attended to his banking, about the cups and the friend, the lover, the brothers and the unlost war because it was obviously an Italian she had before her, 'but you've got blue eyes, so you must be from the north,' she said as she stamped the last page of the new bankbook and handed it over in its jacket through the grille. 'Ciao,' she said, because she liked him and Marietta had said it's what you say.
Friday. Tomorrow, she is going to be married. To a younger man she met three months ago over the counter in the bank. Nina and Father are on their way from Clare. For better or worse. Motte cousins from Toorak and Brighton will be out in force, particularly on the distaff, as the staff is for the most part embroiled in the imbroglio. Richer or poorer. And the man she is going to marry is bringing his uncle. Whether or not it will be all right, Zio Nicola is the only relative he has in the country.
Tomorrow, provided the weather holds, she will be married here, in the Treasury Gardens, beside the ornamental pond, among the pink camellias and the palms, behind which the white Treasury Buildings are as close to a palace as Melbourne affords. If you exclude Government House. But you do exclude it because if the Treasury Gardens are damp, the Botanic Gardens around Government House are even damper. Nobody else she knows has been married out of doors by a registrar and not a minister. When Nina said why not a church, Stella said, _'God?_ What's God got to do with anything?' 'In the sight of,' said Nina. 'Bunny, we are gathered in the sight of.'
But for Nina's information, there's something she'd better get straight, the man's an Italian. He's asked, he's here, there are children to be born to replace the men who've died. You'd marry an Australian, but your experience of Australians is they don't last. Twenty thousand taken at Singapore; it doesn't bear thinking about. Henry wants to stay here, so you're marrying him. Anyway, who knows how long 'here' will last? If Nina doesn't like it she needn't come.
She doesn't like it, but she's coming all the same. From behind the clump of bushes she is coming, hurtling along in a splendid temper. Green hat. Green suit. Green crocodile handbag. Green crocodile shoes. Storms across the lawn towards the elm-lined bitumen – you can see her now as if it were yesterday. You can see her from the flat where you're about to put on your wedding dress, because it's tomorrow already, the day you're getting married.
It takes her seven minutes fast and furious walking from the moment you first see her to the moment she rat-a-tat-tats on the door and you get your friend Marietta, who has lent her flat for the occasion, to go and greet her. Nina makes you nervous, but you've decided she's not going to get the better of you today.
She pulls herself up to her majestic five feet five (easily two inches taller than you) and from under the Garbo brim of her hat announces, 'You must call it off at once.'
'Oh, Neen,' you say.
'I can't have my brothers' sister...'
'Now, Neen. Fair go, old girl.'
'My husband is a war hero.'
'Your second husband.'
'Perhaps I...?' Marietta intervenes. 'Now I, Mrs Furlonger, am an Italian. It is one of the oldest and richest cultures in the world. Think of the Roman Empire.'
'What about the British Empire?' goes Nina.
'War puts the stamp of nobility on those who have the courage to meet it,' thus Marietta. It sounds like something Nina ought to believe. 'The English have lost their glorious past. They have made a religion out of eating and games. When the war is lost, who will protect you? You have been abandoned.'
'The Allies have taken Tripoli,' Nina sniffs.
'The Japanese will take Australia.'
'I'm getting married, Neen, and that's that.'
'To a painter,' goes the fervent Marietta, 'to an Italian painter! Italians are the best painters in the world. Think of Da Vinci, Mrs Furlonger. Think of Raphael.'
Nina thinks she is thinking of these, though in fact she's thinking of Michelangelo and of the Americans on whom she is pinning her hopes, and Stella, too, is thinking of Michelangelo, though she thinks she's thinking of Da Vinci and Raphael and all this thinking is making her very tired and causing her to see things.
Weddings.
Brides divide to reveal other brides.
Everywhere you look, there are leaves falling and people marrying. Stella's mind has travelled half a century and she sees attendants at Italian weddings in the Treasury Gardens in the 1990s wearing red, Yugoslav attendants in cream, attendants from Essendon in apricot, from Hong Kong in mauve and Mauritius in taupe. Everyone gets married in the gardens now and no one makes a fuss about it. Brides, brides, the air is the shifting of their veils. The leaves fall yellow and the leaves fall red. The cameras click. Many of them are Japanese cameras with Japanese film in them taking photos of Australian weddings. The leaves fall. The wedding parties gabber and jabber and chatter and talk in tongues.
Things looked bad the week my parents married. The Germans were fighting back savagely in North Africa and the Luftwaffe was active. The Japanese were in Java and were bombing Darwin aerodrome; the day of the wedding, they sank five allied merchant ships in Australian waters. Leslie Hallett painted a picture that weekend called _Glory,_ in which a little Australian soldier is drowning in a great green sea. When it was finished he walked into the sea at Indented Head and kept on walking. His body was washed up five days later.
If Leslie Hallett caught her eye, it's something that slipped through the crevices of my mother's memory long ago. He painted her not as she was, but as she appeared, with her pretty rounded brow, the soldier's friend. The brushwork broke with tradition, laying itself down on the canvas with a simplicity pure in its core. Two months before he died, this painting was part of Siècle's first big show. If Reg Sorby had had the money then, he tells me, he would have bought the entire exhibition. But Reg was hardly more than a boy, taken along by his mother to whet his appetite. He was seventeen. Three years later he was shooting to kill, and the hatred of it never left him.
The Hallett show at Siècle was where Reg met Dadda, and Dadda, he tells me, was Harry Laurington's rival for the hand of Leslie Hallett's sister, who'd come home from France before the war broke out, bringing Dadda with her.
Hallett. A name made famous by the young painter who suicided at Indented Head on the Bellarine Peninsula in May 1943. His sister would capitalise on it some years later.
The Midnight Knitter wakes up again.
She's at Marietta's flat now, in a wedding frock. White, modest wartime pattern. Nina, having conquered her initial repugnance, is now in a tizz because a mouse got in under the icing of the cake they brought from Clare and perished, drunk with rum, under the fondant. Oh well, at least it was drunk. Stella doesn't think the guests will notice if they slice off the block of cake containing the mouse. They'll pretend it's being set aside for poor children.
The future Granpa is seated on a sofa grinding his teeth and just able to see through foggy irises the source of sperm that will cause him to gain a name in ten months' time. The soon-to-be-grandparental shoes are exceptionally shiny and the hose on the well-turned ankles matching. Midway between the manly spread of the rheumatic knees a handsome, no-nonsense bamboo cane is recording the pulse rate of a heart that is ticking off the seconds between now and doomsday when at last that calculator, History, will be called to account.
The source of sperm is lounging on a cocktail cabinet in an attitude at once indolent and elegant. The legs are long, the hands unavailable for scrutiny, being buried in the trouser pockets. There is a paisley waistcoat being worn beneath a dark jacket with satin lapels. The trousers do not match the jacket. They have red cords down the side seams and look for all the world like a Salvation Army bandleader's pants. It is amazing what you can see through foggy irises when hungering for detail. The source of sperm is making quite a lot of roguish noise to cover deficiencies in the occasion. He is laughing with the bride and the unlikely matron of honour.
The future Granpa casts around for consolation. He thinks of the joining of certain bulls with certain cows and decides that the source of sperm, being sound of limb, could well have been a Hereford/ Hereford cross, his favourite. There's a bit of height there, a decent shoulder girdle. Good brow, fair teeth. He'd like to put his ear to the rib cage to check the heart and lungs, but the voice, at least, has the kind of timbre a man would expect in a better class of beast. You would probably not sink this particular cut of Italian in the same boat as you'd sink the Hun, given half a chance.
The senior sister is restraining her gaiety in fitting fashion. As a person who has blundered openly at least once at marriage, it is perhaps not hers to do more than address her thoughts to the Almighty and dispose of the mouse.
There is a knock at the door of the flat. The hostess, Marietta, wife of the interned consular official, Lamberto, opens the door. 'Oh,' she says, and her formerly smiling mouth becomes a line of chagrin. She does not know the name of the old gentleman standing in the doorway, but she has seen him before and heard him spoken of by the Melbourne fascist elite, the Sansepolchristi, in less than glowing terms. He doffs his grey top hat. Full wedding regalia, white carnation in the button hole of his frock coat.
'Zio Nicola,' says the groom, with a fond cadence in his voice. There is general embracing. Marietta retires surreptitiously to her bedroom, takes a little book from the bedside table and notes down the names of everyone present.
Meanwhile, Mottes have gathered in the Treasury Gardens, where the registrar is practising among the shrubs, pacing up and down, mouthing the words, the book behind his back. Mr Southwell. Every few paces he turns on his heel, licks his fingers and plasters down long strands of hair across his bald pate.
Bunny Motte, whose great-grandfathers assisted at the circumambulation of the continent, perished of thirst surveying the Ord, advised their daughters not to marry madmen and invented a medicinal powder which would cure everything if taken under medical supervision – Bunny Motte is about to foul the line. They wouldn't have missed the occasion for the world. Gilchrists, Bloomfields, Beauchamps and Taylors are also in attendance. They mutter to each other that the chappie, though Italian by birth, is a French national. He is a painter and there are those among them who hope, on the quiet, that Bun hasn't been taking her clothes off in draughty places.
The leaves fall red and the leaves fall yellow. The bride and the groom, the sister and the friend arrive in a taxi together. Most unusual. The two old men are in a taxi just behind. Tap, strut, tap, strut, they follow the bridal party over the lawn. Does she take this fellow for her wedded what-not? Yes, she does. And why not? Her lover and her brothers are dead. The leaves fall yellow and the leaves fall red.
Afterwards, there was quite a lot of working out to do at the reception. Who was this Marietta character? A pseudo-Frenchman was all very well, but a real Italian at such a time, just quietly, was a bit much. No wonder old Bob Motte wasn't in the best of humours. And who was the old fellow in fancy dress?
As for Bun, what a scream – handing round the sandwiches at her own wedding, saying, 'The ones without the parsley are the ones without the poison.'
The books beside me here on the table tell a story, but it isn't that one. They only talk of lamentable stresses and strains. They say that Dadda's exhibition, called 'Where the Nice Girls Live', the one that Viva Laurington mounted at Siècle in 1961, marked the start of the meteoric rise in his reputation and, less importantly, the failure of the highly irregular marriage contracted in the Treasury Gardens in May 1943, the purpose of which was to replace a lover and three brothers who had fallen from a devastated family tree, and to give a displaced person half a chance.
SIX
The Nice Girls
ALLEGRA HAS SHOVED so much stuff on her face she practically had to gum it on to make it stay there. She has teased her hair till not a single strand of it could come adrift in a hurricane. Her fingernails are painted green to go with her eye make-up. She is wearing fishnet stockings and our mother is furious. There are times when our mother would rather we wore party frocks and built-up shoes, and this is one of them. You never know who you will run into from Rumpton and Rudge's at this sort of do. Our mother would have invited numerous people to run into, except that Viva Laurington stayed her hand, saying that Siècle had its own guest list and Stella mustn't put herself to so much trouble. Our mother has had her nose put out of joint.
From the car park where Mum, Dadda and I are still debating whether or not to go in, Siècle looks like a low, malicious grin with the candy stick of the covered way protruding from it. I am wondering what it is like inside, where Allegra is already, playing it cool.
There wasn't time to obey our mother's command and wash the muck off her chops before Mrs Laurington called for her. The Brolga didn't turn a hair, just said, 'What a clever little thing.'
'You and Bel go in, Henry. I'll wait out here.' Mum is not only angry about the guest list and Allegra's make-up, she is also angry with Dadda because he is wearing a very loud tie with a dark blue shirt and a crumpled brown jacket. Allegra made him the tie. It has flying saucers and rocket ships embroidered on it and, across the bottom, she has written Ka-Boom! in stick-on glitter gold. She and I think it's very good, but Mum is furious because we spoilt a perfectly nice tie she'd helped Allegra choose from the place where Rumpton buys his shirts.
Dadda doesn't say anything, but he is frowning. He frowns as if all history were written on his face and had occurred for no other reason than to arrive at the moment of this impasse.
'No. It's your night. I'd only muck things up for you.'
'Come on, Mum,' I say to her, but she just whistles a tune and taps her fingernails on the dashboard.
Dadda sighs. At length, he opens his door. 'Come on,' he murmurs to me. He doesn't take my hand in his growling palm as he used to, but shoves me in the back and pushes me along in front of him.
'Shake a leg, Isobel,' he says as we pass under the awning. I am shaking a leg. I'm shaking both of them and his fingers twitching in the middle of my back are really irritating me. He's like a gladiator, using me as his shield. We climb a little ramp to the foyer of the gallery-restaurant.
Checkie Laurington is leering at us from the receptionist's desk. 'It's an absolutely wonderful show, Henri,' she croons.
I hate her! She is nineteen years old and you can see her nipples through her skivvie. Dadda smiles flickeringly and forces me forward under The Brolga's smooching beak. 'Where's Stella?' The Brolga warbles. 'Nothing wrong is there, Henri?' She doesn't wait for an answer, but says, 'I'd like you to meet...' Gobble gobble gobble gobble and gobble, it sounds like to me. If you want to be upper middle class in this country, you have to develop a dewlap and wear your biscuits firmly poised between index finger and thumb just a bite's length from your cherry-red lips. Similarly, you have to wear your drink cheek high and trot like a show pony all around the VIP. The gush is a thing I must practise as it is not taught at our school.
This first part of the restaurant hasn't any tables and chairs in it tonight. They are all arranged in the second part, down a small flight of stairs, where there is a little parquet dance floor and a stage for a jazz band. The jazz band players are not here yet and the baby grand piano is closed. The lighting down there is dim, while up here, where the paintings are hung, it's brighter, though very smoky.
Allegra is slouching around crankily and shoving trays in people's faces as if she were carrying a circular saw and about to slice off their heads. Everyone's noticed her fishnet stockings, particularly as one of the seams is crooked.
'G'day,' she says to me blackly. She thumps a platter down on a table in a corner and sloshes some claret from a flagon into a glass for me, 'Get this inside you. Where's Mum?'
'Out in the car. Won't come in.'
'Don't blame her,' says Allegra and yawks her mouth down in the corners. 'What about this lot, eh? There'd be about as much genuine feeling here as there'd be on a numb budgie.'
We sock down a couple of clarets each and she tells me that earlier on Checkie Laurington refused admittance to Bridget and Maggie Kelly. 'I told her they were Dadda's friends,' says Allegra (she's a bit tipsy), 'and she just said, "Ew, well, thet's regrettible, but I'm afred we corn't allow people in without an inviteetion".' The Melbourne private schoolgirls' accent is one of the most abominable ever foisted on the human race and we are glad we have good, ordinary, four-square voices in which we can express the full range of our passions instead of being stranded somewhere between the genuine and the fake, so worried about the sounds that are going to come out of our mouths that we can't decide whether we're throwing a tantrum or accepting a knighthood.
'Gotta go,' she says at length, and swings the tray up onto her shoulder. She weaves away through the fumes and the crawlers, crooning sensually, 'Chartreuse? Anisette? Cognac?' which are not the drinks on her tray, but the opening lines of a play by Pirandello.
For a man who collects rubbish, Dadda can be surprisingly neat. This exhibition is called Where the Nice Girls Live: Paintings and Drawings by Henry Coretti. Aunt Nina would be disappointed as there are no nudes.
In his paintings, Dadda sometimes uses little bits of machinery, like the wheels from inside clocks or little cast metal blocks from gutted engines and radios _. O the Mind, Mind_ is the inside of a neatly dissected female head in profile, filled with gadgets and pulleys and miniature light globes making a series which coils from the shaft of the nostril round the inside of the skull in a snail shell till it reaches the centre. There Dadda has put a doll's house stove with its door off and a turkey in its oven. It is stuck onto an oblong of gold leaf, surrounded by exquisite squiggles in egg tempera such as you would find in an illuminated manuscript. I know he uses egg tempera sometimes, because it goes bad and the smell drives us insane. The rest of the woman is sitting up whole and clothed. The Dunlopillo of her armchair overlays springs which give in the correct fashion, demonstrated by arrows.
The notes to the exhibition read: _'Where the Nice Girls Live, Numbers 1_ and _2_ make a pair of paintings in which the indentation of real tyre marks has been used to great effect. _Number 1_ depicts the aftermath of a hooligan "wheelie" over the gutter of a suburban street, recognisably outside Coretti's own home. _Number 2_ is also painted in Coretti's neighbourhood and shows the lower margins of a lavatory block in a local park. Here the tyre marks make an arc in front of a broken sapling, which is eerily casting the shadow of a mature tree on the wall behind it.'
The notes are signed at the bottom, Viva H. and Cecilia V. Laurington. Cecilia V., now in the first year of a Fine Arts degree at the University of Melbourne, is currently trying to hold a conversation in Italian with Dadda. She's obviously practised it beforehand and Dadda is going, 'O-oh', in that sing-song way he has, his blue eyes moving about wickedly under his dancing brows. 'Very goo-ood, Checkie,' he says.
I've just about had enough.
Allegra has slung herself into a dark spot where the gallery stairs go down to the dance floor. She is emitting gobfuls of smoke and breathing very audibly behind them. People swan past gooing, gushing, kissing and congratulating. I squeeze in beside her.
'Giss a puff.' She hands me the weed. I bop out seven rings straight. 'I'm going to see how Mum is,' I say to her.
'Don't let her come in.'
She won't be there anyhow. She will have gone home. She will have wrecked her shoes with angry walking.
From the covered way I can see that our mother isn't in the car. She will have walked down to the main road and caught the tram home. She might still be waiting at the tram stop. But I hope not because, if she is, she will ignore me. She will go and sit up one end of the tram, facing the other way; I will go and sit up the other end, watching her. She will get off a stop after the one I get off at and be home earlier than I am and out of sight by the time I get there. I will lie down fully clothed in the dark in my unmade bed and stay awake till all of them are in, each making a separate set of sounds.
Someone has to look after them.
I like riding on trams up the end, in the dark, the loneness of it, watching my mother's back up the other end, hating to be watched.
She stares rigidly back at herself from the glass over the advertisements on the driver's cabin.
If I were to have feelings tonight, what a rag-bag collection they would make; they would be helter-skelter all over me with little sharp teeth digging into the veins that run over bones, their eyes twisting in their rotten little sockets. Just as well I am drunk.
I like this hurtling round on a tram at night and the steel wheels on the steel tracks turning time into abracadabra. Ding! My stop. Glass on the road. One street light illuminating one tree and the rest just black forms with wind in them. See. My mother's back rattles away over the rise.
The clouds have coalesced to a mountainous thickness and the moon is being crushed in them. Enormous rain is falling down slowly, like dinner plates. I like this back alley bluestone street into which the trees come over the fences like thieves with a foot caught in a backyard crime. Cartons and household waste are piled up here and sludging into the stormwater channel, beginning to rush away now as the rain comes pelting. At the end of the alley, where the lane leads off to our street, a tree is churning violently against a light pole heaped with rubbish at the base. There's a bat slung dead in the wires, been there for weeks, and the rain is plummeting down so violently now it bounces back off the stones for several feet.
I run. As I reach the alley's end, the rubbish at the base of the light pole reels up before me. Say, little girl, will you be my partner? And it's do-si-do and sashay with him with your left hand...
Numb nose and numb ears. In all this waiting for you I have polished off a whole bottle of white on my own. Frontignac. Bridget Kelly used to call it 'Frontie-yack', so my mother always calls it 'Frontie'.
I have been told the story of my parents' marriage break-up several times by different people in different places. In that beach house where Leslie Hallett sheltered during the war, for instance. By people who knew or thought they knew or thought I ought to know or thought the telling would erase the antipathy I would feel for Viva Laurington, née Hallett, for many, many years.
SEVEN
Correspondence
IN THE GATHERING dark, my ears strain for the sound of a car on the lonely road. Lucky to hear it over the din of rain on the roof. Cars, cars are crossing Australia and the greater world in all directions. A destination for every car, and for every destination, a departure.
When someone leaves, the ones left behind loll over the edge of the emptied space and speculate. They gauge its depth, they think about whether to fill it in, they imagine filling it in and try to see themselves doing so through various sets of eyes. For instance, would it trouble Aunt Nina if we filled Dadda's space in? Probably not in the slightest. Would it trouble Mum? No doubt – but, being one of the left-behind, she had to be included among the speculators.
We sat around slackly for a while, curling our lips, then Allegra said she thought it was time I had a studio, because after all I'd begun my art course by the time Dadda disappeared. Our mother was dubious. 'His things are in there,' she said.
'We'll just move them so there's room for Bel. Come on, Mum, we'll clean it out together.'
She was reluctant, but we talked her into it, a joint exorcism; it would mean more room in the house.
The green door creaks open. Three people on the doorstep. Allegra dressed in a cleaning tutu, pink net, long white tights with holes in them, brand new pink satin ballet slippers without toe blocks. 'Where did you get those?' asks Stella, greying auburn fuzz sprouting from the leg holes of a pair of knickers worn on her head. She means the ballet slippers: we all know the tutu comes off the back of a St Vincent de Paul truck; Bridget Kelly sorts for the Paulines on Wednesdays and bears us in mind. 'Leone,' Allegra explains. Leone? Leone hasn't been heard of for eight or nine years. 'She gave them to me. They were meant for Isobel to make a house in, like the one she made in Dadda's shoe. But I knew she wouldn't make one for Leone, so I kept them.' Allegra is standing in third position, mop rampant. I have the pan and broom, top hat and tails.
We peer in. It is dark inside, but a warm dark, the dark of an afternoon nap in Uncle Garth's den at Clare with the brown velvet curtains drawn, fingers of sunlight drumming on their backs. Stella and Allegra stand each side of the doorway and peer at me peering in.
Helped up the two wooden steps by their eyes, I go first with my folded easel and my paint box. They wait outside.
The studio smells of oils and Dadda. Along the neighbours' side, his work bench with the shadow of the window transom starkly on it. One old wooden easel. He's taken the good ones with him. Mine is a small metal one. I spread its rubber-tipped tripod and stand it beside his. Father and daughter. Daughter doesn't quite fit, so she goes on the map drawers he's left behind him. I open the third drawer to see if he's left his spoon collection. He has: plastic, silverplate and wood. In another drawer, old jewellery. He has taken his collection of gutted clocks that used to be in the top drawer, though.
I'm overwhelmed by the time that's passed since I was last here. As I sit down, the snoring of the dump sirens trembles by my ear. They are sleeping on the bench, lounging on the couch and chair he kept beneath the window facing our room, near his sink. Genoa cut brown velvet stuffed with horse hair. Apple green blind. Eight years without me setting eyes on them.
Sun-drenched Modigliani women snooze under the sills in the places where he might have put them, postage stamp sized, in a painting: minute, but full of allure. Oh, though they are not here, they warm the dark, they take the solitariness from the shadow of the window transom on the benchtop.
I hear the mop being leant against the wall and Stella murmuring. Keep cathedrals, grazed knees in the Lady Chapel, Scunthorpe Anglicans. We should lock this place with its stilled ghosts, as it is.
We should lock the last expression on his face in here. _Don't be like that, Isobel,_ his face said.
But what way was I meant to be? Joyful? Carefree? Do-as-you-like-Dadda-it's-all-the-same-to-me? It isn't all the same to me.
I grab my stupid little easel off the map drawers, ready to charge out, but something comes thudding down on my foot. Something light and black, fuzzed by tears, sounding and smelling like a... shoe.
He has left me a shoe.
I pick it up and rub the dust off it with the cuff of my dress shirt. It takes my hand in my father's foothold.
By me on his bench, a little sharp knife. I pull the tongue out through the laceless upper and slit it so it sits upright. On this upper I shall build my church.
Soon my tears are the wine of consecration. They fall into the heel where his heel was as I carve my cross. In the beginning was the word and the word was: _Here. Now. With me. Always._
I carve my way through the smell and feel of him, his foot my tutor. Quietly, Allegra is beside me. She rests her head next to mine and I feel the cushion of her frothy hair. She strokes my back and kisses the tear on my cheek. She has brought me a sherry in a peanut butter jar.
She knows that when I lift myself from my work my life becomes an unrelieved absence, a gone-for-goodness. A tyranny of love.
Quietly she moves about behind me, flicking dust with her duster-wand. She is doing her cleaner's dance as I carve. Every so often the shuffle of pink taffeta feet. Her wraith spins along the wall, flits over the bench. She is weaving a spell of peace around me.
He had been away six months before we began to close his tracks over. He was on our house like a system of wounds, everywhere: in the gable over the front verandah, inset with stone and bottle-top eyes from his trade with Bridget Kelly; in his bell and miniature drum collection, which visitors shook to rouse us at the front door; in the passage and the parlour, his nooks and crannies, assemblages of saints and martyrs pinched from Catholic churches; in the neglected woodwork and uselessness of all contraptions. He was even in the toilet with his photo of D'Annunzio's bedroom. For a long time we held his place.
Then he rang us on a dodgy line from London. We were to understand he loved us. Whatever he did, we were to remember that. He would like us to write to him and would we please redirect his letters to a London _poste restante_? His tone was apologetic and we could put The Brolga's voice to every word he said.
At our end, indignation – at his, play-acting, pretending to be the victim of circumstances. Allegra and I tumbled the receiver back and forth between us as if it had been scorching, while Mum took her fury out on doors. She slammed, she banged, she battered with her fists in frustration. He could hear her twelve thousand miles away in England. 'What's that noise?' he asked, though he knew full well what it was, having heard it before. 'I think there's something wrong with the line.' Perhaps he was enjoying himself. We suspected he was, both of us grabbing at our stomachs as though we'd been kicked. Our faces reddened, our heads sang, we could no longer hear what he was saying. We had lost his protection, if indeed we had ever had it. To score with Dadda, we would have to become tough, rich and shallow.
But we didn't ask for money. We were too slow. The Motte streak in us made it too difficult. He paused at the logical spot where we should have asked, but we didn't name our sum. So he made an unspecified offer, which, when it came, was not enough.
Our scholarships didn't run to allowances. Identical letters came from the Department of Education. _Dear Miss Coretti,_ they said, _respecting your recent correspondence with our office, living allowance is payable on a Commonwealth Scholarship when the holder is female, single and under 21 only where the holder can demonstrate three years' continuous full time employment, is living away from home, or the family breadwinner's income is below that specified on the attached schedule. As it appears none of these conditions applies to you, living allowance will not be paid._
We thought of talking our mother into taking divorce proceedings, but knew we'd be in more trouble with her than it was worth. She would accuse us of mercenary motives and would feel we were suggesting that she had failed.
We knew you had to wait two years for a divorce on the grounds of desertion. If Dadda set foot over the threshold in that time, a suit would fail. To prove adultery, you had to force an admission out of the adulterer or catch him, preferably photograph him, in the act and name the other party. Technically that was beyond us, as Dadda and his paramour were so far away.
So nothing was done.
What did he see in Viva Laurington that we had to suffer for her existence in this way? The more we thought about her, the worse we found her to be. We concentrated on hating her, hatched plans for her murder in our spare moments. It didn't occur to us to take our feelings out on Dadda. What was Dadda but putty in a wicked witch's hands?
We were not the only ones to be left, of course, but we had no fellow feeling for Harry and Checkie Laurington.
Harry Laurington took the blow by shifting Siècle from its _déclassé_ restaurant setting to something cooler, a disused bond store just two streets from the bluestone brontosaur. There he hung art, pumped a cappuccino machine with his good arm and allowed the young to recite new poems, perform dance routines and sing their heavy-laden, guitar-strumming songs.
The opening of the new Siècle coincided with Checkie's twenty-first birthday. Rumour had it that incense was burnt and a dancer from the Alvin Aley Company of New York performed for her delectation.
We suspected that Checkie Laurington, being made of a different metal, didn't feel as we did. We were curious to know, all the same. We pried at edges and lifted lids and found out from those people who attend life's dramas that, in her mother's absence, Checkie was being groomed to run Siècle. Remote and unassailable as a character in a novel, she was being inducted into a social rite from which we, two willy-willies of emotion, would be forever banned.
Love. There's nothing rational about it. A lifetime isn't long enough even to know oneself, let alone another person, so why is a year, a month, a day, deemed long enough acquaintance by a woman to allow a man access to her body? If it weren't that this silent script were evoked by circumstance, what woman in her right mind would lie down under a man and let him have her? Without the script, which is virtually saying _Go Mad,_ lovemaking would be an appalling pastime.
Love began for me when Rumpton turned sixty and decided to play golf three days a week. He left it up to Rudge, the junior partner, to choose his eventual successor. Alive to alliteration, Rudge chose a young man of twenty-six called Russell. Arnie Russell. A young married man of twenty-six, father of two with a third on the way. Wonderful handicap.
Rumpton, Rudge and Russell, one day to be Rudge and Russell. What could be better? Rumpton the heron, Rudge the late ape and Russell the great white whale. Blue eyes. Not light and luscious like Dadda's, but the product of a blue-blue cross, 'the gold standard of blue' Rudge called it in doctor's parlance.
In the latter half of 1964, after Dadda had been gone nine months and a brass plaque bearing the name Arnold F. Russell had been added to those of Rumpton and Rudge at the entrance to the stairwell, the _petit fours_ were got up for afternoon tea by Rudge's daughters, Rosemary and Sue, whose eyes, close set under heavily boned brows, did not stir Rumpton in the way Allegra's and mine had when we were their age.
Because Rosemary and Sue could not be relied upon to choose suitable fare, Stella would ring down to the cake shop and order beforehand. Stella considered the Rudge girls talentless and unexceptional and had given Allegra and me glamorous press prior to our meeting Arnie.
One afternoon when Marjorie Rudge and daughters were paying a visit, I trudged onto the aspidistra'd verandah, my folio on the point of spilling on the floor. I launched myself into one of Rudge's rattan chairs to wait for my mother, who was busy with a dictaphone at the new electric typewriter. Above the buzzing and clackety-clack, Marjorie Rudge was being nasal and prolonged to some friend on the phone. Like a toy tiger, she rubbed herself up the lintel of the door separating the office from the verandah. It was _petit four_ time and her daughters had been dispatched. She regarded me nonchalantly as she twanged along, slinging words carelessly from her red-painted mouth. Her eyes, mauve-lidded, were at half-cock and the sun kept catching her pink woollen bosom as it lurched to her breathing, in and out of her fake fur coat. I'd never seen her looking wanton before: it came as a surprise. My drawings slewed onto the floor. I began to shuffle them back into the folio when a pair of exceptionally well shined shoes with a considerable weight being borne on the outsides of the heels slid into view like the water police into a mooring. Jaunty flagging of the trouser bottoms, navy blue, betokened something tall and built, as the eye travelled upwards, along the lines of Chesty Bond. Kindly smirk, Gothically arched white eyebrows over slightly drooping blue-blue eyes. Gold standard blue. A long, long way from the floor. And then much closer as Arnie squatted to examine, as it happened, pencil sketches of the female nude.
He said, 'My wife draws,' three words which told me that though his wife drew, she didn't draw professionally, or as well as I did. They also suggested that if he hadn't already had a wife, he would have considered me a candidate for the post. At eighteen, being new to the ways of love, one sees oneself saying _I do, I do_ half-a-dozen times a day to the likes of Arnie, because to Arnie's shape and status adheres the pulp cycle marriage myth, happily-ever-afterness. All that had to happen was for his wife, Marjorie, to die.
Only her name was Margaret and, interestingly, it wasn't Arnie who let that cat out of the bag. His strategy of seduction involved naming his children, but keeping the post of wife speculative for as long as he was able. This was throughout afternoon tea, during which his eyes never strayed from my person. To the self-satisfaction in his voice, I ascribed kindliness and doctorly concern. It is true that Arnie was a funny man, even a parody of himself, self-consciously glamorous, wishful in his conventionality, unreliably reliable. With the collusion of Marjorie Rudge and Stella Coretti, he painted himself as a provident father and a good husband, a man who in other circumstances would have been ideal for me.
Marjorie Rudge, who knew a vibe when she saw one, said, when he'd disappeared into his consulting room, that he'd left a trail of carnage behind him. 'A _trail_ of carnage! But he'll never leave Marg.'
I cannot say I wasn't warned. But who would believe Marjorie Rudge? Stella called her 'Ersatz Personified'. How could the fake ever be real, and therefore capable of uttering the truth?
When, after a respectable lapse of time, my mother told me she thought I had 'an admirer', her tone was complicit, even encouraging. Perhaps she wanted to live a love affair with Arnie through me. Soon the focal point of her days became my afternoon visits.
Then Arnie got her to rearrange his Thursday afternoon appointments and I stopped coming on Thursday afternoons. Because... I had discovered what my father already knew about the Genoa cut velvet couch in the studio. Between ten past four and five twenty-nine in the afternoons, it was deliciously warmed by the sun through the window on the neighbours' side. When the apple-green blind was drawn the people upon the couch could not be seen.
Thursday afternoons were all the more delicious for being 'illicit', though there were those who knew about it. Stella was jealous. As long as she was a party to what was going on it was all right with her, but the affair between Arnie and me was passionate, out of her control and pretty soon out of ours.
Allegra, who had taken up the politics of the left and was beginning to shape up on the side that would soon be vigorously opposing the draft for the war in Vietnam, had to make an exception of Arnie and put him in a special category for my sake, though the dismal facts were that Arnie had done his National Service, no one in his family had ever voted anything but Liberal, and he was completely in favour of conscription for Vietnam.
He would make brief appearances at the pub where I met with Allegra and her student friends – Prince Charming and the Existentialists – and he would kiss me in front of them, a naked warning to the young men present that I was spoken for. Then he would leave. Naturally the student friends wanted to know who the hell the weirdo in the suit was and Allegra, between salient points she was putting to them about the iniquities of the draft, would say, 'Oh, he's an eye doctor.'
Consider the eye. There are those who insist that you only have to consider its complexity to prove the existence of a divine architect. And yet few would maintain the thousand and one things that go wrong with the eye to be the work of a divine saboteur. Granpa didn't blame the Devil for his blindness, he blamed sheep dust.
Consider the oyster. Much more open to invasion than the human eye, it hasn't the machinery of a highly organised human body to protect it. It has but its shell.
Seventy-one oyster shells lay scattered on the sand of Camp Cove, Sydney, in March 1965, the result of thirty-five oysters having been devoured between gulps of cold Riesling by two people, the thirty-sixth being held aloft, a smelly little pleasure dome on its home-made plate resting on the fingertips of Arnie's clean pink hand. Framed in the cup formed by the hand and the oyster's shell, a sunset over the Sydney Harbour Bridge.
'It's got glaucoma,' Arnie said. 'Its vitreous humour can't escape through its ducts. They must be clogged. If this oyster came to me for help, I would have to puncture it with this oyster knife to let its humour escape. Terrible thing to have to be punctured to let your humour out.' And so saying, he drove the point of his oyster knife into the dome and released a spray of sea water that dived into the ruddy liquid sky. 'I don't know what we're going to do, Isobel.'
The beach chilled a degree. He was lying on his belly, still holding the oyster up, watching me with his left blue eye. He was smiling, but he was also crying. We had decided against an abortion on the threshold of the abortionist's.
It was an illegal place, of course, they all were then: up some fire-stairs, above Woolworth's in King's Cross. Arnie couldn't countenance it.
At this kind of point you think it's love that's going through their heads, but it isn't. You only have to read what they write to know that behind the tears they think you're a boggy fen into which they, in their innocence, have stumbled. You are the scarlet woman against whom someone has been warning them all their lives.
Or is that a harsh judgement? When I put it much later to Reg Sorby, he agreed with me.
From the abortionist's, through a trip in a bus up South Head, to the purchase of the oysters, Arnie was going to leave his wife and children for me. During the eating of the oysters and the drinking of the wine, he changed his mind. But the interval had attached me hopelessly to the vision of Arnie, me and my baby. We would live up high in Sydney, always looking out over the sublime view of ferries and sailing boats on laughing bays. He would be the father and husband he had described himself to be.
But there was his wife. The children he had already. Margaret hadn't dropped dead as the jettisoned woman does in so much fiction. She wasn't evil or mentally ill. By the thirty-sixth oyster, ominously rotten, it was I who was to be sacrificed.
I felt it keenly. I had no notion of myself as a boggy fen into which an innocent man had blundered and was now stuck fast. I thought I was the woman Arnie loved. I did not see myself as the Viva Laurington of his family: I would at all times leave Arnie free to choose. And how beautifully he chose, with his noble, uninhibited weep.
What I did not think was that I had no choice. We flew back to Melbourne and he went back to his family, who never even knew he had been away.
Lucky for Arnie Russell I was far too young and poor to cause him trouble.
He and his family returned to their original home in Sydney in a hurry. 'Offered something better up there,' said Rumpton, gloomily shaking his head as he explained Russell's sudden departure to 'Motto'. An acceptable myth. Rumpton was piqued; it meant he had to give up his golf while Rudge found someone else.
The replacement was called Plant, not alliterative, but proper, didn't everyone think, Plant being a bit of a weed? By the way, while we were on the subject, that Arnie Russell seemed to have landed himself something in California. Marjorie Rudge looked at me and said again, 'A trail of carnage.'
In July, not knowing what else to do, I ran away from home. I went to Sydney, put my age up, and took a job in an Oxford Street pub that served beer to art students, like me. I wrote a note to Allegra before I left, telling her I was pregnant and going to Sydney. I didn't know what I'd do with the baby, perhaps I'd give it up for adoption, but I didn't think so.
Oxford Street was a street of junk shops, Greek and Lebanese cafes, fruit shops, butchers, and in one place they sold fish – fish that were still alive. Eels writhed in the windows, bream and bass and shark flapped out their last, even the skirt of a manta ray once rippled feebly, the creature's eyes sunk like cannon shot deep in their sockets. Torpid crabs tapped on this baffling substance, glass, and the whiskery edges of the lobsters were bent around no-room-to-move in a highly concentrated sea.
Shabby, tacky, dark brown Oxford Street with its bespoke tailors, their dummies clad in double-breasted Mafia suits, the pawnbrokers full-to-bursting. Walls stacked with unclaimed articles, maroon glitter electric guitars, grimy radios, Electroluxes, Sunbeam Mixmasters in yellow, in blue, in white, no bowl, one beater, patched with skin-coloured Elastoplast, with black insulation tape. Binders, folders, bed ends, pots and pans, trays, chairs, irons with frayed electric cords wound round them.
The main window of my pub was a head-on scene of racing jockeys, whips flailing, haunches high, one in the lead with his eye on the winning post, one behind looking over his shoulder. Inside it wasn't as salubrious as out. The milky backs of the window pictures were covered in tobacco murk. Old cream tiles along the walls at wainscot height were falling off, leaving here and there a quincunx of cement dabs. The top tiles were trimmed with a dull green glaze where they bevelled into the wall and pictures advertising Resch's Pilsener and Dinner Ale were so begrimed as to be only marginally present. One thing I learnt from being a barmaid: no New South Welshman worth his salt drank Dinner Ale. Dinner Ale was a myth on the wall being handed to an apron by a suit and hat.
The students drank Bloody Marys, beer and Turkish coffee. There was more flash and dash to them than there'd been among Melbourne students. Melbourne took its painting very seriously, and on the whole worked harder at it than Sydney. But Sydney had colour and fire. Everything was lively and rude. No one brooded and fought with themselves in Sydney. They fought with each other. Beers were emptied over contemptuous heads. Tussles were quick to flare up and just as quick to die down. I loved it. It pulled me out of myself till I forgot that one day a baby would be born to me. My tummy became what seemed a permanent source of delight and ridicule.
I got along well with my boss, an insomniac called 'Wednesday' Monday – his name was actually Ashley Monday – and, since I was good for custom and unable to partake even of sherry without feeling nauseated, he let me have time off to go rampaging with the students.
Wednesday Monday had inherited his pub laterally from a pair of aged aunts, 'relicts', as he called them, of a no-good husband and brother. Wednesday was a lawyer – the aunts had put him through Sydney University – but he'd never practised, because, he said, he couldn't bear to wear shoelaces and they were _de rigueur_ in public courts. He was perennially cheerful, sat in the pub all day 'ready to spring', as he called his overwound appearance, but he never drank anything. Every now and again he had to wear dark glasses to ease the eye strain from being awake all the time.
We came to be such good friends that sometimes he'd come on the student rampages with me and we'd leave someone else in charge. The rampage was a circuit through people's rooms and houses to gallery openings and parties. I noticed that whenever I was introduced, people knew the name. 'Coretti,' they would say, their eyebrows shooting up their faces. And then they sang, 'Oh,' an invitation to say yes, I was his daughter.
One who sang it had grown a Van Dyke beard to distract attention from the nose he had broken as a bantamweight boxing champion in the fifties. His eyes were two grey glitters in asterisks of mirth. His S's sizzled and he hurdled his adjectives with long, bounding vowels. His name was Bart Turner. He had a gallery; his primary interests were the young and the new. Of courssse he could help me. And hadn't I better phone my mother? He'd phone my mother. Then and there. She must be worried sssick. Though it was the middle of someone's launch, he was asking the exchange to put him through to my mother's number in Melbourne before he had finished talking to me.
'Hullo, Stella? Stella, you won't remember me. I'm calling from Sydney. The name's Bart Turner. I met you once at Horrible Laurington's... Yes, I know. I know. They came swanning through here on their way to wherever it was they were going. Sssick! But I don't want to talk to you about that. I've got your daughter here... No, she hasn't been taken by the white slave trade. It's much much simpler than that... Yes, ye-es. Well, I don't have to tell you, then, do I, since you've already guessed correctly? Well, what I was ringing to say was, it's quite all right if she stays in Sydney. She's got a room and a job in a pub... Well, it's a students' haunt, Stella, I don't think there's much risk of that, and anyway, Isobel doesn't look like a prostitute, she wouldn't be likely to do a roaring trade either, in her condition. Anyway, what I rang to say was, we can look after her... Well, there's not much risk of that, either, Stella, I'm forty years old and my friend Len wouldn't like it... Look, I'll take her along to Crown Street myself... Well, I suppose that's up to Isobel to decide.
'Going to keep it?' he mouthed, his hand over the receiver. I nodded. 'Well, at this stage, the answer's yes, Stella... I quite agree... Me too.' He handed me the phone. 'She wants a word with you.'
'Keep it, darling, keep it!' she cried. 'I'd be looking into prams for the rest of my life!'
And so the fate of Eli the Unexpected was sealed. And so was my fate, and so, though we didn't know it at the time, was the fate of Allegra.
Bart Turner came and stood with me in the prenatal mother's line until a sister told him fathers weren't wanted and he went away.
Yellow cards in a plastic folder. White booklet. Orange plastic seats in the middle of the grey-cream waiting room. A system of wood veneer cupboards along one wall. These were the consulting rooms. I was to go and sit on the orange seats after seeing the nursing bra sister. Obviously I had done the right thing. Nobody came racing out to question me more closely. If I played it right nobody would come and dispossess me of the baby when the time came.
The nursing bra sister had no teeth. She came in three sections, shoe'd shanks, white uniformed thighs and the rest of her, white veiled and blue-cardiganed. The sections were arranged slant-wise on a chair. She knew her product, could tell a good pair of nipples at a hundred paces and recommended, at the price of a guinea, the lace-up Hilton rather than the clip-front Berlei for a bust as young as mine.
As she gave me my fitting, 'Gawd,' she said, 'there's nothing much to fit.'
I went and sat on the orange chairs and felt the laces on my nursing bra slip and get stuck under my rib cage. I remembered the dirndl and bodice I had dreamed of in my youth and opened Allegra's letter.
She wrote to say she had taken a job.
Thanks for the shoes. I particularly like the ones with the Mercury wings on the heels. I've just thought of a design for ones with FJ Holden fronts. I got a job in Newman College serving breakfast to the boys. Fat sausages, greasy bacon and rubber eggs. A couple of days ago I shot a monseigneur with a sausage that slipped out of the tongs. Got him fair in the bib. To think I got my inaugural waitress training from The Brolga!
The only females you ever see in Newman dining hall are drudges like me, there isn't even a picture of Mary. They say it's the same in Trinity and Ormond, and, of course, there are no pictures of Mary there. It gives me a bad feeling, really really bad, so bad I always make sure I serve things up with lashings of fat, so they'll die of heart attacks.
A very reverend gentleman told me he didn't think my footwear was altogether suitable, but I pointed out that I was not wearing thongs and nor did my shoes have open toes through which the scalding porridge might plop, and thus, they met the union regulations. The reverend gentleman was nonplussed. Then someone whispered in his ear the magical words 'Henry Coretti', and presto! all objections to my footwear were swept aside and a special cordiality has set in. Now I serve on high table. Oh, Lucky Me!
Be it said that it was not I who revealed my paternity, but another very senior reverend gentleman with an Italian-sounding name. Meanwhile the exceptionally reverend Cardinal Mannix watches proceedings from his spot on the wall, while I serve the sausages under the pope's nose opposite, humming 'Underneath the Arches' all the while.
When asked by the very senior reverend gentleman why I hum this particular song, I pointed out the arches of the dining hall. They're wonderful. They make the ceiling into a dome.
Walter Burley Griffin must have liked goblins and trolls – everything in Newman is snug and strong, even the chairs he designed to go with his architecture. It's like being under the roots of a great big tree. God's tree. It only sees fit to shelter the male of the species.
An occasional pipe is taken at high table during the clearing away of the plates. Recently there was talk of a dog being found in a student's bed. I suppose he mistook it for a girl on his way upstairs in the dark.
I hope you are eating your greens.
I was having lunch with Wednesday Monday in the Greasy Spoon when they went past. Arm in arm down Darlinghurst Road. Dadda and Viva Laurington. Eli the Unexpected decided he'd had enough cooped up under the lace-up bra and the rib cage. It was time to decamp. The waters broke and Wednesday called a cab.
Eli was frowning when he was born and I can't say I blame him.
EIGHT
If
WHEN ELI WAS born it took two days for the interested parties to find us. We'd been whisked away from Crown Street, where the action was, to the tip of Darling Point, where the impecunious mothers recuperated in a splendour so out of keeping with their social status that the wards were designed to face away from Sydney Harbour, lest the view overwhelm them and make them forget themselves.
If Arnie Russell had been the mother of Eli Coretti and not the father, he would have been given a quilted coathanger by the rich ladies from St Mark's Church of England, Darling Point, while he was having a sun lamp shone on his stitches. If Arnie had been the mother, not the father, he would have waited in the queue for the toilet in a gingham nightie, bent at the waist and calculating how to perform his offices without disturbing the doctor's needlework. If Arnie Russell had been Eli's mother, he would have had to think up white lies about where the father was to tell the three mothers who shared his ward. The father was in California.
When Eli was born, he had to be called something. There was a Jasmin in the ward, a Stewart and a Stephen. If Arnie Russell had been Eli's mother, he would have wondered how these names were arrived at. He would have looked at the baby's caved-in face and his pulsating head with its tufts of hair like the Devil's horns, and been bewildered. Suddenly, his chest would have swelled up to three times its accustomed size and he would have found himself bathed in milk. He would have had to shake the milk out of his newborn baby's nose and towel it out of his ears. He would have had a sister come with a breast pump and been milked so that babies with dry mothers could share in the bonanza. But he, unfortunately, was in California.
Wednesday Monday would have brought him a packet of nipple protectors and a horn with a rubber bulb for the baby's stroller. He would have heard Bart Turner laughing at the baby's hair horns outside the viewing window. And then he would have received a shock.
Because it was not Bart Turner who put his head around the door of the ward, making the mothers of Jasmin and Stewart and Stephen gasp in surprise at his sartorial magnificence, and making the mother of the as-yet-unnamed Eli cry out, 'Dadda!'
When Dadda came to see his grandson, he came in his grandfatherly outfit, a double-breasted cherry-red coat with vertical blue and white stripes, black and white inch-by-inch check trousers. Dress shirt with lace frill front. Long hair and sideburns to mid-ear, but his face was hairless. He came bearing gifts – a pillar-box red Mary Quant helmet from London, a packet of madeleines from Paris and an ounce of very special tea. The madeleines were still quite fresh.
If Arnie Russell had been Eli's mother, he would have sat up in bed in tears and a Mary Quant helmet, drinking lapsang souchong and eating madeleines with a dissolute reprobate who, rendered down, wouldn't have yielded enough glue to stick on a postage stamp. Hopelessly fond of the man, he would have eaten more than his fair share of the madeleines, partly out of gluttony and partly out of having nothing much to say and therefore playing the rule of not speaking with one's mouth full. But Arnie was in California and I had to do all these things for him.
'Well,' said Dadda presently, 'how was the birth? Did you stay awake all through it?'
I nodded my head and stuffed my face. 'It was like poohing a football,' I managed when the silence extended into the post-swallow period.
'Really? I must say that's a fine description for it. Those in the know, however, tell me there is a...' Dadda makes a tulip of his hand, '... sublime moment at the actual moment, as it were.'
I put two madeleines in at once and suppressed the urge to say it no doubt depended on whether or not the baby had fuzzy hair like Checkie Laurington's. I managed at length to say, 'I never thought I'd get to like lapsang souchong, Dadda. I used to think you and Mum were mad sitting up there having lapsang souchong parties. But I think I'm beginning to understand.'
'And... and...' Dadda screwed his face into an expression of concern, 'how,' he sipped, 'is...?' He cocked his eyebrow.
'Allegra is very well, Dadda, thank you. She has taken a job as a kitchen maid to defray costs. I, of course, am a barmaid.'
'Now, now, you know I meant...'
'Your wife?'
'Ye-es.'
I stuffed in another madeleine and semaphored so-so. I wanted to say _give me your address and phone number and you might find out,_ but I could see Bart Turner standing in the doorway of the ward behind Dadda and he winked at me. Address and phone number were probably all under control.
'Well, w-w-what are you going to call the baby?' Dadda asked in a rising tone, as if I would, of course, have an answer to that one.
'How did you arrive at our names?' I said. 'You've had more practice at this kind of thing than I have.'
'Well, yes, now you mention it. We had a funny time finding a name for Allegra. Your mother wanted to call her Belinda, but your silly old Granpa said, _Oh, you can't do that, Daught, too German. Berlinda._ So we called Allegra after my mother, instead. I'm surprised you didn't know that.'
'I didn't even know you had a mother, Dadda.'
'Well, darling I wasn't found under a cabbage leaf, you know. My parents got up to the same sort of pranks your mother and I got up to, the same sort of pranks you and... Tell me, darling, this isn't a virgin birth, is it?'
The question made me cry a good deal harder. Dadda put his arms around me and soothed. 'There, there,' he said. 'There, there, Sibella. Did he run away?' I nodded; my hair creaked on his gaudy coat sleeves. 'Mustn't worry.' He held me back so he could look into my face. 'Bart tells me you've been doing some very clever paintings. Ye-es. He's going to put them in a show for you.' He sat beside me on the high hospital bed, then suddenly rolled back and pulled me onto his chest.
I beat him with my fists. 'You're so cruel, Dadda! So cruel!' I cried.
'I tell you what.' He sat up. 'Let's play a game. I've got the toe of my shoe stuffed with Scrabble letters.' He took his shoe off and held it up in the air. 'Pick a letter,' he said, 'and whatever you pick will be part of the baby's name.'
I drove my hand down the length of his yellow shoe. 'Yellow?' I asked him.
'I bought them in Carnaby Street,' he said.
'There are only three letters in this shoe cap, Dadda.'
'Oh, are there? Are there really? There ought to be five.'
'L. I. E.'
'Oh dear, I wonder what happened to the F and the X. I thought it was so auspicious, too. I found them on a bus seat in New Beach Road yesterday.'
'Fixel? Fexil? They're terrible names, Dadda.'
'Felix, darling. It's Latin for happy.'
'You wouldn't catch me calling a son of mine Felix, Dadda. And I can't call him Lie; he's a fact.'
'Well, call him Eli, carissima.'
Eli the Unexpected.
Eli the taciturn.
Eli the stern, who, if he could have spoken would have recited a long list of newborn baby's rights. He had drawn a bead on each of us as he lay waiting in his amniotic world.
I had been reading Lawrence Durrell when Eli was born. I thought the world was going to turn into an imitation Alexandria and I would learn what some of Durrell's similes meant – how could Justine's profile be 'lit by a painful academic precision', for instance? On paper I could get 'academic precision' into women's faces, but I couldn't for the life of me make it painful. Among the young women I knew, the academically inclined positively revelled in their superiority. 'Academic hauteur' was the closest I could come to pain – a kind of affected pain which no doubt was deeply pleasurable to the wearer.
I thought the world would continue along the road towards Turkish sobranis for breakfast and away from tea and cornflakes. I had only melted one telephone table with the Birko and was prepared to melt more in my search for the perfect cup of black coffee. But the search was cut short. I was in error. The world into which Eli was born was not imitation Alexandria. It was imitation London.
The drink was Nescafé. The cigarettes were Benson and Hedges. I felt cheated. It had all been going on behind my back. When I arrived home in Melbourne, Allegra was sucking in her cheeks, pouting and walking pigeon-toed. She was shoving her curly hair into berets and hats. When she tried to iron it straight, she burnt an ear. In the end she had the front part straightened and cut à la Quant and left the long, incorrigible back long and incorrigible. She wore men's jeans and skinny rib jumpers. She wore boots. What she wouldn't wear was the stripy tie, the lace blouse and bracered mini skirt Dadda had bought for her in Chelsea. She wouldn't let me wear the ones he'd bought for me, either. The Brolga had probably chosen them, she said.
She was steeping herself in Comrade Marx. She laughed at Lawrence Durrell and said I was living in a dream world. Furthermore, art should be _demotic,_ a word I'd come across in _Justine,_ which I at first thought was a misprint for _demonic,_ though demonic was an odd adjective to go with Greek.
Newman College banned her for joining a political club that spurned the Newman Society and had as its slogan: More Concerned with Kafka than Wordsworth. She was spoken to for wearing jeans in a certain lecture hall on campus. 'Fuck them,' she said. Our mother was stunned. 'Fuck, fuck, fuck,' Allegra repeated, 'it's only a word.'
She was very bad tempered. 'You know nothing,' she said to me, 'just nothing.' All the same, she was good to me. She took me out on Saturday nights to the pub where her friends gathered. Her ex-boyfriend, Macka, had been called up for the army in 1964, but was rejected because of acne on the back. Now he was a photojournalist. He practised fashion shots, the 'in' thing, on Eli and me. The model in those days wore her legs as if striding and stuck her bottom lip out. Allegra was above this capitalistic nonsense, but not beyond wearing the clothes.
Out of teenage sloth was born an angry brilliance – there's a Durrell sentence for you – but by brilliance I don't mean being able to quote Marx verbatim, nor do I mean that she'd been dousing her eyeballs, _à la Justine,_ in belladonna. Marx's prose style left a lot to be desired, I decided, in the midst of her long harangues. It was her delivery and the animation it imparted to her that made her brilliant.
She was little, fabulously pretty and inspiring. Whatever she'd said, the legions would have followed her. Most of us didn't know what she was talking about, perhaps she didn't know herself particularly, except that there was something wrong, and she made us feel it.
Something was wrong, but not everything was dreary.
She has been invited to be among the first women to sit at high table at Trinity College. She has assembled herself a suitable costume for the occasion. Jodhpurs, she feels, teamed with long brown riding boots left over from Stella's youth. A navy blue Voluntary Aid Detachment overcoat with a Red Cross on the sleeve and captain's stripes. Military buttons. Stella isn't sure. In the first place, we thieved the coat from Clare on the recent trip we took to show Aunt Nina Eli. (Babies have the same sort of effect on Nina as the Bible had on Granpa. She didn't criticise, though she would have preferred a girl and was clearly pleased the baby was mine rather than Allegra's.)
'It's against the law, you know,' says Stella. She means wearing army drag when you aren't in the army. She also means that the coat has a deep significance for Nina, who took her wartime duties seriously.
'Fuck the law,' says Allegra.
'If You Please,' says Stella primly. Poor Stella. In this day and age of straight hair, she has had most of hers burnt off by an overzealous hairdresser who must have thought the curlier the hair, the more lotion required to straighten it. She has had to resort to wearing a turban day and night. Rumpton and Rudge call her The Begum.
'Well, I don't care,' says Allegra.
Sizzling with discontent, Stella gets on with mopping the kitchen floor. Then the drum and bell collection makes its announcing noises at the front door.
It is too late for Allegra to change costumes. I play the butler. 'Hey!' I call when I open the door. 'There's a sod in a suit out here!'
I'd been expecting a sod with a beard, but it is none other than Denzil Rumpton, the incredibly good-looking son of Gloom Bottom, the one who put his mother into a health farm after his birth. He is known to the cognoscenti as Gregory Impeccable. Ex Melbourne Gram. He's crackers about Allegra, but his face drops when he sees what she is wearing. 'Aren't you ready yet?' he asks hopefully. 'I'm ready,' says Allegra.
'But... But... It isn't fancy dress, Allegra. I can't take you looking like that!'
Our mother rests on her mop and evokes her Family Tree. 'Allegra,' she says grandly, 'is related to the Queen Mother.' Meaning, I suppose, that Allegra can wear anything she likes. Oddly, this does not lead Denzil into the wilderness. He attributes Prince Charles's jug ears to the Bowes-Lyons.
'Fuck!' shrieks Allegra when things look like developing into a family tree–lopping contest. 'Are you taking me, Denzil, or aren't you? If you don't take me, I'll storm the dining hall.'
Stella, tribal in her wrath, brandishes her mop as Denzil and Allegra sweep out, furious with each other.
'Fu!' says Eli, who cannot yet pronounce his K's.
That is not where the evening ends. Allegra and Denzil have had such a fight by the time they reach Trinity, Denzil won't take her in. True to her word, Allegra storms the dining hall in his wake and is about to fling off her VAD coat in full view of the assembled company when she suddenly realises she's forgotten to put a blouse on under it.
Our front door bursts open and in sweeps Allegra, laughing her head off. In hot pursuit, Denzil. Stella is lying on the couch in the parlour with her hat off and her teeth out. She is wearing her nightie. The family dachshund, Puglia, wakes up, excited. Backs up the nightie. Wags her tail.
Allegra fails to make it to her first important engagement as a Feminist. Denzil doesn't understand.
It is 1966. Allegra cries in her sleep. When I ask what the matter is, she tells me of the many, many wrongs and I realise I am ignorant.
_No vote, No voice, No choice!_ says the placard she bears to the Kew Town Hall, where she goes to boo the Liberals.
How, she asks, is it possible for the government of Australia to draft into the army, for the purpose of fighting a foreign war, young men who do not have the vote?
Why, she asks, does it take the government a week and two ministers to decide that Australians can see, unexpurgated, a BBC documentary on Vietnam?
Why is it that Australian conscripts have died in defence of a corrupt Vietnamese regime?
'But there is no front!' she screams when someone says the Americans are winning. And there is no front. Johnson names Ho Chi Minh and the Viet Cong as his targets, but it is the futile war of sledgehammer and ant. America is flexing its military muscles for the benefit of the Russians and Chinese. What are Australians doing there? And what are Australians doing here?
At last, a woman has made it onto a jury in the state of Victoria. And how did the papers report it? They said she was wearing a bright red suit and sunglasses. They said she actually asked some relevant questions. Could this be the first step in the evolution of a female brain?
While conscripts die abroad, margarine is at war with butter in Australia. While you can fight and die in a trumped-up war at eighteen years of age, you cannot, at any age, see, unexpurgated, four of the feature films at the Melbourne Film Festival.
How is it that section 127 of the Australian Constitution excludes Aborigines from the census? And from the vote? Irony of ironies, Aborigines have to go to war to get the vote! How come they pay tax if they are not represented?
And if Asians are considered fit to study here, why are they not considered fit to live here?
In 1966, the only country with stricter censorship laws than Australia is Catholic Ireland.
In Australia you can't read _Lady Chatterley's Lover_ or _The CarpetBaggers_ unless you buy them under the counter in a brown paper bag from certain shady establishments. One of these belongs to the daughters of the dump caretaker, Bridget Kelly. In Maggie and Kelly Kelly's second-hand and salvage shop, which comes into being in the March of this strenuous year just a grenade's throw from our house, banned books aren't the only things you can buy.
You can also buy that substance no one can spell yet, or pronounce, marijuana.
You can smell Maggie and Kelly's Pantechnicon a block away. Incense, beeswax candles and sandalwood soap are sold here. You can hear it a block away, too. Donovan keeps singing: _He's the universal soldier and he really is to blame./His orders come from far away no more./They come from here and there and you and me/And brothers, can't you see,/This is not the way to put the end to war._ Between stanzas Indian bells and wind chimes tinkle and clang. Some of the wind chimes are made by Maggie Kelly from bits and pieces collected for her by Bridget from the dump. Caftans and buffalo-hide thongs are sold here. Large Op and Pop Art posters. Army, navy and air force surplus. Old badges. Hinges. Bicycle reflectors and clips for trouser cuffs. Second-hand books. Vases. Windscreen wipers. LPs. And much, much more, upstairs.
The floor is covered with sea grass matting, which very soon gets tatty from the traffic. The goods are set out in tubs, or hung from the ceiling, or draped around the walls. Hovering between occident and orient, the blonde shop assistants, Rapunzels in caftans, smoke dope, cut their toenails behind the counter and occasionally make a sale.
Maggie Kelly, though eighteen, is too young to own a second-hand dealer's licence, so the licence for the shop, which bears its name in comely art nouveau writing (though I say so myself) on the window, is held by the redoubtable Bridget.
Maggie Kelly might be too young to hold a second-hand dealer's licence and too young to vote, but she is not too young to be a mother.
She came by her baby from a schoolfriend, Johnny Green, who wrote himself into the Book of Man just before writing himself off in a stolen panel van at a place where the proposed Tullamarine Freeway will, in due course, run. He is missed sentimentally by Maggie, who peers out of her hair constantly in order to find his features in her baby, Chantal.
Bridget Kelly thinks Johnny Green is better off dead.
Maggie Kelly is not alone in looking surreptitiously for evidence of the other side in her baby. Eli, of course, has blue, blue eyes, 'gold standard blue', and he is blond. But he is also a stern, gruff baby, given to ordering everyone around. I have no idea where that comes from. Too efficient to learn how to speak until absolutely necessary, Eli cries out, 'Fu!' and makes frantic finger movements at whatever it is he wants to be provided with. Ignoring his requests results in loud, prolonged complaint. Chantal Kelly, twice his size, is his mortal enemy. This is a shame, as the legal proprietor of Maggie and Kelly's Pantechnicon is the only mortal, enemy or friend, kind enough to offer me remunerative employ.
Eli is a big baby, Chantal is bigger. He is a loud baby, she is louder.
She walked when she was ten months old; Eli seems set to sit on his bottom for the rest of his life. While she is framing sentences of intricate vocabulary, he sticks with his one word, 'Fu!'
Yet we know what he means: we are not to ignore his manhood at any cost.
We don't come across many of his sex during waking hours. It tends to be women who come to rummage among the bits and pieces in the Pantechnicon. Eli is saving himself for the day when.
Dadda.
Has just walked into the shop.
'Sibella!' he says, as if we were old friends. Eli, sitting on the counter with Stella's recently discarded turban in his hands, stares. Dadda crouches and levels his luscious blues with Eli's gold standards. Eli puts Stella's hat on Dadda's head. Dadda puts the hat on Eli's head. Eli to Dadda. Dadda to Eli. Eli, who rarely laughs, laughs. Dadda laughs. Comrades. It's enough to make you sick.
Dadda has been overseas again and has come back to Melbourne this time, rather than Sydney. Yesterday, his picture was all over the papers. He has sold a painting to the Tate. Beside him in the photos, the featherless Brolga in fur, calling herself his agent.
I say nothing, determined to stick by Allegra's rule of giving bastards a wide berth.
'He's a very handsome fellow, Sibella. Ye-es,' says Dadda.
Dadda has been talking to our mother. He tried to give her some money, but she wouldn't take it. Allegra will have nothing to do with him.
'Would you take the money, Sibella?' he asks. 'I don't want you girls and the baby to suffer just because your parents don't get on.'
'It isn't that, Dadda. If you're happy with her, that's your business. But we don't want to be pensioned off. We don't want to hear what you're doing without us, either. It would be better if you kept it to yourself.'
'But I don't want to have hurt you.'
'I suppose murderers don't want to have committed murder, either, Dadda. But there's no going back.'
'That isn't a fair comparison, Sibella.'
'Why did you marry our mother if you didn't love her?'
'I did love your mother. She was unusual, pretty, kind. She tried very hard for me. I appreciate it. I owe your mother a great deal, but equally, I owe a lot to Viva.'
'Not equally!'
'She has done a lot for me.'
'You're her trophy.'
'Isobel!'
'You are, Dadda. She's triumphant at your side.'
'I knew her first, Sibella, before I knew your mother.'
'Why didn't you marry her, then?'
'There are many things that could happen that don't. I might equally say to you, why didn't you marry the father of your son?'
'He was married already.'
'There, you see, there are reasons for everything.' Having cleared that one up, he adopts an air of quiet confidentiality. 'I was in love when I was your age, too.'
'I don't want to hear.'
But he leans forward on the counter, bends under my hung head and looks me in the eye. 'In Paris when I was nineteen.'
'I'll block my ears if you tell me,' I say, but he holds my wrists firmly and continues looking me in the eye.
'When your mother and I met, Sibella, I wasn't the only one with another story. Your mother was wearing an engagement ring when I met her. She's still wearing it. I had neither the right nor the inclination to stop her wearing it. She called you Isobel. Perhaps it wasn't after your Motte grandmother at all, perhaps it was for the daughter she might have had with this man. Your mother lived another life in her head. It was a life I couldn't share.'
'Maybe you didn't _want_ to share it.'
'No. I couldn't share it. It was your mother's way of coping.'
_'Your mother, your mother._ Why don't you call her by her name?'
Dadda lets go my wrists. He puts his hands in his pockets and then slowly turns around, looking upwards at a wind chime. 'You can't make people over again. They're only there once. I tried to love Stella partly because she was sweet and partly because I thought she'd been through an experience similar to my own, and we could comfort each other. It didn't work. We married without knowing each other. Our family backgrounds, our beliefs, just weren't compatible.'
'Well, what good's Viva Laurington to you?'
'She knows about me. She wants to know.'
'She's so horrible and heartless.'
'That's how you feel,' he says. 'She knows art. I'm afraid your mother... Stella, doesn't. I've looked for other things in myself, but art seems to be all I have.'
'God! What about us?'
'You and art, then. You and Eli and Allegra and art.'
'But you can't have us _and_ her.'
'Then all I have is art.'
He does not look me in the eye again, but caresses Eli's head, then kisses it and kisses me and then he is gone. He has left behind a teapot on the counter. It is a strange teapot, brown, with two spouts. Inside, a cellophane butterfly on a wire spring and a lump of money.
NINE
Respect: the Odyssey of the Walnut Table
UNCLE GARTH USED to lilt his way through _The Windhover_ when I was small and on the hottest of those hot summer days, I would shiver when he came to _'blue-bleak embers, ah my dear,/Fall, gall themselves, and gash gold-vermilion'._ I could never explain to myself the power of _vermilion,_ how it was greater, warmer, more marvellous than gold, how it released me from oppression and made of my uncle's voice a sound striving to lose itself in the universe. This word, this thought, vermilion, must be the Resurrection. The drunken body of Garth would not redeem him, but his passionate voice, though it resounded in numb isolation, afforded him his glimpse of paradise. I seemed to find God when I listened to him reciting Hopkins, and God was profound. We were not made in God's image, but could only catch flashes of something infinitely more beautiful than we could comprehend.
God was beyond me, and I knew on earth we'd got it wrong. From childhood upwards I found I could not walk into a religious place without feeling defrauded and ashamed. In adulthood, although I could feel the sincerity of the people who believed, I was horrified by the stupidity of belief itself. Churches made me feel self-conscious, sermons made me angry. There was no genuine comfort in the world, no grace. It was all just an outpouring of rote-learned words for the occasion. Catholicism, with its plaster representations of divinity, was laughable. Protestantism was parsimony carried over into everyday life.
Where was the essence in religion? We could name its component parts, but inevitably, in the naming, we besmirched that which it is beyond our power to describe. Was the human race, by its make-up, then, condemned to an eternal naming of the parts? In personality, I, like my father, wavered towards those who believe that to search for the essence of things is a waste of time; such activity leads to infinite regression, anything might be called an essence, so it seems the human lot to be stuck with naming things and demonstrating how they work as things. I began my life as a painter immersed in this mental set. It was a bleak one – works of Dadda's, such as his refrigerator chapel, demonstrated that meaning was easily lost or subverted and suggested that it might be more honest to worship fridges than to worship God.
There was something in Dadda's work too difficult for me when I was a young painter; while his outlook was bleak, it was mocking rather than angry. I felt I belonged among people who were more passionate than Dadda; I felt that if I did belong among those people, Dadda would be stung. He would be sorry. So it was with my generation, we turned against images, and we turned in anger – _you want images, we'll give you images_ was how it started – and we gave them American hegemony, blood, ugliness, effluence and greed; we gave them sex and guns and lies. In time, we would deny the image altogether and, with us, the image would take from, or sink through, the canvas, leaving only its minimal representation.
In time, we would learn that minimalism is not a virtue in itself, that there are times in history when simplification increases the power of statement, and times when it is merely trite. On the way to that understanding, however, our search would lead us to an altered understanding of religion. We would expose ourselves to Eastern orthodoxies. Unfortunately, we would do it from a position of affluence and, in time, we would make connivers of gurus, and gurus of connivers.
So we exposed emptiness: an emptiness in which our own footsteps would come back to us. And if we had read history we would have known it was not the first time a generation had found itself in a sepulchral cul-de-sac.
Notwithstanding my experiences as a painter, I still find churches either grandiose or banal. However, there is one Lady Chapel, attached to a college in Melbourne, in which the concept of grace seems to have been deeply understood by its creator. Pews adzed from heavy beams lie horizontal in a mothering space. The atmosphere is steeped in motherhood, apprehended as a holy idea.
I came on this place when I had no one to turn to, when I was lonely, sick and unhappy. I am not religious, yet I know the context of being is immeasurably larger than the self. I know we are contained. Before I was in this Lady Chapel I had never incanted a prayer that gave me comfort. But in there, instead of dogma, I found symbols, guides to a way of thinking, to choosing hope, even when in despair – and I remembered Uncle Garth reciting Hopkins, _'Not, I'll not, carrion comfort, Despair, not feast on thee.'_ For Hopkins, Christ's mother symbolised the life force. _'Wild air, world-mothering,'_ he called her. And in her, _'God's infinity/Dwindled to infancy.'_
The chapel embodies kindliness and bears the weight of life within it. The mother of this chapel is both doomed and redeemed. She is God's essential partner, and he is hers. Without her, God cannot be.
I had a heated discussion about religious aesthetics once: it was in this room, with Reg, at this old walnut table. Reg was decrying the pope because he wouldn't sanction contraception. I thought of the pope, I thought of the breeding millions and I thought of the unbridgeable gap between the holy man making pronouncements from his palace and the people who are often so poor, a baby is an asset to beg with or to sell, and I lost my temper with Reg and said he didn't understand poverty. For most of the world's poor, contraception was a luxury they knew nothing of and had no access to. Their lot was to wait for death, and their chance of staying alive probably increased when they had children.
I had never believed Reg capable of a lofty thought, and continued on to say that though much religious dogma was manifestly stupid, it wasn't all bad, in fact it was the source of some good thinking. As an example of good religious thinking captured by art, I described to him the Lady Chapel I'd found. I had thought to teach the earthy, self-indulgent Reg a lesson in humility, but it was I who learnt. 'Go on,' he said, not unkindly – I was aggravated into deeper righteousness by his tone – 'tell me more.' And he heard me out, adjusting his glasses to look at me over the top of them. Then he looked away and as I continued my description, he pouted and grunted, 'Mmm, Mmm,' at intervals into his clasped hands.
'You know,' he said, when I'd finished, 'I like you, Isobel. You're a passionate, talented woman. And I know what you think of me, but I have a confession to make. Irreverent and irreligious as I might be, I was the person who created that chapel.' And he patted my knee and chuckled as if his laughter were bouncing on a spring.
It was totally unexpected. I had either to reject a hard-won notion of religion or to accept that Reg, for all his seeming lack of refinement, had a spiritual side to his nature that went deep. I did not think that such an achievement might be accidental, and even supposing it were, it had to be deeply instructive.
At night in Reg's house the spaces are low, wide and embracing. His ceilings rest on broad beams. The hanging lights commemorate his wives – from the giant Chinese lantern in the kitchen, tasselled and ceremonial, with a picture show in silhouette on every panel, to the milk glass chandelier with translucent centres in the drops. One wife adorned the ceiling with corn dollies from America. It might have been the same one who hung globes of _Helichrysum_ from it, giving this room a joyful yellow light.
If houses can be said to reflect their owners' natures, then Reg is a stout beam on whom a wide and motherly ceiling rests.
I think Reg's chapel is his finest achievement. It goes much deeper than a man's idea of what motherhood ought to be. It's not about sex, but about gravity and grace and the natural purpose in female life, which isn't there in the lives of men. Reg made the place in genuine homage; he made it in spite of himself.
When I think of Clare, I think its every intention was motherhood, but it spoke more of Granpa than it did of Euphrosyne. It was a place of opulent comfort, its verandahs generously deep, its windows wide and reaching from floor to ceiling. In the front of the house, the inside and outside flowed into each other. Most of the furnishings, too, were supplied by Granpa, the monster table symbolising amplitude, generosity and conviviality, but also luxury, the luxury of walnut, and the further luxury of its having come, in sections, from Ireland by ship.
When Allegra and I knew Clare, we were conscious of its having been tragically maimed; what's more, its maiming was synonymous with Granpa's. It was the maiming of an honourable, well-intentioned man, a provider brought low by circumstances beyond his control. Allegra and I were encouraged to imagine what Clare had been in the past: a paradise under the reign of Euphrosyne. Granpa had lain all bounty at her feet. She was his Virgin Mary. Yet we knew damned well that Granpa was a rash man with a violent temper and deeply stained by prejudice and, furthermore, Euphrosyne, who had withrawn from life when she fell upon hard times, did not translate into our world. We could not withdraw; the hard times were there from the time our mother was born; withdrawing was a luxury you came across in Chekhov and Charlotte Brontë; it implied a labyrinth of rooms and someone rich enough to keep them, even in decay.
The keeping of rooms at Clare not only required money, of course; it also required flair, a flair our mother had and tried to apply to our less-than-salubrious home. It was not surprising that by the time we were young women, Allegra and I began to find a lot to criticise in our mother's proud, flamboyant housekeeping. As little children, it hadn't struck us as anomalous, and we would side with her when frugal people tried to hem her in. But then came Marx, and everything in our mother's attitude began to seem wrong. We upbraided her constantly for her love of possessions and her inability to accept anything synthetic, or of what she called 'inferior workmanship'. While Allegra and I admired things for their potential to do work, our mother attached a different set of values to them, a set we believed to be both aesthetically and morally obsolete. Our mother held plastic and vinyl in contempt, just as Euphrosyne before her had looked down on electroplating. The cutlery at Clare was either stainless steel or solid silver; people of discernment would know it was and use it as a measure of Granpa's capacity to provide. Against this standard, of course, our own father had demonstrated no capacity.
Allegra and I understood there was a historical aspect to our inheritance, but we couldn't have named it in 1967, when Aunt Nina died. We lived a world away from her Australia. To us she was very dear, very quaint, but everything except her upright manner was irrelevant to the future, and we had often been deeply irritated by her.
By the end of November 1967, when Aunt Nina was dying, farmers were preparing for a terrible summer drought. It had been the driest year on record. The dams at Clare were evaporating and there wasn't enough water for the garden.
Tony Furlonger had taken Aunt Nina into Scunthorpe Hospital before we arrived. When we opened the house, blowflies reverberated in the long, bright rooms.
Stella cleaned up as if in conversation with her home. She oiled away its dust and polished its silver as if massaging its bed-sore muscles. The crystal sang with her care. She was trying to work up the courage to go and see Aunt Nina, but she couldn't do it. Nina had terminal cancer. The thought of her vacating life was almost more than Stella could bear. Afraid of those recesses of her soul that were filled with tears for her family, she stayed behind, with Eli to keep her company, while Allegra and I drove into Scunthorpe in Allegra's battered Kombi.
We found Aunt Nina giving the staff hell at Scunthorpe Hospital. Her Chinese doctor was in tears. 'I know what you've got in that needle, young man,' she kept announcing in her tremulous, most forbidding contralto. 'I've heard the name Furlonger whispered round this hospital, don't think I don't know what you're up to! My relatives will hear about this!'
Her relatives could hear her well before they reached her side. When they did, she seized Allegra with one hand and me with the other in a grip of iron. 'They're trying to put me down,' she quavered.
She didn't want to be treated by an Asian doctor. We were to take her home immediately. The poor doctor was inconsolable. It was morphine he had in his needle. All he wanted was to ease her pain. 'H'mph,' said Nina, gathering strength, 'a likely story.' She wouldn't let go of us. Her brothers hadn't given their lives so she could be killed by this fellow.
The doctor wept, the matron soothed, Aunt Nina clung.
I noticed an old woman in a wheelchair abandoned in the middle of a gaggle of crones on the way to the bathroom. She was dead, one arm slung out over the side of the chair, round face staring up at me, goggle-eyed dead. She was dead, but I didn't like to say so. She was dead, I could see she was, and it was an embarrassment. You didn't expect to see her sitting there in that condition. Life had just passed out of her on the way to the bathroom and now the need for a bath had been bypassed. The nurse who'd been wheeling her had probably seen she was dead and thought it best to leave her among the other old women where she mightn't be noticed. Who knows but that the nurse was hoping someone else would find her?
The doctor had been saying in a low voice between sobs that Aunt Nina would die soon and nothing could be done for her except the relieving of pain. I undid her fingers from my wrist, thinking life could pass out of her by accident in this cheerless place and we would not live with ourselves if it did. On my way to phone Stella, I passed the dead woman and her company of jaw-chumping white heads into which life had shrunk down past the periphery of knowing.
'We're bringing her home,' I told my mother, allowing her no say in the matter. 'It won't be for long,' I tried to reassure her. 'It's horrible in here. Too horrible.' I knew she would understand this message, though she would be afraid of it. I felt it being delivered to me by people who were already dead; I was under instructions.
When she saw me returning, Aunt Nina seemed to know what had just passed between me and my mother, and sprang up as if there'd been nothing wrong with her. She flung on her dressing-gown and filled the spaces that had sagged in it. Pink bloomed in her cheeks.
She remembered the date. When we arrived home, she had us bring out a bagful of Christmas presents she was saving for December along with cards, cellophane and ribbons. Sherry was fetched. We opened her windows onto the rose garden where, in spite of the drought, her standard reds were in full bloom and filled the room with their scent. She wrapped, she tied, she inscribed. She sent us shopping for more sherry.
Then she thought she heard a baby crying. A baby? She frowned – surely not a baby? Our mother took quick breaths and went to say words, but her little jaw clacked up under her top teeth before she could frame what she had to say. Didn't Nina remember Eli? we asked. He wasn't such a baby now. He was two.
'Oh, well, in that case...' said Nina. Having a baby in the house made her get up and attempt to take charge again. Obviously we were unfit to run things. There was nothing to sterilise the nappies in, of course. And no proper facilities for boiling bottles. I didn't know whether or not to construe this as an invitation to leave. I, like the doctor, was cowed by her.
I did penance in the passenger seat of her ancient Pontiac, 10 m.p.h. for a distance of fifteen miles to Scunthorpe cemetery in hundred-degree heat – object, to choose her plot. When I offered to drive, her retort was, 'You wouldn't like another person to drive your car, would you? It's like using someone else's hanky.' I refrained from comment. Mercifully, we had left Eli at home, or he would surely have boiled away. Even though I had my window wide open, we were going so slowly no breeze at all was coming in on us.
The whole exercise took three and a half hours, and by the time we arrived home, Eli, demanding my presence, could be heard at the front gate, just under a mile from the house. That night, because the atmosphere had grown rather tense at Clare, Eli and I went back to Melbourne with Allegra, who had to submit a thesis for her master's degree. This was the part my mother had been dreading from the outset. I'd been hoping we wouldn't have to go through it.
Nina's grip on life was so tenacious it became a competition between life and her. She had it by the throat, daring it to atone. The top was off the sherry bottle way before the sun had crossed the yardarm every morning. 'Men,' she would say at random, 'are brutes.'
Then, just before Christmas, she was laid low by a stroke. All faculties went except the ability to point to the sherry bottle and then to her lips. A nurse was summoned for the daily bath.
She did not die till the last day of 1967. Rain was falling steadily and drenchingly, making its longed-for sound on the roof and releasing from the earth the aroma of abundance.
We had not only to relinquish Aunt Nina, but Clare as well. Mountshannon belonged already to the Furlongers, who had cut a path to it from Coolang and were keeping their traps and guns in it.
There were no horses to ride over on and bid it farewell, and in any event we hadn't been there since Granpa died. Then it had been overgrown, shadowy and mournful. The only happiness we'd had was on the beautiful track leading down to it from Clare. We had had to dismount to allow a herd of grey kangaroos to cross the path. There'd been more of them grazing around the house, and ringtailed possums huddled in the wainscoting. There was the smell of long-dead fires in the grates, and in the kitchen, the stench of bush rats.
A couple of days after we had buried Aunt Nina, a strange car came jolting over the pot holes from the gate to the front driveway. It was a Landrover containing a man and a woman. The man turned out to be Reg Sorby, whom we only knew by reputation at that stage. The woman was his second wife, Helene. They had heard we had a giant walnut table for sale.
'It isn't for sale,' said our mother indignantly.
'Oh, well, would you mind if we just had a look?' asked Helene Sorby, who was a short wasp of a woman with the determination of a pneumatic drill.
Stella barred the steps.
'You wouldn't have a cool drink?' asked Reg from the driver's seat. Another woman, Aunt Nina, for instance, would have been affronted by his gall, but Stella, being Stella, said, 'Well, you'll have to have it without ice. There isn't any.'
When we said to her _Don't give him a drink!_ her answer was that the poor man had probably driven his bitch of a wife for miles in the heat and was on the verge of desiccation. She returned to the verandah with two ginger ales in the best crystal glasses.
Allegra and I both knew that wasn't the tactic you used with frightful people like the Sorbys, but our mother appeared to be under the impression they'd be so cowed by the glassware they'd hightail it out the gate.
Predictably, when Helene Sorby had finished her drink, she flicked a finger against the side of her glass and said, 'Got any more crystal like this?'
'Of course we have!' carolled our mother. 'There's a whole set, even down to proper whisky and martini glasses.' She supposed Helene Sorby would have no answer to that.
But, 'Decanters?' asked Helene Sorby.
'Of course.'
'How much do you want for them?'
Save for our sentimentality, Helene Sorby would have robbed us of our every small adornment. She was the sort of person whose family appreciates her only after her death, when the thought of highway robbery gives way to a feeling of legitimised possession.
The least liked of Reg's wives, Helene was the most acquisitive. Reg says he must be the only person in the world who ever liked her. She was responsible for many of the painting commissions from which he made his name. Some time after she robbed us of our table, she divorced Reg for someone else but then regretted it and ever since has kept in close contact with him, telling him how fond of him she is – fond, says Reg, in a preposthumous way.
Among other things, Helene Sorby dealt in antiques. She had circled the Scunthorpe district like a hawk when 'a friend' had told her of the imminent demise of Aunt Nina. She'd sniffed out 'the relatives', having been advised that the most vulnerable point of attack was Tony Furlonger, who could probably make use of her talent for cannibalising deceased estates. Tony had come and asked what we were going to do with all the furniture, then buttered us up and told us to be on the lookout for an antique dealer. He was, said Reg, 'too honourable' to take a commission.
We hated Reg for buying our beautiful table; we hated Tony Furlonger for letting him. With a sense of occasion that made us put our ideological soundness into mothballs, we declared that a real man would have offered to keep the table for us until such time as we could come by a mansion we could fit it in. But we were a few shekels short of a mansion and, instead, the table came here, across the Alps like Hannibal's elephants, to end up in the room where I am now at Reg's retreat in New South Wales.
It was last seen leaving Clare, two of its handsome legs tethered to the bull bar of Reg's Landrover, the other two tethered by a very long length of rope to the back bumper bar. We exhausted ourselves with its ordeal. And as we exhausted ourselves, we packed.
We had objected with energy to the table taking a long journey practically naked, but Helene Sorby had remarked scornfully that the table was in need of new French-polishing anyway. But what about stones, crashes, injury, rain? Well, that was a risk she was prepared to take, didn't we realise we were lucky to fetch anything at all for such a white elephant? Somebody else might have made us pay transport costs and we could imagine what they would have come to, considering our isolation.
Chagrined, we charted the table's course across Aunt Nina's 1935 Broadbent road maps. After wandering through the baffling scenery of the south-west, where nothing changes but the position of elephantine outcrops on flat horizons, we had the table calling in at Warracknabeal to freshen up at the Ladies' Restroom. We'd just come across the ancient letter announcing Aunt Nina's election to Life Membership of this establishment. Her name was to be inscribed on the frosted mirror behind the manageress's desk.
Daughters of life members were admitted free of charge to the Warracknabeal Ladies' Restroom, while foreign bottoms had to pay for the privilege. We had been a special case for exemption, being nieces, when we passed through Warracknabeal in childhood. An almanac had been consulted, the passage on childless aunts being referred to, and we'd been given the nod. To partake of the privilege we had had to walk down an aisle of assorted chairs, some dating back to the Boer War, before going through a low-lintelled Alice in Wonderland door. This gave onto a semi-circle of quaintly painted outside conveniences. We pictured Helene Sorby being turned back for bad manners at Warracknabeal, and derived a modicum of satisfaction.
As we riffled through the requests for payment of upkeep of graves and the thank you notes from cemetery caretakers, it occurred to us that many a Motte lay under a tended sod, thanks to Aunt Nina, and there was a short debate as to whether charity dictated that we take over the sod-tending as her legatees. We got over that difficulty by finding a bigger one. Aunt Nina's beeswax preparation for polishing the table had been left behind on the front steps of Clare, along with the special polishing rags made from Granpa's singlets. The contempt! We bundled them into a parcel with a short sharp note from Stella. This was not just any table, she wrote. This was an heirloom upon whose surface nothing but the traditional polish should be applied. She included a time-honoured recipe and insisted that no common rag be used in the buffing. This table required pure cotton that had been worn next to the skin of a well-bred man.
The next day, when we came across Euphrosyne's stud book and the pedigrees of Oberon and Countess, we supposed the table would be passing through Wodonga and would continue on to Corryong. There, it would have to cross the now overbrimming Murray and begin the perilous but beautiful climb up Kosciuszko. Here, fist-sized rocks were so numerous as to fling every twentieth car into the fragrant sub-alpine forests or down the gorgeous honey-scented valleys above the snow line. If our table had to come to rest, we prayed, let it be in this heavenly locality.
The orphaned chairs were beside themselves.
We packed the hand-embroidered tablecloths and guest towels, the monogrammed dinner service with its indigo rims, the breakfast china, the Minton, the Doulton, the Staffordshire and what was left of the Waterford after Helene Sorby's lust had been satisfied by purloining the glasses from which she and Reg had had their drinks.
Allegra observed that the table ought just about to be in the tablelands by now. We took sherry and heart. From there it was a mere spin across flat surfaces.
The black wedding dress was loaded into the Kombi, the letters from the front, three chairs, the trinkets, the tantalus. The carving knives. Countess's saddle – no, not the saddle – well, all right then, why not? But why? Because if the saddle, then not the tablecloths. The serviette rings, the silver. The portrait of the stud bull by A. Twine.
And under the old camp bed where Uncle Garth used to recline drunkenly in his den, _Mad Meg._ She was mounted on a piece of hardboard, framed, the glass in front of her broken. Stuck on the back, in his scrawl on a Memorandum from one of his practices in Adelaide, was written:
_For this, be sure, tonight thou shalt have cramps, Side-stitches that shall pen thy breath up; urchins Shall forth at vast of night, that they may work All exercise on thee: thou shalt be pinch'd As thick as honeycomb, each pinch more stinging Than the bees that made them._
Reg had refused to ring us when he arrived in New South Wales. All this fuss over something we no longer owned got on his nerves.
We had to wait more than twenty years to find that our table had made the journey not over the Alpine Way as we imagined, but blamming up the Hume in a cattle truck onto which Reg had loaded it in Horsham. When it reached here, Reg was at pains to tell us, Helene had made him dismantle a wall to get it inside. He'd nearly sent it back to us. The walnut table was not his favourite piece of furniture. All the same, when Helene claimed it in the divorce settlement, he refused to relinquish it on the grounds that it couldn't be moved.
The old woman reaches into a pocket and hands me a fistful of poppy seeds, wrapped up in an embroidered linen handkerchief. 'Why these?' I ask her.
'Well, since your garden's full of trenches...' She refers to a plumbing disaster at my house, and also to the Somme and the Marne and the connection between soldiers and poppies. If I did not know my mother, this gesture would be without significance. But even the hanky is meaningful. She is living the end of the embroidered linen era, when flowers were large and generous, china was fine, the silver solid and polished – never a quick-dry tablecloth, but threads, painstakingly pulled by Aunt Nina, who spoke so beautifully and announced, as she lay dying, that there was a cake nearly ready in the oven. We could eat it at the funeral.
In the drawers of mahogany and rosewood chests we were to find, lovingly labelled and left for us, a half-a-century-old batiste of gentlest yellow with nasturtium sprays in the corners and the edges fluted and French turned.
I can't help loving my mother and Aunt Nina for that: all that faultless ironing, all that care. The Red Cross labels from their Aid Detachment coats, the regimental colours kept in the button box, the photos in nursing uniforms, the ambulance driver, the debutante whose dress was remodelled from Vere's crepe-de-chine nightie. And our little letters, tied with Christmas ribbon, kept as treasures. Yet, how furious they made us! How small we sometimes felt and how unwilling to carry the burden their respectability thrust on us. How we despaired for the props of their endurance!
How solid Aunt Nina seemed, and at death still was – unshakeably loyal. While we, young and yellow as the dandelions nodding stupidly from her grave, wanted to yell, _Stop it! Cut it out! Quit it! You can't do this to us! Pass the family brain, Aunt Nina, it looks like custard, needs more gelatine!_
Too 'well bred' to hear, she went on stitching, delicate eyebrows raised, served us off the best china, the proper table, and never a serviette without its silver ring. Now the Midnight Knitter holds down bills with them, wraps poppy seeds in the hankies... But they're ironed, they're clean, they're polished all the same, and never a vase with something dead in it. Respect for things: things worthy of respect.
TEN
Our Father's Daughters
IN THE DAYS when men were men and women were their wives, O Best Beloved, in the way out, far off times when The Leader of the Opposition, Mr Whitlam, who was formerly the Deputy Leader of the Opposition, was, according to the Government, full of 'gall', 'recklessness', 'desperation', 'malice' and 'lying' (and his wife's name wasn't Marjorie), in those days, girls were 'raven-haired', 'attractive' and 'pert' and were the daughters of men. It was as if the men, and not their wives, had laboured hard and long to give their daughters life.
Our mother was known as 'Mrs Henry Coretti', but during the year of 1968, had she been newsworthy, like the 'attractive blonde from Sydney' who became Miss World, she would have become 'The Former Mrs Henry Coretti', because 'The Former Mrs Harry Laurington' became 'The Present Mrs Henry Coretti', the two years for desertion being up.
How did our mother take it? When the phone rang at home, she would give our number and say, 'This is the former Miss Motte speaking. How may I help you?' At work she would say 'Rudge and Plant, formerly Rumpton, Rudge and Russell, O'Rourke speaking.'
The troubled and destitute were still beating a path to our front door, among them an increasing number of Indians. The chap who had failed to levitate the stoppers from our decanters, name of Krishna, belonged to an obscure sect and called our mother Mummy. 'Ooo, Mummy,' he would begin his tale of woe, 'in the Punjab it is not possible to purchase proper medicine.' And Stella Coretti, née Motte, known variously as Motto, O'Rourke and Mummy, would twist the arm of some chemist she had befriended in the course of her labours and 'proper medicine' would appear in the Punjab. In our letterbox, letters in longhand from the grateful recipients, accompanied by new requests. Clothing required in Amritsa.
Our mother, a grazier's daughter, did not like synthetics. I, unmarried mother and inheritrix-in-waiting, had little money for clothes. My father bought me a new black boucle jumper. It went missing. A photograph from Amritsa turned up on our mantelpiece in which a grateful Indian was wearing it. I said, 'My jumper!' My mother said, 'Oh, that old thing. I was going to put it in the poor bag.'
Indians came, and priests whose vestments needed ironing. And not only ironing, but laundering too. The steam and dry iron, given me for my birthday by my father, caused such a sweaty pong to arise from the vestment armpits that the iron forever after wore a brown and bubbled skin on its ironing surface.
My toothbrush tasted of soap. Was it an accident? Money, presents and salves for Dadda's conscience were costing me.
'You shouldn't accept things from Dad,' Allegra said.
I felt like bursting. Little things such as a jumper meant a lot to me. There were also toys for Eli. Sometimes I sensed The Brolga's hand behind them and couldn't take them. Then they would come circuitously, ostensibly from Bart, turning up in the foyer at Figments, Bart's brother's gallery in Melbourne. Figments was right next door to Siècle, so I couldn't imagine Viva having a hand in getting them there.
Bart Turner continued to be very good to me. He came to Melbourne and organised a show of his painters' works at Figments. I got my picture in the paper. The caption read, 'Isobel Coretti (21), the pert, raven-haired daughter of artist Henry Coretti, shows that she, too, can paint. Several of Isobel's works are included in the "Turner at Figments" show in Harcourt Lane.'
Not one of the works was photographed. The photographer didn't even look.
Allegra said, 'It'd make you chuck.'
Dapper, asterisk-eyed Bart Turner, eating mandarins at the front desk of the gallery with his brother Miles, made a noise like a grasshopper jumping up and down in a paper bag (that was his way of laughing), and said, 'Well, don't chuck here, Allegra, or you might make headlines: "Artist's Daughter Chucks in Gallery Foyer – Part of Daring Show – Not Only Pornographic, but Sick As Well." Never mind. It's something to get a mention at all, considering how young and unknown we are.'
Bart's brother, Miles, was a hairy giant of a man, slow spoken and impenetrable, a weight in a doorway, reassuring when you considered what lay behind the door. In the 'Turner at Figments' show, naked women masturbated in murals, parallels were drawn between the bayonet and the penis, and frightfully injured Vietnamese women and children were pasted into very tasteful collages.
Harry Laurington was believed, temporarily, to have lost the power of speech. He was rumoured to be thinking of regaining it in order to henpeck Miles over pinching one of his painters. Less explicit works by the Master of Masturbation had once graced the walls of Siècle. Miles Turner made a noise like a doormat being thumped (that was his way of laughing), and said, 'Well, technically it's still on the same wall it was last time, except it's on the other side.'
Siècle was the third of five buildings along a dead-end lane. The lane took its name from the second building, called the Harcourt-Wilson, an elaborate monster left over from the gold rush and now used as a detoxification centre and a home for destitute men. The Methodists who ran it owned the unprepossessing church on the corner of Harcourt Lane. Harry Laurington had half an old bond store and Miles Turner had the other half. Harry had the better part of the deal because Siècle's steps faced in the direction from which people came, whereas the steps to Figments faced the dead end, giving onto the derelict back of a factory.
On the far side of Harcourt Lane, downhill, was a children's playground, hemmed in by Maggie and Kelly Kelly's Pantechnicon on the street and by the back of a panel-beater's at the dead end. With Maggie and Kelly close by, there was plenty of grass available for those who wished to partake. I still worked for Maggie and Kelly two days a week, but because the income from the Pantechnicon wasn't enough to pay the rent on the shop and to put Chantal and Eli into preschool, I was often lumbered with the babies. I became a very good one-handed, hip-laden painter and they became multicoloured children.
My contribution to 'Turner at Figments' had to do with the ironies of indoor/outdoor life in Paris, a city you might have imagined would be far from my thoughts as I mashed and nappied and trundled trikes.
However, ten million people were on strike in Paris and students were rioting in the streets, while indoors, the peace negotiations for Vietnam were stalled because the delegates couldn't agree on the shape of the table they were to sit around. Chantal and Eli couldn't agree to sit around a table, either, and I, feeling very much inclined to stage my own riot, was struck by the coincidence.
Another artist chose as his theme the assassination of Martin Luther King. Race riots, peace walks and the starving in Biafra also featured in 'Turner at Figments'. Yet the regular press showed next to no interest in what one reviewer termed 'infantile rage', until the arrival of the police and the peaceful, not to say debonair, exits of Miles and Bart to the waiting paddy wagon.
This occurred on the morning of the mandarins. Our show was five days old and we were in time to make the mid-week papers. Bart was stilled on the page calling out, 'Remember to feed the fish!' to me and Allegra as we waved him goodbye and the driver of the paddy wagon wondered how, having got the vehicle down Harcourt Lane, he was to execute a turn and get it out again. In the end he had to back, while I leant out into the main street, Chantal on one hip and Eli on the other, to see when it was all right for him to merge.
The charge was obscenity. People might still have been talking about it at the end of the week, but Sirhan Sirhan got in with his gun and on the Thursday, Bobby Kennedy was dead. It looked as though our art show would pass relatively unnoticed into history.
However...
Summoned to attend and defend Bart and Miles, the Brothers Turner, was one Mr Ashley Monday LIB, known to his friends as Wednesday, who arrived from Sydney by Fokker Friendship at Essendon Airport in the company of the elder accused. 'In spite of its being my first trip in an aeropla-yne,' Wednesday told the press in his slow, joy-riding voice that crackled in the troughs and relished the bends, 'I'm thrilled to annou-ounce that I didn't have to use the paper bag kindly supplied by TAA-ee to be sick in.'
His suit was creased and he wore no laces in his shoes. Sunken but bright of eye, craggy browed but perpetually amused, of all creatures on earth he resembled most the tawny frogmouth.
'Gosh,' he said amiably, when presented with the offending material in the courthouse, 'If you think that's obsce-ene, what about this?'
And he festooned his sparsely haired head with plastic boils, which, he pointed out, you could buy at any trick shop you cared to enter. Furthermore, there was nothing to prevent you from wearing them on public transport; he'd done so on many an occasion. 'Ver-y efficaycious in getting a seat,' he explained. 'But seriously,' it was hard to take Wednesday seriously with a head covered in fake boils and p-sounds that exploded in his mouth like small incendiary bombs, 'I've seen picture books on European cities available in public libraries on unrestricted lo-oan in which little or no effort has been made to conceal the public display-ee of genitals faithfully rendered in marble by persons who are considered to be patricians among the fay-mous.' Pausing while the word 'patrician' sank into his audience, Wednesday clasped his skinny, veinous hands to his chest. The top of his skull, with its illustrative boils, weighed heavily on the bottom so that his jaw stuck out like a saucer and his terrible teeth could be seen in his glistening gums.
'Obscenity,' he summed up, 'is a matter of taste. By some, I am considered an obscenity, and yet I walk the streets of Sydney every da-ay... in fact, the three days of this trial are the only days of my life, since I found my feet, that I have _not_ walked the streets of Sydney. And I have walked them unmolested. War is obsce-ene, and yet there is war. _Assassination_ is obscene, and yet there is assassination. Is the painting of a penis or a masturbating girl a more disturbing thing to see than the filmed slaying of innocents in war or the cutting down by assassins of those who strive for peace? In 1963 President Kennedy died in living rooms all over the world on slow videotape that was shown over and over and over again. Two months ago-o it was his brother's turn. I saw Bobby Kennedy die in a hamburger joint in King's Cross. I don't know where you saw it, but you're bound to have.
'You cannot condo-one the nightly showing of footage of the Vietnam War, and in the same spirit, conde-emn the "Turner at Figments" show – alas that it was ripped untimely from the wa-alls, because here were young people speaking out, here were young people trying to point out that hypocrisy has reached a high pitch in this country. Girls masturbate, why shouldn't they? There is no civil law against masturbation. Rockets and bombs are shaped like penises, and it seems to me there is quite a lot of point in saying so. While the middle class in this country decorates its walls with so-called "tay-asteful" art, people are dying foul deaths in Vietnam and Biafra; why shouldn't you look into a collage and be given pause to reflect on what it's made of?'
Wednesday removed his boils. 'Plastic,' he said, 'these sores are made of plastic. Girls are made of flesh and bloo-ood. They blee-eed. Physically and mentally they bleed. All the bedspreads and the decency in history won't alter that. This country is guilty of insisting on plastic blood, on pretending that Uncle Sam and the women's magazines know be-est. Yet we cherish the right to free speech. You cannot take yourselves seriously and at the same time prevent the expression of controversial views in your midst.'
You could tell by the expression on the magistrate's face that Wednesday was a dead man. There was a short, tense interval during which Miles was heard to mutter crossly, 'Why did you have to bring periods into it?'
'Oh no-o, Milesss,' sang Bart at his most operatic, 'why no-ot? I mean, why no-ot?'
The reason why not was pretty soon all too evident. The magistrate concluded, 'If a person does something offensive, their motives for doing it are immaterial.' Evidently the magistrate was no lover of poetic justice: Bart and Miles each got a two-hundred-dollar good-behaviour bond and Wednesday was charged with contempt of court. 'The court ought to be charged with contempt of _me,'_ he said.
Bart Turner sat shut up on himself like an umbrella. He was biting the mound of his thumb to stop himself from laughing. His eyes had disappeared into their asterisks and a tear or two slid feebly down his creases. Miles, too big for his chair, didn't think it was quite so hysterical. Large, fed-up breaths escaped from him and made the hairs in his nostrils quake as they passed.
'Oh, come along, Milesss,' Bart managed. 'The point isss, we've made a point.'
'You don't live in Melbourne,' Miles grunted. 'Make a point in Melbourne and they'll put a hat over it to stop it showing.'
There was a new show on the walls of Figments, a hard-edged and expressionless show. Bart and Allegra didn't like to say so, but managed to say at length that its Marxist content wasn't very high. Miles accused them of being ignorant, this kind of deadpan art was international and what was Marx about if he wasn't about internationalism? The old insistence on figuration and immediate accessibility smacked of 'bourgeois sentimentalism'. That was what Australia was suffering from, they had only to go next door and see Reg Sorby's show at Siècle for proof positive.
Siècle was one place in Melbourne where Allegra would not set foot. She was nevertheless curious to know what was happening there and somehow worked out that I would not be compromising my principles by going. Perhaps she thought I was thoroughly unprincipled, but I had an uneasy feeling that my principles, unattended by textbooks, were being overlooked.
Next door at Siècle Reg Sorby, according to one of his critics, was being 'led astray from an individual expressiveness into the desert of dullness'. I mounted the stairs a step or two behind Bart. 'Oh, Bart!' sang Checkie Laurington from the front desk, and then 'Oh, Isobel!' and she turned her back. Harry's eye glinted through a door at the far end of the gallery, then disappeared.
My footsteps made a very loud noise on Siècle's floor. Bart kept giving me sideways glances, pointing at the pictures and making a tragedy mask of his face. Reg had swatted wildflowers and bird wings to his canvases. It didn't improve them. The best-looking thing in Siècle, alas, was Checkie Laurington. She was tall and someone had cut her hair into a marvellous golden bob, shaping it into the nape of her long neck. Her mother, I hated to think it, had a neck like that.
Our inheritance was a long time coming down to us. It took eight months to clear the probate on Aunt Nina's will and auction off the house as a deceased estate. Tony Furlonger's cousin Jim ran the auction; there were only two bidders. Tony Furlonger bought the house. He bought everything except the two Kombi vanfuls Allegra and I had scuttled off with. The only large piece of furniture we were able to take had been a bed. We needed one for Eli.
Tony complained that Clare homestead was too big to be useful to him. Coolang had its own homestead, after all. Nevertheless, he moved his daughter Jocelyn and her husband Ralph in there. Ultimately they took it on a separate title from Coolang and renamed it 'Welaragang', which is what the Aborigines knew it as before great-grandfather Motte took it from them. That was perhaps the only decent fact to emerge from the whole tacky transaction.
After debts and duties, the combined inheritance of Allegra, our mother and me was $27 000. The walnut table alone is now worth five times that.
It was Bart Turner who put it to Allegra that in a world biased against women, it was time a woman owned an art gallery and exercised a female bias in the selection of the art to be shown. Why should Henry Coretti, he asked, be given all the advantages and have all the doors opened for him, when Isobel Coretti, who could probably paint just as well, had no such advantages lying in wait for her?
There was a house opposite the Pantechnicon which was reputed to be the oldest standing in the main street. Its demolition had been mooted several times, but was opposed by those with an interest in local history. It was a little wooden box; its metre-wide verandah ran the length of its front and was abutted at the street by an old picket fence. The door was in the middle of the front, just as in a picture book, and there was a chimney at either end of the pitched roof. It came up for auction just as probate was being cleared on Aunt Nina's will.
The door opened into a corridor on either side of which was a bedroom. Behind the left bedroom was a third bedroom. The corridor wall stopped behind the right bedroom, where there was a little sitting room. On a closed-in back verandah was a delapidated kitchen and a token bathroom. The taps didn't turn on. The toilet was up the backyard near the hen house, where there was a cage for a prize rooster who had been called Charlie. His vital statistics were painted on a slab of wood proudly mounted over his door.
Also up the back, which was really pretty small, was a shed that had obviously served as a sleepout in times of overcrowding. It was chock-a-block with junk: hand-made toys; several of Charlie's cups; a great many empties, which, if set up in chronological order would have told a history of beer-making in the city of Melbourne; and a handwritten copy of an advertisement placed in the _Argus_ at the turn of the century which read: 'Range of whips available for discerning gentleman customer. Most requests handled, apply P.O. Box...'
Allegra called the gallery Mad Meg, after the Bruegel we'd found under Uncle Garth's bed. It was with enthusiasm and a rebellious heart that I copied Meg onto a wooden sign which we eventually hung outside our door, like a sign outside an English pub.
We ripped up the aged floral carpet and gave it to Bridget Kelly, who reckoned she could make some waistcoats out of it. The boards under the carpet were eight inches wide and made of Baltic pine. We tried to lime the varnish off and made a mess of it. Bridget Kelly came and said, 'Nah. What you need on that's a floor sander. I got a floor sander down at my place, or half a floor sander, I should say.'
'It doesn't necessarily follow that a half-varnished floor has need of half a floor sander, Bridget,' our mother opined from behind her bucket in a slimy corner. She had stood by at the auction when Allegra bid for Mad Meg, fiercely ruing a transaction which was about to change an eight-bedroom mansion with servant's quarters into a three-bedroom timber cottage in inner Melbourne which had probably been owned by a servant. In spite of Allegra's having outbid several men for the place, we were barely on speaking terms with our mother. Bridget Kelly reckoned she knew a bloke who could come up with the other half of the floor sander. The bloke was Big Ernie, her husband.
With the aid of extension cords and an overloaded fuse box, Big Ernie attempted to drown us in chivalrous, varnish-removing sound. It was some time coming and during its imminence Big Ernie consumed what can only be called a volume of beer larger than his external dimensions would have seemed to allow. For Big Ernie was not so much big as large in presence.
Big Ernie was given to song rather than to speech, and of song, it seemed, he knew but one, this being 'The Ball at Kirrimuir'. 'Oh the mother superior, she was there,' he warbled between blasts, 'Sittin' by the fire/Knittin' contraceptives from/A worn-out rubber tyre...'
At the utterance of the word 'contraceptive' our own mother became extremely superior. Belonging, as she did, to the late medieval age, she took a dim view of condoms and diaphragms. Given the dim view, it was probably not too surprising that Dadda left her. As for the pill, it was another hideous trick worked on womankind. Furthermore, it was an incitement to adultery. Allegra and I had to go to extreme lengths to disguise any evidence of indulgence in the ordinary activities of our generation. There had been The Great Shoulder-Bag Emptying scene, which took place in our family kitchen and was not unlike the cigarette blitzes they used to carry out at school. Bridget Kelly used to reckon the only reason kids had to wear school uniform in Australia was so that everyone had the same set of pockets to empty out on fag blitz days.
During The Great Shoulder-Bag Emptying, it was revealed that Allegra Coretti (23), the parthenogenetically descended daughter of Henry, had in her possession a plastic and aluminium pill dispenser in which twenty-eight pills were arranged in a telltale ring. Our mother said, grimly, 'I thought you said it was only a word.'
'Who'll do me this time?/Who'll do me now?' sang Big Ernie as Bridget helped us whitewash the walls in another room. 'Change the record, for Chrissake, Ernie. It's killin' the cockroaches,' Bridget yelled.
The knickers protecting our mother's head from the ravages of paint and dust waggled at the leg holes. She spoke of enunciation and breeding. Allegra threatened to give her a face full of paint. Our mother's scraper paused on the wall. 'Madam,' she said to her elder child, 'I am fed up to the gills with paint. You can take your scraper and slice your tongue off with it.'
Ludicrous in her headgear, she stood her ground like a Maori warrior, her hand clenched on the paint scraper. Allegra set to attack. Her eyes flashing, her thumbs grinding with rage.
'Fuck off!' Allegra screamed and our mother flung the scraper in a wild arc. It stuck in the door by its corner.
'Shit!' said Bridget, flushing pink.
'My family home!' our mother yelled. 'This dump!'
'Don't, Mum, don't.' I tried to put my arms around her, but she clouted me. 'Vulgar tramps,' she said darkly as she left the room.
Then Allegra said, without compassion, 'Don't cry, Bel, just don't cry.'
That was the day the Russians rolled their tanks into Czechoslovakia, the day the National Gallery of Victoria opened, the day Reg Sorby's old dictum about figurative art being the only valid way an artist could express himself went out the window.
It was also the day we were told we were our father's daughters and our mother resolved never to have anything to do with us again, the day we had to find new living quarters for Allegra, me and my son.
On the evening of that day, I slept on the floor at Bridget Kelly's. Allegra didn't sleep at all.
Allegra and Kelly Kelly sat among the junk cartons and stacks of salvaged books smoking pot. The smell of marijuana mingled with what was wafting off the dump and vied for dominance. It was an awful fight. Maggie Kelly strummed her guitar and sang political songs in an extraordinarily gravelly voice. Chantal slept on the sheep dog under a table. Eli, having crawled into the sheep dog's kennel and erected a barbecue griddle over the doorway, sat up half the night growling, 'Wild ammimal. Wild ammimal.' Bridget Kelly snored in a hammock in the kitchen and her son, Little Ernie, sat in the backyard cleaning his rifle. Big Ernie spent the night plastered on the back steps of Mad Meg.
PART TWO
MAD MEG
ELEVEN
The Art of Bamboozlement
BETWEEN REMEMBRANCES, the old woman with the white hair is knitting. She knits with a morose doggedness and grinds her teeth. Every night she takes an extraordinary multicoloured article out of a bag she leaves in the armchair when she goes to bed. That can be at any time, as the whim takes her. I sit up with her, because if I don't, she wanders and loses her way.
'What's that you're knitting?' I ask her.
'Oh, you should know,' she mumbles. 'You're the one with all the bright ideas.'
The drawers of her mind fly open by default; a thief has been there and removed selected pieces – maybe he's the man in the shortie pyjamas.
Her profile is lit up by the fire: head forward, neat little nose and stubborn chin. She feels for her daughters love, hate, jealousy, pride, spite. Sometimes she says her genes have been diluted.
Her fight with Allegra over Mad Meg was spirited. Battle lines were soon firmly drawn. There were certain roads Allegra would not cross. She would not cross the quadrant of streets that surrounded Stella's house. Rudge and Plant's was out of bounds. Since Dadda and The Brolga shared premises close to the city but well south of Rudge and Plant's, any shop or gallery south of the Yarra was unvisitable. It took Allegra an inordinate amount of time to get from either Mad Meg or our house to the university where she worked, such were the boundaries, ethical, moral and ideological, that impeded the progress of the psychedelic Kombi.
Allegra and I rented an Edwardian house on the brontosaurian heights of our neighbourhood. It faced west, so its ambience was dominated by the setting sun. It exhaled terracotta, red, purple and blue, and threw velvet shadows behind it as the night came up.
Over the hump of the brontosaurian heights, facing north, was Harcourt Lane. On the high side, the warehouse containing Figments and Siècle was like the plain member of a pair, as the Harcourt-Wilson seemed to have been turned in the manner of a fig, so that its insides were on its outside. It was hung with generous balconies, on several of which were to be seen little weatherboard boxes with louvre windows, which presumably housed the overflow of derros and winos under Methodist care. On the corner of the main road on the high side, the pre-unification Methodist Church faced east, a squat building with a grey front. A couple of steps up from the footpath, and the once-monthly Uniting congregation was face to face with a pair of shut doors that were almost never in sunlight. The church, being only one storey tall and without a steeple, was robbed of its shadow by the Harcourt-Wilson behind it in the lane. In the passage of a day, its existence went unrecorded, shadow-wise, in the main street.
On the downhill corner of Harcourt Lane, the Pantechnicon faced east into the main street, watching the morning shadow-play on the road. Pedimented, spiked and cupola'd, this shadow-play was scored across by tram lines, or lost in the unevenness of the road, or it vied there in the rain with reflections. In the mornings, yellow arrived first, then pale grey. Mad Meg was white: its little bullnosed front verandah would frown across at the Pantechnicon opposite, like a person in bed who doesn't want to get up yet. Its mood correlated strongly with that of the Mad Meg collective, and it was generally unoccupied in the early hours.
I was the only painter in the collective, which was seven strong. Maggie Kelly joined when she discovered a talent for photography. Bridget had assembled a Pentax for her from collected bits, quite a feat when one considers the intricacy of Pentax cameras in 1970. In addition to Maggie, there were three critics, an art historian and Allegra, who was by this time a fully fledged social scientist. Everything we did was put to the vote, and often the outcome was six to one, the out-of-step party being me. I was into Caution in a big way.
Many women wanted to show with us. Apart from painters, there were potters, sculptors, printmakers and performance artists. We were pretty soon booked up, but as ever with this kind of enterprise, the cash flow was hardly commensurate with the enthusiasm. Though we could operate on Fridays and Saturdays, on other days of the week we were often all working, and if the artist couldn't caretake the gallery would be shut. This led Allegra and me to squabble because she thought I was the ideal person to caretake. I could live off the inheritance, hers as well as mine, if I wanted to. I didn't want to. I wanted to earn more money, not fritter away what we had already. Allegra tried to assure me that by the time the inheritance money ran out, private property would have been taken over by the state anyway, the political left being in the offing. But I was cautious: there was Eli to think of and I wasn't about to go begging Dadda for his upkeep; furthermore, although the words 'Property is Theft' were often bandied around, nobody except Allegra was of the opinion that the ALP was going to disinherit them in the name of a fair go.
Our openings, for which Maggie, Kelly and I had a roster for childminding, were held on Friday nights. It was good of Kelly Kelly to take on something that wasn't her responsibility, but she was a loving aunt. Quiet and mysterious, she hid behind her hair, which was very long, very fair and very straight – that she should have come into the world through a rubbish tip was unbelievable.
On opening nights, perhaps because we were at the base of the brontosaurian hill and not at the apex, a certain group of dishevelled men would congregate on the footpath outside, trying by various nonchalant manoeuvres to look like guests and merit a glass of plonk. If things were especially crowded they would get their glass of plonk by accident and regale the company with philosophy and song. These were the gentlemen who rotated through the detox, changing places with each other down to the last cubic centimetre of liver. Periodically among them was Big Ernie Kelly.
One night when Kelly was babysitting and Mad Meg was bursting at the seams, Big Ernie did us the honour and rolled round with the district's most famous drunk, Courtly Tom. Six feet four inches tall, Courtly Tom had slicked back his hair with Californian Poppy all his life. He wore his beer belly high, which meant he had a creased crutch. Surrounding him was an ether; he said it was his aura and that it was so strong, other people could smell it – they probably could have lit it with a blue flame, too.
Courtly Tom had once played rugby for Wales and had appeared on Melbourne television in the 1950s, dressed as a sultan in a children's show. Whether in or out of detox he could be seen on Wednesday afternoons dressed in a tux, refulgent with use, getting onto a tram, bound, it was said, for Pentridge Gaol where he taught the inmates the intricacies of trombone playing.
On the evening of his appearance at Mad Meg he favoured the company with the toreador song from _Carmen_. Maggie fled, embarrassed, to the back room. We followed her and found her furiously smoking. She turned her back on us, tensely shredding a match between her thumbnails.
'Come on, Maggie, it's all right,' Allegra coaxed. 'It's nothing to get upset over. They're just two men wanting to have a good time, but they haven't got enough money to go to the pub. Let's face it, most people's problems would be solved if you gave 'em a fistful of dollars.'
'Every time someone gives my old man a fistful of dollars,' said Maggie, still with her back turned, 'he spends it on the TAB or winds up in the detox.'
'It's because he never had a chance, Mag, that's all. He was born into poverty. It's not his fault. But things are going to change, Mag, they're going to change. Poverty is going to be eliminated, believe me.'
Maggie flushed and tossed her head. 'God, you know what you look like when you say that, Allegra? You look like an imported Italian doll. You might have lived here all your life, but what do you know about it, except what it looks like?' She burst into tears and flung herself down at our desk, burying her head in her hands. The cigarette she was holding burnt a hole in her long fair hair.
'There are causes, Mag, economic causes. Money isn't properly distributed.'
Maggie sat up wet-faced and laughed. 'You want to know what causes poverty, Allegra? I'll tell you. The more you hunt, the more causes you'll find. There's a cause for every example, there are individual and collective causes, minor and major causes, accidental and deliberate causes, external and internal causes. University faculties get built in the name of causes, but poverty goes on forever.'
'You ought to go to uni, Maggie. You'd be a natural.'
But, as things stood, Maggie couldn't go to uni. She'd been busy having Chantal when girls her age at better schools were matriculating.
If Maggie often felt patronised by Allegra, so did I. Yet, except when I felt it imperative to have my way, we didn't fight. Allegra was as protective as she had always been and turned a blind eye to my falling a long way short of the Marxist feminist ideal. I was always the exception to the rule, and, when the pack closed in on me, baying for blood, Allegra would stand between them and me until I could safely get myself out of the road.
But, while Allegra was protective, neither she nor my mother was forgiving. I carried the war dispatches to and fro with gusto, over the brontosaurian hill, through the grounds of the brontosaur itself, and down the primrose path to the hobs of hell, where I would shake the bell and drum collection at the door and it would whoosh open and there she'd be, Stella Motte in the attitude of a valkyrie.
_Allegra_ | (owner of the vacuum cleaner) to _Stella_ (borrower of the vacuum cleaner): Where's the vacuum cleaner?
---|---
_Stella_ | (who needs the iron) to _Allegra_ (who has the iron): Cleaning the vacuum. Who said you could take the iron?
_Allegra_ | (who gutsed the entire cream cake over three days) to _Stella_ (who baked the cream cake): The iron is Isobel's. The cream on that cake you sent was _off_!
They were turning the humble huff into a high art form. I was sure there was a knack for getting them back together, but I couldn't think what it was. I was to and fro like Balashev from the Czar to Napoleon in _War and Peace_.
The problem, of course, was the dwindling down of Clare and all its delusions of grandeur to Mad Meg and its left-wing social conscience. Stella had ideas of lineage and breeding which had not leapt the generation gap. They lay in the intergenerational ditch stunned, but still adept at attaining the moral high ground.
While her daughters spent the family fortune on alien ideas, Stella filled our erstwhile home with the dispossessed. Members of a certain faith played strange stringed instruments on her sitting-room floor. A girl with a baby and a musk ox bladder was living in Dadda's studio. Her dog fouled the Genoa cut velvet couch. A woman poet and some kind of Tibetan holy man slept together in our former bedroom. Stella ordered me home to identify a substance the holy man had wrapped up in a handkerchief. 'Heroin,' I said, and she asked how it was I knew. Had I been injecting myself? Perhaps I was injecting her grandson, Eli? For all I knew, the 'heroin' could have been talcum powder, but I was so annoyed with her predilection for bizarre strangers, I was determined to bash back some of the guilt she was peddling my way.
When illness broke out in her commune, she consulted Aunt Nina's first aid book from 1910. She diagnosed the musk ox girl as having corpora lutea. I was summoned to the showdown. No doubt the musk ox girl had come by her corpora lutea from drugs or sex. 'Probably sex,' I told my mother. 'I've had them, too.'
She looked at me with a hung face. And just what, she would like to know, did I take to get rid of them? 'Nothing,' I told her. 'I get them every time I ovulate.' She flew to the first aid book. Did 'ovulate' mean what she thought it did? She looked at me, horrified, with a crimped top lip. 'I dare say you get them, too,' I said. 'They're hereditary. Mothers pass them on to their daughters.'
'But you children never had anything wrong with you when you were little!'
I was about to feel sorry for her, but she added, 'Except for you,' as if I was defective. 'You had asthma.'
'Yellow bodies, Mum,' I said, checking my rising spleen. 'That's what corpora lutea are, little deposits of fat left in your ovaries after you've shed an egg. Every functioning female has them. And most of us have neurones, too. Presumably you do. Presumably they're in working order!'
That being the case, why could I not use my father's studio to paint in? Why was some hippie allowed to stink the place out with shitty nappies and a musk ox bladder? I was told that I had been allowed to stink the place out with shitty nappies and paint. Now it was someone else's turn. Somehow or other, the hippie and her baby were being blamed on me.
The poet who shared our bedroom with a saffron-robed monk earned herself some money occasionally by playing a guitar and singing in a folk attic. As I fought with my mother in her kitchen, squalid with her boarders' messes, this person in the back room sang wanly:
_Then by and come the king himself Looked up with pitiful eye, 'Come down, come down, Mary Hamilton, Tonight ye'll dine with me.'_
My mother sat on one of the two unencumbered kitchen chairs and played Mary Hamilton for all she was worth.
The light in the kitchen in those days was lugubrious. One of her boarders, unaware of the intricacies involved in turning on the sink heater, had allowed too much gas to accumulate before lighting the jet and there had been an explosion. The bottom of the blind had been singed off on the diagonal, and one of the window panes had broken, necessitating a brown paper square to be cut and placed over the broken glass. A less lenient landlady would have required the gas igniter to pay for the damage, but the standard of tenant led an outsider to suspect that the landlady would be lucky even to be collecting the rent.
There were no kings to come and cast their eyes on her. Boo-boos other people would have been thankful to relinquish years ago had begun to dance to her tune like mindless clowns. _See that I am good,_ she seemed to be saying, _and rescue me from this hellish pit._
But I doubt, if you'd dangled a rope of any kind into that pit, that she would have seen it in the chaos of her despair. Or, if she saw it, she would say the mindless clowns needed her and she couldn't come today.
Off she would trudge to Rudge and Plant each morning; back she would trudge each night. With resignation and more good nature than was warranted, she extended her bounty to all who were in need. She'd had her instructions in goodness from rural Anglicans. The philanthropy of country vicars had taken root and grown into an earth-hugging, wide-spreading tree whose fruit was plentiful, but whose shade prevented small things under it from growing into more than stunted suggestions of what they might have been. Lurking by habit in the shade of her embrace, they were sun-shy and behaved very badly in winter when they were reached by the day's cold light. The silhouette of Stella's tree in winter was formed of hybrid heads, lop-eared, crook-snouted, cock-jawed. They bayed and mewed and lamented at discovery, but seldom scuttled out to fend for themselves in the world; for the tree, in its way, was strong and all-providing. However, it had come to occupy too much lateral territory at the expense of height, and the weight of its branches was bringing Stella down.
She saw her task as the fostering of intrinsic goodness in her foundlings. Sometimes she succeeded. She hid adolescents from violent or drunken parents and battered wives from brutal husbands. These days people would find a degree of self-abuse in her actions, but then, when the only havens available were for alcoholic men, Stella became a nucleus for the weak and the unlucky and, in time, many people owed her for her kindness.
But if Stella could shore up people's hopes, she could also approve or disapprove of the nature of hope. For her and for Allegra, I was somewhere between an ally and a cause. In this position, naturally, I could always be an enemy or a failure if things went wrong. I had never been welcome to talk about Arnie, even though I needed to. I was to put Arnie by into the drawer of things for which history, at a later date, would be expected to atone.
Stella's belief in goodness was a belief that she was good and therefore worthy of the family motto, 'Never fail another' (if indeed that was the family motto). Allegra's escutcheon was not borne with anything like the same assurance. Allegra was alive to our mother's unintentional tyranny. Somehow, Dadda's leaving had come to be my fault. Perhaps I had conspired with him on those traipses through the dump. Or I hadn't kept my eye on him properly. Whatever I'd done, I had proved myself useless as a saviour. Allegra vacillated: there were times when it seemed Stella's reasoning was right and I was a lousy saviour, and times when she was most definitely and categorically wrong and Dadda's leaving was not my fault.
I tried to imagine myself as my mother and Allegra would have had me: I ought to have been a comical, charismatic giant, a Mad Meg who, whenever danger threatened, would have made everybody laugh instead of being afraid. Somehow or other, I would have fetched in money and bestowed it, while Allegra and my mother would have spent it, giving me a reason to go out and fetch in more.
Allegra lived in the dormer-windowed ceiling of our house, guarded at the roof ridge by a terracotta dragon with wings. Eli and I had all the ground floor to ourselves. We had hardly any furniture and the place was large, so Eli was able to invent himself a jungle.
The house had a high, tiled verandah that gave it the feeling of a ship breasting the waves of roofs and trees that fell away to the city below. In sunsets the interior walls were rosy pink, and sometimes cloud shadows sarabanded through them. Our old cat, Silk, had a favourite resting place at the top of the front steps, where she sat with her eyes just shut, purring the last of the colours from the day. I set up my easel in the front room and every evening I tried to follow the colours into night, but they would stay on my eye even after Allegra had turned on the lights upstairs and Eli was hugging my legs, wanting to be fed and put to bed.
In accordance with feminist principles, Eli was given dolls for his birthdays. While Chantal Kelly reshaped the back garden with her graders and bulldozers, Eli put his dollies in a wagon, made himself a gun out of a piece of wood, wore a crash helmet given him by Bridget Kelly and sang out, 'Come on, men!' as he staged charges and raids up and down the hall and in and out of bedrooms. Sometimes the place would be thick with trees, and sometimes it would be a sandy desert. One day I heard him mumble, 'Biggles lit another cigarette,' and I knew he was under outside influence.
Four days a week in their last year before school, Eli and Chantal attended a red brick kindergarten with a fenced-in, tanbark yard. The weekly cost of this privilege would have kept a child in South East Asia for several months. My inheritance being largely spoken for by Mad Meg, the rent on our house, and my future plans for Eli, meant I had to take a job. I potted orchids in a public garden. This left me little time to examine the content of stories read to innocents in the kindergarten. If Allegra had heard Biggles alluded to in play, she would have been all for fronting Mr Goodfellow, who owned the kindergarten, and demanding that only ideologically sound stories be read. As it was, because Mr Goodfellow called Eli Mr Blue-eyes, Allegra said he was teaching him how to use his good looks as a weapon. But Eli was just as likely to go round with a bag over his head as he was to go round wearing a crown, a fact Allegra happily overlooked.
According to feminist principles, the absence of Eli's father from our lives ought to have been looked upon as a blessing rather than a curse, but, from the time Eli knew there were fathers in the world, he had set out on a father hunt. Ask Eli what he wanted to be when he grew up and he'd say, 'Normal.'
Along the road to normality, he passed a bride in full regalia coming out of a Greek Orthodox church. He asked Allegra, whose hand he was holding, what the lady was doing. 'She's getting married,' Allegra said. 'That man she's with now is going to be nice to her for the rest of her life.'
'Why?' asked Eli.
'Because he has to. Those are the rules. He's made a promise.'
Eli watched the bride over his shoulder, and plucked his rosy cheek in thought.
'Do kids get married?' he asked as the couple drove off.
'No. You have to wait till you're grown up,' said Allegra. 'Why doesn't the chap wear something fancy?'
When Eli and Chantal Kelly were married, Eli wore the dress. He wore a latterday hat of Aunt Nina's and carried a bouquet of nasturtiums. All the other kids on the block came to look him over and offer their suggestions. Someone brought net for a train. Chantal wore a pair of shorts, a plastic moustache and the crash helmet. Maggie took photos and Allegra performed the ceremony.
First Eli stood by the garbage tin with his belly poking out, trying to tie a sash around his middle. Then Chantal had to do it for him. Then he put his nasturtiums on top of the garbage tin and bent over to fix his hem. Then he posed with his legs wide apart, his thumbs in his ears and a wild look on his face. Chantal stood back, poked her belly forward and blew her cheeks out. Eli rolled on the lawn, Chantal wafted the bouquet overhead. Eli had his pyjamas on under the dress and was wearing gumboots. The marriage celebrant failed to have the wedding taken seriously, due to an overflow of spirits. Later, attempts were made to impress the gravity of the situation on the 'bride'. Allegra thought it was important for Eli to understand the nature of marriage.
'Where's Mum's husband, then?' Eli was heard to ask.
'Well, that's just it. If he loved you, he'd be here.'
'My dad?'
I snaffled him up and took him to his room. 'It doesn't matter your Dadda isn't here,' I said. 'You can have mine.'
But Arnie's absence did matter. Time passed, but there was no forgetting him. There'd been a few attempts at boyfriends since, but nothing worked. I tried to dismantle him, but I harboured an objection deep inside myself. Certainly he had been a bad choice as a lover, but I wouldn't have exchanged Eli for any other child, even given a wide choice of father, and that meant to me there had been something right about Arnie.
In spite of herself, Allegra seemed to understand my quandary. We never talked about Arnie. If he loved you, he'd be here, was the closest I had heard to an opinion. I had constructed a story for Arnie that was the opposite: he stayed away because he loved us. His presence would have interfered in our lives. He was nothing like me. He had loved me across a boundary that isn't normally crossed. I was some sort of tragic exotic and he, an Edward Rochester, chained in marriage.
I imagined Arnie must have yearned for me as I yearned for him, but I'm sure now that he didn't. I'm sure if he thought of me, he was in a hurry to put me out of his mind. In his imagination, my forbidden child was probably no more than a poignant embellishment of himself.
At my job, I learnt about orchids. The nursery where I worked was in a public park. It had a herbarium and a chief botanist, a Dr Beryl Blake, who, although not a mother in fact, had an Earth Mother dimension in her character, and was a mother by proxy. No man had clattered by to claim her on his steed of fire and it wasn't hard to guess why, for Beryl was as wide as she was tall, her largest dimension being that around which a tape measure might have been at pains to meet itself, that part known in other women as the waist. Her bulk here made it impossible for her to see where she was putting her feet, so that rougher pieces of ground and stairs were known to seriously thwart her progress. This progress, to avoid the embarrassment of frequent prostration, was conducted by and large over a single course which covered the distance from her office to the compactus in the herbarium where a bottle of whisky was kept, no doubt to bamboozle the cockroaches. Checking the bamboozling involved frequent visits from this gruff but kindly woman under whose auspices unmarried mothers, illegal immigrants and homosexuals were paid to play poker in the potting shed and pot the occasional plant.
At my interview she sat with difficulty on her chair, its plankiness not being particularly friendly to her sphericity. 'We're putting you on orchids,' she said. 'And I understand you draw. Ultimately I might be able to persuade the powers that be to let you team up with Loyola in the herbarium, though whether she'll let you do orchids or not's another question. She likes to do the orchids herself.'
Loyola? Someone less devout than the little grey-haired draughtsperson who was queen of the herbarium might have dropped the o's and called herself Lyla when she discovered the world of Colleens, Pats, Carmels and Marys into which she had been born, but Loyola O'Flynn took her name very seriously. A crucifix was the dominant item of her clothing.
In the younger days of Beryl Blake, those before her eyelids had everted, showing a sorrowful pink interior to the world, there had been, I was told, 'a man', so I didn't have to tell her what they were like, or what I must have gone through. She knew. She knew. Further, it was a sorry world that judged a salary by the gender of its recipient, but there was nothing much she could do about that.
The delivery of this opinion was accompanied by some rather dangerous lurching, corrected by feet that had seemed to be having difficulty with the angle of the floor. To the eye of a civilian the floor seemed horizontal enough, but then, the civilian's eye was a little less bloodshot than that of the chief botanist.
I was directed to take the path from the herbarium to the potting shed and to introduce myself to the conservatory ganger, whose name was Abdul bin Hadji.
I noticed, on my way to the potting shed, a dark fold in the garden into which a green painted hut seemed to have been slid, like something stolen, into a giant's pocket.
For all the public knew, this was the place where rakes and barrows were kept. The more curious among the public, however, might have wondered at the staccato sounds coming out of it, an _ack ack ack_ of human voices, as opposed to a _yack yack yack_. They might also have wondered at the smell of garlic, ginger and soy sauce. If those curious members of the public had worked for the Department of Immigration, they might have done more than pause and observe. Because the contents of the Green Hut, or Crin Hu, as it was known to the cognoscenti, was a somewhat fluid population of Malaysian Chinese and Singaporean Malays who, for fear of the race riots going on, did not wish to return to their home countries.
For the most part, they were students or trainees in botany, agriculture or forestry. They tended to spend the daylight hours indoors, due to visa expiry and the aftermath of the White Australia Policy, which made them conspicuous on the street. I understood that, apart from unintentionally providing a roof over their heads, the government of Victoria was not underwriting their stay. A certain bishop was known to drop into the 'Herb' and consult with the cockroach-bamboozling oracle of the compactus in the company of the chief botanist. Money would change hands.
Orchids, I discovered, like a tight pot. They like a soupçon of compost and aren't above lounging around up to their petals in filth. Some even smell like dung heaps and attract stercoraceous insects to them to perform cross-pollination. Perhaps the original orchid was a coprophile, though some smell sweet and bring the honey-eaters. Others are blue, and therefore, highly attractive to bees. And others yet mimic a female wasp in shape so males will try to mate with them and thereby perform the proxy ceremony on which all, or certainly, most, orchids rely to set their seeds.
Orchids are hermaphrodites, but outward-looking. If they were narcissists, they'd never have been able to form such an array. Some of them even bloom below the ground. It may seem unlikely, but orchid mutations could be hastened by delaying tactics. Just as a knitter turns a heel on a sock by stalling on a round needle and interrupting the even flow of stitches, so may orchid petals fuse to make hoods. And just as a cabler translocates within a row, orchids may translocate to turn their heads. Mother Nature is a knitter, prone to err.
My mother drives her needle into a row of purl, lolls her old white head to one side, sighing, and says of the garment, 'It's the ravelled sleeve of care.'
Stitches are added in or dropped or knitted back-to-front. Every now and then I have to untangle the dozen balls of different wool. For a long time, her knitting was precise and very correct, but she only knitted one thing, over and over. It was a Fair Isle pattern jumper front. She lost it in stages. First, she would switch to plain in the middle of a purl row, and that would put her Fair Isle out of sync. Half the neck would appear too soon and she'd be bamboozled by the asymmetry of the other half. She'd ask me had I tampered with it, or had I seen anyone tamper with it? Nowadays she'll be going along splendidly, knitting at random, and then she'll suddenly disown what she's done. 'I didn't do that,' she'll say. 'That looks like nothing on earth.' Other times, she'll be sad and say, 'I think they've taken away my pattern.'
TWELVE
Encroachment
I AM PAINTING, but doing it all wrong. I don't obey the rules of the collective. I am riven. At night in my dreams I practise Arnie. I make good his deficiencies and perfect him, making myself whole again. By day, my thinking self does him to death, dissolves him in wells of light. Removes and replaces him with something bigger than he is.
Whatever I say to the collective is greeted by scorn. We sit on the polished floor of Mad Meg in a circle, where the cigarette smoke curls itself into itself, like lovers tumbling. We bicker over feminist semantics: an art historian, three critics, a photographer, a political scientist and me. Six Marxists and a woman who paints, or a painter who's a woman. I am dragged from my shelf, laid out, dissected and improved upon. When I paint a cup on a table, I'm painting a female symbol – womb and nourishment – when I thought I was painting loneliness.
I am told I shouldn't want heroes, but I want them. I want life to be bigger than it is. Obey the rules, the Marxist sisters seem to say, and life will be bigger for everyone. But artists, like lovers, are mad people, and painting, like love, is inextricable from madness. Remove painting from a painter and you have someone who renders the world in imagined strokes of paint; remove loving from a lover and you have someone who endows every threshold with the expectation of a beloved step, a longed-for shadow cast into the room.
But when love fails, there is yet more life. I have to agree with the collective: it is my condition. I am weighed down. I paint an open wardrobe, stuffed with female clutter on one side, on the other, a wire coat hanger and a man's dropped shoe. The collective loves it. I call it Equal Hanging Space.
I try not to practise Arnie by molesting my bedclothes every night. A man is not an assemblage of pillows. I try to think of something or someone else, but my love, alas, is steadfast. It is with me every day, tormenting me. I have learnt him, he is on me like my skin and in me, an unremitting pain. When I look for new love, I find my old love: a machine swings round to measure whoever I am looking at. Blue eyes, it says, and straight fair hair; tall, it says, with plump, sensual lips. Unreasonably good-looking, it says, and I try to get it to say something else. I move its calipers to fit some short, dark, brown-eyed man with an interest in art, but I find his speech pattern is identical to Arnie's. Every time I choose, the machine finds a parameter; it surfaces after a little acquaintance, it says, 'See, you fooled yourself. You're always fooling yourself, Isobel. You think you've hooked onto something special that sets a man apart and all you've done is gather in another aspect of Prince Charming. He is going to crop up like a paper chase all your life.'
Then I have to ask myself was Arnie all that bad? If he'd had a different component in the faith compartment, would he have betrayed his wife and stayed with me? And it seems to me that men are just machines. You can fool a machine it's in love and it will behave as if it's in love until some displacing factor gets fed into its network, and then it will behave as if it isn't in love anymore. Or not with you, anyway.
While I am having these thoughts, the calipers start measuring a middle-sized, hazel-eyed actor with a beard. There is nothing, on scrutiny, that seems familiar there...
I marched with Roscoe in the May Moratorium of 1970. He was a moral veteran from 1968 and proudly pinned his Moratorium badges all over Eli's dark blue top coat. At four and a half, Eli came on the march, even though certain eminent religious gentlemen had prophesied the imperilment of life and limb. He had to come, I couldn't find a sitter. Everyone I knew was marching and I was too proud to ask my mother.
Stella took no interest in the war, but when someone wrote PEACE in big red letters on the pillars of the cenotaph, and was accused of desecration by the premier, she declared herself all the way with the desecrater.
I decided if Chantal Kelly could march, then so could Eli.
The worst happened, and we lost him in the crush and, worse still, we found him. He was discovered, badgeless, with thirty-five dollars in his pocket, handing over, in exchange for a further $3.60, a sixty-eight badge to terrible Reg Sorby. Reg, ignoring the family kafuffle, made his way, pinning on the badge, to the front of the now-seated crowd and grabbed a microphone. 'And they thought we were a mob of ratbags,' he proclaimed.
People carpeted the Treasury Gardens, where Dadda and Mum once, in another month of May, plighted their troth among the elms and claret ash. More people carpeted the roads as far as the eye could see. There was laughter and cheering for Reg, because the Federal Police had just been to his rural retreat to arrest a cocky he'd entered in the draft.
'I can't stand that man,' said Roscoe, and stepped out of our lives in his buffalo-hide sandals, over the seated throng. Then we spotted Dadda. He was wearing pale blue denims and a waistcoat. His hair was shoulder length and he had pink laces in his blue suede shoes. As if drawn to a magnet, Eli plastered himself to Dadda's front. They loved each other with all the zeal that was lacking in my life.
_Dilly dilly, dilly dilly, come and be killed, Dilly dilly, dilly dilly, come and be killed._
I am singing to Eli. Outside the moon is riding in her chariot of clouds: moon, the reflected glory of the sun.
Down the hall, in the low, white kitchen, Allegra is eating eggs. She clouts the spoon on her teeth at every bite, evoking the wraith of Nina, tut-tutting our appalling manners. I hear her peanut-buttering her toast, then her chair teeters back on the ill-laid lino and I hear the foot of her boot clunk onto the table among the plates. Usually it takes more than one foot swing to perform this action.
She is taking on her feminist self for the sake of the collective tonight, but her boots are turquoise mock-croc plastic. They lace up the side. We saw them on a revolving mirrored disc in a window in town and knew that one of us had to have them. They're hand-made French, a demonstration pair for the opening of a boutique. She bought one boot and I the other.
She is telling the assembled Marxists, with her mouth full, that she thinks my work is ideologically sound after all. It is an incidental statement made during a collective meeting held at our house as a dispensation to my maternity. My presence is being awaited and I am pretending that Eli is not yet asleep.
Tonight, the Mad Meg collective debates the desirability of sending three hundred dollars to the communists in Cambodia. I hoped I would be excused on account of maternal duties and no sitter, but the collective decided to do me the great honour of coming to me so I would not have to go to them.
I keep wanting to pinch Eli to wake him up, but his little red mouth is fluttering in sleep. 'Dilly dilly, dilly dilly,' I sing, but Allegra's footstep sounds in the hall and she says, _sotto voce_ but impatiently, 'Come on Bel, we haven't got all noight.'
She has developed a very proletarian way of saying 'nate'. Not 'nace' at all. It is just as well Aunt Nina is not in the 'hice' to hear her. Furthermore, her deportment could do with some attention: she drags her heels. 'Well, you would, too, if your heels were as high as these,' she says, and I remind her that one of the heels is mine. She says, 'Oh yeah. Sorry. I'll get 'em mended.'
The collective crunches up its egg shells and shoves them into the brickette hot water heater. It chucks the knives and the teaspoons into the sink and paws the tabletop to remove the crumbs. Teeth are noisily sucked. The fridge, on its last legs, shudders to a standstill. Out come the fags.
As I take my seat Allegra stands behind me, her hands lightly on the back of my chair. 'I think we'll save the Cambodian vote till the end,' she says, drumming her fingers. And then, guiltily, 'God, Bel, don't look so abject. You're sitting there all scrunched up in your caftan like a lump of hair at a Hare Krishna convention.'
'It's the Cambodian vote,' I say. 'I don't think we ought to send the money.'
'We're saving that till last,' she says in my ear.
'Well, you might be, but I've been reading about it in the paper.'
'Oh?' Imperturbably. 'Well, it probably ain't true, Bel, what you read.'
'I think it is.' And in this kitchen in suburban Melbourne, a tacked-on room with a low, white, smoke-stained ceiling, I get emotional about Cambodian rivers choked with butchered bodies. I think I can see them, I think I can smell them.
'Listen,' she says, sitting down beside me and looking up into my face, 'how do you know the communists did it?'
'Because the report I read said so. The bodies were South Vietnamese. They were rubber workers. They worked for the French. I suppose you think that's a double sin – wrong nationality, wrong sponsors? The people who saw them were on a ferry. They counted four hundred bodies. Some of them were bound together with wire. The stink was so bad the people who saw them couldn't stop vomiting.'
'American propaganda,' says one of the art critics. There is a collective stabbing out of butts.
'So are we to assume,' I cry with venom, 'that everything coming out of Vietnam and Cambodia that doesn't come from the communists or doesn't support Marx is treacherous lies?'
Allegra slowly swivels round to me with her face lowered. 'You didn't say this was how you felt when the issue was first raised.'
I stare at the passive, mottled hands in my lap. They look like newborn rats. 'I didn't fully understand. It's one thing to be sending money to the National Liberation Front because we don't believe in American puppet governments, but quite another to be sending money to butchers.'
'Everyone makes mistakes in war,' she says.
'Oh, feeble, Allegra, feeble!' I feel the words hurdling my lips to make their mark. She swivels slowly away, her head still down.
'Shit!' goes the collective, and someone mutters, 'We're not even up to the first item on the agenda yet and you two are at each other's throats.'
Allegra and I crouch back into our chairs.
'We'll cross that one off, then,' says Allegra, quietly. 'We'll deal with it another time.' One of the critics then flies in from the left. 'No! I think it's important. I mean, where does Isobel stand? We're talking about a national struggle here.'
'No, we're not,' croaks Maggie Kelly. 'We're talking about the first item on the agenda.' After all those years of standing on school benches in disgrace, Maggie Kelly has a better idea of what constitutes injustice than any of us. The Nouveau Proletarian wing of the collective can talk till it's black in the face about being a gallery with a social conscience and adopting positions on Vietnam and Cambodia, but for Maggie local justice takes precedence.
There is a short war between Maggie and the Nouveau Proletarians but Maggie, who shoots from the hip and has a voice like a lawnmower, wins, and the Troika lapses into sulks. Maggie puts on her glasses, which replace her eyes with a pair of Op Art kitchen interiors, and takes from her leather-fringed Pantechnicon bag a wad of complaints.
For years our local tip has neighboured a swamp, a swamp that is slowly claiming a one-time billabong of the Yarra. In its puddles and marshy recesses mosquitoes buzz, and not only mosquitoes but also gnats and dragonflies, among the marginal, emergent weeds.
A cat with six kittens lives in a box at its drying edge. Eels writhe through its mud and lizards sun themselves in its mottles of light. Mynah birds drive ducks away by stuffing rubbish into their nests.
Among the swamp's visitors, the crane and the kookaburra, mating pairs of green parrots, wrens and wagtails, silvereyes and wattle birds, magpies, ravens and ibis.
Bacteria ooze in the ooze. Phyto- and zoo-plankton plank on the bacteria. Browsers browse on the plankton. Everything with a mouth goes for everything without a mouth, and those without mouths multiply and give off the odour characteristic of swamps, an odour which has drawn to it two waves of vigilant men.
The first wave consisted of men in Bermuda socks with clipboards and theodolites. Their arrival aroused in Bridget Kelly, Caretaker Grade 2 (female), preparations for war. The men were surveyors, their object, landfill, the demolition of Bridget's house, the end of her livelihood and the beginning of a large, expensive freeway.
The Mad Meg collective is to design the protest banners. FREEWAY NO WAY they'll say, and SAVE OUR HOMES. SAVE OUR TIP, two smaller banners will declare, to be picked out in psychedelic rubbish colours by Eli and Chantal.
There are two projected routes for the freeway. Both annihilate Bridget Kelly and one would take Mad Meg and the Pantechnicon with her. There are barricades to be womaned and parliament to be marched upon.
Child-minding contingencies and surety in the event of jail are to be left to the neurotic quarter of the Mad Meg collective. The minute-taker notes down my name and Maggie Kelly's.
Maggie Kelly tells of graders and bulldozers that are creeping from the east with caution and many tea breaks, and of parliamentarians scheming.
These creep and scheme while Bridget Kelly feeds bread to the displaced ducks and throws stones at the mynahs. If the ducks are too slow to take the bread, it's taken by a flock of seagulls to whom the solidified sea of the dump is Mecca.
These seagulls have caused the arrival of the second wave of men. They do not come from the Department of Main Roads as did the surveyors. They come under the auspices of the Federal Department of Aviation. They are scientists from Canberra. Among them the large-bottomed, pinheaded variety, who stands head and shoulders above his team and keeps the sun from his beak with a towelling hat. Bridget Kelly clobbers him amidships with dirt clods when he isn't looking. Another, who is tall from the small of the back to the top of the head, but short from the small of the back to the bottoms of his feet, can't find a hat large enough to accommodate both his head and his hair, and so goes without. Bridget Kelly likes him because he thinks what he's doing is a heap of governmental bullshit. She has sold him a filing cabinet and a shade to protect his nose from sunburn. Also in this wave of men is a woman, a Direct Descendant of the Man who Engineered the Sewer Outlet at Bondi Beach, Sydney. The photo of her forebear, proudly straddling the outlet on its inaugural flush, has made its way into the Melbourne papers in her stead. Bridget Kelly keeps her in tadpole nets.
Water has been sampled. Mud. The gut contents of eels. The kittens have been stroked and put back in their box. The general conclusion of the large-bottomed pinhead is that a cyclone fence should be erected around the swamp and notices prominently displayed to warn the public, under threat of hefty fines, not to feed the gulls. A further problem lies in the plethora of gnats and dragonflies, mosquitoes and other insects upon which the gulls are known to dine. It can only be solved by insecticides of known efficacy, deposited in efficacious amounts. This opinion has been ventured in the wake of the downing (without casualty) of an American DC7 by a flock of seagulls last December at Mascot Airport, Sydney.
The Melbourne gulls may or may not be in the projected flight paths of DC7s which are about to take off from the just-opened International Airport of Melbourne at Tullamarine.
The collective notes, with considerable scorn, that despite a brand spanking new freeway from the city centre to the very airport doors, sightseers on Day One caused a traffic jam that lasted from nine o'clock in the morning till seven o'clock at night.
And anyway, the swamp is miles from Tullamarine, so why are certain burghers of the City of Melbourne being asked to take their pick between Annihilation and Poisoning? One or two of them have already taken their pick and also their mattock to the encroachment from the east and done some damage to progress under cover of night. But they are no match for governmental bullshit. All the complainants can hope for is a stay of execution. Just now, execution is staying. It is lounging in orange jerkins beside the billabong while mosquitoes sting it, rabbits burrow into its towering sides and Annihilation does battle with Poison for the upper hand.
The blood of the Kelly–Coretti faction of the Mad Meg collective is up. Canvas is being unrolled in imagination down the hall of this house. Allegra is mixing paint. The canvas is rolled up again when the art historian, a Catholic from our local diocese (and our natural ally), thinks she can get us cut-price calico from the mercery section of her dad's emporium. But will the paint take on the calico? Maggie Kelly mentions a circus tent currently on offer to the Pantechnicon. We could cut it up with Big Ernie Kelly's electric saw, but that seems excessive. We debate canvas and calico. Calico wins because it's lighter to carry; the letters can be silk-screened on.
An interruption. Eli, who sleepwalks, wanders into the kitchen with his bear under his arm. He opens the door of the saucepan cupboard, pees inside it, closes the door and goes back to bed. The Kelly–Coretti faction has hysterics. Saucepans are hastily doused and the cupboard cleansed of the offending liquid.
During this unexpected occurrence, the Nouveau Proletariat regroups and the question of lead poisoning from car exhausts is raised. The Nouveau Proletariat consists of the three critics, all of whom went to school with Checkie Laurington.
And then (between haughty cigarette puffs) there's the cost of double-glazing the houses at the freeway outlet. These are nice little single-fronted Victorian houses, one of which is owned by a member of the Nouveau Proletariat. Shouldn't we widen our agenda?
Maggie Kelly ponders lead poisoning and double-glazing and raises the fact that Mad Meg and the Pantechnicon have withstood the effluent from bumper-to-bumper traffic since traffic became bumper-to-bumper and if the freeway were to take the proposed alternate route, the route which would take it to the nice little Victorian cottages, the aforementioned would in fact be better off than they are now.
This information is not received kindly by the Nouveau Proletariat, which uncurls itself from its superior sulky poses, raises its languid brows and plumps its hair, or cleans the long fingernails of its left hand with its thumbnail, or sighs and expostulates, 'Look, either we're serious or we're not. If we're serious, then lead poisoning and double-glazing are issues.'
'I would have thought public transport was,' retorts the Catholic art historian.
'That's being smug,' says the fingernail cleaner.
Our collective is split along class lines.
Maggie Kelly, who has taken the cutlery out of our sink and is sitting on the draining board washing her feet, says, 'And that's a put-down.'
'Oh God,' says the hair-plumper, looking at her watch, 'is that the time?'
And so the dust is swept behind the door. All the fags are smoked. It's time to go home. The Nouveau Proletariat kisses. It is something we have to get used to. Maggie Kelly, who is not kissed, plants a book with a brown paper cover in Allegra's hands before she stomps off into the night.
The Pantechnicon has had a recent boost to its trade. A goodly portion of the 71 000 copies of _Portnoy's Complaint_ not seized by the vice squad (400 languish in custody) has come into the possession of the Kellys. All over Melbourne, in trams and buses, trains and toilets, the book with the brown paper cover is being read. The populace is up to its neck in smut.
As we close the door on Maggie and decamp into the murky interior of our house, Allegra says, 'I wonder if I like those women.'
'Wonder? It's plain to see what they're like.'
'Is it? Or are they a threat to you?' She cartwheels effortlessly down the hall, our mock-croc boots like a pair of hedge trimmers lopping off a branch.
'Noight,' she says from halfway up the stairs.
I retire, fuming, to my rumpled, Arnie-less bed. A streetlight shines through the leadlight, casting coloured patterns on the walls. Silk comes in through the window, her shadow tiger-sized. She curls up behind my knees, purring and kneading a nest for herself.
Allegra knows the Nouveau Proletariat from university. I feel, I know, in my heart of hearts that these are people whose only concern is what they look like and whom they're seen with. Allegra Coretti has been queen of the campus. Being Henry's daughter only strengthens her appeal in the eyes of her followers. For these people, politics is a form of high fashion. They are the antithesis of Maggie Kelly, whose politics are those of experience. As for Allegra, she marches with the candour and grace of a saint. Others stride, but Allegra strolls. Others are angry, but Allegra is radiant. It's strange to think that she once was too scared to go and leave a message with her boyfriend's father. She's magnificent now: small, but lithe, her hair is a buoyant, electric mass. Life moves through her like a seal through water.
THIRTEEN
The Politics of Kite-Flying and the Elusiveness of Cash
WHEN ELI WENT to school, his attitude towards his clothes was cavalier. On his first day he lost a shirt and a shoe. I had to take him aside and tell him that now he was at school, he must remember school clothes cost much more than dressing-up clothes. Shoes, in particular, were expensive items. The next day he came home with three left shoes and a shirt I'd never seen before, into which you could have stuffed two Elis. I pointed out that all the shoes he'd brought home were for the same foot. Didn't his right foot feel strange in a left shoe? 'No,' he said, 'it feels bourgeois.' Bourgeois was his expression for terrific. He'd spent his lunch money on them; they'd come from the lost property box. He'd bought three shoes so he'd have one in reserve.
I sewed name tags into everything, but it wasn't long before there wasn't an item of school clothing with his name on it. Friday was my day to pick the children up. I enquired of his teacher just why it was that my son's uniform kept changing. Surely it wasn't that he stripped off every day? This boy whose ambition it was to grow up Normal?
'Oh, he's not normal,' said Mrs Hildebrand. 'Children have to pay a ten-cent fine for losing clothes. I put him in charge of the lost property box. He holds up the contents of the box piece by piece every afternoon, but nobody knows who owns what at this age and a lot of the stuff remains unclaimed, so he buys it, ten cents an item. I make him bring things back if a parent complains, and he's not allowed to take things that are clearly marked.'
'Well, what's he done with his own clothes?'
'No idea. If I send Eli to run a message, he comes back looking quite different from when he left. He's not normal, dear, whatever gave you the idea he was normal? He's one of the funniest children I've ever taught. Someone told me he was Henry Coretti's grandchild. Is he?'
'Yes, he is. Henry's my father.'
'Is your father eccentric?'
'Yes. I suppose he is.'
'Well, then, that'll be where it comes from.'
'My mother's eccentric too.'
'Oh yes, I've seen your mother. That very handsome woman who drives the Daimler. I envy her that beautiful sable coat.'
'Think of all the poor little creatures who died so it could be made. I wouldn't be surprised if she went out and killed them all personally.'
'You don't get on?'
'That person is not my mother. And most especially, she is not my son's grandmother.'
Normal Eli might not have been, but he hankered in his own way for conformity, and did things like bring men home for me to marry. The first was a cyclops.
Eli had taken his affairs in hand and joined the Church of England Boys who met for games after school on Tuesdays. The cyclops, sweaty after basketball, would bring him home. Because this was something of a kindness, I felt I had to make him a cup of coffee. Then he began to bring his chess set. Despite my best efforts to be checkmated in the minimum number of moves, he was slow to leave on Tuesday evenings, one cup of coffee might now and then stretch to four. Eli could be heard singing happy songs in the bath, 'She loves you, yeah, yeah, yeah', and 'Love, love me do'. Since I was such an excellent loser, the cyclops came up with the game of suicide chess in which she who is checkmated first is the winner.
While Allegra was upstairs editing Mad Meg's news sheet, I was downstairs perpetrating, and indeed perpetuating, the faults and follies against which the news sheet railed. If I did not perpetrate and perpetuate I was afraid my son would have to give up basketball, which he loved and was very good at.
When he turned seven he took up philately on alternate Thursday nights, and began to bring home a stamp-loving but deserted father. This sorrowful being, who had balefulness down to an even finer art than my mother had, sought to increase the range of his chaperonage and to include within it the unoccupied side of my bed, so that on alternate Thursday nights I was forced to institute a life drawing class in the empty sitting room where some artist friends and I would gather around the naked Bridget Kelly and draw her from the left, from the right, with feet facing or with head.
The chaperone would come in after stamps and drink coffee sorrowfully by the sitting-room wall. He and Bridget knew each other from being on the Parents and Friends of the local school together. Bridget, often in a most indecorous pose, would yell out such choice morsels as, 'Is that fucker Brannigan still the president?' 'Yes,' the baleful chaperone would go. 'Needs his balls put through a mangle. What about that bitch, McIver, she still the secretary?' 'Yes.' 'Geez, nothin' changes.' If he hadn't gone by ten, we gave him a stick of charcoal and a drawing pad on which to work off his frustrations.
Around about this time, Eli decided to be christened. Since his mother had not taken charge of his affairs and he was already seven years old, a list of potential godparents was submitted and I was required to choose. I could not. Needless to say, in my wanderings I did not come across many Anglicans; almost all persons of my acquaintance belonged to the Atheist persuasion, though there were a few Muslims at the gardens. Apart from conducting a doorknock, I could think of no way of bagging an Anglican. Eli suggested we stand outside a church on a Sunday. This seemed a good idea, so Allegra and I dressed obscurely in hats and mingled round the church door Eli had pointed out to us as the one belonging to his boys' club. As the congregation issued out after the sermon, we realised we had made a mistake. It was several years since the hat was popular garb among Anglicans and our hats were of the large, floppy, Mavis Bramston kind. They were black and we were wearing dark glasses to save ourselves from being noticed. Everyone noticed us. The vicar bolted for the vicarage even though Eli, dressed in long trousers, a shirt and tie, was lounging on the low bluestone wall just outside the church and waved to him. Eli had taken his grandmother shopping for suitable Sunday attire. It was Eli who darted into the crowd of parishioners, bagged his Anglican and brought her back to us. She was, we discovered, the organist, and Eli had worked his blue-eyed magic on her.
She was tall, blonde and blushing and the owner of standard blues herself. Her name was Belinda Bloomfield. Perfect. As for a godfather, because I had refused all Eli's suggestions, Allegra said she'd be it.
On the day of the christening, we were sitting in our kitchen and I had just told Belinda the joke about Berlin-da when she blushed brilliantly, gasped and an elegant hand flew to her bosom. She was staring, bewitched, at the kitchen doorway. It was Dadda, resplendent in velvet. He was carrying Eli and in his free hand he had a Black Forest torte.
Allegra stood up immediately, turned her back on him and stared fixedly at the fridge. Dadda put the cake on the table and stood between Allegra and the door so that when she went to rush past him, Eli and Dadda gathered her in. She started to cry. Dadda and Eli sat her at the table, Dadda keeping his arm around her and Eli staying on his lap. I introduced him to Belinda Bloomfield, who, had she been a Christmas tree, would have burst into flames.
'I like Eli's suit, Sibella. He cuts a very fine dash. Is it a special occasion, then?'
'My christening,' said Eli.
'Oh, then a very special occasion, it's just as well I brought a cake. You can eat it to celebrate.' He kissed Allegra's forehead, then said to Eli, 'Well, then, if you're being christened I mustn't hold you up, must I?'
'That's right, Dadda, piss off,' sobbed Allegra. Dadda was about to do just that when a white handkerchief on the end of a broom handle was fluttered round the door. It disappeared and our mother, in her christening robes, materialised in the doorway. It was her first visit to our house since the Mad Meg war began.
'Heck,' she said, then to me, with narrowed eyes, 'To whom do we owe this honour?'
It was clear that Belinda Bloomfield had rarely, if ever, witnessed a scene of this type, but one thing was for certain, when she set her eyes on Dadda, it was love. She was heard to mutter, 'Lord! I didn't think they made them like that in the flesh.' Then she sprang to her feet, looking at her watch and saying, 'Lord, it's getting late.' The Lord was figuring prominently on this holiest of days.
Then Stella said, as if nothing had changed in a decade and Dadda were requiring a part in a pantomime. 'Well, Dadda, we haven't got a father-in-law.'
'You mean godfather,' snapped Allegra, blowing her nose. 'I'm going to be godfather. Come if you're coming, Dadda. Go if you aren't.'
Dadda came. His presence was such a shock all round that when we arrived at the church we couldn't find Eli, so Dadda had to go back and look for him. The vicar tried to take it all in his stride. The other christenees were babies. Eli, having had a trouser leg caught up in the bike chain, skidded up on his bike sporting a chocolate goatee. Dadda, looking sheepish in the daisy and dream-strewn Kombi, arrived just after him.
It only took about ten minutes for Eli to be christened and we were invited to repair to Belinda Bloomfield's for afternoon tea. Dadda said he had to go. 'Where are you going?' I yelled, but he was already gone, hurtling away in a taxi, away, away, and but for the signature on the christening certificate forced out of him by Eli, I would have said I'd dreamed him being there.
When we went to collect the Black Forest torte from the kitchen table, we found that some sinner had hacked a piece from it.
It was a day of coincidences. Belinda Bloomfield lived in splendour by the Yarra. A house with a garden of magnolias, rhododendrons and camellias, and a double garage, was not the sort of house into which a Coretti foot would normally stir, but on this occasion, as we mounted the stair to the picture-windowed sitting room, a voice already in there breathed, 'Bunny Motte.'
The response was, 'Baby Beauchamp.' Our mother's cousin's daughter.
The last time Baby Beauchamp had seen Bunny Motte was in 1942 when Hitchcock's _Rebecca_ hit the Scunthorpe Roxy, so it was an attestation to our mother's youthfulness that Baby Beauchamp/ Bloomfield could still recognise her and a remarkable feat of her own memory to have named her Beauchamp at ten paces, when the last time they'd met, Baby had been twelve years old.
The family flower beds were dug, but with some caution, we noticed. The dead were resurrected, the living were reported on and the likenesses of the young to the old, both in feature and in habit. Scant but dignified reference was made to Bunny's three brothers, lost in the war. And to Geoffrey Latimer, her fiancé, what a shame, he would have been... oh well. Baby Beauchamp stood and exercised her spine by making chick wings of her elbows, her face displaying an implacability Nina would have been proud of. She removed the crumb-blighted and cream-smirched plates from the afternoon tea table as Belinda arrived with the absolutely unblemished teapot. A perfect silver teapot: rotund, with a lid that fitted and a spout so placed that the pourer never spilt a drop.
At the sight of such perfection, Stella, shrill with champagne, said she'd have coffee and at the same time lamented, provocatively, the dwindling of the prototypical Motte to the impecunious, dented teapot present. One was given to suppose the Beauchamps thrived at the cost of Mottes, who were doddering off the planet in some confusion, having to be satisfied with rather less than the overcrowded family vaults in the better class of cemetery. Indeed, the name Motte was increasingly to be seen on those little plaques in brick walls in memorial gardens, after their rather plain coffins had descended to the furnace to the accompaniment of musak.
While Beauchamps watched, unmoved.
Beauchamps who had played the organ for generations, for whom it would be no great trouble to thrum out a threnody on an organ such as the one at Scunthorpe, donated – had Baby ever noticed? – by Augustus Motte, illustrious grandfather of she who took three sugars and several dollops of cream to sweeten a bitter cup.
As for us, who had been coughed up, diluted, by that viper Time, we fired cross volleys to the Mottean lamentations, intending to remind the lamenter of place and occasion. The new Christian took his godmother to the kitchen to have her instruct him in the art of chocolate snowball-making, for not only was his mother a heathen, she kept nothing worthy of a male child in her pantry.
In a neighbourhood short of greenery and parks, children will innovate. For his seventh birthday, my mother bought Eli a bright pink kite with a long tail, but Eli was at a loss to know where to fly it. Allegra condemned the kite as just the sort of frivolous present our mother would give, bourgeois old bitch. If it got caught in electric wires, Eli would be electrocuted. That was our mother's trouble, she never took the local environment into account, and, as a consequence, always lived in an inappropriate way. Eli eyed his kite, chap-fallen, for some days. It was Kelly Kelly who knew the solution. Our anti-freeway protests had been made loudly enough to put the freeway project into abeyance for an indefinite time and, since the encroachment was wire-free and easily reached through the dump, it provided an ideal surface from which to learn to fly the kite.
Eli loved his kite and took it with him whenever he could. It was something of a nuisance on trams, but Eli insisted that we take it to show his grandfather. He wouldn't dismantle it because, he said, people on the tram might like to look at it.
We used to meet Dadda in a South Yarra cafe. On the day of the kite, we ambled down Toorak Road to Fawkner Park to fly it.
After all his practice with Kelly Kelly on the encroachment, Eli's kite-flying was showy. The kite looped and swooped, trembled and flapped rapidly and audibly at the edges. When Dadda tried, however, it described a course like a heartbeat on a cardiogram and, rather than embarrass himself by learning in public from a seven-year-old, he gave it back to Eli, around whom there was soon a small crowd watching the pink tail furling in the clear blue sky.
Eli laughed and cheered and danced with the kite strings. 'Look at this! Look at this!' he cried and put the kite to zigzagging tricks. 'Isn't it bourgeois, Granpa! It's so bourgeois!'
Dadda started laughing. 'Oh my, ye-es. Ye-es, Eli, it is bourgeois, now you mention it. It's as bourgeois as anything.'
'Gran's bourgeois, she's bour-geois!' And he drew a big S for Stella in the sky.
'Really?' Dadda responded, stroking his chin.
'She's my bourgeois Grandma! She gave me my kite.'
It was hard to say which of his grandparents Eli loved more. He loved my mother for her kindheartedness and would wax furious if Allegra and I put her down. He told us we were horrible to her and we ought to give her a few hugs from time to time. But for me, working out when I ought to give the first hug kept me from giving it, and all my troubles seemed due to her. Both Allegra and I kept telling Eli he'd find out in time why we were antagonistic towards her.
We were on a football field. It was autumn, white moths among glossy greenery, swallows cruising, Dadda directly under the kite with his head turned up. I became aware of someone standing beside me. Unusually for an autumn day in a Melbourne park, she was carrying a white lace parasol. Under it, I discerned the doll pink cheeks and doll bright eyes of Rose Hirsch, Siècle's one-time right-hand woman.
'Zees is your little boy, Isobel?' she asked.
'Yes.'
''Is grandpère says 'is name is Eli, no? 'E is very clever wis the _cerf volant_. 'Ow do you say it in English?'
'Kite,' I said, thinking the accent a little far-fetched after thirty years residence down among the drab vowels and the antipodean fondness for elision. I was to discover, nonetheless, that Rose Hirsch never lost her very French accent; it was as much a part of her as the beauty spot on her chin.
'Is zis some kind of animal, zis kite?'
'I'm not sure, I don't know. I think there's some sort of bird called a kite.'
'Oh, really? In French it is like a... somesing wis antler?'
'A reindeer, maybe?'
'Yes, zat is right.' She called to my father, 'Ey! Henri!' Entranced by the kite, he waved without looking at her. Rose and he were obviously in frequent contact.
''E is a champion,' said Rose of Eli. 'Tell me, 'ow is sings wis you and your beautiful sister at zis Mad Meg gallery?'
'Fine, Rose.'
'I came to your last show zere, your own paintings. It is very good work. Very nice gallery.' She smiled at me. 'No, seriously, Isobel. You are a very talented painter. But I sink it is hard for you to find enough time to do your work, no?'
I was surprised Rose knew anything about me. Everyone in Melbourne knew about Rose, who was famous for her eccentricities. I hadn't thought that someone as devoted to being extraordinary could possibly be interested in anyone else.
'I love your little boy. 'E 'as such a beautiful chin,' she ran the back of her hand up under her own chin. 'So pretty. Delicate. I love it where ze veins go. Tell me, 'ow do you like ze job you 'ave in zat garden?'
'How did you know about that?'
'A little vulture told me.'
'Oh? The second Mrs Henry Coretti?'
'Oui.'
'How does she know? I didn't want Dadda to be told.'
'Ze second Mrs Henri Coretti knows everysing, my de-ah. She 'as a spy ring operating. Your fuzzer would 'elp you if you asked. Why don't you ask 'im?'
'I don't want to, I'd rather do it myself.'
'Well, I don't mean to interfere, Isobel, but you must plan for ze future. Your fuzzer is quite a rich man zese days. You ought to know 'ow rich. You ought to sink about zese sings if only for ze sake of your muzzer. After all, your muzzer was Henri's wife zrough ze years of 'is poverty. Do you 'ave any of 'is paintings?'
'No.'
'None at all? But zis is terrible!'
'It would be too difficult for my mother. She finds it painful to look at them. And if Allegra and I had one in our house, it would make us feel bad on our mother's account.'
'But I sink zat your muzzer and your sister are at war, no?'
'What, the vulture told you that, too, did she?'
'No. Your fuzzer told me zis.'
'Well, they are fighting. But that doesn't mean they hate each other. They just believe in different things. Our mother is old fashioned.'
'But what about you, Isobel? Why should you not 'ave some of your fuzzer's paintings?'
'I told you. It would make my mother and Allegra feel bad.'
'Zat is not sufficient reason. Henri is a famous and beautiful painter, and 'e is famous because 'is work merits it. You are 'is daughter, 'e was never able to give you much before, but now... You 'ave earned it. It is what 'e owes to your muzzer and your sister. 'E feels it; 'e says so. We must see each ozzer sometimes, Isobel. I know many sings about your fuzzer. Your parents' marriage didn't last, but zere are many reasons to be proud of Henri Coretti.'
'We hate his wife.'
''Ate. I don't like zis word 'ate. People should not cut zere arms off for a broken finger.'
'What about people with broken hearts?'
'Ah, now zat is different.'
'Well, what's the cure, Rose?'
'Alas, zere is no cure for a broken 'eart. Come to see me, Isobel. We 'ave much to talk about.'
Eli's pink kite hovered in a sky of gold standard blue. Eli was its eyes, searching out its prey.
' _A plus tard_ , Henri!' called Rose to my father, and her face, when she turned to say goodbye to me, broke into a complicit smile.
I was waking every day with a pain in my back. It was as if I had spent all night standing. It was the pain of picking up heavy loads, as when Eli and Chantal were babies and I carried one on each hip.
The pain went when I was working at my easel, but it came back at my potting job. Eight hours a day I worked, four days a week, sitting in the greenhouse, potting and labelling. I didn't have to be so industrious, but I was preoccupied and didn't want to have to talk. I tried to imagine all the orchids in the world, so I could pot and label the lot simultaneously. For this I would be given a lump sum and told I could retire. Then I could leave my job, buy our house from the owner, pick up Eli from school and live.
When I thought about Dadda and The Brolga, my heart hardened. If he was rich, as no doubt he was, then his money was contaminated.
The pain in my back grew severe when I thought of my mother. I loved her and hated her; she was wonderful and terrible. I remembered how proud of her I had felt when I was a child. She had been the pretty person with the swinging step, the mother to end all mothers. She had loved and laughed and done her best for all of us. But her load had become impossible and she herself had done nothing to stop it becoming so. I couldn't bear to think of it; I tried not to think of it. Whenever I started to think of it, I allowed myself to be carried off by Arnie. Saved. Loved. Cared about.
I moved away from the other potters and sat behind a mountain of black pots. I potted, I labelled, I wept. Abdul bin Hadji, the short and shiny ganger, perved on me. He thought he could draw. He wanted to draw me. He asked me to come to his bungalow home so he could. He wanted to see me without my clothes.
I moved back to the other potters. Ferdie, the tall and very thin Egyptian, fended off the short and shiny ganger but asked for my hand in marriage in return. I said, 'Fancy wanting to marry my hand!' and I showed him one of them with its filthy, broken nails. Ferdie was amiable enough. He was a part-time uni student. He had come to Australia to do an engineering degree.
Then bin Hadji told me that Ferdie was camp, and ordered me to distant parts of the garden, following me. 'You have a child,' he kept saying. 'You should get married. I will marry you.'
Before I could ask Beryl Blake for a change of job, I learnt that bin Hadji was a project of hers. She had saved him from deportation three times. 'If he could find an Australian wife,' she said, 'he could stay.' I thought of this woman who had nothing in her life but Botany, good deeds and a hopeless passion for the man she called 'The Bish', and though I had grievances, I held my hush.
When 'The Bish' was coming, the herbarium and its neighbouring rooms were cleaned. Important notices appeared on the bulletin board, such as 'Nun Cures Migraine', 'Catholics in Niu Gini' and 'India, Land of Miracles'. We were paraded and the sins of the world against us were given audience. 'The Bish' in his Catholic caftan jerked slowly past us, crushing his right fist into the mortar of his left hand. Grinding sin. He hissed, 'Yess, yess,' to Beryl Blake's recital of woe. There was a fervid glint in the whites of his eyes and spittle on his lower lip. His hair fell in licorice straps over his forehead. Perhaps he knew of a Jesuit solution to the world and would reveal it in the fullness of time.
Visits from 'The Bish' were pretty nearly always followed by a bout of falling off chairs under the compassionate gaze of Loyola O'Flynn.
This garden, then, was a catafalque for a woman's ruined mind. I could see her pain. I drew her, feeling the pull of her indecorous rubber thongs between her toes, the foot sweat in summer weather, the shanks she'd probably never even looked at. Long, straight grey hairs grew beneath her knees. Her gut was the size of a full-term pregnancy. No neck, just a head stuck on like a dob of plasticine. I felt for her what I'd felt for Uncle Garth. Because of her I painted a crowd of bottom-heavy people with tiny, ineffectual wings. Their faces were turned upwards, unable to reach the dazzling lightning in the sky.
All Allegra could say in reaction to the picture was, 'I'm not sure about this. It's a bit metaphysical. Do you think you ought to change your job?'
I was almost Allegra's opposite; in love with my painting but timid of my fate. Thin and boylike, I crept about as a cat does, looking for a place to soothe myself. I was lovesick. Happy moments passed like lightning; pain, like gravity, tied me to my course.
Maggie Kelly was the only person to whom I could unburden myself. We shared our photos: two of Johnny Green and one of Arnie, climbing the Acropolis. It was almost as if we were sharing an illicit secret, yet these were the fathers of our children, an out-of-focus hood on a motorbike and a striding colossus beside whose head a guard further up the mountain hovered like a toy.
I hadn't known Johnny Green and Maggie hadn't known Arnie, but we looked for their features in our children and were happy when we found them. This child-scanning we indulged in brought us close.
Maggie had no doubt at all that the art critical troika of the Mad Meg collective were sham leftists; 'Nouveau Proletariat' was her name for them. They had brought with them a linguistic code that ordinary people never use, except in jest or mimicry. Their game was to adopt key words like 'issue' and then to construct a hierarchy of 'issues' to make those problems that didn't affect them look unimportant and therefore not worth bothering about.
But Maggie held her ground. The bourgeois stratagems involved in making the freeway did not escape her. First she drew our attention to the type of person whom the freeway would displace. On the most probable route, there were no middle-class constructions under threat of demolition. The houses at the freeway outlet, for which the Nouveau Proletariat were not alone in wanting protection from lead and noise pollution, had become middle-class dwellings, having belonged, at the time when the plans were drawn up, to the working class. All along the plan had been to move the poor and, to satisfy the consciences of those who were going to move them, they would be shunted into high-rise flats or pushed to the city fringes where they would become isolated. In this situation of shabby newness and bad service, the voice of the poor would effectively be stifled, while the problems of poverty would be magnified in outer suburban ghettos.
It was easy to see that what Maggie said was right, and when you looked into it in any depth you could see the importance of solidarity among poor people. This led me to take comrades Marx, Engels and Lenin from the library, to learn how solidarity could be brought to the poor of Melbourne.
Ours was an area in which migrants established themselves through the fruit shops, the takeaways, the electrical shops, the butchers and the corner stores. It was cosmopolitan and kept on renewing itself. Melbourne's extensive Italian, Greek and Jewish communities had all had a foothold here at some time. Most had started out as petty bourgeois. To my dismay, Comrade Marx called petty bourgeois socialism both reactionary and utopian, and on these grounds to be deplored. Worse still, Comrade Lenin didn't see why the petty bourgeoisie shouldn't be killed outright by the wage slaves, among whom he counted doctors and lawyers. Rumpton, Rudge, Russell and Plant would be saved in the red revolution, probably against their wills, while the proprietors of the Pantechnicon and Mad Meg, who would both manufacture and carry the leftist banners, could expect the guillotine. Maybe the Manifesto and The State and Revolution had suffered in translation to the extent that everywhere you looked in them, the hatred of shops and shopping was writ large.
For all my misgivings, the ALP came to power at the end of 1972 and there was much rejoicing in the Mad Meg quarter. Within three months of the swearing in of 'twenty-seven more state funerals', as one minister called the ALP cabinet, the Attorney-General had raided his own department and set the pace. One Catholic member in the House said he didn't mind if homosexuality was legalised so long as it wasn't made compulsory. Bill after bill after bill would go from the House of Representatives to the Senate and come back again, unpassed, because the ALP didn't have a Senate majority. They were heady times, and might have been headier had Victoria not had a Liberal state government. The fight against the freeway was far from over.
I had been foolish enough to raise my interpretation of Marx, Engels and Lenin at a collective meeting. Yes, there was no getting away from it, my political scientist of a sister averred, now the Left was in power, we would have to examine the morality of the ownership of Mad Meg being concentrated in so few hands.
The moral examination was done on the spot and it was decided, four votes to three, that Mad Meg should not only be run as a collective, but owned as one as well. The dissenters were Maggie, the Catholic art historian (whose name was Cathy) and me. The official reason given for dissent was that whereas the Nouveau Proletarian troika had access to capital, Maggie and Cathy hadn't. Maggie and Kelly paid rent for the Pantechnicon. The people who owned the building were active and valuable anti-freeway campaigners because they owned a cluster of shops which would all go if one of the freeway proposals came into effect. On the other hand, they were enemies of Maggie and Kelly, who were doing modestly well out of their shop and didn't want it to be demolished so the owners could build a projected shopping mall. The Pantechnicon had not yet come under direct threat because the owners were locked in a battle with the Historic Buildings Trust. The Trust wanted to preserve a beautiful Victorian awning, behind which the Pantechnicon and its neighbours lay low, staving off hikes in the rent and being asked to feel privileged by their continued, leaseless occupancy.
Cathy had bought herself a house and had nothing extra to spare, although she wanted to help run the gallery, however it was owned. Allegra would have broken down the ownership to a fifth each, but I said no; if she wanted co-operative ownership, then I wanted Maggie in and was prepared to buy a sixth on her behalf.
'But that'll leave you with a third of the vote,' the Troika decided.
'I'll forfeit a sixth to Maggie.' At the back of my mind, as I voiced this thought, was a feeling, firstly, that that's what they meant, since Maggie and I were often in agreement, and, secondly, a dread for the day when we would sell Mad Meg on what looked like remaining an open property market, despite Allegra's prophecy that property would be abolished. I could see the bickering now, and if I'd had a more forceful character, might have bought out of the gallery there and then, but it was Allegra's dream. She wanted a new egalitarian world: not an armchair paradise but a real place towards which she was certain she had started out.
Allegra had an optimist's attitude to money. Every dollar spent was a dollar earned, but she seemed to think that if she did the spending, she also did the earning. This strange trait caused her cheques to bounce and meant she had fairly frequent and voluble meetings with bank managers. She had inherited the rationale for this behaviour from our mother, but she lacked our mother's invention in dealing with it.
The belief in her own unassailable goodness and the tale of her having been a pioneering teller had the staffs of banks at our mother's little feet. (Her feet were little because Euphrosyne had wanted it that way and refused to buy her larger shoes after her feet had reached size six.) Our mother wielded her debts in much the same way as she wielded the family table linen, with flair. There was never her equal in sincerity when faced with a bank manager or a credit official in a department store. No high-handed Ninaisms for her, just breeding and tragic indigence. The pastoral acres, the bushfires, the brothers and their drunken replacement – she made the girl with the smallpox in _Bleak House_ look inexperienced.
I had two thousand dollars in the bank. There it stayed and slowly, slowly grew to three thousand. Property prices were always that bit ahead of me and my yearly income that bit too low to think of being able to buy a house. If the Troika didn't buy into Mad Meg I'd probably have to get some help from Dadda, or marry someone for a mortgage. I felt dragged down by the thought of marriage and of Eli being the tolerated child from a former relationship. I had visions of his much-loved nature giving way to surliness and the profound discomfort children feel when an outsider moves in on their most important bonds.
My conversation with Rose Hirsch kept coming back to me. I wondered how she got along with The Brolga. I imagined talking to her and explaining our position about Dadda. He'd never been rich when he lived with us; it was our mother's earnings we relied on most: to survive without Dadda was truly to survive. We would purge ourselves of him and punish him with our survival. Why should he come off looking like a benevolent fund? He was cruel and we did not want him to feel redeemed. And yet there was Eli, and me at his age, and the thought of a heap of lolly ending up with The Brolga or, worse still, Checkie. All we would have proven was our own stubbornness.
These considerations joined the circuit of my thoughts, and I began to need something to relieve the monotony. So I saw myself loved by someone as yet unknown. I saw myself, free of torment, painting. I would pot and fret, paint and forget, and I dreamt two futures, love and art.
FOURTEEN
Dadda's Beginnings
AT THE PARTING of the ways, it was said, Harry Laurington had bestowed on the Hirschs a collection of art and a pair of semidetached houses in lieu of superannuation. Being Harry's minions hadn't made them rich, nor had it given them much room, for their houses were minuscule, so minuscule that, up to the time of his death from cancer about a year before I started to visit Rose, Laurent had lived in one and Rose in the other.
It was a hostile street. The narrow bitumen paths either side of the road were steep or precipitous depending on the direction you were travelling. Being a thoroughfare, it was impossible to park there and hard to cross at any time of day. The dirty, busted-off street trees raised the bitumen as if their roots had been crude tin openers, the sort that make the palms of your hands feel vulnerable. On the Hirschs' side, the single-fronted Edwardian duplexes staggered up the hill, their serrated roof ridges stacked, each one higher than the last. They had no view of anything whatever except the merciless brick and dull glass of neglected flats opposite. Squat brick fences with tough wire gates in them and plots of vegetation, tiny and wan, kept the houses from the street. It was as if they had turned their backs on life; their windows were blinded, their doors unwelcoming.
I circled Rose's house for several days, ashamed of myself for having to put in so much effort when she'd seemed, after all, to be a friend. I kept on saying to myself, _She can't live here_ , kept thinking, _This must be wrong, there's another street of the same name somewhere else_ , for when I thought of Rose, it was not of someone who lived in a brick duplex. I thought it more likely she'd live in a turret or a gazebo.
On Rose's verandah, if indeed it was her verandah I found myself standing on at last, a dogged white geranium held its single flower out to the sun from an earth-filled crack in the tiles. I rang. Rose called, ' _J'arrive!_ ' and so I thought she might be expecting someone French and nearly ran away; but when she opened the door, she was effusive and kind and I wondered how I could have thought she was otherwise.
Rose was short and, in the days of fashionable sun-tanned anaemia, plump and white. Her clothing was several exquisite petticoats worn one on top of another under a dark waistcoat. Done up in black gym boots, her feet pointed straight forward when she walked, like a doll's. She had a doll-like jauntiness and rubbed the knuckles of her right hand across the flat palm of her left in a vigorous, friendly way. It reminded me of frotting silk and taffeta in childhood, of the warmth my fingers imparted to the cold material, and of the way the material replied with its gritty feel, its warpy, wefty voice.
She slid her jaw from side to side as she talked, letting her voice slink out as if it were lingerie kept in a delicious little drawer. Speech was an adventure in sound-making for her; she craned forward in flashy-eyed excitement, saying, 'Is-o-bel, Is-o-bel,' feeling the shape of my name, like a taste, in her mouth.
In two rooms off her tiny hall, her dolls sat in darkness; large, small, caucasian, oriental, black, fantastic. There were boys, girls, babies and parents, china dolls with wooden bodies, dolls in bisque with bodies of stuffed cloth. They were wide-eyed and expectant. Speech seemed about to come from the open mouths in their faintly luminescent faces. They seemed intent, interested; they might have had something intelligent to say.
In the sitting room where the light was on although it was daylight outside, more dolls – on every surface and bookstack, dolls, and Rose's creatures, antlered, erotic, furred, scaled, bewinged, beringed. They sailed in boats or flew or rocked or went on wheels, and all had Rose's eyes, dark eyes into which you could not see without first seeing yourself.
I fingered my tea cup with the forgotten relish of childhood, as when Allegra and I had cleared ourselves a space and spread our play house cloth, set out our tea cups and poured the magic brew, the brew of little women: little women who'd pinched liqueur glasses and imagined we were drinking wine; little women who delved into the lascivious mysteries. Civilised little women. Except when it was just after weddings, and then all hell broke loose: with cellophane and sausages, with wild, wild berries and drawings of Leone and a man, with a beard, dark glasses and a stick. We would skid from impeccable manners to what we thought was pagan debauchery. We'd made a card: Leone with her juicy thighs on one side and the mystery man on the other, and then we imagined what happened when the card was shut and thwocked the sausage till it split and the inferior meat came smearing out of it.
Rose, incredibly, was in her sixties; in her words she had reached six ten times and was happy to be there. She said if you were a European Jew, six was a sensible age to be, and there was great wisdom in remaining six.
'Some people don't become six easily,' she confided. 'Ze second Mrs Coretti 'as probably been fifty all 'er life. Fifty was written inside 'er like a commandment. Zhou shalt be fifty, Mrs Viva Hallett, Laurington, Coretti; somebody 'as to be, and you are ze one who 'as been picked.' She made a gesture as if catching a nit in her strong white fingers. 'But Henri. Henri is different. I don't know how old Henri is; 'e might be six like me or seven like 'e was when I first met 'im. Zen again 'e might be very old, but I don't sink so. I sink 'e's permanently at ze age when young men fall in love.'
It surprised me greatly that Rose had known Dadda so long. With the exception of Uncle Nicola, whose personality was locked away from us behind his appearance and a language we didn't speak, we knew of no one who'd known Dadda in childhood, and, though we were inundated with our mother's family history, we knew virtually nothing about Dadda's.
Rose had a double-spouted teapot just like the one Dadda had given me on the day he'd tried to sweet-talk me in the Pantechnicon. Rose had been given hers by Dadda as well, but her tea was broken orange pekoe. We took it black in an assortment of elegant cups and saucers.
It was obvious just from his art sales and the growing popularity of his work that Dadda was becoming increasingly rich and, as Rose said, Viva hadn't any money when he and she eloped. Harry had contributed nothing, so it must be Dadda who could afford sable marten for Viva to wear and a new Daimler for her to drive.
Since his second marriage, Dadda had had to change galleries in both Sydney and Melbourne. He was the darling, now, of the upper middle class. He and Viva swanned around in diplomatic and old money circles, where even Lauringtons extended them a brittle politeness.
Dadda had told Rose he often felt uncomfortable. To get away from goo and gush, apparently, he had begun to undertake expeditions to map the intrusion of the technological age into remote regions. The sable marten would have no one to see it and comprehend what it meant in the places Dadda was choosing to go, so he was choosing to go alone.
When Viva was young she had thrown Dadda over for Harry Laurington because Harry had had money and plans, whereas Dadda had been in difficulties. Viva had even said that, married to Harry, she could be of more help to Dadda than she would have been had she married him. But Dadda hadn't wanted to be helped by Viva and had turned his back on her.
'Your fuzzer never loved Viva, not in my opinion,' said Rose. 'Probably 'e loved your muzzer because your muzzer was uncritical and kind.'
'Funny way of showing it. Why did he run off with Viva if he loved our mother?'
'Your muzzer is very stubborn, Isobel. You know zis. And she belongs to ze group of people who don't understand art and what it demands of artists. She 'as taste, of course, but ze driving force behind your muzzer's character is not an affinity wis art but a sympathy for people who suffer, for potential artists.
'Your muzzer also suffers because zere are no rewards for kindness zat goes unseen. Your muzzer is afraid to let go of people who 'ave benefited from 'er kindness. It seems a slap in ze face to 'ave sheltered someone when zat person no longer needs shelter. Zere are women who mourn because zeir children 'ave grown up.'
'Tell me actually, Rose. Do you mean that Dadda grew out of our mother? That all the possibilities between them had been exhausted?'
'Yes. Your fuzzer and muzzer came to an impasse. She was a zoroughgoing muzzer and felt unrewarded, because your fuzzer cannot be – 'ow shall I put it? – corrected. Your muzzer 'as a very strong desire to correct things. But people can't be corrected, zey can only agree to conform, or not agree. Who zey are is written into zem.'
'And he's a deserter!'
'No. 'Is goal was very different from your muzzer's. Some artists are little more zan ze brushes zey 'old, but your fuzzer is also a clever man. 'Is lesson is zat 'istory never stops, and yet it 'as its rhythms. Our lives are just footnotes in an unimaginably large story. But for all zis, zough our paths cross and recross, we will never again be who we were or where we were.'
'He used to be affectionate and funny, but now he's shallow and thick-skinned!'
'No, Isobel, 'e just refuses to alter course, 'e is no one wizzout 'is work.'
'That doesn't mean he had to run away with a piranha.'
'You're only seeing it from ze place you're standing, Isobel.'
'Why should I see it from anywhere else?'
'Look, Henri told me your muzzer wanted four sons, but she 'ad two daughters. Ze men never came home from ze war, but ze children started a new kind of life. We all wish that 'istory stopped needing more to 'appen when we came along, because we seem to ourselves to be perfect. We wish we could live forever because of our goodness, our intelligence, our beauty or our strength, but 'istory can only echo.
'We wish our marvellous moments would keep us brilliant all life long, if only to show our children zat we 'ad zose moments, zat we belonged, zat ze people we loved were worth loving and ze sings we believed in were worth our belief. If we were 'appiest at nineteen, we seek what reminds us of zat 'appiness. We want zat story told again. Your muzzer wishes in a naive way. She 'as invested ze past wis more glamour zan it could 'ave 'ad. What she is remembering is 'er own innocence. But innocence can never recur; we can only savour again ze circumstances of our innocence. Viva 'as a hold over your fuzzer because, whatever else she is, she is not naive and she doesn't require Henri to change.'
Rose paused and shrugged – her bottom lip, though full and red, drawn straight. She dropped her heavy-lidded eyes. 'I must tell you about 'ow I met your fuzzer, Isobel,' she said, looking my way again. 'In many ways, I know 'im better zan Viva ever could.'
When Rose talked, she engaged her audience completely. She was intent and intense. A panorama of expression lit her face; she leant over her story and drew it from her breast with her hands, her arms, her spine. Never quickly, always as if she were shaping clay. She made you feel, smell, wear and be in her adventure, her evocations were so strong.
She began by telling me that when she was twelve, her father had taken her to Milan on a holiday.
'We went by train from ze Gare de Lyon, late at night. I felt excited to be leaving Paris behind, and I imagined zat everysing I could no longer see no longer existed; ze deserted boulevards were all being folded up in two dimensions and disappearing into an encyclopaedia wis gold tops and bottoms on ze pages. It would stay like zat until I came 'ome again and brought it back to life.
'We travelled through France all night and most of ze next day, and towards evening, we started to go zrough the Alps. Never before 'ad I seen mountains. Zey shot up in ze air like a petrified catastrophe.' Rose made an operatic face. 'I suppose zey were a petrified catastrophe. And in between zem was a big, fat moon which was mirrored in a long lake.
'Zat night we stayed in Turin in an 'otel where ze wallpaper reminded me of endpapers in books; ze design was always ze same but ze colour changed from room to room: dark red, dark blue, brown. So I imagined now I was entering my own story, separate from ze one I had left in Paris. A story told in a musical language of which I could not understand a word.
'Ze next day we went on to Milan. When we arrived at Milano Centrale, it was ze middle of a hot day. I was so sirsty I wanted a great big bubbling drink I could see advertised in neon lights at a food booth. I did not know who ze man was who 'ad come to pick us up, I wanted so much for ze two-dimensional drink to become zree dimensional. I zought the advertisement said it was radioactive as well as sparkling, so I zought it must be wonderful to drink zis drink; my whole guts would be sparkling! But my Papa and zis man were so busy talking Italian zere was no chance to ask, and I felt maybe I was being punished for shutting my little sisters up in two dimensions and leaving zem behind.
'I kept feeling zis wonderful drink getting furzzer and furzzer away from me, so all ze way in zis car of my Papa's friend, which smelt rather good to me – I suppose it was ze petrol – I imagined I was dying of TB. And zen, Isobel, I saw somesing astonishing, a castle soaring up out of a moat, and ze drawbridge was down! For a moment I wondered if ze man who was driving ze car was going to take us into zat castle, and I wondered if we were going to meet ze King of Italy!'
It was the summer of 1924, not quite two years since Mussolini's march on Rome. The dictatorship had not yet been declared, but elections held in April, giving Mussolini sixty-five per cent of the vote, were known to have been rigged. Rose's father, an architect, went to Milan to draft a project for a friend of the Coretti family. My grandfather, Emilio Coretti – who was driving the car – and Rose's father, Lev Katz, knew each other through having attended the same socialist congresses. Both had been minor office holders in their respective parties and, to my surprise, Uncle Nicola was actually quite a prominent socialist, a union secretary.
I was astonished to learn that my grandfather made shoes. I'd been led by my mother to believe he was a journalist, but the journalism he did was for a socialist magazine, whereas the shoes he made were not just any shoes, but the best of all possible shoes, hand-made Milanese.
The Corettis lived opposite the Sforsa Castle in Piazza Castello. On the ground floor of the house was a fitting room, which Lev Katz had built some years before. It was actually a room within a room that could be made by pushing panels and bookshelves around. It was very clever; two completely different rooms could be made with the rearrangements, either a large sitting room or a hexagonal booth lined with mirrors and shelves that had Emilio Coretti's customers' lasts on them. The customers only ever saw the inside of the hexagon.
Rose asked my grandfather if he made shoes for Mussolini. He said he didn't, but if he did, he thought he would make explosive boots to make the marching all the more spectacular. The most famous feet my grandfather attended to were those of the poet D'Annunzio, whose florid style was Mussolini's inspiration. D'Annunzio was bombastic and courageous. He drove the latest cars and flew planes. 'All part of the pantomime,' said my grandfather. 'He couldn't come to Mussolini's investiture because he fell off a balcony during a love tryst.'
My grandmother, Allegra Coretti the First, was perhaps the best shod woman in Europe. Allegra Coretti the Second apparently had the face, but the hair was different. Our grandmother had a Marcel wave. Her hair was short and fair. She was a slim woman, little, like Allegra and me, but whereas our grandfather was dark and very energetic, our grandmother was languid and graceful. She had wonderful clothes. The shoes she was wearing when Rose met her were purple suede with musical staves embroidered in gold onto the tongues. She spoke French very well, and English. She was an opera singer and had to know several languages. She'd fallen in love with English; she said it was the language of love, and spoke it with our father from the time he was a baby – hence his gorgeous, lilting accent.
Upstairs our grandmother had a wardrobe as big as a small country. It was there Rose met Dadda, who was known in the family as Henri. The wardrobe had an electric light in it – Milan, a very advanced city in the twenties, even had electric streetlights – and full-length mirrors so our grandmother could see herself in all her glory. Henri was lying on a mink stole under her dresses, wearing an imitation bald patch with grey hair sewn around the edges and a pair of spectacles attached to a warty nose underneath. He was pretending to be dead. In the space between the dresses and the mirrors, he had made two pyramids out of his mother's shoes. Rose was already inside the wardrobe when he said, in a deep and ghostly voice, ' _Lasciata ogni Speranza, voi ch'entrate._ ' (Abandon all hope ye who enter here.) He nurtured a passion for Dante from a very early age and it was Uncle Nicola, a passionate Dante scholar, who taught him.
''E was so sweet, Henri,' said Rose. ''E was only seven and 'e 'ad little veins under 'is chin, like Eli. And little pretty legs and tiny ankles and tiny knees. We spent hours making masks and turning ourselves into ozzer selves. I couldn't speak Italian; 'e couldn't speak French, so we just made everysing up. Henri 'ad a box full of all sorts of zings: feazzers, thimbles, dry grass, little bottles, beads, sequins, fur and satin and offcuts from ze shoes your grandfuzzer made.
'We dressed each other up. 'E was wearing a cape and 'e 'ad circlets of 'orse 'air bristles tied around 'is legs like a red Indian. And 'e 'ad a real sword, but it was so big it trailed on ze ground from ze scabbard 'e had tied under 'is armpits. I used to love the way Henri walked. 'E still walks like zat: long jaunty strides from ze 'ip. A parody of marching, so gallant! Like a person who is going to fix everysing up wis some powerful joke 'e 'as waiting up 'is sleeve.
'Zat castle over ze road from your fuzzer's house was really extraordinary. Henri and I visited it on zat first day, in our fancy dress clothes. I went as a pear tree because we found a velvet bag full of little glass pears in Henri's box of odds and ends. I made myself a crown and a necklace and a belt. Zere were even some artificial leaves left over from some shoes, so I made myself bracelets and anklets from zem. Everybody had to watch for us coming! I tell you, we were mighty beautiful rascals!
'When we got inside, ze walls were _gi-gant-tesque_! Zere were little markets set up at zeir bases under stripy umbrellas, but zey were dwarfed. Henri was stooping and rolling 'is eyes under the bigness. Even if you ran, it took too long to get anywhere. Zere seemed to be so much sky above ze courtyard and I could feel ze ghosts of soldiers in suits of armour. It was very scary. Zere was even an 'awk in ze sky whose shadow was swooping zrough ze sunlit places.
'Ze courts at ze back of ze castle were much more friendly. You could walk out into an open park behind zem. I remember zinking zat ze grass in zat park was very brown. In fact, all of Milan was dusty and dry. Zey said it was ze sirocco blowing straight up Italy from Africa.
'Ze ceilings inside ze duke's court at ze back of ze castle were painted wis angels and stars and vines. It made me zink zat once upon a time ze skies above Milan must 'ave been blue because ze skies in ze frescos were very blue, but zat summer, I tell you, ze real sky outside was brown and gritty.
'In one room Henri and I lay on our backs on ze floor and zere was God, sitting in a circle on ze ceiling vault, surrounded by angels. And zere also was Jesus, rising out of a tomb, being carried up to 'Eaven on a sort of gold plate rimmed wis little snakes.
'Milan 'ad quite a different feel from Paris. Zat castle was part of a wall surrounding ze old city. It was ze place where people went to shelter when ze city was under attack. And it was attacked many times, going right back to ze Goths.
'At ze 'ub of ze old city is ze Square of ze Duomo, and ze Duomo is a very wonderful building, I can tell you. Your grandmuzzer took us zere. When I first saw it, I zought it was a building wis a ballroom inside, because it was covered all over with figures on pinnacles and I zought zey were dancers. I didn't know much about saints, you see, being Jewish. We climbed up inside a tower and came out on ze roof, and Henri said we were coming out into 'Eaven. Perhaps 'e zought getting to 'Eaven meant being 'eld up 'igh in ze air on a very long stick, because zat's what it looked like to me. Zere were a couple of sousand saints being 'eld up on long sticks on zat roof, all at different heights, depending on 'ow good they were. 'Ighest of all was ze golden virgin. After seeing 'er, I made my first doll, an angel wis real chicken wings who would fly about ze saints on ze Duomo's roof and ask zem if zey were hungry. I 'ad to throw my angel out after a day, because ze wings went bad. Zat was ze first time I fully realised zat living things go rotten when zey die.
'Right next to the square of ze Duomo was an elegant arcade wis a domed glass roof. It was very, very exciting for me, because your grandmuzzer knew someone who lived zere, and your grandfuzzer 'ad a very beautiful name for her. 'E called 'er "Ze Apparition of Light". I'd 'eard about 'er quite often before I met 'er. She was an old Russian lady who told fortunes. You would go up ze stairs to 'er apartment, and when ze door was opened, all you could see at first was a swirl of smoke and sunlight streaming zrough ze windows zat overlooked ze square. Zen you would see 'er reclining on a green chaise longue in ze middle of ze room, a cigarette in a long 'older and ze light seeming to pour zrough 'er long fair 'air, so it looked like an 'alo.
'We went to visit 'er quite a few times. Your grandmuzzer seemed to bring her reams of paper, or piles of journals. Zere were papers and journals everywhere, 'eaped up on desks and chairs and on ze floor. And Ze Apparition of Light was always surrounded by a bevy of adoring young men.
'She could speak French, and when I first met 'er she read my palm. She said it made 'er laugh; ze story was very funny and she liked me. She especially liked me because I 'ave six fingers on my right 'and.' Sure enough, on Rose's right hand, the little fingers were perfect duplicates of each other, plump, white and well made, but fused to the first knuckle. 'I am really two people,' she said, 'but one of me is just a little finger.'
It was in The Apparition's salon that Rose first came to suspect there was something very wrong in Italy. It must have been in early June that the normally buzzing salon had taken on an air of gravity and tension. Someone had disappeared, been abducted in Rome, jostled into a car apparently and taken away at high speed. Enemies were putting it about that he'd gone abroad, but everyone in the salon knew he hadn't. They were certain he'd been murdered.
'My fuzzer and both your grandparents were really shocked by zis news, and when someone was arrested in Rome carrying a pair of zis man's bloodstained trousers in an attache case, zey felt zere was no room for doubt.
'Sure enough, 'e 'ad been murdered, but 'is body wasn't found for quite a while, and ze police seemed to be arresting ze wrong people to give ze people who were really involved time to get out of ze country. One of zem was caught leaving Genoa in a speedboat, and 'e was only caught wis reluctance because a journalist recognised 'im. It showed everybody zat zere was an 'igh level of corruption in ze police force, and got zem saying zere was an 'igh level of corruption everywhere.
'It seemed zere was a gang involved and zat zey'd received orders from someone in ze Fascist 'ierarchy to silence ze murdered man, who was Mussolini's chief opponent in ze Chamber of Deputies.
'It was Matteotti, Isobel. Perhaps you know ze story?'
I had to confess my ignorance.
''E was secretary of ze Socialist Party. After 'is murder Mussolini gained total control.
'Ze Apparition's lover was ze most prominent of ze socialist deputies in Rome, Filippo Turati. 'E tried 'ard, but could not convince Matteotti's widow and 'is muzzer to take ze decisive public action zat was needed to bring Mussolini down. Zese women 'ad no sense of being part of a larger web. Zey'd been protected all zeir lives and channelled into domesticity. Lavish mourning was socially acceptable, ze death and ze grief were everysing.'
And so the moment for action was lost. Mussolini's opposition withdrew from the Chamber. They called themselves the Aventine Secession. They fought the brutality and open corruption with newspaper campaigns until the newspapers were either forcibly taken over by the Fascists, or crushed.
FIFTEEN
Reason, Religion and Folly
I WANTED ALLEGRA to befriend Rose Hirsch, but it was a long time happening. She couldn't see how any friend of The Brolga's could be a friend of ours. But Rose knew all about Dadda, I protested.
She knew our Italian grandparents. Didn't Allegra want to know what they were like, what they did, where they lived?
Allegra pretended she was indifferent. Families were unimportant, anyway, and the sooner family structures disappeared and gave way to the collective rearing of children the better. I had only to look at Eli to see corruption in the making. It was not that she didn't love Eli, it wasn't his fault, just that he'd been born a bit too soon. Now there were co-operative, community play centres run by the parents, the whole approach to child-rearing would be revolutionised.
Actually, there was only one co-operative play centre we knew of, and they were continually being left in the lurch by parents who had to race off to courses, or were single and found it hard to make up the quota of parent days.
When I told Allegra about The Apparition, she said, 'Huh! Those Milan socialists! Reactionaries! They wrecked the Second International. In their view the middle class existed for the sake of the working class. They were bourgeois themselves. They had a vested interest in the survival of the free market and parliamentary democracy.'
'What's wrong with that?'
We were lying on blankets on our damp back lawn. Overhead the spring of 1975 was trying to move in by cramming the winter into one dark lumpy cloud in the only bit of sky I could see, framed by guttering, paling fences and a large old oak tree from which the occasional acorn smote our picnic. The washing plapped on our rusty hoist. Failed cauliflowers were raising white fists at the ends of long green arms, challenging me to modify my horticultural methods. Allegra was trying to read five books and a newspaper all at once. Her hair fizzed out, long and wild, from her little heart-shaped face. Every now and again she gathered it into a bundle as thick as a hay bale and tried to tie it in a knot. She moved rapidly from book to book, now sitting up, now lying down; sunglasses on, sunglasses off; biting her pencil, jotting in a margin. She was researching something, I didn't quite know what. I was trying to look like Jackie Onassis.
'You don't understand, Bel,' she said, still jotting away. 'Socialism is about working-class rule. It's about the destruction of the bourgeois state and the so-called free market. Not everybody's capable of doing great things, but everybody needs the wherewithal to live. Look at the situation in art. As things stand, gallery directors do better than artists out of artworks.'
'Not at Mad Meg,' I said.
'But Mad Meg's only one gallery.' She sat up straight, cross-legged, specs on the end of her nose, forehead wrinkled. 'Why should that be the case? By the time art ends up in the public galleries, it's been bought and sold many times. Do people go to galleries to worship art icons for what they are, or for what they cost?'
'I don't know. I haven't a clue why some people paint and others want to see their work. It's just part of culture, aesthetics. I couldn't say what use it has. None I can put a name to. Maybe its original function was to identify people and places. I suppose it has to do with God, too, striking fear into people or eliciting respect from them, or just trying to show God's presence in, or absence from, the world. And then, western art is used to signify taste. Once God has fallen away, I suppose it serves Mammon.'
'Precisely. In a free market system, people are wage slaves. In a socialist system, it isn't money that rules people's lives. A good socialist system aims to free people from the constraints of capitalism so they can live life creatively.'
'How can you live life creatively in the absence of mundane reasons for living it at all?' I was plucking grass, blade by blade, and nibbling on the luscious white bits.
'You're too introspective and cynical, Bel.'
'You still haven't told me why socialism is better than two-party democracy.'
'Socialism is about workers saying how, when and where they will work. It's about ownership by the people. It's about co-ordination and co-operation, equality of women with men. It isn't about centralised rule by the puppets of capitalism in Canberra. Better to know the face on your representative, because then any complaints you have will be real to that person. Truly socialist government is a one-party system with a vested interest in the future of all people in common.'
'What if you don't like the party?'
'The party is open to change and progress, Bel. Communes elect their own representatives. What could be more democratic than that?' 'Might lead to stagnation. I don't see how two-party systems are less democratic.'
'Well, they are, you just get the swapping of one power elite for another. In a one-party system you're not voting for some distant ideology or lobby group there to sustain a money-driven economy, you're voting for what you want to happen around you.'
'But Marx and Lenin talk about violent overthrow, Allegra. As a pacifist, I don't take to that. They stress the necessity for violent change.' 'Yes. Well, things aren't as they were. Now it involves a radical change from within and people are changing, you only have to look about you to see it. There are communes being started everywhere. There are co-operative food shops. People are learning to live simply and use environmentally friendly products. It's all happening.'
To some extent I had to say she was right. Environmentally friendly products, however, hadn't scored a mention from comrades Engels, Lenin and Marx, and conventional people of our parents' generation still sat rigid in their suburban houses, keeping the television polished and making sure the demand for dead chooks was so high special shrines with refrigerated altar aisles had to be built for them.
All the same, our generation was rejecting this way of life, preferring food that was less processed and came in recyclable containers, and a lifestyle that was far more open and socially active. Petty niceties like polished footwear and two-piece suits had been dispensed with. It was not uncommon to see men in caftans. Hair was worn in all varieties of long. Babies were suckled when and where their mothers pleased. The government had brought in a universal health scheme and a legal aid scheme. Australian art, film, writing and theatre were having a field day and everyone began to take them seriously, instead of holding them up against English or American creations and comparing them unfavourably.
It seemed that since the vote had been given to eighteen-year-olds, the voice of the young was being heard in parliament. Something had happened to the pecking order among females, too: it wasn't there. Everybody was talking and everybody had a sex life; the word 'spinster' was an hilarious anachronism. It was only my vegetables that weren't behaving like good Marxists – their behaviour was distinctly biblical: succumbing to plague and blight, bringing forth tares and breaking off at the base with something Jesus doesn't mention.
The ambience suited Dadda. It was rumoured that at a certain party he was raped in a thistle patch by an English pop singer (female) of some note. The Dadda/Brolga quarter was said to be shaky, but solidified once again when it was realised that in this climate, Dadda was a phenomenal commodity; through him, The Brolga could come by just about anyone she pleased.
Dadda took to going alone to deepest, darkest Japan. North of Japan, he confided, was a confluence of current that mingled Soviet with Japanese rubbish. It had given him some trouble in Customs on his return and some of it had been confiscated.
I pictured Dadda getting through checkpoints with his luggage full of detritis while other people were being blown up at airports and hijacked in planes in the name of someone else's liberation, and I wondered how he avoided arrest. But he seemed content and the world was more and more delighted with his output. He had begun concerning himself with the cycles and destinies of global rubbish.
I would have to sneak off to galleries to look at his work, as I didn't want to be seen by people who would tell on me. Even so, the proprietor of the stuffy gallery where he showed in Melbourne would follow my progress with an ancient eyebrow raised. Either she thought I was the wrong sort of client, or she knew who I was.
Once I ran into The Brolga. She hovered in the foyer in a bat-winged coat, humming and ha-ing before oh-well-why-notting over the threshold. 'Hello, Isobel,' she said to me grudgingly. I nodded curtly, but when I looked back at Dadda's painting, I couldn't see it, there was so much interference in my head. I left before seeing everything, mumbling to the eyebrow raiser as I went that she ought to get in the pest control people as I'd just seen a rat.
It's stopped raining. The Midnight Knitter is snoring in her armchair by the fire. Firelight livens the whiteness of her hair with a colour verging on red. The garment has tumbled from her hands and covers her old feet, misshapen from a lifetime of wearing shoes too small for her.
_Mad Meg_ 's silent bellow issues from the dark wall of the kitchen as I pad through to the balcony for some fresh air.
Choughs come here in the mornings to eat bread from Reg's hands, squabbling and beating each other to it. The rain has left large blebs of water on the balcony rails. It is cold out here, the smell of eucalyptus keen in the nose.
Above, the sky has cleared in patches, giving it a rinsed look behind sudsy clouds. Somewhere, unimaginably far away, there are pockets in the universe so dense that even light cannot escape them. There, gravity rules. If two such pockets should collide, out of phase, I have heard it said the impact would cause an immense explosion, reversing the direction of entropy and sending matter back into the field where light moves fastest and gravity is the weakest force. Thus, the history of the universe, a history told in deep time, might be a never-ending series of bangs and crunches.
I was told this a while ago by an astronomer from Edinburgh on the overnight train from Paris to Milan. He was going to the Isle of Elba via Turin for a conference on interstellar gases. 'There's an equation for it,' he said.
And I'd imagined black holes were evidence of a celestial mother, vacuuming.
It was the only trip I've ever made to Milan. I took myself off in 1991, going first to Paris. We left Paris an hour or so before midnight. The astronomer and I made ourselves as comfortable as we could among the bags and knapsacks and other bodies in our carriage and tried to sleep, roasted by a heater on one side and frozen by the draught from a jammed window on the other. I woke once to see a full moon skating on an alpine lake. When I woke again, the train was ramming, loud and vulgar, through the veils of morning.
After Turin, where the astronomer left me, I journeyed on through an industrial landscape to Milano Centrale, Mussolini's monstrous marble railway station with its liver-coloured veins. It compared favourably, I thought, with the tombstone towers, those monuments to the still-born businesses of the 1980s, along St Kilda Road at home.
I had come to look at the city where Dadda was born. I stayed in a pensione on the fifth floor of a run-down building in Corso Buenos Aires, where the air was far from _bueno_. To reach my room, I had to hail the caretaker to open the street door and then cross an echoing courtyard, negotiate another locked grille with a key too small for the lock, take a two-person lift that had three directions in which it could open and dismount in the right orientation on what I had calculated was the fifth floor, there being no lighted panel to say so. And even then, there was the room key to collect.
The room was like a prison cell with a view of similar cells stretching into the far-flung smog. The window was barred, no doubt to deter intending suicides. Blue bedspread, white sink, reading light, writing table. In the event of fire, no instructions, no fire escape, no stairs.
In Piazza Castello, where Dadda was born, the houses are tall and bourgeois and bound to each other wall to wall. Though it is a tree-lined crescent, there are no external gardens. Aerial photographs of the castle show the houses open onto common courtyards that are grassed but have no shrubs. The Milanese don't seem to care for plants, or perhaps they don't thrive in the polluted air.
If Cupid had been leaning out of the firing slits in the castle when Dadda was a baby, he could have shot him through the heart. The arrow would have crossed the grassy remains of the moat, in which cats of every variety (all thin) commune, mate, produce more cats and get fed by a madwoman, a bent little beetle, who passes by on a bike in the afternoons. The moat is about fifteen or twenty feet deep, and the sound of mudguards rattling is a signal for the hardier cats to scale the walls and congregate on the drawbridge. The drawbridge is forbidden the public by a ridiculously thick chain between metal bollards set in concrete in the ground. All you have to do is step over the chain.
The madwoman has had her bicycle adapted to her calling, and in a large wooden crate affair affixed to the back wheel are meals-on-wheels for cats. She scuttles with her saucers of offal, back and forth, back and forth. She has to be quick, because the moment the smell of liver hits the air, rottweilers and alsatians come loping towards her through the park. For the cats who don't make it to the drawbridge, she shunts meat off a platter over the side into the moat.
In the moat, as well as the cats, I saw boots, long modern boots, boots that had belonged on feet but were now cast aside: a long black one, a long studded brown one with a cuban heel, another black with a high heel, and a green. Left over, perhaps, from the War of the Boots?
But the Square of the Duomo is where it happens in Milan, where the strikers gather, the effigies are burnt, the victims of terrorists are laid out and the corteges of the famous, the infamous, the assassinated and the merely dead leave for the final resting place. From Piazza Castello you go there down via Dante, street of the mellifluous poet, clanging, banging and racketing with trams. And there it is, the gala-ball Duomo, an apparition in pink and white marble, once gazed on by that human Apparition from her apartment above the square in the glass-domed arcade. Dadda spent much of his first year there.
The Apparition of Light had thrown her weight behind the war effort. She and her comrades, including Comrade Coretti (first without, and then with her baby son), spent the war furiously knitting socks for the soldiers. These were probably more of a hindrance than a help, as they had no heels. Ostensibly the heellessness was to save on time and wool, and if the sock got holed, it could just be turned around a bit. On the other hand, maybe no one knew how to turn a heel.
I don't suppose it was socks that accounted for the rout of the Italian army on the 31st of October in 1917 when Dadda was newly in his basinette and the Austrians invaded Italian soil. Ten thousand Italians were killed, thirty thousand wounded and almost three hundred thousand captured. The rest fled, most of them leaving behind their weapons. Certain members of the military blamed the defeat on socialist insubordination and sabotage, but letters from the front describing pitiful conditions, hunger, cold and lack of supplies told the real story.
Grandfather Emilio Coretti, dapper, moustachio'd and in the pink of his youth, avoided the front because he was a bootmaker. It didn't seem to matter that his specialty was Cinderella footware, leather was in short supply. Three hundred and ten thousand pairs of boots had been lost at the front along with the killed and the captured. The Apparition, who only just forgave him for his nonparticipation in the fray, wrote to him, when he was in a distant part of Italy on a leather-finding mission, that certain people were getting private exemptions from service. Worse, she wrote admonishingly, prisoners were being sent to the front. Since she couldn't persuade him to go sit in a trench and sacrifice himself for Italy, she bade him instead to look at the conditions in factories and see for himself the savage and arbitrary regulations that were being enforced on men, women and children, day workers and night workers, in the name of the war, but in reality for the benefit of the factory owners. 'Il Coretti' must do his best to save and protect the bootmaking industry for its workers: he could, for example, refuse to supply owners he found in breach of the laws.
This was just what Emilio was doing. He was compiling a dossier on work practices in the bootmaking industry for Uncle Nicola, who had been researching the conditions of the workers and peasants for years for the Humanitarian Society of Milan.
Uncle Nicola, older than our grandfather by a decade or more, served with pride in the Alpini up until the defeat at Caporetto, during which he'd been peppered with shrapnel and, afterwards, honourably discharged. He was a graduate in law from the University of Bologna, where the assassinated Matteotti had been a fellow student. His aim in life was to revolutionise the revolution. He was probably quite correct when he asserted that there would have to be traders and merchants before a socialist economy could work; these people were a true class, without whom one of the vital factors of production, selling produce, would be overlooked. He saw no inconsistency in playing the stock exchange while awaiting the downfall of capitalism; the stock exchange told you where things were, and leather, it seemed, could be had from Argentina or Australia. So it was that an order for leather was placed with a Melbourne merchant.
To Uncle Nicola's surprise, Australian leather was well and truly spoken for. Not only was it lying all over the slopes of Gallipoli on the feet of Australians living and dead, a fact which flabbergasted the Coretti brothers to whom the Dardanelles seemed totally irrelevant, but it was also keeping Russian soles from the mud, a Melbourne bootmaker having received an order for four hundred thousand pairs of boots from the powers that were, might and could have been in Russia. It was 1917.
The Italians had made a hesitant entry into the First World War. The Italian left was noninterventionist, and the International hailed them as heroes. But there was nothing heroic about it, Italy couldn't provision an army. When one was eventually mounted, underprovisioned, the left invented a policy of 'Relative Neutrality', according to which it was all right for the proletariat to fight a defensive war on Italian soil, but not an offensive war against the proletariat of another country on foreign soil. So to fight a war at all, the socialists required an invasion: no wonder the Austrians obliged.
I found Dadda's city a place of contradictions. At the other end of the Square of the Duomo in 1991, face-to-face with the gold madonna on the Duomo's highest spire and reflected in every window in the square, was the black, white and yellow holy family of Benetton. Behind it hid the Palace of Reason.
SIXTEEN
Mad Meg's Basket
A MAN IS 'nesting' on the roof of Mad Meg. He's going to be there for a week. In the mornings, we're to send him up a bucket of food and he's to send us down a bucket of waste. If it rains, he will put up an umbrella, a red umbrella as specified on his instruction sheet, pinned up by the back door. To see him, you have to go out into the backyard. Initially he was furious that we wouldn't let him make his nest on the front part of the roof because it was too near the electricity lines. He accused us of censorship and discrimination against people in wheelchairs who wouldn't be able to see him because you have to go down steps to get into the backyard.
He's an installation, part of a joint effort by Figments, Siècle and us, though Siècle doesn't realise it's participating. Siècle is guilty of all modern crimes. Because their paintings are consciously made with the buying public in mind, they are guilty of producing 'art objects', of turning artworks into dollars.
At Mad Meg, on the other hand, we have given up painting. What's the point? In our day and age, culture ceases to be property. Property only reinforces the bourgeois myth and the hard sell through a hedge of pompous words, behind which lurk the slight, the superficial, the fatuous and the vacuous.
Institutional art is not art. You might as well exhibit money – in fact, Mad Meg has. One of our exhibits was the tapestry of a ten-dollar note and somebody knitted us a heap of chequebooks.
The art object has become the enemy of art. We live in our heads. If you can't make art that isn't permanent, you can't make art. Part of the beauty of art is its ephemeral nature.
Inside Mad Meg on this occasion we have a total environment called Little Red Riding Whom? On your right as you come in the door, there's an installation called Manhood, in which a frieze of the same suburban street scene, taken over and over from early morning till late night, surrounds a Harley-Davidson mounted on a revolving disc in the centre of the room. Visitors are invited to sit on the bike and enjoy the resultant effect, which is that of making the street scene move. People record their experiences afterwards in a book by the door: some say it looks as though the houses are flying through twenty-four hours, others that the scene is repeated ad nauseam in the same way the suburbs are, morning, noon and night, so when you're riding, in a sense you're going nowhere and the bike loses all its prestige. A few feel powerful when they sit on the bike and there are those, of course, who say the whole thing's shit, by which I suppose they mean it's the deconstruction of motion.
On your left there's an installation called Womanhood which consists of a carefully set kitchen table using the sausage as an art object. Viewers are invited to sit on the chairs around the table and watch a video on melon balling in which several experts show how to do it their way.
In the right-side back room is an installation called Manhole which didn't require anything of the artist apart from artistically removing the manhole cover from the ceiling. Needless to say, across the corridor is an installation called Womanhole, which, in years gone by, before schoolgirls mastered the Art of Tampon Insertion, would have been fair game for the vice squad, but in these enlightened times anything goes.
At Figments, Miles is doing Objecthood. Room One: white paintings on white walls. Room Two: continual release of bubbles from a bubble-blowing machine over a sand drawing by Miles's wife, Anita, on the floor. Room Three: Absence – the huge painting of a black pedestal, abraded so the struts behind the canvas show through. The process of making art is part of the art object.
In the back room off Miles's foyer is a show of pictures painted in the dark by me and Miles's nephew, David Silver. They are called 'The Birth and Extinction of Red'. Mine map the departure of red from a bowl of roses into darkness; his are much more minimal, done like a Dulux colour chart, from black to red to white. I painted mine in the evenings; he, an insomniac, painted his in the mornings. I was told that mine were far too figurative until I came up with the idea of them representing the eclipse of the bourgeois image.
Both galleries have Siècle invitations on trays by the doors. When Checkie Laurington came to the opening she asked Miles, 'What are they doing there?' When she came over to see the Mad Meg part of the show, she said, 'Don't you realise, Allegra, you won't sell any of this?'
Later, she joined us for drinks upstairs in Miles's flat.
Miles's wife, Anita, is shy and never appears at openings. She hasn't set foot in Mad Meg, though we always send her invitations. We (the politically correct plural) like her work, and Miles has suggested that Mad Meg might be a good place to show since we're mainly women and, these days, we mainly show conceptual or performance art. But Anita will only show with Miles and then only in a group.
Allegra has gone out of her way to befriend Anita, but Anita doesn't like to be wooed. Whenever anyone from the Mad Meg collective is around, Anita lets Miles do the talking.
She and Miles live above Figments on two barn-like floors. It's a steep climb up three sets of steps that are more like ladders. The first set takes you to the space where Miles hangs his picture stock on vertical floor-to-ceiling racks. Also on this floor are his collections of antiquarian books and Aboriginal artefacts, so there are a great many dusty bookstacks and spooky shapes to negotiate in the half light on the way up to the flat.
Anita generally hangs around the kitchen, brewing coffee in the shadows. Whenever Allegra makes to go and ask if she can help, Miles says, 'No, no, it's all right. It's Anita's custom.'
'Brewing the coffee and doing the washing up,' says Allegra, 'are not customs in anyone's language, and I'm sure Anita's ancestors did neither.' Anita is descended from the Bunurong and Wurundjeri Aboriginal people.
Bridget Kelly says the Bunurong haunt the dump: it was one of their corroboree sites. Whether they haunt it or not, they have just made it part of a land claim. Before white settlement, they were granted 27 acres of (their own) inner Melbourne suburban land, which these days takes the form of two recreation reserves, the billabong and what has become the dump. A couple of weeks ago, Kooris from several tribes marched on state parliament and presented a petition for their return. The petitioners were duly thanked, but the dump and the billabong were declared non-negotiable. They, according to state parliament, had been on a road construction reserve since 1957.
No one can take rubbish to the dump anymore because it is no longer officially a dump. There is a lot of pressure for the freeway to go through. Even the Workers' Union has agreed to it and we're thoroughly shat off with them. Bridget wouldn't give in without a fight, though. When the bulldozers came to wreck her house, she barricaded herself in and loaded Little Ernie's shotgun. After a standoff that took up most of the daylight hours, hours during which thwarted bulldozer drivers had sought advice, had taken their tea and lunch breaks in the regulation hut, to and from which they had had to be driven in a regulation vehicle. When night fell and the dozer drivers had knocked off, the police went in. They had her gas and electricity cut off.
Bridget, short of neither candles nor firewood, soon had the house better lit than it had been by the dust-smirched globes. The house seemed a little wooden ark set to sail through the night o'er the rubbishy sea.
When the police turned on a loud hailer, Bridget hurled up a window and yelled, 'Piss off, you fuckers! You're disturbing the peace!'
Kellys and Corettis called encouragement from the fence.
When the television crews arrived, the whole place was floodlit like a stage on which a sinister glint had been imparted to the barrel of Little Ernie's rifle at a front window. Lights on police cars flashed; the players in the drama threw long-legged shadows with little bodies riding around on top of them. 'She's got a gun,' they said. The spectators were required to withdraw to a safe distance. We stood shoulder to shoulder and the shadow we threw behind us was a long monster with many heads and twice as many legs.
About eight o'clock the police, with the quiet aid of the fire brigade, managed to get themselves over the ribbon wire (a draconian precaution ostensibly taken against seagull feeders, but in reality to deter freeway bombers) at the back of the house and burst into it from the rear, but in their bursting, they upset several of the great many candles Bridget had burning all over the extremely messy and tinder-dry interior. The flames immediately found things to feed on and got raging: the police burst out the back again and Bridget burst out the front. So fierce was the fire, it took less than fifteen minutes for the house to go up like a bomb, windows exploding outwards and a great whoompf of smoke rising in a column in the floodlit sky.
The fire engines, having been engaged at the rear of the house down a dead-end lane, didn't reach the access for a good five minutes (perhaps they were under instructions not to), and the house continued to give itself up in white smoke as wide as the house was long and as thick as the house was deep.
The column sank under the brigade hoses, but then it rose again with another whoompf, this time grey.
The television crews hardly knew what to report. It looked as though one of the portals to hell had been opened; when the flames had conquered the smoke, the muck mounds behind the house grew tipsy in the light. Then they, too, caught fire, the flames tiptoeing rapidly over them, hither and thither, little devils dancing.
We stood away from the heat, our monster shadow giving way to leaping witches. People began to say, 'Methane, methane,' to account for the spot fires.
Soon the smoke had dispersed and the entire neighbourhood was out of doors, gaping. 'That's not your land!' screamed Bridget at the policeman who was busy arresting her. 'It belongs to the Abos. And the stuff in the house belongs to me! All my stuff! It's me livin', understand?'
But hardly anyone understood Bridget. The police impounded the rifle and locked her up for twenty-four hours while we found the money to bail her out. She was a bit of a hero to the local blacks, who'd seen her screaming on television, but being a people who keep to themselves, they didn't contribute to her release and soon let her heroism pass. They didn't want a white woman taking action on their behalf.
We rang Wednesday Monday and he took his second trip to Melbourne by air, this time without a pocket full of fake boils. The magistrate was instructed to remember that, although Bridget had been armed, she hadn't actually fired the rifle, nor was she intending to fire it.
Bridget's crime, if indeed it was a crime, was to resist having her home of more than twenty years flattened to make way for a freeway.
Were the cars of outer suburbia more important than the people of inner suburbia? Was it not true that Bridget and many people like her in inner suburbia were being dispossessed, their wishes overruled by those whom money made powerful? One need only document the Aboriginal history of the area to see that it was not possession, but dispossession, that was nine-tenths of the law.
This time the magistrate was sympathetic. One had the feeling the case was being wrapped up as quickly as possible, probably to keep the land rights issue and the freeway controversy out of the papers. Since Bridget had no money to pay a fine, he gave her three months with a parole period, contingent on a bond, of two weeks. We took picnics in to her and decent books to read. She found jail restful and was a bit sorry to leave when her two weeks were up.
Kellys were billeted out at various places until they were found a two-bedroom Housing Commission flat into which they moved, all six and two dogs without a blanket or a stick of furniture to their name. Inside a month, however, it was all one could do to open the front door and squeeze down the passage, to find Bridget pottering about among her cartons of this and her cartons of that, looking for a lens here, a doorknob there. 'Want a jumper? Got a jumper'd fit you. People throw out anythin', anythin' at all.'
Maggie and Kelly decided they'd be better off living above the Pantechnicon, although they had no permit to use the shop as living quarters. As for the land rights, they would be investigated by the appropriate department in Canberra. Which meant, to put it in lay terms, they disappeared into a rigmarole of words as sugar dissolves in a cup of tea.
The burning of the dump was a stroke of luck for the freeway lobby and made the favoured route a certainty. If nothing else, we at Mad Meg and the Pantechnicon could breathe easy.
Since there are more of us than will fit on the Turners' settees, we are sitting on the floor on Persian prayer cushions. Among us for the first time since Aldo's Italian lessons, Checkie. We are playing cards.
Checkie has gallery-going down to a fine art. The feminists and the art school mafia came to this one. After Checkie clopped through Mad Meg and staged an irreproducible laugh, like the repeated stabbings of a stiletto through someone else's open parachute, over Little Red Robinhood on our roof, she clopped up to Figments and latched on to Miles. Obviously her father has warned her to steer clear of the seventies avant garde because its object isn't money. Nevertheless, she was discovered asking Miles intelligent questions loudly. Particularly within earshot of David Silver. It's Love, and the chips are down.
Every so often, cigarette in a holder and hand on hip, Checkie swivels round to address David as if some vital question has just occurred to her. She's tall, she holds her head high, a daring posture, because if anyone should fail to answer her, she would be stuck conspicuously with her chin up and her eyelids at half mast. But her questions are almost always answered. We have to say, though it causes pain, Checkie is considered highly intelligent, even by Miles and Bart. In Motte circles, they would have considered her 'a calculator'.
I know David from Sydney. He used to live in Bart's flat above the gallery. I didn't see very much of him. Apparently he slept all day and spent his nights working and smoking non-stop.
I didn't warm to him right away, because the only time he spoke to me he said he liked that woman my father was with. Perhaps he was teasing, nevertheless, it was a reminder that I'm not the only person entitled to have an opinion about The Brolga. He redeemed himself a bit later when he laughed at Bart's observation that sleeping with her must be like going to bed with a cheese grater.
I knew that he painted, but I didn't know what. He wasn't in 'Turner at Figments', but when he rang up the day Miles and Bart were arrested and I answered the phone, he sounded quite affable and pleasant, as if we were old friends.
His Uncle Bart has been taking care of him since he was sixteen. Apparently he did very well at school, but Bart wasn't able to persuade him to go to university. Instead, he did odd jobs around the gallery. Bart said he was a 'dizzarsster'. 'Honestly, only David could drive a single nail into a wall and bring the wall down. It's really amazing that someone so lacking in manual dexterity should want to become a painter.'
Painter, nevertheless, he has become, and now it's Miles's turn, apparently, to look after him, even though one supposes he could look after himself at twenty-eight or twenty-nine or whatever age he is.
Miles is dealing the cards for pontoon. The stakes are matches. Allegra lies on her belly opposite David, looking ravishing in a mini-dress made out of pale pink and pale green evening gloves. Checkie is seated like the Lorelei beside him. One has to say that Checkie is also looking ravishing, but whereas Allegra is Essence of Ravish, Checkie is Ravishable Chic, her costume a light brown minifrock, the back of which is much longer than the front and makes a train which the wearer can pass between her legs and toss over her shoulder. In addition to being a simple, very good idea, lest the back of the costume look too plain it bears the stencil of a blue Matisse dancer. The rest of us – Maggie, me and two of the Troika – are out of contention.
I look for the lack of manual dexterity that Bart said was David's hallmark, but it isn't in evidence. His hands are very well formed, the graceful brown fingers pink at the tips. Like Miles, he is a chain smoker, but it doesn't show on his hands.
He has been drinking and is just a bit drunker than the rest of us. As the cards are dealt out and he loses yet again, Allegra wins yet again and Checkie resorts to breaking her matches in two so she can stay in the game, he tells us, between bouts of laughter that shake his diminutive body from head to foot, that he's had crabs.
'I didn't get them from having a good time, either,' he laughs. 'I think they must have been on this terrible old mattress I used to sleep on once.'
Checkie descends from her Olympian heights to say, 'I hope you're rid of them now.'
David picks his ear wax with one of her broken matchsticks, 'Oh, it was years ago in Sydney. I didn't have any money. I used to live in an abandoned house that had no electricity or water. I had an old coat and a mattress and a pile of newspapers to keep me warm. I used to eat sugar cubes and people's leftover biscuits in cafes. I got so malnourished I fainted on the way to the doctor's and one of my teeth fell out. I think I must've had scurvy. My skin was so bad you wouldn't believe. You know how your skin goes when you don't eat properly.'
'No, actually,' answers Checkie. 'You mean to say you practically starved to death in the middle of Sydney at the age of... of... how old were you?'
'Fifteen. Sixteen. I ran away from home.'
'Good heavens! But you're Miles's nephew, I thought. Which means you're Bart's nephew. Why didn't you go and get a feed from Bart?'
'I didn't know Bart then.'
'But what about... what about... nous?'
Miles cackles, removing a strand of tobacco from the tip of his tongue. 'Nous isn't a universal quality.'
'I wanted to be a painter,' says David. 'When I told them in the unemployment office, they started calling me van Gogh and got me a job selling magazines from a trolley in a hospital. It was a women's hospital. They used to call me the Book Man. Books! I sold _Post_ and _People_ and at night-time I went home to a condemned house, my crabby mattress and my pile of paper.'
'Ew,' says Checkie. 'Ghastly. How did you end up with Bart, then, if you didn't know him?'
'I knew of him. My mother used to talk about him all the time.'
'The pink sheep of the family,' quips Miles, sending David into another paroxysm of laughter. He borrows half Checkie's halved match collection and bets the lot to Allegra's bank.
She wins. What Allegra hasn't won tonight isn't worth talking about.
David Silver. Though his legs are the same length, he walks stiffly, like a man who wears a built-up boot. Perhaps his feet don't match. Yet when he takes his socks off, he has beautiful feet, just as he has beautiful hands. His socks don't even smell.
His hair is thick and strong and dark, but it grows as if it has been yanked back by someone in a terrible temper. His widow's peak is so deep, his temples seem about to sprout a pair of devil horns. His face is triangular, the sensual mouth taking up most of the width of the chin. He has a beauty spot on his left cheek that gives him a decadent appearance.
One great eyebrow spans his face so his eyes are like soft furry animals sheltering under it. They are green eyes, prettily shaped and dark, the only really dark green eyes I've seen. He is a child-man with a childish speech impediment: he says 'wiv' instead of 'with'. He is done up in soft brown skin that has no freckles. He devours cigarette smoke as if it were mother's milk. Because he keeps crossing his eyes to look at his nose, Miles says he has a Pinocchio complex. He seems to have trouble breathing up his nose because when he crosses his eyes, he also tries to breathe in, but he can't seem to get enough breath through. He seems like a thwarted thing to me, curiously beautiful, but kept back from a larger plan of himself.
The kitchen shadows are enlivened by the television. Anita is listening to the late, late news. There's a deadlock in federal parliament, and things are tending drastically towards anarchy.
SEVENTEEN
History and Moving into It
WHITLAM'S SACKING WAS one of those occasions you remember. Work stopped in the potting shed and everyone converged on the compactus in the herbarium where Beryl Blake was shamelessly pouring tots for us, even one for Loyola O'Flynn. 'It's a travesty of justice,' Beryl kept saying, a pink-mouthed groper in a hot aquarium. 'A travesty of justice,' her gills blood-mottled with her fourth or fifth tot. She relished the bite and hiss of this word, justice, the rhythm of which kept her from falling backwards off her stool. The radio was turned right up. The herbarium smelt of whisky and feet. Thongs plapped and squealed on the lino. Wattle birds jabbered in the red grevillea just outside.
Afterwards, the Mad Meg collective joined thirty thousand other people on the streets of Melbourne. We thought we were in the majority but we weren't. Barely a month later, the vote went so far the other way, we were stunned into silence.
There were a couple of half-baked, day-long general strikes in which participation was optional. That was all. The shakers and movers were gathered in impartially by History.
Eggs are cracking all around _Mad Meg_. The housewives, who have followed her to Hell, flay the monsters around her with sticks and swats, but the monsters are our imperfections and failures made flesh and they are perpetual; they swarm up walls and out of water neverendingly. They occupy the sky and burn our horizons. The eggs crack: the newborn could be with us or against us, but ideal people are never born. We come without memory, and the consequences of our coming are immediate, ongoing and always unexpected.
Dadda died in 1976. It still seems strange to me to say that Dadda died. I still think there must be some mistake. How could he die? How could he hurt us to begin with and then die?
Eli and I were weeding the garden when the phone went. It was a Sunday. Allegra was still in bed with David Silver: David was meant to be living with Miles but, little by little, it seemed he was moving in with us. I walked from the sun-bright garden into the blue-blackness of the house. Eli and I both loved sun-blindness and would feel our way along the cool hall of the house as if we were underground and searching for treasure. I picked up the receiver. There was silence for a moment and then The Brolga Laurington said, 'Oh, hullo, Isobel. Thank God it's you. I wouldn't have been able to say this to your sister. Henry's dead.' Allegra said later it would have been more civil of her to have kicked me in the heart.
She said there was no point going to the hospital where Dadda had been taken with a heart attack, but Eli and I went immediately, leaving Allegra and David to tell our mother.
'I'm not good at grief,' said The Brolga when I entered the room where Dadda was lying dead. I was horrified to see Checkie Laurington remove Uncle Nicola's diamond ring from his finger and give it to her mother. 'Leave my father alone,' I hissed. 'Leave my father alone.'
'Oh dear,' said The Brolga with dry-eyed chagrin. 'I told you not to come, Isobel. Really, what is there for you to do here?'
I hissed at her, 'Get out.' I hissed at Checkie Laurington, 'Get out.' Viva interposed a hand over her daughter's shoulder and murmured in her ear. At the door of the room she said to me, 'I'll give you five minutes, Isobel, and then I don't want you here.'
Just then Allegra arrived, running. We locked ourselves in. Not only did we lock the door in which, because it was a special room, there was a key, but we barricaded ourselves and Eli in there with the metal chest that had been by Dadda's bed.
There was no doubt that Dadda was dead: though we pulled the tubes out of him and tried to revive him by sitting him up, he was dead. His skin was blotched, his hands were yellow, his forehead was like polished stone. When we opened his eyes, his eyeballs were turned upwards. We tried to move the blue bit down so it would see us, but it wouldn't come. Eli climbed up on him and tried to make his mouth talk, but his mouth would only hang open a bit. We stared at his flesh, all its pores, the down on his cheeks, the uselessness of it with no life in the body. How shabby it was! The blotched chest with man's hair on it, the growl gone out of his thumbs and palms, the incapacity of his legs now to weigh shapes into his shoes. Eli stared at me and I at him and Allegra. We howled and howled. The Brolga outside the door said, 'Oh dear, really.'
Then she got angry and started to call us pagans. After all, she was his wife. We didn't really hear her. We didn't hear her till weeks afterwards. Our siege lasted a couple of hours; sometimes we sat under the bed, sometimes on it, sometimes on top of Dadda, sometimes at his sides. Then, because they couldn't get the door open as they couldn't locate another key, a locksmith arrived. It was the first case of its type. We were banned from the funeral. She spread his ashes, but wouldn't tell us where.
PART THREE
GOOD INTENTIONS
EIGHTEEN
Christmas and an Explanation
THE PAGES OF history are heavy and hard to turn. Beyond entertainment and fashion, history resists what is new. We pretend that politics will correct our faults and eliminate evil from our midst, but to think we can eradicate undesirable aspects of ourselves is nonsense. Yesterday's evil is today's expedience and today's good is tomorrow's calumny.
Reality goes on and on whether we are here or not. People in our midst try to tell us there is no reality, all is a figment of thought. Lucky, then, the starving child with fly-infested eyes, because its pain, its tears, its whimpering are illusory. But the children in the deserts who look like worn-out shoes off a rubbish tip are real. The laws of nature bind them as they bind ourselves, their pain is human pain. They have always been there, their crouching mothers so emaciated their veins and arteries throw shadows, their fathers, maybe, legless in some makeshift hospital in a war zone.
According to Reg Sorby, the human race, rich or poor, is despicable: it runs on avarice and greed and it's destroying the planet. Women, Reg maintains, would be much better at running things than men are. To 'prove' this is so, he has constructed a phantasmagoria of past societies where women were in power and godhead was female. When his phantasms turn up in paint, however, women don't so much have heads as winking vaginas and vigilant breasts. His paintings of nudes seem relics of the sixties, when intelligence was running a poor second to sexuality.
But in the nineties, the West has bourgeoisified, the working class has aspirations not recognisably different from the middle. All around us are palaces of conspicuous consumption, household glut, women's wardrobes bursting, scent bottles lying forgotten everywhere, fake faces in bottles, fake shoulders hanging around in coats: this cult of appearances, this female cult which now is dominant and extends itself to men. This is where we've come. Lenin and Marx lie smashed in public squares. The centrepiece of this life is an eighty-dollar haircut with a Rolex watch, a pant, a Tee and coloured glasses in which we see/ don't see ourselves in our own coloured glasses, for which, in turn, the have-nots clamour and hanker.
Advertising hoardings have more effect on behaviour than nuclear stockpiles, those masculine playthings which, their bluff now called, have been moved aside to reveal – not the Palace of Reason – but the Republic of Unintended Consequences.
The Midnight Knitter is clicking away again, driving her needles into a web of wool and mumbling about scruffiness and boot polish. Even if only in her mind, there are still men in shiny shoes somewhere. They chat affably with one another, as if the answer lay in shiny shoes. But then they stall, they break down, they get deadlocked. The moment passes. Catastrophe bowls along, a strapless, backless prima donna whose price is beyond all reckoning.
God is back. The widow Coretti has been seen among the Anglican flock, at one with the Christmas miracle. Her brolgaisms quelled by the swell of song, at her side an undersized young gentleman from whose shuddering shoulder-length curls fervour has shaken the dandruff. Beside the undersized, an oversized young lady bursting from a clinging cotton garment lavishly decorated with sunflowers and clapped to her midriff by a wide black shiny belt. Beside her, a tweed coat that might have been made for a schoolboy shop dummy encases an erstwhile schoolboy with wisps of mouse-brown hair. Between carols the oversized young lady sits forward on the pew, eagerly and piously listening to the efforts of the choir above. As she listens, she blinks false eyelashes and the undersized young gentleman gives the tweedy young gentleman a look behind her back which says, 'See what I have to put up with'.
God has struck. The organist strikes back. The Brolga rises. The Brolga kneels. _Once in royal David's city_ , sings The Brolga. _Stood a lowly cattle shed_ , sing her three young friends. When the time comes for the body and the blood, The Brolga has a way of looking as though she belongs, whereas the undersized young man rushes forward far too eagerly and has to jig about to shake nonchalance into his person as the queue heads for the altar. The oversized young lady would look wrong anywhere, her presence increasing the insignificance of the tweedy boy.
This line of humanity, stuck between a fat chap with naturally wavy hair and his jumper on inside out and the sexton, who wears his suit like a coffin, is the directorship of a new gallery, south of the Yarra, called Viva Hallett-Coretti. Not only but also. _O come all ye faithful_. On the way back from communion, the undersized young man throws his dramatic but amateur gaze on the cross before entering the pew.
How do we know this? Belinda Bloomfield, Eli's godmother, is the organist. We at Mad Meg feel that the undersized young man, known as Jerry Gosper to intimates and Jerry Gospel to us, has suggested the God tactic. He finds it good for his own malaise, which goes clothed in the tweed coat, even in warm weather. To this tweed ambitious parents have attached the name Nyle Laidlaw. It is Jerry Gospel's sister Pattie who interposes her sun-flowered girth between the would-be lovers (though it is difficult to think of Jerry Gospel in an attitude of sprawled scrawn inviting the reluctant Nyle to join him on the couch). Jerry and Pattie Gospel share premises, while Nyle lives alone and is pursued around parties forbearing to be kissed.
God can count himself lucky, having found employment. He resides in the large black onyx cross that once belonged to Allegra Coretti the First and ought now to belong to her namesake, but instead is to be found ostentatiously flattening the already flat chest of Viva Hallett-Coretti. Can we stand it? And what's more, is it legal? Where is the will? Is there a will? Dadda was not a careful man in earthly matters. The first solicitor we hire tells us to go for a settlement out of court. But we do not know the extent of the estate. Wednesday Monday looks up the rules and tells us to contest.
We lodge objections. Probate is deferred. The Brolga is ordered to furnish reasons why Allegra and I cannot co-administer the estate with her.
An inventory is produced and prophylaxis offered in the form of two paintings, a drawing of ourselves as children and the annotated photograph of 'An Ingenious Sewerage Plant'. This is not a third of his estate. We are entitled to a third of his estate.
She furnishes reasons why we cannot co-administer, citing our hospital siege as grounds.
There is humming and hah-ing. We are furiously impatient. What has she done with his mess? I demand his shoes and his collection of clock parts at point-blank range. 'Oh, really, Isobel,' she says through her flywire door, 'really!'
Allegra, armed with a colander, hooks a whole cauliflower out of a pot where it is softening for the jardinière in a kitchen at someone's party and hurls it at her as she passes by. It misses and sieves itself through a window screen.
We are not increasing our chances of being permitted to co-administer. There are grounds for The Brolga to suppose she might be done grievous bodily harm. A restraining order is issued, but probate is not cleared. We slap caveats on her house, her gallery and Dadda's estate. Negotiations stall. We are pushing as hard on our side as she is on hers.
When Harry Laurington turned up on his own in Mad Meg one Saturday morning, Allegra and I gave each other sidelong glances and slid from behind our desk to stand before him, a couple of bodgies, our arms folded across our chests. Harry swung his good arm down, palm upwards, in a conciliatory gesture. We shifted our centres of gravity. 'Please...' he said in a tone that told us he was finding the going hard.
Then, 'Coffee, Harry?' Allegra asked à la Lauren Bacall. After all, except on an artistic level, our quarrel was not with him.
'Thank you.' Harry dipped his head and prodded his middle momentarily in a courteous little bow. We showed him into Mad Meg's closet-sized kitchen, put on the percolator and sat him down.
Suspicious Motte gazes wandered upon him from Coretti faces. 'It's about your father,' he said, making the kindly gesture again. 'I haven't come here about the will, because I know nothing about it, but I think you have a right to know where Viva spread Henry's ashes.'
Allegra said aggressively, 'Of course we have. He was, after all, our blood relation. It may seem strange to you, but Isobel... we... both of us loved Dadda. We didn't ask him to throw our mother and us over for that... Mink Murderer!'
Harry nodded. 'Quite. Quite. Believe me, there are several people who wish your father had not run off with Viva. Checkie and I, for instance, share your pain.'
We doubted it in silence.
'But as I was saying... I have a house on the Bellarine Peninsula. It's been in my family since it was built in the 1890s. I am very attached to it. When I was small, my parents used to take me there. I had good parents, I loved them very much. It's a peaceful place, my mother had everything done in white, and while it is many years now since she died, I've kept up her custom. The yard is planted with rare Australian plants; again, it was my mother's idea. She loved the native plants of this country and would go on expeditions to find rare species in order to try and propagate them. She was a great friend of Elinor Sorby, Reg's mother, who used to collect orchids. Anyway, such plants as survived on the peninsula my mother nurtured to maturity. Elinor drew them, particularly the orchids, which my mother used to classify. Reg has a collection of sketchbooks somewhere. They're quite special.
'My mother was a kind and careful woman, very dignified, but not unadventurous. When I was still quite young, she and my father were drowned in a freak yachting accident. The weather seemed perfect for sailing, but treachery can wear the calmest face. There was a rip, the boat was carried out, they tried to get back in but capsized and were carried out themselves. They were never found. As you can imagine, I was devastated. The beach house came to represent my mother and her marriage for me; it was my solace. I don't let anyone alter it.
'During the war I met up with a young chap – Viva's brother, Leslie Hallett – as you no doubt know. A terrifically gifted painter. I kept him down in the beach house AWOL for much of the war. He was sensitive, given to depressions, quite unlike Viva as it turns out. He detested war, yet felt guilty about having evaded it. Young friends of his had gone off to die or be captured. He and I shared this guilt of not having gone; I'd been considered unfit because of my arm.
'You probably know Leslie's final painting – everyone does – the soldier drowning in a vast green sea?
'He'd been reading Auden's poem on Bruegel's _Landscape with the Fall of Icarus_ , that's why there are two unattached wings falling out of the sky. In shape, they mimic the sails of the ship in the Bruegel. Of the ship, Auden says _and the expensive delicate ship that must have seen/ Something amazing, a boy falling out of the sky/Had somewhere else to get to and sailed calmly on._ Leslie has cast the sea as the overwhelming aspect of war. You, Isobel, once had a related idea: your painting of heavy people with miniature wings; but whereas your figures had never been able to fly, Leslie's soldier had flown, if only in thought.
'Perhaps there's even more to it than that. I sometimes think the "expensive, delicate ship" to which the wings refer is how Leslie imagined my life. When he'd finished that painting, he must have walked into the sea and kept on going. He was found washed up along the coast towards Point Lonsdale. He was in uniform. There were no marks on him to suggest there'd been foul play: Viva and I took him back to the beach house and buried him under the grevilleas. We chose the grevilleas because they're wonderful in spring, tumbling with nectar eaters: wattle birds, silvereyes and the like – real, not mythical wings.
'When Henry died, Viva begged me to let her bury him near Leslie. I tired to explain to her that nowadays you need all kinds of permits. When I rang to find out, I was told no one had been buried on private land since the 1950s. She badgered me and dragged Reg Sorby into it, too, but Reg told her to get some sense and have Henry cremated and scatter his ashes. Ultimately, that's what she did. I understand Reg arranged it all. I gave her permission to scatter the ashes at the beach house. She would have done that, I dare say, quite probably near Leslie's grave.
'If you would like to go down to the house, either of you or your mother, or all three, whatever you like, you can just tell me in advance and I'll give you instructions and tell you where to find the key. It takes a couple of hours to get there; you may like to stay in the house for a day or so, however long you wish.'
The percolator hicked its coffee-scented fumes. As I set out three mugs on the sink, I was overtaken by the feeling I'd done it exactly this way before. I glanced at the kitchen light; someone had stuck a Japanese bookmark to the shade – flypaper, I thought of flypaper, and Granpa's terry towelling hat. I almost expected to hear the squabbling of doomed chooks as I poured the first coffee and handed it to Harry. I almost laughed, but Harry's tone had pulled me away from the farcical and all I could do was contemplate how we rob ourselves of feeling when we laugh too soon.
'I don't hate your father,' he said. His voice was low and quavery; our mother would have called it 'well bred'. 'He was a terrific painter, among the best: highly original, very perceptive. His world was a system of interconnecting signs; his approach was something like a highly eccentric doctor's. Your father saw worldly malaise, but it was incurable.'
'Fatalistic bastard,' said Allegra, but Harry ignored it and went on.
'When Viva came home from Europe in 1937, she came with Henry. I knew neither of them then, nor did I know Leslie. I had heard of Rose and Laurent Hirsch and seen them in various places, and to tell you the truth I thought Rose was a terrible exhibitionist, but there was no doubting she was lovely. I was happy to meet her, ultimately, when Leslie came into my life, and it was through Rose I met Viva. Naturally I was interested in Leslie's sister. Though Viva didn't paint herself, she had a terrific eye for art. Leaving aside all else, Viva is one of these people with a highly developed sense of aesthetics. She certainly brought out my own latent sensibilities. I still believe we made a better pair than Viva and Henry. And, anyway, Henry's heart wasn't in it, not then.
'I know there is no love lost between you and Viva, but I should tell you she's nobody's fool: glib, yes, but stupid, never.'
'H'mph!' went Allegra. 'That's a matter of opinion. There are other ways to be. Why did she choose that one?' This drew no reaction from Harry. 'Are you quite sure you don't know anything about Dadda's will?' Allegra then asked sharply.
'No, I don't. But you were quite right not to settle out of court. You must press her to disclose the whole estate. It could be that she had things salted away in her name. It wouldn't surprise me; she's extremely shrewd.'
'Extremely hard! How could you marry her?' Allegra cried, suddenly passionate. 'How could anyone marry her?'
'You might well ask, Allegra. Once, you know, we were all great friends: Rose, Laurent, your father, Leslie, Viva and me. Rose is completely beside herself about Henry. She can't bear losing friends, so she doesn't; she makes a doll or an image and that person stays alive as long as she does.
'I'm sorry we can't be better friends, you two and me. People say I'm contemptuous of the sort of art you show. Not at all. But you will find in time, as I have, that in art you are destined all too often to become what you oppose. In the forties and fifties Siècle was truly radical, believe me, but Bart and Miles took over then and they were younger. Some of my artists had become rich and famous, when not every work of theirs merited adulation. Bart doesn't like me, just as I didn't care for the conservatives who preceded me. You do your best. By my own rood, I've succeeded; my artists eat. They've shocked Australia out of its hide-bound provinciality from time to time. Bart would have had nothing to complain of had I not gone in first.
'I don't dislike the work you show at all. It's wonderfully witty, but wit alone can't give you an income. You'll either have to rely on government grants to keep going or you'll have to compromise if you want to be running in ten years' time.'
'Well, there's always our share of our father's paintings,' said Allegra.
'That's what I mean. Compromise. Let me know when you want the beach house key.' And with that he rose. Allegra pouted and sulked as I walked him towards the door. As he left, he squeezed my elbow. 'I'm sorry about it all,' he said.
Harry Laurington seemed pretty civilised to me, but Allegra said he was a manipulating hypocrite. She was sure he was trying something on. I was forbidden to take up his offer, kind though it had seemed.
Until his visit, we'd always treated Harry as beneath contempt for having married The Brolga and inflicted Checkie on the world. I'd never talked to him before and found myself surprised, even hurt, by his sensitivity and intelligence. I didn't want Harry Laurington to be clever, compassionate and far-sighted. I wanted him to be a fraud.
I had not seen Rose since Dadda died. She sent a little note to say how sorry she was and how Allegra and I must be brave, though she herself was feeling like a coward. Allegra said Rose probably knew where the ashes were and deliberately didn't tell us. 'I wouldn't trust her as far as I could kick her,' she said, bitterly. But I trusted Rose and eventually went to see her, remembering what Harry had said about her making a doll or an image in remembrance of someone who'd died. There was a blue-eyed boy with wings and a curly serpent's tail watching her now. We'd laughed together over how the artists in the Sforsa Castle had dealt with the problem of haloes and perspective, and now her boy had a halo stuck around his head, shoulder to shoulder, the yoke of sainthood.
Of course she knew where Dadda's ashes were, and would have told us had Harry not decided to do so. Furthermore, Harry had given Rose, to give to me, a little stack of letters Uncle Nicola had written to and received from the Corettis after he came to Australia. Harry thought Viva had borrowed them from Dadda to read and then forgotten them when she and Dadda eloped.
NINETEEN
Uncle Nicola
AS WELL AS leaving me his shoe to make a church from when he eloped, Dadda had left a mysterious package in a cupboard in his studio. At first I thought it was just an old newspaper in a plastic bag, part of one of his constructions, but it was pretty heavy when I picked it up, and I noticed the newsprint was in Italian. I put it in my bedroom to open later, and promptly forgot about it. A couple of years afterwards, Dadda asked if I knew where it was. I recognised the package from his description, but I didn't know where I'd put it, and it wasn't until after he died that I found it. The wrapping was a disintegrating copy of _Critica sociale_ , the journal run by The Apparition and Filippo Turati, and inside was a brace of duelling pistols (essential luggage for a man of Uncle Nicola's stamp) and three diaries.
Uncle Nicola had a fine, neat, beautifully sloping hand. Afterthoughts and postscripts were added into his diaries in exquisite little boxes, making the record a work of art. Roughly translated, his frontispiece reads, 'We cannot assume good outcomes from good intentions.' Elsewhere he makes the comment, 'All acts, regardless of intention, have consequences, the preponderance of which cannot be foreseen.' I think these are admirable observations; they impose nothing on those who read them, but the moral gravity of choice.
In 1926 he wrote that life in Italy had become sad and joyless. City councils had been bullied out of existence and fascist _Podestà_ , (mayors), appointed in their places. The large, high circulation non-fascist press was broken into submission, several papers having had their offices ransacked and their printing presses violently destroyed. According to the Fascist's official historian, Giovanni Gentile, the Fascist Party had recently become the organ of the 'self-perfecting state', which at maturity would express the moral will of the people, so that anyone not wishing to serve it would be its enemy. The most depressing feature of the whole farce was the 'investigation' into Matteotti's death. It had taken place in a provincial town where communication was difficult and witnesses could be watched. The murderers were tried for manslaughter (principal culprit unknown) as the result of 'a practical joke gone horribly wrong'. Though the order for the abduction had come from high up in the Fascist hierarchy, quite possibly from Mussolini himself, the hierarchy was exculpated, and the assassins were free men inside a year.
Uncle Nicola had fallen into a grey and mortifying isolation in which his work had lost all meaning. Respect for people was less and less possible each day. Friends were deserting each other and, one by one, those who had been prominent antifascists were dropping from sight. When Giovanni Amendola, leader of the Aventine Secession, died as a result of injuries inflicted on him during a third bashing, Uncle Nicola wrote that he despaired of men.
In August 1927 Uncle Nicola became seriously ill with influenza, and went to Sorrento to recuperate. There, he received a telegram from a friend telling him not to come back to Milan because the Fascists wanted him dead. The offices of his legal practice, famous for being the place to which socialists could go for legal help and where free aid was available for peasants and workers, had been devastated.
Apart from my grandfather, Uncle Nicola had no family to worry about. He'd never married; perhaps it was something that had slipped by him in his peregrinations, but more likely it was because he'd invested what he had in Emilio's life. My great-grandmother Coretti had TB. There'd been three girls between Nicola and Emilio, but the first had died of measles, and the next two of some hereditary disease, a lung disorder that no one understood. Great-grandmother was nearly fifty when Emilio Coretti was born and she died not long afterwards. Great-grandfather paid housekeepers to take care of Emilio when he was a small child, but he was quite an old man, and when Nicola graduated from Bologna Law School he took over, paying housekeepers first in Milan, then in Bologna, then in Genoa and finally back in Milan, the circuit on which his socialist and trade union interests took him. There had been some family money, which tided them over while Emilio was still a student, but the family had never been more than modestly prosperous. They dealt in hides and leathergoods and this was how Emilio had his start.
With his talent for shoe design and his brother's contacts in the Milanese upper middle class, Emilio had done very well and, of the brothers, was the richer man. But it worried Nicola that Emilio, too, might have to get out of Italy, and he wrote saying he ought to consider going to Paris, where he had friends and where his talent was already appreciated. As for himself, it was obviously growing urgent for him to leave the country.
Indeed, hardly had he sent the letter when he was clubbed in his sickbed, a sure sign that there was more assault waiting for him. If he stayed, he would die by violence and by degrees, a walking advertisement of what happened to the people who, though they had once set the standards, had now become the dissenters.
He was such an energetic and enthusiastic man, it was difficult to think of Uncle Nicola laid low in Sorrento wondering where on earth he could go, but in 1927, even for someone as resourceful as he was, Italy had become a difficult country to escape. The Alps to the north were an effective barrier, and within Italy Mussolini had begun to crack down on emigration, promoting a policy of _italianità_ (Italianisation), policed by blackshirts with instructions to shoot on sight.
By this time, the socialist leadership had decamped to Paris. The Apparition had succumbed to TB and Filippo Turati had been 'rescued' by the flamboyant Carlo Rosselli from the underground Florentine group, _Non mollare!_ (Don't weaken!). Uncle Nicola was impressed when Rosselli publicly returned to Italy to be thrown into jail. Like a great many Italians of his time, Uncle Nicola had a taste for public heroism, both real and operatic.
I wondered why Uncle Nicola did not go to France, but apparently three quarters of a million Italians had already gone there, half a million of whom were in Paris, cooped up in cheap hotels and pensions among increasingly resentful Parisians. The problem for the socialist leadership in exile had become not so much how to assimilate the refugees into Paris as how to get them out of France altogether.
When Uncle Nicola decided to emigrate, he was thinking of America. He went to Genoa, where he was very well known, particularly to the Federation of the Workers of the Sea, whom he had helped out of legal difficulties on several occasions. The second attack on his life came in Genoa when the offices housing the Federation were stormed. He was not hurt, but it was clear that he was out of time. There was a ship leaving Genoa in a matter of hours and a passport was procured for him with a bribe. It was an Australian ship bound for Fremantle, Melbourne and Sydney – he was given his choice. He chose Melbourne. For its climate, the average daily temperature suggesting it was mild and warm. And so, unaware that the weather is invented in Melbourne by a god who has frequent changes of mind, Uncle Nicola breasted the sea for eight weeks in a boat of Australian design that lumbered onward at a fair pace under a regime, not of Fascists this time, but of chops and chips.
Before leaving Italy he had taken out all the money he had access to through the Bank of Milan in Genoa and had spent a good deal of it on a diamond ring, a gold-tipped walking stick and some clothes made by a Milanese tailor, thinking that if he had to go into exile, at least he'd go in Milanese style. He pocketed the rest of his money for whatever exigencies lay ahead. One of these was the requirement by Australia that he arrive with forty pounds in his pocket. He had not severed his political ties; he was going to make himself the Australian correspondent for _Avanti!_ in exile.
We only knew Uncle Nicola in his old age when he was touchy, but still effusive within the boundaries imposed on him by arthritis. I remembered, as Rose did, his pale, washed-out blue eyes that did not betoken the energy of his body or his mind. Nevertheless, they were eyes in which the pupils would flare and contract as he spoke, as if some dynamic image was before him, flashing with colour, motion and light.
A true eccentric, he arrived in Melbourne knowing absolutely no one except the Italian Consul, with whom he had been at university. The Consul, a one-time socialist, had joined up with Mussolini at Piazza San Sepolcro in Milan and was thus a first-hour Fascist. Uncle Nicola, with nothing to lose but his entire credibility and future prospects, approached this man to see if he could get him some work commensurate with his talents. Besides his degree in law, Uncle Nicola spoke three languages, none of them English, and had been an organiser of peasant leagues and a trade union secretary. When the Consul told him he could get work in a restaurant or a fruit shop, Uncle Nicola was suitably appalled.
He had thought to escape _italianità_ by going to the ends of the earth, but insofar as there were only a few people to swing it, _italianità_ was in full swing in Australia. Nowhere did it swing more energetically than from the pulpit in Melbourne, whence it sought to belabour the ears of Italians with the news that they were the most noble of ears on the planet, easily as noble as those lent to Mark Antony in _Julius Caesar_. Such ears ought now not just be lent, but given, along with gold wedding rings and generous donations to the establishment of the greatest society the world could ever hope to see.
The cardinal beneath whose immortalised gaze Allegra, in her capacity as a waitress, was ultimately to shoot a Catholic clergyman in the bib with a breakfast sausage, had an adviser from Rome, a clerical chappie whose enthusiasm for Mussolini was infectious, not to say terminally so, for the Catholic hierarchy. For the most part, nonetheless, the noble ears of the Italian parishioners seemed somehow to have been inured.
Uncle Nicola found himself in Australia, the antipodean correspondent for _Avanti!_ , with no immediately apparent way of penetrating the community whose representative he, with his qualifications and credentials, supposed he ought to be. The Italian community in Melbourne in 1927 was of moderate size, the largest non-British group in the country; too small, all the same, to have much influence in policy making without having the sympathy of Australians and Australian institutions. The _fascisti_ were concentrated in the consulates and disseminated their propaganda through such outlets as the Dante Aligheri Society (hence Dadda's disdain for it) and Italian language schools like the one run by Stella's matron of honour, Marietta. As in Italy, there was disunity in the expatriate Left. Communists, socialists and anarchists, most of them with very little education, stuck to their respective prejudices, making the only recognisable antifascist focus the red-shirted members of the Matteotti Club. These, by and large, were labouring men.
Though anti-clerical and atheist in belief, Uncle Nicola celebrated Christmas, and Christmas was bearing down upon him twelve hours earlier this year than was its custom, and in blazing sunshine rather than in snow. Thinking himself alone in Melbourne's Botanic Gardens, he was attempting to divest a fir tree of a branch with which to decorate the meagre room he had found himself in South Melbourne, when he was apprehended, fortunately not by a policeman, but by a Belgian with a very full and very black beard. Uncle Nicola, mistaking him for a Frenchman, said, ' _Pardon_ ,' and proffered the branch guiltily.
'French?' asked his assailant.
The other didn't speak Italian, but through a combination of languages the pair found themselves able to communicate. The Belgian said it was against the law to strip trees in public parks. Uncle Nicola was not surprised. He had attacked the tree in a burst of absentmindedness, he said, though he had done no such thing, and at that moment the din of cicadas started up without warning and terrified the superstitious wits out of him.
Then they got talking, and as they talked, they walked. A very long way, as it happened, because they found they had a great deal to say to each other. The Belgian told Uncle Nicola he'd concocted an English name for himself out of Melbourne's streets. He was known as Spencer Lonsdale, and he explained that this was necessary in a country such as Australia where nobody could pronounce anything that wasn't a part of the daily vocabulary. His real name had five consonants in a row and it was just too much to expect ever to be called by it. He'd been in Australia twice, the first time when he was a young boy. His name made difficulties for him at school, so, at his father's suggestion, he changed it. His father was a botanist who counted Professor Baldwin Spencer among his associates, hence the name Spencer had come Lonsdale's way from two directions.
A Belgian botanist seemed something of a rarity to Uncle Nicola, botany not being a major preoccupation of Belgians as far as he was aware. Spencer Lonsdale explained he'd had a German grandmother who insisted her Belgian offspring be educated at the University of Kiel in Schleswig-Holstein. His father had developed his fondness for botany there, at the university where the famous Baron von Mueller had taken his degree. As soon as he was able, he had set sail with his wife and children for the fabulous Antipodes where the aged von Mueller, having lost his directorship of the Botanic Gardens because he'd developed them as a systems garden rather than as a public park, lived in sorrow, wrapped up in scarves and wearing gloves and clogs, complaining bitterly of tuberculosis, which he thought he ought to have but probably didn't, and surrounded by mountains of specimens and hillocks of letters.
While his father went on specimen-collecting expeditions funded by South Australian gentry, Lonsdale endured an Australian education. This was supplemented by botanical conversations pursued in German in the company of von Mueller who turned out to be not as old as he looked: some years previously he had proposed to a lady from Phillip Island and then regretted it and had a medical friend tell her the engagement had to be called off because he was impotent and in poor health. As a consequence he cultivated a cough and imagined he was old and had a weak chest.
While Lonsdale's father had found Australia enchanting and experienced it as an endless opening-up of possibilities, his mother, a German like his grandmother, detested it. She felt she'd been dumped in a parochial, narrow-minded city, a city in which only a bastard brand of English was spoken, a city, if indeed you could dignify the straggly environment with such a name, in which there was either flood or drought, which smelt of sewage and was full of rough people and almost altogether without culture.
When Spencer reached fifteen, he was taken 'back to Germany'. Never having been in Germany in the first place, 'back' was a disappointment for a boy who was used to a more rugged and less circumscribed life. He completed his education, sat out the Great War in enemy territory at his father's university and then returned to Australia in 1920.
He had of Australia a double opinion. On one hand he could sympathise with his mother's complaint that there wasn't enough intellectual life; on the other, he had inherited the view of Baron von Mueller that Australia could be the liberation of the thinking man's imagination. For von Mueller, Australia was endlessly enriching, the only restrictions being social and physical. These, particularly the social restrictions, hampered him mightily in later life, but he had the worldwide distinction, nevertheless, of being the quintessential botanist of his era, having had a whole botanical kingdom to describe.
The great age of collecting was over when Lonsdale returned to Australia, so he, also passionate about botany, had to content himself with holiday forays and field naturalist clubs. German teaching had been resumed at the University of Melbourne, and, being a Belgian with an English name, he was given one of only two foreign language lectureships, the other being in French.
The university had recently expressed an interest in enlarging its foreign language facilities, and, being very strapped for funds, thought of offering unpaid tutorships in a number of modern languages, one of which was Italian. The idea was that the students would pay for their instruction. Alas, the courses would not carry degree status, but why did Uncle Nicola not apply for a tutorship?
The question was asked from a chair on which Lonsdale was standing, hammering a paper decoration into place on Uncle Nicola's wall, while Uncle Nicola found a tub for his fir branch. Unable to afford much in the way of food, and finding the Australian diet not remotely to his taste (the lamb chop, a status symbol to us, was an object of loathing to him), Uncle Nicola had bought a barrel of wine and four glasses from which to drink it with three as yet putative friends. On Christmas Eve 1927, one of the glasses found the lips of the first one.
Approached from the south, Uncle Nicola wrote to the Corettis, one could just see the university on a far green knoll, above a hand-mown field. The lone mower was pretty nearly always to be seen on one or other of the vertical or horizontal mown strips. There was a duck pond. The professor of medicine rode a horse to and from his lectures. One of the university buildings was known, for want of a name, as the 'Gothic Structure'. There were female students, but not many, and any young man seen frequently in the company of any particular young woman could count on a reprimand for moral laxity. While the library had a complete set of the works of Zola, they were kept under lock and key in a special reading room because of their dubious moral tone. Any young man wanting handy hints on how to do in his lover's fiancé or prospective mother-in-law would have to brook the librarian's censure first.
Uncle Nicola taught Italian in Italian, as English, with its absence of arm movement and its lack of a natural crescendo, proved just too difficult. Spencer Lonsdale's Italian, on the other hand, was improving every day, and the more he knew about Uncle Nicola and the reason for his being here, the more he was convinced that Uncle Nicola needed both protection and support. Lonsdale left an archive at the university in which Uncle Nicola's activities and personality are documented. They were to become close friends. It's quite doubtful that Uncle Nicola could have found an outlet for his views and for what he considered his utmost duty, intelligent opposition to the Italian regime, had he not made this most important of contacts.
The Australian summer was hot, Uncle Nicola wrote to Milan, but not a great deal hotter than the Italian. He could still go decently clad, which was just as well, for he hadn't the wardrobe to go indecently clad, though he was thinking of laying out five and sixpence on a bathing costume when he'd found there'd just be time enough for swimming in his schedule.
The state of Victoria had a very long coastline and it was common for families to have beach houses within striking distance of Melbourne. He had been with Lonsdale and his family to their beach house on the Mornington Peninsula and had found the sea clean, clear and wonderful to swim in: indeed, he had swum so far out, Lonsdale had been terrified he'd be taken by a shark. The beach sand was white and very fine. While the Lonsdales entered the bush like people disappearing into the chapters of a book in search of its theme, and returning later with amazing collections of wildlife, Uncle Nicola sat organising his future in the cicadian din, which, he wrote, was a formless opera, awaiting extraction into tonal range.
Behind him now, in Italy, were the beatings and cod liver oil treatments, the book burnings, the sack of masonic lodges and of Liberal and Socialist offices and headquarters, the summary, mindless killings and the arbitrary alterations to the law. He relaxed a bit and waited to receive replies to his letters. And waited.
TWENTY
Displacement and Despoliation
HERE, AT REG'S retreat, October is the month when grasses burst, fat with seed, and the progeny fall or are thrown or carried across the land by winds or birds or animals or rain.
Some seeds stick, being covered in spurs or exudate, and others, barbed flints embodying the word 'pierce', work downward, downward, into earth or flesh. Beautiful are the seeds that fly, plumbobs under fluff. So too, the lantern cases bobbling on stalks. Spiral seeds have fine hairs, moist in the ovary so their shafts lie straight as arrows in a quiver; when the hairs dry out, the seed heads spring, and rapidly the shafts coil after them to end in a hook for feathers or fleece.
Nature's imagination is vast and travels in any direction it can. All is strategy, occupation, exploitation, driven by error and fashioned by chance; a new stitch here, a new loop there. The sky's blue mantle wraps us into our earthen cradle, and here, where we dwell, dispossession is nine-tenths of the law.
There was a trust, The Brolga maintained, though she was tardy in maintaining it. It dated back to the early days of Siècle. It was called the Siècle Trust, and set up in memory of her brother, Leslie Hallett.
When questioned by Wednesday Monday, Harry Laurington disclosed that he had given to Rose and Laurent Hirsch a collection of paintings to use as they wished, either lending them out or selling them, until such time as either they or Harry died. Following Laurent's death, Harry had left all the pictures with Rose, permitting her to sell up to half of them. At the core of the collection, however, there were fifteen Halletts, twelve Corettis, eight Sorbys, and a dozen or so paintings by other artists which were not to be sold; these were earmarked for the Siècle Trust and Rose was paid a small fee for taking care of them. They were either to form part of a Siècle collection or to be sold to raise money for a scholarship. In the event of there being enough money to finance a scholarship without selling the works, they were to be hung as a permanent collection either in the Siècle Gallery or in the beach house, where the scholarship-holder would live and work for a year. Eventually, if Checkie Laurington should find herself in a secure enough position, Siècle would become the property of the state of Victoria, the collection being used to finance the Trust scholarship.
The Brolga said Dadda had willed the bulk of his estate to the Trust. If he were to predecease Harry or the Hirschs, two-thirds was to go to the Trust so the scholarship could be founded as soon as sufficient income was realised. The other third of the estate he willed to Viva, except for a third of the third, which was to come to us. Hence the parsimonious out-of-court offering.
Harry Laurington said it was quite plausible. In the early days of Siècle, when they'd all been dependent on each other for help, they'd dreamed of their influence extending beyond their lives, though it was true that Dadda had severed his connections with Siècle before Leslie had died and they'd come up with the idea of the Trust.
The Brolga had the paperwork to back her up, but it came as no surprise that the father of her new curator, Jerry Gospel, was a legal gent, specialising in trusts. He was a partner in the firm which had drafted the agreement between Harry and the Hirschs. Over breakfast with David and Miles in the foyer of Figments, Wednesday rued that there was little he could do. The will looked legitimate and we'd have a job ahead of us to prove it wasn't.
We cooked crumpets in a two-doored silver toaster, plugged into a lump of plugs, on the floor. Five people, five plates with round, sweaty crumpet-smudges, Miles at his desk near the door where the fire escape served as an entrance. It was impossible to get ordinary chair legs to sit evenly on Figments' floor, so Miles had Bauhaus chairs from the Pantechnicon; the involuntary lurching of days gone by had given way to irresistible jigging. Up and down went Allegra and, every time he turned a page of the Saturday morning _Age_ , up and down went Miles.
Wednesday Monday didn't jig; he never sat right back in a chair, and these could only be jigged if sat right back in. He crouched forward, clutching the canvas where it met the chrome of his seat, pink mouth chewing out the bad news as if it were a low variety of cud and he were crushing it thoroughly in his molars: the only masticating tawny frogmouth in existence.
I sat with my back to the cool, white-washed wall, my backside gingerly on the floor. Above me was a new all-white artwork by a chap called Barrie Bull. The smoke curled from David's cigarette. As Miles shook the paper straight, delighted crease wings formed about his eyes. 'Someone's been done in with an umbrella,' he said.
'What's it say about the Bull?' asked David, meaning an article for which a journalist had been around to Figments to interview Barrie Bull. Through the anxiety in his voice, it was clear that David regarded Barrie Bull as the competition.
The light was surfing through Allegra's hair. While Miles went on about Zen qualities and the simultaneity of creation and destruction, I contemplated Allegra as the ideal model for anyone having difficulties with the concept of the halo, although there was Rose's solution to make a yoke of it. I've always found it hilarious how the Virgin's gold plate swivels round from the back to the side when she's depicted in profile. I thought I'd get Allegra to pose for me for a few views from the rear. I shifted my seat to the front doorstep to place myself between Allegra and the light, but the halo effect was lost; instead the light travelled up and down tendrilling hairs, through dark places, leaping from hair to hair, a light monkey. I wondered if Jesus had ever stood on his head, and thought of asking Eli, now adolescent, whose hair was such a mix of cowlicks it was impossible to trim, to stand on his head under the clothesline. What would happen at the interface between the ground and the halo-haired head in the event of Eli ever becoming a saint?
I was taken with Eli's blondness but also other varieties of fair, like red. Kelly Kelly had wonderful hair, long, straight and silky, like a gum trunk. It luminesced at twilight. Maggie's was much the same, but had a bit of yellow in it and didn't fall as straight as Kelly's. Allegra's hair had a touch of red, just like Aunt Nina's and our mother's; my hair was much darker, heavily dark, I thought, too heavily dark for my rather pale complexion.
Allegra jigged, playing with a cord in her lap as if she'd been spinning at a loom and her prince had just arrived. Much better than a pallid pre-Raphaelite's, her face had a high colour over the gorgeous architecture of her bones. She was wearing long black boots under a dark red pregnancy pinafore.
I thought as I sat there on the step of Figments that Mad Meg wasn't going to supply the six members of its co-operative with much in the way of security. A one-sixth share was worth about five thousand dollars. I'd given Maggie hers, and was happy to have done so. We were like-minded about many things and the years had brought us close. If my ambition was to buy a house, Maggie's was to get an education. She'd gained a mature-age entry to tech and was now three or four years along the track towards a diploma in art and photography. Because I'd forfeited a share in Mad Meg to Maggie, that brought me only five thousand dollars when the Troika bought in. The sooner I could raise the cash to buy a house for me and Eli and be free of the interminable financial dearth of our household, which now included David and would soon extend to the expectee, the better.
Allegra had begun to suggest that I was mean, whereas I thought of myself as frugal. She'd never minded my frugality in the past, but was now of the opinion that Eli's only decent clothes hadn't been bought for him by me. It was true enough that my mother and Allegra bought most of Eli's clothes and, before that, Dadda had, but I'd always thought this was a reflection of my poverty and their generosity rather than my selfishness.
David had let it be known he didn't like the smell of corned beef in the process of corning, so it was a question of whether Eli and I were to forgo a major source of protein for the sake of the olfactory sensibility of a chain smoker.
In toto I had nine thousand, five hundred dollars saved and needed only two thousand more to afford a deposit and raise a mortgage. As two thousand dollars was more than a year's saving for me, I was very tempted to call in my portion of Dadda's estate, but had to close my eyes and grit my teeth. I was as certain as Allegra that the will was shonky.
As there'd been nowhere else to take my fears and woes, I'd eventually taken them to Beryl Blake. Between us, there was an unspoken understanding for which I have always been grateful. Instinctively each of us knew where the other stood and that a bond between us was not only possible, but mutually beneficial. She'd moved me up to herbarium work with Loyola, where I was better paid and doing something aesthetically quite satisfying. It had been several years now since female pay rates had been scrapped, but I was still ineligible for superannuation as I was technically a part-timer.
'Ah,' she said, and then, 'Ah,' again. 'How would you like to illustrate a book?' She was watching herself rotate a biro, almost conscious of its slim elegance in her stout red hands. 'There's a chap I know who's an expert on orchids. He's a painter and he's putting together an orchid book. He wants someone to do the botanical drawings for it. I rather think he wouldn't be Loyola's cup of tea. You're more the sort of person he'd be looking for.'
She swivelled around on the plump seat of her office chair, which held her aloft, a beach ball balancing act, lifted her phone receiver and dialled emphatically so that the mottled but still firm flab of her arms swung briskly, a half-saucer stain of perspiration on her dress beneath.
'Hullo,' she said, the weight of command and a decent education in her emphases. 'It's Beryl Blake from Bunurong Gardens here. Could I speak to Mr Sorby, please? Oh, it is you, Reg.'
My heart sank.
I was told he had a townhouse not far from the gardens. The inhabitants of this area were either wretched or rich, but the wretched were not overwhelmingly so. They were near a hospital and the local chemist's reflected their requirements: crepe bandages, Agarol, cough medicine, Rectinol, Alka Seltzer, disposable nappies, Mercurochrome, Bandaids, bunion pads, pensioner discount, moccasins; very little in the way of expensive adornment, but they did stock rainbow Kleenex tissues under a prominent sign saying: NO DRUGS OF ADDICTION OR MONEY KEPT ON THESE PREMISES. The garden, too, with its park benches under mature Moreton Bays offered a pleasant if fig-sticky spot for them to congregate, their bottles discreetly wrapped in brown paper bags. The bottle contents were consumed from enamelled mugs, giving the appearance of innocent tea-taking. They communed in good weather, coralled behind a picket of walking frames, crutches and sticks, and would yell 'G'day!' to me and drink my health when they started to recognise me on my way to see Reg.
His house was behind a high brick wall in which, on my first visit, I found a small door ajar.
Bricked courtyard. Double-fronted, whitewashed Victorian abode. Door open. 'Come in,' cheerily from the depths.
I see myself, little, too dark-haired, shy, reflected on all sides in picture glass. Confidence uninspired, bedroom to the right of me. I am going past a doorway to the left of me, when 'In here,' comes out of it.
Jolly, rotund, become urbane since he supervised the sack of Aunt Nina's table, he steps back from a large bushscape, brush in hand, the quintessential painter in a white paint-pocked coat.
'Isobel,' he says, without turning round, 'I poured you a glass of champagne.' Behind him on the table, an open bottle of champagne, two flutes – one a sip's distance from full.
He sniffs, purses his lips, adjusts his glasses, dabs. Then he turns around to greet me. 'Well, aren't you pretty?' he says, beaming, the voice a tyre, neither over-nor under-inflated, rolling on a bitumen road.
He starts toddling forward and says, 'Follow me,' drink in one hand, bottle in the other. Through a kitchen, a courtyard and into an orchidarium. 'They say your sister's prettier,' still toddling along in front, ever the tactful one, 'but she doesn't use her face as well as you do. Sweetly made, but sourly expressed. Looked good in the Moratorium but something's happened to her. As for me,' navigating himself into a blue-cushioned cane chair at the far side of an occasional table and managing at the same time to run the index finger of his flute-holding hand along my cheek, 'I like the shy ones.'
My heart sinks further. 'Sit down,' he says, and I take the opposite chair.
Crosses knees, cocks head, amiable smile. 'You have an interesting face, do you know that?' I open my mouth to say something but he bats my intentions (whatever they may be) aside. 'No, no, I mean it. It's a distillation.' He makes a distillery ring with his chubby fingers. 'Everything I know about you comes together in that face.' Again, the smile, like a contented baby's. There are worse things than being happy, but I wonder if portrait painters should be exempted from the niceties of this liberationist world? 'There's only one problem.' Moving forward, elbows on knees, thumbs in conversation. 'When you smile sometimes your mouth goes down at the corners and it looks like a smirk. Why do women of your age do that?'
'I didn't know they did,' I say.
'You're in-vol-ved in a plot,' he says, stressing the 'vol'; tests my knee for go-on-you-know-you-are, like palpating a mango for ripeness. 'A smirk is the look of a guilty person, or a sly person, or a person whose aim is to set another person up.' Swig. 'Now you're looking bored; mental shutters over your nice blue eyes.'
Not bored, terrified! In the back of my mind, the bedroom off the hall, bed neatly made, coverlet straight. Grog, compliments and a dreadful reputation.
'All right, Isobel.' His laughter like suspension over pot holes. He slaps his own baby-bottom knee. I think he sees through my irises: no alps, no snow, no speckled trout; just house, bedroom, lecherous old man. No wife at present to keep him in check.
'The orchids,' I remind him. His dendrobiums are out, pink and, alas, labiate.
'Some of them only germinate after fire, did you know that? Tiny little ground orchids. There's a bright red one turns black after mating.' He gives me a twinkling eye and says, 'You're blushing.'
Allegra was disgusted, 'Reg Sorby!' she forced the 'Sorby' out as if divesting her mouth of a boot. 'You've got to be joking!'
'I'm not,' I said. 'He's going to pay me. I'll get royalties.'
'That's probably not all you'll get! The old lech! If you let him anywhere near your duds, Isobel, that's it! I'll divorce you.'
I was determined not to let either Allegra or David, or worse still the pair of them combined, wreck my act. But that meant practising self-defence on two fronts simultaneously. And on the other front not only my virtue was at risk, but my integrity as a painter of my generation, an integrity built on values that were in opposition to Reg's.
Was it that he told you how he thought you ought to feel, and then tried to castigate you into his power when it was obvious you didn't feel it? Had he no suavity at all? I told him I wasn't just a Sunday painter, nor simply a plant anatomist. I was serious about my work.
Just because he was rich didn't mean he had anything over me. I could say no; it wouldn't be the first time I'd knocked back money.
He liked that. He said so. And I wondered, as the platitudes gallumphed across the glass tabletop, if Reg was like this because he was naturally crass or because he despised the habit of finesse.
In the world outside the orchidarium, women were being taught that Reg was the enemy. More and more of them were taking assertiveness courses to learn the verbal defences against manipulation. I was never vigilant in this regard and consequently found myself in ludicrous situations. But I got out of them.
I told Reg he was made of leftovers from the sixties. Of course, he had fought the battle against pomposity, hypocrisy and prudery; furthermore, he had stood up and been counted over Vietnam and lent his name and reputation to conservation and alternative lifestyle movements. Lately, however, he'd been touting himself as the champion of women, while concurrently painting them naked with no heads.
I said I believed personal philosophies ought to be thorough-going but self-critical, or it's just huffing and puffing. Our ideas have to be able to co-exist without cancelling each other out. For instance, should the enemy of prudery be the friend of sexual exploitation?
Harking back to the sixties, did Reg remember how women and their children were deserted by their liberated husbands, how husbands became sex-driven and wives, ignorant of their own best interests, countenanced adultery, even indulged in it themselves, though most probably not with comparable relish?
'Do you know,' he responded, 'I didn't know women had orgasms until last year?'
'God! Pull the other one, Reg!'
'Then I realised what my problem was,' he continued, nodding seriously, 'I've only ever been married to Catholics.'
Thus was the outrageous strategy of Reg. A strategist would have been able to cope with it, but strategies require space in minds and I was taken up with other things. 'Catholicism's the root of all evil,' he said. Reg's philosophy seemed to be that he could believe what he pleased. It didn't have to make sense, if he wanted to think he'd just found out about the female orgasm, he could think so, though what kind of a dash he thought he was cutting, God only knows. The years when it was victory to turn your back on the world were now well in the past. General prudery was dead, but this was the age of women calling the tune and the tune was self-protective and conservative, not predominantly sexual, as Reg depicted it.
When Allegra knew I'd been to a meeting with Reg, she wanted a blow-by-blow account. I told her I wasn't blowing him. 'Well, just watch you don't,' she said. 'Nobody knows where that old bugger's been. He's probably a walking time bomb.' Was I protecting myself? Weren't there other ways I could earn money? Was money necessary? Why squander my honour just for a place to live?
But I had always wanted a house for myself and Eli. I did not regard it as a sin, nor was I going to apologise for my desires. A boy from Eli's school came home with him one afternoon and, finding it odd that we had no furniture, asked, 'Is this your city house?' Eli, who took our relative poverty in a happy spirit, answered, 'Well, actually no. It's our space museum.' Eli ran a book on the football at school to keep himself in funds and bought his clothes, as ever, from lost property. When Reg arrived one night to take me out for a meal, he said, 'Have her home by twelve, son.' He was thirteen; his voice hadn't even broken. Reg took a shine to him at once.
Eli led him on a tour of inspection. 'Needless to say,' he said, 'I won't be showing you where she sleeps.' That cracked Reg up.
On Eli's bedroom wall, he kept a blown-up photograph of Dadda. It was slightly misty, as if Dadda were really there, seen through uneven glass.
'Interesting man, your grandfather,' said Reg, peering at it. 'Didn't like his work, though,' he pursed his lips and made as if to rub something out with his hand.
'That's all right,' Eli said, unabashed. 'He didn't like yours much, either.' Reg's shoulders bounced with laughter. He patted Eli on the back. 'You'll do me, boy. You'll do me.'
Allegra kept craning pregnantly round doors. When I came home from my evening out, she and David were waiting up for me in the kitchen, a committee of two.
'Home early?' Allegra teased. 'It isn't even midnight.'
'Oh, get off my back, what I do's my business.'
'Oh, oh, listen to the lady,' said David adenoidally. 'What do you want wiv that messy old sentimentalist, Isobel?'
'I'm working with him.'
'Working wiv him? I mean to say, do you call what he does work?' He gave a derisory, face-wetting laugh.
'He lives from it, which is more than the minimalists can claim.'
'Oh, I don't know,' Allegra said. 'Barrie Bull lives from his work, so do some others. It isn't impossible.'
'So am I a traitor for doing botanical drawings for a book on orchids?'
'We are what we do, Isobel,' said David.
'Well, I'm not a botanical drawing.'
'That's not what I meant.'
'I'm sure it's not what you meant. You mean I'm debasing my skills for the sake of a mortgage. Well, you can think what you bloody like! I'm going to bed.'
In my bed I found Eli, yawning, 'How did it go then, Mum?' he asked. It was clear he rated Reg about eight and a half out of ten.
Reg's presence in my life brought David into sharper focus. If Reg was prone to dispossessing me of the things I was about to say, then David was prone to talking disparagingly over them, putting what was otherwise a soft and charming voice to ill use.
As for Allegra, I'd thought Reg was being unobservant when he said her expression had changed since the Moratorium; after all, pregnancy had brought out her marvellous complexion. From that night onwards, however, I began to see what he meant. Allegra had become a co-inquisitor; David was describing the arena in which he would allow her intellect to wander. It wasn't so much that Reg was manipulating me as that David was manipulating Allegra. It was obvious she was under strain; they weren't the quietest couple to live beneath, there were lots of quarrels, squabbles and tantrums, no explanations offered, though they would have known they were being overheard. When they decided to get married, I couldn't see how they were going to be happy. Perhaps things would alter after the baby.
The baby was pretty close as we sat lamenting the Siècle Trust in Figments. David and Allegra were in concord for once; perhaps because Allegra was tired and relying on him to do the daily chores, though, less cynically, it could have been that they were truly in love. You would have said so to see the tableau of David leaning on the back of her chair, biting his bottom lip in a shy smile, and Allegra rolling her head across his chest.
Since there was virtually nothing we could do to combat The Brolga at this stage, we had turned our attention to the freeway. It was well on its way to completion and the Board of Works was maintaining that no one whose property did not abut the construction reserve was entitled to lodge a complaint. Furthermore, would-be complainants ought to have complained earlier. It had been pointed out to the Board of Works that people had been complaining for a decade, but the Board said it had nothing on file. We could only conclude that was because it didn't keep a file.
To pay for the privileges of two daily peak-hour traffic jams (buses, trucks and semi-trailers not excluded), and many weekendly traffic jams so that people who lived in Melbourne's well-heeled east could drive to work in and around the city or go to the races in their ridiculous hats and white Rolls Royces stuffed with chicken and champagne lunches, the town councils of inner suburbia were supposed to levy a charge on their rate payers.
The rate payers were not only reluctant, they were downright furious. If there were to be no compensation for traffic snarls, noise, air pollution and the corresponding diminution in the quality of life, then there could be no payment of costs by residents.
We took Wednesday down to look at the construction, a couple of blocks away. There was a three-quarter cyclone-wire stockade into which the freeway nosed like a whale beached in a diminutive inlet. Buildings to either side of the central road plantation, up which the freeway was slowly nudging, were freshly daubed with slanders against the state premier and the Board of Works. Peripheral issues, such as the freeing of jailed union men and the liberation of their sisters, bristled with rage in the interstices, and under that which was topical and urgent lay the complaints of yesteryear, the peace signs: VICTORY TO THE NLF, NIXON OUT OF CAMBODIA, WE WANT GOUGH and NO NUKES.
Wednesday crouched and leapt around the encroachment, seemingly in a tangle of competing desires. Laws and the breach thereof flew about him, an invisible flock over which, by arching himself forward and flinging his arms high, he cast his net. The fence around the road-widening was incomplete, he exclaimed. Victorian law said a construction zone couldn't be declared until it was fully fenced.
Though it was a Saturday, we rang and arranged to meet the mayor at the Town Hall. When we arrived there, having picked up all the extra people we could on the way, we found the mayor spying through a hole in the door on a taxidermy class in one of the ground-floor rooms. The Town Hall was remarkably grand for a neighbourhood that had, until recently, been poor; the corridors were wide, the door lintels massive and the doors positively baronial in their sweep. It had been built by the rate-paying but absentee landlords of last century, so they could hold balls in their electorate without actually having to resurface any roads other than those coming from the eastern suburbs where they chose to live. Those were the days when subtenants paid rent to tenants and Labor members of council weren't above being bought off in little matters such as which tender would be successful for enlarging the Town Hall and furnishing its Mayoral Library.
In the Mayoral Library the first impression one had was not of the mahogany-shelved room with a high, elaborate ceiling rose off which hung a thirty-globed candelabrum, a room whose windows, boasting more than generous sills, were draped in claret velvet and on whose walls were honour rolls, inscribed in gold leaf with the names of those who had been councillors in the year of the Town Hall's construction. The first impression, on the contrary, was of files of ancient manila folders tied in brown tape rather than red, of teetering piles of well-thumbed, food-stained books of regulations, abandoned coffee cups, and a quantity of ash and dust, lying in such a way as to suggest the presence of desks beneath it. As our mayor consulted a vellum-backed volume of building regulations snatched from one of the lower piles, Allegra directed my eye to the frieze of wallpaper just visible above the picture rails of the one wall not covered in shelves: it was the same lyrebird pattern as had been on the walls of our bedroom at Clare.
As we repaired streetwards down handsome granite stairs, it appeared that Wednesday was right; we could legally delay the freeway construction by preventing the fence from being completed.
On the following evening the mayor donned his regalia and we rustled up a posse of squatters who would seal themselves into the building site behind a barricade of old car bodies. Though David, the Kellys and I squatted, Allegra couldn't very well do so. Instead, she gave her psychedelic Kombi to be overturned onto the barricade. Reg Sorby supplied us with champagne and two beautiful picnic hampers, saying he couldn't join us as he was off to Sydney to paint a portrait. 'Well,' he said as he was leaving, 'I hope no one gets diarrhoea.'
Diarrhoea did not turn out to be a problem but although it took the police some time to dismantle the barricade, it only took them as long as it had taken us to erect it, and we were soon being thrown or carried out, our picnic uneaten. A wall of police was posted round the compound to protect the road workers. We were pretty shat off with the workers. They weren't with us. We tried to persuade them: this had been a working-class suburb, we said, but they didn't care. It was 'Yuppieland' now, best thing for it was to slam a freeway through.
Trendy though many of them may have been, the indigenes of inner suburbia did not give in lightly and came together as never before. The genuine poor, the nouveau poor, the green, Opposition MPs state and federal, boutique and gallery owners, botanists, detoxificants, secondary, primary and preschool children gathered in the evenings in the Mayoral Library of the Town Hall to receive instructions.
Courtly Tom, in his shiny tux, was seen one evening with numerous wick-like extensions hanging from his pockets. Some time later, he was seen again, inebriated, with the news that the freeway was so solidly constructed it would take a team of jack hammerers considerable time to make a hole large and deep enough to bury a bomb so the freeway could be blown up.
The threatened median strip at the debouchment, the only piece of green within walking distance for most of us, was turned one night into a forest. At three o'clock in the morning there was a clash between protesters and police, who had concealed their numbers and therefore couldn't be reported for any violence they might commit. By then, however, most of the forest had been planted and it was going to be considerably more difficult to remove than the car-body barricade had been.
Pine trees, palm trees, wattle trees and young red river gums throve in abundance around picnickers to whom wandering minstrels sang, no longer of war, but of displacement and despoliation. Two of the singers had very long, very straight flaxen hair and played recycled guitars; the voice of one was gravelly and deep, of the other, silvery and high. Listeners were clad in all colours of the rainbow, some in the red, black and yellow of the Aboriginal land rights flag, others in tie-dyed cheesecloth and hooded caftans from the Pantechnicon. Many wore jeans. The short- and long-sighted, by and large, corrected their focal lengths through John Lennon glasses.
Banners of all types, the canvas, the calico, the paper and the plastic proclaimed WRONG WAY, GO BACK!, WHAT'S FREE ABOUT THE FREEWAY? and PUT IT THROUGH TOORAK! When the co-operative kindergarten had to be shifted because of high projected levels of air pollution, there were several banners saying KIDS, NOT CARS.
One way of stopping a freeway is to replace the dirt that the workmen remove in the process of widening the road at the freeway exit. Professors, clergypersons, members of the Ananda Marga, bartenders, mechanics, fitters, turners, breadbakers, sock producers, Orange People, clerical assistants, tram drivers, connies, podiatrists, in short, the polyglot, shovelled it, shunted it, biked it, triked it, spaded, prammed and strollered it back into the holes it came out of. The anti-uranium lobby set up a stall selling yellow cake.
As we were unable to bomb it out of existence, creating a permanent block at the exit seemed the way to go. A Tower of Babel built of tip-up trays, old cars and extraordinary scrap metal (some of it courtesy of Bridget Kelly) was set in concrete across the freeway mouth to instructions given in a plethora of tongues, some of them Heaven sent, we noticed, for God had bidden his minions, 'Go to the Barricade.'
The Church of the Freeway was founded amid many halleluiahs. The Brolga, naturally enough, did not number among these parishioners, but a fervent black-eyed lady in her fifties, strong in the ankles and teeth, was heard every now and then to call out loudly, 'I'm saved! I'm saved! Halleluiah!' From another, secular, direction came The Freeway Survival Kit, featuring the gas mask and the ear plug. The Community Rape Board erected a trestle table and handed out printed matter. Bridget Kelly had a stall called The Toxic Tadpole. She was available for psychic consultation.
Poets circulated, one with a peacock feather hanging from his ear. A semi-permanent camp was established and a cracker night held. The camp kitchen doled out vegetarian curry in pita bread which children of the campers refused to eat, causing not a little tension.
A motorised rally drove to the home of the Victorian state premier and opened a freeway across his drive.
From the Stella Coretti-née-Motte quarter came a petition with numerous pages of signatures from people who lived as far afield as Mudgee, New South Wales, who had put their eyesight in the care of Rudge and Plant and had probably signed while their pupils were still dilated by drops.
One evening, at the instruction hour, Stella arrived with a contingent of Chinese. This caused a temporary thawing in diplomatic relations between her and Allegra, as we had a show of revolutionary Chinese posters currently on the walls of Mad Meg. The Chinese contingent smiled and nodded frantically when taken on an inspection tour. 'Ah, ah, Rittre Red Book,' they said every now and again.
I was ordered to find out surreptitiously how our mother had come by her new boarders. Had it entailed a noble act?
I was able to report back that Stella had struck up a friendship with a Chinese child on the back of a bicycle. The child had a mother who was walking the bicycle. Not too surprisingly, she was Chinese, and a member of the China–Australia Friendship Association. The baby's father, however, was Mexican, a professor of mathematics. The baby's name was Angel.
Angel and his mother were touring Melbourne by bike, looking for accommodation for three Chinese chaps who were staying with them. The poet, the Tibetan monk and the musk ox girl having moved on to higher, if not greener, pastures, Stella was in a position to oblige. Luckily for Stella, since 'breeding' prevented her from bargaining, the Chinese woman examined the prospective accommodation and named a reasonable sum. And so it was that Liu, Yip and Wang came to inhabit our former bedroom, our former dining room and our father's one-time studio.
Allegra invited them to come one evening and talk Marx to the collective. The talk consisted of Allegra firing questions while Liu answered either, 'In China it is very much like this', or 'In China it is not very much like this'. Yip and Wang responded with polite giggles, which they screened with their hands.
Then, in spite of ideologically sound boarders, Stella and Allegra fell out all over again. This time over tablecloths.
She has cut a hole in a tablecloth. Doesn't she realise those cloths took years to tat? Why couldn't she, as happens no doubt in plenty of other shotgun marriages, have bought herself a dress? There are cheap dresses aplenty to be had at that place, the Whatsamaycallit, the Thingummyjig where those girls from the dump work, and they'd fit anybody, even a hippopotamus.
Allegra sticks her head through the hole and tells Stella to piss off. 'Only things with pizzles piss,' says Stella, and then, announcing that Nina would be churning in her urn, she sweeps out of the Edwardian house where the bride is being outfitted.
'She was buried!' shrieks Allegra after her, as Maggie and I fix the cloth into a wedding drape. Kelly has run a strip of gathering along the edge of another antique cloth and attached it to a crown of gardenias to make the veil. Allegra, in spite of her fuming, is the most beautiful lace-clad thing in the world: she and her attendants start out for her nuptials on a tram.
We are all dressed up in swathes of this and that. Chantal Kelly is wearing a crocheted bedspread. Maggie and Kelly are in cheesecloth saris. Kelly and Cathy, the art historian, are playing improvised flute music and one of the Troika is twanging a zither as we accompany Allegra down the median strip. We are wafting through gardenia scent and neck-to-knee trees. Even the grass is excited at the procession and leans towards Allegra, trembling glossily where it can and waving little daisies. The fifteen-year-old Eli, in a caftan and peacock-feathered turban, is holding a stunning parasol from the Pantechnicon over Allegra's head. It is red silk, hung round with a golden fringe and tassles. Life on the barricade comes to a hushed, admiring stop.
The poet with the peacock feather, a porcine gent, is wearing a white wedding toga, one plump shoulder completely revealed. He is reciting and, as he recites, a bruised and laddered dancing troupe is performing a pas de deux that could do with quite a deal more practice.
As our parents did before her, Allegra has hired a marriage celebrant. She is a very large person in a colossal garment of purple and red. Eli calls her the Marriage Elephant. In her exceedingly long and bedraggled hair she wears two tufts of wilting bougainvillea either side of her central part. She is barefooted, and it is interesting to speculate how close together she could get those feet to stand, given what appear to be massive thighs inside the very large garment.
David is waiting at the freeway barricade, clad, one might think, for a funeral in a dark suit, hand-made for his father, apparently, in the dark ages. He is also sporting a yarmulka and prayer shawl. We have all commented that it didn't occur to us to think he was Jewish.
Well, they plight their troth as so many have, with mixed success, before them. And then, as they are exchanging rings, the contractions start and the bride has to lie down with her head in Eli's lap, crumpled up in the makeshift forest, while the groom looks round for a public phone in a pharmacy.
When Nin was born, it was October. The grasses were splitting open and nature's cunning array of seeds was sticking, burrowing, pirouetting, putting down roots and raising its flags. A blackbird on the front verandah of my soon-to-be-acquired house kept trying to make her nest, but wasn't able to complete it. Every day we would find her tapping at the eaves, nest-building materials draped and spread around, but no nest.
Buzzing in the house's hall was a European wasp, the first one Reg had seen in Australia. He made me a nectar trap; the wasp count would keep increasing year to year.
When Nin was born, her father was amazed by her and stood for hours looking into her cot, tugging and tugging his lower lip.
'I hope she doesn't turn out like your mother, dear,' he'd say.
TWENTY-ONE
David
THE DEATH OF Silk meant the cat niche in the Edwardian house was vacant until a stra black cat took u residence under the verandah. She was pregnant and her eyes were full of gooey stuff. She duly had kittens, was attacked by a passing dog, lost most of her milk, and all the kittens died except a ginger one. David had rescued her from the dog, and thereafter walked round with her under his arm and the kitten in his pocket. He called the mother cat Melanie and the kitten Lemontina.
He had stopped staring at Nin and started to say she wasn't interesting. It was difficult to know what to make of this. We'd called David's lack of tact courageous outspokenness in the past; perhaps we should call this a clumsy form of teasing? Reg's unreassuring opinion was that most men aren't interested in babies. I felt it might have been nearer the mark to say some were jealous. Whatever was up with David, his infatuation with the cats pretty soon meant both he and Lemontina had gooey eyes like Melanie's. 'Well, go and see a doctor,' Allegra would say when he complained, but the only time David had been to see a doctor was when he'd thought he had scurvy and the doctor had offended him by being noncommittal.
Periodically he would come out of the bathroom with eye goo wrapped around a cotton bud, expecting Allegra to take it to a clinic and have it analysed.
'That's not how analysis works,' she'd say. 'You have to go to them so they can take a sample under sterile conditions.'
David knew nothing about sterility. His approach to germs was romantic: he spoke of infusoria and phlogiston. If these terms were good enough for Goethe they were good enough for him. He would continue to chain smoke and eat apple pips to stop himself from getting cancer.
He abluted towards sundown rather than sun-up because he worked all night long. He made it known that his ablutions came before the baby's and included a period of arm-flinging exercise which he pursued with the vigour of an executioner practising. It was true he worked very hard, but everyone around him paid the price. His catchcry, 'We are what we do', ignored the consequences of deeds on other people.
As far as his art was concerned, he was terrifically vain and saw himself in competition with all the painters round him. He hated another reputation to overtake his own. This had happened with the minimalist master, Barrie Bull, who'd had a bust-up with Miles over his personal differences with David. Barrie Bull now claimed that Miles wasn't interested enough in selling him and had taken an offer from Hallett-Coretti, or Not-Only-But-Also, as it was known in our quarter. Not the sanest of men, Barrie Bull had been sighted roaming the streets of Melbourne, even promenading the St Kilda Pier, with a girl and a Great Dane on one arm and Jerry Gospel not far away from the other. 'Not only but also,' Allegra muttered, when told.
To add to her problems money was, as ever, short. She had taken six months' paid leave from her university tutoring job, but was ineligible for more and, until David won a grant he'd applied for providing money to work at home for a year, it was looking as though Nin would have to go into a crèche from the age of six months on. Hesitantly, Allegra suggested he might look after her. He could paint at the same time. She pointed out that I'd done it looking after two children.
David was insulted to the core by the imputation that his work was only as important as mine and no more important than the bringing-up of a baby. He hadn't had the baby; it probably would have been born whether he'd been there or not.
The looming tension in Allegra broke bounds with a vicious 'Fuck you! I only had her in the first place because you said you wanted her!'
'No, you didn't,' he said. 'You had her because you're getting on and you thought it was time.'
The cynicism brought Allegra low and she swept past me and from the room in a flurry of tears, holding the frightened baby close as if to absorb her paroxysms. Such occasions were not uncommon, and when our mother was there in addition to David, Allegra was so tense she'd grow aggressive with Nin and stop just short of doing her harm. Our mother, the wound healer, aggravated Allegra's wounds like nothing and no one else.
Nin was a dear little thing, alert and dainty, but this wasn't the first time she'd broken into body-shuddering wails while her parents yelled at each other.
David's attitude to the women in our family was appalling, and we were incredulous that he seemed unaware of it. We functioned on the myth that he'd straighten out in time, that it was damage done to him in childhood, that tender loving care would reverse it, and time and again we gave him the benefit of the doubt. But hardly a day would go by when he didn't provoke a situation that teetered dangerously towards violence. Rescuing the situation wasn't easy and we seemed to live our whole lives around his whims and his sudden fury. I was very glad to have moved from the house, away from his constant hectoring.
He was worse to Stella than to Allegra or me. There was almost love in the antagonism between David and Stella; they dominated everything with the intense feelings each set off in the other. No light bulb could be changed by him in her presence without a major crisis, David standing on makeshift ladders, attacking the remnants of a broken bulb in a socket with a pair of nail scissors, impervious even to simple advice: and no wonder impervious, because it was advice acrimoniously given and the intent was not to get the job done but to reproach David with his ignorance and ineptitude. Here was someone who knew nothing whatever about the practicalities of changing light bulbs, no one had ever taught him, and indeed somewhere along the line, he'd picked up the attitude that it was beneath a man's dignity to ask. Above all, his manhood had to be respected, and part of respect for manhood was never to give advice to a man if you were female. I hadn't realised just how deep went Bart's statement that David was maladroit.
So we had David being pugnacious on the one hand and Stella riling him on the other and we sincerely believed neither of them was accountable for their actions. Allegra and I huddled like a pair of waifs on the periphery of our own lives.
David was Bart's sister's son. We didn't know anything about the sister, except her name was Dorothy and she was the one daughter in a family of five. I asked Miles why David was so uptight and aggressive, but Miles was very circumspect with his near and dear, and would only say that David was talented and intelligent and that was all that mattered. Since it affected my near and dear, I decided it wasn't all that mattered, and when Bart came to Melbourne on one of his trips, he came to see my new house and I made him tell me what he knew.
'Isobel, Isobel,' he called, nimbly skipping across the planks over the ditches in my front yard where pipes were being laid. All he needed was a top hat and a black cape lined with scarlet and he could have been on the run from a Fellini film set: above him, black clouds crackled into halos of lightning – I'd never seen it before, nor have I seen it since. When raindrops hit the dirt around my trenches they made fifty-cent-sized craters, the harbingers of a weird storm that was to throw its tin lids down echoing stairwells all around us for fifteen minutes before it stopped short as if summoned urgently away.
We watched in awe from the back rooms of my house which, though sunken in a garden, commanded a sweep of sky large enough to contain the extensive drama. In another age, people would have imagined the descent of angry angels: Heaven at war and tumbling out of itself. I could hear Uncle Garth reciting, _How art thou fallen from Heaven, O Lucifer, son of the morning! How art thou cut down to the ground_ , the rich voice entombed in its groggy, breath-starved torso. He was in Hell, poor Uncle, a Hell in which Heaven would suddenly flash on his eye, making it all the harder to bear.
Bart had taken up yoga. From a series of bizarre poses on my sitting-room floor, he related David's story. 'Shouldn't really talk while you're doing this,' he would say from a lotus or a cobra or headstand, only to launch into a new spiel.
His sister, Dorothy, had been the cause of her parents' union, and God must have waxed exceeding wrath with the antics of Cec and Therese, for he visited unmarried motherhood thrice upon their daughter, a most unfair series of visitations for any young girl, but particularly bad for Dorothy, who was highly intelligent. 'But one of those people who can't believe in themselves,' said Bart. 'It was almost as if her brain sapped her of vital energy and left her exposed, begging for an ordinary life where she could revel in commerce as other women do, instead of being overwhelmed by vulgarity to the point of being unable to push a stroller through a department store.'
Dorothy was pregnant when she sat her Intermediate Certificate at the age of fifteen. She came fourth in her school overall and dux of Latin, Latin being the subject that marked a student as among the intellectual elite. Then she gave birth to her first baby, a boy, who was adopted out. She reappeared in the ranks of high school students after a four-month absence which was covered over by saying she'd been out working and trying to decide whether or not she was going to continue in school. Her examination results that year were no less impressive than they'd been the year before. Her mother, Therese, felt vindicated in the stand she'd taken against Cec, insisting that Dorothy was clever and her prospects would be ruined if she did not finish school.
Bart grew up under the cover of Dorothy's sinfulness. She was just ten months older than he was. Whenever she did well at school, Cec would say, 'Well, what about Bart?' as if Bart's cleverness were more important than Dorothy's. But Bart was fond of Dorothy and didn't like it when his father tried to put her in his shade.
Bart was sorry about Dorothy's first baby. He was detailed to find out who the father was so Cec could do him grievous bodily harm. But he didn't try. Instead, he took boxing lessons, saying to Cec that he would avenge the family. Cec approved and Bart became a champion schoolboy boxer.
The only trouble for Bart in being a champion was the embarrassing interest his father took in his prowess. Whenever he flattened someone, Bart said it was the image of his father he flattened.
Cec Turner was a nightwatchman. In Bart's childhood the family home, a brick semi-detached bungalow on the tar-covered sand dunes of Maroubra, was often filled with strange consignments of things like frozen chickens or bottles of Asti Spumante or cutlery with hotel monograms on it. They seemed to have been purloined from the hotels, night clubs and hospitals watched over by Cec at nights. They would remain in the house for a day or two and then someone would arrive with a truck and they'd disappear.
The family crockery and cutlery was all monogrammed. The sheets on the beds bore hotel and hospital markings, as did the towels. Therese asked no questions about them and was told no lies. Only the children wondered why the linen and accoutrements had strange names woven in or baked, moulded and graven onto them. As they grew older, they learnt their mother's knack of turning a blind eye. It was their father's business. They felt themselves to be an only-too-average average Australian family.
The year Dorothy sat her matriculation, Cec hounded and teased her right up to her exams. She matriculated very well, but any quiet joy the family might have stolen behind Cec's back was muted because by the time the results were published, Dorothy was obviously pregnant for the second time.
'No wonder she went looking for love,' said Bart, 'It was probably the only way she had of soothing herself.'
She had matriculated so well, the papers had wanted a photograph. Bart had handled the enquiries and said he'd provide a photograph of Dorothy himself because she was a very shy person and couldn't handle a newspaper interview. Without telling his father, he borrowed some photographic equipment from his school and set Dorothy up in the lounge room in front of an arrangement of family sheets made so that none of the hospital labels showed. It was a good photograph and was used in several newspapers.
Cec's wrath, like God's, was visited on his pregnant daughter. How dare she show her face in her condition? She was a slut, that's what she was, pure and simple. You couldn't fish her out of a canal with a grappling hook.
Bart, who was sixteen and more agile than powerful, called his father a bully and, for good measure, a thief as well. Cec's answer had been to flatten Bart's nose with a mighty punch to the middle of the face. 'Hence my delightful appearance: two asterisks around an exclamation mark.'
From that time on, Bart became his father's victim. But Bart wasn't particularly afraid. He had no other opinion of his father than that he was a bastard. He finished school, left home and sought the company of gentler men.
Dorothy's child was a daughter, once again adopted out. She did not even ask about university but found a job with a jeweller in Bondi. The jeweller was a liberal Jew with a wife and family. He taught Dorothy intaglio work, at which she became very good. But amorous proceedings took place between engravings, and soon Mr Silbermann had implanted a third child in Dorothy's womb.
Dorothy, who had not been well loved by the father, or fathers, of her other two children, felt she was loved by Mr Silbermann. He said he could not marry her, but he rented a little house for her and made provisions for the boy, who was named David, a name with both British and Jewish connotations. Dorothy took the name Silver, which was close enough to Silbermann to sound like it and far enough away not to interfere with Mr Silbermann's previous arrangements.
Mr Silbermann enrolled the boy early in an Anglican grammar school and put aside money for his education, so that in the event of anything happening to him, the boy would at least be partly catered for.
Alas, something did happen to Mr Silbermann, and it happened when David Silver was only eighteen months old. One morning in his Bondi bed, beside his legitimate wife, he suffered a stroke and died. Dorothy, who opened the shop for him in the mornings and that morning wondered if she hadn't made some mistake for which, although he was not the punishing type, he was punishing her by staying away and not calling to tell her what kept him, sat at her work in the little cubicle beside the shop counter until midday, when the son of Mr Silbermann's marriage arrived with his sad news. This lad, who was eighteen or so, had not an inkling that little David, who would sit beside his mother in a high chair arranging stones and building blocks on its tray, was his half-brother. He thought Dorothy was a young woman, fortuitously named Silver, on whom his father had taken pity.
Bart hesitated in his telling of the story here. 'Poor Dorothy,' he said, his eyes growing red, 'I suppose she was unbelievably lonely, unable to say what the man had meant to her, unable even to tell who she was. So lonely, and I dare say frightened, too.'
Aside from the trust account he'd left for David, a suit, a prayer shawl and a yarmulka which she had taken to the cleaners for him and, deciding to keep them, collected the day after he died, Dorothy had nothing of Mr Silbermann. Since her father had barred her from the family home, her mother, from whom she received furtive letters with a friend's address to which she could reply, was her only recourse.
Overcoming her fear of repercussions, Therese went to find her daughter. She found her in a miserable state, stocktaking while the Silbermanns were negotiating a buy-out. At nights she was nestling up to Mr Silbermann's drycleaning. She had entered a slow, merciless decline into a breakdown, for which she ultimately had to be hospitalised.
Dorothy remained in the psychiatric ward for several months, during which, though Cec objected violently, Therese took David to live with them. It was very stressful but the child came first with Therese and if Cec put on a performance, that was just an additional load she would have to carry. David was three when he went back to his mother, but had already been wheedled into Cec's ignorant, sadistic cycle. Here was the dominant male in his life.
Dorothy was eventually well enough to take work, but no longer as a jeweller's assistant, since the fine work brought on attacks of grief during which she could neither see nor make a mark. She took work, instead, in a clothing factory whose employees were mainly Italian women. There she was befriended by a Sicilian girl whose brother was about to start out as a market gardener in northern New South Wales. Dorothy was invited to join the family celebration of his twenty-first birthday.
'I suppose she found it very inviting,' said Bart. 'They were friendly people who celebrated lavishly, and I dare say they liked her; Dorothy was pretty, peachy cheeks, wavy brown hair, green eyes, but the head was always down, the smile had to be fished up. A meek, demure person; the family probably liked that, and they knew she had a matric – it used to count in those days. Maybe they set out to rescue her and install her as Joe's wife. Anyway, that's what happened.'
And within another two years, Dorothy had two more children, and David became stranded in a stepfamily. The husband turned out to be vindictive. He didn't like having a wife who was cleverer than he was, even though she was an asset among friends and family, where her intelligence reflected favourably on him. He was not very keen on Dorothy's son being cleverer than he was, either.
Dorothy had thought she was doing the right thing by David, marrying and establishing a family of which he could be part, but she was to make the sad discovery that vindictive people do not change unless they stop getting their kicks from vindictiveness. Her child came under violent attack from which she was too frail to shield him. Her husband began trying to beat the signs of intelligence out of David and so, to keep him from physical harm, he was sent to boarding school from the age of eight. Paying the fees was a considerable family sacrifice, in spite of the money put by for it by Mr Silbermann. Therese Turner contributed what she could from a qualification in accountancy, surreptitiously gained by correspondence during David's infancy, when he had lived with her.
Back then, Therese had studied while David slept. She studied on Cec's darts nights, on his football afternoons and on the interminable television evenings in the square house with its clop-clop lino floors over which the chairs skidded on tubular, rubber-tipped legs, where the sharp ringing of the phone sliced conversations into thick chunks, and the buffalo grass lawn bristled front and back inside sawtooth borders of red brick. She studied to the smell of cigarettes and beer and through the steam of boiling vegetables. With a background of racing commentaries, she studied, and even tried to interest Dorothy in studying as she sat beside her hospital bed or walked with her through the shrubbery around fibro bungalows where the impoverished mad recuperated. But Dorothy's mind was dark with pain; she walked through charred memories, every discovery revealing more loss. Her intelligence had been gutted and all that remained were fragile, peripheral things, dwelling places that afforded little cover and almost no comfort at all.
Therese with a heap of account books or tax returns, at the kitchen table after the evening meal, became a familiar sight. She made quite a tidy sum from it, would never tell Cec how much, but made sure it was put by for David, to be used in the long run for uniforms and books and a supplement to his school fees.
When David was nine, his mother and stepfamily moved to Queensland, exacerbating his isolation. His school holidays were problematic. During term breaks he mostly stayed at school. Therese then learnt to drive, bought a car, and would take him on little holiday expeditions. One time, she told him to ask a friend to come too, but David said he hadn't any friends. He was a peculiar child, of course. How could he be anything else?
One Christmas, when David was thirteen, his grandmother decided to have him for the holidays. There'd be plenty of room for him since Miles, who was studying at university, was the only one of her children still at home. It wasn't fair on David to leave him at school or make him go home to a teasing and violent stepfather when there was a bed and food and a grandmother who loved him in Sydney. Cec could do what he liked, but she was going to have the boy for Christmas.
As she was leaving the house to go and pick him up from school, she asked Cec to back his car out of the drive so she could get hers out. Cec, who could do nothing with a good grace, roared backwards down the drive and ran over a beer bottle he himself had left there, puncturing his tyre. Later, with circuitous logic, this was found to be David's fault.
At the dinner table in the evening, the child was made to wear that look on the face that plays havoc with the search for an appropriate expression. He bent his head, his eyes slid from side to side, his top lip wanted a legitimate reaction to give it shape. He was told to 'get that expression off his face'. He was asked 'What are you looking like that for, anyway?' And as he was 'looking like that' because the accident wasn't his fault but his grandfather's, he was hit. Had Therese been sitting at the table and not standing at the kitchen counter serving up the sweets, she could have undone the chain of events. But Therese could not be everywhere at once.
Miles loomed from his seat and let fly.
Therese shepherded David out of the room and tried to distract his attention from the thudding and yelling in the kitchen, where the china was being broken and the sweets she'd prepared ground into a couple of angry faces. But the symptoms of family illness were running riot and the child, who was no fool, knew they were. He was, however, accustomed to accepting his fate, not as a child for whom the best arrangements possible were being made but as a child who was to be punished for accidents that could have been avoided had he not existed. He valued himself less highly than a beer bottle and a car tyre.
Worse still, as a result of David's existence his Uncle Miles left home and went to live with his Uncle Bart, whom his grandfather called 'that poofter'. This had made his grandmother sad because Miles had been her solace. The message David got was that if you wanted a man's approval, you'd better not be a poofter, nor identify with the problems of a woman.
Miles had guarded Therese against the tyrannies of Cec and now she was without protection. It was a maternal crime to have brought three exceptional children into a world that paid homage to mediocrity and looked down its nose at instability and homosexuality. It was a crime only partially expiated by the two sons between Bart and Miles, one of whom ran a fishing tackle shop on the north coast and the other of whom was a mechanic. These sons were Brendan and Damon. It was Therese, who'd been brought up a Catholic, who'd given them their names. After David came for Christmas, Cec began to say that Catholic was one thing, but Jewish was another.
David's youth was lived hardly realising any part of him was Jewish. He was brought up Anglican with a dash or two of Catholic during his holidays. Though he scorned his grandmother, she was his only source of love, and fondness for her meant fondness for the myth of Christ's divinity. He believed in resurrection, in sinning and atonement. An overlooked fact of his childhood was the Jewishness of an unremembered father.
The year after the fight between Cec and Miles, Miles drove David home to Queensland at the end of term.
In spite of having had jewellers for parents, David had no prowess with his hands. He was clumsy; he couldn't hold a ball without dropping it down the only culvert for miles, and once he kicked a football so off skew it bounced up the school roof and lodged in a chimney. It was almost as if he had a psychological set against success so strong that his failures were spectacular. There was a subject at his school called Mechanical Ability; every year his report showed him to be below average in Mechanical Ability. He might come top of several subjects, but every year he was at the bottom of the class in Mechanical Ability and every year, it became apparent to Miles, his stepfather would take him to the field opposite the family house and torture him with tackles and drop kicks. When Miles tried to intervene in this process of degradation, he found that David himself was stubborn and the charade would go on until the boy was thoroughly disgraced and the stepfather's cruelty was triumphant.
In time, David had become obsessed with physical activities and sport. Almost every moment of his waking day was spent vicariously standing on a podium, a medal round his neck. Bart, who had been a hero, was filled with pity when he learnt this. It seemed that David longed for nothing more than a man to take pride in him.
It was after Miles had moved to Melbourne that David ran away from school and went to squat in a derelict house in Woolloomooloo.
At first he was too shy to go and introduce himself to Bart, whom he didn't really know, but when he did introduce himself, Bart was protective, just as he was later towards me, and immediately took David into his care. He wrote to Dorothy to say David was with him and he would do his best to make sure he finished school.
When David was eighteen he was a lightly built boy with a big head. He had just matriculated well enough to go to any university in the country and do whatever course he liked, but, Bart discovered, he didn't even know the role of universities in the world and was contemptuous of them, probably to cover fear and ignorance, and Bart couldn't penetrate the defence. Therese would come to visit every now and then and would try to interest him in one course or another, saying how lucky he was to have all these options, but David was even more defensive with Therese. Therese would say it was probably a good sign, as he wasn't going to be pushed into anything easily.
Ultimately he decided on art, the one thing matriculation hadn't prepared him for. 'He wasn't going to go to tech and do it, though. He was going to sit at home and be a painter, which meant sitting at home with me. He was ruining my sex life, but whenever I felt like reading him the riot act, I would look at him and feel what he suffered inside myself. And Isobel,' Bart said, wearily, 'he couldn't draw. I mean, literally. He had no sense of a picture plane at all. He just liked to sit there contemplating the pristine condition of the drawing pad, with his pencils sharpened and in a nice neat row. When he did draw eventually, the drawings were absolutely minute and fenced in with little frames. They were so weird, really, that even though the tech knocked him back, I began to think there might be something in him. And he was obsessive about it, too. Once he started, you couldn't stop him. He would draw and smoke all day, all night. Then he started to go out and buy little hard-covered books with marbled endpapers and he literally filled them with this amazing stuff. Private little self-contained worlds. Beautiful. He mightn't have been able to draw on an ordinary scale, but his tiny drawings were compact and laden with out-thrusting energy.
'At first I thought they were copies of other people's work, but then he clearly got his own imagery going. Stiff little men, strange little dogs, straight roads, round suns and moons, everything geometric. It made me decide to do a show on artist's books. I wasn't wrong, he really took off from there.'
Bart untwined himself from the cobra and lay for five minutes, 'toe-tally relaxed' on my sitting-room floor.
I hadn't known he and Miles came from such a traumatised family. Their mother was still alive and immensely proud of them both. I'd met her once and would never have guessed at a husband like Cec. She was calm, well spoken and obviously intelligent. Her body was neat, strong and snug. A vertical gold filling in one of her front teeth added character to her smile. Though she had no background in art, she had instinct, and it wasn't an instinct for ballerinas or swans. Bart, Miles and David had it, too: a sophisticated appreciation of what it is that artists do and why they do it.
As for Dorothy, once in a while there would be a fleeting contact, a wish to hear news of David, but as far as Bart understood, she now lived imprisoned in an outer suburban dream home with a huge bunch of keys for the many locks on sliding doors that opened and closed with the precision of guillotines.
While Bart relaxed, I sat there thinking about David's radical appearance, his very broad forehead and ferocious widow's peak, his very narrow chin, spanned by a thick, purplish mouth. Physically he seemed the illustration of his mother, starved of sustenance by his intelligence. It was a sad story.
I remembered the first night he spent at our house. He was very drunk. 'He-h!' he said to me. 'You gotta paint all the time, no matter what, if you're going to be a great painter.' So saying, he fell asleep and subsided down the wall. We started laughing, Allegra and I. We were willing to forgive anything in a relative of Bart and Miles. Asleep, he was as innocent as a baby. The inside of his arms was soft and young. His hands were very beautiful, the fingers straight, pink on the tips, the nails white-mooned and, ironically for a maladroit man, beautifully clipped. We brought him a pillow and put a rug over him.
Bart opened one eye and, ten seconds later, the other. 'I can't help loving David,' he said. 'All that aggression in him is fear and self-defence. It's as if he's harbouring a soft little creature inside him, done up in that wonderful satiny skin. Maybe that's his obsession with the kitten. I'm sorry for Allegra. You never know what's going to cause a catastrophe with David; he's been so hurt, so messed about by life. I have to confess I'm afraid for the baby, part of David himself is still as small as she is, he probably feels she's a usurper.'
Then he sat up and said, brightly, 'Tea time!' and tossed me a packet of lapsang souchong out of his bag.
'What's Allegra to do?' I asked, in my kitchen now, submerged in the back garden where a challis lily grew, its as yet unopened flowers python heads on a propeller-leafed tree.
'Well, not what she's doing now,' Bart answered. 'Oh, I don't really know there's much she can do. She could leave him, but he's very persistent, very tenacious, and he does believe in his life with her. That it's gone this far has given him something to hold. I have to confess what surprises me is not so much his behaviour – he's as mad as a hatter – but Allegra's. I mean, David's weird, it's obvious he's weird. He can be very well intentioned towards the people round him, but then he mucks it up completely, does his block and says it's the fault of the people round him, so that whatever it is in him that needs care and attention escapes the notice of most people. He's hard going. Allegra's given herself a heavy load.'
TWENTY-TWO
A Table in the Presence of Enemies
THE MAD MEG collective met in the back two rooms of Mad Meg. It was pretty cramped, for despite our preference for art of the 'event' kind, which was meant to disappear like the dust behind the door once it was over, or travel somewhere else as a 'total environment', we accumulated many an extraordinary article: a revolving desert sunset with accompanying birdsong, bird beak helmet and feathered breast, an 'event' sculpture, collectively constructed, of a witch on a broomstick, whose vapour trail was made by children tying anything taking their fancy onto long strings attached to the witch's rear. There were boulders wrapped in orange plastic, a flagon full of Murray River water, boxes of sticks, a heap of bricks, a stuffed carp.
A sack of sand doubled as seating for the coordinator, a functionary role supposed to rotate on a yearly basis; squabbling, however, led to its being either Allegra or Maggie, they being better at keeping the peace than the rest of us.
The domination of Maggie and Allegra naturally led to accusations of factionalism, but whenever the orb and sceptre looked like going the way of the Troika, there would be an in-depth examination of principles, culminating in the Troika being unable to accept because of its anarchist conscience.
Cathy the historian did not think of herself as coordinator, but she was an archivist by habit, and by habit kept our records.
I was deemed politically to the right of Genghis Khan, and therefore ineligible for the post. Putting aside the slur on my character, this suited me fine, as I had plenty else to do.
In the late 1970s the Mad Meg _Broad Sheet_ had become a widely read item on the art scene and had latterly distinguished itself by going into a shiny cover, through the spine of which largish staples were required to contain the collective outpouring. While the collective had voiced no objection to the publication of letters from those of the Other Gender on the letters page, indeed, they welcomed the opportunity of reply, recent strongly felt opposition had been raised to the inclusion of articles and reviews by the said gender on the hallowed territory of The Art Debate. The Kelly–Coretti faction of the Mad Meg collective believed we should invite men to criticise us with the same rigour and respect they accorded male practitioners. The Troika was adamantly against us: it had to be a female critique (all three, of course, were critics) in which a female world view was brought to the discussion of the work.
'But,' said Maggie, who wanted to include a review we'd received from a man, 'this review's interesting because he says "the work is informed by a female sensibility".'
A collective, wet and lip-vibrating laugh leapt out of the Troika.
'Well, he goes on to what he thinks a female sensibility is,' said Maggie defensively.
'How would he know?' the Troika snapped.
Maggie replied that she thought the chap had put it very sensitively and well.
'Well, we don't,' came the Troika's reply. 'We think it's unethical.'
'How, for God's sake?'
'It's politically suspect.'
Behind the conglomerate, co-operative-approved personality, the Troika managed to be about as supportive as a perished rubber band. Nin's presence at the meeting was being tolerated with dense clouds of cigarette smoke and ill-concealed chagrin. Earlier, she'd had a bellow and was given to me to soothe and entertain in another room. 'I'm a dreadful mother,' Allegra had said, but motherhood was not yet an 'issue' for the Troika.
The Kelly–Coretti faction began arguing that the result of a wholly female critique could be, and often unfortunately was, mushy self-congratulation which lowered the tone and made us look ridiculous.
Well, then, was the Troika's riposte, didn't men look pretty ridiculous, profoundly more ridiculous for their pompous, elitist, perpetual possession? Should we not fight elitist tactics with elitism, give the men a taste of their own stuff?
Maggie and Cathy bridled at this; elitism was a crime in itself. Mad Meg didn't exist to commit crimes; it existed to bolster the image of women in art. This was a good piece of writing; someone had taken the time and had the insight, and insight was an individual gift, not an ideology. Getting men interested in the women's art movement was important if women were going to be considered in the mainstream. Teaching men to consider women as practitioners rather than objects in art was very important.
Here Allegra wiped a herring over our trail: men were unaware, too, of how women saw them, she said sadly, and for the most part, they didn't think it was important to find out. But it was important; men were arrogantly overriding the requirements of women in the world and using marriage as a form of tyranny. Just let them try to see life through female eyes! Let them admit female critics to their realm and let them reciprocate by coming to look at women's work with impartiality and honesty.
I had no doubt it was her own marriage she was calling a tyranny, but the generality lit a spark in me. I saw a whole set of pictures in my mind: ironies of the female nude, crisply painted all-too-peachlyflesh-bound little women crouched in corners, overburdened by Janus-faced men, whose upper faces would be kissing a Blakean God or male angel leaning down from a cloud and whose lower faces would be seeking out the woman. Everything but the woman in each picture would be painted in stone-like Blakean fashion: all would be massive weight, bearing downwards. In one canvas, the woman would be pressed up to the front of the picture as if against the glass. The very window existed in my house; I could hardly wait to ask Maggie to pose.
But the meeting was getting on. I was being cursed and asked questions. They were taking a vote. They were going to run an issue of the _Broad Sheet_ on a series of women's shows, each one reviewed by both a woman and a man. Seemed fair enough to me. Would I be one of the shows? Of course; this thing I was thinking of now had me in its tow, I had only to take charge of it, and I'd be away.
The Troika wasn't sure it approved of the twin review idea and felt it could only go ahead if one of their number were allowed to do a deconstruction to sum up at the end. Would we? Wouldn't we? We would, but not with burning enthusiasm, and Allegra, I could see, was biting her tongue.
I painted bourgeois interiors where the chairs looked twitchy and sick of sitting about in rooms. I painted them romping in paddocks. I painted vigorous messes, plumbing that was tending towards anarchy, shoes, a silver glitter pair of high-heeled platforms belonging to Allegra, which seemed to announce her feet slung, crossed, off a chair above and to the right of them. It seemed to me you could sketch the rest of the picture from this token. You can tell a person by their feet. I took photos of The Brolga's and Pattie Gospel's at an opening at the National Gallery, long pointy claws in deluxe brown and short, stubby fatties in blunt-nosed slingback blacks. I had a lot of fun with shoes. There were little pictures taking shape all over the house. But these were just things I painted on the side, inspired by my house and my daily life. More importantly, my ironies of the nude were slowly taking shape and resolving themselves into something poised, something I had never attained before. I kept them in a separate room – nothing else in there but a tape deck and classical guitar music. I painted uninterruptedly over Easter 1983, and each night before I fell asleep, I read Uncle Nicola's diaries.
At the beginning of 1928 Uncle Nicola wrote, 'I do not know what I shall do for money, but I shall live as if I had it and were an ascetic. I could cast myself as a person with a priestly vocation, a variety of missionary whose task it is to establish my faith at the ends of the earth. But how to establish my faith? I picture myself in my frock coat, being cooked in a cauldron. By whom? I don't have to invent an opposition to myself because all I have is an opposition. On my own, I am ludicrously opposable.
'The Consul-General has tried to intervene to have me removed from the university because I am, according to him, "a Communist whose views are well known in Italy" and I am supposed to have "come here with the sole purpose of spreading propaganda and promoting unrest among the labouring classes". Would that I could penetrate the labouring classes to spread unrest! But the labouring classes tend to treat me as a joke. As for spreading propaganda among them, the mechanics of doing so all but prevent it being done. There are no secretaries at the university, the professors do their own typing. Duplication is by way of a bed of gelatine yielding, with luck, ten fair copies at a time, and the supply of gelatine, paper and ink is one's own responsibility.
'There are few people of culture who share my politics and I count myself very lucky indeed to have come across _il dottore professore_ Lonsdale! Without him – _mortificazione!_ I am so poor, I have to wear carpet slippers to save my shoes when at home in my room, and even then, I am in trouble with the landlady ( _una donnetta molto grassa e molto orribile_ ) for walking about at nights, because the floorboards creak. Today she wanted to know, and in English too, what all the scratching and creaking was about. The other tenants think I pace around all night for the express purpose of getting on their nerves, when it is they getting on my nerves that makes me pace. Well, I tried to tell her, in English, that I was muttering _The Inferno_ to calm myself, as I find there is nothing more calming than to have, stretching and preening in my throat, the verbal cat of _The Inferno_. Such a creature needs the legs and arms to keep it company. As for the scratching, what can the woman mean, unless it is that even in uncouth surroundings, a dignified and cultured person replies to the many letters he receives and is at times perhaps given to assailing the paper and causing the ill-made furniture in his room to complain of its poor condition. But a man must write, and with elan, to those whose spirits are oppressed and need his expressions of hope and goodwill.
'No news from Milan. I have to throw myself into inventing my work here in Australia to stop dwelling on it. They would write if they could.
'So, I hear from Paris from the Antifascist Concentration just about every day, but only recently have I begun to hear from anyone in Australia. They are very short of money in Paris, and I've promised to do what I can to raise money here, but the Catholic Italians, who have most of the wealth, have been ordered to steer clear of me. Their monsignor, who is from Rome, publishes a weekly bulletin called _The Angel of the Family_ in which I am periodically decried.
'Spencer says my support is growing, because my last lecture was packed out. Certainly it was packed out: it was packed out with fascists. I'm more attractive to them than the opera. They come to watch me for my curio value. I am famous for my clothes and my oratory. There are few people to whom antifascism means anything here, and I can't communicate with these because my English is so poor. The more I worry about it, the worse it becomes. Spencer says that's because I want to speak sophisticated English without learning ordinary English first. But the level of education of people to whom I really want to talk exceeds the use of ordinary English, and, in any event, I don't think in ordinary words. When it comes to educated Italian speakers, the icons of their culture are often not the icons of mine, and conversations with them are restricted to a one-way flow of words from me to them.
'I open my mouth and flow over, but what effect does it have? I am a one-man fountain – quite refreshing to paddle in my waters, but knowing why I'm here is irrelevant. This is an island of somnambulists in which there is no temptation to wake.
'Oh well, I have been appointed as an instructor by the university, however much the Consul-General has tried to assassinate my character. He wrote and told them they should have asked him about me first and warned them I was a dangerous communist. The best he can do now is warn the Italian community against enrolling in my course. Quite amusingly, I notice, he has his own troubles. The fascist old guard is saying the party should be more powerful than the consulates, since the Italian state is now deemed in Italy to have reached its self-perfecting phase. Ah, what a paragon is the perfection of Italy – a rambling poetaster who has read Nietzsche, a man who believes, with the help of his friends, that the arrival in the world of himself, Benito Mussolini, has been immanent in all history. Just as the expulsion of Nicola Coretti to the ends of the earth has. I race out, bombard Melbourne with my orotund sorrow, and people come along to the performance. Then they go straight to the Consul-General and tell him what I've just done and what I am about to do. I'm a sideshow.
'By now the Consul-General knows of my intention to set up an antifascist club in Melbourne and is trying to convince Australian politicians that I'm dangerous, a political subversive of the worst stripe. It's a view the Catholic branch of the Labor Party would probably endorse. A strange country, this. You have to pass a spelling test to get into it. Strange to say, though they could have excluded me by giving me a test in any European language but my own, my test was in Italian. I have heard of 'undesirables' being tested in Gaelic. All I can think is the Italian Consul-General wanted to flex his muscles by having power over my life. Thank God for the consul-curbing potential of British colonial arrogance!
'There are two Australias, the British and the Irish, and there's no question but that the British is in the ascendency. The working class and the present government are like different species of being, their interests so thoroughly opposed as to seem irreconcilable. To succeed in this country is to succeed by displacement and exclusion. But, if they belong anywhere, the Italians belong in Irish Australia, with the Catholic church its way of competing for ascendency.
'Clearly, there is something wrong in Milan. It might be that the mail is being intercepted. If I could get up, go there, sort it out! But of course, I cannot. I could do all the mechanical things, step onto a ship, return, find out what is going on, but I would be stepping into my death, and quite possibly that of my brother's family as well. Thinking about anything else takes enormous effort, and sometimes the meaning I attach to everything I do just falls away.'
As well as chronicling his life, Uncle Nicola was constantly ruminating over politics. From these ruminations I was able to diagnose myself as safely unamenable to Genghis Khan and to reproach the Troika with an ideological joust of my own: I was against dictatorships and tyrannies and all forms of fanaticism. I was for democracy and a degree of central control: people ought to be able to see the political debate in progress and choose their side. In response to a thorough mocking on the grounds that it didn't matter what side you picked, democracy based on capitalism meant prosperity for the first world at the cost of the third and was simply the substitution of one interest group for another, I was able to reply that, while I had no answer to the exploitation of the third world by capitalists, people in a democracy could, at least in theory, participate in the political debate themselves, and this extended to their right to speak out against capitalist exploitation. In a democracy, furthermore, people could lobby, form minor parties or use ballot papers to express a view not put by either major party: such protest votes were being counted in Australia, even as I spoke. The Tasmanian electors, in reaction to proposals by their electricity commission to build a dam in the wilderness for a hydroelectric scheme, chose to invalidate their votes by writing 'No Dams' across their ballot papers, while famous people like Reg Sorby chained themselves to the landscape and got covered in leeches. The scrutineers counted the protest vote and it was reported Australia-wide.
In Fascist Italy what Mussolini said went, even if it was total bunkum like the war in Abyssinia, ostensibly fought to unite the Fascist State in that noblest of all expressions of nationhood: War. In reality it was a ridiculous beat-up in which a great many more Italian soldiers died of malaria than died fighting.
If Fascism was idiotic I'd always believed Nazism was the scourge of the twentieth century, but to my horror, reading the diaries, I found Hitler to be the great disclaimer, for he was the perfect product of his times: racism was in place before he was, and it was virulent and pretty well universal. Australian 'socialists' went as far as condemning the idea, emanating from China, of a pan-Pacific International which would strive to unite the proletariats of China, India, Indonesia, Korea, the Philippines and Latin America against the oppressive classes in those countries. Only the white man was worth solidarity in Australia.
Uncle Nicola did not hear from his family until well into 1928, and then it was through a letter from Rose's father, Lev Katz. The Corettis had fled to Paris and were now living with the Katzes, but the story was complicated.
In December 1927, when Uncle Nicola was ripping a branch from a fir tree in Melbourne's Botanic Gardens, the house with the trompe l'oeil fitting room was raided and ransacked by blackshirts looking for, and finding, articles Emilio and Allegra had been gathering for _Non mollare!_ Emilio was beaten up and had his leg broken badly in two places.
Allegra and Dadda had been in the house when the bashing occurred, and both were deeply affected by it. Lev wrote, 'Allegra barely sleeps, and even now, sometimes seems dazed by the rapidity and cruelty of the attack. They arrived here in January, Allegra bringing with her a case full of smashed possessions. She thought she might raise something with them in Paris but, in truth, there was nothing we could salvage.
'They escaped via the sea route, Emilio doing the best he could with his leg splinted, but he was in agony when he arrived here, and it was pretty soon evident to us that he had gangrene, and there wasn't much that could be done for him.
'We didn't know where you were, and, never thinking of Australia, lived in expectation of you arriving any day. We did not locate you until April and by then, I have to say, Emilio had died from overwhelming infection. I'm afraid he died in so much pain, we hesitated to tell you. It has been harrowing for us all.
'We buried him at Montparnasse, in the family vault of my father-in-law, Dan Grafman. I never got on very well with my father-in-law before this happened (he's Orthodox and wouldn't let me marry Rivkah unless I consented to a kosher household), but he showed me something. He showed me the value of tradition in times of indescribable anguish. You search for the words and the deeds and it's a comfort to know people have searched before you and found something you recognised as emotionally appropriate. So, even though old Dan has endless set-tos with my father, Isaac, who's liberal and doesn't even believe in God, I was grateful to him on this occasion. I hope you don't mind your brother being buried Jewish? There's the custom, in Jewry, of taking in the stranger and treating him as a family member. A tradition was better than no comfort at all. It was a demonstration of love and care, an opportunity to reclaim good memories.
'Old Filippo Turati came to the funeral. He lives near us these days, and he asked me to see him home; he'd collected some money to help Allegra and Henri out and wanted to hand it over. He's taken a fancy to Henri, who's impressed us all with the courage he showed looking after his mother, shielding her from Emilio's pain. Turati asked if he'd like to do some work for him – running messages, that kind of thing. As Henri was keen, I said later he could do it and now, it seems, there's work for Allegra too, if she's up to it. She says she is and that the discipline will be good for her. It's translating work, I believe. Though she has moments of anguish still, Allegra is strong-minded and very practical. I did not know this side of her before.
'I'm sorry to have to communicate this grim news to you, far away as you are, but you can be proud of both Henri and Allegra; neither of them has lost sight of Emilio. It's strange how weeks of pain can substitute itself in the mind for a full and vigorous life, how hard it is to remember back before the pain, but both Allegra and Henri are determined to remember and keep on reminding us to remember too.
'I try to picture Emilio as the happy inventive person he was, gregarious and thoughtful. He was my best friend.'
Uncle Nicola wrote, begging Allegra to come to Australia at once. But he did not have citizenship, and Australia was wary of accepting refugees. The Americans had imposed quotas on Southern Europeans, stating a preference for 'Nordic Types', and, following suit, Australia began to assert its preference for Britons. The Italians were quite visible in Australia, and because a high proportion of them came to labour for low wages, people thought of them as a threat to living standards. A widow with an eleven-year-old child didn't stand a chance unless she had money, and, if at all possible, British citizenship.
When I was in Paris on my way to Milan, I went to the Cemetery of Montparnasse and photographed the aisles of the dead. An idea was growing in my head. Back in Australia I'd seen a suite of photographs documenting the journey of a European Jew to the unknown land of Australia between the years of 1920 and 1950. The suite was not simply a range of photographs taken over those years, it was also a layered journey from a distant past to a futuristic present, and this did not happen from left to right. Each photo was constructed to represent a present of its own; each was a layered history. I thought to do a similar thing in paint and collage and, as I began my journey in search of the Grafman family vault, which contained the ashes of my grandfather, I was thinking of the formal problems such a work would entail. Perhaps I would build up a three-dimensional surface, a sophisticated kind of peep show, making an alley of the Grafman vault and its history. I was about to head down the first pathway, but fifty yards off a face peered out from behind one of the tombs and fixed me with an intense, crooked gaze.
I had been going, simply, to stand at the place where my father had stood, watching the entombment of his father's ashes. I was going to savour the wretched irony of Emilio Coretti occupying a space in the Grafmans' vault, while Rose's grandfather, Dan Grafman, through whose courtesy he lay there, had been gassed in the Auschwitz 'shower block' and thrown onto the heap of bodies which were pillaged for their hair, teeth and jewellery before being carted off for cremation. All this because his daughter was married to a German Jew who was an active socialist. Every now and then, human fat clogged the ovens at Auschwitz, due to insufficient stoking of the bodies.
I never found the vault; the predator anticipated me at every turn.
TWENTY-THREE
The Crushing
I CALLED MY exhibition for the double criticism edition of the Mad Meg _Broad Sheet_ 'The Crushing'. Some of it was mildly pornographic and meant to convey the idea that men's sexuality is a lot more ambivalent than women's. Men seemed to me to have a decided ability to conduct their sexual activities, not with their eye on their female partner, but on a dominant man whose good opinion they were seeking. They lived in a hierarchical pyramid at the apex of which sat God the Judge of Men and under whose base Women, the Unworthy Ones, were crushed. To this God women didn't matter, they were without souls; the best they could aspire to was the 'versatile, comely beast' of Granpa's imaginings. Breeding stock.
I stood around on opening night with Rose and Maggie, guessing tipsily what might be said. The chap would probably say something like an 'admirable tension is generated by the juxtapositioning of stone and flesh', or, as had happened to one poor woman before me, 'my reaction is better left unsaid'. The _Broad Sheet_ 's woman reviewer would probably say something like 'there are things in life too complex for oral transmission (R. D. Laing, op. cit.)'. Given half an opportunity, I myself would have said, 'The females in this work are undeniable, and they are crushed females, distorted, flattened and, in one case, pushed up against the front of the picture plane as if into a pane of glass...'
Maggie's hair had curled up against the glass like the hair of a dead woman in a locket.
At one stage I thought of an all-male guest list, since the pictures addressed men more than they did women, but I was glad I decided against it when David, who'd been quite chatty and affable, was suddenly very drunk and began to say what I was doing had been done already a thousand times, no, a million times better, by Lucien Freud. It wasn't only women who were oppressed in his pictures, men also suffered. Lucien Freud hadn't exactly crossed my mind in connection with this exhibition. William Blake had, as part of the question of rendition, not to say theme, since Blake was concerned to represent a bearded God and women in attitudes of prostration. Furthermore, rotting furniture played no part in my pictures, as it did in Freud's, and my women were not yellowish green, their flesh was undeniable.
It was still and warm outside. David, however, reeled from room to room as though he were out in high seas on a stormy night. At first people took it with good grace, but then David's rival, Barrie Bull, came with his entourage from Not-Only-Hallett-But-Also-Coretti, and that meant two madmen in a confined space with an audience.
As David lurched about, he began to chant: 'Barrie Bull, Barrie Bull, Barrie Bull Bull Bull/Barrie Bull, Barrie Bull, Barrie Bull Bull Bull.'
There were olives, cheese and taramasalata on trays in Mad Meg's kitchen. Barrie Bull took a knife, albeit a bread and butter knife, out of the taramasalata and started challenging David to a duel.
Allegra, unaware of what was going on inside, sat out in the mint patch, talking Proletarian to some students. Maggie and Kelly had gone and that left Rose and me to defend ourselves from behind the doors of Mad Meg's office cupboard. Barrie Bull's entourage gasped and sucked in air as their hero circled David menacingly.
'You're a cunt!' snarled David, at his most delightful. Barrie Bull's wife (who was a different person from his girlfriend) circled with the duellers (David's weapon was a full wine glass), saying to David, 'You never say that to Barrie! You never say that to Barrie!', while David kept his free hand in his pocket and allowed his shoulder to be pummelled by the very much more nuggety Bull.
'You better watch it!' Barrie Bull was yelling. 'You better watch it, cunt! Next time I see you, you're dead meat!'
Rose and I took some heart from that 'next time', but the circling went on. I wished I'd had a video recorder; what I meant in my work could not have been illustrated with greater clarity. It was art's young _homme célèbre_ , Barrie Bull, and his challenger acting out their animal ritual as if Mad Meg were a territorial breeding ground for crocs. But they were worse at it than crocodiles, there wasn't that healthy _thwack!_ of chainmailed breasts, that natural economy of thrashing which endangers only those within a body's length of the combatants. Elizabeth Bull took an accidental blow to the jaw as Barrie raised his pummelling arm and retired, weeping, into the dumbfounded band of spectators.
I was beginning to be sickened myself. These two bastards didn't care what was on the walls – or perhaps they did, and either out of jealousy or contempt, were determined it wouldn't stay there. Red wine was going everywhere as David, now armed with a newly opened bottle, fended off his assailant. When they staggered into the room where my ink washes were, I'd had enough. I fled out the back, from where Allegra was pushing her way in through the students, and sat huddled in the mint, praying to God, whether Blake's or mine, that my work survived. 'Please,' I said, 'please, God, so much work has gone into this. It can't be that these bastards are going to wreck it all.'
_Thwack!_
_Chop!_
Breaking glass, shrieking, and 'Davey, Davey,' said repeatedly, in a level, placating voice.
The room with my ink washes in it, Allegra leaning into it, saying, 'Davey, Davey, come on now, stop it.' Barrie Bull has backed off, the bread and butter knife in the breast pocket of the waistcoat he has on over his tee shirt. The rooms are crowded now with people pressing in to lean on my paintings and take a look. Red wine all over my drawings. Checkie Laurington gasping over my shoulder. A rip in one of my acrylics where one of them pushed the other against the wall. Fat Pattie Gospel stares at it with her lip curled back. Jerry Gospel struts about insofar as the two-strut perimeter of the fight allows, ostentatiously waiting for Barrie Bull, who has now turned to me and is saying, 'Isobel, you're so sad. Why are you so sad?'
A lot of work and thought went into my sketches. I tried to do a lot of things: to portray stasis and life, the ambivalent sexuality of men, the vulnerability of women, to convey brute strength in the skimming of an ink mark across paper and weight in the lick of a brush. David is still reeling and rolling and clumsily batting off Allegra, who is trying to calm him.
I shall go home to my house where Nin and Eli are. Rose is saying, 'Zis is terrible to destroy your work, Isobel. I sink zat young man is very difficult.'
Difficult! I've been robbed, gutted, destroyed. Difficult! I should murder him! As I walk towards my house in a huddle of arms, I suppose to murder David I would have to murder history, kill the poor mad sister of Bart and Miles, and their father. I'd kill their father, if I could kill their father. I would eradicate him and those who made him the way he was.
My hatred is intact. And it is not hatred of David but of the history that has culminated in brutality thrusting itself out of a velvet-skinned boy, of the historical usurper who has taken a naturally beautiful voice and twisted it around ugliness and spite.
_Crack!_ goes the sky, so loud my house shakes, then it is doused in light. Two hours ago it was still and warm. There is another rending crack, the light goes off and the sky, a giant's sack of silver, rips dazzling apart.
Eli pads from his bedroom and sticks his head into the sitting room. 'Hear that?' he croaks, sleepily.
I ask, 'You all right? Nin all right?'
'She didn't wake up,' says Eli, 'but her hair's all drenched from sweat.'
_Crack!_ goes the sky again. 'God, it's like _The Wizard of Oz_ ,' says Eli. The light flickers off and then on. 'You all right, Mum? What's the matter?'
'They've wrecked my show,' I sob.
'What?'
'David and Barrie Bull had a fight. They didn't hurt each other, but they wrecked my show.'
'Oh, Mum.' Eli sits down and puts his arms around me. 'What happened?'
'I don't know really. David got drunk and started taunting Barrie Bull. It was just a brawl.'
'Did they wreck everything?'
'My ink washes, one of my acrylics. I didn't stay to count the rest.'
Rose comes out of the kitchen, with tea and a brandy for each of us. She sits on the floor, shaking her head.
'Jesus!' says Eli, leaping up, suddenly angry. He punches himself in the hand. 'Is he still at Mad Meg?'
'I don't know, darling. Don't make it worse.'
'I'll kill him.'
'So would your muzzer, given 'alf a chance,' says Rose.
'Let it be, boy. I'm too upset for more drama tonight.'
And we are saved from more drama by the phone. Eli answers. 'It's Gran, Mum, she wants to talk to you.'
'Jesus, it's midnight!'
'She just wants proof you're home.'
I clap the receiver into my ear. 'Hello,' I say, unencouragingly.
'Just checking.'
'God, Mum...'
'Well, darling, you never know. I just didn't like to think Eli might have been there on his own.'
'Good grief, Mum, he's sixteen! He can look after himself.'
'Well, darling, I can't help it. I am his grandmother, after all. He could've been hit by lightning or something.'
'What a grotesque imagination you have!'
'I can't help it. The light's been off here. I s'pose it's been off there, too, has it?'
'Well, of course it has! We're on the same circuit, you silly old faggot!'
'There's something making noises in my back garden.'
'Probably the plants.'
'Oh, you are hard. I think it's a stunned rat or something; it's kind of crying. My dogs might get it.'
'Well, if it's a rat you'd be better off if the dogs did get it. Sool them onto it and make sure.'
'It might be a cat. And even if it was a rat, rats can't help it. They didn't ask to be born rats. Anyway, they're quite intelligent; they can be quite tame.'
'Well, why don't you take some cheese out to the back door and whistle it up? I'm sorry, but I'm not about to court divine electrocution for the sake of a stricken rat!' I have to yell as the thunder is exploding and rolling all around us now.
'It's a funny thing it isn't raining yet,' my mother yells back, terrified.
'Well, don't go poking your head out into it, it might be struck off!'
'Oooh, oooh, think of all those poor young unemployed people out in it!'
'Oh, Mum, go back to bed! Go on, get yourself a brandy and go back to bed.'
'Oh, don't be a rat...'
'You just said yourself that rats are quite intelligent. The best of vermin are happily sitting in their holes right now, waiting for the cataclysm to pass. I'm sure they won't hold it against you if you do the same.'
'Oh, you are hard-hearted. There are lots of kids out in it, or don't you read the papers? They're being kicked out of their homes all over the place.'
'Well, I'm not kicking Eli out, and I've got Nin here, too. So you can rest in peace.'
'Why've you got Nin there?'
'Eli was minding her. We had an opening at Mad Meg.'
'Oh, did it go well?'
'Not very.'
'Whose was it?'
'Mine.'
'Oh. Nobody tells me anything.'
'Sorry. I'm just cross about something that happened. Two blokes got drunk and some of the work was damaged.'
'Oh. I'm sorry, darling.'
'Well, you wouldn't have liked the work, anyway. You'd probably have cheered.'
'Oh, don't say that.'
'Well, it's true. Now, just go to bed, all right? Just go to bed and go to sleep and forget about rats and cats and plants and other people's problems, okay?'
'Okay,' and the phone goes down and you can tell by the resigned clicking it makes that exhortations to common sense will have no effect.
'God, you're cruel to Grandma,' breathes Eli. He puts his hands on my shoulders and peers clownishly into my face. 'Couldn't you be a bit _nicer_? I mean she is old.'
'She's always been like that.'
'Well...' he puts his head on one side, 'if there was a cat out our back, wouldn't you...?'
'No!'
He laughs through his nose. 'Liar,' he says and points to the black cat Yorta who is knocking on the window to get in.
'That was different. I couldn't stand the look on her face. And anyway, Yafta needed a companion.' Yafta is our tabby, won four years ago by Eli in a Nature Study test; the teacher's cat had had kittens.
Suddenly it's raining and a terrific wind has begun to blow. We rush for the open windows and batten everything. Just as we are battening, there is a deafening crash. We turn in from our windows and gape at each other, tingling with fear. 'It must be the blue gum at the front of the house, the plumbers who fixed my pipes warned me it had shallow roots.'
And it is the gum, the electric wires swinging like a skipping rope above it, the current pulsing in concert through the lights inside the house.
'Did I hear someone calling?' cries Rose, above the din. Eli thinks he's heard it, too.
'Bel! Bel!' It's Allegra in the street in a maelstrom of hair. She can't get across the yard. We can just see her, pointing at the tree and shouting, 'David's under it!'
The lights are out now and Eli has our biggest torch. We follow him out into the roaring, raining garden. And indeed, David is lying on the lawn under some of the branches, but he doesn't seem hurt, he's reciting. ' _Natheless I had been a tree within the wood_ ,' he chants, hooting and laughing against the rain. ' _And many a new thing understood_ / _That was rank folly to my head before_.' Then he lies there, chuckling.
'Can you stand up?' asks Eli.
'I am standing up,' answers David. Inside the house, poor Nin pits her lungs against the storm.
Eli treated David with a mixture of high regard and scorn. David knew every sporting statistic there had ever been and this went down extremely well with Eli, who also watched David's dedication to his work with admiration.
But as Eli grew so did the ambiguity in his feelings. He needed men in his life and David was one of them. He would talk football and life with David in a way he could never talk to Allegra or me. This kind of conversation was not available for him with Reg, as Reg was more interested in hearing what he himself had to say, and as for Miles, he existed on a different plane.
Eli's need for men's company was not an absolute need for the dominance hierarchy. For all the rubbishings we'd taken over my failure to thwart his warlike and chauvinistic instincts, it was clear that inside his imaginings was a basic need to care, to lead and to protect. This was how he used his increasing size; his feats of manhood were written into him. It was clear that Eli was going to be a paternalist; he was bound to aggravate many women and bound to be told one day that the need to care, lead and protect is also part of a woman's psyche. Eli's character was quite forceful, verging sometimes on overbearing, but never violent. He was kind, and his kindness extended to David.
In turn, David had two types of feeling towards Eli. On the one hand he considered him lazy and not particularly intelligent; on the other, he admired Eli's marked ability at sport.
Be it said that, like David, Eli couldn't change a light bulb either. It was not that he did not know how to, but that something would inevitably interrupt him mid-change and he'd forget all about it. Strips were mown off the lawns around our new house, but there was always something better to do than mow.
We left David out in the rain that night, having helped Allegra in over the fallen tree with great difficulty in the darkness, the wind and the wet. She gashed her leg and while Rose was seeing to it I put a sleeping bag on the verandah for David, weighed down with a brick so it wouldn't blow away. But about an hour afterwards he was bashing on the doors and windows belligerently. Eli went and stood challengingly in the front doorway and David quietened down. When told he'd better not come inside, however, he manipulated Eli into going outside and then back to the Edwardian house with him.
There, after more great difficulty clambering over the fallen tree, he kept Eli up all night, showing him first of all his old school reports – 'He was drunk, Mum,' was Eli's explanation of that – and then, to demonstrate his industriousness, his latest work. When Eli came home at last, there was a war in him: care and respect for me on one side, admiration of David, mad and all as he was, on the other. 'You have to hand it to him,' Eli said, 'he works.'
'So do I,' I answered.
'Yes,' said Eli, 'and your work attracts more attention than his, even though you work full time as well. He'll suffer for this in more ways than you might think.'
'Why are you defending him? He ruined months of my work. That show was a departure for me, a new start. Don't defend him, Eli, he doesn't deserve defence. He's going to try to sweet-talk his way back into Allegra's life and never make amends for what he's done. He can't make amends. It's disgraceful, he's disgraceful, there's no honour in him at all.'
'I'm not defending him. What he did was despicable, but he will suffer.'
We left my show on the walls of Mad Meg so people could see the destruction for themselves. It was abhorred by the critics in the papers. A couple of reviewers who'd seen it before the fight were full of praise for it. For once, no one said I was Dadda's daughter.
As for David, reviews of his work, which heretofore had usually been excellent, ceased to appear. He sold nothing, had made enemies and no one wanted to handle him, even Miles for the time being. But if he suffered, Barrie Bull, being worth too much, did not. Though it was well known that he had provoked the fight, people still rolled up at Not-Only-But-Also to buy him, even people who'd personally expressed their outrage to me: sincerity has the depth of a ten-dollar bill held sideways in Melbourne.
It was a storm-stricken crow my mother had found floundering around her back verandah. Must've had an injured wing, she thought. It looked at her, she looked at it. It went _caw!_ she went _caw!_ She went inside, took a towel from the bathroom, terrified of it drowning in so much rain. She eyed it, it eyed her; it went _caw!_ again, she went _caw!_ Only a crow, she thought, didn't ask to be born a crow, though remembered crows on Clare picking the eyes out of newborn lambs, but a city crow, she thought, it wouldn't have. Her brothers would have shot it, that made her... She wasn't afraid anymore, just determined. Only pecked her once... see, and laughingly showed me the hole in her finger. Put it in a box and covered it in net till morning. Then rang. When they said, 'Where are you?' gave the address, but they asked for her membership number.
Member? Didn't know you had to be a member. Car number plate, then? Don't own a car. But, madam, this is the RACV, more laughing, wanted the RSPCA.
A girl came by eventually and took it away; such a nice girl, spoke well, thought she couldn't wear green, however, and had equine teeth... s'pose that's what comes. Rang up later, couldn't stop thinking about it, 'Hey, remember me? I'm the old bat who rescued the crow.' Rescued the crow to catch the bat, swallowed the bat to catch the rat; remember, poor old lady thought she'd die? They said, 'Sorry to tell you this, madam, but we had to put it down.' Poor old lady cried.
Rudge and Plant had given her a handshake when she turned sixty-five, but it wasn't a golden one, 'Not even brass,' she said. It was a marcasite watch, chosen by Marjorie Rudge of the fake fur coat. Stella had been awarded our family house in the divorce settlement, so she had a small amount of security. Charming as ever with bank managers, she worked to a system of mortgage raising and repayment which leant more to the raising side of things than the repaying.
During the Allegra–Stella war (which was beginning to make the thirty-years war look short, and was being fought in fits and starts around the exigencies of babysitting) there were unavoidable coincidences. For instance, queuing for a teller, one behind the other, in the bank where both of them had accounts resulted in Allegra closing hers for no reason that was apparent to the teller.
Our mother, short, rigid and supercilious in her daughter's presence, had reached him first and was forced to invoke her family tree when he wasn't willing to honour the cheque she wanted to cash without first clearing it. When our mother had been a teller in a bank she had cashed the cheques of persons known to her on the spot, without requiring an alibi. And she had been a teller in this very bank: head office, what's more. The calibre of bank tellers had definitely gone off since her day. Could she see the manager?
But the manager had already seen her and called, 'Cash it! Cash it!' in dulcet tones from his cubicle. 'Anything for Mrs Coretti!' Which meant either he had a special soft spot for our mother, as a great many people had, or he was acquainted with her time-tested technique.
During this performance, the queue, one person long, went from sighing loudly and folding its arms across its chest to drilling two holes in the back of Stella's head with gimlet eyes.
Performance over, the ex-teller gave a perfect demonstration of the Alexander Technique, looking neither to her right nor to her left – where her other misbegotten daughter was innocently bystanding – and making a balletic exit.
The first misbegotten daughter was then heard by the second to be addressing the teller rather more icily than he deserved. 'Well, then,' she was heard to say in a voice both solemn and composed, 'I shall just have to move my account elsewhere, shan't I?' I thought the use of 'shan't' a trifle unproletarian, but once on the street, Allegra began to brandish the thirty-two dollars and sixty cents' worth of Mad Meg kitty, narrowing her eyes and grimacing. 'Fucking men!' she said. 'They always think they know best!' She'd wanted to take forty dollars out of the account.
I ask her, 'Is thirty dollars the entire Mad Meg kitty?' and immediately rage turns to despair. 'What are we going to do?' she howls, and well she might howl, because in the wake of David destroying my show, the Troika want to be bought out. 'We'll have to sell,' she sobs. 'Oh, I'm sorry Bel, I'm so sorry.' And we find ourselves sitting in the gutter outside the Pantechnicon, the Mad Meg kitty in Allegra's lap, weighed down by the coins, and our arms around each other, howling our heads off. Then Maggie and Kelly are sitting each side of us. 'That's not all,' Maggie croaks, and addresses our bloodshot eyes to the awning over our heads. A truck has driven into it and buckled it so badly that the owners of the building have had the caveat on demolition removed. Maggie, Kelly and Chantal are going to have to find other premises or go out of business.
We repair to the aromatic Pantechnicon interior. Maggie thinks they might take a stall at the Victoria Market on Sundays to start with, but the problem remains of where they will live, as they are now living illegally above the shop. It's gloom all round. Allegra has decided to get divorced, but David is still living with her and, though Miles has taken his clothes, his paints remain. In spite of exhortations from Miles and threats not to allow him access to his clothes, he is going to be awfully hard to dislodge. He has said he'll fight Allegra tooth and nail to get custody of Nin, and although it's most unlikely he'll ever be granted custody of anything, he says he's suing for his share of Mad Meg, in which he has no shares but could succeed in taking half Allegra's. We're bankrupt, the Kellys are evicted, and though the details of Dadda's will haven't yet been finalised, The Brolga has succeeded in getting the Siècle Trust up and running, the first artist-in-residence being ensconced in Harry's beach house where Dadda's ashes lie, unvisited, under the grevilleas. We sit around with our heads hung, listening to the distant din of the cars now whizzing off and onto the freeway, as if there had never been a protest at all.
PART FOUR
THE PROMISED LAND
TWENTY-FOUR
Translating
ITALIAN BARBERS WERE, and still are, part of Australian life; they once out-numbered hairdressers and advertisements datin back fifty years can still be seen in their windows, offering Gent's Hair Cut, Australian or Continental, 5/-, Ladies' Style Cut, 7/6. It was in such a shop that the Antifascist Concentration of Melbourne had its headquarters.
The barber, Enzo Caruso, was once a member of the Matteotti Club. Well before I was born, Uncle Nicola's friend Spencer Lonsdale, authority residing in his big black beard, would sit in the barber's chair on Wednesday nights and preside over the excitable and volatile left-wing Italians.
Uncle Nicola would try to keep the agenda up to date by reporting on the activities of the Antifascist Concentration in Paris, but apart from liking the sound of Uncle Nicola's voice, a sound behind which they felt it justified to rally, the Matteotti Club members were more concerned with the dominance of one man's red-shirted biceps over another's. Life was an opera: you rallied, sang lustily, strong lips giving shape to the patriotic reverberations in your breast. You skited, maybe you slept with a pistol under your pillow or dealt in scrap metal about whose origins you knew better than to ask. Perhaps you had connections with the Mafia – after all, the Mafia were antifascist too.
Poor Uncle Nicola. Crane his well-educated head into their midst as he would, he could not get much more across than that Enzo Caruso trimmed his beard to look like Lenin's. Yet he strove to tell his audience what he thought about their position in Australia. Australia, he would say, was the living demonstration of Marxist class conflict. If ever an antifascist voice was needed it was here, where the clash between government and workers was head on. Australia was a country of possessors and the dispossessed. _'I possessori e i spossessori!'_ But though he rallied and rounded up his listeners with passionate rhetoric, Wednesday night after Wednesday night the Antifascist Concentration of Melbourne would dissolve, not so much inspired by his ideas as having learnt how to imitate his delivery of them. They'd make tulips of their hands and declare they'd have, with eyeballs peeled, _'onore e soddisfazione morale'_ , as if they were off to fight duels. _'Ardore!'_ they'd exclaim with Uncle Nicola's timbre and emphasis, _'entusiasmo!'_
Uncle Nicola tried to involve influential figures from the Australian Left in the activities of his group, but these people were quite often above barber-shop politics, their racial chauvinism extending to Italians. To them, Italy was an overpopulated place wanting to get rid of its surplus of unemployed peasants, a place where labour had little value, and the labourer, accordingly, went cap in hand to his prospective employer, unable to negotiate. This attitude must have brought Uncle Nicola low, as his major contribution to Italian leftist politics had been the organisation of peasant leagues. However, instead of seeing the Italians as a pool of workers waiting to be organised and converted, the Australian labour movement preferred to leave well alone and to regard the Italians as intractable peasantry. For their part, all but a few of the Italians were politically apathetic, having come to Australia to rid themselves of politics.
It was something of a victory then, when the Australian Antifascist Concentration, less than thirty strong, helped rustle up several hundred supporters, enough to fill a large theatre, in order to commemorate Matteotti on the fourth anniversary of his death. Uncle Nicola was a star attraction in his native tongue.
Several Australians attended the memorial: communists, socialists, anarchists and Labor men, and messages of support came from Trades Hall. The occasion was widely reported in both the English language press and the small but active left-wing Italian press. Only one Australian, a retired Labor MP, spoke out against the rally, saying Australians should refrain from showing solidarity with a group of people about whom they did not have enough information.
While the Australian public might have known little about the Italians in their midst, their government was constantly informed – over-informed on occasions. There was certainly plenty of adverse information on Uncle Nicola available to people in power and as a result he was constantly denied citizenship, while rabid and active fascists were granted it.
He sent a copy of the speech he made at the Matteotti rally to the socialist directorate in Paris and sustaining words of encouragement came back to him. He was now receiving so much propaganda from Europe, he had to take a post-office box. He would redirect what came his way to places where it might have had some effect. He lobbied politicians and prominent members of the Italian community, whether they were fascists or not. Night after night he spent as many hours as he could at his creaky desk, roasted in summer, frozen in winter (in spite of long johns, knee pads and flannel tied round his chest), making sure the appropriate information reached the right people. Not even the Depression, which hit when Australia was in a very shaky state and was to have a drastic effect on numbers in the Antifascist Concentration of Melbourne, would prevent Uncle Nicola from pursuing his single-minded course.
In June 1929, Uncle Nicola wrote: 'Come to the aid of my understanding, Lord. There have been coal miners and timber workers on strike all year all over the land, and the Prime Minister says it's time the Commonwealth Government pulled out of industrial arbitration. He announced his plan, then took off immediately to play golf at the seaside while his office was being redecorated.'
In October: 'Well, he got what he deserved. A man from his own party brought him down over a technicality on the floor of the House. The gentleman golfer was then forced to an election in which his party was soundly thrashed, with the former secretary of Trades Hall, who had been fined fifty pounds for inciting the timber workers' strike, taking his seat.'
Uncle Nicola had been filled with hope when the government changed hands, and wrote to Paris saying he'd probably have his citizenship soon. But he was disappointed once again. His application was deferred, the structure of the Australian Labor Party being such that they actually had a bias in favour of fascists. Uncle Nicola was at pains to make this clear to them, but the party apparatus was cumbersome, the wheels turned slowly, and the fascists had some fellow travellers among the turners.
The Depression threw most of the Italian left-wingers out of work, so Uncle Nicola lost practically all his followers in a general migration north to the Queensland canefields. Times were really tough for him. To save him from complete penury, the university found him some money for a research project and in the next few years he was to make copious annotations to _The Inferno_ , showing a disdain for many of the inferences made by Dante scholars before him. For a socialist he was a fine theologian, and, like Uncle Garth, knew the Bible inside out.
He fretted for Allegra and Dadda, for whom he felt full responsibility now Emilio was dead. In Paris the Katzes, like so many other people, were finding the going hard as there was no real work for architects in such straitened times. Rose's recent marriage to Laurent Hirsch was a blessing, however, since Laurent came from a wealthy family and also had money of his own.
His family hadn't been able to interest Laurent in the family brokerage firm and were trying to interest him in medicine. There was a relative in the medical profession living in Melbourne, Australia. Having washed their hands of Laurent, they hoped he might be influenced into a lucrative career on the other side of the world. Although it was so very far away, the Katzes, less capitalistically inclined than the Hirschs, were excited that Rose was going to Melbourne to live, and Uncle Nicola was warned in advance that friends were on the way.
It was December 1931. Uncle Nicola, awaiting Rose and Laurent at Port Melbourne Dock in striped trousers and a morning coat, was easily recognised among the good British stock awaiting reinforcement in their far-flung clime. The once-applauded Labor government, bedevilled by the Depression and the refusal of the state government of New South Wales to meet its overseas loans repayments, was on the brink of being trounced in its turn by a reconstituted Opposition. Though Rose didn't know any of this, her first impressions of her new homeland were those of doom.
As they drove through Melbourne by taxi to Uncle Nicola's lodgings, Rose was amazed by the absence of tall buildings and couldn't help thinking Melbourne had suffered some catastrophe they hadn't yet heard about, such as an earthquake. It was the late afternoon of a mild day, the sky suffused with pink, gold and grey, through which gulls were wheeling clamorously, one among them had an eye missing, and another, a foot, which added to the feeling of devastation yet to reveal itself.
Instead of devastation, however, Rose found horticulture, agriculture, slowness. 'Plants, everywhere, plants. And everywhere birds. Not just seagulls, but birds I had never dreamed existed. I thought maybe they had been made up, invented, dreams on wings.'
Uncle Nicola's room in South Melbourne served him in every capacity. He cooked, washed, slept, read, wrote, declaimed and entertained there. He'd latterly put a rug on the floor to muffle the sound of the nightly promenade he took for the sake of his punctuation, although the other tenants still insisted he only promenaded to annoy them. In the area that served as a makeshift kitchen he'd installed two wine barrels for the supply of bulk Rutherglen, without which, he assured Rose, his digestion would have been seriously impaired.
For the evening meal, he had invited guests. Spencer Lonsdale and his Australian wife Grace, the Pro-Consul for Paraguay, Señor Gil-Serrano, who taught Spanish at the university and was as penurious as Uncle Nicola, and lastly a shell-shocked French-speaking mathematician from the university called Cedric Barnes, who avoided treading on cracks. Uncle Nicola's floor, which was largely bare boards in the places to which his rug did not extend, demanded intricate negotiation of him.
Gil-Serrano, a Spaniard from Madrid, had never been to Paraguay, but had become the Pro-Consul because he was one of the few people in Melbourne who spoke Spanish. The small stipend he received for his consular activities was an essential adjunct to the money he earned from his Spanish classes. The mathematician, Cedric Barnes, was terrified of Germans, who were all spies as far as he was concerned, and he spoke continually about them, warning Rose and Laurent to avoid them wherever possible. He and Laurent had in common their war service around Pozières. Laurent was distinguished even further by having been among the six thousand 'saviours of Paris', ferried to the front in taxi cabs from Paris for the Battle of the Marne.
Uncle Nicola had gone to great pains. Along with his menu went a performance in which the entire making of the meal was mimed, capped off at the dessert by an illustration of fascists avoiding him on Melbourne streets. He was excited and delighted to have company, and even more excited and delighted when he discovered that Laurent was just as enthusiastic a cook as he, and able to perform the menu fluently in French for the benefit of those whose Italian was shaky. The truffle was an item they discussed, lamented and found no substitute for.
Grace Lonsdale, not typical of Australian womanhood, was a naturalist and a painter. Her French was fair. A wavering socialist to start with, she had become devout and well informed since her husband had made a close friend of Uncle Nicola. An intense woman, who dressed severely and had worn-down, smoke-stained teeth in a round face, she was very interested to know what was going on among the Italians in Paris. She smoked furiously while Rose told her.
The influence of the grand old men of the Left was vanishing in Paris. Allegra, who had spent a number of months working for Filippo Turati, missed working for him now he was incapacitated by age and illness. Allegra had become very homesick, while Dadda, almost fifteen, having been in Paris nearly four years, was well on the way to becoming a Parisian. With his spring-jointed stride he was marching around Paris, quoting over and over the motto of the new Italian Left, now under the leadership of Carlo Rosselli: _'Non mollare mai! Neppure un momento, neppure un pollice.'_ (Don't ever weaken! Not for a moment, not for an inch.)
For his first three years in Paris, Dadda had had no regular schooling, but he had been coached on Sundays by an Italian woman, a Signora Regina Pecora. Signora Pecora, in her sixties, didn't run to sophisticated instruction. She tutored several children for the money on the strength of a teaching qualification she had not used for many years. Dadda's maths was never marvellous and, though his knowledge of science used to impress me, he must have picked it up from his own interested reading, for he certainly didn't learn it in Paris. What he did learn was how to draw, and not from Signora Pecora, but from Lev Katz, who taught him about perspective in architecture and showed him drafts he'd done for buildings.
When it became clear that Dadda was talented, Lev arranged to have him enrolled at a technical school. He was chosen out of his class there to attend an advanced session in the evenings, in a loft above the school where he met with professional painters and discovered that art was what he burned for.
Uncle Nicola was delighted to hear this news of Dadda, though he fretted inwardly as it seemed to mean that for Dadda to have a profession he would have to stay in Paris. Laurent didn't agree. 'An artist is an artist wherever he is,' he said. 'My family wanted me to be a broker, and they think they've sent me out here so I'll get interested in the medical profession. They don't know I've come of my own accord. I want to be a chef. And I will be. Rose wants to be an artist, and she will be. Henri will paint no matter where he is. He might as well come and paint here.'
Uncle Nicola took heart. It was something to be a pioneer. There seemed a lot to hope for in this country, after all. He put by his fears that it would not survive and began to think of its survival – his annotations to _The Inferno_ now seemed weighty with promise. His gift to Australia would be the intellectual heritage of Italy. In his mind's eye, he saw Dadda, a cultured and gifted man, a modern man with respect for the genius of men like Dante, Goethe, Voltaire and Shakespeare, but a man with ideas for a new and better world.
The evening was a great success. Laurent and Rose befriended Gil-Serrano, who had been stranded in Australia by the First World War and led a bizarre and lonely life, being infrequently called upon to translate or to act on behalf of Paraguay. He'd seen all manner of things, from Spanish speakers trapped in a derailed train to a boatload of grandees moored in Westernport, unable to come ashore because Customs wouldn't clear them. Cedric Barnes, the mathematician, was still living out the war, on the lookout all the time for trip-wires and dead men, and he did not cultivate friends. He used all his energy just keeping his maths safe in a head full of horror. Rose and Grace Lonsdale liked each other, though they were very different women. For Rose, plants were the things that grew behind railings. They adorned the way but didn't intrude; she was fascinated to learn from Grace that they had a sex life. But Rose wasn't much attached to the world of things and beings. She preferred to think of life as a mysterious sojourn between states.
Rose and Laurent spent their first night in Australia in a nearby hotel room Uncle Nicola had reserved for them. Their first accommodation was to be a rented flat in South Melbourne where they stayed a month or so before deciding to buy a boarding house in St Kilda.
Laurent, in due course, called on his relatives in Melbourne. After all, at a time when it was hard to get into Australia, they had agreed to underwrite his stay for five years, and he was grateful, but he knew in his heart of hearts that Rose would be exasperated by them and they would not approve of the course of action he and Rose had taken.
True to predicted form, Uncle Julius and Aunt Orpah were offended that they hadn't been asked to meet Laurent at the ship. Laurent tried to explain about the Corettis in Paris and how important it was that Rose should make contact with Uncle Nicola as soon as she was able. Priding themselves on their rise to solid respectability in Australian society – after all, Australia even had a Jewish Governor-General – the relatives weren't too moved by the story of refugee Italians and were even less impressed that Laurent had been in Melbourne some time before looking them up. Laurent, ambushed by tea, cakes and Toorak, knew immediately and precisely where he stood in their regard. There could be no question that there was anything wrong with Laurent; it had to be those who surrounded him who were at fault. The task of Laurent's uncle, his mother's brother, Dr Julius Lichtblau, would be to see him safely into the medical faculty, and of Laurent's Aunt Orpah (Australian born and bred) either to bring Rose into line or to get her nephew out of an ill-considered marriage.
Laurent did not tell them he'd just bought a boarding house in which he and Rose intended not only to let rooms, but also to create a 'salon' in the basement. He did, however, leave his address with his aunt, who had then presumed her plump frame up the crumbling steps of the boarding house, encountered Rose in the front room sewing 'some ghastly article', declared the house 'completely unsuitable' and telephoned an estate agent she knew, making an appointment for Rose and Laurent to go and see him right away and 'get yourselves out of this mess'.
That was not the last they saw of Aunt Orpah, for though she considered Laurent and Rose to be in a mess, they made such an exotic pair that she was intrigued in spite of herself. Invitations to Passover were to come every year thereafter and were generally accepted, because Laurent had a deep family feeling and he knew Aunt Orpah would thrive on the legend of his inimitable Rose.
Each year Rose grumbled at having to go to Passover, and each year Aunt Orpah used the occasion to ask when the children were going to appear, so the questions could be asked of Julius, the Hebrew could be taught and read, the traditions passed on. Aunt Orpah's grandchildren were all too old to ask the questions now; it was up to Rose. But though Rose often tried, she did not fall pregnant. It was something she would regret at a later time, almost as much as she would regret the horrendous loss of her family.
When Rose arrived in the bosom of the Australian middle class, people feared her. Her speech was unexpurgated, her clothes were highly eccentric and she didn't have a proper bathroom in her house. Those who admired her did so in secret until she might have proved herself one way or the other. The more daring among the women she met came knocking timidly at her door with jars of home-made jam, or some peculiarly Australian knick-knack such as a swizzle stick for cocktails she once showed me, which folded away in the shape of a golf club.
In Australia, women thought they were making perfectly good mayonnaise with a one-to-three mixture of vinegar and condensed milk. Drop by drop, Rose hand-whisked olive oil into egg yolk. People who ate with her weren't certain whether to take up the practice themselves or to condemn it as 'something with oil in it'.
On Rose's breath the visitor was apt to find garlic. The Aunt Ninas of the time probably would have done her the courtesy of not mentioning it, but the pillars of wisdom who ran the ceaseless campaign to correct Australia's mores might have told her discreetly that a good Australian breath smelt of toothpaste.
Rose, now in her eighties, is still feared in some circles in Australia, but, having visited Paris, I now know she would have been feared there, too. For while the sturdy Australian toilet, immaculately clean, bears testimony to generations of Australian women who have seen their role as continually having to guard the home against, or rid it of, invaders with pestilential propensities, the condition of Parisian toilets reveals a nation of women quite above cleaning them, and that is not to say the men do it. There is now a two-franc public loo on the streets of Paris which completely cleans itself. Charming city though it is, Paris, like Melbourne, is thoroughly bourgeois.
In that charming city, in the winter of 1931 when Dadda was fifteen, the great artistic rebellion against the bourgeoisie was over. Former rebels such as Picasso were now mainstream artists. Robert Delaunay had turned from his colour studies of window forms back to the formality of circular forms in which he explored 'colour, the fruit of light'. Dadda's countryman De Chirico had made his mark as a metaphysical painter and was doing sets for the Ballet Russe. There was also Derain, steadfastly treading his own path.
However, while Dadda's career was shaped through close acquaintance with the work of Picasso, Delaunay and Derain, and even more acquaintance with De Chirico, whose picture structure, imagery and technique made a deep impression on him, the person who seeped into his soul as no other was Modigliani. Not only did he know Modi's story well, having grown up with it, but now, in Paris, he was able to see the work.
He followed up the glowing women, the friends whose portraits Modi had painted, as he became drunker and drunker, exploring the lineaments of their being. Modi would sing to the women in Italian to soothe them when they became frightened of him, so excited by the growth of his portraits that on one occasion, as he worked feverishly on the bottom of a canvas, the top took leave of the easel and fell on his head.
The Paris in which Rose had spent her childhood was the Paris from which Modigliani wrote to his mother in Livorno saying he would repay the money the family had been sending him just as soon as his fortunes turned the corner, and this should be any day. Though they boasted a deputy in parliament, the Modiglianis were not well off and it must have broken Modi's heart when the only one-man show of his life was ordered from the walls of a Paris gallery because the nude in its window was judged offensive to the public view.
Modi loved his family and they loved him, but not without reservations respecting his talent. Indeed, after his death they did not even salvage one painting, though they could have rescued several from the clutches of avaricious dealers.
How much of Modigliani's antibourgeois attitude was genuine and how much an affectation dictated by the bohemians of Paris is open to question, but he certainly painted without regard for 'taste'. His generation wanted to fire the imagination of their audience, to provoke rebellion against middle-class conservatism and conceit. Now, of course, you can buy a Modigliani print in any suburban print and paper shop and it wouldn't be out of place in many middle-class homes. We mass-produce Modigliani and sell Jeanne Hébuterne, his tragic lover, as part of our 'tasteful' range of stationery.
If Modigliani were to materialise as a young painter today, I feel certain feminists would despise him. People with alcoholism, schizophrenia or depression in their families would despair of him as incapable of living the way most of us live. Most of us live, however, by doing trivial work. Our empty weekday houses, stuffed to the gunwales with modern accoutrements, are symbols not so much of achievement as of wasted time. What the socialists despaired of for the proletariat has become the lot of the bourgeoisie. This was Dadda's theme; it was through Modigliani and antibourgeois Paris that he came to it.
TWENTY-FIVE
Modernity and Obsolescence
THE BOURGEOIS MENTALITY runs deep in Australia. Ironically, since feminism and Marxism are more often associated than not, it was probably the bourgeois mentality more than anything else which drove young women to react against the revolution in social mores that had its beginning in the sixties. Having themselves acknowledged as human beings rather than pleasure toys and slaves led those who, a few years before, would have been destined for bourgeois housewifery and pedantic toilet-cleaning into the public arena. The former housewife now had an education. If children were to be had from her, they would come at the cost of some effort from men, and it would be effort expended on the very tasks decried by Marxists in the past. If men wanted to remain married, they had now to address the practical problems of running and keeping a house and family and, in Australia, the land where people own their own homes, these were strongly bourgeois concerns.
By 1983 it was no longer good enough for David Silver to insist on his entitlement to an uninterrupted life as a painter when there was a child to bring up and a wife with a career. Though few offered their sympathy, everyone understood Allegra's position, while David's had become completely reprehensible.
His violence went much deeper than his public outburst against me. Allegra, too proud even to have turned to me for help, eventually admitted to having taken regular beatings from David during their relationship. As is characteristic of bashers, he would hit her where the bruises wouldn't be obvious – the back of the head, over the ears, the sides of her face that could be hidden by her hairline – that way she, like other victims of this kind of violence, could spare herself the agony of making any admissions to her family or friends. Afterwards, remorse knew no bounds.
Surely there is no one more generous and flattering than a basher trying to crawl back into his hole. Allegra was plied with oil paintings, cake, long-stemmed roses and cheques, but she was past giving in now. David might be the father of her daughter, but he was a wrecker, and there was no excuse for his behaviour.
Primitively, deep down, he had destroyed my show because he hadn't wanted me, a woman, to succeed. Feeling his efforts at justifying himself were falling short, he then tried to condemn my work as bourgeois. He said it was 'tasteful' and tastefulness was anathema in real art. To him, leaving the show on the walls was spelling out contempt for the growing tenor of the times; whereas for us it was the exposure of ingrained misogyny, one of the factors contributing to the absence of women practitioners in the history of art.
The only correctness in David's point of view was that the world had moved back into money, and the slickness and suavity that went with it. My work was never slick and suave, however, and it was farcical to suggest it was.
At this time there was widespread graffiti in Melbourne, presumably the work of young feminists, saying, IF HE BEATS YOU, LEAVE. This presupposed there was sympathy for the victims of violence in the outside world, somewhere to go and something to go with, and that leaving would not aggravate already serious problems. The feminist movement consisted largely of bourgeois women, like The Troika, who knew little about poverty, alienation and fear, and unless these things were foisted upon them, did not want to know. To know would have been to expose their own vulnerability, as the world in which they lived was owned and controlled by men. They had been brought up to avoid knowing, and the lives of many of them were lived on bluestone foundations bequeathed to them by mothers who kept poverty, alienation and fear at bay by occupying the tops of the low peaks of Australian society and creating space in which their daughters could at least be independent for a while. They probably did not realise that leaving can be the most dangerous part of a violent relationship. It is the time when desperate men go mad, and it can mean distress and material loss for a woman to the extent that personal gain is ever after under threat.
David would launch himself like a kamikaze bomber at my house, where Allegra had come to stay, and where, to Eli's chagrin, we had had the locks changed. Eli thought we were going too far, but David, bearing wine and roses, would burst into restaurants where we were eating or having coffee, or he would follow us along streets, ducking into alcoves, hiding behind bushes, until, overwhelmed by his need, he would rush us, brandishing bank cheques for large sums of money. Whenever Allegra went out without me, she was mobbed, cajoled, entreated and begged. The worst moments were those when he'd throw himself on the bonnet of my car – Allegra no longer had a car – in heavy traffic.
Eli, who thought we were being hard and suffered to see David suffer, took ear-bashings from him that went on for hours: how Allegra was a fool to toss away everything they'd built together, and how it would spell the end of her friendship with Bart and Miles. Then he'd have a belated rush of love for Nin and then express his remorse for having hit Allegra once or twice when he ought not to have hit his child's mother.
When Allegra heard this, she was nearly driven insane. 'Once or twice!' she screamed. 'Once or twice! It was every fucking Friday! Beat her up on Friday, she can go back to work on Monday! Why are you feeling sorry for him, Eli? And why is the only reason he shouldn't have hit me be that I'm the mother of his child?'
'He's two people,' Eli answered.
'Yes, and both of them treat women abominably. One of them grovels while the other slugs.'
'You ought to have told us, Allegra,' I said. 'You could have come and stayed with us before now.'
'You knew,' she said, and I immediately felt smug, guilty and self-righteous, because I had known that the fights could be fierce and physical, but after them Allegra would seem to be bristling with a hostility that said _Just leave me alone! Don't even ask._ So I wouldn't ask, and the one time I tried to comfort her, she pushed me away. It had seemed that my role was to take care of Nin.
I'd never had to ask Allegra to understand my problems; she'd protected me as a matter of course. She had said nothing because she was incredulous that David would hit her and certain every time that it couldn't possibly happen again. She tried to describe it, to decipher whatever it was that triggered him, but could only arrive at self-loathing. 'I let him beat me in my home,' she sobbed. 'I don't know why he does it. I don't understand.'
I could feel her pain in me. I felt degraded by him, but how much worse it was for Allegra. I thought that surely, confronted with it, Eli would feel this pain and never make that sort of mistake himself. But something else was happening to Eli.
On the threshold of manhood, he was very susceptible to David. He liked the feeling of power the situation conferred on him and, instead of seating himself on our side, he occupied the ground he thought did him credit, the middle. When he came asking Allegra if David could see Nin, she exploded.
'Can't you see past your nose, Eli?'
'He loves Nin.'
'No! No! He does not love Nin. He disowned her. He found that blasted little cat more interesting. The fucking cat ran away, Eli, and you know what he has the gall to say to me? I killed it! I drove it away! Or I ran over it in secret and put it in the garbage or some such. He doesn't know what love is. The only two people he cares about in the whole world are Bart and Miles. No, he can't see Nin. I won't let him lay a finger on Nin. Oh God...' She began to sob deeply, the tears making her lips glisten in the cave of her hair. 'Mothers are supposed to keep the wolf from the door, not allow the wolf into their wombs! I'm hopeless, I'm hopeless.'
Then Eli tried to understand. Back he went to David with the reasons he couldn't see Nin. That one resulted in a siege, and Eli, trying to keep the peace, became David's envoy.
'Can't you see he's manipulating you?' Allegra would say, but Eli couldn't see it. As if David were a wounded soldier out on the slopes at Gallipoli, Eli would not abandon him. He sat with David through his self-recriminations, his spite, his self-pity, and though David brought himself very low in Eli's presence, Eli didn't stop respecting him. Rather, he admired him for bringing himself low.
'Oh, Eli!' Allegra despaired. 'You think you've got him in the palm of your hand, and you like to feel bigger, older and wiser than he is. Your mother's there watching, and you wish it was your father!'
Eli didn't deny it. 'David's spelling himself out,' he said. 'He needs someone with him when he does that. I'm the only person available.'
'You mean you're the only person who'll listen. You encourage him by listening.'
'He needs encouragement. And, yes, I am bigger than he is.'
'But, for God's sake, he used to call you an oaf!'
'Well, he was wrong, wasn't he? He probably won't change much by talking it out, but he may become more conscious that what he does affects people. David is a small man with a soft voice. He doesn't realise that what he does affects everybody round him; he's suffering from the delusion that he's weak. He's never felt separated off from women, initiated into manhood, if you like.'
'Well, thanks very much, Eli! That's just what we wanted to hear. He hates us, but he doesn't hate us in the way he should!'
'No, that's not it,' said Eli. 'If a man isn't accepted by other men, it's just as hard on him as it is on a woman not accepted by other women.'
With David, History played the puppeteer. It jerked him mercilessly round the central axis of his being without allowing him to assume a natural identity. Bart had been right about him; he was more the child than Nin. He cried on every doorstep, believing he would never survive without Allegra. His despair was terrible, and genuine, too. We had to act as though it were not.
He even went to our mother to ask if she could put him up, or 'talk sense' to Allegra, but though she fed him a hot meal, she would not take him in.
For a while, David slept in Miles's flat; but that was only a temporary arrangement, since Anita was expecting a child and would need the room. Miles said he didn't sleep, anyway, but paced about all night smoking, and when Anita didn't think she could stand it any longer, he had had to ask David to find somewhere else.
There are in Melbourne hotel-hostel type establishments, as I dare say there are in all big cities, where drifters and addicts and forsaken people go when they are not destitute enough to qualify for doss houses nor wealthy enough to rent a bed-sitting room. Such people are usually men in middle age whose families have broken up. They are defeated domestically and socially, so they go to these places where they might be offered a meal a day or provided with an electric jug and a hot plate in the same room they sleep in. It was Miles who told me that David had gone to live in one of these. He did not do it as some sort of trick designed to fill us with pity; he did it because, though he had enough money to rent a house had he wanted one, he knew no better than this.
We did pity him, and for half an hour even considered having him back. In the long run, however, we did not need Miles to tell us that we would only revive a hideous moral situation by having him. David did not need us; he needed the world in which to face himself as an adult. He started to talk to Miles about buying a house with enough room in it for Allegra, Nin and him. Miles would tell him to stop pretending, but David was incapable of that.
Our biggest agony came when we had to begin the process of selling Mad Meg. We were informed we had a caveat on our title. It was in David's name, but as he would never have come up himself with an idea like that, it had to be that he'd befriended someone who would. He was arming himself with every available weapon; this just about had to be a retaliatory act by The Brolga, on whom we'd served caveats. When David began to say he'd have it lifted as soon as we agreed to a final settlement of Dadda's will, we knew it was The Brolga. Then he came knocking on my door with a new, more generous offer for us; we were not to ask him how the offer materialised, but we would, of course, realise that David was entitled to half Allegra's share and if we wanted to get out of our financial difficulties, we'd accept.
'Strange how the will keeps altering and probate keeps being deferred though Dadda's been dead six years,' Allegra commented. We had had the caveat lifted, our advice being that David couldn't block the sale although he could sue for half Allegra's share of the proceeds. That, however, would barely yield him the cost of litigation and it did not cross his mind that if he could sue Allegra, she could sue him. We didn't know how much money David had but, according to Miles, it was probably quite a deal more than Allegra had, because Therese Turner had given him the proceeds from the sale of the Turner family house in Sydney as a wedding present. It was something Allegra hadn't known and was cause for more humiliation. David could have saved Mad Meg with less than a sixth of Therese's gift.
My ruined show had to come off the walls at last. It had become the topic of much art criticism, had been photographed many times and visited as an Incidental Installation. Troupes of young girls, dressed in several oversized sweaters with gaping necks, came chucking their tangled tresses over an eye as if their heads were clogged salt shakers and saying, 'God! Bor-ing!' 'Boring' was enjoying a brief vogue as an expression of disgust.
It was like seeing ourselves again in our uniform of youth. They painted their lips, they smoked, they chewed their nails. Their world was aggressive, competitive and heartless; the ones without contacts would blossom into unemployment, the ones with, into chronically junior positions. It was not surprising they dressed like tramps, deliberately making holes in their clothes and sticking safety pins through the lobes of their ears. 'It makes me feel greedy,' said Allegra, 'when I thought by this time, I'd be feeling proud of something.'
In the eighties climate of extravagance for some and defeat for others, when obnoxious references to the 'downwardly mobile, genteel poor' were being made on talkback radio, Viva Hallett-Laurington-Coretti, The Brolga, was having the time of her life. No longer did her thin hair bristle across her scalp when one of her stepdaughters was nearby. She simply ignored us on those few occasions fate threw us together.
Repeated bouts of ill health were keeping Harry Laurington from work. It was thought he had some sort of slow virus, multiple sclerosis or muscular distrophy, and he was often confined to bed. Eventually, he signed Siècle over to Checkie. Mother and daughter now joined forces to become the _femmes formidables_ of art dealing in Melbourne.
When the government semi-privatised alcoholism in a bid to tighten up per capita grants to charitable institutions, the ex-Methodists, who had long ago amalgamated with the Presbyterians and the Congregationalists into the Unitings, decided to put the Harcourt-Wilson and its neighbouring church on the market. The price of real estate was rocketing up.
Our caveat on Not-Only-Hallett-But-Also-Coretti having been withdrawn because we could not afford to bring an action against her, The Brolga made a killing by selling it at the peak of a crazy market. A couple of weeks previously she had stood to one side of the vulgar horde at the auction of the Harcourt-Wilson, waited until the other bidders were all done and put in a bid no one could top. She did it on an asset base that someone somewhere must have considered more than ample.
Evidence that Christians once used to worship in the corner church was to remain only in the form of their concrete floor, upon which the purchaser had painted neat, even chic, black strips, between which the art-buying public could now happily park their Mercedes and execute lyrical turns in keeping with the delight engendered by the conclusion of a deal at Siècle.
Not-so-Mercedes still nosed their way up the lane to Figments, though when The Brolga opened a neo-Mussolinian coffee lounge on the ground floor of what had been the Harcourt-Wilson and was now renamed the Siècle Trust, the local council, thoroughly Brolga-ised by her attendance at their meetings during which she would toss in suggestions from the public gallery, had produced a plan for a pedestrian concourse. It was to be paved with pale gold bricks and graced here and there by a dainty Cootamundra wattle of the rare dark variety which appeared at the rate of about one per thousand germinations, and consequently was about a thousand times as expensive as its dust-grey siblings. Miles Turner was furious. Not only did it mean no car access to Figments, but it meant accepting second-fiddle status to Viva Laurington (the Turners never got around to calling her Coretti) and a rate increase.
Miles lodged objections but the issue fell short of splitting the community. The plans, drawn up by The Brolga's suggestion of the ideal architectural draughtsperson, yet another young man called Gosper who was by happy coincidence related to The Brolga's own assistant, Jerry, were put up in the bowels of the town hall for any interested rate payer to come and see. Memories of the days when our mayor had himself locked into the freeway building site with the protesters were stuffed away in files in old grey Brownbuilt filing cabinets for which no further use could be found, the offices being recarpeted and the affairs of the city now being stored on computer. The Brownbuilts were stored in the obsolete furniture repository on the town hall roof with the vague idea that they might be auctioned off to the public at a later date. In the meantime, several new artworks had come to grace the mayoral chambers.
A debate began in the letters page of the local paper. 'Anonymous Person, Name Supplied', wrote to complain of projected 'improvements' to the local environs, which served no purpose other than pushing those who could not afford the subsequent rate increases out of their homes and their businesses. 'Visionary, Name Not Very Difficult to Guess', replied, writing of the crime being perpetrated against the city of Melbourne by those who insisted on occupying valuable space to which no improvements were being made. People must live at higher density, Melbourne was far too sprawled, the city must be livened up and young career people encouraged to come back to Melbourne's heart. WHERE WILL WE GO? ASK PENSIONER RESIDENTS, was a headline beneath which Big Ernie Kelly, now divorced by Bridget and rendered homeless through the sale of the Harcourt-Wilson, was to be seen smiling ruefully; it had been his home off and on for thirty years or more. Seen also, avoiding direct eye contact with the camera, was the one-time coordinator of the detox, saying his welfare group had bought up some houses a little bit further out in which they were going to try some experiments in community living. Detoxificants would have bathroom and kitchen facilities where they would be afforded more privacy than they'd had in the past and would be able to prepare their own meals.
Courtly Tom, who must have known he was not to be 'prioritised' by the new order of people in charge, 'impacted' upon the remaining cubic centimetre of his liver with a bottle of gin, taken neat on the Harcourt-Wilson steps the day before takeover. He now lay in state in a local funeral parlour, where a 'For Auction' sign blocked the muted in-flow of light through the tasteful beer-bottle brown Religious Windows. These windows would have overlooked the tramscape on the main street had it not been for the sign and the hearse, hired from another firm for the ride out to the crematorium with the corpse. Inside the parlour it was not unlike what being inside a bottle of beer must be like. The mourners, who had once strummed their guitars, praised the Lord and never lost hope in the reverberating concrete chamber on the corner, consecrated in the name of the Most High (before becoming a car park), stood around with their shoes shined and their hair slicked down, talking about the good old days (before the Most High saw fit to turn a blind eye on the escalating price of real estate in the inner suburbs, sending the Most Low out under sheets of newspaper in public parks).
Beryl Blake, chief botanist at Bunurong Gardens, would bring the Most Low gifts of warm hand-me-down coats; she attended the Drug Rehabilitation Clinic herself, and they were mutually acquainted. 'It's their self-esteem,' she'd say to me, 'their self-esteem.' She knew how they felt, but was on a salary and had a warm bed and a home. Sometimes she'd go out and lecture them on nutrition, saying if they must drink, they must also eat some protein. Luncheon sausage was known to change hands.
The Brolga's black onyx cross was never to be seen these days, bashing against her breastbone. She was out of mourning with a vengeance. No doubt seeing an opportunity, she took an interest in David. She had known him in Sydney in the pre-Dadda days and, in public, had always maintained he was intelligent. Sometimes he would be seen lunching with her and Checkie in the Siècle Trust coffee lounge, oblivious of the fact that Miles might find it objectionable in any way. He would tell Miles he had the wrong impression of Viva.
The friendship blossomed enough for David to be seen quite often in the company of Pattie and Jerry Gospel, and even – how times change! – in the company of Viva's star artist, Barrie Bull. On the opening nights at other galleries, the coterie would swoop in, The Brolga in the lead and Checkie not far behind her. But, regaled as he may have been by those who fawned on Viva, David still went home to his wretched room at nights.
So suited was The Brolga to the times that she accused Rose of having a set of Henry Corettis belonging to the Trust stowed away in some unknown nook. In fact Rose and Laurent, in accordance with the agreement they'd made with Harry, had sold the Corettis not earmarked for the Trust in order to have something to live on. Accordingly, when Viva produced a document which purported to name the Corettis due to come to the Trust and included among them the ones that had been sold by the Hirschs, Rose was appalled. Eventually, she was to receive a letter with the letterhead of Gosper and Co., demanding restitution.
Even supposing the demand had not been outrageous, Rose had no money. She'd never had money, and Laurent's had gone on buying the boarding house years ago. They had run it in such a way they'd lost money while gaining friends. I sat with Rose in her little labyrinthine house, reading the letter which seemed so out of place among the dolls of her imaginings as to be made doubly ludicrous. Her financial history was not that of a profiteer.
When they first came to Melbourne, Rose and Laurent hadn't been slow attracting friends, even though these were not the people they met in the normal course of a day. Their source was Grace Lonsdale. Grace painted, therefore she knew artists; she was political, and therefore knew politicians; she was fascinated by the natural world around her, and therefore knew natural scientists; and she spoke French and German, was married to a linguist, and therefore knew some of the more sophisticated local Europeans.
Though artists had had studios, the concept of the salon was not really known in Melbourne, and it took two highly eccentric minds to think of setting one up. It had begun with a show, in the basement, of paintings by Rose, to which came Grace Lonsdale and a bevy of friends, and Uncle Julius and Aunt Orpah (in spite of everything and remembering that moral support for the boy was important, even if he never changed his mind and took up a suitable profession). Some of the boarding-house residents came, and among them the latest arrival, a good-looking but aloof young girl about whom nothing much was known, Viva Hallett.
She said she was twenty, but Rose knew she couldn't possibly be. She arrived on Rose's doorstep one Friday evening, wearing a new summer dress and a charming sun hat and carrying a smart case. She had a somewhat proud air about her, which might also have been taken as defiance or defensiveness, but she wasn't twenty. She had asked for a room and Rose had taken her up six flights to the third floor, where there was a cool, west-facing room with a view of St Kilda Beach. Viva said she would take it and was able to pay a week's rent in advance. But in truth, Viva had no job, nor even enough money to buy food, and Rose soon noticed that the hat and dress were all she appeared to own in the way of clothes. The smart suitcase had sounded suspiciously empty when placed on the floor and the hangers in the wardrobe remained tellingly unencumbered after she'd settled in.
She had been going out each day as if to a job, but Rose knew there was no job and climbed the stairs to see her one evening. Would she not come down and share a meal with Rose and Laurent and their friend Signor Coretti?
It was probably the first time Viva had received such an invitation and it took her some time to frame an acceptance. She said eventually, and as though she had studied it in front of a mirror, 'Oh, I'd be charmed.' When Rose took her by the arm to go downstairs, Viva stiffened a bit, but she was quick to pick up on politesse, and Rose said that by the time she was halfway down she'd adopted the comportment of a Marlene Dietrich. At the meal she behaved as if everything in her life were in order, but it was obvious to Rose that she was a child still, and in need of help.
She had said she would like to learn French so she could join in the conversation better, and Rose had said she would teach her, adding that Viva could stop pretending to go to work each morning and come to her for a French lesson. With a sangfroid that was to become characteristic, Viva said, 'Oh, you've guessed I haven't a job. Well, you might as well know also, I can't pay for my room.' In a situation in which most young women would have been moved to cry, Viva remained dry-eyed.
'You know, I admired 'er for zat,' said Rose. 'She 'ad substance.'
Concerning Viva's unemployment, Uncle Julius was to come in handy. On the night Rose and Laurent opened their salon, Uncle Julius was very taken with Viva and asked whether she would come and work for him. He needed a secretary. He said it out of earshot of Aunt Orpah, who would never have allowed him to employ anyone female under fifty.
From this time on, Viva took a passionate interest in France and everything French and, with her instinct for elegance, wore her inexpensive clothes with panache. Apparently Uncle Julius was entranced, because in no time at all Viva had the money to go to France, a journey Aunt Orpah was only too happy for her to take. Going meant there were some admissions Viva had to make. She was sixteen, not twenty. Her mother was dead, but somehow she was able to get, or fake, her father's permission and obtain a passport. Rose heard that the Lichtblaus allowed her to name them as next of kin, and in the off-peak travelling season of 1934, she sailed by herself for France. She docked at Le Havre and went directly to Paris.
'Fancy,' said Rose, 'she lived wis my family, made friends of my friends and regarded our boarding 'ouse as a 'ome she could return to. Why is she suing me?'
Eventually, Rose approached Harry over the matter. He was ill in bed in his Toorak house. Here he lived, as he put it, with his shadow and his reflection. Checkie had arranged a nurse, a housekeeper and a gardener. He told Rose the only part of the day he looked forward to was his morning coffee, which the nurse, Jean, a delicately made Eurasian of about thirty-five, would make for him, following his directions, after she'd given him his morning wash.
When he had heard Rose out, he asked her to go and fetch me, as he had something to say that he'd like both of us to hear.
Harry lay in a single bed with his head in a window niche so he could see his azaleas, red, pink and white, without having to move more than was comfortable. He told us he'd put the Siècle Trust into Viva's hands entirely, abrogating any responsibility for it the moment Henry Coretti returned to the Siècle fold. But 'Don't give her anything,' was his advice. 'Split up everything you've got, even what's promised to the Trust and still to come. Give it to your friends to take care of – Reg, even Allegra if she has somewhere to keep them, because one thing's for certain, Allegra will never give anything to Viva. The ownership of those paintings isn't clear-cut. It's quite possible I could sue her successfully for some of them, though I'm tired and I won't live forever.
'In my opinion Allegra and Isobel ought to have the Corettis you still have, Rose, the ones that were earmarked.
'You should get that lawyer friend of yours and Bart's from Sydney onto it, Isobel, he knows what he's doing. He's got a sense of honour as well as a sense of humour. I'd be prepared to will the Corettis to you and Allegra as if I owned them. And it could be said that I do. It might be all I have the time to do. Then, Isobel, leave your father's paintings to the Trust if you want to, if it comes to mean anything to you. Viva and Checkie already have more than enough.
'I suppose you can't forestall the sale of your gallery, can you, Isobel?' he asked. He did not realise that Mad Meg was already sold. It had gone at auction just a few days previously.
It was a Saturday morning, Allegra in such a state she developed a migraine and had to run out and spew in the mint patch halfway through the bidding. When I went to help her she was reeling, her top lip beaded with sweat.
The men who came to auction the place were not the people she had spoken to, they were some other seedy crew who'd just taken over the agent's where she'd gone to arrange things. The first one, a man of about twenty-eight, dressed like an undertaker, had bowled up in a Daimler just like The Brolga's. Allegra had been there when he'd come and paced around the rooms, criticising.
'Ha! Outside toilet,' he'd say. 'You know, you wonder why people don't fix these places up – or knock 'em down. You could build a nice shop here.'
When the actual auctioneer arrived, taking the sign from the back of his powder-blue Mercedes, the younger man asked how he was. 'Oh, been at another one earlier on this morning.' 'How'd you go?' 'Oh, you know; you sell the shit to the shit.' Allegra had had to bite her hand to stop herself from having a fit of fury.
Mad Meg was bought by a silk-cravatted man and a bauble-choked wife who were so tanned that Maggie Kelly said they looked like a couple of duty-free handbags you'd pick up on your way back from Acapulco. The wife's mouth, gaudily lipped, twisted to one side of her face. When Mad Meg was knocked down to her husband, she squealed, 'Oh! That's the second one we've bought this week!'
Beaten in the bidding were a young Greek family wanting to start up an electrical goods business, and some Vietnamese looking for a shop from which to sell material.
We had to attend the document signing, where we were given twenty-eight days to vacate. After the exchanging of the copies, the red-lipped wife said to me, 'You wouldn't like to rent it back from us, would you? We probably won't be knocking it down for a while yet.'
I nearly spat on her, but instead said, 'Why don't you try to rent it to one of the other bidders?'
'Oh, we like the look of you.'
'The others probably need it more than we do.'
'Yes, but they're... you know.'
'We're you know too. We're half-Italian.'
'Yes, I could see that from the names. You don't look it, though.'
Allegra listened to all this with her back turned. When the appalling woman had gone, she said we ought to burn the place down and pour ten tons of concrete on the site.
The demise of Mad Meg was very painful for Allegra. She wept for days and couldn't eat anything. I made little bowls of food for her and bought piles of fruit, but she didn't touch any of it. She was sinking into herself and David wasn't making matters any easier. I found him in the house twice when I came home after work. The first time he must have taken our altered telephone number, because the calls began again the following day. We would let him rave while the receiver dangled down the wall. The second time I found him he was rifling through Allegra's letters and papers. When I told him to leave, he kept at what he was doing with surly determination. It took Eli to dislodge him, but he wouldn't go unless Eli went with him.
About this time, I had a call at the gardens from the people Allegra worked with at the university. 'She's practically catatonic,' the woman said. 'She needs help.' I explained to Beryl Blake and she let me go early to collect Allegra and Nin, who was in the university crèche. I brought them back to the 'Herb', and Beryl told us of a psychiatrist she knew. He was available for emergencies, took drink cases, among other things. By that she was saying it was her psychiatrist, the person who kept her able enough for work. I left Nin drawing on Beryl's whiteboard with a felt pen and took Allegra to the shrink.
The waiting room was taken up and we had to wait in a pink corridor, confronted by a hideous Japanese print of zebras on the facing wall. The place looked like a converted massage parlour, with a deadlock on the back of the front door.
Every so often a secretary flounced by us in the hall. 'Just checking my cherubs,' she'd say. There seemed to be people in drugged or comatose states shut up all over the place. Down the hall we could hear the psychiatrist's every word as he interviewed someone in a far room with the door open.
Allegra was silent, weeping the whole time. She sat on the only chair, rolling her head from side to side across the wall behind.
'Shitty joint,' I said. But Allegra was beyond noticing her surroundings.
A policewoman came through the front door, her shadow thrown down on the floor as the door opened. 'Yoo-hoo,' she called, blinking at the sudden change of light. The secretary stomped out of her room, saying, 'Up here,' in a whisper while the floorboards cracked beneath her tread. Perhaps there were sleeping babes with rosy faces tucked up in cots behind the doors, or the puppies of some rare breed, dopey, in skins too big for them. But directly afterwards a dazed old lady, smartly dressed, was hefted down the hall on the policewoman's arm. Behind her toddled an old man, carrying his hat. Perhaps the old lady had run away from home like that, in her Sunday best.
At last the secretary opened the waiting-room door and a dishevelled youth appeared in the interior, his cardigan filthy and full of holes. The interviewee from up the corridor could have been his mum; she sighed at the waiting-room door, stringy and downcast. He jogged a bit in his runners, adjusted his penis, hands in pockets, and flicked a foot so his balls hung comfortably.
The waiting room was small and Edwardian, painted pink like the corridor, Seurat's _Grande Jatte_ on a wall by the casement that gave onto the street, the roof of my battered car, a busy corner and a toylike railway station. It was like the room we all used to know about above Woolworths in Darlinghurst Road, King's Cross, something wrong somehow. The kind of place where panicking women go, a dread room, a room where you paid, not just for your sins, but for the sins inflicted on you.
Allegra still sat in the corridor, head bent now, face hidden in hair.
The psychiatrist was a man in his mid-fifties, looking terribly tired, hung-over and dandruffy. His suit was crumpled, his eyes red-rimmed. He took Allegra by the elbow and she got up mechanically and went with him into the consulting room.
No, I hadn't made a mistake, I thought in the hallway, at a point where the wallpaper was mismatched, the renovations hasty, feeble light at the leadlight panes around the door, pink and green with random cracks, a brown stain threatening at the roofline. It _is_ an illness, I thought, she _does_ need help.
Minutes passed. As if the house were tilting at sea, the door to the consulting room clacked open and shut.
The secretaries began a tea-break conversation down the hall. The phone went. 'Doctor is very busy, I'm afraid.' 'Doctor won't be fitting in any new patients until...'
The doctor called me in. 'I'd like to put her in hospital,' he said. Later Allegra said she couldn't even hear him, she'd had a head full of muffled machinery.
She was having a breakdown. I took charge of Nin, which meant having her at work with me every day. She had turned four and was a handful, loving to climb and draw and make little rooms in empty shelves in the compactus. I explained to Beryl that leaving her with my mother or at the crèche would mean leaving her open to being taken by David, who had even begun to badger me at work, so that Loyola started taking all the calls and saying I wasn't available.
Allegra didn't improve at first, because being in a hospital bed meant having David or our mother at close quarters. I told the doctor to ban David until she had recovered. 'He's the mad one,' I said. 'He's the one you should be seeing.' But the doctor answered he wasn't out to punish anyone, just to help Allegra. Nevertheless, he acceded to my request and had the ward sister prevent both David and our mother from visiting.
Our mother fought with me over it, as if Allegra's illness were deliberate and we were blaming her for it. 'Conspirators!' she called us. 'You're just like that man was. Sly.' It was no use saying Allegra needed time and space, our mother, who had never had time and space granted her, wanted to find out whose fault the madness was and perform a correction. 'There was no madness in my family,' she said, and I thought of Granpa, the horse-whipping of Weightly Lisle, the constant inebriation of Uncle Garth and Aunt Nina's pathological goodness. 'No, there was no madness in your family,' I said.
When he thought Allegra was not acutely ill anymore the doctor had her moved to a public psychiatric hospital. He couldn't stop David from coming there and eventually it was I who had to confront him and tell him to desist. He quizzed and hectored Allegra the whole time. 'What did the psychiatrist say about me?' he'd ask. 'What did you tell him about me? You see, dear, it isn't me who's mad, it's you.'
'I've got money for her,' he said when I confronted him, brandishing the same bank cheque he'd been carrying around for weeks. 'She could go and stay in a private hospital, I'd pay for it. I'll go and see the psychiatrist myself. I will, I will. You can come, you can come with me and be my witness.'
The hospital was such a drear place that our mother, who only visited once because Allegra wouldn't respond to her, commented if you weren't depressed when you went in there, you would be by the time you came out. It was wet on the grass between the wards. They were shacks, a sick pale green inside. Steam rose from an underground drain by the black, shining road to the main building where the administration was.
Up a wooden ramp, be merry, I would tell myself, be bright. A giant man would be creeping back and forth, back and forth, over and over the same worn track of corridor. I would see his profile through a high panel, left side, right side, left side, right side. Be bright, be cheerful, bringing clean socks in my bag and new nightclothes. The woman who sold me the nightgowns said, 'Sorry to hear she's in hospital. I'll pray for her. That's what I'll do, pray.' They all knew Allegra there, it was Cathy's father's shop.
Smile, I would say to myself, be gentle. There was a woman in a floral dress and cardigan who leapt at me, the synapses bulging under skin that was choking her; a nurse's arm came down on her shoulder. 'Everyone loves you, Agnes. See, the lady loves you, all of us, all...'
I love you, Agnes, yes, I love you. Looking into her eyes was like looking down railway tunnels.
Once there was a squabble in the occupational therapy room, between a large, high-pitched Dutch girl with enormous fists and an ex-Jesuit who was hunched up on himself, jacked upright by the stake of his backbone. And there was a woman in there who knew me once, and turned away.
'Allegra?' She would seem much more well than all the other people, but it was because she wouldn't take her medication. She had decided not to believe in psychotropic drugs. She was a focus of anger and destructiveness in a world of zombies.
The doctor said he could take no more responsibility for her if she wouldn't take her medication. And so she was discharged.
TWENTY-SIX
Allegra
_The weatherman says clear today, He doesn't know you've gone away, And it's raining, raining in my heart..._
Slowly she circles my living room, her arms wrapped round herself. Allegra in Aunt Nina's dress, sad and creased from a long night's dancing, flowers drooping in her waist-length hair. A kitten plays with her shadow on the floor, leaping empty champagne bottles and ashtrays spilling over.
_Oh misery, misery-ee, What's gonna become of me-ee?_
The black lace of her skirt sends patterns swirling through the room.
Up the hall, Cathy and her boyfriend wave to me and open the front door. Pale light slips briefly down the hall, then folds on itself and is gone. Car doors slam, an engine comes grumpily to life and the gears crash twice as they drive away.
Allegra has given me a New Year present, a pink stone in the shape of an embryo. It's 1984, George Orwell's year, the sun now rolling blocks of light across the floor.
She has been weeping off and on for days. She tries to come out of it, only to start weeping all over again. Her cheeks and her lips sting and the folds of her nose are sore.
She and Nin still live here, but we are besieged by David. When he isn't trying to heave windows up from the outside or kicking at doors, he's on the phone. He wants to talk forever, so we leave the phone dangling down the wall and every fifteen minutes or so, Allegra says yes or no into the mouthpiece.
He's off his rocker. Sometimes I find him huddled on my front doorstep when I leave to go to work in the mornings. He threatens to kick the door down if I don't let him in. The door is covered in kick marks. Then it's red roses all over again, one a day, first with a card saying he'll leave her alone, then with a plea, then with a farewell, then with the announcement that someone else wants to marry him.
No real attempts have been made to kidnap Nin, though once he took her from our mother and sat with her on my front verandah until Allegra came home from work. There, she was only just holding on to her job after having been away so long and coming back before she was well enough. He began, 'I don't want my child anywhere near that woman, do you hear me? I said, DO YOU HEAR ME?'
'I heard you.'
'Well, what are you going to do about it?'
'Perhaps you could mind her?'
'What do you mean, I could mind her? You know fucking well I can't mind her.'
'You could if you tried.'
'I can't. I'm not living anywhere.'
'Why not?'
'Why not, she asks!'
'Well, she'll have to stay here with me then, won't she? If she's not worth finding a home for.'
'I'll contest Allegra. I warn you.'
Maggie, Kelly and I have decided to take Allegra on a holiday. Maggie is brewing coffee behind me in the kitchen and Kelly is shunting the remains of last night's party into garbage bags. There's a young chap cleaning up in the yard outside. I don't know who he is – a gate-crasher, maybe, or someone Eli knows, except that Eli's away with his friends. Anyway, he seems to be polishing the shrubs. He's taken the fallen python lilies and made stacks of them like party hats in my rockery. He seems to have wrapped up the outside rubbish in Christmas paper and tied it with bows. It's poking gaily out of the tin.
I suppose if I were a proper modern woman, I'd stride firmly out into the yard in search of an ulterior motive in the cleaning, stuff the fact he's actually found room for the rubbish.
'What are you?' I decide to demand from where I am, jacked up on the wall beside the open back door. 'Some kind of...' I can't think of the word, '... altruist,' I say at last. He is carefully extracting a green pea from a lily trumpet: now he holds it in the palm of his large hand and stands before me, a castigated child flexing his knees. What knees! They are round as newborn baby heads under his twitching thighs. From the back (I tell him to turn round so I can have a look), his knees have that bland sleepy expression knee backs tend to have on people who are disinclined to strut. His back is very straight and his shoulders broad – he is the stamp of boy a Motte would be proud to own. I can't think why I didn't notice him earlier. I tell him he can turn back round again now. He is hiding his eyes away under a big forehead, and when I fish them out, they are light green. 'Sorry,' he says, and lowers his head.
'Why's that?'
'Sorry I'm an altruist.' When he smiles, his teeth are big, white and gappy, but he moves his head to keep me away from those green fish under their ledge.
This isn't meant to happen at all. Ideologically most unsound. You're meant to have them quaking at the knees by now, and confessing drug habit. Then you're meant to say the habit is no concern of yours in a voice that sounds as if you were chewing a fistful of rubber bands. But I say instead, 'It's 1984. Matching thongs are in. You people without them are meant to go and sleep under bridges and content yourselves with cheap drink. Would you like to have some coffee?'
'Okay,' he says and I ask him his name.
He says he is called Link, because his name is Philip Chaney. Not related to either Lon or the senator. His dad is called Newsome, a fly-by-night, present whereabouts unknown. Sitting on the back step while Maggie tries to tempt Allegra from her trance with coffee, Kelly and Link and I talk of fathers and whether it's worth chasing them up if they abandon you, until it occurs to me to ask him if he often goes round vomiting in strangers' rockeries uninvited.
'Well, um, actually, I was invited. Allegra asked me to come, but I think she forgot. I mean I think she forgot she asked me, and she forgot who I was, and as I didn't have anyone much to talk to, I got drunk. It was a good party, though.'
He used to drive a delivery van, he tells me. A while ago it seems he dropped off some paintings to their owners after we cleaned out the storeroom at Mad Meg. 'Usually I just pick up the pictures from the galleries and shoot through,' he says, 'but I wanted to ask Allegra something, so I hung around. I didn't quite know how to look casual. First of all I just sort of stood around with my arms crossed. Then I leant against the upright of a door. Allegra was busy packing the paintings into cardboard. I kept thinking she must've thought I'd gone already, so in the end, I kind of coughed. Then she looked up and she sort of said something like did she have to sign a docket. And I said, uh, no, she didn't have to do that, but would she mind telling me – because I'd noticed her name was Coretti and the signwriter who'd written it on the gallery window did a really bad job – the letters were all the wrong size. Anyway, I wanted to know if she was a relative of that man whose pictures I'd seen in the National Gallery?' He rubs the back of his head, screwing up his eyes at the complications.
'And I found out he was her father. She said you were a better painter than him, but she didn't have any of your work to show me. I ran into her again last week at the market and I asked if there was anywhere I could go to see your paintings. She said if I wanted to see for myself, why didn't I come to this party? She said you'd be having a New Year's party, and that's why I came.' He takes a deep breath, stretching his magnificent arms like a little boy about to jump into a swimming pool, then blows out his breath and relaxes again, glad to have got that lot off his chest. 'But I don't think you could possibly be a better painter,' he adds in a scarcely audible voice, 'and anyway,' he continues, 'I saw some of your paintings here. I wandered into a few of the rooms. And you're not, so there.'
Casually, I empty a bottle of soda water over his head, but he is hardly even surprised. It splashes down over his shoulders and fizzes through the fuzz on his legs without him doing anything to stop it. He just sits there, unimpressed, his arms jacked up on his knees and folded across his chest. 'Those ones in the front room upstairs, for instance...'
'There's only one in the front room upstairs. A couple of Stone's Green Ginger Wine bottles lying at the base of a wall.'
'Yeah. That's the one. I mean, I can't see the point of it. There's hardly anything in it.'
'How much do you know about Australian painting, Link?'
'Not much, I'll admit. But I trust my own judgement. I like to think I have some judgement... though I probably don't have.'
'It's one of my father's most famous paintings. It's the subject of litigation at this very moment. I've had to have a burglar alarm installed. I'm not given to securing my possessions,' I explain, 'and if _Where the Nice Girls Live No. 2_ belonged to me, it wouldn't be protected. It belongs to my son and my niece, however – or will do one of these days. I'm not much of a guardian. I have always lived with an open door.
'Not so long ago a show of mine was vandalised and some of my work was destroyed. Other people urged me to be angry and sue, but I was sorry for the person who did it, really sorry. To him it must've seemed that my work was directed against him. But one's artistic work is only ever about what one knows. I was sorry for having caused someone unconquerable pain, sorry he had to lard his actions by roundly disparaging my talent, but if I fail to paint, even in my head, there's nothing left for me. I didn't take the matter any further, though people said I should have.'
'That means that person gets away with what he's done.'
'No. He probably lives with it all the time. He needs help. Not my help, but some kind of help. No one will help him. Perhaps no one knows how. I don't suppose anybody wants to help, either. You go seeking help for people like that and doctors turn round and tell you you're sick. People have the attitude that it's a waste of time to try to salvage vandals and bashers, but what they're really saying is such people are inconvenient. It's passive extermination. Like poor people are inconvenient.'
We drink our coffee slowly, letting the new sunlight warm the blood in our eyelids, like lizards on a rock. Then Link wants me to explain Dadda's painting to him so I take him upstairs to where the painting is hanging.
Reg Sorby brought it here and had it bolted to my wall. The seeing eye blinks on and off above the door behind us. It isn't an ideal place to hang the painting, a small, front-facing room giving onto a balcony through curtained French doors. There's a lot of natural light, which means the painting will deteriorate over time if I don't cover it in summer. Reg's insurer suggests a large pane of bullet-proof glass anchored into the wall with steel pins. It seems a bit extreme to me and would cost me more than the guardianship is worth. Furthermore, although that would protect it from attack, it would hardly cope with the light problem and my walls are of such poor quality that anything heavy falls off them, taking chunks of plaster with it.
Dadda's painting is a seeming blank, and yet it is a hive of riches. I explain the surface qualities and the latent messages in what is depicted. Little ironies in graffiti and subtleties in the application of the paint that cause the image of a girl to come and go as the viewer moves. I explain how it is a transitional painting, making reference to abstraction and hard-edge. It is an unusual painting for its time, clean and puritanical in execution but depicting something unclean and licentious. In the middle of the wall, in neat little letters, is written: LET HER R.I.P.
We dawdle downstairs. The summer morning has come up sleepy and still. It isn't hot, but long in shadow and very green. Water is in the leaves of things and great hydrangeas, mysteriously blue, are being born from leaf wombs. It is the season of blue in Melbourne, the season of petal skulls, of agapanthus and intense lobelia, of purple marguerites growing in globes. Allegra is swaying slowly now, from foot to foot, as Link and Kelly and I stroll about the house, talking in low voices, waiting for it to be time to pack our car and go on our holiday. In the sitting room now, with Allegra, we stand each side of the fireplace, me swivelling on my right heel, grasping the mantelpiece; Link swivelling on his left, also grasping the mantelpiece; Kelly leaning back towards the fireplace.
We are talking about Mad Meg and what our hopes were for it, when a key is thrust noisily into the front door lock and loud, angry footsteps sound in the hall.
'My God, it's David,' I whisper, my heart knocking in my chest. He was supposed to have gone to spend New Year with Bart in Sydney. I run into the hall, flailing my arms about and crying out, 'My God, this is pretty ridiculous!' But David thuds on up my stairs and tries to wrench Dadda's painting off the wall. Why has David got a key? Why does he know the painting's here? Is it Eli who's told him?
'It's bloody stupid, David, you know I'll only have to get the police! Don't be so silly! Please!'
I keep going for his arm like a puppy, and he, single-minded, having worked himself up into a fury, keeps pushing me off.
I yell for Link. When he reaches the upstairs room, David is still yanking away at the canvas with seemingly indomitable determination. Link lays his great hands on the door lintels. 'I think you'd better leave it alone,' he says. And it's extraordinary – men are baboons – David takes one look at big strong Link and changes his tack completely. He stands there, to one side of _Where the Nice Girls Live No. 2_ , like the sober subject of a Renaissance allegory choosing between a life of action and a life of contemplation and, sensing that the only advantage he has is age, he enquires almost genially, 'And who are you?'
Link lets go of the door lintels. He strolls into the room, keeping his knees stiff. He folds his hands across his chest and stands, legs apart, with his weight evenly distributed. 'I think you'd better leave,' he says.
Then David says, 'This is my painting,' and does a John the Baptist appealing to God with his left hand.
The gesture is too much. 'God, I'm sick of you, David!' I scream. 'Get out of my house!'
I have cost Link the situation. Archly David descends the stairs, but instead of heading for the front door, makes for the kitchen. He pours himself a cup of coffee.
'You bastard!' I roar. 'I don't want to talk to you! This isn't your house! Get out!'
David gazes out at the challis lilies, plucking his purple lower lip with his superb hand – how can he have such beautiful hands? 'Those lilies are taking over,' he says smarmily. 'What do you call them again? Python lilies? Isn't that your name for them? Your house is being strangled, Isobel. Snakes outside and snakes wivin.'
'Oh, how ridiculous you are! Why don't you wake up to yourself? If the painting belongs to anyone, it belongs to Eli and Nin.'
David turns his back on the garden, faces me and folds his arms, elbows to his sides, like a bat. He is smoking, the smoke draws a sophisticated squiggle up his shirt and over his face. Casually he blows it to one side. 'You're going to seed, Isobel,' he says in a low voice, so Link, who is straddling the stairs to block the way, can't hear. 'Your skin's all dry. You have cicatrices in your chin. You're also a talentless bore. You come from a mad family. You're mad yourself. Your father made a dreadful mistake when he married your mother and he spent the latter part of his life trying to unmake it. That's why you inherited nothing. He left his work to the people who knew what it was worth.
'That painting you have upstairs belongs to a series all collected and owned by one person. You and your sister were most injudicious in having it removed. It belongs to the Siècle Trust and that is where it will end up.'
Link comes into the kitchen and says, 'You'd better go.'
'Who is this bore?' David snaps.
'Go,' says Link, without raising his voice.
'How old are you?' David harps at him, and it's such a pathetic ploy that I feel a weary laugh hump in my chest.
Link raises an eyebrow, takes a sharp little breath and repeats, 'Go.' 'Are you a painter?' David pesters. 'Haven't I seen you around before? When I find out who you are, you're done for. Do you realise that?'
'Go,' says Link and casually saunters into my sitting room where he sits on the arm of a chair and begins to whistle noiselessly. Her back to us, Allegra is standing stock-still.
'Isobel, if you don't tell me who this person is, I'll sue for my share of your ridiculous gallery.' David has begun to hiss and jab.
'That would be an odd pretext for suing,' I mumble.
'I think you said his name. It was something like Link. Yes, it was Link, wasn't it? You said Link...'
'No, it was Michael, actually,' I answer on an impulse. 'Michael Beatty from Ballarat, remember? You said you rather liked his water-colours once. Miles showed them about three years ago at Figments. He can draw, too.'
'That's not Michael Beatty.'
'It is.'
'Yeah, I'm Michael Beatty,' says Link from the sitting room.
David sits down on the edge of the kitchen table, slaps his forehead a few times with the heel of his hand and starts to laugh. 'Oh dear, oh dear,' he goes, rocking back and forth. 'That isn't Michael Beatty. You're insane. That man's name's Link. I remember where I've seen him. He delivers paintings for shows. He's a truckie! Isobel's taken up with a truckie! Ha ha! Ha ha!' And then he barks, 'You're a liar!' and he dances up to me and does the John the Baptist thing at my face as if he were about to beckon my brains out of my nostrils. 'You've never told the truth in your whole life!'
'Keep trying, David,' I sigh, ruing the terrible waste of time. I leave the kitchen slowly and go to sit on the other arm of the chair where Link is sitting. It is as if we agree at all costs that Allegra is to be protected. David thunders after me. All Link has to do is stand. He is much the bigger man. David, faced with physical defeat, suddenly struts out of the house, slamming the front door heartily as he goes.
The very molecules of air unclump themselves as we relax.
'Why does he have the key?' Link asks.
I'm worn out and my voice begins to shake. 'I don't know. Maybe it was Eli, or perhaps I left one lying round inadvertently, I'm careless with things like keys.'
'I'm going to ring a locksmith.' He looks at me for assent.
'It'll cost a fortune on New Year's Day,' I say.
'He's a maniac,' says Link simply. 'He could kill you.'
'Only by accident,' says Allegra at last, still turned away from us but meeting our eyes in the mirror over the mantelpiece. 'He judges his bashings. If I were as strong as you I don't suppose his slaps would even feel hard.'
Link raises his large palm as if to warm it at Allegra's hair. I think he is madly in love with Allegra. 'What you're saying is preposterous,' he whispers.
'But there's something else wrong!' she says.
'No. It's plain what's wrong.'
'But it matters where the violence came from,' she says. 'He learnt it. He learnt to blame women for everything that goes wrong in his life and to punish them. He keeps on doing it because it's a catharsis for him. He goes away, gets all worked up over what he thinks is a real situation where people are pernicious and out to destroy him. No one will take him aside and show him there are other, more effective ways of behaving. The person you hate can't teach you love. It needs to be someone else, someone with a baritone and muscles. Men who hit are men who learnt it from their fathers. Only men they respect more than their fathers can unteach them. They're terribly injured and they inflict terrible injuries as a consequence.'
'I'm going to stay here tonight,' says Link. 'And then I'm going to call a locksmith.' He says to me, 'You ought to get ready now and take Allegra for her holiday.'
'It's so squalid, so ridiculous, so stupid.' I shoot my hands out, dolefully. 'It isn't just happening here. All over Melbourne there are people chucking themselves about in an aggrieved stupor. I don't want to involve you. You can't imagine how tiresome and stupid it all is! Allegra and I used to ask each other why it is that people paint; and now it seems that Dadda painted to cause wars. His wife pretends his memory is sacred and sets herself up to make millions by saying so. At Mad Meg we thought we were fighting her kind of evil and inventing a new way of life, when all we were doing was fighting pettiness with obscure gestures. Rose Hirsch – you know her, she's the embroiderer – well, she is being dragged into something that isn't her fault at all. Reg Sorby – everyone knows Reg Sorby – he's playing Robin Hood with Dadda's paintings, and David is locked into a fight that none of us comprehend. Why get involved in a mindless melee, Link?'
'Where do you stand?'
'Peace! All I want is peace. To paint. To buy my materials and continue with my thoughts and my life. I've had enough.'
'Would you be calm down at the beach tonight if I didn't stay here?'
'No. I'd lie awake in a lather of anxiety. You're right about David. He is dangerous, and he's also a tool in other people's hands.'
Link says, 'Well, I'm going to stay and guard the house and the painting... If you want me to. Do you want me to? It'll save me from having to sleep under a bridge in my unmatching thongs.'
I accepted his offer and, anyway, I'd begun to wonder why all mothers didn't call their sons Link. It suddenly seemed a noble name. My love calibrator was obviously measuring him, poor boy, because his was obviously measuring Allegra.
As we drove, Allegra slept on my shoulder and Nin on my lap, while Kelly read us strange, sad poems by John Crowe Ransom. It set us hankering after Browning, a neglected favourite of ours from our school days. We did a patch job on 'A Toccata of Galuppi's', but when Kelly tried to finish it with, _Dear dead women, with such hair, too – what's become of all the gold/Used to hang and brush their bosoms? I feel chilly and grown old,_ the three of us wept as though our golden age had passed.
We were going down the Great Ocean Road, the tragic headlands folding and unfolding towards Lorne, tragic because in February '83, while I was formulating my politics with the aid of Uncle Nicola's diaries and the Mad Meg collective was deciding to subject our shows to both male and female criticism, fire had swept down from the hills, surrounding Lorne and cutting it off in all landward directions. A town further up the coast towards Melbourne was burnt out almost completely, but as we'd driven through it, there was reconstruction going on everywhere, and the headlands, so many Golgothas, were already sprouting greenery.
In April, I had taken a trip with Beryl Blake to collect and photograph the ground orchids that are only ever seen after fire. 'Cruel flowers,' she had said. 'Cruel, cruel little flowers that need mountains to be razed before they'll show themselves. That's vanity for you, Isobel, and the doggedness of frailty.'
We passed the Lorne pub, over whose crowded balcony a helicopter had hovered during the blaze, reporting it back to a stunned Melbourne, a hundred or so kilometres away. Melbourne suffered, first, a thick and shocking rain of topsoil blown over by strong winds from the Western District, and then flaming branches, singed leaves and coals, the outfall from Hell.
That summer Melbourne's magnificent deciduous trees had suffered. To keep them alive it had been necessary to dig trenches in the gorgeous parks and gardens and water the roots directly with reclaimed sewage water. There had been losses and the filthy outfall from the storm complicated things further by coating the vital surfaces of leaves.
When we arrived in Lorne the house we'd rented was a disappointment; the tenants before us had obviously been roistering on New Year's Eve. There was rubbish everywhere and sick in all three of the bedrooms.
'Well, at least they didn't stick around,' Maggie said. Kelly had to race outside and retch into the bushes. Nin played with the kitten on a patch of garden by the house's back door. It had wandered into our party off the street. Kelly decided to keep it and had brought it along, tied in a sock, where it had squirmed and parped in her lap and stuck its miniature claws through the mesh. Now it was happy to be out, skittering through grass tussocks.
Maggie and I, mothers both and quite unmoved by vomit, got busy cleaning the place up. Maggie, ever the sensible one, had brought a spade, so we were able to bury the rubbish. The sick was another matter. To wash it off under the shower would have clogged the drains, so we took bedclothes and mats out and beat the solid bits against a post and rail fence, then chucked on buckets of water.
Allegra, meanwhile, did meaningless tasks in slow motion in the kitchen. 'God,' said Maggie, 'we'll have to watch her, Bel. She's completely out of her tree, poor thing. We'd better make sure someone's with her all the time.'
And so we did make sure. She didn't put on her swimming costume to go to the beach but, still in Aunt Nina's wedding dress, sat cross-legged in the sand, sifting handful after handful of it. Every so often, Nin would peer up at her face through her hair and say, 'Mummy.' She would crouch in front of Allegra, grinning enticingly. Sometimes Allegra would be lost somewhere; sometimes, though, she would stroke Nin's hair, but when she tried to speak the tears would splash and contract into tiny craters on the sand. If Nin put her arms around her, Allegra would cry even more, though she didn't ward her off.
We had taken the place for five days but Maggie said, 'We mightn't be able to stay that long, not if Allegra doesn't get any better.'
Maggie and Kelly had planned to go bushwalking the next day. I said I'd stay with Allegra and Nin as it was obvious Allegra wouldn't be up to it. That night we slept in our own sheets and burnt mosquito coils, which lessened the stale smell in the house. Allegra had seemed to go to sleep long before the rest of us, who sat up playing cards. She still hadn't changed from Aunt Nina's dress, but we thought it best to let her be. In the morning, I would run a bath for her; she could have a good soak and I'd make her breakfast. We had brought grapefruit, which we now passed round to one another, pressing the cool yellow globes against our cheeks and enjoying the fresh, light smell.
Maggie was off to Sydney in February, having graduated in photography and scored a good job in a Sydney tech. The Pantechnicon had folded before Mad Meg and the Kellys had run a market stall for a while, but then they sold up and Kelly and Chantal had taken up apprenticeships; Kelly in cabinet making and Chantal in electricity. Chantal was one of only two girls in the state to qualify.
As we had feared, the building housing the Pantechnicon had been demolished and now, freeway-free, the inevitable marble-coated shopping mall was on its way to becoming... what? What, after all, is a shopping mall, particularly one with more marble in it than Milano Centrale?
Though, on the whole, the new architecture was excessive, I had to confess a sneaking admiration for some of it; some of it really was new and surpassed the ugly and often gerry-built sixties boxes. I liked the clothes, too – they had regard for shape and finish, as seventies clothes never had, and certain designers had their tongues firmly in their cheeks. But often enough, the new stuff was extraordinarily expensive, and that which was made for poor tarts like us was generally thrown together in as hideous a fashion as ever before; only the price tags had risen. There were so many mirrors now in shops, you ran the risk of banging right into yourself, but one thing they demonstrated was that although the shops dripped opulence, the populace looked just as daggy as ever.
We yacked as we played, all of us nicotine abstainers now, breaking off the ends of our matchstick stakes and chewing on them. We sipped a bit of cask white while Kelly told us about her new bloke. 'His name's Guitar,' she said, pronouncing it Geet-ar. 'He's a Hell's Angel.'
We laughed at the thought of Kelly, slender, blonde and pretty, as a bikie's moll. 'Oh, it's not like that,' she said, 'not quite. Only rides at weekends. He's doing the apprenticeship. We've started up a little business and we're going to call it Shadow Cabinets. We're going to be really careful where we get the wood from. It'll be Australian, as far as that's possible, and crafted by Australians, too. None of this wood-chip shit. We're going to have a workshop up in the hills and train kids how to make stuff and design it. There's a bit of a market for handcrafted stuff up in the hills, what with the mudbrick houses and the mudbrick hippies.'
'When are you going to start?' I asked.
'Well, we've started. Mum's got us a dance floor and a heap of other recycled stuff, and guess who's renting us the workshop? Your mate, Reg Sorby. Christ, that guy's got a finger in every bloody pie, I can tell you. This workshop's next to a factory he owns that makes snail bait. I tell you, he's the snail bait king of Victoria.'
Talk of Reg made me wary. Before he came round to my house with _Where the Nice Girls Live No. 2_ , he called in with his easel and paint box to paint my python lily. We hadn't sold Mad Meg back then and were still trying to work out ways to save it. 'H'mm,' he hummed, his back to me as he dabbled the python heads onto the canvas. 'You two are in a fix, aren't you?'
'We have to sell,' I'd said.
'So it would seem,' he answered, standing back and smiling coyly at a nymphette he'd slipped in behind the bush.
'She's pretty fat in the thighs, Reg,' I said. He pursed his lips, but then sniffed and smilingly added nipples.
'Well, I'll tell you what,' he said. 'I'm going to do you a favour. Since you were kind enough to let me paint these lilies, I'll buy the others out of your collective so you and your sister can hang on to your gallery.'
'Christ!' I said. 'We'd be the laughing stock of the universe if we sold out to you! Imagine! You in charge of Mad Meg!'
'Mussolini paid thirty-three thousand lire for a Modigliani nude once,' he said.
'Yes, Reg. But Modigliani was dead.'
'How about if I made you a gift of some of your father's paintings, then? I've got a potential buyer. You could sell some of them and get yourselves out of trouble.'
'Well, that's an idea! So you've got some of Dadda's paintings, have you?'
'Well, no-o. But I can get some. You see, I own some of the paintings that are in dispute, but I own them jointly with Harry and Viva. Viva's forgotten that I bought from Rose, and then, because I'm not stupid with money and never have been, not like you and your sister, I deeded them back to Siècle – a gift, you see, that I could write off my tax. But it was partly a gift to myself, because I'm part of the Siècle Trust. Your father wasn't, because he went off and married your mother before the Trust had been properly thought out.
'I thought you said you didn't like Dadda's work, Reg?'
'I don't like it much.'
'Why?'
'Oh, I just don't. But I'm a rich man now, Isobel. I don't need Henry Coretti. I have a feeling that if I opt out of that Trust I'm going to be worth at least a quarter, if not a third of its assets. You ought to have your father's paintings. I look upon them as yours. And, as I said, I might have a buyer. How much do you need to save your bacon?'
'Fifteen, twenty thousand dollars. Preferably twenty.'
'Ah. Well, two of your father's paintings would probably fetch thirty.'
But though _Where the Nice Girls Live_ materialised, Reg's buyer didn't, and our troubles, instead of diminishing, seemed to multiply. I didn't know whether we had come by the painting legally or not. Harry Laurington thought it was probably legal that we had the painting, but Reg was not above getting hold of it by illegal means. I sighed to think of the mess our lives would be in if Reg had stolen it.
'Pontoon,' said Maggie. We were all pretty tired by this and lacking the coherence of mind to play further rounds, so I went to bed in the same room as Allegra and the slumbering Nin, and Maggie and Kelly bunked down in the sitting room.
We woke about half past eight or nine the next day. Nin had been sitting beside my head for an hour or more, pretending to read from a pile of books we'd brought along for her. Kelly was already in the kitchen, cutting the juicy, golden fruit into halves and brewing up tea. We'd decided the night before to cut down on coffee and go for health- and strength-giving tea. There was a mint patch in the so-called garden at the back of the house, so the smell of mint was in the air, and the house was in better odour than it had been. Allegra was asleep, so I decided to let her be until after Kelly and Maggie had gone and I could take care of her and Nin without distractions.
We ate brown toast dripping with honey while Nin sloshed around in a plate of muesli. Kelly had put on shorts and kept her clodhoppered feet on the rung under her chair seat. Maggie came out of the bathroom in a hat with corks hanging round the brim and we laughed. They were about to go when Maggie put her head around the bedroom door where Allegra lay sleeping. 'Do you think she's all right?' she said. 'She's awfully still.' She went into the room with a jacket on her arm and stooped over the bed. I saw her take Allegra's wrist and shake it, and I buried my head in my arms as if I were about to be in a crash. 'No pulse,' Maggie said. The worst of it was I'd known it subliminally for hours.
An ambulance came and took her to the local hospital. There, they said there'd have to be an autopsy, but I was freaked out, I couldn't bear them to touch her. The blood sample already showed an overdose and there was an empty pill bottle in her bag.
Kelly had taken Nin swimming to distract her. Whenever my mind turned to Nin I felt the uncomprehending ache of a little child and knew this terrible new reality would break on her and do her great harm. Surely Allegra had thought of that; but as I pictured her weeping on the beach, I realised that she had gone beyond coping even with her child's love. She had gone to the point of thinking her life would ruin her child's. And she had gone to needing only to die.
There would have to be an autopsy in Melbourne. 'It's the law,' they said. 'How would we know if it wasn't the barbiturates that killed her? We know it's the most likely thing, but we don't know if it's the actual thing. We have to rule out any suggestion that she didn't die by her own hand.' So I had to sign the wretched paper giving them permission, though I felt my hand being clawed back and I felt her nails wounding me deep between the bones that run to my fingers.
I went back to Melbourne with her in the ambulance. I would have to tell my mother. I knew I would break down in front of my mother, so to help me tell it, I would ask Beryl to be with me. Beryl would be the best person to ask. A kind person, someone with authority. How would I even open my mouth to tell my mother over the phone I was coming to see her? I would ask Beryl; Beryl might have been a drunk, but she was kind and calm. She understood the need for grace in life.
In the ambulance, I wept, held Allegra's hand and bit her wrist as if I were a baby trying to rouse her. Tears and saliva bound us. There was a nurse in the cabin with me. She kept talking gently and patting my head, but I couldn't hear a word she said and the pats felt as though I were being lightly pelted with little bags of sand. I fell into a kind of doze with my head on Allegra's belly, and dreamt a little bit that when I woke the world would be transformed, Allegra would wake up and I would stop crying. But too soon the ambulance was in the parking bay of the morgue, the doors were being opened, the temperature and light in the cabin were changing and I had to let her go. Instead of a sweet, peaceful awakening, the most horrible duty of my life to perform, when I myself was stricken almost to dumbness.
They were good and rang Beryl for me; and so I wept and waited, waited and wept until she arrived – good Beryl, ugly, real, compassionate.
'I think we won't ring first,' she said, squatting in front of me and patting my knee, so I caught a little whiff of her, female and motherly. 'We'll just call around. It'll be hard, but I'll be there.'
My face a cascade of tears, I rang the bell and drum collection. She opened the door and I took her in my arms: my mother, my dear mother, for whom we had had such years of anger. 'Allegra,' I said in her ear and her embrace grew firmer. Then we came apart and she stepped back, tears popping from her eyes, wiping her arthritic index finger upwards under her nose. 'I can't cry,' she said. 'I've never been able to cry.' She blew her nose, not on a Kleenex, never on a Kleenex, but on a proper hanky. 'One of Neen's,' she said, dangling it and making a little laugh.
'This is going to be hard for you to believe,' she said, 'but I heard your grandfather singing his round-up song last night and I knew it was Allegra. I've even been expecting you, my poor darling, poor little darling. I suppose she did it...?'
'To herself! Yes! To herself!' I cried, suddenly feeling that she ought to have stabbed herself or jumped from a high place. 'Fucking sleeping tablets! What a coward! What a coward!'
'Oh, darling, people like Allegra don't grow old. They bloom brilliantly and drop.'
'No, no, no, no, no!' I writhed around what she had said. 'No! She was sad. She was unhappy. It didn't mean she had to kill herself.'
'Darling, there are some people...'
'Not Allegra!'
'Oh well, not Allegra then,' she said with a kind of resignation, nodding her head. I'd touched her in a place that hurt, but I was suddenly furious with her for assigning Allegra so easily to death. It didn't occur to me till much later that my mother had had to assign a lot of people to death while being expected to soldier on herself. I didn't stop to think that she'd done her best with what had been available. I became passionate and accusing. 'You never believed in her!' I cried.
'I did. I admired her courage and the way she took up causes.' And then she broke out, 'Hell! Didn't I get down on my hands and knees trying to fix up the floor of that bloody gallery? And where did that get you both? I wanted you girls to have good lives. I didn't want you carrying the can for everybody else, as I've had to.'
'You haven't had to,' I roared. 'You chose to. You tried to reform Dadda when it would have been better if he'd reformed us!' Suddenly, here I was in the middle of a pitched battle, not knowing why I was there and why my mother and I were instantly at each other's throats at a time when we ought to have been supporting each other. I was sure the fault lay with her, but, of course, I'd gone to give terrible news to my mother and had expected her to react impeccably, to suddenly have words and wisdom she'd never had before, to be dignified in her sorrow, when where is the dignity in the suicide of your child? Instead of catching her emotional intentions, which were to bring comfort and express sadness, I fell into an old habit of my own: I took her at her word. Behind the statement that meant she believed Allegra was the type of person who'd kill herself was something evil, a kind of maternal curse.
Afterwards Beryl said she hadn't known what to do, both my mother and I had closed ourselves off to reason. And then she said, 'There's such a thing as forgiveness, Isobel. It's not just a Catholic precept, it's an actual gesture, but you can't make it unless you leave yourself open to feeling it. I'm sure your mother loves you and I'm sure she loved Allegra. I wouldn't be surprised if her grief went very, very deep.'
'You don't know her,' I said.
In the next few days, my mother and I hung around each other, hugging then hurting, hugging then hurting, until she said she couldn't bring herself to look at Allegra dead, and didn't have the strength to go to the funeral. Beryl and Eli were both with us then, and when I went to attack her, Beryl said, 'No, Isobel, no.'
'But Allegra's her daughter!'
'Be merciful,' said Beryl, 'this is Allegra's mother.'
Eli said, 'Go, Mum. I'll take care of Grandma. Otherwise, she's all alone, and even if she went she'd be all alone.'
What about me being all alone? I thought. The scene of Arnie and me on the beach came back to me, surrounded by oyster shells. Once again, I was the one to be sacrificed. I would have to be my father and mother. I would have to reconcile this war of opposites both in myself and by myself.
Bouquets in cellophane wilted along the hall of my house, while outside, the streets never seemed so ordinary: shut grey weekday terraces, waiting dogs by rubbish bins. No wind. Onion weed flowered in the bluestone lanes.
You were a trap for the light, Allegra. Beautiful as a spider's web, your hair arranged to hide the coroner's work. A shadow fought itself in the stillness under your thumb.
David came with Miles and stood around, surly and speechless. He stroked the back of his hand down your cheek, but that was his only moment. Nin clung to my hand.
Following the hearse in my battered car, we could talk, and did, as if there were no funeral, hoping it wouldn't rain, hoping we could keep up with the fast-moving cortege. I put my foot down flat on hills, I shot through intersections. An empty yoghurt cup rolled around the floor.
At the cemetery we couldn't find the gate, and walked with the other mourners in our best high heels or polished boots in a quagmire beyond the fence, helping each other over puddles. Nin ducked in ahead of everyone through a broken bit of fencewire, grinning, but she soon came back to me, a grimy little posy of grave weeds in her hand.
We threw in after you flowers, clods of earth, a stone – a river stone, smooth, pink, shaped like an embryo. I heard it thump on the coffin lid. There are only hard things to cling to in this life: facts have edges and cut into your palm, mysteries are slippery.
We filed back to the cars. The paths were narrow, runnelled and pot-holed. It seemed ridiculous, a hole in the ground, the knocking of clay clods on the lid of a box. A ritual without solace.
A prayer worth saying would be made of stone. Disbelief is the first reaction. You wake in the dark, expecting something to happen. It doesn't. There is no knock on the door, 'Sor-ry!' and the arms around your neck, the voice rocking in your ear, real.
You tumble into daylight and do things, then tumble out of it into a sleep you think was dreamless in the recalling of it. Why dream? If you want to dream, you might as well stay awake. Those times you wake up at two o'clock, three o'clock, in pitch dark, can probably be accounted for by worry – you've a lot on your plate – or a full bladder, too much instant coffee, not enough physical exertion. Cut the coffee. Jog. Go to yoga classes, and say your mantra over and over till none of it matters.
Allegra, when I talk to you, doors bang, taps run, and radios talk back to me. A rat scratches in the ceiling, there are ticks, clicks, squeaks; the floorboards shift and the dust falls under them.
I try to draw you, but your image baulks at my hand.
I try to write you. The words come down on the page. Word, word, word, word: nose, eyes, lips, hair. I scrape, I rake, I assemble lines in the dark, and they feel nothing like you. They are small, pointy, rutted, but your hair was crisp, electric.
I chop. I grope. I live in the forest now, where my axe rings cries from the trees. The chips fly out from the divots: wretched, broken, dangerous little things that, stuck together, cannot make a tree.
A hot wind bashes the house and sets the flyscreens thudding. Hydrangeas sock their moppet heads against the fence slats. The ground cracks, the lobelias rust like wire. The summer dances wildly with itself. Is it you? Oh, if it's you, Allegra, come into my room.
Autumn, and the torn tired gardens reel giddily. The leaves come down without changing colour. They are crisp underfoot, then soaking.
Then the winter falls in with an immense, laboured groaning, bending the branches back like fingers in an iron tango, snapping against the torsion of the trunks. Great crows, like incinerated pages, ride the currents, and clatter on the roof.
There is a false spring. Buds appear along the branches, growing where they can, but the sun, treacherous, early, abandons them to storms. Then there is lightning, then rain.
Listen, Allegra. The rattling of the last tram has stopped. The distant boom of an all-night tanker passes. In a far room the soft head of a dog bumps under a chair. Green digits in the dark go 3.00, 3.00, 3.00. You can hear the night pressing onwards with a high-pitched sound that varies in amplitude like the wheels of millions of minute wagons turning.
Listen. The sound you hear is the sound of mortal time. You hear it on the skin and under it; you hear it on the fringe of sense. Rhythmic, fearful, it is measuring life in years lived, things owned, children had, marriages, mortgages and love run out of. Sometimes it seems contained like a dead weight in a flour sack, and other times it is as insubstantial as a bead of light rising in the sky. It is the booming of air in arid valleys where progress is the passage of light through moving clouds and your voice is a light thing, dispersing. It is that music, that mud music. It is the birth of moments, hours, days and years to come.
The fender of night is about to bump softly on its mooring. It is a vast, scudding shipful of sound, and what if... it does not bump? Are we poured into it, dreamless, forever?
I try to shape the dark into your image, dancing slow, sad and quiet. But you are black air moving on useless hands.
I couldn't bear our mother's pity. It was real, sweet, and when she offered it, I wept. But I couldn't bear it. For the time being, I was set against my mother and longed to drive my fingers through the leaf litter under the trees at Harry Laurington's beach house to feel my father on my skin.
TWENTY-SEVEN
Relations
MY MOTHER IS waiting to be driven somewhere. In the room of the house that is now hers, all hers except for the mortgage she has raised on it, and has been all hers now for years, except for the mortgages she has raised on it year after year as if she were a small, stout dog chasing its docked tail slowly. I can see her through the thief-proof flywire door. The lock has been changed three times in the past twelve months, for fear that people who have been in the house when the keys were mislaid have in fact taken the keys and may come back with them for her furniture and colour television set and electric foot-warmer (a present from one of her boarders), leaving her bereft of her comforts
There aren't any boarders now, just dogs and furniture and photographs.
She is waiting to be driven to the cemetery. It is now over a year since Allegra died. We are going to visit Allegra. I can see her through the slats of her new trendy fence, which was put up on part of the mortgage money, and the flywire, thief-proof door with its three-times-altered lock, all of which, locks and door, were also purchased on the mortgage money, and she is bending over a table at the end of the hall. Her back is turned to me and she does not know yet that I am here, because her dogs have gone deaf and are probably, in any event, still in bed, and the noise of my car pulling up has not entered the slow traffic of their neurones. Perhaps I should drive round the block and pull up again, making more noise this time, so that their familiar racket will not shock her into thinking I am a terrorist.
She was not always stout, but she was never still, and is not still now. She is grabbing things, and I can tell by the shudder of her moth-holed cashmere jumper across her back that she is whistling. She has spent a lifetime not quite knowing how to whistle: part of her seems to say that a person who whistles is jolly, another part says that whistling is unladylike.
She does not look up as I click the gate. Sometimes she slinks through her house as if she were an intruder, apologising to photographs of people bearing features similar to, but sterner than, her own. These two-dimensional Mottes watch her from walls and tables as if they had the power to condemn. Nina casts the sternest glance of all. She seems to be saying that no Motte worth her salt would ever have gone off to be married to an enemy alien under the bushes in a public park, in a frock not even run up by a dressmaker.
Irregular pieces of chipped granite border her eccentric garden, in which she seeks to induce the life in every twig. The boozers and derelicts of the plant world sprawl and falter wherever there is ground to be remotely upright in. I take a piece of the granite that won't be missed and place it in the pocket of my jacket, so it knocks against my thigh as I climb the steps to the verandah where Allegra and I once married each other in front of a chest-of-drawers altar.
I ring the bell and drum collection, and immediately the dogs jerk into chorus. My mother gives a brief shriek and leaps into action, grabbing a coat, a bag, a brolly and two bunches of flowers from a vase as she comes down the hall. Clock, clock, clock.
'Just a minute,' she says as she fiddles for quite some time with the wrong keys in the latch, her features dulled under the sheen of the wire door.
'Your dogs need tuning,' I say.
She is through the door, dangling a live mouse in her arthritic fingers. 'What'd you say? Take this mouse, will you?' I take the mouse. She shuts the doors. The inner door, the outer door. 'Just put it over next door's fence, there's a love.' I chuck the mouse over the fence, and she places a bunch of freesias between her teeth as she negotiates the gate, stepping through like a ballroom dance student. On the street she starts to laugh. 'Sorry. Poor thing. I found it in the kitchen. I do hope it will be all right in there.' And she puts her nose over her neighbour's fence. 'Can't see it... ooh, there it is. It's all right.'
'Come on.'
'It's probably frightened. It's walking sideways. I hope you didn't throw it hard. Oh, there we are, it's straightening up. It should be okay. I fed it a biscuit.'
I am in the car and opening the door for her. 'Hurry up.'
As she clambers in, gathering up her wayward dimensions to fit in the seat, she's already apologising for putting me through what we haven't gone through yet. She kisses my cheek. 'Now I promise, cross my heart...' a feat performed with difficulty, considering her irregular burden and her ample left breast, '... I'll behave myself.' She starts to whistle and claps on her cemetery-visiting face as I start the car.
She tells me about some money-saving cake she baked this morning. 'You only use one egg,' she says. 'You'll have to tell me what you think. I've got a surprise this afternoon. Someone's coming. But she's exactly the sort of person who'll be able to tell how many eggs there are in a cake. We're related. You bake it for thirty-five minutes exactly. By my stove, anyway. I'm going to write a book called "Stoves I Have Known", because gas stoves burn your bottoms. I haven't got any cream to put on it. I couldn't afford cream. I don't suppose you've got any?'
'I have. You can have it.'
'Will it be enough?'
'Three bottles.'
'What sort?'
'Cream.'
'Thickened?'
'I don't know.'
'I don't like thickened cream. It's a con. Allegra used to say it was poisoned. Remember that cake I sent over? She said the cream was off, but it wasn't, it was thickened. Did you buy that rose tree you were going to buy?'
'Yes.'
'Standard Red?'
'Yes.'
'Good. Nina'll be pleased.'
'We're not planting it over Aunt Nina.'
'Remember when we did? Allegra wouldn't come, had something else on, someone's opening. It was after she came back from Adelaide that time. "God!" she roared, "Tony Furlonger's selling our furniture! And he's let Nina's grave go to pot!" We decided to plant the rose tree then. I don't suppose anyone's been to see it since?'
'Not that I know of.'
'It's probably dead. Remember how hot it was? Roses need a lot of watering when you put them in. And you, poor thing, you had to drive all that way.'
'I didn't have to. I chose to. I thought it would be nice to plant a rose tree on the grave.' And I think of the word 'nice', and Aunt Nina liking it and us planting the tree because we wanted to make a gesture to her, even if she wasn't there, not to assuage our guilt but because we'd loved her and we missed her.
Just as I'm tripping over Dadda and our incapacity to make a gesture there, my mother pats my wrist hesitantly and says, 'It's a shame we can't visit Dadda. It's funny, but I brought these freesias because he always liked freesias, your poor old father.'
'Yes. All right.'
'He wasn't such a bad man. There've been worse. He did his best.'
I can't help myself from saying, 'I'm sure it's quite respectable to bring freesias in memory of Dadda. And as for his best, it was a hell of a lot better than most other people's best.'
'Yes, Bel.' She's quiet for a while and I feel castigated by my own irritation. 'Well, she'll be glad you brought that rose. Allegra always loved standard roses. Did you ever find out...?'
'No.' She is asking whether we ever discovered where Dadda's ashes were. Somehow or other, Allegra and I decided we wouldn't tell her. We felt something shameful in it and decided it was better to be hostile than forthcoming. Our mother would almost certainly have wanted to go to Harry's beach house and pat the ground.
'Well, I'm just asking,' she says, and why shouldn't she want to know where Dadda's buried? After all, she was his wife for all those years. She'd sooner pat the earth he lay in than own something he left behind, but it rankles that she doesn't understand his stature as an artist. He was a failure to her, he 'did his best', as if art were an occupation for mental defectives.
I think sometimes there's no point in talking to anybody; it's a myth that people listen to what you say. They hear you, the pattern of noises is familiar; it falls on them like a blanket, new and exciting at first, but by degrees faded, thinner, till they start to think it's ready for the charity bag. What does she think of me? She thinks I am a failure, an unknown painter, at least I don't paint flowers. An Old Maid.
There are pencil pines bordering the cemetery, like distorted pompadours, bewitchingly green in the morning light, stuffed with yapping mynahs. Ha ha. Ha ha. Ha ha! Great are the passions of human kind! Ha ha! To be a bird... to have a brain the size of a walnut and to live automatic, but flyingly. To take fright and easily burst from the green, get it over with quickly, with an easy, slithering bird crap, heedless of fouling the fence wire, dart back and settle in again, all predestined. Instead of being ungainly, fleshy, adulterous and long lived. Well, inclined towards adultery, or men half my age: there's not an awful lot of choice when you're nudging forty. Allegra will never have to reach it.
I take the spade and the little rose tree from the hatch of my car and cross the verge as my mother swings the turnstile in the cemetery fence. A dog squeezes by us, its feet drumming up the broken path, scattering the little skinks into the crevices of old graves. Body bobbing, ears back, it leaps among the weeds and monuments, startling a squad of black birds into flying up briefly in formation, flapping once like a pennant, and settling out on telegraph wires above the castellated tower of the caretaker's house. Cabbage moths flit, taking up the wan light on the whiteness of their wings. Clock, clock, clock. My mother is following me, more like child than mother.
Here I am again on the pot-holed path, bordered by storm-racked cypresses from which the winter wind has wrenched great branches. Allegra's grave. The earth still humped up over her and growing dandelion buds, like bandaged heads. Here and there a yellow petal splits away, an emblem of pain. My mother takes her weeder out and a drear and mournful rain begins to fall. She looks up, fidgeting and biting her still-red lip. I dig the hole in the centre of the grave and plant the rose.
When it's done my mother turns away to toddle up the path, God knows what she's feeling. I take the jagged stone out of my pocket and drive it into the earth as hard as I can. 'There,' I say to my sister. 'Sweet dreams.' Because I feel as if I'm carrying a jagged stone in my belly and that all the juice of my young life is stemmed.
'I couldn't afford cream,' my mother says, smudging a hanky up her face. 'It was seventy cents a thingo.'
It is the parallel lines in my house I like: the louvres on the bedroom cupboard, the flow of green and white stripes on my counterpane. Lines that, in theory, never meet are always meeting as they come in under windows, through the slats in my bamboo blinds and the uprights of doors. They shift and spill themselves on the paintings all over the badly plastered walls.
My mother is waiting in the car while I fetch the cream. It is lines I like, the superimposition of form on form... the straight edge of a shelf shadow where it falls on the multicoloured spines of books. Against the bedroom wall my paintings are stacked back-to-front. Once they said I was a bold, adventurous painter, but now they say there's a difference between troubling and troubled. I have nothing to say as I drop the colours into them, the shadow into shadow, plumbing the dark.
She is standing inside me, looking out. As I close the door the little woven mat in the hall undulates from one side to the other as slime comes up in my eyes and the tears splash down on the round orange lid of the cream jar in my hands.
In old age my mother still picks up people with abandon, on trams, in shops, at the meals-on-wheels. As she grows more and more short-sighted, this habit of hers has expanded to include Nauruans, Vietnamese, Czechs, Lithuanians, along with her horde of Australians who happen to bear one or other of the half-dozen surnames such as Taylor, Emery, McVicar or Bloomfield, who could by accident of birth be related to the Mottes.
These people, netted at bus stops, cornered in banks, trapped in supermarkets, generally end up in her parlour, gazing at the pictorial record of her good works. The giant, aged photo of her grandmother Gilchrist presides from its own chair, the plaster not being of good enough quality to take its nail. On the mantelpiece, in her motherly arms, godchildren of great variety, full-faced, round-eyed, smirk and scowl. She, her sister and brothers, ponder the parade in stages of their youth and age from her sideboard. Granpa and Euphrosyne, their wedding portrait neatly sundered by the press of time, pass their eternal condemnation on their frivolous daughter, their eyes just visible above the throng. From Greece and Palestine, from Tokyo and Ottawa, the faces, the faces.
These days, whenever she has occasion, I am exhibited as her pièce de résistance, the single mother. Delivered as the facts are to perfect strangers, my blood immediately leaps to the boil.
The eyes of the most recent Bloomfield strain, bones creak in her old neck. 'Such a pity. Such a pretty girl. I'm sorry she was _soiled_.'
I leap from my seat.
'Oh, she's had a good few baths since then,' my mother sniggers. I feel like hitting her. 'Sit down, darling. You're spoiling the view.' A smirk spirals through her and screws her fat old bottom to the chair. She shunts the old woman's claw conspiratorially as if I were the prize slide in a slide night. 'Passionate nature,' she giggles. 'Comes from the Italian side of the family. They take umbrage easily. In fact... sit down, darling, there's a girl... if umbrage were a plant, I doubt whether you could find a pot large enough. But I always think Isobel's like my sister. Do you remember my sister? Round the jaw especially, here, look.' And she claws me down to my seat, holds up a studio portrait of Nina beside me and indicates the repetition of her one dubious feature in my face.
'Ye-es. And the nose, if straightened.'
'Not Isobel's nose, dear. Isobel's nose is as straight as a die; gave it a good pull every morning for years so it wouldn't be snub. It was Nina's nose that needed straightening. Pulling down, I should have thought. She always wore it a mite too high for me. Nina heartily disapproved of poor old Henry, I can tell you.'
'My father was not poor. My father was not old.'
'Exotic of you to have married an Italian. Especially during the war.'
'He was destitute, poor wretch. Didn't have anywhere to go or anyone to turn to.'
I recite to myself my litany of patience: she is old, forgetful, doesn't know what she's saying.
'When I die, Isobel, I want to be buried whole.'
I mutter under my breath, 'Is that with or without your tongue?'
'M. B. Bloomfield here is eighty-eight.'
'And every year of it lived!' the Bloomfield booms, her great voice reverberating deeper than would seem possible in her sparrow's chest.
'She taught me French.'
'She was the worst pupil I ever had. Stella Motte! With a name like Motte, fancy!'
'Well, they hadn't spoken French since the Norman conquest, M.B.'
'Honked like a swan with its neck caught in a bottle.' The old bag has enormous teeth. The eyes are shrewd and dark. 'You were always silly, Stella. Silly, but charming. She never did her homework. Used to bring me cakes to cover up.' She points her toe to show that, at eighty-eight, she still has an ankle.
My mother starts reciting, ' _La plume de ma tante est sur le bureau de mon oncle. Où est le canif de Charles? Le canif de Charles est dans la poche de Robert. Charles pleure. Pauvre Charles.'_
'My, what a memory!' the Bloomfield croaks.
'Your fault, M.B. That blasted book... What was it called?'
'Don't look at me, Mum. I don't know.'
'Yes, you do. You do. You had it yourself at school.'
'I can't remember.'
'Well, try!' she snaps and raps me on the wrist.
' _French without Tears_ ,' the Bloomfield drawls and I laugh, but my mother is cross with me.
'It'll happen to you one day, Isobel. You'll be wanting to say something and you won't be able to retrieve it.' Under pressure, the words fly out of her reach. Like a child in a roomful of wayward balloons, she stamps her foot. Beyond these walls, old women to whom I owe no allegiance get along splendidly without me.
The Bloomfield perches forward in her chair, her dark eyes narrowed to slits. 'Have you ever had your portrait painted, Isobel?'
'No.'
'Such a pity. You have a classical face.'
'I've had mine painted,' my mother pipes up. 'During the war. A young chap used to come to the canteen and draw me.'
'Oh, sugar and spice! A waste of paint!' the Bloomfield cries. 'Chocolate box features, Stella. You're a pair of pretty eyes in a dizzy head.'
'Well, it's true,' says my mother, on her dignity.
'I had my portrait painted,' whinnies the Bloomfield. 'By a real live man.'
'Well, a dead man would be hard put to do it,' says my mother.
'I have a classical face,' says the Bloomfield, 'otherwise, Bunny, how would you have recognised me after all these years? Of course, I knew it was you at once. You said, "Well, if it isn't M. B. Bloomfield! Bet you don't know who I am?" I didn't hesitate for a moment. "Bunny Motte," I said. I'm sorry your poor old Henry died young and rich. I liked Henry. A most intelligent man, even if he was a communist. I knew him before you were married, did you know?'
'No!' my mother yells. 'No, indeed I did not. He was always doing things behind my back!'
'I'd hardly call it behind your back, dear. He was friendly with a neighbour I used to have before I was put in the old folks' unit. Now, she's someone you'd know, Isobel. She's the lady who does the embroidery, Rose Hirsch. She's getting on now, of course, although she doesn't look it. Perfect skin.'
'Oh, her,' says my mother, archly. 'She used to be on the desk at Lauringtons'. I suppose Henry would have known her from there. How long ago was it, M.B.?'
'I think he was a frequent caller, even after his marriage with that other person.'
I thrust another slice of cake at my mother, and she gobbles it down, distracted, sticking her arthritic knuckle through chumping lips.
'I had a funny phone call from Eli yesterday,' she says, by way of a diversion.
'Who's Eli?'
'My grandson.'
'Isobel's son? Or Allegra's?'
I say, 'My son. Allegra had a daughter.'
'How old?'
'Eli's twenty-one,' I continue. 'Nin'll soon be six. She'd be here, but she's gone to swimming classes this morning and then she's going out to a birthday lunch with a friend.'
'Cold day for swimming,' says M.B.
'Heated pool,' I say.
'Good Heavens! In my day it was the river or the beach.'
'He said your phone was looking lonely, Bel,' says my mother, still talking about Eli. 'So he rang me up. That's him, M.B.' My mother hands around a plastic frieze of Eli, aged eight, in a rat mask.
'Very handsome,' says M.B. 'Father was a rat, I suppose?'
'Too true,' rejoins my mother. 'Tell Isobel about those books you write.'
'Ha! Novels! Four of them are out of print, and the fifth you can only buy in New Zealand.'
'She uses a _nom de plume_ ,' says my mother, pronouncing it with care. She rolls her eyes. 'Julia Drake.'
'Oh, hush, Bunny. I'm old-fashioned. It's my business. I can see the name means nothing to your daughter, but it satisfies my artistic streak – that's all it is, a streak, but it's human endeavour. I feel I will have left my mark. Oh, blast, I've dropped my hearing aid now. Wretched thing...' on her hands and knees, she strokes my mother's carpet, '... they make them so small. Can you see it? I'll never be able to get up. Your carpet smells, Stella. Have those dogs of yours been inside again?'
Two of them are coralled out in the laundry and one is injured and asleep in my mother's bed, its bitten head 'suppurating' on her pillow.
Temporarily cut off from communication, M.B. cannot hear my mother's diverting history of the dogs, how they were all strays, how she feeds them on fruitcake and vichyssoise, how their bowels are affected and the injured one has had its ear bitten off by the other male, cranky with eczema.
'Have you got eczema?' M.B. creaks to her feet with some help, slapping the 'pill of a thing' back into her ear. 'I get it in summertime. I have to take off all my clothes and go round _nude!'_ She feels for the seat of her chair, bent almost double, her backbone like a lizard's frill which skews her partially undone zipper to one side. 'A young man rang my doorbell last time. Gave him quite a surprise. Ah, life isn't so very unendurable; it has its compensations. What's that tune you're whistling, Stella? Makes me want to _go._ Stop it!'
My mother composes her mouth in a prim, sarcastic smile. She hates being told to stop it. 'My children were Truby King babies. Truby King mothers were instructed to whistle rather than hit. It was supposed to breed placid babies. But my two got around Truby King. They learnt to criticise in their cots. Allegra particularly. Twiddle your thumbs in a distant room and she'd yell out, "Stop it!"'
'I don't wonder, the way you twiddle yours, Stella. You'll curdle the air.'
'Stop it, stop it, stop it! All my life, stop it!' My mother slaps the arm of her chair repeatedly. 'Parkfuls of exhausted women with brutal babies!'
M.B. thinks it is hilarious. 'It's a wonder they survived with you as a mother!'
'One didn't.' Like the last wrench of a dripping tap that has stopped the drips entirely and the tap forever. 'Ah well,' shrilly, 'time for more tea. Bad for the bowels. It's all bad for the bowels. If Allegra had been a dietitian we'd never have got anything through to the other side of our faces, would we?' And she gives me a vicious, hysterical tap on the head. 'Ah, don't, can't, mayn't, stop! I draw my breath at the same bank you do, M.B. Do you like what the plasterers did to my ceiling? Cost the earth, but there are still cracks in it.'
So we are reduced to home improvements, how Nina used to criticise: _'Such a waste of money, when you could be sending the girls to decent schools._ It's always the ones who haven't any children. She'd send practical things, like a plastic picnic set for Christmas, when we never went on picnics.'
M.B. loses her hearing aid again. This time it has landed in the cream of my mother's cake; she plucks it out and licks it. 'Poo! Tastes of wax!'
M. B. Bloomfield shakes her iron-grey head. 'You a mother! Bunny Bloody Motte! Whatever would Truby King think? Licking my hearing aid!' She stares at the carpet, bemused, fluttering her long, gnarled hand. 'No matter.' She plants her feet on the floor and rises. 'Time to go.'
Julia Drake, authoress, all pride residing in her robust nose, bobs for a higher perch on her front doorstep, flings her key at the slot and is clouted on the claw by the beaten plaque that holds it. 'Ouch! I hate old age!' She stamps. She glares at me. 'Don't fidget. I won't keep you long. What I want you to see is just through here...' She leads the way into her box of a maisonette that makes no noise, casts blurry shadows and smells as impersonal as a new refrigerator. It makes no effort at all to greet her, but just contains, like the weary idea of a young architect with an old mother. She opens a little white door, with some difficulty, over the thick carpet.
And there, over the neatly made bed with its white, tight coverlet, is the portrait she spoke of, in half profile, looking down.
'He made me sit still for what seemed ages. Fed me whisky for my aching neck. What do you think?'
'I think it was painted by Leslie Hallett. It's lovely. Really adventurous.' He has made her lay a hand by her chin and incline her head above it. Though the hand and face are ageing, the attitude is somehow young, so the portrait conveys the passage of life. 'How long did it take him?'
'An afternoon. Leslie was a very quick painter.'
'Did you like him?'
'I didn't know him very well. He was Harry Laurington's friend and I'd known Harry since childhood. It was painted down at Indented Head in Harry's beach house. I seemed to be the only person with money in those days. I told Harry that if this young man Hallett, who was Harry's "find'', painted my portrait, I'd pay him.
'You know, it's fate that our paths should cross, Isobel. I first met your father on the day that was painted. Your father was very charming. At that time he was with Viva. You know, it's always fascinated me why Viva married Harry, or to put it another way, why he married her. There was no doubt in my mind that, at that stage of his life anyway, Harry was homosexual. I was under the impression, a very strong impression, that he and Leslie were having an affair. Harry used to tell me about his affairs, you see.'
TWENTY-EIGHT
At Harry's Beach House
THIS NEWS OF Harry Laurington galvanised me. I spent the rest of the afternoon with M. B. Bloomfield, and what I heard caused me to pay Harry a visit.
'Thank God I can still handle a coffee cup,' he said, as he half sat, half lay, propped up in his azalea niche. The nurse, Jean Higgs, set his coffee tray the way he liked it.
I still felt I had to establish grounds for a visit. Although I liked Harry and I felt he liked me, we hadn't yet become friends. I took my cup in my hands and wondered how to begin. Once Jean had left the room I said, 'Harry, I want to ask you something very personal. I've met someone you know. She's a friend or relative of my mother's, so I more or less met her by chance. Her name's Bloomfield, M. B. Bloomfield.'
'Oh, Molly! She's still alive then? Must be getting on now, she has a year or two on me. Our families knew each other and one of her brothers was in my house at school. Ted. I think I heard that Ted had died not so long ago?'
'Yes, he's dead,' I said, 'And so is her other brother...'
'Sandy. Yes, I knew about Sandy.' He pre-empted me a little too quickly to seem innocent of the question that lay behind the prompt. I was going to hinge my conversation on Sandy Bloomfield because M.B. had told me she was pretty certain there'd been an affair between him and Harry. Harry had chosen her as his confidante, probably in order to cover Sandy's tracks. He had sworn to M.B. that he was in love with another young man, thinking no doubt such risqué information would divert M.B's attention.
M.B. believed that Sandy had managed to keep his homosexuality under wraps all through a long, prosperous society marriage, because his wife would hear nothing against him and was thus protected from knowing what she would rather not.
I was about to devise a leading question to recover lost ground when Harry started ringing his bell for Jean and saying, 'Poor Jean. She'd probably like a cup of coffee herself. I'll ask her to join us.'
I wondered whether he had just betrayed himself, but I was too slow to force the moment while Jean was away fetching herself a cup.
Harry was faster. He said, 'You know, Molly Bloomfield went to Europe in the thirties. Doing the grand tour. She went to England first, of course. She was one of those people who spoke of it as "home", though I should think even her grandparents were born in Australia. Then she went to Germany – she prides herself on her German – and she sent back such glowing accounts of England and Germany that we rather thought we might be rid of her. And I say rid of her because, although she's a forthright person, she's very critical of everyone and everything and she's inclined to pry. But you might understand Molly Bloomfield – funny that her name is almost the same as Joyce's Molly Bloom – when I tell you that when she got to France she stayed with a family whose sons were in the _Action française_ , and she was damned proud of it. She's as anti-Semitic as they come. Doesn't even believe in the Holocaust – or didn't the last time I spoke to her, some years ago now. I didn't keep up with her much because her attitude was hurtful to Laurent and Rose. She was a clever woman, but obtuse.'
By this time, Jean was back in the room, and I could see just how adept Harry was at protecting himself with mental and social barriers. But the fact that I knew what he was doing did not escape him and he coloured a little bit as he drank. 'Tell me, Isobel,' he said, 'are things looking up for you? I understand Reg has cornered the whole "Nice Girls" series for you.'
I didn't feel like answering him. I got up and strolled around the room, looking at the artworks on his walls but without being able to take them in. He was treating me in a shallow, evasive, manipulative way. Perhaps he was desperate, but what about me? Inside my chest I felt there was an iron stake.
I thought but was not able to say, 'It's my relationship to Checkie that is worrying me.'
'You've never been down to the beach house,' Harry said. His tone was kind enough, but it was not balm to me. 'Would you like to go, Isobel? There's a chap down there painting at the moment, but the place is easily big enough for two. You could stay there for a while and get some serenity into your life. Reg says you haven't been painting. I can understand that, but you must remember that gifts like yours are unique. Gifted people often don't have easy lives because they are impelled to go where others won't and think what others dare not think. Allegra used to protect you, didn't she?'
As I nodded my head the stake in my chest moved up and down dangerously. Then he said, very gently, 'Come here, Isobel,' and he held out his good arm for me to come to him. Maybe he was opening his heart to me; I stopped at his hand, as though to come just within the ambit of a gate, but he slipped his arm around me, inviting me to put my head on his chest. It was too much, the stake inside me churned and I wept bitterly. I knew that Harry's arm around me was warm, but I was in a hard and terrible place of my own where his embrace was like trying to shift a boulder with a feather.
I accepted Harry's invitation to the beach house. Even if it was a red herring, I needed my father and it was the only place I could think of where he was. Reg drove me down.
On the journey, I asked him, 'Is Harry gay?' and Reg said, 'Why do you ask?'
'Because I want to know.'
'What do you want to know that for? It doesn't matter.'
'It matters to me.'
'What difference would knowing that make?'
'Well, if he's gay, how come he's Checkie's father?'
'He might be bi.'
'Stop avoiding the question, Reg.'
'How would I know the answer? I don't go around asking people whether they're gay.'
'You just don't want to answer me, do you?' The flat acres were passing us by. Indented Head seemed a million miles away, and I as far as ever from an answer to my question. Then, 'Is he Checkie's father?' I demanded.
'Well, I suppose so. He's spent all that money and time on her, so I guess she's his daughter.' Then he laughed. 'What? Are you afraid of being related to her or something?'
'Wouldn't you be?'
'Checkie's not bad.'
'You're outrageous, Reg. For God's sake, tell me what I want to know!'
'Allegra was your sister.'
'Fuck! I'm asking you if my father was also Checkie's father? For God's sake, tell me!'
But all he would say was, 'There's fatherhood and fatherhood. And then, there's motherhood, and, believe me, motherhood is much more important. Women are weightier characters than men.'
'You're being cruel and unfair, Reg. I want a straight answer.'
'Well, I'll try to make sure you get it, but it isn't mine to tell, Isobel.'
I was not a David. I couldn't keep up a barrage until someone weakened or took pity on me. The evasions of both Harry and Reg seemed to add up to an unpalatable truth, one that would partially explain the wrangle over the will. Even Wednesday had wondered, albeit speculatively, whether Checkie might not have a legal claim equal to Allegra's and mine.
I thought over Rose's description of Allegra the First, Dadda's mother, and her blonde hair in its Marcel wave. Latterly, Checkie had resorted to a hairdo similar to a Marcel wave; the frizzy hair she'd had as a child had given way to a smoother look. She'd come to the funeral but kept her distance. Someone said even she was weeping. Later, she'd sent me a card saying she was thinking of me. I didn't want her to think of me and threw it away. The idea that I might be closely related to her made me feel sick.
I couldn't eat the lunch Reg bought for us in Queenscliff at Mietta's. However, the old hotel, lovingly restored to opulence, full of light and air and respect for things well done, conjured up visions of Euphrosyne breaking a long journey here, or simply coming to have some time at the sea, and for a while I felt better and downed a couple of glasses of white wine.
We arrived at the beach house at about three in the afternoon. Just as I had heard, the beach front had very little to recommend it; a smear of sand and pebble kept the tilting greenish sea with its scrolling straps of kelp from a low sea wall, then there was some unkempt grass, a straggle of road and a spruce hedge, behind which the house disclosed itself.
What I had heard about the house was correct, too. It was of unambiguous lines, the work of a person or persons with a highly developed sense of proportion. White painted weatherboard, large without seeming so, it sat on flat ground, an elegant iron roof sweeping down from the ridge to create a deep, shaded verandah around all that could be seen from the front. There were three large, white, wooden sun lounges in the verandah's shade. I had to say I liked it, it was Harryish, and in spite of his evasion of the truth, I wanted to think well of Harry. Inside it was as he had said, furnished and carpeted all in white, the bathroom pristine with its immaculate old ball and claw, the bedrooms featuring beds so soft they might have illustrated 'The Princess and the Pea', their taffeta-covered eiderdowns just like Nina's, only white. It was an interior upon which anyone might impress their story without feeling blotted out. Perhaps Harry's mother had collected rare stories as well as rare plants.
The artist-in-residence, an Englishman called Jack Ives, was stretching canvases on the lawn. He stood up to greet us, holding out his hand. He was a smallish man though well built, a bit self-conscious, freckly faced with thinning red hair. Reg had said he took himself far too seriously, which made me think he was probably a very good and very committed painter and Reg, who confessed to being more self-indulgent than anyone else he knew, felt threatened.
'How're you going to get that inside?' asked Reg, probably thinking that the 'Pom' wouldn't have thought out the practicalities of stretching a large canvas out of doors. But he had thought of them. There was a converted stable at the back of the house with a large number of floor-to-ceiling racks, and Reg's blood probably curdled further when he saw that there were a great many large paintings hanging on them.
And they were superb paintings, superb. Reg probably didn't like it at all when I couldn't stop myself from saying so.
Jack Ives was addressing all the academic concerns we had given thought to at Figments and Mad Meg, and yet he was an expressive painter. All else aside, the Siècle Trust, The Brolga and Checkie and whatever might be the truth concerning them, here was a brilliant painter. A wizard with a powerful, sweeping arm; a creator; a destroyer.
And a man of few words, very few. He excused himself and continued working. Reg and I sat on the sun lounges and drank more wine. He'd brought me half-a-dozen good reds and half-a-dozen good whites. He'd also brought food and blankets and the sort of painting materials he knew I liked. I was touched. Reg didn't have to take an interest in me, but it was his care, as much as anything else, that brought me through this most difficult time of my life.
After Reg went Jack Ives left me alone, sitting on the verandah, pleasantly numb around the lips and cheekbones from drinking. Harry's mother's Australian native garden was planted in an odd, almost military fashion inside the spruce hedge. I could see the grevilleas and the raised rectangle of ground that was probably Leslie Hallett's grave. This grave had found its way into some of Jack Ives's paintings as a white geometric shape which excluded and repelled the living forms around it.
I was curious to see the work again and stole, barefooted, around the back of the house to take another peep. He was squatting on the ground, still absorbed in his task and oblivious of me. I stood, glass in hand, on a mat of flattened golden grass, in an angle of shade where I wouldn't be seen.
Though the landscapes were expressively painted, both vibrant and alive, it seemed to me he was painting the self as observer, the blank, mechanical, impersonal self, the orderer. These paintings acknowledged the existence of the human machine, like the calipers that turned Arnie Russell into the gold standard of love. It had certainly been an automatic, sexually driven interaction. There was an inveterate philistine wrapped up in that Gold Standard. I'd fallen for him in the way Granpa might have fallen for a cow.
Jack Ives's calipers imposed the rules of observation on the human aesthete, rules both painter and viewer shared. As a painter, he seemed to be up very high, using a brush with a very long handle and a very wide sweep, so while he was the inventor of these pictures, he did not invent the rules by which they were pictures. In those he was a participant.
As I made my way back to the verandah through the slants of afternoon light, I began to think about Link and laughed at myself. I hadn't exactly started a love affair with him, but he'd been there through the worst of Allegra's suicide. When I finally started sleeping at night, he'd stayed with me, slept in the same bed and consoled me when I fell from sleep into horrible awakenings. It was several months now, and he was talking of getting a girlfriend, so I had better stop my deluded mind from wandering. Link was beautiful, but it was I who needed him, not he who needed me.
I couldn't guess what Eli thought about it all. He'd just said, 'Don't hurt Link, Mum. He's only young.'
I was feeling cut off from Eli. I wondered if I knew him or understood him at all. We had the problem of David between us now, and it seemed insurmountable. I was at a complete loss to know what his feelings for David consisted of. Was it a man-to-man thing? A father–son thing? I couldn't guess.
I hated myself because I did not even long for Eli anymore, as I had when he was a small child and the dearest of all people to me. I know now that I was feeling as Allegra had felt, as though I had run out of life, convinced, absolutely certain, I had run out of life. Those brilliant paintings around the back of the house represented everything I myself had ever attempted in art; I'd never paint as well as that. No. I would stick around a little longer, until everything was in order and my mother and Nin and Eli would be catered for, and then: I could see only good resulting from my death.
But before I died I would go and peer more closely at the rare Australiana, find Leslie Hallett's grave and sift the dirt that lay over it through my fingers.
I was about to swing myself off the sun lounge and do just that when something totally unexpected happened. The Brolga came to pay Jack Ives a visit.
The Daimler slid up over drying fronds of spruce recently cut from the hedge. My first impulse was to go inside and lock myself in, but I fought it and stayed where I was, making myself virtually the first thing she saw. Jack Ives was still around the back of the house.
She sat there for a while, eyeing me, her hand on the knob of the car radio which was spewing out news. Then, with precision, she clicked the radio off, swung from her seat and opened the car door in the same movement, put her handbag on the roof of the car and shut the door with care, still watching me.
She did not avoid me but crossed in front of her car, came over to me and sat on the end of the sun lounge. I was turning away from her, that stake in my chest starting to do its horrible work all over again.
'May I have a drink with you, Isobel?' she asked. I didn't move but she got up, skirted the lounge, took up the glass from which Reg had been drinking, poured herself some wine and sat facing me so I couldn't avoid her.
'I didn't know you'd be here,' she said, managing sincerity. 'Had I known, I would have left you in peace. I'm just down here to organise some transport for Jack's paintings. We'll be showing him at the Trust next week. Are you able to talk to me?'
I nodded.
'You've been to see Harry.'
I nodded.
'He told me.' She sipped, holding me in a gaze that seemed to have much more to do with the person behind it than with me. She was going to tell me the truth and she wasn't afraid. She was dressed in white, the saint of the charlatans. She had dressed to match the house. All kinds of accusations coursed through me, only to jam like logs in a river. I was in great pain.
Meeting my eye only occasionally, she said, 'It's quite strange I should find myself sitting here talking to you on this lovely, deep verandah of Harry's mother's house. I might have lived an entirely different life, but I was always attracted by this kind of poise.'
_Poise!_ I thought. Then, _Poison! 'Thank God it's you and not your sister, Isobel. Henry's dead. I'm not good at grief_.' The log jam was making me curl forwards in my desire to attack, but I could not sit up; there was a heavy slab of grief preventing me.
'You look uncomfortable.' But she made no effort to relieve me, no kindly gesture, she might as well have been preparing to tell her life's story to the sun lounge.
'I've come a long way from my beginnings, Isobel. Do you know that?'
Finding myself unable to speak, I shook my head.
'Do you want to know?'
I nodded again.
'Tell me if you want me to stop.' How could I tell her? My voice box was jammed.
She stood up and stretched, running a hand over the small of her back. I felt like saying, 'You've seen _Gone with the Wind_ too many times.'
'I'm a timber feller's daughter,' she said, not knowing, nor even inclined to read what was in my mind. Turning towards me, she leant against the verandah post, dressed in white, certainly, but blocking the light, her shadow long and attenuated – like a brolga's, I thought grimly. As she settled into her story, I felt she'd lied about not knowing I was here. She must have known – Harry would have told her. Perhaps, all the same, her original sincerity was meant, though whether for her own sake or mine I couldn't tell. She would have thought it all out, what she would tell and what she would not – perhaps she would tell it all.
'East of Melbourne there's a group of little towns,' she went on. It was not a spur-of-the-moment story; she'd spent a long time composing it. There began a hot, live pain in my viscera, better than the seizure in my chest, at least.
'They used to call them the timber towns. They're really quite picturesque to drive through. Very green. Very green because they're very wet – in winter, anyway. They're in a mountain ash forest. I don't really know who lives there these days, but I like to drive through sometimes in the Daimler and to look out and to say, "Aren't these picturesque little villages?" as if I knew nothing about them. Sometimes I go to a certain pub up there and I might have a drink with the men in the bar. If they know who I am, they don't acknowledge me.
'Last time I went I had a conversation with a young man, a Koori, not a full blood. A nice, garrulous young man – a little bit drunk – who told me he wanted to write, to write poetry. He'd given up working with the Department of Conservation, Forest and Lands, even though he said these days they planted as much timber as they felled and there weren't, strictly speaking, any straight timber fellers anymore; people these days were trained. They had more skills, they planted as many trees as they culled and knew how to work the modern machinery, designed to reduce damage to forests. Still, this young man didn't want to work with his brawn anymore. He'd had enough of it; he wanted to work with his brain.
'We were sitting at the bar, and on the other side of me was an old man, well, ostensibly old, though I know for a fact he's no older than me. I went to school with this old-looking man.'
I couldn't feel any sense in what she was saying. I could see a seagull raking through Dadda on Leslie Hallett's grave, and thought maybe a goodly proportion of Dadda was seagull by now. And ant, perhaps, and cruel little curled-up clover seeds.
'He didn't recognise me and I didn't enlighten him, and not only because at an alcove table behind us there was a woman who'd been to school with us, too. A toothless antique with a shopping trolley full of port.'
Then I began to register The Brolga as an elderly woman. Her hair, once dark brown, was now predominantly grey; there was virtually no colour left in her once-blue eyes, her slimness, which had made her look so chic, had become thinness. Her legs were so skinny that her pantihose was bagging a bit at one of the ankles. She caught me staring. 'Seventy next birthday,' she said, and with a shock I realised that Dadda, too, would have been seventy and, though I felt like a child in her presence, I myself was now middle-aged and had seen the faces of my friends take on their kinder, older lineaments as well.
She continued. 'The old man laughed at the young one's saying he wanted to work with his brains, but it wasn't necessarily cruel laughter.
'When we were children there was a horrific fire through the area. About twenty locals were killed. Those of us who were lucky enough to make it in time were protected by sheltering in a dugout, which is a series of underground passages cut in a zigzag into a hill. Over the entrance you hang a blanket that dangles in a trough of water. The blanket takes the water up, as an oil wick takes up oil. Where there are great, out-of-control fires there is usually wind, and the idea of the blanket is to filter out smoke, because most people die of smoke inhalation in fires.
'Now, those who have never been in a raging fire cannot know that it travels as fast or faster than a speeding car, so people on foot are done for. The reason why my old schoolfellow was laughing was that the state government in its wisdom has seen fit to bulldoze the dugouts in the timber towns and replace them with an emergency plan. People now are required to congregate in local school halls, some of which are weatherboard and raised off the ground, and none of which has been built with any particular regard to protecting them from fire.
'You can imagine the heat of a fire in tall timber – well, you can't imagine it – it's a lethal heat, attended by smoke overhead. How are people to be rescued from above-ground buildings? They would be better off sitting inside a hill. But, as the old man pointed out, this country is full of experts and academics who know better than anyone else. They have even dispensed with fire trails cut through timber to allow easy access to spot-fires. I suppose you would call it desk or computer forestry.
'The reason I tell you this is to point out that people in timber towns are still regarded with contempt, as if no wisdom resided there among them.
'I grew up in that atmosphere. My mother was a minister's daughter. After my father came back from the First World War he left an unviable soldier settlement in the Mallee for a logging job.
'He became a tree feller, but really he wasn't robust enough for that kind of life and he started drinking, believing his fortunes had fallen so far he'd never be able to pull himself out of the mess.
'There was another problem with my father. I was born during the war while he was away on active service in the navy. It seems he doubted that I was his child. I don't know what grounds he had for doing so, and I shall never know myself whether he was or was not my father.
'Tree felling took its toll on him physically. He was often injured: strained muscles, split hands, that kind of thing, so he spent quite a large amount of time at home. And then came the timber strike in 1929, when he couldn't have worked even if he'd wanted to.
'I was eleven when my father went on strike. They were out for months and things were very hard indeed in towns such as the one where we lived. There was not enough food, and people like my mother walked to richer towns to beg for it.
'There was no food, but somewhere or other there was drink, because as things got worse my father and several other men alleviated their misery with alcohol. You've no idea how it was to be in a family with a father who drank. The families where the men didn't drink either looked down on us or pitied us. There is a kind of respectability among those people; they're intensely practical and very knowledgeable about the bush. Men who are suited to the life have no need to drink. So you can understand it when I say that we, who had neither brawn nor practicality nor stoicism, were the lowest of the low.
'We lived in what you could only call a shack. Outside it was unpainted weatherboard with a tin roof, and inside there were four rooms, all unlined. My little brother and I slept in the same bed. Sometimes it snowed in winter. The bedclothes were never adequate, nor were the clothes we wore in the day.
'My father, having no work to do, became vindictive and began to use me to reproach my mother with adultery. He destroyed her spirit by degrees. You think I'm a cold woman, Isobel, but I've felt things. When I was a child, I felt my body was full of worms. I hated the way we lived. I would lie in bed at night with my life tying itself in knots in my stomach. I was physically cold, I was afraid, I couldn't sleep. My nights were an endurance of dark hours.
'My father was genuinely not well enough to go to work during the Depression. All the same, he would have to queue up with the trainloads of unemployed men who would arrive in our town and huddle round the boiler in the sawmill from about five in the morning onwards, hoping for work. A boss would come, choose the ones who looked fit and send the others packing back to Melbourne or wherever else they might have come from. For rejected locals it was very demoralising.
'Then my mother got a job, through a friend, in a Melbourne clothing factory. She'd be up before dawn to catch the train and never home before eight at night. I think her plan was to take Leslie and me to Melbourne to live, but she had to establish herself first.
'Her marriage was over, but in those days it was ridiculously difficult to get a divorce. Anyway, her long absences during the day had bad repercussions for me. My father began to get interested in me physically. I won't go into the sordid details, which are better forgotten anyway, but suffice it to say, my first experience of sex was with him, and when I was fifteen he made me pregnant. I didn't know how to tell my mother, or even what to tell her. All I knew was that if your periods stopped it could mean you were pregnant. I couldn't even picture my insides. I was very ignorant. Anyway, I told my mother as best I could. She was appalled, but had the sense and intelligence not to blame me in any way, and to take me up to Melbourne immediately and help me get an abortion. It cost us every bit of money we had, including some her friends at work had collected for her, and all of my father's she could lay her hands on in the house. She only had enough left for fares.
'I spent the first week after the abortion in Melbourne with the friend of my mother's who'd collected the money for her. She was an Italian woman whose daughter, of my age, had just died from heart disease. I was given this girl's clothes.
'My mother decided to go back home and collect Leslie so the three of us could live in Melbourne, away from my father. But my father was waiting for her and beat her for taking all his money. She tried to sneak away with Leslie on the early morning train, but he came after her. There was a scene at the station. People who saw it couldn't say whether my mother threw herself in front of the train or fell into its path while struggling to get Leslie away from my father. The train driver had put his brakes on, but my mother was hit and it was fatal. The only thing the coroner was finally sure of was that my father hadn't pushed her.' Here she laughed bitterly. 'In my view, even if she did commit suicide, he pushed her.
'While the coronial enquiry was going on I stayed in Melbourne with my mother's Italian friend, but I was afraid of my father coming and also of becoming this woman's substitute daughter, so I tried to find somewhere I'd feel safe. That was how I came across Rose.
'I know you disapprove deeply of my behaviour towards Rose, but I might have met anyone in the position I was in. Many untold futures lay ahead of me. I met Rose; Rose was important, but she wasn't cardinal. Soon after I met Rose, I met two other people – through Rose, it is true, but there were many other ways I might have met them.
'The first was your Uncle Nicola, and while I couldn't speak his language, I could appreciate for the first time in my life that there were men of culture and grace in the world. Reasonable men who were also men of soul. He fascinated me. I loved him in a reverential kind of way. I used to ache for him to put a hand on me and bless me with dignity like his.
'I felt I had to learn to talk to him, and so I did. We conducted our conversations in French and for the French, it is true, I have Rose to thank. But I also have Rose to thank for something else.
'My ticket to freedom was not Rose, but Laurent's uncle, Julius Lichtblau. Rose set it up. "Why not?" she said to me, "He's so rich!" Well, he was rich, very, at a time when ordinary people were being reduced to working in sweatshops, or not working at all. My own mother had been reduced to beggary for a while.
'I was a whore, if you like, Isobel. Fucking for money at fifteen years and eleven months of age – and I knew I was a whore. Rose seemed to think it was something in the general run of life. French men had mistresses – Julius was French by birth, he knew how to be careful, I was young and pretty and needed the money.'
'She told me you were his secretary.'
'Yes, she would.'
'Rose might not know that was the case.'
'Then again, Isobel, she might. As far as I was concerned I just prayed it would stay a secret. If it did not, I would deny it. I was going to get out of there, after all, and, furthermore, I was going to rescue my little brother. I was going to help him become what your Uncle Nicola was. You see, Leslie was talented, very, but he had no way of expressing himself. Freedom isn't a birthright, it's a social invention.
'I kept my life in separate compartments. Julius and his surgery during the day and three evenings a week; your Uncle Nicola on Sunday afternoons and evenings. Before going to see your uncle I would take a bath and then a shower, as if I could wash myself clean of my rottenness. I would try to send Nicola mental messages that what I needed was his hand on my shoulder or on my head, or the nape of my neck, to cleanse me. I felt riddled with filth.
'I would tell him about my little brother and how beautifully he could draw and he would tell me about Henry in Paris. My dream was to get to Paris and find a way of getting Leslie there, too. In Paris we'd be safe. Our father wouldn't find us there and Leslie would be in a milieu that would favour him. He needn't grow up into a life of relentless poverty. But, of course, I had a problem: I was too young to be eligible for a passport, and Leslie was not only too young, he was way too young. I had to ask Julius for help.
'At first he was terribly reluctant. He didn't want to lose his screw, and then he was supposed, by the Hirschs anyway, to be in love with me. That was my introduction to blackmail. Julius wasn't very careful. He gave me presents. He even bought them for me in shops, when I was with him. I started to say I'd tell Orpah about his goings-on if he wouldn't help me to get to Paris. So he helped me. He won custody of me in a court case against my father. My father didn't appear; it's quite possible he knew nothing about it. All Julius was obliged to do was put a notice in the paper. It was enough for the Italian woman to testify. It wasn't quite what I wanted, but I was learning: it got me to Paris. I planned to send for or come back for Leslie when I'd fixed something up.'
Her head was bowed in what seemed like remembered pain. For the first time in my life I felt something for her and wanted to know more. I wanted to hear about Paris. I wondered if it could possibly have been Viva about whom Dadda was talking that day long ago in the Pantechnicon, the day of the double-spouted teapot and the cellophane butterfly, when he said he had been in love in Paris when he was my age. When I was nineteen and he was fifty.
Surreptitiously I'd been seeking the identity of this person, piecing together the Paris of the time, comparing my speculations with fact, thinking that if I should discover who it was, my father might be redeemed, or at least reprieved from callousness. I didn't understand how he could have left us the way he did and still be judged a good man.
TWENTY-NINE
Paris
ONE THING I will say of Viva is that she knew herself. She was very well aware of the frailty of her emotions. I have thought over our conversation for a long time, and I can see that there was more depth to her than the words of the story told that day at the beach house would have implied.
When Viva arrived in Paris in November 1934, the sixth French government in four years had just fallen. Standing on the steps of his residence in Avenue Foch was Gaston Doumergue, ex-president and ex-member of the _Cartel des Gauches_ , attired not in presidential nor ex-presidential garb, nor indeed in any manner of dress which would have identified him as the left-wing Liberal he had always claimed to be, but in the _beret basque_ of the _Croix de Feu_ , whose ranks were parading past him. The fingernails of his right hand grazed his forehead in a gesture of salute. History might have pulled his arm out straight, with palm extended, catching up with the new fashion in Germany, but History was too busy to worry about the details.
A wave of happiness swept over Viva, not because of the duplicity of Gaston Doumergue, whose name she did not even know, but because Paris was Paris, undeniable even in winter, both lived in and pampered, spat upon and reviled. Here was the Paris of comfort and the Paris of pain, Balzac's city where people plotted and played each other false, where love sent unlikely couples scurrying to trysts, where fates were sealed and feelings were trifled with, where there were intractable confrontations and, most of all, where there was art.
Viva was in love with Paris before she reached it. Her visions came from talking to Grace Lonsdale and borrowing books from her and her francophile friends. Viva had studied the _Indicateur des rues_ with such close scrutiny that Baron Haussmann's city grid lay in her mind like a cradle from which she would one day rise up, civilised and sophisticated. She had already worked out how to get from where the Katzes lived in Boulevard Raspail, Montparnasse, to St Germain des Prés, Notre Dame, the Louvre, the Opera, the Madeleine and the Arc de Triomphe, all of which had figured one way or another in the books she'd read.
She had arrived at the place Rose had left on her journey to Milan twelve years before, and just as Paris had folded away for Rose into a two-dimensional city contained in an encyclopaedia with gold-tipped pages, so it opened out for Viva, an elaborate greeting card into which she could pass from dream to reality.
Though the Katzes had written telling her to let them know when and where her train would be arriving, Viva had only forwarded the date to them, saying she would use the opportunity of her arrival to see if her French could get her from one side of the city to the other. She had practised her opening lines on the train journey, and was proud of herself when she arrived at mid-afternoon in Boulevard Raspail at one in particular of the ubiquitous six-storey stone buildings under grey Mansard roofs, the one housing the Katzes and Corettis.
Not only had she given the directions correctly, she'd also made a phone call to tell her hosts she'd arrived and was coming and, further, she'd managed a better than rudimentary conversation with the cab driver on the way.
It was Rose's sister Simone who had taken her phone call and it was she who was waiting in the lobby of the apartment building when Viva arrived. Simone, though quite good-looking, was not as attractive as Rose. There was a heaviness about her, accentuated while she was helping Viva heft her luggage into the dark, tiny birdcage of a lift. Simone chatted the whole time, eyeing Viva up and down in smiling appraisal, as if to say 'so this is what you look like'.
So old was the apartment building, and the parquet floor so walked upon, that it dipped in valleys from its height at the skirting boards to its trough underfoot. It was sepulchral on the landing. Thin light leaked up the light well and the stairs that spiralled around it. There was a double door to the apartment that lost some of its grandeur as Simone, rattling, sighing and twisting knobs, ran a gamut of locks to open it.
The apartment, when Simone had managed the door, was full of light, the shutters thrown open to reveal struggling greenery on the balconies and glimpses of the winter filigree of treetops in the Luxembourg Gardens beyond. Rivkah Katz, a short, rotund, spirited person, leant out the door of her kitchen. Then, crying, 'Ooh la la!' came pattering out, wiping her hands on her apron, to greet Viva affectionately and show her the room she was to share with Allegra.
It was in a suite of three traversing the end of the hall that divided the front of the apartment from the back. Bridging Allegra's room at the back with Henri's at the front was a small bathroom. The suite was almost self-contained, its three doors opening into a little rounded vestibule that could be closed off, although it never was. Henri slept in what had been a library and, before that, at the time of the Second Empire, probably a servants' preparation room. It had front-facing windows, visible through the slightly open door.
Allegra was not there but the room, which looked west and thus into one of those shafts down which plumbing hurtles into the abyss between back windows, was quite large, large enough for two beds, two wardrobes, one writing desk and a great many boxes. These, Rivkah explained, contained papers Allegra was proofreading or translating, or letters that had come through the Antifascist Concentration to which she had to compose answers, letters asking for money or, far less frequently, offering it. The paperwork was relatively safe in the Katz household – whereas the Concentration was subjected to occasional raids, either by police at the behest of the Italian Embassy or by secret service people trying to sabotage the antifascist effort in various ways.
Between the sitting and dining rooms was a pair of grey marble fireplaces, back to back, above each of which were large, gilded mirrors, somewhat spotty with age and inclined to impart a champagne tint to whatever was reflected in them. They made the apartment look warmer than it felt, there being no fires in the grates due to a scarcity of fuel.
The sitting room was walled with books, among which Viva noted a section for contemporary works: Colette, whose works she'd read in English, and Gide and Valéry, whom she promised herself she'd read in French.
'Do you have anything in Italian?' she asked, thinking that if she could read French, what was to stop her learning to read Italian? She could probably find out more about Henri and Allegra if she could read Italian.
The Italian books were kept in Henri's room, she was told. They were largely political and better kept out of sight of people who might be more curious than was a good thing. Rivkah thought to warn Viva that Grandfather Grafman was a case in point, not being a fellow traveller. With him, it was better to sit in the dining room out of sight of books in any language, so if he wanted to argue he could, without Grandfather Katz resorting to the printed word to catch him out.
Viva was delighted with the dining room, not that it was anything more than a space filled by a large table and chairs for ten people. There was a Jewish candlestick at one end of the otherwise empty tabletop, and at the other side of the table from the fireplace was a substantial, mirrored sideboard. What charmed Viva were the drawings and paintings on the walls. They were the work of Henri, beautiful, inventive, light, bright work that someone had taken the care to have mounted and hung.
As Viva marvelled and Rivkah crowed about Henri's artistic adventurousness and growing finesse, the gamut of locks was run again at the front door, this time disclosing Claude, Rose's little sister.
Viva had not prepared herself for Claude to be beautiful. Next to Rivkah and Simone she had felt herself to be elegant, graceful and accomplished; all the work she had put into herself was borne out. Rivkah and Simone, though friendly, were none of these things. Claude, on the contrary, shone like a bright dark stone. She was tiny and beautifully made, a happy, smiling girl on the brink of womanhood. More exquisite than Rose and easy in her manner, Claude gave Viva the feeling of having been beaten at something. Perhaps at life. For Claude was blessed. Viva had recognised it straight away and had betrayed herself to the others, momentarily aghast. She would need to learn how to handle Claude.
Thus confronted, she had to admit to herself that she'd been planning to sweep Henri Coretti off his feet. It made her realise in her heart of hearts that Henri had played the leading role in her imagined Paris, that everything she'd accomplished, she'd accomplished to show him whose reputation had come to her so glowingly from Uncle Nicola.
How easily her subconscious plans unmasked themselves and could make her, if she let them, the object of her own derision.
She would meet Henri that evening although it would be another day and a half before she met Allegra, who had gone to Brussels on some work for the Paris Concentration. Lev and Henri turned up after dark, Lev to eat and Henri to bolt down food before going to his painting class. He greeted Viva quite warmly, but was obviously anxious to get moving, his thumbs fidgeting wildly and making their growling sound. What was it to him where she'd come from, what languages she spoke and how many social barriers she'd crossed just to be here? He was at ease with his life as it was. Her appearance on the scene hadn't pulled him up short. She knew herself to be in love with him without knowing who he was. She thought, somewhat enviously, that perhaps he was still ignorant in love. No doubt Simone and Claude were too, though they were probably not, as he was, sophisticated in their understanding of the world in which they moved. His knowledge of politics was particularly noticeable in his exchanges with Lev across the table. They spoke of things French, in particular of Doumergue's seeming treachery; they spoke of things Italian, in particular of the new inane regulation that Italian schoolteachers had to teach in fascist uniforms. Though Viva was only a couple of months younger than Henri, she felt outclassed and, though she could get the gist of the conversations, it was a strain to keep up with the aural ambush of French.
Uncle Nicola had told her of Henri's work with the Concentration. Since the death of Filippo Turati and Carlo Rosselli's arrival in Paris, Dadda had become very useful, particularly as he was becoming an expert at forging documents which helped to get some of the refugees out of France and into countries where there was some future for them. Even if he had been ugly it would have made no difference to Viva's feelings, but he was beautiful, and not just beautiful but charming, amusing, passionate and optimistic as well. Instead of his seeming like a human being in the thick of life and practically unaware of her existence, he appeared to Viva as an almost impossible challenge.
His days and nights, she discovered, were very full. By day, he was all over Paris collecting information, distributing money, taking material to and from printing presses, and so forth. By night, in addition to studying art he also proofread manuscripts and, she was to learn, served as one of the Paris contacts for those involved in smuggling propaganda into and out of Italy.
When Viva's dream faltered in the face of Claude's undeniable beauty and Henri's daunting elan, she felt the smallness of her life attempting to fold her into it as if into a walnut shell. But she resisted, and two days later was in bliss once again, for Allegra had returned from Brussels, and Allegra lived up to all that had been said of her.
It was Allegra from whom Henri acquired his fair-haired, clear blue-green-eyed looks. By way of personality, Viva hadn't known quite what to expect. She knew Allegra had been an opera singer, but that conjured up loudness and considerable poundage, whereas Allegra was slim and small, just as Viva had been told. Hers was a soprano voice, clear but not very strong. Before Henri was born she had been in demand for child parts. Afterwards Maestro Toscanini, not very pleased that she wanted to keep on working and be a mother at the same time, told her she was only good enough for the front rows of choruses. This Allegra told Viva without the slightest embarrassment, adding that on one occasion when he had lost his temper he made her stand at the back of the stage on a barrel, because when she was in the front row her smallness emphasised the rotundity of everyone else.
Despite her size Allegra dominated the life of the women in the Katz household. Not that she was loud, forward, brisk or sharp. Equipoise was what she had; everyone else took up their position relative to her. And through her body language, which consisted both of quiet invitation and of subtle territoriality which would always keep this particular princess in her tower, she enticed Viva to sit beside her and be her student.
Allegra seemed somehow able to read Viva's situation and, without exchanging any words at all about it, to give a perfect demonstration of the tactics to which Viva should resort to overcome her qualms.
Poise was the thing, a certain front of maturity and wisdom in the presence of which Claude assumed her status as an adolescent. Allegra did Viva a favour by showing her how to French-plait Claude's hair for school. Viva learnt that one confessed to Claude's beauty and treated it as part of the pride of the house. She was relieved to see that Henri treated Claude this way, too, as if she were his little sister. Weaving the thick, dark hair into its plait, Viva could seem to conspire with Henri, to share something with him from which Claude, being just too young, was excluded.
Viva knew she must learn, and learn quickly, to give substance to these mock conspiracies. She must both court Henri with discretion and learn as much as she could about his situation. She began to pick up Allegra's mannerisms, not very subtly she supposed, but she would work hard to carry them off. Allegra seemed to like her, particularly as she had news of Uncle Nicola, and Uncle Nicola in turn had written to Paris about her and about her little brother, Leslie.
Allegra was less enthusiastic than Uncle Nicola had imagined about coming to Australia. Nevertheless she asked Viva all about it, how Rose was settling in, whether Uncle Nicola was in good health and what means Viva would use to bring her brother to France, particularly since she herself would have to become a French citizen first. It was all very well to be here under the auspices of a French guardian, but she was an Australian citizen and would need to do more if she wanted to become French.
In the meantime, said Allegra, since Viva had picked up French so quickly she must now learn Italian. If she wanted to work for the Concentration, people with Italian, French and English were in demand, and there was a particular project, with a small salary attached, for which Viva might be very well suited. It would be fairly hard work involving a lot of perseverance, but she was sure Viva would cope with it, since she had already coped with what Allegra understood to be considerable travail.
Allegra set aside a small amount of time each day to teach Viva the basics. These learnt, she set her up with an English–Italian dictionary and put her to work directly on a project of growing importance since Hitler's accession to power in the March of 1933, that of alerting the international community to what the accession meant.
Though no one would say where the heap of papers in Allegra and Viva's bedroom came from, the content of the material spelt out the message that Carlo Rosselli's group, _Giustizia e libertà_ , wanted delivered to the world.
Carlo was now the person the refugees thought of as their leader. His brother Nello was still inside Italy, spending his time in and out of jail. He was a young historian who felt himself being marginalised by the fascists every time he put forward an idea. At one stage he had wanted to establish an international journal for young historians to debate politics of any kind. For a while he was indulged by established pro-fascist historians, but he soon learnt that antifascism was only tolerated from one quarter, that of the eminent professor Benedetto Croce, whose international stature had to be respected on pain of criticism from the rest of Europe.
The translating would be a test for Viva, but Allegra showed her some knacks to it. She learnt to accustom herself to style and found a good deal of truth in what Allegra said about people using certain recurring word patterns. Viva would translate literally into English from the papers and would pass them on to Allegra, who would then put some shape into them, translate them into French, and hand them back to Viva with the object of her learning for herself what the work was all about.
The finished translations went to Henri, and from Henri to a press in Paris, another in London and a third, Italian, one, in Geneva. There they would be typeset before being distributed clandestinely by the International League for the Rights of Man.
So Viva was drawn into the deep waters of a highly political life where, essentially, it was swim or drown. She had invested too much in this venture to give it up now, nor would she have given it up for it was fascinating to a young woman who hungered after a life of the mind. Here were people who believed in something, who were not bowed down by the things that oppressed them, who did not give in to their infirmities or defeat at the hands of providence. People like this had their lives on the line.
Viva dared a little happiness and was surprised to find such a thing in herself. It was a sensation that could bathe her from head to foot, a River of Jordan. She felt sometimes that she'd been freed from unjust imprisonment. Sometimes she even felt cleansed of Julius Lichtblau, though there were nightmares from which she would wake up feeling contaminated all over again, for she couldn't have survived in Paris if Julius hadn't sent her a small allowance.
The allowance came to her every four weeks, accompanied by a letter saying how Julius missed her and hoped she would soon come back. She would reply there and then in the post office, so she wouldn't have the duty hanging over her, studiously avoiding giving away the happiness of her life in Paris because she might yet need Julius to help her rescue Leslie.
She could not think how to write to Leslie to let him know her life had changed, and his prospects with it. She knew of no one whom she could trust to act as an intermediary. She felt quite helpless as far as he was concerned and could only hope he would go looking for her in Melbourne, starting with the Italian woman, of whose existence he knew. She would quite probably direct him on to Rose. The Italian woman had no idea Viva had gone to Paris and Viva was not going to let her know for fear of her father finding out and destroying her plans. Some days she would think that Leslie had the courage to run away from their father, something he could legally do when he turned fifteen, and that he would have the sense to trace her to Rose. At other times she despaired, because he was a soft, introspective boy who tended not to act things out.
With Leslie almost always in the back of her mind and the thought that he may feel she had abandoned him, she found herself battling certain jealousies of which she might otherwise have been more in control. On the walls of the room she shared with Allegra were photos of a past of which Viva knew next to nothing. She envied family photographs, mementos of times that appeared to have been happy; there'd been nothing like that where she came from. She recognised Uncle Nicola, Henri, Allegra, Lev and Rose, snapped in the summer of 1922 in Milan, and she felt excruciatingly jealous of them and the occasion. She knew she had to cast off jealousy if ever she were to free herself of the past, but the effort was enormous and, combined with her anxieties, there were times when she was desolate.
She tried very hard to actually be impressive, instead of just dreaming that she would be one day, but there was a stiltedness to her character she was very rarely rid of. Consciousness of it made her envy the people she was now sharing her life with, not one of whom was given to daily self-crucifixion.
Sometimes Viva found her only escape from these feelings in work, but in work she had only one source of self-esteem, Allegra. If Allegra found she'd been mistaken in construing the meaning of something she would feel mortified, knowing at the same time that mistakes were part of learning and experiences to build on rather than regret. She would say these things over and over to herself as if repetition would build something into her character, but it seemed sometimes that all she was capable of was building an unassailable moat around herself.
Much of her energy, she knew, was expended in suppressing negative feelings about herself. These tended to grow and encompass others, particularly Claude, whose beauty never disappeared no matter what she was doing. Viva would look for imperfections, attempt to see them and make them subtly known, but Claude could not be brought down. The family loved her summer freckles and in France women were much less conscious than Australians of body hair, of which Claude's was very visible, being black. There were no rewards in looking for imperfections in Claude, so she would discover herself trying to find them in Allegra.
Love would teeter on the edge of hate for this woman who, without seeming to, ruled the roost. This was how Viva would have liked herself to be. She put up a good facade. Simone deferred to her, recognising in her a better brain, something, more aplomb, maybe. Viva tried to make it sustain her.
She did not spend her whole life translating, of course. There was Paris to be out and about in. At the nearby Café Dôme the artists used to congregate. Picasso was to be seen there, and once Matisse, sitting alone at a small table, his back turned on Picasso and his entourage. Sometimes Simone, Claude and Viva would have coffee there. Simone and Claude knew a lot of the locals and even had a nodding and waving acquaintance with Picasso. On those occasions Viva could lose herself in the sheer excitement of being alive and would know, as she came down to earth again, that excitement had made her attractive, playful and young. She would flirt and be flirted with and get caught up in the breezy talkative moods of Claude and Simone.
Even so, Claude was an obvious favourite with the cafe society and Viva wondered how she would go among these people on her own. Artists drew her just as frequently as they drew Claude, and told her she had an interesting face, but she could feel herself being pushy on the occasions when Claude had all the attention, and the sight of Simone being pampered brought out the worst in her. Viva could be wittily and caustically sarcastic, and the artists would draw her being so, a sharp-edged but not unattractive harpy. Claude, on the other hand, was all flowing lines, the child-woman, the beauty ready to bloom. Once a painter came up to her at the Dôme and kissed her for being beautiful.
Viva feared what she took to be an inevitability: Henri and Claude. The more she feared it, the more it seemed to happen before her eyes. They, innocently happy and marked out for one another by virtue of innate beauty; she in an agony of suppressed spite. She felt compelled to learn not just about Henri's life but about his art as well. She talked art to him when and where Claude could not. She kept up with the movements, read copiously and had opinions. She could see things in Henri's work, and he soon noticed her sensitivities. He would invite her to come and look at work in progress in his attic over the technical school. There she talked to other young painters as well and they took notice, suddenly wanting her near for her acuity.
But the more sophisticated she became about art, the more her relationship with Henri resolved itself into something she didn't really want. He respected her good eye and her intelligence, but he did not love her or even appear to think of loving her. He loved Claude, she could tell. When Claude dropped in to see his work he'd romp around with her, bringing out her dazzling laughter. Claude did nothing wrong, her every act enhanced her, so that Viva felt the pain of unregarded love.
THIRTY
Faking It
IN 1935 THE mealtime talk in the Montparnasse apartment, when it was not of Mussolini and his African ambitions, was of Hitler and his program to sterilise undesirables. On one occasion Grandfather Grafman was there, and he couldn't see why it shouldn't be done. If idiots didn't breed, he said, there wouldn't be idiots.
It was a good job Grandfather Katz wasn't there at the time but even so, Lev weighed in with examples of intelligent couples with idiot children, and by the dessert had Grandfather Grafman claiming that such children were the offspring of faithless women and it was God's will being done. That evening ended with the diners storming away from the table in eight different directions, resulting in much door-slamming and the vigorous slinging of plates through the sink in the kitchen. Rivkah was angry with Lev for inciting her father, Lev was angry with her father, her father would not back down from his opinion, Allegra stormed off at the arrogance, and Henri and the three girls took to their heels, each thinking of something else they had to do.
When Viva joined Allegra in their room Allegra did not look up from her work, at which she was jabbing her nib and sighing, her lips in a firm straight line with the blood pressed out of them. Some minutes went by, during which water slapped and gurgled down the pipes outside and the sounds of the house jangled and scraped against each other, as though striving to make quirky, percussive music. By and by, this was added to from outside. Some extraordinary clicking and baying noises started coming up from the area of the yard where the bins were kept. Looking down four storeys, Viva saw a drunk, drenched in moonlight, gesticulating deftly as if conducting the whole soundscape. Hearing the hullaballoo, Allegra clacked her pen down on the wooden desktop, rubbed her weary eyes and joined Viva at the window.
'That's a sane kind of insanity,' she said, as the lunatic continued to caterwaul and bow. 'He's getting a lot of pleasure from it: like a footnote in a high drama.'
It was September. At the vast rally just held in Nuremberg Hitler had forbidden Jewish–Aryan marriages and there were reports that some Aryan girls found walking in the streets with Jewish boys had been carted off to a concentration camp. Shopkeepers in Germany would no longer take the risk of serving Jews in case they were victimised for doing so. Jews were being deprived of German citizenship and their rights to pensions. They were being banned from public service, journalism, and the entertainment industry. Anti-Jewish graffiti was beginning to surface even in France.
'He's a silly, rigid old man,' said Allegra of Dan Grafman. 'He doesn't realise that what's happening in Germany is a threat to him.' Allegra was uncharacteristically frazzled and feeling deeply frustrated that the Italians, though they worked hard at it, hadn't managed to get across the meaning of Fascism. People were too apathetic to take any notice.
'It's a disease,' said Allegra, 'it kills our capacity to live with each other.' She despaired of the world which, she said, was full of ordinary people who didn't seem to care who governed them. The trouble with Democracy was that few people used it, indeed were capable of using it, with the vigilance it demanded.
'It's wrong to think our lives are insignificant,' she continued. 'We're all part of the fabric, and all of us have a moral duty to the future, even that poor clown down there, reminding us not to take ourselves seriously all the time. Even so, each person's actions matter, and that is the only constant in history.'
There was no archetype for good behaviour or good society, she went on. The human race was not degenerating from some master plan towards chaos and ruin, nor was it evolving towards perfection. It was just a species of animal whose survival depended on one of its characteristics more than on others. The means to adapt were in human heads. Humans had to be both against nature and with it at the same time.
Viva found this an odd observation for Allegra to make, since she was a creature of cities and her professional concerns had been artifice. But she'd read Darwin, as many educated Marxists had, and had come to see cities as accretions of the human habitat. Viva found it hard to guess what was actually in her mind, since the ideas she had of the animal kingdom seemed to come mainly from lithographs in books. Allegra did study the human condition, nevertheless, and saw that it was often wretched, much more wretched than the condition of pigeons, cats, dogs, sparrows and horses, practically the only animals with which she was acquainted.
Earlier in the month she had gone to the ski resort of Chamonix in the Massif, near the Italian border. Ostensibly the trip had been a skiing holiday, but the real reason she'd gone was to pick up a drop of propaganda material that had come in from Italy, hidden in the seat of an imported sleigh. She had taken Viva with her so as not to attract attention as a woman travelling on her own. Viva posed as her niece. Her orders were to go to a particular shop to get herself and Viva fitted out with rented skis. While she was there, she was to ask to see the shop's Italian proprietor on the pretext of an old friendship, and he would hand the material on.
The man wasn't there. According to his assistant, he'd been out skiing the previous day with a group of Italian tourists, but the group had failed to turn up at nightfall. The locals had become worried that they had had some accident or become lost, and so a party had gone out searching for them as soon as it was light. There was no news of them yet.
Allegra and Viva rented their skis and returned to their chalet to think out what might have happened and what they ought to do. Supposing it were an innocent misadventure, the best thing would be to wait and behave as though on holidays; but if it were sinister, then the last place Allegra would want to be found was in Chamonix enquiring after the ski-shop owner, let alone displaying an interest in sleighs. She and Viva were covered to a certain extent by counterfeit French identification papers, bearing different Paris addresses, but it was worrying that they themselves were obviously not French. They'd been taken for Englishwomen at the chalet, yet according to the papers they'd had to show to identify themselves at the hotel, they were Parisians. They'd had to invent a story about both of them having spent long periods in England, hence their accents, and Viva's occasional difficulties in conversation.
The man who ran the chalet was inquisitive about them. At another time, he might have seemed charming, but his curiosity about England stemmed from his having spent a long time in London himself, and Allegra, who didn't know London at all, had made the mistake of saying they had lived there. When asked where, she'd said Kensington, an area the chalet owner knew well, while all Allegra knew about it was that it had some gardens.
It was out of season and there were few guests staying in the chalet. The owner had done them the honour of inviting them to take their evening meal at his table. He was alone, which meant there would be no opportunity to engage someone else in conversation, and Allegra was nearly caught out several times. During his English reminiscences she had to resort to statements such as 'I much prefer Paris', saying 'All that smog!' for the inevitable next question.
Viva, though she'd never set foot in London, knew a lot more about it than Allegra did, and was able to tackle the ins and outs of the royal family to what seemed like the man's satisfaction. To date, it was the one occasion when Viva had really counted, and she felt it. She was with Allegra now and not just beside her, thanks to a valued talent for dextrous, calculating talk, errors of expression aside. There was the resultant problem, nevertheless, of the man taking a fancy to her. When they came back unexpectedly to the chalet on the morning of the skis, he was there, red-faced and breathless, gesticulating at a coffee table in the morning room where he was wanting to treat them to coffee and almond pastries. It fell to Viva to get them out of it by saying she was feeling cold, and the only reason they were back was to go to their room for warmer clothes. With her useful cool, she thanked the man for his offer and said perhaps they might join him later.
Allegra was grateful for Viva's sangfroid, without which, she knew, her own preoccupation would have been more obvious. In their present situation the worst case would be if the contact in Chamonix had been roughed up or murdered by the so-called 'Italian tourists'. If there were spies around, they could well be on the lookout for anybody trying to find the ski-shop owner. As Allegra saw it, she and Viva could take one of two courses of action. They could either stay and pretend to be bona fide holiday makers, come what may, or they could wait till the afternoon coach and get out as fast as they were able.
Again it was Viva who provided the solution which would cover their tracks. They spent the rest of the morning on the beginners' slope, Allegra teaching, and Viva learning. On her way downhill on one occasion, Viva saw someone fall and sprain an ankle, and it occurred to her that such an accident could serve as a pretext for getting them out of their situation. For a couple of hours she spent her time trying and failing to hurt her ankle and mystifying Allegra by her incapacity to carry out simple alterations to her technique which would keep her ankles intact. Time was running out when she launched herself at a big fat masculine learner and collided, managing not only to sprain an ankle but also to knock herself out. She had calculated very well: the large fat learner was hardly hurt at all.
Viva had to be carried back to the chalet at about noon, when the hungrier guests were gathered in the foyer for lunch. The talk was of the search party, half of whom had returned, having found only one ski cap which might not have belonged to any of the lost group at all.
Viva was taken to her room, where her ankle was bound by the chalet owner in person. He kept saying the injury wasn't very serious, there was no need to go racing back to Paris on the afternoon coach, they could be his guests for a couple more days. Allegra was insistent, however. Since Viva was her niece and not her daughter, she must take her home, it was the only responsible course of action. What if a bone were broken? Allegra would be unable to face Viva's mother for not getting expert attention as quickly as possible, and then there was the further inconvenience of having already instructed the porter to arrange transport to the coach and to book the tickets.
Just as they were leaving, with Viva's ankle lovingly wrapped in crêpe, the news of the ski-shop owner's murder began to break. The rest of the search party had found his body in the snow, stabbed and shot through the head for good measure.
Viva had been unsure how to take this death, not quite able to draw the correct conclusion about the murderers, but on the night of the altercation among the Katzes Allegra left it in no doubt. She was on edge because they hadn't picked up the information drop at Chamonix, and she was still afraid of being traced.
Furthermore, they could not now re-use the faked papers for fear of being identified as the two 'Englishwomen' present at Chamonix looking for the ski-shop owner on the day of his murder; and faked papers represented a lot of time and risk for the Concentration.
'We ought to move, you, me and Henri,' Allegra said. 'We could put the Katzes in great danger by staying here.'
Between them, Viva, Henri and Allegra had very little money. Staying at the Katzes had been cheap, comfortable living, so a move would be made with reluctance. Rivkah wouldn't hear of them going. Emilio had been Lev's best friend and besides, Lev was a known socialist, so the family was at no greater risk having the Corettis than they would have been not having them. If it was Dan Grafman Allegra was worried about, it had always been this way with him, and no doubt it always would be. He just liked to be annoying.
But Allegra didn't want to be responsible for harm coming to the Katzes, whose kindness and courage had already meant so much, and the problem exceeded the boundaries of family friendship. Antifascist French could get into trouble for harbouring Italian refugees. Rivkah must not forget that Mussolini had his first handout from pro-fascist French and the longer he was in power, the more support there would be for him in countries outside Italy. Some people – many people – came to confuse an entrenched state of affairs with a moral one.
A couple of miles to the east of the Katzes there was a hotel full of newcomers and some of the poorest families of refugees. The Concentration was seeking help to relocate, clothe and feed them, a job better done where they lived than at the Concentration, which was already inundated with more requests than it could handle.
The hotel was called the Icebox by one and all because it was in rue de la Glacière, the Street of the Icebox, and it lived up to its name in winter.
When not preoccupied with the refugees, Viva and Allegra could carry on with their translating work at the relatively salubrious headquarters of _Giustizia e libertà_ in rue du Val-de-Grâce, and Henri could carry on his work distributing propaganda.
Because this move was bound to identify Viva and Allegra by their place of work, it was up to Viva now to decide whether to commit herself to the antifascist cause, when she could instead spend the rest of her time in Paris doing something less risky.
Allegra didn't consider it fair to expect Viva to take on the commitment without first knowing what else she might do to keep herself. Emilio's name in the shoe trade meant Allegra knew certain boutique owners in Paris, particularly those who worked for Mario Fortuny, selling the wonderful pleated-silk Delphos dresses and the famous capes and tunics of printed velvet, imported from the Fortuny workshops in Venice. Just before leaving the Katzes Allegra took Viva to visit the Fortuny boutique, where they tried on some of the world's most sought-after garments and had themselves made up by a woman who'd attended to the _maquillage_ of both Lilian and Dorothy Gish. Viva remembered it as a happy day in her life, since it helped her realise the sort of beauty she possessed: a severe, classical, high-cheek-boned elegance, even at the young age of eighteen, an elegance to cultivate from that day on.
The demand for glorious clothes was not high in 1935; Fortuny was struggling and couldn't pay his models much money. Viva, however, had the right appearance for a model and could have taken a job, but she was swayed by the prospect of living in the same building as Henri at a healthy distance from Claude, and now she'd seen what she looked like when splendidly arrayed, she had something in her armoury.
As the winter of 1935–6 came on, the rue de la Glacière plunged ever deeper into darkness and its denizens grew bronchial, not to say tubercular in a few cases. Indeed, there was a death from a lung haemorrhage in one of the basement rooms soon after Viva and the Corettis moved in. The undertakers couldn't get the coffin down to the corpse so they had to get the corpse up to the coffin, the staircases being so precipitous, narrow and sharp-angled. It seemed almost that the hotel had been built around its furniture rather than that anyone had been able to perform the miracle of provisioning the cramped rooms with bed, closet and dressing table.
The living was just a step above squalid: crowded, cheap, evil-smelling, noisy and, every now and then, pestiferous. It was certainly a shock to the system after living with Rivkah and Lev. The noise went on all day long: tramping, rattling, shouting and non-stop talking.
The cacophony in the Katz household had been of an altogether different intellectual order.
To cater for the huge influx of Italians there were cafes and restaurants selling Italian food and wine. Locally, the refugees went to La Popote, down a dark alley, a large room with tables lined up in the middle, covered in floral oilcloths. The food was greasy, the service bad and the wine next to undrinkable. Viva, Henri and Allegra preferred to eat French and as their main eating spot chose La Rotonde, some distance away in Boulevard Montparnasse. There they shared a meal almost every evening, and a couple of times a week, Viva would go with Henri to the loft to talk art and see it in the making.
Though Viva's cheque from Julius Lichtblau came every month to the local post office where it could be cashed, all the other mail was delivered at the Katzes'. Allegra now wrote that it should come to La Rotonde instead. Uncle Nicola was not to worry: it was being done for safety's sake.
Uncle Nicola did worry, of course, and wrote back to say that as he was at long last an Australian citizen, he would sponsor them to come to Australia as soon as they could. They must not involve themselves in dangerous work; they must come as soon as their passages to Australia could be arranged. Perhaps they'd like to bring Viva with them. He had good news for Viva. Her brother had turned up, Rose was looking after him, and yes, he certainly was a talented young man. Rose and Laurent were doing everything they could to nurture an artistic consciousness in Australia. It was really very interesting, a great challenge.
Uncle Nicola was excited about a speech he had given at the Congress for World Peace in Melbourne, and even though he'd given it in Italian, an Australian newspaper had reported his eloquence and commented that even his political enemies came to enjoy hearing him talk. He stood for something. Australia wasn't quite sure what, but he had style.
Soon Viva was writing regularly to Leslie, and receiving regular replies. He supposed their father was still mooching around the Upper Yarra, getting progressively more pickled. Leslie himself was happy to be with Rose, who took great care of him. By the way, Uncle Nicola gave the impression that the change of address in Paris meant the place was about to blow up. Leslie hoped it wasn't so, and sent Viva a little gouache he'd done of St Kilda Beach for her to remember him by. Henri was most impressed and would pick it up and examine it as a matter of course almost every time he came into Viva's room.
Viva's liberation from Julius Lichtblau now seemed a possibility, so she was able to cash his monthly cheques without a feeling of total revulsion. If she let herself reflect on him, however, she would want to cast off the parts of her body he had touched, and she would be driven from reflection towards a determination to have Henri at all costs. Now more appealing than getting Leslie to Paris was the prospect of getting Henri to Melbourne.
The distributors of aid in the Icebox made a desk of Viva's dressing table, a prized article of furniture which boasted a three-winged mirror. With the wings turned in certain directions it could be used both for rear vision, taking in the occupants of the inner staircase, and for street vision, policing the outdoor queue when one happened to form. In the event of an outdoor queue a buzzer for the concierge, herself an immigrant Italian, would be pushed and the fat old lady would waddle out, saying, 'Shoo, shoo! No betting today!' as if the overflow from the queue reaching up to Viva's third-floor room consisted of inveterate gamblers rather than refugees in desperate situations. In this way the attention of the police, be they French or Italian, was avoided.
On the third floor Henri had excelled himself in the concoction of identification papers and passports. Often the passports came from people who'd obtained them legally and had either died, disappeared or lost them. Blank passports were got in, one or two at a time, from England, France and Italy, and Henri had made his own range of rubber stamps and embossers to give these a legitimate appearance. He collected specimen signatures of customs and immigration officials from passports and dockets and reproduced them with an accuracy that made him the first choice of refugees coming for help at the Icebox.
The Icebox also distributed money, which was kept in Henri's room and removed in daily instalments. Some of the refugees would come again and again, barely able to cope with the day-to-day contingencies since they were not only unemployed but unemployable. Lice, impetigo and VD were common, so the Icebox was also in the business of handing out disinfectants, iodine and modicums of arsenic, the latter to deal with rats on the one hand and syphilis on the other.
Everyone knew it was only a matter of time before the Icebox was raided and they would have to safeguard the passports, the papers and the money. It was fortunate that the building was subject to dry rot, as a certain wainscot in Henri's room which appeared sound on the surface was in fact hollow inside, allowing flat parcels to be concealed in it when the beading was lifted.
Viva was in charge when the first raid occurred. Two members of the Italian secret police had joined the queue in an effort to catch her red-handed. It was one time in her life she had something to thank Mussolini for – his hatred of scalps covered in redundant outcrops of human hair. From the moment she saw these close-cropped men cross the street outside to join the queue, she knew what to do. All incriminating evidence was hidden behind the wainscoting and a fairly presentable but obviously sick young man was selected from the queue and made to go and lie down, shivering, on the bed in Allegra's room. Viva put the dressing-table mirrors in an innocent position then, turning back requests for passports and papers, saying the Icebox didn't do that kind of thing, she dealt out only medication.
The two men arrived at the 'desk' together, asking for passports and papers. 'No, no,' she said, 'only medication. If you want passports, you'll have to see the Italian embassy,' and she handed them the embassy's address, which Henri had had roneoed for exactly this purpose.
Finding themselves not getting anywhere, the men brandished some identification papers that meant nothing to Viva and insisted they had information which led them to believe the Icebox was the source of false documents. Viva shrugged and allowed them to go over her room, in which nothing incriminating was stored. Then she opened the door to Henri's room while looking nervously at Allegra's door – an invitation to the men to open it instead. This they did, ignoring Henri's room. They grew excited by the presence of the sick young man. Viva said, 'He's a very sick man, and can't be moved. He's very well known in the USA, and any attack on his person would cause the worst possible publicity for the regime.' They had to believe her, but searched Allegra's room nevertheless and, finding nothing, but very suspicious of the bed, they went away.
Viva found that she could distance herself from her emotions so efficiently, she sometimes wondered whether she had any. Henri had only to show up, however, and she would instantly be back doing battle with both hope and jealousy. The Icebox offered virtually no privacy, and Viva would sometimes glimpse Claude coming to see Henri or going from his room, leaving little doubt as to what was happening. Claude was not a good dissembler; she tried to give the impression that she came to see Henri on sisterly errands, but no sister ever glowed so sleekly from brotherly love, nor threw such apologetic looks to her brother's would-be lover on the stairs.
Viva tried to reassure herself and exhort herself to patience. She drew comfort from the thought that Claude might not be too bright – it was rather hard to tell because people seldom asked cleverness of her. At every opportunity, during the occasional meals they took with the Katzes among the artists at the Dôme, Viva would try to show Claude up, but knew at the same time she ran the danger of appearing sententious. She would feel sententiousness rising in her, and get rid of it with a laugh she'd borrowed from Allegra, which made the person laughing seem full of genuine delight. There being little quarter for delight in Viva's life, however, she knew it was a pitiable situation to have to mimic it.
When the Spanish Civil War began, Carlo Rosselli, furious with Léon Blum for not offering the Spanish socialists his support, formed a column of soldiers from among the refugees and took them to Madrid. The Italians in Paris had to try to struggle along without their leader. For Allegra, Viva and Henri it meant increasing responsibilities, and they found themselves working often as long as from 6 a.m. to 10 p.m. on endless tasks and difficulties, both grave and petty. They soon learnt how to plummet into sleep in those crucial pauses in the life of a great overcrowded city.
In the spring of 1937, posing as apprentices to a French couturier, Viva and Henri did two courier runs into Italy, their propaganda wrapped up in cut-out dress patterns. Then they were sent with a consignment of newspapers to Bagnoles de l'Orne, a town in Normandy.
They sat opposite each other next to the windows, Viva with her back to the engine, her hair in a new sort of chignon Allegra had shown her. She knew she looked striking, if not captivating, older than her twenty years. If she were to win Henri's heart, it would have to be with her own best cards and not somebody else's. They were to stay overnight in Bagnoles and, for once, she felt she had a strong hand.
They were sharing the carriage with three youngish and rather dishevelled men who were wearing top coats, though the weather was warm and pleasant. Viva could smell the perspiration of one of them, which wafted past her face every time he leant forward, causing her to skip a breath in disgust. Henri, too, must have smelt something, because he started to wince at Viva sidelong. Shortly, he tossed his head, as if to say let's go for a walk in the corridor. Out of sight of the men, they started laughing and blowing and shaking the smells out of their clothes.
'What about the coats?' Viva loved the distinctive angles of Henri's teeth as he talked, the eye teeth clambering over the front four like children wanting a better view. 'Maybe they're geologists,' he said, his top lip dipping into the question.
'Why do you say that?'
'Their pockets look as though they're loaded with chisels and mallets. But equally, I suppose, they could be farmers. Maybe they're carrying potatoes and turnips.' His eyes danced as he leant against the windows. He was enjoying the trip, but just as he seemed about to reappraise the situation and enfold Viva into it more closely, he looked out the window and started talking about Normandy's black cows.
The smelly men had boarded the train in Paris, just when Viva and Henri thought they had a carriage to themselves. Beyond a little staleness, they hadn't smelt much at first, but the increasing heat of the day began to draw out disgusting odours.
Viva could see the two bags of journals in the racks over the men's heads. She and Henri had been told not to allow them out of their sight. 'I wonder why they want so many copies in Bagnoles?' she asked. Even Henri, with his beautiful strong arms, had found them heavy to hoist onto the racks. He didn't speculate; he said his sense of smell had been dulled by living in the Icebox. If he could smell properly, he supposed, the carriage would be unbearable to sit in. The atmosphere in the Icebox was so oppressive in summer they called it the Sweatbox. Capable of creating draughts and indoor hurricanes all winter long, the air went to sleep in summer, and couldn't be inveigled in or out of windows from its stale resting places under beds.
Back in the compartment the three men were sitting still and silent as before, their coats open, betraying nothing of what made them bulky. But for the bored jiggling of one man's foot, they might have been dummies.
Henri told the jiggler his girlfriend wanted to know what he had in his pockets but, with slow insolence, the man just folded his tongue between his teeth and passed his eyes over Henri, smirking.
Viva, proud to have been called Henri's girlfriend, began to think this smirk and the stares of the other two men were masking some nefarious plot in which she and Henri seemed to have a part, however small.
As they came into Briouze, where they were to take a branch line to Bagnoles, the men left the carriage ahead of them, while Henri hefted down the papers. Strange to say, most of the twenty-odd people who alighted at Briouze also boarded the train for Bagnoles. Perhaps it was a bigger place than they'd thought.
The men in the smelly coats had gone and instead they were sharing their carriage with a mature, elegant woman who was sporting a man's white suit and matching beret. She went to some lengths to beg Henri not to trip over her legs. The slightest knock, apparently, could have maimed them irreparably. On her dignity, Viva gave the woman a levelling stare she'd been cultivating. The woman was travelling with a companion who was either a nurse or a dutiful daughter. Whichever, she snapped at Viva, 'There's a risk of gangrene.'
The woman's legs looked perfectly all right to Viva. She didn't walk with sticks or crutches; rather, you would have noticed her for her elegant deportment.
'Wretched trains,' the woman said, 'you ring ahead for a first class carriage only to find there aren't any. Don't breathe a word to anyone you've seen me here.'
Perhaps she was someone famous.
Then, 'Is one of you afflicted? Or are you just going to visit someone?'
'We're on our honeymoon,' said Viva.
'Oh, really? You could have gone to the Côte d'Azure at this time of year. What? Haven't any money? Shame. There's a casino at Bagnoles. The place was highly recommended to me. There's a battery of doctors. Terrible shame if either of you is afflicted. You're both so young.'
A mystified silence fell on the carriage while the woman crossed her legs with care and put on more lipstick, gazing lovingly into the mirror of her compact. Snapping it shut, she said, 'They told me a story about a Capuchin monk who came here crippled. After he took the waters at Bagnoles, he was able to leap between two pinnacles of rock way up in the air. It was a miracle. And there's another story about an eighty-year-old man whose old horse came back to him rejuvenated after being lost in the forest. He followed the horse to Bagnoles, bathed in the waters, got married and had a dozen children. He built a tower in the shape of an erection to show his appreciation of whatever power it is that makes the waters miraculous.'
The way to Bagnoles was wooded and very green. As the woman squished her lips about, marshalling the lipstick into the cracks, Viva identified beech and chestnut trees. She had always identified trees, right from childhood, thinking out their timber content and the uses to which they might be put. The woods were scumbled darkly with pines and larches. Hazel, cotoneaster and blackberry sprouted on the margins of the trainline. So, Bagnoles was a health spa. Yet it gave her an unhealthy feeling, because the way there was not unlike the way to the timber towns from Melbourne. By contrast, however, the people who went to Bagnoles seemed well heeled and, this being very unusual in a courier run, the adventure had taken on an air of mystery that had Henri hunting about for clues. His eyes were roving and his face was cracked with uneasy anticipation.
Because of the delicate state of their fellow traveller's legs, Henri caused much toe-pointed perching when he left the carriage for the corridor, beckoning Viva to follow him.
They were supposed to take their consignment to a Hôtel Cordier and book in for the night under the name of Picard. They were to behave as if they were a young married couple, but no one had told them in what capacity they were going to Bagnoles, whom they were going to meet or what was to become of the papers they were carrying. All they knew was they were to ask for a Monsieur Genet at their hotel and all would be made clear to them. Allegra had given Henri her engagement and wedding rings for Viva to wear. Up until this, he had kept them in his wallet. 'Better put them on,' he said, taking them out, 'but be careful, won't you, because they're quite valuable.'
Viva had fantasised this moment the whole trip, imagining it would mark a turning point in Henri's affections. He was just slipping the rings onto her hand and realising that they were going to be rather tight on her when the woman's companion came outside looking for the lavatory. Henri covered Viva's ring-encumbered knuckle by folding her hand into his and placing it over his heart. Viva, nearly dying of sexual excitement, stood on tiptoes and kissed him passionately. But when the woman had passed, Henri held her off and shook his head slowly. 'We can't,' he said. Viva read that as We can, We must and We will. Heaving with desire, she returned to the carriage and did her best to flash Allegra's rocks to advantage before the invalid. But the ego opposite had the knack of flattening any story besides her own, making of her companions two-dimensional creatures who might have been decorations on the lining of a box around a Christmas present.
The sea at Indented Head sloshed through Viva's story like a photocopier telling off the pages. It was getting too late for her to go back to Melbourne, and I was in that state between wakefulness and sleep when the irrelevancies brush like trailed coats over snags, bits of them coming off and lodging on the groundscape of one's thoughts. Viva's talk had set me vaguely thinking about arterial trees and what it might be that blocks them; clots, like marbles, rolled onto T-junctions and sat there, producing downward starvation and upward glut. Limbs went numb and withered, while all around in her story the earth most gorgeously held up its green bounty, stiff with water pressure to the intricate tips, and I could not help wondering what Dadda might have been feeling.
THIRTY-ONE
Justice and Liberty
VIVA AND I were inside the beach house now. It had been dark for some time Jack Ives had made us both some dinner which we'd eaten on our laps. Then he'd gone off to sketch out what he would do the next day. She was reclining on Harry's white couch, the Fortuny model in decline. On her head was a thick white headache band. Her face strained for dignity, a withered bird's face, the skin very dark around the eyes.
'There were several cars parked around the station at Bagnoles,' she said. 'They all had their hoods back and their windows were wound down and rattling. There was a lad in a cap beside one of them, with a sign clasped to his chest saying "Cordier". We'd heard about runners from the woman with the legs, who said there was a class of person at these resorts who lived on nothing but tips.
'Your father didn't approve of this sort of thing at all, and I felt very self-conscious what with our two sacks of paper and a couple of sugar bags for our clothes. Henri said he abhorred servitude of the sort expected from the boy, who was lumping bundles into the trunk of this throbbing black Renault sent up to fetch us. When Henri tipped him a good many more francs than was customary, he got a chagrined look in return, probably for his lack of _savoir faire_.
'There were three of us for the Cordier. The third was an old man in a homburg, with a bandaged foot. He had to be lifted into the car by the runner and the chauffeur. His tip got a respectful raising of the cap from the boy.
'I felt cross with Henri for being grumpy. Here we were lifted out of the slums and plonked in the lap of luxury. There was the casino on an artificial lake, all flag-poled and cupola'd and obviously very much in use. Henri called it an abomination. He had a look on his face which I hated – his eyes used to disappear into dark caves under his forehead so I couldn't see anything except the sun shining on the fringe of his lashes. I used to think he did it on purpose. He could be very unreliable when he was young, full of self-righteousness. His mother used to say it was just "his way", but I thought he was over-concerned with being in the right.
'The Cordier was just as sumptuous as the car. It was built into a hillside; you drove down to it off the road. In keeping with many of the other establishments in Bagnoles it was of grey-white brick, picked out in terracotta round the windows and cornices. A flight of white stairs led from a sun terrace to the driveway, where two porters were ready to lift the crippled man from the car and carry him up to the lobby.
'I felt very shabby and disappointed. After all, I had worked out and practised my seduction technique night and day for a week. I'd primed Henri up properly before the trip and I was saturated with love. Sodden with it – and here was he in a bad mood because Bagnoles hadn't been designed by Emile Zola. It wasn't my fault.
'I slapped my ringed hand down on the desk to spite him, and he gave the name we were supposed to give. He didn't say everything we were supposed to say, though. We were supposed to be meeting someone there called Genet. Since he didn't say it, I did. Your father was holding his mouth in that disapproving way he used to have...'
'Disapproving?' I asked. 'Dadda disapproving?'
'Well, of course, you never saw it, I suppose. He used to hold his mouth in a certain way, with its little side kinks crimped upwards and his lips pursed forward. And, of course, he was grinding his thumbs, making that annoying noise with them. It made me think he was feeling for the fabric of the place. The clerk picked up the phone, began dialling and asked, quite unexpectedly, "Which Monsieur Genet?"
'Well, we didn't know there was more than one, but apparently the original Monsieur Genet's brother had arrived at the weekend. The first Monsieur Genet was convalescing from an illness and the second had come to visit him.
'I said we wanted to see the first Monsieur Genet, and although that was probably the right thing to do, it made Henri scowl all the more. The concierge dialled and gave us a room number, saying the porter would take our luggage to our room while we visited Monsieur Genet. Well, we'd been told not to let the journals out of our sight, so I said we had a special reason for wanting to meet Monsieur Genet in the lobby.
'The concierge dialled up again and said we wanted to meet the first Monsieur Genet in the lobby, but apparently he was taking the rest cure, so his brother was on the way instead.
'We were quite nervous, because we'd been told always to follow instructions to a T, and to be very wary of changed instructions. Contrary to expectation, however, the second Monsieur Genet was a roly-poly, beaming sort of man who bounced off the stairs like a cat on springs. He was disarmingly pleasant, and the light kept flashing off his little round spectacles, making him look a bit like a clown.
'He tossed our hessian bags up over his shoulder as if they'd been full of loose-packed straw rather than bundled papers, and he bounded on up the stairs, motioning us to follow.
'I remember the room we went into was quite sumptuous and had open French windows facing out over the gardens at the back of the hotel. There was a trellis smothered in white roses, leading the way to what I supposed was a race. We could see water slipping behind a wall. And there was a huge sequoia on the lawn. I remember thinking it was wearing its shadow like a skirt that had slid from its hips and lay around its feet. As I said, I was soggy with love.
'I didn't know who it was lying on the bed with his leg up on pillows. Henri told me later it was Carlo Rosselli himself, and the other man was his brother, Nello. Apparently Carlo had developed phlebitis soon after going to Spain. He had fought on in the trenches for six or seven months, but at the end of May he'd come back to France to undergo a cure. But we didn't know this till afterwards and all Henri could do was sound off about decadent luxury.
'The meeting didn't last long. They told us to go and have fun in Bagnoles. Well, we wandered off, me trailing after Henri, who wasn't even looking at the surroundings, so I had to imagine my idyll rather than have it.
'The situation had snaked out of my control, and it became a lost cause altogether when we saw an eagle pause in the sky, drop and come up out of a cedar with a squirrel in its beak. Henri was disgusted. He turned away and said he wanted to follow the race and see where it went.
'I thought I'd brought a curse on myself, but I couldn't think what I was doing wrong. Claude was far away but she seemed to be exerting long-distance control.
'And then I realised it wasn't only Claude who was influencing my mood, because when the race became a stream and was surrounded by weeping willows that were halfway to showing their full leaf, I was reminded of my horrible childhood. Weeping willows beside creeks were a sign of the invader in our town. They symbolised rural pacification. They were where city people had their picnics, spreading their rugs on skerricks of soft green grass in the hope that the place was used enough to consign the stinging ants to its margins. They'd bring wads of newspaper to stun the horseflies and mosquitoes. The idea was you ate your soggy tomato sandwich, drank your thermos of tea, had a bit of a ramble and then went home, supposedly renewed.'
But Viva knew better than this. She knew that at a similar time of year to June in France, the Australian forests would be alive with rosellas and green parrots, there would be cockatoos, magpies and a banquet of textures, aromas and sounds in which you could lose yourself completely. It could enter you and make you disappear, along with all your fear and anguish.
She would have liked to tell Henri, but felt her way of telling would be inadequate and would not induce the intended mood. Gates would shut in her mind when she reached for the tender part of herself, and her thoughts could only be conveyed in banalities. Though this was the occasion, if any, to conquer Henri, she was being forbidden to do it. She felt herself to be like the eagle, but trapped too high in air currents to discern her prey, the creature Claude, whose beauty was the perfect camouflage.
As she was thinking these thoughts, they came on a camp. A rough-looking man was sitting by a tent, skinning a rabbit. He had his back to them but Viva recognised the coat. 'It's one of those men from the train,' she whispered. 'Maybe they're poachers and they come here to catch hares and rabbits to sell.' There were such people in Paris; Viva had seen them at Les Halles, displaying the rabbits hung up inside their open coats.
The man's back seemed to mark the boundary of their walk and they turned back without disturbing him.
'We had to sleep in the same bed that night. I tried very hard to be vulnerable and seductive, but we were treated to a champagne and pheasant dinner and your father let himself be disgusted. Oh, the arrogant assumptions of twenty-year-olds! When we went to bed he lined up the pillows in between us.'
Viva held her devastation at arm's length so that it seemed like someone else's. She imagined weeping herself to shreds, but didn't do it. For years now, the only legitimate reaction she could rouse in herself had been jealousy.
The next day they took the train back to Paris. Henri had a hangover that lasted the whole trip. When he apologised for his bad mood, she made of it what little she could.
'We went back to Bagnoles years later,' she continued. 'I wanted to undo the pain of Henri's rejection. We spent a few days there but it was winter, and the feeling of the place was quite different. It was all boarded up. The Cordier wasn't a hotel anymore and we had to stay in a place that was nothing like it. We'd make love in the evenings, but I'd wake up later and find Henri crying. No doubt it was for his lost Claude. But I was not lost, I was there. And then I thought perhaps it wasn't just for Claude he wept, but for the Rossellis. They were assassinated the day we left Bagnoles all those years ago. The bodies weren't found until a couple of days after we got back to Paris.
'For a while, Henri and I thought we might have led their killers to them, maybe they were the rabbiters. But that wasn't it at all. They'd been under the surveillance of a group of right-wing French, a splinter group that formed when the _Action française_ was disbanded. They called themselves the _Cagoulards_ , the Hooded Ones. Apparently a few of them were staying at a hotel opposite the Cordier called the Bel Air. Henri and I looked for it when we went back, but it had been knocked down. Anyway, it was up the hill from the Cordier and had a good view of it.
'Apparently, while we were on our way back to Paris, Nello and Carlo went for a drive to Alençon, a town down the road going to Couterne. They visited a patisserie there and had some coffee and cakes. The killers overtook them on their way back to Bagnoles.
'They got some distance ahead and faked a breakdown, with the car blocking the road. They had their bonnet up, and when the Rossellis stopped and got out of their car, they shot them. Nello must have survived the shooting, because when the bodies were found, he also had stab wounds.
'At first the police suspected Italian fascists, but bit by bit it became clear they were French. A hairdresser from Bagnoles had been riding her bike past the place where it happened and she saw a Peugeot take off at high speed, and blood on the road. The bodies weren't far away, but they weren't found until some man was caught short on the road and nipped in behind the Rossellis' parked car to relieve himself.
'The hairdresser knew the killer's car, of course, and she started to receive threatening mail. That led the police to the _Cagoulards_. Apparently they'd done it in return for the delivery of a hundred semiautomatic Berettas from Italy on Mussolini's say-so. They were going to stage a fascist-style coup d'état in France. It was fixed to take place in early November, but they couldn't raise enough support from inside the army and someone dobbed them in. The leaders were arrested. Their trial dragged on until well after Henri and I had left France. By the time the war broke out they were in jail, but they were let out because they said they wanted to fight. More than one of them ended up in Vichy fighting for Hitler. There's a certain brand of make-up I never buy because one of the Rossellis' assassins escaped to Spain and was instrumental in setting up the firm. Some of the others found work through him after the liberation.
'The first inkling Henri and I had that anything was wrong was when we arrived back at rue de la Glacière from Bagnoles and found it milling with gendarmes. The Icebox had been ripped apart and Allegra and the concierge had been badly hurt. We never knew if the two attacks were linked. Perhaps they weren't at all, but someone else from _Giustizia e libertà_ was roughed up at the same time and it was obvious someone wanted to silence us. It didn't stop the next edition of _Giustizia e libertà_ carrying a front-page denunciation of Mussolini, all the same.
'Later that night – it was terribly hot, I remember, and the nurses in the hospital waiting room were ill-tempered and unsympathetic, they said we were terrorists and deserved what we got – the concierge died. Allegra had been beaten around the head and had contusions. She was drifting in and out of consciousness, but it seemed she would pull through. She managed to tell us to get out and go to Nicola in Australia as soon as we could.
'Even though your grandmother was very ill,' she said, her voice under strain, 'all I could think of was I'd won. It sounds horrible, but the thought of Henri coming to Australia with me put every feeling I had for anyone else out of my mind. I was almost gay. Your grandmother was dangerously ill, and I was gay!
'We went to live in a cheap hotel near the Katzes. I went through the motions of disgust and outrage. I even felt generous letting Henri say goodbye to Claude.
'He wanted to marry her and bring her to Australia, but Lev Katz said no. When Henri got his Australian citizenship she could go out, but not before. It was so far away. Henri was twenty, but Claude was still only seventeen.
'Only seventeen! I thought, only seventeen! – I've been only seventeen, I was only fifteen once. I was so jealous of Claude, so jealous! She had everything: beauty, protection, love. Then, when I went to see Allegra before leaving for Australia, she said, "Don't lose your heart to Henri, Viva." It put me in an agony, knowing I must keep to myself till he felt he'd said goodbye to Claude. If I really had won, the prize would be a while coming.
'Well, we went to England, took ship and sailed. Your father was downcast for the whole journey. I had to pretend sympathy. However, after we'd arrived in Melbourne to a very warm welcome, I felt the tide had turned and I'd mastered something. Henri was smiling again. My only problem seemed to be Julius, but I told him I was in love with Henri and he could see I was: or perhaps Orpah had extracted a promise from him. Whatever, he let me go with no fuss.
'We lived at Rose's. It was really good for a time, and of course Henri became a great favourite with the Lonsdales and other people we knew in Australia, but still I hadn't seduced him. Sometimes he would hold my hand or put his arm around me affectionately, but only because he thought I was showing sisterly concern.
'He wrote to Claude for a long time, but he had to wait five years before he could become a citizen and by then, of course, we were in the middle of a war, and France had fallen.
'One letter got through for Rose from Lev, when the invasion of France was imminent. He said the Germans would never bring the French to heel. The national climate in France was something they wouldn't understand. The will to fight had evaporated since the First World War and the loss of two million Frenchmen, which, when you think of it, was a third of the male population between the ages of twenty and fifty.
'Allegra had made quite a good recovery, despite having had her skull fractured, and was living back with them at their insistence. She was still working for what was left of the Concentration, but the Rossellis were an appalling loss and everything was in dreadful disarray.
'It was after France had fallen that Henri found it all too much to take and began sleeping with me, but it was like a palliative to him, and if he could have turned me into Claude he would have.
'The only person who loved me for myself was Leslie. He was called up during 1941, though he hated the whole idea of it, as did we all. Leslie didn't show much interest in women. I kept thinking it was because he was young, but Rose, whose morality was much freer than mine, said she thought he might be homosexual. I dismissed this at first but then Harry Laurington, whom I did not then know, virtually abducted him, coerced him to desert and set him up as a painter, here in this house.
'Rose sat by and allowed it to happen. Homosexuality was, of course, illegal then. It carried terrific social opprobium and severe penalties for those who were caught.
'I was outraged. I suppose as much because of the danger involved as because I saw Leslie's actions in the light of my own. I'd been a rich man's moll, after all, and quite a lot of that was driven by the thought of saving Leslie for a better life. Now he was a rich man's moll, and the situation was ever so much more dangerous than Julius Lichtblau's fling with me.
'Your father tried to persuade me that Leslie's sexuality didn't matter, but to me, and to most people at that time, the thought of an older man keeping a younger one to have sex with him was criminal. And then, what would happen when the affair ended? Did your father or any of those people care? Leslie was a poor boy, there was no rich family for him to fall back on. When Harry grew tired of him, there would only be me. I can tell you, I tormented them. And the more I did, the more they closed ranks on me, determined to make me capitulate.
'In the meantime, Harry joined forces with Rose and Laurent and went ahead with his cafe-restaurant. The first Siècle was in the basement of the boarding house. They opened three or four nights a week. It wasn't ever going to be legal because it didn't conform with fire regulations, so they ran it as a kind of open-house bistro for their friends. Laurent and Rose did all the cooking, but the war made the menu rather unpredictable, with rationing and what have you. And, of course, they didn't have a wine licence though they served wine. A certain group of people came, and Siècle was known among the painters. I worked there infrequently. They would give me a share of the take but basically I earned my living doing a war job.
'People liked Siècle, I suppose because it was exciting, being illegal, and it took their minds off fear. Sometimes they couldn't run to a main course and once Rose did wonderful things with tripe, but none of the Australians could eat it.
'On Sundays Henri and I would come down here to see Leslie. I could see everyone else "making do" and "making the best of it", and sometimes I'd feel part of the whole thing, but at other times I'd think, "at whose expense?" Because it seemed in a way to be at my expense. Henri was using me as a substitute for Claude and Leslie was letting Harry have sex with him in return for being fed and hidden from the Military Police.
'We spent Christmas Day 1941 down here; there was me, your father, Leslie and Harry. They say Christmas is the time when family members murder each other. Well, I felt hostile towards everyone. Harry and Leslie tried to placate me, but I wasn't going to condone their affair. And then Henri told me I was being impossible, and I had a huge fight with him, following which he hitchhiked back to Melbourne, went to Rose's and told her he was leaving. He said he wanted to clear his head and he was going to go where his feet took him for a while. He told Rose of his love for Claude and of his affair with me, and that he didn't think it would be very helpful to stay around me because he didn't feel for me what he felt for Claude.
'Well, I remembered his protracted farewells to Claude in Paris and was mortified by the memories. To me, my lovemaking with Henri was the consummation of my feelings for him. For him, it was substitute love. I felt I had nothing. No thing. Except, my period didn't come. I didn't know where the hell Henri had gone, he'd just vanished, but my period didn't come. Second month, my period didn't come. Third month.
'We thought we were about to be invaded any day by the Japanese and here was I, going to have a baby.
'I had things out with Harry. I told him all my fears. At first, I hated him and said so. But Harry was a kind man. He is a kind man. If I could not patch things up with Henri, he said, he would marry me and adopt the child. It seemed he and Leslie had decided something, perhaps they had decided things would work out better if Harry married me. I asked what would happen about sex between them. Harry said he didn't know, the attraction between them was very powerful. I said he would ruin Leslie if the sex continued, he ought to give him a chance, Leslie hadn't even experienced sex with a woman yet. And Harry agreed. Leslie ought to be given the chance to live an orthodox life.
'I gave Henri six more months, by the end of which I had only days to go before my confinement.
'We were married here, in this place, Harry and I, on the 4th of September 1942. Checkie was born in Geelong, on the 11th. She was christened Cecilia Viva Laurington, Cecilia being your grandmother's confirmation name. Henri didn't even know she existed.
'There are worse fates than being born into the Laurington family, Isobel, and Harry's relatives were absolutely delighted to think he had a wife and child, so I left it at that.
'Henri was still in Australia. Nicola would get cards from here and there and everywhere, but there was never a forwarding address. He said he'd be back, but wasn't sure when. Nicola didn't know that Checkie was Henri's child. I don't suppose he ever thought about things like that, taken up as he was with his politics. I didn't think it was in anyone's interests to tell him, so he always assumed Checkie was Harry's.
'Rose knew, of course, and told Henri when he got back to Melbourne early in '43. He gave Rose the paintings he'd done while he was away and she gave them to me for Checkie, but he wasn't going to interfere with our lives.
'The next I heard of him, he was going to marry some bank teller!'
This obviously hurt Viva a great deal, as she had to clutch herself and reel around and make as if to clap her hands to her ears.
But I also felt something. I was suddenly overcome with a deep surge of love for my mother. 'She's not some teller,' I said to Viva. 'You have a life, Stella has a life. You've shown courage when you've been called upon to show it, but you haven't much compassion. Stella has. No doubt it's cleverer to be courageous than it is to be kind, but Stella's whole life is kindness, even when she bungles it.'
'Is that enough?' she asked bitterly.
'No more than courage is. I guess it takes courage to overcome the taboo on killing, yet people kill in the name of ideals. At least the motive for kindness is kindness itself; courage may have the vilest of motives.'
'And kindness, the worst of outcomes!' she cried. There were tears in her eyes now. 'No, I'm not kind. I never was kind, but I did love Henri, and I had to bear your grandmother saying to me, "Don't lose your heart to Henri, Viva. Henri will always be in love with girlishness, and you are a woman." She said it as if I were capable of sharing the sort of dignity she had. All through my life, I'll remember our visit to Fortuny's. When I said I wouldn't model for them in spite of the huge temptation, they gave me a blue silk slip and a scarf. A magic scarf, made with secret dyes. I used to dance with that scarf in your father's loft, and they'd draw me while that slip clung to me as if it were a lover. Or a child. I stole a bit of Henri to make Checkie, Isobel. She was a sweet baby. I compensated myself with that; she had your grandmother's gorgeous blonde hair, but the eyes were mine, the face was mine. I could read history in Checkie, the coming together of separate strands, the expunging of the pain of my past.
'We met once on the street, by accident, Henri and I, just before his wedding. I cursed that I didn't have Checkie with me, it might have won him away from your mother. He said he was going to marry someone who had experienced much the same sort of loss as he had. We knew Claude was dead by this time. The same sort of loss.' Viva shook her head and wept.
'And then to have him call his first child Allegra! I hated him! Oh, God, how I hated him! But he'd left some paintings with us and I was determined to keep them. I was going to show the world who his true wife was. I understood him. I knew him. What did your mother know? And then, when I met her by accident, you two sitting there with his eyes copied into your faces, and she so heedless, so damned blithe! The day she married the man I loved, my brother walked into the sea and drowned himself because his own situation seemed to him to be morally impossible.'
THIRTY-TWO
Homage to Bruegel
LESLIE'S DEATH WAS Viva's business. It certainly did not happen through any fault of my mother's, nor did my mother wish to do anyone harm by marrying my father. All she knew was an antiquated sense of honour that she took on as an obligation. Mottean honour was clumsy and crashed into things, apologising as it destroyed. With foresight and self-restraint, however, these people might not only have been honourable but provident as well.
However, if respect for other people and their stories doesn't follow on from protection, then what is protection for, if not self-aggrandisement? The Mottes restricted their respect to things that had been well made. They imagined themselves to be the kings and queens of taste and that the world awaited a pat on the head from them for its skill and obedience.
Our father knew he wasn't the world's spindle and that destiny could depend on more than kindness. Sometimes your fate could be in other people's hands entirely. And that's what happened eventually to the Katzes.
I did not understand how Dadda knew Claude was dead before he married our mother. I'd never spoken to Rose about the loss of her family; it was probably too painful for her to tell, but sometime after my night with Viva she let me piece the story together with her help. I filled in the gaps later, in Paris, when I visited the place where the Italians held most of their important functions, the Masonic Lodge of the Grand Orient of France. I went there to look up the Italians, but found instead books commemorating the fiftieth anniversary of the round-up of the Parisian Jews.
The news that Claude Katz was dead reached Uncle Nicola late in 1942. Uncle Nicola, undaunted by the inevitable collapse of the Paris Concentration, was attempting, with his friends and other interested Italians and Australians, to form a new movement which would dedicate itself to the establishment of a free and democratic Italy at the end of the war. I read in his diaries that there was a big meeting about it in September at Melbourne's Savoy Theatre. Uncle Nicola proposed the movement be called the _Movimento Garibaldi_ , but there were some newly arrived Jewish-Italian refugees who said the name Garibaldi, associated as it was with anti-clericalism, would alienate the Catholics; a better name was _Movimento Italia libera_ , a name not tainted by past politics.
These Jewish-Italians were members of the Paris Antifascist Concentration who had been smuggled into England by the Resistance at the end of July and given jobs on a ship carrying Italian prisoners of war to Australia. One of them, Moishe Petrucelli, knew Lev Katz and his family fairly well. He was also acquainted with Allegra through the Concentration, had known Emilio and recognised immediately who Uncle Nicola was.
He told how he and his wife and children had gone to a meal at the Katzes soon after the invasion. Allegra was there. She had made a remarkable recovery, though there was a long, shallow depression in her forehead where her skull had been fractured. At the meal also were the two grandfathers. While Grandfather Isaac Katz was deploring the way the government of Paul Reynaud had just fled Paris while urging everyone else to stay behind, Grandfather Dan Grafman was saying it was the only good move the Reynaud government had ever made, specially as the scoundrels had taken their mistresses with them. Fate had decreed that Reynaud's mistress, the unspeakable Mme de Portes, had got what was coming to her. On his flight into southern France, Reynaud had had to slam on the brakes and de Portes had had her neck broken, fatally, by a flying piece of her own luggage.
As Grandfather Dan was splitting his sides over the fate of Mme de Portes, the others spoke tentatively about Vichy and Pétain and the appointment of Xavier Vallat to the Committee for Jewish Affairs. The only time he was publicly anti-Semitic was when he'd called Léon Blum a Talmudist. That was not too rabid, and now he was insisting that internment ghettos had no place in modern life, so there were some grounds for hoping French Jews might not endure the fate of Germans. It was nerve-racking for the Katzes nevertheless, since Lev and Grandfather Isaac were German by birth and it said so on their French citizenship papers.
It was not until May 1941, when the round-up of foreign-born Jews began, that the Katzes and Petrucellis started living in dread. Grandfather Grafman received a terrific shock in December when he was one of a thousand Jewish notables rounded up in reprisal for attacks on German soldiers. Fifty-seven were shot, but Pétain expressed his outrage and the rest were let go. In exchange for their freedom, a fine of one billion francs was levied on the Jewish community. There was now reason to fear the next step, which, if Germany was any guide, would be the freezing of bank accounts and the seizing of property.
Then Vallat was sacked and replaced by Darquier de Pellepoix, whose noble name was invented but whose anti-Semitism was not, and there were real grounds for fear. The great irony was that the safest place for a Jew in France was in the Italian-occupied sector, where the Italians were actually protecting Jews. Synagogues were operating and Jews were going about their lives as normal.
Moishe Petrucelli knew there'd be a round-up but he wasn't alone in not knowing when. Fearing it would be soon, Lia Petrucelli sent her two youngest children to stay with the Katzes, since the Katzes at least had French citizenship and that had to be regarded as some protection. The first hint came on July 15th. The Committee for Jewish Affairs had co-opted Jewish women in Paris to prepare thousands of labels by attaching to them wrist-sized pieces of string. Believing foreign men would be rounded up first, Petrucelli went into hiding in a Masonic lodge with several other Italian Jews. He was wrong about the men being rounded up. The door-knocking started at 4 a.m. and Lia Petrucelli, with her two remaining children, must have been among the first people detained.
It was still dark and the streets were deserted as the gendarmes and authorised civilians hassled the families from their homes. Youths from the _Parti populaire français_ , wearing blue shirts, were among the groups mobilised for the job. Some civilian 'helpers' had even been issued with uniforms; the women's were striped patriotically in red, white and blue. In areas where the Jewish communities were large, armed gendarmes blocked off the streets. Buses were parked in ranks at collection points.
People were told to dress quickly and take enough provisions for the family for two days. Many a door had to be forced. More than once, women whose husbands had been arrested threw their children through windows and jumped after them. People cried out to warn each other.
At five o'clock the round-up gangs were in the fifth arrondissement. There was banging on the Katzes' door. It had been a night of little sleep and much worry. The Petrucelli children were in Allegra's room, but how did the police know they were there? Or were they knocking up French nationals? 'Police! Police!' they called on the other side of the door. Allegra went. The Katzes weren't there, she said, they'd gone away. She herself was an Italian citizen, an Aryan.
The police appeared to accept her story and went away. Everyone dressed frantically; little loads of essential possessions were packed. More knocking. What could it be? Another gang to deal with? The Katzes huddled in Allegra's room while she went to the door.
An amiable gendarme stood there, a little embarrassed, with him two zealous youths, chucking their chins and cocking their heads. Yes, he was sorry to say, the Katzes were on the list, and he had information they were home. Allegra mustn't conceal them or he had orders to take her too. It'd be all right. Yes, they'd probably be deported, but they'd be free to come back to Paris after the war. They were lucky. Other people were having their apartments boarded up, but they wouldn't need to, would they? Not with Allegra living there.
Why the Katzes? Allegra cried. They were French. French, like the gendarme himself. Behind her the youths had found the family in her room. They stood around politely, gawky in their shiny clothes, while the Katzes and the Petrucelli children filed into the hall. Her dearest friends.
Well, where were they to be taken? Where? To a _lycée_ , what _lycée_? Where after that? When?
Come along now, Madame, don't complicate things. It'll all go much smoother if you do as we say. And as they left they kissed her, each of them, on both cheeks. Rivkah, Lev, Simone, Claude and the two young children. She tried to absent each of them, to separate them out, isolate and hide them, but what she could do in her head, she could not do in life.
'It's because of me!' she cried after the gendarme, clawing his arm. 'It has to be because they've been helping me, hasn't it? I'll go in their place, let me go in their place! They're French, don't you understand, French like yourself!'
'The Winter Velodrome,' he said, 'that's all I know.'
Moishe Petrucelli, in his hiding place, had heard the screaming and shouting of women and children – so it would be all of them, his own, and the families of the men he was with. At eleven in the morning one of their protectors came to tell them Allegra Coretti had phoned. There were buses from all over Paris converging on the Winter Velodrome and others, carrying only adults, headed for the concentration camp at Drancy.
Outside the Winter Velodrome rue Nélaton had been milling with buses since 6 a.m. Allegra had taken the Métro to La Motte-Piquet and done her best to find a vantage point where she could watch them disembarking. She was certain she must have missed the Katzes as she explored the streets around for somewhere to conceal herself and watch. Rue Nélaton was out of the question; it was guarded by mounted police and the houses there opened right onto the street. There was a railway bridge to one end of the street, but how would she climb that and watch unseen from there?
It was about 8 a.m. when some workers at the Citroën factory in nearby rue du Docteur-Finlay started turning up for work. One of them told her they could see into the Velodrome from the factory, and let her come up with him. Three-quarters of an hour later she saw the Katzes, all of them, arrive. She waited another hour. No sign of Lia Petrucelli, but since everyone arriving appeared to have children with them, it seemed likely that family groups were going to be held here.
She tried to think out what to do. The men at the Citroën factory were sympathetic and volunteered to take things into the Velodrome if they could.
It was worth trying, but a better idea started forcing its way forward. She was going to try to convince a sympathetic lawyer she knew to bluff his way into the Velodrome, saying he needed the Katzes and Petrucellis to testify in court. They'd witnessed a crime, that was it, a crime.
She rang up from the factory, but the man couldn't be found and she would have to ring again later. She had a place from where she could see the buses arrive and another from where she could see into a small yard where there was a tap and people were queuing for water. She could see lots of people she knew, families she'd helped. She watched all day. By the time the factory was ready to close its doors for the night, the little yard with the tap in it was a quagmire. She had seen every sort of person arrive, those who couldn't walk came on stretchers. Surely not? Surely not! People learnt later that even the dead had been rounded up.
It did not last for a day, or the two days for which the families had been requested to prepare. Those detained in the Winter Velodrome on the 16th of July 1942 had to stay there for a week. The Winter Velodrome was built to take fifteen thousand spectators for an afternoon's sport, not seven thousand people required to live there for a week. Of twelve lavatories and twenty-four urinals, only half could be used from the outset. At the end of the first day, battered by full summer sun on a glass roof, the place was stinking, the urinals were overflowing, the lavatories blocked. The tap in the yard overlooked by the Citroën factory was the only source of water. Children started to go down with infections and adults began to lose their minds.
On the third day, by which the Citroën workers were throwing bread into the yard and trying, but rarely succeeding, to deliver packages from relatives and friends during their lunch break, Claude Katz was evacuated with appendicitis.
The Rothschild Hospital had long been a source of information about French concentration camps and deportations, because it was there that men from the camps were taken in cases of acute illness. At the end of a day during which Allegra felt she was being stonewalled whenever she tried to contact the lawyer, somebody brought the news that Claude Katz was there. Claude lived on for four more days, too ill to do more than struggle for her life. Allegra was torn between being at her side and trying to find the lawyer to get the Katzes out. But the shutters had come down on the lawyer; people pretended they didn't know where he was or that he would be unable to see her, and she felt cast out and helpless. More so when Claude, beyond help, died of overwhelming infection.
There was indecision among the Germans and bickering between them and the Vichy French over what was to happen to the children. Were they to be deported or not? Vichy said no, then said if children were to be deported, their mothers must be allowed to stay with them. There was also the question of seven hundred francs per deportee to be paid by the French to the Germans, some of whom were of the opinion that the French didn't deserve to have their Jews removed for them. While great men fought over the niceties of deportation, conditions in the Winter Velodrome deteriorated to hell on earth and officials were refusing to work inside. Relocation of the inhabitants became a necessity.
Whether to appease Vichy or simply to solve their own problems, the Germans ordered that both adults and children be taken to the concentration camps at Pithiviers and Beaune-la-Rolande. From there the parents would be taken a few at a time to Drancy, and from Drancy they would go direct to Auschwitz. French officials would accompany them to the borders of the Reich. As for the children, they would follow when permission was granted by Eichmann. On the 29th of July that permission came.
After the war there were two opinions given on the decision of the Vichy French to agree to the deportation of the children. If they were ignorant of the extermination program, then it may have been done as a humanitarian measure to unite the children with their parents at the fabled Jewish state in the east. If they knew of the existence of Auschwitz and had any idea of its purpose, then Vichy wanted the Germans to do for them a job they might well have considered doing themselves, had the fascists won the war.
That German concentration camps were brutal was known seven years before the fall of France; people had been shot attempting to escape, they'd been shot for being disorderly or disobedient. Foreign journalists had visited Dachau and knew these things to be true. By the end of 1933, Hitler's first year in office, the Nazis had established sixty-five camps, but Himmler complained of inefficiencies. Inefficiencies in camps for political prisoners and stateless people? What was meant by that? In what way could such camps be inefficient? Were there not enough mail bags being made, or whatever it was that political prisoners and stateless people made? Or did inefficiencies mean that the inmates weren't getting enough food, shelter, warmth or equipment to keep the camps running smoothly? Or were the inefficiencies in that area which is forever with us, the area of processing people? Perhaps it wasn't the inmates who were inefficient at all, but their jailers.
Moishe Petrucelli was smuggled out of France by the Resistance the day Eichmann passed a death sentence on his wife and children. Allegra Coretti had left Paris by then. Her aim was to stay alive long enough to see her friends delivered from hell. It was not to be. With her Italian identification papers, she headed for the Italian occupied zone. Who knows whether she made it? More is known about the SS and the trouble they had commandeering closed trains, which, they complained, were being used to take soldiers to the eastern front instead of for the important job of deportation. Even supposing Allegra did reach the Italian sector, the fall of Italy saw a sell-out of the Jews and political subversives in the area. Informers were paid well by the SS, and it doesn't take much to rationalise the forfeiture of someone else's life for your own survival.
The fate of those in the Winter Velodrome, of course, was more Rose's story than Viva's. Viva said she resisted knowing about it until she and Dadda eloped and they went to Europe.
'We tried to find out about the ones whose fates weren't known,' she said, 'but there was no trace of them.'
It was well after midnight in Harry's beach house when Viva finished telling me her part of the story. And Checkie, I asked, did she know the things Viva had told me? Did she know that Dadda was her father?
Viva said Checkie grew up thinking she was Harry's child. She knew now she was not, but when we first knew her she had understood herself to be Harry's child.
'Harry's very proud of Checkie,' said Viva, 'and she loves him.' Checkie was not told of her paternity until Viva left with Dadda.
Harry told her, but he told her in a protective, gentle, fatherly way. Harry had then groomed Checkie to take over Siècle from him. As far as the Trust was concerned, Harry declared himself out of it when Dadda and Viva married, but before he withdrew, unknown to Viva, he and Reg Sorby had revised the Trust charter. Exempt now from the core collection were the works by Henry Coretti. This was because Harry and Reg thought Stella, Allegra and I should at least have a claim on them. Among these were the paintings Dadda had left with Rose for Checkie.
When Harry said Viva and Checkie already had enough, he meant they had all the Halletts, the Sorbys and key pieces by other painters.
Viva tried to convince Dadda to join the Trust in Harry's place, but Dadda wouldn't. He agreed with Reg and Harry that Stella, Allegra and I should now have the paintings that had been destined for Checkie, because Checkie was already extremely well provided for, her future assured as the eventual trustee for the Siècle collection. Not only would she be trustee, she would inherit a house in Toorak and the beach house when Harry died.
Dadda hadn't left an official will: rather, Gosper and Co. had cobbled his intentions together from letters and other things 'in black and white'. We, being hot-headed, impatient and hurt, had never had our demand to see a will granted. In fact, in spite of some good advice, we'd compromised ourselves again and again legally.
Viva felt bitter about the Corettis. It was one thing to have the most important of the Halletts, but quite another to have Dadda's first Australian paintings.
The Trust collection had been the joint idea of Viva and Harry, and between them they had decided it should include Dadda's paintings. This was partly to keep Dadda's presence in their lives low-key: they could distance themselves from him by including important pieces of his work with important pieces from other painters. In this way his work would always be available to Checkie, should she want to give it special attention. And on the whole, it seemed a good deal more worthwhile to create a collection such as this than to sit on important works of Australian art, knowing what they represented and yet doing nothing with them. The Trust and the story of Siècle and its painters would make a valuable cultural gift to Australia.
When Siècle was the restaurant with the lollipop awning, Viva and Harry had amassed quite a lot of work, but they didn't have a suitable place to store it. Then Laurent fell ill with cancer, and a decision had to be made about the restaurant. They decided to scrap it and buy up something for Rose and Laurent to live in so they could also caretake the paintings. This had prompted Harry to buy the brick duplex where Rose now lived, the collection being housed next door. Because Siècle was richer in assets than in cash, Rose and Laurent were given the right to sell certain of the paintings, always excepting the Halletts and the original collection of Corettis.
When Dadda first showed at Siècle in 1961, Reg Sorby bought out the entire show and made it over to the Siècle Trust, partly as a gesture to his patrons and partly to avoid paying tax. There were already so many paintings in the collection, however, that the house next to Rose was overflowing.
When Viva and Dadda eloped and Harry moved to Harcourt Lane, they began to store the overflow of paintings above Siècle.
Then Rose and I made friends. Dadda had already told Rose that we didn't have any of his work and he didn't know how to give us any without causing offence. Rose talked it over with Reg, and Reg, who didn't give a damn about offending people, started to manoeuvre work away from the Trust. He hid the core collection of Corettis up here, in his rural retreat. His idea was to hide them until some solution could be agreed upon. But The Brolga didn't want me on the Trust, she wanted Dadda's work to go to Checkie alone.
Checkie told Viva that Reg was hiving off the Corettis and this culminated in a row during which Dadda started taking trips to Japan and to the desert. Dadda was on Reg's side, but Viva was adamant about the paintings he had left for Checkie and, indeed, had a letter he'd written at the time saying that the paintings were all he had to give.
When I asked Viva if it was true she had scattered Dadda's ashes on Leslie Hallett's grave, she said, 'Yes. And I've written in my will that my ashes are to be spread there too.'
'Who by?' I found myself asking, suddenly angry with her. 'Not by me.' I thought of Checkie turning Bridget Kelly away from Dadda's exhibition, of her rallying to the cause of David Silver when it suited her, of Viva manipulating David into taking legal action against us instead of helping Allegra salvage her dream, of Dadda's death and the stupidities over the will – we had read the situation as the human drama it was, they had read it as a brittle testament. Technically, they were right, but emotionally, what were they? And then, too, I thought of Checkie's tears at Allegra's funeral. What were they all about? I thought of her taking Uncle Nicola's diamond ring off Dadda's finger and handing it to her mother. 'I don't want your ashes to lie with Dadda's.'
I could see I'd made her furious. She was bottling in a rage, it twitched in her fingers and shuddered in her chin. Her eyes were skidding. She thumped down her glass and strode outside, her arms folded. I looked at the shoes she'd left behind on the floor; long, skinny, Ferragamo Brolga shoes.
I walked over to the French doors and looked out. I could see her lying face-down on her brother's grave, her shoulders heaving in grief. So, she had some real emotions after all. I couldn't hate her, didn't hate her anymore. She commanded, at the very least, my respect. Nor did I hate Checkie. I didn't really know Checkie. I didn't think I ever could know her. There had been occasions when I'd heard her being droll, occasions, too, when she'd laughed at some funny thing I'd said. But I could not like her; she was mercenary, she and Allegra were natural antagonists, and I had loved Allegra. I loved her still. I could feel her inside me, looking out.
It was Reg who sent Viva to the beach house to have things out with me. At first, she said she'd come only on condition Dadda's paintings went to Checkie, but Reg would give no undertaking, and she came quite possibly in the hope of manoeuvring me into relinquishing my claim. I have to say she almost succeeded, but if she loved Dadda then so did I, and if he meant so much to her, then he meant easily as much to me. She had an idée fixe that Checkie would become the Australian expert on Henry Coretti; maybe she'd write a detailed biography. She wanted the story of herself and Dadda known, but she wanted it told in her terms. I had known for quite some time that Jerry Gospel's friend, the reluctant Nyle, had sat at her feet in the role of Checkie's tape recordist, but I'd imagined Viva told a rather different story from the one she told me. I think she surprised herself with the extent of her own admissions to me. She'd held me in contempt formerly, as if I'd had no entitlement to know and was quite unworthy of knowing. It hadn't been beyond Viva to feed David with some vicious one-liners about my work. Knowing this history didn't help my relationship with Viva, and when, in defiance of Reg, she tried to bully the decision she wanted out of me, I told her to remember that Nin and Eli, being Dadda's descendants, also had a claim on the paintings. When she came up with a plan that was reasonable and suited the three of us, we would acquiesce.
Harry Laurington died instructing Viva to make my mother, Eli, Nin and me members of the Trust. It was a bit late as far as my mother was concerned, as her memory was, by this, quite impaired, but I would consent to the release of the paintings if the children and I were made full members of the Trust.
Now Viva is old and frail, bringing to mind dear old Beryl's comment about orchids and the doggedness of frailty, and I am here at Reg's, still waiting for a reasonable proposal from her – or for news of her death. Without Harry's wisdom to guide Checkie, she is inclined to take her mother's part, but Reg says he is sure he can talk her out of it – for I also believe it is important for the work and its history to be handed down together.
Kelly Kelly and her bloke Guitar have opened a branch of their business up there, and between us we give Nin what we can of a family life. We take her out painting. She is learning to use natural textures. There is much you can do with a limited palette around here, where nature's calligraphy is at once intricate and vast. David has legal access to Nin, but we are blessedly far away from Melbourne for the time being, and there is little risk of him pressing a demand.
Eli went to America a while ago to find his father and came back saying he hadn't missed much and nor had I. He liked his father all the same, and thought he'd keep in touch. Arnie was shipwrecked on his third divorce. Eli is in South-East Asia now, sending home articles on Cambodia. I'm trying to place them in newspapers, but not having much luck. He doesn't keep in contact with David, who is still hanging round with Jerry and Pattie Gosper, the reluctant Nyle and Barrie Bull.
I have thought for a long time about the breach between me and Eli. I think I am in the category of women once mentioned by Rose, who mourn when their children grow up. I missed my little child, but now, in his place, I have a man, and the love I feel is different. Eli was never the person I was expecting; he was himself. All a mother can do is provide an environment for the self. The environment Eli grew up in had some deficiencies, but he learnt to use those in order to know himself better. I think if I had adhered to some strict notion of upbringing, if I had been a textbook Marxist or militant feminist, Eli would have had to wear it and he may have been a good deal longer finding his way through life's trials than he has been, though his childhood wish to be normal was never granted. Normality, after all, is only academic.
It is morning here now. The Midnight Knitter is stirring in her chair. You, whoever you are, are about to arrive. What do you want to know? What can I tell you? About a lonely cup on a tabletop I once painted? Womb and nourishment, but through art – the gift the world would not allow me to bestow. I didn't think it mattered if a painter was male or female. I started out thinking a woman could be an artist, but now I know the artist is also a woman.
The rain has stopped and everything is clear.
About the author
Scientist, writer, biographer and historian, Sally Morrison's many interests are reflected in a dense, layered writing style that has captured the attention of critics and readers over a long and rich career.
Born in Sydney, raised and educated in Canberra, and now living in Melbourne, Sally originally trained as a molecular biologist before beginning her writing career in the 1970s. Her work includes the play _Hag_ , short story collection _I Am a Boat_ , a biography of Clifton Pugh, _After Fire_ , and novels _Who's Taking You to the Dance?, Against Gravity_ and _The Insatiable Desire of Injured Love_. Her most recent novel, _Window Gods_ , is set in the same world as the award-winning _Mad Meg_.
www.sallymorrison.com
www.facebook.com/SallyMorrisonWriter
First published in 1994
This edition published in 2015 by Hardie Grant Books
Hardie Grant Books (Australia)
Ground Floor, Building 1
658 Church Street
Richmond, Victoria 3121
www.hardiegrant.com.au
Hardie Grant Books (UK)
5th & 6th Floor
52–54 Southwark Street
London SE1 1UN
www.hardiegrant.co.uk
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publishers and copyright holders.
The moral rights of the author have been asserted.
Copyright text © Sally Morrison
A Cataloguing-in-Publication entry is available from the catalogue of the National
Library of Australia at www.nla.gov.au
Mad Meg
eISBN 978 1 74358 3425
Cover design by Nada Backovic
Cover image courtesy of Miguel Angel Munoz/Arcangel Images
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\section{Introduction}
Solutions to important real world problems from science and technology turn out to realize minimal energy of suitable nonlinear functionals. Finding these solutions and examining closely their qualitative properties is a central problem of the Calculus of Variations, and the machinery of the nonlinear functional analysis is what serves to pursue that study. On the other hand, each minimizer of a variational functional solves weakly the corresponding Euler--Lagrange equation and this fact allows to rely on the powerful theory of PDEs as additional tool in the Calculus of Variations. The Euler--Lagrange equations are divergence form PDEs of elliptic type, usually nonlinear, and their weak solutions (the minimizers) own some basic minimal smoothness. The \textit{regularity theory} of general (non necessary variational) divergence form elliptic PDEs establishes how the smoothness of the data of a given problem influences the regularity of
the solution, obtained under very general circumstances. Once having better smoothness, powerful tools of
functional analysis apply to infer finer properties of the solution and the problem itself. The importance of these issues is even more evident in the settings of variational problems if dealing with discontinuous functionals over domains with
non-smooth boundaries when many of the classical nonlinear analysis techniques fail.
Starting with the deep results of Caffarelli and Peral (\cite{CP}), a notable progress has been achieved in the last two decades in the regularity theory of nonlinear divergence form elliptic PDEs (see also \cite{AM, BDM, BK, BW1, CM, KM, MP} and the references therein). On the base of suitable $L^p$-estimates for the gradient $Du$ of the weak solution a satisfactory Calder\'on--Zygmund type theory has been developed, firstly for equations with principal term depending only on $Du,$ and later also dependence on the independent variables $x$ has been allowed. Moreover, the minimal regularity requirements have been identified for the nonlinear terms of the equations and the boundary of the underlying domain in order the Calder\'on--Zygmund theory still holds true for large class of equations with generally $x$-discontinuous ingredients. In all that context, the possibility to deal with equations with general nonlinearity with respect to the solution $u$ is a rather delicate matter, and the reason of this lies in the fact that such equations are not invariant under particular scaling and normalization, whereas these are crucial ingredients of the perturbation approach in \cite{CP}.
\medskip
We deal here with the Dirichlet problem
\begin{equation}\label{1}
\begin{cases}
\mathrm{div\,}\ba(x,u,Du)=\mathrm{div\,}(|F|^{p-2}F) & \textrm{in}\ \Omega\\
u=0 & \textrm{on}\ \partial\Omega,
\end{cases}
\end{equation}
where $\Omega\subset\mr^n,$ $n\geq2,$ is a bounded and generally irregular domain,
$\ba\colon\mr^n\times\mr\times\mr^n\to\mr^n$ is a Carath\'eodory map,
$p>1$ is arbitrary exponent and $F\in L^p(\Omega,\mr^n).$
Our main goal is to obtain a Calder\'on--Zygmund type regularizing effect for \eqref{1}. Namely, assuming $F\in L^{p'}(\Omega,\mr^n)$ for $p'>p,$
under rather general structure and regularity hypotheses on $\ba(x,z,\xi)$ and $\partial\Omega,$ we derive global $L^{p'}(\Omega)$-gradient estimate for any bounded $W^{1,p}_0(\Omega)$ weak solution of \eqref{1} in terms of $\|F\|_{L^{p'}(\Omega,\mr^n)}$ showing this way that $F\in L^{p'}(\Omega,\mr^n)$ implies $Du\in L^{p'}(\Omega,\mr^n).$
In the case when $\ba=\ba(x,Du),$ similar results have been obtained in \cite{BW2} in the settings of classical Lebesgue spaces and in \cite{BR} for weighted Lebesgue spaces, assuming the standard ellipticity condition and allowing discontinuity of $\ba$ with respect to $x,$ measured in terms of small-BMO seminorm. In the recent paper \cite{NP}, the authors succeeded to obtain \textit{interior} gradient estimates for \eqref{1} also in the case when $\ba$ depends on the solution $u.$ The problems arising with the scaling and normalization in that situation are cleverly avoided by including the nonlinear differential operator into a \textit{two parameter} class of elliptic operators, that turns out to be invariant with respect to dilations and rescaling of domain. In order to run the approximation procedure of \cite{CP}, a \textit{uniform} control with respect to these two parameters is necessary, and the authors of \cite{NP} carry out it by means of a delicate compactness argument relying on the Minty trick. This approach, however, strongly requires \textit{uniqueness} for the approximating equation, that is why, $\ba(x,z,\xi)$ is assumed to be \textit{Lipschitz continuous with respect to $z$} in \cite{NP}.
Here we suppose that $\ba(x,z,\xi)$ is small-BMO function with respect to $x$ and it satisfies the standard uniform ellipticity condition in $\xi$ but, in contrast to \cite{NP}, $\ba$ is assumed to be \textit{only H\"older continuous} with respect to the variable $z.$ To get our main result, we combine the two-parameter approach from \cite{NP} with correct scaling arguments in the $L^q$-estimates for the maximal function of the gradient and Vitali type covering lemma. However, we rely here on the \textit{higher gradient integrability} in the spirit of Gehring--Giaquinta rather than on the uniqueness of the approximating equation, and this allows us to weaken the $z$-Lipschitz continuity of $\ba$ to only H\"older one.
We start with considering two appropriate reference problems with \textit{only gradient nonlinear terms}, given by the $z$-compositions of $\ba(x, z, \xi)$ first with the weak solution $u(x)$ and then with its local average $\bar{u}.$ Thanks to the uniform ellipticity of the associated
nonlinearities, the reference solutions support higher integrability results and H\"{o}lder continuity properties. We then combine these
properties with the $z$-H\"older continuity of $\ba$ and the comparison estimates of \cite{BR}, regarding nonlinear terms like $\ba(x,Du),$ in order to obtain the desired comparison estimates. Once having these, standard maximal
function approach and a Vitali type covering lemma give the main result.
It is worth noting that we need $\ba(x,z,\xi)$ to be
H\"older continuous in $z$ \textit{only in the case when $p<n.$} Otherwise, the weak solution of \eqref{1} is itself a H\"older continuous which implies that the nonlinear term in \eqref{1}, fixed at the solution $u(x),$ that is $\mathbf{A}(x,\xi):= \ba (x,u(x),\xi),$ is a small-BMO with respect to $x$ if $\ba (x,z,\xi)$ is required to be merely \textit{continuous} in $z.$ This suffices to run our procedure and to get the Calder\'on--Zygmund property assuming only $z$-\textit{continuity} of $\ba$ when $p\geq n.$
Another advantage of the approach here adopted is that it works also near the boundary of $\Omega$ and this allows to obtain \textit{global} gradient estimates for the solutions of \eqref{1}. Indeed, this requires some ``good'' geometric properties of $\partial\Omega$ and these are ensured when $\Omega$ belongs to the class of the Reifenberg flat domains.
The paper is organized as follows. In Section~\ref{Sec2} we list the hypotheses imposed on the data and state the main result, Theorem~\ref{Thm1}. Some comments about the structure and regularity assumptions required are given as well. Section~\ref{Sec3} provides an analysis of how the equation in \eqref{1} and the hypotheses on the nonlinear term behave under the two-parameter scaling and normalization.
Section~\ref{Sec4} forms the analytic heart of the paper. We derive there good gradient estimates for solutions to appropriate limiting problems to which \eqref{1} compares. With these estimates at hand, we
employ in Section~\ref{Sec5} a Vitali type covering lemma and scaling arguments in order to prove Theorem~\ref{Thm1} by obtaining suitable decay estimates for the level sets of the Hardy--Littlewood maximal function of the gradient. The last Section~\ref{Sec6} is devoted to the refinement of the main result in the case when $p\geq n.$ The H\"older continuity of $\ba(x,z,\xi)$ with respect to $z$ is relaxed to only continuity and we combine our recent results \cite{BPS,BPS-arxiv} with these of \cite{BR} to get the refined version of Theorem~\ref{Thm1} when $p\geq n.$
\bigskip
\noindent
{\bf Acknowledgements.}
S.-S.~Byun was supported by the National Research Foundation of Korea grant funded by the Korea government (MEST) (NRF-2015R1A4A1041675).
D.K.~Palagachev is member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
P. Shin was supported by National Research Foundation of Korea grant funded by the Korean government (MEST) (NRF-2015R1A2A1A15053024).
\section{Hypotheses and main results}\label{Sec2}
Throughout the paper, we will use standard notations and will assume that the functions and sets considered are measurable.
We denote by $B_\rho(x)$ (or simply $B_\rho$ if there is no ambiguity) the $n$-dimensional open ball with center $x\in\mr^n$ and radius $\rho,$
and $\Omega_\rho(x):=\Omega\cap B_\rho(x)$
for an open set $\Omega\subset \mr^n.$
The Lebesgue measure of a measurable set $A\subset\mr^n$ will be denoted
by $|A|$ while, for any integrable function $u$ defined on $A,$
$$
\overline{u}_A:=\Xint-_A u(x)\; dx = \frac{1}{|A|}\int_A u(x)\; dx
$$
stands for its integral average. If
$u\in L^1_{\text{loc}}(\mr^n),$ then the Hardy--Littlewood maximal function of $u$ is given by
$$
\m u(x) : = \sup_{\rho>0} \Xint-_{B_\rho(x)} |u(y)|\; dy,
$$
while $\m_A u:=\m\left(\chi_A u\right)$ when
$u$ is defined on a measurable set $A,$ with the characteristic function $\chi_A$ of the set $A.$
We will denote by $C^\infty_0(\Omega)$ the space of infinitely differentiable functions over a bounded domain $\Omega\subset\mr^n$ with compact support contained in $\Omega,$ and $L^p(\Omega)$ stands for the standard Lebesgue space with a given $p\in[1,\infty].$ The Sobolev space $W^{1,p}_0(\Omega)$ is defined, as usual, by the completion of $C^\infty_0(\Omega)$ with respect to the norm
$$
\|u\|_{W^{1,p}(\Omega)}:=\|u\|_{L^p(\Omega)}+\|Du\|_{L^p(\Omega)}
$$
for $p\in [1,\infty).$
In what follows we will consider a bounded domain $\Omega\subset\mr^n$ with $n\geq2,$ the boundary $\partial\Omega$ of which is \textit{Reifenberg flat} in the sense of the following definition.
\begin{definition}\label{defin2.1}
The domain $\Omega$ is said to be $(\delta,R)$-Reifenberg flat if there exist positive constants $\delta$ and $R$ with the property that for each $x_0\in\partial\Omega$ and each $\rho\in(0,R)$ there is a local coordinate system $\{x_1,\cdots,x_n\}$ with origin at the point $x_0,$ and such that
$$
B_\rho(x_0)\cap\{x:x_n>\rho\delta\}\subset B_\rho(x_0)\cap\Omega \subset B_\rho(x_0)\cap\{x:x_n>-\rho\delta\}.
$$
\end{definition}
Turning back to problem \eqref{1}, the nonlinear term is given by the Carath\'eodory map $\ba\colon\Omega\times\mr\times\mr^n\to\mr^n$ where
$\ba(x,z,\xi)=\big(a^1(x,z,\xi),\cdots,a^n(x,z,\xi)\big).$ We suppose moreover that
$\ba(x,z,\xi)$ is differentiable with respect to $\xi,$ and $D_\xi\ba$ is a Carath\'eodory map.
Throughout the paper the following structure and regularity conditions on the data will be assumed:
\medskip
$\bullet$ \textit{Uniform ellipticity:} There exists a constant $\gamma>0$ such that
\begin{equation}\label{3}
\begin{cases}
\gamma|\xi|^{p-2}|\eta|^2\leq\langle D_\xi\ba(x,z,\xi)\eta,\eta\rangle,\\
|\ba(x,z,\xi)|+|\xi||D_\xi\ba(x,z,\xi)|\leq\gamma^{-1}|\xi|^{p-1}
\end{cases}
\end{equation}
for a.a. $x\in\Omega$ and $\forall(z,\xi)\in\mr\times\mr^n,$ $\forall\eta\in\mr^n.$
It is worth noting that the uniform ellipticity condition \eqref{3} implies easily the following monotonicity property:
\begin{equation}\label{Mo1}
\big\langle \ba(x,z,\xi_1)-\ba(x,z,\xi_2),\xi_1-\xi_2 \big\rangle \geq
\begin{cases}
\widetilde{\gamma}|\xi_1-\xi_2|^p & \textrm{if}\ p\geq 2,\\
\widetilde{\gamma}|\xi_1-\xi_2|^2(|\xi_1|+|\xi_2|)^{p-2} & \textrm{if}\ 1<p<2,
\end{cases}
\end{equation}
where $\widetilde{\gamma}$ depends only on $\gamma,$ $n$ and $p.$
\medskip
$\bullet$ \textit{H$\ddot{o}$lder continuity:} There exist constants $\Gamma>0$ and $0<\alpha<1$ such that
\begin{equation}\label{4}
|\ba(x,z_1,\xi)-\ba(x,z_2,\xi)|\leq \Gamma|z_1-z_2|^\alpha|\xi|^{p-1}
\end{equation}
for a.a. $x\in\Omega,$ $\forall z_1,z_2\in\mr$ and $\forall\xi\in\mr^n.$
\medskip
$\bullet$ \textit{$(\delta,R)$-vanishing property:} For each constant $M>0$ there exist $R>0$ and $\delta>0,$ depending on $M,$ such that
\begin{equation}\label{5}
\sup_{z\in[-M,M]}
\sup_{0<\rho\leq R}\sup_{y\in\mr^n}\Xint-_{B_\rho(y)}\Theta\big(\ba;B_\rho(y)\big)(x,z)dx \leq \delta,
\end{equation}
where the function $\Theta$ is defined by
$$
\Theta\big(\ba;B_\rho(y)\big)(x,z):=\sup_{\xi\in\mr^n\setminus\{0\}}\frac{|\ba(x,z,\xi)-\overline{\ba}_{B_\rho(y)}(z,\xi)|}{|\xi|^{p-1}},
$$
and $\overline{\ba}_{B_\rho(y)}(z,\xi)$ is the integral average of $\ba(x,z,\xi)$ in the variables $x$ for a fixed couple $(z,\xi)\in\mr\times\mr^n,$ that is,
$$\overline{\ba}_{B_\rho(y)}(z,\xi)=\Xint-_{B_\rho(y)}\ba(x,z,\xi)\ dx.
$$
\medskip
To make clear the meaning of the above assumptions, we should note that, thanks to the scaling invariance property of $\partial\Omega,$ $R$ could be any number greater than $1$ in Definition~\ref{defin2.1}, while $R$ could be taken equal to $\text{diam\,}\Omega$ in \eqref{5}. For what concerns
$\delta$ instead, the definitions of $(\delta,R)$-Reifenberg flatness and $(\delta,R)$-vanishing property are significant only for small values, say $\delta\in(0,1/8).$ Roughly speaking, the Reifenberg flatness of $\Omega$ means that $\partial\Omega$ is well approximated by hyperplanes at every point and at every scale. In particular, domains with $C^1$-smooth boundary or with boundary that is locally given as graph
of a Lipschitz continuous function with small Lipschitz constant are Reifenberg flat.
Actually, the class of the Reifenberg flat domains is much wider and contains sets with rough fractal boundaries such as the von Koch snowflake that is a Reifenberg flat when the angle of the spike with respect to the horizontal is small enough. As for the $(\delta,R)$-vanishing property \eqref{5}, it exhibits a sort of smallness in terms of BMO for what concerns the behaviour of $\ba(x,z,\xi)$ with respect to the $x$-variables. For instance, \eqref{5} is satisfied when $\ba\in C^0_x$ or even VMO${}_x.$ This way, \eqref{5} allows $x$-\textit{discontinuity} of the nonlinearity which is controlled in terms of small-BMO.
\medskip
Turning bach to the Dirichlet problem \eqref{1}, recall that a function $u\in W^{1,p}_0(\Omega)$ is said to be a \textit{weak solution} if
$$
\int_\Omega \big\langle \ba(x,u(x),Du(x)), D\phi(x)\big\rangle\; dx = \int_\Omega \big\langle|F(x)|^{p-2}F(x), D\phi(x)\big\rangle\; dx
$$
for each test function $\phi\in W^{1,p}_0(\Omega).$
\medskip
Our main result is as follows.
\begin{theorem}\label{Thm1}
Suppose \eqref{3} and \eqref{4}, and let $u\in W^{1,p}_0(\Omega)\cap L^\infty(\Omega)$ be a bounded weak solution of \eqref{1}. Assume that $\|u\|_{L^\infty(\Omega)}\leq M$ and $|F|^p\in L^q(\Omega)$ for some $q\in (1,\infty).$ Then there is a small constant $\delta=\delta(\gamma,\alpha,n,p,q,\Gamma,M)$ such that if $\ba$ is $(\delta,R)$-vanishing and $\Omega$ is $(\delta,R)$-Reifenberg flat, then $|Du|^p\in L^q(\Omega)$ with the estimate
$$
\int_\Omega |Du|^{pq} dx \leq C \int_\Omega |F|^{pq} dx,
$$
where $C>0$ depends only on $\gamma,$ $\alpha,$ $n,$ $p,$ $q,$ $\Gamma,$ $M$ and $|\Omega|.$
\end{theorem}
\section{Scaling and normalization properties}\label{Sec3}
In this section, we will show how the scaling and normalization reflect on the structure conditions and regularity assumptions imposed on the data.
Recall that that $\Omega$ is assumed to be a $(\delta,R)$-Reifenberg flat domain and the nonlinearity $\ba$ satisfies the conditions \eqref{3}, \eqref{4} and the $(\delta,R)$-vanishing property \eqref{5}. Let $\sigma$ be a large enough positive constant which is to be determined later in a universal way so that it will depend only on the given data such as $n,$ $p,$ $q,$ $\gamma,$ $\Gamma$ and $M.$ Then for each fixed $\lambda>0$ and $0<r\leq \frac{R}{\sigma},$ we define a bounded domain
$$
\widetilde{\Omega}=\left\{\frac{1}{r}x:x\in\Omega\right\},
$$
a Carath\'eodory map $\widetilde{\ba}\colon\mr^n\times\mr\times\mr^n\to\mr^n$ by
$$
\widetilde{\ba}(x,z,\xi)=\frac{\ba(r x,\lambda r z,\lambda\xi)}{\lambda^{p-1}},
$$
a Sobolev function $\widetilde{u}\in W^{1,p}_0(\widetilde{\Omega})$ and a measurable function $\widetilde{F}\in L^{p}(\widetilde{\Omega})$ by
$$
\widetilde{u}(x)=\frac{u(rx)}{\lambda r}\quad \mathrm{and}\quad \widetilde{F}(x)=\frac{F(r x)}{\lambda}.
$$
Straightforward calculations yield the following properties:
\begin{itemize}
\item[$\bullet$]
$\widetilde{\ba}$ satisfies the uniform ellipticity condition \eqref{4} with the same constant $\gamma.$ That is,
\begin{equation}\label{S}
\begin{cases}
\gamma|\xi|^{p-2}|\eta|^2\leq\langle D_\xi\widetilde{\ba}(x,z,\xi)\eta,\eta\rangle,\\
|\widetilde{\ba}(x,z,\xi)|+|\xi||D_\xi\widetilde{\ba}(x,z,\xi)|\leq\gamma^{-1}|\xi|^{p-1}
\end{cases}
\end{equation}
for a.a. $x\in\mr^n$ and $\forall(z,\xi)\in\mr\times\mr^n,$ $\forall\eta\in\mr^n.$
Moreover, the monotonicity
\begin{align}\label{Mo}
\langle \widetilde{\ba}(x,z,\xi_1)-&\widetilde{\ba}(x,z,\xi_2),\xi_1-\xi_2 \rangle\\
\nonumber
&\geq
\begin{cases}
\widetilde{\gamma}|\xi_1-\xi_2|^p & \textrm{if}\ p\geq 2,\\
\widetilde{\gamma}|\xi_1-\xi_2|^2(|\xi_1|+|\xi_2|)^{p-2} & \textrm{if}\ 1<p<2,
\end{cases}
\end{align}
does follow with the same constant $\widetilde{\gamma}.$
\item[$\bullet$]
$\widetilde{\ba}$ satisfies
\begin{equation}\label{S1}
|\widetilde{\ba}(x,z_1,\xi)-\widetilde{\ba}(x,z_2,\xi)|\leq \Gamma(\lambda r)^\alpha|z_1-z_2|^\alpha|\xi|^{p-1}
\end{equation}
for a.a. $x\in\widetilde{\Omega},$ $\forall z_1,z_2\in\mr$ and $\forall\xi\in\mr^n$ with the same constants $\alpha$ and $\Gamma.$
\item[$\bullet$]
$\widetilde{\ba}$ is $(\delta,\frac{R}{r})$-vanishing. Namely,
$$
\sup_{z\in[-\frac{M}{\lambda r},\frac{M}{\lambda r}]}
\sup_{0<\rho\leq \frac{R}{r}}\sup_{y\in\mr^n}\Xint-_{B_\rho(y)}\Theta\big(\widetilde{\ba};B_\rho(y)\big)(x,z)dx \leq \delta.
$$
\item[$\bullet$]
$\widetilde{\Omega}$ is $(\delta,\frac{R}{r})$-Reifenberg flat.
\item[$\bullet$]
If $u\in W^{1,p}_0(\Omega)$ is a weak solution of \eqref{1}, then $\widetilde{u}\in W^{1,p}_0(\widetilde{\Omega})$ is a weak solution of the problem
\begin{equation}\label{UP}
\begin{cases}
\mathrm{div\,}\widetilde{\ba}(x,\widetilde{u}(x),D\widetilde{u}(x))=\mathrm{div\,}(|\widetilde{F}|^{p-2}\widetilde{F}) & \textrm{in}\ \widetilde{\Omega}\\
\widetilde{u}=0 & \textrm{on}\ \partial\widetilde{\Omega}.
\end{cases}
\end{equation}
\end{itemize}
\section{Comparison estimates}\label{Sec4}
A crucial step in the proof of the main result is ensured by appropriate comparison of the weak solution to \eqref{UP} with these of the associated reference problems \eqref{HP}, \eqref{FP} and \eqref{VP} below. Throughout the section, for the sake of simplicity,
we will use the notations $u,$ $F,$ $\ba$ and $\Omega,$ instead of $\widetilde{u},$ $\widetilde{F},$ $\widetilde{\ba}$ and $\widetilde{\Omega},$ respectively.
We start with the following useful lemma.
\begin{lemma}\label{Lem1}
Let $\Omega\subset\mr^n$ be a bounded open set. Assume that $\ba:\Omega\times\mr\times\mr^n\to\mr^n$ satisfies \eqref{Mo} for a.a. $x\in\Omega$ and for some $p\in(1,2).$ Then, for any $\zeta_1,$ $\zeta_2\in W^{1,p}(\Omega_\rho),$ any non-negative function $\eta\in C^\infty(B_\rho),$ any bounded function $\phi$ defined on $\Omega_\rho$ and any constant $\tau>0,$ we have
\begin{align*}
\int_{\Omega_\rho} |D\zeta_1-D\zeta_2|^p\eta\; dx \leq &\tau\int_{\Omega_\rho} |D\zeta_1|^p\eta\; dx\\
&+ C\int_{\Omega_\rho} \langle \ba(x,\phi,D\zeta_1)-\ba(x,\phi,D\zeta_2),D\zeta_1-D\zeta_2\rangle\eta\; dx
\end{align*}
with $C>0$ depending only on $\gamma,$ $p$ and $\tau.$
\end{lemma}
\begin{proof}
See \cite[the proof of Lemma~3.7]{BR}, \cite[Lemma~3.1]{NP}.
\end{proof}
Let $\sigma > 6$ be a universal constant which will be chosen later in Lemma \ref{Lem3}, and consider a localized solution $u$ in $\Omega_\sigma$ of the problem
\begin{equation}\label{LUP}
\begin{cases}
\mathrm{div\,}\ba(x,u(x),Du(x))=\mathrm{div\,}(|\widetilde{F}|^{p-2}\widetilde{F}) & \textrm{in}\ \Omega_\sigma,\\
u=0 & \textrm{on}\ \partial \Omega\cap B_\sigma,
\end{cases}
\end{equation}
with
\begin{align}\label{31}
\|u\|_{L^\infty(\Omega_\sigma)}\leq \frac{M}{\lambda r}, \quad \frac{1}{|B_\sigma|}\int_{\Omega_\sigma} |Du|^p dx \leq 1,\quad \mathrm{and}\quad \frac{1}{|B_\sigma|}\int_{\Omega_\sigma} |F|^p dx \leq \delta^p.
\end{align}
Assume further that
\begin{align}\label{32}
\frac{1}{|B_6|}\int_{\Omega_6} |Du|^p dx \leq 1.
\end{align}
We let next $h\in W^{1,p}(\Omega_\sigma)$ to be the weak solution of
\begin{equation}\label{HP}
\begin{cases}
\mathrm{div\,}\ba(x,u(x),Dh(x))=0 & \textrm{in}\ \Omega_\sigma,\\
h=u & \textrm{on}\ \partial \Omega_\sigma,
\end{cases}
\end{equation}
and $f\in W^{1,p}(\Omega_5)$ the weak solution of
\begin{equation}\label{FP}
\begin{cases}
\mathrm{div\,}\ba(x,\overline{u}_{\Omega_5},Df(x))=0 & \textrm{in}\ \Omega_5,\\
f=h & \textrm{on}\ \partial \Omega_5,
\end{cases}
\end{equation}
with
\begin{equation}\label{33}
\frac{1}{|B_5|}\int_{\Omega_5}\Theta\big(\ba;\Omega_5\big)(x,\overline{u}_{\Omega_5})dx \leq \delta
\end{equation}
and
\begin{equation}\label{34}
B_5^+\subset \Omega_5 \subset B_5\cap\{x:x_n>-10\delta\}.
\end{equation}
We consider finally the limiting problem
\begin{equation}\label{VP}
\mathrm{div\,}\mathbf{A}(Dv(x))=0 \quad \textrm{in}\ U,
\end{equation}
where $U=B_4$ for the interior case and $U=B_4^+$ for the boundary case, where the map $\mathbf{A}:\mr^n \to \mr^n$ is given by
$$
\mathbf{A}(\xi):=\frac{1}{|U|}\int_U \ba(x,\overline{u}_{\Omega_5},\xi)\; dx.
$$
\medskip
The following is the main result of this section.
\begin{lemma}\label{MainLem}
For any small constant $\varepsilon\in(0,1),$ there exist a large constant $\sigma=\sigma(\gamma,\alpha, n,p,\Gamma,M,\varepsilon)> 6$ and a small constant $\delta=\delta(\gamma,\alpha, n,p,\Gamma,M,\varepsilon)>0$ such that if $u\in W^{1,p}(\Omega_\sigma)$ is a weak solution of \eqref{LUP} with
\eqref{31}, \eqref{32}, \eqref{33} and \eqref{34},
then there exists a weak solution $v\in W^{1,p}(U)$ of \eqref{VP} such that
$$
\|D\bar{v}\|_{L^\infty(\Omega_3)}\leq N_0\quad \text{and}\quad\int_{\Omega_4} |Du-D\bar{v}|^p dx \leq \varepsilon^p
$$
for some constant $N_0=N_0(\gamma,n,p)>1.$ Here, the function $\bar{v}\in W^{1,p}(\Omega_4)$ is equal to
$v$ when $U=B_4,$ and $\bar{v}$ is the zero extension of $v$ from $B_4^+$ to $B_4$ when $U=B_4^+.$
\end{lemma}
The proof is based on the following Lemmas~\ref{Lem2}, \ref{Lem3} and \ref{Lem4}.
\begin{lemma}\label{Lem2}
Let $\sigma>6$ and $0<\varepsilon_1<1.$ Then there exists a small constant $\delta=\delta(n,p,\gamma,\sigma,\varepsilon_1)>0$ such that if $u\in W^{1,p}(\Omega_\sigma)$ is a weak solution of \eqref{LUP} and
$h \in W^{1,p}(\Omega_\sigma)$ is the weak solution of \eqref{HP} with \eqref{31},
then
\begin{align}\label{42}
\frac{1}{|B_6|}\int_{\Omega_6} |Du-Dh|^p dx \leq \varepsilon_1^p.
\end{align}
\end{lemma}
\begin{proof}
The proof will be divided into two cases.
\medskip
$\mathbf{Case\ 1:}\ 1<p<2.$
Taking $u-h$ as a test function for equations \eqref{LUP} and \eqref{HP}, it follows from the Young inequality with $\tau_1>0$ that
\begin{align}\label{43}
\int_{\Omega_\sigma} \langle \ba(x,u,Du)-\ba(x,u,Dh),Du-Dh\rangle\; dx = \int_{\Omega_\sigma} \langle |F|^{p-2}F, Du-Dh \rangle\; dx\\
\nonumber \leq \tau_1\int_{\Omega_\sigma}|Du-Dh|^p\; dx + C(\tau_1)\int_{\Omega_\sigma} |F|^p\; dx.
\end{align}
Then Lemma~\ref{Lem1} implies
\begin{align*}
\int_{\Omega_\sigma} &|Du-Dh|^p\; dx\\
&\leq \tau\int_{\Omega_\sigma} |Du|^p\; dx + C_0(\tau)\int_{\Omega_\sigma} \langle \ba(x,u,Du)-\ba(x,u,Dh),Du-Dh\rangle\; dx\\
&\leq \tau\int_{\Omega_\sigma} |Du|^p\; dx + C_0\tau_1\int_{\Omega_\sigma}|Du-Dh|^p\; dx + C(\tau,\tau_1)\int_{\Omega_\sigma} |F|^p\; dx.
\end{align*}
Setting $\tau_1=\frac{1}{2C_0}$ in the above inequality, we obtain
\begin{align*}
\int_{\Omega_\sigma} &|Du-Dh|^p\; dx \leq 2\tau\int_{\Omega_\sigma} |Du|^p\; dx + C(\tau)\int_{\Omega_\sigma} |F|^p\; dx,
\end{align*}
and so \eqref{31} yields
\begin{align*}
\frac{1}{|B_6|}\int_{\Omega_6} |Du-Dh|^p\; dx &\leq \left(\frac{\sigma}{6}\right)^n\frac{1}{|B_\sigma|}\int_{\Omega_\sigma} |Du-Dh|^p\; dx\\
&\leq \left(\frac{\sigma}{6}\right)^n\frac{2\tau}{|B_\sigma|}\int_{\Omega_\sigma} |Du|^p\; dx + C(\tau,\sigma)\frac{1}{|B_\sigma|}\int_{\Omega_\sigma} |F|^p\; dx\\
&\leq 2\tau\left(\frac{\sigma}{6}\right)^n + C(\tau,\sigma)\delta^p.
\end{align*}
Now, taking the constants $\tau$ and $\delta$ sufficiently small so that
$$2\tau\left(\frac{\sigma}{6}\right)^n \leq \frac{\varepsilon_1^p}{2}\quad \mathrm{and} \quad C(\tau,\sigma)\delta^p \leq \frac{\varepsilon_1^p}{2},$$
we obtain the conclusion \eqref{42} when $1<p<2.$
\medskip
$\mathbf{Case\ 2:}\ p\geq 2.$
Having in mind \eqref{Mo} and \eqref{43}, we get
\begin{align*}
\int_{\Omega_\sigma} |Du-Dh|^p\; dx &\leq \widetilde{\gamma}^{-1}\int_{\Omega_\sigma} \langle \ba(x,u,Du)-\ba(x,u,Dh),Du-Dh\rangle\; dx\\
&\leq \widetilde{\gamma}^{-1}\tau_1\int_{\Omega_\sigma}|Du-Dh|^p\; dx + C(\tau_1)\int_{\Omega_\sigma} |F|^p\; dx
\end{align*}
for any $\tau_1>0.$ Taking $\tau_1=\frac{\widetilde{\gamma}}{2}$ in the above inequality, it follows from \eqref{31} that
\begin{align*}
\frac{1}{|B_6|}\int_{\Omega_6} |Du-Dh|^p\; dx &\leq \left(\frac{\sigma}{6}\right)^n\frac{1}{|B_\sigma|}\int_{\Omega_\sigma} |Du-Dh|^p\; dx\\
&\leq \frac{C_1(\gamma,p)\sigma^n}{|B_\sigma|}\int_{\Omega_\sigma} |F|^p\; dx \leq C_1\sigma^n\delta^p.
\end{align*}
We choose now the constant $\delta$ small enough to have $C_1\sigma^n\delta^p \leq \varepsilon_1^p,$ and this gives the claim
of Lemma~\ref{Lem2}.
\end{proof}
We need the following higher integrability result for the equation \eqref{HP}.
\begin{proposition}\label{LemHIG}
(\cite[Theorem~1.1]{KK}, \cite[Theorem~2.2]{BW2} \cite[Lemma~3.2]{BR})
Let $h\in W^{1,p}(\Omega_\sigma)$ be a solution of \eqref{HP}. Then there is a positive constants $p_0>p$ depending only on $\gamma,$ $n$ and $p$ such that for any $p_1\in (p,p_0),$
$$
\left(\int_{\Omega_5} |Dh|^ {p_1}\ dx\right)^\frac{1}{p_1} \leq C\left(\int_{\Omega_6} |Dh|^p\ dx\right)^\frac{1}{p}
$$
holds, where $C>0$ depends only on $\gamma,$ $n,$ $p$ and $p_0.$
\end{proposition}
We also need the following oscillation theorem for the equation \eqref{HP}.
\begin{proposition}
\label{LemHOL}
(\cite[Theorem~4.2]{T}, \cite[Theorem~7.7]{G})
Let $h\in W^{1,p}(\Omega_\sigma)$ be a solution of \eqref{HP}. Then there is a positive constant $\beta>0$ depending only on $\gamma,$ $n$ and $p$ such that
$$
\osc_{\Omega_5} h \leq C\left(\frac{5}{\sigma}\right)^\beta \|h\|_{L^\infty(\Omega_\sigma)}
$$
holds, where $C>0$ depend only on $\gamma,$ $n$ and $p.$
\end{proposition}
Now, we compare the weak solution $h\in W^{1,p}(\Omega_\sigma)$ of \eqref{HP} with the weak solution $f\in W^{1,p}(\Omega_5)$ of \eqref{FP} to have the following result.
\begin{lemma}\label{Lem3}
For any $\varepsilon>0,$ there are two constants $\delta\in\left(0,\frac{1}{8}\right)$ and $\sigma>6$ depending only on $\gamma,$ $n,$ $p,$ $\Gamma,$ $\alpha,$ $M$ and $\varepsilon,$ such that if $u\in W^{1,p}(\Omega_\sigma)$ is a weak solution of \eqref{LUP} and $f\in W^{1,p}(\Omega_5)$ is the weak solution of \eqref{VP} with \eqref{31}, \eqref{32} and \eqref{34}, then
$$
\int_{\Omega_5} |Dh-Df|^p dx \leq \varepsilon^p.
$$
\end{lemma}
\begin{proof}
The proof will be divided into two cases.
\medskip
$\mathbf{Case\ 1:}\ 1<p<2.$
We first prove the following inequality:
\begin{align}\label{45}
\int_{\Omega_5} |Dh-Df|^p dx \leq C_0\int_{\Omega_6} |h|^p dx
\end{align}
where $C_0$ depends only on $\gamma,$ $n$ and $p.$
Let $\eta\in C^\infty_0(B_6)$ be a cut-off function with the properties $0\leq\eta\leq1,$ $\eta\equiv1$ on $B_5$ and $|D\eta|\leq 2.$ Taking $\eta^ph$ as a test function for the equation \eqref{HP}, we have
\begin{align*}
\int_{\Omega_6} \langle \ba(x,u,Dh)-\ba(x,u,0),\eta^pDh\rangle\; dx &= \int_{\Omega_6} \langle \ba(x,u,Dh),\eta^pDh\rangle\; dx\\
&= -p\int_{\Omega_6} \langle \ba(x,u,Dh),\eta^{p-1}hD\eta\rangle\; dx\\
&\leq \gamma^{-1}p\int_{\Omega_6} \eta^{p-1}|h||Dh|^{p-1}|D\eta|\; dx\\
&\leq \tau\int_{\Omega_6} \eta^p|Dh|^p\; dx + C(\tau) \int_{\Omega_6} |h|^p|D\eta|^p\; dx
\end{align*}
as consequence of the Young inequality with $\tau>0.$ By Lemma~\ref{Lem1}, we have
\begin{align*}
\int_{\Omega_6} \eta^p|Dh|^p\; dx &\leq \frac{1}{4}\int_{\Omega_6} \eta^p|Dh|^p\; dx + C\int_{\Omega_6} \langle \ba(x,u,Dh)-\ba(x,u,0),\eta^pDh\rangle\; dx\\
&\leq \frac{1}{2}\int_{\Omega_6} \eta^p|Dh|^p\; dx + C\int_{\Omega_6} |h|^p|D\eta|^p\; dx,
\end{align*}
and so
\begin{align}\label{46}
\int_{\Omega_5} |Dh|^p\; dx \leq C\int_{\Omega_6} |h|^p\; dx.
\end{align}
Further on, taking $h-f$ as a test function for \eqref{HP} and \eqref{FP}, we obtain
\begin{align}\label{47}
\int_{\Omega_6} \langle \ba(x,u,Dh),Dh-Df \rangle\; dx = \int_{\Omega_6} \langle \ba(x,\overline{u}_{\Omega_5},Df),Dh-Df \rangle\; dx.
\end{align}
In view of Lemma~\ref{Lem1} with $\eta\equiv 1,$ \eqref{S} and the Young inequality, we obtain that
\begin{align*}
\int_{\Omega_5} |Dh-&Df|^p dx\\
&\leq C\int_{\Omega_5} |Dh|^p dx + C\int_{\Omega_5} \langle \ba(x,\overline{u}_{\Omega_5},Dh)-\ba(x,\overline{u}_{\Omega_5},Df),Dh-Df\rangle dx\\
&= C\int_{\Omega_5} |Dh|^p dx + C\int_{\Omega_5} \langle \ba(x,\overline{u}_{\Omega_5},Dh)-\ba(x,u,Dh),Dh-Df\rangle dx \\
&\leq C\int_{\Omega_5} |Dh|^p dx + C\int_{\Omega_5}|Dh|^{p-1}|Dh-Df| dx\\
&\leq \frac{1}{2}\int_{\Omega_5} |Dh-Df|^p dx + C\int_{\Omega_5} |Dh|^p dx.
\end{align*}
Thus, the claim \eqref{45} follows
by \eqref{46}.
Recalling $\|u\|_{L^\infty(\Omega_6)}\leq \frac{M}{\lambda r},$ the maximum principle implies $\|h\|_{L^\infty(\Omega_6)}\leq \frac{M}{\lambda r}.$ Therefore, \eqref{45} yields
$$
\int_{\Omega_5} |Dh-Df|^p\; dx \leq C_0\int_{\Omega_6} |h|^p\; dx \leq C_0|B_6|\left(\frac{M}{\lambda r}\right)^p.
$$
If $C_0|B_6|\left(\frac{M}{\lambda r}\right)^p \leq \varepsilon^p,$ then we get the conclusion.
So, assume alternatively that $C_0|B_6|\left(\frac{M}{\lambda r}\right)^p > \varepsilon^p.$
In view of \eqref{S1}, \eqref{47} and Lemma~\ref{Lem1}, we have
\begin{align*}
\int_{\Omega_5} |Dh-&Df|^p dx\\
&\leq \frac{\tau}{2}\int_{\Omega_5} |Dh|^p dx + C(\tau)\int_{\Omega_5} \langle \ba(x,\overline{u}_{\Omega_5},Dh)-\ba(x,\overline{u}_{\Omega_5},Df),Dh-Df\rangle dx\\
&= \frac{\tau}{2}\int_{\Omega_5} |Dh|^p dx + C(\tau)\int_{\Omega_5} \langle \ba(x,\overline{u}_{\Omega_5},Dh)-\ba(x,u,Dh),Dh-Df\rangle dx \\
&\leq \frac{\tau}{2}\int_{\Omega_5} |Dh|^p dx + C(\tau)\int_{\Omega_5} (\lambda r)^\alpha|u-\overline{u}_{\Omega_5}|^\alpha|Dh|^{p-1}|Dh-Df| dx.
\end{align*}
The Young inequality gives
\begin{align*}
\int_{\Omega_5} &|Dh-Df|^p dx\\
&\leq \frac{\tau}{2}\int_{\Omega_5} |Dh|^p dx + C(\tau)\int_{\Omega_5} (\lambda r)^\frac{\alpha p}{p-1}|u-\overline{u}_{\Omega_5}|^\frac{\alpha p}{p-1}|Dh|^{p} dx + \frac{1}{2} \int_{\Omega_5} |Dh-Df|^p dx,
\end{align*}
and this implies
\begin{align}\label{48}
\int_{\Omega_5} |Dh-Df|^p dx \leq \tau\int_{\Omega_5} |Dh|^p dx + C(\tau)\int_{\Omega_5} (\lambda r|u-\overline{u}_{\Omega_5}|)^\frac{\alpha p}{p-1}|Dh|^{p} dx.
\end{align}
To estimate the second term in the above inequality, we first take constants $\alpha_0$ and $p_1$ such that
$$
0<\alpha_0<\min\{\alpha,\ p-1\},\quad p<p_1<p_0, \quad \mathrm{and}\quad p_1=\frac{p(p-1)}{p-\alpha_0-1},
$$
where $\alpha$ is given in \eqref{S1} and $p_0$ is as in Proposition~\ref{LemHIG}.
We then use the H\"older inequality, \eqref{31} and Proposition~\ref{LemHIG}, to find that
\begin{align*}
\int_{\Omega_5} &(\lambda r|u-\overline{u}_{\Omega_5}|)^\frac{\alpha p}{p-1}|Dh|^{p} dx\\
&\leq \left(\int_{\Omega_5} (\lambda r|u-\overline{u}_{\Omega_5}|)^\frac{\alpha pp_1}{(p-1)(p_1-p)} dx\right)^\frac{p_1-p}{p_1}\left(\int_{\Omega_5}|Dh|^{p_1} dx\right)^\frac{p}{p_1}\\
&\leq (2M)^\frac{(\alpha-\alpha_0)p}{p-1}\left(\int_{\Omega_5} (\lambda r|u-\overline{u}_{\Omega_5}|)^\frac{\alpha_0 pp_1}{(p-1)(p_1-p)} dx\right)^\frac{p_1-p}{p_1}\left(\int_{\Omega_5}|Dh|^{p_1} dx\right)^\frac{p}{p_1}\\
&\leq C \left(\int_{B_5} (\lambda r|u-\overline{u}_{\Omega_5}|)^\frac{\alpha_0 pp_1}{(p-1)(p_1-p)} dx\right)^\frac{p_1-p}{p_1}\int_{\Omega_6}|Dh|^{p} dx\\
&= C \left(\int_{\Omega_5} (\lambda r|u-\overline{u}_{\Omega_5}|)^p dx\right)^\frac{p_1-p}{p_1}\int_{\Omega_6}|Dh|^{p} dx,
\end{align*}
and then by \eqref{48}, we have
\begin{align}\label{49}
\int_{\Omega_5} |Dh-Df|^p dx \leq \Bigg[\tau+C(\tau)\bigg(\underbrace{\int_{\Omega_5} (\lambda r|u-\overline{u}_{\Omega_5}|)^p dx}_{I_1}\bigg)^\frac{p_1-p}{p_1}\Bigg]\underbrace{\int_{\Omega_6}|Dh|^{p} dx}_{I_2}.
\end{align}
It follows from the triangle inequality that
\begin{align*}
I_1=\int_{\Omega_5} (\lambda r|u-\overline{u}_{\Omega_5}|)^p dx \leq&\ C \int_{\Omega_5} (\lambda r|u-h|)^p dx+C\int_{\Omega_5} (\lambda r|h-\overline{h}_{\Omega_5}|)^p dx\\
&+ C \int_{\Omega_5} (\lambda r|\overline{h}_{\Omega_5}-\overline{u}_{\Omega_5}|)^p dx\\
\leq&\ C \underbrace{\int_{\Omega_5} (\lambda r|u-h|)^p dx}_{I_3}+C\underbrace{\int_{\Omega_5} (\lambda r|h-\overline{h}_{\Omega_5}|)^p dx}_{I_4}.
\end{align*}
Remembering that $C_0|B_6|\left(\frac{M}{\varepsilon}\right)^p > (\lambda r)^p$ and using the Poincar\'e inequality and Lemma~\ref{Lem2}, we have
\begin{align*}
I_3=\int_{\Omega_5} (\lambda r|u-h|)^p dx &\leq C\int_{\Omega_5} (\lambda r|Du-Dh|)^p dx \\
&\leq C \left(\frac{1}{\varepsilon}\right)^p\int_{\Omega_5} |Du-Dh|^p dx \leq C \left(\frac{\varepsilon_1}{\varepsilon}\right)^p.
\end{align*}
On the other hand, Proposition~\ref{LemHOL} yields
\begin{align*}
I_4=\int_{\Omega_5} (\lambda r|h-\overline{h}_{\Omega_5}|)^p dx \leq C(\lambda r)^p\left(\frac{5}{\sigma}\right)^{p\beta}\|h\|^p_{L^\infty(B_\sigma)} \leq \frac{C}{\sigma^{p\beta}},
\end{align*}
because of $\|h\|_{L^\infty(B_\sigma)}\leq \frac{M}{\lambda r}.$ Consequently,
\begin{align*}
I_1 \leq C\left(\left(\frac{\varepsilon_1}{\varepsilon}\right)^p+\frac{1}{\sigma^{p\beta}}\right).
\end{align*}
Further on, using Lemma~\ref{Lem2} and \eqref{31}, we find
$$
I_2=\int_{\Omega_6}|Dh|^{p} dx \leq C\left(\int_{\Omega_6}|Du-Dh|^{p} dx + \int_{\Omega_6}|Du|^{p} dx\right) \leq C(\varepsilon_1^p+1) \leq C_1,
$$
while \eqref{49} gives
\begin{align*}
\int_{\Omega_5} |Dh-Df|^p dx &\leq C_1\left(\tau+C_2(\tau)\left(\left(\frac{\varepsilon_1}{\varepsilon}\right)^p+\frac{1}{\sigma^{p\beta}}\right)^\frac{p_1-p}{p_1}\right)\\
&\leq C_1\tau+C_1C_2(\tau)\left(\frac{\varepsilon_1}{\varepsilon}\right)^\frac{p(p_1-p)}{p_1}+C_1C_2(\tau)\left(\frac{1}{\sigma}\right)^\frac{p\beta(p_1-p)}{p_1}.
\end{align*}
Taking $\tau,$ $\varepsilon_1$ sufficiently small and $\sigma$ sufficiently large such that
$$C_1 \tau = \frac{\varepsilon^p}{3},\quad C_1C_2(\tau)\left(\frac{\varepsilon_1}{\varepsilon}\right)^\frac{p(p_1-p)}{p_1}\leq \frac{\varepsilon^p}{3} \quad \mathrm{and}\quad C_1C_2(\tau)\left(\frac{1}{\sigma}\right)^\frac{p\beta(p_1-p)}{p_1} \leq \frac{\varepsilon^p}{3},$$
we get the claim.
\medskip
$\mathbf{Case\ 2:}\ p\geq2.$
By using \eqref{Mo} instead of Lemma~\ref{Lem1} in the above proof, we can obtain the conclusion in a similar manner.
\end{proof}
\begin{lemma}\label{Lem4}
Under the hypotheses of Lemma $\ref{Lem3},$ we further assume \eqref{33} and
\eqref{34}. Then there exists a weak solution $v\in W^{1,p}(U)$ of \eqref{VP} such that
\begin{align*}
\|D\bar{v}\|_{L^\infty(\Omega_3)}\leq N_0\quad \mathrm{and}\quad\int_{\Omega_4} |Df-D\bar{v}|^p dx \leq \varepsilon^p
\end{align*}
for some constant $N_0=N_0(\gamma,n,p)>1.$ Here, the function $\bar{v}\in W^{1,p}(\Omega_4)$ is equal to
$v$ if $U=B_4,$ and $\bar{v}$ is the zero extension of $v$ from $B_4^+$ to $B_4$ if $U=B_4^+.$
\end{lemma}
\begin{proof}
According to Lemma~\ref{Lem2}, Lemma~\ref{Lem3} and \eqref{31}, it follows from the triangle inequality that
$$
\int_{\Omega_5} |Df|^p \ dx \leq C \int_{\Omega_5} |Du|^p + |Du-Dh|^p + |Dh-Df|^p dx \leq C,
$$
where $C$ depends only on $n$ and $p.$ Then we proceed in doing a comparison estimate from standard perturbation argument, as in Lemma~3.1 and Lemma~3.7 of \cite{BR}, in order to obtain the desired conclusion.
\end{proof}
\begin{proof}[Proof of Lemma \ref{MainLem}]
The proof follows directly from the triangle inequality and Lemmas~\ref{Lem2}, \ref{Lem3} and \ref{Lem4}.
\end{proof}
\section{Global gradient estimates}\label{Sec5}
This section is devoted to the proof of the main result, Theorem~\ref{Thm1}. We start with a modified Vitali covering lemma for the problem \eqref{1}.
\begin{proposition}
{\em (see \cite{BW1, NP})}
\label{prop1}
Let $\mathcal{C}$ and $\mathcal{D}$ be measurable sets with $\mathcal{C}\subset \mathcal{D}\subset \Omega.$ Assume that $\Omega$ is $(\delta,R)$-Reifenberg flat. Suppose that there exist $0< \varepsilon<1$ and $\sigma >1$ for which
\begin{enumerate}
\item $|\mathcal{C}| <\varepsilon |B_{R/\sigma}|;$
\item for all $x\in \Omega$ and $r\in (0,\frac{R}{\sigma}]$ with $|\mathcal{C}\cap B_r(x)|\geq \varepsilon|B_r(x)|,$ there holds $\Omega\cap B_r(x)\subset \mathcal{D}.$
\end{enumerate}
Then we have
$$|\mathcal{C}|\leq \left(\frac{10}{1-\delta}\right)^n\varepsilon|\mathcal{D}|.$$
\end{proposition}
We now return to the scaled and normalized problem \eqref{UP}.
\begin{lemma}\label{Lem10}
Assume that $\widetilde{\ba}$ satisfies \eqref{S} and \eqref{S1}. Let $\widetilde{u}\in W^{1,p}(\widetilde{\Omega})$ be a bounded weak solution of \eqref{UP} with $\|\widetilde{u}\|_{L^\infty(\widetilde{\Omega})}\leq \frac{M}{\lambda r}.$ Then there exists a constant $N_1=N_1(\gamma, n, p)>1$ so that for any small $\varepsilon\in(0,1),$ there exist a small constant $\delta=\delta(\gamma, \alpha, n, p, \Gamma, M, \varepsilon)>0$ and a large constant $\sigma=\sigma(\gamma, \alpha, n, p, \Gamma, M, \varepsilon)> 6$ such that if $\widetilde{\ba}(x,z,\xi)$ is $(\delta, \sigma)$-vanishing and $\widetilde{\Omega}$ is $(\delta, \sigma)$-Reifenberg flat, and if
\begin{equation}
\label{51}
\left\{x\in\widetilde{\Omega}_1:\m(|D\widetilde{u}|^p) \leq \left(\frac{6}{7}\right)^n\right\}
\cap \left\{x\in\widetilde{\Omega}_1:\m(|\widetilde{F}|^p)\leq\left(\frac{6}{7}\right)^n \delta^p\right\}\neq\varnothing
\end{equation}
then
$$
\left|\left\{x\in\widetilde{\Omega}_1: \m(|D\widetilde{u}|^p)(x)>\left(\frac{6}{7}\right)^n N_1^p\right\}\right|<\varepsilon|B_1|.
$$
\end{lemma}
\begin{proof}
Remembering \eqref{51}, there is a point $\widetilde{x}\in \widetilde{\Omega}_1$ such that for all $\rho>0,$
\begin{equation}\label{52}
\frac{1}{|B_\rho|}\int_{\widetilde{\Omega}_\rho(\widetilde{x})} |D\widetilde{u}|^p dx \leq \left(\frac{6}{7}\right)^n\quad\mathrm{and} \quad \frac{1}{|B_\rho|}\int_{\widetilde{\Omega}_\rho(\widetilde{x})} |\widetilde{F}|^p dx \leq \left(\frac{6}{7}\right)^n\delta^p.
\end{equation}
Let $\sigma>6.$ Since $\widetilde{\Omega}_\sigma \subset\widetilde{\Omega}_{\sigma+1}(\widetilde{x}),$
we have
$$
\frac{1}{|B_\sigma|}\int_{\widetilde{\Omega}_\sigma}|\widetilde{F}|^p\; dx \leq \left(\frac{7}{6}\right)^n\frac{1}{|B_{\sigma+1}|}\int_{\widetilde{\Omega}_{\sigma+1}(\widetilde{x})}|\widetilde{F}|^p\; dx \leq \delta^p.
$$
Similarly, it follows that
$$
\frac{1}{|B_\sigma|}\int_{\widetilde{\Omega}_\sigma}|D\widetilde{u}|^p\; dx \leq 1 \quad \mathrm{and}\quad \frac{1}{|B_6|}\int_{\widetilde{\Omega}_6}|D\widetilde{u}|^p\; dx \leq 1.
$$
Thus, we are under the hypotheses of
Lemma~\ref{MainLem}, which implies that there exist a big constant $\sigma=\sigma(\gamma,\alpha, n,p,\Gamma,M,\varepsilon)> 6$ and a small constant $\delta=\delta(\gamma,\alpha, n,p,\Gamma,M,\varepsilon)>0$ such that the conclusion of Lemma \ref{MainLem}
holds for such $\bar{v}$ and $N_0.$
Further on, we will show that there exists a constant $N_1=N_1(\gamma,n,p)>1$ such that
\begin{equation}\label{53}
\left\{ y\in \widetilde{\Omega}_1:\m(|D\widetilde{u}|^p)>\left(\frac{6}{7}\right)^n N_1^p\right\}
\subset \left\{y\in \widetilde{\Omega}_1:\m_{\widetilde{\Omega}_4}(|D\widetilde{u}-D\bar{v}|^p)>N_0^p\right\}.
\end{equation}
To do this, let $\widetilde{y}\in\{y\in \widetilde{\Omega}_1:\m_{\widetilde{\Omega}_4}(|D\widetilde{u}-D\bar{v}|^p)\leq N_0^p\}.$ Then
$$\frac{1}{|B_\rho|}\int_{B_\rho(\widetilde{y})} \chi_{\widetilde{\Omega}_4}|D\widetilde{u}-D\bar{v}|^p\; dx \leq N_0^p $$
for any $\rho>0.$ If $\rho>2,$ then $\widetilde{\Omega}_\rho(\widetilde{y})\subset \widetilde{\Omega}_{2\rho}(\widetilde{x})$ and it follows from \eqref{52} that
\begin{align*}
\frac{1}{|B_\rho|}\int_{\widetilde{\Omega}_\rho(\widetilde{y})} |D\widetilde{u}|^p\; dx \leq \frac{1}{|B_\rho|} \int_{\widetilde{\Omega}_{2\rho}(\widetilde{x})} |D\widetilde{u}|^p\; dx \leq 2^n.
\end{align*}
On the other hand, if $\rho\in(0,2],$ then $\widetilde{\Omega}_\rho(\widetilde{y})\subset \widetilde{\Omega}_3$ and so we have
\begin{align*}
\frac{1}{|B_\rho|}\int_{\widetilde{\Omega}_\rho(\widetilde{y})} |D\widetilde{u}|^p\; dx &\leq 2^{p-1}\frac{1}{|B_\rho|}\int_{\widetilde{\Omega}_\rho(\widetilde{y})} |D\widetilde{u}-D\bar{v}|^p+|D\bar{v}|^p\; dx \\
&\leq 2^{p-1}N_0^p + 2^{p-1}\frac{1}{|B_\rho|}\int_{\widetilde{\Omega}_\rho(\widetilde{y})} |D\bar{v}|^p\; dx \leq (2N_0)^p.
\end{align*}
Taking $N_1^p=\left(\frac{7}{6}\right)^n\max\left\{2^n, (2N_0)^p\right\},$ the claim \eqref{53} follows.
We now use \eqref{53}, the weak $(1,1)$-estimate for the Hardy--Littlewood maximal function and Lemma \ref{MainLem}, to observe that
\begin{align*}
\left|\left\{y\in\widetilde{\Omega}_1: \m(|D\widetilde{u}|^p)>\left(\frac{6}{7}\right)^n N_1^p\right\}\right| &\leq \left|\left\{y\in \widetilde{\Omega}_1:\m_{\widetilde{\Omega}_4}(|D\widetilde{u}-D\bar{v}|^p)>N_0^p\right\}\right|\\
&\leq \frac{C(n,p)} {N_0^p} \int_{\widetilde{\Omega}_4} |D\widetilde{u}-D\bar{v}|^2\; dx \leq C \varepsilon^p|B_1|.
\end{align*}
Thus, the claim follows in view of the arbitrariness of $\varepsilon>0.$
\end{proof}
Turning back to the problem \eqref{1}, scaling and normalization give
\begin{corollary}\label{Cor1}
Assume that $\ba$ satisfies \eqref{3} and \eqref{4}. Let $u\in W^{1,p}(\Omega)$ be a bounded weak solution of \eqref{1} with $\|u\|_{L^\infty(\Omega)}\leq M.$ Then for any small constant $\varepsilon\in(0,1),$ there exist a small constant $\delta=\delta(\gamma,\alpha, n,p,\Gamma,M,\varepsilon)>0$ and a big constant $\sigma=\sigma(\gamma, \alpha, n, p, \Gamma, M, \varepsilon)> 6$ such that if $\ba(x,z,\xi)$ is $(\delta, R)$-vanishing and $\Omega$ is $(\delta, R)$-Reifenberg flat, and if
$$
\left\{x\in\Omega:\m(|Du|^p) \leq \left(\frac{6}{7}\right)^n\lambda^p\right\} \cap\left\{x\in\Omega:\m(|F|^p)\leq \left(\frac{6}{7}\right)^n \lambda^p\delta^p\right\}\neq\varnothing
$$
then
$$
\left|\left\{x\in\Omega_r(y): \m(|Du|^p)(x)>\left(\frac{6}{7}\right)^n\lambda^p N_1^p\right\}\right|<\varepsilon|B_r|
$$
for any $r\in\left(0,\frac{R}{\sigma}\right]$ and any $y\in \Omega.$
\end{corollary}
We now take $N_1,$ $\varepsilon$ and the corresponding $\delta$ and $\sigma$ from Corollary \ref{Cor1}.
\begin{lemma}\label{Lem11}
Assume that $\Omega$ is $(\delta,R)$-Reifenberg flat, and $\ba$ satisfies \eqref{3}, \eqref{4} and \eqref{5}. Let $F \in L^p(\Omega,\mr^n)$ and $u\in W^{1,p}_0(\Omega)$ be a bounded weak solution of \eqref{1} with $\|u\|_{L^\infty(\Omega)}\leq M.$ Then
\begin{align*}
&\left|\left\{ x\in\Omega: \m(|Du|^p) (x)> \left(\frac{6}{7}\right)^n N_1^{p(k+1)} \right\}\right|\\
& \qquad \leq 20^n\varepsilon \left(\left|\left\{x\in\Omega:\m(|Du|^p)(x) > \left(\frac{6}{7}\right)^n N_1^{pk}\right\}\right|\right.\\
& \qquad\qquad\qquad +\left.\left|\left\{x\in\Omega:\m(|F|^p)(x)> \left(\frac{6}{7}\right)^n N_1^{pk}\delta^p\right\}\right|\right)
\end{align*}
for all integer $k\geq k_0,$ where $k_0$ is an integer satisfying
\begin{equation}
\label{55}
\frac{C_4}{N_1^{p(k_0+1)}}\int_\Omega |F|^p\;dx < \varepsilon|B_{R/\sigma}| \leq \frac{C_4}{N_1^{pk_0}}\int_\Omega |F|^p\;dx
\end{equation}
with $C_4=C_4(n,p,\gamma).$
\end{lemma}
\begin{proof}
Taking $u$ as a test function for \eqref{1}, we have
\begin{align*}
\int_{\Omega} \langle \ba(x,u,Du)-\ba(x,u,0),Du\rangle\; dx &= \int_{\Omega} \langle \ba(x,u,Du),Du\rangle\; dx\\
&= \int_{\Omega} \langle |F|^{p-2}F,Du\rangle\; dx\\
&\leq \int_{\Omega} |F|^{p-1}|Du|\; dx\\
&\leq \tau\int_{\Omega} |Du|^p\; dx + C(\tau)\int_{\Omega} |F|^p\; dx.
\end{align*}
Lemma~\ref{Lem1} and \eqref{Mo} give
\begin{align*}
\int_\Omega |Du|^p\; dx &\leq C_3\int_{\Omega} \langle \ba(x,u,Du)-\ba(x,u,0),Du\rangle\; dx\\
&\leq C_3\tau\int_{\Omega} |Du|^p\; dx + C(\tau)\int_{\Omega} |F|^p\; dx
\end{align*}
and selecting $\tau=\frac{1}{2C_3},$ we obtain
\begin{align*}
\int_\Omega |Du|^p\; dx \leq C \int_{\Omega} |F|^p\; dx.
\end{align*}
This estimate and the weak type $(1,1)$-estimate for the maximal function yield
\begin{align*}
\left|\left\{x\in\Omega:\m(|Du|^p)(x)>\left(\frac{6}{7}\right)^n N_1^{p(k+1)}\right\}\right| \leq &\ \frac{C}{N_1^{p(k+1)}}\int_\Omega |Du|^p\; dx \\
\leq &\ \frac{C_4}{N_1^{p(k+1)}}\int_\Omega |F|^p\; dx,
\end{align*}
for some positive constant $C_4=C_4(n,p,\gamma).$ Selecting the integer $k_0$ for which \eqref{55} holds, we find that for all $k\geq k_0,$
\begin{align*}
\left|\left\{x\in\Omega:\m(|Du|^p)(x)>\left(\frac{6}{7}\right)^n N_1^{p(k+1)}\right\}\right| \leq &\ \frac{C_4}{N_1^{p(k+1)}}\int_\Omega |F|^p\; dx \\
\leq &\ \frac{C_4}{N_1^{p(k_0+1)}}\int_\Omega |F|^p\; dx < \varepsilon|B_{R/\sigma}| .
\end{align*}
We define now
$$
\mathcal{C}=\left\{ x\in\Omega: \m(|Du|^p)(x)> \left(\frac{6}{7}\right)^n N_1^{p(k+1)} \right\}
$$
and
\begin{align*}
\mathcal{D}=&\ \left\{ x\in\Omega:\m(|Du|^p)(x) > \left(\frac{6}{7}\right)^n N_1^{pk}\right \}\\
&\ \cup\left\{x\in\Omega:\m(|F|^p)(x)> \left(\frac{6}{7}\right)^n N_1^{pk}\delta^p\right\}
\end{align*}
in order to apply Proposition~\ref{prop1}. Then the first assumption of Proposition~\ref{prop1} follows directly, while the second one comes from Corollary~\ref{Cor1}. Consequently, we obtain the conclusion of Lemma~\ref{Lem11}.
\end{proof}
With all these tools in hand, we are in a position now to prove Theorem~\ref{Thm1}.
\begin{proof}[Proof of Theorem~\ref{Thm1}]
Straightforward calculations yield
\begin{align*}
\int_\Omega \m(|Du|^p)^q\ dx =&\ q\int_0^\infty t^{q-1}|\{x\in\Omega:\m(|Du|^p)(x)>t\}|\ dt\\
\leq&\ q\int_0^{\left(\frac{6}{7}\right)^n N_1^{pk_0}} t^{q-1}|\{x\in\Omega:\m(|Du|^p)(x)>t\}|\ dt\\
&\ +q\sum_{k=k_0}^\infty\int_{\left(\frac{6}{7}\right)^n N_1^{pk}}^{\left(\frac{6}{7}\right)^n N_1^{p(k+1)}} t^{q-1}|\{x\in\Omega:\m(|Du|^p)(x)>t\}|\ dt \\
\leq&\ \left(\frac{6}{7}\right)^{nq} N_1^{pqk_0}|\Omega|\\
&\ +\left(\frac{6}{7}\right)^{nq}(N_1^{pq}-1)\sum_{k=k_0}^\infty N_1^{pqk}\left|\left\{x\in\Omega:\m(|Du|^p)(x)>\left(\frac{6}{7}\right)^n N_1^{pk}\right\}\right|,
\end{align*}
where $N_1$ is given in Lemma \ref{Lem10} and $k_0$ is given in Lemma \ref{Lem11}.
Keeping in mind \eqref{55}, we have
\begin{equation}
\label{M1}
N_1^{pqk_0} \leq \left(\frac{C}{\varepsilon|B_{R/\sigma}|}\int_\Omega |F|^p\;dx\right)^q \leq C \int_\Omega |F|^{pq}\;dx.
\end{equation}
Further on, Lemma~\ref{Lem11} yields
\begin{align*}
\sum_{k=k_0}^\infty N_1^{pqk}\bigg|\bigg\{&x\in\Omega:\m(|Du|^p)(x)>\left(\frac{6}{7}\right)^n N_1^{pk}\bigg\}\bigg|\\
\leq &\ \sum_{k=k_0}^\infty N_1^{pqk}(20^n\varepsilon)^{k-k_0}\bigg|\bigg\{x\in\Omega: \m(|Du|^p)(x)>\left(\frac{6}{7}\right)^n N_1^{pk_0}\bigg\}\bigg|\\
&\ +\sum_{k=k_0}^\infty\sum_{l=k_0}^k N_1^{pqk}(20^n\varepsilon)^{k-l}\bigg|\bigg\{x\in\Omega:\m(|F|^p)(x)>\left(\frac{6}{7}\right)^n N_1^{pl}\delta^p\bigg\}\bigg|\\
= &\ \sum_{k=k_0}^\infty N_1^{pqk}(20^n\varepsilon)^{k-k_0}\bigg|\bigg\{x\in\Omega: \m(|Du|^p)(x)>\left(\frac{6}{7}\right)^n N_1^{pk_0}\bigg\}\bigg|\\
&\ +\sum_{l=k_0}^\infty\sum_{k=l}^{\infty} N_1^{pqk}(20^n\varepsilon)^{k-l}\bigg|\bigg\{x\in\Omega:\m(|F|^p)(x)>\left(\frac{6}{7}\right)^n N_1^{pl}\delta^p\bigg\}\bigg|\\
=&: S_1+S_2.
\end{align*}
We take now $\varepsilon>0$ small enough to have $0 < 20^nN_1^{pq}\varepsilon<\frac{1}{2},$ and observe that \eqref{M1} gives
\begin{align*}
S_1=&\ \sum_{k=k_0}^\infty N_1^{pqk}(20^n\varepsilon)^{k-k_0}\bigg|\bigg\{x\in\Omega: \m(|Du|^p)(x)>\left(\frac{6}{7}\right)^n N_1^{pk_0}\bigg\}\bigg|\\
\leq&\ 2 N_1^{pqk_0}|\Omega| \leq 2|\Omega|\left(\frac{C}{\varepsilon|B_{R/\sigma}|}\int_\Omega |F|^p\;dx\right)^q \leq C \int_\Omega |F|^{pq}\;dx.
\end{align*}
On the other hand,
\begin{align*}
S_2=&\ \sum_{l=k_0}^\infty\sum_{k=l}^{\infty} N_1^{pqk}(20^n\varepsilon)^{k-l}\bigg|\bigg\{x\in\Omega:\m(|F|^p)(x)> \left(\frac{6}{7}\right)^n N_1^{pl}\delta^p\bigg\}\bigg|\\
\leq&\ 2 \sum_{l=k_0}^\infty N_1^{pql}\bigg|\bigg\{x\in\Omega:\m(|F|^p)(x)>\left(\frac{6}{7}\right)^n N_1^{pl}\delta^p\bigg\}\bigg|\\
\leq&\ C\int_\Omega \m(|F|^p)^q(x)\ dx.
\end{align*}
Therefore, we conclude that
$$
\int_\Omega \m(|Du|^p)^q(x)\ dx\leq C\int_\Omega |F|^{pq}(x)\ dx + C\int_\Omega \m(|F|^p)^q(x)\ dx.
$$
At this point, applying the strong type $(pq,pq)$-estimate for the maximal function, we complete the proof of Theorem~\ref{Thm1}.
\end{proof}
\section{Refinements of the gradient estimates}\label{Sec6}
It turns out that, for values of the exponent $p$ greater than or equal to the space dimension $n,$ the result of Theorem~\ref{Thm1} continues to hold under weaker assumption on the $z$-behaviour of $\ba(x,z,\xi)$ than the H\"older continuity \eqref{4}.
Precisely, assume hereafter that $p\geq n$ and
\begin{equation}\label{4'}
|\ba(x,z_1,\xi)-\ba(x,z_2,\xi)|\leq \omega(|z_1-z_2|)\xi|^{p-1}
\end{equation}
for a.a. $x\in\Omega,$ $\forall z_1,z_2\in\mr$ and $\forall\xi\in\mr^n,$ where $\omega\colon \mathbb{R}^+\to\mathbb{R}^+$ is a modulus of continuity, that is, $\lim_{\tau\to0^+}\omega(\tau)=0.$
We will make use of our results from \cite{BPS,BPS-arxiv} where global H\"older continuity is proved for the weak solutions of general quasilinear elliptic equations with Morrey data over domains with \textit{uniformly $p$-thick complements.} Since any Reifenberg flat domain has uniformly $p$-thick complement (cf. \cite{BPS-arxiv}), the above mentioned results hold true in the situation here considered.
Using the assumption \eqref{3} on uniform ellipticity and its outgrowth \eqref{Mo1}, it is not hard to check that
\begin{align}
\label{6.2}
\langle \ba(x,z,\xi)+|F(x)|^{p-2}F(x),\xi\rangle \geq&\ C_1|\xi|^p -C_2\big(|F(x)|^p \big)^{\frac{p}{p-1}},\\
\label{6.3}
\big| \ba(x,z,\xi)-|F(x)|^{p-2}F(x)\big| \leq &\ \gamma^{-1}|\xi|^{p-1} + |F(x)|^{p-1}
\end{align}
for a.a. $x\in\Omega,$ $\forall (z,\xi)\in \mr\times\mr^n,$ and where $C_1$ and $C_2$ are positive constants depending only on $n,$ $p$ and $\gamma.$
Further on, assuming $|F(x)|^{p}\in L^q(\Omega)$ with $q>1$ as did in Theorem~\ref{Thm1}, it follows $|F(x)|^{p-1}\in L^{\frac{pq}{p-1}}(\Omega).$ We have
$$
\frac{pq}{p-1}>\frac{p}{p-1}\geq \frac{n}{p-1}
$$
because of $q>1$ and $p\geq n,$ and therefore, we first choose a number
$$
p'\in \left(\frac{p}{p-1},\frac{pq}{p-1}\right)
$$
and consequently define
$$
\lambda'=n\left(1-p'\frac{p-1}{pq}\right).
$$
It is obvious that $\lambda'\in(0,n)$ and the choice of $p'$ and $\lambda'$ guarantees that the Lebesgue space $L^{\frac{pq}{p-1}}(\Omega)$ is embedded into the Morrey space $L^{p',\lambda'}(\Omega).$ Thus $|F(x)|^{p-1}\in L^{p',\lambda'}(\Omega)$ with
$$
p'>\frac{p}{p-1},\quad (p-1)p'+\lambda'>n,
$$
and the inequalities \eqref{6.2} and \eqref{6.3} ensure the validity of the results from \cite{BPS,BPS-arxiv}. In other words, the weak solution of the Dirichlet problem \eqref{1} is H\"older continuous function up to the boundary and
\begin{equation}\label{6.4}
\sup_{\overline{\Omega}}|u(x)|+ \sup_{x,y\in \overline\Omega,\ x\neq y} \frac{|u(x)-u(y)|}{|x-y|^\alpha} \leq H
\end{equation}
with $\alpha\in(0,1)$ and $H$ depending on the data of \eqref{1} and on $\|Du\|_{L^p(\Omega)}.$
Let us note at this step that the above arguments are relevant only if $p=n,$ because \eqref{6.4} is direct consequence of the Sobolev embeddings and the Morrey lemma when $p>n.$ Moreover, in case of \textit{Lipschitz continuous} domain $\Omega,$ \eqref{6.4} follows from the classical results of Ladyzhenskaya and Ural'tseva \cite[Chapter~IV]{LU}.
For a fixed weak solution $u\in W^{1,p}_0(\Omega)$ of \eqref{1}, we define now the Carath\'eodory map
$$
\mathbf{A}(x,\xi):= \ba (x,u(x),\xi)
$$
and use \eqref{5}, \eqref{4'} and \eqref{6.4} to infer, through \cite[Lemma~1]{PRS}, that it obeys the $(\delta,R)$-vanishing property
(see \cite[Definition~2.2]{BR}). Moreover,
\eqref{3} implies uniform ellipticity of $\mathbf{A}(x,\xi)$ and $u\in W^{1,p}_0(\Omega)$ solves
$$
\begin{cases}
\mathrm{div\,}\mathbf{A}(x,Du)=\mathrm{div\,}(|F|^{p-2}F) & \textrm{in}\ \Omega\\
u=0 & \textrm{on}\ \partial\Omega.
\end{cases}
$$
This way, \cite[Theorem~2.6]{BR} yields the following refinement of Theorem~\ref{Thm1} in the case $p\geq n$ where the H\"older continuity \eqref{4} of $\ba(x,z,\xi)$ with respect to $z$ is relaxed to only continuity. The constant $H$ below is the one appearing in \eqref{6.4}.
\begin{theorem}
Suppose \eqref{3} and \eqref{4'}, and let $u\in W^{1,p}_0(\Omega)$ be a weak solution of \eqref{1} with $p\geq n.$ Let $|F|^p\in L^q(\Omega)$ for some $q\in (1,\infty).$ Then there is a small constant $\delta=\delta(\gamma,n,p,q,\omega(\cdot),H)$ such that if $\ba$ is $(\delta,R)$-vanishing and $\Omega$ is $(\delta,R)$-Reifenberg flat, then $|Du|^p\in L^q(\Omega)$ with the estimate
$$
\int_\Omega |Du|^{pq} dx \leq C \int_\Omega |F|^{pq} dx,
$$
where $C=C(n,p,q,\gamma,\omega(\cdot),H,|\Omega|).$
\end{theorem}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,131 |
Warmun(Turkey Creek) is one of the hottest places in Australia. It experiences exceptionally high daytime temperatures and overnight temperatures along with far below average humidity levels. Numbers of clear days are above average and wind speed is below average. Rainfall and numbers of cloudy days are average.
Warmun(Turkey Creek) has a wet and a dry season. The wet season in Warmun(Turkey Creek) begins around October and ends around April. Over the course of the wet season, Warmun(Turkey Creek) receives around 680.6mm of rain. By comparison, in the dry season from May to September, less than 47.6mm of rain falls in total. At the height of the wet season in February it rains on average 12.8 days in that one month. Indeed, Warmun(Turkey Creek) has experienced as much as 167.9mm in a single April day and as much as 612.6mm in a single month (January).
Temperatures in the wet season average between 35.1 and 39.3OC during the day and 20.6 and 25.1OC overnight. Temperatures in the dry season average between 29.5 and 36.1OC during the day and 12.5 and 19.4OC at night. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,496 |
{"url":"https:\/\/www.dpmms.cam.ac.uk\/person\/jes98","text":"# Department of Pure Mathematics and Mathematical Statistics\n\n## Publications\n\nFloer cohomology of Platonic Lagrangians\nJE Smith\n\u2013 Journal of Symplectic Geometry\n\nE1.04\n\n01223 764270","date":"2019-10-13 22:27:43","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\": 0.8981125354766846, \"perplexity\": 12481.572577130755}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986648343.8\/warc\/CC-MAIN-20191013221144-20191014004144-00295.warc.gz\"}"} | null | null |
New for 2018 are the Popti Cornish Bakehouse's crackers for cheese range. These rich and buttery crackers are accented with a flavourful crunchy mix of seeds, and have been designed specifically to pair with a range of cheeses.
These crackers work beautifully with an aged cheddar. If you are looking for an additional wine paring, try a rich Malbec wine.
Approx 110g with a shelf life of 12 months. Store in a cool, dry place. Once opened store in an airtight container. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,528 |
Q: Connection cannot be null when 'hibernate.dialect' not set having more issues with seting up hibernate with spring3. this time it is saying that connection is nul as the dialect is not set which it is on my hibernate.cfg.xml file.
here is the full exception:
org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'mySessionFactory' defined in URL [file:war/WEB-INF/datasource-config.xml]: Invocation of init method failed; nested exception is org.hibernate.HibernateException: Connection cannot be null when 'hibernate.dialect' not set
at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.initializeBean(AbstractAutowireCapableBeanFactory.java:1455)
at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.doCreateBean(AbstractAutowireCapableBeanFactory.java:519)
at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.createBean(AbstractAutowireCapableBeanFactory.java:456)
at org.springframework.beans.factory.support.AbstractBeanFactory$1.getObject(AbstractBeanFactory.java:294)
at org.springframework.beans.factory.support.DefaultSingletonBeanRegistry.getSingleton(DefaultSingletonBeanRegistry.java:225)
at org.springframework.beans.factory.support.AbstractBeanFactory.doGetBean(AbstractBeanFactory.java:291)
at org.springframework.beans.factory.support.AbstractBeanFactory.getBean(AbstractBeanFactory.java:193)
at org.springframework.beans.factory.support.DefaultListableBeanFactory.preInstantiateSingletons(DefaultListableBeanFactory.java:567)
at org.springframework.context.support.AbstractApplicationContext.finishBeanFactoryInitialization(AbstractApplicationContext.java:913)
at org.springframework.context.support.AbstractApplicationContext.refresh(AbstractApplicationContext.java:464)
at org.springframework.test.context.support.AbstractGenericContextLoader.loadContext(AbstractGenericContextLoader.java:96)
at org.springframework.test.context.support.AbstractGenericContextLoader.loadContext(AbstractGenericContextLoader.java:44)
at org.springframework.test.context.TestContext.buildApplicationContext(TestContext.java:198)
at org.springframework.test.context.TestContext.getApplicationContext(TestContext.java:233)
at org.springframework.test.context.support.DependencyInjectionTestExecutionListener.injectDependencies(DependencyInjectionTestExecutionListener.java:126)
at org.springframework.test.context.support.DependencyInjectionTestExecutionListener.prepareTestInstance(DependencyInjectionTestExecutionListener.java:85)
at org.springframework.test.context.TestContextManager.prepareTestInstance(TestContextManager.java:231)
at org.springframework.test.context.junit4.SpringJUnit4ClassRunner.createTest(SpringJUnit4ClassRunner.java:95)
at org.springframework.test.context.junit4.SpringJUnit4ClassRunner.invokeTestMethod(SpringJUnit4ClassRunner.java:139)
at org.junit.internal.runners.JUnit4ClassRunner.runMethods(JUnit4ClassRunner.java:51)
at org.junit.internal.runners.JUnit4ClassRunner$1.run(JUnit4ClassRunner.java:44)
at org.junit.internal.runners.ClassRoadie.runUnprotected(ClassRoadie.java:27)
at org.junit.internal.runners.ClassRoadie.runProtected(ClassRoadie.java:37)
at org.junit.internal.runners.JUnit4ClassRunner.run(JUnit4ClassRunner.java:42)
at org.eclipse.jdt.internal.junit4.runner.JUnit4TestReference.run(JUnit4TestReference.java:50)
at org.eclipse.jdt.internal.junit.runner.TestExecution.run(TestExecution.java:38)
at org.eclipse.jdt.internal.junit.runner.RemoteTestRunner.runTests(RemoteTestRunner.java:467)
at org.eclipse.jdt.internal.junit.runner.RemoteTestRunner.runTests(RemoteTestRunner.java:683)
at org.eclipse.jdt.internal.junit.runner.RemoteTestRunner.run(RemoteTestRunner.java:390)
at org.eclipse.jdt.internal.junit.runner.RemoteTestRunner.main(RemoteTestRunner.java:197)
Caused by: org.hibernate.HibernateException: Connection cannot be null when 'hibernate.dialect' not set
at org.hibernate.service.jdbc.dialect.internal.DialectFactoryImpl.determineDialect(DialectFactoryImpl.java:97)
at org.hibernate.service.jdbc.dialect.internal.DialectFactoryImpl.buildDialect(DialectFactoryImpl.java:67)
at org.hibernate.engine.jdbc.internal.JdbcServicesImpl.configure(JdbcServicesImpl.java:172)
at org.hibernate.service.internal.StandardServiceRegistryImpl.configureService(StandardServiceRegistryImpl.java:75)
at org.hibernate.service.internal.AbstractServiceRegistryImpl.initializeService(AbstractServiceRegistryImpl.java:159)
at org.hibernate.service.internal.AbstractServiceRegistryImpl.getService(AbstractServiceRegistryImpl.java:131)
at org.hibernate.cfg.SettingsFactory.buildSettings(SettingsFactory.java:71)
at org.hibernate.cfg.Configuration.buildSettingsInternal(Configuration.java:2270)
at org.hibernate.cfg.Configuration.buildSettings(Configuration.java:2266)
at org.hibernate.cfg.Configuration.buildSessionFactory(Configuration.java:1735)
at org.hibernate.cfg.Configuration.buildSessionFactory(Configuration.java:1775)
at org.springframework.orm.hibernate4.LocalSessionFactoryBuilder.buildSessionFactory(LocalSessionFactoryBuilder.java:184)
at org.springframework.orm.hibernate4.LocalSessionFactoryBean.afterPropertiesSet(LocalSessionFactoryBean.java:314)
at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.invokeInitMethods(AbstractAutowireCapableBeanFactory.java:1514)
at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.initializeBean(AbstractAutowireCapableBeanFactory.java:1452)
... 29 more
Here is my dataSource-config.xml thats ets up the sessionfactory
<?xml version="1.0" encoding="UTF-8"?>
<beans xmlns="http://www.springframework.org/schema/beans"
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:p="http://www.springframework.org/schema/p"
xmlns:tx="http://www.springframework.org/schema/tx" xmlns:context="http://www.springframework.org/schema/context"
xsi:schemaLocation="
http://www.springframework.org/schema/beans
http://www.springframework.org/schema/beans/spring-beans-3.0.xsd
http://www.springframework.org/schema/tx
http://www.springframework.org/schema/tx/spring-tx-3.0.xsd
http://www.springframework.org/schema/context
http://www.springframework.org/schema/context/spring-context-3.0.xsd
">
<bean id="myDataSource" class="org.apache.commons.dbcp.BasicDataSource"
destroy-method="close">
<property name="driverClassName" value="${database.driver}" />
<property name="url" value="${database.url}" />
<property name="username" value="${database.user}" />
<property name="password" value="${database.password} " />
</bean>
<bean id="mySessionFactory"
class="org.springframework.orm.hibernate4.LocalSessionFactoryBean">
<property name="dataSource" ref="myDataSource" />
<property name="packagesToScan" value="com.jr.freedom"/>
<property name="hibernateProperties" value="classpath:hibernate.cfg.xml"/>
</bean>
<!-- Declare a transaction manager -->
<bean id="transactionManager"
class="org.springframework.orm.hibernate3.HibernateTransactionManager"
p:sessionFactory-ref="mySessionFactory" />
</beans>
And below is the hibernate.cfg.xml file
<?xml version='1.0' encoding='utf-8'?>
<!DOCTYPE hibernate-configuration PUBLIC
"-//Hibernate/Hibernate Configuration DTD 3.0//EN"
"http://hibernate.sourceforge.net/hibernate-configuration-3.0.dtd">
<hibernate-configuration>
<session-factory>
<!-- JDBC connection settings -->
<property name="hibernate.connection.driver_class">com.mysql.jdbc.Driver</property>
<property name="hibernate.connection.url">jdbc:mysql://localhost:3306/freedom</property>
<property name="hibernate.connection.username">root</property>
<property name="hibernate.connection.password">password</property>
<!-- JDBC connection pool, use Hibernate internal connection pool -->
<property name="connection.pool_size">25</property>
<!-- Defines the SQL dialect used in Hiberante's application -->
<property name="hibernate.dialect">org.hibernate.dialect.MySQLDialect</property>
<!-- Enable Hibernate's automatic session context management -->
<property name="current_session_context_class">thread</property>
<!-- Disable the second-level cache -->
<property name="cache.provider_class">org.hibernate.cache.NoCacheProvider</property>
<!-- Display and format all executed SQL to stdout -->
<property name="show_sql">true</property>
<property name="format_sql">true</property>
<!-- Drop and re-create the database schema on startup -->
<property name="hbm2ddl.auto">create</property>
<!-- Mapping to hibernate mapping files -->
<!--mapping resource="org/kodejava/example/hibernate/app/Label.hbm.xml"/-->
</session-factory>
</hibernate-configuration>
As you can see, the dialect is being set.
edit: my database.properties file
# DB properties file
database.url=jdbc:mysql://localhost:3306/freedom
database.driver=com.mysql.jdbc.Driver
database.user=root
database.password=password
database.maxConnections=25
edit: here is a full stack trace. accessing the database could be the issue but i can succefully access it via command prompt?
2288 [main] WARN org.hibernate.engine.jdbc.internal.JdbcServicesImpl - HHH000342: Could not obtain connection to query metadata : Cannot create PoolableConnectionFactory (Access denied for user 'root'@'localhost' (using password: YES))
2289 [main] INFO org.springframework.beans.factory.support.DefaultListableBeanFactory - Destroying singletons in org.springframework.beans.factory.support.DefaultListableBeanFactory@4f549ceb: defining beans [org.springframework.web.servlet.mvc.method.annotation.RequestMappingHandlerMapping#0,org.springframework.format.support.FormattingConversionServiceFactoryBean#0,org.springframework.validation.beanvalidation.LocalValidatorFactoryBean#0,org.springframework.web.servlet.mvc.method.annotation.RequestMappingHandlerAdapter#0,org.springframework.web.servlet.handler.MappedInterceptor#0,org.springframework.web.servlet.mvc.method.annotation.ExceptionHandlerExceptionResolver#0,org.springframework.web.servlet.mvc.annotation.ResponseStatusExceptionResolver#0,org.springframework.web.servlet.mvc.support.DefaultHandlerExceptionResolver#0,org.springframework.web.servlet.handler.BeanNameUrlHandlerMapping,org.springframework.web.servlet.mvc.HttpRequestHandlerAdapter,org.springframework.web.servlet.mvc.SimpleControllerHandlerAdapter,org.springframework.web.servlet.config.viewControllerHandlerMapping,userService,myDataSource,mySessionFactory,transactionManager,propertyConfigurer,org.springframework.web.servlet.mvc.annotation.DefaultAnnotationHandlerMapping#0,viewResolver,org.springframework.web.servlet.mvc.annotation.AnnotationMethodHandlerAdapter#0,hello,userController,org.springframework.context.annotation.internalConfigurationAnnotationProcessor,org.springframework.context.annotation.internalAutowiredAnnotationProcessor,org.springframework.context.annotation.internalRequiredAnnotationProcessor,org.springframework.context.annotation.internalCommonAnnotationProcessor,org.springframework.context.annotation.internalPersistenceAnnotationProcessor,org.springframework.context.annotation.ConfigurationClassPostProcessor$ImportAwareBeanPostProcessor#0]; root of factory hierarchy
2289 [main] DEBUG org.springframework.beans.factory.support.DisposableBeanAdapter - Invoking destroy method 'close' on bean with name 'myDataSource'
2289 [main] DEBUG org.springframework.beans.factory.support.DefaultListableBeanFactory - Retrieved dependent beans for bean '(inner bean)#14': [org.springframework.web.servlet.config.viewControllerHandlerMapping]
2289 [main] DEBUG org.springframework.beans.factory.support.DefaultListableBeanFactory - Retrieved dependent beans for bean '(inner bean)#8': [org.springframework.web.servlet.mvc.method.annotation.ExceptionHandlerExceptionResolver#0]
2289 [main] DEBUG org.springframework.beans.factory.support.DefaultListableBeanFactory - Retrieved dependent beans for bean '(inner bean)#1': [org.springframework.web.servlet.mvc.method.annotation.RequestMappingHandlerAdapter#0]
2290 [main] DEBUG org.springframework.beans.factory.support.DefaultListableBeanFactory - Retrieved dependent beans for bean '(inner bean)': [org.springframework.web.servlet.handler.MappedInterceptor#0]
A: I had this error with JPA, then I just added this line to application.properties:
spring.jpa.database-platform=org.hibernate.dialect.PostgreSQLDialect
based on this post: org.hibernate.HibernateException: Access to DialectResolutionInfo cannot be null when 'hibernate.dialect' not set
A: I had the same problem. Switching from the Apache Commons DataSource:
<bean id="myDataSource" class="org.apache.commons.dbcp.BasicDataSource"
destroy-method="close">
<property name="driverClassName" value="${database.driver}" />
<property name="url" value="${database.url}" />
<property name="username" value="${database.user}" />
<property name="password" value="${database.password} " />
</bean>
To C3P0:
<bean id="dataSource" class="com.mchange.v2.c3p0.ComboPooledDataSource">
<property name="driverClass" value="com.mysql.jdbc.Driver" />
<property name="jdbcUrl" value="jdbc:mysql://ipadress:3306/schema" />
<property name="user" value="root" />
<property name="password" value="abc" />
</bean>
made my problem go away. Maybe someone can explain why, I am just happy it works and wanna share what little I know :)
A: I think its the problem of not loading your hibernate.cfg.xml file in your application.
I too got the same problem. I have tried with the following code in my project:
Configuration cfg = new Configuration();
cfg.configure("hibernate.cfg.xml");
Then it has worked fine.
Hope this will help.
A: This issue comes if your mysql-connector version and mysql-installer version is different.
Internally it will throw Caused by:
com.mysql.jdbc.exceptions.jdbc4.MySQLNonTransientConnectionException: Client does not support authentication protocol requested by server; consider upgrading MySQL client.
So you need make your mysql-connector version and mysql-installer version same...
A: The database connection is missing. Add to your hibernate.cfg.xml file a lines like this
<property name="connection.driver_class">com.mysql.jdbc.Driver</property>
<property name="connection.url">
jdbc:mysql://localhost:3306/your_database
</property>
<property name="connection.username">your_user</property>
<property name="connection.password">your_password</property>
(replace localhost if the database is not installed on your computer, and set the values beginning with your_ in my example).
A: I think your datasource-config.xml is not in classpath
instead of having this file in WEB-INF/datasource-config.xml
copy it to WEB-INF/classes/datasource-config.xml
A: I you are using hibernate and InnoDB why don't you set the hibernate.dialect to org.hibernate.dialect.MySQL5 or org.hibernate.dialect.MySQL.
and the connection property to hibernate.connection.url.
A: i think you are not configure hibernate.cfg.xml.
Configuration configuration = new Configuration();
configuration.configure("hibernate.cfg.xml");
SessionFactory factory = configuration.buildSessionFactory();
A: Added the following in persistence.xml file,
<property name="hibernate.dialect" value="org.hibernate.dialect.Oracle10gDialect"/>
A: You did't post your session factory initialization code. But I guess this is the same problem with this one
From Hibernate 4.3.2.Final, StandardServiceRegistryBuilder is introduced. Please follow this order to intialize, e.g.:
Configuration configuration = new Configuration();
configuration.configure("com/jeecourse/config/hibernate.cfg.xml");
ServiceRegistry serviceRegistry = new StandardServiceRegistryBuilder().applySettings(
configuration.getProperties()).build();
sessionFactory = configuration.buildSessionFactory(serviceRegistry);
Please double check your code.
A: I think you don't create the Configuration like below.
Configuration conf = new Configuration().configure();
Best
A: Creation of ServiceRegistry instance separately and pass it to buildSessionFactory may cause this error. Try to create SessionFactory instance as follows:
private static SessionFactory factory;
factory = cfg.buildSessionFactory(
new ServiceRegistryBuilder().applySettings(cfg.getProperties())
.buildServiceRegistry());
A: I think the error cased by unloading hibernate.cfg.xml file after you have created the Configuration Object like this:
Configuration config = new Configuration();
config.configure("hibernate.cfg.xml");
A: I had the the same problem and the issue was, that my computer was not entered in the pg_hba.conf of the postgres database that allows which Clients(IP-Adresses) are allowed to speak with the database.
btw: Other users tells that the error raises too if they simply forget starting database service locally
A: We were recently facing the same issue with one of our deployment, as we were also using hibernate and dialect is null is a vague error.
I use was with connection string itself and did the changes below to make it work
connectionUrl :- jdbc:mysql://127.0.0.1:3306/{{dbName}}?serverTimezone=UTC&allowPublicKeyRetrieval=true&useSSL=false
Steps how did we solve it
1) We ran simple DB connection with JDBC driver that will gave the exception of time zone.
2) we found the issue with time zone and had to add more properties in connection string as show above.
3) We also added 2 other properties as still it was not working.
Dialect is not set is vague to took lot of our time, I hope this helps.
A: This worked for me added the dialect property...
<?xml version="1.0" encoding="UTF-8"?>
<persistence version="2.2" xmlns="http://xmlns.jcp.org/xml/ns/persistence" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://xmlns.jcp.org/xml/ns/persistence http://xmlns.jcp.org/xml/ns/persistence/persistence_2_2.xsd">
<persistence-unit name="project_name" transaction-type="RESOURCE_LOCAL">
<properties>
<property name="javax.persistence.jdbc.url" value="jdbc:mysql://localhost:3306/"/>
<property name="javax.persistence.jdbc.user" value="root"/>
<property name="javax.persistence.jdbc.password" value="root"/>
<property name="javax.persistence.jdbc.driver" value="com.mysql.jdbc.Driver"/>
<property name="javax.persistence.schema-generation.drop-source" value="script-then-metadata"/>
<property name="dialect" value="org.hibernate.dialect.MySQLDialect"/>
</properties>
</persistence-unit>
</persistence>
A: I face this problem when I project update , the error is javax.persistence.jdbc.url', 'hibernate.connection.url', or 'hibernate.dialect'
and I found the solution. it work fine.
here is the details solution
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,803 |
Tracking rapid genetic changes will help researchers engineer ethanol- and antibiotic-producing microbes.
Using rapid new DNA-sequencing technologies, University of California-San Diego researchers followed evolutionary changes in E. coli grown under stressful conditions. They were able to identify which genes mutated, when, and what the effects were on the bacteria's growth. The researchers say the technique, called experimental evolution, will help those trying to learn how to genetically engineer bacteria to churn out high concentrations of ethanol and other useful chemicals (see "Bacterial Factories").
Bacteria such as E. coli evolve relatively quickly: they divide rapidly and sloppily, passing on error-filled copies of their genetic information to the next generation. Using new microarray technology, Bernhard Palsson, a professor of bioengineering, and his colleagues studied this rapid evolution over very short timescales at a high level of detail.
Experimental evolution could prove a powerful tool for researchers working on metabolic engineering, says Herring. The genetic networks of metabolism are complex and include elements that researchers would have a hard time predicting. Herring says that experimental evolution "can show connections between different physiological systems [that we] didn't know about before." Herring is now a research scientist at Cambridge, MA-based Mascoma, which hopes to design microorganisms that efficiently convert biomass into ethanol (see "Redesigning Life to Make Ethanol").
Collins says metabolic engineering is "often done irrationally." When researchers introduce new parts to bacteria or yeast, they don't know whether other mutations have been introduced, "let alone how other pathways may be involved." Comparative genome sequencing could provide this kind of information, allowing researchers to better predict the effects of genetically engineered changes and to rapidly identify which changes lead to favorable attributes.
Still, Stephanopoulos and the others say that comparative genome sequencing can now help researchers attribute changes in microbes' traits (such as the ability to thrive on glycerol) to changes in genotype. In doing so, the technology could help microbiologists and pharmaceutical companies study how strains of antibiotic-resistant bacteria, a major health problem, emerge, and what mutations are responsible. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,379 |
package org.asciidoctor.extension;
import org.asciidoctor.ast.ContentModel;
import org.asciidoctor.ast.StructuralNode;
import java.util.Arrays;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
public class YellStaticBlock extends BlockProcessor {
private static Map<String, Object> configs = new HashMap<String, Object>() {{
put(Contexts.KEY, Arrays.asList(Contexts.PARAGRAPH));
put(ContentModel.KEY, ContentModel.SIMPLE);
}};
public YellStaticBlock(String name) {
super(name, configs);
}
public YellStaticBlock(String name, Map<String, Object> config) {
super(name, configs);
}
@Override
public Object process(StructuralNode parent, Reader reader, Map<String, Object> attributes) {
List<String> lines = reader.readLines();
String upperLines = null;
for (String line : lines) {
if (upperLines == null) {
upperLines = line.toUpperCase();
} else {
upperLines = upperLines + "\n" + line.toUpperCase();
}
}
return createBlock(parent, "paragraph", Arrays.asList(upperLines), attributes, new HashMap<Object, Object>());
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,870 |
What is the Finders Keepers?
The Finders Keepers is a design market, that features the work of independent makers and designers from across Australia. Founded in 2008, the Finders Keepers has now grown to be Australia's leading design market. With events held in Sydney, Brisbane and Melbourne, the market promotes and supports over 1200 sellers each year.
More of a festival than a market, the events combine design, art, good food, live music and a fun community spirit under one awesome roof in major city locations. Our markets are a unique experience to shoppers with each visit, with new sellers being showcased at each event. We have always sought out beautiful venues with historical significance to host our events, further reinforcing the sense of wonder to shoppers and stallholders alike.
Attracting tens of thousands of visitors to each event, our markets provide a retail shopping event to the general public like no other. With something for everyone, our events provide a solid channel for shoppers mindful of conscious consumerism, and those seeking to actively support small, local businesses.
We seek to encourage creativity, to connect, inspire and empower the creative community; this is always at the heart of what we do!
The Finders Keepers was founded in the summer of 2008 in Sydney, by young designers and friends, Brooke Johnston and Sarah Thornton. The markets quickly expanded to Brisbane in 2009, with Melbourne hosting their first event in 2010. We're very proud to announce that 2016 will see our very first event to be held in Adelaide in August!
The aim for the market was simple; to create a supportive environment for like-minded, independent designers to sell their work. Brooke and Sarah had a vision of an event that didn't exist at the time; somewhere they wanted to shop, which combined everything they loved under one roof.
They wanted to create something different to the ordinary market trade so prevalent at the time, instead seeking to produce an inspiring market experience more like a design festival! They had a vision to create a nurturing space to encourage young emerging designers at the rise of DIY. To not only sell their work but also connect directly with customers, meet the makers and share their stories, combining music, food and wine in a fun atmosphere.
It turns out Brooke and Sarah set something in motion that local makers and shoppers were also seeking. With zero business experience and absolutely no funding (but with massive doses of self-produced soul and hard-working heart) the markets were born.
Realising their vision in 2007 with a modestly-sized market of just 20 stalls (originally named "Hope Street Markets"), the ladies expected only friends and family to turn up. The event, however, was an overwhelming success, and the markets have continued to expand and draw huge crowds ever since!
Each year the events have grown substantially, with Brooke and Sarah gaining loads of market know-how and using their newfound wisdom to produce bigger and better events. From managing the full magnitude of business tasks between them to forging a flourishing company that now employs 8 people part-time, in addition to a passionate and hard-working event team based in each city, the best is yet to come. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,750 |
Archive for June 5th, 2012
Celebrate La Salle ready to go for a third year
Posted by wlpo on June 5, 2012
You'll hear music and smell food cooking in Hegeler Park this weekend. The festival "Celebrate La Salle" begins Thursday and runs through Sunday.
It's the third year for the fest. Food stands will be serving tacos, hamburgers, brats and sweets like cotton candy and lemon shake-ups. Kids games will be set up Friday, Saturday and Sunday. There will also be live music and a beer garden.
Friday night features a free Dream Wave Wrestling event from 6-9pm.
La Salle residents should expect electric service related letter
You've been hearing about electric aggregation and you've probably received things in the mail trying to get you to change power companies. If you live in La Salle you'll be receiving another letter. This one will be on City of La Salle letterhead.
Mayor Jeff Grove says they were mailed out Tuesday. You'll only need to respond to the letter or taken action if you want to opt out of the new electricity program. You're expected to save money on power since the city has gone out looking for the best electric rate. You'll still be billed by Ameren.
LPHS approves TIF agreement with Peru
TIF Districts are used to spur development. A TIF that includes the old motels and truck stops and the new Holiday Inn Express in Peru hasn't seen much development, just a lawsuit.
Still, the boards at LP High School, Illinois Valley Community College and Peru Elementary District have reached an agreement for the commercial TIF district. The city council approved it Monday night.
Dimmick School District and La Salle County oppose it and are fighting it in court.
Peru Mayor Scott Harl says Peru provides one-third of La Salle County's sales taxes. He just doesn't understand why they would be against the TIF.
Marseilles Man Dies In Crash
A man from Marseilles a has died after a crash on the Morris Blacktop.
A representative of Good Samaritan Hospital in Downers Grove says 23-year old William Shelton of Marseilles died after being flown to the hospital Monday afternoon. La Salle County Sheriff's Deputies think Shelton was south on East 22nd Road just before noon Monday when he pulled in front of a car driven by Brandon Jensen of Ottawa.
Jensen was taken to St. Elizabeth Medical Center in Ottawa.
Why Is Gas More Expensive In Mendota?
How can gas prices be a quarter cheaper just 15 minutes down the road? That's what a former alderman is asking in Mendota.
According to the NewsTribune, John Pierson brought the discrepancy up to the Mendota City Council last night. He says he knows the council can't really lower prices but could put pressure on service station operators. Mayor David Boelk agreed with Pierson and said he would like to talk to Greg Gromann who runs two gas stations and convenience stores in Mendota.
Right now, unleaded gas is 14 to 20 cents more expensive in Mendota than in most other places in the valley.
Potential Kinzinger Challenger Turns In Needed Signatures
An Ottawa woman looks like she's done all that's required of her to run for Congress.
Democrat Wanda Rohl turned in well more than the 600 signatures needed to be on the November ballot. Last month, she received unanimous support from county chairpersons in the 16th Congressional District.
Rohl admits she doesn't have political experience but thinks she can overcome that against incumbent Republican Adam Kinzinger. She was injured in an ATV crash nine years ago which has left her paraplegic. She says safety net programs like Medicare and Medicaid helped keep her alive and keeping these programs funded is one of her top issues.
Cement Factory Shows No Signs Of Bouncing Back In Oglesby
If you were holding out hope that Buzzi Unicem would someday return to full production in Oglesby, you were dealt another setback.
The Oglesby City Council and cement maker have reached an agreement on what infrastructure around the plant belongs to the city. Buzzi is dismantling electric service and as part of the agreement, acknowledge some lines and transformers belong to the city.
Only a handful of people still work at the plant that began major layoffs in 2008.
La Salle House Fire Quickly Put Out
A small house fire had La Salle and Peru fire departments busy.
The firefighters were sent to a home in the 1100 block of Canal Street in La Salle just after five Monday evening. Smoke was showing from the house when crews arrived but they had it out in about a half hour.
No one was hurt. The cause of the fire is under investigation. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,508 |
Jill Jackson (født 24. januar 1979) er en pop/folk/country singer/songwriter fra Skotland, Storbritannien. Jill Jackson er blandt andet kendt som forsanger i bandet Speedway.
Sangskrivere
Sangere fra Skotland | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 761 |
I guess my first post was deleted. Sold my GMG and had a new grill. New grill had some issues so I am looking for a new pellet smoker.
Welcome and good luck on your hunt for a new grill!
Welcome to the forum! There are a few really good pellet grill brands out now. If you want a truly great pellet grill/smoker, you should at least take a serious look at the MAKs. You are, without a doubt, going to pay a little more, but you will also be getting your money's worth. They lead the market in innovation and quality.
Welcome aboard! +1 on the MAK. They are the best in the industry. Yup they cost a bit more but the advantage is you won't be caught in the mode of buying cheap, replacing to get something better only to then find you need something even better. That process will end up costing you more than if you bought the MAK to begin with. Ergo the saying: buy once cry once.
I have/had a MAK (arrived 12/7/18 and it will be leaving by end of 2018). I disagree with you on quality. I have GMG and looking to step up. Ruled out Yoder so far. Going back to take a look at Memphis, Myron Mixon, Blaz'n Grill right now. I agree with everyone on buying quality.
Just wondering what you didn't like about the MAK and especially in the quality area?? Did you have a 1 star or 2 star?
2 Star. This thread was titled because my first post disappeared. So, I better be careful of what I type here because it will be censored.
Sorry, but that's not a fair statement. We do not censor honest discussion of pellet smoking/grilling related things. You can discuss anything (preferably pellet-related). The only thing we ask, is since this forum is sponsored solely by Big Poppa Smokers, please do not post where to buy the types of products which BPS sells. This is very clearly stated in the forum rules.
Also, some forums allow for selling personal items on their forum, and some do not. Again, because this forum is sponsored by BPS, we do not provide for personal items to be sold here.
Your first post was listing a grill for sale. We get a lot of folks, who come in and their first and only post is trying to sell something. That's simply not the purpose of this forum.
Point taken. I bought a MAK 2 that was brand new and purchased from Big Poppa Smokers the end of November 2018. I have not posted anything about my experience. I gave my email to wolverines in case they wanted to reach me directly. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,516 |
\section{Introduction}
Categories, which were invented by Samuel Eilenberg and Saunders Mac Lane, form a very
high-level abstract mathematical theory that unifies all branches of mathematics. The
standard reference on category theory is Mac Lane's book-length introduction to
categories~\cite{mclane98}. Category theory plays a central role in modern mathematics and
theoretical computer science, and, in addition, it is used in mathematical
physics (e.g., see~\cite{flori10}), in software engineering~\cite{fiadeiro04}, etc.
However, what makes category theory even more interesting is that it is an alternative to set
theory as a foundation for mathematics. Indeed, as Mac~Lane~\cite{mclane86} pointed out
\begin{quote}
It is now possible to develop almost all of ordinary Mathematics in a well-pointed topos
[i.e., an ordinary category that has some additional properties] with choice and a natural
number object. The development would seem unfamiliar; it has nowhere been carried out yet in
great detail. However, this possibility does demonstrate one point of philosophical interest:
The foundation of Mathematics on the basis of set theory (ZFC) is by no means the only
possible one!
\end{quote}
Roughly, a category is a universe that includes all mathematical objects of a particular form
together with maps between them. These maps must obey a few basic principles. For example, the
collection of all sets and functions between them forms the category that is traditionally
denoted by $\mathbf{Set}$. Obviously, one can form categories of fuzzy structures.
For instance, Carol Walker~\cite{walker04} and Siegfried Gottwald~\cite{gottwald06} have
defined such categories. However, some other categories of fuzzy structures emerged from efforts
to define fuzzy models of linear logic. In particular, Michael Barr~\cite{barr91}, Basil Papadopoulos
and this author~\cite{pap-syr00,pap-syr05,syropoulos06} have introduced such categories.
Not so surprisingly, a category of fuzzy structures is not a fuzzy structure itself. Indeed,
Alexander \v{S}ostak~\cite{sostak99} was the first researcher who had realized this. In order to
remedy this situation, \v{S}ostak introduced a new structure that mimics the way fuzzy
sets are defined as is evident from the following definition:
\begin{definition}
An L-fuzzy category (where $L$ is a GL-monoid) is a quintuple $\mathcal{C} = (\mathrm{Ob}(\mathcal{C}),\omega,
M(\mathcal{C}), \mu, \circ)$ where $\mathcal{C}_{\bot} = (\mathrm{Ob}(\mathcal{C}), M(\mathcal{C}),
\circ)$ is a usual (classical) category called the bottom frame of the fuzzy category $\mathcal{C}$;
$\omega : \mathrm{Ob}(\mathcal{C}) \rightarrow L$ is an $L$-subclass of
the class of objects $\mathrm{Ob}(\mathcal{C})$ of $\mathcal{C}_{\bot}$ and
$\mu:M(\mathcal{C})\rightarrow L$ is an $L$-subclass of the class of morphisms
$M(\mathcal{C})$ of $\mathcal{C}_{\bot}$ . Besides $\omega$ and $\mu$ must satisfy the following conditions:
\begin{enumerate}
\item if $f : X \rightarrow Y$ , then $\mu(f ) \le\omega(X) \wedge ω(Y)$;
\item $\mu(g \circ f )\ge \mu(g) \ast \mu(f)$ whenever composition $g\circ f$ is defined,
where $\ast$ is a binary operator that obeys a number of rules;
\item if $e_{X} : X \rightarrow X$ is the identity morphism, then $\mu(e_{X}) = \omega(X)$.
\end{enumerate}
\end{definition}
For reasons that will become clear later on, \v{S}ostak's approach is not the ideal
solution to the problem of the ``fuzzification'' of category theory. Instead, by following
a different path one should be to define an alternative form of fuzzy categories. In particular,
since any category can be identified with a graph (the inverse is not true), I am using the
results of fuzzy graph theory to define, what I think are, real fuzzy categories.
This approach is justified by the fact that (meta)categories are introduced using notions from (meta)graph theory.
Interestingly, G{\'e}rard Huet and Amokrane Sa{\"\i}bi~\cite{huet00} have followed a similar line
of thought in order to define constructive categories and categorical structures.
One may wonder why someone should get into the trouble to define fuzzy categories. Indeed,
this is a quite reasonable question since there is no room for more meaningless generalizations.
However, if fuzziness, in particular, and {\em vagueness}, in general, are fundamental properties
of this cosmos, then we should use this fact to define deviant Mathematics.\footnote{Obviously,
according to quantum mechanics our world is vague, but probabilistic systems do not fundamentally
differ from their deterministic counterparts (e.g., nondeterministic and deterministic Turing
machines have exactly the same computational power). Nevertheless, it seems that this does not
apply to fuzziness.} Thus, if this is true, then we could use fuzzy categories as a tool to
develop the foundations for this deviant Mathematics.
\section{Basic Ideas and Concepts}
As was explained in the introduction, I will define fuzzy categories using known notions
from fuzzy graph theory. Thus, it is important to recall the notions that are necessary to
define fuzzy categories. Let us start with fuzzy graphs:
\begin{definition}
A {\em fuzzy graph}~\cite{lee05} consists of a collection of nodes and a collection of arrows
between these nodes. Each arrow must have a specific {\em domain} node (i.e., its source),
a {\em codomain} node (i.e., its target), and a plausibility degree, which expresses the grade
to which it is possible to go from the domain to the codomain of an arrow.
\end{definition}
Obviously, it is possible to have two or more different arrows that have the same domain
and codomain, but, clearly, different plausibility degrees. The notation
``$f:A\overset{\rho}{\rightarrow}B$'' means that $f$ is an arrow that goes from $A$ (the domain)
to $B$ (the codomain) with plausibility degree that is equal to $\rho\in[0,1]$. Alternatively,
one can use the following notation:
\begin{displaymath}
A\overset{f}{\underset{\rho}{\longrightarrow}}B.
\end{displaymath}
\begin{definition}
Let $k>0$. In a fuzzy graph $\mathcal{G}$ a path from a node $X$ to a node $\Psi$ of length
$k$ is a sequence $(f_1,f_2,\ldots,f_k)$ of arrows, which are not necessarily distinct, such that
\begin{displaymath}
X\overset{f_k}{\underset{\rho_k}{\longrightarrow}} A_{k-1}
\overset{f_{k-1}}{\underset{\rho_{k-1}}{\longrightarrow}}\ldots A_{1}
\overset{f_{1}}{\underset{\rho_{1}}{\longrightarrow}} \Psi
\end{displaymath}
and the plausibility degree of the path is $\min_{i=1}^{k}\rho_{i}$.
\end{definition}
\begin{remark}
We can use any t-conorm, but for reasons of simplicity we use min.
\end{remark}
Having defined fuzzy arrows and fuzzy arrow composition, we can proceed with the definition
of fuzzy categories.
\begin{definition}
A fuzzy category $\mathscr{C}$ comprices
\begin{enumerate}
\item a collection of entities called {\em objects};
\item a collection of entities called {\em arrows} or {\em morphisms};
\item operations assigning to each $\mathscr{C}$-arrow $f$ a $\mathscr{C}$-object
$A=\mathop{\mathrm{dom}} f$, its domain, a $\mathscr{C}$-object $B=\mathop{\mathrm{cod}} f$,
its codomain, and a plausibility degree
$\rho=\mathop{\mathrm{p}} f$. Typically, the plausibility degree is a real number
belonging to the unit interval. These operations on $f$ are indicated by displaying $f$ as an arrow
starting from $A$ and ending at $B$ with plausibility degree $\rho$:
\begin{displaymath}
A\overset{f}{\underset{\rho}{\longrightarrow}}B\quad\mbox{or}\quad
f:A\overset{\rho}{\rightarrow}B;
\end{displaymath}
\item an operation assigning to each pair $(g,f)$ of arrows with $\mathop{\mathrm{dom}} g=
\mathop{\mathrm{cod}} f$, an aror $g\circ f$, the {\em composite} of $f$ and $g$, having
$\mathop{\mathrm{dom}}(g\circ f)=\mathop{\mathrm{dom}} f$,
$\mathop{\mathrm{cod}}(g\circ f)=\mathop{\mathrm{cod}} g$, and
$\mathop{\mathrm{p}}(g\circ f)=\min\{\mathop{\mathrm{p}} f, \mathop{\mathrm{p}} g\}$. This
operation and the previous three are subject to the {\em associative law}: Given the
configuration
\begin{displaymath}
A\overset{f}{\underset{\rho_1}{\longrightarrow}}B
\overset{g}{\underset{\rho_2}{\longrightarrow}}C
\overset{h}{\underset{\rho_3}{\longrightarrow}}D
\end{displaymath}
of $\mathscr{C}$-objects and $\mathscr{C}$-arrows, then $h\circ(g\circ f)=(h\circ g)\circ f$;
\item an {\em assignment} to each $\mathscr{C}$-object $B$ of a $\mathscr{C}$-arrow
$\mathbf{1}_{B}:B\overset{1}{\rightarrow}B$, called the {\em identity arrow on} $B$, such
that the following {\em identity law} holds true:
\begin{displaymath}
\mathbf{1}_B\circ f=f\quad\mbox{and}\quad g\circ\mathbf{1}_B=g
\end{displaymath}
for any $\mathscr{C}$-arrows $f:A\overset{\rho_f}{\rightarrow}B$ and
$g:B\overset{\rho_g}{\rightarrow}A$.
\end{enumerate}
\end{definition}
\begin{remark}
Obviously, every ordinary category is a fuzzy category with arrows that have plausibility
degree equal to 1.
\end{remark}
\begin{example}
Let us give a relatively simple example of a fuzzy category that has as objects sets.
Assume that $f:X\rightarrow Y$ is function. Then we say that $f$ is an
arrow from $X$ to $Y$ with plausibility degree $\lambda$ if there are fuzzy subsets that
are characterized by the functions $A:X\rightarrow\mathrm{I}$
and $B:Y\rightarrow\mathrm{I}$ such that $B(f(x))-A(x)\ge\lambda$, for all $x\in X$. Assume that
$f:X\overset{\lambda_1}{\rightarrow}Y$ and $g:Y\overset{\lambda_2}{\rightarrow}Z$ are two
arrows. Then since $B(f(x))-A(x)\ge\lambda_1$ and $C(g(f(x)))-B(f(x))\ge\lambda_2$, for all $x\in X$,
one concludes that $C(g(f(x)))-A(x)\ge\lambda_3$, for all $x\in X$ and where
$\lambda_3=\min\{\lambda_1,\lambda_2\}$. Thus, the
composite arrow $g\circ f:X\overset{\lambda_3}{\rightarrow}Z$ exists and can be defined
form its constituents. It is not difficult to see that the associative law holds also. Clearly,
the identity arrow for some object $X$ is the identity function of this set. Also, it
is almost trivial to see that the identity law holds. The resulting fuzzy category
will be called \textbf{FSet}.
\end{example}
\begin{remark}
The objects of a fuzzy category can be fuzzy structures, but this is something that should
not concerns us. After all, category theory is about arrows and their properties and
not about objects.
\end{remark}
\begin{example}
Assume that $R:X\times X\rightarrow[0,1]$ is a fuzzy relation such that
$R(x,x)=1$ for all $x\in X$ and $R(x,y)\mathbin{\ast}R(y,z)\le R(x,z)$ for
all $x,y,z\in X$ and where $\ast$ is a t-norm. A fuzzy relation with these properties
is called a {\em $\ast$-fuzzy preorder}. When the t-norm is function $\min$, then it will
be called just fuzzy preorder. Any fuzzy preorder relation $R$ determines a fuzzy preorder
category $P$ in which the arrows $p\overset{\rho}{\rightarrow}p'$ are exactly those pairs
$\langle p,p'\rangle$ for which $R(p,p')=\rho$. The fuzzy relation is transitive, which
implies that there is a unique way of composing arrows. Also, the fuzzy relation is
reflexive and so there are the necessary identity arrows.
\end{example}
\begin{example}
Let $\wedge$ be the only object of a category and let us identify all arrows of this category
with their plausibility degrees. For example $\wedge
\overset{\lambda}{\underset{\lambda}{\longrightarrow}}\wedge$
is the arrow $\lambda$ whose plausibility degree is obviously $\lambda$. Given two arrows
$\lambda_1$ and $\lambda_2$, and assuming that $\wedge$ denotes the minimum, as is usually the
case, then $\lambda_1\circ\lambda_2=\lambda_1\wedge\lambda_2$. Assume there is an arrow
$1_{\wedge}$ such that $1_{\wedge}\circ\lambda=\lambda$ and $\lambda\circ1_{\wedge}=\lambda$
for all arrows $\lambda$, then according to the definition of arrow composition this arrow
is the identity arrow, that is, $1_{\wedge}=1$. Note that it is quite possible to have more than
one arrow that has plausibility degree equal to one, nevertheless, for our purposes these
arrows will behave exactly like $1$ does. In different words, they will be isomorphic,
but we will say more about isomorphisms in a while.
\end{example}
\begin{example}
A category is a {\em deductive system} (for example, see~\cite{lambek94} for a thorough
description of this categories-as-deductive-systems paradigm). In this paradigm, objects
are seen as {\em formulas}, arrows as {\em proofs} (or deductions), and an operation
on arrows as a {\em rule of inference}. In particular, each arrow $f:A\rightarrow B$ is
thought of as the ``reason'' why $A$ entails $B$. Thus, the identity law is the reason
why $A$ entails $A$, for all $A$ objects (formulas) and the associative law becomes the
following rule of inference:
\begin{prooftree}
\AxiomC{$f:A\overset{}{\longrightarrow}B$}
\AxiomC{$g:B\overset{}{\longrightarrow}C$}
\BinaryInfC{$f\circ g:A\overset{}{\longrightarrow}C$}
\end{prooftree}
Similarly, a fuzzy category is a {\em fuzzy deductive system} in which objects may be
{\em fuzzy formulas} (remember, the objects of any fuzzy category are not necessarily ``crisp''),
arrows are {\em fuzzy deductions}, and the associative law is the following fuzzy inference:
\begin{prooftree}
\AxiomC{$f:A\overset{\rho_f}{\longrightarrow}B$}
\AxiomC{$g:B\overset{\rho_g}{\longrightarrow}C$}
\BinaryInfC{$f\circ g:A\overset{\rho_{f\circ g}}{\longrightarrow}C$}
\end{prooftree}
\end{example}
\paragraph{Commutative diagrams} In category theory {\em commutative diagrams} play the role
equations play in algebra. In the simplest case a commutative diagram can be identified with
two different paths starting from the same object $A$ and ending with the same object $B$ in
which the composition of the arrows that make up the first path and the composition of the arrows of
the second path yield two arrows that have the same effect (i.e., when applied to the same
object(s), they yield the same result). In general, when dealing with fuzzy arrows we need
to stick to this requirement, but we distinguish at least two different cases. In the first
case, the plausibilities must be exactly the same while in the second case, they must be
greater than a specific minimum. In particular, assume that
\begin{align*}
A&\overset{f_n}{\underset{\lambda_n}{\longrightarrow}}\ldots
\overset{f_{1}}{\underset{\lambda_{1}}{\longrightarrow}}B\\
A&\overset{g_m}{\underset{\rho_m}{\longrightarrow}}\ldots
\overset{g_{1}}{\underset{\rho_{1}}{\longrightarrow}}B
\end{align*}
are two paths. Then these paths form a {\em strong} commutative diagram provided that
\begin{displaymath}
\min\Bigl\{\lambda_1,\ldots, \lambda_n \Bigr\}=\min\Bigl\{\rho_1,\ldots, \rho_m \Bigr\}.
\end{displaymath}
Otherwise, we say that the two paths commute with plausibility degree $\nu$, where
\begin{displaymath}
\nu=\min\Bigl\{\min\{\lambda_1,\ldots,\lambda_n\},\min\{\rho_1,\ldots,\rho_m\}\Bigr\}.
\end{displaymath}
\begin{example}
The identity law can be expressed with the following strong commutative diagrams:
\begin{center}
\begin{tabular}{lcr}
\begin{diagram}
& & B \\
& \ruTo^f_{\lambda_1} & & \rdTo_1^{\mathbf{1}_B} \\
A & & \rTo_{\lambda_1}^{\mathbf{1}_{B}\circ f=f} & & B\\
\end{diagram} & \qquad &
\begin{diagram}
& & B \\
& \ruTo_1^{\mathbf{1}_B} & & \rdTo^g_{\lambda_2}\\
B & & \rTo_{\lambda_2}^{g\circ\mathbf{1}_{B}=g} & & A\\
\end{diagram}
\end{tabular}
\end{center}
Obviously, $\lambda_1\le1$ and $\lambda_2\le1$.
\end{example}
\paragraph{Isomorphisms} In ordinary mathematics two entities of the same kind can be
isomorphic or not. In the fuzzy setting, they can be isomorphic up to some degree and
the following definition follows this principle:
\begin{definition}
Two objects $A$ and $B$ are isomorphic to some degree
$\lambda$ if there are arrows $f:A\overset{\lambda_1}{\longrightarrow}B$
and $g:B\overset{\lambda_2}{\longrightarrow}A$ such that the following diagrams
\begin{center}
\begin{tabular}{lcr}
\begin{diagram}
& & B \\
& \ruTo^f_{\lambda_1} & & \rdTo^g_{\lambda_2} \\
A & & \rTo_{1}^{\mathbf{1}_{A}} & & A\\
\end{diagram} & \qquad &
\begin{diagram}
& & A \\
& \ruTo^g_{\lambda_2} & & \rdTo^f_{\lambda_1}\\
B & & \rTo_{1}^{\mathbf{1}_{B}} & & B\\
\end{diagram}
\end{tabular}
\end{center}
are commutative with degree that is equal to $\lambda$, where
$\lambda=\min\{\lambda_1,\lambda_2\}$.
\end{definition}
Arrows can be {\em monic} or {\em epic} up to some degree. In particular, an arrow
$f:A\overset{\lambda_1}{\longrightarrow}B$ is monic up to $\nu$ if there are two arrows
$g:C\overset{\lambda_2}{\longrightarrow}A$ $h:C\overset{\lambda_3}{\longrightarrow}A$
such that the following diagram
\begin{diagram}
C & \rTo^{g}_{\lambda_2} & A\\
\dTo_{\lambda_3}^{h} & & \dTo_{\lambda_1}^{f}\\
A & \rTo^{f}_{\lambda_1} & B
\end{diagram}
commutes with degree equal to $\nu=\min\{\lambda_1,\lambda_2,\lambda_3\}$. In
addition, $g\mathrel{=_{\nu}}h$, that is, $g$ and $h$ are equal with degree $\nu$.
Similarly, an arrow $f':A\overset{\kappa_1}{\longrightarrow}B$ is epic up to $\nu'$
if there are two arrows $g':B\overset{\kappa_2}{\longrightarrow}C$ $
h:B\overset{\kappa_3}{\longrightarrow}C$
such that the following diagram
\begin{diagram}
A & \rTo^{f'}_{\kappa_1} & B\\
\dTo_{\kappa_1}^{f'} & & \dTo_{\kappa_2}^{g'}\\
B & \rTo^{f}_{\kappa_3} & C
\end{diagram}
commutes with degree equal to $\nu'=\min\{\kappa_1,\kappa_2,\kappa_3\}$. In
addition, $g'\mathrel{=_{\nu'}}h'$, that is, $g'$ and $h'$ are equal with degree $\nu'$.
\paragraph{Initial and Terminal Objects}
An object $T$ of a fuzzy category $\mathscr{C}$ is called terminal if
there is exactly one arrow $A\overset{1}{\longrightarrow}T$ for each object $A$ of
$\mathscr{C}$. An object of a fuzzy category that has a unique arrow with plausibility
degree equal to one {\em to} each object (including itself), is called an {\em initial object}.
\section{Conclusions}
I have introduced fuzzy categories and some fuzzy categorical structures. There is
much work ahead! First one needs to define fuzzy functors, then fuzzy natural
transformations. However, the most important of all is to see whether these categories
have interesting properties and whether they can be used to solve interesting problems.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,224 |
ADatP (Abk. für ) ist eine Gruppe von NATO-Standards (STANAG). Darunter sind unter anderem auch Standards für den Aufbau von Daten für die Übertragung und Verarbeitung durch automatisierte Systeme.
Beispiel
ADatP-3 NATO Message Text Formatting System (FORMETS) - Concept of FORMETS (CONFORMETS): In diesem Standard sind Regeln zur Formatierung von Meldungen festgelegt. Seit 2008 basiert dieser Standard auf einer Beschreibung in XML Schemas und ist damit vergleichbar mit zivil genutzten Standards, wie zum Beispiel UN/EDIFACT.
Link 1 (NATO-Originalbezeichnung: ) ist ein NATO-Standard für taktische Datenlinks der Luftverteidigung.
Literatur
Weblinks
NATO Homepage mit frei verfügbaren ADatP zum Herunterladen
Einzelnachweise
Datenformat
Abkürzung
Kommunikation (NATO)
NATO-Verfahren | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,640 |
Pushpa Trailer – Release Date, Cast, and Review – Watch Online
December 7, 2021 by Ariadna Pandey
Pushpa Trailer: Pushpa Official Trailer on Mythri Movie Makers. Written and Directed by Sukumar. Produced by Naveen Yerneni and Y. Ravi Shankar of Mythri Movie Makers in association with Muttamsetty Media, the film stars Allu Arjun, Fahadh Faasil, and Rashmika Mandanna, Dhanunjay, Rao Ramesh, Suneel, Anasuya Bharadwaj & Ajay Ghosh.
The film's music is composed by Devi Sri Prasad, with cinematography and editing were performed by Miroslaw Kuba Brozek and Karthika Srinivas respectively. The film is scheduled to release on 17 December 2021, in Telugu along with dubbed versions in Malayalam, Tamil, Hindi, and Kannada languages.
Allu Arjun is one of those versatile actors who know very well to entertain their audience in the best possible way and once again, Allu Arjun is on our screen with one of the most anticipated movies, Pushpa.
Pushpa Official Trailer
GORKHA Official Trailer, Release Date, Story, and Cast Details
Pushpa Story
The story of Pushpa revolves around a red sand wood smuggler who smuggles red sand wood for their living. Pushpa, the main character played by Allu Arjun who is a truck driver. Police found the scam and decided to stop all this stuff in the deep jungle but Pushpa and the team managed to skip over all the difficulties.
Although We can see Pushpa being tortured by the police but in the next scene we can see him free and destructive.
The movie is full of action and romance at the same time. The hero and villain both are powerful that making the movie much entertaining and enjoyable.
JOKER Official Trailer – Featuring Hrithik Roshan and Priyanka Chopra
Pushpa: The Rise Cast
Allu Arjun as Pushpa Raj
Fahadh Faasil as Bhanwar Singh Shekhawat IPS
Rashmika Mandanna as Srivalli
Pushpa: The Rise Release Date
Pushpa: The Rise will be released on 17th Dec 2021 to watch worldwide.
Matsya Kaand REVIEW, Cast and Release Date – Watch Online
Other Details of Pushpa
Release date: 17 December 2021 (India)
Director: Sukumar
Budget: 250 crores INR
Produced by: Naveen Yerneni, Y. Ravi Shankar
Production companies: Mythri Movie Makers; Muttamsetty Media
FAQs about Pushpa
What is the release date of Pushpa?
The Pushpa Movie will be released in two parts. Pushpa: The Rise will release on 17th December whereas the second part is scheduled for a 2022 release.
Who are the lead characters of Pushpa?
Allu Arjun and Rashmika Mandanna are the lead characters of Pushpa. The movie also features Fahadh Faasil (in his Telugu debut).
Categories Movies, Trailers, Trending Tags Action, Thriller
Radhe Shyam: Soch Liya song teaser is out now – Full song on Dec 8, 2021
Kangana Ranaut's Tejas will be released on this date – Check all the details | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,874 |
In the realm of dinner choices, pizza is always a favorite. Did you know that pizza is the lunch and dinner choice over all other food groups for kids ages 3 to 11? We like to think that pizza is a fantastic choice for all ages, and something that is easy for everyone to agree on. Orlando with all its attractions, including Walt Disney World and Sea World, has created dizzying excitement all day. Now it's time to wind down and relax. You're back in your room changing into your comfortable clothes, and you hope to find excellent pizza delivery to Westgate Blue Tree Resort Lake Buena Vista Florida.
Caprino's Italian Restaurant is the premiere resource for some of the best pizza choices in Lake Buena Vista, delivered fresh and hot right to your room. A truly great authentic Italian pizza begins with a crisp crust, and Caprino's creates that by using imported water and flour. Next your pizza delight is topped with a delectable tomato sauce, or a white sauce created from light Ricotta cheese or a garlicky cream style sauce.
After that, selected crisp vegetables are placed on your pizza along with the freshest meats Caprino's can find, depending on what toppings you've chosen to have on your Caprino's Italian pizza delivery to Westgate Blue Tree Lake Buena Vista 32836. Why order from a large chain pizza place when you can have a delightfully different local pizza option from Caprino's Italian Restaurant? With a passionate focus on creating the best pizza possible, Caprino's has become a first-rate pizza choice in and around Orlando.
What truly makes Caprino's unique and sets it apart is the wide variety of topping choices you have. Pizza connoisseurs who adore the exotic along with a great variety will love having choices like spicy banana peppers, Capicola, Italian Soppressata, calamari, chopped figs, shrimp, chicken, anchovies, salmon, Prosciutto Di Parma, roasted red peppers and arugula. No matter what your topping choice is, it will never be boring when you choose Caprino's!
Traditionalists will love plain cheese pizzas hand-crafted with cheeses like provolone, mozzarella, Asiago, Ricotta and Parmesan Reggiano. Other traditional toppings might include spicy or sweet sausage, pepperoni, bacon, mushrooms, onions, green peppers and extra cheese. We think you'll agree that the best Pizza delivery near Lake Buena Vista FL 32836 comes expertly prepared and delivered straight to you from Caprino's.
Caprino's was started by Chef Stefano Tedeschi to become a classical Italian restaurant. Chef Steff knew he was interested in cooking at a young age when he began learning by working with his father in a family pizza restaurant. From there, he went on to develop true chef skills and a level of artistry that is most often fostered through total immersion into the art of cooking. He did this by working next to true Italian Chefs in Italy. His signature style and delightful recipes are used today in the creation of Caprino's delicious pizza.
Chef Steff has also cultivated a passionate interest in sports and has friends from both the sports and culinary worlds. He's appeared on NFL television programs as well as on Food Channel programs, to the delight of friends and family. He takes the art of fine pizza creation over to another level, establishing Caprino's as the best pizza near Westgate Blue Tree Resort Lake Buena Vista.
We think you'll be amazed and thrilled by the specialty pizza choices available through Caprino's. The Chef specials include The Caprino Pie, a delectable blend of goat cheese, spinach, chopped figs, a light cream base and specially created olive oil garlic. Chef Stefano's Special comes with roasted red peppers, spicy sausage, mozzarella, provolone and ricotta cheese. The Chef Stefano's special is then topped off with Prosciutto Di Parma for a tasty finish.
Embark on a virtual trip to the islands with the Hawaiian Pie, featuring sweet and salty ham and pineapple. Meat lovers adore the Hog Wild, with Italian Soppressata, which is a special type of dried Italian salami. Add to that spicy sausage and bacon and you see where this pizza got its name. The Hog Wild is then topped with Parmesan Reggiano, Mozzarella and Provolone for extra melted cheese goodness.
One taste of this delightful pizza and you'll see why Caprino's is consistently considered the best pizza restaurant near Westgate Blue Tree Resort Lake Buena Vista FL 32836. One experience with Caprino's delivery and you'll also see why they're the preferred, reliable and fast Pizza delivery to Westgate Blue Tree Resort Lake Buena Vista Florida. | {
"redpajama_set_name": "RedPajamaC4"
} | 8 |
// Copyright (c) Microsoft Corporation. All rights reserved.
// Licensed under the MIT License.
using System.Collections.Generic;
using System.Net;
using System.Threading.Tasks;
using Azure.Core.TestFramework;
using Azure.ResourceManager.Insights.Models;
using NUnit.Framework;
namespace Azure.ResourceManager.Insights.Tests.BasicTests
{
[AsyncOnly]
public class DiagnosticSettingsCategoryTest : InsightsManagementClientMockedBase
{
public DiagnosticSettingsCategoryTest(bool isAsync)
: base(isAsync)
{ }
[Test]
public async Task GetDiagnosticSettingsCategoryTest()
{
var ExpectDiagnosticSettingsCategoryResource = new DiagnosticSettingsCategoryResource("ID1", "Name1", "type1", CategoryType.Logs);
var content = @"
{
'id': 'ID1',
'name': 'Name1',
'type': 'type1',
'properties':
{
'categoryType': 'Logs'
}
}
".Replace("'", "\"");
var mockResponse = new MockResponse((int)HttpStatusCode.OK);
mockResponse.SetContent(content);
var mockTransport = new MockTransport(mockResponse);
var insightsClient = GetInsightsManagementClient(mockTransport);
var result = (await insightsClient.DiagnosticSettingsCategory.GetAsync("resourceUri", "name1")).Value;
AreEqual(ExpectDiagnosticSettingsCategoryResource, result);
}
private void AreEqual(DiagnosticSettingsCategoryResource exp, DiagnosticSettingsCategoryResource act)
{
Assert.AreEqual(exp.Id,act.Id);
Assert.AreEqual(exp.Name, act.Name);
Assert.AreEqual(exp.CategoryType.Value, act.CategoryType.Value);
}
[Test]
public async Task ListDiagnosticSettingsCategoryTest()
{
var ExpectDiagnosticSettingsCategoryResource = new List<DiagnosticSettingsCategoryResource>() { new DiagnosticSettingsCategoryResource("ID1", "Name1", "type1", CategoryType.Logs) };
var content = @"
{
'value':[
{
'id': 'ID1',
'name': 'Name1',
'type': 'type1',
'properties':
{
'categoryType': 'Logs'
}
}
]
}
".Replace("'", "\"");
var mockResponse = new MockResponse((int)HttpStatusCode.OK);
mockResponse.SetContent(content);
var mockTransport = new MockTransport(mockResponse);
var insightsClient = GetInsightsManagementClient(mockTransport);
var result = (await insightsClient.DiagnosticSettingsCategory.ListAsync("resourceUri")).Value.Value;
AreEqual(ExpectDiagnosticSettingsCategoryResource, result);
}
private void AreEqual(IReadOnlyList<DiagnosticSettingsCategoryResource> exp, IReadOnlyList<DiagnosticSettingsCategoryResource> act)
{
for (int i = 0; i < exp.Count; i++)
{
AreEqual(exp[i], act[i]);
}
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,688 |
Q: What is this smiley-with-beard expression: "<:]{%>"? I came across the following program, which compiles without errors or even warnings:
int main(){
<:]{%>; // smile!
}
Live example.
What does the program do, and what is that smiley-expression?
A: The program is using digraphs, which allow C++ programming with keyboards (or text encodings) that may not have the characters C++ typically uses.
The code resolves to this:
int main(){
[]{}; // smile!
}
A:
int main(){
<:]{%>; // smile!
}
It's basically a Lambda expression (Lambda expression is one of C++11 features) using digraphs (both digraphs and trigraphs works on C++):
[] {};
Using only digraphs:
<:]<%};
<:]<%%>;
[:>{%>; // like my cubic hat?
[:><%};
[:><%%>;
Mixing them with Trigraphs:
<:??)<%??>; // popeye
??(:>{??>; // pirate
A: The program uses digraphs to represent the following:
[] {};
This is a lambda expression that does nothing. The corresponding symbols have these equivalents:
<: = [
%> = }
Though they are generally unneeded today, digraphs are useful for when your keyboard lacks certain keys necessary to use C++'s basic source character set, namely the graphical ones. The combination of the characters that make up a digraph are processed as a single token. This in turn makes up for any insufficiently-equipped keyboards or other such hardware or software.
A: That's an empty lambda using a digraph disguise. Normal lambdas don't have beards.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,060 |
import {join} from 'path';
export const PROJECT_ROOT = join(__dirname, '../..');
export const SOURCE_ROOT = join(PROJECT_ROOT, 'src');
export const DIST_ROOT = join(PROJECT_ROOT, 'dist');
export const DIST_COMPONENTS_ROOT = join(DIST_ROOT, 'md2');
export const SASS_AUTOPREFIXER_OPTIONS = {
browsers: [
'last 2 versions',
'not ie <= 10',
'not ie_mob <= 10',
],
cascade: false,
};
export const HTML_MINIFIER_OPTIONS = {
collapseWhitespace: true,
removeComments: true,
caseSensitive: true,
removeAttributeQuotes: false
};
export const NPM_VENDOR_FILES = [
'@angular', 'core-js/client', 'rxjs', 'systemjs/dist', 'zone.js/dist'
];
export const COMPONENTS_DIR = join(SOURCE_ROOT, 'lib');
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,363 |
Q: ZFS root pool free space I have an openindiana (oi_151a7) box where I used 30GB for the root partition during installation. This has become rpool1. I cannot figure out how to make the remaining 202 GB available. Looking for suggestions. My partition table as follows:
Current partition table (original):
Total disk cylinders available: 26469 + 2 (reserved cylinders)
Part Tag Flag Cylinders Size Blocks
0 root wm 1 - 3912 29.97GB (3912/0/0) 62846280
1 unassigned wm 0 0 (0/0/0) 0
2 backup wu 0 - 3912 29.98GB (3913/0/0) 62862345
3 reserved wm 1 - 26468 202.76GB (26468/0/0) 425208420
4 unassigned wm 0 0 (0/0/0) 0
5 unassigned wm 0 0 (0/0/0) 0
6 unassigned wm 0 0 (0/0/0) 0
7 unassigned wm 0 0 (0/0/0) 0
8 boot wu 0 - 0 7.84MB (1/0/0) 16065
9 unassigned wm 0 0 (0/0/0) 0
Slice 3, was created by me, by hand, using the format utility. This is what I'm trying to turn into a usable file space.
When I run format, and select disk 0, I'm told c4t0d0s3 is part of the root pool:
root@oi01:~# format
Searching for disks...done
AVAILABLE DISK SELECTIONS:
0. c4t0d0 <ATA-VB0250EAVER-HPG7 cyl 26469 alt 2 hd 255 sec 63>
/pci@0,0/pci103c,1609@11/disk@0,0
1. c4t1d0 <ATA-INTEL SSDSC2CT06-300i-55.90GB>
/pci@0,0/pci103c,1609@11/disk@1,0
2. c4t2d0 <ATA-Maxtor 7L300S0-1G10-279.48GB>
/pci@0,0/pci103c,1609@11/disk@2,0
Specify disk (enter its number): 0
selecting c4t0d0
[disk formatted]
/dev/dsk/c4t0d0s0 is part of active ZFS pool rpool1. Please see zpool(1M).
/dev/dsk/c4t0d0s3 is part of active ZFS pool rpool1. Please see zpool(1M).
[...]
A:
When I run format, and select disk 0, I'm told c4t0d0s3 is part of the root pool:
This would be because you've screwed up the slice - it is overlapping with c4t0d0s0. You would need a slice starting at cylinder 3913 to remove the overlap. Then you should be able to add it as a vdev to another pool (if this is what you are after).
If you just want to resize your root pool to use all available disk space, boot up a live CD, edit the root and backup slices accordingly to occupy the full disk and restart your system - your zpool should adapt automagically. If not, try toggling the "autoexpand" property of the rpool.
Further reading: http://www.distrans.org/wiki/unixsystems/opensolaris/growing_root_pool
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,588 |
Q: HorizontalScrollView inside Movable View I have a movableView(moves with finger touch movement) of 400*400, it has a horizontal view at bottom (400*100). When I want to scroll horintalScrollView to see the list, my movableView is moving, but horintalSrollView is not taking any touchEvent.
I want it like, when I am touching horizontalScrollView, only it should scroll. And when not touching it and want to move my movableView it should move.
Any suggestions would be helpful.
Thanks
A: Try this logic...
scrollView.setOnTouchListener(new OnTouchListener() {
// Setting on Touch Listener for handling the touch inside movableView
@Override
public boolean onTouch(View v, MotionEvent event) {
// Disallow the touch request for parent move on touch of child view
v.getParent().requestDisallowInterceptTouchEvent(true);
return false;
}
});
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,640 |
{"url":"https:\/\/repository.uantwerpen.be\/link\/irua\/155466","text":"Publication\nTitle\nCost-effectiveness of bone SPECT\/CT in painful total knee arthroplasty\nAuthor\nAbstract\n The purpose of this study was to quantify the economic value of bone SPECT\/CT versus CT or metal artifact reduction sequence (MARS)MRI for the diagnostic assessment of recurrent moderate-to-severe pain after total knee arthroplasty (TKA). Methods: An Excel-based simulation model was developed to compare bone SPECT\/CT versus CT or MARS-MRI from a payer perspective. Clinical endpoints (diagnosis- delayed or otherwise, and the subsequent treatment and complications) and their corresponding cost data (2017 U. S. dollars) were obtained by performing a best evidence review of the published literature. Studies were pooled and parameters weighted by sample size. A cost-utility analysis was performed estimating the incremental cost per quality-adjusted life years gained between bone SPECT\/CT and the comparative scans. One-way (+\/- 25%) sensitivity analysis was performed to gauge the model robustness. Results: For every 1,000 TKA patients, diagnostic bone SPECT\/CT was expected to lead to 3-y cost savings up to $1,867,695 versus CT (or$ 622.6 per patient per year) and $1,723,435 versus MARS-MRI (or$ 574.5 per patient per year) for a payer. With corresponding incremental quality-adjusted life years gains of 39.7 and 41.0 against CT and MARS-MRI, SPECT\/CT can be considered as a cost-saving and dominant strategy in the workup of persistent\/recurrent pain in TKA patients. The model was limited by the still sparse literature data, was most sensitive to imaging-related sensitivity\/specificity, but proved robust for varying prevalence of surgical\/nonsurgical causes of pain. Conclusion: Bone SPECT\/CT is a potentially highly cost-saving and dominant imaging intervention versus CT or MARS-MR scanning in patients with recurrent and persistent knee pain after TKA.\nLanguage\nEnglish\nSource (journal)\nThe Journal of nuclear medicine. - New York\nPublication\nNew York : 2018\nISSN\n0161-5505\nVolume\/pages\n59 :11 (2018) , p. 1742-1750\nISI\n000448962500019\nPubmed ID\n29602816\nFull text (Publisher's DOI)\nFull text (open access)\nFull text (publisher's version - intranet only)\nUAntwerpen\n Faculty\/Department Research group Publication type Subject Affiliation Publications with a UAntwerp address","date":"2022-10-04 07:24:57","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2603224217891693, \"perplexity\": 13677.67239334016}, \"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-2022-40\/segments\/1664030337480.10\/warc\/CC-MAIN-20221004054641-20221004084641-00033.warc.gz\"}"} | null | null |
Working At Marin County Office of Education
Marin County Office of Education overview
Marin County Office of Education is a small professional organization based in California with only 120 employees and an annual revenue of $30.0M.
marinschools.org
The team at Marin County Office of Education
Mary Burke (Administrator)
What do people say about Marin County Office of Education
Do you work at Marin County Office of Education ? Anonymously review Marin County Office of Education as an Employer.
What Do You Like About Working For Marin County Office of Education?
Marin County Office of Education Rankings
Marin County Office of Education is ranked #5 on the Best Companies to Work For in San Rafael, CA list. Zippia's Best Places to Work lists provide unbiased, data-based evaluations of companies. Rankings are based on government and proprietary data on salaries, company financial health, and employee diversity.
#5 in Best Companies to Work For in San Rafael, CA
#16 in Biggest Companies in San Rafael, CA
Marin County Office of Education careers
On average, employees at Marin County Office of Education stay with the company for 3.2 years. Employees most commonly join Marin County Office of Education after leaving San Francisco Public Library. When they leave Marin County Office of Education, they most frequently get their next job at San Francisco Unified School District.
Boulder Valley School District3.9 years
FRONTIER REGIONAL SCHOOL DISTRICT3.5 years
Nassau BOCES2.8 years
Top Employers Before Marin County Office of Education
San Francisco Public Library10.0 %
Menlo Park10.0 %
California Academy of Sciences7.5 %
University of California Press5.0 %
Top Employers After Marin County Office of Education
San Francisco Unified School District9.1 %
San Rafael Elementary School9.1 %
Kindred Healthcare6.8 %
The Speech Pathology Group6.8 %
Do you Work At Marin County Office of Education?
Marin County Office of Education Financial Performance
$10M - $100M
How Would You Rate The Company Culture Of Marin County Office of Education?
Have you worked at Marin County Office of Education? Help other job seekers by rating Marin County Office of Education.
Marin County Office of Education Competitors
Variety Child Learning CtrServices
ISD 622 North St. Paul-Maplewood-OakdaleServices
JOHN ESSEX HIGH SCHOOLServices
LAKELAND HIGH SCHOOL BUILDINGServices
Arundel High SchoolServices
Frequently Asked Questions about Marin County Office of Education
How many Employees does Marin County Office of Education have?
Marin County Office of Education has 120 employees.
How much money does Marin County Office of Education make?
Marin County Office of Education generates $30.0M in revenue.
What industry is Marin County Office of Education in?
Marin County Office of Education is in the professional services industry.
What type of company is Marin County Office of Education?
Marin County Office of Education is a education company.
Who are Marin County Office of Education's competitors?
Marin County Office of Education competitors include Nassau BOCES, Progressus Co, FRONTIER REGIONAL SCHOOL DISTRICT, Boulder Valley School District, LIVE OAK ELEMENTARY SCHOOL DISTRICT SANTA CRUZ COUNTY CALIFORNIA, Irvine High School, Long Beach High School, Johnson City High School, Redwood School, ISD 622 North St. Paul-Maplewood-Oakdale, Centennial BOCES, LISBON COMMUNITY SCHOOL DISTRICT IOWA, Antioch Unified School District, Exceptional Education, Modesto City Schools, Spring-Ford Area School District, Variety Child Learning Ctr, LAKELAND HIGH SCHOOL BUILDING, JOHN ESSEX HIGH SCHOOL, Arundel High School.
Who works at Marin County Office of Education?
Where is Marin County Office of Education's headquarters?
Marin County Office of Education's headquarters is in San Rafael, CA.
What is Marin County Office of Education's website?
You can find the Marin County Office of Education website at marinschools.org.
Are you an executive, HR leader, or brand manager at Marin County Office of Education?
You can find out what it is like to work at Marin County Office of Education, also known as Marin County Office of Education and Marin County School District.
Zippia gives an in-depth look into the details of Marin County Office of Education, including salaries, political affiliations, employee data, and more, in order to inform job seekers about Marin County Office of Education. The employee data is based on information from people who have self-reported their past or current employments at Marin County Office of Education. While we have made attempts to ensure that the information displayed are correct, Zippia is not responsible for any errors or omissions, or for the results obtained from the use of this information. The data presented on this page does not represent the view of Marin County Office of Education and its employees or that of Zippia. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,259 |
JUCO BASEBALL: Trojans hold off Shelton State in opener
ALEXANDER CITY | Shelton State, the defending Alabama Community College Conference baseball champion, rallied with four runs in the eighth inning on Friday but lost its opener 6-5 to Central Alabama.
'It was a typical first game,' said Shelton State coach Bobby Sprowl, who directed his team to the runner-up spot in the 2008 JUCO World Series in Colorado. 'The pitchers were probably ahead of the hitters. Each of us had a big inning.
'We were able to play 14 or 15 guys, and it's about what we expected. We wanted to win, obviously, but it's nothing to be frustrated about.'
Jeff Parton started for Central Alabama and pitched four scoreless innings to get the win. He allowed one hit. Freshman left-hander Michael Carden started for Shelton State and took the loss. He allowed two hits and two runs and struck out three.
Jeff Judah hit a two-run homer off Carden in the second inning. The Trojans added a run in the third and scored three off relief pitcher Junior Gray, who gave up three hits in the seventh.
'Carden was as good as I've seen him in the two innings he pitched,' Sprowl said. 'He just gave up a home run.'
Shelton State got its first run in the fifth inning. Matt Kirkwood doubled and scored on a double by Logan Pierce.
In the Buccaneers' eighth, Jon McKinney walked and John David Smelser followed with a run-scoring double.
Justin Diliberto singled to drive in Smelser. After Brett Whitaker walked, Cayleb Coker and Coy Arrowood each doubled in a run.
Shelton State makes its home debut on Sunday, facing Central Alabama at 1 p.m. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,935 |
{"url":"https:\/\/artofproblemsolving.com\/wiki\/index.php?title=2021_JMPSC_Sprint_Problems\/Problem_11&diff=prev&oldid=161828","text":"# Difference between revisions of \"2021 JMPSC Sprint Problems\/Problem 11\"\n\n## Problem\n\nHow many numbers are in the finite sequence of consecutive perfect squares $$9, 16, 25, \\ldots , 2500?$$\n\n## Solution\n\nThe perfect squares are from $3^2$ to $50^2$. Therefore, the answer is the amount of positive integers between $3$ and $50$, inclusive. This is just $50-3+1=\\boxed{48}$.\n\n## Solution 2 (General Method)\n\nThe set is $$S=\\{3^2,4^2,5^2,\\cdots,50^2\\}$$ Notice that the cardinality of $$\\{a_1,a_2,\\cdots,a_n\\}$$ is equal to the cardinality of $$\\{f(a_1),f(a_2),\\cdots, f(a_n)\\}$$ For all functions $f$ with domain containing $a_1, \\cdots, a_n$. In our case, apply $f(x)=\\sqrt{x}$ to get $$S_1=\\{3,4,5,\\cdots,50\\}$$ Now apply $f(x)=x-2$ to get $$S_2=\\{1,2,3,\\cdots,48\\}$$ Which clearly has cardinality $\\boxed{48}$.\n\n~yofro","date":"2022-08-08 15:43:36","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\": 16, \"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\": 0.5739536285400391, \"perplexity\": 238.35283000829534}, \"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-2022-33\/segments\/1659882570868.47\/warc\/CC-MAIN-20220808152744-20220808182744-00294.warc.gz\"}"} | null | null |
"The Car" (1977) (Elliot Silverstein)
1977 / 96mins. / English Language / Original Aspect Ratio (2.35:1) / Color / USA
Picture Quality: A++ (Source: DVD)
not because I think it is a perfect movie, but because of all the people who have severely underrated this movie. This is a very well crafted movie. No, its not the best acted movie, but for this type of movie, it is. You have to look at these movies in a relative manner. Its a movie about a mysterious car that goes around wreaking havoc. That is the story. A ridiculous premise but the makers pull it off. No small feat. The actors do a good job and I really enjoyed the direction, especially the long shots out in the desert. See this in widescreen, if possible. It adds a lot to the feel of the movie. James Brolin is very good as the hero and there are a lot of memorable scenes. No, its not a 10. More like an 8.5, but its far better than a 5 average. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,922 |
Torkeh Dārī (persiska: ترکه داری) är en ort i Iran. Den ligger i provinsen Östazarbaijan, i den nordvästra delen av landet, km nordväst om huvudstaden Teheran. Torkeh Dārī ligger meter över havet och antalet invånare är .
Terrängen runt Torkeh Dārī är platt åt nordost, men åt sydväst är den kuperad. Den högsta punkten i närheten är Tak Ālatī, meter över havet, km söder om Torkeh Dārī. Runt Torkeh Dārī är det ganska glesbefolkat, med invånare per kvadratkilometer. Närmaste större samhälle är Khvājeh, km nordväst om Torkeh Dārī. Trakten runt Torkeh Dārī består i huvudsak av gräsmarker.
Inlandsklimat råder i trakten. Årsmedeltemperaturen i trakten är °C. Den varmaste månaden är juli, då medeltemperaturen är °C, och den kallaste är januari, med °C. Genomsnittlig årsnederbörd är millimeter. Den regnigaste månaden är maj, med i genomsnitt mm nederbörd, och den torraste är augusti, med mm nederbörd.
Kommentarer
Källor
Orter i Östazarbaijan | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,552 |
Home News Header Sri Lankan students win big at Intel ISEF 2016
Sri Lankan students win big at Intel ISEF 2016
Team Sri Lanka won five awards at this year's Intel International Science Engineering Fair. Better known as ISEF, this is the world's largest high school science and research competition, which concluded in Phoenix, Arizona. The competition is funded jointly by Intel and the Intel Foundation with additional awards and support from dozens of other corporate, academic, governmental and science-focused organizations. This year, approximately US$ 4 million was awarded and few prestigious ones went to three talented students from Sri Lanka.
Abishekh Gomes from Belvoir College International, Colombo won a grand award and two special awards while Chamindu Jayasanka from Hanwella Rajasinghe Central College and P.M. Lochana Piyumantha from Madampe Senanayake Central College clinched two special awards for their innovations.
Abishekh's wearable device which converts American Sign Language (ASL) into English received the third place grand award of US$ 1000 in the Embedded Systems category at the Intel ISEF Grand Awards Ceremony. He also won a special award of US$ 2000 from Synaptics USA, a world-leading human interface solution developer which bases on touch, display, and biometrics technologies, and US$ 500 from the Patent and Trademark Office Society of the United States.
Chamindu's 'Modified and Adjustable Crutches' won a special award of US$ 1000 from the Patent and Trademark Office Society of the United States for his outstanding originality and clarity in science and technology.
Lochana's nano-technology-based solution for resistant Endometrial cancer cells also won another special award of US$ 1000 from Qatar Foundation for the excellence he shown in human life sciences.
Speaking to the media, Indika de Zoysa, Country Business Manager, Intel EM Ltd., Sri Lanka Liaison Office said, "Intel believes young people are key to future innovation and that in order to confront the global challenges of tomorrow, we need students from all backgrounds to get involved in science, technology, engineering, and math. I hope this year's achievements of Abishekh, Lochana and Chamindu will inspire other young people in Sri Lanka, to pursue their interest in these fields and apply their curiosity, creativity, and ingenuity to the common good. I wish them the very best of luck in their future endeavors."
The 2016 Intel International Science and Engineering Fair featured more than 1,700 young scientists selected from 419 affiliate fairs in 77 countries, regions, and territories. In addition to the top winners, approximately 600 finalists received awards and prizes for their innovative research, including 22 "Best of Category" winners, who each received a US$5,000 prize.
We congratulate Abishekh, Chamindu, and Lochana on their amazing victory. Once again, Sri Lanka shines on the global stage. Hopefully, we'll see (and write) other great achievements by these talented students and other great Sri Lankan innovators in the near future.
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ReadMe is your guide to the Sri Lankan IT industry. We give you unparalleled insights, accurate, local tech news, thoughtful features and sometimes scathing opinions on where things are headed. Stay tuned for the best of Sri Lanka! | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,286 |
\section{Introduction}
A peculiar feature of general relativity is that the theory
admits a nontrivial topology of a spacetime.
A solution of the Einstein equation that connects distant points
of space--time was introduced by \citet{ein35}.
This solution called the {\it Einstein--Rosen bridge}
was the first solution to later be referred to as a wormhole.
Initially, this type of solution was just a trivial or
teaching example of mathematical physics.
However, \citet{morr88} proved that some wormholes are "traversable";
i.e., space and time travel can be achieved by passing
through the wormholes.
They also showed that the existence of a wormhole requires
exotic matter that violates the null energy condition.
The existence of wormholes, though they are very exotic,
has not been ruled out in theory.
Inspired by the Morris--Thorne paper, a number of theoretical works
(see \citet{vis95, lob09} and references therein) on wormholes
have been done.
The curious nature of wormholes such as time travel,
negative energy conditions, space--time foams, and
growth of a wormhole in an accelerating universe have been studied.
In spite of enthusiastic theoretical investigations,
studies searching for observational evidence of the existence
of wormholes are scarce.
Only a few attempts have been made to show the existence
or nonexistence of wormholes in our universe
\citep{cram95, tor98, saf01, bog08}.
A possible observational method that has been proposed to detect
or exclude the existence of wormholes is the application
of optical gravitational lensing, since the light ray propagation
is sensitive to a local spacetime geometry.
The gravitational lensing of wormholes was pioneered
by \citet{cram95}, who inferred that some wormholes show
"negative mass" lensing.
They showed that the light curve of the negative-mass lensing event
of a distant star has singular double peaks.
Several authors subsequently conducted theoretical studies
on detectability \citep{saf01, bog08}.
Another gravitational lensing method employing gamma rays was
proposed by \citet{tor98}, who postulated that the singular
negative-mass lensing of distant active galactic nuclei causes
a sharp spike of gamma rays and may be observed
as double-peaked gamma-ray bursts.
They analyzed BATSE data to put an upper limit to
the negative mass density $O(10^{-36}) \:\mbox{g cm}^{-3}$
in the form of wormholelike objects.
There have been several recent works
\citep{sh04, per04, nan06, rah07, dey08, abe10, asada11}
on the gravitational lensing of wormholes as structures of
space--time.
Such studies are expected to unveil lensing properties
directly from the space--time structure.
One study \citet{dey08} calculated the deflection angle of light due
to the Ellis wormhole, whose asymptotic mass at infinity is zero.
The massless wormhole is particularly interesting
because it is expected to have unique gravitational lensing effects.
This type of wormhole was first introduced by \cite{ell73}
as a massless scalar field.
Later, \citet{morr88} studied this wormhole and proved it
to be traversable.
The dynamical feature was studied by \cite{shi02},
who showed that Gaussian perturbation causes either explode
to an inflationary universe or collapse to a black hole.
\cite{das05} showed that the tachyon condensate can be a source
for the Ellis geometry.
\citet{abe10} provided a method to calculate light curves
of the gravitational microlensing of the Ellis wormhole
in the weak-field limit.
This result has been discussed as one example of
corrections to the standard formula of the deflection angle
by \cite{asada11}.
The main results of this paper are:
(1) We derive analytic expressions for
calculations of the wormhole lensing.
(2) We show the astrometric image centroid trajectory
by the Ellis wormhole lens.
Studies of centroid displacements of lensed images
have been limited within the Schwarzschild lens
\citep{Walker,MY,HOF,SDR,JHP,asada02,HL}.
In Section 2, we discuss gravitational lensing by the Ellis wormhole
in the weak-field limit.
We use a suitable coordinate transformation,
such that the lens equation to determine image positions
can take a much simpler form than the previous one derived by
\citet{abe10}.
By using this method, calculations of the gravitational microlensing
by the Ellis wormhole are shortened.
In addition, we prove that two images always appear
for the weak-field lens by the Ellis wormhole.
In section 3, we discuss astrometric image centroid displacements due to
gravitational microlensing by the Ellis wormhole.
The results are summarized in Section 4.
\section{Gravitational lensing by the Ellis wormhole:
Weak-field approximation}
\subsection{Ellis wormhole lens}
Magnification of the apparent brightness of a distant star by the
gravitational lensing effect of another star was predicted by
\citet{ein36}. This kind of lensing effect is called "microlensing"
because the images produced by the gravitational lensing are very
close to each other and are difficult for the observer to resolve.
The observable effects are not only
the changing apparent brightness of the source star
but also the shift in the image centroid position.
The brightness changing effect
was discovered in 1993
\citep{uda93, alc93, aub93} and has been used to detect
astronomical objects that do not emit observable signals
(such as visible light, radio waves, and X rays) or
are too faint to observe.
Microlensing has successfully been applied to detect
extrasolar planets \citep{bon04, bea06} and brown dwarfs
\citep{nov09, gou09}.
Microlensing is also used to search for unseen black holes
\citep{alc01, ben02, poi05} and massive compact halo objects
\citep{alc00, tis07, wyr09}, a candidate for dark matter.
The gravity of a star is well expressed by the Schwarzschild
metric. The gravitational microlensing of the Schwarzschild metric
\citep{ref64, lieb64, Pacz86} has been studied in the weak-field
limit. In this section, we simply follow the method used for
Schwarzschild lensing. Figure \ref{fig1} shows the relation between
the source star, the lens (wormhole), and the observer. The Ellis
wormhole is known to be a massless wormhole, which means that the
asymptotic mass at infinity is zero.
The Ellis wormhole is expressed by the line element
\begin{equation}
ds^2 = dt^2 - dr^2 - (r^2 + a^2) (d\theta^2 + sin^2(\theta) d\phi^2),
\end{equation}
where $a$ is the throat radius of the wormhole.
However, this wormhole deflects light by gravitational
lensing \citep{cle84,che84,nan06,dey08} because of
its curved space--time structure.
The deflection angle $\alpha(r)$ of the Ellis wormhole
was derived by \cite{dey08} to be
\begin{equation}
\alpha(r) = \pi \left\{\sqrt \frac{2 (r^2 + a^2)}{2 r^2 + a^2} -1 \right\},
\end{equation}
where $r$ is the closest approach of the light.
In the weak-field limit ($r \rightarrow \infty$), the deflection angle becomes
\begin{equation}
\alpha(r) \rightarrow \frac{\pi}{4}\frac{a^2}{r^2}
+ o\left(\frac{a}{r}\right)^4. \label{eqn:defl}
\end{equation}
Note that
\cite{dey08} treatment is true of the weak-field region
but it may be corrected in the strong-field one,
because they assume that $r=0$ were singular and it
could be excluded as is the case for the Schwarzschild metric.
For the Ellis wormhole, however, $r=0$ is not a singularity
but a regular sphere and hence
$r=0$ cannot be excluded in the strong-field lensing.
The angle between the lens (wormhole) and the source $\beta$ can then be written as
\begin{equation}
\beta = \frac{1}{D_L} b - \frac{D_{LS}}{D_S}\alpha(r),
\end{equation}
where $D_L$, $D_S$, $D_{LS}$, and $b$ are the distances from the observer to the lens, from the observer to the source, and from the lens to the source, and the impact parameter of the light, respectively. In the asymptotic limit, Schwarzschild lensing and massive Janis--Newman--Winnicour (JNW) wormhole lensing \citep{dey08} have the same leading term of $o\left(1/r\right)$. Therefore, the lensing property of the JNW wormhole is approximately the same as that of Schwarzschild lensing and is difficult to distinguish. As shown in Equation (\ref{eqn:defl}), the deflection angle of the Ellis wormhole does not have the term of $o\left(1/r\right)$ and starts from $o\left(1/r^2\right)$. This is due to the massless nature of the Ellis wormhole and indicates the possibility of observational discrimination from the ordinary gravitational lensing effect. In the weak-field limit, $b$ is approximately equal to the closest approach $r$. For the Ellis wormhole, $b = \sqrt{r^2 + a^2} \rightarrow r (r \rightarrow \infty)$. We thus obtain
\begin{equation}
\beta = \frac{r}{D_L} - \frac{\pi}{4} \frac{D_{LS}}{D_S}\frac{a^2}{r^2}\hspace{1cm}(r > 0). \label{eqn:proj}
\end{equation}
The light passing through the other side of the lens may also form images. However, Equation (\ref{eqn:proj}) represents deflection in the wrong direction at $r < 0$. Thus, we must change the sign of the deflection angle:
\begin{equation}
\beta = \frac{r}{D_L} + \frac{\pi}{4} \frac{D_{LS}}{D_S}\frac{a^2}{r^2}\hspace{1cm}(r < 0).
\label{eqn:proj2}
\end{equation}
It would be useful to note that a single equation is suitable both for $r > 0$ and $r < 0$ images in the Schwarzschild lensing. However, such treatment is applicable only when the deflection angle is an odd function of $r$.
If the source and lens are completely aligned along the line of sight, the image is expected to be circular (an Einstein ring). The Einstein radius $R_E$, which is defined as the radius of the circular image on the lens plane, is obtained from Equation (\ref{eqn:proj}) with $\beta = 0$ as
\begin{equation}
R_E = \sqrt[3]{\frac{\pi}{4}\frac{D_L D_{LS}}{D_S} a^2}. \label{eqn:re}
\end{equation}
The image positions can then be calculated from
\begin{equation}
\beta = \theta - \frac{\theta_E^3}{\theta^2} \hspace{1cm} (\theta > 0) \label{eqn:imgeq}
\end{equation}
and
\begin{equation}
\beta = \theta + \frac{\theta_E^3}{\theta^2} \hspace{1cm} (\theta < 0), \label{eqn:imgeq2}
\end{equation}
where $\theta = b / D_L \thickapprox r / D_L$ is the angle between the image and lens, and $\theta_E = R_E / D_L$ is the angular Einstein radius. Using reduced parameters $\hat{\beta} = \beta / \theta_E$ and $\hat{\theta} = \theta / \theta_E$, Equations (\ref{eqn:imgeq}) and (\ref{eqn:imgeq2}) become simple cubic formulas:
\begin{equation}
\hat{\theta}^3 - \hat{\beta} \hat{\theta}^2 -1 = 0 \hspace{1cm} (\hat{\theta} > 0) \label{eqn:poly}
\end{equation}
and
\begin{equation}
\hat{\theta}^3 - \hat{\beta} \hat{\theta}^2 +1 = 0 \hspace{1cm} (\hat{\theta} < 0). \label{eqn:poly2}
\end{equation}
Following Abe (2010), first, we briefly summarize how to obtain
the roots of the above equations.
As the discriminant of Equation (\ref{eqn:poly}) is $-4\hat{\beta}^3 - 27 < 0$, Equation (\ref{eqn:poly}) has two conjugate complex solutions and a real solution:
\begin{equation}
\hat{\theta} = \frac{\hat{\beta}}{3} + U_{1+} + U_{1-} ,
\label{theta+}
\end{equation}
with,
\begin{equation}
U_{1\pm} = \sqrt[3]{\frac{\hat{\beta}^3}{27} + \frac{1}{2} \pm \sqrt{\frac{1}{4}\left(1 + \frac{2 \hat{\beta}^3}{27}\right)^2 - \frac{\hat{\beta}^6}{27^2}}}.
\end{equation}
The real positive solution corresponds to the physical image.
The discriminant of Equation (\ref{eqn:poly2}) is $4\hat{\beta}^3 - 27$. Thus it has a real solution if $\hat{\beta} < \sqrt[3]{27/4}$:
\begin{equation}
\hat{\theta} = \frac{\hat{\beta}}{3} + U_{2+} + U_{2-},
\label{theta-}
\end{equation}
where,
\begin{equation}
U_{2\pm} = \omega \sqrt[3]{\frac{\hat{\beta}^3}{27} - \frac{1}{2} \pm \sqrt{\frac{1}{4}\left(1 - \frac{2 \hat{\beta}^3}{27}\right)^2 - \frac{\hat{\beta}^6}{27^2}}} ,
\end{equation}
with $\omega \equiv e^{(2\pi/3)i}$.
This solution corresponds to a physical image inside the Einstein ring. For $\hat{\beta} > \sqrt[3]{27/4}$, Equation (\ref{eqn:poly2}) has three real solutions. However, two of them are not physical because they do not satisfy $\hat{\theta} < 0$. Only the solution
\begin{equation}
\hat{\theta} = \frac{\hat{\beta}}{3} + \omega U_{2+} + U_{2-}
\end{equation}
corresponds to a physical image inside the Einstein ring.
In both cases of $\hat{\theta} > 0$ and $\hat{\theta} < 0$,
careful treatments of $\hat{\beta}$ inside the square roots
and the cube ones are required in order to
know which they are a real positive, real negative or complex.
\subsection{Simplified expressions of lensed image positions}
Next, let us consider an appropriate coordinate transformation as
\begin{equation}
u \equiv \frac{1}{\hat{\theta}} ,
\label{u}
\end{equation}
so that Eqs. (\ref{eqn:poly}) and (\ref{eqn:poly2}) can be
rewritten respectively as
\begin{equation}
u^3 + \hat{\beta} u -1 = 0 \hspace{1cm} (u > 0)
\label{eqn:cubic1}
\end{equation}
and
\begin{equation}
u^3 - \hat{\beta} u +1 = 0 \hspace{1cm} (u < 0).
\label{eqn:cubic2}
\end{equation}
Note that these equations take exactly the standard form
called {\it depressed cubic} for using Cardano's method
to find analytic roots of a cubic equation.
In next subsetion, it is proven that
there exists the only one true root for each equation.
Hence, we immediately get the unique real root for $u > 0$ as
\begin{eqnarray}
u_{1} &=& \frac{1}{\hat{\theta}_{1}}
\nonumber\\
&=& \sqrt[3]{\frac12 + \sqrt{\frac14 + \frac{\hat{\beta}^3}{27}}}
- \sqrt[3]{-\frac12 + \sqrt{\frac14 + \frac{\hat{\beta}^3}{27}}} ,
\label{u+}
\end{eqnarray}
and the unique real one for $u < 0$ as
\begin{eqnarray}
u_{2} &=& \frac{1}{\hat{\theta}_{2}}
\nonumber\\
&=& - \sqrt[3]{\frac12 + \sqrt{\frac14 - \frac{\hat{\beta}^3}{27}}}
- \sqrt[3]{\frac12 - \sqrt{\frac14 - \frac{\hat{\beta}^3}{27}}} .
\label{u-}
\end{eqnarray}
Clearly it can be shown by direct but lengthy calculations
that Eqs. (\ref{u+}) and (\ref{u-}) agree with
Eqs. (\ref{theta+}) and (\ref{theta-}), respectively.
Note that they are much simpler
than Eqs. (\ref{theta+}) and (\ref{theta-}).
In particular, these improved expressions make it
much easier to see the sign of the argument of
the square root and the cube one compared with
Eqs. (\ref{theta+}) and (\ref{theta-}).
For $\hat{\beta} > \sqrt[3]{27/4}$,
the argument of the square root in Eq. (\ref{u-})
is always negative, so that
Eq. (\ref{u-}) can be rewritten as
\begin{eqnarray}
u_{2}
&=& -2 \sqrt{\frac{\hat{\beta}}{3}}
\cos \left[ \frac13
\arctan\left(2 \sqrt{\frac{\hat{\beta}^3}{27} - \frac14}\right)
\right] .
\label{u+2}
\end{eqnarray}
Here we used a relation for two real numbers $p > 0$ and $q < 0$ as
\begin{eqnarray}
\sqrt[3]{p + \sqrt{q}} + \sqrt[3]{p - \sqrt{q}}
&=&
2 \sqrt[3]{r}
\cos\left[ \frac13 \arctan\left( \frac{\sqrt{-q}}{p} \right)\right] .
\label{relation}
\end{eqnarray}
This relation can be shown as follows.
For $p > 0$ and $q < 0$, we can put
$p + \sqrt{q}
= r \exp(i\phi)$
by introducing a radial distance defined as
$r \equiv \sqrt{p^2 - q}$
and an angle coordinate defined as
$\tan\phi \equiv p^{-1} \sqrt{-q}$.
{}From its complex conjugate we obtain
$p - \sqrt{q} = r \exp(- i\phi)$.
Hence we get
\begin{eqnarray}
\sqrt[3]{p + \sqrt{q}}
&=&
\sqrt[3]{r} \exp(i\phi/3) ,
\\
\sqrt[3]{p - \sqrt{q}}
&=&
\sqrt[3]{r} \exp(-i\phi/3) ,
\end{eqnarray}
both of which are combined to get Eq. (\ref{relation}).
\subsection{Number of images lensed by the Ellis wormhole}
It is not convenient to use Cardano's formulas
to know how many images appear for the Ellis wormhole lens,
since the formulas include a combination of square roots
and cube ones
and therefore straightforward but lengthy calculations
are required to know the sign of the argument
of the roots.
In order to bypass such difficulties,
we use Descartes' rule of signs (e.g., \cite{Waerden}),
which states that the number of positive roots either equals
that of sign changes in coefficients of a polynomial
(ignoring powers which do not appear) or
less than it by a multiple of two.
This theorem tells that Eqs. (\ref{eqn:poly}) and (\ref{eqn:cubic1})
have the only one positive root,
because the sign of the coefficient of each power
(ignoring powers which do not appear)
is $+$, $-$, $-$ for Eq. (\ref{eqn:poly})
and $+$, $+$, $-$ for Eq. (\ref{eqn:cubic1}).
For Eqs. (\ref{eqn:poly2}) and (\ref{eqn:cubic2}),
we make a parity transformation as
$\hat{\theta}^{'} = - \hat{\theta}$ and
$\hat{u}^{'} = - \hat{u}$,
so that we can directly apply the Descartes' theorem.
After the parity transformation is made for
Eqs. (\ref{eqn:poly2}) and (\ref{eqn:cubic2}),
the sign of the coefficient of each power is
$-$, $-$, $+$ for Eq. (\ref{eqn:poly2})
and $-$, $+$, $+$ for Eq. (\ref{eqn:cubic2}).
Therefore, we have the only one negative root
for each of Eqs. (\ref{eqn:poly2}) and (\ref{eqn:cubic2}).
The L.H.S. of Eq. (\ref{eqn:cubic1}) becomes $-1$ and $\beta \geq 0$
for $u=0$ and $1$, respectively.
The continuity of the L.H.S. thus means that
the positive root $u_1$ and $\hat\theta_1$ satisfy
\begin{eqnarray}
&& 0 < u_1 \leq 1 ,
\\
&& 1 \leq \hat\theta_1 < +\infty ,
\end{eqnarray}
respectively.
The L.H.S. of Eq. (\ref{eqn:cubic2}) becomes
$-\infty$ and $\beta \geq 0$
for $u=-\infty$ and $-1$, respectively.
Hence, the continuity of the L.H.S. means that
the negative root $u_2$ and $\hat\theta_2$ satisfy
\begin{eqnarray}
&& u_2 \leq -1 ,
\\
&& -1 \leq \hat\theta_2 < 0 ,
\end{eqnarray}
respectively.
The above inequalities on $\hat\theta_1$ and $\hat\theta_2$
hold also for the Ellis wormhole
similarly to the Schwarzschild lens.
\section{Astrometric image centroid displacements by the Ellis wormhole}
The light curve of Schwarzschild lensing was derived by
\citet{Pacz86}, whereas
the counterpart by the Ellis wormhole was calculated by \citet{abe10}.
The magnification of the brightness for each image
by the Ellis wormhole lens is
\begin{eqnarray}
A_1 &\equiv&
\left|\frac{\hat{\theta}_1}{\hat{\beta}}
\frac{d\hat{\theta}_1}{d\hat{\beta}}\right|
\nonumber\\
&=&
\frac{1}{\left(1 - \frac{1}{\hat{\theta}_1^3}\right)
\left(1 + \frac{2}{\hat{\theta}_1^3}\right)}
\nonumber\\
&=&
\frac{1}{(1 - u_1^3) (1 + 2u_1^3)} ,
\label{A1}
\\
A_2&\equiv&
\left|\frac{\hat{\theta}_2}{\hat{\beta}}
\frac{d\hat{\theta}_2}{d\hat{\beta}}\right|
\nonumber\\
&=&
\frac{1}{\left(1 + \frac{1}{\hat{\theta}_2^3}\right)
\left(\frac{2}{\hat{\theta}_2^3} - 1\right)}
\nonumber\\
&=&
\frac{1}{(1 + u_2^3) (2u_2^3 - 1)} ,
\label{A2}
\end{eqnarray}
where $A_1$ and $A_2$ are magnification of the outer and inner images,
$\hat{\theta}_1$ and $\hat{\theta}_2$ correspond to
outer and inner images, respectively.
Here, we use
$0< u_1 \leq 1$and $u_2 \leq -1$.
Hence, the total magnification of the brightness $A$ is
\begin{eqnarray}
A &\equiv& A_1 + A_2
\nonumber\\
&=&
\frac{1}{(1 - u_1^3) (1 + 2u_1^3)}
+
\frac{1}{(1 + u_2^3) (2u_2^3 - 1)} .
\label{eqn:a}
\end{eqnarray}
The relation between the lens and source trajectory in the sky is
shown in Figures \ref{fig2} and \ref{fig3}.
The time dependence of $\hat{\beta}$ is
\begin{equation}
\hat{\beta}(t) = \sqrt{\hat{\beta}_0^2 + {(t -t_0)^2/t_E}^2}, \label{eqn:hbeta}
\end{equation}
where $\hat{\beta}_0$ is the impact parameter of the source trajectory and $t_0$ is the time of closest approach. $t_E$ is the Einstein radius crossing time given by
\begin{equation}
t_E = R_E / v_T, \label{eqn:te}
\end{equation}
where $v_T$ is the transverse velocity of the lens relative to the
source and observer. The light curves obtained from Equations
(\ref{eqn:a}) and (\ref{eqn:hbeta}) are shown as thick red lines in
Figure \ref{fig4}. The light curves corresponding to Schwarzschild
lensing are shown as thin blue lines for comparison.
\cite{abe10} found that
the magnifications by the Ellis wormhole are generally less than those of
Schwarzschild lensing.
The light curve of the Ellis wormhole for $\hat{\beta_o} < 1.0$ shows
characteristic gutters on both sides of the peak immediately
outside the Einstein ring crossing times ($t = t_0 \pm t_E$).
The depth of the gutters is about 4\% from the baseline.
Amazingly, the star becomes fainter than normal in terms of
apparent brightness in the gutters. This means that
the Ellis wormhole lensing has off-center divergence.
In conventional gravitational lensing theory \citep{sch92},
the convergence of light is expressed by a convolution of
the surface mass density. Thus, we need to introduce negative mass
to describe
a diverging lens (like a concave lens in optics)
by the Ellis wormhole.
However, negative mass is not a physical entity.
Since the lensing by the Ellis wormhole is
converging at the center,
lensing at some other place must be diverging
because the wormhole has zero asymptotic mass
and hence converging and diverging effects are compensated each other
in total.
For $\hat{\beta_o} > 1.0$, the light curve
of the wormhole has a basin at $t_0$ and no peak.
Using these features, discrimination from Schwarzschild lensing
can be achieved. Equations (\ref{eqn:re}) and (\ref{eqn:te}) indicate
that physical parameters ($D_L$, $a$, and $v_T$) are degenerate
in $t_E$ and cannot be derived by fitting the light-curve data.
This situation is the same as that for Schwarzschild lensing.
To obtain or constrain these values, observations of
the finite-source effect \citep{nem94} or parallax \citep{alc95}
are necessary.
Astrometry gives a method for breaking the degeneracy
as discussed later.
In analogy with the center of the mass distribution,
the centroid position of the light distribution of
a gravitationally microlensed source is given by
\begin{eqnarray}
\hat{\theta}_{pc}
&=&
\frac{A_{1} \hat{\theta}_{1} + A_{2} \hat{\theta}_{2}}{A} .
\label{pc}
\end{eqnarray}
In making numerical figures, we employ $x-y$ coordinates
in the way that the center is chosen as the location of the Ellis wormhole,
$x$-axis is taken along the direction of the source motion
and $y$-axis is perpendicular to the source motion.
See Figure \ref{fig5} for
the image centroid trajectories for
$\hat\beta_0 = 0.2, 0.5, 1.0, 1.5$.
For each $\hat\beta_0$,
the maximum difference between the image centroid position
by the Ellis wormhole and that by the Schwarzschild one is
$-0.03, -0.08, -0.16, -0.20$
in the units of the Einstein ring radius,
respectively.
This implies that the astrometric lensing by the Ellis wormhole
is relatively weaker than that by the Schwarzschild one.
In the weak-field region,
the suppression of the anomalous shift of the image centroid position
is because the bending angle by the Ellis wormhole
is proportional to the inverse squared impact parameter,
whereas that by the Schwarzschild lens depends on
the inverse impact parameter.
Figure \ref{fig6} shows the relative displacement
of the image centroid with respect to the source position
that is assumed to be in uniform linear motion.
The maximum vertical displacement is
$0.06, 0.14, 0.18, 0.15$
for $\hat\beta_0 = 0.2, 0.5, 1.0, 1.5$,
respectively.
Here, a key question is whether
the Ellis lensing and the Schwarzschild one are distinguished from
the centroid displacement curve.
The relative displacement trajectory by the Schwarzschild lens
is known to be an ellipse \citep{Walker,JHP}.
It is natural to ask whether the displacement curve
by the Ellis wormhole lens is also an ellipse.
Figure \ref{fig6} shows that
the relative trajectory by the Ellis lens
looks like an ellipse but has a small difference.
The shape is symmetric along the $x$-axis but
slightly asymmetric along the $y$-axis
like a tree leaf,
particularly for $\hat\beta_0 = 0.2$.
Figure \ref{fig6}, however, shows that
such a deviation of the relative trajectory from elliptic orbits
is very small.
Another difference is that the relative displacement
at large $t$, for instance $t=-20$ or $20$,
is dependent strongly on the Ellis lens or the Schwarzschild one.
This is because the asymptotic behavior of the centroid
displacement is different
($\hat\beta^{-2}$ or $\hat\beta^{-1}$).
In other words, the displacement effect by the Ellis lens
goes away faster.
This suggests that a long-term observation including a tail part
of the centroid curve is required to distinguish
the Ellis lenses by astrometric observations alone.
The detectability of the image centroid displacements of the background star
depends on the timescale called
{\it the Einstein radius crossing time} $t_E$
that depends on the transverse velocity $v_T$.
There is no reliable estimate of $v_T$ for wormholes.
Following Abe (2010),
we assume that the velocity of the wormhole is approximately
equal to the rotation velocity of stars ($v_T = 220 km/s$)
if it is bound to the Galaxy.
If the wormhole is not bound to our Galaxy,
the transverse velocity would be much higher.
We assume $v_T = 5000 km/s$ \citep{saf01} for the unbound wormhole.
Table 2 shows the Einstein radius crossing times of
the Ellis wormhole lensings for the Galactic bulge and LMC
in both bound and unbound scenarios.
As the frequencies of current microlensing observations are limited
to once every few hours, an event for which the timescale is
less than one day is difficult to detect.
To find very long timescale events ($t_E \geq 1000 days$),
long-term monitoring of events is necessary.
The realistic period of observation is $\leq 10 years$.
Thus, the realistic size of the throat that we can search
for is limited to $100 km \le a \le 10^7 km$ both
for the Galactic bulge and LMC if wormholes are bound to our Galaxy.
If wormholes are unbound, the detection is limited to
$ 10^5 km \le a \le 10^9 km$.
The detectability depends also on the angular shift
due to the Ellis wormhole lens.
The typical angular scale is $O(\theta_E)$.
See Table 1 for the size of $\theta_E$
corresponding to various values of the throat radius.
Near future astrometry space missions such as Gaia
and JASMINE are expected to have angular sensitivity
of a few micro arcseconds,
for which the detection is limited as
$a \geq 10^2$ km.
This limit is much weaker than that by
the mission life time as
$10^5 \mbox{km} \leq a \leq 10^9 \mbox{km}$
for unbound wormholes.
Note that there is a small difference in the image centroid position
(and its motion with time) between the Scwarzschild lensing and
the Ellis wormhole one.
In practice, therefore, it is unlikely to detect
the wormhole by astrometric observation alone.
It is safe to say that
the astrometric lensing provides a supplementary method
of supporting a photometric detection:
First, the impact parameter of the source trajectory $\hat\beta_0$
is determined from light curve observations.
By using the obtained $\hat\beta_0$,
one can fit the astrometric observations
with wormhole lensing templates.
If astrometric data show a better fit with a wormhole case,
the detection by light curves will be reinforced.
What is more important is that
astrometric observations give an additional information
such as the angular size of the image centroid position shift,
so that the degeneracy among $(D_L, a, v_T)$ can be partially broken.
Before closing this section, let us briefly summarize
the chain of logic for identifying Ellis lenses.
{}From light curves, first,
we distinguish the Ellis lenses from the Schwarzschild ones.
The best fit values of the model parameter combinations
are obtained from them.
Next, the image centroid observations are used
to partially break the parameter degeneracy.
\section{Summary}
We studied the gravitational microlensing
effects of the Ellis wormhole in the weak-field limit.
First, we performed a suitable coordinate transformation,
such that the lens equation
and analytic expressions of the lensed image positions
can become much simpler than the previous ones.
Second, we proved that two images always appear
for the weak-field lens by the Ellis wormhole.
By using these analytic results,
we investigated astrometric image centroid displacements due to
gravitational microlensing by the Ellis wormhole.
The anomalous shift of the image centroid by the Ellis wormhole lens
is smaller than that by the Schwarzschild lens,
provided that the impact parameter and the Einstein ring radius
are the same.
Therefore, the lensed image centroid by the Ellis wormhole
moves slower.
Studies of astrometric image centroid displacements
due to another type of wormholes or nontrivial topology
of spacetimes are left as a future work.
\acknowledgments
We would like to thank Professor Volker Perlick for invaluable
comments on the Ellis wormhole lensing in
the strong-field region.
This work was supported in part (H.A.)
by a Japanese Grant-in-Aid
for Scientific Research from the Ministry of Education,
No. 21540252.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,030 |
\section{Introduction}
ACM's consolidated article template, introduced in 2017, provides a
consistent \LaTeX\ style for use across ACM publications, and
incorporates accessibility and metadata-extraction functionality
necessary for future Digital Library endeavors. Numerous ACM and
SIG-specific \LaTeX\ templates have been examined, and their unique
features incorporated into this single new template.
If you are new to publishing with ACM, this document is a valuable
guide to the process of preparing your work for publication. If you
have published with ACM before, this document provides insight and
instruction into more recent changes to the article template.
The ``\verb|acmart|'' document class can be used to prepare articles
for any ACM publication --- conference or journal, and for any stage
of publication, from review to final ``camera-ready'' copy, to the
author's own version, with {\itshape very} few changes to the source.
\section{Template Overview}
As noted in the introduction, the ``\verb|acmart|'' document class can
be used to prepare many different kinds of documentation --- a
double-blind initial submission of a full-length technical paper, a
two-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''
journal article, a SIGCHI Extended Abstract, and more --- all by
selecting the appropriate {\itshape template style} and {\itshape
template parameters}.
This document will explain the major features of the document
class. For further information, the {\itshape \LaTeX\ User's Guide} is
available from
\url{https://www.acm.org/publications/proceedings-template}.
\subsection{Template Styles}
The primary parameter given to the ``\verb|acmart|'' document class is
the {\itshape template style} which corresponds to the kind of publication
or SIG publishing the work. This parameter is enclosed in square
brackets and is a part of the {\verb|documentclass|} command:
\begin{verbatim}
\documentclass[STYLE]{acmart}
\end{verbatim}
Journals use one of three template styles. All but three ACM journals
use the {\verb|acmsmall|} template style:
\begin{itemize}
\item {\texttt{acmsmall}}: The default journal template style.
\item {\texttt{acmlarge}}: Used by JOCCH and TAP.
\item {\texttt{acmtog}}: Used by TOG.
\end{itemize}
The majority of conference proceedings documentation will use the {\verb|acmconf|} template style.
\begin{itemize}
\item {\texttt{acmconf}}: The default proceedings template style.
\item{\texttt{sigchi}}: Used for SIGCHI conference articles.
\item{\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.
\item{\texttt{sigplan}}: Used for SIGPLAN conference articles.
\end{itemize}
\subsection{Template Parameters}
In addition to specifying the {\itshape template style} to be used in
formatting your work, there are a number of {\itshape template parameters}
which modify some part of the applied template style. A complete list
of these parameters can be found in the {\itshape \LaTeX\ User's Guide.}
Frequently-used parameters, or combinations of parameters, include:
\begin{itemize}
\item {\texttt{anonymous,review}}: Suitable for a ``double-blind''
conference submission. Anonymizes the work and includes line
numbers. Use with the \texttt{\acmSubmissionID} command to print the
submission's unique ID on each page of the work.
\item{\texttt{authorversion}}: Produces a version of the work suitable
for posting by the author.
\item{\texttt{screen}}: Produces colored hyperlinks.
\end{itemize}
This document uses the following string as the first command in the
source file:
\begin{verbatim}
\documentclass[sigconf]{acmart}
\end{verbatim}
\section{Modifications}
Modifying the template --- including but not limited to: adjusting
margins, typeface sizes, line spacing, paragraph and list definitions,
and the use of the \verb|\vspace| command to manually adjust the
vertical spacing between elements of your work --- is not allowed.
{\bfseries Your document will be returned to you for revision if
modifications are discovered.}
\section{Typefaces}
The ``\verb|acmart|'' document class requires the use of the
``Libertine'' typeface family. Your \TeX\ installation should include
this set of packages. Please do not substitute other typefaces. The
``\verb|lmodern|'' and ``\verb|ltimes|'' packages should not be used,
as they will override the built-in typeface families.
\section{Title Information}
The title of your work should use capital letters appropriately -
\url{https://capitalizemytitle.com/} has useful rules for
capitalization. Use the {\verb|title|} command to define the title of
your work. If your work has a subtitle, define it with the
{\verb|subtitle|} command. Do not insert line breaks in your title.
If your title is lengthy, you must define a short version to be used
in the page headers, to prevent overlapping text. The \verb|title|
command has a ``short title'' parameter:
\begin{verbatim}
\title[short title]{full title}
\end{verbatim}
\section{Authors and Affiliations}
Each author must be defined separately for accurate metadata
identification. As an exception, multiple authors may share one
affiliation. Authors' names should not be abbreviated; use full first
names wherever possible. Include authors' e-mail addresses whenever
possible.
Grouping authors' names or e-mail addresses, or providing an ``e-mail
alias,'' as shown below, is not acceptable:
\begin{verbatim}
\author{Brooke Aster, David Mehldau}
\email{dave,judy,steve@university.edu}
\email{firstname.lastname@phillips.org}
\end{verbatim}
The \verb|authornote| and \verb|authornotemark| commands allow a note
to apply to multiple authors --- for example, if the first two authors
of an article contributed equally to the work.
If your author list is lengthy, you must define a shortened version of
the list of authors to be used in the page headers, to prevent
overlapping text. The following command should be placed just after
the last \verb|\author{}| definition:
\begin{verbatim}
\renewcommand{\shortauthors}{McCartney, et al.}
\end{verbatim}
Omitting this command will force the use of a concatenated list of all
of the authors' names, which may result in overlapping text in the
page headers.
The article template's documentation, available at
\url{https://www.acm.org/publications/proceedings-template}, has a
complete explanation of these commands and tips for their effective
use.
Note that authors' addresses are mandatory for journal articles.
\section{Rights Information}
Authors of any work published by ACM will need to complete a rights
form. Depending on the kind of work, and the rights management choice
made by the author, this may be copyright transfer, permission,
license, or an OA (open access) agreement.
Regardless of the rights management choice, the author will receive a
copy of the completed rights form once it has been submitted. This
form contains \LaTeX\ commands that must be copied into the source
document. When the document source is compiled, these commands and
their parameters add formatted text to several areas of the final
document:
\begin{itemize}
\item the ``ACM Reference Format'' text on the first page.
\item the ``rights management'' text on the first page.
\item the conference information in the page header(s).
\end{itemize}
Rights information is unique to the work; if you are preparing several
works for an event, make sure to use the correct set of commands with
each of the works.
The ACM Reference Format text is required for all articles over one
page in length, and is optional for one-page articles (abstracts).
\section{CCS Concepts and User-Defined Keywords}
Two elements of the ``acmart'' document class provide powerful
taxonomic tools for you to help readers find your work in an online
search.
The ACM Computing Classification System ---
\url{https://www.acm.org/publications/class-2012} --- is a set of
classifiers and concepts that describe the computing
discipline. Authors can select entries from this classification
system, via \url{https://dl.acm.org/ccs/ccs.cfm}, and generate the
commands to be included in the \LaTeX\ source.
User-defined keywords are a comma-separated list of words and phrases
of the authors' choosing, providing a more flexible way of describing
the research being presented.
CCS concepts and user-defined keywords are required for for all
articles over two pages in length, and are optional for one- and
two-page articles (or abstracts).
\section{Sectioning Commands}
Your work should use standard \LaTeX\ sectioning commands:
\verb|section|, \verb|subsection|, \verb|subsubsection|, and
\verb|paragraph|. They should be numbered; do not remove the numbering
from the commands.
Simulating a sectioning command by setting the first word or words of
a paragraph in boldface or italicized text is {\bfseries not allowed.}
\section{Tables}
The ``\verb|acmart|'' document class includes the ``\verb|booktabs|''
package --- \url{https://ctan.org/pkg/booktabs} --- for preparing
high-quality tables.
Table captions are placed {\itshape above} the table.
Because tables cannot be split across pages, the best placement for
them is typically the top of the page nearest their initial cite. To
ensure this proper ``floating'' placement of tables, use the
environment \textbf{table} to enclose the table's contents and the
table caption. The contents of the table itself must go in the
\textbf{tabular} environment, to be aligned properly in rows and
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detailed instructions on \textbf{tabular} material are found in the
\textit{\LaTeX\ User's Guide}.
Immediately following this sentence is the point at which
Table~\ref{tab:freq} is included in the input file; compare the
placement of the table here with the table in the printed output of
this document.
\begin{table}
\caption{Frequency of Special Characters}
\label{tab:freq}
\begin{tabular}{ccl}
\toprule
Non-English or Math&Frequency&Comments\\
\midrule
\O & 1 in 1,000& For Swedish names\\
$\pi$ & 1 in 5& Common in math\\
\$ & 4 in 5 & Used in business\\
$\Psi^2_1$ & 1 in 40,000& Unexplained usage\\
\bottomrule
\end{tabular}
\end{table}
To set a wider table, which takes up the whole width of the page's
live area, use the environment \textbf{table*} to enclose the table's
contents and the table caption. As with a single-column table, this
wide table will ``float'' to a location deemed more
desirable. Immediately following this sentence is the point at which
Table~\ref{tab:commands} is included in the input file; again, it is
instructive to compare the placement of the table here with the table
in the printed output of this document.
\begin{table*}
\caption{Some Typical Commands}
\label{tab:commands}
\begin{tabular}{ccl}
\toprule
Command &A Number & Comments\\
\midrule
\texttt{{\char'134}author} & 100& Author \\
\texttt{{\char'134}table}& 300 & For tables\\
\texttt{{\char'134}table*}& 400& For wider tables\\
\bottomrule
\end{tabular}
\end{table*}
Always use midrule to separate table header rows from data rows, and
use it only for this purpose. This enables assistive technologies to
recognise table headers and support their users in navigating tables
more easily.
\section{Math Equations}
You may want to display math equations in three distinct styles:
inline, numbered or non-numbered display. Each of the three are
discussed in the next sections.
\subsection{Inline (In-text) Equations}
A formula that appears in the running text is called an inline or
in-text formula. It is produced by the \textbf{math} environment,
which can be invoked with the usual
\texttt{{\char'134}begin\,\ldots{\char'134}end} construction or with
the short form \texttt{\$\,\ldots\$}. You can use any of the symbols
and structures, from $\alpha$ to $\omega$, available in
\LaTeX~\cite{Lamport:LaTeX}; this section will simply show a few
examples of in-text equations in context. Notice how this equation:
\begin{math}
\lim_{n\rightarrow \infty}x=0
\end{math},
set here in in-line math style, looks slightly different when
set in display style. (See next section).
\subsection{Display Equations}
A numbered display equation---one set off by vertical space from the
text and centered horizontally---is produced by the \textbf{equation}
environment. An unnumbered display equation is produced by the
\textbf{displaymath} environment.
Again, in either environment, you can use any of the symbols and
structures available in \LaTeX\@; this section will just give a couple
of examples of display equations in context. First, consider the
equation, shown as an inline equation above:
\begin{equation}
\lim_{n\rightarrow \infty}x=0
\end{equation}
Notice how it is formatted somewhat differently in
the \textbf{displaymath}
environment. Now, we'll enter an unnumbered equation:
\begin{displaymath}
\sum_{i=0}^{\infty} x + 1
\end{displaymath}
and follow it with another numbered equation:
\begin{equation}
\sum_{i=0}^{\infty}x_i=\int_{0}^{\pi+2} f
\end{equation}
just to demonstrate \LaTeX's able handling of numbering.
\section{Figures}
The ``\verb|figure|'' environment should be used for figures. One or
more images can be placed within a figure. If your figure contains
third-party material, you must clearly identify it as such, as shown
in the example below.
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{sample-franklin}
\caption{1907 Franklin Model D roadster. Photograph by Harris \&
Ewing, Inc. [Public domain], via Wikimedia
Commons. (\url{https://goo.gl/VLCRBB}).}
\Description{A woman and a girl in white dresses sit in an open car.}
\end{figure}
Your figures should contain a caption which describes the figure to
the reader.
Figure captions are placed {\itshape below} the figure.
Every figure should also have a figure description unless it is purely
decorative. These descriptions convey what's in the image to someone
who cannot see it. They are also used by search engine crawlers for
indexing images, and when images cannot be loaded.
A figure description must be unformatted plain text less than 2000
characters long (including spaces). {\bfseries Figure descriptions
should not repeat the figure caption – their purpose is to capture
important information that is not already provided in the caption or
the main text of the paper.} For figures that convey important and
complex new information, a short text description may not be
adequate. More complex alternative descriptions can be placed in an
appendix and referenced in a short figure description. For example,
provide a data table capturing the information in a bar chart, or a
structured list representing a graph. For additional information
regarding how best to write figure descriptions and why doing this is
so important, please see
\url{https://www.acm.org/publications/taps/describing-figures/}.
\subsection{The ``Teaser Figure''}
A ``teaser figure'' is an image, or set of images in one figure, that
are placed after all author and affiliation information, and before
the body of the article, spanning the page. If you wish to have such a
figure in your article, place the command immediately before the
\verb|\maketitle| command:
\begin{verbatim}
\begin{teaserfigure}
\includegraphics[width=\textwidth]{sampleteaser}
\caption{figure caption}
\Description{figure description}
\end{teaserfigure}
\end{verbatim}
\section{Citations and Bibliographies}
The use of \BibTeX\ for the preparation and formatting of one's
references is strongly recommended. Authors' names should be complete
--- use full first names (``Donald E. Knuth'') not initials
(``D. E. Knuth'') --- and the salient identifying features of a
reference should be included: title, year, volume, number, pages,
article DOI, etc.
The bibliography is included in your source document with these two
commands, placed just before the \verb|\end{document}| command:
\begin{verbatim}
\bibliographystyle{ACM-Reference-Format}
\section{Introduction}
Coping with global climate change-related challenges, including air pollution, is one of the most pressing concerns that civilizations confront today~\cite{FERRERO2016450,hampshire2018electric,Zhang_2020}. The transportation industry has the most significant cause of air pollution as the major contributor to greenhouse gas emissions~\cite{hickman2007looking,kufeoglu2020emissions}. Transportation accounted for 27\% of the UK's total greenhouse emissions in 2019 as the largest emitting sector. A vast majority of the emission, 91\%, comes from automobiles~\cite{Millen2021}. Hence, the transportation industry and academic communities are increasingly interested in developing alternative energy vehicles to reduce emissions. Automobile manufacturers are introducing a new generation of electric vehicles (EVs) which often employs connected and automated driving functions~\cite{gersdorf2020mckinsey}.
EVs are considered one of the most promising means to reduce emissions and dependency on fossil fuels. Along with environmental benefits, EVs provide superior energy efficiency to conventional vehicles~\cite{hensley2009electrifying}. As the cost of batteries continues to decrease, the large scale adoption of EVs is becoming more viable~\cite{gao2014road}. Despite the advantages and competitive cost, many customers remain concerned about running out of battery power before reaching their destination or waiting for their EVs to charge. The primary obstacles to EV adoption are the availability of chargers and the range that can be travelled on a single charge, often referred to as \emph{range anxiety} in the literature~\cite{lombardi2018electric}.
In addition to the rising need and relevance of EVs, with the recent technological development and advancements in machine learning and various other kinds of techniques in data analysis, the value of personal data has increased manifold and, almost parallelly, the threat to attack one's private information and the risks of violation of the users' privacy are becoming more and more significant and concerning~\cite{nist,lemetayer:hal-01420983}. As a consequence, there has been widespread awareness about the importance of protecting the privacy of personal data while exploiting their utility and analyzing them. One of the most successful approaches to address this issue of privacy-preserving protocols to analyse and utilize sensitive data is along the lines of \emph{differential privacy} (DP)~\cite{DworkDP1,DworkDP2}, which mathematically guarantees that the query output does not change significantly regardless of whether a specific personal record is in the dataset or not.
However, the classical central version of DP requires a trusted curator who is responsible for adding noise to the data before publishing or performing analytics on it. A major drawback of such a central model is that it is vulnerable to security breaches because the entire original data is stored in a central server. Moreover, there is the risk of having an adversarial curator. To circumvent the need for such a central dependency, a local model of DP, a.k.a. \emph{local differential privacy} (LDP)~\cite{DuchiLDP}, has been in the spotlight of late, where the users apply the LDP mechanism directly to their data and communicate the locally perturbed data to the server.
LDP is particularly suitable for situations where the users need to communicate their personal data in exchange for some service. One such scenario is the use of location-based services (LBS), where a user typically reports her location in exchange for information like the shortest path to a destination, points of interest in the surroundings, traffic information, friends nearby, etc. One of the recently popularized standards in location privacy is \emph{geo-indistinguishability} (GeoI)~\cite{AndresKostasCatuscia_GeoInd}, which optimizes the quality of service (QoS) for the users while preserving a generalized notion of LDP on their location data. The obfuscation mechanism of GeoI depends on the distance between the original location of a user and a potential noisy location that they report~\cite{Bordenabe:14:CCS,Fernandes:21:LICS}.
GeoI can be implemented directly on the user's device (tablet, smartphone, etc.). The fact that the users can control their explicit privacy-protection level for various LBS makes it very appealing. However, a drawback of injecting noise locally to the datum is that it deteriorates the quality of service (QoS) due to the lack of accuracy of the data.
On the other hand, future vehicles are getting more sophisticated in sensory, onboard computation and communication capacities. Furthermore, the emergence of Mobile Edge Computing (MEC) also changes the Intelligent Transportation Systems (ITS) by providing a platform to assist computationally heavy tasks by offloading the computation to the Edge cloud~\cite{gillam2018exploring}. This architecture often employs three tiers, with the vehicle on the first tier, MEC on the second tier, and standard cloud services on the third tier~\cite{maple2019connected}. Figure~\ref{fig:system_arch} shows the system architecture for the location privacy framework proposed in this paper.
ITS provides a platform containing distributed and resource-constrained systems to support real-time vehicular functions where these functions' efficacy relies on the data shared across entities. However, the risk of privacy disclosure and tracking increases due to data sharing~\cite{hahn2019security}. Privacy-preserving schemes are developed using established techniques such as group signature, anonymity, and pseudonymity~\cite{lin2013achieving,kumar2015intelligent}. However, it is possible to identify privatised data with adequate background information. Hence, DP approaches have emerged as the gold standard of data privacy because they provide a formal privacy guarantee independent of a threat actor's background knowledge and computing capability~\cite{zhao2020survey}.
GeoI is the state-of-the-art method for location privacy-preserving with LDP. It can preserve one's location privacy among a set of locations with similar probability distributions without requiring a trusted third-party. It provides rigorous privacy for location-based query processing and location data collection by modelling the location domain based on the Euclidean plane. However, vehicles are located on the road network under normal circumstances. For vehicular location queries, GeoI mechanism may result in publishing unrealistic privatised locations such as houses, parks or lakes. Thus, there is a need for an adapted model of GeoI for vehicular application. This paper proposes a novel privacy model called Approximate Geo-Indistinguishability (AGeoI), based on the notion of GeoI by using a discrete road network graph. Our key contributions in this paper are outlined as follows.
\begin{enumerate}
\item We presented the notion of AGeoI, a formal standard of location-privacy in a bounded co-domain, by generalizing the classical paradigm of GeoI, and adapting it to a graphical environment. We illustrate its applicability by proving that it satisfies the compositionality theorem. Moreover, we show that the truncated Laplace mechanism satisfies AGeoI by deriving the appropriate $\epsilon$ and $\delta$ as privacy parameters.
\item We proposed a two-way privacy-preserving navigation method for the EVs dynamically querying for charging stations on road networks -- we protect against threats on individual locations of their queries by ensuring AGeoI, and we provide protection against adversaries tracing the trajectories of the EVs by interpolating their query locations in an online setting.
\item We performed experiments on real vehicular journey data from San Francisco with real locations of charging stations from the area
under two settings of their sparsity in the road network, and show that our method ensures a very high fraction of EVs who enjoy \emph{privacy for free} and illustrate that the cost of utility-loss for the EVs is very low compared to the formal gain in privacy.
\end{enumerate}
The rest of this paper is organised as follows. Section \ref{sec:related_work} reviews some of the related work in this area. Section \ref{sec:preliminaries} introduces some fundamental notions on DP and GeoI. Section \ref{sec:approx_geo_ind} develops the mathematical theory of AGeoI. Section \ref{sec:system_model} elucidates the model of our proposed mechanism by formalizing the problem we are tackling, thoroughly discussing system architecture, and laying out the privacy-threat landscape we are addressing in this work. Section \ref{sec:cost_of_privacy_analysis} draws some insight into the cost of privacy on the EVs induced by our mechanism. Section \ref{sec:experiments} presents the experimental results to illustrate the working of our mechanism, and, finally, Section \ref{sec:conclusion} concludes the work.
\section{Related Work}\label{sec:related_work}
Both corporate and academic communities have recently piqued the interest in advancing EVs and charging infrastructure to improve the transportation system's sustainability. Despite the advancements, the EV sector confronts challenges that delay the adoption process, such as range anxiety, an absence of convenient and available charging infrastructure, and waiting time to charge~\cite{franke2013understanding,kumar2020adoption}.
An offline static map of charging stations is insufficient to resolve these obstacles since EVs may need to reserve a charging station when a trip is planned or query the available stations based on their battery state, and charging stations must be reservable. Thus, live vehicular and charging station data is utilised in querying and reservation/scheduling mechanisms~\cite{tian2016real,zhang2021intelligent,flocea2022electric}. Encryption techniques can be used in such mechanisms to prevent external intrusions, but they cannot preserve users' privacy from malicious servers and third-party providers.
Several data types are considered in these mechanisms, including real-time location, intended route, battery level, and station availability, to ensure the drivers are not detoured from their intended route\cite{tian2016real,wang2019sharedcharging}. Although disclosing such information poses privacy concerns for the driver's location and vehicle tracking, the privacy requirements of such mechanisms are not sufficiently studied in the literature. Existing methods for planning charging points for EV journeys are considered mechanisms for confidentiality and integrity, but the drivers' location privacy is regarded as an issue of trust in the third-party service provider~\cite{plugshare,chargepoint}.
This problem can be addressed by several approaches, based threat model of the system. Location anonymity is achieved through cloaking an area~\cite{chow2009casper,niu2014achieving}. These approaches can be only applied to the Edge of our system model to provide anonymity to a group of EVs, but we consider the Edge as an honest-but-curious threat actor and aim to preserve individual vehicles' privacy locally. Thus, these techniques are not applicable to our considered threat model. Furthermore, anonymity techniques do not provide a formal privacy guarantee~\cite{DworkDP_Compositionality}. Similarly, mix-network approaches cannot be applicable because it is not guaranteed that multiple vehicles will be presented in an Edge's coverage in any timestamp due to vehicles' movement~\cite{guo2018independent}. An applicable approach is to download the charging station's live map on EVs to search for the nearest or on-the-route available charging station; however, the communication overhead of this technique is predicted to be much higher than the vehicles' location-based inquiry since it will require downloading a recent snapshot of the map for each query.
DP methods are increasingly being deployed to preserve location privacy in a variety of domains, including automotive systems. The studies in~\cite{zhou2018achieving,luo2019geo} proposed models by deploying a GeoI-based mechanism on the Edge for location-based services. However, their approach did not consider preserving vehicles' location privacy against the Edge. An approach that compliments the problem we aim to address in this paper was proposed by Qiu et al. in \cite{qiu2020location} where the authors proposed a technique to crowd-source a task in vehicular network while preserving GeoI of the location of the vehicles offering Mobility as a Service (MaaS) in the spatial network to solve a task at a publicly known location in the map (e.g. taxi services). The problem formulation in this work is the inverse of what we aim to achieve in this paper. As a result, the work in \cite{qiu2020location} cannot be extended to address the privacy concerns induced my multiple queries dynamically made along the journey.
In \cite{Cunningham2021Trace}, Cunningham et al. studied the problem of trajectory sharing under DP and proposed a mechanism to tackle it. However, this work assumes the setting of an offline trajectory sharing which breaks down in the practical environment where the trajectories are being shared online as there is no prior information or limitation on the number of queries made by an EV during a journey, and their respective locations. Therefore, the method proposed by the authors in \cite{Cunningham2021Trace} cannot be directly adapted to our dynamic environment closely simulating the real-world scenario for such a use case.
\section{Preliminaries}\label{sec:preliminaries}
The most successful approach to formally address the issue of privacy threats is DP, mathematically guaranteeing that the query output does not change significantly regardless of whether a specific personal record is in a dataset or not. Most research performed in this area probes two main directions. One is the classical central framework~\cite{DworkDP1,DworkDP2}, in which a trusted third party (the curator) collects the users' personal data and obfuscates them with a differentially private mechanism.
\begin{definition}[Differential privacy~\cite{DworkDP1,DworkDP2}]
\label{def:dp}
For a certain query, a randomizing mechanism $\mathcal{R}$ provides \emph{$\epsilon$-DP} if, for all neighbouring\footnote{differing in exactly one place} datasets, $D$ and $D'$, and all $S \subseteq$ Range($\mathcal{R}$), we have
\begin{equation*}
\mathbb{P}[\mathcal{R}(D) \in S] \leq e^{\epsilon}\,\mathbb{P}[\mathcal{R}(D') \in S]
\end{equation*}
\end{definition}
A major drawback of central model is that it is vulnerable to security breaches because the entire original data is stored in a central server. Moreover, there is the risk that the curator may be corrupted. Therefore, a local variant of the central model has been widely popularized of late~\cite{DuchiLDP}, where the users apply a randomizing mechanism locally on their data and send the perturbed data to the collector such that a particular value of a user's data does not have a major probabilistic impact on the outcome of the query.
\begin{definition}[Local differential privacy~\cite{DuchiLDP}]
\label{def:ldp}
Let $\mathcal{X}$ and $\mathcal{Y}$ denote the spaces of original and noisy data, respectively. A randomizing mechanism $\mathcal{R}$ provides \emph{$\epsilon$-LDP} if, for all $x,\,x'\,\in\,\mathcal{X}$, and all $y\,\in\,\mathcal{Y}$, we have
\begin{equation*}
\mathbb{P}[\mathcal{R}(x)=y] \leq e^{\epsilon}\,\mathbb{P}\left[\mathcal{R}(x')=y\right]
\end{equation*}
\end{definition}
Recently, GeoI~\cite{AndresKostasCatuscia_GeoInd}, a variant of the local DP capturing the essence of the distance between locations~\cite{Bordenabe:14:CCS,Fernandes:21:LICS} has been in focus as a standard for privacy protection for location based services, being motivated by the idea of preserving the best possible quality of service despite the local obfuscation operated on the data.
\begin{definition}[Geo-indistinguishability~\cite{AndresKostasCatuscia_GeoInd}]
\label{def:geoind}
Let $\mathcal{X}$ be a space of locations and let $d_{\text{E}}(x,x')$ denote the \emph{Euclidean distance} between $x\in \mathcal{X}$ and $x'\in \mathcal{X}$. A randomizing mechanism $\mathcal{R}$ is \emph{$\epsilon$-geo-indistinguishable} if for all $x_1,\,x_2\in\mathcal{X}$, and every $y\in \mathcal{X}$, we have
\begin{equation*}
\mathbb{P}[\mathcal{R}(x)=y] \leq e^{\epsilon d_{\text{E}}(x_1,x_2)}\,\mathbb{P}\left[\mathcal{R}(x')=y\right]
\end{equation*}
\end{definition}
\begin{table}
\caption{List of Notations}\label{table:notations}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{cc}
\hline
Notation & Description \\ \hline
$\mathcal{X}$ & Domain of original locations \\
$d_{\mathcal{X}}$ & Distance on $\mathcal{X}$\\
$\mathcal{Y}$ & Domain of obfuscated locations \\
$d_{\mathcal{Y}}$ & Distance on $\mathcal{Y}$\\
$\mathbb{P}_{\mathcal{K}}\left[y\vert x\right]$ & Prob. that mechanism $\mathcal{K}$, applied to value $x$, reports $y$\\
$I$ & Fixed Edge in the network\\
$R(I)$ & Area of coverage by $I$\\
$m$ & Number of locations reported by each EV\\
$l_u$ & Vector of locations reported by EV $u$\\
$\mathcal{L}(t)$ & Set of location vectors received by $I$ at time $t$\\
$\mathcal{L}'(t)$ & Shuffled set of all individual locations queried at time $t$\\
$\mathcal{R}(t)$ & Set of nearest charging stations for $\mathcal{L}'(t)$\\
$G$ & Road network graph\\
$d_G$ & Travelling distance in graph $G$\\
\hline
\end{tabular}%
}
\end{table}
\begin{table}
\caption{List of Acronyms}\label{table:acronyms}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{cc}
\hline
Acronym & Description \\ \hline
DP & Differential Privacy\\
LDP & Local Differential Privacy\\
GeoI & Geo-Indistinguishability\\
AGeoI & Approximate Geo-Indistinguishability\\
EV & Electric Vehicle\\
IoV & Internet of Vehicles\\
RSU & Roadside Unit\\
MEC & Mobile Edge Computing\\
CoP & Cost of Privacy\\
QoS & Quality of Service\\
CS & Charging Station\\
\hline
\end{tabular}%
}
\end{table}
\section{Approximate geo-indistinguishability}\label{sec:approx_geo_ind}
In the classical framework of GeoI~\cite{AndresKostasCatuscia_GeoInd}, the space of the noisy data is, in theory, unbounded under the planar Laplace mechanism. Under a certain level of GeoI that is achieved, planar Laplace mechanism ensures a non-zero probability of obfuscating an original location to a privatized one which may be quite far, thus inducing a possibility of a substantial deterioration to the QoS of the users. This loss of QoS can be more sensitive in the context of the navigation of EVs, where it is extremely important to prioritize a bounded domain where a user is willing to drive -- this may be a result of time constraint, rising cost of fuel, geographical boundaries (e.g. international borders), etc. -- giving rise to an idea of \emph{area of interest} for each EV. This motivated us to extend the classical GeoI to a more generalized, approximate paradigm, inspired by the approach of the development of approximate DP from its pure counterpart.
Let $\mathcal{X}$ and $\mathcal{Y}$ be the spaces of the real and nosy locations equipped with distance metrics $d_{\mathcal{X}}$ and $d_{\mathcal{Y}}$, respectively. In general $\left(\mathcal{X},d_{\mathcal{X}}\right)$ and $\left(\mathcal{Y},d_{\mathcal{Y}}\right)$ may be different and unrelated. However, for simplicity, here we assume $\mathcal{X}\subseteq \mathcal{Y}$ and, therefore, $d_{\mathcal{X}}=d_{\mathcal{Y}}=d$, and we proceed to define the notion of \emph{approximate geo-indistinguishability}. Note that $d$ may not necessarily need to be symmetric, i.e., there may exist $x1,\,x_2\,\in\,\mathcal{Y}$ such that $d(x_1,x_2)\neq d(x_2,x_1)$.
\begin{definition}[Approximate geo-indistinguishability] A mechanism $\mathcal{K}$ is \emph{approximately geo-indistinguishable (AGeoI)} or $\emph(\epsilon,\delta)$-\emph{geo-indistinguishable} if for any set of values $S\subseteq\mathcal{Y}$ and any pair of values $x,x'\,\in\,\mathcal{X}$, there exists some $\epsilon,\,\delta\in\mathbb{R}_{\geq 0}$ such that $\delta e^{d(x,x')}\in[0,1]$, we have:
\begin{equation}\label{eq:approxGeoI}
\mathbb{P}_{\mathcal{K}}\left[y\,\in\, S\vert x\right]\leq e^{\epsilon\,d(x,x')}\mathbb{P}_{\mathcal{K}}\left[y\,\in\, S\vert x'\right]+\delta\,e^{d(x,x')}
\end{equation}
\end{definition}
One of the biggest advantages of DP and all of its variants that are accepted by the community is the property of compositionality, where the level of privacy can be formally derived with repeated number of queries. Thus, we now enable ourselves to investigate the working of the compositionality theorem with the approximate geo-indistinguishability which we defined, to stay consistent with the literature~\cite{DworkDP_Compositionality}.
\begin{restatable}{theorem}{compositionality}\label{th:compositionality}[Compositionality Theorem for AGeoI]
Let mechanisms $\mathcal{K}_1$ and $\mathcal{K}_2$ be $(\epsilon_1,\,\delta_1)$ and $(\epsilon_2,\,\delta_2)$ geo-indistinguishable, respectively. Then their composition is $(\epsilon_1+\epsilon_2,\,\delta_1+\delta_2)$-geo-indistinguishable. In other words, for every $S_1,\,S_2\subseteq \mathcal{Y}$ and all $x_1,\,x'_1,\,x_2,\,x'_2\,\in\,\mathcal{X}$, we have:
\begin{align}\label{eq:composition}
\mathbb{P}_{\mathcal{K}_1,\mathcal{K}_2}\left[(y_1,y_2)\,\in\, S_1\times S_2\vert (x_1,x_2)\right]\nonumber\\
\leq e^{\epsilon_1\,d(x_1,x'_1)+\epsilon_2\,d(x_2,x'_2)}\nonumber\\
\times\mathbb{P}_{\mathcal{K}_1,\mathcal{K}_2}\left[(y_1,y_2)\,\in\, S_1\times S_2\vert (x'_1,x'_2)\right]\nonumber\\
+\left(\delta_1+\delta_2\right)\,e^{d(x_1,x'_1)+d(x_2,x'_2)}
\end{align}
\end{restatable}
\begin{proof}
Let us simplify the notation and denote: $$P_i=\mathbb{P}_{\mathcal{K}_i}\left[y_i\in S_i\vert x_i\right]$$ $$P'_i=\mathbb{P}_{\mathcal{K}_i}\left[y_i\in S_i\vert x'_i\right]$$
$$\tilde{\delta}_i=\delta_i\,e^{d(x_i,x'_i)}$$
for $i\,\in\,\{1,2\}$.
As mechanisms $\mathcal{K}_1$ and $\mathcal{K}_2$ are applied independently, we have:
\begin{align}
\mathbb{P}_{\mathcal{K}_1,\mathcal{K}_2}\left[(y_1,y_2)\,\in\, S_1\times S_2\vert (x_1,x_2)\right]=P_1.P_2 \label{eq:simp1}\\
\mathbb{P}_{\mathcal{K}_1,\mathcal{K}_2}\left[(y_1,y_2)\,\in\, S_1\times S_2\vert (x'_1,x'_2)\right]=P'_1.P'_2 \label{eq:simp2}
\end{align}
Therefore, we obtain:
\begin{align}
\mathbb{P}_{\mathcal{K}_1,\mathcal{K}_2}\left[(y_1,y_2)\,\in\, S_1\times S_2\vert (x_1,x_2)\right]=P_1.P_2\nonumber\\
\leq\left(\min\left(1-\delta_1,e^{\epsilon_1\,d(x_1,x'_1)}P'_1\right)+\tilde{\delta}_1\right)\nonumber\\
\times\left(\min\left(1-\delta_2,e^{\epsilon_2\,d(x_2,x'_2)}P'_2\right)+\tilde{\delta}_2\right)\nonumber\\
\leq m_1m_2+\tilde{\delta}_1m_2+m_1\tilde{\delta}_2+\tilde{\delta}_1+\tilde{\delta}_2\nonumber\\
\left[\text{where }m_i=\min\left(1-\delta_i,e^{\epsilon_i\,d(x_i,x'_i)}P'_i\right)\right]\nonumber\\
\leq e^{\epsilon_1\,d(x_1,x'_1)+\epsilon_2\,d(x_2,x'_2)}P'_1P'_2\nonumber\\
+\tilde{\delta_1}-\tilde{\delta_1}\tilde{\delta_2}+\tilde{\delta_2}-\tilde{\delta_1}\tilde{\delta_2}+\tilde{\delta_1}\tilde{\delta_2}\nonumber\\
\leq e^{\epsilon_1\,d(x_1,x'_1)+\epsilon_2\,d(x_2,x'_2)}\nonumber\\
\times\mathbb{P}_{\mathcal{K}_1,\mathcal{K}_2}\left[(y_1,y_2)\,\in\,
S_1\times S_2\vert (x'_1,x'_2)\right]\nonumber\\
+\left(\delta_1+\delta_2\right)\,e^{d(x_1,x'_1)+d(x_2,x'_2)}\nonumber
\end{align}
\end{proof}
We now proceed to generalize the conventional planar Laplace mechanism~\cite{ChatzikokolakisElSalamounyPalamidessi+2017+308+328} to define the \emph{truncated Laplace mechanism} extended to a generic metric space.
\begin{definition}[Truncated Laplace mechanism]\label{def:TruncatedLaplace}
The \emph{truncated Laplace mechanism} $\mathcal{L}$ on a space $\mathcal{X}$ equipped with, not necessarily symmetric, distance metric $d$ truncated to a radius $r$, is defined as:
\begin{align}\label{eq:TruncatedLaplace}
\mathbb{P}_{\mathcal{L}}[y\vert x]
=\begin{cases}
c\,e^{-\epsilon\,d(y,x)} & \text{if }d(x,y)\leq r\\
0 & \text{otherwise}
\end{cases}
\end{align}
where $c$ is the truncated normalization constant defined such that $\int\limits_{y\in\mathcal{Y}}\mathbb{P}_{\mathcal{L}}[y\vert x]dy=1$, and $\epsilon$ is the desired privacy parameter. Let us call $r$ to be the \emph{radius of truncation} for $\mathcal{L}$.
\end{definition}
Note: In case of a discrete domain $\mathcal{Y}$, $c$ is defined by normalizing $\sum\limits_{y\in\mathcal{Y}}\mathbb{P}_{\mathcal{L}}[y|x]=1$, and in this case $\mathcal{L}$ is an extended truncated geometric mechanism \cite{ghosh2009universally} extended to a generic metric space.
\begin{restatable}{lemma}{positivedelta}\label{lem:positivedelta}
$\frac{e^{-\epsilon\,d(x_1,x_2)}\mathbb{P}\left[y\vert x_1\right]-\mathbb{P}\left[y\vert x_2\right]}{e^{(1-\epsilon)\,d(x_1,x_2)}}\,e^{r}\leq 1$, where $r$ is the radius of truncation for $\mathcal{L}$, as in \eqref{eq:approxGeoI}.
\end{restatable}
\begin{proof}
\begin{align}
\frac{e^{-\epsilon\,d(x_1,x_2)}\mathbb{P}\left[y\vert x_1\right]-\mathbb{P}\left[y\vert x_2\right]}{e^{(1-\epsilon)\,d(x_1,x_2)}}\,e^{r}\leq 1\nonumber\\
\iff c\left(e^{-\epsilon d(x_1,x_2)+d(x_1,y)}-e^{-\epsilon d(x_2,y)}\right)\nonumber\\
\leq e^{(1-\epsilon)d(x_1,x_2)-r}\label{eq:legaldelta}
\end{align}
Now we observe that $d(x_1,x_2)+d(x_1,y)\geq d(x_2,y)$ due to the fact that $d$ is a metric and it satisfies triangle inequality. Immediately, we have $e^{-\epsilon d(x_1,x_2)+d(x_1,y)}-e^{-\epsilon d(x_2,y)}\leq0$ for any $\epsilon\in\mathbb{R}_{\geq 0}$. Therefore, as $c\geq 0$, \eqref{eq:legaldelta} is trivially satisfied as the RHS is always non-negative.
\end{proof}
\begin{restatable}{theorem}{truncatedLap}\label{th:truncatedLap}
$\mathcal{L}$ satisfies $(\epsilon,\delta)$-geo-indistinguishability where
$\delta=\max\left\{\max\limits_{\substack{S\subseteq \mathcal{Y}\\x_1,x_2\in\mathcal{X}}}\frac{e^{-\epsilon\,d(x_1,x_2)}\mathbb{P}\left[y\vert x_1\right]-\mathbb{P}\left[y\vert x_2\right]}{e^{(1-\epsilon)\,d(x_1,x_2)}},0\right\}$.
\end{restatable}
\begin{proof}
Trivially $\delta e^{d(x_1,x_2)}>0$ for any $x_1,x_2\in\mathcal{X}$ as $\delta>0$. Moreover, Lemma \ref{lem:positivedelta} ensures that $\delta e^{d(x_1,x_2)}<1$.
Now observe that for every $S\subseteq\mathcal{Y}$ and for all $x_1,x_2\,\in\,\mathcal{X}$, we have:
\begin{align}
e^{-\epsilon\,d(x_1,x_2)}\mathbb{P}_{\mathcal{L}}\left[y\vert x_1\right]-\mathbb{P}_{\mathcal{L}}\left[y\vert x_2\right]\leq \delta\,e^{(1-\epsilon)\,d(x_1,x_2)}\nonumber\\
\implies \mathbb{P}_{\mathcal{L}}\left[y\vert x_1\right]-e^{\epsilon\,d(x_1,x_2)}\mathbb{P}_{\mathcal{L}}\left[y\vert x_2\right]
\leq \delta\,e^{d(x_1,x_2)}\nonumber
\end{align}
\end{proof}
The explicit process of sampling private locations satisfying AGeoI from a given set of original locations through truncated Laplace mechanism on a discrete location space has been described in Algorithms \ref{alg:AGeoIMech} and \ref{alg:AGeoISamp}.
\begin{algorithm}
\SetAlFnt{\small}
\textbf{Input:} Discrete domain of original locations: $\mathcal{X}$, Discrete domain of private locations: $\mathcal{Y}$, Desired privacy parameter: $\epsilon$, Desired truncation radius: $r$;
\textbf{Output:} Channel $C$ satisfying \eqref{eq:TruncatedLaplace}\;
\SetKwFunction{Fmain}{DTLap}
\SetKwProg{Fn}{Function}{:}{}
\Fn{\Fmain{$\mathcal{X},\mathcal{Y},\epsilon,r$}}{
Set $C\leftarrow \text{empty channel}$\;
Set $Y\leftarrow \text{empty list}$\;
\For{$x \in \mathcal{X}$}
{
$c_x=\frac{1}{\sum\limits_{\substack{y\in\mathcal{Y}\\d(x,y)\leq r}}e^{-\epsilon\,d(x,y)}}$ \;
\For{$y \in \mathcal{Y}$}
{
\eIf{$d(x,y)\leq r$}
{$C[x,y]=0$}
{$C[x,y]=c_x\,e^{-\epsilon\,d(x,y)}$}
}
}
\textbf{Return:} $C$\;
}
\caption{Discrete and truncated Laplace mechanism (DTLap)}\label{alg:AGeoIMech}
\end{algorithm}
\begin{algorithm}
\SetAlFnt{\small}
\textbf{Input:} Discrete domain of original locations: $\mathcal{X}$, Discrete domain of private locations: $\mathcal{Y}$, Desired privacy parameter: $\epsilon$, Desired truncation radius: $r$; Vector of original locations: $X$\;
\textbf{Output:} Corresponding vector of private locations: $Y$\;
\SetKwFunction{Fmain}{DTLapSamp}
\SetKwProg{Fn}{Function}{:}{}
\Fn{\Fmain{$\mathcal{X},\mathcal{Y},\epsilon,r,X$}}{
$C=$ \Call{DTLap}{$\mathcal{X},\mathcal{Y},\epsilon,r$}\;
Set $Y\leftarrow \text{empty list}$\;
\For{$x \in X$}
{
Randomly sample $y\in\mathcal{Y}\sim C[x,:]$\;
Append $y$ to $Y$
}
\textbf{Return:} $Y$\;
}
\caption{Sampling private locations with DTLap (DTLapSamp)}\label{alg:AGeoISamp}
\end{algorithm}
\section{System Model}\label{sec:system_model}
This section details our privacy-preserving model for finding an optimal charging station in IoV as a use case of the proposed AGeoI technique. We begin with a discussion of the location privacy problems inherent in finding optimal charging stations in the IoV, followed by road networking modelling, a description of the system architecture with the differentially private location sharing algorithm, defining the system tiers' trust relationship, and the privacy threat actor model.
\subsection{Problem Statement}
EVs have been identified as a critical component of future sustainable transportation systems to reduce CO2 emissions and have garnered significant attention from academia and business~\cite{kumar2020adoption}. Due to the limited vehicle onboard battery capacity, EVs may be required to visit charging stations (CSs) during journeys. It causes some drivers to have range anxiety, which is the vehicle has insufficient battery power to cover the travel needed to reach its intended destination. It is highlighted as one of the barriers to EVs' widespread adoption~\cite{bulut2017mitigating}. CSs can be not always readily available since it often takes a while to be sufficiently charged for EVs. A CS booking service can help to overcome range anxiety.
EVs may access such services through third-party providers to discover the nearest and readily available CSs to minimise charging wait times by static or live location queries. However, location sharing raises privacy challenges, such as vehicle tracking. GeoI technique provides a formal privacy guarantee for location queries. However, it is not highly applicable to this use case for two reasons. It does not consider the feasible locations where a vehicle can be present, and it does not stop vehicle tracking in the case of linked queries during the vehicle trajectory. Thus, a tailored privacy-preserving mechanism is facilitated by combining the proposed AGeoI technique and dummy location generation.
\subsection{Road Network Model}
Similar to~\cite{qiu2020location}, the road network $G$ is represented as a weighted directed graph $G=(N,E,W)$, where $N$ is the set of nodes, $E\subseteq N^2$ is the set of edges, and $W:N^2\rightarrow \mathbb{R}^{+}$ is the set of weights representing the minimum travelling distance between any two nodes. The nodes and edges correspond to junctions and road segments of the network, respectively. Each edge $e \in E$ is addressed by the pair of respective starting node, ending node, and a weight representing the travelling distance through that edge, i.e., $e=(N^s_e, N^e_e, w_e) \in N$, where the direction of the traffic is from $N^s_e$ to $N^e_e$ on $e$. For any $i\in N$ and $j\in N$, let the sequence of edges $(e_1,\ldots,e_r)$ denote a \emph{path} from node $i$ to node $j$ if $N^s_{e_1}=i$ and $N^e_{e_r}=j$. Hence, let $C(i,j)$ represent the set of paths that connect node $i$ to node $j$. Then $W$ is a $N\times N$ matrix, where
\begin{equation*}
W_{ij}=
\begin{cases}
\min\limits_{p\in C(i,j)}\sum\limits_{e\in p}w_e & \text{ if $C(i,j)\neq \phi$}\\
\infty & \text{ otherwise}
\end{cases}
\end{equation*}
Essentially $W_{ij}$ is the shortest travelling distance from node $i$ to node $j$ in the network. We shall address the quantity $W_{ij}$ as the \emph{traversal distance} between nodes $i$ and $j$ in the graph $G$ and denote it as $d_{G}(i,j)$ for every $(i,j)\,\in N^2$. Note that, as $G$ is a directed graph, $d_{G}$ may not be symmetric.
\subsection{System Architecture}
\begin{figure}[ht!]
\centering
\includegraphics[width=0.48\textwidth]{Figures/EV_system_architecture.png}
\caption{System Architecture (EV:Electric Vehicle, RSU: Roadside Unit, MEC: Mobile-Edge Computing Unit)}
\label{fig:system_arch}
\end{figure}
The Internet of Vehicles (IoV) applications are reshaping transportation systems by minimising human error, advancing travel convenience, and decreasing energy, operational and environmental costs~\cite{omar2016wireless,Duan2020emerging,ji2020Artificial}. EVs have emerged as a viable technology for reducing carbon emissions and travelling costs~\cite{patil2020grid}. However, range anxiety is one of the major challenges of their wide adoption. Vehicular location data can be utilised to optimise the vehicle charging plan and mitigate range anxiety. Third-party services such as Chargemap\footnote{\url{https://chargemap.com/}}, Ampeco\footnote{\url{https://www.ampeco.com/}}, and PlugShare\footnote{\url{https://www.plugshare.com/}} can recommend available and the closest charging stations for the users. However, the users are required to trust these third-party service providers to use these services, which presents significant privacy concerns in the honest-but-curious service provider threat model.
ITS integrates connected vehicles, cloud computing, and IoV with transportation infrastructures for advantages on reduced fuel and energy use, $CO_2$ emission, traffic congestion, and road safety. Future transportation system would be a distributed computing platform which enables autonomous driving functions and cooperation between the vehicles based on vehicle to vehicle (V2V), vehicle to infrastructure (V2I), and vehicle to cloud (V2C) communications with methods for big data analysis with high reliability and ultra-low-latency. The wireless communication is enabled through technologies such as IEEE 802.11p DSRC/WAVE (Dedicated Short Range Communication/Wireless Access in Vehicular Environments) and cellular advances such as C-V2X~\cite{hasrouny2017vanet,vukadinovic20183gpp}. MEC technology is an industrial specification from ETSI for providing cloud computing capabilities at the Edge of a mobile network which is also applicable for autonomous driving tasks~\cite{Lee2018exploring}.
As the system architecture is depicted in Figure~\ref{fig:system_arch}, the vehicles are part of a three-tier cloud computing architecture and connected to Roadside Units, MEC Server, and the Core Cloud Network through a secure communication channel, where these components are considered trusted. The Core Cloud Network enables the connection between the vehicle and third-party services, including charging station recommender, as the main consideration of this paper. However, it is not possible to ensure fully trusted third-party services that they will not utilise vehicular location data for further objectives. Thus, the honest-but curious threat model is considered for the third party service provider and only privatised vehicular location data is shared in our proposed architecture. The roles of the system components are described in the following.
\subsubsection{Vehicle Tier}
In this paper, we fix a road network $G$ with nodes $G(N)$ and edges $G(E)$. We choose an arbitrary edge $I\in G$, and focus on the queries made by the EVs in $I$'s range of coverage, $R(I)$, provided by its \emph{road side unit (RSU)}. When an EV moves from the area of coverage of one Edge cloud to another, we can assume the queries and the privacy threats against the Edge to reset as each Edge communicates with the Cloud-based services and the third-party service providers. At any given time, an EV $u$ locally obfuscates its true location $x^u\in R(I)$ to $x^u_1\in R(I)$ using truncated Laplace mechanism guaranteeing AGeoI and generates $m-1$ feasible dummy locations $\{x^u_2,\ldots,x^u_m\}\in R(I)^{m-1}$ in the coverage area of the respective Edge. Then $u$ reports the vector of $m$ locations, $l_u=\left(x^u_1,\ldots,x^u_m\right)$, to $I$ for the Edge to process and communicate the query to the Cloud services and the third-parties to find the nearest available charging stations in $R(I)$.
Figure~\ref{fig:dummylocations} shows an example of reported $10$ dummy locations with the privatised location for two respective time windows, where the dummy locations in the following time window can have a feasible link at least one of the preceding dummy locations. Toy examples are depicted on Figures~\ref{fig:toy1} and \ref{fig:toy2} as more simplistic explanations of the proposed privacy-preserving mechanism, where the first represents a static query on an exampler discrete road network, and the following one represents linked dynamic queries on the same exampler discrete road network.
\begin{figure*}[ht!]
\centering
\includegraphics[width=1\textwidth]{Figures/folium-timestamps.png}
\caption{Reported dummy and privatised locations for two respective time windows (White Pins: Privatised locations, Orange Pins: Dummy locations in $1st$ Time window, Blue Pins: Dummy locations in $2nd$ Time window)}
\label{fig:dummylocations}
\end{figure*}
\begin{figure}[ht!]
\centering
\includegraphics[width=0.48\textwidth]{Figures/toy-example.png}
\caption{A toy example for a static location query on discrete road network}
\label{fig:toy1}
\end{figure}
\begin{figure}[ht!]
\centering
\includegraphics[width=0.48\textwidth]{Figures/toy-example2_V1.png}
\caption{A toy example for linked 3 location queries on discrete road network}
\label{fig:toy2}
\end{figure}
\subsubsection{Edge Tier} Given the large volume of data generated and shared between vehicles and infrastructure, installing Edge cloud close to the vehicles are needed to host the off-board vehicular services, which require a low access latency from the onboard vehicular services~\cite{Lee2018exploring}. Along with essential data processing and forwarding functions, the Edge tier provides another layer for data aggregation and deploying additional privacy-preserving measures before sharing the data with any third parties. The Edge is responsible for forwarding the location vectors received from the EVs at any time-step by disassociating the links between the location vectors and the corresponding vehicles, inducing an additional layer of anonymity for the Cloud and the third-party. Suppose at time $t$, Edge $I$ receives the set of location vectors from EVs $u_1,u_2,\ldots,u_{k_t}$ in $R(I)$. Therefore, $I$ receives the collection $\mathcal{L}(t)=\{l_{u_i}: i\in[k_t]\}$. Keeping a record of the IDs to link the location vectors with their respective senders, the Edge shuffles all the individual locations from every $l_u\in\mathcal{L}(t)$ and communicates this collection of all the reported locations to the Cloud and the third-party service provider. In particular, the Edge forwards the shuffled collection of locations $\mathcal{L}'(t)=\left\{x^u_i\colon u\in\{u_1,\ldots,u_{k_t}\}, i\in[m]\right\}$ to the cloud and the third-party service provider(s).
\subsubsection{Cloud Tier} It is expected to provide computation and storage capabilities for top-level processes, including data sharing interfaces for third-party services.
\subsubsection{Third-party Service Provider} It is the external party to ITS and is expected to enhance the quality of the function for finding the available charging stations for the vehicles by receiving search queries compromised of privatised and dummy location vectors for the respective vehicles.
\subsubsection{Communication Channel} ITS comprises a network of roadside units (RSU), vehicle on-board electronic control units (ECU), and distributed cloud computing and storage services. Wireless communications are enabled for V2V (Vehicle-to-Vehicle), V2I (Vehicle-to-Infrastructure) and V2X(Vehicle-to-Everything) by the technologies such as IEEE 802.11p DSRC/WAVE (Dedicated Short Range Communication/Wireless Access in Vehicular Environments), cellular advances such as C-V2X, and the long-term evolution for vehicles (LTE-V)~\cite{seo2016lte}. Confidentiality of the wireless communication channel is secured by public key infrastructure (PKI) encryption methods which is beyond the scope of this work.
\subsection{Privacy Threat Landscape}
In real-time IoV location-based applications, vehicle users who wish to receive services tailored to their current locations are needed to share their location with the service provider. Different applications may have varying needs for data sensitivity. Data perturbation techniques are often used in private data sharing by introducing uncertainty in the data, which entails a trade-off in terms of the related application's quality of service in general. For example, EV users who want to locate the nearest available charging station during travel send a query with their current location to a service provider using the Intelligent Transportation Systems (ITS) infrastructure.
Charging an EV can take up to several hours. Due to the lengthy charging cycle, making a bad choice of charging station, such as one with no empty charging piles, may result in increased range anxiety and waiting time to charge the vehicle. Thus, choosing the nearest available charging station is necessary to alleviate range anxiety and wait time. Such information can be provided by real-time location data mining systems which often will be a third-party operator for a specific area~\cite{tian2016real}.
We consider the system into three categories: (i) the vehicle users (data subject), (ii) ITS infrastructure, including the MEC and Cloud Tiers (data controller and data processor), and (iii) the third party that receives the privatised data from the deployed deployment privacy-preserving mechanisms. The third party is assumed to be an EV charging management system, which may operate in a registration-based approach for a specific area.
The system includes several categories of data flow with different sensitivity: (i) publicly available data such as road network and location of the charging stations, (ii) not-private data such as availability of the charging stations, and (iii) private data as the real-time location of the vehicle users.
The vehicular location data may also be subject to unauthorised use, data inference, retaining or disclosure besides its primary use. Thus, a privacy-preserving mechanism can be utilised for location data sharing to provide a privacy guarantee while the users can receive the same or similar quality of service. Thus, the third-party provider is considered an honest-but-curious adversary model, which assumes it is honest in accurately executing the protocol required to provide location data. However, it may be curious to infer users' private information based on the obtained location data~\cite{paverd2014modelling}.
\subsubsection{Privacy Requirements}
In distributed systems, privacy guarantee and trust establishment remain open research challenges since multiple actors are involved in data share and computation. Vehicular location privacy can be ensured by considering three privacy properties in the settings of the proposed system. These are indistinguishability, unlinkability and confidentiality~\cite{wagner2018technical}. Indistinguishability means that the threat actor cannot distinguish between two outcomes of a privacy mechanism, which is provided by deploying the approximate geo-indistinguishability into the Vehicle Tier. The second property, unlinkability, refers to the threat actor's feature that cannot link the released data. It is ensured by deploying the shuffle model at the Edge Tier at each timestamp before releasing data to the third party. The last property, confidentiality, can be ensured by securing the communication channel by the appropriate encryption mechanisms, which is beyond the scope of this paper.
\subsubsection{Privacy Threats}
In this work, we mainly address two notable sources of threat that might put the EV's privacy at risk: a) a potential identification of the individual locations of the queries made by the EV, and b) tracing the journeys of the EVs by dynamically exploiting the queries made along their journeys.
\paragraph{Location identification} It is essential that the individual locations of the queries made by the EVs are protected from being identified. For the use case of location charging stations, it is essential that the privatized version of the true location of the EV is within a certain radius of interest w.p. 1, making sure that the reported location is within a feasible and drivable distance away and, most importantly, within the area of coverage of the Edge where its true location lies. Therefore, we proceed to define AGeoI as an extension of GeoI, which is the state-of-the-art standard for location privacy. Therefore, to ensure the privacy of the position of any given query in the road network, the EVs locally obfuscate their true locations using the truncated Laplace mechanism with their desired parameter $\epsilon$ and the radius of truncation $r$, which, in turn, decide the value of $\delta$.
\paragraph{Journey tracing}
It is often the case that a certain EV might enquire about the nearest available charging station, but choose to proceed with the journey at that point of time, and raise further queries along the journey. In our model we capture this realistic setting of allowing multiple queries to be made by the EV in a given journey. This immediately leads to a threat of approximately tracing the trajectory of the journey of an EV by interpolating the locations of query, despite having each individual location being AGeoI.
This is due to the fact that the obfuscated location of each query is not distinguishable from the real location, but they are not too far off from each other with a very high probability. Therefore, if the number of queries made in a single journey is large, it could be fairly straightforward to approximately deduce the trajectory of the journey.
Cunningham et al.~\cite{Cunningham2021Trace} proposed a mechanism to securely share trajectories under LDP. However, the authors in \cite{Cunningham2021Trace} assumed a model of offline sharing of the entire trajectory and, hence, sanitizing it with the proposed mechanism to engender a LDP guarantee. In our setting, this method cannot be directly implemented as we are working in a dynamic environment where the queries made by the EVs are in real time, with the server not having any prior knowledge on the number or the location of the queries made by a certain EV. Therefore, the mechanism of \cite{Cunningham2021Trace} cannot trivially be extended in the online location-sharing environment, and hence, the threat of the adversaries able to reconstruct the journey of a particular EV with a high number of queries remains as a concern.
To mitigate this problem, for the $k^{th}$ query with $t>1$, an EV generates $m-1$ dummy locations in the area of coverage of the Edge $I$ and reports the vector of $m$ locations $l^{k}_u=\left(x^u_1,\ldots,x^u_m\right)$ to the RSU of $I$, where $x^u_1$ is the privatized version of the true location $x^u$ of $u$, and $x^u_2,\ldots,x^u_m$ are randomly generated dummy locations from $R(I)$ such that every dummy location corresponds to at least one of the locations in $l^{k-1}_u$ which ensures a feasible~\footnote{w.r.t. speed, travelling conditions, etc.} journey in between the times of the $k^{th}$ and $k-1^{st}$ queries, as illustrated by Figure \ref{fig:dummylocations}, making it impossible for the Edge to exclude some of the dummy locations as being unrealistic as far as tracing the journey is concerned. For $k=0$, i.e., an EV can generate any random $m-1$ dummy locations in $R(I)$. This procedure recursively ensures that with every given query the adversary will have at least $m$ possible trajectories that the EV could have realistically followed, making it improbable to be able to conclude about the actual one. We shall omit the superscript of the number of the query made by the EV if it is clear from or insignificant in the context.
Note that we assume that the principle adversary lies at the third-party service provider, and despite having individually geo-indistinguishable locations, it is possible to trace the entire trajectory with a fair bit of accuracy by interpolating the individual geo-indistinguishable locations of queries. At any given time, the Edge collects all the reported locations from the querying EVs, shuffles them by effacing the links between the location vectors and the corresponding EVs, and sends this jumbled collection of all the reported locations in the network for knowing their respective nearest available charging stations to the third-party service provider. After receiving the response, the Edge, which internally keeps the record of the IDs of the EVs against their queried locations, assigns the corresponding vector of locations of the nearest available charging stations to each EV and communicates them back to the respective vehicles.
In other words, at time $t$, if the Edge receives the location vectors from $k_t$ querying EVs as $\mathcal{L}(t)=\{l_{u_1},\ldots, l_{u_{k_t}}\}$, the Edge is responsible for shuffling all the individual location points in these reported vectors and forward the scrambled collection of locations $\mathcal{L}'(t)=\{x^u_i\colon u\in\{u_1,\ldots,u_{k_t}, i\in[m]\}$ to the Cloud and the third-party, while internally keeping a track of the IDs of the EVs to reconnect the query-response back to the corresponding vehicles. Setting $\hat{x}$ as the location of the nearest available charging station from location $x$ in $R(I)$, the Edge receives $\mathcal{R}(t)=\{\hat{x}^u_i\colon u\in\{u_1,\ldots,u_{k_t}, i\in[m]\}$ as the response from the third-party. After this, matching the stored IDs of the EVs with the locations of the charging stations, the Edge communicates the response vector $\hat{l}_u=(\hat{x}^u_i:i\in [m])$ back to the corresponding EV $u$. Then the EV can choose to navigate to $\argmin\limits_{x\in\hat{l}_u}\{d_G(x,x_u)\}$, where $x_u$ is the real location of EV $u$. The overview of this mechanism is explained in Figure \ref{fig:system_arch}.
\section{Cost of privacy analysis} \label{sec:cost_of_privacy_analysis}
\begin{definition}[Cost of privacy]\label{def:CoP}
Suppose an EV $u$ at location $x^u$ chooses to locally obfuscate its real location of query as $x^u_1$ using the truncated Laplace mechanism $\mathcal{L}_{\epsilon,r}$ satisfying $(\epsilon,\delta)$-geo-indistinguishability with a corresponding radius of truncation $r$. Then we define the \emph{cost of privacy (CoP)} of EV $u$ as $\operatorname{CoP}(u,\mathcal{L}_{\epsilon,r})=\vb{c}(x^u,\hat{x}^u_1)-\vb{c}(x^u,\hat{x}^u)$, where $\hat{x}^u$ and $\hat{x}^u_1$ are the nearest available charging stations in the network to $x^u$ and $x^u_1$, respectively, and $\vb{c}:G(N)^2\mapsto \mathbb{R}^+$ is any cost function that reflects the cost of commute from locations $x$ to $y$ in the network.
\end{definition}
In other words, CoP, as in Definition \ref{def:CoP}, essentially captures the \emph{extra} cost that an EV needs to cover as a result of the privatized location it reports to the Edge satisfying AGeoI, as opposed to its true location. For the purpose of this paper and simplicity of the analysis, we considered the cost function as the travelling distance in the network, i.e., $\vb{c}=d_G$. However, in practice, any suitable cost function could be used (e.g. fuel efficiency, time, etc.) could be used as $\vb{c}$, depending on the context and requirement of the architecture.
To formally characterize and analyze the CoP of the EVs in the network, inspired from the classical version of \emph{Voronoi decomposition}, we extend the concept in the setting of our road network in the network coverage for a fixed Edge w.r.t. graph-traversal distance, $d_g$.
\begin{definition}[Voronoi decomposition]\label{def:Voronoi}
Let $G$ be the graph representing the road network equipped with travelling distance $d_G$. Let the set of charging stations in $G$ be $C_G=\{c_1,\ldots,c_{n_G}\}$. Then the \emph{Voronoi decomposition} on $G$ w.r.t. $C_G$ is defined as $\vb{V}_G=\{V_i\colon\,i\in[n_G]\}$ such that $V_i\cap V_j=\phi$ for any $i\neq j$ and $\bigcup\limits_{i\in[n_G]}V_i=G$, where
\small
\begin{equation*}
V_i=\{x\in G\colon\,d_G(x,c_i)\leq d_G(x,c_j)\,\forall\,j\in[n_G],j\neq i\}
\end{equation*}
\par
\end{definition}
\begin{definition}[Closed ball around a location]\label{def:ClosedBall}
For any $x \in G$ and $r\in\mathbb{R}_{\geq 0}$, the \emph{closed ball} of $x$ of radius $r$ is defined as $\beta_r(x)=\{y\in G\colon\,d_G(x,y)\leq r\}$
\end{definition}
\begin{definition}[Fenced Voronoi decomposition]\label{def:FenceVoronoi}
For any $r\in\mathbb{R}_{\geq 0}$, and for charging station $i$, let the \emph{$r$-fenced Voronoi decomposition} on road network $G$ be defined as $V^{-r}_G=\{V_i^{r}\colon\,i\in[n_G]\}$ such that $V^{-r}_i\cap V^{-r}_j=\phi$ for any $i\neq j$ and $V_i^{-r}=\{x\in V_i \colon\, B_r(x)\subseteq V_i\}$. In other words, $V_i^{-r}$ is essentially constructing an area which is contained within $V_i$ being restricted by a \emph{fence} at a distance $r$ from the edge of $V_i$.
\end{definition}
\begin{restatable}{theorem}{zeroCoP}\label{th:zeroCoP} Suppose an EV $u$ positioned at $x^u$ on $G$ obfusucates its location using AGeoI with any radius of truncation $r\in\mathbb{R}_{\geq 0}$. Let $\hat{x}^u$ be the location of the nearest available charging station to the true location $x^u$. Then we have $\mathbb{P}\left[\operatorname{CoP}(u,\mathcal{L}_{\epsilon,r})=0\right]=1$ for every $x^u\in V_{\hat{x}^u}^{-r}$. In other words, if an EV $u$ lies in the $r$-fenced Voronoi decomposition for its nearest available charging station, we get \emph{zero cost for privacy} for $u$ w.p. 1.
\end{restatable}
\begin{proof}
Immediate from the definition of fenced Voronoi decomposition.
\end{proof}
\begin{restatable}{theorem}{zeroCoPProb}\label{th:zeroCoPProb} Suppose an EV $u$ lies in $V_{\hat{x}^u}\setminus V_{\hat{x}^u}^{-r}$ and it uses AGeoI to obfuscate its true location $x^u$ to $x^u_1$ with a radius of truncation $r$ and privacy parameter $\epsilon$ for making a private query to the Edge. Then $$\mathbb{P}\left[\operatorname{CoP}(u,\mathcal{L}_{\epsilon,r})=0\right]=1-\sum\limits_{x^u_1\in V_{\hat{x}^u}^c}ce^{-\epsilon d_G(x^u,x^u_1)}$$ where $c$ is the normalizing constant of the truncated Laplace mechanism as in Definition \ref{def:TruncatedLaplace}.
\end{restatable}
\begin{proof}
To compute $\mathbb{P}\left[\operatorname{CoP}(u,\mathcal{L}_{\epsilon,r})=0\right]$, we only need to exclude the possibilities where the reported location of the EV lies outside the Voronoi decomposition of the station $\hat{x}^u$, which, essentially, is $1-\sum\limits_{x^u_1\in V_{\hat{x}^u}^c}ce^{-\epsilon d_G(x^u,x^u_1)}$.
\end{proof}
\section{Experimental Study}\label{sec:experiments}
In this section, a series of experiments are conducted to evaluate the proposed method in Edge cloud-assisted intelligent transportation systems settings for EV charging station finding use case. The objectives of experiments are (i) to validate proposed theoretical claims and solutions through experiments, (ii) to apply the method to find the nearest available charging station for EVs as a use case study, and (iii) to investigate the cost of privacy for the use case. The experiments are deployed in a laptop with Intel core I7 processor, 16 GB random access memory, and Ubuntu 20.04 operation system using standard Python packages.
\subsection{Dataset Preparation}
The regional map, exported from OpenStreetMap\footnote{\url{http://www.openstreetmap.org}}, is utilised in this study. The cost of privacy is analysed by calculating the extra routing distance to the identified optimal charging station due to additional noise in vehicular locations in the queries. Thus, the cost of privacy depends on the sparsity of the charging stations. Two datasets with different densities are prepared to comprehend the impact of this sparsity. The United States Department of Energy allows downloading up-to-date data of the existing and planned alternative fuel stations with their geographical coordinates~\cite{chargingstations}. The first dataset is exported, containing the locations of 404 existing EV charging stations in San Francisco. For the second dataset, we followed the assumption that EV charging stations will be more spread in the future, and parking locations are likely to deploy charging facilities. Thus the second dataset is prepared by merging the existing and planned charging station locations with on-street and off-street parking locations provided by DataSF~\cite{chargingstations2}. It contains 716 independently distributed locations.
The EPFL (Swiss Federal Institute of Technology - Lausanne) mobility dataset contains the GPS records of the 536 taxi trajectories in San Francisco for four weeks. The data format includes a taxi identifier, latitude, longitude, state of the taxi (vacant or occupied), and UNIX epoch timestamp~\cite{piorkowski2009crawdad}. It is possible to split the entire taxi trajectory into realistic trajectories of each customer because the dataset contains the information for the occupancy of the taxis. By doing so, we could export over 450 thousand trajectories. 536 trajectories, one randomly chosen from each taxi, are utilised in this study.
\subsection{Experimental Setup}
The edges of the road network $G$ are truncated into discrete segments with equal $k$ travel distance, similar to the work in~\cite{qiu2020location}. DTLap (e.g., Algorithm 1) is utilised to generate the privacy channel by using the Laplace mechanism for the user's desired values for privacy budget $\epsilon$ and truncation radius $r$. Following this, DTLapSamp (e.g., Algorithm 2) is used to generate privatised locations with respect to the users' real locations. The generation of discrete road segments is done by setting the parameter $k$ to 100 metres. From the EPFL mobility dataset, 536 random traces, one per taxi, are selected. Each trace contains a series of GPS coordinates and 3 randomly selected along each for the real locations of the queries. As a result, the experiments are carried out for 1608 locations where each 3 of these are linked. The parameters of $\epsilon$ and $r$ are varied in the range of 0.2 to 2, and 1 to 20, respectively.
The experimental scenario considers the steps; (i) vehicles made three location queries during their journeys to find the nearest available charging stations where each location query contains a privatised location and $10$ dummy locations as a vector, (ii) the Edge cloud collects all privatised locations and dummy locations with to send the third-party through the core cloud as a vector of all locations, (iii) the third-party responds the vector of locations with the nearest available CS for each, (iv) the Edge responds vectors from each vehicle. Hence, the Edge cannot distinguish the privatised locations among the dummy locations. The experimental scenario is executed on two different CSs densities.
\begin{figure*}[t]
\centering
\includegraphics[width=0.8\textwidth]{Figures/Figure1.png}
\caption{Cost of privacy for varying $\epsilon$ or $r$ of AGeoI ($1st$ row is for sparse CSs, $2nd$ row is for dense CSs)}
\label{fig:cost_of_privacy}
\end{figure*}
\begin{figure*}[ht!]
\centering
\includegraphics[width=1\textwidth]{Figures/Figure2.png}
\caption{Fraction of zero cost of privacy for varying $\epsilon$ or $r$ of AGeoI ($1st$ row is for sparse CSs, $2nd$ row is for dense CSs)}
\label{fig:privacy_for_free}
\end{figure*}
\subsection{Results}
DP approaches introduce a trade-off between privacy and data utility, with a higher level of privacy requiring a greater level of noise. Due to the fall in data utility, the efficacy of the respective service may correspondingly decrease. It is termed the cost of privacy in this study. However, there is a finite number of cases with the maximum efficacy for many applications. Thus, there must be a sweet spot of privacy for free where the user will receive the maximum efficacy with some level of privacy. It is also valid for our use case, in which a vehicle must be able to utilise a finite number of unique CSs.
The following results are achieved by carrying out the experiments for $3$ linked queries of $536$ randomly selected vehicle trajectories for varying values of $\epsilon$ or $r$ ranging from $0.2$ to $2$, and $1$ to $20$, respectively. Figure~\ref{fig:cost_of_privacy} demonstrates the plot of cost of privacy in terms of the extra travelling distance due to privacy-preserving mechanism, where a similar pattern is observed for both of the datasets.Another observation is that a high incidence of queries resulted in having zero cost for privacy-preserving. Figure~\ref{fig:privacy_for_free} demonstrates fraction of the queries with privacy for free, where two datasets followed similar patterns. Vehicle queries contains dummy locations with their privatised locations. It is possible that the dummy locations can sometimes provide a better utility, but our experiments consider the utility of privatised location as the worst-case for analysis.
Figure~\ref{fig:cost_of_privacy} demonstrates that our mechanism provides a negligible cost of utility-loss for the formal privacy-gain enjoyed by the EVs. By increasing the truncation radius, it is observed that an abrupt drop in the distance between the location of the nearest available charging station for the true location of the query and that of the privatised one, implying that the cost of the extra travel distance needed to be taken due to the AGeoI mechanism is almost negligible. A similar trend is seen for the varying $\epsilon$ with a fixed radius. As the level of privacy decreases, the fraction of EVs in the network enjoying \emph{privacy for free} grows to be more than $60\%$ for a radius of truncation of merely 10 road segments, where each segment is 100 meters long, for $\epsilon\geq 0.5$. However, more than $90\%$ of the EVs achieve a zero cost of privacy for $\epsilon\geq 1.5$, irrespective of the truncation radius in Figure~\ref{fig:privacy_for_free}. The user can enhance the level of perturbation in queries by decreasing the $\epsilon$ and increasing the $r$. Due to increasing perturbation in disclosed location, the width of the confidence interval for zero cost of privacy increases, as seen in Figure~\ref{fig:privacy_for_free}. The likelihood of achieving zero cost of privacy fluctuates over a wider range, and it does not monotonically decrease with the growing radius due to rising randomness.
\section{Conclusion}\label{sec:conclusion}
This paper studied a fundamental problem on the risk of privacy violations for EVs dynamically querying for charging stations along their journeys. The problem's setting has not been addressed in the literature explicitly, and some of the related techniques along the lines of privacy-preserving vehicle routing cannot be adapted directly in practical model we considered in this paper.
To resolve this, firstly, we have theorized the notion of approximate geo-indistinguishability and justified its mathematical soundness and applicability by proving the compositionality theorem, which is a novelty in itself, allowing us to attain GeoI in a strictly bounded space of noisy data. We derived the appropriate privacy parameters to prove that the truncated Laplace mechanism satisfies approximate geo-indistinguishability and, therefore, used it to propose a location-privacy preserving method for EVs querying for charging stations. Our method induces two-way privacy protection: on the individual locations of the queries and for the trace of the entire journeys.
We performed experiments on datasets with real vehicle traces and locations of charging stations in San Francisco, and illustrated the privacy-utility trade-off for our method for two different settings of the sparsity of charging stations in the network. We observed a consistent trend across all the experiments which suggests that a substantial majority of the EVs have privacy for free, i.e., suffer no loss of utility even for fairly high-level approximate geo-indistinguishability attained. In general, we observe that the cost of privacy induced by our method is fairly low across settings, thus, ensuring formal privacy protection for the EVs without incurring a high price for that as far as the journey is concerned. We analyzed this cost of privacy under our method using Voronoi decomposition to draw insight into the working of our method. Construction Voronoi diagram of the road network for a location-based service will allow us to analyse further the privacy-utility trade-off, which is a promising avenue of research.
\begin{acks}
The authors would like to thank Professor Graham Cormode for the insightful discussions that supported the development of this study. Ugur Ilker Atmaca and Sayan Biswas are shared co-first authors who have worked together on a publication and contributed equally. The authors would also like to extend their sincere thanks to the ERC (European Research Council ) Advanced Grant of Catuscia Palamidessi which supported Sayan Biswas' research visit at WMG, the University of Warwick, which enabled this successful collaboration.
\end{acks}
\bibliographystyle{ACM-Reference-Format}
\section{Introduction}
Coping with global climate change-related challenges, including air pollution, is one of the most pressing concerns that civilizations confront today~\cite{FERRERO2016450,hampshire2018electric,Zhang_2020}. The transportation industry has the most significant cause of air pollution as the major contributor to greenhouse gas emissions~\cite{hickman2007looking,kufeoglu2020emissions}. Transportation accounted for 27\% of the UK's total greenhouse emissions in 2019 as the largest emitting sector. A vast majority of the emission, 91\%, comes from automobiles~\cite{Millen2021}. Hence, the transportation industry and academic communities are increasingly interested in developing alternative energy vehicles to reduce emissions. Automobile manufacturers are introducing a new generation of electric vehicles (EVs) which often employs connected and automated driving functions~\cite{gersdorf2020mckinsey}.
EVs are considered one of the most promising means to reduce emissions and dependency on fossil fuels. Along with environmental benefits, EVs provide superior energy efficiency to conventional vehicles~\cite{hensley2009electrifying}. As the cost of batteries continues to decrease, the large scale adoption of EVs is becoming more viable~\cite{gao2014road}. Despite the advantages and competitive cost, many customers remain concerned about running out of battery power before reaching their destination or waiting for their EVs to charge. The primary obstacles to EV adoption are the availability of chargers and the range that can be travelled on a single charge, often referred to as \emph{range anxiety} in the literature~\cite{lombardi2018electric}.
In addition to the rising need and relevance of EVs, with the recent technological development and advancements in machine learning and various other kinds of techniques in data analysis, the value of personal data has increased manifold and, almost parallelly, the threat to attack one's private information and the risks of violation of the users' privacy are becoming more and more significant and concerning~\cite{nist,lemetayer:hal-01420983}. As a consequence, there has been widespread awareness about the importance of protecting the privacy of personal data while exploiting their utility and analyzing them. One of the most successful approaches to address this issue of privacy-preserving protocols to analyse and utilize sensitive data is along the lines of \emph{differential privacy} (DP)~\cite{DworkDP1,DworkDP2}, which mathematically guarantees that the query output does not change significantly regardless of whether a specific personal record is in the dataset or not.
However, the classical central version of DP requires a trusted curator who is responsible for adding noise to the data before publishing or performing analytics on it. A major drawback of such a central model is that it is vulnerable to security breaches because the entire original data is stored in a central server. Moreover, there is the risk of having an adversarial curator. To circumvent the need for such a central dependency, a local model of DP, a.k.a. \emph{local differential privacy} (LDP)~\cite{DuchiLDP}, has been in the spotlight of late, where the users apply the LDP mechanism directly to their data and communicate the locally perturbed data to the server.
LDP is particularly suitable for situations where the users need to communicate their personal data in exchange for some service. One such scenario is the use of location-based services (LBS), where a user typically reports her location in exchange for information like the shortest path to a destination, points of interest in the surroundings, traffic information, friends nearby, etc. One of the recently popularized standards in location privacy is \emph{geo-indistinguishability} (GeoI)~\cite{AndresKostasCatuscia_GeoInd}, which optimizes the quality of service (QoS) for the users while preserving a generalized notion of LDP on their location data. The obfuscation mechanism of GeoI depends on the distance between the original location of a user and a potential noisy location that they report~\cite{Bordenabe:14:CCS,Fernandes:21:LICS}.
GeoI can be implemented directly on the user's device (tablet, smartphone, etc.). The fact that the users can control their explicit privacy-protection level for various LBS makes it very appealing. However, a drawback of injecting noise locally to the datum is that it deteriorates the quality of service (QoS) due to the lack of accuracy of the data.
On the other hand, future vehicles are getting more sophisticated in sensory, onboard computation and communication capacities. Furthermore, the emergence of Mobile Edge Computing (MEC) also changes the Intelligent Transportation Systems (ITS) by providing a platform to assist computationally heavy tasks by offloading the computation to the Edge cloud~\cite{gillam2018exploring}. This architecture often employs three tiers, with the vehicle on the first tier, MEC on the second tier, and standard cloud services on the third tier~\cite{maple2019connected}. Figure~\ref{fig:system_arch} shows the system architecture for the location privacy framework proposed in this paper.
ITS provides a platform containing distributed and resource-constrained systems to support real-time vehicular functions where these functions' efficacy relies on the data shared across entities. However, the risk of privacy disclosure and tracking increases due to data sharing~\cite{hahn2019security}. Privacy-preserving schemes are developed using established techniques such as group signature, anonymity, and pseudonymity~\cite{lin2013achieving,kumar2015intelligent}. However, it is possible to identify privatised data with adequate background information. Hence, DP approaches have emerged as the gold standard of data privacy because they provide a formal privacy guarantee independent of a threat actor's background knowledge and computing capability~\cite{zhao2020survey}.
GeoI is the state-of-the-art method for location privacy-preserving with LDP. It can preserve one's location privacy among a set of locations with similar probability distributions without requiring a trusted third-party. It provides rigorous privacy for location-based query processing and location data collection by modelling the location domain based on the Euclidean plane. However, vehicles are located on the road network under normal circumstances. For vehicular location queries, GeoI mechanism may result in publishing unrealistic privatised locations such as houses, parks or lakes. Thus, there is a need for an adapted model of GeoI for vehicular application. This paper proposes a novel privacy model called Approximate Geo-Indistinguishability (AGeoI), based on the notion of GeoI by using a discrete road network graph. Our key contributions in this paper are outlined as follows.
\begin{enumerate}
\item We presented the notion of AGeoI, a formal standard of location-privacy in a bounded co-domain, by generalizing the classical paradigm of GeoI, and adapting it to a graphical environment. We illustrate its applicability by proving that it satisfies the compositionality theorem. Moreover, we show that the truncated Laplace mechanism satisfies AGeoI by deriving the appropriate $\epsilon$ and $\delta$ as privacy parameters.
\item We proposed a two-way privacy-preserving navigation method for the EVs dynamically querying for charging stations on road networks -- we protect against threats on individual locations of their queries by ensuring AGeoI, and we provide protection against adversaries tracing the trajectories of the EVs by interpolating their query locations in an online setting.
\item We performed experiments on real vehicular journey data from San Francisco with real locations of charging stations from the area
under two settings of their sparsity in the road network, and show that our method ensures a very high fraction of EVs who enjoy \emph{privacy for free} and illustrate that the cost of utility-loss for the EVs is very low compared to the formal gain in privacy.
\end{enumerate}
The rest of this paper is organised as follows. Section \ref{sec:related_work} reviews some of the related work in this area. Section \ref{sec:preliminaries} introduces some fundamental notions on DP and GeoI. Section \ref{sec:approx_geo_ind} develops the mathematical theory of AGeoI. Section \ref{sec:system_model} elucidates the model of our proposed mechanism by formalizing the problem we are tackling, thoroughly discussing system architecture, and laying out the privacy-threat landscape we are addressing in this work. Section \ref{sec:cost_of_privacy_analysis} draws some insight into the cost of privacy on the EVs induced by our mechanism. Section \ref{sec:experiments} presents the experimental results to illustrate the working of our mechanism, and, finally, Section \ref{sec:conclusion} concludes the work.
\section{Related Work}\label{sec:related_work}
Both corporate and academic communities have recently piqued the interest in advancing EVs and charging infrastructure to improve the transportation system's sustainability. Despite the advancements, the EV sector confronts challenges that delay the adoption process, such as range anxiety, an absence of convenient and available charging infrastructure, and waiting time to charge~\cite{franke2013understanding,kumar2020adoption}.
An offline static map of charging stations is insufficient to resolve these obstacles since EVs may need to reserve a charging station when a trip is planned or query the available stations based on their battery state, and charging stations must be reservable. Thus, live vehicular and charging station data is utilised in querying and reservation/scheduling mechanisms~\cite{tian2016real,zhang2021intelligent,flocea2022electric}. Encryption techniques can be used in such mechanisms to prevent external intrusions, but they cannot preserve users' privacy from malicious servers and third-party providers.
Several data types are considered in these mechanisms, including real-time location, intended route, battery level, and station availability, to ensure the drivers are not detoured from their intended route\cite{tian2016real,wang2019sharedcharging}. Although disclosing such information poses privacy concerns for the driver's location and vehicle tracking, the privacy requirements of such mechanisms are not sufficiently studied in the literature. Existing methods for planning charging points for EV journeys are considered mechanisms for confidentiality and integrity, but the drivers' location privacy is regarded as an issue of trust in the third-party service provider~\cite{plugshare,chargepoint}.
This problem can be addressed by several approaches, based threat model of the system. Location anonymity is achieved through cloaking an area~\cite{chow2009casper,niu2014achieving}. These approaches can be only applied to the Edge of our system model to provide anonymity to a group of EVs, but we consider the Edge as an honest-but-curious threat actor and aim to preserve individual vehicles' privacy locally. Thus, these techniques are not applicable to our considered threat model. Furthermore, anonymity techniques do not provide a formal privacy guarantee~\cite{DworkDP_Compositionality}. Similarly, mix-network approaches cannot be applicable because it is not guaranteed that multiple vehicles will be presented in an Edge's coverage in any timestamp due to vehicles' movement~\cite{guo2018independent}. An applicable approach is to download the charging station's live map on EVs to search for the nearest or on-the-route available charging station; however, the communication overhead of this technique is predicted to be much higher than the vehicles' location-based inquiry since it will require downloading a recent snapshot of the map for each query.
DP methods are increasingly being deployed to preserve location privacy in a variety of domains, including automotive systems. The studies in~\cite{zhou2018achieving,luo2019geo} proposed models by deploying a GeoI-based mechanism on the Edge for location-based services. However, their approach did not consider preserving vehicles' location privacy against the Edge. An approach that compliments the problem we aim to address in this paper was proposed by Qiu et al. in \cite{qiu2020location} where the authors proposed a technique to crowd-source a task in vehicular network while preserving GeoI of the location of the vehicles offering Mobility as a Service (MaaS) in the spatial network to solve a task at a publicly known location in the map (e.g. taxi services). The problem formulation in this work is the inverse of what we aim to achieve in this paper. As a result, the work in \cite{qiu2020location} cannot be extended to address the privacy concerns induced my multiple queries dynamically made along the journey.
In \cite{Cunningham2021Trace}, Cunningham et al. studied the problem of trajectory sharing under DP and proposed a mechanism to tackle it. However, this work assumes the setting of an offline trajectory sharing which breaks down in the practical environment where the trajectories are being shared online as there is no prior information or limitation on the number of queries made by an EV during a journey, and their respective locations. Therefore, the method proposed by the authors in \cite{Cunningham2021Trace} cannot be directly adapted to our dynamic environment closely simulating the real-world scenario for such a use case.
\section{Preliminaries}\label{sec:preliminaries}
The most successful approach to formally address the issue of privacy threats is DP, mathematically guaranteeing that the query output does not change significantly regardless of whether a specific personal record is in a dataset or not. Most research performed in this area probes two main directions. One is the classical central framework~\cite{DworkDP1,DworkDP2}, in which a trusted third party (the curator) collects the users' personal data and obfuscates them with a differentially private mechanism.
\begin{definition}[Differential privacy~\cite{DworkDP1,DworkDP2}]
\label{def:dp}
For a certain query, a randomizing mechanism $\mathcal{R}$ provides \emph{$\epsilon$-DP} if, for all neighbouring\footnote{differing in exactly one place} datasets, $D$ and $D'$, and all $S \subseteq$ Range($\mathcal{R}$), we have
\begin{equation*}
\mathbb{P}[\mathcal{R}(D) \in S] \leq e^{\epsilon}\,\mathbb{P}[\mathcal{R}(D') \in S]
\end{equation*}
\end{definition}
A major drawback of central model is that it is vulnerable to security breaches because the entire original data is stored in a central server. Moreover, there is the risk that the curator may be corrupted. Therefore, a local variant of the central model has been widely popularized of late~\cite{DuchiLDP}, where the users apply a randomizing mechanism locally on their data and send the perturbed data to the collector such that a particular value of a user's data does not have a major probabilistic impact on the outcome of the query.
\begin{definition}[Local differential privacy~\cite{DuchiLDP}]
\label{def:ldp}
Let $\mathcal{X}$ and $\mathcal{Y}$ denote the spaces of original and noisy data, respectively. A randomizing mechanism $\mathcal{R}$ provides \emph{$\epsilon$-LDP} if, for all $x,\,x'\,\in\,\mathcal{X}$, and all $y\,\in\,\mathcal{Y}$, we have
\begin{equation*}
\mathbb{P}[\mathcal{R}(x)=y] \leq e^{\epsilon}\,\mathbb{P}\left[\mathcal{R}(x')=y\right]
\end{equation*}
\end{definition}
Recently, GeoI~\cite{AndresKostasCatuscia_GeoInd}, a variant of the local DP capturing the essence of the distance between locations~\cite{Bordenabe:14:CCS,Fernandes:21:LICS} has been in focus as a standard for privacy protection for location based services, being motivated by the idea of preserving the best possible quality of service despite the local obfuscation operated on the data.
\begin{definition}[Geo-indistinguishability~\cite{AndresKostasCatuscia_GeoInd}]
\label{def:geoind}
Let $\mathcal{X}$ be a space of locations and let $d_{\text{E}}(x,x')$ denote the \emph{Euclidean distance} between $x\in \mathcal{X}$ and $x'\in \mathcal{X}$. A randomizing mechanism $\mathcal{R}$ is \emph{$\epsilon$-geo-indistinguishable} if for all $x_1,\,x_2\in\mathcal{X}$, and every $y\in \mathcal{X}$, we have
\begin{equation*}
\mathbb{P}[\mathcal{R}(x)=y] \leq e^{\epsilon d_{\text{E}}(x_1,x_2)}\,\mathbb{P}\left[\mathcal{R}(x')=y\right]
\end{equation*}
\end{definition}
\begin{table}
\caption{List of Notations}\label{table:notations}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{cc}
\hline
Notation & Description \\ \hline
$\mathcal{X}$ & Domain of original locations \\
$d_{\mathcal{X}}$ & Distance on $\mathcal{X}$\\
$\mathcal{Y}$ & Domain of obfuscated locations \\
$d_{\mathcal{Y}}$ & Distance on $\mathcal{Y}$\\
$\mathbb{P}_{\mathcal{K}}\left[y\vert x\right]$ & Prob. that mechanism $\mathcal{K}$, applied to value $x$, reports $y$\\
$I$ & Fixed Edge in the network\\
$R(I)$ & Area of coverage by $I$\\
$m$ & Number of locations reported by each EV\\
$l_u$ & Vector of locations reported by EV $u$\\
$\mathcal{L}(t)$ & Set of location vectors received by $I$ at time $t$\\
$\mathcal{L}'(t)$ & Shuffled set of all individual locations queried at time $t$\\
$\mathcal{R}(t)$ & Set of nearest charging stations for $\mathcal{L}'(t)$\\
$G$ & Road network graph\\
$d_G$ & Travelling distance in graph $G$\\
\hline
\end{tabular}%
}
\end{table}
\begin{table}
\caption{List of Acronyms}\label{table:acronyms}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{cc}
\hline
Acronym & Description \\ \hline
DP & Differential Privacy\\
LDP & Local Differential Privacy\\
GeoI & Geo-Indistinguishability\\
AGeoI & Approximate Geo-Indistinguishability\\
EV & Electric Vehicle\\
IoV & Internet of Vehicles\\
RSU & Roadside Unit\\
MEC & Mobile Edge Computing\\
CoP & Cost of Privacy\\
QoS & Quality of Service\\
CS & Charging Station\\
\hline
\end{tabular}%
}
\end{table}
\section{Approximate geo-indistinguishability}\label{sec:approx_geo_ind}
In the classical framework of GeoI~\cite{AndresKostasCatuscia_GeoInd}, the space of the noisy data is, in theory, unbounded under the planar Laplace mechanism. Under a certain level of GeoI that is achieved, planar Laplace mechanism ensures a non-zero probability of obfuscating an original location to a privatized one which may be quite far, thus inducing a possibility of a substantial deterioration to the QoS of the users. This loss of QoS can be more sensitive in the context of the navigation of EVs, where it is extremely important to prioritize a bounded domain where a user is willing to drive -- this may be a result of time constraint, rising cost of fuel, geographical boundaries (e.g. international borders), etc. -- giving rise to an idea of \emph{area of interest} for each EV. This motivated us to extend the classical GeoI to a more generalized, approximate paradigm, inspired by the approach of the development of approximate DP from its pure counterpart.
Let $\mathcal{X}$ and $\mathcal{Y}$ be the spaces of the real and nosy locations equipped with distance metrics $d_{\mathcal{X}}$ and $d_{\mathcal{Y}}$, respectively. In general $\left(\mathcal{X},d_{\mathcal{X}}\right)$ and $\left(\mathcal{Y},d_{\mathcal{Y}}\right)$ may be different and unrelated. However, for simplicity, here we assume $\mathcal{X}\subseteq \mathcal{Y}$ and, therefore, $d_{\mathcal{X}}=d_{\mathcal{Y}}=d$, and we proceed to define the notion of \emph{approximate geo-indistinguishability}. Note that $d$ may not necessarily need to be symmetric, i.e., there may exist $x1,\,x_2\,\in\,\mathcal{Y}$ such that $d(x_1,x_2)\neq d(x_2,x_1)$.
\begin{definition}[Approximate geo-indistinguishability] A mechanism $\mathcal{K}$ is \emph{approximately geo-indistinguishable (AGeoI)} or $\emph(\epsilon,\delta)$-\emph{geo-indistinguishable} if for any set of values $S\subseteq\mathcal{Y}$ and any pair of values $x,x'\,\in\,\mathcal{X}$, there exists some $\epsilon,\,\delta\in\mathbb{R}_{\geq 0}$ such that $\delta e^{d(x,x')}\in[0,1]$, we have:
\begin{equation}\label{eq:approxGeoI}
\mathbb{P}_{\mathcal{K}}\left[y\,\in\, S\vert x\right]\leq e^{\epsilon\,d(x,x')}\mathbb{P}_{\mathcal{K}}\left[y\,\in\, S\vert x'\right]+\delta\,e^{d(x,x')}
\end{equation}
\end{definition}
One of the biggest advantages of DP and all of its variants that are accepted by the community is the property of compositionality, where the level of privacy can be formally derived with repeated number of queries. Thus, we now enable ourselves to investigate the working of the compositionality theorem with the approximate geo-indistinguishability which we defined, to stay consistent with the literature~\cite{DworkDP_Compositionality}.
\begin{restatable}{theorem}{compositionality}\label{th:compositionality}[Compositionality Theorem for AGeoI]
Let mechanisms $\mathcal{K}_1$ and $\mathcal{K}_2$ be $(\epsilon_1,\,\delta_1)$ and $(\epsilon_2,\,\delta_2)$ geo-indistinguishable, respectively. Then their composition is $(\epsilon_1+\epsilon_2,\,\delta_1+\delta_2)$-geo-indistinguishable. In other words, for every $S_1,\,S_2\subseteq \mathcal{Y}$ and all $x_1,\,x'_1,\,x_2,\,x'_2\,\in\,\mathcal{X}$, we have:
\begin{align}\label{eq:composition}
\mathbb{P}_{\mathcal{K}_1,\mathcal{K}_2}\left[(y_1,y_2)\,\in\, S_1\times S_2\vert (x_1,x_2)\right]\nonumber\\
\leq e^{\epsilon_1\,d(x_1,x'_1)+\epsilon_2\,d(x_2,x'_2)}\nonumber\\
\times\mathbb{P}_{\mathcal{K}_1,\mathcal{K}_2}\left[(y_1,y_2)\,\in\, S_1\times S_2\vert (x'_1,x'_2)\right]\nonumber\\
+\left(\delta_1+\delta_2\right)\,e^{d(x_1,x'_1)+d(x_2,x'_2)}
\end{align}
\end{restatable}
\begin{proof}
Let us simplify the notation and denote: $$P_i=\mathbb{P}_{\mathcal{K}_i}\left[y_i\in S_i\vert x_i\right]$$ $$P'_i=\mathbb{P}_{\mathcal{K}_i}\left[y_i\in S_i\vert x'_i\right]$$
$$\tilde{\delta}_i=\delta_i\,e^{d(x_i,x'_i)}$$
for $i\,\in\,\{1,2\}$.
As mechanisms $\mathcal{K}_1$ and $\mathcal{K}_2$ are applied independently, we have:
\begin{align}
\mathbb{P}_{\mathcal{K}_1,\mathcal{K}_2}\left[(y_1,y_2)\,\in\, S_1\times S_2\vert (x_1,x_2)\right]=P_1.P_2 \label{eq:simp1}\\
\mathbb{P}_{\mathcal{K}_1,\mathcal{K}_2}\left[(y_1,y_2)\,\in\, S_1\times S_2\vert (x'_1,x'_2)\right]=P'_1.P'_2 \label{eq:simp2}
\end{align}
Therefore, we obtain:
\begin{align}
\mathbb{P}_{\mathcal{K}_1,\mathcal{K}_2}\left[(y_1,y_2)\,\in\, S_1\times S_2\vert (x_1,x_2)\right]=P_1.P_2\nonumber\\
\leq\left(\min\left(1-\delta_1,e^{\epsilon_1\,d(x_1,x'_1)}P'_1\right)+\tilde{\delta}_1\right)\nonumber\\
\times\left(\min\left(1-\delta_2,e^{\epsilon_2\,d(x_2,x'_2)}P'_2\right)+\tilde{\delta}_2\right)\nonumber\\
\leq m_1m_2+\tilde{\delta}_1m_2+m_1\tilde{\delta}_2+\tilde{\delta}_1+\tilde{\delta}_2\nonumber\\
\left[\text{where }m_i=\min\left(1-\delta_i,e^{\epsilon_i\,d(x_i,x'_i)}P'_i\right)\right]\nonumber\\
\leq e^{\epsilon_1\,d(x_1,x'_1)+\epsilon_2\,d(x_2,x'_2)}P'_1P'_2\nonumber\\
+\tilde{\delta_1}-\tilde{\delta_1}\tilde{\delta_2}+\tilde{\delta_2}-\tilde{\delta_1}\tilde{\delta_2}+\tilde{\delta_1}\tilde{\delta_2}\nonumber\\
\leq e^{\epsilon_1\,d(x_1,x'_1)+\epsilon_2\,d(x_2,x'_2)}\nonumber\\
\times\mathbb{P}_{\mathcal{K}_1,\mathcal{K}_2}\left[(y_1,y_2)\,\in\,
S_1\times S_2\vert (x'_1,x'_2)\right]\nonumber\\
+\left(\delta_1+\delta_2\right)\,e^{d(x_1,x'_1)+d(x_2,x'_2)}\nonumber
\end{align}
\end{proof}
We now proceed to generalize the conventional planar Laplace mechanism~\cite{ChatzikokolakisElSalamounyPalamidessi+2017+308+328} to define the \emph{truncated Laplace mechanism} extended to a generic metric space.
\begin{definition}[Truncated Laplace mechanism]\label{def:TruncatedLaplace}
The \emph{truncated Laplace mechanism} $\mathcal{L}$ on a space $\mathcal{X}$ equipped with, not necessarily symmetric, distance metric $d$ truncated to a radius $r$, is defined as:
\begin{align}\label{eq:TruncatedLaplace}
\mathbb{P}_{\mathcal{L}}[y\vert x]
=\begin{cases}
c\,e^{-\epsilon\,d(y,x)} & \text{if }d(x,y)\leq r\\
0 & \text{otherwise}
\end{cases}
\end{align}
where $c$ is the truncated normalization constant defined such that $\int\limits_{y\in\mathcal{Y}}\mathbb{P}_{\mathcal{L}}[y\vert x]dy=1$, and $\epsilon$ is the desired privacy parameter. Let us call $r$ to be the \emph{radius of truncation} for $\mathcal{L}$.
\end{definition}
Note: In case of a discrete domain $\mathcal{Y}$, $c$ is defined by normalizing $\sum\limits_{y\in\mathcal{Y}}\mathbb{P}_{\mathcal{L}}[y|x]=1$, and in this case $\mathcal{L}$ is an extended truncated geometric mechanism \cite{ghosh2009universally} extended to a generic metric space.
\begin{restatable}{lemma}{positivedelta}\label{lem:positivedelta}
$\frac{e^{-\epsilon\,d(x_1,x_2)}\mathbb{P}\left[y\vert x_1\right]-\mathbb{P}\left[y\vert x_2\right]}{e^{(1-\epsilon)\,d(x_1,x_2)}}\,e^{r}\leq 1$, where $r$ is the radius of truncation for $\mathcal{L}$, as in \eqref{eq:approxGeoI}.
\end{restatable}
\begin{proof}
\begin{align}
\frac{e^{-\epsilon\,d(x_1,x_2)}\mathbb{P}\left[y\vert x_1\right]-\mathbb{P}\left[y\vert x_2\right]}{e^{(1-\epsilon)\,d(x_1,x_2)}}\,e^{r}\leq 1\nonumber\\
\iff c\left(e^{-\epsilon d(x_1,x_2)+d(x_1,y)}-e^{-\epsilon d(x_2,y)}\right)\nonumber\\
\leq e^{(1-\epsilon)d(x_1,x_2)-r}\label{eq:legaldelta}
\end{align}
Now we observe that $d(x_1,x_2)+d(x_1,y)\geq d(x_2,y)$ due to the fact that $d$ is a metric and it satisfies triangle inequality. Immediately, we have $e^{-\epsilon d(x_1,x_2)+d(x_1,y)}-e^{-\epsilon d(x_2,y)}\leq0$ for any $\epsilon\in\mathbb{R}_{\geq 0}$. Therefore, as $c\geq 0$, \eqref{eq:legaldelta} is trivially satisfied as the RHS is always non-negative.
\end{proof}
\begin{restatable}{theorem}{truncatedLap}\label{th:truncatedLap}
$\mathcal{L}$ satisfies $(\epsilon,\delta)$-geo-indistinguishability where
$\delta=\max\left\{\max\limits_{\substack{S\subseteq \mathcal{Y}\\x_1,x_2\in\mathcal{X}}}\frac{e^{-\epsilon\,d(x_1,x_2)}\mathbb{P}\left[y\vert x_1\right]-\mathbb{P}\left[y\vert x_2\right]}{e^{(1-\epsilon)\,d(x_1,x_2)}},0\right\}$.
\end{restatable}
\begin{proof}
Trivially $\delta e^{d(x_1,x_2)}>0$ for any $x_1,x_2\in\mathcal{X}$ as $\delta>0$. Moreover, Lemma \ref{lem:positivedelta} ensures that $\delta e^{d(x_1,x_2)}<1$.
Now observe that for every $S\subseteq\mathcal{Y}$ and for all $x_1,x_2\,\in\,\mathcal{X}$, we have:
\begin{align}
e^{-\epsilon\,d(x_1,x_2)}\mathbb{P}_{\mathcal{L}}\left[y\vert x_1\right]-\mathbb{P}_{\mathcal{L}}\left[y\vert x_2\right]\leq \delta\,e^{(1-\epsilon)\,d(x_1,x_2)}\nonumber\\
\implies \mathbb{P}_{\mathcal{L}}\left[y\vert x_1\right]-e^{\epsilon\,d(x_1,x_2)}\mathbb{P}_{\mathcal{L}}\left[y\vert x_2\right]
\leq \delta\,e^{d(x_1,x_2)}\nonumber
\end{align}
\end{proof}
The explicit process of sampling private locations satisfying AGeoI from a given set of original locations through truncated Laplace mechanism on a discrete location space has been described in Algorithms \ref{alg:AGeoIMech} and \ref{alg:AGeoISamp}.
\begin{algorithm}
\SetAlFnt{\small}
\textbf{Input:} Discrete domain of original locations: $\mathcal{X}$, Discrete domain of private locations: $\mathcal{Y}$, Desired privacy parameter: $\epsilon$, Desired truncation radius: $r$;
\textbf{Output:} Channel $C$ satisfying \eqref{eq:TruncatedLaplace}\;
\SetKwFunction{Fmain}{DTLap}
\SetKwProg{Fn}{Function}{:}{}
\Fn{\Fmain{$\mathcal{X},\mathcal{Y},\epsilon,r$}}{
Set $C\leftarrow \text{empty channel}$\;
Set $Y\leftarrow \text{empty list}$\;
\For{$x \in \mathcal{X}$}
{
$c_x=\frac{1}{\sum\limits_{\substack{y\in\mathcal{Y}\\d(x,y)\leq r}}e^{-\epsilon\,d(x,y)}}$ \;
\For{$y \in \mathcal{Y}$}
{
\eIf{$d(x,y)\leq r$}
{$C[x,y]=0$}
{$C[x,y]=c_x\,e^{-\epsilon\,d(x,y)}$}
}
}
\textbf{Return:} $C$\;
}
\caption{Discrete and truncated Laplace mechanism (DTLap)}\label{alg:AGeoIMech}
\end{algorithm}
\begin{algorithm}
\SetAlFnt{\small}
\textbf{Input:} Discrete domain of original locations: $\mathcal{X}$, Discrete domain of private locations: $\mathcal{Y}$, Desired privacy parameter: $\epsilon$, Desired truncation radius: $r$; Vector of original locations: $X$\;
\textbf{Output:} Corresponding vector of private locations: $Y$\;
\SetKwFunction{Fmain}{DTLapSamp}
\SetKwProg{Fn}{Function}{:}{}
\Fn{\Fmain{$\mathcal{X},\mathcal{Y},\epsilon,r,X$}}{
$C=$ \Call{DTLap}{$\mathcal{X},\mathcal{Y},\epsilon,r$}\;
Set $Y\leftarrow \text{empty list}$\;
\For{$x \in X$}
{
Randomly sample $y\in\mathcal{Y}\sim C[x,:]$\;
Append $y$ to $Y$
}
\textbf{Return:} $Y$\;
}
\caption{Sampling private locations with DTLap (DTLapSamp)}\label{alg:AGeoISamp}
\end{algorithm}
\section{System Model}\label{sec:system_model}
This section details our privacy-preserving model for finding an optimal charging station in IoV as a use case of the proposed AGeoI technique. We begin with a discussion of the location privacy problems inherent in finding optimal charging stations in the IoV, followed by road networking modelling, a description of the system architecture with the differentially private location sharing algorithm, defining the system tiers' trust relationship, and the privacy threat actor model.
\subsection{Problem Statement}
EVs have been identified as a critical component of future sustainable transportation systems to reduce CO2 emissions and have garnered significant attention from academia and business~\cite{kumar2020adoption}. Due to the limited vehicle onboard battery capacity, EVs may be required to visit charging stations (CSs) during journeys. It causes some drivers to have range anxiety, which is the vehicle has insufficient battery power to cover the travel needed to reach its intended destination. It is highlighted as one of the barriers to EVs' widespread adoption~\cite{bulut2017mitigating}. CSs can be not always readily available since it often takes a while to be sufficiently charged for EVs. A CS booking service can help to overcome range anxiety.
EVs may access such services through third-party providers to discover the nearest and readily available CSs to minimise charging wait times by static or live location queries. However, location sharing raises privacy challenges, such as vehicle tracking. GeoI technique provides a formal privacy guarantee for location queries. However, it is not highly applicable to this use case for two reasons. It does not consider the feasible locations where a vehicle can be present, and it does not stop vehicle tracking in the case of linked queries during the vehicle trajectory. Thus, a tailored privacy-preserving mechanism is facilitated by combining the proposed AGeoI technique and dummy location generation.
\subsection{Road Network Model}
Similar to~\cite{qiu2020location}, the road network $G$ is represented as a weighted directed graph $G=(N,E,W)$, where $N$ is the set of nodes, $E\subseteq N^2$ is the set of edges, and $W:N^2\rightarrow \mathbb{R}^{+}$ is the set of weights representing the minimum travelling distance between any two nodes. The nodes and edges correspond to junctions and road segments of the network, respectively. Each edge $e \in E$ is addressed by the pair of respective starting node, ending node, and a weight representing the travelling distance through that edge, i.e., $e=(N^s_e, N^e_e, w_e) \in N$, where the direction of the traffic is from $N^s_e$ to $N^e_e$ on $e$. For any $i\in N$ and $j\in N$, let the sequence of edges $(e_1,\ldots,e_r)$ denote a \emph{path} from node $i$ to node $j$ if $N^s_{e_1}=i$ and $N^e_{e_r}=j$. Hence, let $C(i,j)$ represent the set of paths that connect node $i$ to node $j$. Then $W$ is a $N\times N$ matrix, where
\begin{equation*}
W_{ij}=
\begin{cases}
\min\limits_{p\in C(i,j)}\sum\limits_{e\in p}w_e & \text{ if $C(i,j)\neq \phi$}\\
\infty & \text{ otherwise}
\end{cases}
\end{equation*}
Essentially $W_{ij}$ is the shortest travelling distance from node $i$ to node $j$ in the network. We shall address the quantity $W_{ij}$ as the \emph{traversal distance} between nodes $i$ and $j$ in the graph $G$ and denote it as $d_{G}(i,j)$ for every $(i,j)\,\in N^2$. Note that, as $G$ is a directed graph, $d_{G}$ may not be symmetric.
\subsection{System Architecture}
\begin{figure}[ht!]
\centering
\includegraphics[width=0.48\textwidth]{Figures/EV_system_architecture.png}
\caption{System Architecture (EV:Electric Vehicle, RSU: Roadside Unit, MEC: Mobile-Edge Computing Unit)}
\label{fig:system_arch}
\end{figure}
The Internet of Vehicles (IoV) applications are reshaping transportation systems by minimising human error, advancing travel convenience, and decreasing energy, operational and environmental costs~\cite{omar2016wireless,Duan2020emerging,ji2020Artificial}. EVs have emerged as a viable technology for reducing carbon emissions and travelling costs~\cite{patil2020grid}. However, range anxiety is one of the major challenges of their wide adoption. Vehicular location data can be utilised to optimise the vehicle charging plan and mitigate range anxiety. Third-party services such as Chargemap\footnote{\url{https://chargemap.com/}}, Ampeco\footnote{\url{https://www.ampeco.com/}}, and PlugShare\footnote{\url{https://www.plugshare.com/}} can recommend available and the closest charging stations for the users. However, the users are required to trust these third-party service providers to use these services, which presents significant privacy concerns in the honest-but-curious service provider threat model.
ITS integrates connected vehicles, cloud computing, and IoV with transportation infrastructures for advantages on reduced fuel and energy use, $CO_2$ emission, traffic congestion, and road safety. Future transportation system would be a distributed computing platform which enables autonomous driving functions and cooperation between the vehicles based on vehicle to vehicle (V2V), vehicle to infrastructure (V2I), and vehicle to cloud (V2C) communications with methods for big data analysis with high reliability and ultra-low-latency. The wireless communication is enabled through technologies such as IEEE 802.11p DSRC/WAVE (Dedicated Short Range Communication/Wireless Access in Vehicular Environments) and cellular advances such as C-V2X~\cite{hasrouny2017vanet,vukadinovic20183gpp}. MEC technology is an industrial specification from ETSI for providing cloud computing capabilities at the Edge of a mobile network which is also applicable for autonomous driving tasks~\cite{Lee2018exploring}.
As the system architecture is depicted in Figure~\ref{fig:system_arch}, the vehicles are part of a three-tier cloud computing architecture and connected to Roadside Units, MEC Server, and the Core Cloud Network through a secure communication channel, where these components are considered trusted. The Core Cloud Network enables the connection between the vehicle and third-party services, including charging station recommender, as the main consideration of this paper. However, it is not possible to ensure fully trusted third-party services that they will not utilise vehicular location data for further objectives. Thus, the honest-but curious threat model is considered for the third party service provider and only privatised vehicular location data is shared in our proposed architecture. The roles of the system components are described in the following.
\subsubsection{Vehicle Tier}
In this paper, we fix a road network $G$ with nodes $G(N)$ and edges $G(E)$. We choose an arbitrary edge $I\in G$, and focus on the queries made by the EVs in $I$'s range of coverage, $R(I)$, provided by its \emph{road side unit (RSU)}. When an EV moves from the area of coverage of one Edge cloud to another, we can assume the queries and the privacy threats against the Edge to reset as each Edge communicates with the Cloud-based services and the third-party service providers. At any given time, an EV $u$ locally obfuscates its true location $x^u\in R(I)$ to $x^u_1\in R(I)$ using truncated Laplace mechanism guaranteeing AGeoI and generates $m-1$ feasible dummy locations $\{x^u_2,\ldots,x^u_m\}\in R(I)^{m-1}$ in the coverage area of the respective Edge. Then $u$ reports the vector of $m$ locations, $l_u=\left(x^u_1,\ldots,x^u_m\right)$, to $I$ for the Edge to process and communicate the query to the Cloud services and the third-parties to find the nearest available charging stations in $R(I)$.
Figure~\ref{fig:dummylocations} shows an example of reported $10$ dummy locations with the privatised location for two respective time windows, where the dummy locations in the following time window can have a feasible link at least one of the preceding dummy locations. Toy examples are depicted on Figures~\ref{fig:toy1} and \ref{fig:toy2} as more simplistic explanations of the proposed privacy-preserving mechanism, where the first represents a static query on an exampler discrete road network, and the following one represents linked dynamic queries on the same exampler discrete road network.
\begin{figure*}[ht!]
\centering
\includegraphics[width=1\textwidth]{Figures/folium-timestamps.png}
\caption{Reported dummy and privatised locations for two respective time windows (White Pins: Privatised locations, Orange Pins: Dummy locations in $1st$ Time window, Blue Pins: Dummy locations in $2nd$ Time window)}
\label{fig:dummylocations}
\end{figure*}
\begin{figure}[ht!]
\centering
\includegraphics[width=0.48\textwidth]{Figures/toy-example.png}
\caption{A toy example for a static location query on discrete road network}
\label{fig:toy1}
\end{figure}
\begin{figure}[ht!]
\centering
\includegraphics[width=0.48\textwidth]{Figures/toy-example2_V1.png}
\caption{A toy example for linked 3 location queries on discrete road network}
\label{fig:toy2}
\end{figure}
\subsubsection{Edge Tier} Given the large volume of data generated and shared between vehicles and infrastructure, installing Edge cloud close to the vehicles are needed to host the off-board vehicular services, which require a low access latency from the onboard vehicular services~\cite{Lee2018exploring}. Along with essential data processing and forwarding functions, the Edge tier provides another layer for data aggregation and deploying additional privacy-preserving measures before sharing the data with any third parties. The Edge is responsible for forwarding the location vectors received from the EVs at any time-step by disassociating the links between the location vectors and the corresponding vehicles, inducing an additional layer of anonymity for the Cloud and the third-party. Suppose at time $t$, Edge $I$ receives the set of location vectors from EVs $u_1,u_2,\ldots,u_{k_t}$ in $R(I)$. Therefore, $I$ receives the collection $\mathcal{L}(t)=\{l_{u_i}: i\in[k_t]\}$. Keeping a record of the IDs to link the location vectors with their respective senders, the Edge shuffles all the individual locations from every $l_u\in\mathcal{L}(t)$ and communicates this collection of all the reported locations to the Cloud and the third-party service provider. In particular, the Edge forwards the shuffled collection of locations $\mathcal{L}'(t)=\left\{x^u_i\colon u\in\{u_1,\ldots,u_{k_t}\}, i\in[m]\right\}$ to the cloud and the third-party service provider(s).
\subsubsection{Cloud Tier} It is expected to provide computation and storage capabilities for top-level processes, including data sharing interfaces for third-party services.
\subsubsection{Third-party Service Provider} It is the external party to ITS and is expected to enhance the quality of the function for finding the available charging stations for the vehicles by receiving search queries compromised of privatised and dummy location vectors for the respective vehicles.
\subsubsection{Communication Channel} ITS comprises a network of roadside units (RSU), vehicle on-board electronic control units (ECU), and distributed cloud computing and storage services. Wireless communications are enabled for V2V (Vehicle-to-Vehicle), V2I (Vehicle-to-Infrastructure) and V2X(Vehicle-to-Everything) by the technologies such as IEEE 802.11p DSRC/WAVE (Dedicated Short Range Communication/Wireless Access in Vehicular Environments), cellular advances such as C-V2X, and the long-term evolution for vehicles (LTE-V)~\cite{seo2016lte}. Confidentiality of the wireless communication channel is secured by public key infrastructure (PKI) encryption methods which is beyond the scope of this work.
\subsection{Privacy Threat Landscape}
In real-time IoV location-based applications, vehicle users who wish to receive services tailored to their current locations are needed to share their location with the service provider. Different applications may have varying needs for data sensitivity. Data perturbation techniques are often used in private data sharing by introducing uncertainty in the data, which entails a trade-off in terms of the related application's quality of service in general. For example, EV users who want to locate the nearest available charging station during travel send a query with their current location to a service provider using the Intelligent Transportation Systems (ITS) infrastructure.
Charging an EV can take up to several hours. Due to the lengthy charging cycle, making a bad choice of charging station, such as one with no empty charging piles, may result in increased range anxiety and waiting time to charge the vehicle. Thus, choosing the nearest available charging station is necessary to alleviate range anxiety and wait time. Such information can be provided by real-time location data mining systems which often will be a third-party operator for a specific area~\cite{tian2016real}.
We consider the system into three categories: (i) the vehicle users (data subject), (ii) ITS infrastructure, including the MEC and Cloud Tiers (data controller and data processor), and (iii) the third party that receives the privatised data from the deployed deployment privacy-preserving mechanisms. The third party is assumed to be an EV charging management system, which may operate in a registration-based approach for a specific area.
The system includes several categories of data flow with different sensitivity: (i) publicly available data such as road network and location of the charging stations, (ii) not-private data such as availability of the charging stations, and (iii) private data as the real-time location of the vehicle users.
The vehicular location data may also be subject to unauthorised use, data inference, retaining or disclosure besides its primary use. Thus, a privacy-preserving mechanism can be utilised for location data sharing to provide a privacy guarantee while the users can receive the same or similar quality of service. Thus, the third-party provider is considered an honest-but-curious adversary model, which assumes it is honest in accurately executing the protocol required to provide location data. However, it may be curious to infer users' private information based on the obtained location data~\cite{paverd2014modelling}.
\subsubsection{Privacy Requirements}
In distributed systems, privacy guarantee and trust establishment remain open research challenges since multiple actors are involved in data share and computation. Vehicular location privacy can be ensured by considering three privacy properties in the settings of the proposed system. These are indistinguishability, unlinkability and confidentiality~\cite{wagner2018technical}. Indistinguishability means that the threat actor cannot distinguish between two outcomes of a privacy mechanism, which is provided by deploying the approximate geo-indistinguishability into the Vehicle Tier. The second property, unlinkability, refers to the threat actor's feature that cannot link the released data. It is ensured by deploying the shuffle model at the Edge Tier at each timestamp before releasing data to the third party. The last property, confidentiality, can be ensured by securing the communication channel by the appropriate encryption mechanisms, which is beyond the scope of this paper.
\subsubsection{Privacy Threats}
In this work, we mainly address two notable sources of threat that might put the EV's privacy at risk: a) a potential identification of the individual locations of the queries made by the EV, and b) tracing the journeys of the EVs by dynamically exploiting the queries made along their journeys.
\paragraph{Location identification} It is essential that the individual locations of the queries made by the EVs are protected from being identified. For the use case of location charging stations, it is essential that the privatized version of the true location of the EV is within a certain radius of interest w.p. 1, making sure that the reported location is within a feasible and drivable distance away and, most importantly, within the area of coverage of the Edge where its true location lies. Therefore, we proceed to define AGeoI as an extension of GeoI, which is the state-of-the-art standard for location privacy. Therefore, to ensure the privacy of the position of any given query in the road network, the EVs locally obfuscate their true locations using the truncated Laplace mechanism with their desired parameter $\epsilon$ and the radius of truncation $r$, which, in turn, decide the value of $\delta$.
\paragraph{Journey tracing}
It is often the case that a certain EV might enquire about the nearest available charging station, but choose to proceed with the journey at that point of time, and raise further queries along the journey. In our model we capture this realistic setting of allowing multiple queries to be made by the EV in a given journey. This immediately leads to a threat of approximately tracing the trajectory of the journey of an EV by interpolating the locations of query, despite having each individual location being AGeoI.
This is due to the fact that the obfuscated location of each query is not distinguishable from the real location, but they are not too far off from each other with a very high probability. Therefore, if the number of queries made in a single journey is large, it could be fairly straightforward to approximately deduce the trajectory of the journey.
Cunningham et al.~\cite{Cunningham2021Trace} proposed a mechanism to securely share trajectories under LDP. However, the authors in \cite{Cunningham2021Trace} assumed a model of offline sharing of the entire trajectory and, hence, sanitizing it with the proposed mechanism to engender a LDP guarantee. In our setting, this method cannot be directly implemented as we are working in a dynamic environment where the queries made by the EVs are in real time, with the server not having any prior knowledge on the number or the location of the queries made by a certain EV. Therefore, the mechanism of \cite{Cunningham2021Trace} cannot trivially be extended in the online location-sharing environment, and hence, the threat of the adversaries able to reconstruct the journey of a particular EV with a high number of queries remains as a concern.
To mitigate this problem, for the $k^{th}$ query with $t>1$, an EV generates $m-1$ dummy locations in the area of coverage of the Edge $I$ and reports the vector of $m$ locations $l^{k}_u=\left(x^u_1,\ldots,x^u_m\right)$ to the RSU of $I$, where $x^u_1$ is the privatized version of the true location $x^u$ of $u$, and $x^u_2,\ldots,x^u_m$ are randomly generated dummy locations from $R(I)$ such that every dummy location corresponds to at least one of the locations in $l^{k-1}_u$ which ensures a feasible~\footnote{w.r.t. speed, travelling conditions, etc.} journey in between the times of the $k^{th}$ and $k-1^{st}$ queries, as illustrated by Figure \ref{fig:dummylocations}, making it impossible for the Edge to exclude some of the dummy locations as being unrealistic as far as tracing the journey is concerned. For $k=0$, i.e., an EV can generate any random $m-1$ dummy locations in $R(I)$. This procedure recursively ensures that with every given query the adversary will have at least $m$ possible trajectories that the EV could have realistically followed, making it improbable to be able to conclude about the actual one. We shall omit the superscript of the number of the query made by the EV if it is clear from or insignificant in the context.
Note that we assume that the principle adversary lies at the third-party service provider, and despite having individually geo-indistinguishable locations, it is possible to trace the entire trajectory with a fair bit of accuracy by interpolating the individual geo-indistinguishable locations of queries. At any given time, the Edge collects all the reported locations from the querying EVs, shuffles them by effacing the links between the location vectors and the corresponding EVs, and sends this jumbled collection of all the reported locations in the network for knowing their respective nearest available charging stations to the third-party service provider. After receiving the response, the Edge, which internally keeps the record of the IDs of the EVs against their queried locations, assigns the corresponding vector of locations of the nearest available charging stations to each EV and communicates them back to the respective vehicles.
In other words, at time $t$, if the Edge receives the location vectors from $k_t$ querying EVs as $\mathcal{L}(t)=\{l_{u_1},\ldots, l_{u_{k_t}}\}$, the Edge is responsible for shuffling all the individual location points in these reported vectors and forward the scrambled collection of locations $\mathcal{L}'(t)=\{x^u_i\colon u\in\{u_1,\ldots,u_{k_t}, i\in[m]\}$ to the Cloud and the third-party, while internally keeping a track of the IDs of the EVs to reconnect the query-response back to the corresponding vehicles. Setting $\hat{x}$ as the location of the nearest available charging station from location $x$ in $R(I)$, the Edge receives $\mathcal{R}(t)=\{\hat{x}^u_i\colon u\in\{u_1,\ldots,u_{k_t}, i\in[m]\}$ as the response from the third-party. After this, matching the stored IDs of the EVs with the locations of the charging stations, the Edge communicates the response vector $\hat{l}_u=(\hat{x}^u_i:i\in [m])$ back to the corresponding EV $u$. Then the EV can choose to navigate to $\argmin\limits_{x\in\hat{l}_u}\{d_G(x,x_u)\}$, where $x_u$ is the real location of EV $u$. The overview of this mechanism is explained in Figure \ref{fig:system_arch}.
\section{Cost of privacy analysis} \label{sec:cost_of_privacy_analysis}
\begin{definition}[Cost of privacy]\label{def:CoP}
Suppose an EV $u$ at location $x^u$ chooses to locally obfuscate its real location of query as $x^u_1$ using the truncated Laplace mechanism $\mathcal{L}_{\epsilon,r}$ satisfying $(\epsilon,\delta)$-geo-indistinguishability with a corresponding radius of truncation $r$. Then we define the \emph{cost of privacy (CoP)} of EV $u$ as $\operatorname{CoP}(u,\mathcal{L}_{\epsilon,r})=\vb{c}(x^u,\hat{x}^u_1)-\vb{c}(x^u,\hat{x}^u)$, where $\hat{x}^u$ and $\hat{x}^u_1$ are the nearest available charging stations in the network to $x^u$ and $x^u_1$, respectively, and $\vb{c}:G(N)^2\mapsto \mathbb{R}^+$ is any cost function that reflects the cost of commute from locations $x$ to $y$ in the network.
\end{definition}
In other words, CoP, as in Definition \ref{def:CoP}, essentially captures the \emph{extra} cost that an EV needs to cover as a result of the privatized location it reports to the Edge satisfying AGeoI, as opposed to its true location. For the purpose of this paper and simplicity of the analysis, we considered the cost function as the travelling distance in the network, i.e., $\vb{c}=d_G$. However, in practice, any suitable cost function could be used (e.g. fuel efficiency, time, etc.) could be used as $\vb{c}$, depending on the context and requirement of the architecture.
To formally characterize and analyze the CoP of the EVs in the network, inspired from the classical version of \emph{Voronoi decomposition}, we extend the concept in the setting of our road network in the network coverage for a fixed Edge w.r.t. graph-traversal distance, $d_g$.
\begin{definition}[Voronoi decomposition]\label{def:Voronoi}
Let $G$ be the graph representing the road network equipped with travelling distance $d_G$. Let the set of charging stations in $G$ be $C_G=\{c_1,\ldots,c_{n_G}\}$. Then the \emph{Voronoi decomposition} on $G$ w.r.t. $C_G$ is defined as $\vb{V}_G=\{V_i\colon\,i\in[n_G]\}$ such that $V_i\cap V_j=\phi$ for any $i\neq j$ and $\bigcup\limits_{i\in[n_G]}V_i=G$, where
\small
\begin{equation*}
V_i=\{x\in G\colon\,d_G(x,c_i)\leq d_G(x,c_j)\,\forall\,j\in[n_G],j\neq i\}
\end{equation*}
\par
\end{definition}
\begin{definition}[Closed ball around a location]\label{def:ClosedBall}
For any $x \in G$ and $r\in\mathbb{R}_{\geq 0}$, the \emph{closed ball} of $x$ of radius $r$ is defined as $\beta_r(x)=\{y\in G\colon\,d_G(x,y)\leq r\}$
\end{definition}
\begin{definition}[Fenced Voronoi decomposition]\label{def:FenceVoronoi}
For any $r\in\mathbb{R}_{\geq 0}$, and for charging station $i$, let the \emph{$r$-fenced Voronoi decomposition} on road network $G$ be defined as $V^{-r}_G=\{V_i^{r}\colon\,i\in[n_G]\}$ such that $V^{-r}_i\cap V^{-r}_j=\phi$ for any $i\neq j$ and $V_i^{-r}=\{x\in V_i \colon\, B_r(x)\subseteq V_i\}$. In other words, $V_i^{-r}$ is essentially constructing an area which is contained within $V_i$ being restricted by a \emph{fence} at a distance $r$ from the edge of $V_i$.
\end{definition}
\begin{restatable}{theorem}{zeroCoP}\label{th:zeroCoP} Suppose an EV $u$ positioned at $x^u$ on $G$ obfusucates its location using AGeoI with any radius of truncation $r\in\mathbb{R}_{\geq 0}$. Let $\hat{x}^u$ be the location of the nearest available charging station to the true location $x^u$. Then we have $\mathbb{P}\left[\operatorname{CoP}(u,\mathcal{L}_{\epsilon,r})=0\right]=1$ for every $x^u\in V_{\hat{x}^u}^{-r}$. In other words, if an EV $u$ lies in the $r$-fenced Voronoi decomposition for its nearest available charging station, we get \emph{zero cost for privacy} for $u$ w.p. 1.
\end{restatable}
\begin{proof}
Immediate from the definition of fenced Voronoi decomposition.
\end{proof}
\begin{restatable}{theorem}{zeroCoPProb}\label{th:zeroCoPProb} Suppose an EV $u$ lies in $V_{\hat{x}^u}\setminus V_{\hat{x}^u}^{-r}$ and it uses AGeoI to obfuscate its true location $x^u$ to $x^u_1$ with a radius of truncation $r$ and privacy parameter $\epsilon$ for making a private query to the Edge. Then $$\mathbb{P}\left[\operatorname{CoP}(u,\mathcal{L}_{\epsilon,r})=0\right]=1-\sum\limits_{x^u_1\in V_{\hat{x}^u}^c}ce^{-\epsilon d_G(x^u,x^u_1)}$$ where $c$ is the normalizing constant of the truncated Laplace mechanism as in Definition \ref{def:TruncatedLaplace}.
\end{restatable}
\begin{proof}
To compute $\mathbb{P}\left[\operatorname{CoP}(u,\mathcal{L}_{\epsilon,r})=0\right]$, we only need to exclude the possibilities where the reported location of the EV lies outside the Voronoi decomposition of the station $\hat{x}^u$, which, essentially, is $1-\sum\limits_{x^u_1\in V_{\hat{x}^u}^c}ce^{-\epsilon d_G(x^u,x^u_1)}$.
\end{proof}
\section{Experimental Study}\label{sec:experiments}
In this section, a series of experiments are conducted to evaluate the proposed method in Edge cloud-assisted intelligent transportation systems settings for EV charging station finding use case. The objectives of experiments are (i) to validate proposed theoretical claims and solutions through experiments, (ii) to apply the method to find the nearest available charging station for EVs as a use case study, and (iii) to investigate the cost of privacy for the use case. The experiments are deployed in a laptop with Intel core I7 processor, 16 GB random access memory, and Ubuntu 20.04 operation system using standard Python packages.
\subsection{Dataset Preparation}
The regional map, exported from OpenStreetMap\footnote{\url{http://www.openstreetmap.org}}, is utilised in this study. The cost of privacy is analysed by calculating the extra routing distance to the identified optimal charging station due to additional noise in vehicular locations in the queries. Thus, the cost of privacy depends on the sparsity of the charging stations. Two datasets with different densities are prepared to comprehend the impact of this sparsity. The United States Department of Energy allows downloading up-to-date data of the existing and planned alternative fuel stations with their geographical coordinates~\cite{chargingstations}. The first dataset is exported, containing the locations of 404 existing EV charging stations in San Francisco. For the second dataset, we followed the assumption that EV charging stations will be more spread in the future, and parking locations are likely to deploy charging facilities. Thus the second dataset is prepared by merging the existing and planned charging station locations with on-street and off-street parking locations provided by DataSF~\cite{chargingstations2}. It contains 716 independently distributed locations.
The EPFL (Swiss Federal Institute of Technology - Lausanne) mobility dataset contains the GPS records of the 536 taxi trajectories in San Francisco for four weeks. The data format includes a taxi identifier, latitude, longitude, state of the taxi (vacant or occupied), and UNIX epoch timestamp~\cite{piorkowski2009crawdad}. It is possible to split the entire taxi trajectory into realistic trajectories of each customer because the dataset contains the information for the occupancy of the taxis. By doing so, we could export over 450 thousand trajectories. 536 trajectories, one randomly chosen from each taxi, are utilised in this study.
\subsection{Experimental Setup}
The edges of the road network $G$ are truncated into discrete segments with equal $k$ travel distance, similar to the work in~\cite{qiu2020location}. DTLap (e.g., Algorithm 1) is utilised to generate the privacy channel by using the Laplace mechanism for the user's desired values for privacy budget $\epsilon$ and truncation radius $r$. Following this, DTLapSamp (e.g., Algorithm 2) is used to generate privatised locations with respect to the users' real locations. The generation of discrete road segments is done by setting the parameter $k$ to 100 metres. From the EPFL mobility dataset, 536 random traces, one per taxi, are selected. Each trace contains a series of GPS coordinates and 3 randomly selected along each for the real locations of the queries. As a result, the experiments are carried out for 1608 locations where each 3 of these are linked. The parameters of $\epsilon$ and $r$ are varied in the range of 0.2 to 2, and 1 to 20, respectively.
The experimental scenario considers the steps; (i) vehicles made three location queries during their journeys to find the nearest available charging stations where each location query contains a privatised location and $10$ dummy locations as a vector, (ii) the Edge cloud collects all privatised locations and dummy locations with to send the third-party through the core cloud as a vector of all locations, (iii) the third-party responds the vector of locations with the nearest available CS for each, (iv) the Edge responds vectors from each vehicle. Hence, the Edge cannot distinguish the privatised locations among the dummy locations. The experimental scenario is executed on two different CSs densities.
\begin{figure*}[t]
\centering
\includegraphics[width=0.8\textwidth]{Figures/Figure1.png}
\caption{Cost of privacy for varying $\epsilon$ or $r$ of AGeoI ($1st$ row is for sparse CSs, $2nd$ row is for dense CSs)}
\label{fig:cost_of_privacy}
\end{figure*}
\begin{figure*}[ht!]
\centering
\includegraphics[width=1\textwidth]{Figures/Figure2.png}
\caption{Fraction of zero cost of privacy for varying $\epsilon$ or $r$ of AGeoI ($1st$ row is for sparse CSs, $2nd$ row is for dense CSs)}
\label{fig:privacy_for_free}
\end{figure*}
\subsection{Results}
DP approaches introduce a trade-off between privacy and data utility, with a higher level of privacy requiring a greater level of noise. Due to the fall in data utility, the efficacy of the respective service may correspondingly decrease. It is termed the cost of privacy in this study. However, there is a finite number of cases with the maximum efficacy for many applications. Thus, there must be a sweet spot of privacy for free where the user will receive the maximum efficacy with some level of privacy. It is also valid for our use case, in which a vehicle must be able to utilise a finite number of unique CSs.
The following results are achieved by carrying out the experiments for $3$ linked queries of $536$ randomly selected vehicle trajectories for varying values of $\epsilon$ or $r$ ranging from $0.2$ to $2$, and $1$ to $20$, respectively. Figure~\ref{fig:cost_of_privacy} demonstrates the plot of cost of privacy in terms of the extra travelling distance due to privacy-preserving mechanism, where a similar pattern is observed for both of the datasets.Another observation is that a high incidence of queries resulted in having zero cost for privacy-preserving. Figure~\ref{fig:privacy_for_free} demonstrates fraction of the queries with privacy for free, where two datasets followed similar patterns. Vehicle queries contains dummy locations with their privatised locations. It is possible that the dummy locations can sometimes provide a better utility, but our experiments consider the utility of privatised location as the worst-case for analysis.
Figure~\ref{fig:cost_of_privacy} demonstrates that our mechanism provides a negligible cost of utility-loss for the formal privacy-gain enjoyed by the EVs. By increasing the truncation radius, it is observed that an abrupt drop in the distance between the location of the nearest available charging station for the true location of the query and that of the privatised one, implying that the cost of the extra travel distance needed to be taken due to the AGeoI mechanism is almost negligible. A similar trend is seen for the varying $\epsilon$ with a fixed radius. As the level of privacy decreases, the fraction of EVs in the network enjoying \emph{privacy for free} grows to be more than $60\%$ for a radius of truncation of merely 10 road segments, where each segment is 100 meters long, for $\epsilon\geq 0.5$. However, more than $90\%$ of the EVs achieve a zero cost of privacy for $\epsilon\geq 1.5$, irrespective of the truncation radius in Figure~\ref{fig:privacy_for_free}. The user can enhance the level of perturbation in queries by decreasing the $\epsilon$ and increasing the $r$. Due to increasing perturbation in disclosed location, the width of the confidence interval for zero cost of privacy increases, as seen in Figure~\ref{fig:privacy_for_free}. The likelihood of achieving zero cost of privacy fluctuates over a wider range, and it does not monotonically decrease with the growing radius due to rising randomness.
\section{Conclusion}\label{sec:conclusion}
This paper studied a fundamental problem on the risk of privacy violations for EVs dynamically querying for charging stations along their journeys. The problem's setting has not been addressed in the literature explicitly, and some of the related techniques along the lines of privacy-preserving vehicle routing cannot be adapted directly in practical model we considered in this paper.
To resolve this, firstly, we have theorized the notion of approximate geo-indistinguishability and justified its mathematical soundness and applicability by proving the compositionality theorem, which is a novelty in itself, allowing us to attain GeoI in a strictly bounded space of noisy data. We derived the appropriate privacy parameters to prove that the truncated Laplace mechanism satisfies approximate geo-indistinguishability and, therefore, used it to propose a location-privacy preserving method for EVs querying for charging stations. Our method induces two-way privacy protection: on the individual locations of the queries and for the trace of the entire journeys.
We performed experiments on datasets with real vehicle traces and locations of charging stations in San Francisco, and illustrated the privacy-utility trade-off for our method for two different settings of the sparsity of charging stations in the network. We observed a consistent trend across all the experiments which suggests that a substantial majority of the EVs have privacy for free, i.e., suffer no loss of utility even for fairly high-level approximate geo-indistinguishability attained. In general, we observe that the cost of privacy induced by our method is fairly low across settings, thus, ensuring formal privacy protection for the EVs without incurring a high price for that as far as the journey is concerned. We analyzed this cost of privacy under our method using Voronoi decomposition to draw insight into the working of our method. Construction Voronoi diagram of the road network for a location-based service will allow us to analyse further the privacy-utility trade-off, which is a promising avenue of research.
\begin{acks}
The authors would like to thank Professor Graham Cormode for the insightful discussions that supported the development of this study. Ugur Ilker Atmaca and Sayan Biswas are shared co-first authors who have worked together on a publication and contributed equally. The authors would also like to extend their sincere thanks to the ERC (European Research Council ) Advanced Grant of Catuscia Palamidessi which supported Sayan Biswas' research visit at WMG, the University of Warwick, which enabled this successful collaboration.
\end{acks}
\bibliographystyle{ACM-Reference-Format}
\section{Introduction}
ACM's consolidated article template, introduced in 2017, provides a
consistent \LaTeX\ style for use across ACM publications, and
incorporates accessibility and metadata-extraction functionality
necessary for future Digital Library endeavors. Numerous ACM and
SIG-specific \LaTeX\ templates have been examined, and their unique
features incorporated into this single new template.
If you are new to publishing with ACM, this document is a valuable
guide to the process of preparing your work for publication. If you
have published with ACM before, this document provides insight and
instruction into more recent changes to the article template.
The ``\verb|acmart|'' document class can be used to prepare articles
for any ACM publication --- conference or journal, and for any stage
of publication, from review to final ``camera-ready'' copy, to the
author's own version, with {\itshape very} few changes to the source.
\section{Template Overview}
As noted in the introduction, the ``\verb|acmart|'' document class can
be used to prepare many different kinds of documentation --- a
double-blind initial submission of a full-length technical paper, a
two-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''
journal article, a SIGCHI Extended Abstract, and more --- all by
selecting the appropriate {\itshape template style} and {\itshape
template parameters}.
This document will explain the major features of the document
class. For further information, the {\itshape \LaTeX\ User's Guide} is
available from
\url{https://www.acm.org/publications/proceedings-template}.
\subsection{Template Styles}
The primary parameter given to the ``\verb|acmart|'' document class is
the {\itshape template style} which corresponds to the kind of publication
or SIG publishing the work. This parameter is enclosed in square
brackets and is a part of the {\verb|documentclass|} command:
\begin{verbatim}
\documentclass[STYLE]{acmart}
\end{verbatim}
Journals use one of three template styles. All but three ACM journals
use the {\verb|acmsmall|} template style:
\begin{itemize}
\item {\texttt{acmsmall}}: The default journal template style.
\item {\texttt{acmlarge}}: Used by JOCCH and TAP.
\item {\texttt{acmtog}}: Used by TOG.
\end{itemize}
The majority of conference proceedings documentation will use the {\verb|acmconf|} template style.
\begin{itemize}
\item {\texttt{acmconf}}: The default proceedings template style.
\item{\texttt{sigchi}}: Used for SIGCHI conference articles.
\item{\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.
\item{\texttt{sigplan}}: Used for SIGPLAN conference articles.
\end{itemize}
\subsection{Template Parameters}
In addition to specifying the {\itshape template style} to be used in
formatting your work, there are a number of {\itshape template parameters}
which modify some part of the applied template style. A complete list
of these parameters can be found in the {\itshape \LaTeX\ User's Guide.}
Frequently-used parameters, or combinations of parameters, include:
\begin{itemize}
\item {\texttt{anonymous,review}}: Suitable for a ``double-blind''
conference submission. Anonymizes the work and includes line
numbers. Use with the \texttt{\acmSubmissionID} command to print the
submission's unique ID on each page of the work.
\item{\texttt{authorversion}}: Produces a version of the work suitable
for posting by the author.
\item{\texttt{screen}}: Produces colored hyperlinks.
\end{itemize}
This document uses the following string as the first command in the
source file:
\begin{verbatim}
\documentclass[sigconf]{acmart}
\end{verbatim}
\section{Modifications}
Modifying the template --- including but not limited to: adjusting
margins, typeface sizes, line spacing, paragraph and list definitions,
and the use of the \verb|\vspace| command to manually adjust the
vertical spacing between elements of your work --- is not allowed.
{\bfseries Your document will be returned to you for revision if
modifications are discovered.}
\section{Typefaces}
The ``\verb|acmart|'' document class requires the use of the
``Libertine'' typeface family. Your \TeX\ installation should include
this set of packages. Please do not substitute other typefaces. The
``\verb|lmodern|'' and ``\verb|ltimes|'' packages should not be used,
as they will override the built-in typeface families.
\section{Title Information}
The title of your work should use capital letters appropriately -
\url{https://capitalizemytitle.com/} has useful rules for
capitalization. Use the {\verb|title|} command to define the title of
your work. If your work has a subtitle, define it with the
{\verb|subtitle|} command. Do not insert line breaks in your title.
If your title is lengthy, you must define a short version to be used
in the page headers, to prevent overlapping text. The \verb|title|
command has a ``short title'' parameter:
\begin{verbatim}
\title[short title]{full title}
\end{verbatim}
\section{Authors and Affiliations}
Each author must be defined separately for accurate metadata
identification. As an exception, multiple authors may share one
affiliation. Authors' names should not be abbreviated; use full first
names wherever possible. Include authors' e-mail addresses whenever
possible.
Grouping authors' names or e-mail addresses, or providing an ``e-mail
alias,'' as shown below, is not acceptable:
\begin{verbatim}
\author{Brooke Aster, David Mehldau}
\email{dave,judy,steve@university.edu}
\email{firstname.lastname@phillips.org}
\end{verbatim}
The \verb|authornote| and \verb|authornotemark| commands allow a note
to apply to multiple authors --- for example, if the first two authors
of an article contributed equally to the work.
If your author list is lengthy, you must define a shortened version of
the list of authors to be used in the page headers, to prevent
overlapping text. The following command should be placed just after
the last \verb|\author{}| definition:
\begin{verbatim}
\renewcommand{\shortauthors}{McCartney, et al.}
\end{verbatim}
Omitting this command will force the use of a concatenated list of all
of the authors' names, which may result in overlapping text in the
page headers.
The article template's documentation, available at
\url{https://www.acm.org/publications/proceedings-template}, has a
complete explanation of these commands and tips for their effective
use.
Note that authors' addresses are mandatory for journal articles.
\section{Rights Information}
Authors of any work published by ACM will need to complete a rights
form. Depending on the kind of work, and the rights management choice
made by the author, this may be copyright transfer, permission,
license, or an OA (open access) agreement.
Regardless of the rights management choice, the author will receive a
copy of the completed rights form once it has been submitted. This
form contains \LaTeX\ commands that must be copied into the source
document. When the document source is compiled, these commands and
their parameters add formatted text to several areas of the final
document:
\begin{itemize}
\item the ``ACM Reference Format'' text on the first page.
\item the ``rights management'' text on the first page.
\item the conference information in the page header(s).
\end{itemize}
Rights information is unique to the work; if you are preparing several
works for an event, make sure to use the correct set of commands with
each of the works.
The ACM Reference Format text is required for all articles over one
page in length, and is optional for one-page articles (abstracts).
\section{CCS Concepts and User-Defined Keywords}
Two elements of the ``acmart'' document class provide powerful
taxonomic tools for you to help readers find your work in an online
search.
The ACM Computing Classification System ---
\url{https://www.acm.org/publications/class-2012} --- is a set of
classifiers and concepts that describe the computing
discipline. Authors can select entries from this classification
system, via \url{https://dl.acm.org/ccs/ccs.cfm}, and generate the
commands to be included in the \LaTeX\ source.
User-defined keywords are a comma-separated list of words and phrases
of the authors' choosing, providing a more flexible way of describing
the research being presented.
CCS concepts and user-defined keywords are required for for all
articles over two pages in length, and are optional for one- and
two-page articles (or abstracts).
\section{Sectioning Commands}
Your work should use standard \LaTeX\ sectioning commands:
\verb|section|, \verb|subsection|, \verb|subsubsection|, and
\verb|paragraph|. They should be numbered; do not remove the numbering
from the commands.
Simulating a sectioning command by setting the first word or words of
a paragraph in boldface or italicized text is {\bfseries not allowed.}
\section{Tables}
The ``\verb|acmart|'' document class includes the ``\verb|booktabs|''
package --- \url{https://ctan.org/pkg/booktabs} --- for preparing
high-quality tables.
Table captions are placed {\itshape above} the table.
Because tables cannot be split across pages, the best placement for
them is typically the top of the page nearest their initial cite. To
ensure this proper ``floating'' placement of tables, use the
environment \textbf{table} to enclose the table's contents and the
table caption. The contents of the table itself must go in the
\textbf{tabular} environment, to be aligned properly in rows and
columns, with the desired horizontal and vertical rules. Again,
detailed instructions on \textbf{tabular} material are found in the
\textit{\LaTeX\ User's Guide}.
Immediately following this sentence is the point at which
Table~\ref{tab:freq} is included in the input file; compare the
placement of the table here with the table in the printed output of
this document.
\begin{table}
\caption{Frequency of Special Characters}
\label{tab:freq}
\begin{tabular}{ccl}
\toprule
Non-English or Math&Frequency&Comments\\
\midrule
\O & 1 in 1,000& For Swedish names\\
$\pi$ & 1 in 5& Common in math\\
\$ & 4 in 5 & Used in business\\
$\Psi^2_1$ & 1 in 40,000& Unexplained usage\\
\bottomrule
\end{tabular}
\end{table}
To set a wider table, which takes up the whole width of the page's
live area, use the environment \textbf{table*} to enclose the table's
contents and the table caption. As with a single-column table, this
wide table will ``float'' to a location deemed more
desirable. Immediately following this sentence is the point at which
Table~\ref{tab:commands} is included in the input file; again, it is
instructive to compare the placement of the table here with the table
in the printed output of this document.
\begin{table*}
\caption{Some Typical Commands}
\label{tab:commands}
\begin{tabular}{ccl}
\toprule
Command &A Number & Comments\\
\midrule
\texttt{{\char'134}author} & 100& Author \\
\texttt{{\char'134}table}& 300 & For tables\\
\texttt{{\char'134}table*}& 400& For wider tables\\
\bottomrule
\end{tabular}
\end{table*}
Always use midrule to separate table header rows from data rows, and
use it only for this purpose. This enables assistive technologies to
recognise table headers and support their users in navigating tables
more easily.
\section{Math Equations}
You may want to display math equations in three distinct styles:
inline, numbered or non-numbered display. Each of the three are
discussed in the next sections.
\subsection{Inline (In-text) Equations}
A formula that appears in the running text is called an inline or
in-text formula. It is produced by the \textbf{math} environment,
which can be invoked with the usual
\texttt{{\char'134}begin\,\ldots{\char'134}end} construction or with
the short form \texttt{\$\,\ldots\$}. You can use any of the symbols
and structures, from $\alpha$ to $\omega$, available in
\LaTeX~\cite{Lamport:LaTeX}; this section will simply show a few
examples of in-text equations in context. Notice how this equation:
\begin{math}
\lim_{n\rightarrow \infty}x=0
\end{math},
set here in in-line math style, looks slightly different when
set in display style. (See next section).
\subsection{Display Equations}
A numbered display equation---one set off by vertical space from the
text and centered horizontally---is produced by the \textbf{equation}
environment. An unnumbered display equation is produced by the
\textbf{displaymath} environment.
Again, in either environment, you can use any of the symbols and
structures available in \LaTeX\@; this section will just give a couple
of examples of display equations in context. First, consider the
equation, shown as an inline equation above:
\begin{equation}
\lim_{n\rightarrow \infty}x=0
\end{equation}
Notice how it is formatted somewhat differently in
the \textbf{displaymath}
environment. Now, we'll enter an unnumbered equation:
\begin{displaymath}
\sum_{i=0}^{\infty} x + 1
\end{displaymath}
and follow it with another numbered equation:
\begin{equation}
\sum_{i=0}^{\infty}x_i=\int_{0}^{\pi+2} f
\end{equation}
just to demonstrate \LaTeX's able handling of numbering.
\section{Figures}
The ``\verb|figure|'' environment should be used for figures. One or
more images can be placed within a figure. If your figure contains
third-party material, you must clearly identify it as such, as shown
in the example below.
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{sample-franklin}
\caption{1907 Franklin Model D roadster. Photograph by Harris \&
Ewing, Inc. [Public domain], via Wikimedia
Commons. (\url{https://goo.gl/VLCRBB}).}
\Description{A woman and a girl in white dresses sit in an open car.}
\end{figure}
Your figures should contain a caption which describes the figure to
the reader.
Figure captions are placed {\itshape below} the figure.
Every figure should also have a figure description unless it is purely
decorative. These descriptions convey what's in the image to someone
who cannot see it. They are also used by search engine crawlers for
indexing images, and when images cannot be loaded.
A figure description must be unformatted plain text less than 2000
characters long (including spaces). {\bfseries Figure descriptions
should not repeat the figure caption – their purpose is to capture
important information that is not already provided in the caption or
the main text of the paper.} For figures that convey important and
complex new information, a short text description may not be
adequate. More complex alternative descriptions can be placed in an
appendix and referenced in a short figure description. For example,
provide a data table capturing the information in a bar chart, or a
structured list representing a graph. For additional information
regarding how best to write figure descriptions and why doing this is
so important, please see
\url{https://www.acm.org/publications/taps/describing-figures/}.
\subsection{The ``Teaser Figure''}
A ``teaser figure'' is an image, or set of images in one figure, that
are placed after all author and affiliation information, and before
the body of the article, spanning the page. If you wish to have such a
figure in your article, place the command immediately before the
\verb|\maketitle| command:
\begin{verbatim}
\begin{teaserfigure}
\includegraphics[width=\textwidth]{sampleteaser}
\caption{figure caption}
\Description{figure description}
\end{teaserfigure}
\end{verbatim}
\section{Citations and Bibliographies}
The use of \BibTeX\ for the preparation and formatting of one's
references is strongly recommended. Authors' names should be complete
--- use full first names (``Donald E. Knuth'') not initials
(``D. E. Knuth'') --- and the salient identifying features of a
reference should be included: title, year, volume, number, pages,
article DOI, etc.
The bibliography is included in your source document with these two
commands, placed just before the \verb|\end{document}| command:
\begin{verbatim}
\bibliographystyle{ACM-Reference-Format}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,467 |
The European Platform of Regulatory Authorities (EPRA) goes back to 1995. Currently, it has 52 members coming from 46 Member States of the Council of Europe. These regulators come together twice a year for an exchange of information and good practices. From 23 to 25 May 2018 this meeting will take place in Luxembourg for the first time in EPRA's history. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,689 |
Just sharing a quick post with you and it's all about Darcy's Postcard Challenge for 2012. Intrigued well check it out HERE or check out the link on the right hand side bar for you to see why I am sooooo excited about this.
My name is down for the challenge so today I am showing you my first lot of pics for the album I have nearly created so that my 52 postcards can be stored.
This is a first for me painting with Gesso … will be using this more and more me thinks, have to thank my dear friend Brenda for giving me the confidence to start using this product … thanks sweetie.
Just a few more close up pics of the completed folders. First pic shows the front of the folders and 2nd pic shows the folders turned over, 3rd is a close up. Each folder will house two postcards.
and more pics, I am sooo enjoying this project that I just want to share!!!! Loads and loads of stamping my fave pastime the postcard stamp is one from Penny Black called … wait for it … POSTCARD.
To find out the names I have given my characters and where they met you will have to pop back on 2nd January and all will be revealed.
You will hopefully see the album all put together … if not every page completed with embellishments. That's the beauty of this challenge you have all year to complete. yayyyyy.
The Stamp Man Challenge – Techniques … first for me using Gesso.
This entry was posted on Wednesday, December 28th, 2011 at 12:42 pm and is filed under 2012 Postcards ChallengeAlbumPenny BlackStampingVintage. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.
All looks absolutely brilliant Shirl, love the painting with the Gesso, and all that gorgeous stamping, can't wait to see it all put together.
OMG your pockets are amazing….wow!
Wow, an awesome project, it will be an amazing collection as it grows.
Wow, wow, wow Shirl, what an amazing collection of pockets, you have been so busy. I just adore the colours, stamps and images and all your creative ideas. Such a fabulous style. I can't wait to see your postcards and how your story progresses. Thanks for the nod. I have been over to take a look and hope I will be able to join in (some if not all!!). Woo hoo I still can't get over all of these.
Wow, that's an outstanding project. So fantastic.
Lovely project. Thanks for entering The Stamp Man technique challenge.Hope you have a lovely week. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,077 |
Q: How to remove the color picker code from users-edit.php I know I am not supposed to alter Wordpress core. But if I want to use the dashboard user profile page at users-edit.php and remove big chunks of the code (like the color picker) how do I do it. From line 259 to 336 - I want to remove all of it.
<?php if ( ! ( IS_PROFILE_PAGE && ! $user_can_edit ) ) : ?>
<tr class="user-rich-editing-wrap">
<th scope="row"><?php _e( 'Visual Editor' ); ?></th>
<td>
<label for="rich_editing"><input name="rich_editing" type="checkbox" id="rich_editing" value="false" <?php checked( 'false', $profileuser->rich_editing ); ?> />
<?php _e( 'Disable the visual editor when writing' ); ?>
</label>
</td>
</tr>
<?php endif; ?>
<?php
$show_syntax_highlighting_preference = (
// For Custom HTML widget and Additional CSS in Customizer.
user_can( $profileuser, 'edit_theme_options' )
||
// Edit plugins.
user_can( $profileuser, 'edit_plugins' )
||
// Edit themes.
user_can( $profileuser, 'edit_themes' )
);
?>
<?php if ( $show_syntax_highlighting_preference ) : ?>
<tr class="user-syntax-highlighting-wrap">
<th scope="row"><?php _e( 'Syntax Highlighting' ); ?></th>
<td>
<label for="syntax_highlighting"><input name="syntax_highlighting" type="checkbox" id="syntax_highlighting" value="false" <?php checked( 'false', $profileuser->syntax_highlighting ); ?> />
<?php _e( 'Disable syntax highlighting when editing code' ); ?>
</label>
</td>
</tr>
<?php endif; ?>
<?php if ( count( $_wp_admin_css_colors ) > 1 && has_action( 'admin_color_scheme_picker' ) ) : ?>
<tr class="user-admin-color-wrap">
<th scope="row"><?php _e( 'Admin Color Scheme' ); ?></th>
<td>
<?php
/**
* Fires in the 'Admin Color Scheme' section of the user editing screen.
*
* The section is only enabled if a callback is hooked to the action,
* and if there is more than one defined color scheme for the admin.
*
* @since 3.0.0
* @since 3.8.1 Added `$user_id` parameter.
*
* @param int $user_id The user ID.
*/
do_action( 'admin_color_scheme_picker', $user_id );
?>
</td>
</tr>
<?php endif; // End if count ( $_wp_admin_css_colors ) > 1 ?>
<?php if ( ! ( IS_PROFILE_PAGE && ! $user_can_edit ) ) : ?>
<tr class="user-comment-shortcuts-wrap">
<th scope="row"><?php _e( 'Keyboard Shortcuts' ); ?></th>
<td>
<label for="comment_shortcuts">
<input type="checkbox" name="comment_shortcuts" id="comment_shortcuts" value="true" <?php checked( 'true', $profileuser->comment_shortcuts ); ?> />
<?php _e( 'Enable keyboard shortcuts for comment moderation.' ); ?>
</label>
<?php _e( '<a href="https://wordpress.org/support/article/keyboard-shortcuts/" target="_blank">More information</a>' ); ?>
</td>
</tr>
<?php endif; ?>
<tr class="show-admin-bar user-admin-bar-front-wrap">
<th scope="row"><?php _e( 'Toolbar' ); ?></th>
<td>
<label for="admin_bar_front">
<input name="admin_bar_front" type="checkbox" id="admin_bar_front" value="1"<?php checked( _get_admin_bar_pref( 'front', $profileuser->ID ) ); ?> />
<?php _e( 'Show Toolbar when viewing site' ); ?>
</label><br />
</td>
</tr>
A: Found it - just added this action hook to the function file.
remove_action( 'admin_color_scheme_picker', 'admin_color_scheme_picker' );
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,516 |
Q: How to find 2x3 pattern in matrix Java This algorithm has me stumped. I'm pretty novice to data structures and alogithms, I understand what the code needs to do, but can't wrap my brain on how to actually code it.
The problem is as follows:
Determines whether the matrix includes a golden ticket. A golden ticket consists of 6 upper-case 'G' where three pairs of Gs are right above each other as shown below. Note that I left out the single quotes to improve readability.
G G
G G
G G
E.g.,
[A b - - C d m]
[- G G Z G G -]
[H o - r G G D] this matrix returns true
[H o - r G G D]
E.g.,
[R g G - C d m W]
[- G G Z G G - r] this matrix returns false
[o G G G r G G D] because the Gs are not in the specified position
[S t C - G G a -] relative to each other
The matrix should be 'rectangular', which means, all rows should have the same number of elements. If that is not the case, an IllegalArgumentException should be thrown.
This is what I have written:
public static boolean goldenTicket(char[][] matrix) {
if (matrix == null) return false;
if (matrix.length == 0) return false;
char char1, char2;
int matchCount = 0;
int indexOne = 0, indexTwo = 0, prevIndex1 = 0,prevIndex2 = 0;
int rows = matrix.length;
int columns = matrix[0].length;
for (int i = 0; i < rows; i++) {
if (matrix[i].length != columns)
throw new IllegalArgumentException("Length of row doesn't match");
if (matrix[i].length == 0) return false;
if (matchCount == 3) return true;
for (int j = 1; j < columns; j++) {
if (matrix[i][j] == 'G' && matrix[i][j - 1] == 'G') {
if (prevIndex1 == 0 && prevIndex2 == 0) {
indexOne = j - 1;
indexTwo = j;
matchCount++;
} else {
prevIndex1 = j - 1;
prevIndex2 = j;
matchCount++;
}
}
}
if (prevIndex1 == indexOne && prevIndex2 == indexTwo) {
matchCount++;
}
}
return false;
}
However, the problem is the code passes both example one and two as above, instead of only passing example 1.
I've already turned in the assignment with only passing 24/25 tests, I just really want to understand how this should work and maybe a better way to code it for future reference.
Thanks in advance!
A: This is the type of problem where divide and conquer can help simplify the code.
The task is to find a 2 x 3 matrix of the letter "G" in a larger matrix.
So, we write a method to check each 2 x 3 matrix within a larger matrix.
Here are the results of one of my tests.
[a, G, G, a]
[a, G, G, a]
[a, G, a, a]
[a, a, a, a]
false
[a, G, G, a]
[a, G, G, a]
[a, G, G, a]
[a, a, a, a]
true
[a, a, a, a]
[a, a, G, G]
[a, a, G, G]
[a, a, G, G]
true
I wrote two methods.
The goldenTicket method checks to see if the matrix is null, less than a 2 x 3 matrix, or if the column lengths are the same. The last code block goes through the matrix looking for the golden ticket.
The isTicket method checks to see if the current 2 x 3 section of the matrix consists of all "G" letters.
Breaking up the code into two methods makes each method smaller and easier to read, and hides somewhat the fact that it takes 4 nested for loops to solve the task correctly.
Here's the complete runnable code.
import java.lang.reflect.Array;
import java.util.Arrays;
public class GoldenTicketExample {
public static void main(String[] args) {
char[][] matrix1 = { { 'a', 'G', 'G', 'a' }, { 'a', 'G', 'G', 'a' },
{ 'a', 'G', 'a', 'a' }, { 'a', 'a', 'a', 'a' } };
displayOutput(matrix1);
char[][] matrix2 = { { 'a', 'G', 'G', 'a' }, { 'a', 'G', 'G', 'a' },
{ 'a', 'G', 'G', 'a' }, { 'a', 'a', 'a', 'a' } };
displayOutput(matrix2);
char[][] matrix3 = { { 'a', 'a', 'a', 'a' }, { 'a', 'a', 'G', 'G' },
{ 'a', 'a', 'G', 'G' }, { 'a', 'a', 'G', 'G' } };
displayOutput(matrix3);
}
private static void displayOutput(char[][] matrix) {
for (char[] row : matrix) {
System.out.println(Arrays.toString(row));
}
System.out.println(goldenTicket(matrix));
}
public static boolean goldenTicket(char[][] matrix) {
if (matrix == null) {
return false;
}
if (matrix.length < 3) {
return false;
}
int columns = matrix[0].length;
if (columns < 2) {
return false;
}
for (int row = 1; row < matrix.length; row++) {
if (matrix[row].length != columns) {
throw new IllegalArgumentException(
"Length of row doesn't match");
}
}
for (int row = 0; row < matrix.length - 2; row++) {
for (int column = 0; column < matrix[row].length - 1; column++) {
if (isTicket(matrix, row, column)) {
return true;
}
}
}
return false;
}
private static boolean isTicket(char[][] matrix, int row, int column) {
for (int r = row; r < row + 3; r++) {
for (int c = column; c < column + 2; c++) {
if (matrix[r][c] != 'G') {
return false;
}
}
}
return true;
}
}
A: It could be solved with O(ncol*nrow), where ncol is the number of columns and nrow is the number of rows.
You should generate a matrix with the same rows and columns as your original matrix, and for each item you consider a pair of number which first element represent number of sequential G's including this item in the row, and second element represent number of sequential G's including this item in the column.
To fill this matrix(I named it onlyGMatrix) we check if item (i,j) is G, if not we simply put pair (0,0) otherwise for i>0 and j>0 we put min {onlyGMatrix(i-1,j)[0] , onlyGMatrix(i,j-1)[0]} + 1 as first element and min {onlyGMatrix(i-1,j)[1] , onlyGMatrix(i,j-1)[1]} + 1 as the second element.
Using your first matrix we would reach the below onlyGMatrix:
(0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0)
(0,0) (0,0) (0,0) (0,0) (1,1) (2,1) (0,0)
(0,0) (0,0) (0,0) (0,0) (2,1) (2,2) (0,0)
(0,0) (0,0) (0,0) (0,0) (3,1) (3,2) (0,0)
And for your second matrix we would have:
(0,0) (0,0) (1,1) (0,0) (0,0) (0,0) (0,0) (0,0)
(0,0) (1,1) (2,2) (0,0) (1,1) (2,1) (0,0) (0,0)
(0,0) (1,2) (2,3) (3,1) (0,0) (1,2) (2,1) (0,0)
(0,0) (0,0) (0,0) (0,0) (1,1) (2,1) (0,0) (0,0)
now your algorithm simply would end successfully when you see pair (3,2) when you building this onlyGMatrix, and if you reach the last element of matrix and you didn't see this pair then you should return false.
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Q: AsyncTaskLoader onLoadFinished with a pending task and config change I'm trying to use an AsyncTaskLoader to load data in the background to populate a detail view in response to a list item being chosen. I've gotten it mostly working but I'm still having one issue. If I choose a second item in the list and then rotate the device before the load for the first selected item has completed, then the onLoadFinished() call is reporting to the activity being stopped rather than the new activity. This works fine when choosing just a single item and then rotating.
Here is the code I'm using. Activity:
public final class DemoActivity extends Activity
implements NumberListFragment.RowTappedListener,
LoaderManager.LoaderCallbacks<String> {
private static final AtomicInteger activityCounter = new AtomicInteger(0);
private int myActivityId;
private ResultFragment resultFragment;
private Integer selectedNumber;
@Override
public void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
myActivityId = activityCounter.incrementAndGet();
Log.d("DemoActivity", "onCreate for " + myActivityId);
setContentView(R.layout.demo);
resultFragment = (ResultFragment) getFragmentManager().findFragmentById(R.id.result_fragment);
getLoaderManager().initLoader(0, null, this);
}
@Override
protected void onDestroy() {
super.onDestroy();
Log.d("DemoActivity", "onDestroy for " + myActivityId);
}
@Override
public void onRowTapped(Integer number) {
selectedNumber = number;
resultFragment.setResultText("Fetching details for item " + number + "...");
getLoaderManager().restartLoader(0, null, this);
}
@Override
public Loader<String> onCreateLoader(int id, Bundle args) {
return new ResultLoader(this, selectedNumber);
}
@Override
public void onLoadFinished(Loader<String> loader, String data) {
Log.d("DemoActivity", "onLoadFinished reporting to activity " + myActivityId);
resultFragment.setResultText(data);
}
@Override
public void onLoaderReset(Loader<String> loader) {
}
static final class ResultLoader extends AsyncTaskLoader<String> {
private static final Random random = new Random();
private final Integer number;
private String result;
ResultLoader(Context context, Integer number) {
super(context);
this.number = number;
}
@Override
public String loadInBackground() {
// Simulate expensive Web call
try {
Thread.sleep(5000);
} catch (InterruptedException e) {
e.printStackTrace();
}
return "Item " + number + " - Price: $" + random.nextInt(500) + ".00, Number in stock: " + random.nextInt(10000);
}
@Override
public void deliverResult(String data) {
if (isReset()) {
// An async query came in while the loader is stopped
return;
}
result = data;
if (isStarted()) {
super.deliverResult(data);
}
}
@Override
protected void onStartLoading() {
if (result != null) {
deliverResult(result);
}
// Only do a load if we have a source to load from
if (number != null) {
forceLoad();
}
}
@Override
protected void onStopLoading() {
// Attempt to cancel the current load task if possible.
cancelLoad();
}
@Override
protected void onReset() {
super.onReset();
// Ensure the loader is stopped
onStopLoading();
result = null;
}
}
}
List fragment:
public final class NumberListFragment extends ListFragment {
interface RowTappedListener {
void onRowTapped(Integer number);
}
private RowTappedListener rowTappedListener;
@Override
public void onAttach(Activity activity) {
super.onAttach(activity);
rowTappedListener = (RowTappedListener) activity;
}
@Override
public void onActivityCreated(Bundle savedInstanceState) {
super.onActivityCreated(savedInstanceState);
ArrayAdapter<Integer> adapter = new ArrayAdapter<Integer>(getActivity(),
R.layout.simple_list_item_1,
Arrays.asList(1, 2, 3, 4, 5, 6));
setListAdapter(adapter);
}
@Override
public void onListItemClick(ListView l, View v, int position, long id) {
ArrayAdapter<Integer> adapter = (ArrayAdapter<Integer>) getListAdapter();
rowTappedListener.onRowTapped(adapter.getItem(position));
}
}
Result fragment:
public final class ResultFragment extends Fragment {
private TextView resultLabel;
@Override
public View onCreateView(LayoutInflater inflater, ViewGroup container, Bundle savedInstanceState) {
View root = inflater.inflate(R.layout.result_fragment, container, false);
resultLabel = (TextView) root.findViewById(R.id.result_label);
if (savedInstanceState != null) {
resultLabel.setText(savedInstanceState.getString("labelText", ""));
}
return root;
}
@Override
public void onSaveInstanceState(Bundle outState) {
super.onSaveInstanceState(outState);
outState.putString("labelText", resultLabel.getText().toString());
}
void setResultText(String resultText) {
resultLabel.setText(resultText);
}
}
I've been able to get this working using plain AsyncTasks but I'm trying to learn more about Loaders since they handle the configuration changes automatically.
EDIT: I think I may have tracked down the issue by looking at the source for LoaderManager. When initLoader is called after the configuration change, the LoaderInfo object has its mCallbacks field updated with the new activity as the implementation of LoaderCallbacks, as I would expect.
public <D> Loader<D> initLoader(int id, Bundle args, LoaderManager.LoaderCallbacks<D> callback) {
if (mCreatingLoader) {
throw new IllegalStateException("Called while creating a loader");
}
LoaderInfo info = mLoaders.get(id);
if (DEBUG) Log.v(TAG, "initLoader in " + this + ": args=" + args);
if (info == null) {
// Loader doesn't already exist; create.
info = createAndInstallLoader(id, args, (LoaderManager.LoaderCallbacks<Object>)callback);
if (DEBUG) Log.v(TAG, " Created new loader " + info);
} else {
if (DEBUG) Log.v(TAG, " Re-using existing loader " + info);
info.mCallbacks = (LoaderManager.LoaderCallbacks<Object>)callback;
}
if (info.mHaveData && mStarted) {
// If the loader has already generated its data, report it now.
info.callOnLoadFinished(info.mLoader, info.mData);
}
return (Loader<D>)info.mLoader;
}
However, when there is a pending loader, the main LoaderInfo object also has an mPendingLoader field with a reference to a LoaderCallbacks as well, and this object is never updated with the new activity in the mCallbacks field. I would expect to see the code look like this instead:
// This line was already there
info.mCallbacks = (LoaderManager.LoaderCallbacks<Object>)callback;
// This line is not currently there
info.mPendingLoader.mCallbacks = (LoaderManager.LoaderCallbacks<Object>)callback;
It appears to be because of this that the pending loader calls onLoadFinished on the old activity instance. If I breakpoint in this method and make the call that I feel is missing using the debugger, everything works as I expect.
The new question is: Have I found a bug, or is this the expected behavior?
A: In most cases you should just ignore such reports if Activity is already destroyed.
public void onLoadFinished(Loader<String> loader, String data) {
Log.d("DemoActivity", "onLoadFinished reporting to activity " + myActivityId);
if (isDestroyed()) {
Log.i("DemoActivity", "Activity already destroyed, report ignored: " + data);
return;
}
resultFragment.setResultText(data);
}
Also you should insert checking isDestroyed() in any inner classes. Runnable - is the most used case.
For example:
// UI thread
final Handler handler = new Handler();
Executor someExecutorService = ... ;
someExecutorService.execute(new Runnable() {
public void run() {
// some heavy operations
...
// notification to UI thread
handler.post(new Runnable() {
// this runnable can link to 'dead' activity or any outer instance
if (isDestroyed()) {
return;
}
// we are alive
onSomeHeavyOperationFinished();
});
}
});
But in such cases the best way is to avoid passing strong reference on Activity to another thread (AsynkTask, Loader, Executor, etc).
The most reliable solution is here:
// BackgroundExecutor.java
public class BackgroundExecutor {
private static final Executor instance = Executors.newSingleThreadExecutor();
public static void execute(Runnable command) {
instance.execute(command);
}
}
// MyActivity.java
public class MyActivity extends Activity {
// Some callback method from any button you want
public void onSomeButtonClicked() {
// Show toast or progress bar if needed
// Start your heavy operation
BackgroundExecutor.execute(new SomeHeavyOperation(this));
}
public void onSomeHeavyOperationFinished() {
if (isDestroyed()) {
return;
}
// Hide progress bar, update UI
}
}
// SomeHeavyOperation.java
public class SomeHeavyOperation implements Runnable {
private final WeakReference<MyActivity> ref;
public SomeHeavyOperation(MyActivity owner) {
// Unlike inner class we do not store strong reference to Activity here
this.ref = new WeakReference<MyActivity>(owner);
}
public void run() {
// Perform your heavy operation
// ...
// Done!
// It's time to notify Activity
final MyActivity owner = ref.get();
// Already died reference
if (owner == null) return;
// Perform notification in UI thread
owner.runOnUiThread(new Runnable() {
public void run() {
owner.onSomeHeavyOperationFinished();
}
});
}
}
A: Maybe not best solution but ...
This code restart loader every time, which is bad but only work around that works - if you want to used loader.
Loader l = getLoaderManager().getLoader(MY_LOADER);
if (l != null) {
getLoaderManager().restartLoader(MY_LOADER, null, this);
} else {
getLoaderManager().initLoader(MY_LOADER, null, this);
}
BTW. I am using Cursorloader ...
A: A possible solution is to start the AsyncTask in a custom singleton object and access the onFinished() result from the singleton within your Activity. Every time you rotate your screen, go onPause() or onResume(), the latest result will be used/accessed. If you still don't have a result in your singleton object, you know it is still busy or that you can relaunch the task.
Another approach is to work with a service bus like Otto, or to work with a Service.
A: Ok I'm trying to understand this excuse me if I misunderstood anything, but you are losing references to something when the device rotates.
Taking a stab...
would adding
android:configChanges="orientation|keyboardHidden|screenSize"
in your manifest for that activity fix your error? or prevent onLoadFinished() from saying the activity stopped?
| {
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Qatar: 'Almost Impossible' to Replace Russia's Europe Gas Flows
Saad Al-Kaabi, Qatar's Energy Minister (photo from archive).
Live News - Middle East - News - Qatar - Top
Qatar's Energy Minister Saad Al-Kaabi said on Tuesday that it would be "almost impossible" for Europe to replace its gas imports in the event Russia decides to clamp down on supplies amid a worsening Ukraine crisis.
Moscow's gas flows account for a sizable portion of Europe's supplies – in 2021, 38 percent of the European Union's natural gas was imported from Russia, according to economic think tank Bruegel.
"There is no single country that can replace that kind of volume, there isn't the capacity to do that from LNG," Kaabi said, referring to liquefied natural gas – natural gas transformed into a different state of matter in order for the resource to be transported or stored.
Additionally, Kaabi said that "Most of the LNG is tied to long-term contracts and destinations that are very clear. So, to replace that sum of volume that quickly is almost impossible."
On Tuesday, Germany also announced it is halting the Nord Stream 2 Baltic Sea gas pipeline project, and the news caused gas prices to spike.
Washington approached both Qatar and Japan in the past on the possibility of supplying gas to Europe should Russia decide to cut its flows.
Moscow already scaled back on its gas exports to Europe in recent months – Russia reduced its supplies to the region by 23 percent during the last quarter of 2021, according to the International Energy Agency.
Though Japan rerouted a portion of its LNG cargoes to the region, there was no incremental added supply because the deliveries were already scheduled through a joint initiative with Japan's JERA and France's EDF, according to Reuters.
Source: Agencies
Qatar Russia Russian gas
How, Not When: Lavrov Lays Out Russia's Priorities for Ending Ukraine Crisis | {
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<manifest xmlns:android="http://schemas.android.com/apk/res/android"
package="net.lemonsoft.lkaspectj">
<application android:allowBackup="true" android:label="@string/app_name"
android:supportsRtl="true">
</application>
</manifest>
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Acuity Link Appoints John Shermyen to Board of Directors
Healthcare Business Leader Brings Extensive NEMT Experience to the Position
Acuity Link
BOSTON, May 1, 2019 /PRNewswire/ -- Acuity Link, a comprehensive non-emergency medical transportation (NEMT) communications and logistics management platform provider, today announced that John Shermyen has been appointed to the company's board of directors. He joins existing board members, Alex Theoharidis, Demos Kouvaris, Richard A. Spencer, and Charles T. Lelon, all whom are actively involved in shaping and supporting the company's strategy to replace today's outdated and inefficient non-emergency medical transportation systems. This includes enabling seamless automated transportation scheduling for all levels of care and modes of transport while providing valuable and actionable insights into patient movement to drive efficiencies, cost savings, and improved patient experience.
Shermyen has over 25 years of experience in the ambulance and NEMT industry, and was involved in the first wave of public company ambulance acquisitions and mergers in the early 1990's. He began his career in the space leading the development of a computer-aided dispatch (CAD) system for the private ambulance industry with EMTrack, followed by the launch of a business process outsourcing (BPO) business specific to managing ambulance and NEMT/ADA providers. In 1986, Shermyen founded LogistiCare, Inc., a provider of specialized transportation network management where he was president and CEO for over a decade and credited with helping to develop the NEMT broker model. Most recently, he served as a senior vice president of health care innovations at Almost Family, Inc., a provider of home health nursing, rehabilitation and personal care services.
"I'm excited to be joining forces with Alex and the executive team at such a pivotal time for the organization and the NEMT industry in general, which is growing at such a significant pace," said Shermyen. "While at first glance it looks like the marketplace is crowded with different offerings, I've been in the medical transportation space for some time and truly believe Acuity Link is unique in its ability to meet patients' comprehensive medical transportation needs while empowering healthcare organizations to make more informed operational and clinical decisions."
Acuity Link was launched to address the mounting need for a comprehensive communications and logistics management platform that could link healthcare systems with emergency and non-emergency medical transportation providers and ambulance crew members for all levels of care and modes of transport. As a result of using Acuity Link's technology, hospitals today benefit from instant access to the closest available and most suitable transportation resources, and a reduction in bottlenecks that impact patient flow, leading to shorter discharge times, and in turn, improved care and experience. The company's software goes beyond the standard transport request procedures, allowing for streamlined interactions, more efficient and effective data collection and operational performance evaluation which can be used to make more informed decisions to drive patient flow through the healthcare ecosystem.
"On behalf of the entire Acuity Link team, I would like to welcome John to our company's board of directors," said Alex Theoharidis, CEO of Acuity Link. "His breadth of experience in entrepreneurship and leading businesses, and in particular, his extensive knowledge of the emergency and non-emergency medical transportation space, make John a strategic addition to our team. I look forward to working closely with him as we continue to scale the reach and impact of the company and enhance our platform to fully maximize the true potential of NEMT, ensuring that patients get the care and convenience they require throughout the entire transport process."
About Acuity Link
Acuity Link is a comprehensive communications and logistics management platform that links healthcare systems with non-emergency medical transportation (NEMT) providers and ambulance crew members for all levels of care and modes of transportation. With Acuity Link's technology, organizations benefit from instant access to the closest available and most suitable transportation resources, reducing bottlenecks that impact patient flow, leading to shorter discharge times, and in turn, enhanced patient care and experience. The company's HIPAA-compliant, customizable, easy-to-use Software-as-a-Service solution can be implemented into any healthcare setting to streamline NEMT and patient flow processes which provides a new level of data critical to enhancing efficiencies within the healthcare ecosystem. This previously unavailable intelligence generates accurate, actionable insights needed to make more informed operational and clinical decisions. For more information about Acuity Link, please visit https://www.acuity-link.net/.
Jessy Green or Sarah Larrow
SVM Public Relations
SOURCE Acuity Link
https://www.acuity-link.net
Acuity Link Announces Transportation Logistics Management... | {
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World·CBC IN RUSSIA
Inside a former Russian spy's info wars course. Today's lesson — the nerve agent poisonings in the U.K.
The Skripal poisonings, which resulted in recriminations and sanctions against Russia, are now a case study at Moscow State University. CBC News goes inside a former spy's classroom, where students learn very different "facts" about the international scandal.
CBC visits Moscow State University, where students learn very different 'facts' about the Skripal scandal
Chris Brown · CBC News · Posted: Dec 23, 2018 4:00 AM ET | Last Updated: December 23, 2018
Prof. Andrey Manoilo, a former spy, answers students' questions about the Skripal affair at Moscow State University. Former Russian intelligence officer Sergei Skripal and his daughter, Yulia, were poisoned with a military-grade nerve agent in Salisbury, England, back in March. (Kirill Zarubin)
The verdict of the former Russian spy standing at the front of the classroom at Moscow State University is blunt: the Kremlin blew it.
He tells the students that their government was outmanoeuvred by a disinformation campaign mounted by the British and Americans that embarrassed Russia in the aftermath of one of the most shocking events of the past year — the attack on former Russian intelligence officer Sergei Skripal and his daughter, Yulia.
The two were found incapacitated on a park bench in Salisbury, England, in March. They'd been poisoned by Novichok, a military-grade nerve agent developed by the Soviet Union. Both survived.
The West blamed Russia for the attack. But Prof. Andrey Manoilo teaches his students that all the evidence pointing to Russian involvement — whether it was released by British police or uncovered by journalists — is part of a Western offensive.
"I lecture the students in what I know best — the technology behind information wars, " said Manoilo, who wouldn't give his age but says he worked for Russia's security service, the FSB, for a "long time."
Moscow State University's main building is one of the most iconic in the capital. (Kirill Zarubin)
Manoilo teaches students how to recognize what he says are the tell-tale signs of a disinformation operation aimed at denigrating the Russian government and its president, Vladimir Putin.
And he says he's also teaching a younger generation of Russians how to fight back.
He invited a team from CBC News to sit in on two of his lectures. The poisoning of the Skripals was one of the case studies students examined.
British 'tricks'
Manoilo tells the students unequivocally that everything about the poisoning — from the smearing of Novichok on the Skripals' door handle to the investigation by British police to the subsequent revelations about the links between the two men named as suspects and the Russian security services — was built on information falsified by the British authorities.
His assertions are diametrically opposed to the facts laid out by British prosecutors, who have meticulously retraced the steps of the suspects from the time they arrived in London on a flight from Moscow to their departure roughly 48 hours later. They were travelling under the aliases Alexander Petrov and Ruslan Boshirov.
Manoilo's main criticism of Russia's handling of the case is that Kremlin officials weren't fleet-footed enough to spot all the British disinformation "tricks" and successfully head them off.
Sergei Skripal and his daughter, Yulia, both survived being poisoned. (Misha Japaridze/AP; Yulia Skripal/Facebook via AP)
According to Manoilo's theory, the British goal was to try to provoke Russia into producing the two men accused of the attempted hit on the Skripals. He says they did that by releasing an "information dump" of faked photos showing the two men near the Salisbury crime scene. That, he says, pushed the Russian government to put the men on a Kremlin-supported TV channel.
In the Q&A on RT television, they claimed to be travelling vitamin salesmen who had long wanted to see the spire on Salisbury cathedral — not far from the Skripals' home — so they booked a last-minute trip in the middle of winter to check it out.
Many outside of Russia who saw the interview felt their explanation was laughable and obviously untrue.
After being identified as suspects in the Salisbury poisoning, Ruslan Boshirov, left, and Alexander Petrov gave an interview to the Kremlin-funded RT channel in Moscow saying they had merely been tourists in the British town. (RT channel via Associated Press)
Manoilo didn't offer an explanation for why the men's alibi seemed so absurd.
In the aftermath, rather than engaging in trolling by trying to laugh off all the revelations, Russia should have systematically engaged in "counter operations" to push back, Manoilo said. He wouldn't elaborate on what he had in mind.
Independent verification
For their part, British investigators released closed-circuit television photos of the suspects arriving in Britain, travelling to an area near the Skripals' house, and then quickly leaving Salisbury without going anywhere near the town's cathedral.
Independent media outlets later accessed publicly available sources such as school yearbooks and certain Russian government databases to track the men to the towns where they were born.
The journalist collective Bellingcat and its Russian counterpart, The Insider, spoke to relatives of the suspects who confirmed their identities and that they had been decorated by Putin for their work in Russia's military security service, the GRU.
Russia's handling of Skripal affair shows 'propaganda machine is not working'
In September, British prosecutors charged the two, in absentia, with attempted murder, use of chemical weapons and other offences.
At the time, British Prime Minister Theresa May said her government had concluded, based on British intelligence, that the operation had "almost certainly" been approved "at a senior level of the Russian state."
'Doctored' footage
Still, Manoilo tells the students not to be duped by any of this. He says outlets such as Bellingcat are clearly agents of Western intelligence agencies and are working in lockstep with them.
"Any footage, any photos, can be doctored. For this, there are special programs … they can be made to look very realistic," he said. "So, therefore, this compilation of video and photos is not reliable evidence either."
In this still from closed-circuit television footage from March 4, 2018, Alexander Petrov and Ruslan Boshirov are seen walking down Fisherton Road in Salisbury. (London Metropolitan Police via Reuters)
Russian state TV has echoed this narrative from the beginning. But now that same explanation is being taught as part of a curriculum at one of Russia's most prestigious institutions of higher learning.
"If you know the scheme, you can predict the actions of those who organize it," Manoilo told his students, who picked his course on information warfare as one of their electives.
'Untrue statements'
Vienna-based journalist Christo Grozev worked on many of the exposés of the suspected Skripal poisoners, including with Bellingcat, and vehemently denies any links to any intelligence agency.
"These are demonstrably untrue statements," he said of Monoilo's claims.
"Everything we have published has been validated by Russian media who go out to villages ... find relatives who proudly show photographs of [the suspects] with Putin and boast about them being GRU spies.
"So are all these [people] part of the big conspiracy? It doesn't make sense."
Manoilo, a former FSB operative, tells his students the poisoning of former Russian intelligence officer Sergei Skripal and his daughter, Yulia, was all part of an elaborate Western disinformation campaign to embarrass Russia. (Kirill Zarubin)
It's unclear precisely what students in the class think of what they're being taught. Several who spoke to CBC News said the course was interesting and gave them a lot to think about.
"Every country has a truth," said 19-year-old Maria Schpak.
"This helps us in understanding both countries and opinions."
'Two truths in their heads at once'
Mark Galeotti, a London-based professor with Prague's Institute of International Relations, says the curriculum reflects a widely held view in Russia.
"There is definitely a strong constituency — especially among those with a background in the security apparatus — who believe the West 'is out to get us,'" he told CBC News in an interview.
However, Galeotti says people outside Russia should be cautious about being too alarmist or reading too much into the anti-Western messaging.
"Today's Russians, much like Soviet citizens, have retained the capacity to hold two truths in their heads at once," he said.
"Young Russians, many are patriotic and some see that the West is antagonistic and hypocritical. But nonetheless, they nod along with the official government line to the point that they can get to a beach in Turkey or start a high-tech company in Silicon Valley."
Russians accused of poisoning ex-spy say they were in Salisbury as tourists
Chemical weapons body, over Russia objections, votes to assign blame to some attacks
During an interview with CBC News, Manoilo was asked repeatedly whether it was logical or realistic that so many different Western governments, institutions, journalists and organizations could all have co-ordinated and conspired to frame Russia for the Skripal poisonings.
"I never told the students who exactly attempted to assassinate the Skripals," he said.
"The thing is, it could have been the British. It could have been someone else. [Students] should believe in the facts."
Moscow Correspondent
Chris Brown is a foreign correspondent based in the CBC's Moscow bureau. Previously a national reporter for CBC News on radio, TV and online, Chris has a passion for great stories and has travelled all over Canada and the world to find them.
Russian spy chief dies after 'long illness'
British police release new video of Skripal poisoning suspects | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,036 |
import SimpleHTTPServer
import SocketServer
PORT = 8000
def main():
"""Create a web server and serve the gui."""
Handler = SimpleHTTPServer.SimpleHTTPRequestHandler
httpd = SocketServer.TCPServer(("", PORT), Handler)
print "serving at port", PORT
httpd.serve_forever()
if __name__ == '__main__':
main()
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,450 |
Het Koninklijk Circus (Frans: Cirque Royal) is een evenementenzaal in het centrum van Brussel. Het gebouw dateert van 1878, toen het dienstdeed voor het enige permanente Brusselse circus. Het was een periode van architectonische vernieuwingen (onder meer het Justitiepaleis) in Brussel onder impuls van Leopold II.
Geschiedenis
De merkwaardige architectuur van Wilhelm Kuhnen suggereert een cirkelvormig gebouw, hoewel het eigenlijk een gelijkzijdige 20-hoek is. Het was voor die tijd groot en modern opgevat, want zo'n 3.500 toeschouwers hadden elk van op hun plaats toch een behoorlijk zicht op de piste. De orkestbak boven de toegang van de artiesten en circusdieren bood plaats aan 40 muzikanten. Onder de tribunes waren stapelruimtes voorzien, evenals huisvesting voor meer dan 100 (circus-)dieren, voornamelijk paarden. De arena was zo gebouwd dat ze kon worden gevuld met water, zodat een zwembad ontstond. De openingsshow werd verzorgd door "La troupe équestre Renz". De glorietijd rond de eeuwwisseling maakt, naast circusvoorstellingen, ook melding van balletten, grote evenementen en zelfs de eerste films.
Aan de vooravond van de Eerste Wereldoorlog hield Jean Jaurès een grote toespraak in het Koninklijk Circus, de laatste vóór hij werd vermoord. Na de oorlog diende het circus nog enige tijd als gevangenis voor Duitse krijgsgevangenen. Een nieuwe directie herstelde de traditie van vooral grote paardenshows vanaf 1920. Maar ook variétéartiesten als Maurice Chevalier stonden geprogrammeerd.
Na de Tweede Wereldoorlog stond het gebouw enige tijd leeg. Het werd in 1953 volledig gerenoveerd naar plannen van architect Charles Van Nueten. Enkel de funderingen en metalen structuur zijn nog origineel. Daarop werd een modernistisch bouwwerk geconstrueerd. Dit was lange tijd de thuishaven van Maurice Béjart met zijn Ballet van de twintigste eeuw.
Cultuurcentrum
Sedert de regionalisering van de cultuur is de zaal eigendom van de stad Brussel. Men doet een poging om de programmatie zowel op Nederlandstalig als Franstalig publiek te richten, ondanks het feit dat het samen wordt gerund met een andere zaal Botanique, die tot de Franse Gemeenschap behoort.
Behalve optredens van individuele artiesten, zangers of muziekgroepen, blijven grote spektakels of evenementen graag gebruikmaken van de zaal, vanwege zijn bijzondere, bijna cirkelvormige opstelling. Zo komt het Cirque du Soleil regelmatig langs.
Externe link
Officiële website
Cultuur in Brussel
Bouwwerk in Brussel (stad) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,914 |
"use strict";
const async = require("async");
const Boom = require("boom");
const Security = require("../tools/security");
const Validator = require("../tools/validator");
const Enhancer = require("../tools/enhancer");
const DependenciesHandler = require("./dependenciesHandler");
const execUpdate = (Model, logger, params, payload, options, callback) => {
const dbOptions = {
new: true,
runValidators: true,
context: "query"
};
const doUpdate = () => {
DependenciesHandler.handleDependencies(payload, options, (err) => {
if (err) {
logger.warn(err);
return callback(err, "badRequest");
} else {
Model.findByIdAndUpdate(params.id, { $set: payload }, dbOptions).lean().exec((err, dbResource) => {
if (err) {
return callback(err.errmsg || err, "badRequest");
} else if (dbResource) {
return callback(null, dbResource);
} else {
return callback(`${Model.modelName} with id ${params.id} not found`, "notFound");
}
});
}
});
};
if (payload.password) {
if (options.password.validate === true) {
const passwordError = Validator.validatePassword(payload.password, options.password);
if (passwordError) {
return callback(passwordError, "badRequest");
}
}
if (options.password.encrypt === true) {
Security.hashPassword(payload.password, (err, hash) => {
if (err) {
logger.error(err);
return callback("Could not generate encrypted password", "badImplementation");
}
payload.password = hash;
doUpdate();
});
} else {
doUpdate();
}
} else {
doUpdate();
}
};
const exec = (Model, logger, request, reply, options) => {
execUpdate(Model, logger, request.params, request.payload, options, (err, data) => {
if (err) {
logger.warn(err.message || err);
if (!data) {
data = "badImplementation";
}
return reply(Boom[data](err.message || err));
} else {
return reply(Enhancer.addMetaResource(data, request.connection.info, options));
}
});
};
const execBulk = (Model, logger, request, reply, options) => {
let updatedResources = [];
let boomType;
const execUpdates = (next) => {
async.eachSeries(request.payload, (payload, next) => {
const params = { id: payload.id };
execUpdate(Model, logger, params, payload, options, (err, data) => {
if (err) {
boomType = data || "badImplementation";
return next(err);
} else {
updatedResources.push(data);
return next();
}
});
}, next);
};
const execCount = (next) => {
Model.count().lean().exec(next);
};
async.parallel([
execUpdates,
execCount
], (err, data) => {
if (err) {
logger.warn(err.message || err);
if (!boomType) {
boomType = "badImplementation";
}
return reply(Boom[boomType](err.message || err));
} else {
const query = {
"total": data[1]
};
return reply(Enhancer.addMetaCollection(updatedResources, query, request.connection.info, options));
}
});
};
module.exports = {
exec,
execBulk
};
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,304 |
Q: ACF: Why can't I display all the images at once? Good day. 'sertificates_gallery' is the gallery ID from the Advanced Custom Fields. Tried also through foreach as in documentation, but it didn't work out. My code now looks like this:
$image = get_field('сertificates_gallery');
$size = 'thumbnail';
if( $image ) {
echo wp_get_attachment_link( $image, $size );
}
A: Actually, you are trying to get the array, so you will need to do in the loop, Please check this link
$image = get_field('сertificates_gallery');
$size = 'thumbnail';
foreach( $image as $single_image ):
echo wp_get_attachment_link( $single_image['ID'], $size );
endforeach;
A: UPD:
<?php
$images = acf_photo_gallery('сertificates_gallery', get_the_ID());
if( $images ): ?>
<div class="images">
<?php foreach( $images as $image ): ?>
<?php echo $image['thumbnail_image_url']; ?>
<?php endforeach; ?>
</div>
<?php else : ?>
no certificates yet
<?php endif; ?>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,879 |
Massage therapists all over the earth love developing their practices. Massage therapists like these work hard to educate everyone regarding the benefits of massage. Their situation is however, that competition is rising constantly. Most massage practitioners have got to continuously be on top of the strategies used to sell their abilities to the world to get more potential clients. Often, the most frequent issues that therapists constantly ponder on is this: What's the best way to develop additional customers that can enable me to start earning more money and boost up my sales? You should understand that there are exploration you can get various ways that allow you to meet your aspirations, although the key here is to have knowledge of the way to develop your business by educating yourself on both advertising and marketing and hands on skills.
You may be stating to yourself that you really grasp the competencies that is going to allow you to be a great massage therapist, but education is continuous. There is no stopping, specifically in this field, and it will be in your best interest to find out whenever possible.
In the recent past, scientists have done a considerable amount of new study relating to massage. There's more and more facts not only with regard to the validated benefits of massage, but also concerning the techniques that are being used to give you these types of results. An example of a website that gives program in this area is: http://erikdalton.com/products/homestudy-i/. With this kind of program you can start at the very first level of Myoskeletal Alignment program that takes you into the basics of spotting and correcting cervical and lower back problems before they end up becoming really serious pain issues. After you are done with this massage training, you can then move to the second level that is going to bring you a deeper knowledge regarding injuryrelief for your clients.
Along with the illustrated books, you'll be able to get a a number of videos that you can study over and over again so that you can perfect the treatments. You develop a priceless level of proficiency that you have at your disposal. After you have finished the massage course you are able to take a certification via the web which will provide you with a certificate of completion. You're also included as part of a directory of certified massage therapists in that specialty and that will facilitate your advertising and marketing activities.
Beside taking a training course to increase your expertise and skill in massage techniques, it's also important to develop your massage practice skills. You don't have to be a advertising expert to be able to make your massage business enterprise a highly visible one and gain credibility among others. One can learn these types of relevant skills through massage business program that are available online like BodyworkBiz can give you promotional strateg that will help you create a even better massage practice with a wide range of online marketing courses each with a particular goal. You don't have to be anxious about your business health because you lack the know how. At BodyworkBiz you'll be able to realize that promoting your massage practice isn't that hard. Besides handy marketing and advertising approaches, you will also be able to find out exactly ways you can attract a broader market to gain prospects and set yourself up for a successful future.
You don't have to be troubled about your routine getting wrecked when taking these massage home study training course. You have the opportunity to study on your own time in a place that suits you. You can schedule one or two hours regularly to go through your books and then take some time implementing your new understanding. It is such a easy way to develop your massage education and get the knowledge you need to build your massage business without being fearful about wasting your personal time. That is the great thing of massage home study you are able to do it when it's convenient for you, enjoying your time while at the same time developing techniques that will be so important to your continued success. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,529 |
\section{Conclusion}
\label{sec:conclusion}
In this paper, we present a domain adaption based training strategy to estimate in-bed human poses across varying illumination and occlusion conditions. Our results in two standard baseline pose estimation algorithms clearly indicate the reduction in cross-domain discrepancy and model agnostic characteristic of the method. This opens the door for future researches to invest effort on model agnostic pose estimation methods specific to in-bed setting. Looking forward, our work also opens up new directions for the largely unexplored and practically important cross-domain in-bed human pose estimation.
\vspace{-1em}
\section{Acknowledgements}
The authors extend their gratitude towards Dr. Ranga Rodrigo and the National Research Council of Sri Lanka for providing computational resources for conducting our experiments.
\section{Introduction}
\label{sec:intro}
An individual spends roughly a third of their lifetime at rest in the bed. Human behavioral monitoring during sleep is crucial for prognostic, diagnostic, and treatment of many healthcare complications, where in-bed postures are the fundamental factor. As an example, the in-bed postures of patients who have undergone surgeries should be monitored continuously long-term, where maintaining correct postures will lead to quick recovery \cite{seeingunder}. Visual inspections by the caretaker \cite{Pressureulcers} are widely used for in-bed pose estimations, but this is labor intensive, and the interpretations will be subjective. Wearable devices have been utilized for in-bed pose estimation, but their obtrusive nature towards subjects degrades sleep quality.
Recent advancements in computer vision have enabled contactless camera-based human pose estimation. Camera-based methods are less expensive, comfortable for the subject, and require less maintenance. Since the introduction of convolutional pose machine \cite{CNNposemachine}, there have been many works related to 2D human pose estimations \cite{simplebaseline, stackHG,sunetal,Yangetal} and 3D human pose estimations \cite{3d}, where the algorithms have achieved high performance. Some works have addressed human pose estimation under general settings such as multi-person pose estimation \cite{multisurvey} and wild setting \cite{wild}. However, the majority of the studies related to pose estimation are based on RGB imaging modality. Considering the in-bed pose estimation setting, RGB imaging modality-based algorithms becomes ineffective due to: 1) illumination conditions, 2) presence of heavy occlusions due to the blankets (covered subjects) and 3) privacy of the subjects, which could cause problems during large scale data collections. As a result, only a few researches have focused on in-bed pose estimation \cite{seeingunder,liu2020}.
The thermal diffusion-based long-wavelength infrared (LWIR) imaging approach for in-bed human behavioral monitoring proposed in \cite{seeingunder,liu2020} overcomes the aforementioned drawbacks in RGB imaging modality-based in-bed pose estimation. However, obtaining annotations (ground truth poses) for the covered images is infeasible in real-world applications as the occlusions and illumination conditions may vary. The realistic way to tackle this challenge is to develop a robust in-bed human pose estimation algorithm by only utilizing labeled LWIR images of uncovered (without blankets) subjects and unlabeled LWIR images of covered (with blankets) subjects during training. The trained algorithm should be able to achieve accurate pose estimation on real-world covered LWIR images.
This challenge can be addressed by utilizing unsupervised domain adaptation techniques in in-bed human pose estimation algorithms. An inherent drawback of deep neural networks is their lack of ability to learn representations from out-of-distribution data \cite{domainshift}. Domain adaptation overcomes this challenge by transferring the knowledge learned in the source domain to the target domain. Several works \cite{DASurvey} have been proposed in the classification paradigm to learn the representations of out-of-distribution data. Particularly \cite{uda_kd1, uda_kd2} utilize a knowledge distillation strategy in image classification to learn out-of-distribution data. However, to the best of our knowledge only \cite{regda} has proposed a domain adaptation strategy in a regression setting to improve keypoint detection. In contrast, our novel data augmentation along with knowledge distillation-based domain adaptation in the in-bed pose estimation setting is the \emph{first} of such works. In this paper, we propose a novel learning strategy to adapt existing human pose estimation algorithms to achieve our end goal. Our key contributions are as follows: 1) two-fold data augmentation strategy to reduce discrepancies between different domains, 2) self-supervised knowledge distillation for cross-domain in-bed pose estimation and finally 3) component wise quantitative performance analysis.
\section{Methodology}
\label{sec:method}
\begin{figure}[t!]
\vspace{-1.5em}
\centering
\includegraphics[width = 0.9 \linewidth]{Images/overall.pdf}
\caption{Overall architecture of the proposed method. We augment the uncovered images using the two-fold data augmentation strategy to reduce the cross domain discrepancy. Knowledge distillation in a self-supervised manner further improves the performance by learning transferable features.}
\label{fig:overall}
\vspace{-1.32em}
\end{figure}
We formulate the problem statement in Sec. \ref{sec:problem}; then we introduce our specific data augmentation protocol in Sec. \ref{sec:data_aug}, and finally we present our pose estimation technique and knowledge distillation in Sec. \ref{sec:pose_estimation} and \ref{sec:kd} respectively.
\vspace{-0.5em}
\subsection{Problem Definition} \label{sec:problem}
In this paper, we aim to tackle the problem of cross-domain in-bed pose estimation. In standard 2D human pose estimation tasks, we have access to a dataset $\mathcal{D}$ with $\mathnormal{n}$ labeled samples $\{x_i, y_i\}_{i=1}^{n}$, where $(x_i, y_i) \in \mathcal{X} \times \mathcal{Y}_K$. Here, $\mathcal{X}$ is the input space and $\mathcal{Y}_K$ $\in$ $\mathbb{R}^{2 \times K}$ is the output space and \textit{K} is the number of keypoints under consideration. The goal is to learn a function $\textit{f}_\theta$ which minimizes the error rate $\textit{E}_\mathcal{D}=\mathbb{E}_{(x,y\sim\mathcal{D})}\mathcal{L}(f_{\theta}(x),y))$, where $\mathcal{L}$ is the loss function. In a cross-domain setting, we have access to a labeled source domain dataset $\mathcal{D}_s = \{x_{i}^s, y_{i}^s\}_{i=1}^{n^s}$ and an unlabeled target domain dataset $\mathcal{D}_t = \{x_{i}^t\}_{i=1}^{n^t}$. The objective of standard unsupervised domain adaptation, is to minimize $\textit{E}_{\mathcal{D}_t}$ through the representations learnt from $\mathcal{D}_s$. In the context of this work, we consider the labeled uncovered LWIR imaging modality data as $\mathcal{D}_s$ and the unlabeled covered LWIR imaging modality data as $\mathcal{D}_t$. To demonstrate the generalizability of our proposed method under challenging conditions, we evaluate our method on the following sub-types of the target datasets: $\mathcal{D}_{t_1}$ which consists of thin covered LWIR images (a thin sheet with approximately $1$ mm thickness) and $\mathcal{D}_{t_2}$ which consists of thick covered LWIR images (a thick blanket with approximately $3$ mm thickness).
\begin{figure}[t!]
\vspace{-1.2em}
\centering
\includegraphics[width = 0.92 \linewidth]{Images/cycaug.pdf}
\vspace{-0.3em}
\caption{CycAug: Translations from uncovered to covered and covered to uncovered are done separately from networks $G$ and $F$. Both networks are trained to learn the mappings in unsupervised manner.}
\label{fig:cycaug}
\vspace{-1em}
\end{figure}
\subsection{Two-fold Data Augmentation for Cross-Domain Discrepancy Reduction} \label{sec:data_aug}
To reduce the domain gap between $\mathcal{D}_s$ and $\mathcal{D}_{t_1}$, $\mathcal{D}_{t_2}$, we propose a data augmentation pipeline comprises of two key components; \textit{CycAug} and \textit{ExtremeAug}.
\vspace{0.4em}
\noindent \textbf{CycAug -- Unpaired Image-to-Image Translation based Data Augmentation:}
The problem becomes challenging when there are no labeled-covered LWIR images available to learn the mapping between poses and covered LWIR images. Here we propose \textit{CycAug}, based on the work done by Zhu \emph{et al.} \cite{Zhu2017UnpairedIT}, to convert a given image sampled from $\mathcal{D}_s$ to thin and thick covered LWIR images. The generated thin and thick covered images have minimal domain discrepancies with both the distributions $\mathcal{D}_{t_1}$ and $\mathcal{D}_{t_2}$, respectively. Through this, we are able to obtain a separate dataset that contains labeled thin covered LWIR images and labeled thick covered LWIR images. This dataset is further utilized to learn a mapping between covered LWIR images and the underlying pose in a supervised manner with the additional domain discrepancy reduction techniques explained in the following sections.
The overview of the proposed \textit{CycAug} data augmentation is shown in Fig. \ref{fig:cycaug}. The objective of this is to find the mapping between $\mathcal{D}_s$ and $\mathcal{D}_t$ without having any data pairs. This can be formally defined as follows,
\begin{small}
\begin{equation}
\begin{aligned}
\mathcal{L}\left(G, F, D_{X}, D_{Y}\right) &=\mathcal{L}_{\mathrm{GAN}}\left(G, D_{Y}, X, Y\right) \\
&+\mathcal{L}_{\mathrm{GAN}}\left(F, D_{X}, Y, X\right) \\
&+\lambda \mathcal{L}_{\mathrm{cyc}}(G, F) +\lambda_{id} \mathcal{L}_{\mathrm{identity}}(G, F),
\end{aligned}
\end{equation}
where,
\vspace{-0.5em}
\begin{equation}
\begin{aligned}
\mathcal{L}_{\mathrm{GAN}}(G, D_{Y}, &X, Y) =\mathbb{E}_{y \sim Y}\left[\log D_{Y}(y)\right] \\
&+\mathbb{E}_{x \sim X}\left[\log \left(1-D_{Y}(G(x))\right]\right.
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\mathcal{L}_{\mathrm{GAN}}(F, D_{X}, &Y, X) =\mathbb{E}_{x \sim X}\left[\log D_{X}(x)\right] \\
&+\mathbb{E}_{y \sim Y}\left[\log \left(1-D_{X}(F(y))\right]\right.
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\mathcal{L}_{\mathrm{cyc}}(G, F) &=\mathbb{E}_{x \sim X}\left[\|F(G(x))-x\|_{1}\right] \\
&+\mathbb{E}_{y \sim Y}\left[\|G(F(y))-y\|_{1}\right]
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\mathcal{L}_{\text {identity }}(G, F)&=\mathbb{E}_{y \sim Y}\left[\|G(y)-y\|_{1}\right]\\
&+\mathbb{E}_{x \sim X}\left[\|F(x)-x\|_{1}\right].
\end{aligned}
\end{equation}
\end{small}
Here $X, Y, G, F$ correspond to the uncovered images ($\{x_{i}^s\}_{i=1}^{n^s}$), covered images ($\{x_{i}^t\}_{i=1}^{n^t}$) and the generator networks utilized to do image to image translation from covered to uncovered and uncovered to covered, respectively. $D_X, D_Y$ correspond to discriminators utilized to make the $F$, $G$ better at generating realistic images. $\|.\|_{1}$ indicates the $\ell_1$ norm. Objective functions, $\mathcal{L}_{\mathrm{GAN}}(.)$, $\mathcal{L}_{\mathrm{cyc}}(.)$, $\mathcal{L}_{\mathrm{identity}}(.)$, $\mathcal{L}(.)$ represent the adversarial training of generator and discriminator to ensure the realistic image generations \cite{gan}, cycle consistency loss \cite{Zhu2017UnpairedIT} to learn the mapping between two image domains uncovered ($\{x_{i}^s\}_{i=1}^{n^s}$) and covered ($\{x_{i}^t\}_{i=1}^{n^t}$) in an unsupervised manner, identity mapping regularization loss \cite{Zhu2017UnpairedIT} to regularize the learning and make convergence more efficient and the final loss as a weighted combination of above losses, respectively. For the training, we choose weighting factors $\lambda$ and $\lambda_{id}$ as $10$ and $5$, experimentally.
After the training of networks $G$ and $F$, for both thin and thick cover LWIR images separately, $G_{\mathcal{D}_{t_1}}, F_{\mathcal{D}_{t_1}}, G_{\mathcal{D}_{t_2}}$ and $F_{\mathcal{D}_{t_2}}$ are obtained. Then we utilize $G_{\mathcal{D}_{t_1}}$ and $G_{\mathcal{D}_{t_2}}$ to generate thin and thick covered LWIR images separately given the uncovered images. The obtained generated covered images are further processed to reduce the domain discrepancies with $\mathcal{D}_{t}$.
\begin{figure}[t!]
\centering
\includegraphics[width = 0.92 \linewidth]{Images/extreme_aug.pdf}
\caption{ExtremeAug: After obtaining the generated thin covered and thick covered images, a series of image processing techniques are applied to generate covered images with more occlusions.}
\vspace{-1.5em}
\label{fig:dataaug}
\vspace{+0.4em}
\end{figure}
\vspace{0.4em}
\noindent \textbf{ExtremeAug -- Extreme Occlusion based Data Augmentation:} Using \textit{ExtremeAug}, the generated thin and thick covered LWIR images are further augmented to have more covering artifacts / occlusions. By introducing different versions of covered images this way, we speculate that, the pose estimation model is expected to learn a robust mapping between covered images and pose, without becoming too sensitive to a certain single cover type (thick or thin).
In \textit{ExtremeAug} as shown in Fig. \ref{fig:dataaug}, a random vertical point is selected in the second $\left(\frac{1}{8}\right)^{\text{th}}$ of the generated covered image from the top, and intensity of all the pixels below that value is decreased to mimic covered area of the body. The resultant image is further distorted by adding dark kernels (kernels with zero values) with the size of $20 \times 20$ on random areas of the image. Then the erosion morphological operation is applied using a kernel size of $3 \times 3$. Finally the image is blurred using Gaussian blurring to reduce visible information.
\vspace{-0.5em}
\subsection{Pose Estimation} \label{sec:pose_estimation}
For an input image $x_{i}^s$ sampled from $\mathcal{D}_s$, we perform the two-fold data augmentation as described in Sec. \ref{sec:data_aug}. We denote the generated images from \textit{CycAug} as $x_{i}^{t_1}$ and $x_{i}^{t_2}$, where $t_1$ and $t_2$ corresponds to their respective domain distributions. The augmented image with \textit{ExtremeAug} are denoted as $x_{i}^{\Tilde{t}}$. We qualitatively validate (Fig. \ref{fig:dataaug}) that the domain discrepancy between distributions $\mathcal{D}_s$ and $\mathcal{D}_t$ are reduced by our two-fold data augmentation. Our intuition is that by reducing the domain discrepancy, the model tends to learn domain invariant features, which then facilitates in decoding covered domain features during inference. Further, by doing augmentations, we hypothesize the geometry of $x_{i}^s$, the augmented versions $x_{i}^{t_1}$, $x_{i}^{t_2}$ and $x_{i}^{\Tilde{t}}$ also have a ground truth label $y_{i}^s$.
We extend the input space with augmented images and optimize the pose estimator $f_\theta$ using the standard mean squared error loss function $\mathcal{L}_{sup}$:
\begin{equation}
\mathcal{L}_{sup}(f_\theta(x_i), y_i) = \frac{1}{\textit{K}}\sum_{j=1}^{\textit{K}} \vert\vert f_\theta(x_i) - y_i\vert\vert_{2}^{2}.
\end{equation}
\vspace{-0.5em}
\subsection{Knowledge Distillation} \label{sec:kd}
Knowledge Distillation \cite{kd} is a procedure of transferring the knowledge embedded in the teacher model to a student model. To this end, we take two clones of the best model trained previously and name them each as: $f_{\theta}^T$ and $f_{\theta}^S$, respectively. The weights of the teacher model, $f_{\theta}^T$ is frozen and used only during inference. $f_{\theta}^{T}(x_{i}^{t})$ is used as soft labels for the student model $f_{\theta}^S$. Through this, we enforce the student model to learn the distribution embedded with $\mathcal{D}_t$. In contrast to traditional knowledge distillation methods \cite{kd}, we employ standard mean squared error measure to enforce the knowledge distillation since our task is in a \emph{regression perspective}. Hence, the loss during this stage is given by:
\begin{equation}
\mathcal{L}_{kd}(f_{\theta}^{S}(x_{i}^{t}), f_{\theta}^{T}(x_{i}^{t})) = \frac{1}{\textit{K}}\sum_{j=1}^{\textit{K}} \vert\vert f_{\theta}^{S}(x_{i}^{t}) - f_{\theta}^{T}(x_{i}^{t})\vert\vert_2^{2}.
\end{equation}
We hypothesize that by learning the distilled knowledge from the teacher in a self-supervised manner, the student is expected to perform better than the teacher \cite{smooth}.
\section{Experiments and Results}
\label{sec:experiments}
\subsection{Dataset}
We evaluate our proposed method on the dataset provided at the IEEE VIP Cup Challenge 2021\footnote[2]{\url{https://www.2021.ieeeicip.org/VIPCup.asp}}, which was extracted from the Simultaneously-collected multi-modal Lying Pose (SLP) dataset \cite{liu2020}. SLP dataset was collected from 109 participants under two different settings: 102 participants under home setting and 7 participants under hospital setting. During the experiment, the participants were asked to lie on the bed and allowed to randomly change their pose under three main sleep posture categories: left side, right side and supine. For each category, 15 poses were collected under 3 cover conditions (no cover, thin sheet $\sim$ 1mm and thick blanket $\sim$ 3mm) using 4 imaging modalities including RGB, depth, LWIR and pressure map, simultaneously.
In our study, training dataset consists of 1) uncovered and pose labeled LWIR images from $30$ subjects, 2) thin covered and pose unlabeled LWIR images from $25$ subjects and 3) thick covered and pose unlabeled LWIR images from $25$ subjects. There are no existing pairs of uncovered and covered images in the training data. Testing dataset consists of labeled thin and thick covered LWIR images from $10$ subjects.
\vspace{-0.5em}
\subsection{Implementation Details}
For the implementation of our approach, we utilized stacked hourglass \cite{stackHG} and simple baseline \cite{simplebaseline} models as the baseline pose estimation networks. Stacked hourglass model consists of hourglass shaped networks stacked together. Each hourglass comprises of convolution, max pooling and upsampling layers to process the visual data and extract the important features. At the end of each hourglass network, two consecutive rounds of $1\times1$ convolutions are applied. Simple baseline model, on the other hand, comprises of an encoder which extracts visual features followed by a decoder to estimate poses. Following previous works \cite{stackHG, simplebaseline}, final network predictions are the heatmaps depicting the confidence of each joint in the given pixel coordinate. We use Adam optimizer with initial learning rate of $0.00025$, and the learning rate decays with a factor of $0.1$ in epochs $45$ and $60$. We train the standard supervision model for $100$ epochs while the knowledge distillation model is trained for $30$ epochs with a constant learning rate of $0.00025$.
\vspace{-0.5em}
\subsection{Results and Discussion}
\begin{table}[t]
\caption{PCKh@$0.5$ values on the test dataset. $^\dagger$ Results extracted from \cite{liu2020}.}
\centering
\begin{tabular}{p{5cm}|p{1.1cm}|p{1.2cm}}
\hline
\textbf{\multirow{2}{*}{\textbf{\hspace{2cm} Method}}} &\textit{\textbf{\small Newell et al. \cite{stackHG}}} & \textit{\textbf{\small Xiao \hspace{1cm} et al. \cite{simplebaseline}}} \\
\hline
Fully Supervised $^ \dagger$ & 94.0 & 92.5 \\
Source (Uncover) data only & 39.52 & 24.12\\
Uncover + CycAug & 72.44 & 70.43\\
Uncover + CycAug + ExtremeAug & 73.38 & 71.87\\
Uncover + CycAug + ExtremeAug + Knowledge Distillation & \textbf{76.13} & \textbf{73.84} \\
\hline
\end{tabular}
\label{tab:results}
\end{table}
The quantitative analysis on the different methods is shown in Table \ref{tab:results}. We also compare our performance with the fully-supervised method proposed in \cite{liu2020}. The experimental results clearly state the effectiveness of the proposed method in improving the performance by $36.61\%$ over the baseline in stacked hourglass model, showing the reduction of domain discrepancy between source and the target domains. The component-wise study of \textit{CycAug} and \textit{ExtremeAug} in Table \ref{tab:results} demonstrates that the heavy occlusion based data augmentation enhances the model to be robust to varying real world conditions. Similar improvement achieved in simple baseline \cite{simplebaseline} model ensures that our approach is \emph{agnostic to backbone architecture}.
Our proposed knowledge distillation improves the performance further by a margin of $2.75\%$ and $1.97\%$ in stacked hourglass and simple baseline respectively, in the PCKh@$0.5$ metric \cite{pck}. We attribute that the pseudo labels generated by the teacher model contribute to the performance enhancement of the student model. Since the knowledge distillation was performed in a self-supervised manner, it demonstrates that learning representations from unlabeled target domain is also crucial to boost the performance of the pose estimator. We believe that our approach motivates the extensive real-world data collection effort instead of focusing on costly manual annotations.
| {
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} | 3,668 |
Oil firms awarded Iraq contracts
Auction for some of world's largest fields under way amid tight security in Baghdad.
Firms successful in the bidding will still have to
deal with Iraq's security problems [EPA]
Shell and Petronas had proposed a per-barrel fee of $1.39 and a plateau production target of 1.8 million barrels per day (bpd), compared to the current 46,000.
Soon afterwards, a consortium led by China's CNPC was awarded the contract for the Halfaya oilfield.
The consortium requested fees of $1.40 per barrel of oil extracted and projected that it would produce 535,000 barrels per day from the field, which has proven reserves of 4.1 billion barrels of oil.
Bloody reminder
With a total of 45 firms vying for contracts on 15 oilfields, including three of the world's largest, the deals have the potential to boost Iraq's capacity by millions of barrels per day and make it a rival to top oil producers Saudi Arabia and Russia.
But security remains a key concern and a series of car bombs across Baghdad on Tuesday, which killed 112 people, served as a bloody reminder of the security threats that will face employees of the successful oil firms.
Angolan firm Sonangol had its bid for the Qayara oilfield rejected after it refused to lower its remuneration fee.
It was the only bidder for the field which is in the northern province of Nineveh, where Sunni fighters are known to operate and disputes between Arabs and Kurds have led to considerable tension.
There were no bids recorded for the East Baghdad field, part of which lies under the Sadr City area of the capital. The field has proven reserves of 8.1 billion barrels of oil.
'National duty'
Hussain al-Shahristani, Iraq's oil minister, said that Iraq would find ways to develop oilfields that were not awarded to international oil firms
"The Iraqi oil ministry will develop ... all the fields that were not awarded today with national capabilities," he said.
"This is our national duty and the Oil Ministry's responsibility. We will develop these fields either as a national administration or by other means that will be decided later by the ministry."
The 20-year contracts will give the oil firms, which range from large Western companies to state-owned giants from India and China, access to cheap Middle East oil reserves.
But Iraq too needs the billions of dollars the oil contracts will provide after decades of war and international sanctions.
Nuri al-Maliki, Iraq's prime minister, opened the auction, hailing the transparency with which the bidding was being conducted.
"The old way was in darkened rooms, behind closed doors. But today, it is clear to everyone," he said.
The auction is only the second held since the US-led invasion of Iraq in 2003.
Iraq's first postwar bidding round in June received only lukewarm support as firms balked at the financial provisions imposed by the Iraqi government. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,973 |
Rob McCracken rejoins the fold in Anthony Joshua's training camp as he prepares for Oleksandr Usyk
Anthony Joshua is eager for his longtime trainer to get to work and sort out his sparring as he gets ready to defend his world titles.
By Wil Esco@wil_esco Aug 12, 2021, 8:00am EDT
Share All sharing options for: Rob McCracken rejoins the fold in Anthony Joshua's training camp as he prepares for Oleksandr Usyk
With Olympic boxing officially wrapped up, trainer Rob McCracken will get back to work in the corner of Anthony Joshua as the unified heavyweight champion prepares for his upcoming title defense against mandatory challenger Oleksandr Usyk.
Earlier this year we reported on the scheduling conflict between McCraken and Joshua with McCracken serving as a coach for Team Great Britain which was running concurrent to Joshua's training camp leading to this upcoming title fight. The Toyko Games are now finished though, and Anthony Joshua is eager to get his trainer back to work putting the finishing touches on his camp to have him as prepared as possible.
McCracken tells Sky Sports that he feels good about where Joshua is currently at and says they still have plenty of time to put in the necessary work together.
"He is doing fine, he's where he needs to be. I'll get back and there will still be seven weeks [before his fight], which is a long time in a 36-minute fight. We'll get back, me and the team, the others [at Tokyo 2020] who work with AJ, so we can put the work together, get the sparring right, get the prep right."
With McCracken now fully focused on Anthony Joshua's task at hand, he'll have to find fighters who can emulate Oleksandr Usyk's southpaw style which won't be easy consider Usyk is one of the very best technical fighters in the sport. Joshua will certainly have the size advantage, however, and we'll just have to see if he tries to impose his physicality to offset the craft of the Ukrainian. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,208 |
Q: 200 ok response to request before work is done in aws lambda Im trying to make a 200 ok response to a request before the work is done, however the work i need to do takes longer than the 3 seconds i need to make the response in.
Im working in aws lambda and the way i approached this was through threading:
t = threading.Thread(target=worker, args=(xml,))
t.start()
# So that you can return before worker is done
return response(200)
However, even when I threaded the work to be done in the background, it seems that aws lambda won't finish the work. It seems that as soon as the response is made, lambda just shuts down. For example, if the work takes 2 seconds to be done, then the following will not work:
t = threading.Thread(target=worker, args=(xml,))
t.start()
# So that you can return before worker is done
return response(200)
but if we sleep for 2 seconds, the work will be done:
t = threading.Thread(target=worker, args=(xml,))
t.start()
time.sleep(2)
# So that you can return before worker is done
return response(200)
If so, what can I do to make a 200 ok response to the request with aws lambda, but also have the work be done in the same lambda function?
A: A threading solution won't work, because once you send a response your Lambda environment will be shutdown. You need to trigger an AWS Lambda function in async mode instead of event. See this guide for one possible solution.
Alternatively you could have your current Lambda function simply call another Lambda function with an async event type that does all the actual work, and then return the 200 response code.
A: I will assume you're using AWS API Gateway to call your lambda function (hence the 200 status code return).
There are two ways to call a lambda, the sync way and the event or async way. API Gateway by default uses the first way unless you explicitly tell it otherwise. In order to make your lambda integration async by default you should add a custom header:
X-Amz-Invocation-Type: Event
In your API Gateway console. You can also add a
InvocationType: Event
header in you client calls
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,822 |
The Pulitzer Prize-winning author of Cosmos and renowned astronomer Carl Sagan's international bestseller about the discovery of an advanced civilization in the depths of space remains the "greatest adventure of all time" (Associated Press).
The future is here...in an adventure of cosmic dimension. When a signal is discovered that seems to come from far beyond our solar system, a multinational team of scientists decides to find the source. What follows is an eye-opening journey out to the stars to the most awesome encounter in human history. Who--or what--is out there? Why are they watching us? And what do they want with us?
One of the best science fiction novels about communication with extraterrestrial intelligent beings, Contact is a "stunning and satisfying" (Los Angeles Times) classic. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,384 |
Q: PBC on Windows: CryptGenRandom() I have been using PBC on Windows after a lot of effort with MinGW and now I noticed that (I have started using BLS signatures) I always get the following warning when running a statically MinGW-compiled version of PBC:
"warning: could not open /dev/urandom, using deterministic random number generator"
which according to PBC manual shouldn't be happening on Windows, as there PBC claims to use CryptGenRandom(). So, I believe that the --host=i386-pc-mingw32 option I used is wrong, however I tried --host=i386-pc-winnt, but with no success.
Finally, I have to make an observation: when examining the final "problematic" .exe using ExeInfoPE I saw NO imports from advapi32.dll which is where CryptGenRandom() lies, whereas the MinGW-compiled DLL in PBC website actually HAS such imports.
Therefore, I believe I have to use some special command to PBC when "./configure"-ing it, so as to guide it to compile the file "arith\init_random.win32.c" instead of "arith\init_random.c" (you can find these in PBC source). So, my question is: how can I correct the above warning? (which, according to the above PBC manual, means that no random output from PBC functions should be cryptographically used)
Note: You can see my compilation options in my answer in Running PBC in Windows - Visual Studio and download PBC binaries and source from here.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,206 |
Pontala calpe är en fjärilsart som beskrevs av Felder 1874. Pontala calpe ingår i släktet Pontala och familjen tandspinnare. Inga underarter finns listade i Catalogue of Life.
Källor
Tandspinnare
calpe | {
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} | 7,008 |
{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/precalculus\/precalculus-concepts-through-functions-a-unit-circle-approach-to-trigonometry-3rd-edition\/chapter-11-sequences-induction-the-binomial-theorem-section-11-3-geometric-sequences-geometric-series-11-3-assess-your-understanding-page-844\/13","text":"Chapter 11 - Sequences; Induction; the Binomial Theorem - Section 11.3 Geometric Sequences; Geometric Series - 11.3 Assess Your Understanding - Page 844: 13\n\nSee below.\n\nWork Step by Step\n\nWe need to substitute $1, 2, 3,$ and $4$ for $n$ into the given equation to find the first four terms. $c_1=\\dfrac{2^{(1-1)}}{4}=\\dfrac{1}{4}$ $c_2=\\dfrac{2^{(2-1)}}{4}=\\dfrac{1}{2}$ $c_3=\\dfrac{2^{(3-1)}}{4}=\\dfrac{2^2}{4}=1$ $c_3=\\dfrac{2^{(4-1)}}{4}=\\dfrac{2^3}{4}=2$ We see that the the next term is equal to $2$ times the current term and this implies that the sequence is geometric with a common ratio of $2$\n\nAfter you claim an answer you\u2019ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide\u00a0feedback.","date":"2021-11-28 08:49:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7548724412918091, \"perplexity\": 394.43346301167264}, \"config\": {\"markdown_headings\": false, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964358480.10\/warc\/CC-MAIN-20211128073830-20211128103830-00327.warc.gz\"}"} | null | null |
Igor Vladimirovitch Vassiliev (en ), né le à Volgograd, est un joueur soviétique puis russe de handball.
Palmarès
En équipe nationale
Médaille d'or aux Jeux olympiques de 1992 à Barcelone,
Médaille d'or au Championnat du monde 1993
Médaille d'argent au Championnat d'Europe 1994
Médaille d'argent aux Goodwill Games de 1994
Médaille d'or au 1990
En club
Championnat de Russie (1) : 1996
Notes et références
Liens externes
Naissance à Volgograd
Naissance en avril 1964
Naissance en RSFS de Russie
Handballeur soviétique
Handballeur international russe
Handballeur aux Jeux olympiques d'été de 1992
Champion olympique de handball
Champion du monde russe de handball
Champion olympique de l'Équipe unifiée | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,673 |
HomeArticles Awards & NominationsQuotesRIAA certificationsSupport acts AlbumsSingles & EPsDVDs & VideosSoundtracksMusic Videos BootlegsPhoto gallery Songs & LyricsTour
RIAA certifications
Songs & Lyrics
/ Articles / Interview
Take 5 with Robby Takac
LENOX — With hits "Iris" and "Slide," the Goo Goo Dolls' sixth studio album, "Dizzy Up the Girl," established the rock band as one of the biggest acts in music by the end of the 20th century. The songs are still plenty popular; in 2018, John Rzeznik and Robby Takac celebrated the record's 20th anniversary with a North American tour. Now, the band is the midst of some more concert dates with Train and singer-songwriter Allen Stone after releasing its latest single, "Miracle Pill." The song is the title track of the group's next full-length album, which will be out in September, according to Takac. Before the Goo Goo Dolls, Train and Stone perform at Tanglewood on Monday (7 p.m.), The Eagle asked the bassist about the band's new music, its old hits and his affinity for PEZ dispensers. The interview has been edited for length.
1. You're in the midst of this tour with Train. How much interaction is there between you guys and Pat [Monahan] and his crew?
Everybody's so unbelievably nice on this trip. Really, it's crazy. It's like a love fest. A lot of these summer tours, there's a lot of personalities, man. You know, it's three bands, and it's their crews and all the PA people and the lighting people and the stage managers and blah, blah, blah, blah. There's just so many people running around backstage, and it's always pretty evident in the first week how the next few months of your life are going to be. This has been great since the beginning, really. Everybody's super kind. Everybody's been doing it for a long time and just trying to make it easy for everybody. ... John goes out every once and a while and sings a song with those guys, and Allen Stone actually sings a song with Train sometimes, too.
2. You have this new single out, "Miracle Pill." For you, what is this song about?
Well, John wrote the song with a friend of his, but the song basically attacks the need for immediacy that everybody seems to have right now. Everybody's looking for a quick fix these days. And more than that, everybody's offering a quick fix. I think it's just an overview of that whole idea.
3. What are some other themes and topics that will be explored on the rest of the record?
As with most of our records, it's just the things that have been happening to us since the last time we put pen to paper. But I think that there's a parallel that happens between people's experiences, as these things go on, that people can relate to, just based on the larger things that go on in our country and our world. I think it's a lot of relatable topics: the need for people to fit in, the inevitable passing of time and the things that come with that, those types of things.
4. We're now more than 20 years removed from the release of "Dizzy Up the Girl." In past interviews, John said he doesn't mind playing the hits from that album. How do you feel about it?
Well, it was interesting playing the whole record. We had never done anything like that before; we went out and did a whole U.S. tour of just playing that album and had a whole bunch of deep cuts, kind of revisiting that record. We play an awful lot of it every night, anyway. You know, half that record did pretty well for us, so we play a lot of that record, anyway. But going deep on the record sort of puts you more in the headspace of where you were at the time when you're going deeper into the songs, some that we had never ever played live before. So, I think it really gave the songs that we had been playing for an awfully long time, nightly, night after night after night, I think it gave them a different feel when we combined them with the rest of the thoughts that we had as we were putting that record out.
5. I read somewhere that you collect PEZ dispensers. Is that true?
Yeah! I've been collecting them for years, actually, but I had a kid about six years ago, so I started to slow down a little bit. So, I'm not quite the rabid collector that I was, but I still have them all in my life in almost every drawer of my house. So, yes.
How many do you have?
Thousands, probably 3,000, I have. Most of them are in boxes now, though, unfortunately. They've been in jail for the last few years. I've been moving around a lot, and I'm pretty busy with a lot of stuff. But I sure do love PEZ, man.
Originally posted on The Berkshire Eagle
Robby's Lobby #149
Goo Goo Dolls: Pop's Blue Collar Heroes
Record: Dizzy Up the Girl
Record: Miracle Pill
Song: Miracle Pill
Event: Tanglewood Music Center, Lenox, MA, August 5, 2019
NEW DELUXE EDITION
IT'S CHRISTMAS ALL OVER
Released November 5!
Released June 25!
NEW EP
Released April 16!
Upcoming tour dates
Lake Nona Golf & Country Club
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Crownsville, MD
The Anne Arundel County Fairgrounds
Ford Idaho Center Amphitheater
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GooGooFans.com is a Goo Goo Dolls fan site run by fans for fans.
We do not represent the Goo Goo Dolls.
For booking/press information, feedback or questions
visit our contact page.
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Past support acts | {
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Q: how do I delay or control queries? I planning to release an Api for public. what I am looking into is since it is free I need to control the usage. there are couple of options in my mind. one is delay number of queries per second. second one is fix queries to certain number per day. but second option seems to do more work on scripting which I am planning to avoid now.
so my question is how can I delay number of queries per second limit to 2 and rest put in Queue.
I am working on php so the same script would be appreciated.
or else any other suggestions welcome
A: You can use php's sleep function to hold the execution in terms of time.
$query = "....................";
// wait for 10 seconds
sleep(10);
// Then execute
$result = mysql_query($query);
//queue other queries in the same way
A: Why don't you use cron for time/resource critical queries? You can fine-tune your settings at anytime to fit your needs at the best.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,076 |
{"url":"https:\/\/www.limulo.net\/website\/coding\/graphics\/animators-sine-ramp.html","text":"# Animators pt2, Sine and Ramp\n\n## Sine Animation\n\nAnd here we are again to see another example of procedurally generated animation.\n\nThis time we would like to create an animator which would use a sin function to do the job.\n\nBelow is the final result: you can interact with it by dragging the mouse from left to right in order to change the oscillation frequency from 1.0 to 4.0 cycles per second.\n\nHere\u2019s the animator code (as usual we are usign Processing here as the base of the prototyping):\n\nclass Animator_Sine\n{\nfloat freq;\nfloat phase;\nfloat t, t0, dt;\nfloat y;\n\nAnimator_Sine ( float _freq, float _phase )\n{\nfreq = _freq>0.0?_freq:1;\nphase = abs(_phase)%(2*PI);\n\/\/ At the beginning 't0' and 't' are equal.\nt0 = t = (millis() * 0.001);\ndt = t - t0;\n}\n\nvoid update()\n{\nt = (millis() * 0.001);\ndt = t - t0;\ny = sin( phase + 2*PI*freq*dt );\ny = (y+1)*0.5;\n}\n\nvoid changeFreq( float _freq )\n{\n\/\/ calculate the phase for the\n\/\/ upcoming sinusoid\nt = (millis() * 0.001);\ndt = t - t0;\nphase = phase + 2*PI*freq*dt;\nphase = phase % (2*PI);\n\/\/ define the new 't0' which is the\n\/\/ new reference for counting time\nt0 = t;\n\/\/ Finally, set the new frequency\nfreq = _freq;\n}\n\nfloat getY() {\nreturn y;\n}\n}\n\n\nHere is where we use the Animator:\n\nclass Circle\n{\nint diameter;\ncolor c = color(255, 150, 50);\nAnimator_Sine sine;\nfloat y_sine;\n\nCircle( int _d )\n{\ndiameter = _d;\nsine = new Animator_Sine( 1, 0.0 );\ny_sine = 0.0;\n}\n\nvoid update()\n{\nsine.update();\ny_sine = sine.getY();\n}\n\nvoid draw()\n{\npushStyle();\nfill( c );\nnoStroke();\nellipse(width\/2, height\/2,\ndiameter*(1+y_sine),\ndiameter*(1+y_sine)\n);\npopStyle();\n}\n\nvoid mouseDrag( float x )\n{\nfloat newFreq = constrain(x, 0.0, width);\nnewFreq \/= width;\nnewFreq = newFreq*3 + 1;\nsine.changeFreq( newFreq );\n}\n}\n\n\nFinally this is our main program:\n\nCircle c;\n\nvoid setup()\n{\nsize(300, 300);\nc = new Circle(100);\n}\n\nvoid draw()\n{\nbackground( 120 );\nc.update();\nc.draw();\n}\n\nvoid mouseDragged()\n{\nc.mouseDrag( mouseX );\n}\n\n\nWhat\u2019s the math behind that? So let\u2019s consider this simple illustration:\n\nHere two different oscillations are shown (different frequencies and intial phases):\n\nNote: we have used single quote and double quotes to identify times belonging respectively to the first and the latter reference system.\n\nNow suppose we want to change the frequency in run time, it is like if we want to move from the oscillation on the left to the one shown in the right side of the illustration above.\n\nThe frequency must be updated to the new desired value and we need also to take care of the phase we are at in order not to brake the animation coherence.\n\nHow can we calculate all the needed parameters?\n\nFirst we know that the two oscillations must be equal so let\u2019s force $y_{1} = y_{2}$ and we obtain:\n\nwhich implies:\n\nNow let\u2019s think for a moment to the second oscillation: when doeas it assume this output?\n\nThe second oscillation will assume this exact value when its time starts counting from its relative beginning, i.e. when its time starts from its $t_{0}''$ reference, in other word when $\\Delta t'' = (t - t_{0}'') = 0$.\n\nThis yields to the following result:\n\nYou can see this passage in code at the following line inside the changeFreq method:\n\nphase = phase + 2*PI*freq*dt;\n\n\nNow that we have the new phase we must re-initilize the time we will start counting from now on.\n\nTo do this we set $t_{0}''$ (the initial time for the second reference system) to the elapsed time $t$.\n\nHere\u2019s the line where we do this (it is always inside the same chageFreq method):\n\nt= millis()*0.001;\n[...]\nt0 = t;\n\n\n## Ramp animations\n\nSomething similar can be done for another kind of animation. Basically what changes here is that we are not using the sin function anymore, instead we use a ramp oscillation (sawtooth if we want).\n\nHere you can play the interactive example: drag from left to right to change the frequency from 1.0 to 4.0 cycles per second.\n\nNote: there are two animated circles intead of one because there are two different animations: on the left you can see a rising ramp while on the right we have a falling one.\n\nHere\u2019s the code for this particular animator; we are using the boolean variable inverse to select if we should use the rising or the falling animation.\n\nclass Animator_Ramp\n{\nfloat freq;\nfloat phase;\nfloat t, t0, dt;\nfloat y;\n\/\/ Set this boolean if you want the ramp\n\/\/ going the opposite direction.\nboolean inverse = false;\n\nAnimator_Ramp ( float _freq, float _phase, boolean _inverse )\n{\nfreq = _freq>0.0?_freq:1;\nphase = abs(_phase)%(2*PI);\n\/\/ At the beginning 't0' and 't' are equal.\nt0 = t = (millis() * 0.001);\ndt = t - t0;\ninverse = _inverse;\n}\n\nvoid update()\n{\nt = (millis() * 0.001);\ndt = t - t0;\n\/\/ Support variable to convert\n\/\/ the 0-2PI phase to a 0-1 one.\nfloat phaseT = phase\/(2*PI);\nif( !inverse )\ny = (phaseT + dt*freq) % 1;\nelse\ny = 1.0 - ( (phaseT + dt*freq) % 1 );\n}\n\nvoid changeFreq( float _freq )\n{\n\/\/ calculate the phase for the\n\/\/ upcoming sinusoid\nt = (millis() * 0.001);\ndt = t - t0;\nphase = phase + 2*PI*freq*dt;\nphase = phase % (2*PI);\n\/\/ define the new 't0' which is the\n\/\/ new reference for counting time\nt0 = t;\n\/\/ Finally, set the new frequency\nfreq = _freq;\n}\n\nfloat getY() {\nreturn y;\n}\n}\n\n\nThe Circle class; it isn\u2019t changed much from the previous example, we have only added two thin circles in order to show the maximum and minimum extensions of the animation:\n\nclass Circle\n{\nfloat x, y;\nint diameter;\ncolor c = color(255, 179, 125);\nAnimator_Ramp ramp;\nfloat y_ar;\n\nCircle( float _x, float _y, int _d, boolean _inverse )\n{\nx = _x;\ny = _y;\ndiameter = _d;\nramp = new Animator_Ramp( 1, 0.0, _inverse );\ny_ar = 0.0;\n}\n\nvoid update()\n{\nramp.update();\ny_ar = ramp.getY();\n}\n\nvoid display()\n{\npushStyle();\nfill( c );\nnoStroke();\nellipse(x, y, diameter*(1+y_ar), diameter*(1+y_ar));\nnoFill();\nstroke(255, 200);\nellipse(x, y, diameter*1, diameter*1);\nellipse(x, y, diameter*2, diameter*2);\npopStyle();\n}\n\nvoid mouseDrag( float x )\n{\nfloat newFreq = constrain(x, 0.0, width);\nnewFreq \/= width;\nnewFreq = newFreq*3 + 1;\nramp.changeFreq( newFreq );\n}\n}\n\n\nTime for the main program:\n\nCircle c[];\n\nvoid setup()\n{\nsize(600, 300);\nc = new Circle[2];\nc[0] = new Circle(150, height*0.5, 100, flase);\nc[1] = new Circle(450, height*0.5, 100, true);\n}\n\nvoid draw()\n{\nbackground( 125, 168, 255 );\n\nc[0].update();\nc[1].update();\nc[0].display();\nc[1].display();\n}\n\nvoid mouseDragged()\n{\nc[0].mouseDrag( mouseX );\nc[1].mouseDrag( mouseX );\n}\n\n\n## Future experiments\n\nWe are going to make some future experiments to create new different kinds of animator. At the end we will have a complete collection of these and we will be albe to chose the best one according to the project needs. Stay tuned!\n\nIf you find this article useful and you like it, please leave a comment below: let us know what do you think about it, we'd really appreciate it. Thank you very much and, as always, stay tuned for more to come!","date":"2020-06-04 16:16:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 5, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4386286735534668, \"perplexity\": 5812.3244628800785}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590347445880.79\/warc\/CC-MAIN-20200604161214-20200604191214-00300.warc.gz\"}"} | null | null |
Welai Timur is een bestuurslaag in het regentschap Alor van de provincie Oost-Nusa Tenggara, Indonesië. Welai Timur telt 2526 inwoners (volkstelling 2010).
Plaats in Oost-Nusa Tenggara | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,664 |
For most pilots drawn into battle at Northern Coalition's staging Keepstar in X47L-Q, the fight dragged on in soul-crushing time dilation for hours, waiting for targets to lock and modules to activate.
Those lucky few able to fire their guns might not have seen the fruits of their efforts aside from a reinforced citadel and a few extra killmarks. Likely none saw the kind of harrowing action and commendable bravery shown by Imperium Director and FC Jay Amazingness, whose conspicuous gallantry saved titans and triggered a meme of the week-style destruction of an NC Avatar.
While at the controls of Sir Edmund, a revered supercapital pilot of the Goonswarm Federation, Jay jumped into X47 with the first wave of attacking titans and was immediately disconnected from the game. Unfortunately for his Erebus, Sir Edmund had bumped far away from his fleet and found himself outside the protective warp disruption bubbles that prevent disconnected capitals from warping away to an emergency safe spot.
In this case, however, the spot wasn't safe, as roving gangs of enemy pilots were counting on this game mechanic. They found Sir Edmund, tackled the expensive titan and began lobbing everything they had at its hull.
"Luckily I had logged back in at this point and wasn't going down without fighting,"Jay said on August 3, as Imperium forces planned their final attack on the Keepstar's hull timer early on August 8.
Pandemic Legion, an ally of Northern Coalition(dot), dropped dreadnoughts on the hapless titan, triggering flashbacks of another dreadbomb Jay experienced less than two years ago. Back then, Sir Edmund was inside a Revenant, the faction supercarrier that is as rare as its own death. Only 12 have ever been destroyed.
Unlike the Revenant's shields, however, the titan's massive armor tank held long enough for support to arrive. A friendly 60-man Jackdaw fleet cleared away interdictors, and Jay refit his Erebus from a focused doomsday weapon to a Bosonic Field Generator (BFG).
The BFG, or Boson as it's more commonly known, fires a cone of hellfire that rips apart most ships in its path. Luckily for Jay, his first shot killed about half a dozen dreadnoughts in what he could only describe as a server error.
"(It) surprised me since one Boson isn't enough to kill them normally, but after looking at their killmails it looks like none of their modules loaded so the server thought it was an empty hull," he said.
As the Boson fired, even more dreads from Northern Coalition jumped onto his position.
In a video Jay recorded, he switches to another character that's waiting in another system along with other response fleet dreads. He asked for 15 volunteers, including force auxiliary (FAX) logistics capitals, to jump into what could have been their own slaughter. These volunteers included members of Legacy Coalition, who were led by Progodlegend, one of Legacy Co.'s most well-known and storied fleet commanders. Unfortunately, Progodlegend was unavailable for comment regarding his part in this incredible story.
"By the time the friendly dreads and FAXes loaded I was down to 60 percent armor, but my Boson was ready again so I fired it off on the NC dreadgang, putting them into mid to low armor and shield. Our dreads then went to town on them and reps were holding me around 55 percent," Jay said.
The server issues weren't finished, however. He disconnected, and again warped to a "safe" spot. It was two hours before he could log in again, and found that his Erebus armor had dropped to 44 percent. He immediately jumped out to another system, fully repaired his ship and jumped back into X47 in time to see the Keepstar's own armor tick down to 0.
That may have been enough action for most people, but Jay also used his personal FAX with amazing results. At one point, Northern Coalition titan pilot Sn8kez found himself drifting near the edge of the Keepstar's protective tether. The only way for it to become vulnerable was for it to lock a target or be pushed from safety. Upon realizing this, Jay stepped into a Lif, the Minmitar FAX, and fit it purely for one mission.
That interplay between two massive ships happened during a relatively functional period for Eve Online's servers.
"I highly doubt I could have pulled it off an hour before," Jay said.
ProGodLegend of the Legacy Coalition helped bump the titan out of tether range by coordinating a fleet of Munnins, calling when to approach and when to break off in time for Jay's Lif to come screaming past.
That FAX was destroyed in the ensuing blast wave from the exploding titan and the odd misfired Imperium Doomsday, but he had another ready to go during The Imperium's extraction. An Erebus belonging to Kelama of the Bastion Alliance had disconnected in the midpoint system of O-N8XZ. It was tackled by a Black Legion Munnin gang that was quickly followed by a squad of enemy dreadnoughts. Jay and five other force auxiliaries were the heroes, jumping in to save the titan while an Imperium subcap and dread fleet forced off the attackers.
The adventures of Jay Amazingness on Aug. 1-2 are something any pilot can look up to. Without regard for his own life and with a calm, steady hand, he showed what good can come from throwing oneself into a fight head first and eyes forward.
When I heard Jay Amazingness was tackled at a safe and that NC. was up on the Titan count I felt it was going to be a bloody day for Goonswarm. I could not believe it when I heard that during max Tidi he was able to get out. This feat was a major turning point in the fight. Well done.
It is a good read by all means. However the Goons has a good 30k characters. It should be able to find some content that isnt about a very few «celebrities». Jay A and Kendarr and Mittens and all the others are cool dudes I am sure, but when it comes to it they are eve players just as any of us.
Absolutely, which is why we run stories when we have them on the adventures of smaller sigs and individuals. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,541 |
Sidenius är en svensk släkt med stamfader Nicolaus Sidenius (cirka 1565–1641), kyrkoherde i Sidensjö, Ångermanland, varifrån släktnamnet antogs. Äldste kände anfader Björn Skeppare (död 1504) från Stockholm.
Medlemmar i urval
Nicolaus Sidenius (cirka 1565–1641), kyrkoherde i Sidensjö, Ångermanland
Daniel Sidenius (1592–1666), professor, assessor vid Svea hovrätt och rektor vid Uppsala universitet.
Catharina Sidenia (1636–1700), genom giftermål med professor Johannes Rudbeckius d.y. (1623-1667) stammoder till adliga ätten Rudbeck nummer 1366
Helena Sidenia (cirka 1680–1759), genom giftermål med komminister Olof Artedi (1670–1728) mor till biologen Peter Artedi (1705–1735)
Källor
Matrikel Härnosands län 1646
Sidensjö och Skorped - släkten och gårdar
Härnösands stifts herdnaminne
Släkter och gårdar i Nordingrå 1535-1890
Prästsläkter
Svenska släkter | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,308 |
What's a "snap pack" you might ask?
These are a unique form of Japanese packaging to house Japanese 3″ CD singles and come from the word "tanazaku" meaning 'long strip of paper'. Formed from a standard 6″ x 3″ card sleeve, this is folded over a (relatively flimsy it has to be said) plastic frame; this frame holds the 3″ CD single in place. The outer part of the frame can be "snapped" down to make a smaller 3″ x 3″ size pack, and the sleeve is designed to be folded and tucked around it. There's your 'snap' package. Their unique design means you can choose to keep them intact, or 'snap' the pack and display one like a picture, if you get the picture.
The last one of its generation made its royal appearance in 2004 as Queen's I Was Born To Love You.
For collectors, unsnapped packs are obviously the most desirable but many Japanese collectors did as the package suggested and snapped these frames so they could store their CDs in a more compact way. This doesn't necessarily mean they are less desirable to collectors but they often command a slightly lower price simply because they aren't in their unadulterated form. We will always mention if a pack has been 'snapped' in our description.
3″ CD singles will play in any CD player that has a shallow indent to the centre of the sliding CD tray. But no popping them into your car CD player, you'll likely to never see it again.
They should be treasured and sung along to, hence the lyrics on the back of nearly all the packs. The Japanese love a bit of karoake.
Lovely to look at/fragile as hell but still great. I have a few of these but, unfortunately, missed out on getting the Propaganda "P:Machinery" one. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,206 |
\section{Introduction}
\vskip .6 cm In the past 70 years the theory of random
matrices had an impressive development in theoretical
physics and in a variety of disciplines. It seems
natural that the study of ensembles of matrices with
a variety of structures is important to assess the
possible use of random matrix theory to deal with
specific problems.\\
In this paper we study an ensemble of sparse real
symmetric block matrices. Every matrix $A$ of the
ensemble has dimension $Nd \times Nd$ and it has $N$
rows and columns. The $N^2$ entries are random matrices
$\alpha_{ij}X_{ij}$, $i,j=1,..,N$, as depicted in
eq.(\ref{d.3}).
The set of random variables
$\{\alpha_{ij}\}$ makes the random block
matrix sparse, the set of blocks $\{X_{ij}
\}$ are real symmetric random matrices
of dimension $d \times d$. Our goal is
the limiting, $ N \to \infty$, spectral
distribution of the Adjacency matrix.\\
Early studies of ensembles of random block matrices
are \cite{bre}, \cite{gir}. The entries of the
blocks $X_{ij,\alpha \beta}$, $i,j=1,..,N$, $\alpha,
\beta=1,..,d$, are often chosen to be Gaussian random
variables (possibly with block-dependent variance) and
the limit of infinite size of the blocks, $d \to \infty$,
and $N$ finite, is performed. This leads to a system of
coupled equations for the resolvent, which may be solved
only in few cases.\\
More recently, several classes of patterned,
or structured ensembles of random matrices with
random blocks were studied. Some examples are the
band matrices or tridiagonal random block \cite{fiod},
\cite{moli}, \cite{sher}, random block Toeplitz or Hankel
\cite{Li}, \cite{basu} , block circulant \cite{kologlu},
structured real symmetric block matrices \cite{oraby1},
\cite {rogers}, \cite{black}, etc.\\
Again the limit of infinite size of the blocks, $d \to \infty$,
is performed.\\
The case of sparse matrices is quite different. Possibly the most
studied model of sparse random matrix ensemble is the Adjacency matrix
of a random graph with $N$ vertices of average vertex degree or average
connectivity $Z$.
\begin{eqnarray} A=\left( \begin{array}{cccccccc} 0 & \alpha_{1,2}
&\alpha_{1,3} & \dots &\alpha_{1,N} \\ \alpha_{2,1}& 0 &\alpha_{2,3}
& \dots & \alpha_{2,N} \\ \dots & \dots & \dots & \dots & \dots\\
\alpha_{N,1} &\alpha_{N,2} & \alpha_{N,3} &\dots & 0 \end{array}\right)
\qquad , \qquad \alpha_{i,j}=\alpha_{j,i} \qquad \qquad
\label{d.1} \end{eqnarray}
The set of $N(N-1)/2$ random variables $\{\alpha_{i,j} \}$
, $i>j$, is a set of independent identically distributed
random variables, each one having the probability density
\begin{eqnarray} P(\alpha)=\left(\frac{Z}{N}\right) \delta (\alpha
-1)+\left(1-\frac{Z}{N}\right) \delta(\alpha)\qquad \qquad \label{d.2}
\end{eqnarray}
The random matrix ensemble of eqs. (\ref{d.1}), (\ref{d.2}), was
considered a basic model of disordered system in statistical mechanics. It
was then analyzed for decades from the early days of the replica approach
\cite{pio} up to more recent cavity methods \cite{mod}.\\
More pertinent to this paper, the moments of the spectral density of
the limiting ($N \to \infty$) Adjacency matrix were carefully studied
and recursion relations for them were obtained \cite{bau}, \cite{khor}.
Remarkably, the knowledge of all the spectral moments, at least in
principle, was not sufficient to obtain the spectral density.\\
In recent years, an ensemble of sparse random block matrices was
considered, where the entry $A_{i,j}$ of the random matrix is a real
symmetric $d \times d$ random real symmetric matrix $X_{i,j}$
\begin{eqnarray} A&=&\left( \begin{array}{cccccccc} 0 & \alpha_{1,2}
X_{1,2} &\alpha_{1,3} X_{1,3}& \dots &\alpha_{1,N} X_{1,N}\\
\alpha_{2,1}X_{2,1}& 0 &\alpha_{2,3} X_{2,3} & \dots & \alpha_{2,N}
X_{2,N}\\ \dots & \dots & \dots & \dots & \dots\\ \alpha_{N,1}X_{N,1}
&\alpha_{N,2} X_{N,2}& \alpha_{N,3}X_{N,3}&\dots & 0 \end{array}\right)
\qquad , \nonumber\\
&& X_{i,j}=X_{j,i}, \quad i,j=1, 2,..,N, \nonumber\\
&& X_{ij,\alpha \beta}= X_{ij, \beta\alpha }, \quad
\alpha,\beta=1,2,..,d \qquad \qquad
\label{d.3} \end{eqnarray}
The generic block $X_{i,j}$ may be considered a matrix weight associated
to the non-oriented edge $(i,j)$ of the graph. One may say that the set of
random variables $\alpha_{i,j} =\alpha_{j,i}$ encodes the architecture
of the non-oriented graph. In this case, see eq.(\ref{d.2}), it is the
Erd\"os-Renyi random graph, with average vertex degree (or connectivity)
$Z$.\\
In the papers \cite{pari}, \cite{cic1}, \cite{pern}, \cite{benet},
the
$d \times d$ real symmetric random matrices $X_{i,j}$ are independent
(except for the symmetry $X_{i,j}=X_{j,i}$) identically distributed
projectors with rank one depending on a unit vector $ \vec{n} \in
R^d$ chosen with uniform probability on the sphere. This probability
distribution for the set $\{X_{i,j} \}$ is unusual in random matrix
literature. It is motivated by physics : the random Laplacian block
matrix associated to the random block matrix $A$ in eq.(\ref{d.3}) is
the Hessian of a system of points of random locations, connected by
springs. \footnote{Actually, in the papers \cite{pari} and \cite{benet},
the connectivity is fixed, rather than average.}
A surprising result is that it is possible to study analytically the model
in the limit $d \to \infty$, with $t=Z/d$ fixed. The limiting resolvent is
solution of a cubic equation, previously obtained in a different model and
different approximation, the Effective Medium Approximation \cite{semer}.
This random matrix ensemble will be briefly recalled in section 5.1,
where a compact derivation of the cubic equation
of the limiting resolvent follows from a functional transform.\\
In this paper we analytically evaluate the limiting ($N \to \infty$)
spectral moments of the Adjacency random block matrix ensemble in
eq.(\ref{d.3}) and the $d \times d$ random matrix blocks $X_{i,j}$
both for finite $d$ and for $d=\infty$. \\
To be definite, for finite $d$, the random matrix blocks $X_{i,j}$
are chosen to belong to the Gaussian Orthogonal Ensemble but in the $d
\to \infty$ limit the results apply to the GUE ensemble as well. \\
The motivation of the present work is more theoretical then
practical. Indeed the sparse random block matrix ensemble of
this paper has two ingredients : the probability distribution of
eq.(\ref{d.2}) of the Erd\"os-Renyi random graph and the Gaussian
Orthogonal Ensemble of the blocks $X_{i,j}$. Both ingredients
seem a most natural choice in the random matrix theory.\\
In section 2 we recall the method of evaluating spectral moments
as weighted paths on a complete graph. In the present case the
weight of any path is the trace of the product of the matrices
associated to the edges. Our analytic result, presented in
eq.(\ref{A.1}), up to $\mu_{12}$, holds for any probability
choice for the random matrices $X_{i,j}$ and any dimension $d$. We
display this rather lengthy equation because it is basic to any
random block matrix ensemble with the Erd\"os-Renyi structure
and the matrix blocks are i.i.d.\\
The sections 2.2 , 2.3 and 3 recall methods to evaluate expectations
of traces of products of random matrices both for $d$ finite and
for $d\to \infty$. They follow traditional methods of quantum
field theory.\\
The limiting moments $\mu_{2k}=\lim_{d \to \infty}\left(\lim_{N \to
\infty} \frac{1}{Nd}<\texttt{tr} A^{2k}> \right)$ are polynomials of
degree $k$ in the average degree $Z$. They are identical for the choice
of the random blocks $X_{ij}$ belonging to GOE or to GUE.\\ Expectations
of trace of products of random matrices, such as $\lim_{d \to \infty}
<\texttt{tr} (X_1)^{p_1} (X_2)^{p_2}..(X_j)^{p_j}>$, where some of the
blocks are repeated and $p_r$ are positive integers,
were called multimatrix Gaussian correlators, colored maps, or words
in several independent Wigner matrices.\\
Sect.3.1 displays our results for the limiting moments $\mu_{2k}$
obtained with the above mentioned methods and with the functional
methods described in sect.4 and 5.\\
Section 4 uses some tools of free probability and describes
a generalization of a functional relation, sometimes called
non-crossing partition transform, or free cumulants. It is indeed
possible to express the infinite set of weighted paths in which one is
interested in terms of irreducible weighted paths. This allows the
evaluation of moments of higher orders and it shows relevant
relations among different ensembles of random matrices. \\
Section 5 makes use of the traditional knowledge recalled in Sections
$2$, $3$, and the functional transform of Sect.$4$, to bear on different
ensembles of sparse block random matrices.\\
There exist probability measures for the matrix blocks $X_{i,j}$ and
limits such that all spectral moments are evaluated and the non-random
spectral density is determined. In other cases the finite number of
spectral moments analytically evaluated is not sufficient to determine
the spectral density, but may be used to develop approximations.\\ In
Sect. 2 we indicate that these different outcomes are related to the
dominant set of closed paths on the graph.
\section{The moments of the Adjacency block matrix}
The method of moments to determine the spectral distribution of random
matrices is one of the oldest and best known techniques, see, for
instance, \cite{kirs}, \cite{greg}, \cite{bai}. More pertinent to the
present work are \cite{bau}, \cite{khor}.\\ Every real symmetric, random
or constant, $n \times n$, matrix $m$ corresponds to a non-oriented graph
with $n$ vertices, the edge from vertex $i$ to vertex $j$ corresponds to
the matrix entry $m_{i,j}$. The matrix entry $(m^p)_{v,w}$ corresponds
to the sum of weighted paths of $p$ consecutive steps on the graph from
vertex $v$ to vertex $w$.\\
Our goal is the determination of the limiting moments $\mu_p$
\begin{eqnarray} \mu_p =\lim_{N \to \infty} \frac{1}{Nd} < \texttt{tr}
A^{p}>\qquad \qquad \label{d.4} \end{eqnarray} where the matrix $A$
is the block matrix in eq.(\ref{d.3}), the expectation is evaluated
from the probability distribution of the set of $\{\alpha_{i,j}\}$ in
eq.(\ref{d.2}) and the probability distribution of the random blocks
$X_{i,j}$ is discussed later in the paper.\\ Every power $p$ of the
block matrix $(A^p)_{v,v}$ , is a homogeneous polynomial of degree
$p$ in the set $\{ \alpha_j \}$ and in the matrices $ X_{i,j}$. Since
the weights of the edges are non commutative, some recursive tools are
not available. It seems useful to consider first the expectation over
the random variables $\{ \alpha_j \}$ and the $N \to \infty$ limit. We
postpone to subsection $2.2$ the evaluation of the expectation of the
trace of products of $p $ random matrices $X_{i,j}$.\\
Let us call $S_1$ the set
of closed paths relevant to the $N \to \infty$ limit of spectral
moments of the matrix ensemble in eq.(\ref{d.1}). These paths are walks
on a tree graph, because walks which include a loop do not contribute
to the moments, in the $N \to \infty$ limit. In a walk, an edge is
traversed in one direction then later on is traversed in the opposite
direction. In the product of matrix blocks which is the weight of the
walk, the distinct matrix blocks appear an even number of times.
These paths were enumerated in \cite{bau}, \cite{khor}. The
powerful recurrence relations may be generalized to accommodate
commutative weights for the edges, but it seems difficult to
accommodate our non-perturbative matrix blocks $X_{i,j}$.\\
The set $S_1$ is the basic set of paths relevant to the moments of sparse
random matrix ensembles. It may happen, for certain probability measures
of the matrix blocks and for certain limits of parameters, that a subset
$S_2 \subset S_1$ of paths is dominant. Each product of matrix blocks in $S_2$
corresponds to a non-crossing partition (as we explain in Sect.4). The
walks in $S_2$ are enumerated by Narayana polynomials. This situation
is most interesting because, with different probability measures of
the blocks $X_{i,j}$, it leads to different algebraic equations for
the resolvent.\\
Finally, in the dense limit ($Z \to \infty$), the further subset
$S_3\subset S_2 \subset S_1$ of the Wigner paths is dominant \cite{molch},
\cite{sinai}, \cite{pastur}, \cite{tao} , \cite{metz}. In these
paths, there is the maximum number of distinct edges. In the products
of matrix blocks corresponding to a Wigner walk, each distinct block
occurs twice. Wigner walks
are in one-to-one correspondence with Dyck paths and are counted by
the Catalan numbers. They are well known and we do not discuss the
set $S_3$ in this paper.\\
\subsection{Closed paths on tree-graphs with weighted edges}
Because of the probability distribution of the $\{ \alpha_{i,j} \}$ random
variables, eq.(\ref{d.2}), in the limit $N \to \infty$, only closed paths
on trees contribute to the limiting moments, \cite{bau}, \cite{khor}.\\
This is true independent from the probability distribution of the random
blocks $X_{i,j}$ and their finite dimension $d$.\\ For a closed path on
a tree, every edge is traversed an even number of times. The limiting
moments of odd order vanish.\\ The weight of a path is the product of
the weights of the traversed edges. For a closed walk of $2p$ steps,
the number $l$ of distinct matrices occurring in the weighted path is $1
\leq l \leq p$.\\ Since the blocks are i.i.d., the identification of a
block $X_{i,j}$ in a product is irrelevant. It is useful a relabeling
of the products of the blocks that only records if the blocks are equal
or different to other ones in the product. For instance :
\begin{eqnarray} &&X_{1,3}X_{3,1}X_{1,3}X_{3,4}
X_{4,7}X_{7,4}X_{4,3}X_{3,1}\qquad \qquad \texttt{is relabeled} \qquad
\nonumber\\ && (X_1)^3 X_2 (X_3)^2 X_2 X_1 \qquad \qquad \label{d.6}
\end{eqnarray}
As the order of the spectral moments increases, the number of relevant
products also increases. \\ In \cite{pern} we analytically evaluated,
with help of computer symbolic enumeration, the spectral moments
of the ensemble of Adjacency matrix up to $\mu_{26}=\lim_{N \to
\infty}\frac{1}{Nd}<\texttt{Tr}\,A^{26}>$. The ensemble of random block
matrices of ref.\cite{pern} is depicted in eq.(\ref{d.3}). In paper
\cite{pern} the blocks $X_{i,j}$ are $d \times d$ rank-one projectors,
whereas in this paper we consider different probability laws, GOE for $d$
finite, GOE or GUE for $d$ infinite.\\ The set of closed walks on trees,
relevant to the spectral moments, does not depend on the probability
distribution of the blocks $X_{ij}$.\\ We list in eq.(\ref{A.1}) in the
Appendix the relevant products of matrices up to the order $\texttt{Tr}\,
A^{12}$.\\ As a simple check, one may replace every block $X_{i,j}$
with one and obtains the table $1$ of the moments in \cite{bau}.\\
\subsection{The average over the products of random blocks $X_{i,j}$,
for finite $d$}
In this paper we choose the $d \times d$ random matrices $X_{i,j}$ as
independent Gaussian real symmetric matrices. The ensemble of these
random matrices is called Gaussian Orthogonal Ensemble , GOE. The
joint probability distribution of the entries of a random matrix $X$
is described by the integration measure, see for instance \cite{pand}
\begin{eqnarray} DX&=& \left( \frac{1}{\sqrt{2}}\right)^d
\left(\frac{1}{\sigma \sqrt{2 \pi}}\right)^{\frac{d(d+1)}{2}}
e^{-\frac{\texttt{tr}\,X^2}{4\sigma^2}} \left(\prod_{\alpha>\beta}
dX_{\alpha,\beta} \right)\left(\prod_{\gamma } dX_{\gamma, \gamma}\right)
\quad ,\nonumber\\ \int DX &=& 1 \qquad \qquad \label{d.7} \end{eqnarray}
The off-diagonal entries $X_{\alpha,\beta}=X_{\beta,\alpha}, \alpha \neq
\beta$, are i.i.d. centered normal variables with variance $\sigma^2$. The
$d$ diagonal entries $X_{\gamma,\gamma}$ are i.i.d. centered normal
variables with variance $2 \sigma^2$.\\
The expectations of products of several matrices are evaluated by Wick
theorem with the basic Wick contraction (or propagator)
\begin{eqnarray} < X_{\alpha,\beta}
X_{\gamma,\delta}>&=&\left(\delta_{\alpha,\gamma}\delta_{\beta,\delta}+\delta_{\alpha,\delta}\delta_{\beta,\gamma}\right)\sigma^2
\quad , \nonumber\\
\frac{1}{d}<\texttt{tr} X^2>&=& (d+1)\sigma^2
\qquad \qquad \label{d.8} \end{eqnarray}
Graphical techniques are useful to keep track of the matrix indices. They
were developed and used long ago to evaluate group factors of Feynman
graphs. \cite{ger},\cite{cvi}, \cite{cann}, \cite{but}. \\
Eq.(\ref{A.2}) displays the evaluation of the averages over the GOE
ensemble of the contributions displayed in eq.(\ref{A.1}). The evaluation
of monomial powers \begin{eqnarray}
\frac{1}{d}<\texttt{ tr}\,X^{2p}> \qquad \qquad
\label{d.8b} \end{eqnarray}
was studied in the past because of its relation with the average resolvent
$g(z)$ $$g(z)=\frac{1}{d} < \texttt{tr} \frac{1}{z I-X}>=\frac{1}{zd}
\sum_{r=0}^\infty < \texttt{tr} \left( \frac{X}{z}\right)^r >$$ Graphical
techniques are efficient and there exist a $5$ terms recursion relation
\cite{ledo} for the monomials of eq.(\ref{d.8b} ) for any finite $d$.\\
We report in Appendix in eq.(\ref{A.2}) the limiting, $N \to \infty$, $d$
finite, first spectral moments obtained by evaluating the expectations
of the terms in eq.(\ref{A.1}) by graphical methods. They provide a check
of the power of a functional transform we describe in section $4$.\\
Simple consistency checks may be performed on the evaluations in
eq.(\ref{A.2}).
Let us suppose that $\{ X_j \}$, $j=1,2,..k$, is a set of i.i.d. random
matrices of the GOE ensemble. Then the random matrix $X=\frac{1}{\sqrt{k}}
\sum_{j=1}^k X_j$ also belongs to the same GOE ensemble, for every $k$
and every $d$. This generates an infinite number of sum rules. For
instance \begin{eqnarray} &&<\texttt{tr}(X_1+X_2+X_3)^6> = <\texttt{tr}
\Bigg[3(X_1)^6+36(X_1)^4(X_2)^2+36(X_1)^3X_2X_1X_2+\nonumber\\
&&+18(X_1)^2X_2(X_1)ìX_2+12(X_1)^2(X_2)^2(X_3)^2+18(X_1)^2X_2(X_3)^2X_2+\nonumber\\
&&+36(X_1)^2X_2X_3X_2X_3+ 6X_1X_2X_3X_1X_2X_3+18 X_1X_2X_3X_1X_3X_2
\Bigg]> \nonumber\\ &&=27< \texttt{tr} (X_1)^6> \label{d.10}
\end{eqnarray}
The $d=1$ case can be computed using recursion relations, which are a small
modification of those in \cite{bau}, as written in \cite{pern2}, for $k=2$;
in these equations there is a sum over $\bar m$, the number of times an
edge $a$ is run up and down; here there is a Gaussian random variable $X_a$
associated to each time the edge is traversed, so there is a factor
$X_a^{2\bar m}$;
since in these relations $X_a$ appears only here, one can take the
average
\begin{equation}
< X_a^{2{\bar m}} > = 2^{\bar m} (2{\bar m}-1)!!
\label{d1av}
\end{equation}
so one has
\begin{equation}
\mu_{2j} = H_{j,1}(1,Z)
\label{fpsiH}
\end{equation}
where $H_{j,1}(y,Z)$ is given by the recursion relations are
\begin{eqnarray}
&&H_{j, 1}(y, Z) = Z \sum_{\bar m = 1}^j < X^{2{\bar m}} > y^{\bar m}
\sum_{j_1 + j_2 = j - \bar m}
H_{j_1,\bar m}(y,Z)H_{j_2,\bar m}(1,Z) \nonumber\\
\label{psiH1} \\
&&H_{j,\bar m}(y, Z) = \sum_{m=1}^j y^m [y^m]H_{j, 1}(y, Z) \left( \begin{array}{cc} m+\bar m - 1 \\ \bar m - 1 \end{array} \right)
\label{psiH}
\end{eqnarray}
and $H_{0,\bar m}(y, Z) = 1$.
The moments for generic $d$, which we computed till $\mu_{12}$, agree
in $d=1$ with the moments obtained with these recursion relations.
\subsection{Factorization}
In the product of the random matrices $\{X_j\}$ there can be a subproduct
of a subset of random matrices, none of which appear anywhere else
in the product. Because of the independence of the blocks $\{X_j\}$
with different index $j$, it follows a factorization, which may be
repeated until the factors cannot be further reduced.
Such factors will be called irreducible . For instance \begin{eqnarray}
&&<\texttt{tr} X_1^2 X_2^2 X_1^2 X_3 X_4^4 X_3 X_2^2>=<\texttt{tr} X_1^2
X_2^2 X_1^2 X_3^2 X_2^2>\frac{1}{d}<\texttt{tr}X_4^4>=\nonumber\\
&&=<\texttt{tr} X_1^2 X_2^2 X_1^2
X_2^2>\frac{1}{d}<\texttt{tr}X_3^2>\frac{1}{d}<\texttt{tr}X_4^4>
\label{d.12} \end{eqnarray} and \begin{eqnarray} <\texttt{tr} X_1 X_2^2
X_1^2 X_3^2 X_4^2 X_3^2 X_4^2 X_2^2 X_1 >=<\texttt{tr}X_1 X_2^2 X_1^2
X_2^2 X_1> \frac{1}{d}<\texttt{tr} X_3^2 X_4^2 X_3^2 X_4^2> \nonumber\\
\label{d.13} \end{eqnarray}
The factorization property holds for arbitrary dimension $d$ and any
probability measure of the random matrices $X_j$.\\ In sect.4 we present
a functional relation which reduces the infinite set of expectations to
the smaller, still infinite set of irreducible expectations.\\
\section{The large-$d$ limit} We now evaluate the first moments of the
spectral density of the Adjacency matrix (\ref{d.3}) in the limit $d\to
\infty$. This is the goal of the present paper.\\
We make the usual choice $\sigma^2 = \frac{1}{d}$ in the Gaussian measure
Eq. (\ref{d.7}).
More precisely, we evaluate the first limiting moments \begin{eqnarray}
\mu_{2p}&=&\lim_{d \to \infty} \bigg(\lim_{N \to \infty}\frac{1}{N d}
\texttt{tr}< A^{2p}> \bigg) \quad \nonumber\\
&=& c_1^{(2p)} Z+c_2^{(2p)} Z^2+ \dots +c_p^{(2p)} Z^p\normalfont
\qquad \qquad \label{f.3} \end{eqnarray} where the Adjacency matrix $A$
is given in eq.(\ref{d.3}), the expectations are evaluated over the
set of i.i.d. random variables $\{\alpha_{i,j}\}$, see eq.({\ref{d.2})
which define the sparsity of the block matrix and the GOE measure for
the entries of the i.i.d. blocks $X_{i,j}$ is in eq.(\ref{d.7}).\\
The spectral distribution of each GOE block $X$, in the $d \to \infty$
limit, is the semi-circle $$\rho(\lambda)=\frac{1}{2\pi}\sqrt{4-\lambda^2}
\quad , \qquad -2 \leq \lambda \leq 2$$
Then the coefficients $c_1^{(2p)}$ are known for every $p$ to be the
Catalan coefficients $C_p$ \begin{eqnarray} c_1^{(2p)}&=& \lim_{d \to
\infty} \left( \ \frac{1}{d}< \texttt{tr} X^{2p}>\right)=\nonumber\\
&=&\frac{1}{2\pi}\int_{-2}^2 \lambda^{2p} \sqrt{4-\lambda^2} \,
d\lambda=\frac{1}{p+1} \left(\begin{array}{cc} 2p \\p\end{array} \right)
=C_p \qquad \qquad \label{f.4} \end{eqnarray}
Before discussing further coefficients $c_j^{(2p)}$ . $1\leq j\leq p$,
let us recall methods of evaluation.\\
In the language of quantum field theory, in order to evaluate a term
like $\lim_{d \to \infty}\frac{1}{d}<\texttt{tr}[X_1^6X_2^2X_1^2X_2^2]>$,
one would draw a closed convex line, with $12$ points on it, with labels
$X_1$, six consecutive points, then $X_2$, two consecutive points,
etc. The evaluation of the expectation is provided by the Wick
theorem and would be called a Gaussian two matrix expectation. In
this example, there are many distinct ways, or graphs, where six
"`propagators"' couple points with the same label. Each graph,
where the six propagators are drawn inside the closed line has a
contribution to the evaluation for finite $d$. In the limit $d \to
\infty$, the only graphs which contribute are the so called planar
graphs, that is the drawings where the propagators do not cross inside
the border line. Furthermore to evaluate the contribution of a planar
graph, it is sufficient to retain only one part of the propagator
(\ref{d.8}) \begin{eqnarray} < X_{\alpha,\beta} X_{\gamma,\delta}>&\sim
&\delta_{\alpha,\delta}\delta_{\beta,\gamma}\frac{1}{d}
\qquad \qquad
\label{f.5} \end{eqnarray}
This approximate procedure, which is valid to obtain the leading
contribution in the limit of the large matrices $X$, is identical to the
procedure for the case of an ensemble of complex self-adjoint matrices
$X$ belonging to the GUE ensemble. Then the table of limiting moments
in eq.(\ref{t.1}) holds for both classical ensembles GOE and GUE.\\
Returning to our example, with the above rules, the planar graphs
are only of two disjoint sets : either the two propagators of the two
pairs $X_2^2$ cut across the graph (in only one planar way) or they
couple pairs of adjacent $X_2$. That is
\begin{eqnarray}
\lim_{d \to \infty}\frac{1}{d}<\texttt{tr}[X_1^6
X_2^2 X_1^2 X_2^2]>&=&\lim_{d \to \infty} \left(
\frac{1}{d^2}<\texttt{tr}[X^6]><\texttt{tr}[X^2]>
+\frac{1}{d}<\texttt{tr}[X^8]>\right)\nonumber\\ &=& 19 \nonumber
\end{eqnarray}
The same argument leads to the functional generator \begin{eqnarray}
&& F(g_1,g_2)=\lim_{d \to \infty}\frac{1}{d}<\texttt{tr}[
\frac{1}{1-g_1X_1^2}X_2^2\frac{1}{1-g_2X_1^2}X_2^2]>=\nonumber\\
&&\quad =\lim_{d \to \infty}\frac{1}{d}<\texttt{tr}[
\frac{1}{1-g_1X_1^2}\frac{1}{1-g_2X_1^2}]>+ \lim_{d \to
\infty}\frac{1}{d^2}<\texttt{tr}[ \frac{1}{1-g_1X_1^2}]><\texttt{tr}[
\frac{1}{1-g_2X_1^2}]> \nonumber \end{eqnarray} More general and more
complex recursion relations were obtained by M. Staudacher \cite{staud}.\\
\subsection{The limiting moments}
We present in eq.(\ref{t.1}) the analytic evaluation of the limiting
moments (first $N \to \infty $ then $d \to \infty$) of the rescaled
Adjacency matrix, up to $\mu_{22}$.\\
They were determined for the random blocks $X_j$ being random matrices
of the Gaussian orthogonal ensemble. The table holds as well for $X_j$
being members of the Gaussian unitary ensemble.\\
The moments from $\mu_2$ to $\mu_{12}$ were evaluated after taking the
large $d$ limit of the expectation of the Table 1 of weighted closed
paths on trees in Appendix A, eq.(\ref{A.1}). The moments from $\mu_2$
to $\mu_{22}$ were evaluated by the more powerful methods described in
the next sections.
\vskip 0.4 cm
$\mu_{2p}=\lim_{d \to \infty} \bigg(\lim_{N \to \infty}\frac{1}{N d}
\texttt{tr}< A^{2p}> \bigg)$
\vskip 0.6 cm
\begin{eqnarray} \mu_2 &=& Z \nonumber\\ \mu_4 &=& 2\,Z+2\,Z^2
\nonumber\\ \mu_{6} &=& 5\,Z+12\, Z^2+5\, Z^3 \nonumber\\ \mu_{8}
&=& 14\, Z+62 \,Z^2+ 56\, Z^3+14\, Z^4 \nonumber\\ \mu_{10} &=&
42\, Z+310\,Z^2+465\,Z^3+240\,Z^4+42\,Z^5 \nonumber\\ \mu_{12} &=&
132 \,Z+1542 \,Z^2+3454 \, Z^3 +2816\, Z^4 +990\,Z^5 +132 \, Z^6
\qquad \nonumber\\ \mu_{14} &=& 429 \, Z+7700\, Z^2+24325 \,Z^3+28182 \,
Z^4+15197\, Z^5 \nonumber\\ &&\quad +4004\, Z^6+429\, Z^7 \nonumber\\
\mu_{16} &=& 1430\,Z+38726\,Z^2+ 166536\,Z^3+259090\, Z^4+192760\,
Z^5 \nonumber\\ &&\quad +76440\,Z^6+16016\, Z^7+1430\,Z^8 \nonumber\\
\mu_{18} &=& 4862\, Z+196374\, Z^2+1122569\, Z^3+2264256\,Z^4+2197188\,Z^5
\nonumber\\ &&\quad +1178712\, Z^6+366996\, Z^7+63648\, Z^8+4862\,Z^9
\nonumber\\ \mu_{20} &=& 16796\, Z+1004126\, Z^2+7503800\, Z^3+19162220\,
Z^4+23427294\,Z^5 \nonumber\\ &&\quad 16071948 \, Z^6+6676410\,
Z^7+1705440\, Z^8+251940\, Z^9+16796\,Z^{10}\nonumber\\ \mu_{22} &=&
58786 \, Z+5175874\, Z^2+49956456 \, Z^3+158795516 \, Z^4 \nonumber\\
&&\quad +238949832\, Z^5+202538688 \, Z^6+106069733\, Z^7+35787906\,
Z^8\nonumber\\ &&\quad +7738434 \, Z^9+994840\, Z^{10}+58786\, Z^{11}
\nonumber\\ \label{t.1} \end{eqnarray}
\section{Non-crossing partitions, irreducible partitions, the
non-crossing transform. }
The role of the combinatorics of non-crossing partitions (definitions here
below), functional transforms for the spectral measures of some random
matrices ensembles in the large dimension limit were studied in a series
of papers, see for instance \cite{bani}, \cite{nica}, \cite{mingo}.\\
In this section we describe these methods as suited for this paper but
also provide a non-crossing transform at the level of product of matrices,
more general than the more usual transform for expectations. This
generality is useful to see relations among different but related matrix
ensembles.\\
We begin by recalling some definitions.\\
\subsection{Basic definitions }
Let \textsl{S} be an ordered set. Then $\pi =\{ B_1,..,B_p\}$ is
a partition of \textsl{S} if the $B_i \neq \emptyset $ are ordered
and disjoint sets, called blocks of the partition, whose union is
\textsl{S}.\\ In this paper the elements of \textsl{S} are real
symmetric $d \times d$ matrices $X$, the blocks of the sparse random
block matrix ensemble (\ref{d.3}). The contribution of a path of length
$2l$ on a tree graph is the product of $2l$ matrices. For instance the
product of eight matrices, contributing to the asymptotic moment $\mu_8$,
$X_1 X_2^2 X_1^2 X_2^2 X_1$ , corresponds to the partition in two blocks
$\pi=\{1,\,4,\,5,\,8\} \{2,\,3,\,6,\,7\}$.\\ Elements of \textsl{S} may be
drawn as points on a oriented line, labeled with the block they belong.
A partition is said to be \textsl{irreducible} if there is no set of
points, all belonging to the same block, which forms a proper interval
on the line. The above partition $\pi$ of a set of $8$ elements into
two blocks is irreducible.\\ A partition with a block such that its
elements are a set of consecutive points is \textsl{reducible}. The
corresponding product of matrices $\{X_j\}$ contains a monomial factor
of a matrix that does not appear elsewhere in the product. In this case,
as it was described in section 2.3, the expectation of the trace of the
product of matrices factorizes in two traces of products and the result
may further be reduced. Then a factorizable partition may be reduced to
a product of irreducible partitions.\\ To any product of $2p$ matrices we
may associate the set of $2p$ products obtained by a cyclic permutation of
the matrices. We call this set the \textsl{orbit} of anyone product. The
size of the orbit is the number of distinct products in the orbit,
after relabeling the matrices.\\ For instance the orbit of the product
$X_1^8 X_2^2 X_3^2$ has size 12, the orbit of $X_1^4 X_2^4 X_1^4 X_3^4$
has size $8$, the orbit of $X_1^{12}$ has size $1$.\\
Let us define the insertion of a partition in a partition. Let us consider
the product of $8$ matrices $X_1 X_2^2 X_1^2 X_2^2 X_1$
corresponding to $\pi=\{1,\,4,\,5,\,8\} \{2,\,3,\,6,\,7\}$. The expanded
form of the product $X_1 X_2 X_2 X_1 X_1 X_2 X_2 X_1$ corresponds to
the sequential form $\{1,2,2,1,1,2,2,1\}$.\\ We now want to insert
the monomial $M$ corresponding to a irreducible partition, e.g.
$M=X^2$ in the case of $\{1,2\}$, inside
this product. We denote such insertion $I(M)$. Then
the label of all matrices after that place remains unchanged if the
matrix also appear in positions before the insertion but are shifted
by the number of integers of the distinct matrices of the insertion,
otherwise. The labels of the matrices of the insertion are shifted by
the number of distinct labels before the insertion.
For instance
\begin{eqnarray} X_1 I(X^2) X_2^2 X_1^2 X_2^2 X_1 &=& X_1^2 X_2^2 X_3^2
X_1^2 X_3^2 X_1 \nonumber\\ X_1 X_2^2 X_1^2 I(X^2) X_2^2 X_1 &=& X_1
X_2^2 X_1^2 X_3^2 X_2^2 X_1 \nonumber\\ X_1 X_2^2 X_1^2 I(X_1^2 X_2^2
X_1^2 X_2^2) X_2^2 X_1&=& X_1 X_2^2 X_1^2 X_3^2 X_4^2 X_3^2 X_4^2 X_2^2
X_1 \qquad \nonumber\\ \label{t.2} \end{eqnarray}
\subsection{The functional transform}
We define $A(X)$ as the sum (finite or infinite) of all irreducible
products of $X_j$ matrices corresponding to possible closed paths, and
define $F(X)$ the sum of all products of $X_j$ matrices corresponding
to possible closed paths.
(The notation might be a bit confusing, we have used the symbol $A$
for the Adjacency block matrix; here it is the sum of products of $X$'s,
which are considered to be noncommuting variables).
Following \cite{cal} and \cite{pern2}, one can obtain all partitions
inserting all partitions between consecutive elements of the
irreducible partitions, or before their first element.
The insertion operator allows to obtain
in a recursive way the full sum $F(X)$ from the irreducible products
\footnote{The reader familiar with Feynman graphs will notice the analogy
of eq.(\ref{t.3}) to the Dyson equation expressing the two point function
in terms of the self-energy, the Bethe Salpeter equation expressing the
four-point function in terms of a two-particle irreducible kernel and
more generally the relation of connected Green functions to skeleton
expansions \cite{sym}.}
\begin{eqnarray} F(X)=1+A\left( I(F) X\right)
\label{t.3} \end{eqnarray}
In this equation\footnote{On the right side of the equation, the number
$1$ stands for the Identity matrix. We reserved the notation I for
the insertion operator} every matrix $X$ in the sum of the irreducible
products in $A(X)$ is replaced by $I(F)X$. The equation can be used
to construct all the partitions with $p$ elements from the irreducible
partitions of order less or equal to $p$.\\
Let us illustrate how one obtains $F$ till order $p=6$, from the
irreducible terms $A=X_1^2+X_1^4+X_1^6$. At order $2$, $X_1^2 \to
I(F)X_1 I(F)X_1=X_1^2$.\\ So $F=1+X_1^2$ at order $2$.\\ At order $4$,
the replacements
\begin{eqnarray} X_1^2 &\to & I(F)X_1 I(F)X_1=\nonumber\\ &&I(X^2)X_1
I(1)X_1 +I(1)X_1 I(X^2)X_1=\nonumber\\ &&X_1^2 X_2^2+X_1 X_2^2 X_1
\nonumber\\
X_1^4 &\to & I(1)X_1 I(1)X_1 I(1)X_1 I(1)X_1 =X_1^4
\nonumber \end{eqnarray} \noindent So $F=1+X_1^2+X_1^2 X_2^2+X_1 X_2^2
X_1+X_1^4=1+F_2+F_4$ including order $4$.\\ Next, at order $6$, the
replacements
\begin{eqnarray} X_1^2 &\to & I(F)X_1 I(F)X_1=\nonumber\\ && I(F_4)X_1
I(1)X_1+I(1)X_1 I(F_4)X_1+I(F_2)X_1 I(F_2)X_1\, ; \qquad\nonumber\\
&&I(F_4)X_1^2=I\left(X_1^2 X_2^2+X_1 X_2^2 X_1+X_1^4\right)
X_1^2=\nonumber\\ &&\quad = X_1^2 X_2^2 X_3^2+X_1 X_2^2 X_1 X_3^2+X_1^4
X_2^2 \, ;\nonumber\\ && X_1 I(F_4)X_1=X_1 I\left(X_1^2 X_2^2+X_1 X_2^2
X_1+X_1^4\right) X_1=\nonumber\\ && \quad = X_1 X_2^2 X_3^2 X_1+X_1
X_2 X_3^2 X_2 X_1+ X_1 X_2^4 X_1\, ;\qquad\nonumber\\ &&I(F_2)X_1
I(F_2)X_1=I(X_1^2)X_1 I(X_1^2) X_1=\nonumber\\ &&\quad =X_1^2 X_2
I(X_1^2)X_2=X_1^2 X_2 X_3^2 X_2 \nonumber \end{eqnarray}
\begin{eqnarray} X_1^4 &\to & I(F)X_1 I(F)X_1I(F)X_1 I(F)X_1=\nonumber\\ &
=& I(F_2)X_1^4+ X_1 I(F_2)X_1^3+ X_1^2I(F_2)X_1^2 + X_1^3 I(F_2)X_1=\qquad
\nonumber\\ &=& X_1^2 X_2^4+ X_1 X_2^2 X_1^3+X_1^2 X_2^2X_1^2+X_1^3
X_2^2 X_1 \nonumber \end{eqnarray}
\begin{eqnarray} X_1^6 \to I(F)X_1 I(F)X_1I(F)X_1
I(F)X_1I(F)X_1I(F)X_1=X_1^6\qquad \nonumber \end{eqnarray}
$F_6$ is the sum of the $12$ terms here obtained. After
taking the trace, using the cyclic property and relabeling
one has \begin{eqnarray} \texttt{tr} F_6=\texttt{tr}X_1^6 +6\,
\texttt{tr} \left( X_1^4 X_2^2 \right)+2\, \texttt{tr }(X_1^2 X_2^2
X_3^2)+3\,\texttt{tr}(X_1^2X_2X_3^2X_2) \qquad \nonumber \end{eqnarray}
in agreement with the evaluation of $\texttt{tr}A^6$ in the third line
of eq.(\ref{A.1}).\\
Furthermore, after expanding the polynomials and by keeping terms
up to products of six matrices, one may check that \begin{eqnarray}
\frac{1}{d}< \texttt{tr} F>=1+\frac{1}{d}<\texttt{tr}\Bigg(X_1^2
\left(\frac{<\texttt{tr}F>}{d}\right)^2+X_1^4\left(\frac{
<\texttt{tr}F>}{d}\right)^4+X_1^6\left(\frac{
<\texttt{tr}F>}{d}\right)^6\Bigg)> \nonumber \end{eqnarray}
The reason of this rearrangement is that the matrices inserted with $I(F)$
in all factors of product of matrices have labels different from any
other factor of the irreducible partition, therefore factorize in the
expectation.\\ Equation (\ref{t.3}) then implies the simpler equation
\begin{eqnarray} \frac{1}{d}< \texttt{tr} F>=1+\frac{1}{d}<\texttt{tr}
A\left( X \frac{<\texttt{tr}F>}{d} \right)> \label{nct0} \end{eqnarray}
Let us multiply each block $X_j$ by the variable $x$, then,
in this example, $A(x X)=x^2 X_1^2+x^4 X_1^4+x^6 X_1^6$,
$$a(x):=\frac{1}{d}<\texttt{tr}A(x X)>=x^2\frac{1}{d}<\texttt{tr}X^2>+
x^4\frac{1}{d} <\texttt{tr}X^4>+x^6\frac{1}{d}<\texttt{tr}X^6> $$
$f(x):=\frac{1}{d}<\texttt{tr}F(x X)>$ solves the simpler equation
\begin{eqnarray} f(x)=1+a\left(x f(x) \right) \label{t.5} \end{eqnarray}
This equation is known as non-crossing partition transform \cite{bei},
\cite{cal}, or it is called the relation between spectral moments and free
cumulants \cite{spei}.\\ It is also useful to introduce a parameter $p$,
counting the number of blocks of the partition.\\
$A(x X,p)$ is the sum of all irreducible products of $X_j$ matrices
corresponding to possible closed paths on trees. Each product in the
sum has the factor $x^{2l} p^h$ if the product has $2l$ matrices and $h$
are distinct.\\
\begin{eqnarray} a(x,p)&:=&\frac{1}{d}<\texttt{tr}A(x X,p)> \quad , \quad
f(x,p):=\frac{1}{d}<\texttt{tr}F(x X,p)> \label{t.6a}\\ f(x,p)&=&1+a\left(x
f(x,p),p \right) \label{t.6} \end{eqnarray}
In the next section, these simple functional equation will be used to
obtain the moments of a limiting spectral density for different matrix
ensembles.\\ More remarkably, the resulting different models only depend
upon the same set of irreducible partitions, the set of closed paths
on trees and the probability density to evaluate the expectations.\\ It
is then useful to recall the irreducible products, from eq.(\ref{A.1})
up to the products of $12$ matrices; each of these monomials corresponds
to a representative partition of an orbit; the integer coefficient
is the size of the orbit.
\begin{eqnarray}
&& p\Bigg( X_1^2+X_1^4+X_1^6+X_1^8+X_1^{10}+X_1^{12} \Bigg) \nonumber\\ &&
p^2 \Bigg( 2\, X_1^2 X_2^2 X_1^2 X_2^2 +10\, X_1^4 X_2^2 X_1^2 X_2^2+12\,
X_1^6 X_2^2 X_1^2 X_2^2+12\, X_1^4 X_2^4 X_1^2 X_2^2+\nonumber\\ && \qquad
+6\,X_1^4 X_2^2 X_1^4 X_2^2+2X_1^2 X_2^2 X_1^2 X_2^2 X_1^2 X_2^2\Bigg)
\nonumber\\ && p^3 \Bigg(6\, X_1^2 X_2^2 X_1^2X_3^2 X_2^2 X_3^2+3\,X_1^2
X_2 X_3^2 X_2 X_1^2 X_2 X_3^2 X_2+2 \, X_1^2 X_2^2 X_3^2 X_1^2 X_2^2
X_3^3 \Bigg) \nonumber\\ \label{t.7} \end{eqnarray}
We recall that the $2l$-th moment of the Adjacency matrix of
eq.(\ref{d.1})-(\ref{d.3}), in the $N \to \infty$ limit, is given by
the set of closed walks on tree graphs, associating a factor $Z=p$
for each distinct edge of a walk.
In ref.\cite{pern2} such set
is proved to be isomorphic to $P_2^{(2)}(l)$, the set of $2$-divisible
partitions with $2l$ elements, with the restriction that, given two
distinct blocks $B_i$ and $B_j$ of a partition, there is an even number of
elements of $B_j$ between any pair of elements of $B_i$.
This isomorphism has been found first in \cite{bose}, in which this
set of partitions, defined in an equivalent way, is called $SS(2l)$.
\\ Furthermore,
the set $P_2^{(2)}(l)$ has the property that it can be factorized in
irreducible partition under non-crossing. Then the generating function
$f$ of the number of partitions with given number of elements is related
to the generating function $a$ of the number of irreducible partitions
with given number of elements \cite{bei}.\\
It may happen that the generating function $a(x,p)$ of the irreducible
partitions is not known in closed form, but only its truncated
Taylor expansion. The eq.(\ref{t.6}) is then used to evaluate the
coefficients of the Taylor expansion of the generating function $f(x,p)$.
\begin{eqnarray} a(x,p)&:=& \sum_{n \geq 1} a_n(p) x^n\quad , \quad
f(x,p):= 1+\sum_{n \geq 1}f_n(p) x^n \nonumber\\ \sum_{n \geq 1}f_n(p)
x^n &=& \sum_{n \geq 1} a_n(p) x^n \left(1+\sum_{m \geq 1}f_m(p) x^m
\right)^n \label{t.8} \end{eqnarray}
By comparing the same power of $x$, one obtains the coefficients $f_n(p)$
as functions of the $a_j(p)$ and viceversa. We display in Table 3 the
lowest order relations.\\
\section{Different and related matrix ensembles}
Depending on the probability law of the random $d \times d$ random blocks
$X_{i,j}$, the functional equations of the previous section may lead to an
algebraic equation for the resolvent of the matrix ensemble or they are
only useful to express the spectral moments in terms of free cumulants.
In all cases they show similarities and differences among different
matrix ensembles.\\
\subsection{The elastic springs, matrix block $X_{i,j}$ is a rank-one
projector}
As it was mentioned in the Introduction, a sparse random block matrix
ensemble was proposed for the study of vibrations of amorphous solids
\cite{pari}, \cite{cic1}, \cite{pern}, \cite{benet}. The
ensemble of Adjacent matrices has the form of eq.(\ref{d.3}) , the
variables $\alpha_{i,j}=\alpha_{j,1}=$ are i.i.d. random variables
with the probability law in eq.({\ref{d.2}), the $d \times d $ blocks
$X_{i.j}$ are rank-one projectors $$X_{i,j}=X_{i,j}^{t}=X_{j,i}=\hat
n_{ij} \hat n_{ij}^t $$ where $\hat n_{ij}$ is a $d$-dimensional
random vector of unit length, chosen with uniform probability on the
$d$-dimensional sphere and $\hat n_{ij} \hat n_{ij}^t $ is the usual
matrix product of a column vector times a row vector originating a
rank-one matrix.\\
It was proved in \cite{pern} that the equations for the generating
functions of the spectral moments are greatly simplified in the $d \to
\infty$ limit, with $t=Z/d$ fixed, then obtaining a cubic equation for
the limiting resolvent. The proof in \cite{pern} allows the use of the
equation (\ref{t.6}) to obtain the limiting resolvent in few lines. Indeed
in this limit, out of the whole set $S_1$ of the closed paths on trees,
relevant for finite $d$, only the subset $S_2 \subset S_1$ survive,
each one with weight one.\\
The evaluation of the limiting moments becomes the counting
of non-crossing partitions. The sum $A(X,Z)$ of all irreducible
products of $X_j$ matrices corresponding to possible closed paths $
A(X,Z)=Z\sum_{l \geq 1} (X_1)^{2l}$.
It is easy to see also in this model from Eq. (\ref{t.3}) one
obtains Eq. (\ref{nct0}), where $F(X, Z)$ depends on $Z$, counting the number
of blocks, and on $d$; as shown in \cite{pern}, taking the average
there is a factor $\frac{1}{d}$ for each block of the partition
(i.e. for each distinct matrix block $X$), so that the dependence
of $F$ on $Z$ and $d$ is through $t=\frac{Z}{d}$. Therefore in
Eq.(\ref{t.6}) one has $p=t=\frac{Z}{d}$.
\begin{eqnarray} a(x,p)&=& p\sum_{l \geq 1}x^{2l}=p \frac{x^2}{1-x^2}
\nonumber\\ f(x,p) &=& 1+p \frac{ x^2 f(x,p)^2}{1- x^2 f(x,p)^2}
\label{t.10} \end{eqnarray}
This cubic equation was the result of reference \cite{pern} for the
model of random elastic springs, in the limit $d \to \infty$ with
$p=Z/d$ fixed.\\ Very recently A. Dembczak-Kolodziejczik and A. Lytova,
by methods of the resolvent, provided an alternative proof of this result
and the analogous one for the Laplacian ensembles \cite{aa}.\\
The function $f(x,p)=\sum_{n=0}^\infty x^{2n}\mu_{2n}$ is a generating
function of the moments of the spectral density. The moments are
Fuss-Narayana polynomials $$ \mu_{2n}=\sum_{b=1}^n \frac{1}{b} \left(
\begin{array}{cc} n-1\\b-1 \end{array}\right) \left( \begin{array}{cc}
2 n\\b-1 \end{array}\right) \,p^b$$
The equivalent cubic equation for the resolvent was obtained by
Semerjian and Cugliandolo long ago \cite{semer} as an approximation,
called the Effective Medium Approximation, on the replica formalism for
a \textsl{different} matrix ensemble. It was also obtained by Slanina
with the cavity method \cite{sla}. Very recently different methods of
obtaining the spectral density ensembles of sparse real symmetric random
matrices have been reviewed \cite{vivo}.\\
The spectral density defined by the generating function $f(x,p)$ is a
member, $\pi_{2,t}$, of a remarkable family of measures, called Free
Bessel laws \cite{bani}. The authors discussed the combinatorial
relevance of these measures and the relation of $\pi_{2,t}$ to
non-crossing partitions where all blocks have an even number of
elements. It was not shown a random matrix ensemble related to this
spectral density.\\
\vskip 2cm \subsection{Matrix block $X_{i,j}$ is element of GOE ensemble,
$d$ finite}
In sections 2.2 and 2.3 we summarized rules to evaluate expectations of
products of block matrices $X_j \in$ GOE, for finite $d$. The products
in which we are interested are the terms in eq.(\ref{A.1}) contributing to the
spectral moments in the limit $N \to \infty$. The spectral moments of
low order are reported in eq.(\ref{A.2}).\\
Here we test the usefulness of the functional transform in a case where
the generating function of irreducible partitions $a(x,p)$ is not known
in closed form, but merely its truncated Taylor expansion.\\
Up to the product of $12$ block matrices, the irreducible terms
are listed in eq.(\ref{t.7}). The expectation of the trace
of the monomial powers are evaluated by the recursion relation
\cite{ledo} \begin{eqnarray} &&\frac{p}{d} <\texttt{tr} \Bigg(x^2
X_1^2+x^4X_1^4+x^6X_1^6+x^8X_1^8+x^{10}X_1^{10}+x^{12}X_1^{12}
\Bigg)> = \nonumber\\ &&p\,\Bigg( \sigma^2 x^2(d+1)+\sigma^4 x^4
(2d^2+5d+5)+\sigma^6 x^6 (5d^3+22d^2+52d+41)+\nonumber\\ &&\sigma^8 x^8
(14 d^4+93 d^3+374 d^2+690 d+509)+\nonumber\\ && \sigma^{10}x^{10}
(42 d^5+386d^4+2290d^3+7150 d^2+12143 d+ 8229)+ \nonumber\\ &&
\sigma^{12}x^{12} (132 d^6+1586 d^5+12798 d^4+58760 d^3+167148 d^2+258479
d+166377) \Bigg) \nonumber \end{eqnarray}
Other terms are reduced to evaluation of pure powers, by Wick
contractions of the $X_2$ fields. For instance \begin{eqnarray}
&&\frac{1}{d}<\texttt{tr} (X_1)^{2n} X_2^2 X_1^2 X_2^2>= \nonumber\\
&& \sigma^4 \Bigg( (d^2+3d+5)<\texttt{tr} (X)^{2n+2}>+\sigma^2
(d+2)(d^2+d+4n) <\texttt{tr} (X)^{2n}>\Bigg)
\nonumber
\end{eqnarray}
The remaining irreducible terms, up to order $12$ are also evaluated
by graphic methods. It is easy to use the evaluated irreducible terms
$a_j$ in the series (\ref{t.8}) and Table $4$ to evaluate the moments and
verify Table $2$ or extend it to higher orders. Indeed, in this ensemble,
both free cumulants and spectral moments of odd order are zero. Then,
for example
$$f_8=a_8+8 a_2 a_6+4 a_4^2 +28 a_2^2 a_4+14 a_2^4 $$ This subsection
shows the the functional transform may be convenient even for finite
$d$.\\
\vskip 2cm \subsection{Matrix block $X_{i,j}$ is element of GOE or GUE
ensemble, $d$ infinite}
To evaluate the spectral moments in the limit $d \to \infty$ one chooses the
variance $\sigma^2=1/d$ in eq.(\ref{d.8}).\\ As is well known, the
leading order expectations is the same for random matrices in the GOE
or in the GUE ensembles.\\
Expectations of trace of products of random matrices in the $d \to
\infty $ limit are simpler than for finite $d$.\\ We report here our
evaluation of the generating function of irreducible terms, $a(x,Z)$, in
the $d \to \infty $ limit, up to order $22$. \begin{eqnarray} a(x,Z)&=&
Z x^2 + 2\, Z x^4 + 5\, Z x^6 + (14 \,Z+6\, Z^2) x^8 +(42\, Z+70 \,Z^2)
x^{10}+ \nonumber\\
&&(132\,Z+552 \, Z^2+50\, Z^3) x^{12}+(429 \,Z+3696 Z^2+1204 \,Z^3)
x^{14}+ \nonumber\\
&& (1430\, Z+22710\, Z^2+17272\, Z^3+618 Z^4) x^{16}+\nonumber\\
&&(4862\,Z+132726\, Z^2+193289 Z^3+23808 Z^4) x^{18}+\nonumber\\ &&
(16796\,Z+752186\,Z^2+1869200\,Z^3+518600\,Z^4+9606\,Z^5)x^{20}+\nonumber\\
&&(58786 Z+4181034\,Z^2+16446826 \,Z^3+8459308\,Z^4+ 524040\, Z^5)x^{22}
\nonumber \end{eqnarray}
This truncated generating function and the perturbative functional
transform lead to the spectral moments up to $\mu_{22}$, reported
in eq.(\ref{t.1}).
The values of the moments $\mu_2$ up to $\mu_{12}$ were independently
computed from the table of moments in the Appendix, eq.(\ref{A.1}).\\
It is interesting to compare these moments with a work by Z.D. Bai,
B. Miao, B. Jin , \cite{bai2007}. The authors study an ensemble of random
matrices, each being the product of two random matrices, $S_n W_n$, where
$S_n=\frac{1}{n} X_n X_n^{\dag}$, $X_n$ is a rectangular $p \times n$
matrix with complex i.i.d. entries, $W_n$ is a Wigner matrix.\\ The
authors prove that in the limit $n \to \infty, p \to \infty$, $p/n=y$
finite, the spectral distribution of the random ensemble $S_n W_n$
converges to a non-random density function, eq.(1.6) of their paper,
and the spectral moments converge to the non-random moments $\beta_k$
we reproduced in Table 4 in the Appendix.\\
Let us show that $Z \beta_k(y=Z)$ is equal to an approximation of the moment
$\mu_k$ of the Adjacency block matrix model in the limit $d\to \infty$,
considered in this paragraph, where the approximation consists in
considering only walks whose sequence of steps correspond to
noncrossing partitions, i.e. considering only partitions belonging to
$NC^{(2)}(h)$, instead of those in $P^{(2)}_2(h)$.
The generating function of the irreducible terms of the latter model
is the series of the power monomials,
and the expectations are the limiting, $d \to \infty$, values for GOE
or GUE ensembles. Let us use $p = Z$ in the following.
\begin{eqnarray} a(x,p)&=& \lim_{d \to \infty} \sum_{k \geq
1}\frac{p}{d}<\texttt{tr} X^{2k}> x^{2k} = p\sum_{k \geq 1} x^{2k}
\frac{1}{k+1}\left( \begin{array}{cc} 2k\\k \end{array}\right)=\nonumber\\
&=& p\left(\frac{ 1-\sqrt{1-4x^2}}{2 x^2 }-1\right) \nonumber
\end{eqnarray}
This closed form may be used in eq.(\ref{t.6}) to obtain the
generating function $f(x,p)$ of the spectral moments \begin{eqnarray}
&&f(x,p)=1+p\left( \frac{1-\sqrt{1-4 x^2 f^2(x,p)}}{2 x^2 f^2(x,p)}-1
\right) \qquad , \qquad \texttt{or}\nonumber\\ && x^2 f^2(x,p) \bigg(
f(x,p)+p-1 \bigg)^2-p f(x,p)+p=0 \qquad \qquad \label{t.30} \end{eqnarray}
Changing variables $f = 1 + p z g$, one gets
\begin{eqnarray}
&&g(z) = {\cal A}(z g(z)) \nonumber \\
&&{\cal A}(z) = (1 + p z)^2 (1 + z)^2
\end{eqnarray}
where $z = x^2$.
From Lagrange inversion formula,
\begin{eqnarray}
\beta_{2n} = [z^{n-1}] g(z) &=& \frac{1}{n}[z^{n-1}] {\cal A}(z)^n =
\frac{1}{n}[z^{n-1}] ((1 + p z)^{2n} (1 + z)^{2n}) \nonumber\\
&=&\frac{1}{n}\sum_{i=0}^{n-1} p^i \left( \begin{array}{cc} 2n \\i \end{array}\right)\left(\begin{array}{cc} 2n \\n-1-i \end{array}\right)
\nonumber
\end{eqnarray}
With $k=2n$, $y=p$ and $i=j+1$ one obtains Eq. (\ref{A.6}).\\
The generating function $f(x,p)$ of the spectral moments is a
solution of the quartic equation (\ref{t.30}). We reproduce in
Table $4$ the spectral moments and compare them with the ones in
eq.(\ref{t.1}). In the latter case, the spectral moment $\mu_{2l}$ is a
polynomial $\mu_{2l}=\sum_{k=1}^l b_k^{(2l)}Z^k$, the lowest coefficient
$b_1^{(2l)}$ and the two highest ones $b_{l-1}^{(2l)}$, $b_{2l}^{(2l)}$
are the same of the corresponding ones in Table $3$ whereas the remaining
$b_k^{(2l)}$ are bigger integers than the analogous ones in Table $4$
because the latter ones have contributions only from closed paths
in $S_2$.
\vskip 2cm \section{Conclusions}
The moments of spectral density of sparse random block matrices, with the
architecture of the Erd\"os-Renyi random graph, are analytically studied
in two steps. In the first step we list the products of blocks pertinent
to the limiting spectral moments, up to $\mu_{12}$ for any probability
distribution of the i.i.d. blocks $X_{i,j}$, random matrices of finite or
infinite dimensions. In the second step, we evaluate the expectations of
the products, for several probability laws of the i.i.d. blocks $X_{i,j}$,
GOE for $d$ finite, GOE or GUE for $d$ infinite.\\ With limited effort,
our list may be adapted to the evaluation of sparse random block matrices
with the architecture of a random regular graph.\\
Three sets of set partitions $S_j$ , $j=1,2,3$, defined in Sect.2,
correspond to sets of closed walks on trees of the weighted graph
which add up to the spectral moments. The relevant set $S_j$ depends
on the probability law chosen for the blocks and further limits of the
parameters.\\
We do not study here the well known set $S_3$ , $S_3 \subset S_2 \subset
S_1$, of the Wigner closed paths. Two examples are provided where the
relevant set is $S_2$. It corresponds to the non-crossing partitions. In
this case, it is possible to evaluate all the moments and the spectral
density, with help of the functional transform.\\
For other important probability laws of the blocks and limiting values of
the parameters, the moments are evaluated by the largest set $S_1$. Then
it is possible to evaluate the spectral moments at high order, but we
know no example where the spectral density was evaluated.\\
\vskip 3cm \section{Appendix}
Table 1. The weighted closed paths on trees relevant to the limiting
moments (that is $N \to \infty $) of the Adjacency matrix.
\vskip 0.8 cm
\footnotesize
\begin{eqnarray}
\frac{1}{N} tr\,A^2 &=& Z\,tr\,(X_1)^2 \qquad \qquad \qquad
\nonumber\\
\frac{1}{N} tr\,A^4 &=& Z\,tr\,(X_1)^4+Z^2\,2\,tr\,(X_1)^2(X_2)^2 \qquad
\qquad \qquad
\nonumber\\
\frac{1}{N} tr\,A^6 &=& Z\,tr\,(X_1)^6+Z^2\,6\,tr(X_1)^4(X_2)^2
+ Z^3\,tr\,\left[ 2\,(X_1)^2(X_2)^2(X_3)^2 +3\,(X_1)^2
X_2(X_3)^2X_2 \right] \qquad \nonumber\\ \frac{1}{N}
tr\,A^8 &=& Z\,tr\,(X_1)^8+Z^2\,tr\,\left[ 8\,(X_1)^6(X_2)^2
+4\,(X_1)^4(X_2)^4+2\,(X_1)^2(X_2)^2(X_1)^2(X_2)^2\right]+\nonumber\\
&+& Z^3\,tr\,\left[8\,(X_1)^4(X_2)^2(X_3)^2+8\,(X_1)^4X_2(X_3)^2
X_2+8\,(X_1)^3(X_2)^2 X_1(X_3)^2+ \right. \nonumber\\
&& \quad
\left. +4\,(X_1)^2(X_2)^2(X_1)^2(X_3)^2\right]+\nonumber\\
&+& Z^4\,tr\, \left[8\,(X_1)^2(X_2)^2 X_3(X_4)^2 X_3+
4\, (X_1)^2 X_2X_3(X_4)^2 X_3X_2+2\, (X_1)^2(X_2)^2
(X_3)^2(X_4)^2 \right] \qquad \nonumber\\ \frac{1}{N}
tr\,A^{10} &=& Z\,tr\,(X_1)^{10}+Z^2\,tr\,10\,\left[
(X_1)^8(X_2)^2+(X_1)^6(X_2)^4+(X_1)^4(X_2)^2(X_1)^2(X_2)^2\right]
+\qquad \qquad \qquad \nonumber\\ &+& Z^3\,tr \,\left[
10\,(X_1)^6(X_2)^2(X_3)^2+ 10\,(X_1)^6X_2(X_3)^2X_2+
10\,(X_1)^5(X_2)^2X_1(X_3)^2 + \right. \nonumber\\ && \quad
+10\,(X_1)^4(X_2)^4(X_3)^2+10\,(X_1)^4(X_2)^2(X_1)^2(X_3)^2+10\,(X_1)^4(X_2)^2
(X_3)^2(X_2)^2 + \nonumber\\ && \quad
+10\,(X_1)^4(X_2)^3(X_3)^2 X_2+10\,(X_1)^4 X_2(X_3)^2(X_2)^3+10\,
(X_1)^2(X_2)^2(X_1)^2X_2)^2(X_3)^2+\qquad \qquad \qquad
\nonumber\\ && \quad \left. +10\,(X_1)^2 (X_2)^2(X_1)^2 X_2(X_3)^2
X_2+5 (X_1)^4X_2(X_3)^4X_2+5(X_1)^3(X_2)^2(X_1)^3(X_3)^2
\right] +\nonumber\\ &+& Z^4\,tr \,10\,\left[(X_1)^4X_2
X_3(X_4)^2X_3X_2+(X_1)^4(X_2)^2X_3(X_4)^2X_3+
(X_1)^4X_2(X_3)^2X_2(X_4)^2+ \right. \qquad \qquad \qquad
\nonumber\\ && \quad +(X_1)^4 X_2(X_3)^2(X_4)^2X_2+
(X_1)^4(X_2)^2(X_3)^2(X_4)^2+ (X_1)^3(X_2)^2 (X_3)^2
X_1(X_4)^2 +\nonumber\\ && \quad +(X_1)^3 (X_2)^2
X_1(X_3)^2(X_4)^2+(X_1)^3(X_2)^2 X_1 X_3 (X_4)^2 X_3+ (X_1)^3
X_2(X_3)^2 X_2 X_1(X_4)^2 +\nonumber\\ && \quad +(X_1)^2 (X_2)^2
(X_3)^2X_2(X_4)^2X_2+ (X_1)^2(X_2)^2(X_1)^2X_3(X_4)^2X_3+\qquad
\qquad \qquad \nonumber\\ && \quad
+\left.(X_1)^2(X_2)^2(X_1)^2(X_3)^2(X_4)^2\right]+\nonumber\\
&+& Z^5\,tr \,\left[ 2(X_1)^2(X_2)^2(X_3)^2(X_4)^2(X_5)^2+
5\,(X_1)^2(X_2)^2 X_3 (X_4)^2(X_5)^2X_3+\right.\qquad \qquad
\qquad \nonumber\\ && \quad + 5\,(X_1)^2 X_2X_3X_4(X_5)^2
X_4X_3X_2+10\, (X_1)^2(X_2)^2(X_3)^2 X_4 (X_5)^2 X_4 +\nonumber\\
&& \quad \left. + 10\,(X_1)^2 X_2(X_3)^2 X_4(X_5)^2 X_4X_2+10\,
(X_1)^2(X_2)^2X_3X_4(X_5)^2X_4X_3 \right] \nonumber\\
\frac{1}{N} tr\,A^{12} &=& Z\,tr\,(X_1)^{12}+Z^2\,tr\,2\,\left[ 6
(X_1)^{10}(X_2)^2+6(X_1)^8(X_2)^4+3(X_1)^6(X_2)^6+\right.\nonumber\\
&& \quad
+6(X_1)^6(X_2)^2(X_1)^2(X_2)^2+6(X_1)^4(X_2)^4(X_1)^2(X_2)^2+3(X_1)^4(X_2)^2(X_1)^4(X_2)^2+\qquad
\qquad \qquad \nonumber\\ && \quad \left.
+(X_1)^2(X_2)^2(X_1)^2(X_2)^2(X_1)^2(X_2)^2\right]+\nonumber\\
&& +Z^3 \,tr \Bigg[ 12 \left[
(X_1)^8(X_2)^2(X_3)^2+(X_1)^8X_2(X_3)^2X_2+(X_1)^7(X_2)^2X_1(X_3)^2+
\right. \qquad \qquad \qquad \nonumber\\ && \quad
+(X_1)^6(X_2)^4(X_3)^2+(X_1)^6(X_2)^2(X_3)^4+(X_1)^6X_2(X_3)^2(X_2)^3+(X_1)^6(X_2)^3(X_3)^2X_2
+ \nonumber\\ && \quad
+(X_1)^6X_2(X_3)^4X_2+(X_1)^6(X_2)^2(X_3)^2(X_2)^2+(X_1)^6(X_2)^2(X_1)^2(X_3)^2+\nonumber\\
&& \quad \left. +(X_1)^5(X_2)^4X_1(X_3)^2+
(X_1)^5(X_2)^2X_1(X_3)^4+(X_1)^5(X_2)^2(X_1)^3(X_3)^2 \right]+\nonumber\\
&& \quad +4 (X_1)^4(X_2)^4(X_3)^4+12 \left[
(X_1)^4(X_2)^4(X_1)^2(X_3)^2+(X_1)^4(X_2)^4(X_3)^2(X_2)^2+\right.
\nonumber\\ && \quad \left.+(X_1)^4(X_2)^3(X_3)^2(X_2)^3+
(X_1)^4(X_2)^3(X_3)^4 X_2 \right]+ 6 \left[
(X_1)^4(X_2)^2(X_1)^4(X_3)^2+ \right.\qquad \qquad \qquad
\nonumber\\ && \quad \left. +(X_1)^4(X_2)^2(X_3)^4(X_2)^2 \right]+12
\left[ (X_1)^4(X_2)^2(X_3)^2(X_1)^2(X_3)^2+\right. \nonumber\\ &&\quad
+(X_1)^4(X_2)^2(X_3)^2(X_1)^2(X_2)^2+(X_1)^4(X_2)^2(X_3)^2(X_2)^2(X_3)^2+(X_1)^4(X_2)^2(X_1)^2(X_3)^2(X_2)^2+
\nonumber\\ && \quad +(X_1)^4(X_2)^2(X_1)^2(X_2)^2(X_3)^2 +
(X_1)^4(X_2)^2(X_1)^2 X_2 (X_3)^2 X_2+ (X_1)^4 X_2 (X_3)^2
X_2(X_1)^2(X_2)^2 +\qquad \qquad \qquad \nonumber\\ && \quad
+(X_1)^4(X_2)^2 X_1 (X_3)^2 X_1(X_2)^2 + (X_1)^4 X_2 (X_3)^2(X_2)^2(X_3)^2
X_2 +
\qquad \qquad \nonumber\\ && \quad +(X_1)^3(X_2)^2 X_1 (X_3)^2
(X_1)^2(X_3)^2 +(X_1)^3(X_2)^2 (X_1)^2 (X_2)^2 X_1(X_3)^2 +\qquad
\qquad \nonumber\\
&& \quad \left. +(X_1)^2(X_2)^2 (X_1)^2 (X_2)^2
(X_1)^2(X_3)^2 \right]+6\, (X_1)^2(X_2)^2 (X_1)^2 (X_3)^2
(X_2)^2(X_3)^2+\qquad \qquad \nonumber\\ && \quad +2\,
(X_1)^2(X_2)^2 (X_3)^2 (X_1)^2 (X_2)^2(X_3)^2 +3\, (X_1)^2
X_2 (X_3)^2 X_2 (X_1)^2 X_2(X_3)^2 X_2 \Bigg]+\qquad
\qquad \nonumber\\
&& +Z^4 \,tr \Bigg[ 12\,\Bigg( X_1^6X_2^2X_3^2X_4^2 +
X_1^6X_2 X_3^2X_4^2X_2 + X_1^6X_2X_3^2X_2X_4^2 +
X_1^6 X_2X_3X_4^2X_3X_2 \nonumber\\
&+& X_1^6X_2^2X_3X_4^2X_3 + X_1^5X_2^2X_3^2X_1X_4^2 +
X_1^5X_2^2X_1X_3^2X_4^2 + X_1^5X_2X_3^2X_2X_1X_4^2\nonumber\\ &+&
X_1^5X_2^2X_1X_3X_4^2X_3 + X_1^4X_2^4X_3^2X_4^2 + X_1^4X_2^4 X_3 X_4^2
X_3 + X_1^4 X_2 X_3^4 X_2 X_4^2\nonumber\\
& +& X_1^4X_2X_3^4X_4^2 X_2 + X_1^4 X_2^3 X_3^2 X_2 X_4^2 +
X_1^4X_2^2X_3^3X_4^2 X_3 + X_1^4 X_2 X_3^2 X_4^2 X_2^3 \nonumber\\
&+& X_1^4 X_2^2 X_3 X_4^2 X_3^3 + X_1^4X_2^3 X_3^2 X_4^2X_2 + X_1^4
X_2 X_3^2 X_2^3 X_4^2 + X_1^4 X_2X_3 X_4^2X_3X_2^3\nonumber\\ &+&
X_1^4X_2X_3^3X_4^2X_3X_2 + X_1^4 X_2X_3 X_4^2X_3^3 X_2 + X_1^4
X_2^3 X_3 X_4^2X_3X_2 + X_1^4 X_2^2 X_1^2X_3^2 X_4^2 \nonumber\\
&+& X_1^4 X_2^2 X_3^2 X_1^2 X_4^2 + X_1^4 X_2^2 X_3^2 X_4^2 X_3^2+
X_1^4 X_2^2 X_3^2 X_2^2 X_4^2 + X_1^4 X_2^2 X_3^2 X_4^2 X_2^2 \nonumber\\
&+& X_1^4 X_2^2 X_1^2 X_3 X_4^2 X_3+X_1^4X_2X_3^2 X_2 X_1^2 X_4^2 +X_1^4
X_2 X_3^2 X_2^2 X_4^2 X_2 + X_1^4 X_2 X_3^2 X_2 X_4^2 X_2^2\nonumber\\
&+& X_1^4X_2^2 X_3^2 X_2 X_4^2 X_2+X_1^4 X_2^2 X_1 X_3^2 X_1 X_4^2+X_1^4
X_2^2 X_3 X_4^2 X_3 X_2^2 +X_1^4 X_2 X_3^2 X_4^2 X_3^2 X_2\nonumber\\
&+& 6\, \Bigg( X_1^4X_2^2X_3^4X_4^2+X_1^4X_2X_3 X_4^4 X_3 X_2 +
X_1^3 X_2 X_3^2 X_2^3 X_1 X_4^2+ X_1^3 X_2^2 X_1 X_3^3 X_4^2X_3\Bigg)
\nonumber\\
&+& 12\, \Bigg( X_1^3X_2^2 X_1^3 X_3^2 X_4^2+ X_1^3 X_2^2 X_1^3 X_3
X_4^2 X_3+ X_1^3 X_2^2 X_1^2 X_3^2 X_1 X_4^2+ X_1^3 X_2^2 X_1 X_3^2
X_4^2 X_3^2 \nonumber\\ &+& X_1^3X_2^2 X_1 X_3^2 X_1^2 X_4^2+ X_1^3X_2^2
X_3^2 X_2^2 X_1 X_4^2 + X_1^3 X_2^3 X_3^2 X_2 X_1 X_4^2+X_1^2 X_2^2 X_1^2
X_2^2 X_3^2 X_4^2 \nonumber\\ &+& X_1^2 X_2^2 X_1^2 X_3^2 X_2^2 X_4^2+
X_1^2 X_2^2 X_3^2 X_1^2 X_3 X_4^2 X_3 +X_1^2 X_2^2 X_3^2 X_1^2 X_2
X_4^2 X_2 + X_1^2 X_2^2 X_1^2 X_3^2 X_2 X_4^2 X_2 \nonumber\\
&+& X_1^2 X_2^2 X_3^2 X_2^2 X_3 X_4^2 X_3+ X_1^2 X_2^2 X_1^2 X_2^2
X_3 X_4^2 X_3+ X_1^2 X_2^2 X_1^2 X_2 X_3^2 X_4^2 X_2 \nonumber\\ &+&
X_1^2 X_2^2 X_1^2 X_2 X_3 X_4^2 X_3 X_2+ X_1^2 X_2^2 X_1 X_3^2 X_1 X_2
X_4^2 X_2\Bigg)\nonumber\\ &+& 6\,\Bigg( X_1^2 X_2^2 X_1^2 X_3^2 X_4^2
X_3^2+ X_1^2 X_2^2X_3^2X_1^2 X_2^2 X_4^2 + X_1^2 X_2 X_3^2 X_2 X_1^2
X_2 X_4^2 X_2 \Bigg) \nonumber\\ &+& 4\, X_1^2 X_2^2 X_1^2 X_3^2
X_1^2 X_4^2 \Bigg]\nonumber\\ && +Z^5 \,tr
\,\Bigg[12\, \Bigg( X_1^4X_2^2X_3^2X_4^2 X_5^2 + X_1^4X_2^2 X_3^2
X_4 X_5^2 X_4 + X_1^4 X_2^2 X_3 X_4^2 X_3 X_5^2 \nonumber\\ &&
+X_1^4 X_2 X_3^2 X_2 X_4^2 X_5^2 + X_1^4 X_2 X_3^2 X_4^2 X_5^2 X_2 +
X_1^4 X_2^2 X_3 X_4^2 X_5^2 X_3\nonumber\\
&& +X_1^4 X_2 X_3^2 X_4^2 X_2 X_5^2 + X_1^4 X_2 X_3 X_4^2 X_3 X_2 X_5^2 +
X_1^4 X_2^2 X_3 X_4 X_5^2 X_4 X_3 \nonumber\\
&& +X_1^4 X_2 X_3 X_4^2 X_5^2 X_3 X_2 +
X_1^4 X_2 X_3^2 X_4 X_5^2 X_4 X_2 + X_1^4 X_2 X_3 X_4^2 X_3
X_5^2 X_2 \nonumber\\
&& +X_1^4 X_2 X_3^2 X_2 X_4 X_5^2 X_4 + X_1^4 X_2 X_3 X_4 X_5^2
X_4 X_3 X_2 \nonumber\\ && +X_1^3 X_2^2 X_3^2 X_4^2 X_1 X_5^2 +
X_1^3 X_2^2 X_1 X_3^2 X_4^2 X_5^2 + X_1^3 X_2^2 X_3^2 X_1 X_4^2
X_5^2\nonumber\\ && +X_1^3 X_2 X_3^2 X_2 X_1 X_4^2 X_5^2 +
X_1^3 X_2^2 X_3^2 X_1 X_4 X_5^2 X_4 + X_1^3 X_2^2 X_3 X_4^2
X_3 X_1 X_5^2 \nonumber\\
&& +X_1^3 X_2 X_3^2 X_2 X_4^2 X_1 X_5^2 + X_1^3 X_2^2 X_1 X_3^2
X_4 X_5^2 X_4 + X_1^3 X_2^2 X_1 X_3 X_4^2 X_3 X_5^2 \nonumber\\
&& +X_1^3 X_2^2 X_1 X_3 X_4^2 X_5^2 X_3 + X_1^3 X_2 X_3^2
X_4^2 X_2 X_1 X_5^2 + X_1^3 X_2 X_3 X_4^2 X_3 X_2 X_1 X_5^2
\nonumber\\
&& +X_1^3 X_2^2 X_1 X_3 X_4 X_5^2 X_4 X_3 + X_1^3 X_2 X_3^2 X_2
X_1 X_4 X_5^2 X_4 \nonumber\\
&& +X_1^2 X_2^2 X_1^2 X_3^2 X_4^2 X_5^2 + X_1^2 X_2^2 X_1^2 X_3^2
X_4 X_5^2 X_4 +X_1^2 X_2^2 X_3^3 X_1 X_4^2 X_1 X_5^2\nonumber\\
&& +X_1^2 X_2^2 X_1 X_3^2 X_1 X_4^2 X_5^2 +X_1^2 X_2^2 X_1^2 X_3
X_4^2 X_3 X_5^2 +X_1^2 X_2^2 X_3^2 X_1^2 X_4 X_5^2 X_4 \nonumber\\
&& +X_1^2 X_2^2 X_1^2 X_3 X_4 X_5^2 X_4 X_3 + X_1^2 X_2^2
X_1 X_3 X_4^2 X_3 X_1 X_5^2 +X_1^2 X_2^2 X_1^2 X_3 X_4^2 X_5^2
X_3 \nonumber\\
&& +X_1^2 X_2 X_3^2 X_2 X_1 X_4^2 X_1 X_5^2 + X_1^2 X_2^2 X_1 X_3^2 X_1
X_4 X_5^2 X_4 + X_1^2 X_2^2 X_1 X_3^2 X_4^2 X_1 X_5^2\Bigg) \nonumber\\
&& +6 \, X_1^2 X_2^2 X_3^2 X_1^2 X_4^2 X_5^2 +
+6\, X_1^2 X_2 X_3^2 X_2 X_1^2 X_4 X_5^2 X_4+3\, X_1^2 X_2
X_3^2 X_2 X_4^2 X_2 X_5^2 X_2 \Bigg]+ \nonumber\\
&+& Z^6\,tr \,\Bigg[ 12 \Bigg( X_1^2 X_2^2 X_3^2 X_4^2
X_5 X_6^2 X_5+X_1^2 X_2^2 X_3^2 X_4 X_5^2 X_6^2 X_4+X_1^2
X_2^2 X_3^2 X_4 X_5 X_6^2 X_5 X_4 \nonumber\\ && X_1^2
X_2^2 X_3 X_4^2 X_5 X_6^2 X_5 X_3+X_1^2 X_2^2 X_3 X_4^2
X_3 X_5 X_6^2 X_5+X_1^2 X_2^2 X_3 X_4 X_5^2 X_4 X_6^2
X_3 \nonumber\\ && X_1^2 X_2^2 X_3 X_4 X_5 X_6^2 X_5
X_4 X_3+X_1^2 X_2 X_3^2 X_4 X_5 X_6^2 X_5 X_4 X_2 +X_1^2
X_2 X_3^2 X_2 X_4 X_5 X_6^2 X_5 X_4
\Bigg)+\nonumber\\
&& 6 \Bigg( X_1^2 X_2^2 X_3 X_4 X_5^2 X_6^2 X_4 X_3+
X_1^2 X_2 X_3^2 X_2 X_4^2 X_5 X_6^2 X_5+ X_1^2 X_2 X_3
X_4 X_5 X_6^2 X_5 X_4 X_3 X_2 \Bigg)+ \nonumber\\ &&
4\, X_1^2 X_2 X_3 X_4^2 X_3 X_5 X_6^2 X_5 X_2+ 2\,
X_1^2 X_2^2 X_3^2 X_4^2 X_5^2 X_6^2 \Bigg]
\nonumber\\
\label{A.1}
\end{eqnarray} \normalsize
This table of products of blocks $X_j$ allows to evaluate
analytically the limiting ($N \to \infty$) spectral moments for
any sparse ensemble of real symmetric random block matrices
$Nd \times Nd$, where the random $d \times d$ real symmetric
blocks are i.i.d., and the architecture of the Adjacency matrix
corresponds to the Erd\"os-Renyi random graph, with average
degree $Z$.\\
Actually, it is possible to obtain also the limiting spectral
moments of real symmetric random block matrices $Nd \times
Nd$, where the random $d \times d$ real symmetric blocks are
i.i.d., and the architecture of the Adjacency matrix corresponds
to regular random graphs, with fixed degree $Z$. This requires
only the determination of the $Z-$dependent weight, as we show
with one example.\\
Erd\"os-Renyi random graph : \begin{eqnarray} \frac{1}{N}
tr\,A^8 &=& Z\,tr\,X_1^8+Z^2\,tr\,\left[ 8\,X_1^6 X_2^2
+4\,X_1^4 X_2^4+2\,X_1^2X_2^2X_1^2X_2^2\right]+\nonumber\\
&+& Z^3\,tr\,\left[8\,X_1^4X_2^2X_3^2+8\,X_1^4X_2X_3^2
X_2+8\,X_1^3X_2^2 X_1X_3^2+ \right. \nonumber\\
&& \quad
\left. +4\,X_1^2X_2^2X_1^2X_3^2\right]+\nonumber\\ &+&
Z^4\,tr\, \left[8\,X_1^2X_2^2 X_3X_4^2 X_3+ 4\, X_1^2
X_2X_3X_4^2 X_3X_2+2\, X_1^2X_2^2 X_3^2X_4^2 \right]
\nonumber \end{eqnarray}
Random regular graph :
\begin{eqnarray}
\frac{1}{N} \texttt{tr} A^8 &=& Z\,\texttt{tr} \,X_1^8+
Z(Z-1)\,\texttt{tr} \bigg( 8 \,X_1^6 X_2^2+4 \,X_1^4 X_2^4 +2\,X_1^2
X_2^2 X_1^2 X_2^2 \bigg) +\nonumber\\ &&Z(Z-1)(Z-2)\,\texttt{tr} \bigg(8
\, X_1^4 X_2^2 X_3^2+4 \, X_1^2 X_2^2 X_1^2 X_3^2 \bigg) +\nonumber\\
&&Z(Z-1)^2 \,\texttt{tr}\bigg( 8 \,X_1^4 X_2 X_3^2 X_2+8 \, X_1^3
X_2^2 X_1 X_3^2 \bigg)+ \nonumber\\
&& Z(Z-1)^2 (Z-2)\,8\, \texttt{tr}\, X_1^2 X_2^2 X_3 X_4^2
X_3+ Z(Z-1)^3\, 4 \,\texttt{tr} \, X_1^2 X_2 X_3 X_4^2 X_3
X_2\nonumber\\
&& Z(Z-1)(Z-2)(Z-3)\,2\, \texttt{tr} \,X_1^2 X_2^2
X_3^2 X_4^2
\nonumber
\end{eqnarray}
\vskip 1.2cm
Table 2. Limiting moments (that is $N \to \infty $) of the Adjacency
matrix, evaluation of the averages for finite $d$. \vskip 0.8 cm
\begin{eqnarray}
\frac{1}{Nd}< tr\,A^2> &=& Z(d+1) \, \sigma^2 \nonumber\\
\frac{1}{Nd} <tr\,A^4> &=& \Bigg[ Z(2 d^2+5d+5)+Z^2 2(d+1)^2\Bigg]
\sigma^4 \nonumber\\ \frac{1}{Nd} <tr\,A^6> &=& \Bigg[ Z(5d^3+22 d^2+52
d+41)+Z^2 6(d+1)(2d^2+5d+5)+Z^3 5(d+1)^3\Bigg]\sigma^6 \nonumber\\
\frac{1}{Nd} <tr\,A^8> &=& \Bigg[ Z(14 d^4+93d^3+374 d^2+690 d+509)+
Z^2 \left[ 8(d+1)(5d^3+22d^2+52d+41)+\right.\nonumber\\
&+& \left. 4(2d^2+5d+5)^2+
6(d^4+5d^3+13 d^2+18d+11) \right] +\nonumber\\
&+& Z^3 28(d+1)^2(2d^2+5d+5) + Z^4 \,14(d+1)^4 \Bigg]\sigma^8
\nonumber\\
\label{A.2}
\end{eqnarray}
\vskip 1cm
Table 3. Spectral moments $f_n$ in terms of free cumulants $a_n$.
\vskip 0.4 cm \begin{eqnarray} f_1 &=& a_1 \nonumber\\ f_2 &=&
a_2+a_1^2 \nonumber\\ f_3 &=& a_3+3a_1a_2+a_1^3 \nonumber\\ f_4 &=&
a_4+4a_1 a_3+2 a_2^2+6a_1^2 a_2+a_1^4 \nonumber\\ f_5 &=& a_5+5a_1a_4
+5 a_2 a_3+10 a_1^2 a_3 +10 a_1 a_2^2+10 a_1^3 a_2+a_1^5\nonumber\\
f_6 &=& a_6+6 a_1 a_5+6 a_2 a_4+15 a_1^2 a_4+3 a_3^2+30 a_1 a_2 a_3+20
a_1^3 a_3+5 a_2^3+30 a_1^2 a_2^2+15 a_1^4 a_2+a_1^6 \nonumber\\ f_7 &=&
a_7+7 a_1 a_6+7 a_2 a_5+21 a_1^2 a_5+7 a_3 a_4+ 42 a_1 a_2 a_4+35 a_1^3
a_4+21 a_1 a_3^2+21 a_2^2 a_3+\nonumber\\
&&\quad 105 a_1^2 a_2 a_3+35 a_1^4 a_3+35 a_1 a_2^3+70 a_1^3 a_2^2+21
a_1^5 a_2 +a_1^7 \nonumber\\
f_8 &=& a_8+ 8 a_1 a_7+8 a_2 a_6+28 a_1^2 a_6+8 a_3 a_5+56 a_1 a_2 a_5+56
a_1^3 a_5+4 a_4^2+56 a_1 a_3 a_4+28 a_2^2 a_4+\nonumber\\
&& \quad 168 a_1^2 a_2 a_4+70 a_1^4 a_4+28 a_2 a_3^2+84 a_1^2 a_3^2
+168 a_1 a_2^2 a_3+280 a_1^3 a_2 a_3+56 a_1^5 a_3+14 a_2^4+\nonumber\\
&& \quad 140 a_1^2 a_2^3+140 a_1^4 a_2^2+28 a_1^6 a_2+a_1^8
\nonumber \end{eqnarray} \begin{eqnarray} f_n=a_n+\sum_{k=2}^n\frac{1}{k}
\left( \begin{array}{cc} n \\ k-1 \end{array} \right) \sum_{Q_k} a_{q_1}
a_{q_2} ..a_{q_k} \qquad \label{m.1} \end{eqnarray} where $Q_k$ is
the set of integers : $$ Q_k=\Bigg\{ (q_1,..,q_k)\in N^k \, \Bigg| \,
\sum_{i=1}^kq_i=n \Bigg\} $$
Notice that, if all $a_j=1$ then $f_n= C_n=\frac{ 1}{n}\left(
\begin{array}{cc} 2n\\n-1 \end{array}\right)$.\\
The inverse relation, irreducible elements $a_j$ in term of spectral
moments is the following.
\begin{eqnarray} a_1 &=& f_1 \nonumber\\ a_2 &=& f_2-f_1^2 \nonumber\\ a_3
&=& f_3-3 f_1 f_2+2 f_1^3 \nonumber\\ a_4 &=& f_4-4 f_1 f_3+10 f_1^2 f_2-2
f_2^2 -5 f_1^4 \nonumber\\ a_5 &=& f_5-5 f_1 f_4-5 f_2 f_3+15 f_1^2 f_3+15
f_1 f_2^2-35 f_1^3 f_2+14 f_1^5 \nonumber\\ a_6 &=& f_6-6f_1f_5-6f_2
f_4+21 f_1^2 f_4-3 f_3^2+42 f_1f_2 f_3-56 f_1^3 f_3+7 f_2^3-84 f_1^2
f_2^2+\nonumber\\ && \quad 126 f_1^4f_2-42 f_1^6 \nonumber\\ a_7 &=&
f_7-7 f_1f_6-7 f_2 f_5-7 f_3 f_4+28 f_1^2 f_5+56f_1f_2f_4-84 f_1^3
f_4+28 f_1 f_3^2+\nonumber\\ &&\quad 28 f_2^2f_3-252 f_1^2f_2f_3+210
f_1^4f_3-84f_1f_2^3+420 f_1^3f_2^2-462f_1^5f_2+132 f_1^7 \nonumber\\ a_8
&=& f_8- 8 f_1f_7-8 f_2 f_6-8 f_3 f_5+36 f_1^2 f_6+72 f_1f_2f_5 -120
f_1^3 f_5-4 f_4^2+72 f_1f_3 f_4+ \nonumber\\ && \quad 36 f_2^2 f_4-360
f_1^2 f_2 f_4+330 f_1^4f_4+36 f_2 f_3^2-180 f_1^2 f_3^2-360f_1f_2^2 f_3+
\nonumber\\ && \quad 1320 f_1^3 f_2 f_3-792 f_1^5f_3-30 f_2^4+660 f_1^2
f_2^3-1980 f_1^4f_2^2+1716 f_1^6f_2-429 f_1^8 \nonumber \end{eqnarray}
\begin{eqnarray} a_n=f_n+\sum_{k=2}^n \frac{ (-1)^{k-1}}{k} \left(
\begin{array}{cc} n+k-2 \\ k-1 \end{array} \right) \sum_{Q_k} f_{q_1}
f_{q_2}..f_{q_k}\qquad \label{m.2} \end{eqnarray}
Equations (\ref{m.1}), (\ref{m.2}) appear in \cite{motte}. Summations
over the set of non-crossing partitions are in \cite{spei}, \cite{nica}.
\vskip 1cm
Table 4. Limiting Moments of the product of two random matrices
\cite{bai2007}.
\vskip 0.4 cm \begin{eqnarray} \beta_2 &=& 1 \nonumber\\ \beta_4
&=& 2+2\,y \nonumber\\ \beta_{6} &=& 5+12\,y+5\, y^2 \nonumber\\
\beta_{8} &=& 14+56 \,y+ 56\, y^2+14\, y^3 \nonumber\\ \beta_{10} &=&
42 +240\,y+405\,y^2+240\,y^3+42\,y^4 \nonumber\\ \beta_{12} &=& 132 +990
\,y+ 2420 \,y^2+2420\,y^3+ +990\,y^4 +132 \, y^5
\nonumber
\end{eqnarray}
\begin{eqnarray}
\beta_k =\Bigg\{ \begin{array}{ccccccccc} \sum_{j=1}^{k/2} \left(
\begin{array}{ccc} k\\ j-1 \end{array}\right)\left( \begin{array}{ccc}
k\\ k/2-j \end{array}\right)\frac{2}{k} \,y^{j-1} \qquad \texttt{if $k$
is even},\\ 0 \qquad\qquad \texttt{if $k$ is odd}.
\end{array} \qquad \label{A.6} \end{eqnarray}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,741 |
{"url":"https:\/\/www.nature.com\/articles\/s41377-020-00358-9","text":"Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.\n\n# Early detection and classification of live bacteria using time-lapse coherent imaging and deep learning\n\nEarly identification of pathogenic bacteria in food, water, and bodily fluids is very important and yet challenging, owing to sample complexities and large sample volumes that need to be rapidly screened. Existing screening methods based on plate counting or molecular analysis present various tradeoffs with regard to the detection time, accuracy\/sensitivity, cost, and sample preparation complexity. Here, we present a computational live bacteria detection system that periodically captures coherent microscopy images of bacterial growth inside a 60-mm-diameter agar plate and analyses these time-lapsed holograms using deep neural networks for the rapid detection of bacterial growth and the classification of the corresponding species. The performance of our system was demonstrated by the rapid detection of Escherichia coli and total coliform bacteria (i.e., Klebsiella aerogenes and Klebsiella pneumoniae subsp. pneumoniae) in water samples, shortening the detection time by >12\u2009h compared to the Environmental Protection Agency (EPA)-approved methods. Using the preincubation of samples in growth media, our system achieved a limit of detection (LOD) of ~1 colony forming unit (CFU)\/L in \u22649\u2009h of total test time. This platform is highly cost-effective (~$0.6\/test) and has high-throughput with a scanning speed of 24 cm2\/min over the entire plate surface, making it highly suitable for integration with the existing methods currently used for bacteria detection on agar plates. Powered by deep learning, this automated and cost-effective live bacteria detection platform can be transformative for a wide range of applications in microbiology by significantly reducing the detection time and automating the identification of colonies without labelling or the need for an expert. ## Introduction The rapid and accurate identification of live microorganisms is of great importance for a wide range of applications1,2,3,4,5,6,7,8, including drug discovery screening assays1,2,3, clinical diagnoses4, microbiome studies5,6, and food and water safety7,8. Waterborne diseases affect more than 2 billion people worldwide9, causing a substantial economic burden; for example, the treatment of waterborne diseases costs more than$2 billion annually in the United States (US) alone, with 90 million cases recorded per year10.\n\nAmong waterborne pathogen-related problems, one of the most common public health concerns is the presence of total coliform bacteria and Escherichia coli (E. coli) in drinking water, which indicates fecal contamination. Analytical methods used to detect E. coli and total coliforms are based on culturing the obtained samples on solid agar plates (e.g., the US Environmental Protection Agency (EPA) 1103.1 and EPA 1604 methods) or in liquid media (e.g., Colilert test), followed by visual recognition and counting by an expert, as described in the EPA guidelines11,12,13. While the use of liquid growth media for the detection of fecal coliform bacteria provides high sensitivity and specificity, it requires at least 18\u2009h for the final read-out. The use of solid agar plates is a relatively more cost-effective method and provides flexibility for the volume of the sample to be analysed, which can vary from 100\u2009mL to several litres to enhance the sensitivity. However, this traditional culture-based detection method requires the colonies to grow to a certain macroscopic size for visibility, which often takes 24\u201348\u2009h in the case of bacterial samples. Alternatively, molecular detection methods14,15 based on, e.g., the amplification of nucleic acids, can reduce the assay time to a few hours, but they generally lack the sensitivity for detecting bacteria at very low concentrations, e.g., 1 colony forming unit (CFU) per 100\u20131000\u2009mL, and are not capable of differentiating between live and dead microorganisms16. Furthermore, there is no EPA-approved nucleic acid-based analytical method17 for detecting coliforms in water samples.\n\nOverall, there is a strong and urgent need for an automated method that can achieve rapid and high-throughput colony detection with high sensitivity (routinely achieving, e.g., 1 CFU per 100\u20131000\u2009mL in less than 12\u2009h) to provide a powerful alternative to the currently available EPA-approved gold-standard analytical methods that (1) are slow, take ~24\u201348\u2009h and (2) require experts to read and quantify samples. To address this important need, various other approaches18,19,20 have been investigated for the detection of total coliform bacteria and E. coli in water samples, including solid phase cytometry21, droplet-based micro-optical lens array measurements22, fluorimetry23, luminometry24, and fluorescence microscopy25. Despite the fact that these methods provide high sensitivity and some time savings, they cannot handle large sample sizes (e.g., \u2265100\u2009mL) or cannot perform the automated classification of bacterial colonies.\n\nTo provide a highly sensitive and high-throughput system for the early detection and classification of live microorganisms and colony growth, we present a time-lapse coherent imaging platform that uses two different deep neural networks (DNNs) for its operation. The first DNN is used to detect bacterial growth as early as possible, and the second DNN is used to classify the type of growing bacteria based on the spatiotemporal features obtained from the coherent images of an incubated agar plate (see Fig. 1). In this live bacteria detection system, which is integrated with an incubator, lens-free holographic images of the agar plate sample are captured by a monochromatic complementary metal\u2013oxide\u2013semiconductor (CMOS) image sensor that is mounted on a translational stage. The system rapidly scans the entire area of two separate agar plates (~56.52\u2009cm2) every 30\u2009min and utilizes these time-resolved holographic images for the accurate detection, classification, and counting of the growing colonies as early as possible (see Fig. 2a). This unique system enables high-throughput periodic monitoring of an incubated sample by scanning a 60-mm-diameter agar plate in 87\u2009s with an image resolution of <4\u2009\u03bcm; it continuously calculates differential images of the sample of interest for the early and accurate detection of bacterial growth. The spatiotemporal features of each nonstatic object on the plate are continuously analysed using deep learning to yield the count of bacterial growth and to automatically identify the type(s) of bacteria growing on the different parts of the agar plate.\n\nWe demonstrated the efficacy of this platform by performing the early detection and classification of three types of bacteria, i.e., E. coli, Klebsiella aerogenes(K. aerogenes), and Klebsiella pneumoniae(K. pneumoniae), and achieved a limit of detection (LOD) of ~1 CFU\/L in \u22649\u2009h of the total test time. Moreover, we achieved detection time savings of more than 12\u2009h compared to the gold-standard EPA methods26, which usually require at least 24\u2009h to obtain a result. We also quantified the growth statistics of these three different species and provided a detailed growth analysis of each type of bacteria over time. Our detection and classification neural network models were built, trained and validated with ~16,000 individual colonies resulting from 71 independent experiments and were blindly tested with 965 individual colonies collected from 15 independent experiments that were never used in the training phase. In our blind testing, the trained models demonstrated an 80% detection sensitivity within 6\u20139\u2009h, a 90% detection sensitivity within 7\u201310\u2009h, and a >95% detection sensitivity within 12\u2009h, while maintaining ~99.2\u2013100% precision at any time point after 7\u2009h, also achieving correct identification of 80% of all three the species within 7.6\u201312\u2009h. In terms of the species-specific accuracy of our classification network, within 12\u2009h of incubation, we achieved ~97.2%, ~84.0%, and ~98.5% classification accuracy for E. coli, K. aerogenes, and K. pneumoniae, respectively. These results confirm the transformative potential of our platform, which not only enables the highly sensitive, rapid and cost-effective detection of live bacteria (with a cost of \\$0.6 per test, including a culture plate) but also provides a powerful and versatile tool for microbiology research.\n\n## Results\n\nWe demonstrated our system by monitoring bacterial colony growth within 60-mm-diameter agar plates and quantitatively analysed the capabilities of the platform for early detection of the bacterial growth and classification of bacterial species. To demonstrate its proof-of-concept, we aimed to automatically detect, classify, and count E. coli and coliform bacteria in water samples using our deep learning-based platform. Throughout our training and blind testing experiments, we used water suspensions spiked with coliform bacteria, including E. coli, K. aerogenes, and K. pneumoniae, and chlorine-stressed E. coli. A chromogenic agar medium designed for the specific detection and counting of E. coli and other coliform bacteria in food and water samples was used as a culture medium for specificity (see the \u201cMethods\u201d section for details). This chromogenic medium results in a blue colour for E. coli colonies and a mauve colour for the colonies of other coliform bacteria (e.g., K. aerogenes and K. pneumoniae). In addition, the medium inhibits the growth of different bacteria (e.g., Bacillus subtilis) or yields colourless colonies in the presence of other bacteria in the sample27.\n\nFollowing the sample preparation method illustrated in Fig. 2a, the sample is placed inside the lens-free imaging system with the agar surface facing the image sensor. After an initialization step, the platform automatically captures time-lapsed holographic images of two separate Petri dishes (covering a total sample area of 28.26\u2009\u00d7\u20092\u2009=\u200956.52\u2009cm2) every 30\u2009min over a duration of 24\u2009h starting from the incubation time; these individual holograms are digitally stitched together and rapidly reconstructed to reveal the bacterial growth patterns on the agar surface (see the \u201cMethods\u201d section). The reconstructed images of the sample captured at different time points are computationally processed using a differential image analysis method to automatically detect and classify bacterial growth and colonies using two different trained DNNs (see Fig. 3), which will be detailed next.\n\n### Design and training of neural networks for bacterial growth detection and classification\n\nWe designed a two-step framework for bacterial growth detection and classification. The first step selects colony candidates with differential image analysis and refines the results with a detection DNN. We designed a pseudo-3D (P3D) DenseNet28 architecture to process our complex-valued (i.e., phase and amplitude) time-lapse image stacks (see the \u201cMethods\u201d section). In each time-lapse imaging experiment, we used 4 time-consecutive frames (4\u2009\u00d7\u20090.5\u2009=\u20092\u2009h) as a running window for the differential image analysis to extract individual regions of interest (ROIs) containing objects that changed their amplitude and\/or phase signatures as a function of time. These initially detected objects that were extracted by the differential analysis algorithm were either growing colonies or surface impurities, e.g., from spreading the sample on the agar surface, evaporation of air bubbles in the agar plate, or coherent light speckles. We then used a DNN-based detection model to eliminate the nonbacterial objects and only kept the growing colonies (i.e., the true positives), as illustrated in Fig. 2b. We used sensitivity (or true positive rate, TPR) and precision (or positive predictive value, PPV) measurements to quantify our results. Sensitivity is defined as\n\n$${\\mathrm{TPR}} = {\\mathrm{TP}}\/{P}$$\n\nwhere TP refers to the number of true positive predictions from our system, and P refers to the total number of colonies resulting from manual plate counting after 24\u2009h (i.e., the ground truth). Precision is defined as\n\n$${\\mathrm{PPV}} = {\\mathrm{TP}}\/\\left( {{\\mathrm{TP}} + {\\mathrm{FP}}} \\right)$$\n\nwhere FP refers to the number of false positive predictions from our system.\n\nIn total, 13,712 growing colonies (E. coli, K. aerogenes, and K. pneumoniae) and 30,000 non-colony objects captured from 66 separate agar plates were used in the training phase. Another 2597 colonies and 13,078 non-colony objects from 5 independent plates were used as validation dataset to finalize our network models and achieved a TPR of ~95% and a PPV of ~95% once the network converged, which took ~4\u2009h of training time. Examples of the training loss and detection accuracy curves are shown in Supplementary Fig. S1.\n\nThe second step further classifies the species of the detected colonies with a classification DNN model following a similar network architecture. To accommodate the different growth rates of bacterial colonies, we used a longer time window in this classification neural network, containing 8 consecutive frames (8\u2009\u00d7\u20090.5\u2009=\u20094\u2009h) for each sub-ROI. Since the bacterial growth detection network uses a shorter running time window of 2\u2009h, there is a natural 2-h time delay between the successful detection of a growing colony and the classification of its species. The network was trained with 7919 growing colonies, which contained 3362 E. coli, 1880 K. aerogenes, and 2677 K. pneumoniae colonies, and it was validated with 340 E. coli, 205 K. aerogenes, and 988 K. pneumoniae colonies from 6 independent plates and reached a validation classification accuracy of ~89% for E. coli, ~95% for K. aerogenes, and ~98% for K. pneumoniae when the network model converged (Supplementary Fig. S2).\n\nAfter these network models were finalized through the training and validation data, we tested their generalization capabilities with an additional set of experiments that were never seen by the networks before; the results of these blind tests are detailed next.\n\n### Blind testing results for the early detection of bacterial growth\n\nFirst, we blindly tested the performance of our system in the early detection of bacterial colonies with 965 colonies from 15 plates that were not presented during the network training or validation stages. We compared the predicted number of growing colonies on the sample within the first 14\u2009h of incubation against a ground truth colony count obtained from plate counting after 24\u2009h of incubation time. Each of the 3 sensitivity curves (Fig. 4a\u2013c) were averaged across repeated experiments for the same species, e.g., 4 experiments for K. pneumoniae, 7 experiments for E. coli, and 4 experiments for K. aerogenes, so that each data point was calculated from ~300 colonies. The results demonstrated that our system was able to detect 80% of the true positive colonies within ~6.0\u2009h of incubation for K. pneumoniae, ~6.8\u2009h of incubation for E. coli, and ~8.8\u2009h of incubation for K. aerogenes. In addition, our platform further detected 90% of the true positives after ~1 additional hour of incubation and >95% of the true positive colonies of all 3 species within 12\u2009h. The results also reveal that the early detection sensitivities in Fig. 4a\u2013c are dependent on the length of the lag phase of each tested bacteria species, which demonstrates interspecies variations. For example, K. pneumoniae started to grow earlier and faster than E. coli and K. aerogenes, whereas K. aerogenes did not reach a detectable growth size until 5\u2009h of incubation. Furthermore, when the tails of the sensitivity curves were examined, some of the E. coli colonies showed late \u201cwake-up\u201d behaviour, as highlighted by the purple arrow in Fig. 4b. Although most of the E. coli colonies were detected within ~10\u2009h of incubation time, some of them did not emerge until ~11\u2009h after the start of the incubation phase.\n\nWe also quantified the false positive rate of our platform with the PPV curve shown in Fig. 4d, which was averaged across all the experiments covering all the species, i.e., 965 colonies from 15 agar plates. The precision can be low at the beginning of the experiments (the first 4\u2009h of incubation) because the number of detected true positive colonies is very small, especially for K. aerogenes. This result means that even a single false positive-detected colony can dramatically affect the precision calculation. Nevertheless, the precision quickly rises up to ~100% within 6\u2009h of incubation and is maintained at 99.2\u2013100% for all the tested species after 7\u2009h of incubation.\n\nWe should emphasize here that the results presented in Fig. 4 represent the lower limits of the detection capabilities of our system since we calculated these sensitivities with regard to the number of true positive colonies after 24\u2009h of incubation, whereas some of these colonies actually did not exist at the early stages due to delayed growth; stated differently in some cases, there were no colonies present at the early stages of the incubation period. We also note that the rising sensitivity curves in our results stand for the emergence of new bacterial colonies, in addition to the growth of colonies. Even though the sensitivity curves converge to flat lines after 12\u2009h, the colonies continue to grow exponentially until much later. Therefore, our system detects emerging colonies at an early stage, when they first appear, forming microscale features invisible to the naked eye.\n\nThese observations also indicate that our system can be very effective and used for high-throughput quantitative studies to better understand microorganism behaviour under different conditions, such as the evaluation of the differences in growth rates between stressed bacteria (e.g., under nutrient deprivation or chlorine treatment) and normal bacteria29,30,31,32,33. There are several reasons to detect and enumerate chlorine-stressed or injured coliform bacteria. First, the detection of injured E. coli or total coliform bacteria is directly related to the sensitivity of the detection platform33. For an effective and sensitive detection platform, false-negative results should be avoided for public health safety. Another important reason is that the detection of injured E. coli or low numbers of E. coli in water samples is correlated with Salmonella outbreaks, a foodborne pathogen causing 1.2 million illnesses and ~500 deaths per year in the United States34, which forms an indirect indicator of contamination in irrigation water35. To evaluate the capabilities of our system to detect injured bacteria, we prepared and imaged 3 agar plates containing chlorine-stressed E. coli (see the \u201cMethods\u201d section) and characterized their growth using our detection workflow, as summarized in Fig. 4e. Our results indicate that we can detect colony formation for chlorine-stressed E. coli on average with an ~2\u2009h delay compared to the regular E. coli strain.\n\n### Blind testing results on the classification of growing bacteria\n\nIn addition to providing significant detection time savings while also achieving very good sensitivity and precision for the early detection of bacterial growth, our method also provides the automated classification of the corresponding species of the detected bacteria using a trained neural network. Therefore, an additional advantage of our system is its capability to further classify the total coliform subspecies, which is not possible with traditional agar plate counting methods. For example, both K. pneumoniae and K. aerogenes colonies appear mauve in our agar plates. However, since our classification neural network not only relies on the byproducts of colorimetric reactions, it can successfully distinguish between different species based on their unique spatiotemporal growth signatures acquired by our platform at the microscale.\n\nFigure 5 shows our blind testing results on species classification using the same experiments reported in the blinded early detection tests, containing 965 colonies of 3 different species from 15 agar plates. In these results, if a colony was not detected in the previous step (i.e., a false negative event compared to the 24\u2009h reading), then it was naturally not sent to the classification neural network. We defined the recovery rate as the number of colonies correctly classified into their corresponding species using our system divided by the total number of colonies counted after 24\u2009h. As the classification of each individual colony is an independent event, we calculated the recovery rate for each bacterial species (reported in Fig. 5a\u2013c) using all of the colonies detected in the previous step, i.e., 336, 280, and 339 colonies of E. coli, K. aerogenes, and K. pneumoniae, respectively. The shaded area in each curve represents the highest and lowest recovery rates found in all the corresponding experiments at each time point. The classification neural network correctly classified ~80% of all of the colonies within ~7.6, ~8, and ~12\u2009h for K. pneumoniae, E. coli, and K. aerogenes, respectively. We once again emphasize that the results presented in Fig. 5a\u2013c represent the lower limits of the classification capabilities of our system since ground truth is acquired after 24\u2009h of incubation. In reality, at various earlier time points within the incubation period, there was no growth for certain regions of the plates, which exhibited significantly delayed growth. To further demonstrate the classification performance of our trained neural network in a manner that is decoupled from the sensitivity of the previous detection network, we report the classification confusion matrix in Fig. 5d for all the colonies that were sent to the classification network for blind testing at 12\u2009h after the start of the incubation. The trained network achieved classification accuracies of ~97.2%, ~84.0%, and ~98.5% for E. coli, K. aerogenes, and K. pneumoniae, respectively.\n\n### Limit of detection as a function of the total test time\n\nWe further quantified the detection limit of our system and compared its performance against both Colilert\u00ae-18, which is an EPA-approved method, and traditional plate counting (Supplementary Table S1, Supplementary Fig. S3). To compensate for the CFU loss during the sample transfer from the water suspension to the filter membrane, we introduced a signal amplification step by preincubating the water sample under test, mixing it with a growth medium for 5\u2009h at 35\u2009\u00b0C before the filtration step (see the \u201cMethods\u201d section for details). For each measurement, two agar plates were prepared and monitored at the same time for comparison, one of which was for the sample amplified with a 5-h preincubation step before filtering, while the other was for the sample directly filtered and transferred to the agar plate (see Supplementary Fig. S3). Both plates were incubated for the same amount of time at each imaging time point to provide a fair comparison between the two. The measurements were repeated using different concentrations of E. coli suspensions; these concentrations were compared to the average of three replicates of the same samples prepared using the Colilert\u00ae-18 method (Supplementary Fig. S3). As shown in Fig. 6a, our system is able to surpass the sensitivity of Colilert\u00ae-18 within ~8\u2009h in total (including the time for signal amplification, sample concentration, and time-lapse imaging, altogether) and reach >2 times the sensitivity of Colilert\u00ae-18 in ~9\u2009h. We also quantified the LOD of our system by preparing and imaging 3 agar plates without bacteria, which show on average <1 CFU count from our setup throughout the test period from 5 to 14.5\u2009h (Fig. 6c), revealing a detection limit of \u00b5\u2009+\u20093\u03c3\u2009=\u2009~2 CFU per test, where \u00b5 and \u03c3 refer to the mean and standard deviation of the detected CFU count, respectively. Due to the effective signal amplification enabled by the preincubation step, even with the lowest bacterial concentration of ~1\u2009CFU\/L, our system was able to detect 2 CFU at 8.5\u2009h and 12 CFU at 9\u2009h; in comparison, for the same contaminated water sample, Colilert\u00ae -18 achieved 1.4\u2009\u00b1\u20091.6\u2009CFU\/L after 18\u2009h of incubation. Furthermore, for all the concentrations we have experimented with (~1\u2013160\u2009CFU\/L), our system successfully detected more than 2 CFU per test in \u22649\u2009h of test time, including all the necessary steps, i.e., the time for signal amplification, sample concentration, and time-lapse imaging; these results reveal that our system with a preincubation step achieves a detection limit of ~1\u2009CFU\/L within \u22649\u2009h of total test time.\n\nWe also observe in Fig. 6b that without the signal amplification enabled by preincubation, the detection performance is negatively affected due to the low transfer rate of bacteria from the container to the agar plate (also see Supplementary Fig. S4). In general, the sensitivity and LOD of our method might be further improved by increasing the preincubation time of the water-broth mixture at the cost of an increase in the total time to achieve automated detection and classification.\n\n## Discussion\n\nWe demonstrated a new platform for the early detection and classification of bacterial colonies, which is fully compatible with the existing EPA-approved methods and can be integrated with them to considerably improve the analysis of agar plates36. The presented approach can automatically detect bacterial growth as early as 3\u2009h and can detect 90% of bacterial colonies within 7\u201310\u2009h (and >95% within 12\u2009h), with a precision of 99.2\u2013100%. The system also correctly classifies ~80% of all of the tested bacterial colonies within 7.6, 8.8, and 12\u2009h for K. pneumoniae, E. coli, and K. aerogenes, respectively. These results present a total time savings of more than 12\u2009h compared to the gold-standard methods (e.g., Colilert test and Standard Method 9222B), which require 18\u201324\u2009h. The presented learning-based bacteria detection and classification framework can potentially be further advanced by training it with a larger number of sample types20, and it can also be applied to other bacteria sensing applications beyond water quality monitoring. In addition to the automated detection of live bacteria and species classification, the rich spatiotemporal information embedded in the holographic images can be used for more advanced analysis of water samples and microbiology research in general.\n\nAnother advantage of this system is its high-throughput imaging capability of agar plates. Our prototype performs a 242-tile scan within 87\u2009s per agar plate, corresponding to a raw image scanning throughput of ~49\u2009cm2\/min. To leave sufficient data redundancy for image postprocessing, we set a relatively large overlap of 30% on each side of the acquired holographic image, which reduces the effective imaging throughput of our platform to ~24\u2009cm2\/min. As our system is based on lens-free holographic microscopy, it does not require mechanical axial focusing at each position and instead autofocuses onto the object plane computationally. We characterized the spatial resolution of our system by imaging a resolution test target, as shown in Supplementary Fig. S5, achieving a linewidth resolution of ~3.5\u2009\u00b5m, roughly equivalent to the performance of a 4\u00d7 objective lens with a numerical aperture (NA) of ~0.1. Compared to our system, which takes 87\u2009s to scan an agar plate, a traditional lens-based bright-field microscope using a 4\u00d7 objective lens would take approximately 128\u2009min to scan a plate with the same diameter (60\u2009mm), owing to the requirement for mechanical axial focusing (see Supplementary Table S2). In addition, the holographic imaging that is at the heart of this system provides better performance for early colony detection over bright-field imaging. Since bacteria can be considered phase objects, growth-related changes in a holographic image are enhanced compared to the bright-field images, enabling the earlier detection of bacterial growth and more sensitive measurements (see Fig. 3b).\n\nAnother important advantage of our system is the minimum requirement for optical alignment; the presented platform is tolerant towards structural changes, such as variations in the sample-to-sensor distance or the illumination angle. Our computational refocusing capability also enables the screening of thick samples, e.g., melted agar plates37. An example of a 3D sample is illustrated in Supplementary Fig. S6, where E. coli colonies are formed at different depths inside the solid culture medium with a thickness of ~5\u2009mm. For example, the colony marked with \u201cA\u201d grew at ~2170\u2009\u00b5m measured from the surface of the agar, whereas the colony marked with \u201cB\u201d was on the agar surface. Our system localizes colonies growing at different depths within a 3D culture medium using a single hologram measurement at each scanning position. However, it is a nontrivial task to image a 3D sample using a conventional lens-based microscope because of the time required for mechanical focusing and the refractive index mismatch between the culture medium and the air, which degrades the image resolution as a result of aberrations. Therefore, the corresponding bright-field microscopy images of the whole plates could only be acquired after 24\u2009h of incubation.\n\nOur platform also employs a modular design that is scalable to a larger sample size and a smaller tile-scan time interval. The monitoring field of view (FOV) of this platform is fundamentally limited by the image acquisition time and the stage moving speed. With further optimization of the hardware and control algorithms, an imaging throughput of >50\u2009cm2\/min can be reached. Alternatively, several image sensors can be installed and connected to a single computer for high-throughput parallel imaging38. In our proof-of-concept implementation, our image processing for each time interval takes ~20\u2009min and fits well into our 30\u2009min measurement period between each scan. In case a shorter time interval is desired, an image processing procedure implemented using MATLAB and Python\/PyTorch programming environments can be further accelerated by programming in C\/C++. With the help of graphic processing units (GPUs), one can expect >10-fold time savings in computation39.\n\nThis unique platform is integrated with an incubator to keep the agar plates at a desired temperature. The incubator is a thermal glass plate that contains uniform lines of optically clear indium tin oxide electrode for heating the sample placed on top. This system is controlled with a controller, which is lightweight. Throughout the experiments, we set the temperature at the agar surface where bacteria grew at ~38\u2009\u00b0C so that all of the tested bacterial species could grow and develop colonies. This temperature was not optimized to promote the growth of a specific species. Therefore, the adjustment of the incubation environment, temperature and humidity can potentially be used to further accelerate colony growth and help us achieve earlier detection and identification of specific bacterial colonies. Another important parameter for the growth of microorganisms is the humidity. Our system can also be integrated with a controlled humidity chamber for better control and analysis of the growth dynamics of various microorganisms40.\n\nIn summary, we presented a deep learning-based live bacteria monitoring system for the early detection of growing colonies and the classification of colony species using deep learning. We demonstrated a proof-of-concept device using 3 types of bacteria, i.e., E. coli, K. aerogenes, and K. pneumoniae, and achieved >12\u2009h time savings for both the early detection and the classification of growing species compared to the gold-standard EPA-approved methods. Achieving an LOD of ~1\u2009CFU\/L in \u2264 9\u2009h, we believe that this versatile system will not only benefit water and food quality monitoring but also provide a powerful tool for microbiology research.\n\n## Materials and methods\n\n### Sample preparation\n\n#### Safety practices\n\nWe handled all the bacterial cultures and performed all the experiments at our Biosafety Level 2 laboratory in accordance with the environmental, health, and safety rules of the University of California, Los Angeles.\n\n#### Studied organisms\n\nWe used E. coli (Migula) Castellani and Chalmers (ATCC\u00ae 25922\u2122) (risk level 1), K. aerogenes Tindall et al. (ATCC\u00ae 49701\u2122) (risk level 1), and K. pneumoniae subsp. pneumoniae (Schroeter) Trevisan (ATCC\u00ae13883\u2122) (risk level 2) as our culture organisms.\n\n#### Preparation of the poured agar plates\n\nWe used CHROMagar\u2122 ECC (product no. EF322, DRG International, Inc., Springfield, NJ, USA) chromogenic substrate mixture as the solid growth medium for the detection of E. coli and total coliform colonies. CHROMagar\u2122 ECC (8.2\u2009g) was mixed with 250\u2009mL of reagent grade water (product no. 23-249-581, Fisher Scientific, Hampton, NH, USA) using a magnetic stirrer bar. The mixture was then heated to 100\u2009\u00b0C on a hot plate while being stirred regularly. After cooling the mixture to ~50\u2009\u00b0C, 10\u2009mL of the mixture was dispensed into Petri dishes (60\u2009mm\u2009\u00d7\u200915\u2009mm) (product no. FB0875713A, Fisher Scientific, Hampton, NH, USA). The agar plates were allowed to solidify, were sealed using parafilm (product no. 13-374-16, Fisher Scientific, Hampton, NH, USA), and were covered with aluminium foil to keep them in the dark before use. The plates were stored at 4\u2009\u00b0C and were used within two weeks of preparation.\n\n#### Preparation of the melted agar plates\n\nCHROMagar\u2122 ECC (3.28\u2009g) was mixed with 100\u2009mL of reagent grade water using a magnetic stirrer bar, and the mixture was heated to 100\u2009\u00b0C. After the mixture cooled to ~40\u2009\u00b0C, 1\u2009mL of the bacterial suspension was mixed with the agar and dispensed into Petri dishes. The plates were either incubated in a benchtop incubator (product no. 51030400, ThermoFisher Scientific, Waltham, MA, USA) or in our imaging platform (for monitoring the bacterial growth digitally).\n\nWe used tryptic soy agar to culture E. coli at 37\u2009\u00b0C and K. aerogenes at 35\u2009\u00b0C and nutrient agar to culture K. pneumoniae at 37\u2009\u00b0C. Twenty grams of tryptic soy agar (product no. DF0369-17-6, Fisher Scientific, Hampton, NH, USA) or 11.5\u2009g of nutrient agar (product no. DF0001-17-0, Fisher Scientific, Hampton, NH, USA) were suspended in 500\u2009mL of reagent grade water using a magnetic stirrer bar. The mixture was boiled on a hot plate and then autoclaved at 121\u2009\u00b0C for 15\u2009min. After the mixture cooled to ~50\u2009\u00b0C, 15\u2009mL of the mixture was dispensed into Petri dishes (100\u2009mm\u2009\u00d7\u200915\u2009mm) (product no. FB0875713, Fisher Scientific, Hampton, NH, USA), which were then sealed with parafilm and covered with aluminium foil to keep them in the dark before use. The Petri dishes were stored at 4\u2009\u00b0C until use.\n\n#### Preparation of the chlorine-stressed E. coli samples\n\nWe used E. coli grown on tryptic soy agar plates and incubated for 48\u2009h at 37\u2009\u00b0C. Disposable centrifuge tubes (50\u2009mL) were used as a sample container, and the sample size was 50\u2009mL. Five hundred millilitres of reagent grade water was filtered for sterilization using a disposable vacuum filtration unit (product no. FB12566504, Fisher Scientific, Hampton, NH, USA). A fresh chlorine suspension was prepared in a 50\u2009mL disposable centrifuge tube to a final concentration of 0.2\u2009mg\/mL using sodium hypochlorite (product no. 425044, Sigma Aldrich, St. Louis, MO, USA), mixed vigorously, and covered with aluminium foil41. Sodium thiosulfate (10% [w\/v]) (product no. 217263, Sigma Aldrich, St. Louis, MO, USA) in reagent grade water was prepared, and 1\u2009mL of the solution was filtered using a sterile disposable syringe and a syringe filter membrane (product no. SLGV004SL, Fisher Scientific, Hampton, NH, USA) for sterilization. Water suspensions were prepared by spiking E. coli into filtered water samples. Fifty microlitres of the chlorine suspension (i.e., 0.2\u2009ppm) was added to the test water sample, and a timer counted the chlorine exposure time. The reaction was stopped at 10\u2009min of chlorine exposure by adding 50\u2009\u00b5L sodium thiosulfate into the test water sample and vigorously mixing the solution to immediately stop the chlorination reaction. CHROMagar\u2122 ECC plates were inoculated with 200\u2009\u00b5L of the chlorine-stressed suspension, were dried in the biosafety cabinet for at most 30\u2009min and then were placed on the setup for lens-free imaging. In addition, three TSA plates and one ECC ChromoSelect Selective Agar plate (product no. 85927, Sigma Aldrich, St. Louis, MO, USA) were inoculated with 1\u2009mL of the control sample (not exposed to chlorine) and 0.2 ppm of the chlorine-stressed E. coli water sample and dried under a biosafety cabinet for approximately 1\u20132\u2009h with the gentle mixing of Petri dishes at some time intervals. After drying, the plates were sealed with parafilm and incubated at 37\u2009\u00b0C for 24\u2009h. After incubation, the bacterial colonies grown on the agar plates were counted, and the E. coli concentrations of the control samples and chlorine-stressed E. coli samples were compared. If the achieved reduction in colony count was between 2.0 and 4.0\u2009log, then the images of CHROMagar\u2122 ECC plates captured using the lens-free imaging platform were used for further analysis.\n\n#### Preparation of the culture plates for lens-free imaging\n\nA bacterial suspension in a phosphate-buffered solution (PBS) (product no. 20-012-027, Fisher Scientific, Hampton, NH, USA) was prepared every day from a solid agar plate incubated for 24\u2009h. The concentration of the suspension was measured using a spectrophotometer (model no. ND-ONE-W, Thermo Fisher), and the suspension was then diluted in PBS to a final concentration of 1\u2013200 CFU per 0.1\u2009mL. One hundred microlitres of the diluted suspension was spread on a CHROMagar\u2122 ECC plate using an L-shaped spreader (product no. 14-665-230, Fisher Scientific, Hampton, NH, USA). The plate was covered with its lid, inverted, and incubated at 37\u2009\u00b0C in our optical platform (Fig. 2).\n\n#### Preparation of a concentrated broth\n\nA total of 180\u2009g of tryptic soy broth (product no. R455054, Fisher Scientific, Hampton, NH, USA) was added to 1\u2009L reagent grade water and heated to 100\u2009\u00b0C by continuously mixing using a stirrer bar. The suspension was then cooled to 50\u2009\u00b0C and filter sterilized using a disposable filtration unit. The broth concentrate was stored at 4\u2009\u00b0C and used within 1 week after preparation.\n\n#### Preparation of samples for comparison measurements\n\nWe evaluated the performance of our method in comparison to Colilert\u00ae-18, which is an EPA-approved enzyme-based analytical method for several types of regulated water samples (e.g., drinking water, surface water, and ground water) to detect E. coli42 and for plate counting using TSA plates and ECC ChromoSelect Selective Agar plates (Supplementary Fig. S3). Two bottles of 1\u2009L reagent grade water were filtered using disposable vacuum filtration units and 0.2\u2009L of the concentrated broth was added into one of the 1\u2009L sample bottles. The bottles were covered with aluminium foil and stored in a biosafety cabinet overnight. A glass vacuum filtration unit was used for the filtration of the 1\u2009L water samples. The components of the unit were covered with aluminium foil and sterilized using an autoclave. The disposable nitrocellulose filter membranes (product no. HAWG04705, EMD Millipore, Danvers, MA, USA) used in the glass filtration unit were also sterilized using the autoclave. A bacterial suspension was prepared by spiking bacteria into 50\u2009mL reagent grade water using a disposable inoculation loop from a TSA plate containing E. coli colonies. The suspension was mixed gently to obtain a uniform distribution of bacteria. Three TSA plates, 3 ECC ChromoSelect Selective Agar plates, and 4 CHROMagar\u2122 ECC plates were removed from the refrigerator and were kept at room temperature for 30\u2009min.\n\nThree bottles of 120\u2009mL disposable vessels with sodium thiosulfate (product no. WV120SBST-200, IDEXX Laboratories Inc., Westbrook, ME, USA) were filled with 100\u2009mL filter sterilized reagent grade water. First, 0.1\u2009mL of bacterial suspension was spiked into a 1\u2009L water sample, a 1.2\u2009L water sample (1\u2009L water\u2009+\u20090.2\u2009L concentrated broth), 3 bottles of 100\u2009mL water samples, 3 TSA plates and 3 ECC ChromoSelect Selective Agar plates, sequentially. The timer was started immediately after adding the spike into the suspensions.\n\nFirst, the suspensions on TSA plates and ECC ChromoSelect Selective Agar were spread using L-shaped disposable spreaders. Then, the water sample with broth was mixed for approximately one minute and then stored at 35\u2009\u00b0C for 5\u2009h. One Colilert\u00ae-18 reagent (product no. 98-27164-00, IDEXX Laboratories Inc., Westbrook, ME, USA) was added into each 100\u2009mL bacterial suspension, and the mixture was shaken. The content of the bottle was poured into a Quanti-Tray 2000 bag (product no. 98-21675-00, IDEXX Laboratories Inc., Westbrook, ME, USA), and after removing bubbles in each well, the bag was sealed using Quanti-Tray Sealer (product no. 98-09462-01, IDEXX Laboratories Inc., Westbrook, ME, USA). Three bags sealed and labelled with the experimental details were incubated at 35\u2009\u00b0C for 18\u2009h. Next, 30\u2009mL filtered reagent grade water was used to moisturize the membrane in the glass filtration unit, and then an E. coli-contaminated 1\u2009L water sample was filtered at a pressure of 50\u2009kPa. The bottle was rinsed using 150\u2009mL of sterilized reagent grade water, and the solution was filtered on the unit (Supplementary Fig. S7). The funnel was rinsed twice using 50\u2009mL of sterilized reagent grade water. After the filtration was complete, the membrane was removed and placed onto a CHROMagar\u2122 ECC plate face down. Gentle pressure was applied on the membrane using a tweezer to remove any air bubbles between the agar and the membrane. Then, 30\u2009g of weight was placed on the membrane to provide continuous pressure during the transfer of bacteria from the membrane to the agar plate (Supplementary Fig. S8). After 5\u2009min of incubation, the membrane was gently peeled off from the agar surface and placed into another agar facing up. The agar containing the membrane was incubated at the benchtop incubator at 35\u2009\u00b0C, and the agar containing the transferred bacteria was incubated at the lens-free imaging platform for time-lapse imaging. After 5\u2009h of incubation, the bottle containing 1.2\u2009L suspension was filtered using the same procedure as described before for filtration of a 1\u2009L sample. The agar plate containing the transferred bacteria was incubated at the second sample tray of the lens-free imaging setup for time-lapse imaging, while the agar containing the membrane was incubated at the benchtop incubator.\n\n### Design of the high-throughput time-resolved microorganism monitoring platform\n\nOur platform consists of five modules: (1) a holographic imaging system, (2) a mechanical translational system, (3) an incubation unit, (4) a control circuit, and (5) a controlling program. Each module is explained in detail below.\n\n1. i.\n\nWe used fibre-coupled partially coherent laser illumination (SC400-4, Fianium Ltd., Southampton, UK), with the wavelength and intensity controlled through an acousto-optic tunable filter (AOTF) device (Fianium Ltd., Southampton, UK). The device was remotely controlled with a customized program written in the C++ programming language and ran on a controlling laptop computer (product no. EON17-SLX, Origin PC). The laser light was transmitted through the sample, i.e., the agar plate that contains the bacterial colonies, and forms an inline hologram on a CMOS image sensor (product no. acA3800-14\u2009\u00b5m, Basler AG, Ahrensburg, Germany) with a pixel size of 1.67\u2009\u03bcm and an active area of 6.4\u2009mm\u2009\u00d7\u20094.6\u2009mm. The CMOS image sensor was connected to the same controlling laptop computer through a universal serial bus (USB) 3.0 interface and was software-triggered within the same C++ program. The exposure time at each scanning position was precalibrated according to the intensity distribution of the illumination light and ranged from 4 to 167\u2009ms. The images were saved as 8-bit bitmap files for further processing.\n\n2. ii.\n\nThe mechanical stage was customized with a pair of linear translation rails (Accumini 2AD10AAAHL, Thomson, Radford, VA, USA), a pair of linear bearing rods (8\u2009mm-diameter, generic), and linear bearings (LM8UU, generic), and it was aided by parts printed by a 3D printer for the joints and housing (Objet30 Pro, Stratasys, Minnesota, USA). The 2D horizontal movement was powered by two stepper motors (product no. 1124090, Kysan Electronics, San Jose, CA, USA)\u2014one for each direction, and these motors were individually controlled using stepper motor controller chips (DRV8834, Pololu Las Vegas, NV, US). To minimize the backslash effect, the whole Petri dish was scanned following a raster scan pattern.\n\n3. iii.\n\nThe incubation unit was built with the top heating plate of a microscope incubator (INUBTFP-WSKM-F1, Tokai Hit, Shizuoka, Japan), and it was housed by a 3D frame printed by a 3D printer. The Petri dish containing the sample was placed on the heating plate with the surface having bacteria facing downwards. The temperature was controlled by a paired controller that maintained a temperature of 47\u2009\u00b0C on the heating plate, resulting in a temperature of 38\u2009\u00b0C inside the Petri dish.\n\n4. iv.\n\nThe control circuit consisted of three components: a microcontroller (Arduino Micro, Arduino LLC) communicating with the computer through a USB 2.0 interface, two stepper motor driver chips (DRV8834, Pololu Las Vegas, NV, US) externally powered by a 4.2\u2009V constant voltage power supply (GPS-3303, GW Instek, Montclair, CA, US), and a metal\u2013oxide\u2013semiconductor field-effect transistor-based digital switch (SUP75P03-07, Vishay Siliconix, Shelton, CT, United States) for controlling the CMOS sensor connection.\n\n5. v.\n\nThe controlling program included a graphical user interface and was developed using the C++ programming language. External libraries including Qt (v5.9.3), AOTF (Gooch & Housego), and Pylon (v5.0.11) were integrated.\n\n### Data acquisition\n\nWe prepared inoculated agar plates of pure bacterial colonies (see the Sample Preparation subsection under the \u201cMethods\u201d for details) and captured images of an entire agar plate at 30-min intervals. The illumination light was set to a wavelength of 532\u2009nm and an intensity of ~400\u2009\u03bcW. To maximize the image acquisition speed, the captured images were first saved into a computer memory buffer and then were written to a hard disk by another independent thread. At the end of each experiment (i.e., after 24\u2009h of incubation), the sample plate was imaged using a benchtop scanning microscope (Olympus IX83) in reflection mode, and the resulting images were automatically stitched to a full-FOV image, used for comparison. Subsequently, the plate was disposed of as solid biohazardous waste. We populated the data (i.e., time-lapse lens-free images) corresponding to ~6969 E. coli, ~2613 K. aerogenes, and ~6727 K. pneumoniae individual bacterial colonies to train and validate our models. Another 965 colonies of 3 different species from 15 independent agar plates were used to blindly test our machine learning models.\n\n### Image processing and analysis\n\nThe acquired lens-free images were processed using custom-developed image processing and deep learning algorithms. Five major image processing steps were used for the early detection and automated classification and counting of colonies. These steps are described in detail below.\n\n#### Image stitching to obtain the image of the entire plate area\n\nFollowing the acquisition of holographic images using the multi-threading approach, all the images within a tile-scan of the whole Petri dish per wavelength were merged into a single full-FOV image. During a tile scan, the images were acquired with ~30% overlap on each side of the image to calculate the relative image shifts against each other. For each image, the relative shifts against all four of the neighbouring images were calculated using a phase correlation43 method, followed by an optimization step that minimized an object function, as defined by\n\n$$\\arg \\mathop {{\\min }}\\limits_{T_{VF}} {\\sum\\limits_{A \\in V\\backslash \\{ F\\}}}\\left( {{\\sum\\limits_{B \\in V\\backslash \\{ F\\}}} {\\left\\| {\\vec t_{AF} - \\vec t_{BF} - \\vec p_{AB}} \\right\\|^2}}\\right)$$\n(1)\n\nwhere V is the set of all tile images, $$F \\in V$$ is a fixed image, e.g., the image captured at the centre of the sample Petri dish, $$\\vec t_{AB}$$ stands for the relative position of image A with respect to image B, and $$\\vec p_{AB}$$ is the local shift between images A and B, calculated by the phase correlation method using the overlapping regions of the two neighbouring images, which can be formulated as\n\n$$\\vec p_{AB} = \\left( {\\Delta x,\\Delta y} \\right) = \\arg \\mathop {{\\max }}\\limits_{(x,y)} {\\cal{F}}^{ - 1}\\left\\{ {\\frac{{{\\cal{F}}\\{ A\\} \\cdot {\\cal{F}}\\{ B\\} ^ \\ast }}{{\\left| {{\\cal{F}}\\{ A\\} \\cdot {\\cal{F}}\\{ B\\} ^ \\ast } \\right|}}} \\right\\}$$\n(2)\n\nwhere $${\\cal{F}}$$is the Fourier transform operator and $${\\cal{F}}^{ - 1}$$ is the inverse Fourier transform operator. The optimal configuration $$T_{VF} = \\left\\{ {\\vec t_{AF}:A,F \\in V} \\right\\}$$ represents the relative positions of all the images with respect to the fixed image F, and it was used as the global position of each tile image for full-FOV image stitching. To eliminate tiles with a low signal-to-noise ratio that lead to incorrect local shift estimation values, a correlation threshold of 0.3 was applied during the optimization, meaning that if the cross-correlation coefficient of the overlapped parts of two images was below 0.3, the shift calculation was discarded. Once the positions of all of the tiles were obtained, they were merged into a full-FOV image of the whole Petri dish using linear blending. We defined a full-FOV image of the whole Petri dish as a \u201cframe\u201d. All the frames were normalized so that the mean value was 50, and they were saved as unsigned 8-bit integer (0\u2013255) arrays.\n\n#### Colony candidate selection by differential analysis\n\nWhen a new frame was acquired at time t, it was cross-registered to the previous frame at time t\u2009\u2212\u20091 and then digitally back-propagated to the sample plane44,45 to obtain the complex light field\n\n$$\\widetilde B_t = {\\mathrm{P}}(F_t,{\\mathbf{z}})$$\n(3)\n\nwhere Ft is the frame at time t, z is a surface normal vector of the sample plane obtained by digital autofocusing46 at 50 randomly spaced positions, and P denotes the angular spectrum-based back-propagation operation44,45, which can be calculated by multiplying the spatial Fourier transform of the input signal and the following transfer function\n\n$$H_k(\\nu _x,\\nu _y) = \\left\\{{\\begin{array}{*{20}{c}} {\\exp \\left[ { - j \\cdot 2\\pi \\frac{{n \\cdot z}}{\\lambda }\\sqrt {1 - \\left( {\\frac{\\lambda }{n}\\nu _x} \\right)^2 - \\left( {\\frac{\\lambda }{n}\\nu _y} \\right)^2} } \\right]} & {\\left( {\\nu _x^2 + \\nu _y^2 \\le \\left( {\\frac{n}{\\lambda }} \\right)^2} \\right)} \\\\ 0 & {{\\mathrm{otherwise}}} \\end{array}}\\right.$$\n\nwhere n is the refractive index of the medium, \u03bb is the illumination wavelength, and vx and vy are the spatial frequencies. This operation was followed by an inverse 2D Fourier transform. The resulting complex-valued reconstruction provides both the amplitude and phase images of the illuminated objects. To accommodate the large FOV of a stitched frame (36,000\u2009\u00d7\u200936,000 pixels), digital back-propagation was performed with 2048\u2009\u00d7\u20092048-pixel blocks, which were then merged together.\n\nFour consecutive frames were taken, i.e., from t\u2009\u2212\u20093 to t, and a differential image was calculated defined by\n\n$$D_t = {\\mathrm{HP}}\\left[ {{\\mathrm{LP}}\\left( {\\frac{1}{3}\\mathop {\\sum}\\limits_{\\tau = t - 2}^t {\\left| {\\tilde B_\\tau - \\tilde B_{\\tau - 1}} \\right|} } \\right)} \\right]$$\n(4)\n\nwhere Dt is the differential image at time t, $$\\tilde B_t$$ represents the complex light field obtained by back-propagating frame t, and LP and HP represent low-pass and high-pass image filtering, respectively. The HP filter removes the differential signal from a slowly varying background (unwanted term), and the LP filter removes the high-frequency noise-introduced spatial patterns. The LP and HP filter kernels were empirically set to 5 and 100, respectively.\n\nFollowing the differential image calculation, we selected regions in the differential image with >50 connective pixels that are above an intensity threshold, which was empirically set to 12. These regions are marked as colony candidates, as they give a differential signal over a period of time (covering four consecutive frames). However, some of the differential signals come from nonbacterial objects, such as a water bubble or surface movement of the agar itself. Therefore, we also used two DNNs to select the true candidates and classify their species.\n\n#### DNN-enabled detection of growing bacterial colonies\n\nFollowing the colony candidate selection process outlined earlier, we cropped out candidate regions of 160\u2009\u00d7\u2009160 pixels (~267\u2009\u00b5m\u2009\u00d7\u2009267\u2009\u00b5m) across the four back-propagated consecutive frames and separated the complex field into amplitude and phase channels. Therefore, each candidate region is represented by a 2\u2009\u00d7\u20094\u2009\u00d7\u2009160\u2009\u00d7\u2009160 array. This four-dimensional (phase\/amplitude\u2013time\u2013xy) data format differs from the traditional three-dimensional data used in image classification tasks and requires a custom-designed DNN architecture that accounts for the additional dimension of time. We designed our DNN by following the block diagram of DenseNet28 and replaced the 2D convolutional layers with P3D convolutional layers47, as shown in Supplementary Fig. S9. Our network was implemented in Python (v3.7.2) with the PyTorch Library (v1.0.1). The network was randomly initialized and optimized using an adaptive moment estimation (Adam) optimizer48 with a starting learning rate of 1\u2009\u00d7\u200910\u22124 and a batch size of 64. To stabilize the accuracy of the network model, we also set a learning rate scheduler that decayed the learning rate by half every 20 epochs. Approximately, 16,000 growing colonies and 43,000 non-colony objects captured from 71 agar plates were used in the training and validation phases. The best network model was selected based on the best validation accuracy. Data augmentation was also applied by random 90\u00b0-rotations and flipping operations in the spatial dimensions. The whole training process took ~5\u2009h using a desktop computer with dual GPUs (GTX1080Ti, Nvidia). The decision threshold value after the softmax layer was set to 0.5 during training, i.e., positive for softmax value >0.5 and negative for softmax value <0.5, which implies equal penalty to false-positive and false-negative events. We adjusted the threshold value to 0.99, empirically based on the training dataset before blind testing, to favour fewer false-positive events.\n\n#### DNN-enabled classification of the bacterial colony species\n\nOnce the true bacterial colonies are selected, they grow for another 2\u2009h to collect 8 consecutive frames, i.e., 4\u2009h, and then are sent to the second DNN as a 2\u2009\u00d7\u20098\u2009\u00d7\u2009288\u2009\u00d7\u2009288 array for the classification of colony species. To perform the classification task, this time, the training data only contain the true colonies and their corresponding species (ground truth). The network follows a similar structure and training process as the detection model, as illustrated in Supplementary Fig. S9. The network was randomly initialized and optimized using the Adam optimizer48, with a starting learning rate of 1\u2009\u00d7\u200910\u22124 and a batch size of 64. The learning rate decayed by 0.9 times every 10 epochs. To avoid overfitting to a specific plate, we discarded colony images extracted from extremely dense samples (>1000 CFU per plate). As a result, approximately 9400 growing colonies were used in the training and validation of the classification model. The whole training process took ~15\u2009h using a desktop computer with dual GPUs (GTX1080Ti, Nvidia).\n\n#### Colony counting\n\nThe respective ground truth information on the growing colonies in each experiment was created after the sample was incubated for >24\u2009h. At the boundary of the plate, the agar always forms a curved surface owing to surface tension, thereby distorting the images of the colonies. Therefore, we limited the effective imaging area to a 50\u2009mm-diameter circle in the centre of the agar plate. In cases where multiple colonies are closely spaced and eventually merge into one large colony (e.g., towards the end of the 24\u2009h incubation period), we then used lens-free time-lapsed images to verify the true colony number when detected by our method to avoid overcounting.\n\n### Calculation of the imaging throughput\n\nIn Supplementary Table S2, we compared the imaging throughput of our system and a conventional lens-based scanning microscope in terms of the space-bandwidth product49 using the following formula:\n\n$$N_{\\mathrm{I}} = \\alpha \\cdot {\\mathrm{FOV}} \\cdot r^2\/\\delta ^2$$\n(5)\n\nwhere NI is the effective pixel count of a frame, \u03b4 is the half-pitch resolution, r is the digital sampling factor along the x and y directions, \u03b1\u2009=\u20092 represents the independent spatial information contained in the phase and amplitude images of the holographic reconstruction, and \u03b1\u2009=\u20091 represents the amplitude-only information contained in an image captured using the standard lens-based bright-field scanning microscope. In the lens-based microscope, we used a colour camera with a pixel size of 7.4\u2009\u00b5m. Therefore, for a 4\u00d7 objective lens, the image resolution is limited to ~3.7\u2009\u00b5m, owing to the Nyquist sampling limit. Without loss of generality, we set r\u2009=\u2009250.\n\n## Data availability\n\nThe data analyzed during the study are available from the corresponding author upon reasonable request.\n\n## Code availability\n\nThe codes used to perform the study are available from the corresponding author upon reasonable request.\n\n## References\n\n1. 1.\n\nSandgren, A. et al. Tuberculosis drug resistance mutation database. PLoS Med.6, e1000002 (2009).\n\n2. 2.\n\nArain, T. M. et al. Bioluminescence screening in vitro (Bio-Siv) assays for high-volume antimycobacterial drug discovery. Antimicrob. Agents Chemother.40, 1536\u20131541 (1996).\n\n3. 3.\n\nJacobs, W. R. Jr. et al. Rapid assessment of drug susceptibilities of Mycobacterium tuberculosis by means of luciferase reporter phages. Science260, 819\u2013822 (1993).\n\n4. 4.\n\nGoodacre, R. et al. Rapid identification of urinary tract infection bacteria using hyperspectral whole-organism fingerprinting and artificial neural networks. Microbiology144, 1157\u20131170 (1998).\n\n5. 5.\n\nLagier, J. C. et al. Culturing the human microbiota and culturomics. Nat. Rev. Microbiol.16, 540\u2013550 (2018).\n\n6. 6.\n\nFierer, N. et al. Forensic identification using skin bacterial communities. Proc. Natl Acad. Sci. USA107, 6477\u20136481 (2010).\n\n7. 7.\n\nKoydemir, H. C. et al. Rapid imaging, detection and quantification of Giardia lamblia cysts using mobile-phone based fluorescent microscopy and machine learning. Lab a Chip15, 1284\u20131293 (2015).\n\n8. 8.\n\nOliver, S. P., Jayarao, B. M. & Almeida, R. A. Foodborne pathogens in milk and the dairy farm environment: food safety and public health implications. Foodborne Pathog. Dis.2, 115\u2013129 (2005).\n\n9. 9.\n10. 10.\n\nDeFlorio-Barker, S. et al. Estimate of incidence and cost of recreational waterborne illness on United States surface waters. Environ. Health17, 3 (2018).\n\n11. 11.\n\nUS Environmental Protection Agency. Method 1604: Total Coliforms and Escherichia Coli in Water by Membrane Filtration Using A Simultaneous Detection Technique (MI Medium). (Environmental Protection Agency, Office of Water, United States, 2002).\n\n12. 12.\n\nCurrent Waterborne Disease Burden Data & Gaps | Healthy Water | CDC. https:\/\/www.cdc.gov\/healthywater\/burden\/current-data.html (2018).\n\n13. 13.\n\nUS EPA. Analytical Methods Approved for Compliance Monitoring under the Long Term 2 Enhanced Surface Water Treatment Rule (US EPA, 2017).\n\n14. 14.\n\nDeshmukh, R. A. et al. Recent developments in detection and enumeration of waterborne bacteria: a retrospective minireview. MicrobiologyOpen5, 901\u2013922 (2016).\n\n15. 15.\n\nAmann, R. & Fuchs, B. M. Single-cell identification in microbial communities by improved fluorescence in situ hybridization techniques. Nat. Rev. Microbiol.6, 339\u2013348 (2008).\n\n16. 16.\n\nKang, D. K. et al. Rapid detection of single bacteria in unprocessed blood using Integrated Comprehensive Droplet Digital Detection. Nat. Commun.5, 5427 (2014).\n\n17. 17.\n\nTitle 40: Protection of Environment. Electronic Code of Federal Regulations Vol. 136.3. https:\/\/www.ecfr.gov\/cgi-bin\/text-idx?node=pt40.1.136 (2020).\n\n18. 18.\n\nHuff, K. et al. Light-scattering sensor for real-time identification of Vibrio parahaemolyticus, Vibrio vulnificus and Vibrio cholerae colonies on solid agar plate. Microb. Biotechnol.5, 607\u2013620 (2012).\n\n19. 19.\n\nChoi, J. et al. A rapid antimicrobial susceptibility test based on single-cell morphological analysis. Sci. Transl. Med.6, 267ra174 (2014).\n\n20. 20.\n\nJo, Y. et al. Holographic deep learning for rapid optical screening of anthrax spores. Sci. Adv.3, e1700606 (2017).\n\n21. 21.\n\nVan Poucke, S. O. & Nelis, H. J. A 210-min solid phase cytometry test for the enumeration of Escherichia coli in drinking water. J. Appl. Microbiol.89, 390\u2013396 (2000).\n\n22. 22.\n\nKim, M. et al. Optofluidic ultrahigh-throughput detection of fluorescent drops. Lab a Chip15, 1417\u20131423 (2015).\n\n23. 23.\n\nTryland, I. et al. Monitoring of \u03b2-D-Galactosidase activity as a surrogate parameter for rapid detection of sewage contamination in urban recreational water. Water8, 65 (2016).\n\n24. 24.\n\nVan Poucke, S. O. & Nelis, H. J. Limitations of highly sensitive enzymatic presence-absence tests for detection of waterborne coliforms and Escherichia coli. Appl. Environ. Microbiol.63, 771\u2013774 (1997).\n\n25. 25.\n\nLondon, R. et al. An automated system for rapid non-destructive enumeration of growing microbes. PLoS ONE5, e8609 (2010).\n\n26. 26.\n\nEPA. EPA Microbiological Alternate Test Procedure (ATP) Protocol for Drinking Water, Ambient Water, Wastewater, and Sewage Sludge Monitoring Methods. (Environmental Protection Agency, Office of Water, United States, 2010).\n\n27. 27.\n\nCHROMagarTM ECC Product Leaflet. http:\/\/www.chromagar.com\/fichiers\/1559127431LF_EXT_003_EF_V8.0.pdf?PHPSESSID=bfb3a740c98b2bf26f8ac5c4d1880fe9 (2020).\n\n28. 28.\n\nHuang, G. et al. Densely connected convolutional networks. 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (IEEE, Honolulu, 2017).\n\n29. 29.\n\nShapiro, J. A. The significances of bacterial colony patterns. BioEssays17, 597\u2013607 (1995).\n\n30. 30.\n\nSu, P. T. et al. Bacterial colony from two-dimensional division to three-dimensional development. PLoS ONE7, e48098 (2012).\n\n31. 31.\n\nFarrell, F. D. et al. Mechanical interactions in bacterial colonies and the surfing probability of beneficial mutations. J. R. Soc. Interface14, 20170073 (2017).\n\n32. 32.\n\nSheats, J. et al. Role of growth rate on the orientational alignment of Escherichia coli in a slit. R. Soc. Open Sci.4, 170463 (2017).\n\n33. 33.\n\nLeChevallier, M. W. & McFeters, G. A. Enumerating injured coliforms in drinking water. J. Am. Water Works Assoc.77, 81\u201387 (1985).\n\n34. 34.\n\nCDC-Salmonella-Factsheet. https:\/\/www.cdc.gov\/salmonella\/pdf\/CDC-Salmonella-Factsheet.pdf (2016).\n\n35. 35.\n\nLiu, H. L., Whitehouse, C. A. & Li, B. G. Presence and persistence of salmonella in water: the impact on microbial quality of water and food safety. Front. Public Health6, 159 (2018).\n\n36. 36.\n\nAlternate Test Procedures in Clean Water Act Analytical Methods. https:\/\/www.epa.gov\/cwa-methods\/alternate-test-procedures (2018).\n\n37. 37.\n\nSanders, E. R. Aseptic laboratory techniques: plating methods. J. Vis. Exp. https:\/\/doi.org\/10.3791\/3064 (2012).\n\n38. 38.\n\nZhang, Y. B. et al. Motility-based label-free detection of parasites in bodily fluids using holographic speckle analysis and deep learning. Light7, 108 (2018).\n\n39. 39.\n\nIsikman, S. O. et al. Lens-free optical tomographic microscope with a large imaging volume on a chip. Proc. Natl Acad. Sci. USA108, 7296\u20137301 (2011).\n\n40. 40.\n\nCobo, M. P. et al. Visualizing bacterial colony morphologies using time-lapse imaging chamber MOCHA. J. Bacteriol.200, e00413\u2013e00417 (2018).\n\n41. 41.\n\nHutchison, J. R. et al. Consistent production of chlorine-stressed bacteria from non-chlorinated secondary sewage effluents for use in the U.S. Environmental Protection Agency Alternate Test Procedure protocol. J. Microbiol. Methods163, 105651 (2019).\n\n42. 42.\n\nColilert 18\u2014IDEXX US. https:\/\/www.idexx.com\/en\/water\/water-products-services\/colilert-18\/ (2020).\n\n43. 43.\n\nPreibisch, S., Saalfeld, S. & Tomancak, P. Globally optimal stitching of tiled 3D microscopic image acquisitions. Bioinformatics25, 1463\u20131465 (2009).\n\n44. 44.\n\nGoodman, J. W. Introduction to Fourier Optics. (Roberts and Company Publishers, Greenwoood Village, 2005).\n\n45. 45.\n\nRivenson, Y. et al. Sparsity-based multi-height phase recovery in holographic microscopy. Sci. Rep.6, 37862 (2016).\n\n46. 46.\n\nZhang, Y. B. et al. Edge sparsity criterion for robust holographic autofocusing. Opt. Lett.42, 3824\u20133827 (2017).\n\n47. 47.\n\nQiu, Z. F., Yao, T. & Mei, T. Learning spatio-temporal representation with pseudo-3D residual networks. 2017 IEEE International Conference on Computer Vision (ICCV) (IEEE, Venice, Italy, 2017).\n\n48. 48.\n\nKingma, D. P. & Ba, J. Adam: a method for stochastic optimization. 3rd International Conference on Learning Representations (ICLR, Ithaca, 2015)\n\n49. 49.\n\nWang, H. D. et al. Computational out-of-focus imaging increases the space\u2013bandwidth product in lens-based coherent microscopy. Optica3, 1422\u20131429 (2016).\n\n50. 50.\n\nGreenbaum, A. et al. Increased space-bandwidth product in pixel super-resolved lensfree on-chip microscopy. Sci. Rep.3, 1717 (2013).\n\n## Acknowledgements\n\nThe authors acknowledge the funding of ARO (Contract # W911NF-17-1-0161), Koc Group and HHMI. The authors would also like to acknowledge IDEXX Laboratories Inc. for loaning the Quanti-Tray Sealer and Drs. Janine R. Hutchison and Richard M. Ozanich from Pacific Northwest National Laboratory for sharing their assistance with the chlorination of bacterial samples.\n\n## Author information\n\nAuthors\n\n### Contributions\n\nH.W., H.C.K., and Y.Q. designed and built the lens-free imaging platform and the computer algorithms. H.C.K. prepared all bacterial samples. B.B., Y.Z., Y.Y., and Y.R. contributed to the system and algorithm design. H.W., H.C.K., and Y.Q. performed the experiments on the presented system. E.C.Y. and H.C.K. captured the images of cultured plates at a benchtop microscope. H.W., Y.Q., B.B., and H.C.K. processed the experimental data. E.G. and H.C.K. prepared the chlorine injured bacterial samples. S.T. and H.C.K. prepared the agar plates for the optimization of image capture settings. H.W., H.C.K., and A.O. wrote the paper. H.C.K. and A.O. formulated the research goals and aims. A.O. supervised the research.\n\n### Corresponding author\n\nCorrespondence to Aydogan Ozcan.\n\n## Ethics declarations\n\n### Conflict of interest\n\nH.C.K, H.W., Y.R., Y.Q., and A.O. have a patent application on the invention reported in this paper.\n\n## Rights and permissions\n\nReprints and Permissions\n\nWang, H., Ceylan Koydemir, H., Qiu, Y. et al. Early detection and classification of live bacteria using time-lapse coherent imaging and deep learning. Light Sci Appl 9, 118 (2020). https:\/\/doi.org\/10.1038\/s41377-020-00358-9\n\n\u2022 Revised:\n\n\u2022 Accepted:\n\n\u2022 Published:","date":"2021-12-01 12:57:53","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\": 1, \"mathjax_display_tex\": 2, \"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\": 0.4619942307472229, \"perplexity\": 4347.89886819246}, \"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-2021-49\/segments\/1637964360803.0\/warc\/CC-MAIN-20211201113241-20211201143241-00381.warc.gz\"}"} | null | null |
Q: powershell select-string -pattern -path finds hits where there are none I´d like to use PS to scan some log files for errors
select-string -pattern '[E]' -path D:\15\s\BAS\VectorNetworkAnalysis.AutomationInterfaceTests\Tests\COMTesting\PythonUnitTests\PyTestsResult*.log
but despite there are NO ERRORS PS lists every line of the logs as a hit, at least the ones containing "[i]".
I tried different variations, with double apostrophes, blanks before/after the [E] and intentionally added one single line containing "[E]" but either I´getting a bunch of lines or none.
Any hint? Thanks a lot.
here is an example of log including the line with the INTENTIONALLY ADDED [E]
2022-03-04 09:22:26 [i] Transmission: Execution State is OK!
2022-03-04 09:22:26 [i] User range calib state is active!
2022-03-04 09:22:28 [i] **** Starting Save/Load calibration test ****
2022-03-04 09:22:29 [i] Transmission: Execution State is OK!
2022-03-04 09:22:29 [i] User range calib was saved!
2022-03-04 09:22:29 [i] User range calib was loaded!
2022-03-04 09:22:31 [i] **** Starting S11 one port meas test ****
2022-03-04 09:26:04 [i] S11OnePortMeas: Execution State is OK!
2022-03-04 09:26:04 [i] S11OnePortMeas: Number of results is OK!
2022-03-04 09:26:05 [E] **** Starting Impedance one port meas test ****
2022-03-04 09:26:06 [i] OnePortMeas: Execution State is OK!
A: Select-String -Pattern uses regex, and in regex, brackets means "match any of the characters in the brackets", so it matches every line that contains an E. You could replace [E] with [qwerty] and it would match all lines that contains any of the letters in "qwerty" (so about all of them in your example).
If you want to match a string literally, you can use the -SimpleMatch parameter which will interpret your input as a string instead and match anything containing that given string.
A: Use a website such as regex101.com to test regular expressions before placing them in a script. This has saved me a lot of time and effort.
(?i)\[e\] matches both [e] and [E].
\[e\] matches [e].
\[E\] matches [E].
The (?i) makes the regular expression case insensitive. Regular expressions have their own case sensitivity rules independent of PowerShell.
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The way I see it, Rupert, every person's life hurtles toward the same fork in the road. Each path has its own successes and failures, based on ever-changing moral and spiritual absolutes. The problem is, most of us have no idea which trail we're on until it's too late. We're judged entirely by the footprints we leave behind. One path leads to self-importance, cynicism and greed. The other path brings you to generosity, sacrifice and humility. But the real difference comes in how your journey affects the ones you love. For they are the ones who ultimately right the travel guides. Let's promise to always keep each other on the right path. Deal. But this ways harder. Exactly. | {
"redpajama_set_name": "RedPajamaC4"
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