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Q: Autocorrelation of noise - negative correlation I am investigating autocorrelation of electrical noise as part of an undergraduate experiment (as detailed in http://physlab.lums.edu.pk/images/a/ab/Correlation.pdf). I sampled noise voltages using an 8-bit AtoD from a noise generator (whose description was not provided). I first sampled the noise directly, then I sampled it after passing it through a low-pass filter. I imported the data into MATLAB and used the autocorr function to autocorrelate it. The no-filter data autocorrelated as expected, being just 1 at t=0 (as in Fig 1 in paper); however, the filtered data decays initially as expected but then dips below 0 before coming back up and staying around 0 as in the image below.
My three issues are:
*
*I am not really sure what this means physically (is the correlation 'direction' changed so-to-speak?),
*Why might this be occurring? I know that a description of the noise generator may be important here but I do not have that information myself.
*Is there any way to correct this issue, and if not, what would be the best compromise?
EDIT:
Here are snapshots of the noise without and with the filter (RC=0.5ms) respectively:
A: Letting $\mathbf{F}$ and $\mathbf{F}^{-1}$ be the forward and inverse discrete Fourier transform, the cyclic autocorrelation of a signal $A$ is given by
$$S(A)=\mathbf{F}^{-1}\left[\mathbf{F}(A)\mathbf{F}(A)^*\right].$$
Let the low-passed signal $A_L$ be
$$A_L=\mathbf{F}^{-1}\left[\mathbf{F}(A)\mathbf{F}(L)\right]$$ where $L$ is the low-pass filter in the time domain.
Then
$$S(A_L)=\mathbf{F}^{-1}\left[\mathbf{F}(A)\mathbf{F}(L)\mathbf{F}(A)^*\mathbf{F}(L)^*\right]=\mathbf{F}^{-1}\left[\mathbf{F}(A)\mathbf{F}(A)^*\mathbf{F}(L)\mathbf{F}(L)^*\right]\\=S(A)*S(L)$$
where the convolution theorem has been applied.
Since $S(A)\approx(A\cdot A,0,0,...)$ for uncorrelated noise, you should roughly be getting back a picture of $S(L)$ when you compute the autocorrelation of the low-pass signal.
So the question becomes: what sort of low-pass filter $L$ did you employ?
| {
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Giunone e il pavone (Juno and the Paycock) è un film del 1929 diretto da Alfred Hitchcock.
Trama
Irlanda, Dublino. Periodo della lotta per l'indipendenza.
Protagonista è la famiglia Boyle, molto povera.
L'ambientazione è tutta in luogo chiuso, casa del pavone; il capofamiglia, che si fa chiamare anche capitano, pur non essendo mai stato su una nave, è in realtà disoccupato e fanfarone (da qui il soprannome). Giunone, la moglie, è una donna giunonica ed autoritaria, ma sensata e con i piedi per terra a differenza del marito; poi c'è Johnny, che ha perso un braccio in guerra per la patria, e Mary, la giovane e graziosa figlia minore. La storia si svolge attorno alle questioni che solleva il mancato ottenimento di un'eredità.
Un avvocato Charlie Bentham avverte la famiglia di un'imminente eredità, ma mentre la moglie è cauta e diffidente il marito si comporta come se avesse già a disposizione questa ricchezza, fa acquisti dispendiosi, abbandona l'amico Joxer con cui trascorreva lunghe giornate al pub.
L'eredità non esiste e la figlia si ritrova incinta dell'avvocato che l'abbandona. Il figlio Johnny è accusato di essere una spia e sarà ucciso. I Boyle si ritrovano in completa rovina e devono lasciare anche la misera abitazione.
Soggetto
John Maxwell, il produttore della British International Pictures, incarica Hitchcock, dopo il grande successo di Blackmail, di portare sullo schermo il dramma omonimo di Sean O' Casey, rappresentato all'Abbey Theatre di Dublino e accolto da un pubblico entusiasta. Saranno utilizzati gli stessi attori che avevano recitato in teatro.
Produzione
Nell'inverno 1929-1930 Hitchcock insieme alla moglie Alma Reville prepara la sceneggiatura. Porta a termine le riprese negli studi di Elstree. La prima a Londra si ebbe il 22 settembre 1930.
Critica
Il film fu recensito molto positivamente. Qualche giornale parlò di un "quasi capolavoro" e furono molto lodati sia la regia sia la recitazione impeccabile degli attori.
Hitchcock invece dichiara addirittura di "aver provato vergogna" perché malgrado considerasse la "commedia stupenda" e gli piacesse la "mescolanza di umorismo e tragedia", non era riuscito a "trasporla in forma cinematografica": si era limitato, a suo dire, a filmare il teatro.
Rohmer pur condividendo con Hitchcock che "l'immagine non risulta di grande interesse" ritiene che "il sonoro sia particolarmente curato: lo scricchiolio dei passi sul pavimento di legno, un uso intensivo e grottesco dell'accento, oltre a una breve raffica di mitragliatrice che fa sobbalzare lo spettatore assopito sulla poltrona".
Giorgio Simonelli si scosta dal giudizio eccessivamente negativo dato dal regista a se stesso e nota che nella prima parte del film cerca "di uscire dalla unità di luogo con l'aggiunta di una scena piuttosto movimentata in un pub" e più avanti "ricerca modalità originali attraverso cui rendere cinematografico il testo" con "un susseguirsi di piani diversi di una stessa inquadratura" tecnica che userà in modo mirabile in un film della sua maturità come Nodo alla gola.
Note
Collegamenti esterni
Film drammatici
Film diretti da Alfred Hitchcock
Film ambientati a Dublino | {
"redpajama_set_name": "RedPajamaWikipedia"
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Continuo, Addio!
Duo Tartini
20 TITRES • 1 HEURE ET 15 MINUTES • SEP 12 2019
Sonata for Violin and Cello in G Major, B. G7: I. Larghetto
Sonata for Violin and Cello in G Major, B. G7: II. Allegro
Sonata for Violin and Cello in G Major, B. G7: III. Allegro
Aria cromatica e variata in A Minor
Capricio for Cello solo in E Minor, No. 6
Duo Tartini & Annabelle Luis
Ricercata 6 in G Minor: I. Adagio
Ricercata 6 in G Minor: II. Allegro
Ricercata 6 in G Minor: III. Adagio
Ricercata 6 in G Minor: IV. Allegro
Capricio for solo Violin in C Minor
Duo Tartini & David Plantier
Sonata for Violin and Cello in C Minor, Op. 5, No. 6: I. Adagio
Sonata for Violin and Cello in C Minor, Op. 5, No. 6: II. Allegro moderato
Sonata for Violin and Cello in C Minor, Op. 5, No. 6: III. Allegro
Duetto III in A Minor: I. Andante
Duetto III in A Minor: II. Fuga
Sonata for Violin and Cello in G Minor, Op. 1, No. 4: I. Allegro
Sonata for Violin and Cello in G Minor, Op. 1, No. 4: II. Cantabile
Sonata for Violin and Cello in G Minor, Op. 1, No. 4: III. Presto
Concertant Duet for Violin and Cello in E Minor, No. 1: I. Cantabile con variazioni
Concertant Duet for Violin and Cello in E Minor, No. 1: II. Rondo. Allegro
℗© 2019: Off The Records | {
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</div>
<div id="impress">
<div class='step' >
<h4>Conversion from Instruction to MachineInstruction</h4>
</div>
<div class='step' >
<pre><code class='prettyprint '>IR -> SelectionDag -> X86TargetLowering [legalize dag operations w.r.t
target ] -> SelectionDAGISel(manual
implementation) || X86FastIsel ->
InstrEmitter:EmitMachineNode[ emit MachineInstructions ] ->
X86DAGToDAGISel::Select[ Register Allocation] ->
X86AsmPrinter::EmitInstruction
</code></pre></div>
<div class='step' >
<h1>Removing instructions</h1>
<h2>Target Independent</h2>
</div>
<div class='step' >
<h4>eraseFromParent()</h4>
<p><code class='inline prettyprint'>llvm/lib/CodeGen/CodeGenPrepare.cpp</code></p>
<p>// Build a truncate instruction.</p>
<p>truncation due to casting may lead to deletion of original instruction.
it has an undo method, which erases the instruction</p>
<p>Codegen preparation. Common to all architectures.
We may loose instruction information here.</p>
<pre><code class='prettyprint '>// OptimizeCmpExpression - sink the given CmpInst
// into user blocks to reduce
// the number of virtual reduce registers
// that must be created and
OptimizeCmpExpression(CmpInst *CI) {}
</code></pre></div>
<div class='step' >
<pre><code class='prettyprint '>OptimizeNoopCopyExpression -
if the specified cast instruction is Noop copy
(e.g, copy from one pointer type to another, i32->i8)
Also adds another
</code></pre></div>
<div class='step' >
<h4>CodeGenPrepare::DupRetToEnableTailCallOpts(BasicBlock *BB)</h4>
<p>Look for opportunities to duplicate return
instructions to the predecessor to enable tail call optimizations. </p>
<pre><code class='prettyprint asm'>/// bb0:
/// %tmp0 = tail call i32 @f0()
/// br label %return
/// bb1:
/// %tmp1 = tail call i32 @f1()
/// br label %return
/// return:
/// %retval = phi i32 [ %tmp0, %bb0 ], [ %tmp1, %bb1 ], [ %tmp2, %bb2 ]
/// ret i32 %retval
/// bb0:
/// %tmp0 = tail call i32 @f0()
/// ret i32 %tmp0
/// bb1:
/// %tmp1 = tail call i32 @f1()
/// ret i32 %tmp1
</code></pre></div>
<div class='step' >
<h4>CodeGenPrepare::DupRetToEnableTailCallOpts(BasicBlock *BB)</h4>
<pre><code class='prettyprint '>// Duplicate the return into CallBB.
(void)FoldReturnIntoUncondBranch(RI, BB, CallBB);
</code></pre></div>
<div class='step' >
<h4>TypePromotion</h4>
<p>creates a truncated instruction</p>
<pre><code class='prettyprint '>TruncBuilder(Instruction *Opnd, Type *Ty) : TypePromotionAction(Opnd) {}
</code></pre></div>
<div class='step' >
<pre><code class='prettyprint '>class SExtBuilder : public TypePromotionAction {}
</code></pre></div>
<div class='step' >
<p>// a = b==b ? 2 : 3 -> if(b==b) 2 else 3</p>
<pre><code class='prettyprint '>CodeGenPrepare::OptimizeSelectInst(SelectInst *SI) {}
</code></pre></div>
<div class='step' >
<p>// Some targets have expensive vector shifts if the lanes aren't all
the same
// (e.g. x86 only introduced "vpsllvd" and friends with AVX2). In
these cases
// it's often worth sinking a ShuffleVectorInstr splat down to its
use so that
// codegen can spot all lanes aren identical.</p>
<pre><code class='prettyprint '>bool CodeGenPrepare::OptimizeShuffleVectorInst(ShuffleVectorInst *SVI) {}
</code></pre></div>
<div class='step' >
<h3>X86/X86FastISel.cpp</h3>
</div>
<div class='step' >
<p>X86InstrInfo.cpp</p>
</div>
<div class='step' >
<p>bool X86InstrInfo::AnalyzeBranch{}</p>
<p>Two eraseFromParent:
1. If the block has any instructions after a JMP, delete them. => dead code.
2. Delete the JMP if it's equivalent to a fall-through. //jmp not required.</p>
<p>Question: can we even instrument this.</p>
<p>only modifies jump instructions which can't even</p>
<p>jCC L1
jmp L2</p>
<p>we could also instrument how many times a particular branch is taken.
This transformation may potentially interfere with that.</p>
</div>
<div class='step' >
<h3>optimizeCompareInstr</h3>
<pre><code class='prettyprint '>Check if an instruction already performs the same compare.
</code></pre></div>
<div class='step' >
<h3>optimizeLoadInstr</h3>
<p>This may effect the instrumentation</p>
<pre><code class='prettyprint '>// Try to remove the load by folding it to a register
// operand at the use. We fold the load instructions if load
// defines a virtual
// register, the virtual register is used once in the same BB,
// instructions in-between do not load or store, and have no side
// effects.
</code></pre></div>
<div class='step' >
<h2>X86ISelLowering.cpp</h2>
<p>MachineBasicBlock *EmitXBegin;</p>
<pre><code class='prettyprint '>// Utility function to emit xbegin specifying the start of an RTM region
// RTM - restricted transactional memory
deletes the previous instruction to insert
</code></pre>
<p>I don't think what's the point of intrumenting some instructions.</p>
</div>
<div class='step' >
<pre><code class='prettyprint '>EmitAtomicLoadArith - emit the code sequence for pseudo atomic
instructions.
EmitAtomicLoadArith6432
</code></pre></div>
<div class='step' >
<pre><code class='prettyprint '>
EmitPCMPSTRM
// When we get size specific XMM0 registers, i.e. XMM0_V16I8
// or XMM0_V32I8 in AVX all of this code can be replaced withth
// that
// in the .td file.file
</code></pre></div>
<div class='step' >
<pre><code class='prettyprint '>
</code></pre>
</div>
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Q: PHP - Recalling information stored in database using sessions I've got problem when using sessions to recall data stored in mySQL database. This is my code:
The login page is simple, the input your username and password kind (i know the password is still plaintext, i plan to change it later).
<?php
$host="localhost";
$user="root";
$pass="";
$db_name="proyek";
$tbl_name="murid";
mysql_connect("$host", "$user", "$pass")or die("Cannot connect to SQL.");
mysql_select_db('$db_name');
?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<title></title>
</head>
<body>
<div id="header">
LOGO
<h1 align="center">TITLE</h1>
</div>
<br/>
<div id="login">
<form id="loginform" name="loginform" method="post" action="checklogin.php">
<table border="0" align="center">
<tr>
<td>NIS</td>
<td></td>
<td><input type="text" name="nislogin" id="nislogin"/></td>
</tr>
<tr>
<td>Password</td>
<td></td>
<td><input type="password" name="pwdlogin" id="pwdlogin"/></td>
</tr>
<tr>
<td colspan="3" align="center"><input type="submit" name="loginbutton" id="loginbutton" value="Login"/></td>
</tr>
</table>
</form>
</div>
</body>
</html>
and this is the code for login check page:
<?php
session_start();
$host="localhost";
$user="root";
$pass="";
$db_name="proyek";
$tbl_name="murid";
mysql_connect("$host", "$user", "$pass")or die("Cannot connect to SQL.");
mysql_select_db('$db_name');
$nis=($_POST['nislogin']);
$pwd=($_POST['pwdlogin']);
$sql="SELECT * FROM murid WHERE nis='$nis' and password='$pwd'";
$result=mysql_query($sql);
$count=mysql_num_rows($result);
if($count==1)
{
$_SESSION['nislogin']=$nis;
$nama=$result['nama'];
$_SESSION['nama']=$nama;
header("location:index.php");
return true;
exit;
}
else
{
echo("Wrong NIS or password.");
return false;
}
?>
i have entered some dummy data in database for testing purposes; id, password, name. how can i recall something from database while user only login with username/id?
i'd like to display something like 'hello, name' in the next page. Help is appreciated.
edit: I've edited my code based on feedbacks and it produces blank; like 'Hello,' with no name.
A: First of all, since you are running a query on your login check page, use that value for your session rather than the post data. Also, whenever you are redirecting, always exit your script.
EDIT:
Since you are in the development stage, you need to display an error if your query fails so you know why. I also realized you need to return an associative array to access the row. Try this.
$sql="SELECT * FROM murid WHERE nis='$nis' and password='$pwd'";
$result=mysql_query($sql);
if (!$result) {
die('Invalid query: ' . mysql_error());
}
$count=mysql_num_rows($result);
if($count==1)
{
$data=mysql_fetch_assoc($result); // since you are only accessing one row,
// otherwise you would put this in a loop to build your array.
session_start();
$_SESSION['nislogin']=$data['nis'];
header("location:index.php");
return true;
exit;
}
I also see this session on your login page, but I don't see that you ever created it and I don't see the purpose.
if(isset($_SESSION['nama']))
{
unset($_SESSION['nama']);
}
Basically all you need from here, is to start the session on index.php and output it.
index.php
session_start();
if(isset($_SESSION['nislogin']))
{
$name = $_SESSION['nislogin'];
} else {
$name = "stranger";
}
<body>
Welcome, <?=$name?>
</body>
| {
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{"url":"https:\/\/cs.stackexchange.com\/questions\/42573\/can-the-concatenation-of-two-non-regular-languages-be-regular?noredirect=1","text":"# Can the concatenation of two non-regular languages be regular? [duplicate]\n\nCan anyone give an example of two non-regular languages $A, B \\subseteq \\{0, 1\\}^\u2217$ for which the language $AB$ is regular?\n\n\u2022 maybe there are none? but dont see an obvious proof.... hmmm, maybe something like, if either language cannot be characterized by the pumping lemma, then neither can the concatenation? \u2013\u00a0vzn May 15 '15 at 3:20\n\u2022 What have you tried? What are your thoughts? Where did you get stuck? We want to help you understand, not do your exercise for you, and we expect you to make a serious effort on your own before asking. What non-regular languages do you know? Have you tried letting $A$ be some non-regular language you know, and then seeing if you can find a choice for $B$ that will work? \u2013\u00a0D.W. May 15 '15 at 4:25\n\u2022 Solutions can be found at \"Is $A$ regular if $A^2$ is regular?\" and \"Are the non-regular languages closed under reverse, union, concatenation, etc?\". \u2013\u00a0Hendrik Jan May 15 '15 at 8:53\n\nHint: Let $C$ be the language of words that have the same number of 1's as 0's. Is $C$ regular? Let $A = \\{0,1\\}^* \\setminus C$. Is $A$ regular? Now, can you find a non-regular language $B$ that will make $AB$ regular?","date":"2020-02-20 03:33:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.4337307810783386, \"perplexity\": 457.73457722990923}, \"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-2020-10\/segments\/1581875144498.68\/warc\/CC-MAIN-20200220005045-20200220035045-00367.warc.gz\"}"} | null | null |
Q: Loading Assemblies There are plenty of discussing that shows how to load assemblies from BIN and from GAC... my question is more general and I would love to know how assembly loading work.
As for example
in the BIN folder we can have
A.dll
A.dll.config
A.dll.config file can look like:
<?xml version="1.0"?>
<configuration>
<startup>
<supportedRuntime version="v2.0.50727"/>
</startup>
</configuration>
to help us setting the correct assembly reference.
I was wondering how can I create a A.dll that needs B.dll but without specify any version so if would always take the B.dll that is in the BIN Folder.
My question is regarding an update of the SDK and all my code still points to the old SDK version, I wanted to have my assembly look for the top most of all versions of the resource, BIN or GAC and use that one...
How can I say that in Visual Studio? I can only Add Reference to a physical file (version) :-(
A: You can use late binding and Reflection. There are many places to read about this, but you can start here.
A: If I understand correctly, I guess you could use the "plugins" way of doing things through the AppDomain object. Perhaps loading the assembly through the AppDomain, since you can set the path for the loaded assembly, you will be able to easily replace the files, unload the assembly and reload the new one in ShadowCopy. The ShadowCopy allows you to copy the latest file at the same location of the current file assembly. Then, build your mechanism to check for a new file, and if it exists, then unload your assembly and reload the latest. This way, everything will be quite transparent for your program, though it requires a bit more programming from your side. Nevertheless, you will gain in long term development.
| {
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"Would I have ever heard of Patti Scialfa, much less bought a CD by her, were I not already a fan of her husband and his famous backing band? Probably not (although I have to admit that I'm not much of a fan of some of the other Jersey Shore bands who were brought to my attention because of their affiliations with or support from Bruce Springsteen). But that doesn't change the fact that this is an excellent album that demonstrates what a powerful and literate singer/songwriter Patti is. She seems to have toned down the vibratto in her vocals that many people find annoying, and her singing has a definite edge and roughness that fits the material very well. She reminds me of a cross between Lucinda Williams and Emmylou Harris, but creates a style that stands on its own independent of any comparisons. "Waiting for Elvis" would make a very good single if artists of Patti and her husband got any regular radio airplay anymore, but the whole album is very accessible and listenable with no filler to be found. I would love to see Patti on her own tour, or have her sing some of these songs on an E Street Band tour. I'd also like for Bruce and Patti to do a "duet" album together, rather than just making background appearances on one another's records. Live renditions of songs like "Mansion on the Hill" and "If I Should Fall Behind" prove that they have great chemistry and that they can achieve gorgeous harmonies. Regardless of your opinions about Bruce Springsteen, however, don't let your views about how she may have gotten a recording contract jade you or prevent you from enjoying what is a genuinely fine album."
"I was so excited to find out that Patti Scialfa was releasing a third CD, that I was worried I might be disappointed ultimately. Not to worry. This is a fantastic CD. With each CD, she changes her perspective some. "23rd Street Lullaby" was told by a woman looking back at her past and how that past had helped to make her the person she had become. On "Play it As It Lays," the woman is a little more world-weary, battered around by life and love, but still optimistic. Her lyrics are like poetry and reflect some of the very things I, a long-married, middle aged mother of teenagers, often think about my life and relationships. She's really put my thoughts into words, and done it beautifully. Also, I love her singing voice; it's tender and tough, playful and sad. Extremely interesting. Like her famous husband said, it's "sawdust and sugar."
Favorite tracks are "Looking for Elvis," "Like Any Woman Would," "Town Called Heartbreak, "Run,Run,Run," and the hauntingly beautiful "Play It As It Lays." I love the message of "Play It As It Lays," that no relationship is perfect, you take the good with the bad and because you love the person, you play it as it lays. The chorus is so beautiful.
I highly recommend this CD. Also recommend Rumble Doll and 23rd Street Lullaby.
PS--I don't know how anyone can say they cannot understand the lyrics. I had no trouble. What I heard was eloquent. I also don't see how anyone can say she wouldn't have a recording contract if it weren't for Bruce. She had this contract BEFORE she was even involved with Bruce, i.e. since the early 1980s."
"Give Patti the attention she is due!!! This new CD is clearly another outstanding example of Patti's gift, in her own right, as singer and songwriter! Simply put, I LOVE IT!
Beautifully crafted tunes with real lyrics, and layered musical composition, Patti delivers her own magic with her unique style. Upbeat and rocking or pensive and introspective, she draws you in and takes you on her journeys. This is one smart and sensitive woman who embraces her experiences and shares them with us through her music! How lucky we are!
Messages full of fire and grace, Patti puts it down for all to hear and you won't be sorry for the listening!!! Oh yes, and the band is full of amazing talent as well!!!
So listen up and beware the nay-sayers who judge her on anything other than her own musical merits!
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"redpajama_set_name": "RedPajamaC4"
} | 2,072 |
import Categories, { getCategories, getCategoriesAsOptions } from './modules.js';
import './server/load_categories.js';
export { getCategories, getCategoriesAsOptions };
export default Categories;
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,852 |
A warm and welcoming dealership that is dedicated to quality and committed to success. We consistently act in a trustworthy manner affirming our integrity and encouraging open communication. At Mahindra PMB our customers are offered a "one stop" shopping experience with skilled sales executives whom will guide you through the process ensuring it's a memorable one. On site we have a fantastic pre-owned division offering good quality vehicles at competitive prices. Each one of them have been meticulously inspected to ensure customers peace of mind. Our service department is host to a fantastic team with highly trained technicians who will make sure your vehicle is looked after and repaired to OEM specifications. We use state of the art diagnostic equipment in a modern workshop environment. Give us a call on 033 345 3692 or pop in at 322 Hoosen Haffejee Street today, we would love to meet you! | {
"redpajama_set_name": "RedPajamaC4"
} | 9,607 |
Other competition work
Competition work on open access
We continue to monitor the market for open access rail services.
Our 2022 monitoring report is the latest of a series of annual publications. In addition to our regular analysis, our latest publication is accompanied by an economic appraisal of historic open access operator entry on the East Coast Main Line.
We will continue to gather and collate data on these metrics and compare outcomes as our monitoring approach evolves.
We continue to keep our monitoring framework under review, both in terms of the metrics that we monitor, and our approach. Specifically, we focus on considering the impact of the post-pandemic recovery and the emerging rail reform agenda on operators.
We intend to continue to monitor the market for open access in a proportionate way. We value stakeholder input into this process and we therefore welcome comments and suggestions from interested parties. These can be submitted by email to competition@orr.gov.uk.
March 2021 update Collapse accordion Open accordion
In February 2020 we published our first report on our open access competition monitoring work. The 2020 report constituted a baseline, and an evidence base to be built on over time. We followed this up with an update published in March 2021.
The metrics that we monitor to assess market performance and passenger outcomes include a number of quality metrics (complaints, delay compensation, passenger satisfaction, punctuality and age of rolling stock).
We will continue to gather and collate data on these metrics, and compare outcomes with the baseline to identify any changes and trends. The sample of operators that we present evidence on will evolve as existing OAOs expand their services and new OAOs commence operations on the network.
As more evidence becomes available, the outputs of our monitoring will feed into our consideration of open access applications and the assumptions underlining our decision making tools. It will also act as an indicator of how well the market is functioning from a competition and regulatory compliance perspective.
We continue to keep our monitoring framework under review, both in terms of the metrics that we monitor, and our approach. A particular issue going forward will be the the COVID-19 crisis and its impact on passenger rail numbers.
Monitoring the impact of, and response, to open access - March 2021 update (pdf 468.51 KB)
Report on our open access competition monitoring work Collapse accordion Open accordion
We have published our first report on our open access competition monitoring work. This report constitutes our baseline, and we will build on this evidence base over time.
The metrics we are monitoring to assess market performance and passenger outcomes include a number of quality metrics (complaints, delay compensation, passenger satisfaction, punctuality and age of rolling stock), and data on fares and revenue.
We continue to keep our monitoring framework under review, both in terms of the metrics that we monitor, and our approach.
ORR publishes update on its competition work on open access Collapse accordion Open accordion
Open access operators play an important role in promoting competition on the railway. The dynamic created by on-rail competition delivers demonstrable benefits to passengers; provides comparators to inform franchising decisions; and better holds Network Rail to account in its role of identifying and allocating capacity.
As such, ORR supports more 'on-rail' competition, where this is in accordance with our statutory duties.
In December 2018, we launched a piece of work to develop a framework for monitoring the impact of, and response to, open access competition. We set out to do this to ensure that the right conditions are created for competition to be fostered, through the presence of open access operators.
Since launching this work, we have engaged with a wide range of stakeholders, including train operators, Network Rail, and other UK competition regulators. The evidence and feedback we gathered has informed the design of our monitoring framework, and our update document, published today, sets out what metrics we intend to monitor on an ongoing basis, and how.
We intend to commence our monitoring activities later in 2019 and will continue to keep our monitoring framework under review to ensure it remains as useful and robust as possible.
Open access update document
Open access monitoring – Update on ORR's plans to monitor the impact of, and response to, open access
ORR launches competition work on open access Collapse accordion Open accordion
ORR supports more 'on-rail' competition where it delivers sustainable benefits to passengers. The dynamic created by on-rail competition delivers demonstrable benefits to passengers; provides comparators to inform franchising decisions; and better holds Network Rail to account in its role of identifying and allocating capacity.
We have launched a piece of work which will deliver a framework for monitoring the impact of, and response to, open access competition. Doing so will help us to better understand the challenges that OAOs face in trying to establish themselves in the GB market, and enable us to spot any problems early.
We will also produce information for industry participants on what, in the context of open access, ORR considers in broad terms to constitute anti-competitive behaviour, as opposed to a competitive response. This will increase transparency for industry on what we expect of them, and inform how we use our competition enforcement powers.
Open access launch document
Open access monitoring - ORR's plans to monitor the impact of, and response to, open access - 3 December 2018
Monitoring the impact of, and response to, open access - April 2022 update (pdf 558.77 KB)
Monitoring the impact of, and response to, open access - April 2022 update - Appendix 1: Economic Appraisal Report by Systra (pdf 873.52 KB)
Open access monitoring: ORR's plans to monitor the impact of, and response to, open access - Update paper - 27 March 2019 (pdf 315.02 KB)
Open access monitoring - ORR's plans to monitor the impact of, and response to, open access - 3 December 2018 (pdf 724.49 KB) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,647 |
\section{Introduction}
Theoretical arguments and experimental evidence
suggest that hadronic matter undergoes a transition to a plasma of quarks
and gluons at high temperature~\cite{Stephanov:2007fk}.
At extremely high temperatures quarks and
gluons are nearly free, and should be described by the Stefan-Boltzmann
law with the appropriate degrees of freedom. When temperature is not much
larger than the critical temperature -- say, $T_c < T < \simeq 2 T_c$ --
strong interactions among the constituents give rise to
non--perturbative effects. In short, at large T the QGP
is a gas of nearly free quarks,
which becomes strongly interacting at lower temperatures
$T=(1-3)\,T_c$~\cite{Shuryak:2007qs,Blaizot:2007sw}.
Several proposals have been made to characterise the properties of
the system in such non-perturbative phase. For instance the above mentioned
strong interactions might be enough to preserve bound states above $T_c$,
while coloured states might appear, deeply
affecting the thermodynamics of the system~\cite{Shuryak:2004tx}.
Analytic techniques are being
refined more and more, so to be able to capture the features of the system
closer and closer to $T_c$~\cite{Ipp:2006ij}.
Model theories of quasi particle physics have been considered
as well~\cite{Bluhm:2004xn,Bluhm:2006av}.
In this work we study this interesting dynamical
region by lattice QCD simulations at $T \simeq 1.1\, T_c$
(the reason for this choice will be clear in the following),
and a nonzero baryon density.
In principle, lattice QCD simulations at non-zero baryon density
are plagued by the sign problem~\cite{Splittorff:2007ck}. However,
it has been realised that this problem can be circumvented
thanks to physical fluctuations, which grow relatively large in
the Quark Gluon Plasma phase. In this work we adopted the imaginary
chemical potential approach~\cite{mpl,hart1,deForcrand:2002ci,D'Elia:2002gd,
Azcoiti:2005tv,Chen:2004tb,Wu:2006su,Cea:2006yd},
which avoids the sign problem and makes
it possible conventional Lattice QCD simulations. The interested reader
might want to consult refs.~\cite{Schmidt:2006us, Philipsen:2005mj}
for recent reviews and
\cite{Lombardo:2004uy,Muroya:2003qs} for more pedagogical introductions
into the subject.
\begin{figure}
\begin{center}
\epsfig{file=diag.eps,width=11 truecm}
\caption{Schematic phase diagram for four flavor QCD in the $T, \mu^2$
plane. The candidate sQGP phase is bound by the chiral (pseudo)critical
line in the negative $\mu^2$ half-plane}
\vskip -0.8 truecm
\label{fig:phaseDIAGR}
\end{center}
\end{figure}
Other studies in the quark gluon plasma phase have addressed
the higher temperature regime~\cite{Fodor:2002km,D'Elia:2004at,D'Elia:2005qu,Ejiri:2005wq,
Kratochvila:2006jx}.
Here we analyze in detail the non-perturbative behaviour close
to $T_c$, in the candidate strongly coupled Quark Gluon Plasma (sQGP)
region (some preliminary results have appeared in~\cite{Lombardo:2006yc}).
We note that in the sQGP region the chiral critical line
lies in the imaginary chemical potential plane, and that such a
chiral line ends in the proximity of the endpoint of the
Roberge--Weiss line~\cite{Roberge}.
We focus our analysis on the particle number and its susceptibility,
on the chiral condensate, and on the Polyakov loop,
and we find that the results are consistent
with those expected of a critical behaviour associated with the critical
line at imaginary chemical potential.
Hence, the numerical results are compatible with simple power law behaviour
of the equation of state as a function of the imaginary chemical
potential $\mu_I$, yielding a modified form of the Stefan-Boltzmann law.
The rest of this paper is organised as follows. Section II is devoted to
the analysis of the particle number and the chiral condensate, which are
related by the Maxwell equation. In Section III we discuss the behaviour of
the Polyakov loop. It turns out that our approach offers a particularly
simple description of an apparent puzzle, and, at the same time, gives
a direct evidence of the phase of the determinant at nonzero, real
chemical potential. The implication on the equation of state are summarised
in Section III, while Section IV discusses our results in the light of
phenomenological proposals, and alternative lattice approaches. Last Section
is a short summing up.
\section{Thermodynamics of the Hot Phase close to $T_c$}
Let us remind ourselves of the critical lines in the phase diagram in the T,
$\mu^2$ plane (Figure~\ref{fig:phaseDIAGR}): at high temperature there is the
Roberge Weiss transition at $\mu = \pi T/3$,
associated with the phase of the Polyakov Loop, ending at $T \simeq T_{RW}$.
At lower temperatures the QGP region is limited by a chiral transition
at negative $\mu^2$, which continues into the physical chiral transition
at positive $\mu^2$, i.e. real chemical potential~\cite{deForcrand:2002ci,D'Elia:2002gd}.
While $\mu$ approaches $\pi T/3$ at a constant temperature
$T \simeq T_{RW}$ the chiral transition
approaches the Roberg Weiss transition. Within the current numerical accuracy
the endpoint of the two transitions cannot be resolved, and the nature
of the critical behaviour around $T = T_{RW}, \mu = \pi T/3$ is
an interesting question in itself. If T is slightly larger than $T_{RW}$
we are approaching the Roberge Weiss transition, if slightly lower we hit
the chiral transition, and at $T = T_{RW}$ we might expect interesting
critical phenomena whose universality class is not known a priori.
Note that in the chiral limit the
$\mu=0$ transition should be of first order, and we do not expect
any tricritical point along the critical line at a real chemical potential.
A possible occurrence of
an endpoint at finite mass in the $T, \mu$ plane depends on dynamical
details, which are not known, and are not relevant for the present study.
We have then carried out simulations on a $16^3 \times 4$ lattice and four
flavor of staggered fermions at $\beta = 5.1$, which, according to
our previous results, yields $T \simeq T_{RW}$,
endpoint of the RW transition,. Fermions are fully degenerate,
with a
bare dimensionless mass ($a$ being the lattice spacing)
$\hat{m} \equiv m a = 0.05$.
For our lattice the value of the (dimensionless) $\hat{\mu} \equiv \mu a$
which is relevant for the Roberg Weiss transition reads $\hat{\mu} = \pi/12$
(the temperature being $T = 1/(a N_t)$ and in our case $N_t = 4$).
With a slight abuse we will omit in the following the
hat-notation, nevertheless
measuring $\mu$ and $T$ in unit of inverse lattice spacing.
First, we check our data for the particle number
against a simple free field behaviour. We have numerically computed
the free field results for real chemical potential
on a $16^3 \times 4$ lattice, and $m_q = 0.05$, and we have
fitted them to an expression motivated by one dimensional QCD~\cite{Bilic:1988rw},
which turns out to be an excellent parametrisation:
\begin{equation}
n(\mu)_{free} = \frac { 3 \sinh ( \mu / T) }{ K + \cosh (\mu / T)} .
\label{eq:nfree}
\end{equation}
yielding the free field results for the number density as a function
of imaginary chemical potential
\begin{equation}
n(\mu_I)_{free} = \frac { 3 \sin( \mu / T) }{ K + \cos(\mu / T)} .
\label{eq:nfreecont}
\end{equation}
We then considered the ratio between the numerical results and such free field
results $R_F (\mu_I) = n(\mu_I)/n(\mu_I)_{free}$ (Figure~\ref{fig:RATIO2free}).
We observe a clear dependence of $R_F (\mu_I)$ on $\mu_I$: the
results are qualitatively different from a free field
and the discrepancy cannot be accounted
for by any simple renormalisation of the degrees of freedom.
This behaviour should be contrasted with that of Fig. 12 of
Ref.~\cite{D'Elia:2004at} where the results at
high temperature did differ from a free field behaviour by a constant factor
very close to one.
\begin{figure}
\epsfig{file=effective_nf_OK.eps,width= 9 truecm}
\caption{$R_F (\mu_I) = n(\mu_I)/n(\mu_I)_{free}$ as a function of $\mu_I$ ,
showing a very clear evidence of a deviation from a free field behaviour}
\label{fig:RATIO2free}
\end{figure}
As a second attempt at interpreting our data in terms of the simplest
parameterisations, we consider an analogy with the Hadron Resonance Gas
model~\cite{Toublan:2004ks,Karsch:2003zq}.
We will discuss more fully these aspects in the last Section.
In a nutshell, the HRG describes the system as a gas of weakly
interacting resonances. The pressure of the HRG model reads:
\begin{eqnarray}
{{P (T, \mu) - P(T, 0)}\over {T^4}}&\simeq& F(T)(\cosh ({{N_c \mu_q}\over T})-1) \\
F(T)&\simeq &\int dm\rho (m) ({m\over T})^2 K_2({m\over T})
\end{eqnarray}
and the argument of the hyperbolic cosine, $N_c \mu_q$,
tells us that in the hadronic phase
one can only excite baryonic degrees of freedom.
Let us remind ourselves here that general arguments guarantee that the
partition function is periodic at imaginary $\mu$,
and that the strong coupling analysis shows that periodicity is smooth
at low temperature. Hence, the number density reads~\cite{D'Elia:2002gd}
\begin{equation}
n(\mu_I) = \sum_k bo_F \sin( k N_c N_t \mu_I)
\end{equation}
When HRG holds true, one term in the Fourier
series should suffice.
\begin{equation}
n(\mu_I) = \frac {\partial P(\mu)}{\partial \mu} = K(T) \sin(N_c N_t \mu_I)
\end{equation}
A cross check with the Hadron Resonance which uses the Taylor approach
requires the computation of an infinite number of derivatives~\cite{Karsch:2003zq},
while the Fourier analysis -- possible with the imaginary
chemical potential approach -- needs only one parameter fit.
In Figure~\ref{fig:Rratios} we display the ratios
\begin{eqnarray}
R_B( \mu_I) &=& \frac{n(\mu_I)}{\sin (3 \mu_I/T)} \\
R_q( \mu_I) &=& \frac{n(\mu_I)}{3 \sin (\mu_I/T)}
\end{eqnarray}
$R_B(\mu_I)$ should be a constant for a simple hadron
gas (cfr. again Ref.~\cite{D'Elia:2004at}), while, mutatis mutandis,
$R_q( \mu_I)$ should be a constant for a ``hadron gas'' made of quarks.
Both $R_B$ and $R_q$ stay constant only for a short interval of chemical
potential, indicating the region where $n(\mu_I) \propto \mu_I$.
The deviation from a linear behaviour (for $\mu_I > 0.05$) , as well
as from the simplest trigonometric parameterisations (for $\mu_I > 0.15$),
are evident from the plot.
\begin{figure}
\epsfig{file=compare_n_hadron.eps,width= 9 truecm}
\caption{$R_x(\mu_I)$ (defined in eqs. 2 and 3) as a function of $\mu_I$ ,
showing the limitations of a linear approximations, as well
as the deviations from the simplest trigonometric parameterisations
motivated by a Hadron Resonance Gas model}
\label{fig:Rratios}
\end{figure}
We now move on and propose to describe the
particle number in the critical region as
\begin{equation}\label{eq:critFIT}
n(\mu_I) = A \mu_I ({\mu_I^c}^2 - \mu_I^2)^\alpha \,.
\end{equation}
This ansatz for $n(\mu_I)$ takes into
account that $n(\mu)$ should be an odd function of $\mu$.
Moreover, it reproduces a singular behaviour of the quark number
susceptibility at a genuine critical point.
Namely, the most divergent
part of the quark number susceptibility $\chi_q$ behaves as
\begin{equation}\label{eq:chi}
\chi_q (\mu_I) \propto \frac{1}{({\mu_I^c}^2 - \mu_I^2)^\gamma}
\end{equation}
where $\gamma = 1 - \alpha $, while $\mu_I^c = \pi / 12$ if the
critical point coincides with the Roberge-Weiss line.
We then fit our data to Eq.~(\ref{eq:critFIT})
with $\mu_I^c$ either open or constrained.
A fit to our entire interval with unconstrained $\mu_I^c$ gives
$A = -0.94(4)$, $ {\mu_I^c}^2 = 0.0804(2)$, $\alpha = 0.28(2)$
with a reduced $\tilde\chi^2 = 2.4$. We checked the stability of these
results by choosing different ranges in chemical potential, and
we obtained the exponent $\alpha$ ranging from 0.34(8) and 0.26(3),
$ {\mu_I^c}^2$ between 0.078(4) and 0.091(12), with reduced $\tilde\chi^2 $
ranging between 1.8 and 5. (see Figure~\ref{fig:nMU} for two representative fits).
If we constraint
$ {\mu_I^c}^2 = (\pi/12) ^2 $ the quality of the fits decreases giving a reduced
$\tilde\chi^2 \simeq 12 $. If we limit the fitting interval
to $\mu_I > 0.15$, we need to add a constant to the function to approximate
the regular component in this interval to obtain a reduced $\tilde\chi^2 = 3.5$,
with $\alpha = 0.12(1)$. All in all, constraining $\mu_I^c$ does not
improve the results.
We now go back to the quark number susceptibility as entailed
in Eq.~(\ref{eq:chi}), for which our results
indicate $\gamma = 1 - \alpha \simeq 0.7$.
Figure~\ref{fig:chiQ} shows $\chi_q$ obtained
by numerical differentiation of $n(\mu_I)$. The numerical quality
is of course poor, but, anyway, a fit to the form of Eq.~(\ref{eq:chi})
with an open $\mu_I^c$ gives $\gamma = 0.66(16)$,
while a fit with constrained $\mu_I^c$ gives $\gamma = 0.44(22)$,
in agreement with the above estimates, within the large errors.
The chiral condensate $<\bar \psi \psi>$ can be inferred from
the chiral equation of state in either phases, and also in the presence
on an explicit chiral symmetry breaking term.
To make a closer contact with thermodynamics, we consider
the Maxwell relation (the temperature is constant and its dependence is
omitted)~\cite{Kogut:1983ia}:
\begin{equation}
\frac{\partial n(\mu, m)} {\partial m } =
\frac{\partial < \bar \psi \psi> (\mu,m) } {\partial \mu}.
\label{eq:max}
\end{equation}
Considering the $m$ dependence in the expression for $n(\mu,T)$
\begin{equation}
n(\mu, m) = F(m) \mu ({\mu_I^c}^2(m) - \mu^2)^\alpha
\end{equation}
where $F(m)$ depends only on the quark mass, and using Eq.~(\ref{eq:max})
we arrive at
\begin{eqnarray}
<\bar \psi \psi> (\mu) &=& H(m) ({\mu_I^c}^2(m) -
\mu^2)^{\alpha + 1} \\ & + & K(m) ({\mu_I^c}^2(m) - \mu^2)^\alpha +
<\bar \psi \psi>_0 \nonumber
\end{eqnarray}
where $H(m), K(m)$ depends only on the mass and
$<~\bar\psi\psi>_0$ is an integration constant
which can be fixed e.g. by the chiral condensate at
zero chemical potential.
We have then fitted the chiral condensate to the leading term
at $ \mu \simeq \mu_c$
\begin{equation}
< \bar \psi \psi> = K ( b - \mu^2)^{\alpha_c} + A
\end{equation}
obtaining a nice fit with a reduced $\tilde\chi^2 = 0.79 , A = 0.552(6) , b =
0.06628(8), K = -0.63(2)$ and $\alpha_c = 0.47 (2)$ in reasonable agreement
with $\alpha$ estimated from the number density. By constraining the
fitting interval $\mu_I > 0.2 $ the sub-leading contributions are less
important, and $\alpha_c = 0.32 (12)$ gets even closer to $\alpha$.
It might be interesting to compare this critical behaviour with that of
the endpoint of QCD from model field theories~\cite{Hatta:2002sj}.
\begin{figure}
\epsfig{file=nmu_finale.eps,width= 9 truecm}
\caption{$n(\mu)$ fitted to the form predicted by the simple critical
behaviour at imaginary $\mu$ Eq.~(\ref{eq:critFIT})}
\label{fig:nMU}
\end{figure}
\begin{figure}
\epsfig{file=chiallOk.eps,width= 9 truecm}
\caption{The quark number susceptibility obtained by numerical
differentiation of the results for the quark number}
\label{fig:chiQ}
\end{figure}
\begin{figure}
\epsfig{file=fitchiral.eps,width = 9 truecm}
\caption{Numerical results for the chiral condensate, and the results of
the fit to a functional form inferred from
the Maxwell equation Eq.~(\ref{eq:max}).}
\label{fig:chiral}
\end{figure}
\section{The Polyakov Loop}
In the same spirit we have fitted the traced Polyakov
loop $L = \mbox{Tr} P $ to
a power law form
\begin{equation}
L(\mu_I) \propto ({\mu_I^c}^2 - \mu_I^2)^\beta
\label{eq:crip}
\end{equation}
Figure~\ref{fig:ABSPoly} displays the results of the fit of the absolute
value of the Polyakov loop, which looks indeed satisfactory.
\begin{figure}
\epsfig{file=N_critical.eps,width= 9 truecm}
\caption{The Polyakov loop fitted to the form predicted by a simple
critical behaviour at imaginary $\mu$ Eq.~(\ref{eq:crip}). }
\label{fig:ABSPoly}
\end{figure}
It is interesting to look in more detail into the behaviour of
$L = \mbox{Tr} P$.
The Polyakov loop $P$ satisfies the same relation as the
quark propagator at nonzero chemical potential~\cite{Kogut:1994eq}
\begin{equation}
P(\mu) = P^\dagger (-\mu)
\label{eq:polsym}
\end{equation}
This relation implies that, while both $L = \mbox{Tr} P $ and $\bar L = \mbox{Tr} P^\dagger $
are real at real chemical potential, $L \ne \bar L$, as noted
in~\cite{Karsch:1985cb, Allton:2002zi, Dumitru:2005ng, Roessner:2006xn,
Kratochvila:2006jx}. We will show that the results at imaginary chemical
potential offer a particularly simple illustration of these ideas,
as well as a direct evidence of the complex phase of the determinant.
The asymmetry at real chemical potential
is easily understood by considering the distribution of the Polyakov loops
in the complex plane: for $\mu=0$,
since the $Z_3$ centre symmetry is broken by the
dynamical quarks, the root corresponding to the phase $\phi = 0$ is preferred,
and the average is non zero.
A non zero, positive chemical potential encourages
forward propagation: the distribution of the phases is further
peaked at $\phi=0$, while the two other phases have the same probability.
Hence, the Polyakov loop remains real, and the
final average is again real, different from zero, and slightly larger
than the one at zero density.
$\bar L$ instead describes backward propagation: again
the Polyakov loop remains real, however its length is reduced,
hence $L (\mu) \ne \bar L (\mu)$.
Notice that at a first na\"{\i}ve look it may sounds strange that,
while configuration by configuration the Polyakov loop and its
hermitian conjugate are always the complex conjugate of each other,
their expectation values, even being real, differ from each other. However it should be
clear that the complex nature of the functional integral measure plays
an essential role in this respect, since the real part of
the expectation value is not just the expectation value of the real
part. In that sense the fact that $L (\mu) \ne \bar L (\mu)$
is directly linked to the complex nature of the fermion determinant,
thus also giving a qualitative feeling about the severeness of the sign problem.
An apparent puzzle then arises when one considers the behaviour at imaginary
chemical potential: there the measure is real and one can show that
the absolute value of $L$ and $\bar L$ are equal as well as their real parts.
What is then the fate of the asymmetry which is present at real
chemical potential?
Consider
\begin{equation}
L_{o/e} (\mu) \equiv L (\mu) \pm L(-\mu) = L (\mu) \pm \bar L(\mu)\, ,
\end{equation}
where Eq.~(\ref{eq:polsym}) has been used in last equality.
$L_{o/e}$ are respectively even and odd in $\mu$. Remember
that the analytic continuation to imaginary chemical potential of
an even function is real, while the analytic continuation of
an odd function is purely imaginary.
Hence, the analytic continuation of the even observable
$ L_e(\mu) = L (\mu) + \bar L (\mu) $ at imaginary chemical
potential is the real part of $L (\mu_I)$; while the analytic
continuation of $ L_o(\mu) = L(\mu) - \bar L (\mu)$
is the imaginary part of $L (\mu_I) $ at imaginary $\mu$.
$L$ itself has no definite $\mu$-parity and
its analytic continuation develops an imaginary part.
We conclude that we must search for the analytic continuation
of the asymmetry $L \neq \bar L$ which is present at real chemical
potential in the imaginary part of the Polyakov loop, which is non-zero
in presence of an imaginary chemical potential.
Figure~\ref{fig:IMPoly} shows the imaginary part of $L(\mu_I)$:
it is different from
zero, offering
a clean, direct evidence of the asymmetry $L \neq \bar L$, hence of the
complex phase of the determinant
at real chemical potential. In the same Figure~\ref{fig:IMPoly}
we have also plotted
$n(\mu_I)$ (both $L$ and $n$ with an appropriate normalisation),
to show its correlation with $L(\mu_I)$. This correlation is
in agreement with the lattice interpretation of the number operator $n$
as 'counting' the links winding forwards minus those winding backwards,
and it should become exact in the heavy quark limit when the model
reduces to a Polyakov loop model~\cite{Blum:1995cb,DePietri:2007ak}.
\begin{figure}
\epsfig{file=n_impol.eps,width= 9 truecm}
\caption{The imaginary part of the Polyakov loop, divided by the coefficient
of the linear term of a fit at small $\mu$, as a function of the
imaginary chemical potential, demonstrating the relevance of the
phase of the determinant for a real chemical potential; in the same plot
we show the number density, again normalised by the coefficient of the
liner term. }
\label{fig:IMPoly}
\end{figure}
Let us now come back to the critical behaviour of $|L|$:
since Im$(L(\mu_I) \propto n (\mu_I)$, we might expect that $|L (\mu_I)| $
approaches zero with a similar power law.
The fit of Figure~\ref{fig:ABSPoly} gives an exponent of 0.45(2), in the same range as
$\gamma$. It would be interesting, of course, the study of the string tension
and other correlators in the same range of chemical potentials.
\section{Analytic continuation to real chemical potential}
Finally, we can analytically continue\footnote{The singularity in the complex $\mu$ plane might well limit
the radius of convergence of the Taylor expansion. However, the
proposed -- Pade' like'-- parametrisation does not suffer from this
problem~\cite{Lombardo:2005ks,Cea:2006yd}
and it should not come as a surprise that we are able to analytically
continue beyond the radius of convergence of the Taylor series: the
analytic continuation is then valid for all the real values of the chemical
potential $\mu$.} our results for n($\mu_I$), obtaining
\begin{equation}
n(\mu) = A \, \mu \, ({\mu_I^c}^2 + \mu^2)^{\alpha}
\label{eq:cont}
\end{equation}
with $\alpha \simeq .3$. The results are shown in Figure~\ref{fig:CONTnmu}.
In the same diagram we also plot the free field results, as an indicative
comparison.
Note that coefficient A has a non trivial dimension, again
indicating that the system is not free.
\begin{figure}
\hskip -0.5 truecm \epsfig{file=fitnew_immu_cont.eps,width= 9 truecm}
\caption{$n(\mu)$ from analytic continuation, together with
a free field behaviour is shown for comparison. The fits suggest that the
slower increase observed in the interacting case with respect to the free case
can be described by on overall exponent smaller than one}
\label{fig:CONTnmu}
\end{figure}
It is then amusing to notice that by using simple
arguments from the theory of critical phenomena we arrive at a modified
(lattice) Stefan-Boltzmann law, which would correspond to $ \alpha = 1$,
and a large ${\mu_I^c}^2 \simeq 0.5$. In this framework a large
$\mu_I^c$ can be interpreted
as a spinodal point at imaginary chemical far away from the Roberge Weiss line.
Obviously, Eq.~(\ref{eq:cont}) accounts for a slower increase of the
particle density closer to $T_c$ than in the free case. This is
expected on physical grounds, as well as
from the behaviour of the susceptibilities~\cite{Gottlieb:1988cq,Bernard:2002yd,Gavai:2003mf,Choe:2002mt,Gavai:2003nn,
Allton:2002zi}, and of course accounts
for the behaviour observed in Figure~\ref{fig:RATIO2free}.
From the results above, we conclude
that the data in the candidate region for a strongly coupled QCD
are accounted for by a conventional critical behaviour:
clearly, a free field behaviour would have been incompatible
with it.
In other words, the nonperturbative features of the plasma are closely
related with the occurrence of the critical line at negative $\mu^2$.
\section {Thermodynamics and particle content}
Early in this note we have contrasted our data with some
very simple parameterisations motivated by the
HRG approach. Here we discuss this point in some more detail,
in particular we wish to examine the proposal that in the deconfined,
strongly interacting region considered here
one might observe either coloured and colourless~\cite{deForcrand:2000jx}
bound states:
in short, at large T the QGP
is a gas of nearly free quarks,
which becomes strongly interacting at lower temperatures $T=(1-3)\,T_c$,
see e.g.~\cite{Shuryak:2006se,Blaizot:2006up} for recent reviews and
a complete set of references.
In Ref.~\cite{Ejiri:2005wq}
the following parametrisation was proposed for the
contribution of the coloured states to the subtracted pressure
$\Delta P_C = P_C (T, \mu) - P_C (T, 0)$
(we slightly simplify the notation):
\begin{eqnarray}
\frac{ \Delta P_C}{T^4} &= & \left( F_q(T)\right)
\left( \cosh(\mu_u/T) +\cosh(\mu_d/T) \right) \nonumber \\
&&+ F_{qq}(T) ( \cosh(2 \mu_u/T) +
\cosh(2 \mu_d/T) \nonumber \\
&&+ \cosh((\mu_u+\mu_d)/T) ) \, .
\label{eq:cologas}
\end{eqnarray}
The susceptibilities at zero chemical potential
can be easily computed from Eq.~(\ref{eq:cologas}),
and we recognise that their ratios allow the identifications
of the relevant degrees of freedom. These prediction for the susceptibilities
ratio was contrasted with the numerical results, finding a poor
agreement.
The imaginary chemical potential approach gives the possibility to
check directly the consistence of various phenomenological models
by analytically continuing
from real to imaginary $\mu$.
We can subject our data to the same analysis by
analytically continuing Eq.~\ref{eq:cologas} from real to
imaginary chemical potential. Setting
$\mu_{isospin} = \mu_u - \mu_d =0$, and including the contribution
from baryons and tetraquarks we get:
\begin{eqnarray}
\frac{\Delta P}{T^4} &=& F_q(T)\cos(\mu/T)) + F_{qq}(T) \cos(2 \mu/T)
\\
&+& F_{qqq}(T) \cos (3 \mu /T)+ + F_{qqqq}(T) \cos (4 \mu /T) \nonumber
\end{eqnarray}
giving in turn:
\begin{eqnarray}
n(\mu_I,T) &=&F_q(T) \sin (\mu_I/T) + 2 F_{qq} (T) \sin (2 \mu_I/T)
\\ & + &
3 F_{qqq} (T) \sin (3 \mu_I /T) + 4 F_{qqq} (T) \sin (3 \mu_I /T) \nonumber
\end{eqnarray}
From the point of view of the imaginary chemical potential analysis,
checking these forms correspond to perform a Fourier analysis of
our results.We have then fitted our data to the form
\begin{equation}
F_K(\mu_I) = {\sum_j}_1^K F_j \sin (j \mu_I/T)
\label{eq:FOURIERnmu}
\end{equation}
The results of the fits are shown in Figure~\ref{fig:nmuFOURIER}.
The reduced $\tilde\chi^2$ ranges
from 84 to 2.85 ($F_1$ to $F_4$) but the errors on the parameters
grow big and the parameters themselves are not stable. We summarise the
results in Table 1, and we conclude
that, even if the trigonometric fits might eventually converge, it is
hard to attach any simple physical interpretation of the parameters
$F_1, F_2, F_3, F_4$ as
contribution from free quarks, diquarks, baryons and tetraquarks.
\begin{figure}
\epsfig{file=nmuFourier.eps,width= 9 truecm}
\caption{Results of the fits of $n(\mu_I)$ to the trigonometric functions
Eq.~(\ref{eq:FOURIERnmu}), see text for details}
\label{fig:nmuFOURIER}
\end{figure}
\begin{table}
\caption{Parameters of the Trigonometric Fits}
\begin{tabular}{ccccc}
\hline
$F_1$ & $F_2$ & $F_3$ & $F_4$ & $\chi^2/d.o.f.$ \\
\hline
-0.110(1) & -- & -- & -- & 84 \\
\hline
-0.071(3) & -0.023(2) & -- & -- & 11.11 \\
\hline
0.028(15) & - 0.114(14) & 0.029(4) & -- & 4.18 \\
\hline
0.43(11) & -0.55(13) & 0.257(66) & -0.049(14) & 2.85 \\
\hline
\end{tabular}
\end{table}
This result is not unexpected, as the Fourier parametrisation
Eq.~(\ref{eq:FOURIERnmu}) is not compatible with the critical fits
Eq.~(\ref{eq:critFIT}).
Does this mean that the occurrence of coloured bound states is ruled out?
Not really: apart from the fact that
extending an Hadron Resonance Gas
description to coloured states is anyway a non-trivial assumption,
given the non-trivial interactions that one could expect.
Consider that the masses
themselves should depend on the chemical
potential~\cite{Liao:2005pa,Bluhm:2004xn} -
the so-called BKS effect. From the perspective of
the present study, the BKS effect is indeed very natural,
in view of the phase diagram in
Figure~\ref{fig:phaseDIAGR}, and the related analysis
of the data in term of critical behaviour which we have presented above.
Remember, in fact, that the coefficients of the Hadron Gas parametrisation
in Eq~(\ref{eq:cologas}) above
represent the contribution of the spectrum of resonances, hence their
temperature dependence is inferred from the one of the masses.
For instance, in Ref.~\cite{Shuryak:2004tx} it was proposed that
\begin{equation}
M_{\rm coloured}\approx 11.5\,T_c\left((T/3T_c)^{0.5}+0.1T_c/(T-T_c)\right)
\end{equation}
yielding the decoupling of the coloured masses at the critical point.
A similar decoupling should take place at the critical line at negative
$\mu^2$, which is not compatible with the simple factorisation of the
terms depending of temperature and fugacities implied by the
HRG model Eq~(\ref{eq:cologas}).
In Ref.~\cite{Liao:2005pa} it was underscored that
if the derivatives of the masses with respect to the chemical
potential
\begin{equation}
M '' (T) = \frac{ \partial ^ 2 M (T, \mu)}{\partial \mu ^2} (T, \mu = 0)
\end{equation}
are large enough, the simple interpretation of the zero chemical potential
susceptibilities as probes of particle contents has to be revised,
and, more generally, the decoupling of the prefactor and the simple
trigonometric factors predicted by the HRG model is no longer true. Hence,
we cannot use simple forms as those of
Eq~(\ref{eq:cologas}) to assess the particle content of the sQGP gas.
Our analysis supports this point of view.
It is interesting to note that indeed a recent direct calculation~\cite{Doring:2007uh} of the
coloured spectrum indicates the survival of heavy coloured states above $T_c$.
\section{Summary and Outlook}
We have studied the critical behaviour of the system in proximity
of the critical endpoint of the chiral and RW line in the negative
$\mu^2$. We have given a simple description of
the non-perturbative features of the sQGP phase, based on the analysis
of the critical behaviour in the imaginary $\mu$ plane.
We have proposed an EOS of the form
\[
n(\mu) = A \, \mu \, ({\mu_I}_c^2 + \mu^2)^\alpha
\]
with $\alpha \simeq 0.3$,
which accounts nicely for all the features of the numerical data. The exponent
would read $\alpha=1$ for a Stefann--Boltzmann-like law.
From a more mathematical point of view, the proposed parametrisation
is a Pade' approximant of order [2,1], as appropriate in the standard
application to critical phenomena. The results thus obtained can be,
in principle, analytically continued within the entire analyticity domain.
Practical limitations - discussed at length in previous work -
do arise because of numerical accuracy.
As for the particle content of the system, our results suggest that
a fit to
\begin{eqnarray}
\frac{\Delta P}{T^4} &=& F_q(T)\cos(\mu/T)) + F_{qq}(T) \cos(2 \mu/T)
\nonumber \\
&+& F_{qqq}(T) \cos (3 \mu /T)+ + F_{qqqq}(T) \cos (4 \mu /T) \nonumber
\end{eqnarray}
cannot afford any definite conclusion. This does not come as a surprise.
The masses themselves, hence the coefficients,
will depend on $\mu$ in some complicated way, which should anyhow
conjure to give the simple behaviour observed in the data.
The fact that the masses themselves should depend on the chemical
potential while approaching the critical endpoint
offers a simple realisation of the BKS mechanism.
It should also pointed out that extending a Hadron Resonance Gas
description to coloured states is anyway a non-trivial assumption,
given the non-trivial interactions that one could expect.
It would be very interesting to confront the numerical results in
a broader range of temperatures with these ideas, as well as
with analytic calculations and phenomenological models.
Future work should hopefully be able to give a coherent account of critical
behaviour, high temperature expansions and particle contents in the
region of the strongly interactive Quark Gluon Plasma.
We hope that the simple description offered here
might be of help in building such a complete picture.
\section{Acknowledgements}
We wish to thank Claudia Ratti, Burkhardt K\"ampfer, Aleksi Vuorinen,
Francesco Becattini and Philippe de Forcrand for helpful conversations
and correspondence. We acknowledge
partial support from the Galileo Galilei Institute in Florence,
under the {\em High Density QCD} program. The numerical simulations were
carried out on the APEmille computer
installed in Milano-Bicocca and we would like to thank our colleagues
in Milano and Parma for their kind support.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,031 |
Roy Halladay
Talking Baseball
February 25, 2011 February 26, 2011 zacharysteele 1 Comment
I just applied to become a full-time baseball nut. MLB.com is going to send someone to NYC for a full baseball season, where they will watch baseball all year (every game to some degree, every day), blog about it, vlog about it, tweet about it, yell about it, talk about it, be interviewed about it, and…well, you get the drift. I'm stoked. A full, non-stop, ridiculously busy year of baseball. To which of the baseball Gods do I need to beg and plead? Anyway, there was a two-part essay, and I thought I would share it here.
The first half, in 500 words or less, was a bit about myself and why I dearly love baseball so. This is what I wrote:
The day that I die, I will bequeath to this world a heart with one seam and two hundred and sixteen stitches. As it is, I'm quite certain that when I was born—I arrived one week early in late June of '72—I did so in a desperate need to avoid closing out the first half of the season in utero. No self-respecting baseball fan wants to be born during the All-Star break. I grew up on a diet of Reggie, complemented that as I aged with sides of Garvey and Cey, spent the glorious span of summer reliving the celebrated games of years past with a whiffle bat and tennis ball, and ultimately found there was no greater joy, no greater love than settling into an uncomfortable seat with a hot dog in one hand and a program in the other. I came alive as spring rolled in, overcoming what most people now refer to Seasonal Affective Disorder. I always just called the Offseason Blues. I lived in Florida. It wasn't cold. There just wasn't any baseball, and the internet wasn't even through Rookie League yet.
I wrote my first short story when I was twelve. It was a heroic tale about a young boy who twisted his ankle while walking to the championship game. It was a horrendous injury, one that left him certain there was no way he would make it to the game, much less play when he arrived. It was heart-wrenching. I poured my soul into that story, and cheered the boy on when he mustered the courage to fight through the pain, make his way to the field, and bring home the deciding run when all seemed lost. I was convinced this was the greatest tale ever told, and no moment in life would ever best it. Four years later, Kirk Gibson hit his limp-legged shot into the seats in the '88 World Series off Dennis Eckersley, and I wasn't entirely sold that he hadn't intentionally stole my thunder. Of course, it was historic, and I became less interested in vengeance with every fist-pump, every painful step he made around the bases. I let it slide, and decided I should at least make do with the chops I'd been given. I might not have to limp (though I could if I needed to impress the girls), but I could string the words together to someday write the best baseball story ever written.
There are no words to adequately express my love for the game. Now two books into a career as an impoverished author, I've decided the only reason I want to make Trump-town cash as a writer is in order to own a franchise. I never evolved as a player—though I've had quite the career in my mind—but I live and breathe this sport. I have to be involved in it every day, every year, and relish every moment of every game I see. I'm Gonzo. Baseball is my chicken.
Right. Part two asked what I believed this year's big story would be. And so sayeth I:
Albert Pujols and his forthcoming pile of Genie's gold is going to be in everyone's ear this year, whether he wants it that way or not. The Yankees are going to sob loudly in their room after being jilted at the Prom by the two-headed stud-monster of Cliff Lee and Andy Pettitte. Adam Wainwright is the latest in what now totals over 150 Major League pitchers who have had, or are scheduled to have, Tommy John surgery. Young phenom Bryce Harper is on the trail to projected glory, soon to join a promising future in the nation's capitol.
All of which will create a generous buzz between now and October. And yet, we're going to spend this year talking about four pitchers and what they mean to the history books, what they mean to the game, and what they mean to a franchise racing against time for one more run to glory. In Roy Halladay, Roy Oswalt, Cole Hamels, and Cliff Lee, the Philadelphia Phillies have a quartet striving to equal, or perhaps best, the pitching staffs of the '93 and '95 Braves, and the '71 Orioles.
In Halladay, Oswalt, and Lee alone, the Phillies have 3 of the top 5 pitchers in highest career winning percentage, with a minimum of 100 starts, in baseball history. They have two (Halladay 2.67, Lee 2.98) of the four pitchers over the last 3 seasons with sub 3.00 ERA's and 600+ innings. And Cliff Lee, well, all he's done over the past three years is rank 6th in wins (48), 7th in ERA (2.98), and 5th in IP (667.1). Toss in his stellar record in the postseason (7-2, 2.13), and his run in the second half of 2009 with Philly (7-4, 3.39, 4-0 in the playoffs), and you have reason to believe the Phillies have the making of something historically special. If that doesn't sparkle your fireworks, and if the idea of pitching in a notably hitter-friendly park makes you squeamish, it's important to note that of the top 6 ERA's in Citizens Bank Park, the Phillies now own 3 of them (Oswalt 2.10, Halladay 2.21, Lee 2.52).
The Phillies head into 2011 with the reigning NL Cy Young winner in Roy Halladay (ahem, no-hitter in the playoffs, ahem), a pitcher in Lee who only walked 18 men last year while striking out 185, Roy Oswalt, who only went 7-1 with a microscopic 1.74 ERA after being traded mid-season, and Cole Hamels, who may be a bit sporadic and reminds one a touch of Patrick Bateman from American Psycho, but had an ERA last season of 3.06 with 211 strikeouts in 208.2 innings.
There may be questions about the Phillies age, whether or not their bullpen can save a frog from jumping, and whether or not they can stay healthy enough, and score enough runs, to win a championship, but one thing is rock solid certain. Everyone is going to be talking about how this rotation stacks up against history.
I WANT THIS JOB! | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,808 |
Folklore is a relatively new term (1846) used to describe a tradition practiced since the beginning of time. Oral tradition was a necessity in society before the use of written communication and, afterward, when very few people were educated in its use. Generations translated their customs, religious beliefs, traditions, behaviors, and more through storytelling. This not only ensured the continuity of these cultural precepts, but was used as a form of entertainment. Today, these tales from all over Europe have survived, and still thrive in the halls of education and in stories for children.
Though the Medieval Period is often overlapping in its timeline with the Renaissance Period, many of today's medieval legends came out of both of these eras, and some began before the Christian Era.
Celtic Mythology grew from ancient lore, both Irish and Welsh, and was ordered into four cycles by Christian monks of the 12th century and later. These cycles, Mythological, Ulster, Fenian, and Kings "Historical," each deal with their own subject matter and contain stories relevant to each.
The Book of Invasions told of the supernatural visitors during the Mythological period, and included people, demons, and divine entities: Partholonians, Nemedians, Fomhóire, Fir Bholg, and Tuatha Dé Denann, and the Milesians.
The Ulster Cycle grew from the ancient lore about heroes in the northern province of Ireland, such as the tale of Connor MacNessa, as told in the Sons of Usnach and Tain Bo Cuailgne.
The Finian Cycle tales deal with Finn mac Cumhal, the leader of an Ireland-roaming warband under the reign of King Cormac mac Art. The stories include The Coming of Finn, The Quest of the Sons of Turenn, The Salmon of Knowledge, The Giant's Causeway, and many others.
While there are more than just legends and sagas from Ireland and Wales, the Welsh Mabinogian (fairy tales) resemble the stories in the Book of Invasions of Irish origin.
The Legend of King Arthur and the Knights of the Round Table characters are also of Celtic nationality, and so there is some mention of them in that mythology. The Arthurian Legends are many, and some of the wars within the tales give credence to the thought that King Arthur may have actually existed around the 6th century A.D.
Camelot was the geographical setting for King Arthur and his Knights of the Round Table, Gawaiin, Perceval, Lancelot, Galahad, and Tristan are as famous as the king himself. The Age of Chivalry they ushered in arrived after the Dark Ages and lent much romanticism to the stories. The French tales of Kid Charlemagne, known as the Song of Deeds, predated Arthurian legend, but have been incorporated into them in most literature as rival tales.
This English poem of old describes events which occurred in Sweden and Denmark and is dated somewhere between the 8th and 11th centuries A.D. It describes a history of epic adventures and is believed to have been handed down through oral tradition. The only manuscript is currently in the British Museum and has survived the destruction of the monasteries under King Henry VIII and the fire that destroyed the personal library of Sir Robert Bruce Cotton (1571 – 1631 A.D.).
This lengthy tale (over 1,300 lines long) depicts a Danish kingdom ravaged by a demon (Grendel) and a young Geatsman hero who comes and defeats the monster, unarmed, by ripping off its arm. Later, Grendel's mother seeks revenge on the village and Beowulf must, once again, destroy the monster. He succeeds in his battle and rids the kingdom of its final threat. The hero returns to his own country and later becomes king of the Geats and reigns for 50 years before another villain, a dragon, rears its head in his native land and he slays it. However, he is slashed in the neck by it and dies shortly after.
This ancient piece of literature remains a topic of study in modern education.
The Tales of Mother Goose, Grimm's Fairy Tales, Hans Christian Andersen, and many more, are stories that are mostly familiar to society today. A look into their history and other writers of children's fantasy reminds us of the oral tradition once used by the ancients and continued through present day parents, as they recite the familiar stories of Little Red Riding Hood, The Three Little Pigs, Rumplestiltskin, and other childhood tales.
Though most of these feel-good stories of moral value engage young children, a look at some of the original texts leaves parents questioning their telling as the story originated. The truth is that the original fairy tales were written for adults, and contained violence and unimaginable evils. Over the years, these stories have been altered to enable their telling to children, as they always had a moral truth in the underlying theme. Through their continued use in books, television, movies, and song, these stories of old have had their place in shaping moral thought in society. And as with all folktales, ancient lore, medieval legends, and fairy tales, they live on through the generations, in the art of oral tradition. | {
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Roger Craig 1988 Topps – Sully Baseball Card of the Day for October 18, 2017
One of the remarkable things about the way the world and information is set up now is in many ways, we can learn information about other places much quicker. There was a time when everything seemed to be regional with a few collectively shared experiences.
In terms of following sports, it takes virtually no effort to find out who plays for what team, what shows are popular where or anything about an area.
When my family moved from Massachusetts to California in 1987, in the pre-internet age, I might as well have been moving to a different country. In many ways it felt like I did.
Among the many things that I didn't know moving to the San Francisco Bay Area in 1987 was some of the details of the sports teams.
Sure, I knew Joe Montana was the quarterback of the 49ers, I knew who Mark McGwire and Jose Canseco was and the Giants still had a Clark at first base, but now it was Will, not Jack.
One thing confused me. One of the stars of the 49ers, whose team was filled with all time beloved fan favorites, was named Roger Craig.
The manager of the Giants? He was named Roger Craig.
Imagine moving to a new place and having THAT to sort through in your head? I met the 49ers Roger Craig a few times. Nice guy. Very funny and self effacing.
I have never met the former manager of the Giants. He seems like a nice guy. He looks like a classic old time manager, which I suppose is a euphemism for "old white guy." He would fit in Bull Durham, possible because he was from Durham, North Carolina.
As a player,he was part of several championship teams including the 1955 Brooklyn Dodgers, 1959 Los Angeles Dodgers and 1964 St. Louis Cardinals. He won a pair of World Series games as a starting pitcher and earned 3 rings.
Craig also pitched on the 1962 Mets, arguably the worst team of all time. Everything balances out.
Craig was the manager for the Padres in the 1970's and became Sparky Anderson's pitching coach in the 1984 World Series.
In 1985, he took over a disastrous Giants team that was coming off a 100 loss season. By 1986, they had a winning record and Roger Craig had his infectious optimism pour out onto the roster. He dubbed players who gave it their all "Hum Babies" and Hum Baby became a rallying cry.
Right around the time we arrived in California, the Giants passed the Reds in the standings and for the first time since 1971, the Giants had a post season team. He managed the Giants to Game 7 of the NLCS. Two years later, the Hum Babies won the National League pennant. The Giants were swept by Oakland in the Earthquake Series, but Craig turned a last place team into a pennant winner.
He remained the Giants manager through 1992 when he retired and Dusty Baker took over.
It is hard to imagine now a time when the Giants making the playoffs seemed so unlikely. But that had not played in October in MY life time until Roger Craig was at the helm.
And once the Giants packed up Candlestick, the 49ers would come in. Either way, there was a Roger Craig there.
Standard | Posted in Card of the Day, Uncategorized | Tagged Roger Craig, San Francisco Giants | 0 comments | {
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path: () => path.join(this.dir.path, 'kelex.json'),
get: () => JSON.parse(fs.readFileSync(this.kelex.dotJSON.path()).toString()),
update: (data) => {
let config = this.kelex.dotJSON.get()
_.merge(config, data)
fsx.writeFileSync(this.kelex.dotJSON.path(), JSON.stringify(config, null, 2))
}
}
}
this.files = {
copyFrom: (templateFolderPath) => {
fsx.copySync(templateFolderPath, this.dir.path)
}
}
}
get name () {
return appName(this.dir.path)
}
fileExists (fileName) {
const file = path.join(this.dir.path, fileName)
return fs.existsSync(file)
}
replacePlaceholders (placeholders) {
const files = rReadSync(this.dir.path)
files.forEach(file => {
if (!file.includes('node_modules') && !file.includes('.idea')) {
let contents = fs.readFileSync(file).toString()
_.each(placeholders, (value, key) => {
contents = contents.replace(new RegExp(`--${key}--`, 'g'), value)
fs.writeFileSync(file, contents)
})
}
})
}
}
function getAppConfig (targetDir) {
let contents = {}
for (let name of config.appConfigName) {
const filePath = path.join(targetDir, name)
if (fs.existsSync(filePath)) {
contents = JSON.parse(fs.readFileSync(filePath).toString())
break
}
}
return contents
}
function appName (targetDir) {
const pkgJsonPath = path.resolve(targetDir, 'package.json')
const appPkg = fs.existsSync(pkgJsonPath) ? require(pkgJsonPath) : {name: _.camelCase(targetDir)}
return appPkg.name
}
function removeNullProperties (obj) {
_.forOwn(obj, (value, key) => {
if (value === null) {
delete obj[key]
}
else if (typeof value === 'object') {
removeNullProperties(value)
}
})
}
module.exports = {
App,
getValidatedAppConfig: getAppConfig,
appName
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,927 |
Q: ColorDialog - How can I check if cancel or X has been pressed In this PowerShell code, the dialog opens but I cannot seem to find a way to know if the cancel or X button has been pressed.
If the user presses cancel then there's a returned value from the Color property.
How can I fix this so that it will return something different if the user presses cancel or X to close the dialog.
$colorDialog = new-object System.Windows.Forms.ColorDialog
[void]$colorDialog.ShowDialog()
# This will always return a value even when cancel is pressed.
$colorDialog.Color
It always returns this if cancel or the X button has been pressed:
R : 0
G : 0
B : 0
A : 255
IsKnownColor : True
IsEmpty : False
IsNamedColor : True
IsSystemColor : False
Name : Black
A: You need to inspect the result returned from ShowDialog():
$colorDialog = new-object System.Windows.Forms.ColorDialog
if($colorDialog.ShowDialog() -eq 'OK'){
$colorDialog.Color
}
else {
Write-Warning "No color was picked"
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 576 |
\section{Introduction}
A fundamental question in galaxy evolution is: what are the consequences of a $\Lambda$CDM cosmology on today's galaxies? Simulations based upon a $\Lambda$CDM cosmology find that major mergers, where large galaxies of roughly equivalent mass merge, were common in the early Universe. These dramatic encounters destroy the underlying structures of the galaxies undergoing the merger, and cause an entirely new type of galaxy to form \citep{Hopkins+06, Chilingarian+10, Borlaff+14, Sonnenfeld+14}. However, while major mergers may have governed galaxy evolution for massive galaxies, these mergers are rare in the nearby Universe. The driving force for the evolution of today's intermediate mass galaxies is from minor mergers, where satellite galaxies are accreted on to host galaxies without destroying the underlying structure of the host. Simulations of minor mergers between a massive spiral and a dwarf satellite find observable evidence for the merger: long-lived tidal streams (a dwarf in the process of being accreted), and a stellar halo \citep[the remnants of multiple mergers that form a halo surrounding the disc;][]{Bullock+Johnston05, Purcell+07, Cooper+10, Purcell+11, Cooper+13}.
The search for the signs of minor mergers, streams and haloes began in the Milky Way. The earliest discovered stream was the gaseous stream from the Large and Small Magellanic clouds \citep{Wannier+72, Mathewson+74}. Subsequently, stellar streams have been detected coming from the Sagittarius \citep{Ibata+94} dwarf galaxy. This led to the discovery of a multitude of streams surrounding the Milky Way \citep[e.g.][]{Belokurov+06, Martin+14}. In the Milky Way's stellar halo, the largest globular clusters, such as $\omega$~Centauri, are thought to be the cores of accreted satellites \citep[e.g.][]{Johnson+10, Marino+10}. These large clusters host multiple stellar populations, which cannot be formed in a simple Milky Way cluster \citep{Mackey+10, Milone+10, Gratton+12}.
Despite the success with the Milky Way, the detection of extragalactic stellar streams and haloes has proven difficult with ground-based intruments. The Milky Way studies have relied on detecting very faint individual stars, a task that is next to impossible for extragalactic sources from the ground without an 8~m class telescope. Thus, most ground-based extragalactic studies beyond the Local Group are limited to studying the surface brightness of streams and haloes. This remains a difficult task; simulations find that the surface brightnesses required to detect streams and haloes are below 28~AB~mag~arcsec$^{-2}$ \citep{Purcell+07, Cooper+10, Cooper+13}. Reaching these extreme depths is complicated by the limited amounts of time available on research-grade telescopes, difficulties in obtaining accurate flat fields, and the extended wings of the point spread functions of reflecting telescopes. The works of \cite{Martinez-Delgado+08, Martinez-Delgado+09, Martinez-Delgado+10, Martinez-Delgado+14} used a small telescope with a clear filter (essentially Sloan $g$, $r$, and $i$ combined) to detect the tidal features of nearby galaxies. Subsequently, these detections justified follow-up observations with larger telescopes \citep{Chonis+11, Martinez-Delgado+12}; these studies derived the physical properties of the tidal streams such as their masses and star formation histories. Many of the above issues with low surface brightness studies are solved by using an array of refracting telescopes. The robotic Dragonfly array \citep{Abraham+vanDokkum14} is currently surveying, to very low surface brightness, a large number of nearby galaxies. Interestingly, \cite{Abraham+vanDokkum14} found only a hint of a stellar halo around M101, down to $\mu_g\sim32$~AB~mag~arcsec$^{-2}$. Finally, by deprojecting, and then stacking, many thousands of SDSS galaxies, \cite{D'Souza+14} have detected the average stellar halo of spiral galaxies with masses from $10^{10}-10^{11}~M_{\odot}$.
Space-based telescopes may be the best way to detect stellar haloes and streams. The Galaxy Halos, Outer disks, Substructure, Thick disks and Star clusters (GHOSTS) survey \citep{Radburn-Smith+11} is investigating the haloes of several nearby spirals with the Hubble Space Telescope. These data have successfully detected the stellar halo of NGC~253 \citep{Bailin+11}, and a mass for the stellar halo has been derived. With the cryogen of the Spitzer Space Telescope exhausted, the telescope has entered a warm mission with only the 3.6~${\rm \mu}$m\ and 4.5~${\rm \mu}$m\ cameras active. These bands are excellent for low stellar mass surface density studies of older stellar populations down to $\sim$0.2~$M_{\odot}~pc^{-2}$ \citep{Krick+11}. This is due to the insensitivity of the 3.6 and 4.5~${\rm \mu}$m\ $M/L$ ratio of the older stellar populations to the effects of extinction, metallicity, and stellar evolution \citep{Bell+deJong01, Oh+08, Meidt+14}. The majority of the uncertainty in the 3.6~${\rm \mu}$m\ $M/L$ ratio comes from contaminants such as emission from polyaromatic hydrocarbons and short-lived asymptotic giant branch stars. Several investigations \citep{vanZee+09, Courteau+11, Krick+11, vanZee+11, Barnes+14, Bush+14} have taken advantage of the warm mission to conduct ultra-deep low surface brightness studies. In particular, \cite{Courteau+11} have measured the mass of the stellar halo in the Andromeda galaxy, and \cite{Barnes+14} have measured the physical properties of a tidal stream near M83.
In this work we analyse the near-infrared (NIR) low surface brightness properties of M63 (NGC5055), a member of the Extended Disk Galaxy Exploration Science (EDGES) Survey \citep{vanZee+11}. Our 3.6~${\rm \mu}$m\ and 4.5~${\rm \mu}$m\ mosaics showcase the results of a minor merger between M63 and a nearby satellite. There is a clearly visible tidal stream and stellar halo associated with this galaxy. We have measured the properties of the stream and stellar halo that we compare to models based on hierarchical galaxy evolution in a $\Lambda$CDM universe.
\section{Near-Infrared Imaging}
\label{sec:nir}
The hallmark of the EDGES survey is ultra-deep (29 AB~mag~arcsec$^{-2}$) wide-field ($>$5 $R_{25}$) Spitzer near-infrared (NIR) mosaics of 92 nearby galaxies. These data were taken during Spitzer's Cycle 8, a warm mission Exploration Science program (van Zee et al. 2011; PID 80025). In the following subsections we detail the steps taken to construct the 3.6~${\rm \mu}$m\ and 4.5~${\rm \mu}$m\ mosaics for M63.
\subsection{Maps}
\label{sec:maps}
The extent of the EDGES M63 mosaic, ($>$5 $R_{25}$), is 31\farcm5 \citep{deVaucouleurs+91}; significantly larger than the extent of the IRAC field of view, $5\farcm2\times5\farcm2$. Therefore, the observing program for M63 was designed to create mosaics from a grid-like mapping pattern of many individual dithered pointings. This mapping pattern was set up to have the 3.6~${\rm \mu}$m\ mosaic centred on the target, with the 4.5~${\rm \mu}$m\ data being taken concurrently. The 4.5~${\rm \mu}$m\ channel is separated from the 3.6~${\rm \mu}$m\ channel by 6$\farcm$5 and thus the 4.5~${\rm \mu}$m\ map is slightly offset from the 3.6~${\rm \mu}$m\ map. The M63 mosaic is composed of 2,568 individual exposures to reach our target depth of 1800~s~pixel$^{-1}$ while minimizing effects of saturation. The images were taken in four Astronomical Observation Requests (AORs), sets of 400 to 800 exposures on March 29, 2012 and August 25, 2012 (Keys: 44244992, 44245248, 44246016, and 44246272). During each date two AORs were taken in quick succession, with a break to begin a new AOR. This strategy assists in asteroid and artefact removal, and also extends the 4.5~${\rm \mu}$m\ coverage by allowing the 4.5~${\rm \mu}$m\ images to be taken at different telescope roll angles.
\subsection{Data Processing}
\label{sec:data_proc}
Before mosaicking, a number of pre-processing operations are applied to the individual frames to remove artefacts common to warm mission data. The first correction removes slew residuals caused by tracking over bright sources without a shutter. These slew residuals create sub-structure in every frame of an AOR. This structure is removed by subtracting a median-frame (constructed by stacking every frame in the AOR into a 3D data cube and taking a median down the frame number axis) from every image in the AOR.
Bright sources also create a complex negative bias in the bright source's columns, the so-called ``column pulldown effect''. In the cryogenic mission this effect creates a single-valued bias over an entire column which may be reliably removed. However, in the warm mission this effect is more difficult to remove as the bias value depends on distance from the centre of the source. The Spitzer Science Center and various third parties have published algorithms which correct for the warm mission column-pulldown, but they are not optimized for the EDGES dataset, which contains diffuse structure and longer exposure times than average. We have developed a pulldown corrector specifically for EDGES. Our pulldown corrector first identifies pulled-down columns by flagging columns with bright sources. Then, it estimates the true value of the column as the average of the flanking columns' rows. This estimated column is then subtracted from the actual column. The result is the functional form of the column pulldown. This result is fit with a linear function and the linear function is subtracted from the actual column, which corrects the pulldown. By using our pulldown corrector along with the Spitzer Science Center's corrector, the majority of column pulldown artefacts are removed automatically. The remaining pulled-down columns are manually flagged and removed. The damage done to the mosaic by their removal is minimized by the excellent coverage of EDGES, where most pixels contain data from 18 separate pointings.
In addition to the column bias issues, there are two sources of frame-wide bias. As discussed in \cite{Krick+11}, the overall bias level of the individual frames depends on the delay time between exposures, the so-called ``first frame effect''. The AORs for M63 have enough frames to measure this effect for each of the four AORs individually. This is accomplished by measuring the background level of every frame with the IDL task {\tt SKY} and fitting a logarithmic profile to the background as a function of delay time with the IDL task {\tt MPFITFUN} \citep{Markwardt09} (see Figure~\ref{fig:first_frame}) of the form:
\begin{equation}
c_{\rm ff}(t_{\rm delay})=c_0{\rm log}(t_{\rm delay})+c_1,
\end{equation}
where $c_{\rm ff}$ is the bias level due to the ``first frame effect'', $t_{\rm delay}$ is the delay time between exposures, and $c_{0,1}$ are the free parameters in the fit. This correction is applied to every frame in an individual AOR.
\cite{Krick+11} also found that the bias level depends on frame number; as an AOR progresses the bias level of individual frames increases. This effect is greatest for the first few frames, and again the M63 AORs are large enough to measure this effect for each AOR (see Figure~\ref{fig:buildup}). Following \cite{Krick+11}, this effect is measured by fitting roots to the background as a function of frame number with the IDL task {\tt MPFITFUN}, of the form:
\begin{equation}
c_{\rm b}(n_{\rm frame})=c_0n_{\rm frame}^{1/4}+c_1n_{\rm frame}^{1/3}+c_2n_{\rm frame}^{1/2},
\end{equation}
where $c_{\rm b}$ is the bias level due to frame buildup, $n_{\rm frame}$ is the frame number, and $c_0$, $c_1$, and $c_2$ are the free parameters in the fit. This correction is applied to every frame in an individual AOR.
Once these artefacts have been removed, the AORs are processed individually with the standard MOPEX pipeline. The plate scale is set to 0.75~arcsec~pixel$^{-1}$, and the AORs are set to a common astrometrical solution. The diffuse background of each AOR is fit as a plane gradient to manually defined regions. This plane gradient is subtracted from each AOR, and because of the common astrometrical solution, the AORs are simply averaged to produce the final mosaic.
Due to the lack of short exposures, the central portions of the M63 mosaics suffer from saturated pixels. To remedy this, archival data from the Spitzer Infrared Nearby Galaxies Survey are used to create a mosaic with the same astrometric solution as the EDGES mosaic. The saturated pixels in the EDGES mosaic are replaced with the archival data after matching background levels.
\section{Analysis}
\label{sec:analysis}
M63 (see Figure~\ref{fig:mosaic}) is a flocculent spiral at 7.9~Mpc \citep{Tully+09, Nasonova+11}. It features an upbending radial profile \citep[][see also Figure~\ref{fig:surf}]{Chonis+11}, which may be a stellar halo \citep{Purcell+07, Purcell+11, Cooper+13}. Additionally, there is evidence of an ongoing accretion event with the presence of a tidal stream \citep{vanderKruit79, Chonis+11}. These features make M63 an excellent test bed for the power of the EDGES survey. We measure the surface brightness profile of M63 and convert that to a mass density profile. We also fit S\'ersic functions to the data and measure the masses of the individual components of M63, the bulge and two discs. In addition, we fit a S\'ersic disc and bulge along with a power-law halo component. The nearby tidal stream is present in the 3.6~${\rm \mu}$m\ mosaic, and we measure the physical properties of the stream, including the stellar mass of the progenitor galaxy, and the time since the progenitor was disrupted.
\subsection{Surface Brightness Profiles}
\label{sec:surf_bright_prof}
We present the 3.6~${\rm \mu}$m\ surface brightness profile for M63 in Figure~\ref{fig:surf}. Before the surface brightness profile was measured, the mosaic was prepared by removing point sources and smoothing. The point sources were removed in an iterative process, starting with a H{\" o}gbom CLEAN algorithm, then masking with SExtractor, and finally removing any remaining sources by hand. The smoothing was accomplished with a Gaussian filter with a standard deviation of 6~pixels, with the IDL function {\tt GAUSS\_SMOOTH}.
After the mosaic was prepared, the surface brightness profile was measured by fitting elliptical isophotes to the data using the IRAF task {\tt ELLIPSE} for the inner 9$\arcmin$. Beyond 9$\arcmin$, {\tt ELLIPSE} fails to find a solution due to low signal to noise. The surface brightness profile beyond 9$\arcmin$ is measured with fixed elliptical annuli defined by the final isophote from {\tt ELLIPSE}. This isophote has an axial ratio of 0.62, position angle of $103$\degr\ east of north, with a major axis of 9\arcmin.
Our surface brightness profiles are aperture corrected with the calibration from the Spitzer Science Center's IRAC instrument handbook \cite[see][]{Dale+09}. This correction takes the form:
\begin{equation}
\frac{f_{\rm true}}{f_{\rm measured}}=A \times {\rm exp}(-R^B)+C,
\end{equation}
where $f$ is the flux density measured within an aperture, $A$, $B$, and $C$ are constants, and $R$ is the radius of the aperture. To use this correction with the annuli of the surface brightness profile, we measure the annuli as the difference between two apertures with a difference in semi-major axis of one pixel.
To convert the surface brightness profile from MJy~sr$^{-1}$, calibrated from the MOPEX pipeline, into a Vega-magnitude based surface brightness, we use the calibration from \cite{Reach+05} ($c_m^{3.6}=$280.9$\pm$4.1~Jy). Furthermore, we calculate a mass surface density via:
\begin{equation}
\Sigma[M_{\odot} {\rm pc^{-2}}] = \Upsilon_{\star}^{3.6} 10^{-0.4(\mu_{3.6} - M_{\odot}^{3.6} - 21.572)},
\end{equation}
where $\Sigma$ is the mass surface density, $\Upsilon_{\star}^{3.6}=0.5$ is the stellar mass-to-light ratio at 3.6~${\rm \mu}$m\ \citep{Oh+08, Eskew+12, Barnes+14, Meidt+14}, $\mu_{3.6}$ is the measured surface brightness in Vega~mag~arcsec$^{-2}$, and $M_{\odot}^{3.6}=3.24$ is the absolute magnitude of the Sun at 3.6~${\rm \mu}$m\ \citep{Oh+08}, and 21.572 is the factor to convert from per square arcseconds to per square parsecs. See Figure~\ref{fig:surf} for the result. Note that the uncertainties are the RMS value reported by {\tt ELLIPSE}, or the standard deviation within the annulus for data beyond 9\arcmin.
\subsection{Profile Fitting and Mass Estimates of Fit Components}
\label{sec:prof_fit}
We have fit two models to our surface brightness profiles; the first is a S\'ersic bulge with two S\'ersic discs, and the second is a S\'ersic bulge and disc with a halo component described by a power law. For the first model we use the following equation \citep{Sersic63, Sersic68} for each individual component:
\begin{equation}
S(R)=S_0{\rm exp}[-(R/h)^{1/n}],
\end{equation}
where $S$ is surface brightness in MJy~sr$^{-1}$, $R$ is the radius in arcmin, $h$ is the scale length in arcmin, and $n$ is the S\'ersic index. We use standard S\'ersic indices: $n=4$ for the bulge, and $n=1$ for the two disc components. The IDL Levenburg-Marquardt fitting algorithm {\tt MPFITFUN} \citep{Markwardt09} fits the model to the data. Table~\ref{table:fit} summarizes the results of the profile fit, and Figure~\ref{fig:disk_fit} shows these fits. Additionally, to illustrate that the profile requires two discdisc components, Figure~\ref{fig:surf} contains a local scale length profile of M63. These local scale lengths are computed by fitting a single disc component to five consecutive points on the surface brightness profile. From the local scale length profile, and the fit, M63 at 3.6~${\rm \mu}$m\ is best described by a two disc up-bending profile, where the slope increases at the bend radius, with the bend at $\sim$4\farcm5.
Up-bending breaks at large radii may also result from the detection of an inner stellar halo. For this scenario we fit our surface brightness profile with a S\'ersic bulge and disc, and the {\it U}-model from \cite{Courteau+11}, a power law designed to measure the inner stellar halo:
\begin{equation}
S_h(R)=S_*\left\{\frac{1+(R_*/a_h)^2}{1+(R/a_h)^2}\right\}^{\alpha},
\end{equation}
where $S_h$ is the surface brightness of the inner halo component in MJy~sr$^{-1}$ at a radius $R$, $R_*$ is the turnover radius (30~kpc), $\alpha=1.26\pm0.4$, $a_h=5.2\pm0.16$~kpc and $S_h(R_*)=S_*$. We fit this model as above. The results of this fit are found in Table~\ref{table:fit} and Figure~\ref{fig:halo_fit}.
From our model fits we measure the mass of the individual components of M63. For the S\'ersic components this is accomplished by first integrating over $R$ to find the total flux density, from \cite{Graham+Driver05}:
\begin{equation}
f(<R)=2 \pi \frac{b}{a} S_0 h^2 n \gamma(2n,x),
\end{equation}
where $f(<R)$ is the total flux density up to a radius $R$, $a$ and $b$ are the semi-major and semi-minor axes respectively, $\gamma$ is the incomplete gamma function, and $x=(R/h)^{1/n}$. We use the average $b/a$ found by our ellipse fitting, and $R=200$~kpc. To find the flux density of the power law fit we use the following from \cite{Courteau+11}:
\begin{equation}
f_h(<R) = 2 \pi \frac{b}{a} S_* R_*^2 \frac{1+a_*^2}{2(\alpha-1)} \times \left\{\left(\frac{1+a_*^2}{a_*^2}\right)^{\alpha-1} - \left(\frac{1+a_*^2}{s_{max}^2+a_*^2}\right)^{\alpha-1}\right\},
\end{equation}
where $a_*=a_h/R_*$ and $s_{max}=R/R_*$. To convert the flux densities to a mass we use:
\begin{equation}
M_{\rm total}[M_{\odot}]= \Upsilon_{\star}^{3.6}\frac{f{\rm[Jy]}}{c_m^{3.6}\rm[Jy]} \times 10^{0.4(\mu + M_{\odot}^{3.6})},
\label{eq:mass_conversion}
\end{equation}
where $M_{\rm total}$ is the total mass, $\Upsilon_{\star}^{3.6}=0.5$ is the stellar mass-to-light ratio at 3.6~${\rm \mu}$m\ \citep{Oh+08, Eskew+12, Barnes+14, Meidt+14}, $c_m^{3.6}=280.9$~Jy is the 3.6~${\rm \mu}$m\ calibration factor to Vega-magnitudes \citep{Reach+05}, $\mu=29.49 \pm 0.35$~mag is the distance modulus \citep{Tully+09, Nasonova+11}, and $M_{\odot}^{3.6}=3.24$~mag is the absolute magnitude of the Sun at 3.6~${\rm \mu}$m\ \citep{Oh+08}. See Table~\ref{table:fit} for the results.
\subsection{Tidal Stream}
\label{sec:tidal_stream}
M63 hosts a nearby tidal stream, first discovered by \cite{vanderKruit79}, and recently studied in-depth by \cite{Chonis+11}. The portion of the stream unobscured by the light from M63's disc and halo is fit well by an ellipse, as seen in Figure~\ref{fig:mosaic}. This ellipse has a position angle of 75\degr\ east of north, an axial ratio of $b/a=0.64$, and a semi-major axis of $a=13\farcm5$ (31~kpc at our fiducial distance to M63 of 7.9~Mpc). To measure the width of the stream, we have defined three rectangular apertures over three sections of the stream, seen in Figure~\ref{fig:mosaic}. To produce a 2-D profile of the stream, the rows perpendicular to north are averaged, see Figure~\ref{fig:box}. A three-parameter Gaussian (peak, position, and standard deviation) and a linear function is then fit to the 2-D profile. From the average of the three apertures, the stream has a peak of $2.0 \pm 0.3$~kJy~sr$^{-1}$, with a standard deviation of $42 \pm 10 \arcsec$. Following the example of \cite{Barnes+14}, where the width of the stream is 20\% of the maximum, results in a width of $75 \pm 18 \arcsec$, or $2.9 \pm 0.7$~kpc.
To measure the total mass of the stream we employ three methods. For the first, we simply measure the total surface brightness within a polygonal aperture, and convert the result to a mass using Equation~\ref{eq:mass_conversion}. To account for the missing light from masked stars and galaxies, the result is multipled by the ratio of total area to missing area (1.29), the total mass is $1.2 \pm 0.5 \times 10^8 M_{\odot}$.
For the second method, we use the ellipse and width of the stream measured above. If the stream is circular with a Gaussian profile, then the total light of the stream is equal to the integrated light within one Gaussian width element multiplied by the circumference of the circle. This is:
\begin{equation}
f_{\rm total}=S_{\rm max}\sigma(2\pi)^{3/2}\eta R,
\end{equation}
where $f_{\rm total}$ is the total flux density of the stream in Jy, $S_{\rm max}$ is the surface brightness at the peak of the Gaussian fit in Jy~sr$^{-1}$, $\sigma$ is the standard deviation of the Gaussian fit in radians, $\eta$ is the number of times the stream loops, and {\it R} is the radius of the circular orbit in radians. With the average standard deviation of the Gaussian fit to the stream, and with the semi-major axis of the ellipse as the radius, we find a mass of (5$\pm 2) \eta \times 10^8 M_{\odot}$.
The final method used to determine the mass of the stream is the dynamical method of \cite{Johnston+01}. This method assumes that the stream's orbit is circular and the stream is within a logarithmic potential. From \cite{Johnston+01}:
\begin{equation}
M_{\rm total}[M_{\odot}] \sim 10^{11} \left(\frac{w}{R}\right)^3 \left(\frac{R_p}{{\rm 10~kpc}}\right) \left(\frac{v_{\rm circ}}{200~{\rm km~s^{-1}}}\right)^2~M_{\odot},
\end{equation}
where {\it w} is the width of stream at the deprojected radius {\it R}, $R_p$ is the pericentre radius of the stream, and $v_{\rm circ}$ is the circular velocity of the host galaxy. The width at 20\% of the maximum is $75 \pm 18 \arcsec$, described above. The circular velocity is 180~${\rm km~s^{-1}}$ \citep{Bosma78, Battaglia+06}. With our assumption that the stream's orbit is circular $R_p=R$, and $R=31$~kpc, shown above. This results in a dynamically derived estimate of the stream's total mass of 2.3$\pm 1.6 \times 10^8 M_{\odot}$.
In addition to an estimate of the mass of the stream, we also calculate the time since disruption with the derivation from \cite{Johnston+01}:
\begin{equation}
t \sim 0.01 \Phi \left(\frac{R}{w}\right) \left(\frac{R_{\rm circ}}{{\rm 10~kpc}}\right) \left(\frac{{\rm 200~km~s^{-1}}}{v_{\rm circ}}\right){\rm Gyr},
\end{equation}
where {\it t} is the time since disruption, $\Phi$ is the angular extent of the stream, and $R_{\rm circ}$ is the circular radius of the stream. Assuming that the stream is circular with some number of loops, $\Phi=2\pi\eta$. The remaining parameters are described above. We find a time since disruption of (2.3$\pm$0.7)$\eta$~Gyr, which agrees with the result of \cite{Chonis+11} of $\sim$1.8$\eta$~Gyr.
\section{Discussion}
\subsection{Inner Stellar Halo}
The surface brightness profile of M63 reveals an extended, low surface brightness feature in the form of an up-bending break beyond $\sim$15~kpc. These up-bending breaks are not unusual for galaxies of M63's Hubble Type \cite[$T=4$,][]{deVaucouleurs+91}; \cite{Pohlen+Trujillo06} have found that $\sim$50\%\ of local SDSS galaxies with $T=2.5-4.4$ feature up-bending breaks), but what could have caused this break?
Classical surface brightness profiles feature either a single disc component \citep{Patterson40, deVaucouleurs59, Freeman70}, or a truncated, down-bending, profile \citep{vanderKruit79, vanderKruit87}. These results are reproduced in simulations of disc galaxies in solitary environments. The simulations work on the principles that gas forms stars after reaching a critical density threshold per the Kennicutt-Schmidt Law \citep{Kennicutt98}, stars may migrate from their original positions through interactions with spiral arms, and that there are no mergers over the galaxy's lifetime. These assumptions result in discs with down-bending breaks at large radii; due to breaks in the gas-density profile \citep{Roskar+08a, Sanchez-Blazquez+09, Martinez-Serrano+09}, or by stars scattered from interactions with spiral arms \citep{Roskar+09}. The only time a solitary disc may naturally form an up-bending break is in the presence of a varying galaxy cluster potential \citep{Moore+96, Moore+99}, or in a dark matter halo with low angular momentum \citep{Herpich+15}. M63 is not within a galaxy cluster, and while there are no measurements of the angular momentum of M63's halo, the simulations that found up-bending breaks from these potentials also have large bulges, which is not seen in M63. Thus far, simulations of galaxies evolving in solitary environments cannot fully reproduce what we observe with M63.
Simulations and analytical models of galaxy mergers in a $\Lambda$CDM universe find that stellar haloes are produced when satellite galaxies are accreted on to larger galaxies. The signature of these haloes appear in surface brightness profiles in the form of up-bending breaks at radii of 15-20 kpc \citep{Tissera+14}, and at surface brightnesses dimmer than $28$~AB~mag~arcsec$^{-2}$ \citep{Purcell+07, Cooper+10, Purcell+11, Cooper+13}; the same radial, and surface brightness, regime where we find the up-bending break in M63. The presence of stellar haloes at these radii and surface brightness is also confirmed observationally for the Milky Way \citep{Carollo+10}, M31 \citep{Courteau+11}, and M81 \citep{Monachesi+13}.
Direct measurements of the observed properties of M63 also suggest that the up-bending break is the signature of a stellar halo. By stacking many late-type SDSS galaxies into a single mosaic, \cite{D'Souza+14} have detected an average stellar halo. They find that the $g-r$ colour profile of the stacked galaxies decreases with radius, until it reaches a break point, where the colour reddens sharply. This is seen in the $B-R$ image of M63 from \cite{Chonis+11}, where the average $B-R$ colour jumps from $\sim$0.8 at the outermost edge of the disc to $\sim$1.2--1.4 beyond the 15~kpc break radius. This reddening suggests a change from a young stellar population within a disc, to older stellar populations within a halo. We assume colour is an analog of age at large radii because metallicity gradients have been found to flatten beyond $R_{25}$ \citep{Bresolin13, Bush+14, Kudritzki+14}. A halo scenario is also supported by the far-ultraviolet map of M63 from \cite{Thilker+07}. This map finds no far-ultraviolet emission past 15~kpc to a star formation rate density of $3\times10^{-4}M_{\odot}$yr$^{-1}$kpc$^{-2}$, the minimum threshold for star formation \citep{Kennicutt98}. While these stars may have formed as part of a disc in the past, this is unlikely as simulations have found that the surface density of discs rapidly declines past the star formation threshold \citep{Roskar+09}. Additionally, there is no detected spiral structure in the $B$, $R$, 3.6~${\rm \mu}$m, and 4.5~${\rm \mu}$m\ data, beyond 15~kpc. The HI maps of \cite{Battaglia+06} contain spiral structure past 15~kpc, however, this structure is inclined in respect to the optical and infrared data in the same radial regime. We therefore assume that the HI gas is not associated with the optical and infrared emission. Both simulations and observational evidence points to the break as the signature of a stellar halo, rather than an up-bending break in a disc. We therefore assume that our halo-model power-law fit is the correct decomposition method for M63.
With the assumption that the break is due to a stellar halo, we may compare our mass measurement of the halo to simulations based on a $\Lambda$CDM cosmology. These simulations operate on the base assumption that galaxy evolution is driven by the hierarchical growth of galaxies via the accretion of small satellite galaxies on to larger host galaxies. A semi-analytical analysis of $\Lambda$CDM based $N$-body simulations \citep{Springel05, Boylan-Kolchin+09} finds that the evidence of this accretion exists within a stellar halo component \cite{Bullock+Johnston05, Purcell+07, Cooper+10, Cooper+13}. We assume that the bulk of M63's stellar halo mass is due to the accretion of smaller satellite galaxies. The stellar halo mass fraction derived from stars formed {\it in situ} in the Milky Way is only $\sim$1$\%$ \citep[e.g.][]{Morrison93, Chiba+Beers00, Purcell+07, Bell+08}, whereas the total stellar mass fraction is found to be near $\sim$2$\%$ \citep[e.g.][]{Law+05, Carollo+10}. If the mechanism for populating the halo with stars born {\it in situ} is similar in the Milky Way and in M63 (they are both spirals of nearly equal mass), then a stellar halo mass fraction above $\sim$1$\%$ is comprised mostly of accreted stars. In Figure~\ref{fig:frac} we plot our result for M63 along with the results for the Milky Way \citep{Carollo+10}, M31 \citep{Courteau+11}, M81 \citep{Monachesi+13}, and M101 \citep{vanDokkum+14}. Additionally, we include the results of the analysis of \cite{D'Souza+14}; they measured the average stellar halo mass fraction over many mass bins by stacking thousands of SDSS galaxies. The model prediction from the numerical simulations of \cite{Cooper+13} is also included. Our result, and most other results (besides M101), fit well within the model prediction of \cite{Cooper+13} and the results from the analysis of SDSS data \citep{D'Souza+14}.
We note that populating stellar haloes with stars from accreted satellites is a stochastic process, where most of the mass comes from a few massive dwarf galaxies \citep{Bullock+Johnston05, Cooper+10}. Thus, a large scatter is to be expected in the stellar halo mass fraction to total stellar mass relation. It is not surprising to have a result such as M101; this galaxy must not have accreted enough massive dwarfs to produce a halo detectable by \cite{vanDokkum+14}. We also note that the result from \cite{D'Souza+14} is for the average stellar halo, the uncertainty quoted is based on detecting an average stellar halo, and in no way describes the intrinsic scatter expected from the stochastic accretion of satellite galaxies. Thus, our results, other observational studies, and model predictions, are consistent with one another within the assumption that the accretion of satellite galaxies is a highly stochastic process.
\subsection{Tidal Stream}
We chose M63 for this study because of the nearby, dramatic, tidal stream. This feature dominates the spatial extent of the 3.6~${\rm \mu}$m\ mosaic; the radius of the disc is roughly half that of the stream. These features are ubiquitous for Milky Way sized spiral galaxies in a $\Lambda$CDM universe, as shown in semi-analytical simulations \citep{Bullock+Johnston05, Purcell+07, Cooper+10, Purcell+11, Cooper+13}. However, only a handful of galaxies show obvious evidence of streams in the EDGES sample, and only M63 and NGC4013 feature prominent streams. Yet streams should be common events for galaxies; the remnants of streams, stellar haloes, can be an order of magnitude more massive than an individual stream. In M63, for example, the halo has $16 \pm 2$\ times more mass than the stream (assuming the stream loops once). Given that the mass of the tidal stream is near the upper limit for dwarf galaxies \citep{Mateo+98, Cook+14}, M63 must have accreted many more galaxies than even the halo-to-stream mass fraction suggests.
A simple explanation for the lack of streams is that the accretion rate of satellite galaxies was greater in the past. To test this explanation, we compare the average accretion rate derived from the stellar halo, to the current accretion rate derived from the stellar stream. To measure the current accretion rate, we make the following assumptions: all of the mass of the tidal stream's progenitor galaxy will be accreted on to M63, the stream loops a single time, and over the timescale since disruption the tidal stream is the only contributor to the accretion rate. Thus, the current accretion rate is the mass of the progenitor divided by the time since disruption, see \S~\ref{sec:tidal_stream} for these measurements. This is an upper limit because the tidal stream will be contributing to the mass of the halo past the time since disruption. To find a lower limit for the average accretion rate, we assume that the halo is comprised entirely of accreted stars, and the age of M63 is the age of the Universe, $\sim$13.5~Gyr. Thus, the average accretion rate is the mass of the stellar halo divided by 13.5~Gyr, see \S~\ref{sec:prof_fit} for this measurement. The ratio of the average accretion rate to the current accretion rate is at least $3 \pm 1$. Given the many assumptions of this analysis, the rate at which M63 is accreting matter from this dramatic tidal event is much lower than in the past. This is in agreement with simulations, which find that the majority of the mass which forms the stellar halo was accreted in the first $\sim$5~Gyr of a galaxy's existence \citep{Bullock+Johnston05, Cooper+10}.
\section{Conclusion}
We present an analysis of the low surface brightness NIR properties of the flocculent spiral M63 as part of the EDGES Survey. We use the 3.6~${\rm \mu}$m\ data to derive the mass of the old stellar population within the individual components of the galaxy including the bulge, disc, halo and nearby tidal stream.
The M63 mosaic consists of data from the EDGES Survey, an ultra-deep (1800~s~pixel$^{-1}$), wide-field ($>5~R_{25}$), Spitzer warm mission survey of 92 nearby galaxies \citep{vanZee+11}. The resulting mosaic reaches a depth of 29~AB~mag~arcsec$^{-2}$ at a distance of $\sim3~R_{25}$ on the surface brightness profile. At this imaging depth the outer component of the galaxy is detected (an outer disc or inner stellar halo) in the form of an up-bending break in the surface brightness profile, along with the nearby tidal stream.
Several factors indicate that the outer component is a stellar halo. An up-bending break is unlikely for a disc as M63 is not within a galaxy cluster, the colour of the feature suggests an old stellar population, there is little active star formation in this region, and there is no evidence of spiral structure past 15~kpc. The halo to total mass ratio is $12 \pm 2 \%$, the largest halo mass ratio recorded thus far for an individual galaxy. This ratio agrees well with the ratios derived from SDSS data for the average galaxy \citep{D'Souza+14}, and is within the envelope predicted by the semi-analytical methods of \citep{Cooper+13}. With the EDGES survey of 92 galaxies we will be able to construct a large sample of stellar halo measurements for a more statistically-grounded result \citep[in prep.]{Staudaher+15}. These new observations will shed light on the interplay between large galaxies and their satellites, and will have interesting implications for the missing satellite problem.
The nearby tidal stream has a mass of (5$\pm 2) \eta \times 10^8 M_{\odot}$, derived from the stream's luminosity and width, and 2.3$\pm 1.6 \times 10^8 M_{\odot}$, measured dynamically. The dynamical method also finds a time since disruption of (2.3$\pm$0.7)$\eta$~Gyr\ for this stream. Despite the prominence of the tidal stream, the $16 \pm 2$\ times more massive halo suggests that the accretion rate of satellites was much larger in the past. This is supported by the ratio of the current to past accretion rates; the average accretion rate is at least $3 \pm 1$\ times the accretion rate derived from the stream alone. This is not surprising given that the Universe was much denser, and galaxies were smaller in the past. Semi-analytical models of Milky Way analogs also find that satellite galaxies were, in general, accreted more rapidly in the past \citep{Bullock+Johnston05, Cooper+10}.
This work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology Support under a contract with NASA. Support for this work was provided by NASA through an award issued by the JPL/Caltech. This research has made use of the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. We would also like to thank the anonymous referee for their constructive feedback.
\clearpage
\bibliographystyle{apj}
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{"url":"http:\/\/dw-suggestions.dreamwidth.org\/1401513.html","text":"peoppenheimer () wrote in 2012-12-09 09:43 am\n\n### Support MathJax in Entries\n\nTitle:\nSupport MathJax in Entries\n\nArea:\nMathJax JavaScript support at site level\n\nSummary:\nI suggest that (in due time!) dw support MathJax <a href=\"http:\/\/www.mathjax.org\/\">MathJax<\/a> for mathematical\/technical formatting in dw entries.\n\nDescription:\nThis suggestion is intended to make it easy for dw people to write beautiful and useful mathematical\/technical content in our entries. MathJax is now a mature and well-supported FOSS extension of html via javascript, with healthy user and developer communities. We've been experimenting with MathJax for the Stanford Encyclopedia of Philosophy, and we've been very pleased with it so far. (This is not an official SEP endorsement.) Code can be written or displayed rendered or in TeX or MathML. This makes it useful also for gacking and modifying, and even for learning more about those markup languages. Anyone who has tried to do serious mathematical or technical typesetting in html will agree, I think, that html is *not* a typesetting language. MathJax goes a long way toward allowing decent technical typesetting in an html context.\n\nIf MathJax can be permitted as a tightly controlled JavaScript layer at the dw site level, which I think it can, then users will be able to write mathematical and technical fragments into their journal entries as easily as any other html. I don't envision putting MathJax support into the rich text editor -- I anticipate that anyone who wants to use MathJax will be comfortable editing their own markup. This is rather an extension of html markup into a wider domain.\n\nIt is possible that I'm overestimating the ease of implementing this suggestion, but I've experimented with MathJax support in my personal webpages and at the SEP site, and it looks as though MathJax makes this as easy as possible. Furthermore, the social\/political aspects look promising, insofar as the MathJax user and developer communities look like just the sorts of folks dw wants to make alliance with, as far as I can tell.\n\nPoll #12341 Support MathJax in Entries\nOpen to: Registered Users, detailed results viewable to: All, participants: 46\n\nThis suggestion:\n\nShould be implemented as-is.\n14 (30.4%)\n\nShould be implemented with changes. (please comment)\n1 (2.2%)\n\nShouldn't be implemented.\n3 (6.5%)\n\n(I have no opinion)\n26 (56.5%)\n\n2 (4.3%)\n\n#### no subject\n\nI've voted for, but my only reservation:\n\nAIUI this would mean that some journals (those with MathJax markup) would require Javascript in order to display correctly. I don't think there's a way around this if we want to be able to display maths. And I think that being able to display maths is an excellent idea. Maybe we need to make sure that it fails to a polite message about how we can't display the maths without JS?\n\n(I don't know how MathML is used, and whether that requires JS, and so whether direct MathML support would be an alternative, but AIUI browsers that support MathML are rather rare at the moment)\n\nRe the \"not in the RTE\" part of the suggestion: Ideally I would like to see a way to put maths markup in the RTE. I wouldn't expect it to display as WYSIWYG, but it would be nice to be able to use maths without having to type raw HTML for everything else.\nEdited (tyop.) 2012-12-16 09:07 (UTC)\n\n#### no subject\n\nI'm pretty sure I don't like the idea of any journal entries that need JS to display correctly.\n\n#### no subject\n\nWell, a Youtube embed needs JS (and\/or Flash) to display correctly... is this substantially different?\n\n#### no subject\n\nI think it is. a video embed is a single piece of code. It's always the same (except for the target video file, which is in turn also a well-defined and well-understood format) and can be carefully vetted by the DW team to make sure it can't be misused.\n\nDisplaying math markup is a lot messier than that. I don't know anything about the particular app\/library we're talking about here. But I assume it would have to be allowed to display arbitrary things all through the post in order to be useful. And that potentially leaves my reading page vulnerable to bad actors (maliciously or incompetently) displaying their garbage over my entire reading page. Does anyone but me remember the \"feature\" that let people display full-page ads over someone's MySpace page through a comment?\n\nI really like the very very conservative nature of DW's approach to outside scripting. I think a richer math markup environment would be cool, but not at the expense of security from harmful or just plain annoying scripts.\n\n#### no subject\n\nFrom the MathJax documentation:\n\nIf you are using MathJax in a community setting, however, like a question-and-answer forum, a wiki, a blog with user comments, or other situations where your readers can enter mathematics, then your readers would be able to use such powerful tools to corrupt the page, or fool other readers into giving away sensitive information, or interrupt their reading experience in other ways. In such environments, you may want to limit these abilities so that your readers are protected form these kinds of malicious actions.\n\nhttp:\/\/docs.mathjax.org\/en\/latest\/safe-mode.html\nhttp:\/\/docs.mathjax.org\/en\/latest\/options\/Safe.html\n\nSo, it does appear that it can be locked down so it can only be used as a rendering engine and not a tool to \"corrupt\" the experience of people reading the page.\n\n#### no subject\n\nI really like the idea of leveraging Dreamwidth's responsive development style to make it an especially good journaling system for smaller groups of people like this. It's something that Facebook can't do, as they have to cater to the mass market. LiveJournal clearly has no intention or ability to do anything like this. But because Dreamwidth is smaller and more flexible, this could be really good.\n\nThat said, is there any way for MathJax to generate readable HTML (even if ugly) as an alternate view for those without JavaScript? It seems like Stanford and all institutions would need that kind of accessibility.\n\n#### no subject\n\ndon't feel like you have to do it! that's part of what we do when we decide whether to put the suggestion into bugzilla or not :)\n\n#### no subject\n\nMathJax \"was design[ed] with accessibility in mind and has several powerful features to make math easier to see and read, both for ordinary users as well as those with print and learning disabilities. Two accessibility features, the ability to scale all math in a page or zoom in on a particular equation, are built in to MathJax. In addition, MathJax works with MathPlayer (http:\/\/www.dessci.com\/en\/products\/mathplayer\/) to make math accessible to screen readers, to screen magnifiers, and to learning disability software.\"\n\nSo, it actually seems like it (deliberately) has excellent accessibility support. I can't vouch for it though as I do not currently use those sorts of software features.\n\nhttp:\/\/www.mathjax.org\/resources\/articles-and-presentations\/accessible-pages-with-mathjax\/\nEdited 2013-09-26 03:44 (UTC)\n\n#### no subject\n\nIn with the crowd who wants to know if it can be done without JS.\n\n#### no subject\n\nMathJax requires Javascript. It can work around browsers that don't support (or users that have not enabled) Web Fonts, but Javascript is an absolute requirement for this package.\n\n#### no subject\n\nI've been doing some quick skim-reading, and thought I'd relay some data-points:\n\n1. Will MathJax work without JS? I can't answer authoratatively, but it seems unlikely given that the big headline text on www.mathjax.org is \"MathJax is an open source JavaScript display engine for mathematics that works in all modern browsers\".\n\n3. As is this one on browser compatibility: http:\/\/www.mathjax.org\/resources\/browser-compatibility\/\n\nWondering about other ways to implement maths support, some other means of editing MathML might be useful in the future, but it appears that browser support for MathML is not sufficiently widespread yet. See https:\/\/en.wikipedia.org\/wiki\/MathML#Web_browsers.\nSo it looks as though MathJax is still about the only widely-accessible cross-browser way of doing this, but it probably requires JS.\n\n#### no subject\n\nMathJax does require that Javascript be enabled on the browser of someone reading the page to see the content. One thought I had is for those who do not wish to enable Javascript, and who also do not want to be frustrated by MathJax content not displaying, that an option could be added to the user configuration that would allow someone to tell Dreamwidth not to show them any page containing MathJax code. This could be made a generic flag to exclude any page that requires Javascript to display properly, and that would allow future enhancements that require Javascript to be automatically included (by the developers) in the show\/no-show setting for those users that do not want Javacript generated content (or content that requires Javacript enabled to work properly).\n\n#### no subject\n\nI have been posting my work in the social sciences on my Dreamwidth blog, but I desperately want to be able to post mathematical solutions I have developed in physics to the same blog. Without something like MathJax (from what I can see, the preferred solution these days as it is supported by organizations like the American Mathematical Society and the American Institute of Physics... and even a few international groups), it is an onerous and ultimately futile effort that requires the conversion of documents into HTML and a collection of unintegrated images of equations and such. It's not really blogging at that point (not quite sure exactly what, but it feels different from blogging). I am perfectly fine with the notion that Javascript be required to read such pages provided the platform is controlled directly by the Dreamwidth team. Given the fact that it's basically a rendering engine and not a generic content generation\/management framework, it meshes well, imho, with the blogging paradigm. By using the \"Safe Mode\" configuration, malicious behaviour (such as linking to uncontrolled Javascript) can be prevented:\n\nIf you are using MathJax in a community setting, however, like a question-and-answer forum, a wiki, a blog with user comments, or other situations where your readers can enter mathematics, then your readers would be able to use such powerful tools to corrupt the page, or fool other readers into giving away sensitive information, or interrupt their reading experience in other ways. In such environments, you may want to limit these abilities so that your readers are protected form these kinds of malicious actions.\n\nhttp:\/\/docs.mathjax.org\/en\/latest\/safe-mode.html\nhttp:\/\/docs.mathjax.org\/en\/latest\/options\/Safe.html\n\nI personally would not need any support in the editor (I use HTML anyway), but it would probably be nice for some if there was a button that allowed the insertion of the raw LaTeX or MathML code into an entry without having to switch back and forth between rich and HTML modes (it could display in the rich editor as the raw text of the math code in a fixed-width font or something... again, just having the support in general would be wonderful even without any sort of editor support).\n\nKeep up the great work!\nEdited 2013-09-26 03:37 (UTC)\n\n#### no subject\n\nActually, how about using forkosh.com's implementation (<- see link)? It doesn't require any JavaScript, and works well, for example:\n\n$x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$\n\nIn fact, what we might desire, is just the installation of cgi-bin script on DW, so people can simply include LaTeX like this:\n\n<img src=\"\/cgi-bin\/mimetex.cgi?\nx=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}\">\n\n\nThen, it just suffices to define a new tag, something like that passes the expression into the CGI script, and you get math formulae as nice images...\n\nInstallation instructions are here.\nEdited 2015-04-26 08:16 (UTC)","date":"2017-05-26 09:22:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 1, \"\/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\": 0.3506351113319397, \"perplexity\": 1996.2340265624578}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-22\/segments\/1495463608652.65\/warc\/CC-MAIN-20170526090406-20170526110406-00436.warc.gz\"}"} | null | null |
China snubs foreign investment
After years of courting foreign investors, Beijing may be losing its ardor for capital, reports Fortune's Clay Chandler.
By Clay Chandler, Fortune senior writer
October 3 2006: 6:10 AM EDT
(Fortune Magazine) -- Henry Paulson got a warm welcome when he returned to China last month in his new role as U.S. Treasury Secretary. But even as Beijing prepared to roll out the red carpet for the former Goldman Sachs (Charts) CEO, it was yanking the rug from under some of his old investment banking buddies. The week before Paulson's visit, China's securities regulator declared a moratorium on all foreign acquisitions of Chinese brokerages, making official a de facto ban that's been in effect for months.
Chinese authorities billed the suspension as a temporary measure to help domestic financial institutions gird for increased competition. But among foreign businessmen in China, the move was widely interpreted as further evidence of a backlash against outside investment in an economy that has eagerly sought it for more than a decade.
China's stock boom
The economy is red-hot, and a flurry of blockbuster IPOs is fueling interest. Here's the smart way to cash in. (more)
Since the beginning of the year, officials have scuttled Citigroup's bid for an 85% stake in Guangdong Development Bank, an insolvent state-owned lender, and rebuffed U.S. private-equity firm Carlyle Group's $375 million bid for an 85% stake in Xugong Construction Machinery, a state-owned heavy-equipment manufacturer.
Beijing has also dialed up pressure on foreign firms to allow their Chinese workers to be organized into state-controlled labor unions, forcing even Wal-Mart, notorious for resisting unions in the U.S., to accept them in China. Finally, in August, China introduced new rules requiring foreign investors to register with the Ministry of Commerce any transactions resulting in foreign control over companies in vaguely defined "key industries" or sectors that could influence state security.
With their economy growing at an annual rate of 10%, China's leaders feel they can afford to turn up their nose at foreign investment these days. As exports surge - China racked up a record $202 billion trade surplus with the U.S. last year - the country is awash in dollars. Indeed, Beijing's stack of foreign currency reserves, already the world's largest, is expected to top the $1 trillion mark within the next few months.
China remains Asia's No. 1 destination for outside investment by a wide margin. From January to August, China drew $37.2 billion in foreign direct investment (FDI), down 2.1% from the same period last year, according to the Ministry of Commerce. In August alone, FDI into China slumped 8.5% compared with the same month in 2005. But Arthur Kroeber, managing editor of China Economic Quarterly, urges keeping the downturn in perspective: "Clearly, there's new concern about M&A deals and foreign takeovers. But they're still quite happy to have you come and build a factory."
Even so, the new ambivalence marks an important shift in Chinese thinking, with economic and political impact. Deng Xiaoping's endorsement of foreign-invested ventures in the early 1990s delivered China from decades of Maoist isolation and poverty; he shrugged off questions about social disruption, inequality, and corruption with the observation that "open windows let in flies." Deng's successors courted foreign investors with gusto, eagerly receiving Global 500 executives for face-to-face chats.
But foreign business leaders are almost never granted an audience with China's current President, Hu Jintao, who has shrewdly consolidated his power base inside the party by positioning himself as a populist, attentive to the plight of ordinary folk buffeted by unruly markets and keen to prevent foreigners from wresting control of the economy. Last month's sacking of Chen Liangyu, the Communist Party boss in Shanghai, is an unmistakable rebuke to former President Jiang Zemin, who rose to power in Shanghai, and the freewheeling, money-mad ethos for which his city has come to stand.
Tsinghua University economist David Li argues that opposition to foreign investment is, in part, a sign of economic maturity. China, he says, now has a cohort of homegrown companies strong enough to compete with foreign giants and ample incentive to keep them at bay. Increasingly, the penchant of local governments to woo foreign investors with tax breaks and subsidies fuels a sense of grievance among these domestic upstarts.
That outrage could be counterproductive. MIT political scientist Yasheng Huang, whose 2003 book, "Selling China," questioned China's dependence on foreign investment and was widely debated in China, now fears an FDI backlash that will only make things worse. The key to improving China's competitiveness, he contends, isn't to slap more restrictions on foreign capital but to scale back restrictions on China's homegrown companies.
One of the benefactors of the current slowdown is Paulson's old firm. Since 2004, Goldman has enjoyed Beijing's blessing on its complicated union with Gaohua Securities, a domestic brokerage it helped bankroll. Goldman's rivals - among them Morgan Stanley (Charts), Citigroup (Charts), and Merrill Lynch (Charts) - are eager to secure their own footholds in China's potentially lucrative domestic capital market. For now, they'll have to wait.
Morgan Stanley buys China's Nan Tung Bank
Four Futures for China Inc.
Paulson: U.S., China agree on economic goals
Yuan-way ticket: Paulson goes to China
From the October 16, 2006 issue
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JPMorgan dramatically slashes Tesla's stock price forecast
Greece is finally done with its epic bailout binge
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January 15, 2011.
Like every year, I cozied up on my living room coach ready to watch that year's Miss America Pageant on TV. The previous year had been huge – Miss Nebraska Brittany Jeffers had made the Top 10, something no Miss Nebraska had done in decades.
The 2011 telecast started… and I fell asleep. I was 8 months pregnant. Large and in charge and exhausted. I woke up a few hours later to about 30 text messages: "Miss Nebraska made the Top 15!" "Holy cow, Nebraska's doing awesome!" "She won – MISS NEBRASKA IS MISS AMERICA!"
LAS VEGAS, NV – JANUARY 15: Miss America 2010 Caressa Cameron (L) prepares to crown Teresa Scanlan, Miss Nebraska, the new Miss America during the 2011 Miss America Pageant at the Planet Hollywood Resort & Casino January 15, 2011 in Las Vegas, Nevada. (Photo by Ethan Miller/Getty Images) Original Filename: GYI0063061136.jpg
All photos courtesy Getty Images
Teresa Scanlan made history that night, becoming the first and only Nebraska representative to date to be crowned Miss America, and the youngest Miss America in more than 80 years. As she began a whirlwind year that would change her life and take her around the world, an 8-year old from Scanlan's hometown of Gering decided she, too, wanted to be Miss America.
Photo courtesy Timmy James Photo
Fast forward 7 years.. and Hope McCoy is now Miss Old West Balloon Fest's Outstanding Teen, crowned last summer at the same pageant Teresa Scanlan competed in. She'll now compete for the title of Miss Nebraska's Outstanding Teen this April.
Thank you to Catered Bowl Premium Pet Food, sponsor of the 2018 There She Is series. Click here to learn more!
"My favorite part has been when I do appearances and I get to enjoy all the little kids saying 'wow! There's a real princess here!'" Hope told me recently. "It warms my heart every time."
Sparked in part by Teresa's success, western Nebraska titleholders hit the ground running with service the moment they are crowned. They volunteer at festivals, parades, fundraisers, fashion shows, and more, all promoting goodwill in their communities and their personal platforms of service. Hope wants to encourage children to read, and she has big plans for book drives and fundraisers throughout Mitchell and Gering.
She's also spreading a message to those children, to her peers, and to others she visits.. to nurture friendships and relationships.
"The biggest issue facing our generation right now is communication, " said Hope. "We can say one thing over text and the person on the other side of it could really misunderstand. We hardly talk or communicate like we should. What we can to prevent it is to put down our devices and talk face to face."
Those connections are often what fuel and inspire us. The same day this photo was taken, when Hope won a crown at the same pageant where Teresa Scanlan started her journey.. she got to meet Teresa herself.
"[She is] my female role model," said Hope. "She has always presented herself super nicely and has always been super kind to me. She was the youngest woman to win the title of Miss America and she still accomplished so many great things that really made me want to do more."
Hope McCoy wants to be a translator in the Navy. She wants to go to college and learn more languages. She wants to travel the world. She also wants to have fun, make new friends and make memories as she follows in her inspiration's footsteps.
"Success can be defined in many ways, but success to most people cannot be defined by someone else," said Hope. "It is defined by your own personal goals, and nobody can change that."
To follow Miss Old West Balloon Fest's Outstanding Teen Hope McCoy on Facebook, click here!
WANT TO LEARN MORE ABOUT HOPE?
CLICK HERE * 2017 * Nebraska Nice
For more information about the Miss Nebraska's Outstanding Teen program or to become a contestant, CLICK HERE to follow the organization on Facebook, or CLICK HERE to follow the organization on Twitter. You can also contact Director Heather Edwards at heatheraloseke@gmail.com or Director Kali Tripp at KaliNicoleTV@gmail.com.
The Miss Nebraska's Outstanding Teen Competition takes place April 28 in North Omaha, Nebraska.
PREVIOUS.. Miss Douglas County Krista Hinrichs
COMING SOON.. Miss Kool-Aid Days Outstanding Teen Emily Lenser
This entry was posted on March 21, 2018, in Miss America and tagged brianna little, brittany jeffers, Catered Bowl, Chloe Blumanthal, gering, getty images, kodi bauman, Miss America, miss america pageant, miss nebraska, miss nebraska's outstanding teen, miss old west balloon fest's outstanding teen, Mitchell, Morgan Baird, navy, Scouting For The Cur, sheridan blanco, teresa scanlan, Timmy James Photo. 1 Comment
All The World's A Stage
I am currently experiencing a CRAZY sense of deja vu..
This week, I'm back on stage at Papillion-La Vista High School for the first time in 17 years, as a special guest in Monarch Theatre's production of Legally Blonde the Musical.
The choir room in this picture.. the theater filled with maroon seats.. the green room where the cast gets ready each night.. it's all SACRED GROUND to me. It was my home away from home while I was in high school, where I took part in plays and performances, formed my dearest friendships, and created art that I was so very proud to showcase to our community. There was something intoxicating about theater to me.. the thrill of seeing your name on a casting sheet, the adrenaline rush when the curtain goes up, the sheer GLEE of hearing the audience applaud at the end.
Many of my fellow 'drama nerds' channeled that love into our current careers: my friend Joe Rohacik now teaches at PLHS, where he is also the PA announcer for the entire district. My friend David Wenzel is now a motivational speaker booked around the country for events (CLICK HERE to read his incredible story!) My 'green room' is now the KETV newsroom, and my stage is 7 Burlington Station.
For others.. performing is simply what they were destined to do. My fellow classmates Audrey Billings and Leanne Hill-Carlson are now professional actresses in Chicago and here in Omaha. Monarch alumni Merle Dandridge and Abbie Cobb are now on national TV, starring in shows ranging from HBO to ABC.
Tyler Rambali is somewhere in between.. constantly learning new skills to teach future performers, while fine-tuning her own talents and seeking out opportunities to perform.
"I love to sing and act!" Tyler told me recently, and backed that up with an impressive resume of work, including awards with her trio, Major minor 3.
"[We] competed in the Galaxy of Stars competition this last summer and the summer before," said Tyler (CLICK HERE to link up to Major minor 3's Facebook page to watch and listen to them sing!). The group won a slew of honors their first year and the Megastar Award last summer, earning a recording session as their prize. Tyler walked away with an additional bonus.
"That is where I met Chelsea Arnold (Miss Kool-Aid Days) and her mother, Paige, who first talked to me about competing in the Miss America system," said Tyler.
Tyler was crowned Miss Chadron last fall, opening up the doors for more performance opportunities across the state… not that she needs them. She's starred in Annie Get Your Gun, Nunsense, Lettice and Lovageas, Defying Gravity, and currently, she's the Cowardly Lion in The Wizard of Oz.
Tyler takes on these roles in addition to classes at McCook Community College where she is also a member of Phi Theta Kappa Honor Society, Phi Beta Lambda, the National Association for Music Educators and Not Your Average Theatre Group. Now as Miss Chadron, she's also making appearances and often, trying to promote change with her personal platform 'Reach Out and Read'.
"Reading has become one of the least important things to my generation," said Tyler. "Reach Out and Read is an evidence-based nonprofit organization of medical providers who promote early literacy and school readiness in pediatric exam rooms nationwide by integrating children's books and advice to parents about the importance of reading aloud [during] well-child visits. I am trying to expand on this by connecting it to Children's Miracle Network: reading to the children there, giving them books and teaching them the importance as well. I also want to start a Reach Out and Read site in my hometown of McCook!"
It's the part of pageants that doesn't end up on reality TV, the service and networking opportunities that open up for these titleholders. THAT is part of what Tyler, a pageant newcomer, hopes to show with her new title.
Tyler painted this to symbolize her new adventure competing for Miss Nebraska.
"I wanted to try something new, and on top of that, I had a chance to make my voice heard, make a difference, and be apart of something really special," said Tyler. "Miss America celebrates women and empowers them and their abilities and accomplishments. It's purpose is to serve others, show your personal style and what you can bring to the table, provide scholarships, and help you to be successful, and that is what it has done for me!"
Tyler also stresses the friendships she's already made, adding to circle of loved ones she describes as the most important in her life.
"I have a huge family (over 20 first cousins on one side!)," said Tyler. "Both of my parents served in the US Army (Dad for 20 and Mom for 7) and my brother leaves for basic training in the Navy in May, but I am so proud of him and his endeavors to be a Navy Air Rescue swimmer. My faith in God is the BIGGEST thing in my life, and it is the only way I will be able to let my brother go off to the military."
Tyler adds that her mother is also one of her role models, teaching her at home from Kindergarten through her senior year.. a tie Tyler compares to ANOTHER of her role models, Miss Nebraska and then Miss America 2011 Teresa Scanlan.
"She has so much in common with me," said Tyler. "She loves to sing and act and she was home schooled. It gives me encouragement and I know that I can do anything I can set my mind to."
For anyone at a Nebraska high school right now, dreaming of someday being on a big time stage or on that screen in front of millions.. just LOOK at the incredible talent coming out of our state. Omaha native Gabrielle Union. North Bend native Marg Helgenberger. Norfolk native Johnny Carson.
Tyler Rambali not only wants to follow their lead, she wants to be CAST as the lead in this incredible production called 'life'.. and she's ready to call Act I: 'Becoming Miss Nebraska.'
"I feel that I am an extremely diverse person who can relate to so many people because I have so many different backgrounds: I am biracial, military, city girl, country girl, stage-fright-girl-turned-performer, and so much more!" said Tyler. "I have been placed where I am, in the position I am for a reason, and I will do my very best to not squander the opportunities that God has given me to be a light for HIM."
Photo courtesy Chris Swasta
CLICK HERE to follow Miss Chadron 2016 Tyler Rambali on FACEBOOK
For more information about the Miss Chadron/Miss Northwest/Miss Fur Trade Days Pageant, visit their FACEBOOK PAGE. For information on becoming a contestant, contact Directors Caitlin Rodiek and Sara Smith by email at misscnwdirectors@gmail.com. You can also contact Caitlin Rodiek by phone at 308-207-0336.
The 2016 Miss Nebraska Scholarship Pageant takes place June 8-11 in North Platte, Nebraska. Learn more on THEIR WEBSITE, FACEBOOK PAGE, or follow ON TWITTER and ON INSTAGRAM.
PREVIOUS.. Miss Sugar Valley's Outstanding Teen 2016 Courtney Pelland!
NEXT.. Miss Heartland 2016 Tosha Skinner!
To read more about this year's contestants, or the Miss Nebraska/Miss Nebraska's OT classes of 2015 & 2014, click the THERE SHE IS link at the top of the page!
This entry was posted on March 17, 2016, in Friends, Hobbies, Miss America, Omaha, TV and tagged 7 burlington station, Abbie Cobb, annie get your gun, army, AUDREY BILLINGS, caitlin rodiek, chelsea arnold, chris swasta, cowardly lion, david wenzel, defying gravity, gabrielle union, hope mccartney, jenn cady photography, joe rohacik, johnny carson, KETV, Leanne Hill-Carlson, Legally Blonde, lettice and lovageas, major minor 3, marg helgenberger, mccook, mccook community college, Merle Dandridge, Miss America, miss chadron, miss kearney, miss kool-aid days, miss nebraska, monarch theatre, nafme, navy, norfolk, north bend, nyatg, papillion-la vista, phi beta lambda, phi theta kappa honor society, plhs, reach out and read, sara smith, stacy pospisil, teresa scanlan, Tyler Rambali, wizard of oz. 3 Comments | {
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The Sukhothai Bangkok reveals the magic of chocolate with a variety of exotic couvertures. Blended from the world's finest cacao beans to tempt guests at its legendary Chocolate Buffet every Friday, Saturday and Sunday from 2pm – 5pm at the Lobby Salon at Baht 990++ per person.
Chef Laurent Ganguillet, a Swiss Chocolatier with more than 27 years of experience, is the man responsible for first launching The Sukhothai's Chocolate Buffet during Christmas of 1992.
At his command is an incredible range of almost 30 different chocolates featured on the Signature "Liquid Chocolate Trolley". The current Best Imported Chocolates are; - P125, Cocoa100%; Ouganda80%; Equateur76%; Ivory coast69%; Lindt milk37%; Aneo34%; Inspiration passion, Inspiration strawberry and Inspiration almond. A cup of liquid hot chocolate gets even better taste with a small add of Hazelnut or Almond Soft Pralinés and top with Whipped Cream; Sea Salt; Cinnamon; Nutmeg; Cracked Pepper; Piment d'Espelette.
Chef Laurent views chocolate as a noble product that must be treated with respect. He is a strong believer in chocolate's medicinal qualities and has formed his own theory that there is one kind of chocolate for every moment: "whatever your mood is, there is a chocolate for it". Chocolate is a source of happiness, as he demonstrates when it comes to blending "Liquid Chocolates". "I can match the flavour to the guests' personalities," he says. "I talk to them, find out things about them, and then blend a chocolate to match their mood and character. Most important is to simply bring a smile to their face." Over the years tastes have changed and matured; Chef Laurent aims to keep the buffet fresh, innovative and reflecting the seasons for our hotel guests as well as our many regular local patrons. Start your afternoon with our new delicate selection of savouries such as Carved Honey Glazed Ham, Japanese appetizers, Tea Sandwich before indulging in Chef Laurent's personalised blends and enticing dessert display of our famous Sticky toffee pudding, Churros, chocolate truffles, macaroons, pastries and cakes. Ice cream and sorbet act as refreshments before moving on to the chocolate fondue and warm pudding. The whole experience is complemented by our wide range selection of tea from SARO, The Oriental Senses of Premium Tea or Lavazza coffee fresh from our Barista.
For reservations, telephone +66 (0) 2344 8888 or email to promotions@sukhothai.com. | {
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Bloemendaal () ist eine Gemeinde in der niederländischen Provinz Noord-Holland. Die Gemeinde zählt Einwohner (Stand ) und hat eine Landfläche von 39,7 km².
Bloemendaal gehört zu den wirtschaftlich reichsten Gemeinden des Landes.
Ortsteile
Zur Gemeinde gehören die Villendörfer Bloemendaal, Aerdenhout, Bloemendaal aan Zee, Overveen (mit dem Sitz der Gemeindeverwaltung), Vogelenzang und Bennebroek.
In den Dünen westlich von Overveen liegt der Ehrenfriedhof Bloemendaal (Eerebegraafplaats Bloemendaal). Hier haben 373 Widerstandskämpfer aus dem Zweiten Weltkrieg ihre letzte Ruhestätte gefunden.
Geschichte
In Vogelenzang war das Landgut De Tiltenberg seit 1930 der Sitz der internationalen katholischen Frauenbewegung De Graal, die 1922 vom Jesuitenpater Jacobus (Jac.) van Ginneken (1877–1945) gegründet worden war. Seit 2007 ist auf dem Landgut das Priesterseminar des Bistums Haarlem untergebracht, nach seinem Patron, dem hl. Willibrord, meist Het Willbrordhuis genannt.
Im Juli/August 1937 fand auf dem Landgut der Familie Vertegaal in Vogelenzang in Anwesenheit von Königin Wilhelmina das 5. weltweite Pfadfinder-Jamboree (Wereld Jamboree 1937) statt. 28.750 Pfadfinder aus 54 Ländern nahmen daran teil.
Politik
Gemeinderat
Seit 1982 ergaben sich bei den Kommunalwahlen in Bloemendaal folgende Sitzverteilungen im Gemeinderat:
Anmerkungen
Bürgermeister
Seit September 2017 ist Elbert Roest (D66) amtierender Bürgermeister. Zu seinem Kollegium gehören die Beigeordneten Nico Heijink (VVD), Ton van Rijnberk (D66), Richard Kruijswijk (GroenLinks) sowie die Gemeindesekretärin Wilma Atsma.
Politische Gliederung
Die Gemeinde wird in folgende Ortsteile aufgeteilt:
Persönlichkeiten
Jan Hendrik de Waal Malefijt (1852–1931), Politiker
Sytse Frederick Willem Koolhoven (1886–1946), Automobilrennfahrer, Flugzeugkonstrukteur und Unternehmer
Mischa Epper-Quarles van Ufford (* 18. August 1901 Bloemendaal – 22. Oktober 1978 Basel), Plastikerin, Porträtplastikerin, Goldschmiedin
Leo Daniël Brongersma (1907–1994), Herpetologe
Diederik Simon (* 1970), Ruderer und Olympionike
Gert Jan Schlatmann (* 1963), Hockeyspieler
Weblinks
Offizielle Website der Gemeinde (niederländisch)
Einzelnachweise
Gemeinde in Noord-Holland
Ort in Noord-Holland | {
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Die reformierte Kirche in Furna im mittleren Prättigau ist ein evangelisch-reformiertes Gotteshaus unter dem Denkmalschutz des Kantons Graubünden.
Geschichte und Ausstattung
Die Kirche wurde in vorreformatorischer Zeit um das Jahr 1490 in spätgotischem Stil errichtet. Renovierungen wurde sie 1860, 1957 und letztmals 1993 unterzogen.
Das Kirchenschiff im Inneren ist von einem holzverkleideten Tonnengewölbe überzogen. Der polygonale Chor wird von einem filigranen Sterngewölbe gedeckt. In seiner Mitte steht ein Taufstein, der nach der Tradition reformierter Konfession auch als Abendmahlstisch genutzt wird.
In den 1930er-Jahren erlangten Furna und seine Kirche Bekanntheit, da die Kirchgemeinde Greti Caprez-Roffler zur ersten Pfarrerin der Schweiz wählte, obwohl die Führung eines Pfarramts durch eine Frau damals rechtlich nicht möglich war.
Kirchliche Organisation
Die Evangelisch-reformierte Landeskirche Graubünden führt Furna als kleine eigenständige Kirchgemeinde, die derzeit (Stand: 2012) von Jenaz aus im Teilzeitdienst betreut wird, innerhalb vom Kolloquium IX Prättigau.
Galerie
Weblinks
Die Kirche Furna mit Fotografien der Aussenansicht und des Kircheninneren auf graubuenden.ch
Dokumentation des Streits um das Frauenpfarramt in Graubünden im Bündner Kirchenboten: Die (illegale) Pfarrerin Greti Caprez in Furna. (PDF; Archivversion)
Furna Gr
Furna Gr, Reformierte Kirche
Furna GR
Kirchengebäude im Kanton Graubünden
Kulturgut von regionaler Bedeutung im Kanton Graubünden | {
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Brikolaż () – majsterkowanie i samodzielne, nieprofesjonalne naprawy różnego rodzaju przedmiotów użytkowych, łączenie niepowiązanych ze sobą elementów. Obecnie występuje jako termin specjalny w wielu różnych dziedzinach wiedzy. Do terminologii naukowej pojęcie brikolażu wprowadził francuski antropolog Claude Lévi-Strauss, który posłużył się nim w swojej strukturalistycznej analizie powstawania mitu, przedstawiając powstawanie mitów właśnie jako formę brikolażu. Wprowadzone przez Lévi-Straussa pojęcie przejęło następnie wielu innych uczonych i myślicieli reprezentujących różne dziedziny wiedzy, w tym np. filozofowie Jacques Derrida, Félix Guattari i Gilles Deleuze, literaturoznawcy Gérard Genette i Cary Nelson, badaczka kultury masowej Deena Weinstein.
Brikolaż u Lévi-Straussa
Lévi-Strauss prezentuje pojęcie brikolażu (bricolage, a także rzeczownik bricoleur i czasownik bricoler) przede wszystkim w pracy Myśl nieoswojona.
Mitotwórczy charakter bricolage'u przedstawiany przez Lévi-Straussa czyni go czymś pośrednim między nauką a sztuką. W myśleniu mitycznym przynależące do różnych kontekstów praktycznych i kulturowych, heterogeniczne elementy łączone są ze sobą i wykorzystywane na nowo tak, że nadaje im się nowy, samodzielny sens. Bricoleur stara się przedstawić kompletny obraz rzeczywistości dysponując jedynie ograniczoną gamą środków intelektualnych – posługując się jedynie takimi środkami, którymi dysponuje na bieżąco (ponowne uporządkowanie treści kultury jest tu wprawdzie warunkowane przez ich oryginalne sensy, ale także odrzuca te sensy, zastępuje je nowymi). Tekst Lévi-Straussa zaczyna się od omówienia komentarzy innych autorów. Szukali oni wyraźnego kontrastu pomiędzy "pierwotnym" bricolage a postawą "cywilizowanego" naukowca i inżyniera, którzy używają jedynie środków intelektualnych uznawanych za adekwatne i którzy postulują niepoznawalność i niewykonalność pewnych rzeczy za pomocą środków bieżących. Według przytaczanych autorów, bricoleur koncentruje się na powierzchni zjawisk, podczas gdy nauka poszukuje ich wewnętrznej struktury. Taka różnica postaw miała kontrastować myśl cywilizowaną i myśl pierwotną. Według Lévi-Straussa nie ma tu jednak ostrej różnicy jakościowej, i nie powinno to prowadzić do uznania tych drugich kultur za "prymitywne" – bardziej stosowne jest określenie "wczesne". Każda kultura w istocie stosuje swoje własne formy bricolage i myślenia analitycznego – takie, na jakie pozwalają jej aktualne potrzeby, środki i ogólna sytuacja. Należy to według Lévi-Straussa postrzegać raczej jako odmienne perspektywy. Bricolage wskazuje tak na ograniczenia myślenia mitycznego, jak i na jego kreatywność w traktowaniu zastanych treści kulturowych.
Znaczenie terminu "brikolaż" w innych dziedzinach wiedzy
Od czasów Lévi-Straussa termin pojawia się w bardzo wielu różnych dziedzinach wiedzy, w tym w antropologii, językoznawstwie, kulturoznawstwie, naukach o sztuce, naukach społecznych, muzykologii, w badaniach nad kulturą masową. W odniesieniu do analiz kultury współczesnej nabiera odmiennego i bardziej ogólnego niż u Lévi-Straussa znaczenia, oznaczając najczęściej montaż nowych stylów i mód poprzez (swobodne) zestawianie elementów zastanego materiału kulturowego. Repetycja, wariacja i rekontekstualizacja zastanych elementów kulturowych prowadzi do powstawania nowych, indywidualnych, osobnych form stylowych i kulturowych. Prócz analiz kultury konsumpcyjnej i popularnej pojęcie brikolażu występuje często także w naukach o sztuce. Punktem wyjścia analiz, w których występuje, jest często sztuka surrealistyczna i dadaistyczna, w tym np. praca Marcela Duchampa Fontanna, w formie porcelanowego pisuaru. W odniesieniu do kubizmu, surrealizmu czy dadaizmu ma się na myśli użycie szerokiego spektrum nietradycyjnych materiałów, od lat 60., zwłaszcza w związku działalnością włoskiego ruchu artystycznego arte povera, termin wiąże się jednak także z kontestacją komercjalizacji świata sztuki, dokonywaną np. poprzez konstruowanie rzeźb ze śmieci. Praktyka taka służy dewaluacji dzieła sztuki i ewaluacji pozaartystycznej zwyczajności, codzienności.
Termin "brikolaż" obecny jest zwłaszcza w analizach kultury konsumpcyjnej występujących w myśli postmodernistycznej. Kładzie się w nich nacisk na badanie zbitek pojęciowych i zestawień różnorodnych przekazów symbolicznych, których eksplorowanie postrzegane jest jako źródło kreatywności, indywidualizmu i niezależności od konsumpcjonizmu. Pojęcie spotyka się jednak z szeroką krytyką, w tym o charakterze teoretycznym – pomimo że stało się symbolem i kluczowym hasłem postmodernistycznego konsumpcjonizmu kulturowego, pozostaje ono w istocie pojęciem typowo strukturalistycznym. Pod wpływem idei poststrukturalistycznych wychodzi ono z mody; zastępuje je zbliżone znaczeniowo pojęcie assemblage.
Zobacz też
polistylizm
kolaż, decollage, asamblaż
Przypisy
Bibliografia
Martyn Hammersley, Bricolage and Bricoleur, artykuł w: Lisa M. Given (red.), The Sage Encyclopedia of Qualitative Research Methods, Sage 2008.
Rebeca Leach, Bricolage, artykuł w: Dale Sotherton (red.), Encyclopedia of Consumer Culture, Sage 2011.
William Ramp, Bricolage, artykuł w: Austin Harrington, Barbara L. Marshall, Hans-Peter Müller (red.), Encyclopedia of Social Theory, Routledge 2012.
Sztuka
Postmodernizm
Etnologia i antropologia kulturowa | {
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\section{Introduction and preliminaries}
In 1912, S.N Bernstein \cite{sbbl1} introduced the following sequence of
operators $B_{n}:C[0,1]\rightarrow C[0,1]$ defined by
\begin{equation}
B_{n}(f;x)=\sum_{k=0}^{n}\binom{n}{k}x^{k}(1-x)^{n-k}f\left( \frac{k}{n
\right) ,~~~~~~~x\in \lbrack 0,1]. \label{s1}
\end{equation
for $n\in \mathbb{N}$ and $f\in C[0,1]$.
In 1950, for $x \geq 0$, Sz\'{a}sz \cite{sbbl4} introduced the operators
\begin{equation} \label{s2}
S_n(f;x)=e^{-nx}\sum_{k=0}^\infty \frac{(nx)^k}{k!} f\left(\frac{k}{n
\right),~~~~~~~f \in C[0,\infty).
\end{equation}
In the field of approximation theory, the application of $q$-calculus
emerged as a new area in the field of approximation theory. The first $q
-analogue of the well-known Bernstein polynomials was introduced by Lupa\c{s}
by applying the idea of $q$-integers \cite{sbbl2}. In 1997 Phillips \cit
{sbbl3} considered another $q$-analogue of the classical Bernstein
polynomials. Later on, many authors introduced $q$-generalizations of
various operators and investigated several approximation properties \cit
{skant, smah1,sradu,smah2,smur3}.\newline
The $q$-integer $[n]_{q}$, the $q$-factorial $[n]_{q}!$ and the $q$-binomial
coefficient are defined by (see \cite{sbbl5})
\begin{align*}
\lbrack n]_{q}& :=\left\{
\begin{array}{ll}
\frac{1-q^{n}}{1-q}, & \hbox{if~}q\in \mathbb{R}^{+}\setminus \{1\} \\
n, & \hbox{if~}q=1
\end{array
\right. \mbox{for $n\in \mathbb{N} $~and~$[0]_q=0$}, \\
\lbrack n]_{q}!& :=\left\{
\begin{array}{ll}
\lbrack n]_{q}[n-1]_{q}\cdots \lbrack 1]_{q}, & \hbox{$n\geq 1$,} \\
1, & \hbox{$n=0$,
\end{array
\right. \\
\left[
\begin{array}{c}
n \\
\end{array
\right] _{q}& :=\frac{[n]_{q}!}{[k]_{q}![n-k]_{q}!},
\end{align*
respectively.
\noindent The $q$-analogue of $(1+x)^{n}$ is the polynomial
\begin{equation*}
(1+x)_{q}^{n}:=\left\{
\begin{array}{ll}
(1+x)(1+qx)\cdots (1+q^{n-1}x) & \quad n=1,2,3,\cdots \\
1 & \quad n=0
\end{array
\right.
\end{equation*
A $q$-analogue of the common Pochhammer symbol also called a $q$-shifted
factorial is defined as
\begin{equation*}
(x;q)_{0}=1,~(x;q)_{n}=\prod\limits_{j=0}^{n-1}(1-q^{j}x),~(x;q)_{\infty
}=\prod\limits_{j=0}^{\infty }(1-q^{j}x).
\end{equation*}
\noindent The Gauss binomial formula is given by
\begin{equation*}
(x+a)_{q}^{n}=\sum\limits_{k=0}^{n}\left[
\begin{array}{c}
n \\
\end{array
\right] _{q}q^{k(k-1)/2}a^{k}x^{n-k}.
\end{equation*}
The $q-$analogue of Bernstein operators \cite{sbbl3} is defined as
\begin{equation}
B_{n,q}(f;x)=\sum\limits_{k=0}^{n}\left[
\begin{array}{c}
n \\
\end{array
\right] _{q}x^{k}\prod\limits_{s=0}^{n-k-1}(1-q^{s}x)~~f\left( \frac{[k]_{q}
}{[m]_{q}}\right) ,~~x\in [0,1], n \in \mathbb{N}.
\end{equation}
There are two $q$-analogue of the exponential function $e^{z}$, defined as
follows :\newline
For $\mid z \mid < \frac{1}{1-q}$ and $\mid q \mid <1$,
\begin{equation} \label{zfr1}
e(z)= \sum_{k=0}^\infty \frac{z^k}{k!}= \frac{1}{1-\left((1-q)z\right)_q
\infty},
\end{equation}
and for $\mid q \mid<1$,
\begin{equation} \label{zfr2}
E(z)= \prod_{j=0}^\infty \left(1+(1-q)q^jz\right)_q^\infty=\sum_{k=0}^\infty
q^{\frac{k(k-1)}{2}}\frac{z^k}{k!} = \left(1+(1-q)z\right)_q^\infty,
\end{equation}
where $(1-x)_q^\infty=\prod_{j=0}^\infty(1-q^jx)$.
Sucu \cite{sbbl9} defined a Dunkl analogue of Sz\'{a}sz operators via a
generalization of the exponential function \cite{sbbl10} as follows:
\begin{equation}
S_{n}^*(f;x):= \frac{1}{e_\mu(nx)}\sum_{k=0}^\infty \frac{(nx)^k}
\gamma_\mu(k)} f \left(\frac{k+2\mu\theta_k}{n}\right),
\end{equation}
where $x \geq 0,~~~f \in C[0,\infty),\mu \geq 0,~~~n \in \mathbb{N}$\newline
and
\begin{equation*}
e_\mu(x)= \sum_{n=0}^\infty \frac{x^n}{\gamma_\mu(n)}.
\end{equation*}
Here
\begin{equation*}
\gamma_\mu(2k)= \frac{2^{2k}k!\Gamma\left(k+\mu+\frac{1}{2}\right)}
\Gamma\left(\mu+\frac{1}{2}\right)},
\end{equation*}
and
\begin{equation*}
\gamma_\mu(2k+1)= \frac{2^{2k+1}k!\Gamma\left(k+\mu+\frac{3}{2}\right)}
\Gamma\left(\mu+\frac{1}{2}\right)}.
\end{equation*}
There is given a recursion for $\gamma _{\mu }$
\begin{equation*}
\gamma _{\mu }(k+1)=(k+1+2\mu \theta _{k+1})\gamma _{\mu
}(k),~~~~k=0,1,2,\cdots ,
\end{equation*
where
\begin{equation*}
\theta _{k}
\begin{cases}
0 & \quad \text{if }k\in 2\mathbb{N} \\
1 & \quad \text{if }k\in 2\mathbb{N}+1
\end{cases
\end{equation*}
Cheikh et al. \cite{sbbl11} stated the $q$-Dunkl classical $q$-Hermite type
polynomials and gave definitions of $q$-Dunkl analogues of exponential
functions and recursion relations for $\mu >-\frac{1}{2}$ and $0<q<1$.
\begin{equation}
e_{\mu ,q}(x)=\sum_{n=0}^{\infty }\frac{x^{n}}{\gamma _{\mu ,q}(n)},~~~x\in
\lbrack 0,\infty ) \label{sr1}
\end{equation
\begin{equation}
E_{\mu ,q}(x)=\sum_{n=0}^{\infty }\frac{q^{\frac{n(n-1)}{2}}x^{n}}{\gamma
_{\mu ,q}(n)},~~~x\in \lbrack 0,\infty ) \label{sr2}
\end{equation
\begin{equation}
\gamma _{\mu ,q}(n+1)=\left( \frac{1-q^{2\mu \theta _{n+1}+n+1}}{1-q}\right)
\gamma _{\mu ,q}(n),~~~~~n\in \mathbb{N}, \label{sr3}
\end{equation}
\begin{equation*}
\theta _{n}
\begin{cases}
0 & \quad \text{if }n\in 2\mathbb{N}, \\
1 & \quad \text{if }n\in 2\mathbb{N}+1
\end{cases
\end{equation*
An explicit formula for $\gamma _{\mu ,q}(n)$ is
\begin{equation*}
\gamma _{\mu ,q}(n)=\frac{(q^{2\mu +1},q^{2})_{[\frac{n+1}{2
]}(q^{2},q^{2})_{[\frac{n}{2}]}}{(1-q)^{n}}\gamma _{\mu ,q}(n),~~~~~n\in
\mathbb{N}.
\end{equation*}
And some of the special cases of $\gamma_{\mu,q}(n)$ defined as:
\begin{equation*}
\gamma_{\mu,q}(0)=1, ~~~~~~\gamma_{\mu,q}(1)=\frac{1-q^{2\mu+1}}{1-q
,~~~~\gamma_{\mu,q}(2)=\left(\frac{1-q^{2\mu+1}}{1-q}\right)\left(\frac{1-q^
}{1-q}\right),
\end{equation*}
\begin{equation*}
\gamma_{\mu,q}(3)= \left(\frac{1-q^{2\mu+1}}{1-q}\right) \left(\frac{1-q^2}
1-q}\right) \left(\frac{1-q^{2\mu+3}}{1-q}\right),
\end{equation*}
\begin{equation*}
\gamma_{\mu,q}(4)= \left(\frac{1-q^{2\mu+1}}{1-q}\right) \left(\frac{1-q^2}
1-q}\right) \left(\frac{1-q^{2\mu+3}}{1-q}\right) \left(\frac{1-q^4}{1-q
\right).
\end{equation*}
In \cite{ckz}, G\"{u}rhan I\c{c}\"{o}z gave a Dunkl generalization of
Kantrovich type integral generalization of Sz\'{a}sz operators. In \cit
{sbbl13}, they gave a Dunkl generalization of Sz\'{a}sz operators via $q
-calculus as:
\begin{equation}
D_{n,q}(f;x)=\frac{1}{e_{\mu ,q}([n]_{q}x)}\sum_{k=0}^{\infty }\frac
([n]_{q}x)^{k}}{\gamma _{\mu ,q}(k)}f\left( \frac{1-q^{2\mu \theta _{k}+k}}
1-q^{n}}\right) , \label{sss1}
\end{equation
for $\mu >\frac{1}{2},~~x\geq 0,~~0<q<1$ and $f\in C[0,\infty ).$
\begin{lemma}
\label{vd}
\begin{enumerate}
\item \label{vd1} $D_{n,q}(1;x)=1$,
\item \label{vd2} $D_{n,q}(t;x)=x$,
\item \label{vd3}$x^2+[1-2 \mu]_q q^{2 \mu}\frac{e_{\mu,q}(q [n]_q(x))}
e_{\mu,q}([n]_q x)}\frac{x}{[n]_q} \leq D_{n,q}(t^2;x) \leq x^2+[1+2 \mu]_q \frac
x}{[n]_q}$,
\item \label{vd4}$D_{n,q}(t^3;x) \geq x^3+(2q+1)[1-2\mu]_q \frac
e_{\mu,q}(q[n]_qx)}{e_{\mu,q}([n]_qx)}\frac{x^2}{[n]_q} +q^{4
\mu}[1-2\mu]_q^2 \frac{e_{\mu,q}(q^2[n]_qx)}{e_{\mu,q}([n]_qx)}\frac{x}
[n]_q^2}$,
\item \label{vd5}$D_{n,q}(t^4;x) \leq x^4+ 6[1+2\mu]_q \frac{x^3}{[n]_q}
+7[1+2\mu]_q^2\frac{x^2}{[n]_q^2} +[1+2\mu]_q^3 \frac{x}{[n]_q^3}$.
\end{enumerate}
\end{lemma}
In this paper we construct Kantrovich type Sz\'{a}sz-Mirakjan operators
generated by Dunkl generalization of the exponential function via $q
-integers. We obtain some approximation results via well known Korovkin's
type theorem and weighted Korovkin's type theorem for these operators. We
also study convergence properties by using the modulus of continuity and the
rate of convergence of the operators for functions belonging to the
Lipschitz class. Furthermore, we obtain the rate of convergence in terms of
the classical, second order, and weighted modulus of continuity.
\section{auxiliary results}
We define a Dunkl generalization of Sz\'{a}sz-Mirakjan-Kantrovich operators via $q$-calculs as follows: \newline
For any $x\in [0,\infty), ~~~n \in \mathbb{N}, ~~0<q< 1,$ and $\mu>\frac{1}{
}$, we define
\begin{equation}
T_{n,q}^{\ast}(f;x)=\frac{[n]_{q}}{e_{\mu ,q}([n]_{q}x)}\sum_{k=0}^{\infty
\frac{([n]_{q}x)^{k}}{\gamma _{\mu ,q}(k)}\int_{\frac{q[k+2\mu \theta
_{k}]_{q}}{[n]_{q}}}^{\frac{[k+1+2\mu \theta _{k}]_{q}}{[n]_{q}}}f\left(\frac{nt+\alpha}{n+\beta}\right)d_{q}t,
\label{snss1}
\end{equation
where $f$ is a continuous and nondecreasing function on $[0,\infty )$.
\begin{lemma}
\label{snlm1} Let $T_{n,q}^{\ast}(.~;~.)$ be the operators given by
\eqref{snss1}. Then we have the following identities and inequalities:
\begin{enumerate}
\item \label{snlm11} $T_{n,q}^{\ast}(1;x)=1$,
\item \label{snlm12} $T_{n,q}^{\ast}(t;x)=\frac{2qn}{(n+\beta)[2]_q}x+\frac{n}{(n+\beta)[2]_q[n]_q}+\frac{\alpha}{n+\beta}$,
\item \label{snlm13}$\frac{n^2}{(n+\beta)^2}\frac{1}{[3]_q[n]_q^2}+\frac{2n\alpha}{(n+\beta)^2[2]_q[n]_q}+\frac{\alpha^2}{(n+\beta)^2}
+\Big(\frac{n^2}{(n+\beta)^2}\frac{3q}{[3]_q[n]_q}+\frac{2n\alpha}{(n+\beta)^2}\frac{2q}{[2]_q}+\frac{3q^{2(\mu+1)}}{[3]_q[n]_q}\\
\times[1-2\mu]_q \frac{e_{\mu,q}(q [n]_q(x))}{e_{\mu,q}([n]_q x)
\Big)x+\frac{n^2}{(n+\beta)^2}\frac{3q^2}{[3]_q}x^2 \leq T_{n,q}^{\ast}(t^2;x) \leq \frac{n^2}{(n+\beta)^2}\frac{1}{[3]_q[n]_q^2}+\frac{2n\alpha}{(n+\beta)^2[2]_q[n]_q}+\frac{\alpha^2}{(n+\beta)^2}
+\Big(\frac{n^2}{(n+\beta)^2}\frac{3}{[3]_q[n]_q}+\frac{2n\alpha}{(n+\beta)^2}\frac{2}{[2]_q}
+\frac{3q^{2(\mu+1)}}{[3]_q[n]_q}[1-2\mu]_q \frac{e_{\mu,q}(q [n]_q(x))}{e_{\mu,q}([n]_q x)}\Big)x\\
+\frac{n^2}{(n+\beta)^2}\frac{3}{[3]_q}x^2 $,
\item \label{snlm14}$\frac{n}{(n+\beta)^3[n]_q}\Big(\frac{n^2}{[4]_q[n]_q^2}+\frac{3n\alpha}{[3]_q[n]_q}+\frac{3\alpha^2}{[2]_q}\Big)+\frac{\alpha^3}{(n+\beta)^3}+
\frac{1}{(n+\beta)^3}\{n\big( \frac{4n^2q}{[4]_q[n]_q^2}+\frac{3n\alpha}{[n]_q}\frac{3q}{[3]_q}+\frac{6\alpha^2q}{[2]_q}\big)+(\frac{2n}{[4]_q[n]_q}
+\frac{3\alpha}{[3]_q})\frac{3n^2q^{2(\mu+1)}}{[n]_q}[1-2\mu]_q\frac{e_{\mu,q}(q [n]_q(x))}{e_{\mu,q}([n]_q x)}+\frac{4n^3}{[4]_q[n]_q^2}[1-2\mu]_q^2
\frac{e_{\mu,q}(q^2[n]_q(x))}{e_{\mu,q}([n]_q(x))}\}x+
\frac{n^2}{(n+\beta)^3}\Big(\frac{6nq^2}{[4]_q[n_q]}+\frac{9\alpha q^2}{[3]_q}+\frac{4nq^3}{[4]_q[n]_q}(2q+1)[1-2\mu]_q\frac{e_{\mu,q}(q[n]_qx)}{e_{\mu,q}([n]_q x)}\Big)x^2
+\frac{4n^3q^3}{(n+\beta)^3[4]_q}x^3 \leq
T_{n,q}^{\ast}(t^3;x) \leq \frac{n}{(n+\beta)^3[n]_q}\Big(\frac{n^2}{[4]_q[n]_q^2}+\frac{3n\alpha}{[3]_q[n]_q}+\frac{3\alpha^2}{[2]_q}\Big)+\frac{\alpha^3}{(n+\beta)^3}
+\frac{1}{(n+\beta)^3}(\frac{4n^3}{[4]_q[n]_q^2}(1+[1+2\mu]_q^2)+\frac{9n^2\alpha}{[3]_q[n]_q}(1+[1+2\mu]_q)+6n(\frac{\alpha^2}{[2]_q}
+\frac{n^2[1+2\mu]_q}{[4]_q[n]_q}))x
+\frac{1}{(n+\beta)^3}\{\frac{6n^3}{[4]_q[n]_q}(1+2[1+2\mu]_q)+\frac{9n^2\alpha}{[3]_q}\}x^2
+\frac{4n^3}{(n+\beta)^3[4]_q}x^3$,
\item \label{snlm15}$T_{n,q}^{\ast}(t^4;x)\leq \frac{n}{(n+\beta)^4[n]_q}(\frac{n^3}{[5]_q[n]_q^3}+\frac{4n^2\alpha}{[4]_q[n]_q^2}+\frac{6n\alpha^2}{[3]_q[n]_q}+\frac{4\alpha^3}{[2]_q})
+\frac{\alpha^4}{(n+\beta)^4}
+\{\frac{n^2}{(n+\beta)^4}(\frac{5n^2}{[5]_q[n]_q^3}+\frac{16n\alpha}{[4]_q[n]_q^2}+\frac{18\alpha^2}{[3]_q[n]_q}+\frac{8n\alpha}{[2]_q})
+\frac{2n^2}{(n+\beta)^4}(\frac{5n^2}{[5]_q[n]_q^2}+\frac{12n\alpha}{[4]_q[n]_q}+\frac{9\alpha^2}{[3]_q})\frac{[1+2\mu]_q}{[n]_q}
+\frac{2n^3}{(n+\beta)^4}(\frac{5n}{[5]_q[n]_q}\frac{8\alpha}{[n]_q}+)\\
\times\frac{[1+2\mu]_q^2}{[n]_q^2}+\frac{5n^4}{(n+\beta)^4[5]_q}\frac{[1+2\mu]_q^3}{[n]_q^3}\}x
+\{\frac{2n^2}{(n+\beta)^4}(\frac{5n^2}{[5]_q[n]_q^2}+\frac{12n\alpha}{[4]_q[n]_q}+\frac{9\alpha^2}{[3]_q})
+\frac{6n^3}{(n+\beta)^4}(\frac{5n}{[5]_q[n]_q}+\frac{8\alpha}{[n]_q})\frac{[1+2\mu]_q}{[n]_q}
+\frac{35n^4}{(n+\beta)^4[5]_q}\frac{[1+2\mu]_q^2}{[n]_q^2}\}x^2
+\{\frac{2n^3}{(n+\beta)^4}(\frac{5n}{[5]_q[n]_q}+\frac{8\alpha}{[n]_q})+\frac{5n^4}{(n+\beta)^4[5]_q}\frac{[1+2\mu]_q}{[n]_q}\}x^3
+\frac{5n^4}{(n+\beta)^4[5]_q}x^4$.
\end{enumerate}
\end{lemma}
\begin{proof}
It is easily seen that
\begin{equation}
\lbrack k+1+2\mu \theta _{k}]_{q}=q[k+2\mu \theta _{k}]_{q}+1. \label{snmn1}
\end{equation}
so we get the followings
\begin{equation} \label{snmn2}
\int_{\frac{q[k+2\mu \theta_k]_q}{[n]_q}}^{\frac{[k+1+2\mu \theta_k]_q}{[n]_q
} 1 d_q t=\frac{1}{[n]_q},
\end{equation}
\begin{equation} \label{snmn3}
\int_{\frac{q[k+2\mu \theta_k]_q}{[n]_q}}^{\frac{[k+1+2\mu \theta_k]_q}{[n]_q
} t d_q t =\frac{1}{[2]_q[n]_q^2}\left(1+2q[k+2\mu \theta_k]_q\right),
\end{equation}
\begin{equation} \label{snmn4}
\int_{\frac{q[k+2\mu \theta_k]_q}{[n]_q}}^{\frac{[k+1+2\mu \theta_k]_q}{[n]_q
} t^2 d_q t =\frac{1}{[3]_q[n]_q^3}\left(1+3q[k+2\mu \theta_k]_q+3q^2[k+2\mu
\theta_k]_q^2 \right),
\end{equation}
\begin{equation}
\int_{\frac{q\lbrack k+2\mu \theta _{k}]_{q}}{[n]_{q}}}^{\frac{[k+1+2\mu
\theta _{k}]_{q}}{[n]_{q}}}t^{3}d_{q}t=\frac{1}{[4]_{q}[n]_{q}^{4}}\left(
1+4q[k+2\mu \theta _{k}]_{q}+6q^{2}[k+2\mu \theta
_{k}]_{q}^{2}+4q^{3}[k+2\mu \theta _{k}]_{q}^{3}\right) , \label{snmn242}
\end{equation
and
\begin{equation}
\int_{\frac{q\lbrack k+2\mu \theta _{k}]_{q}}{[n]_{q}}}^{\frac{[k+1+2\mu
\theta _{k}]_{q}}{[n]_{q}}}t^{4}d_{q}t=\frac{1}{[5]_{q}[n]_{q}^{5}}\left(
1+5q[k+2\mu \theta _{k}]_{q}+10q^{2}[k+2\mu \theta
_{k}]_{q}^{2}+10q^{3}[k+2\mu \theta _{k}]_{q}^{3}+5q^{4}[k+2\mu \theta
_{k}]_{q}^{4}\right) . \label{snmn24}
\end{equation
From the Lemma \ref{vd}
we have the following results:
\begin{equation}
\frac{1}{[n]_{q}}\frac{1}{e_{\mu ,q}([n]_{q}x)}\sum_{k=0}^{\infty }\frac
([n]_{q}x)^{k}}{\gamma _{\mu ,q}(k)}[k+2\mu \theta _{k}]_{q}=x,
\label{snb24}
\end{equation
\begin{equation}
x^{2}+q^{2\mu }[1-2\mu ]_{q}\frac{e_{\mu ,q}(q[n]_{q}x)}{e_{\mu ,q}([n]_{q}x
}\frac{x}{[n]_{q}}\leq \frac{1}{[n]_{q}^{2}}\frac{1}{e_{\mu ,q}([n]_{q}x)
\sum_{k=0}^{\infty }\frac{([n]_{q}x)^{k}}{\gamma _{\mu ,q}(k)}[k+2\mu \theta
_{k}]_{q}^{2}\leq x^{2}+[1+2\mu ]_{q}\frac{x}{[n]_{q}}, \label{snb25}
\end{equation}
\begin{equation} \label{snub26}
\frac{1}{[n]_q^3}\frac{1}{e_{\mu,q}([n]_{q}x)}\sum_{k=0}^\infty \frac
([n]_{q}x)^k}{\gamma_{\mu,q}(k)}[k+2\mu \theta_k]_q^3 \leq x^3+3[1+2\mu]_
\frac{x^2}{[n]_q}+ [1+2\mu]_q^2\frac{x}{[n]_q^2},
\end{equation}
$\frac{1}{[n]_q^3}\frac{1}{e_{\mu,q}([n]_{q}x)}\sum_{k=0}^\infty \frac
([n]_{q}x)^k}{\gamma_{\mu,q}(k)}[k+2\mu \theta_k]_q^3$
\begin{equation} \label{snb26}
\geq x^3+(2q+1)[1-2\mu]_q\frac{e_{\mu,q}(q[n]_qx)}{e_{\mu,q}([n]_qx)}\frac
x^2}{[n]_q}+ q^{4 \mu}[1-2\mu]_q^2\frac{e_{\mu,q}(q^2[n]_qx)}
e_{\mu,q}([n]_qx)}\frac{x}{[n]_q^2},
\end{equation}
and
\begin{equation} \label{snb27}
\frac{1}{[n]_q^4}\frac{1}{e_{\mu,q}([n]_{q}x)}\sum_{k=0}^\infty \frac
([n]_{q}x)^k}{\gamma_{\mu,q}(k)}[k+2\mu \theta_k]_q^4 \leq x^4+[1+2\mu]_
\frac{x^3}{[n]_q}+7[1+2\mu]_q^2\frac{x^2}{[n]_q^2}+[1+2\mu]_q^3\frac{x}
[n]_q^3}.
\end{equation}
\begin{enumerate}
\item From \eqref{snmn2} we have $T_{n,q}^*(1;x)=\frac{[n]_q}
e_{\mu,q}([n]_{q}x)}\sum_{k=0}^\infty \frac{([n]_{q}x)^k}{\gamma_{\mu,q}(k)
\frac{1}{[n]_q}=1$. \newline
\item If $f(t)=t$ then \eqref{snss1}, \eqref{snmn3} and \eqref{snb24} imply
that
\begin{eqnarray*}
T_{n,q}^{\ast }(t;x) =\frac{2qn}{(n+\beta)[2]_q}x+\frac{n}{(n+\beta)[2]_q[n]_q}+\frac{\alpha}{n+\beta}.
\end{eqnarray*}
\item If $f(t)=t^{2}$ then from \eqref{snss1}, \eqref{snmn4}, \eqref{snb24}
and \eqref{snb25} we get (3). \newline
\item If $f(t)=t^{3}$ then from \eqref{snss1}, \eqref{snmn242}, \eqref{snb24
, \eqref{snb25}, \eqref{snub26} and \eqref{snb26} we get (4)
\item If $f(t)=t^{4}$ then from \eqref{snss1}, \eqref{snmn24}, \eqref{snb24
, \eqref{snb25}, \eqref{snub26} and \eqref{snb27} we have (4)
\end{enumerate}
\end{proof}
\begin{lemma}
\label{snlm2} Let the operators $T_{n,q}^{\ast }(.~;~.)$ be given by
\eqref{snss1}. Then
\begin{enumerate}
\item \label{snlm22} $T_{n,q}^*(t-x;x)=\left(\frac{2qn}{(n+\beta)[2]_q}-1\right)x+\frac{n}{(n+\beta)[2]_q[n]_q}+\frac{\alpha}{n+\beta}$
\item \label{snlm23} $T_{n,q}^*((t-x)^2;x)\leq \frac{n}{(n+\beta)^2[n]_q}(\frac{n}{[3]_q[n]_q}+\frac{2\alpha}{[2]_q})+\frac{\alpha^2}{(n+\beta)^2}+\{\frac{n^2}{(n+\beta)^2}\frac{3}{[3]_q[n]_q}
(1+[1+2\mu]_q)+\frac{2n}{(n+\beta)[2]_q}(2\alpha-\frac{1}{[n]_q})-\frac{2\alpha}{n+\beta}\}x+\{\frac{n}{n+\beta}(\frac{3n}{(n+\beta)[3]_q}
-\frac{4n}{n+\beta)[2]_q})+1\}x^2$,
\item \label{snlm24} $T_{n,q}^*((t-x)^4;x)\leq \frac{n}{(n+\beta)^4[n]_q}(\frac{n^3}{[5]_q[n]_q^3}+\frac{4n^2\alpha}{[4]_q[n]_q^2}+\frac{}{[3]_q[n]_q}+\frac{}{[2]_q})+\frac{\alpha^4}{(n+\beta)^4}\\
\{\frac{n^2}{(n+\beta)^4}(\frac{5n^2}{[5]_q[n]_q^3}+\frac{16n\alpha}{[4]_q[n]_q^2}+\frac{18\alpha^2}{[3]_q[n]_q}+\frac{8n\alpha}{[2]_q})
+\frac{2n^2}{(n+\beta)^4}(\frac{5n^2}{[5]_q[n]_q^2}+\frac{12n\alpha}{[4]_q[n]_q}+\frac{9\alpha^2}{[3]_q})\\
\times \frac{[1+2\mu]_q}{[n]_q}+\frac{2n^3}{(n+\beta)^4}(\frac{5n}{[5]_q[n]_q}+\frac{8\alpha}{[n]_q})\frac{[1+2\mu]_q^2}{[n]_q^2}
+\frac{5n^4}{(n+\beta)^4[5]_q}\frac{[1+2\mu]_q^3}{[n]_q^3}-\frac{4n}{n+\beta)^3[3]_q}(\frac{n^2}{[4]_q[n]_q^2}+\frac{3n\alpha}{[3]_q[n]_q}
+\frac{3\alpha^2}{[2]_q})-\frac{4\alpha^3}{(n+\beta)^3}\}x+\{\frac{2n^2}{(n+\beta)^4}(\frac{5n^2}{[5]_q[n]_q^2}+\frac{12n\alpha}{[4]_q[n]_q}
+\frac{9\alpha^2}{[3]_q})+\frac{6n^3}{(n+\beta)^4}\\
(\frac{5n}{[5]_q[n]_q}+\frac{8\alpha}{[n]_q})\frac{[1+2\mu]_q}{[n]_q}+\frac{35n^4}{(n+\beta)^4[5]_q}\frac{[1+2\mu]_q^2}{[n]_q^2}
-\frac{4}{(n+\beta)^3}(\frac{4n^3}{[4]_q[n]_q^2}(1+[1+2\mu]_q^2)\\
+\frac{9n^2\alpha}{[3]_q[n]_q}(1+[1+2\mu]_q)+6n(\frac{\alpha^2}{[2]_q}+\frac{n^2}{[4]_q}\frac{[1+2\mu]_q^2}{[n]_q^2}))
+6(\frac{n^2}{(n+\beta)^2}\frac{1}{[3]_q[n]_q^2}\\
+\frac{2n\alpha}{(n+\beta)^2[2]_q[n]_q}+\frac{\alpha^2}{(n+\beta)^2})\}x^2+\{\frac{2n^3}{(n+\beta)^4}(\frac{5n}{[5]_q[n]_q}
+\frac{8\alpha}{[n]_q})+\frac{5n^4}{(n+\beta)^4[5]_q}\frac{[1+2\mu]_q}{[n]_q}\\
-\frac{4}{(n+\beta)^3}(\frac{6n^3}{[4]_q[n]_q}(1+2[1+2\mu]_q)+\frac{9n^2\alpha}{[3]_q})+6(\frac{n^2}{(n+\beta)^2}\frac{3}{[3]_q[n]_q}
+\frac{2n\alpha}{(n+\beta)^2}\frac{2}{[2]_q})\\
+\frac{6n^2}{(n+\beta)^2}\frac{3}{[3]_q}\frac{[1+2\mu]_q}{[n]_q}-4(\frac{n}{(n+\beta)[2]_q[n]_q}+\frac{\alpha}{n+\beta})\}x^3
+\{\frac{5n^4}{(n+\beta)^4[5]_q}-\frac{16n^3}{(n+\beta)^3[4]_q}\\
\frac{6n^2}{(n+\beta)^2}\frac{3}{[3]_q}-\frac{8n}{(n+\beta)[2]_q}+1\}x^4$.
\end{enumerate}
\end{lemma}
\section{Main results}
We obtain the Korovkin's type approximation properties for our operators
defined by \eqref{snss1}.\newline
Let $C_{B}(\mathbb{R^{+}})$ be the set of all bounded and continuous
functions on $\mathbb{R^{+}}=[0,\infty )$, which is linear normed space with
\begin{equation*}
\parallel f\parallel _{C_{B}}=\sup_{x\geq 0}\mid f(x)\mid .
\end{equation*
Let
\begin{equation*}
H:=\{f:x\in \lbrack 0,\infty ),\frac{f(x)}{1+x^{2}}~~~\mbox{is}~~
\mbox{convergent}~~~\mbox{as}~~~x\rightarrow \infty \}.
\end{equation*}
\parindent=8mmIn order to obtain the convergence results for the operators
T_{n,q}^{\ast }(.~;~.)$, we take $q=q_{n}$ where $q_{n}\in (0,1)$ such that
\begin{equation}
\lim_{n}q_{n}\rightarrow 1,~~~~~~\lim_{n}q_{n}^{n}\rightarrow a
\label{snnas5}
\end{equation}
\begin{theorem}
\label{snth1} Let $q=q_{n}$ satisfying \eqref{snnas5}, for $0<q_{n}<1$ and
if $T_{n,q_{n}}^{\ast }(.~;~.)$ be the operators given by \eqref{snss1}.
Then for any function $f\in C[0,\infty )\cap H$,
\begin{equation*}
\lim_{n\rightarrow \infty }T_{n,q_{n}}^{\ast }(f;x)=f(x)
\end{equation*
is uniformly on each compact subset of $[0,\infty )$.
\end{theorem}
\begin{proof}
The proof is based on the well known Korovkin's theorem regarding the
convergence of a sequence of linear and positive operators, so it is enough
to prove the conditions
\begin{equation*}
\lim_{n\rightarrow \infty }{T}_{n,q_{n}}^{\ast
}((t^{j};x)=x^{j},~~~j=0,1,2,~~~\{\mbox{as}~n\rightarrow \infty \}
\end{equation*
uniformly on $[0,1]$.\newline
Clearly from \eqref{snnas5} and $\frac{1}{[n]_{q_{n}}}\rightarrow
0~~(n\rightarrow \infty )$ we have
\begin{equation*}
\lim_{n \to \infty}{T}_{n,q_n}^*(t;x)=x,~~~\lim_{n \to \infty}{T
_{n,q_n}^*(t^2;x)=x^2.
\end{equation*}
Which completeS the proof.
\end{proof}
We recall the weighted spaces of the functions on $\mathbb{R}^{+}$, which
are defined as follows:
\begin{eqnarray*}
P_{\rho }(\mathbb{R}^{+}) &=&\left\{ f:\mid f(x)\mid \leq M_{f}\rho
(x)\right\} , \\
Q_{\rho }(\mathbb{R}^{+}) &=&\left\{ f:f\in P_{\rho }(\mathbb{R}^{+})\cap
C[0,\infty )\right\} , \\
Q_{\rho }^{k}(\mathbb{R}^{+}) &=&\left\{ f:f\in Q_{\rho }(\mathbb{R}^{+})~~
\mbox{and}~~~\lim_{x\rightarrow \infty }\frac{f(x)}{\rho (x)}=k(k~~~\mbox{is
~~~\mbox{a}~~~\mbox{constant})\right\} ,
\end{eqnarray*
where $\rho (x)=1+x^{2}$ is a weight function and $M_{f}$ is a constant
depending only on $f$. Note that $Q_{\rho }(\mathbb{R}^{+})$ is a normed
space with the norm $\parallel f\parallel _{\rho }=\sup_{x\geq 0}\frac{\mid
f(x)\mid }{\rho (x)}$.
\begin{theorem}
\label{snth2} Let $q=q_n$ satisfying \eqref{snnas5}, for $0<q_n< 1$ and if
T_{n,q_n}^*(.~;~.)$ be the operators given by \eqref{snss1}. Then for any
function $f \in Q^k_\rho(\mathbb{R}^+)$ we have
\begin{equation*}
\lim_{n\to \infty} \parallel T_{n,q_n}^*(f;x)-f \parallel_\rho=0.
\end{equation*}
\end{theorem}
\begin{proof}
From Lemma \ref{snlm1}, the first condition of \eqref{snlm11} is fulfilled
for $\tau=0$. Now for $\tau=1,2$ it is easy to see that from (\ref{snlm12}),
(\ref{snlm13}) of Lemma \ref{snlm1} by using \eqref{snnas5} \newline
\begin{equation*}
\parallel T_{n,q_n}^* \left( t) ^{\tau};x\right) -x ^{\tau }\parallel
_{\rho} =0.
\end{equation*
This completes the proof.
\end{proof}
\section{\textbf{Rate of Convergence}}
Here we calculate the rate of convergence of operators \eqref{snss1} by
means of modulus of continuity and Lipschitz type maximal functions.
Let $f\in C[0,\infty ]$. The modulus of continuity of $f$ denoted by $\omega
(f,\delta )$ gives the maximum oscillation of $f$ in any interval of length
not exceeding $\delta >0$ and it is given by
\begin{equation}
\omega (f,\delta )=\sup_{\mid y-x\mid \leq \delta }\mid f(y)-f(x)\mid
,~~~x,y\in \lbrack 0,\infty ). \label{snson1}
\end{equation
It is known that $\lim_{\delta \rightarrow 0+}\omega (f,\delta )=0$ for
f\in C[0,\infty )$ and for any $\delta >0$ one has
\begin{equation}
\mid f(y)-f(x)\mid \leq \left( \frac{\mid y-x\mid }{\delta }+1\right) \omega
(f,\delta ). \label{snson2}
\end{equation}
\begin{theorem}
Let $T_{n,q}^*(.~;~.)$ be the operators defined by \eqref{snss1}. Then for
f\in \tilde{C}[0,\infty),~~x\geq 0, ~~0<q< 1$ we have
\begin{equation*}
T_{n,q}^*(f;x)-f(x)\leq \left\{ 1+\sqrt{\phi_n(x)}\right\} \omega\left(f;\frac{1}{\sqrt{
[n]_{q}}}\right),
\end{equation*}
where
\begin{eqnarray*}
\phi_n(x)=\frac{n}{(n+\beta)^2[n]_q}(\frac{n}{[3]_q[n]_q}+\frac{2\alpha}{[2]_q})+\frac{\alpha^2}{(n+\beta)^2}+\{\frac{n^2}{(n+\beta)^2}\frac{3}{[3]_q[n]_q}
(1+[1+2\mu]_q)\\
+\frac{2n}{(n+\beta)[2]_q}(2\alpha-\frac{1}{[n]_q})-\frac{2\alpha}{n+\beta}\}x+\{\frac{n}{n+\beta}(\frac{3n}{(n+\beta)[3]_q}
-\frac{4n}{n+\beta)[2]_q})+1\}x^2
\end{eqnarray*}
and $\tilde{C}[0,\infty)$ is the space of uniformly continuous functions
on $\mathbb{R}^+$ and $\omega(f,\delta)$ is the modulus of continuity of the
function $f \in \tilde{C}[0,\infty)$ defined in \eqref{snson1}.
\end{theorem}
\begin{proof}
We prove it by using \eqref{snson1}, \eqref{snson2} and Cauchy-Schwarz
inequality.\newline
$\mid T_{n,q}^{\ast }(f;x)-f(x)\mid $
\begin{eqnarray*}
&\leq &\frac{[n]_{q}}{e_{\mu ,q}([n]_{q}x)}\sum_{k=0}^{\infty }\frac
([n]_{q}x)^{k}}{\gamma _{\mu ,q}(k)}\int_{\frac{q[k+2\mu \theta _{k}]_{q}}
[n]_{q}}}^{\frac{[k+1+2\mu \theta _{k}]_{q}}{[n]_{q}}}\mid f(t)-f(x)\mid
d_{q}(t) \\
&\leq &\frac{[n]_{q}}{e_{\mu ,q}([n]_{q}x)}\sum_{k=0}^{\infty }\frac
([n]_{q}x)^{k}}{\gamma _{\mu ,q}(k)}\int_{\frac{q[k+2\mu \theta _{k}]_{q}}
[n]_{q}}}^{\frac{[k+1+2\mu \theta _{k}]_{q}}{[n]_{q}}}\left( 1+\frac{1}
\delta }\mid t-x\mid \right) d_{q}(t)\omega (f;\delta ) \\
&=&\left\{ 1+\frac{1}{\delta }\left( \frac{[n]_{q}}{e_{\mu ,q}([n]_{q}x)
\sum_{k=0}^{\infty }\frac{([n]_{q}x)^{k}}{\gamma _{\mu ,q}(k)}\int_{\frac
q[k+2\mu \theta _{k}]_{q}}{[n]_{q}}}^{\frac{[k+1+2\mu \theta _{k}]_{q}}
[n]_{q}}}\mid t-x\mid d_{q}(t)\right) \right\} \omega (f;\delta ) \\
&\leq &\left\{ 1+\frac{1}{\delta }\left( \frac{[n]_{q}}{e_{\mu ,q}([n]_{q}x)
\sum_{k=0}^{\infty }\frac{([n]_{q}x)^{k}}{\gamma _{\mu ,q}(k)}\int_{\frac
q[k+2\mu \theta _{k}]_{q}}{[n]_{q}}}^{\frac{[k+1+2\mu \theta _{k}]_{q}}
[n]_{q}}}(t-x)^{2}d_{q}(t)\right) ^{\frac{1}{2}}\left( T_{n,q}^{\ast
}(1;x)\right) ^{\frac{1}{2}}\right\} \omega (f;\delta ) \\
&=&\left\{ 1+\frac{1}{\delta }\left( T_{n,q}^{\ast }(t-x)^{2};x\right) ^
\frac{1}{2}}\right\} \omega (f;\delta ) \\
&&
\end{eqnarray*
if we choose $\delta =\delta _{n}=\sqrt{\frac{1}{[n]_{q}}}$, then we get our
result.
\end{proof}
Now we give the rate of convergence of the operators ${T}_{n,q}^*(f;x) $
defined in \eqref{snss1} in terms of the elements of the usual Lipschitz
class $Lip_{M}(\nu )$.
Let $f\in C[0,\infty )$, $M>0$ and $0<\nu \leq 1$. The class $Lip_{M}(\nu )$
is defined as
\begin{equation}
Lip_{M}(\nu )=\left\{ f:\mid f(\zeta _{1})-f(\zeta _{2})\mid \leq M\mid
\zeta _{1}-\zeta _{2}\mid ^{\nu }~~~(\zeta _{1},\zeta _{2}\in \lbrack
0,\infty ))\right\} \label{snn1}
\end{equation}
\begin{theorem}
\label{snsn1} Let $T_{n,q}^*(.~;~.)$ be the operator defined in \eqref{snss1
. Then for each $f\in Lip_{M}(\nu ),~~(M>0,~~~0<\nu \leq 1)$ satisfying
\eqref{snn1} we have
\begin{equation*}
\mid T_{n,q}^*(f;x)-f(x)\mid \leq M \left(\lambda_{n}(x)\right)^{\frac{\nu}{
}}
\end{equation*}
where $\lambda_{n}(x)=T_{n,q}^*\left((t-x)^2;x\right)$.
\end{theorem}
\begin{proof}
We prove it by using \eqref{snn1} and H\"{o}lder inequality.
\begin{eqnarray*}
\mid T_{n,q}^{\ast }(f;x)-f(x)\mid &\leq &\mid T_{n,q}^{\ast
}(f(t)-f(x);x)\mid \\
&\leq &T_{n,q}^{\ast }\left( \mid f(t)-f(x)\mid ;x\right) \\
&\leq &\mid MT_{n,q}^{\ast }\left( \mid t-x\mid ^{\nu };x\right) .
\end{eqnarray*
Therefore\newline
$\mid T_{n,q}^*(f;x)-f(x) \mid$
\begin{eqnarray*}
&\leq & M \frac{[n]_q}{e_{\mu,q}([n]_qx)}\sum_{k=0}^\infty \frac{([n]_qx)^{k
}{\gamma_{\mu,q}(k)} \int_{\frac{q[k+2\mu \theta_k]_q}{[n]_q}}^{\frac
[k+1+2\mu \theta_k]_q}{[n]_q}}\mid t-x \mid^\nu d_q(t) \\
& \leq & M \frac{[n]_q}{e_{\mu,q}([n]_qx)}\sum_{k=0}^\infty \left(\frac
([n]_qx)^{k}}{\gamma_{\mu,q}(k)}\right)^{\frac{2-\nu}{2}} \\
& \times & \left(\frac{([n]_qx)^{k}}{\gamma_{\mu,q}(k)}\right)^{\frac{\nu}{2
} \int_{\frac{q[k+2\mu \theta_k]_q}{[n]_q}}^{\frac{[k+1+2\mu \theta_k]_q}
[n]_q}}\mid t-x \mid^\nu d_q(t) \\
& \leq & M \left(\frac{[n]_q}{\left(e_{\mu,q}([n]_qx)\right)
\sum_{k=0}^\infty \frac{([n]_qx)^{k}}{\gamma_{\mu,q}(k)}\int_{\frac{q[k+2\mu
\theta_k]_q}{[n]_q}}^{\frac{[k+1+2\mu \theta_k]_q}{[n]_q}} d_q(t)\right)^
\frac{2-\nu}{2}} \\
& \times & \left(\frac{[n]_q}{\left(e_{\mu,q}([n]_qx)\right)
\sum_{k=0}^\infty \frac{([n]_qx)^{k}}{\gamma_{\mu,q}(k)} \int_{\frac{q[k+2\mu
\theta_k]_q}{[n]_q}}^{\frac{[k+1+2\mu \theta_k]_q}{[n]_q}}\mid t-x \mid^2
d_q(t) \right)^{\frac{\nu}{2}} \\
& = & M \left(T_{n,q}^*(t-x)^2;x\right)^{\frac{\nu}{2}}.
\end{eqnarray*}
Which completes the proof.
\end{proof}
Let $C_{B}[0,\infty )$ denote the space of all bounded and continuous
functions on $\mathbb{R}^{+}=[0,\infty )$ and
\begin{equation}
C_{B}^{2}(\mathbb{R}^{+})=\{g\in C_{B}(\mathbb{R}^{+}):g^{\prime },g^{\prime
\prime }\in C_{B}(\mathbb{R}^{+})\}, \label{snt2}
\end{equation
with the norm
\begin{equation}
\parallel g\parallel _{C_{B}^{2}(\mathbb{R}^{+})}=\parallel g\parallel
_{C_{B}(\mathbb{R}^{+})}+\parallel g^{\prime }\parallel _{C_{B}(\mathbb{R
^{+})}+\parallel g^{\prime \prime }\parallel _{C_{B}(\mathbb{R}^{+})},
\label{snt1}
\end{equation
also
\begin{equation}
\parallel g\parallel _{C_{B}(\mathbb{R}^{+})}=\sup_{x\in \mathbb{R}^{+}}\mid
g(x)\mid . \label{snt3}
\end{equation}
\begin{theorem}
\label{snsn2} Let $T_{n,q}^*(.~;~.)$ be the operator defined in \eqref{snss1
. Then for any $g \in C_B^2(\mathbb{R}^+)$ we have
\begin{equation*}
\mid T_{n,q}^*(f;x)-f(x)\mid \leq (\frac{2qn}{(n+\beta)[2]_q}-1)x+\frac{n}{(n+\beta)[2]_q[n]_q}+\frac{\alpha}{n+\beta}+\frac{\lambda_{n}(x)}{2}) \parallel g
\parallel_{C_B^2(\mathbb{R}^+)}
\end{equation*}
where $\lambda_{n}(x)$ is given in Theorem \ref{snsn1}.
\end{theorem}
\begin{proof}
Let $g\in C_{B}^{2}(\mathbb{R}^{+})$, then by using the generalized mean
value theorem in the Taylor series expansion we have
\begin{equation*}
g(t)=g(x)+g^{\prime }(x)(t-x)+g^{\prime \prime }(\psi )\frac{(t-x)^{2}}{2
,~~~\psi \in (x,t).
\end{equation*
By applying linearity property on $T_{n,q}^{\ast },$ we have
\begin{equation*}
T_{n,q}^{\ast }(g,x)-g(x)=g^{\prime }(x)T_{n,q}^{\ast }\left( (t-x);x\right)
+\frac{g^{\prime \prime }(\psi )}{2}T_{n,q}^{\ast }\left( (t-x)^{2};x\right),
\end{equation*
which implies \newline
$\mid T_{n,q}^{\ast }(g;x)-g(x)\mid $
\begin{eqnarray*}
&\leq &\left(\frac{2qn}{(n+\beta)[2]_q}-1\right)x+\frac{n}{(n+\beta)[2]_q[n]_q}+\frac{\alpha}{n+\beta} \parallel g^{\prime }\parallel _{C_{B}(\mathbb{R}^{+})} \\
&+&\frac{n}{(n+\beta)^2[n]_q}(\frac{n}{[3]_q[n]_q}+\frac{2\alpha}{[2]_q})+\frac{\alpha^2}{(n+\beta)^2}+\{\frac{n^2}{(n+\beta)^2}\frac{3}{[3]_q[n]_q}
\times(1+[1+2\mu]_q)\\
&+&\frac{2n}{(n+\beta)[2]_q}(2\alpha-\frac{1}{[n]_q})-\frac{2\alpha}{n+\beta}\}x+\{\frac{n}{n+\beta}(\frac{3n}{(n+\beta)[3]_q}
-\frac{4n}{n+\beta)[2]_q})+1\}x^2
\times\frac{\parallel g^{\prime \prime
}\parallel _{C_{B}(\mathbb{R}^{+})}}{2}.
\end{eqnarray*
On using \eqref{snt1}~~~$\parallel g^{\prime }\parallel
_{C_{B}[0,\infty )}\leq \parallel g\parallel _{C_{B}^{2}[0,\infty )}$
completes the proof from \ref{snlm23} of Lemma \ref{snlm2}.
\end{proof}
The Peetre's $K$-functional is defined by
\begin{equation}
K_{2}(f,\delta )=\inf_{C_{B}^{2}(\mathbb{R}^{+})}\left\{ \left( \parallel
f-g\parallel _{C_{B}(\mathbb{R}^{+})}+\delta \parallel g^{\prime \prime
}\parallel _{C_{B}^{2}(\mathbb{R}^{+})}\right) :g\in \mathcal{W}^{2}\right\}
, \label{snzr1}
\end{equation
where
\begin{equation}
\mathcal{W}^{2}=\left\{ g\in C_{B}(\mathbb{R}^{+}):g^{\prime },g^{\prime
\prime }\in C_{B}(\mathbb{R}^{+})\right\} . \label{snzr2}
\end{equation
There exits a positive constant $C>0$ such that $K_{2}(f,\delta )\leq
C\omega _{2}(f,\delta ^{\frac{1}{2}}),~~\delta >0$, where the second order
modulus of continuity is given by
\begin{equation}
\omega _{2}(f,\delta ^{\frac{1}{2}})=\sup_{0<h<\delta ^{\frac{1}{2
}}\sup_{x\in \mathbb{R}^{+}}\mid f(x+2h)-2f(x+h)+f(x)\mid . \label{snzr3}
\end{equation}
\begin{theorem}
\label{snsn3} Let $T_{n,q}^{\ast }(.~;~.)$ be the operator defined in
\eqref{snss1} and $C_{B}[0,\infty )$ be the space of all bounded and
continuous functions on $\mathbb{R}^{+}$. Then for $x\in \mathbb{R}^{+},$
f\in C_{B}(\mathbb{R}^{+})$ we have\newline
$\mid T_{n,q}^{\ast }(f;x)-f(x)\mid $\newline
$\leq 2M\left\{ \omega _{2}\left( f;\sqrt{\frac{(\frac{4qn}{(n+\beta)[2]_q}-2)x
+\left( \frac{2n}{(n+\beta)[2]_q[n]_q}+\frac{2\alpha}{n+\beta}\right)+\lambda _{n}(x)}{4}}\right)\newline
+\min\left( 1,\frac{(\frac{4qn}{(n+\beta)[2]_{q}}-2)x+\left( \frac{2n}{(n+\beta)[2]_q[n]_q}+\frac{2\alpha}{n+\beta}\right)
+\lambda _{n}(x)}{4}\right) \\
\parallel f\parallel _{C_{B}(\mathbb{R
^{+})}\right\} $,\newline
where $M$ is a positive constant, $\lambda _{n}(x)$ is given in Theorem \re
{snsn1} and $\omega _{2}(f;\delta )$ is the second order modulus of
continuity of the function $f$ defined in \eqref{snzr3}.
\end{theorem}
\begin{proof}
We prove this by using the Theorem \eqref{snsn2}
\begin{eqnarray*}
\mid T_{n,q}^*(f;x)-f(x)\mid &\leq & \mid T_{n,q}^*(f-g;x)\mid+\mid
T_{n,q}^*(g;x)-g(x)\mid+\mid f(x)-g(x)\mid \\
&\leq & 2 \parallel f-g \parallel_{C_B(\mathbb{R}^+)}+ \frac{\lambda_n(x)}{2
\parallel g \parallel_{C_B^2(\mathbb{R}^+)} \\
&+ &\left(\frac{2qn}{(n+\beta)[2]_q}-1\right)x+\frac{n}{(n+\beta)[2]_q[n]_q}+\frac{\alpha}{n+\beta
\parallel g \parallel_{C_B(\mathbb{R}^+)}
\end{eqnarray*}
From \eqref{snt1} clearly we have ~~~$\parallel g
\parallel_{C_B[0,\infty)}\leq \parallel g \parallel_{C_B^2[0,\infty)}$
\newline
Therefore,
\begin{equation*}
\mid T_{n,q}^*(f;x)-f(x)\mid \leq 2 \left(\parallel f-g \parallel_{C_B
\mathbb{R}^+)}+\frac{(\frac{4qn}{(n+\beta)[2]_q}-2)x+\frac{2n}{(n+\beta)[2]_q[n]_q}+\frac{2\alpha}{n+\beta}+
\lambda_n(x)}{4}\parallel g \parallel_{C_B^2(\mathbb{R}^+)}\right),
\end{equation*}
where $\lambda_{n}(x)$ is given in Theorem \ref{snsn1}.\newline
By taking infimum over all $g\in C_{B}^{2}(\mathbb{R}^{+})$ and by using
\eqref{snzr1}, we get
\begin{equation*}
\mid T_{n,q}^{\ast }(f;x)-f(x)\mid \leq 2K_{2}\left( f;\frac{(\frac{4qn}{(n+\beta)[2]_q}-2)x+\frac{2n}{(n+\beta)[2]_q[n]_q}+\frac{2\alpha}{n+\beta}+\lambda _{n}(x)}{4
\right)
\end{equation*
Now for an absolute constant $D>0$ in \cite{scc1} we use the relation
\begin{equation*}
K_{2}(f;\delta )\leq D\{\omega _{2}(f;\sqrt{\delta })+\min (1,\delta
)\parallel f\parallel \}.
\end{equation*
This complete the proof.
\end{proof}
Atakut and Ispir \cite{satk} introduced the weighted modulus of continuity
and defined as, for an arbitrary $f\in Q_{\rho }^{k}(\mathbb{R}^{+})$
\begin{equation}
\Omega (f,\delta )=\sup_{x\in \lbrack 0,\infty ),\mid h\mid \leq \delta
\frac{\mid f(x+h)-f(x)\mid }{(1+h^{2})(1+x^{2})}. \label{snd}
\end{equation
The two main properties of this modulus of continuity are $\lim_{\delta
\rightarrow 0}\Omega (f,\delta )\rightarrow 0$ and
\begin{equation}
\mid f(t)-f(x)\mid \leq 2\left( 1+\frac{\mid t-x\mid }{\delta }\right)
(1+\delta ^{2})(1+x^{2})(1+(t-x)^{2})\Omega (f,\delta ), \label{smd}
\end{equation
where $f\in Q_{\rho }^{k}(\mathbb{R}^{+})$ and $t,x\in \lbrack 0,\infty )$.
\begin{theorem}
Let $T_{n,q}^*(.~;~.)$ be the operators defined by \eqref{snss1}. Then for
f \in Q^k_\rho(\mathbb{R}^+),~~0<q< 1$ and $x\geq 0 $ we have \newline
\begin{equation*}
\sup_{x \in [0,\infty)}\frac{\mid T_{n,q}^*(f;x)-f(x)\mid}{(1+x^2)} \leq
C_\mu\left( 1+\frac{1}{[n]_q}\right)\Omega(f,\frac{1}{\sqrt{[n]_q}}),
\end{equation*}
where $C_\mu$ is constant independent of $n$.
\end{theorem}
\begin{proof}
We prove it by using \eqref{snd}, \eqref{smd} and Cauchy-Schwarz inequality
\newline
$\mid T_{n,q}^{\ast }(f;x)-f(x)\mid $
\begin{eqnarray*}
&\leq &\frac{[n]_{q}}{e_{\mu ,q}([n]_{q}x)}\sum_{k=0}^{\infty }\frac
([n]_{q}x)^{k}}{\gamma _{\mu ,q}(k)}\int_{\frac{q[k+2\mu \theta _{k}]_{q}}
[n]_{q}}}^{\frac{[k+1+2\mu \theta _{k}]_{q}}{[n]_{q}}}\mid f(t)-f(x)\mid
d_{q}(t) \\
&\leq &2(1+\delta ^{2})(1+x^{2})\Omega (f;\delta )\frac{[n]_{q}}{e_{\mu
,q}([n]_{q}x)}\sum_{k=0}^{\infty }\frac{([n]_{q}x)^{k}}{\gamma _{\mu ,q}(k)
\int_{\frac{q[k+2\mu \theta _{k}]_{q}}{[n]_{q}}}^{\frac{[k+1+2\mu \theta
_{k}]_{q}}{[n]_{q}}}\left( 1+\frac{1}{\delta }\mid t-x\mid \right) \left(
1+(t-x)^{2}\right) d_{q}(t) \\
&=&2(1+\delta ^{2})(1+x^{2})\Omega (f;\delta )\frac{[n]_{q}}{e_{\mu
,q}([n]_{q}x)} \\
&\times &\bigg{\{}\sum_{k=0}^{\infty }\frac{([n]_{q}x)^{k}}{\gamma _{\mu
,q}(k)}+\sum_{k=0}^{\infty }\frac{([n]_{q}x)^{k}}{\gamma _{\mu ,q}(k)}\int_
\frac{q[k+2\mu \theta _{k}]_{q}}{[n]_{q}}}^{\frac{[k+1+2\mu \theta _{k}]_{q}}
[n]_{q}}}(t-x)^{2}d_{q}(t) \\
&+&\frac{1}{\delta }\sum_{k=0}^{\infty }\frac{([n]_{q}x)^{k}}{\gamma _{\mu
,q}(k)}\int_{\frac{q[k+2\mu \theta _{k}]_{q}}{[n]_{q}}}^{\frac{[k+1+2\mu
\theta _{k}]_{q}}{[n]_{q}}}\mid t-x\mid d_{q}(t) \\
&+&\frac{1}{\delta }\sum_{k=0}^{\infty }\frac{([n]_{q}x)^{k}}{\gamma _{\mu
,q}(k)}\int_{\frac{q[k+2\mu \theta _{k}]_{q}}{[n]_{q}}}^{\frac{[k+1+2\mu
\theta _{k}]_{q}}{[n]_{q}}}\mid t-x\mid (t-x)^{2}d_{q}(t)\bigg{\}} \\
&\leq &2(1+\delta ^{2})(1+x^{2})\Omega (f;\delta ) \\
&\times &\left( 1+T_{n,q}^{\ast }((t-x)^{2};x)+\frac{1}{\delta }\sqrt
T_{n,q}^{\ast }((t-x)^{2};x)}+\frac{1}{\delta }\sqrt{T_{n,q}^{\ast
}((t-x)^{2};x)T_{n,q}^{\ast }((t-x)^{4};x)}\right) \\
&&
\end{eqnarray*
where $T_{n,q}^{\ast }((t-x)^{2};x)$ and $T_{n,q}^{\ast }((t-x)^{4};x)$ is
defined in (\ref{snlm23}) and (\ref{snlm24}) of Lemma \ref{snlm2}.\newline
If we choose $\delta =\delta _{n}=\sqrt{\frac{1}{[n]_{q}}}$, then we get our
result.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,111 |
Neoleucinodes alegralis is een vlinder uit de familie van de grasmotten (Crambidae), uit de onderfamilie van de Spilomelinae. De wetenschappelijke naam van deze soort is voor het eerst gepubliceerd in 1920 door William Schaus.
De soort komt voor in Guatemala, Panama en Frans-Guyana.
Grasmotten
Dier uit het Neotropisch gebied | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,277 |
\section{Introduction}
Algebraic multigrid (AMG) was designed as a solver for large, sparse linear systems,
typically M-matrices, resulting from the discretization of elliptic PDEs. For many such
problems, AMG has been shown to achieve fast convergence, and scale in parallel to hundreds of thousands of
processors \cite{Baker:2012ko}. Such convergence and scaling properties are desirable
for solvers, and substantial work has been devoted to broadening the applicability of AMG.
Continued research has made for a rich theoretical basis for AMG \cite{Falgout:2004cs,
Falgout:2005hm, Vassilevski:2008wd, Vassilevski:2010vy, MacLachlan:2014di},
as well as many numerical implementations and variations that are either
robust for a larger class of linear systems \cite{Brezina:2004eh,DAmbra:2013iwa,
Manteuffel:2016vd}, or effective at solving a specific problem such as linear elasticity
\cite{Baker:2009va} or Hemholtz \cite{Olson:2010bh}. Nevertheless, the one-size-fits-all AMG
solver remains elusive, in part because many of the theoretical results are
difficult to use in a practical setting.
A novel feature of AMG in contrast to many other linear solvers is
that the setup and solve complexity in terms of floating point
operations (FLOPs) are both typically linear or log-linear in the total number of degrees of freedom (DOFs).
This is fundamental to good scaling of time to solution with increasing
problem size, but also limits the options in algorithm design, particularly when trying
to directly use theoretical results on convergence. Two common aspects
seen in AMG convergence theory are the use of orthogonal projections
onto subspaces and requiring
a given approximation property to hold for all vectors. In both cases,
namely constructing an orthogonal projection or enforcing a constraint for $n$
basis vectors, the complexity of explicitly enforcing such requirements is at least quadratic in $n$
and, thus, not feasible in keeping with the desired linear complexity of AMG. Furthermore,
AMG can only have linear complexity when all operators are sparse, including the coarse-grid
operators constructed for a multilevel algorithm. In the abstract setting of convergence
theory, such sparsity constraints are not accounted for, adding an additional
barrier to the direct use of convergence theory in practical methods.
Here, we review the tension between theory and practice in AMG and propose
a new variant of AMG that aims to directly address these complications. An overview of AMG convergence theory is given in Section \ref{sec:theory}.
Fundamental results on two-grid and multigrid convergence theory are presented in a simple and
consistent manner, to clarify what is required of interpolation operators for effective
AMG convergence, and the so-called ``optimal'' and ``ideal'' interpolation operators
are introduced. Section \ref{sec:practice} proposes a discussion of AMG
interpolation operators used in practice, and how they relate to theoretical results,
along with an examination of how different theoretical results can be approximated
in linear complexity. This leads to the introduction of a general weighted functional
to be minimized in forming interpolation operators in Section \ref{sec:functional}, which is
shown to have a unique solution for a fixed interpolation sparsity pattern.
A conjugate gradient method is developed to approximate the solution, with a novel
preconditioner that is applicable to general equations with a Sylvester- or Lyapunov-like form
(Section \ref{sec:pcg}). Numerical results demonstrate that a constrained
energy minimization \cite{Brannick:2007fb,Mandel:1999wg,Olson:2011fg,Wan:1999ky}
consistently outperforms a weighted energy minimization. Although this may seem intuitive for
AMG researchers, a number of other interesting results also come up that lead to
open questions on interpolation in AMG:
\begin{itemize}
\item Enforcing one constraint vector to be (almost) exactly in the range of interpolation is fundamental to good AMG convergence.
However, adding additional constraint vectors that are not effectively reduced by the current AMG hierarchy does not necessarily
improve convergence, which is at odds with motivation of traditional adaptive AMG methods.
\item Energy-minimization applied to columns of $P$ (while maintaining constraints) is also fundamental to a
convergent AMG method for some more difficult model problems. However, although further iterations of energy minimization
continue to reduce the associated residual, convergence of the
resulting AMG solver does not improve after a small
number of iterations.
\item Using a diagonal preconditioner for energy-minimization iterations applied to columns of $P$ can offer significant improvement
in convergence of the resulting AMG solver.
\end{itemize}
\section{Theoretical framework}\label{sec:theory}
Multilevel solvers come in various forms, including geometric multigrid (GMG),
AMG, finite element algebraic multigrid (AMGe), algebraic multilevel iterations (AMLI),
and the method of subspace corrections. In this work, we focus on AMG as
a general method to solve a linear system $A\vect{x} = \vect{b}$ using
only ``algebraic'' information, in contrast to GMG and
AMGe, which require additional information, on the underlying grid or finite element stiffness matrices,
respectively. Subspace corrections are presented in a more general yet framework than AMG,
but analysis of subspace correction can also be applied to AMG \cite{Vassilevski:2008wd}.
The basis for AMG as an iterative method to solve $A\vect{x}=\vect{b}$ is in reducing error
through two processes: ``relaxation'' and ``coarse-grid correction.'' If designed properly,
these processes are complementary in the sense that they are effective on
different error modes and, together, effectively reduce all types of error.
Relaxation refers to a general iterative method of the form
\[
\mathbf{x}_{k+1} = \mathbf{x}_k + M^{-1}(\mathbf{b} - A\mathbf{x}_k),
\]
and is often chosen to be a simple method such as Jacobi or Gauss-Seidel.
This process is typically efficient at removing ``high-frequency error,'' or error
associated with large eigenvalues of $A$. Convergence of a relaxation scheme
in the ``energy norm'' or ``$A$-norm,'' $\|\mathbf{v}\|_A^2 = \langle A\mathbf{x},\mathbf{x}\rangle$, is equivalent
to bounding the error-propagation matrix in the $A$-norm, namely
$\|I - M^{-1}A\|_A < 1$. Furthermore, $\|I - M^{-1}A\|_A < 1$ if and only if
$M+M^T-A$ is symmetric positive definite (SPD) \cite[Theorem 2.3.1]{Vassilevski:2010vy}. In this
case, we say that $M$ is an $A$-convergent relaxation operator.
Multigrid originated in the geometric setting, where high-frequency error
actually has a high physical frequency. Since standard relaxation schemes
such as Jacobi and Gauss-Seidel are able to capture this error well, the natural
way to capture the converse, low-frequency error, is to recursively coarsen
the underlying grid so that low-frequency modes on the initial fine grid
appear high-frequency on coarser grids. Relaxation on coarser grids will
then reduce this error, and the results can be interpolated back to the fine grid.
The algebraic concept is much the same but low-frequency refers to algebraically smooth
modes, corresponding to large eigenvalues of $I - M^{-1}A$ or, typically, small eigenvalues of $A$,
and vice-versa for high-frequency. Because there is no explicit grid in the
algebraic setting, algebraic coarsening is based on choosing a coarse subspace
which can capture algebraically smooth error from the fine grid.
For an $n\times n$ SPD matrix $A$, consider an $\ell^2$-orthogonal decomposition of
$\mathbb{R}^n$, where any $\mathbf{x}$ can be decomposed
as $\vect{x} = R^T\vect{y}_c + S\vect{y}_f$, where $RS = 0$. Here, $S$ corresponds
to the space on which relaxation is effective, and $R$ defines the coarse space on which a coarse-grid correction
is constructed. Defining the interpolation operator, $P$, we assume that $PR$ is a projection, which requires $RP = I$.
A Galerkin coarse-grid operator is formed, $A_c = P^TAP$, and an exact coarse-grid correction (in the $A$-norm) given
by the $A$-orthogonal projection onto $\Ima(P)$, $\pi_A = P(P^TAP)^{-1}P^TA$.
In this work, a CF-style splitting will be used (or in the case of
aggregation-based coarsening, a root-node approach, where one node in each
aggregate is declared a C-point and the rest F-points \cite{Manteuffel:2016vd}).
A CF-splitting has the useful property that coarse-grid nodes are a subset of
nodes on the current grid, allowing for $A$ to be written
in the block form
\[
A = \begin{bmatrix} A_{ff} & A_{fc} \\ A_{cf} & A_{cc} \end{bmatrix}.
\]
In this case, splitting operators take the form $R = \begin{bmatrix} 0 & I\end{bmatrix}$,
$S = \begin{bmatrix} I & 0\end{bmatrix}^T$, and $P = \begin{bmatrix} W^T & I\end{bmatrix}^T$,
where $RAR^T = A_{cc}$ and $S^TAS = A_{ff}$.
Together, the two-grid error-propagation matrix operator for AMG, with an $A$-symmetric relaxation scheme
based on $M^{-1}$ and $M^{-T}$, is given by
\[
E_{TG} = (I - M^{-T}A)(I - \pi_A)(I - M^{-1}A).
\]
Convergence of $E_{TG}$ is generally considered in the $A$-norm, where each iteration reduces
error in the $A$-norm by at least a factor of $\|E_{TG}\|_A$. Noting that $E_{TG}$ is symmetric in the
$A$-norm, it follows that eigenvectors of $E_{TG}$ are $A$-orthogonal and $\|E_{TG}\|_A = \rho(E_{TG})$.
Thus, optimizing the AMG convergence rate can be viewed
equivalently as minimizing $\|E_{TG}\|_A$ or $\rho(E_{TG})$. Two-grid convergence
can also be considered in terms of the spectral equivalence between a preconditioner, $B_{TG}$, and $A$,
where $E_{TG} = I - B_{TG}^{-1}A$ \cite[Proposition 5.1.2]{Vassilevski:2010vy}; however, here we
bound convergence in terms of $\|E_{TG}\|_A$ for consistency. A multilevel method is implemented
and analyzed as a two-grid method with an inexact coarse-grid solve, where the coarse-grid ``solve'' recursively
calls a two-grid method on the coarse-grid problem. Multigrid convergence is also considered in the $A$-norm,
where we want to bound $\|E_{MG}\|_A \leq K < 1$.
Let $M$ and $M^T$ be $A$-convergent relaxation operators and define the symmetrized relaxation operator
as $\widetilde{M} = M^T(M+M^T-A)^{-1}M$, so that $I-\widetilde{M}^{-1}A = (I-M^{-1}A)(I - M^{-T}A)$.
This symmetrizes the action of $M$ and is used primarily as a
theoretical tool (as $\widetilde{M}$ is rarely easily
computable). Common bounds on two-grid convergence factors come from considering various orthogonal
projections onto the range of the interpolation operator, $P$. Define
$\pi_X := P(P^TXP)^{-1}P^TX$ as the unique $X$-orthogonal projection
onto $\Ima(P)$ for some nonsingular operator $X$, e.g. $A$ or $\widetilde{M}$, and
$Q_P := P(P^TP)^{-1}P$ as the $l^2$-orthogonal projection onto $\Ima(P)$.
For computable bounds, assume that $X$ is spectrally equivalent to $\widetilde{M}$, denoted $X\simeq \widetilde{M}$;
that is, there exists $0<c_1\leq c_2$ such that
\begin{equation}
c_1\mathbf{v}^TX\mathbf{v} \leq \mathbf{v}^T\widetilde{M}\mathbf{v} \leq c_2\mathbf{v}^TX\mathbf{v}. \label{eq:spec_equiv}
\end{equation}
\begin{linenomath}
It follows from \eqref{eq:spec_equiv} and the definition of orthogonal projections that
\begin{align}
\|(I-\pi_X)\mathbf{v}\|_X^2 & \leq \|(I-\pi_{\widetilde{M}})\mathbf{v}\|_X^2 \leq \frac{1}{c_1}\|(I-\pi_{\widetilde{M}})\mathbf{v}\|_{\widetilde{M}}^2 \label{eq:X_M1} \\
\|(I-\pi_{\widetilde{M}})\mathbf{v}\|_{\widetilde{M}}^2 & \leq \|(I-\pi_X)\mathbf{v}\|_{\widetilde{M}}^2 \leq c_2\|(I-\pi_X)\mathbf{v}\|_X^2 \label{eq:X_M2}
\end{align}
for all $\mathbf{v}$.
\end{linenomath}
\begin{linenomath}
Finally, note the following identities with respect to the Frobenius inner product and trace that are used regularly in this work:
\begin{align*}
\langle A,B\rangle_F & = \sum_{ij}A_{ij}B_{ij} = \tr(B^TA) = \tr(A^TB), \\
\tr(ABC) & = \tr(CAB) = \tr(BCA),
\end{align*}
and let $A\circ B$ denote the Hadamard product, defined as the element-wise multiplication of two matrices
$A,B\in\mathbb{R}^{m\times n}$.
\end{linenomath}
\subsection{Two-grid convergence}
Substantial work has been devoted to understanding convergence theory of AMG
in the two-grid setting \cite{Lee:2008iu,Notay:2014uc,Vassilevski:2010vy,
Vassilevski:2008wd,Falgout:2004cs,Falgout:2005hm,MacLachlan:2014di,Zikatanov:2008jp}.
By Lemma 4.1 of \cite{SFMcCormick_1984b}, we can analyze
$\|E_{TG}\|_A$ either directly, or by considering the variants with only
pre- or post-relaxation, as
\[
\|E_{TG}\|_A = \|(I-\pi_A)(I-M^{-1}A)\|_A^2 = \|(I-M^{-T}A)(I-\pi_A)\|_A^2.
\]
One of the simplest two-grid convergence bounds is given by Lemma 2.3
of \cite{SFMcCormick_1985b}:
\begin{theorem}
If there is a $\delta > 0$ such that
\[
\|(I-M^{-T}A)\vect{v}\|_A^2 \leq \|\vect{v}\|_A^2 - \delta
\|(I-\pi_A)\vect{v}\|_A^2,
\]
for all $\vect{v}$, then $\|E_{TG}\|_A \leq 1-\delta$.
\end{theorem}
Following \cite{JWRuge_KStuben_1987a, MacLachlan:2014di}, sufficient
conditions for two-grid convergence are given in the following theorem
\begin{theorem}
\label{thm:classical}
Let symmetric and positive-definite matrix $X$ be given, and assume
that there exist $\alpha,\beta > 0$ such that
$\|(I-M^{-T}A)\vect{v}\|_A^2 \leq \|\vect{v}\|_A^2 -
\alpha\|A\vect{v}\|_X^2$ and $\|(I-\pi_A)\vect{v}\|_A^2 \leq \beta
\|A\vect{v}\|_X^2$ for all $\vect{v}$. Then, $\|E_{TG}\|_A^2 \leq 1-\alpha/\beta$.
\end{theorem}
\begin{proof}
\begin{linenomath}
We prove the bound on $\|E_{TG}\|_A$ by proving the corresponding bound
on $\|(I-M^{-T}A)(I-\pi_A)\|_A$. For any $\vect{v}$,
\begin{align*}
\|(I-M^{-T}A)(I-\pi_A)\vect{v}\|_A^2 & \leq \|(I-\pi_A)\vect{v}\|_A^2
- \alpha\|A(I-\pi_A)\vect{v}\|_X^2 \\
& \leq \|(I-\pi_A)\vect{v}\|_A^2 -
\alpha/\beta\|(I-\pi_A)^2\vect{v}\|_A^2 \\
& \leq (1-\alpha/\beta)\|\vect{v}\|_A^2.
\end{align*}
\end{linenomath}
\end{proof}
The first assumption on Theorem \ref{thm:classical} is commonly
referred to as the smoothing property, since it assumes that
relaxation effectively reduces the error in an approximation when the
residual associated with that error is large (when measured in the
$X$-norm). The second assumption on Theorem \ref{thm:classical} is
referred to as the strong approximation property, since it
assumes that coarse-grid correction is effective at reducing errors
when the associated residuals are small; this is equivalent to
assuming that such errors are well-approximated within $\Ima(P)$.
This assumption is termed the {\it strong} approximation property as
it can clearly be replaced by a weaker one, that
$\|(I-\pi_A)\vect{v}\|_A^2 \leq \beta\|A(I-\pi_A)\vect{v}\|_X^2$ for
all vectors, $\vect{v}$, stating that coarse-grid correction is
effective at reducing errors for which the residual after coarse-grid
correction is small. This latter assumption is commonly referred to
as the {\it weak} approximation property. The difference between the
weak and strong approximation properties comes up in the multilevel
setting, and is discussed in Section \ref{sec:theory:multi}.
In practice, the weak and strong approximation properties are
typically considered in slightly altered forms. For the strong
approximation property, an equivalent statement is that for any
$\vect{v}$, there exists a $\vect{v}_c$ such that
\[
\|\vect{v}-P\vect{v}_c\|_A^2 \leq \beta \|A\vect{v}\|_X^2.
\]
While a similar equivalence could be derived for the weak
approximation property, a more typical bound arises by noting that,
for any $\vect{v}$ and $\vect{v}_c$,
\[
\|(I-\pi_A)\vect{v}\|_A^2 = \langle
A(I-\pi_A)\vect{v},(I-\pi_A)\vect{v}-P\vect{v}_c\rangle \leq \|A(I-\pi_A)\vect{v}\|_X\|(I-\pi_A)\vect{v}-P\vect{v}_c\|_{X^{-1}}.
\]
Thus, a sufficient condition for the weak approximation property to
hold is that for any $\vect{v}$ there exists a $\vect{v}_c$ such that
\begin{equation}
\label{eq:wap_bestkind}
\|\vect{v}-P\vect{v}_c\|_{X^{-1}}^2 \leq \beta\|\vect{v}\|_A^2.
\end{equation}
Note that this is trivially true for $\vect{v} \in \Ima(P)$, but the
weak approximation property is implied by this condition for $\vect{v}
\in \Ima(P)^\perp$. It is in this form that much of the recent
two-grid AMG theory has been developed.
Of particular interest is the result obtained taking $X =
\widetilde{M}^{-1}$, so that the smoothing property in Theorem
\ref{thm:classical} trivially holds with $\alpha = 1$. In this case,
the two-grid convergence bound in Theorem \ref{thm:classical} is
determined entirely by the constant in the weak approximation
property. Indeed, in this setting a sharp bound on convergence is
possible \cite{Falgout:2005hm,Vassilevski:2010vy}.
\begin{theorem}[Weak approximation property]\label{th:wap}
Let $A$ be SPD, $\widetilde{M}= M^T(M+M^T-A)^{-1}M$ for some relaxation
scheme $M$, and $P$ the interpolation operator for a two-grid method. Suppose $\exists$ $K$ such that
for any $\mathbf{v}\neq 0$, there exists a $\mathbf{v}_c$ such that
\[
\frac{\|\mathbf{v} - P\mathbf{v}_c\|_{\widetilde{M}}^2}{\|\mathbf{v}\|_A^2} \leq K.
\]
Then the two-grid method converges uniformly, and $\|E_{TG}\|_A \leq 1- \frac{1}{K}$.
Furthermore, the best (minimal) constant $K$ over all $P$ is given by
\begin{equation}
K_{TG} = \max_{\mathbf{v}\neq\mathbf{0}} \frac{\| ( I - \pi_{\widetilde{M}})\mathbf{v}\|^2_{\widetilde{M}}}{\|\mathbf{v}\|^2_A}, \label{eq:wap1}
\end{equation}
in which case $\|E_{TG}\|_A = 1- \frac{1}{K_{TG}}$.
\end{theorem}
\begin{linenomath}
Equation \eqref{eq:wap1} gives a sharp bound on two-grid convergence,
but can be generalized to any matrix $X$ and corresponding
$X$-orthogonal projection onto $\Ima(P), \pi_X$. Let $X$ be
spectrally equivalent to $\widetilde{M}$ as in equation \eqref{eq:spec_equiv}.
Then, from equations \eqref{eq:X_M1} and \eqref{eq:X_M2},
\begin{align}
c_1\max_{\mathbf{v}\neq0}\frac{\| ( I -
\pi_X)\mathbf{v}\|^2_X}{\|\mathbf{v}\|^2_A} & \leq
c_1\max_{\mathbf{v}\neq0}\frac{\| ( I -
\pi_{\widetilde{M}})\mathbf{v}\|^2_X}{\|\mathbf{v}\|^2_A} \notag \\
& \leq
K_{TG} \leq
\max_{\mathbf{v}\neq0} \frac{\| ( I - \pi_X)\mathbf{v}\|^2_{\widetilde{M}}}{\|\mathbf{v}\|^2_A} \leq
c_2\max_{\mathbf{v}\neq0} \frac{\| ( I - \pi_X)\mathbf{v}\|^2_X}{\|\mathbf{v}\|^2_A} .\label{eq:wap_bounds}
\end{align}
In the case of $X=I$, \eqref{eq:wap_bounds} simplifies to considering
interpolation error in the $l^2$-norm \cite{Zikatanov:2008jp}
\[
\lambda_{\textnormal{min}}(\widetilde{M})\max_{\mathbf{v}\neq0}\frac{\| ( I - Q_P)\mathbf{v}\|^2}{\|\mathbf{v}\|^2_A} \leq K_{TG} \leq
\lambda_{\textnormal{max}}(\widetilde{M}) \max_{\mathbf{v}\neq0} \frac{\| ( I - Q_P)\mathbf{v}\|^2}{\|\mathbf{v}\|^2_A},
\]
\end{linenomath}
motivating the often-used simpler form of the WAP,
\[
\|(I - Q_P)\mathbf{v}\|^2 \leq \frac{K_{TG}}{\|A\|} \|\mathbf{v}\|_A^2.
\]
The necessarily complementary role of relaxation and coarse-grid correction
in AMG is accounted for in the WAP by requiring interpolation accuracy with
respect to $\widetilde{M}$, that is the coarse-grid correction must account
for low-eigenvalue modes of $\widetilde{M}$, which are not effectively
reduced through relaxation with $\widetilde{M}$. A bound on two-grid
convergence can also be formulated as two independent constraints based
on coarse-grid selection and bounding the energy in the range of $P$ as
follows \cite{Falgout:2004cs,Vassilevski:2010vy}.
\begin{lemma}[Two-grid energy-stability]\label{lem:2g_stability}
Let $X$ be spectrally equivalent to $\widetilde{M}$ as in \eqref{eq:spec_equiv},
and define $A_s = S^TAS, X_s = S^TXS$, where
\begin{equation}
\kappa_s \leq \lambda_{\text{min}}(X_s^{-1}A_s) \leq \lambda_{\text{max}}(X_s^{-1}A_s) \leq c_2. \label{eq:cr1}
\end{equation}
If $PR$ is bounded in energy, $\|PR\|_A^2 \leq C$ for some $C$, a WAP in
the $X$-norm, with projection $PR$ is satisfied. These are sufficient
conditions for uniform two-grid convergence, with
$K_{TG} \leq \frac{c_2}{\kappa_s}\|PR\|_A^2$.
\end{lemma}
Lemma \ref{lem:2g_stability} can be seen as an energy-stability constraint
coupled with a compatibility measure of the fine and coarse grids.
Equation \eqref{eq:cr1} measures how well relaxation, $\widetilde{M}$, or
the spectrally equivalent $X$, captures information about the
fine-grid operator, $A_s$.
This is based on the idea of compatible relaxation \cite{Brandt:2000vn,Brannick:2010hz},
which ensures that
the relaxation scheme is able to effectively reduce error on the fine grid. Then,
assuming a compatible choice of grids, interpolation must be stable in energy,
where $\|PR\mathbf{v}\|_A^2 \leq C \|\mathbf{v}\|_A^2$.
Note, Lemma \ref{lem:2g_stability} is equivalent to Theorem \ref{th:wap}, however
the differing explicit conditions make for different approaches to constructing
multigrid hierarchies. One important difference is that in this case, the lemma
is formulated in terms of a general operator norm as opposed to a constraint
for all $\mathbf{v}$. This distinction is discussed in more detail in Section
\ref{sec:practice}.
\subsection{Multilevel convergence}\label{sec:theory:multi}
Now that two-grid convergence theory has been introduced, let us continue by
considering when two-level convergence can be extended to the multilevel
setting. For two-level convergence, either the weak or strong approximation property provide
sufficient conditions for convergence; however, the same is not true when considering the
multilevel case. In the multilevel setting, if the smoothing and strong
approximation properties hold on all levels of the multigrid
hierarchy, with constants that are uniformly bounded (independently of
the level in the hierarchy), then multilevel convergence of the
multigrid V-cycle can be proven \cite{JWRuge_KStuben_1987a}. Even if
the weak approximation property holds uniformly, though, multilevel
convergence still cannot be guaranteed. Since $A(I-\pi_A) =
(I-\pi_A^T)A$, we can easily derive the bound
\[
\|A(I-\pi_A)\vect{v}\|_X \leq \|I-\pi_A^T\|_X\|A\vect{v}\|_X,
\]
showing that if $\|I-\pi_A^T\|_X$ is not uniformly bounded across the
levels in the hierarchy, then a uniform weak approximation property
does not imply a uniform strong approximation property.
The standard example of this is the use of a piecewise constant
interpolation operator, $P$, for any standard discretization of the
Poisson problem. If $\vect{v}$ is a smooth vector, then
$\|A\vect{v}\|_X$ will be small for many reasonable choices of $X$,
such as $X = D^{-1}$, where $D$ is the diagonal of the system matrix, $A$. After
coarse-grid correction, $(I-\pi_A)\vect{v}$ will have jumps induced by
the piecewise-constant interpolation, so $\|(I-\pi_A)\vect{v}\|_A$
will be large, reflecting the high-frequency character of
$(I-\pi_A)\vect{v}$. With this, the strong approximation property can
only be achieved with a large constant, $\beta$. In contrast, since
$\|A(I-\pi_A)\vect{v}\|_X$ will also be large, the weak approximation
property can be fulfilled with a moderate constant, $\beta$. As is
well-known, piecewise constant interpolation is sufficient for good
two-level convergence, but not multilevel, consistent with the
theoretical results.
Let $E_{MG}$ be the error-propagation matrix
for a $V(1,1)$-cycle with a full multigrid hierarchy. The resulting convergence
factor is bounded by the $A$-norm of $E_{MG}$, which can take the form
\[
\|E_{MG}\|_A = 1 - \frac{1}{K_{MG}},
\]
for some $K_{MG} \geq 1$. Multigrid with an arbitrary number of levels can
be thought of as a recursive use of two-grid methods with inexact coarse-grid solves, which is
typically how convergence theory is formulated in the multilevel setting.
The standard multilevel convergence result and some equivalent or
sufficient conditions are stated below. In all cases, we assume that
the multigrid hierarchy is specified by matrices $A^{(k)}$ and
interpolation operators $P^{(k)}$, with the convention that $P^{(k)}$
is the interpolation operator from level $k+1$ to level $k$, and
$A^{(k+1)} = \left(P^{(k)}\right)^TA^{(k)}P^{(k)}$. Furthermore, we
assume that on each level, a relaxation scheme is specified that
satisfies a consistent smoothing property,
\[
\left\|\left(I-\left(M^{(k)}\right)^{-T}A^{(k)}\right)\vect{v}^{(k)}\right\|_{A^{(k)}}^2
\leq \left\|\vect{v}^{(k)}\right\|_{A^{(k)}}^2 - \frac{\alpha}{\|A^{(k)}\|}\left\|A^{(k)}\vect{v}^{(k)}\right\|^2,
\]
for all vectors, $\vect{v}^{(k)}$, with $\alpha$ independent of $k$.
\begin{theorem}[Strong approximation property]
\label{th:sap}
If, for every $\mathbf{v}^{(k)}$, there exists a $\mathbf{v}^{(k+1)}$ such that
\begin{equation}
\left\|\mathbf{v}^{(k)} - P^{(k)}\mathbf{v}^{(k+1)}\right\|_{A^{(k)}}^2 \leq \frac{\beta}{\|A^{(k)}\|} \left\|A^{(k)}\vect{v}^{(k)}\right\|^2,\label{eq:sap}
\end{equation}
for some $\beta$ independent of $k$. Then the multilevel $V(1,1)$-cycle converges uniformly, and
$K_{MG} \leq \beta/\alpha$.
\end{theorem}
Note that the strong approximation property (SAP), is similar to the
multilevel generalization of the WAP of equation
\eqref{eq:wap_bestkind} when $X = \frac{1}{\|A\|}I$,
\[
\left\|\mathbf{v}^{(k)} - P^{(k)}\mathbf{v}^{(k+1)}\right\|^2 \leq \frac{\beta}{\|A^{(k)}\|} \left\|\vect{v}^{(k)}\right\|_{A^{(k)}}^2.
\]
In this form of the WAP, interpolation of an eigenvector, $\mathbf{v}^{(k)}$, must
be accurate in the $l^2$-norm to the order of its corresponding eigenvalue, with
constant $\frac{\beta}{\|A^{(k)}\|}$. A stronger statement is required by the SAP, namely
that interpolation of an eigenvector in the $A^{(k)}$-norm must be accurate to the order of its
corresponding eigenvalue. A
detailed look at the SAP can be found in Theorem 5.6.1
and Chapter 6 of\cite{Vassilevski:2010vy}.
Two sufficient conditions for the strong approximation property are
stated below. Both rely on the smoothing property stated above
holding uniformly across all levels. To simplify notation, we define
$\pi_k =
P^{(k)}\left(A^{(k+1)}\right)^{-1}\left(P^{(k)}\right)^TA^{(k)}$ as
the $A^{(k)}$-orthogonal projection on level $k$ of the hierarchy, and
$Q_k = P^{(k)}\left(\left(P^{(k)}\right)^TP^{(k)}\right)^{-1}\left(P^{(k)}\right)^T$
as the $\ell^2$-orthogonal projection on level $k$ of the hierarchy.
\begin{corollary}[$l^2$-boundedness of $\pi_k$]
\label{cor:sap1}
If, for every $\mathbf{v}^{(k)}$,
\begin{equation}
\left\|(I - \pi_k)\mathbf{v}^{(k)}\right\|^2 \leq \frac{\beta}{\|A^{(k)}\|^2} \left\|A^{(k)}v^{(k)}\right\|^2\label{eq:wap2},
\end{equation}
for some $\beta$ independent of $k$, then the multilevel $V(1,1)$-cycle converges uniformly, and $K_{MG} \leq \beta/\alpha$.
\end{corollary}
\begin{corollary}[WAP$(A^2)$]
\label{cor:sap2}
If, for every $\mathbf{v}^{(k)}$,
\begin{align}
\left\|(I - Q_k)\mathbf{v}^{(k)}\right\|^2 & \leq \frac{\beta}{\|A^{(k)}\|^2} \left\|A^{(k)}v^{(k)}\right\|^2,
\end{align}
for some $\beta$ independent of $k$, then the SAP holds with constant
$\beta$ and $K_{MG} \leq \beta/\alpha$.
\end{corollary}
\begin{proof}
For $\hat{\mathbf{v}}^{(k+1)} = \left(\left(P^{(k)}\right)^TP^{(k)}\right)^{-1}\left(P^{(k)}\right)^T\mathbf{v}^{(k)}$,
\[
\left\|\mathbf{v}^{(k)}-P^{(k)}\hat{\mathbf{v}}^{(k+1)}\right\|_{A^{(k)}}^2 \leq \|A^{(k)}\|\left\|\mathbf{v}^{(k)}-P^{(k)}\hat{\mathbf{v}}^{(k+1)}\right\|^2 \leq \frac{\beta}{\|A^{(k)}\|} \left\|A^{(k)}v^{(k)}\right\|^2.
\]
\end{proof}
\subsection{``Optimal'' and ``ideal'' interpolation}
Returning to the two-level case, alongside bounds on two-grid convergence factors, specific interpolation operators have
been derived as the best interpolation operator in certain contexts.
Let $P = \begin{bmatrix}W\\I\end{bmatrix}$ and $R = \begin{bmatrix}0&I\end{bmatrix}$,
and consider relaxing the numerator of \eqref{eq:wap1} from the $\widetilde{M}$-norm to the following problem:
\begin{equation}
\min_P \max_{\mathbf{v}} \frac{\| ( I - PR)\mathbf{v}\|^2}{\|\mathbf{v}\|^2_A} = \begin{bmatrix} -A_{ff}^{-1}A_{fc} \\ I\end{bmatrix}
:= P_{ideal}, \label{eq:wap_ideal}
\end{equation}
where $P_{ideal}$ is so-called ``ideal interpolation'' \cite{Falgout:2004cs}.
Due to the inverse of $A_{ff}$, $P_{ideal}$ is often a difficult
operator to compute directly and may be dense, neither of which
are compatible with the goals of AMG (see Section \ref{sec:practice}). However, denoting the graph distance
between nodes $i$ and $j$ in $A_{ff}$ by $|i-j|_G$, then the
following decay property is well-known:
\[
[A_{ff}^{-1}]_{ij} \leq C q^{|i-j|_G-1},
\]
for some constant $C$ and $q < 1$, where $q\approx \frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1}$, for condition number, $\kappa$, of
$A_{ff}$ \cite{Brannick:2007fb,DemkoMossSmith}. Thus, for sparse,
well-conditioned $A_{ff}$ (as is expected with
a proper choice of coarse grid), coefficients of $A_{ff}^{-1}$ decay exponentially fast away from the
diagonal. Under this assumption, $A_{ff}^{-1}$ can be approximated well with a sparse matrix, at least in a Frobenius sense.
Approaches to approximating $P_{ideal}$ can be found in \cite{Manteuffel:2016vd,Brannick:2007fb},
each of which contain some variation of the following result:
\begin{lemma}[Ideal interpolation]\label{lem:ideal}
Let $P_{ideal}$ be as in \eqref{eq:wap_ideal} and let $P$ take the form $P = \begin{bmatrix}W\\I\end{bmatrix}$, restricted to a fixed
nonzero sparsity pattern. Then minimizing the difference between columns of $P$ and $P_{ideal}$ in the $A$-norm is equivalent
to minimizing each column of $P$ in the $A$-norm. Furthermore, the solution of this minimization is unique.
\end{lemma}
However, considering that \eqref{eq:wap_ideal} does not provide a sharp bound on convergence, ideal interpolation typically does
not provide optimal (two-grid) convergence factors over all $P$. The optimal $P$ with respect to two-grid convergence
is given in the following lemma \cite[Lemma 1]{brannick2017optimal}.
\begin{lemma}[Optimal interpolation]\label{lem:opt}
Let $0<\lambda_1\leq ... \leq \lambda_n$ and $\mathbf{v}_1,...,\mathbf{v}_n$
denote the eigenvalues and eigenvectors, respectively, of the generalized
eigenvalue problem
\[
A\mathbf{v} = \lambda\widetilde{M}\mathbf{v}.
\]
Then the minimal convergence rate of the two-grid method $\|E_{TG}(P)\|_A$
over all $P$ with dim$(P) = n_c$ is given by
\[
\|E_{TG}(P_{opt})\|_A^2 = 1 - \lambda_{n_c+1},
\]
with corresponding optimal interpolation matrix given by
\[
P_{opt} = \begin{bmatrix} \mathbf{v}_1 & ... & \mathbf{v}_{n_c}\end{bmatrix}.
\]
\end{lemma}
Although results in \cite{brannick2017optimal} suggest that, at times, a sparse approximation to $P_{\textnormal{opt}}$
may be feasible, it is certainly more difficult to develop a cheap, sparse approximation to $P_{\textnormal{opt}}$ compared with
$P_{\textnormal{ideal}}$. That being said, Lemma \ref{lem:opt} does corroborate the general AMG approach of including
eigenvectors of $A$ (actually of $\widetilde{M}^{-1}A$) associated with small eigenvalues in $\Ima(P)$. In fact, it follows
from Lemma \ref{lem:opt} that if the first $n_c+1$ eigenvalues of $A\mathbf{v} = \lambda\widetilde{M}\mathbf{v}$
are all approximately zero, AMG \textit{cannot} achieve strong convergence factors. This highlights the importance of
the distribution of eigenvalues on the performance of AMG.
\section{Interpolation in practice}\label{sec:practice}
AMG is a popular solver largely because of its linear complexity in
the setup and solve phase. Let $A$ be SPD and consider forming a multigrid hierarchy to solve $A\mathbf{x} = \mathbf{b}$. Interpolation operators in
AMG methods are often (implicitly) constructed with the goal of
controlling or minimizing some functional with a theoretical
relation to convergence, such as \eqref{eq:wap1}, \eqref{eq:sap}, \eqref{eq:wap2}, or \eqref{eq:wap_ideal}. However, there
are two important factors that must be considered in practice and are generally absent from theory -- (i) the process used to form
interpolation operators must remain linear in complexity in keeping
with the desired $O(n)$ total cost for AMG methods, and (ii) interpolation operators must
remain sparse in order to construct a sparse coarse-grid matrix that
can be used in a recursive process.
These constraints can
prove difficult to achieve when designing AMG methods, and make
approximating some of the bounds in Section \ref{sec:theory}
more tractable than others. In particular, the operators used in the convergence theory, such as $\pi_A$, $P_{ideal}$, and $P_{opt}$, generally cannot
be easily computed in any practical setting. Furthermore, these convergence results are typically required to hold for all $\mathbf{v}$ or, equivalently, for some basis for the space
such as the eigenvectors of $A$. Constructing interpolation or coarsening based on a full basis of vectors is generally not tractable in linear
complexity and, thus, two forms of approximation are often used, (i) work with a candidate set of $k$ vectors, where
$k \ll n$, or (ii) work in an operator norm, which is a supremum over all vectors.
The first approach is to directly satisfy conditions of a theorem but
only for a set of candidate vectors of dimension $k\ll n$. This
is a standard approach for satisfying the WAP or SAP, and classical
AMG \cite{JWRuge_KStuben_1987a} can be viewed as doing this for only
the constant vector, while smoothed aggregation (SA) \cite{Van:2001bw} may
use a larger basis, such as the rigid-body modes for elasticity
problems. Most adaptive multigrid methods \cite{DAmbra:2013iwa,
Brezina:2004eh, Brandt:2011hb} can also be viewed in this way, where
the candidate vectors arise from the adaptive process. As in \eqref{eq:wap1}, the (two-grid) convergence rate is bounded by the
maximum of $K_{TG}$ over all $\mathbf{v} \neq \mathbf{0}$. Noting the denominator of $\|\mathbf{v}\|_A^2$, the maximum will generally occur for
$\mathbf{v}$ associated with small eigenvalues of $A$, where
$\|\mathbf{v}\|_A^2$ is very small compared to $\|\mathbf{v}\|_{\widetilde{M}}^2$. For differential operators,
it is common to have a zero or near-zero row sum, making the constant a good representation of low-energy modes. Developing and
using additional candidate vectors is the basis for adaptive approaches, which are designed for difficult linear systems beyond the scope of
classical SA or AMG \cite{Brandt:2011hb, Brezina:2004eh,DAmbra:2013iwa}. In such solvers, an adaptive process is used to develop
a set of target vectors representative of low-energy modes of $A$. Interpolation is then constrained to interpolate these
modes either exactly or nearly so, and the process is repeated on coarse grids.
An alternative approach is to formulate the
minimization over all vectors $\mathbf{v}\neq\mathbf{0}$. Consider the
energy-stability constraint in Lemma \ref{lem:2g_stability},
based on controlling $\|PR\|_A^2$. As an induced $A$-norm, $\|PR\|_A$
is defined via a supremum over all $\mathbf{v}\neq\mathbf{0}$. However, $\|PR\|_A$
can be bounded using the Frobenius norm, which gives an indirect
approach to bounding $\|PR\|_A$. Let $R
= \begin{bmatrix}0&I\end{bmatrix}$, then
\[
\|PR\|_A^2 = \|A^{\frac{1}{2}}PRA^{-\frac{1}{2}}\|^2 \leq \|A^{\frac{1}{2}}PRA^{-\frac{1}{2}}\|_F^2 = \tr(P^TAPS_A),
\]
where $S_A^{-1}:= RA^{-1}R^T = (A_{cc} - A_{cf}A_{ff}^{-1}A_{fc})^{-1}$,
is the inverse of the Schur complement of $A$ in $A_{cc}$. Although an interesting
equivalence, the Schur complement is difficult to form in practice. A more
tractable approach is obtained by using an intermediate bound,
\[
\|PR\|_A^2 \leq \|A^{-1}\|\|A^{\frac{1}{2}}PR\|^2 \leq \|A^{-1}\|\|A^{\frac{1}{2}}PR\|_F^2 = \|A^{-1}\|\tr(R^TP^TAPR).
\]
Given the form of $R = (\mathbf{0}, I)$, this simplifies to
\begin{equation}
\|PR\|_A^2 \leq \|A^{-1}\|\tr(P^TAP). \label{eq:ainv}
\end{equation}
Minimizing $\tr(P^TAP)$ was proposed in this form in \cite{Brannick:2007fb}, and is equivalent to minimizing columns of $P$
in the $A$-norm. This approach has been used in smoothed aggregation (SA) \cite{Van:2001bw}, root-node AMG \cite{Manteuffel:2016vd,
Schroder:2012di}, and the general energy-minimization framework proposed in \cite{Olson:2011fg}. Recall from Lemma \ref{lem:ideal} that
minimizing energy in columns of $P$ is also equivalent to minimizing the difference between columns of $P$ and $P_{ideal}$ in the $A$-norm.
It is worth considering the leading constant in \eqref{eq:ainv}, $\|A^{-1}\| = \frac{1}{\lambda_{\text{min}}(A)}$, as this is likely large and
could lead to a poor bound on $\|PR\|_A^2$. Note that the energy constraint in Lemma \ref{lem:2g_stability} can also be formulated as
\[
\mathbf{v}^TR^TP^TAPR\mathbf{v} \leq \eta \mathbf{v}^TA\mathbf{v} \hspace{3ex}
\Longleftrightarrow \hspace{3ex} \mathbf{v}_c^TA_c\mathbf{v}_c \leq \eta\mathbf{v}^TA\mathbf{v},
\]
for all vectors $\mathbf{v}$ with $\mathbf{v}_c = R\mathbf{v}$
\cite[Theorem 5.2]{Falgout:2005hm}. The factor of $\|A^{-1}\|$ in
\eqref{eq:ainv} accounts for the possibility that $P$ is chosen so
that $\mathbf{v}_c^TA_c\mathbf{v}_c$ is an $O(1)$ quantity when
$\mathbf{v}$ corresponds to the smallest eigenvalue of $A$
(corresponding to a bad choice of $P$). In practice, we make
the heuristic assumption that the choice of $P$ will not be so bad
and, thus, minimizing $\tr(P^TAP) =
\sum_i \lambda_i(A_c)$ is an effective way to control $\|PR\|_A$.
\section{Trace-minimization}\label{sec:functional}
As discussed in Section \ref{sec:practice} and can be seen in other AMG methods, AMG interpolation operators are often constructed
based on some combination of ensuring that a given set of candidate
vectors is interpolated exactly, while ensuring energy stability of the
coarse-grid operator. In this direction, we now propose to form $P$ through minimizing a general weighted functional combining these two approaches,
\begin{equation}
\mathcal{G}(P) = (1-\tau)\frac{\|(I - PR)\mathbf{v}\|_{\widetilde{M}}^2}{\|\mathbf{v}\|_A^2} + \tau \tr(P^TAP),
\label{eq:gen_func0}
\end{equation}
for $\tau\in[0,1)$ and candidate vector $\mathbf{v}$. If multiple candidate vectors, $\{\mathbf{v}_i\}$, are available a priori, for example,
the rigid body modes in elasticity, then we minimize over the maximum
$\mathbf{w}\in$ Span$\{\mathbf{v}_i\}$:
\[
\mathcal{G}(P) =
(1-\tau)\max_{\mathbf{w}\in\text{Span}\{\mathbf{v}_i\}\setminus\{\mathbf{0}\}}\frac{\|(I - PR)\mathbf{w}\|_{\widetilde{M}}^2}{\|\mathbf{w}\|_A^2} + \tau \tr(P^TAP),
\]
This is a complementary
approach, focusing on achieving accurate interpolation of the
low-energy modes in the candidate set as well as energy stability on the coarse grid. It is also complementary
in the sense that the first term is defined over a candidate set of vectors, $\{\mathbf{v}_i\}$, while the second term is defined over $P$,
and should improve interpolation regardless of the provided candidate vectors.
\begin{linenomath}
Let $P$ take the form $P = \begin{bmatrix}W\\I\end{bmatrix}$, and consider minimizing \eqref{eq:gen_func0}. Define a set of $n_B$
candidate vectors as $A$-orthonormalized columns of a matrix $B = \begin{bmatrix}B_f\\B_c\end{bmatrix}$, and let $X\simeq\widetilde{M}$ as in \eqref{eq:spec_equiv}.
Then, consider minimizing $K_{TG}$ from \eqref{eq:wap1}, restricted to
unit linear combinations of $\mathbf{v}\in \mathcal{B} =
\{\mathbf{w}\in\Ima(B)\mid \|\mathbf{w}\|_A=1\}$:
\begin{align}
\max_{\mathbf{v\in\mathcal{B}}} \|(I - \pi_{\widetilde{M}})\mathbf{v}\|_{\widetilde{M}}^2 & \leq
c_2\max_{\mathbf{v\in\mathcal{B}}}\|(I - \pi_X)\mathbf{v}\|_{X}^2 \nonumber\\
& \leq c_2 \max_{\mathbf{v\in\mathcal{B}}} \|(I - PR)\mathbf{v}\|_{X}^2 \nonumber\\
& = c_2 \max_{\mathbf{v\in\mathcal{B}}} \Big\langle X \begin{bmatrix} \mathbf{v_f} - W\mathbf{v_c} \\ 0 \end{bmatrix},
\begin{bmatrix} \mathbf{v_f} - W\mathbf{v_c} \\ 0 \end{bmatrix} \Big\rangle \nonumber\\
& = c_2 \max_{\mathbf{v\in\mathcal{B}}} \Big\langle X_{ff}W\mathbf{v_c}, W\mathbf{v_c}-2\mathbf{v}_f \Big\rangle +
c_2\|\mathbf{v_f}\|_{X_{ff}}^2 \nonumber\\
& \leq c_2 \Big\langle X_{ff}WB_c, WB_c-2B_f \Big\rangle_F + c_2\|B_f\|_{X_{ff}}^2 \nonumber\\
& = c_2\Big\langle X_{ff}WB_cB_c^T, W\Big\rangle_F - 2c_2\Big\langle X_{ff}W, B_fB_c^T\Big\rangle_F + c_2\|B_f\|_{X_{ff}}^2. \label{eq:term1}
\end{align}
This approximates the WAP in the $X$-norm using an $l^2$-projection onto $\Ima(P)$ (as opposed to the optimal $\pi_{\widetilde{M}}$-orthogonal
projection). Recall the second term in \eqref{eq:gen_func0} corresponds to minimizing the columns of $P$ in the $A$-norm. Expanding
$\tr(P^TAP)$ gives
\begin{align}
\tr(P^TAP) & = \tr\bigg(\begin{bmatrix}W^T & I\end{bmatrix} \begin{bmatrix} A_{ff} & A_{fc} \\ A_{cf} & A_{cc}\end{bmatrix} \begin{bmatrix} W \\ I\end{bmatrix}\bigg) \nonumber\\
& = \tr(W^TA_{ff}W) + 2\tr(A_{cf}W) + \tr(A_{cc}) \nonumber \\
& = \Big\langle A_{ff}W,W\Big\rangle_F + 2\Big\langle W, A_{fc}\Big\rangle_F + \tr(A_{cc}) \label{eq:term2}
\end{align}
\end{linenomath}
\begin{linenomath}
Substituting equations \eqref{eq:term1} and \eqref{eq:term2} into \eqref{eq:gen_func0} gives a functional of $W$ to minimize in forming
$P$. Dropping terms independent of $W$ and pulling out a factor of two for a more familiar form, define
\begin{align}
\begin{split}
\mathcal{F}(W) & = \frac{\tau}{2}\Big\langle A_{ff}W,W\Big\rangle_F + \frac{c_2(1-\tau)}{2}\Big\langle X_{ff}WB_cB_c^T, W\Big\rangle_F \\
&\hspace{10ex} - \Big\langle W, c_2(1-\tau)X_{ff}B_fB_c^T - \tau A_{fc}\Big\rangle_F. \label{eq:functional}
\end{split}
\end{align}
Observe that \eqref{eq:functional} is a quadratic functional in $W$. Define a bounded linear operator, $\mathcal{L}$, and right-hand-side,
$\mathcal{B}$, as
\begin{align}
\mathcal{L}W & = \tau A_{ff}W + c_2(1-\tau)X_{ff}WB_cB_c^T \label{eq:linop} \\
\mathcal{B} & = c_2(1-\tau)X_{ff}B_fB_c^T - \tau A_{fc}, \notag
\end{align}
in which case $\mathcal{F}(W) = \tfrac{1}{2}\langle \mathcal{L}W, W\rangle_F - \langle W,\mathcal{B}\rangle_F$. Note that
if $A_{ff}$ and $X_{ff}$ are symmetric and positive definite, then $\mathcal{L}$ is self-adjoint and positive definite in the Frobenius norm:
\begin{align*}
\Big\langle \mathcal{L}W, Z\Big\rangle_F & = \Big\langle \tau A_{ff}W, Z\Big\rangle_F + c_2(1-\tau)\Big\langle X_{ff}WB_cB_c^T, Z\Big\rangle_F \\
& = \Big\langle \tau W, A_{ff}Z\Big\rangle_F + c_2(1-\tau)\Big\langle W, X_{ff}ZB_cB_c^T \Big\rangle_F \\
& = \tau\Big\langle W, \mathcal{L}Z\Big\rangle_F, \\
\Big\langle \mathcal{L}W,W\Big\rangle_F & = \tau\Big\langle A_{ff}W, W\Big\rangle_F + c_2(1-\tau)\Big\langle X_{ff}WB_cB_c^T, W\Big\rangle_F \\
& = \tau\Big\langle A_{ff}W, W\Big\rangle_F + c_2(1-\tau)\Big\langle X_{ff}WB_c,WB_c\Big\rangle_F \\
& > 0 \text{ when }W\neq 0.
\end{align*}
Using the symmetry of $\mathcal{L}$, the first and second Frech\'et derivative of $\mathcal{F}$ are given by:
\begin{align*}
\mathcal{F}'(W)[V] & = \lim_{\alpha\to 0} \frac{\mathcal{F}(W + \alpha V) - \mathcal{F}(W)}{\alpha} \\
& = \Big\langle \mathcal{L}W - \mathcal{B}, V\Big\rangle_F \\
\mathcal{F}''(W)[V][U] & = \Big\langle \mathcal{L}U, V\Big\rangle_F.
\end{align*}
\end{linenomath}
Since $\mathcal{L}$ is self-adjoint and positive definite, $\mathcal{F}''(W)[V] \geq 0$ $\forall$ $V$. Thus, the minimum of $\mathcal{F}$
in $W$ is achieved at $W$ such that $\mathcal{F}'(W)[V] = 0$, and $\mathcal{F}'(W) = 0$ $\forall$ $V$ if and only if
$\mathcal{L}W = \mathcal{B}$. This has a unique solution, $W = \mathcal{L}^{-1}\mathcal{B}$. However, it is likely that
$\mathcal{L}^{-1}\mathcal{B}$ is dense and not practical, motivating a constrained sparsity pattern for $W$.
\subsection{Constrained sparsity pattern}
In practice, the sparsity pattern of $W$ must be fixed a priori in order to control the operator complexity of $W$ and $A_c$.
Define a vector space
\[
\mathcal{X} = \Big\{ W \text{ : } W\in\mathbb{R}^{N_f\times N_c}, W_{ij} = 0 \text{ if } (i,j)\not\in \mathcal{N}\Big\},
\]
for a set of indices $\mathcal{N}$ denoting a fixed sparsity pattern for $W$. A Hilbert space $\mathcal{H}$ can be defined
over $\mathcal{X}$ with the Frobenius inner product, $\langle A,B\rangle_F = \sum_{ij} A_{ij}B_{ij}$. It is easily verified that
$\mathcal{X}$ is complete over the norm induced by $\langle\cdot,\cdot,\rangle_F$ due to the completeness of $\mathbb{R}$.
Now define the bounded linear functional $\hat{\mathcal{L}}:\mathcal{H}\to\mathcal{H}$ as
\[
(\hat{\mathcal{L}}W)_{ij} =
\begin{cases} (\mathcal{L}W)_{ij} & (i,j)\in\mathcal{N} \\ 0 & (i,j)\not\in\mathcal{N}\end{cases},
\]
and a corresponding bilinear form
\[
a(W,V) = \Big\langle \hat{\mathcal{L}}W, V\Big\rangle_F.
\]
A quadratic form as in \eqref{eq:functional} restricted over $\mathcal{N}$ can then be defined as
\begin{equation}
\hat{\mathcal{F}}(W) = \frac{1}{2}\Big\langle \hat{\mathcal{L}}W, W\Big\rangle_F - \Big\langle W, \hat{\mathcal{B}}\Big\rangle_F,
\label{eq:sp_func}
\end{equation}
where $\hat{\mathcal{B}}\in\mathcal{H}$ is $\mathcal{B}$ restricted to $\mathcal{N}$. Note that in $\mathcal{H}$,
$\langle W,\mathcal{B}\rangle_F = \langle W,\hat{\mathcal{B}}\rangle_F$. A similar derivation as shown for $\mathcal{L}$ confirms
that $\hat{\mathcal{L}}$ is self-adjoint and $a(W,V)$ symmetric. Then, observe that for $W\in\mathcal{H}, W\neq 0$,
$\hat{\mathcal{L}}$ and $a(W,V)$ are positive:
\[
\Big\langle \hat{\mathcal{L}}W,W\Big\rangle_F = \Big\langle \mathcal{L}W,W\Big\rangle_F > 0,
\]
The following standard lemma of functional analysis can then be
invoked to find a minimizer of $\hat{\mathcal{F}}(W)$ \eqref{eq:sp_func}.
\begin{lemma}
\label{lem:solution}
\begin{linenomath}
Let $a(x,y)$ be a bounded, symmetric, positive-definite bilinear form
on a Hilbert space $\mathcal{H}$, and $\mathcal{G}(x)$ be a
bounded linear functional on $\mathcal{H}$. Then the following are equivalent
\begin{align}
x & = \min_{x\in\mathcal{H}}\hspace{1ex} \frac{1}{2}a(x,x) - \mathcal{G}(x) + C \label{eq:lem1_1} \\
x & \textnormal{ satisfies } a(x,y) = \mathcal{G}(y) \hspace{1ex} \text{ for all }y\in\mathcal{H}
\label{eq:lem1_2}
\end{align}
\end{linenomath}
Furthermore, there exists a unique solution $x\in\mathcal{H}$ satisfying
\eqref{eq:lem1_1}, \eqref{eq:lem1_2}.
\end{lemma}
Based on Lemma \ref{lem:solution}, we seek the unique solution to
\begin{equation}
\hat{\mathcal{L}}W = \hat{B}, \hspace{2ex}W\in\mathcal{H},\label{eq:sp_solution}
\end{equation}
which can be iterated towards using the preconditioned conjugate gradient method
introduced in Section \ref{sec:pcg}.
\begin{remark}
A conceptual limiting case of the proposed weighted energy minimization is to interpolate candidate vectors exactly
and minimize energy based on that constraint. However, this does not directly fit into the framework of \eqref{eq:gen_func0}.
Constrained energy-minimization has been proposed in various forms \cite{Brannick:2007fb,Mandel:1999wg,Olson:2011fg,Wan:1999ky},
and was used as a basis for root-node AMG in \cite{Manteuffel:2016vd}. Defining the affine space
$\mathcal{A} = \{ W \text{ : } W\in\mathcal{H}\text{ and } WB_c = B_f\}$, the constrained minimization problem is given by
\begin{equation}
W = \argmin \Big\langle A_{ff}W + A_{fc},W\Big\rangle_F, \hspace{1ex} W\in\mathcal{A}.\label{eq:constrained}
\end{equation}
Since the linear operator now consists of normal matrix multliplication, $A_{ff}W$ as opposed to the left and right multiplication
in \eqref{eq:linop}, the existence and uniqueness of a solution to \eqref{eq:constrained} can be shown in a linear algebra
setting (see \cite{Olson:2011fg}) along with a CG implementation based on projecting into $\mathcal{A}$.
\end{remark}
\section{Preconditioned conjugate gradient}\label{sec:pcg}
Because $\hat{\mathcal{L}}$ is self-adjoint and positive in $\mathcal{H}$, conjugate gradient (CG) in the Hilbert space setting is a
competitive approach to solving \eqref{eq:sp_solution} in an iterative fashion. It is generally advisable to precondition CG iterations
for optimal convergence. Here, we construct a diagonal preconditioner for \eqref{eq:sp_solution} to make iterations more robust
when $\hat{\mathcal{L}}$ is poorly conditioned at a marginal increase of computational cost.
Unlike with matrices, however, it is not clear what the ``diagonal'' of $\hat{\mathcal{L}}$ is. Let $W\in\mathbb{R}^{N_f\times N_c}$, and
define the operator $\overline{(W)}$ as the columns of $W$ stacked in a column-vector. Note that $\overline{(W^T)}$ then gives the
rows of $W$ stacked as a column-vector. Let $Y$ be the permutation matrix such that $\overline{(W)} = Y\overline{(W^T)}$ and
$YY^T = Y^TY = I$, which can be thought of as a mapping of $W$ from row-major format to column-major format. First note the following
lemma with regards to Kronecker products and the action of $Y$.
\begin{lemma}\label{lem:kron}
Let $Y$ be a permutation matrix mapping $W\in\mathbb{R}^{N_f\times N_c}$ from row-major format to column-major format, that is,
$\overline{(W)} = Y\overline{(W^T)}$. Then, for any $P\in\mathbb{R}^{N_f\times N_f}$ and $Q\in\mathbb{R}^{N_c\times N_c}$,
\[
Y(P\otimes Q)Y^T = Q\otimes P
\]
\end{lemma}
\begin{proof}
\begin{linenomath}
First consider the structure of $Y$. Note the following relations between $W,
\overline{(W)}$, and $\overline{(W^T)}$, i.e. $W$ stored as a standard dense
matrix, a column-major matrix, and a row-major matrix, respectively,
\begin{align*}
\overline{(W)}_{i+jN_f} & = W_{ij} \\
\overline{(W^T)}_{j+iN_c} & = W_{ij}.
\end{align*}
Defining $Y$ such that $Y\overline{(W^T)} = \overline{(W)}$, it
follows that
\[
Y_{i+jN_f, j+iN_c} = 1, \hspace{2ex}\text{ for $i\in[0,N_f],j\in[0,N_c]$},
\]
and the action of $YAY^T$ is then given as
\begin{equation}
[YAY^T]_{i+jN_f,k+lN_f} = A_{j+iN_c, l+kN_c}.\label{eq:lem_y}
\end{equation}
Now consider the element-wise Kronecker products of $P$ and $Q$:
\begin{align}
[P\otimes Q]_{j+iN_c, l+kN_c} = P_{ik}Q_{jl}, \label{eq:kron1} \\
[Q\otimes P]_{i+jN_f, k+lN_f} = P_{ik}Q_{jl}, \label{eq:kron2}
\end{align}
for $i,k\in[0,N_f], j,l\in[0,N_c]$. Combining \eqref{eq:lem_y}, \eqref{eq:kron1}, and \eqref{eq:kron2} gives
\begin{align*}
[Y(P\otimes Q)Y^T]_{i+jN_f,k+lN_f} & = (P\otimes Q)_{j+iN_c,l+kN_c} \\
& = P_{ik}Q_{jl} \\
& = [P\otimes Q]_{i+jN_f, k+lN_f}.
\end{align*}
It follows that $Y(P\otimes Q)Y^T = Q\otimes P$.
\end{linenomath}
\end{proof}
\begin{remark}
Lemma \ref{lem:kron} is a known result that we arrived at inadvertently, where $Y$ is known as the ``Perfect Shuffle'' matrix
\cite{Davio:1981gc}. Its relation to row-major and column-major
storage of matrices is, to our knowledge, a new contribution to the literature.
\end{remark}
\begin{linenomath}
Now consider finding the diagonal of $\mathcal{L}$ by looking at $\overline{\mathcal{L}}$ as an operator on $\overline{(W)}$. To
do so, represent the action of $A_{ff}W$ through $(I_{N_c}\otimes A_{ff})\overline{(W)}$, where $(I_{N_c}\otimes A_{ff})$
gives a block diagonal matrix of $N_c$ $A_{ff}$'s, each to be multiplied by one column of $W$. Recalling the identity
$(A\otimes B)(C\otimes D) = (AB\otimes CD)$ and Lemma \ref{lem:kron},
\begin{align*}
\overline{\mathcal{L}}\overline{(W)} & = \tau(I_{N_c}\otimes A_{ff})\overline{(W)} + c_2(1-\tau)(I_{N_c}\otimes X_{ff})\overline{(WB_cB_c^T)} \\
& = \tau(I_{N_c}\otimes A_{ff})\overline{(W)} + c_2(1-\tau)(I_{N_c}\otimes X_{ff})YY^T\overline{(WB_cB_c^T)} \\
& = \tau(I_{N_c}\otimes A_{ff})\overline{(W)} + c_2(1-\tau)(I_{N_c}\otimes X_{ff}) Y \overline{(B_cB_c^TW^T)} \\
& = \tau(I_{N_c}\otimes A_{ff})\overline{(W)} + c_2(1-\tau)(I_{N_c}\otimes X_{ff})Y(I_{N_f}\otimes B_cB_c^T)Y^TY\overline{(W^T)} \\
& = \tau(I_{N_c}\otimes A_{ff})\overline{(W)} + c_2(1-\tau)(I_{N_c}\otimes X_{ff})(B_cB_c^T\otimes I_{N_f})\overline{(W)} \\
& = \tau(I_{N_c}\otimes A_{ff})\overline{(W)} + c_2(1-\tau)(B_cB_c^T\otimes X_{ff})\overline{(W)} \\
& = \Big[ \tau(I_{N_c}\otimes A_{ff}) + c_2(1-\tau)(B_cB_c^T\otimes X_{ff})\Big] \overline{(W)}.
\end{align*}
\end{linenomath}
This derivation can be naturally extended to $\mathcal{H}$, where $W\in\mathcal{H}$ has a specified sparsity pattern, $\mathcal{N}$, by
setting the $k$th row and column of $\overline{\mathcal{L}}$ equal to zero for all $k$ such that $\overline{(W)}_k := W_{ij}$, and
$(i,j)\not\in\mathcal{N}$. Because $\overline{\mathcal{L}}$ is a block operator with block size $N_f\times N_f$, it follows that there is a
distinct ``diagonal'' in $\mathcal{L}$ corresponding to each $j$th column of $W$,
\begin{equation}
D_j = \tau\cdot\text{diag}(A_{ff}) + c_2(1-\tau)(B_cB_c^T)_{jj}\cdot\text{diag}(X_{ff}).\label{eq:d1}
\end{equation}
A diagonal preconditioning for $\hat{\mathcal{L}}$ is then given by taking the Hadamard product with $\mathcal{D}\in\mathcal{H}$,
where the $j$th column of $\mathcal{D}$ is given by the element-wise inverse of \eqref{eq:d1}:
\begin{equation}
\mathcal{D}_{ij} = \frac{1}{\tau(A_{ff})_{ii} + c_2(1-\tau)(B_cB_c^T)_{jj}(X_{ff})_{ii}}, \hspace{3ex} \text{ for } (i,j)\in\mathcal{N}. \label{eq:diag_pre}
\end{equation}
In the case of $A_{ff}$ having a constant or near-constant diagonal, and letting $X_{ff}$ be the diagonal of $A_{ff}$ (a common
practical choice), $\mathcal{D}$ is constant or near-constant. In practice, preconditioning with $\mathcal{D}$ is important for problems
in which diagonal elements of $A$ or target vectors $B$ consist of a wide range of values.
\subsection{Sylvester and Lyapunov equations}\label{sec:pcg:syl}
In fact, \eqref{eq:diag_pre} can be used to define a preconditioner for general systems of the Sylvestor- or Lyapunov-type:
\begin{equation}\label{eq:gen_mat}
AWB + CWD = F,
\end{equation}
for solution matrix $W$, where $A,B,C$ and $D$ need not be symmetric (of course an appropriate Krylov solver must be chosen based
on properties of the functional). A diagonal preconditioner for \eqref{eq:gen_mat} is given by taking the Hadamard product with
\begin{equation}\label{eq:genp}
\widehat{\mathcal{D}}_{ij} = \frac{1}{B_{jj}A_{ii} + D_{jj} C_{ii}}.
\end{equation}
Systems of the form in \eqref{eq:gen_mat} arise often in the context of optimal control theory. Letting $B = C = I$, \eqref{eq:gen_mat} is
a Sylvester equation; letting $B = A^T$, $C = -I$, and $D = I$, \eqref{eq:gen_mat} is a discrete Lyapunov equation; and letting $B = C = I$
and $D = A^T$, \eqref{eq:gen_mat} is a continuous Lyapunov equation. There have been many efforts at developing Krylov methods and
preconditioners for such systems; for example, see \cite{Dehghan:2010dx,Ding:2010dc,Hochbruck:1995kj,Simoncini:2009ig,Xie:2010cl}.
Here we develop a simple preconditioner for problems of the form \eqref{eq:genp}, that is easy to construct and apply.
\section{Numerical results}\label{sec:numerical}
In this section, we present numerical results for a variety of problems, comparing a weighted energy minimization and constrained
energy minimization, and analyzing the choice of constraint vector. The method proposed here is implemented in the PyAMG library
\cite{Bell:2008}; AMG methods such as strength-of-connection, coarsening, etc., follow that of \cite{Manteuffel:2016vd}, and the reader
is referred there for details. In figures, \textit{RN} refers to a constrained energy minimization using root-node AMG
\cite{Manteuffel:2016vd} and TM$_{10^k}$ refers to weighted energy minimization proposed here with weight $\tau = 10^k$. The test
problems considered are:
\begin{enumerate}\setlength\itemsep{1em}
\item \underline{Anisotropic diffusion:} 2-dimensional rotated anisotropic diffusion, discretized with linear finite elements,
on an unstructured triangular mesh:
\begin{align}
\label{eqn:diffusion}
-\nabla\cdot Q^T D Q\nabla u & = f\quad\textnormal{for ${\Omega} = [0,1]^2$},\\
\label{eqn:diffusionbdy}
u & = 0\quad\textnormal{on $\partial\Omega$},
\end{align}
where
\begin{equation*}
Q = \begin{bmatrix} \cos{\theta} & - \sin{\theta} \\
\sin{\theta} & \phantom{-}\cos{\theta}
\end{bmatrix},
\qquad
D = \begin{bmatrix}1 & 0 \\ 0 & \epsilon \end{bmatrix}.
\end{equation*}
Due to the unstructured mesh, all angles $\theta\in(0,\sfrac{\pi}{2})$ are effectively equivalent
from a solver perspective; thus, moving forward we (arbitrarily) let $\theta = \sfrac{3\pi}{16}$. Mesh
spacing is taken to be $h\approx 1/1000$, resulting in approximately 1.25M DOFs.
\item \underline{Diffusion with an oscillatory coefficient:} 2-dimensional diffusion problem (as in equations
(\ref{eqn:diffusion}-\ref{eqn:diffusionbdy})), discretized with linear finite elements on a structured, regular triangular
mesh with $2N^2$ elements, with a piecewise linear coefficient that oscillates at every other grid point, regardless of mesh size:
\[
Q = I,
D = \begin{bmatrix} f(x,y) & 0 \\
0 & f(x,y)
\end{bmatrix},
\]
where
\[
f(x,y) =
\begin{cases}
K & \mbox{if mod}(Nx,2)=1 \mbox{ AND mod}(Ny,2)=0 \\
K & \mbox{if mod}(Nx,2)=0 \mbox{ AND mod}(Ny,2)=1 \\
1 & \mbox{if mod}(Nx,2)=1 \mbox{ AND mod}(Ny,2)=1 \\
1 & \mbox{if mod}(Nx,2)=0 \mbox{ AND mod}(Ny,2)=0
\end{cases}.
\]
Taking $h = 1/N$, we recognize $Nx = x/h$ as the (integer) index of a
mesh point in the $x$-direction, with a similar interpretation of
$Ny$. For points $(x,y)$ not on the mesh, $f(x,y)$ is interpolated
linearly (as a function in the finite-element space).
This results in a number of coefficient oscillations that
grows proportionally with the number of mesh points, resulting in a checkerboard-like
pattern with alternating large and small coefficients. Diffusion problems
with large and frequent coefficient changes, such as this one,
traditionally make difficult test problems for multigrid methods.
\end{enumerate}
\subsection{Determining $P$}\label{sec:results:P}
Here we look at AMG convergence as a function of the number of
iterations of CG used to determine $P$ and how constraint
vectors are enforced, either exactly, in a constrained energy minimization, or weighted by $\tau\in(0,1)$, in
a weighed energy minimization. For all results, a V-cycle is applied with two iterations of Jacobi pre- and post-relaxation
as a preconditioner for CG. Unless otherwise specified, the constraint vector is chosen as the constant
vector with several Jacobi smoothing iterations applied; weighted energy-minimization uses the
diagonal preconditioning of Section \ref{sec:pcg}; and constrained energy-minimization uses the diagonal
preconditioning of \cite{Olson:2011fg}.
\subsubsection{Anisotropic Diffusion}
Figure \ref{fig:poisson} shows the work-per-digit-of-accuracy (WPD) as
a function of the number of iterations of CG used to determine $P$,
for variations in energy minimization applied to anisotropic Poisson (Problem \# 1), with anisotropy $\epsilon \in\{1,0.001,0\}$.
WPD is defined as
\[
WPD = \frac{-C}{\log_{10}(\rho)},
\]
where $C$ is the cycle-complexity of the multigrid solver and $\rho$ is the average convergence rate of the solver over
all iterations. This metric measures how many work-units, defined as the floating point operations to perform a single matrix-vector
multiply, are required to reduce the residual by one order of magnitude. This metric is particularly useful for cross-comparisons
of solvers with differing sparsity structures. For more detail, see, for instance, \cite{Manteuffel:2016vd}.
Interpolation is fixed to use a degree-four sparsity pattern, that is, the sparsity pattern for each column of $P$ reaches out to neighbors within
graph distance four from the corresponding C-point (see \cite{Manteuffel:2016vd,Schroder:2012di}). This wider sparsity pattern often
leads to better convergence rates for difficult problems \cite{Schroder:2012di}, but also requires more iterations of energy-minimization.
Essentially, wider sparsity patterns create more interpolation coefficients in $P$, which are then determined through energy-minimization.
\begin{figure}[!ht]
\centering
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{poisson_weighting_wpd_theta1_d4.pdf}
\caption{$\epsilon = 1$}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{poisson_weighting_wpd_theta001_d4.pdf}
\caption{$\epsilon = 0.001$}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{poisson_weighting_wpd_theta0_d4.pdf}
\caption{$\epsilon = 0$}
\end{subfigure}
\caption{WPD as a function of number of iterations of energy-minimization applied to $P$ for problem $\#1$ and (a) isotropic diffusion
($\epsilon=1$), (b) anisotropic diffusion ($\epsilon = 0.001$), and (c) totally anisotropic diffusion ($\epsilon = 0$).}
\label{fig:poisson}
\end{figure}
Several immediate results follow from Figure \ref{fig:poisson}. First, there is a limit at which additional
iterations to determine $P$ no longer improve convergence. For the isotropic case $(\epsilon = 1)$, the best convergence
rates are obtained by simply enforcing the constraint with a single constrained smoothing pass; additional
energy-minimization steps do not improve convergence. As the level of anisotropy increases ($\epsilon\to 0$), the number
of iterations of CG required to achieve the best performance increases. However, convergence of the
AMG solver based on a given constraint vector and coarsening scheme remains bounded below, regardless of further
energy minimization of $P$. Second, it is clear that enforcing the constraint exactly or near-exactly is fundamental
to good convergence, even for the simplest isotropic problem. Although theory tells us that interpolating low-energy
modes is necessary for good convergence, the fact that this cannot be achieved through weighted energy minimization
is slightly non-intuitive. Energy-minimization reduces the columns of $P$ in the $A$-norm, which should thus build $P$
to include low-energy modes in its range. Heuristically, it seems that after a handful of CG iterations, the range of $P$
would contain sufficient low-energy modes for good convergence. However, it is clear in Figure \ref{fig:poisson} that even
in the isotropic case, using a large $\tau = 0.1$ to focus on energy minimization over constraints leads to very poor
performance.
Together, these points underline the role of energy minimization in AMG convergence as an acceleration
technique. For some difficult problems, energy minimization is critical to achieving scalable convergence.
Strongly anisotropic diffusion is one such example that typically proves difficult for standard AMG methods,
but can be solved effectively with constrained energy minimization \cite{Manteuffel:2016vd}. Nevertheless,
regardless of energy minimization, strong convergence cannot be obtained without enforcing or nearly-enforcing
an appropriate constraint vector (Figure \ref{fig:poisson}).
\subsubsection{Diffusion with an oscillatory coefficient}
Figure \ref{fig:jumppoisson} shows the WPD as a function of the number
of iterations of CG used to determine $P$ for variations in energy
minimization applied to the oscillating coefficient problem (Problem \# 2), with coefficient oscillations of $K = 10^6$ and $K=10^3$.
For $K = 10^6$, Figure \ref{fig:jumppoisson:left} shows results for diagonal preconditioning
of energy-minimization and Figure \ref{fig:jumppoisson:center} shows
the case of no preconditioning. Figure \ref{fig:jumppoisson:right} shows the case of diagonal preconditioning with
$K=10^3$. Comparing Figures \ref{fig:jumppoisson:left} and \ref{fig:jumppoisson:center},
we see that using preconditioning in weighted energy minimization reduces the number of
iterations necessary to achieve good convergence. Moreover,
preconditioning appears to actually improve
the best achievable AMG convergence factor in practice. For constrained energy-minimization,
energy minimization iterations without preconditioning increases the WPD by $3-5\times$ within a reasonable
number of iterations on $P$ (of course, asymptotically the preconditioned and non-preconditioned results are
equivalent but, in practice, only $O(1)$ iterations are done.) This raises an interesting question as to if better
preconditioners for energy minimization can actually improve the AMG solver's performance in a way that additional
iterations with a diagonal preconditioner cannot in practical time; however, this is a topic for future study.
Focusing on the more practical solvers in Figure \ref{fig:jumppoisson:left}, we also see that the results mirror those in Figure
\ref{fig:poisson}. Overall, the constrained energy-minimization case performs best, with weighted energy-minimization able to
approach the constrained case only for the right $\tau$ values and enough energy-minimization iterations on $P$. Again, there
is a limit beyond which additional energy-minimization iterations no longer improve AMG convergence. For constrained
energy-minimization, relatively few iterations are needed. Lastly, enforcing the constraint exactly or near-exactly is fundamental
to good convergence. Using energy-minimization with larger $\tau$ values leads to poor performance.
The effects of the oscillating coefficient $K$ can be seen by comparing Figures \ref{fig:jumppoisson:left} and
\ref{fig:jumppoisson:right}. Interestingly, the larger $K$ value leads to a need for smaller $\tau$ values for the weighted case,
(compare the curves for $\tau=10^{-7}$). Overall, apart from changing the size of beneficial $\tau$ values, the size of the coefficient
oscillation does not noticeably affect either the weighted or constrained energy-minimization.
A final note of interest is that larger interpolation sparsity patterns do not help here. Thus, a moderate sparsity pattern of degree
three is chosen for these results.
\begin{figure}[!ht]
\centering
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{jumpy_coeff_1000000_w_Pprecon.pdf}
\caption{With diag. precon., $K = 10^6$}
\label{fig:jumppoisson:left}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{jumpy_coeff_1000000_wout_Pprecon.pdf}
\caption{Without diag. precon., $K = 10^6$}
\label{fig:jumppoisson:center}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{jumpy_coeff_1000_w_Pprecon.pdf}
\caption{With diag. precon., $K = 10^3$}
\label{fig:jumppoisson:right}
\end{subfigure}
\caption{Work-per-digit of accuracy, comparing weighted and constrained
energy-minimization, the use of diagonal preconditioning and two different coefficient jumps.}
\label{fig:jumppoisson}
\end{figure}
\begin{remark}
We did not find tracking the CG residual norm during energy-minimization to
be useful and, hence, omit plots of this information. The key difficulty is
that it is not clear how to connect the residual norm to the eventual
multigrid convergence rate. In other words, it is not clear how to use the
residual norm to halt the energy-minimization process. For instance, taking
the cases of constrained energy-minimization from Figures \ref{fig:poisson} and
\ref{fig:jumppoisson}, it is clear that at most five iterations of
energy-minimization are needed. However, the residual norm continues to
decrease monotonically by multiple orders of magnitude from iteration five
to iteration 19. Yet, this extra residual reduction does not speed up
convergence of the resulting multigrid solver. In practice, the
number of iterations needed typically equals the
degree of the sparsity pattern of $P$ plus some small number, usually two or
three. This number of iterations is required to first fill the allowed sparsity
pattern, and then to provide two or three iterations of additional improvement.
\end{remark}
\subsection{Constraint vectors and adaptivity}\label{sec:results:target}
In Section \ref{sec:results:P}, we learned two things: (i) for good convergence, it is important that $P$ exactly or almost exactly
interpolates an appropriate constraint vector, and (ii) coupled with a good constraint, energy minimization can improve convergence,
but only by a fixed amount. This leads to the natural idea of adding an additional constraint vector when further energy minimization
of $P$ no longer improves convergence. Such an approach is the basis
of adaptive multigrid methods, where a set of constraint
vectors are developed that are then included or approximately included in the range of $P$ \cite{Brandt:2011hb,Brezina:2004eh,
DAmbra:2013iwa}. There are multiple ways
to generate constraint vectors; here we take the simple approach of generating a random vector $\mathbf{x}_0$ and applying some
form of improvement iterations (either relaxation or V-cycles) to reduce $\|\mathbf{x}_0\|_A$. Table \ref{tab:poisson} shows results
for constrained energy minimization AMG applied to the anisotropic
Poisson problem, with varying numbers of improvement iterations and
varying numbers of constraint vectors.
\begin{figure}[!h]
\centering
\begin{subfigure}[b]{0.475\textwidth}
\begin{tabular}{c c c c c c c} Vecs & Imp. Iters & OC & CC & CF \\\midrule
1 & 2 & 1.52 & 5.97 & 0.75 \\
2 & 2 & 1.55 & 6.01 & 0.74 \\
3 & 2 & 1.56 & 6.02 & 0.79 \\\midrule
1 & 5 & 1.51 & 5.95 & 0.64 \\
2 & 5 & 1.54 & 5.99 & 0.70 \\
3 & 5 & 1.55 & 6.01 & 0.73 \\\midrule
1 & 10 & 1.50 & 5.95 & 0.53 \\
2 & 10 & 1.54 & 5.98 & 0.67 \\
3 & 10 & 1.55 & 5.99 & 0.69 \\\midrule
1 & 25 & 1.50 & 5.95 & 0.49 \\
2 & 25 & 1.54 & 5.97 & 0.67 \\
3 & 25 & 1.55 & 5.98 & 0.65 \\\midrule
1 & 100 & 1.50 & 5.95 & 0.48 \\
2 & 100 & 1.50 & 5.95 & 0.50 \\
3 & 100 & 1.50 & 5.95 & 0.51 \\\midrule
\end{tabular}
\caption{Two-grid}
\end{subfigure}
\begin{subfigure}[b]{0.475\textwidth}
\begin{tabular}{c c c c c c c} Vecs & Imp. Iters & OC & CC & CF \\\midrule
1 & 2 & 1.64 & 9.39 & 0.76 \\
2 & 2 & 1.67 & 9.52 & 0.81 \\
3 & 2 & 1.67 & 9.50 & 0.85 \\\midrule
1 & 5 & 1.63 & 9.29 & 0.64 \\
2 & 5 & 1.66 & 9.44 & 0.78 \\
3 & 5 & 1.66 & 9.47 & 0.83 \\\midrule
1 & 10 & 1.62 & 9.27 & 0.54 \\
2 & 10 & 1.64 & 9.36 & 0.76 \\
3 & 10 & 1.66 & 9.42 & 0.82 \\\midrule
1 & 25 & 1.62 & 9.23 & 0.51 \\
2 & 25 & 1.66 & 9.32 & 0.69 \\
3 & 25 & 1.66 & 9.42 & 0.78 \\\midrule
1 & 100 & 1.62 & 9.28 & 0.50 \\
2 & 100 & 1.62 & 9.28 & 0.54 \\
3 & 100 & 1.62 & 9.27 & 0.54 \\\midrule
\end{tabular}
\caption{Multigrid}
\end{subfigure}
\caption{Constrained energy minimization applied to a strongly
anisotropic diffusion problem ($\epsilon=0.001$) in a two-grid
and multigrid method. Constraints are initialized as a random vector; for the first constraint, Jacobi iterations are
applied as improvement iterations. After an AMG hierarchy has been formed with one target, a new random vector
is generated and V-cycles are applied as improvement iterations to generate a second target. The hierarchy is rebuilt
using the new constraints, and so on. }\label{tab:poisson}
\end{figure}
Several interesting things follow from the results in Table \ref{tab:poisson}. First, the difference in convergence factor
between two-grid and multigrid is very small. This indicates that we are solving our coarse-grid problem well using V-cycles, and
that convergence is limited by how ``good'' the coarse-grid problem is, and not how accurately we are solving it. Moreover, naively adding
constraint vectors that were not accounted for in the range of $P$ does not improve convergence and, in fact, degrades
convergence in all cases, while increasing the setup complexity. Although more involved processes have been developed for
adaptive multigrid methods, these simple tests give insight that improving convergence is not as simple adding new constraint
vectors.
\section{Conclusions}\label{sec:conclusions}
This paper explores the role of energy minimization in AMG interpolation from a theoretical and practical
perspective. The eventual goal is to develop improved interpolation techniques that are more robust than current
state-of-the-art, without the significant overhead setup cost of fully adaptive methods. A minimization framework is
developed based on a weighted combination of interpolating known low-energy modes with a global energy
minimization over $P$. On one hand, accurately interpolating the constraint vectors proves to be of fundamental
importance to good convergence, as observed where constrained energy minimization consistently performs best,
and weighted energy minimization performs best with the relative weight of interpolating constraints $\gg 0.99$.
However, convergence generally does not improve when additional constraint vectors are added beyond the first.
This either means, for these test problems, (i) accurately interpolating one constraint vector leads to convergence
factors close to the optimal rate for the given coarse grid \cite{brannick2017optimal}, or (ii) there are other factors
fundamental to convergence of AMG that are not being addressed in this framework. Results here do not suggest
the newly proposed algorithm is superior to existing methods, but do provide insight on the convergence of
AMG in the practical setting, as well as the relation to AMG convergence theory.
\section*{Acknowledgments}
The authors gratefully acknowledge the contributions of Ludmil
Zikatanov to the work presented here.
\bibliographystyle{siam}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,415 |
Postal terrorism
American investigators are finding evidence of anthrax contamination in more and more public buildings, and the number of people infected with the disease continues to mount. But senior officials are giving conflicting signals about who might be behind the anthrax attacks, and about how to handle them
IF THE motive behind sending anthrax spores through the post was to cause confusion, anxiety and disruption, the terrorist or terrorists who came up with the idea must be pleased with their efforts. Anthrax attacks have succeeded in disrupting the work of all three branches of government—executive, legislative and judicial—and in making America's leaders seem confused and even helpless in face of the threat. The number of people who have died from the disease is so far tiny compared with the mass slaughter caused by the suicide-hijackings of September 11th. But, in its very different way, the insidious, invisible and anonymous attack is just as potent a source of anxiety.
On October 29th, New York's mayor, Rudy Giuliani, announced that a 61-year-old woman had been hospitalised with what was suspected to be pulmonary anthrax, the most serious form of the disease. The woman worked in a hospital, but anthrax is not contagious. The office where she worked was near the post room. If the initial diagnosis is confirmed, that would take to 16 the number of confirmed cases of the disease identified since September 11th. Of those, nine had pulmonary anthrax, contracted through inhaling large numbers of tiny spores of the disease. Three have died. Seven people have developed the less-serious cutaneous form of the disease, a skin complaint. One, a woman in New Jersey whose case was also revealed on October 29th, was the first victim in this outbreak of the disease whose job was neither in the media nor involved handling the post.
It is the postal service itself that has been most affected by the anthrax outbreak, and where workers are most at risk. This was the cause of the government's biggest mistake in handling the attacks: the failure to take enough precautions after the discovery of a letter containing anthrax in Washington on October 15th. Only after two postal workers died a week later did the authorities realise the full extent of the danger. Even now, a union representing post-office workers in New York has taken legal action to demand the closure of one big sorting office where four mail-sorting machines have tested positive for anthrax spores. The workers feel hard done by in comparison with the bureaucrats who work in federal-government offices in Washington, especially since officials now suggest that some contaminated letters may not have been detected yet.
Supremely empty
So far this week traces of anthrax have been found in four more government buildings in Washington, DC. One, the Supreme Court, was already closed after anthrax was found at an office that handled some of its mail. On October 29th, its judges sat in another building for the first time since the court building opened in 1935. Two weeks ago, the House of Representatives shut for a few days because of an anthrax scare, prompting tabloid newspapers to castigate congressmen as "wimps". Traces of anthrax have been detected in mail-handling offices for, among other arms of government, the White House, the CIA, the State Department and the Department of Justice. American embassies around the world are checking their mail-rooms, after traces were also found in the diplomatic bag in Lima, Peru.
It is not known why anthrax has turned up in so many different places. It was first discovered in the offices of a publisher of tabloid newspapers in Boca Raton, Florida, where a picture editor died on October 5th. No source has been uncovered for that contamination, nor has any link been established between it and subsequent discoveries in Washington, New York and New Jersey. There, the sources of some, if not all, the known traces of anthrax are clear: three letters posted in Trenton, New Jersey, to the New York Post, Tom Brokaw, a famous television newsreader, and Tom Daschle, leader of the Democratic majority in the Senate.
Ridge under high pressure
These letters and their contents have provided investigators with a fund of evidence. But in interpreting it to the public, officials have given confusing and often contradictory accounts. In particular, it was at first suggested that the anthrax found was not "weapons-grade", and so was not as lethal as some of the more hysterical reactions suggested. But now it is accepted that it is milled in such a way as to require sophisticated knowledge of how to "weaponise" bacteria. An attempt has been made to co-ordinate the government's comments through Tom Ridge, holder of the new post of Director of Homeland Security. But Mr Ridge has been candid about the government's perplexity: "There are a lot of theories out there. We just need some facts to turn a theory into reality."
Among those theories is the suggestion that the anthrax originated in Iraq, which is known to have dabbled extensively in the black arts of chemical and biological warfare. That is a useful notion for those hawks in the Pentagon who would like the current campaign against the Taliban in Afghanistan broadened to an onslaught designed to get rid of America's nemesis, Saddam Hussein. Conversely, it might also be helpful to Osama bin Laden and his al-Qaeda network to frame Iraq, thereby encouraging America to broaden its anti-terror campaign: that might weaken the broad but fragile global coalition America has assembled.
Signed, sealed, undelivered
But officials have not been able to produce definitive evidence against either Iraq or Mr bin Laden, though many would agree with President George Bush when he said "I wouldn't put it past him." There remains the possibility that the anthrax attack is unrelated to September 11th, except in providing a pretext for an act of evil, whose perpetrator might be a lone madman, like the Unabomber. The three letters, all similar, all in poor English and all calling for "Death to America and "Death to Israel" could be intended to throw investigators off the scent. As for the anthrax itself, and the chemicals with which it was milled, scientists believe it could have come from Iraq, the Soviet Union or America. In recent days, new chemical tests have apparently strengthened the case of those arguing it was produced in America.
Floundering in their search for the culprits, America's leaders are also finding it hard to strike the right balance in what they say about the disease. Perhaps unfairly, they are criticised both for holding back information, and for confusing people by revealing too much. And they are torn between the need to show they are taking sensible precautions and the importance of preventing panic. The original mistakes over the post office have left them especially vulnerable to the charge that they are not doing enough to safeguard public health. Yet it remains true that anthrax is not contagious, that the chances of ordinary people contracting it through opening their mail are tiny, and that, despite its achievements as a tool of terrorism, as a weapon of mass destruction anthrax is, so far, a failure.
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Gemey est une marque de produits cosmétiques fondée en 1923 aux États-Unis et propriété du groupe L'Oréal depuis 1973.
Historique
La société américaine Gemey est fondée en 1923. .
La société est rachetée en 1973 par le groupe L'Oréal, et en 1976 la marque de mascaras Ricils, également acquise par L'Oréal lui est accolée.
Dans les années 1980, Gemey Paris est développée par la fusion des marques Gemey et Ricils.
En 1998, Gemey Paris et Maybelline New York (également propriété du groupe L'Oréal) fusionnent en France sous le nom de Gemey Maybelline. À l'international, Gemey Paris disparait au profit de Maybelline New York.
Notes et références
Histoire d'une marque : Gemey Maybelline, Le Blog de Cameline,
Autres :
Articles connexes
L'Oréal
Maybelline
Marque de produit cosmétique
L'Oréal | {
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All rights belong to NFL.
NFL Knocked Out Cold Players getting concussions.... Joshua 1:9 Be strong and courageous.
The Vikings decided to let Andrew Sendejo test Free Agency which means it's Anthony Harris time in MN.
All rights to the NFL and Fox.
Michael Thomas Dirty Hit On Andrew Sendejo. Sendejo Knocked Out.
COPYRIGHT DISCLAIMER** The nfl owns no right to this video I've modified it to my own by cutting out announcers voices and any and all NFL/PLAYOFF logos!
Andrew Sendejo puts Marcus Wheaton on blast!
Sendejo Gives Wallace A Concussion. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,195 |
{"url":"http:\/\/math.andrej.com\/category\/homotopy-type-theory\/","text":"# Mathematics and Computation\n\n## A blog about mathematics for computers\n\nPostsby categoryby yearall\n\n# Posts in the category Homotopy type theory\n\n### What is an explicit bijection? (FPSAC 2019 slides)\n\nHere are the slides with speaker notes for the talk What is an explicit bijection which I gave at the 31st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2019). It was the \"outsider\" talk, where they invite someone to tell them something outside of their area.\n\nSo how does one sell homotopy type theory to people who are interested in combinatorics? That is a tough sell. I used my MathOverflow question \"What is an explicit bijection?\" to give a stand-up comedy introduction, after which I plunged into type theory. I am told I plunged a little too hard. For instance, people asked \"why are we doing this\" because I did not make it clear enough that we are trying to make a distinction between \"abstractly exists\" and \"concretely constructed\". Oh well, it\u2019s difficult to explain homotopy type theory in 50 minutes. Anyhow, I hope you can get something useful from the slides.\n\nVideo recording of the lecture is now available.\n\n### A course on homotopy (type) theory\n\nThis semester my colleague Jaka Smrekar and I are teaching a graduate course on homotopy theory and homotopy type theory. The first part was taught by Jaka and was a nice review of classical homotopy theory leading up to Quillen model categories. In the second part I am covering basic homotopy type theory.\n\nThe course materials are available at the GitHub repository homotopy-type-theory-course. The homotopy type theory lectures are also recorded on video.\n\n### Spartan type theory\n\nThe slides from the talk \u201cSpartan type theory\u201d, given at the\u00a0School and Workshop on Univalent Mathematics.\n\n### The real numbers in homotopy type theory (CCA 2016 slides)\n\nI am about to give an invited talk at the Computability and Complexity in Analysis 2016\u00a0conference (yes, I am in the south of Portugal, surrounded by loud English tourists, but we\u00a0are working here, in a basement no less). Here are the slides, with extensive speaker notes, comment and questions are welcome.\n\nSlides:\u00a0hott-reals-cca2016.pdf\n\n### A HoTT PhD position in Ljubljana\n\nI am looking for a PhD student in mathematics. Full tuition & stipend will be provided for a period of three years, which is also the official length of the programme. The topic of research is somewhat flexible and varies from constructive models of homotopy type theory to development of a programming language for a proof assistant based on dependent type theory, see the short summary of the Effmath project for a more detailed description.\n\nThe candidate should have as many of the following desiderata as possible, and at the very least a master\u2019s degree (or an equivalent one):\n\n1. a master\u2019s degree in mathematics, with good knowledge of computer science\n2. a master\u2019s degree in computer science, with good knowledge of mathematics\n3. experience with functional programming\n4. experience with proof assistants\n5. familiarity with homotopy type theory\n\nThe student will officially enrol in October 2015 at the University of Ljubljana. No knowledge of Slovene is required. However, it is possible, and even desirable, to start with the actual work (and stipend) earlier, as soon as in the spring of 2015.\u00a0The candidates should contact me by email as soon as possible. Please include a short CV and a statement of interest.\n\nUpdate 2015-03-28: the position has been taken.\n\n### Seemingly impossible constructive proofs\n\nIn the post Seemingly impossible functional programs, I wrote increasingly efficient Haskell programs to realize the mathematical statement\n\n$\\forall p : X \\to 2. (\\exists x:X.p(x)=0) \\vee (\\forall x:X.p(x)=1)$\n\nfor $X=2^\\mathbb{N}$, the Cantor set of infinite binary sequences, where $2$ is the set of binary digits. Then in the post A Haskell monad for infinite search in finite time I looked at ways of systematically constructing such sets $X$ with corresponding Haskell realizers of the above omniscience principle.\n\nIn this post I give examples of infinite sets $X$ and corresponding constructive proofs of their omniscience that are intended to be valid in Bishop mathematics, and which I have formalized in Martin-L\u00f6f type theory in Agda notation. This rules out the example $X=2^\\mathbb{N}$, as discussed below, but includes many interesting infinite examples. I also look at ways of constructing new omniscient sets from given ones. Such sets include, in particular, ordinals, for which we can find minimal witnesses if any witness exists.\n\nAgda is a dependently typed functional programming language based on Martin-L\u00f6f type theory. By the Curry-Howard correspondence, Agda is also a language for formulating mathematical theorems (types) and writing down their proofs (programs). Agda acts as a thorough referee, only accepting correct theorems and proofs. Moreover, Agda can run your proofs. Here is a graph of the main Agda modules for this post, and here is a full graph with all modules.\n\n### Brazilian type checking\n\nI just gave a talk at \u201cSemantics of proofs and certified mathematics\u201d. I spoke about a new proof checker Chris Stone and I are working on. The interesting feature is that it has both kinds of equality, the \u201cpaths\u201d and the \u201cstrict\u201d ones. It is based on a homotopy type system proposed by Vladimir Voevodsky. The slides contain talk notes and explain why it is \u201cBrazilian\u201d.\n\nGitHub repository:\u00a0https:\/\/github.com\/andrejbauer\/tt\n\nAbstract: Proof assistants verify that inputs are correct up to judgmental equality. Proofs are easier and smaller if equalities without computational content are verified by an oracle, because proof terms for these equations can be omitted. In order to keep judgmental equality decidable, though, typical proof assistants use a limited definition implemented by a fixed equivalence algorithm. While other equalities can be expressed using propositional identity types and explicit equality proofs and coercions, in some situations these create prohibitive levels of overhead in the proof.\nVoevodsky has proposed a type theory with two identity types, one propositional and one judgmental. This lets us hypothesize new judgmental equalities for use during type checking, but generally renders the equational theory undecidable without help from the user.\n\nRather than reimpose the full overhead of term-level coercions for judgmental equality, we propose algebraic effect handlers as a general mechanism to provide local extensions to the proof assistant\u2019s algorithms. As a special case, we retain a simple form of handlers even in the final proof terms, small proof-specific hints that extend the trusted verifier in sound ways.\n\n### Univalent foundations subsume classical mathematics\n\nA discussion on the homotopytypetheory mailing list prompted me to write this short note. Apparently a mistaken belief has gone viral among certain mathematicians that Univalent foundations is somehow limited to constructive mathematics. This is false. Let me be perfectly clear:\n\nUnivalent foundations subsume classical mathematics!\n\n### The elements of an inductive type\n\nIn the HoTT book\u00a0issue 460\u00a0a question by gluttonousGrandma (where do people get these nicknames?) once more exposed a common misunderstanding that we tried to explain in section 5.8 of the book (many thanks to Bas Spitters for putting the book into Google Books so now we can link to particular pages). Apparently the following belief is widely spread, and I admit to holding it a couple of years ago:\n\nAn inductive type contains exactly those elements that we obtain by repeatedly using the constructors.\n\nIf you believe the above statement you should keep reading. I am going to convince you that the statement is unfounded, or that at the very least it is preventing you from understanding type theory.\n\n### The HoTT book\n\nThe HoTT book is finished!\n\nSince spring, and even before that, I have participated in a great collaborative effort on writing a book on Homotopy Type Theory. It is finally finished and ready for public consumption. You can get the book freely at http:\/\/homotopytypetheory.org\/book\/. Mike Shulman has written about the contents of the book, so I am not going to repeat that here. Instead, I would like to comment on the socio-technological aspects of making the book, and in particular about what we learned from open-source community about collaborative research.\n\n### How to implement dependent type theory III\n\nI spent a week trying to implement higher-order pattern unification. I looked at couple of PhD dissertations, talked to lots of smart people, and failed because the substitutions were just getting in the way all the time. So today we are going to bite the bullet and implement de Bruijn indices and explicit substitutions.\n\nThe code is available on Github in the repository andrejbauer\/tt (the blog-part-III branch).\n\n### How to implement dependent type theory II\n\nI am on a roll. In the second post on how to implement dependent type theory we are going to:\n\n1. Spiff up the syntax by allowing more flexible syntax for bindings in functions and products.\n2. Keep track of source code locations so that we can report where the error has occurred.\n3. Perform normalization by evaluation.\n\n### How to implement dependent type theory I\n\nI am spending a semester at the Institute for Advanced Study where we have a special year on Univalent foundations. We are doing all sorts of things, among others experimenting with type theories. We have got some real experts here who know type theory and Coq inside out, and much more, and they\u2019re doing crazy things to Coq (I will report on them when they are done). In the meanwhile I have been thinking how one might implement dependent type theories with undecidable type checking. This is a tricky subject and I am certainly not the first one to think about it. Anyhow, if I want to experiment with type theories, I need a small prototype first. Today I will present a very minimal one, and build on it in future posts.\n\nMake a guess, how many lines of code does it take to implement a dependent type theory with universes, dependent products, a parser, lexer, pretty-printer, and a toplevel which uses line-editing when available?\n\nWhat I find most amazing about the work is that Egbert does not have to pretend to be a homotopy type theorist, like us old folks. His first contact with type theory was homotopy type theory, which impressed on his mind a new kind of geometric intuition about $\\Pi$\u2019s, $\\Sigma$\u2019s and $\\mathrm{Id}$\u2019s. If we perform enough such experiments on young bright students, strange things will happen.","date":"2020-01-22 10:21:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5069921016693115, \"perplexity\": 878.7589685676276}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-05\/segments\/1579250606975.49\/warc\/CC-MAIN-20200122101729-20200122130729-00509.warc.gz\"}"} | null | null |
package io.crate.planner.optimizer;
import io.crate.action.sql.SessionContext;
import io.crate.auth.user.User;
import io.crate.expression.symbol.Literal;
import io.crate.expression.symbol.Symbol;
import io.crate.metadata.CoordinatorTxnCtx;
import io.crate.metadata.NodeContext;
import io.crate.metadata.SearchPath;
import io.crate.metadata.settings.SessionSettings;
import io.crate.planner.optimizer.rule.MergeFilters;
import org.junit.Test;
import java.util.List;
import java.util.Set;
import java.util.function.Function;
import static io.crate.analyze.SymbolEvaluator.evaluateWithoutParams;
import static io.crate.testing.TestingHelpers.createNodeContext;
import static org.hamcrest.MatcherAssert.assertThat;
import static org.hamcrest.Matchers.containsInAnyOrder;
import static org.hamcrest.Matchers.is;
public class OptimizerRuleSessionSettingProviderTest {
private NodeContext nodeCtx = createNodeContext();
private Function<Symbol, Object> eval = x -> evaluateWithoutParams(
CoordinatorTxnCtx.systemTransactionContext(),
nodeCtx,
x
);
@Test
public void test_optimizer_rule_session_settings() {
var settingsProvider = new LoadedRules();
var sessionSetting = settingsProvider.buildRuleSessionSetting(MergeFilters.class);
assertThat(sessionSetting.name(), is("optimizer_merge_filters"));
assertThat(sessionSetting.description(), is("Indicates if the optimizer rule MergeFilters is activated."));
assertThat(sessionSetting.defaultValue(), is("true"));
var mergefilterSettings = new SessionSettings("user",
SearchPath.createSearchPathFrom("dummySchema"),
true,
Set.of(MergeFilters.class));
assertThat(sessionSetting.getValue(mergefilterSettings), is("false"));
var sessionContext = new SessionContext(User.of("user"));
// Disable MergeFilters 'SET SESSION optimizer_merge_filters = false'
sessionSetting.apply(sessionContext, List.of(Literal.of(false)), eval);
assertThat(sessionContext.excludedOptimizerRules(), containsInAnyOrder(MergeFilters.class));
// Enable MergeFilters 'SET SESSION optimizer_merge_filters = true'
sessionSetting.apply(sessionContext, List.of(Literal.of(true)), eval);
assertThat(sessionContext.excludedOptimizerRules().isEmpty(), is(true));
}
}
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Q: Remove overlapping numbers from inside a tuple in python such that no 2 tuples have the same starting or ending number I have a list of tuples. Each tuple consists of a string and a dict. Now each dict in that, consists of a list of tuples. The size of the list is around 8K entries.
Sample data:
dataset = [('made of iron oxide', {'entities': [(12, 16, 'PRODUCT'), (17, 20, 'PRODUCT'), (15, 24, 'PRODUCT'), (12, 19, 'PRODUCT')]}),('made of ferric oxide', {'entities': [(10, 15, 'PRODUCT'), (12, 15, 'PRODUCT'), (624, 651, 'PRODUCT'), (1937, 1956, 'PRODUCT')]})]
From here output expected is:
dataset = [('made of iron oxide', {'entities': [(17, 20, 'PRODUCT'), (15, 24, 'PRODUCT'), (12, 19, 'PRODUCT')]}), ('made of ferric oxide', {'entities': [(10, 15, 'PRODUCT'), (624, 651, 'PRODUCT'), (1937, 1956, 'PRODUCT')]})]
Note: (12, 19, 'PRODUCT') is kept in output as the difference between start to end number is greater than (12, 16, 'PRODUCT'). PRODUCT is just a label and inconsequential.
These numbers are indexes of the sentences whose entities index are being displayed. Random sentences have been put in the example as it is inconseqential and the operation needs to be only on the entities dict. I want to remove overlapping numbers in my list and only keep those index values of entities that have the greatest length i.e., any value of entities cannot have the same starting or end number.
A: class Solution(object): #Ref. https://www.geeksforgeeks.org/merging-intervals/
def merge(self, intervals):
"""
:type intervals: List[Interval]
:rtype: List[Interval]
"""
if len(intervals) == 0:
return []
self.quicksort(intervals,0,len(intervals)-1)
#for i in intervals:
#print(i.start, i.end)
stack = []
stack.append(intervals[0])
for i in range(1,len(intervals)):
last_element= stack[len(stack)-1]
if last_element[1] >= intervals[i][0]:
last_element[1] = max(intervals[i][1],last_element[1])
stack.pop(len(stack)-1)
stack.append(last_element)
else:
stack.append(intervals[i])
return stack
def partition(self,array,start,end):
pivot_index = start
for i in range(start,end):
if array[i][0]<=array[end][0]:
array[i],array[pivot_index] =array[pivot_index],array[i]
pivot_index+=1
array[end],array[pivot_index] =array[pivot_index],array[end]
return pivot_index
def quicksort(self,array,start,end):
if start<end:
partition_index = self.partition(array,start,end)
self.quicksort(array,start,partition_index-1)
self.quicksort(array, partition_index + 1, end)
for i in range(len(dataset)): #Your Solution
arr1 = []
for item in dataset[i][1]['entities']:
arr1.append([item[0],item[1]])
ob1 = Solution()
arr2 = ob1.merge(arr1)
arr3=[]
for item in arr2:
arr3.append((item[0],item[1], 'PRODUCT'))
dataset[i][1]['entities'] = arr3
| {
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\section{Introduction} \label{sec:intro}
The nebular ionization parameter describes the ionizing photon density relative to gas density, and it is a fundamental diagnostic of radiation feedback in photoionized \ion{H}{2}\ regions. In recent years, diagnostics of the ionization parameter such as [\ion{O}{3}]$\lambda5007$/[\ion{O}{2}]$\lambda3727$, [\ion{S}{3}]$\lambda9069$/[\ion{S}{2}]$\lambda6717,6731$, [\ion{O}{3}]/H$\beta$, and [\ion{O}{3}]/[\ion{S}{2}]\ \citep{Pellegrini2012ApJ...755...40P, Zastrow2013ApJ...779...76Z,Keenan2017ApJ...848...12K, Wang2019ApJ...885...57W} have been used to evaluate the nebular optical depth to Lyman continuum (LyC) radiation in both individual \ion{H}{2}\ regions and starburst galaxies. An especially compelling class of objects are the Green Pea galaxies \citep{Cardamone2009MNRAS.399.1191C}, which are selected on the basis of their extreme ionization parameters in [\ion{O}{3}]/H$\beta$. Confirming predictions \citep[e.g.,][]{JaskotOey2013ApJ...766...91J}, the Green Peas have yielded the most consistent detections of LyC-emitting galaxies in the local universe \citep[e.g.,][]{Izotov2016MNRAS.461.3683I, Izotov2018MNRAS.478.4851I}, and are therefore of vital interest to galaxy evolution and cosmic reionization. However, although we noted above a direct link between ionization parameter and LyC optical depth, the exact relationship between these these quantities is not well understood in these starbursts, due to complicating factors like gas morphology, composition, geometry, and density distributions; and also variations in ionizing spectral energy distributions (SEDs) from various candidate stellar populations and other ionizing sources. Nevertheless, the ionization parameter is an easily observed and widely used diagnostic of the nebular conditions in star-forming regions both near and far.
It is well known that [\ion{O}{3}]$\lambda\lambda4959,5007$ emission drops precipitously at oxygen abundances $12+\log\rm(O/H) > 8.4$, \citep[e.g.,][]{Kewley2002ApJS..142...35K}. For example, the well-known abundance diagnostic $R23\equiv($[\ion{O}{2}]$\lambda3727 + $[\ion{O}{3}]$\lambda\lambda4959,5007)/\rm H\beta$ increases monotonically to maximum values around this metallicity. This is caused by strong sensitivity of this line to the electron temperature, which decreases at higher oxygen abundance. Therefore, using line ratios that rely on [\ion{O}{3}]$\lambda\lambda4959,5007$ as a diagnostic of ionization parameter will be unreliable at metallicities where these lines are weak.
In this work, we use the \ion{H}{2}\ regions of the Local Group galaxy M33 to explore the regime where [\ion{O}{3}]$\lambda\lambda4959,5007$ is, and is not, effective as a diagnostic of ionization parameter for the purpose of evaluating radiation feedback and LyC escape. In order to generate the emission-line images required for this analysis, it is necessary to carry out continuum subtraction, and we also further explore this process.
\section{Observations of M33} \label{sec:obs}
The north half of M33 was observed with the MOSAIC-1.1 imaging camera at the Mayall 4-m telescope, Kitt Peak National Observatory, on 2011 October 28--29. We used the narrowband filters for [\ion{O}{2}]$\lambda3727$ (``O2", FWHM 50\AA), [\ion{O}{3}]$\lambda5007$ (``O3", FWHM 50\AA), and [\ion{S}{2}] $\lambda6724$ (``ha16", FWHM 81\AA) for line imaging. For the continuum, we used broadband filters BATC454 ($\sim 4320 - 4700$ \AA) and BATC705 ($\sim 6950 - 7920$ \AA) for continuum subtraction of [\ion{O}{2}]\ and [\ion{O}{3}]; and of {\rm H$\alpha$}\ and [\ion{S}{2}], respectively. The continuum filters are used with the kind permission of R. Windhorst. We also used archive {\rm H$\alpha$}\ observations obtained in 2001 with the same setup by \citet{Massey2007AJ....134.2474M}. There were $6\times 1400$ s, $4\times 600$ s, $5\times600$ s $+1\times900$ s, and $5\times300$ s exposures in [\ion{O}{2}], [\ion{O}{3}], [\ion{S}{2}], and {\rm H$\alpha$}, respectively; and $5\times500$ s and $2\times1050$ s $+3\times 550$ s exposures in the blue and red continuum filters, respectively. All observations were dithered to cover the gaps between the eight MOSAIC CCD chips. After being median-combined, the images have a variety of residual defects, including edge effects generated by the image combining process and a very low-level reflection of the telescope pupil.
In the [\ion{S}{2}]\space image, the giant \ion{H}{2}\space regions NGC 604, NGC 595, IC 131 and the star BD $+$30 243 generated trails likely caused by CTE effects,
causing a band $\sim$1000 pixels wide running through one section of the image. The northeast corner of the [\ion{S}{2}]\ continuum image also has an unusually bright patch, possibly due to reflections. Such bad pixels were trimmed out of the final array when performing the continuum subtraction.
We use routines from the \texttt{astropy photutils} package to identify and mask the foreground Milky Way stars\deleted{, and interpolate over the masked pixels.} \added{We then interpolate over these masked pixels using a bilinear interpolation so that the faint wings of foreground stars do not affect the flux measurement of \ion{H}{2}\ regions.}\explain{Addressing minor comment 5 of the referee.}
We use a sample of 7 objects from \citet{TsC2016MNRAS.458.1866T}, listed in Table \ref{tab:TSC-dered-rered-fluxes}, to carry out flux calibration.
For [\ion{S}{2}]\ and [\ion{O}{2}], we sum the reported flux from the two lines forming these doublets, which are both included in the filter bandpasses.
We recover the un-dereddened fluxes using the reported $c(\rm H\beta)$ and extinction law:
\begin{equation} \label{reddening law}
\log\left(F_{\lambda_{1}} / F_{\lambda_{2}}\right)_{\text{corr}} = \log\left(F_{\lambda_{1}} / F_{\lambda_{2}}\right)_{\text{obs}} + C \left(f_{\lambda_{1}} - f_{\lambda_{2}}\right)
\end{equation}
The values in Table \ref{tab:TSC-dered-rered-fluxes}, columns (6)-(9) are the un-dereddened fluxes ($F_{obs}$) that we adopt for calibration.
\begin{deluxetable*}{cccccccccc}
\tablenum{1}
\tablecaption{Line fluxes after removing reddening correction\tablenotemark{a} \label{tab:TSC-dered-rered-fluxes}\explain{Added uncertainty for literature ratios}}
\tablewidth{0pt}
\tablehead{
\colhead{ID} & \colhead{c(H$\beta$)\tablenotemark{b}} & \multicolumn4c{Reddened} & \multicolumn2c{[\ion{O}{3}]/{\rm H$\alpha$}\ literature} & \multicolumn2c{[\ion{O}{3}]/{\rm H$\alpha$}\ Observed}\\
& & \colhead{{\rm H$\alpha$}} & \colhead{[\ion{O}{2}] $\lambda 3727$} & \colhead{[\ion{O}{3}] $\lambda 5007$} & \colhead{[\ion{S}{2}] $\lambda\lambda 6718, 6732$} & mean & st.d. & mean & st.d.
}
\decimalcolnumbers
\startdata
B2011 b5 & 0.67 & 460 & 89 & 194 & 56 & 0.42 & 0.10 & 0.34 & 0.07 \\
IC 131 & 0.51 & 402 & 116 & 332 & 60 & 0.83 & 0.20 & 1.06 & 0.15 \\
BCLMP 290 & 0.12 & 318 & 188 & 177 & 32 & 0.56 & 0.07 & 0.49 & 0.05 \\
NGC 588 & 0.16 & 315 & 90 & 475 & 19 & 1.51 & 0.24 & 1.25 & 0.06 \\
BCLMP 626 & 0.02 & 292 & 239 & 165 & 43 & 0.57 & 0.06 & 0.66 & 0.06 \\
LGC HII3 & 0.09 & 302 & 171 & 285 & 31 & 0.94 & 0.12 & 1.03 & 0.29 \\
IC 132 & 0.37 & 370 & 46 & 566 & 15 & 1.53 & 0.20 & 1.89 & 0.18 \\
\enddata
\tablenotetext{a}{All fluxes are relative to H$\beta$ = 100.}
\tablenotetext{b}{c(H$\beta$) values from \citet{TsC2016MNRAS.458.1866T} used to remove their reported reddening correction.}
\end{deluxetable*}
We apply rectangular apertures corresponding to the reported slit width at the position for each object given in
Table 1 of \citet{TsC2016MNRAS.458.1866T}. We use the \texttt{photutils} aperture photometry routine in \texttt{astropy} to obtain the photometry.
To account for positional inaccuracies arising from atmospheric seeing, we offset the slit positions by a normally distributed random variable with standard deviation equal to the seeing in pixels. We average over 20 such randomly offset apertures for the integrated photometry.
The computed ratios are then used together with the calibrated flux ratios derived from the data of \citet{TsC2016MNRAS.458.1866T}
to determine the flux calibration, using the {\rm H$\alpha$}/H$\beta$\ ratio of 2.86.
\section{Continuum subtraction} \label{sec:subtract}
Narrow-band imaging data include flux from both line emission and diffuse stellar continuum. To isolate the line flux, an off-line image containing only the continuum is usually subtracted. Due to differing filter transmissions and variation in the continuum spectral energy distribution (SED), the continuum image must be scaled before subtraction, and determining the scale factor is nontrivial. \citet{Hayes2009AJ....138..911H} and \citet{James2016ApJ...816...40J} have utilised stellar population synthesis modeling that compute spatially varying scale factors on a pixel-by-pixel basis. These methods are model-dependent and \citet{Hayes2009AJ....138..911H} discuss their advantages and pitfalls in detail.
In this work, we focus on empirical methods that compute a single characteristic scale factor for a large region or an entire image.
\citet{Keenan2017ApJ...848...12K} describe a method where a single optimal scale factor is found for such a region. They note a slope change in the mode of the pixel values vs scale factor for the continuum-subtracted images. \citet{Keenan2017ApJ...848...12K} show that this transition results when the scale factor induces any oversubtraction. At small values of the scale factor, the mode of the pixel-value histogram is determined by the lowest-value pixels. The change in flux for these pixels is small as the scale factor varies, hence the slope of mode vs scale factor is shallow. At higher values of the scale factor, the mode is dominated by oversubtracted pixels. As the scale factor increases, the first pixels to become oversubtracted are the brightest ones, and their flux has a strong dependence on scale factor. Thus, the slope of the mode vs scale factor is steeper in the over-subtracted regime (see \citet{Keenan2017ApJ...848...12K} for more details).
\citet{Hong2014PASP..126...79H} present a method that similarly identifies the transition to oversubtraction, but based on the skewness of the pixel-value histogram as a function of the scale factor. The above two methods have been shown to work well for images where there are a significant number of continuum-dominated pixels. \deleted{However, we still encountered some difficulties in applying both of these.}
\added{However, we still encountered some difficulties in adequately constraining the best scale factors.
In particular, the slope transition reported by \citet{Keenan2017ApJ...848...12K} can be hard to discern. There can be multiple slope changes and oscillatory behavior in the mode, as \citet{Keenan2017ApJ...848...12K} indicate. The exact point of transition is therefore uncertain.
Similarly, the method used by \citet{Hong2014PASP..126...79H} relies on computing the second derivative of the skewness with respect to the scale factor. Estimating this derivative accurately requires a very fine search through the scale factor space.
Appendix~A demonstrates the functionality of these two methods (Figures \ref{fig:haro-11-fraction}, \ref{fig:eso338-fraction} and \ref{fig:mrk71-fraction}).
}
In this paper, we therefore propose a revised method that uses the mode to obtain the optimal scale factor,
\added{which can provide a narrower confidence interval while also reducing the computing power needed}.
\explain{Addressing comment 1.2 of referee}
\subsection{Revised Mode Method to Identify Scale Factor} \label{subsec:agg-factor}
The line emission represents an excess signal over the continuum, hence, the pixel-value distributions will have a tail to positive values dominated by the real signal. Since we are interested in identifying the diffuse background to carry out the continuum subtraction, we therefore employ a $3\sigma$ filter from the \texttt{scipy} library on the line image. The routine computes the mean and standard deviation $\sigma$ of the data, then rejects any data points that are $>3\sigma$ from the mean. The mean and standard deviation are recomputed and the filtering is done again. This iterative process is continued until there are no more rejections. At each scale factor, we first subtract the scaled continuum from the line image, then invoke the $3\sigma$ filter on the subtracted image. This removes much of the real signal, leaving a residual histogram that is more dominated by the diffuse continuum signal. The resulting pixel-value distribution is better suited for the purpose of identifying the optimal continuum image scale factor.
Similar to \citet{Hong2014PASP..126...79H}, we observe a transition in the skewness of the pixel histogram as the scale factor is increased.
Figure~\ref{fig:histogram} shows how the shape of the pixel-value distribution changes as the scale factor is varied. At low scale factors (Figure \ref{fig:histogram}(a)), the image is undersubtracted. The distribution for the subtracted image resembles the initial flux distribution, where pixels with high continuum values contribute to high-value bins. As a result, the histogram is skewed to the right.
As the scale factor is increased, the flux distribution in the line image approaches the flux distribution in the scaled off-line image.
Ideally, at this point of optimal continuum subtraction, the background flux should be zero. This is characterised by the mode of the pixel histogram being zero. However, this is not always the case, since the sky may have a different SED compared to the diffuse stellar continuum.
This issue is particularly relevant for ground-based observations where the sky background is significant. \added{Once the scale factor has been set by subtracting the stellar emission, the residual sky background can be eliminated while performing aperture photometry, as we have done for the flux measurements in Tables \ref{tab:TSC-dered-rered-fluxes} and \ref{tab:object-list}}
\explain{Addressing minor comment 1 of referee.}
On further increasing the scale factor, the pixels with strong continuum flux now contribute to the negative bins as they are the first ones to become oversubtracted. The spread increases in the negative direction, skewing the histogram to the left (Figure \ref{fig:histogram}(c)). This is a reversal of the behavior seen earlier, and it corresponds to the transition in skewness reported by \citet{Hong2014PASP..126...79H}. It should be noted that the negative tail is less statistically `heavy' compared to the positive tail, due to the continued presence of pixels with emission-line flux.
\citet{Hong2014PASP..126...79H} demonstrate the same effect as a slight positive bias to the skew at the transition point.
At the optimal scale factor, the number of background pixels is therefore maximized in the modal bin, as shown in Figure \ref{fig:fraction-chart}.
Due to the background noise, the optimally subtracted histogram will still have some spread around the mode. We therefore also consider the total number of pixels in the bins adjacent to the modal bin. This is also helpful if the mode falls on the boundary between two bins. Using all three bins provides some robustness against such cases. Thus, the optimal scale factor is that which maximises $({f_0 + f_1 + f_2})/{N}$, where $f_0$ is the number of pixels in the modal bin; $f_1$ and $f_2$ are the number of pixels in the pre-modal and post-modal bins, respectively; and $N$ is the total number of pixels in the image, after applying the $3\sigma$ filter. If the modal bin is the first or last bin, then $f_1$ or $f_2$ are accordingly assumed to be zero. The value of $N$ changes with the scale factor due to the $3\sigma$ filter. Initially, the brightest pixels get rejected, but at the correct scale factor, the flux from these correctly subtracted pixels falls within the $3\sigma$ limit.
\added{The chosen metric, $({f_0 + f_1 + f_2})/{N}$, is computed using only the pixel histogram at the current scale factor. In comparison, the methods of \citet{Keenan2017ApJ...848...12K} and \citet{Hong2014PASP..126...79H} require the histograms at adjacent values of the scale factor to compute the slope of the mode, or the second derivative of skewness which requires finer sampling to reduce the error bound.}
\explain{Addressing major comment 1 of the referee.}
A jagged or scalloped pattern is apparent in Figure \ref{fig:fraction-chart} as the scale factor is increased. The pattern arises due to the binning criteria chosen for the of the modal bin fraction. In particular, it is due to the choice of using only the 3 modal bins to measure of the peak of the pixel distribution. In the case of highly skewed histograms, the peak is inadequately sampled by only 3 bins, and small changes in the pixel distribution are amplified, giving rise to the jagged pattern seen. This behaviour can be smoothed by modifying the criterion to include more bins, or by finer sampling of the pixel distribution by increasing the number of bins. Finer sampling has an added computational cost. For the purposes of identifying the globally optimal scale factor from a symmetric histogram, the chosen metric works well and is unaffected by the local variations due to skewed histograms.
\added{Appendix~A demonstrates this method, with comparisons to the \citet{Keenan2017ApJ...848...12K} and \citet{Hong2014PASP..126...79H} methods (Figures~\ref{fig:haro-11-fraction}, \ref{fig:eso338-fraction} and \ref{fig:mrk71-fraction}).}
\explain{Addressing major comment 1}
\begin{figure*}
\gridline{\fig{M33_factor_0.0000.png}{0.3\textwidth}{(a)}
\fig{M33_factor_0.1800.png}{0.3\textwidth}{(b)}
\fig{M33_factor_0.2600.png}{0.3\textwidth}{(c)}
}
\caption{Histograms for M33 [\ion{O}{3}]\ continuum subtraction. Panel (a) shows undersubtracted pixel histogram with high-value tail. Panel (b) shows the optimally subtracted pixel histogram, where the number of pixels in the 3 modal bins is maximized. Panel (c) shows the emergence of the low-value tail in an oversubtracted pixel histogram. }
\label{fig:histogram}
\end{figure*}
We caution that the modal bin maximisation works to subtract the dominant background component in the image, regardless of whether it is diffuse starlight, sky emission, or diffuse nebular emission. Thus, if the goal is to subtract diffuse starlight, then that component should constitute a significant fraction of the total pixel population.
For example, if sky pixels dominate the variance, then the algorithm produces an undersubtracted image relative to diffuse starlight. This can be caused by two effects. First, if sky emission dominates the flux of the background pixels, then the algorithm identifies the optimal scale factor for the sky background. This is illustrated in Appendix \ref{sec:other-applications}. Similarly, if the random variance of the sky pixels is high in images with low signal-to-noise, then the large variance makes the reduced spread of the stellar continuum pixels harder to detect. This can be explained as follows: the unsubtracted line image has a certain variance arising from the true signal, which is gradually subtracted out by increasing the scale factor. At the same time, we introduce additiional variance through the random noise of the sky pixels, which increases on increasing the scale factor. If the signal-to-noise ratio is low, the reduction in variance of diffuse starlight is drowned out by the noise that we introduce. The spread is minimised at a lower scale factor, resulting in an undersubtracted image. We address this issue in Section \ref{subsec:application} and Appendix \ref{sec:other-applications}.
On the other hand, the presence of widespread diffuse line emission will also bias the pixel-value histogram and the spread towards higher values. Contamination of pure continuum pixels with diffuse line emission therefore also inflates the variance of continuum pixels. The algorithm proposed above
overcompensates for the larger spread, resulting in a scale factor greater than optimum. This is mitigated by ensuring the presence of pure continuum pixels in the image. Therefore, prior knowledge about the emission region characteristics is required. In short, the background must be dominated by diffuse starlight pixels in order for these statistical methods to identify the optimal scale factor to subtract this background component.
\added{See \citet{Hong2014PASP..126...79H} for a discussion on how the various background compositions affect these continuum-subtraction methods.}
\explain{Addressing minor comment 1.}
\subsection{Application of the Method}\label{subsec:application}
Like most spiral galaxies, M33 has a strong color gradient in its diffuse starlight, implying that the scale factor for subtracting this component will vary spatially. We therefore define 5 regions using elliptical isophotes (Figure \ref{fig:el-offb_img}), which are are chosen by visual inspection of the approximate stellar surface brightness. Region 0, the galactic center, is dominated by resolved sources and does not have many background pixels.
When presented with an image like this, our algorithm tends to produce oversubtracted images since there is not enough background to determine the correct scale factor. The opposite holds for Region 4, which is dominated by sky pixels and relatively lacking in diffuse starlight. Sky-dominated images tend to undersubtract diffuse starlight when we apply our method, as described in Section~\ref{subsec:agg-factor}. Regions 1, 2 and 3 have a good mix of emission, continuum, and background pixels, so our method works well for such regions. Due to the different characteristics of the pixel populations of Regions 0 and 4, we assign
the scale factors for Region 0 and 4 by setting the modes of their continuum-subtracted pixel values equal to the modes for Regions 1 and 3, respectively.
In our pipeline, we first employ the $3\sigma$ filtering described above to the emission-line image. The pixels with the strongest fluxes are rejected by the filter.
Next, we generate the pixel-value histogram of the continuum-subtracted image. The bin width is chosen following \citet{Sturges1926}: $\text{The number of bins}\ n \approx 1 + \log_{2}N$, where $N$ is the total number of pixels. For our images, this value is 22-25 bins, depending on the Region. The modal bin is identified and the modal bin fraction calculated as described earlier in Section \ref{subsec:agg-factor}. The exact value of the mode $M$ can be calculated by taking the intersection of two lines that linearly interpolate the data in the modal bin:
\begin{equation}\label{modelines}
M = L + \frac{f_{0} - f_{1}}{2f_{0} - f_{1} - f_{2}}\times h
\end{equation}
where $L$ is the lower limit of the modal bin, and $h$ is the bin width. This exact value is needed to set the scale factors for Regions 0 and 4.
An animation of the continuum subtraction process (Figure~\ref{fig:video}) is available in the online version of this paper.
\added{Table~\ref{tab:object-list} gives our narrowband fluxes for the objects, with
errors estimated by computing the median background around each object during aperture photometry. We confirmed the {\rm H$\alpha$}\ luminosities of three giant \ion{H}{2}\ regions: NGC 604, NGC 595 and IC 131 with the values given by \citet{Relano2009ApJ...699.1125R}.}
\explain{Addressing minor comment 4.2 by referee.}
\begin{figure*}
\gridline{\fig{M33_oiii_el.png}{0.5\textwidth}{(a)}
\fig{M33_oiii_offb.png}{0.5\textwidth}{(b)}
}
\caption{Narrow-band [\ion{O}{3}]\ on-line (a) and off-line (b) images of M33. The five regions with varying background levels are shown.
\label{fig:el-offb_img}}
\end{figure*}
\begin{figure*}
\begin{interactive}{animation}{M33_OIII_movie.mp4}
\plotone{oiii_img_hist.png}
\end{interactive}
\caption{The online version of this figure is an animation of the step-by-step subtraction process, showing how the shape of the pixel histogram changes for the corresponding continuum-subtracted [\ion{O}{3}]\ images as the continuum-image scale factor is increased from 0 to 0.36. The point of optimal subtraction can be identified around scale factor $\approx 0.18$. The video duration is 9 seconds.
\label{fig:video}}
\end{figure*}
\begin{figure*}
\fig{M33_OIII_Fraction_in_modal_bins.png}{0.6\textwidth}{}
\caption{Modal bin fraction, calculated from equation \ref{modelines}, vs the scale factor. This metric is maximised in this example when the scale factor is 0.188, indicating the point of optimal subtraction. The scalloped pattern is caused by the binning of the pixel value distribution (see text).
\label{fig:fraction-chart}}
\end{figure*}
To further test our new method of continuum subtraction, we also apply it to \ion{C}{3}]\ $\lambda1909$ narrow-band imaging of three starburst galaxies, Haro 11 \citep{Micheva2020ApJ...903..123M}, ESO338-IG04, and Mrk 71. Whereas M33 has a strong color gradient and many resolved individual stars, these more distant galaxies are dominated by more uniform, diffuse starlight. These examples also have very faint line emission compared to the M33 data. We find that our new continuum subtraction technique works well on these galaxies. The results are shown in Appendix~\ref{sec:other-applications}.
\begin{deluxetable*}{cccccDDDDDDDD}
\tablenum{2}
\tablecaption{\ion{H}{2}\space Regions in Northern M33 \label{tab:object-list}}
\tablewidth{\linewidth}
\tabletypesize{\scriptsize}
\tablehead{
\colhead{Object} & \colhead{RA (J2000)} & \colhead{Dec (J2000)} &
\colhead{12+log(O/H)\tablenotemark{a}} & \colhead{Opacity\tablenotemark{b}} &
\multicolumn2c{{\rm H$\alpha$}\tablenotemark{c}} & \multicolumn2c{{\rm H$\alpha$}\ err\tablenotemark{c}} &
\multicolumn2c{[\ion{O}{2}]\tablenotemark{c}} & \multicolumn2c{[\ion{O}{2}]\ err\tablenotemark{c}} &
\multicolumn2c{[\ion{O}{3}]\tablenotemark{c}} & \multicolumn2c{[\ion{O}{3}]\ err\tablenotemark{c}} &
\multicolumn2c{[\ion{S}{2}]\tablenotemark{c}} & \multicolumn2c{[\ion{S}{2}]\ err\tablenotemark{c}}
}
\decimalcolnumbers
\startdata
BCLMP 616 & 1h32m54.38s & 30d50m28s & $8.226^{+0.006}_{-0.005}$ & 1 & 14.6 & 0.22 & 7.61 & 0.20 & 2.03 & 0.46 & 2.97 & 0.40 \\
LHK2017 54 & 1h32m56.28s & 30d40m36.7s & $8.383^{+0.004}_{-0.003}$ & 4 & 23.6 & 3.9 & 13.1 & 1.2 & 5.2 & 2.6 & 13.1 & 4.5 \\
BCLMP 289 & 1h32m57.5s & 30d44m27s & $8.208^{+0.003}_{-0.004}$ & 3 & 57.4 & 19 & 31.4 & 0.83 & 66.4 & 4.6 & 18.1 & 21 \\
CPSDP 67 & 1h32m59.21s & 30d41m20.8s & $8.518^{+0.007}_{-0.007}$ & 1 & 10.1 & 1.4 & 4.91 & 0.16 & 2.65 & 1.4 & 3.64 & 2.0 \\
BCLMP 285 & 1h33m02.88s & 30d41m08.1s & $8.238^{+0.003}_{-0.003}$ & 1 & 13.1 & 0.25 & 5.95 & 0.04 & 4.63 & 0.27 & 2.13 & 0.42
\enddata
\tablenotetext{a}{From \citet{Lin2017ApJ...842...97L}.}
\tablenotetext{b}{$1=$ optically thick, $2=$ blister, $3=$ optically thin, $4=$ shock, $5=$ indeterminate.}
\tablenotetext{c}{Luminosities given in $10^{36}$ erg/s.}
\tablecomments{Data for first five objects are shown here. The online version of this paper has data for all 108 objects in a machine readable format.}
\end{deluxetable*}
\section{Ionization-parameter mapping}\label{sec:IPM}
We generate line ratio maps for [\ion{O}{3}]/[\ion{O}{2}]\ and [\ion{O}{3}]/[\ion{S}{2}]\ as described by \citet{Keenan2017ApJ...848...12K} and \citet{Pellegrini2012ApJ...755...40P} to evaluate the optical depth of photoionized \ion{H}{2}\ regions by ionization-parameter mapping (IPM).
IPM is most directly applied to distinct objects, and significant diffuse continuum can mask the signal from individual HII regions. Unsharp masking allows us to locally remove the average diffuse ionized gas emission over a given length scale. This allows for an amplification in the signal for optically thin nebulae, where the low-ionization emission may be dominated by diffuse ambient emission. The residual presence of field stars affects the median smoothing, so we first
\added{carry out a bilinear interpolation}
\explain{addressing minor comment 5 of the referee}
over these across regions three times the stellar PSF. We then mask any emission with intensity greater than 30 counts ($5.8\times 10^{-15}\ \rm erg\ s^{-1}\ cm^{-2}\ px^{-1}$). \added{This is approximately the saturation level of the detector.}\explain{Addressing minor comment 7 of the referee.} We median filter the remaining emission over a length scale 30$\times$ the stellar PSF, corresponding to 83.1 pixels.
Finally, we subtract that medianed image from the original data, smoothed to the seeing to improve S/N.
Following \citet{Pellegrini2012ApJ...755...40P}, we classify the M33 \ion{H}{2}\ regions into 5 classes: (0) indeterminate, (1) optically thick, (2) blister, (3) optically thin, and (4) shocked, based on the ionization structure in the halos of individual regions. Objects are considered optically thick if [\ion{O}{2}]\ and/or [\ion{S}{2}]\ dominates over at least two-thirds of the circumference in projection. Blister and optically thin \ion{H}{2}\ regions are those with the low-ionization species dominating over one-third to two-thirds; and less than one-third of the circumference, respectively.
Shocked nebulae show an ionization structure that is unlikely to have resulted from photoionization alone, with a pocket of low ionization species on the interior surrounded by [\ion{O}{3}]\ on the outside. Objects are categorised as ``indeterminate" due to poor S/N, incomplete data, or abnormal ionization morphology.
Our classifications are given in Table~\ref{tab:object-list} for our sample of 108 objects, and are based on consideration of both [\ion{O}{3}]/[\ion{S}{2}]\ and [\ion{O}{3}]/[\ion{O}{2}]\ ratio maps. The object locations in the disc of M33 are shown in Figure \ref{fig:finder-chart}. We present examples of categories 1--4 in Figures~\ref{fig:bclmp-668} to \ref{fig:bclmp-667}. These figures show the [\ion{O}{3}]/[\ion{S}{2}]\ and [\ion{O}{3}]/[\ion{O}{2}]\ ratio map for these representative objects.
Similar images for all the objects in our sample are provided in the interactive version of Figure~\ref{fig:metal-chart} below.
\begin{figure*}
\fig{M33_Ha_finder.png}{0.5\textwidth}{}
\caption{Continuum-subtracted {\rm H$\alpha$}\ image with the sample \ion{H}{2}\space regions from Table \ref{tab:object-list} marked.
\label{fig:finder-chart}}
\end{figure*}
Figure \ref{fig:bclmp-668} shows ratio maps for BCLMP 668, an optically thick nebula. As evidenced by Figure \ref{fig:bclmp-668}(b) and (c),
this object shows a classic, Str\"omgren sphere structure, with the low-ionization envelope completely surrounding the highly ionized core.
The emission from [\ion{O}{3}] \space is relatively low throughout.
\begin{figure*}
\gridline{\fig{BCLMP_668_halpb.png}{0.3\textwidth}{(a)}
\fig{BCLMP_668_o3o2b.png}{0.3\textwidth}{(b)}
\fig{BCLMP_668_o3s2b.png}{0.3\textwidth}{(c)}
}
\caption{BCLMP 668 in smoothed, unsharp-masked (a) continuum subtracted {\rm H$\alpha$}, (b) [\ion{O}{3}]/[\ion{O}{2}], and (c) [\ion{O}{3}]/[\ion{S}{2}]\space ratio maps.
\added{Color bar units are in image counts.}
This object is classified as optically thick, and displays a shell-like structure of high vs low ionization zones.}
\label{fig:bclmp-668}
\end{figure*}
\begin{figure*}
\gridline{\fig{BCLMP_650_halpb.png}{0.3\textwidth}{(a)}
\fig{BCLMP_650_o3o2b.png}{0.3\textwidth}{(b)}
\fig{BCLMP_650_o3s2b.png}{0.3\textwidth}{(c)}
}
\caption{Same as Figure~\ref{fig:bclmp-668} for BCLMP 650. This object's optical depth is classified as blister, with highly ionized gas subtending between one-third and two-thirds of the object's circumference.}
\label{fig:bclmp-650}
\end{figure*}
Figure \ref{fig:bclmp-650} shows ratio maps for BCLMP 650, a blister object. In Figure \ref{fig:bclmp-650}(b) and (c), we see that the envelope of the low-ionization species [\ion{O}{2}]\space and [\ion{S}{2}]\space extend about halfway around the \ion{H}{2}\space region in projection. There is a break to the west (upper portion in image), where [\ion{O}{3}]\space dominates, allowing the escape of ionizing radiation.
\begin{figure*}
\gridline{\fig{IC_132_halpb.png}{0.3\textwidth}{(a)}
\fig{IC_132_o3o2b.png}{0.3\textwidth}{(b)}
\fig{IC_132_o3s2b.png}{0.3\textwidth}{(c)}
}
\caption{Same as Figure~\ref{fig:bclmp-668} for IC 132. Highly ionized [\ion{O}{3}]\space is seen all around the halo, indicating the escape of ionizing radiation.}
\label{fig:ic-132}
\end{figure*}
IC 132 (Figure \ref{fig:ic-132}) is an optically thin \ion{H}{2}\space region that is extremely bright in [\ion{O}{3}]. Figure \ref{fig:ic-132}(b) shows that the ratio of [\ion{O}{3}]\wavelength{5007}/[\ion{O}{2}]\wavelength{3727} is $>1$ over the entire visible extent of the object.
The ionization structure in Figure \ref{fig:ic-132}(b) is similar to that found by \citet{LopezHernandez2013MNRAS.430..472L} for this object. Comparing Figure \ref{fig:ic-132}(c) with other objects demonstrates how the [\ion{O}{3}]\wavelength{5007}/[\ion{S}{2}]\wavelength{6724} \space morphology changes dramatically for optically thick vs thin regions.
\begin{figure*}
\gridline{\fig{BCLMP_667_halpb.png}{0.3\textwidth}{(a)}
\fig{BCLMP_667_o3o2b.png}{0.3\textwidth}{(b)}
\fig{BCLMP_667_o3s2b.png}{0.3\textwidth}{(c)}
}
\caption{Same as Figure~\ref{fig:bclmp-668} for BCLMP 667. This object is classified as shocked, and the ratio maps reveal the inverted ionization profile.}
\label{fig:bclmp-667}
\end{figure*}
The morphology of BCLMP 667 (Figure \ref{fig:bclmp-667})
shows the lower-ionization species dominating along the inside curve of this object, which is the opposite of what is expected for photoionization from a source driving the shell-like nebular morphology. We therefore classify this as a shock-ionized object.
\subsection{Metallicity dependence of IPM}\label{subsec:metal-dependence}
Figure \ref{fig:metal-chart}(a) shows [\ion{O}{3}]/[\ion{O}{2}]\ excitation vs $12+\log(\rm O/H)$, with optical depth class shown by the symbol colors.
We have used data from \citet{Lin2017ApJ...842...97L} for the values of $12 + \log\rm (O/H)$ for the \ion{H}{2}\space regions in our sample (Table \ref{tab:object-list}). As expected, Figure \ref{fig:metal-chart}(a) shows a strong anticorrelation between optical depth and [\ion{O}{3}]/[\ion{O}{2}]\ ratio. However, there is also a clear trend with metallicity, and there are no objects classified as optically thin (as opposed to blister) having $12 + \log\rm (O/H)> 8.25$.
The strength of the [\ion{O}{3}]\wavelength{5007} line exhibits a strong, non-linear relationship with the O abundance: it gets stronger with increasing metallicity in the metal-poor regime, but at higher metallicities, the greater abundance of metals cools the nebula, rendering it unable to collisionally excite [\ion{O}{3}]\space in the visible-wavelength transitions. Thus, fine-structure lines in the infrared dominate the [\ion{O}{3}]\ emission at higher $12+\log(\rm O/H)$.
Bright-line abundance indices such as $R_{23}\equiv \rm ($[\ion{O}{2}]$\lambda3727 + \rm $[\ion{O}{3}]$\lambda\lambda4959,5007)/\rm H\beta$ and O3N2 $\equiv \log((\rm$ [\ion{O}{3}]$\lambda5007/\rm H\beta)/($[\ion{N}{2}]$\lambda6583/\rm H\alpha$)) are based on this principle, and are thus maximised in the interval $12 + \log\rm (O/H) = 8.0 \text{ to } 8.5$, dropping off steeply at higher values \citep[e.g.,][]{Yin2007A&A...462..535Y, Kewley2002ApJS..142...35K}. For our sample,
\citet{Lin2017ApJ...842...97L} derive the values of $12+\log(\rm O/H)$ using the bright-line indices O3N2, N2 $\equiv \log(\rm [NII]6583/H\alpha)$
\citep[e.g.,][]{Marino2013A&A...559A.114M} as well as direct modelling of the electron temperature $T_{e}$, with a majority derived using methods reliant on the [\ion{O}{3}]\wavelength{5007} emission line. We caution that \ion{H}{2}\ structural evolution effects can shift strong line ratios like N2 by more than an order of magnitude \citep{Pellegrini2020aMNRAS.496..339P}; however, the dominant trends with metallicity are well established.
Since we utilise the [\ion{O}{3}]/[\ion{O}{2}]\ and [\ion{O}{3}]/[\ion{S}{2}]\ ratios as a tracer for degree of ionization, the method of IPM used here is therefore also sensitive to the metallicity. The optical [\ion{O}{3}]\ lines are weak at higher metallicity even though the O$^{++}$ ion may still be prevalent, and therefore the efficacy of IPM is reduced in this regime. Thus, some of the optically thick objects may be mis-classifications on account of [\ion{O}{3}]/[\ion{S}{2}]\ and [\ion{O}{3}]/[\ion{O}{2}]\ being reduced at higher abundances.
As seen in the LMC and SMC, the most luminous \ion{H}{2}\ regions are also the most likely to be optically thin, including blister objects \citep{Pellegrini2012ApJ...755...40P}. This trend is also seen in our sample in Figure~\ref{fig:thinfraction}, which shows the frequency of optically thin and blister \ion{H}{2}\ regions as a function of {\rm H$\alpha$}\ luminosity for objects having $12+\log(\rm O/H) < 8.4$.
In Figure~\ref{fig:metal-chart}(b), we see that the most optically thin objects are tightly clustered at the lowest metallicities. Luminous \ion{H}{2}\ regions with $L(\rm H\alpha) \geq 10^{38}\ \rm erg\ s^{-1}$ at moderate metallicities of $12+\log(\rm O/H)=8.3$ or 8.4 are mostly classified as optically thick, while the opposite is true at lower metallicity. This further supports the likelihood that some of the higher-metallicity, high-luminosity objects classified as optically thick are actually optically thin.
On the other hand, a real trend in increased frequencies of optically thin nebulae must also exist for metal-poor environments. Dust content decreases with metallicity, thereby decreasing the opacity to the Lyman continuum. Also, larger star clusters generating luminous \ion{H}{2}\ regions tend to be more prevalent at lower metallicity, increasing the likelihood of powering objects with the earliest O stars, and enhancing the likelihood of optically thin \ion{H}{2}\ regions. Metal-poor OB atmospheres also tend to be hotter than at solar metallicity \citep[e.g.,][]{MaederMeynet2001A&A...373..555M, MartinsPalacios2021A&A...645A..67M}, driving higher nebular ionization parameters. Thus, without detailed modeling of the individual objects, it is impossible to clarify the locus of optically thin objects in Figure~\ref{fig:metal-chart}, but the IPM classifications in Table~\ref{tab:object-list} likely significantly underestimate the frequency of optically thin nebulae.
\begin{figure*}
\begin{interactive}{js}{metal_chart.zip}
\fig{metal_chart.png}{0.4\textwidth}{(a)}
\fig{metal_luminosity.png}{0.4\textwidth}{(b)}
\end{interactive}
\caption{[\ion{O}{3}]$\lambda5007/$[\ion{O}{2}]$\lambda3727$ (panel (a), left) and {\rm H$\alpha$}\ luminosity (panel (b), right) vs oxygen abundance. Symbol type and color show optical depth classifications.
Objects classified as optically thin and blister have higher [\ion{O}{3}]/[\ion{O}{2}]\ values, which is a function of metallicity.
\added{Median measurement errors for [\ion{O}{3}]/[\ion{O}{2}], $\log L$({\rm H$\alpha$}), and $12+\log(\rm O/H)$ are 0.06 dex, 0.018 dex, and 0.004 dex. Systematic errors are on the order of 20\% for [\ion{O}{3}]/[\ion{O}{2}]\ and $L$({\rm H$\alpha$}), and 0.18 dex for $12+\log(\rm O/H)$.}
An interactive version of panel (b) will be made available in the online version. Users can: \\
i) Filter the objects by $\log$([\ion{O}{3}]/[\ion{O}{2}]) by means of a slider. \\
ii) Select an object by clicking on the symbols or in a drop-down list to view the corresponding ratio maps used for classification as shown in Figures \ref{fig:bclmp-668} -- \ref{fig:bclmp-667}. Both monochrome and colour images are available. \\
iii) Select the different optical depth categories in the legend to plot only objects in the selected categories. \\
iv) Hover with a mouse pointer on the symbols to view the ID, metallicity and [\ion{O}{3}]/[\ion{O}{2}]\ value for each object.}
\label{fig:metal-chart}
\end{figure*}
The classifications are also dependent on the spatial resolution and depth of the ratio maps. Comparing Figure~\ref{fig:thinfraction} with the results of \citet{Pellegrini2012ApJ...755...40P} for the LMC and SMC, the frequency of optically thin objects is much lower, about a factor of 2 at $L(\rm H\alpha)\ \sim10^{38}\ \rm erg\ s^{-1}$. There is no reason to believe that the ISM properties of M33 are substantially different than in these galaxies, and so most likely our classifications are affected by the lower spatial resolution of the M33 imaging data (22 pc in the smoothed images) relative to the imaging of the Magellanic Clouds (1.4 pc).
\added{Given the qualitative similarity of the results from these two studies, the systematic errors generated by the resolution effects do not substantively change the observed trend in Figure~\ref{fig:thinfraction}. However, we caution that absolute interpretation of optical depths is much less reliable.}
\explain{Addressing referee comment 2.2}
\begin{figure*}
\fig{thin_fraction.png}{0.5\textwidth}{}
\caption{Fraction of optically thin \ion{H}{2}\space regions as a function of luminosity for $12+\log(\rm O/H) < 8.4$. Luminosities of the sample objects range from $\log L(\rm H\alpha) = 36.7$ to 39.7, thus the frequencies in bins with $\log L(\rm H\alpha) < 37.5 = 0.$
There are no objects in our sample that have $\log L(\rm H\alpha)$ in the range 38.8 to 39.2. }
\label{fig:thinfraction}
\end{figure*}
We therefore conclude that IPM based on [\ion{O}{3}]/[\ion{O}{2}]\ or [\ion{O}{3}]/[\ion{S}{2}]\ is useful only at lower metallicities, $12 + \log (O/H) \lesssim 8.4$, and it is also dependent on spatial resolution. Other diagnostic lines can extend the use of IPM. For example, \citet{Zastrow2013ApJ...779...76Z} have similarly used [\ion{S}{3}]$\lambda 9069/\rm$[\ion{S}{2}]$\lambda6724$ ratio maps to identify Lyman continuum escape. In general, the underlying principle remains the same: to reveal the nebular ionization structure by differentiating the high vs low ionization zones.
\subsection{A kpc-sized patch of elevated [\ion{O}{3}]}
In general, the [\ion{O}{3}]\wavelength{5007}/[\ion{O}{2}]\wavelength{3727}\space ratio in the diffuse ISM is low, $< 0.4$, and largely invariant. However, in a region bounded approximately by R.A. $\sim$ 1h 33m 40s to 1h 34m 20s and decl. $\sim$ 30\degr 53\arcmin\ to 30\degr 59\arcmin, we observe a large-scale, bi-lobed patch of diffuse emission where the [\ion{O}{3}]\wavelength{5007}/[\ion{O}{2}]\wavelength{3727}\space ratio is $\sim 0.6$, significantly greater than the background (Figure \ref{fig:hii-patch}). This patch corresponds to an area $\sim 9\arcmin\times3\arcmin.5$, implying a structure with dimensions on the order of $\sim$1 kpc for the M33 distance of 840 kpc \citep{Freedman1991ApJ...372..455F}. The integrated [\ion{O}{3}]\wavelength{5007} luminosity of the diffuse patch, excluding other \ion{H}{2}\space regions in the vicinity, is $\sim(4.9\pm 1.5)\times10^{38}$ ergs/s. Such a large scale region of elevated ionization is unusual and difficult to explain.
There is an optically thin \ion{H}{2}\space region, BCLMP 637, at the centre of this patch of excited gas. BCLMP 637 is one of the most highly ionized objects in our sample, with an [\ion{O}{3}]\wavelength{5007}/[\ion{O}{2}]\wavelength{3727}\space ratio of $\sim4$. Its ionization is apparently dominated by [NM2011] J013350.71+305636.7 \citep{NM2011ApJ...733..123N}, a WN3 star. We evaluate whether this WN3 star is responsible for photoionizing the large, excited patch in what follows.
From aperture photometry, we find that the observed [\ion{O}{3}]\wavelength{5007} luminosity of BCLMP 637 is $\sim(1.38\pm 0.08)\times10^{38}$ erg/s, about $4\times$ less than that of the large patch.
The {\rm H$\alpha$}\ luminosity of BCLMP 637 is $(6.75\pm 0.62) \times10^{37}\ \rm erg\ s^{-1}$, corresponding to $Q(\rm H_0) \sim 5\times 10^{49}\ \rm s^{-1}$. We compare this to the PoWR WNE stellar models at LMC metallicity by \citet{Todt2015A&A...579A..75T} and find that this value is roughly an order of magnitude higher than that predicted by the brightest models. In particular, the LMC WNE PoWR models 10-17, 09-16, and 13-21 predict $M_V$ in the range --4.8 to --5.0 for an early type WR star, which agrees with $M_V$ of --4.9 for [NM2011] J013350.71+305636.7 \citep{NM2011ApJ...733..123N}. The ionizing photon flux predicted by these models is in the range $1.1-1.3\times10^{49}$ photons/s, which is much less than the $Q(\rm H_0)$ required to ionize even BCLMP 637. Thus, additional ionizing OB stars are likely present in the nebula. However, there is no further evidence suggesting an unusual stellar population in this object that can be responsible for also ionizing the large, extended patch of elevated ionization.
Thus we also consider candidate high-mass X-ray binaries (HMXBs) within the patch from the X-ray survey of M33 by \citet{Pietsch2004A&A...426...11P}.
We examine two sources: [PMH2004] 192 and [PMH2004] 229. Extrapolating the reported X-ray fluxes in the 0.2--4.5 keV band from \citet{Pietsch2004A&A...426...11P} with typical HMXB power law indices of 1.5-2.5 results in ionizing photon emission rates $Q(H_{0})$ on the order of $10^{46}\ \rm s^{-1}$. Again, this value of $Q(H_{0})$ is orders of magnitude below the required $10^{50}\ \rm s^{-1}$ to explain the origin of the elevated excitation.
Interestingly, \citet{Bigiel2010ApJ...725.1159B} report the existence of unusually hot and bright giant molecular clouds (GMCs) adjacent to this high ionization patch, with a lower inferred CO-to-H$_2$ conversion factor. These GMCs are located between the two lobes, slightly south of center (Figure \ref{fig:hii-patch}). This suggests that the elevated ionization is indeed real and physically associated with M33, and that the source responsible for exciting the diffuse [\ion{O}{3}]\space is also heating up the GMCs. The identity and nature of this source remains unknown.
\begin{figure*}
\fig{o3_o2_power.png}{0.7\textwidth}{}
\caption{Unsmoothed [\ion{O}{3}]/[\ion{O}{2}]\ ratio map. The kpc-sized patch is marked. The location of unusually bright GMCs reported by \citet{Bigiel2010ApJ...725.1159B} is marked with a red X.}
\label{fig:hii-patch}
\end{figure*}
\section{Conclusion} \label{sec:conclusion}
To summarise, we have used narrow-band imaging of M33 to generate ratio maps in [\ion{O}{3}]\wavelength{5007}/[\ion{S}{2}]\wavelength{6724} \space and [\ion{O}{3}]\wavelength{5007}/[\ion{O}{2}]\wavelength{3727} \space to explore the limits of ionization-parameter mapping as a probe of \ion{H}{2}\space region optical depth to ionizing UV radiation. We employ a revised empirical method for continuum subtraction building on methods by \citet{Keenan2017ApJ...848...12K} and \citet{Hong2014PASP..126...79H}. This method uses the pixel histogram distribution for diffuse emission after filtering out bright, resolved emission, and exploits the dispersion around the mode.
We show that, due to the metallicity dependence of the [\ion{O}{3}]\wavelength{5007} emission line, the [\ion{O}{3}]\wavelength{5007}/[\ion{S}{2}]\wavelength{6724} \space and [\ion{O}{3}]\wavelength{5007}/[\ion{O}{2}]\wavelength{3727} \space ratios can only be effective as optical depth diagnostics in the low metallicity regime ($12 + \log\left(O/H\right) \lesssim 8.4$), which is roughly $\lesssim 0.5Z_{\odot}$. Most likely, we are unable to identify a number of optically thin \ion{H}{2} \space regions at higher metallicities due to the weakness of [\ion{O}{3}]\wavelength{5007} emission in this regime. Other emission lines should be used to trace higher ionization species in these conditions.
We report the presence of a peculiar large scale ($\gtrsim 1$ kpc) structure in northern M33 that is excited in [\ion{O}{3}]\wavelength{5007} and conspicuously absent in other bands. The known WR star and HMXBs in the vicinity of this patch cannot provide the required radiation to account for its ionization. Further observations are needed to understand its origin.
\acknowledgments
We thank Rogier Windhorst for use of the continuum filters and Anne Jaskot for assistance with the observing run.
\added{We are also grateful to the anonymous referee for helpful comments.}
This work was supported by NSF AST-1210285, NASA HST-GO-15088, and the University of Michigan.
Data processing was performed using the open-source Python libraries AstroPy, \citep{astropy:2013,astropy:2018}, Photutils \citep{photutilsBradley_2019_2533376}, NumPy \citep{numpy2020NumPy-Array} and SciPy \citep{scipy2020SciPy-NMeth}. Animations of the continuum subtraction process (Figures \ref{fig:video}, \ref{fig:haro-11-histogram}, \ref{fig:eso338-histogram} and \ref{fig:mrk71-histogram}) were prepared by generating individual frames in Matplotlib \citep{matplotlibHunter:2007} and then combined using FFmpeg \citep{ffmpegTomar2006}. Static and interactive versions of Figures \ref{fig:metal-chart} and \ref{fig:thinfraction} were created using the data visualisation libraries Altair \citep{altairVanderPlas2018} and Vega \citep{vegaSatyanarayan2017}.
\vspace{5mm}
\facilities{Mayall(MOSAIC-1), HST(STIS)}
\clearpage
\section{Introduction} \label{sec:intro}
The nebular ionization parameter describes the ionizing photon density relative to gas density, and it is a fundamental diagnostic of radiation feedback in photoionized \ion{H}{2}\ regions. In recent years, diagnostics of the ionization parameter such as [\ion{O}{3}]$\lambda5007$/[\ion{O}{2}]$\lambda3727$, [\ion{S}{3}]$\lambda9069$/[\ion{S}{2}]$\lambda6717,6731$, [\ion{O}{3}]/H$\beta$, and [\ion{O}{3}]/[\ion{S}{2}]\ \citep{Pellegrini2012ApJ...755...40P, Zastrow2013ApJ...779...76Z,Keenan2017ApJ...848...12K, Wang2019ApJ...885...57W} have been used to evaluate the nebular optical depth to Lyman continuum (LyC) radiation in both individual \ion{H}{2}\ regions and starburst galaxies. An especially compelling class of objects are the Green Pea galaxies \citep{Cardamone2009MNRAS.399.1191C}, which are selected on the basis of their extreme ionization parameters in [\ion{O}{3}]/H$\beta$. Confirming predictions \citep[e.g.,][]{JaskotOey2013ApJ...766...91J}, the Green Peas have yielded the most consistent detections of LyC-emitting galaxies in the local universe \citep[e.g.,][]{Izotov2016MNRAS.461.3683I, Izotov2018MNRAS.478.4851I}, and are therefore of vital interest to galaxy evolution and cosmic reionization. However, although we noted above a direct link between ionization parameter and LyC optical depth, the exact relationship between these these quantities is not well understood in these starbursts, due to complicating factors like gas morphology, composition, geometry, and density distributions; and also variations in ionizing spectral energy distributions (SEDs) from various candidate stellar populations and other ionizing sources. Nevertheless, the ionization parameter is an easily observed and widely used diagnostic of the nebular conditions in star-forming regions both near and far.
It is well known that [\ion{O}{3}]$\lambda\lambda4959,5007$ emission drops precipitously at oxygen abundances $12+\log\rm(O/H) > 8.4$, \citep[e.g.,][]{Kewley2002ApJS..142...35K}. For example, the well-known abundance diagnostic $R23\equiv($[\ion{O}{2}]$\lambda3727 + $[\ion{O}{3}]$\lambda\lambda4959,5007)/\rm H\beta$ increases monotonically to maximum values around this metallicity. This is caused by strong sensitivity of this line to the electron temperature, which decreases at higher oxygen abundance. Therefore, using line ratios that rely on [\ion{O}{3}]$\lambda\lambda4959,5007$ as a diagnostic of ionization parameter will be unreliable at metallicities where these lines are weak.
In this work, we use the \ion{H}{2}\ regions of the Local Group galaxy M33 to explore the regime where [\ion{O}{3}]$\lambda\lambda4959,5007$ is, and is not, effective as a diagnostic of ionization parameter for the purpose of evaluating radiation feedback and LyC escape. In order to generate the emission-line images required for this analysis, it is necessary to carry out continuum subtraction, and we also further explore this process.
\section{Observations of M33} \label{sec:obs}
The north half of M33 was observed with the MOSAIC-1.1 imaging camera at the Mayall 4-m telescope, Kitt Peak National Observatory, on 2011 October 28--29. We used the narrowband filters for [\ion{O}{2}]$\lambda3727$ (``O2", FWHM 50\AA), [\ion{O}{3}]$\lambda5007$ (``O3", FWHM 50\AA), and [\ion{S}{2}] $\lambda6724$ (``ha16", FWHM 81\AA) for line imaging. For the continuum, we used broadband filters BATC454 ($\sim 4320 - 4700$ \AA) and BATC705 ($\sim 6950 - 7920$ \AA) for continuum subtraction of [\ion{O}{2}]\ and [\ion{O}{3}]; and of {\rm H$\alpha$}\ and [\ion{S}{2}], respectively. The continuum filters are used with the kind permission of R. Windhorst. We also used archive {\rm H$\alpha$}\ observations obtained in 2001 with the same setup by \citet{Massey2007AJ....134.2474M}. There were $6\times 1400$ s, $4\times 600$ s, $5\times600$ s $+1\times900$ s, and $5\times300$ s exposures in [\ion{O}{2}], [\ion{O}{3}], [\ion{S}{2}], and {\rm H$\alpha$}, respectively; and $5\times500$ s and $2\times1050$ s $+3\times 550$ s exposures in the blue and red continuum filters, respectively. All observations were dithered to cover the gaps between the eight MOSAIC CCD chips. After being median-combined, the images have a variety of residual defects, including edge effects generated by the image combining process and a very low-level reflection of the telescope pupil.
In the [\ion{S}{2}]\space image, the giant \ion{H}{2}\space regions NGC 604, NGC 595, IC 131 and the star BD $+$30 243 generated trails likely caused by CTE effects,
causing a band $\sim$1000 pixels wide running through one section of the image. The northeast corner of the [\ion{S}{2}]\ continuum image also has an unusually bright patch, possibly due to reflections. Such bad pixels were trimmed out of the final array when performing the continuum subtraction.
We use routines from the \texttt{astropy photutils} package to identify and mask the foreground Milky Way stars\deleted{, and interpolate over the masked pixels.} \added{We then interpolate over these masked pixels using a bilinear interpolation so that the faint wings of foreground stars do not affect the flux measurement of \ion{H}{2}\ regions.}\explain{Addressing minor comment 5 of the referee.}
We use a sample of 7 objects from \citet{TsC2016MNRAS.458.1866T}, listed in Table \ref{tab:TSC-dered-rered-fluxes}, to carry out flux calibration.
For [\ion{S}{2}]\ and [\ion{O}{2}], we sum the reported flux from the two lines forming these doublets, which are both included in the filter bandpasses.
We recover the un-dereddened fluxes using the reported $c(\rm H\beta)$ and extinction law:
\begin{equation} \label{reddening law}
\log\left(F_{\lambda_{1}} / F_{\lambda_{2}}\right)_{\text{corr}} = \log\left(F_{\lambda_{1}} / F_{\lambda_{2}}\right)_{\text{obs}} + C \left(f_{\lambda_{1}} - f_{\lambda_{2}}\right)
\end{equation}
The values in Table \ref{tab:TSC-dered-rered-fluxes}, columns (6)-(9) are the un-dereddened fluxes ($F_{obs}$) that we adopt for calibration.
\begin{deluxetable*}{cccccccccc}
\tablenum{1}
\tablecaption{Line fluxes after removing reddening correction\tablenotemark{a} \label{tab:TSC-dered-rered-fluxes}\explain{Added uncertainty for literature ratios}}
\tablewidth{0pt}
\tablehead{
\colhead{ID} & \colhead{c(H$\beta$)\tablenotemark{b}} & \multicolumn4c{Reddened} & \multicolumn2c{[\ion{O}{3}]/{\rm H$\alpha$}\ literature} & \multicolumn2c{[\ion{O}{3}]/{\rm H$\alpha$}\ Observed}\\
& & \colhead{{\rm H$\alpha$}} & \colhead{[\ion{O}{2}] $\lambda 3727$} & \colhead{[\ion{O}{3}] $\lambda 5007$} & \colhead{[\ion{S}{2}] $\lambda\lambda 6718, 6732$} & mean & st.d. & mean & st.d.
}
\decimalcolnumbers
\startdata
B2011 b5 & 0.67 & 460 & 89 & 194 & 56 & 0.42 & 0.10 & 0.34 & 0.07 \\
IC 131 & 0.51 & 402 & 116 & 332 & 60 & 0.83 & 0.20 & 1.06 & 0.15 \\
BCLMP 290 & 0.12 & 318 & 188 & 177 & 32 & 0.56 & 0.07 & 0.49 & 0.05 \\
NGC 588 & 0.16 & 315 & 90 & 475 & 19 & 1.51 & 0.24 & 1.25 & 0.06 \\
BCLMP 626 & 0.02 & 292 & 239 & 165 & 43 & 0.57 & 0.06 & 0.66 & 0.06 \\
LGC HII3 & 0.09 & 302 & 171 & 285 & 31 & 0.94 & 0.12 & 1.03 & 0.29 \\
IC 132 & 0.37 & 370 & 46 & 566 & 15 & 1.53 & 0.20 & 1.89 & 0.18 \\
\enddata
\tablenotetext{a}{All fluxes are relative to H$\beta$ = 100.}
\tablenotetext{b}{c(H$\beta$) values from \citet{TsC2016MNRAS.458.1866T} used to remove their reported reddening correction.}
\end{deluxetable*}
We apply rectangular apertures corresponding to the reported slit width at the position for each object given in
Table 1 of \citet{TsC2016MNRAS.458.1866T}. We use the \texttt{photutils} aperture photometry routine in \texttt{astropy} to obtain the photometry.
To account for positional inaccuracies arising from atmospheric seeing, we offset the slit positions by a normally distributed random variable with standard deviation equal to the seeing in pixels. We average over 20 such randomly offset apertures for the integrated photometry.
The computed ratios are then used together with the calibrated flux ratios derived from the data of \citet{TsC2016MNRAS.458.1866T}
to determine the flux calibration, using the {\rm H$\alpha$}/H$\beta$\ ratio of 2.86.
\section{Continuum subtraction} \label{sec:subtract}
Narrow-band imaging data include flux from both line emission and diffuse stellar continuum. To isolate the line flux, an off-line image containing only the continuum is usually subtracted. Due to differing filter transmissions and variation in the continuum spectral energy distribution (SED), the continuum image must be scaled before subtraction, and determining the scale factor is nontrivial. \citet{Hayes2009AJ....138..911H} and \citet{James2016ApJ...816...40J} have utilised stellar population synthesis modeling that compute spatially varying scale factors on a pixel-by-pixel basis. These methods are model-dependent and \citet{Hayes2009AJ....138..911H} discuss their advantages and pitfalls in detail.
In this work, we focus on empirical methods that compute a single characteristic scale factor for a large region or an entire image.
\citet{Keenan2017ApJ...848...12K} describe a method where a single optimal scale factor is found for such a region. They note a slope change in the mode of the pixel values vs scale factor for the continuum-subtracted images. \citet{Keenan2017ApJ...848...12K} show that this transition results when the scale factor induces any oversubtraction. At small values of the scale factor, the mode of the pixel-value histogram is determined by the lowest-value pixels. The change in flux for these pixels is small as the scale factor varies, hence the slope of mode vs scale factor is shallow. At higher values of the scale factor, the mode is dominated by oversubtracted pixels. As the scale factor increases, the first pixels to become oversubtracted are the brightest ones, and their flux has a strong dependence on scale factor. Thus, the slope of the mode vs scale factor is steeper in the over-subtracted regime (see \citet{Keenan2017ApJ...848...12K} for more details).
\citet{Hong2014PASP..126...79H} present a method that similarly identifies the transition to oversubtraction, but based on the skewness of the pixel-value histogram as a function of the scale factor. The above two methods have been shown to work well for images where there are a significant number of continuum-dominated pixels. \deleted{However, we still encountered some difficulties in applying both of these.}
\added{However, we still encountered some difficulties in adequately constraining the best scale factors.
In particular, the slope transition reported by \citet{Keenan2017ApJ...848...12K} can be hard to discern. There can be multiple slope changes and oscillatory behavior in the mode, as \citet{Keenan2017ApJ...848...12K} indicate. The exact point of transition is therefore uncertain.
Similarly, the method used by \citet{Hong2014PASP..126...79H} relies on computing the second derivative of the skewness with respect to the scale factor. Estimating this derivative accurately requires a very fine search through the scale factor space.
Appendix~A demonstrates the functionality of these two methods (Figures \ref{fig:haro-11-fraction}, \ref{fig:eso338-fraction} and \ref{fig:mrk71-fraction}).
}
In this paper, we therefore propose a revised method that uses the mode to obtain the optimal scale factor,
\added{which can provide a narrower confidence interval while also reducing the computing power needed}.
\explain{Addressing comment 1.2 of referee}
\subsection{Revised Mode Method to Identify Scale Factor} \label{subsec:agg-factor}
The line emission represents an excess signal over the continuum, hence, the pixel-value distributions will have a tail to positive values dominated by the real signal. Since we are interested in identifying the diffuse background to carry out the continuum subtraction, we therefore employ a $3\sigma$ filter from the \texttt{scipy} library on the line image. The routine computes the mean and standard deviation $\sigma$ of the data, then rejects any data points that are $>3\sigma$ from the mean. The mean and standard deviation are recomputed and the filtering is done again. This iterative process is continued until there are no more rejections. At each scale factor, we first subtract the scaled continuum from the line image, then invoke the $3\sigma$ filter on the subtracted image. This removes much of the real signal, leaving a residual histogram that is more dominated by the diffuse continuum signal. The resulting pixel-value distribution is better suited for the purpose of identifying the optimal continuum image scale factor.
Similar to \citet{Hong2014PASP..126...79H}, we observe a transition in the skewness of the pixel histogram as the scale factor is increased.
Figure~\ref{fig:histogram} shows how the shape of the pixel-value distribution changes as the scale factor is varied. At low scale factors (Figure \ref{fig:histogram}(a)), the image is undersubtracted. The distribution for the subtracted image resembles the initial flux distribution, where pixels with high continuum values contribute to high-value bins. As a result, the histogram is skewed to the right.
As the scale factor is increased, the flux distribution in the line image approaches the flux distribution in the scaled off-line image.
Ideally, at this point of optimal continuum subtraction, the background flux should be zero. This is characterised by the mode of the pixel histogram being zero. However, this is not always the case, since the sky may have a different SED compared to the diffuse stellar continuum.
This issue is particularly relevant for ground-based observations where the sky background is significant. \added{Once the scale factor has been set by subtracting the stellar emission, the residual sky background can be eliminated while performing aperture photometry, as we have done for the flux measurements in Tables \ref{tab:TSC-dered-rered-fluxes} and \ref{tab:object-list}}
\explain{Addressing minor comment 1 of referee.}
On further increasing the scale factor, the pixels with strong continuum flux now contribute to the negative bins as they are the first ones to become oversubtracted. The spread increases in the negative direction, skewing the histogram to the left (Figure \ref{fig:histogram}(c)). This is a reversal of the behavior seen earlier, and it corresponds to the transition in skewness reported by \citet{Hong2014PASP..126...79H}. It should be noted that the negative tail is less statistically `heavy' compared to the positive tail, due to the continued presence of pixels with emission-line flux.
\citet{Hong2014PASP..126...79H} demonstrate the same effect as a slight positive bias to the skew at the transition point.
At the optimal scale factor, the number of background pixels is therefore maximized in the modal bin, as shown in Figure \ref{fig:fraction-chart}.
Due to the background noise, the optimally subtracted histogram will still have some spread around the mode. We therefore also consider the total number of pixels in the bins adjacent to the modal bin. This is also helpful if the mode falls on the boundary between two bins. Using all three bins provides some robustness against such cases. Thus, the optimal scale factor is that which maximises $({f_0 + f_1 + f_2})/{N}$, where $f_0$ is the number of pixels in the modal bin; $f_1$ and $f_2$ are the number of pixels in the pre-modal and post-modal bins, respectively; and $N$ is the total number of pixels in the image, after applying the $3\sigma$ filter. If the modal bin is the first or last bin, then $f_1$ or $f_2$ are accordingly assumed to be zero. The value of $N$ changes with the scale factor due to the $3\sigma$ filter. Initially, the brightest pixels get rejected, but at the correct scale factor, the flux from these correctly subtracted pixels falls within the $3\sigma$ limit.
\added{The chosen metric, $({f_0 + f_1 + f_2})/{N}$, is computed using only the pixel histogram at the current scale factor. In comparison, the methods of \citet{Keenan2017ApJ...848...12K} and \citet{Hong2014PASP..126...79H} require the histograms at adjacent values of the scale factor to compute the slope of the mode, or the second derivative of skewness which requires finer sampling to reduce the error bound.}
\explain{Addressing major comment 1 of the referee.}
A jagged or scalloped pattern is apparent in Figure \ref{fig:fraction-chart} as the scale factor is increased. The pattern arises due to the binning criteria chosen for the of the modal bin fraction. In particular, it is due to the choice of using only the 3 modal bins to measure of the peak of the pixel distribution. In the case of highly skewed histograms, the peak is inadequately sampled by only 3 bins, and small changes in the pixel distribution are amplified, giving rise to the jagged pattern seen. This behaviour can be smoothed by modifying the criterion to include more bins, or by finer sampling of the pixel distribution by increasing the number of bins. Finer sampling has an added computational cost. For the purposes of identifying the globally optimal scale factor from a symmetric histogram, the chosen metric works well and is unaffected by the local variations due to skewed histograms.
\added{Appendix~A demonstrates this method, with comparisons to the \citet{Keenan2017ApJ...848...12K} and \citet{Hong2014PASP..126...79H} methods (Figures~\ref{fig:haro-11-fraction}, \ref{fig:eso338-fraction} and \ref{fig:mrk71-fraction}).}
\explain{Addressing major comment 1}
\begin{figure*}
\gridline{\fig{M33_factor_0.0000.png}{0.3\textwidth}{(a)}
\fig{M33_factor_0.1800.png}{0.3\textwidth}{(b)}
\fig{M33_factor_0.2600.png}{0.3\textwidth}{(c)}
}
\caption{Histograms for M33 [\ion{O}{3}]\ continuum subtraction. Panel (a) shows undersubtracted pixel histogram with high-value tail. Panel (b) shows the optimally subtracted pixel histogram, where the number of pixels in the 3 modal bins is maximized. Panel (c) shows the emergence of the low-value tail in an oversubtracted pixel histogram. }
\label{fig:histogram}
\end{figure*}
We caution that the modal bin maximisation works to subtract the dominant background component in the image, regardless of whether it is diffuse starlight, sky emission, or diffuse nebular emission. Thus, if the goal is to subtract diffuse starlight, then that component should constitute a significant fraction of the total pixel population.
For example, if sky pixels dominate the variance, then the algorithm produces an undersubtracted image relative to diffuse starlight. This can be caused by two effects. First, if sky emission dominates the flux of the background pixels, then the algorithm identifies the optimal scale factor for the sky background. This is illustrated in Appendix \ref{sec:other-applications}. Similarly, if the random variance of the sky pixels is high in images with low signal-to-noise, then the large variance makes the reduced spread of the stellar continuum pixels harder to detect. This can be explained as follows: the unsubtracted line image has a certain variance arising from the true signal, which is gradually subtracted out by increasing the scale factor. At the same time, we introduce additiional variance through the random noise of the sky pixels, which increases on increasing the scale factor. If the signal-to-noise ratio is low, the reduction in variance of diffuse starlight is drowned out by the noise that we introduce. The spread is minimised at a lower scale factor, resulting in an undersubtracted image. We address this issue in Section \ref{subsec:application} and Appendix \ref{sec:other-applications}.
On the other hand, the presence of widespread diffuse line emission will also bias the pixel-value histogram and the spread towards higher values. Contamination of pure continuum pixels with diffuse line emission therefore also inflates the variance of continuum pixels. The algorithm proposed above
overcompensates for the larger spread, resulting in a scale factor greater than optimum. This is mitigated by ensuring the presence of pure continuum pixels in the image. Therefore, prior knowledge about the emission region characteristics is required. In short, the background must be dominated by diffuse starlight pixels in order for these statistical methods to identify the optimal scale factor to subtract this background component.
\added{See \citet{Hong2014PASP..126...79H} for a discussion on how the various background compositions affect these continuum-subtraction methods.}
\explain{Addressing minor comment 1.}
\subsection{Application of the Method}\label{subsec:application}
Like most spiral galaxies, M33 has a strong color gradient in its diffuse starlight, implying that the scale factor for subtracting this component will vary spatially. We therefore define 5 regions using elliptical isophotes (Figure \ref{fig:el-offb_img}), which are are chosen by visual inspection of the approximate stellar surface brightness. Region 0, the galactic center, is dominated by resolved sources and does not have many background pixels.
When presented with an image like this, our algorithm tends to produce oversubtracted images since there is not enough background to determine the correct scale factor. The opposite holds for Region 4, which is dominated by sky pixels and relatively lacking in diffuse starlight. Sky-dominated images tend to undersubtract diffuse starlight when we apply our method, as described in Section~\ref{subsec:agg-factor}. Regions 1, 2 and 3 have a good mix of emission, continuum, and background pixels, so our method works well for such regions. Due to the different characteristics of the pixel populations of Regions 0 and 4, we assign
the scale factors for Region 0 and 4 by setting the modes of their continuum-subtracted pixel values equal to the modes for Regions 1 and 3, respectively.
In our pipeline, we first employ the $3\sigma$ filtering described above to the emission-line image. The pixels with the strongest fluxes are rejected by the filter.
Next, we generate the pixel-value histogram of the continuum-subtracted image. The bin width is chosen following \citet{Sturges1926}: $\text{The number of bins}\ n \approx 1 + \log_{2}N$, where $N$ is the total number of pixels. For our images, this value is 22-25 bins, depending on the Region. The modal bin is identified and the modal bin fraction calculated as described earlier in Section \ref{subsec:agg-factor}. The exact value of the mode $M$ can be calculated by taking the intersection of two lines that linearly interpolate the data in the modal bin:
\begin{equation}\label{modelines}
M = L + \frac{f_{0} - f_{1}}{2f_{0} - f_{1} - f_{2}}\times h
\end{equation}
where $L$ is the lower limit of the modal bin, and $h$ is the bin width. This exact value is needed to set the scale factors for Regions 0 and 4.
An animation of the continuum subtraction process (Figure~\ref{fig:video}) is available in the online version of this paper.
\added{Table~\ref{tab:object-list} gives our narrowband fluxes for the objects, with
errors estimated by computing the median background around each object during aperture photometry. We confirmed the {\rm H$\alpha$}\ luminosities of three giant \ion{H}{2}\ regions: NGC 604, NGC 595 and IC 131 with the values given by \citet{Relano2009ApJ...699.1125R}.}
\explain{Addressing minor comment 4.2 by referee.}
\begin{figure*}
\gridline{\fig{M33_oiii_el.png}{0.5\textwidth}{(a)}
\fig{M33_oiii_offb.png}{0.5\textwidth}{(b)}
}
\caption{Narrow-band [\ion{O}{3}]\ on-line (a) and off-line (b) images of M33. The five regions with varying background levels are shown.
\label{fig:el-offb_img}}
\end{figure*}
\begin{figure*}
\begin{interactive}{animation}{M33_OIII_movie.mp4}
\plotone{oiii_img_hist.png}
\end{interactive}
\caption{The online version of this figure is an animation of the step-by-step subtraction process, showing how the shape of the pixel histogram changes for the corresponding continuum-subtracted [\ion{O}{3}]\ images as the continuum-image scale factor is increased from 0 to 0.36. The point of optimal subtraction can be identified around scale factor $\approx 0.18$. The video duration is 9 seconds.
\label{fig:video}}
\end{figure*}
\begin{figure*}
\fig{M33_OIII_Fraction_in_modal_bins.png}{0.6\textwidth}{}
\caption{Modal bin fraction, calculated from equation \ref{modelines}, vs the scale factor. This metric is maximised in this example when the scale factor is 0.188, indicating the point of optimal subtraction. The scalloped pattern is caused by the binning of the pixel value distribution (see text).
\label{fig:fraction-chart}}
\end{figure*}
To further test our new method of continuum subtraction, we also apply it to \ion{C}{3}]\ $\lambda1909$ narrow-band imaging of three starburst galaxies, Haro 11 \citep{Micheva2020ApJ...903..123M}, ESO338-IG04, and Mrk 71. Whereas M33 has a strong color gradient and many resolved individual stars, these more distant galaxies are dominated by more uniform, diffuse starlight. These examples also have very faint line emission compared to the M33 data. We find that our new continuum subtraction technique works well on these galaxies. The results are shown in Appendix~\ref{sec:other-applications}.
\begin{deluxetable*}{cccccDDDDDDDD}
\tablenum{2}
\tablecaption{\ion{H}{2}\space Regions in Northern M33 \label{tab:object-list}}
\tablewidth{\linewidth}
\tabletypesize{\scriptsize}
\tablehead{
\colhead{Object} & \colhead{RA (J2000)} & \colhead{Dec (J2000)} &
\colhead{12+log(O/H)\tablenotemark{a}} & \colhead{Opacity\tablenotemark{b}} &
\multicolumn2c{{\rm H$\alpha$}\tablenotemark{c}} & \multicolumn2c{{\rm H$\alpha$}\ err\tablenotemark{c}} &
\multicolumn2c{[\ion{O}{2}]\tablenotemark{c}} & \multicolumn2c{[\ion{O}{2}]\ err\tablenotemark{c}} &
\multicolumn2c{[\ion{O}{3}]\tablenotemark{c}} & \multicolumn2c{[\ion{O}{3}]\ err\tablenotemark{c}} &
\multicolumn2c{[\ion{S}{2}]\tablenotemark{c}} & \multicolumn2c{[\ion{S}{2}]\ err\tablenotemark{c}}
}
\decimalcolnumbers
\startdata
BCLMP 616 & 1h32m54.38s & 30d50m28s & $8.226^{+0.006}_{-0.005}$ & 1 & 14.6 & 0.22 & 7.61 & 0.20 & 2.03 & 0.46 & 2.97 & 0.40 \\
LHK2017 54 & 1h32m56.28s & 30d40m36.7s & $8.383^{+0.004}_{-0.003}$ & 4 & 23.6 & 3.9 & 13.1 & 1.2 & 5.2 & 2.6 & 13.1 & 4.5 \\
BCLMP 289 & 1h32m57.5s & 30d44m27s & $8.208^{+0.003}_{-0.004}$ & 3 & 57.4 & 19 & 31.4 & 0.83 & 66.4 & 4.6 & 18.1 & 21 \\
CPSDP 67 & 1h32m59.21s & 30d41m20.8s & $8.518^{+0.007}_{-0.007}$ & 1 & 10.1 & 1.4 & 4.91 & 0.16 & 2.65 & 1.4 & 3.64 & 2.0 \\
BCLMP 285 & 1h33m02.88s & 30d41m08.1s & $8.238^{+0.003}_{-0.003}$ & 1 & 13.1 & 0.25 & 5.95 & 0.04 & 4.63 & 0.27 & 2.13 & 0.42
\enddata
\tablenotetext{a}{From \citet{Lin2017ApJ...842...97L}.}
\tablenotetext{b}{$1=$ optically thick, $2=$ blister, $3=$ optically thin, $4=$ shock, $5=$ indeterminate.}
\tablenotetext{c}{Luminosities given in $10^{36}$ erg/s.}
\tablecomments{Data for first five objects are shown here. The online version of this paper has data for all 108 objects in a machine readable format.}
\end{deluxetable*}
\section{Ionization-parameter mapping}\label{sec:IPM}
We generate line ratio maps for [\ion{O}{3}]/[\ion{O}{2}]\ and [\ion{O}{3}]/[\ion{S}{2}]\ as described by \citet{Keenan2017ApJ...848...12K} and \citet{Pellegrini2012ApJ...755...40P} to evaluate the optical depth of photoionized \ion{H}{2}\ regions by ionization-parameter mapping (IPM).
IPM is most directly applied to distinct objects, and significant diffuse continuum can mask the signal from individual HII regions. Unsharp masking allows us to locally remove the average diffuse ionized gas emission over a given length scale. This allows for an amplification in the signal for optically thin nebulae, where the low-ionization emission may be dominated by diffuse ambient emission. The residual presence of field stars affects the median smoothing, so we first
\added{carry out a bilinear interpolation}
\explain{addressing minor comment 5 of the referee}
over these across regions three times the stellar PSF. We then mask any emission with intensity greater than 30 counts ($5.8\times 10^{-15}\ \rm erg\ s^{-1}\ cm^{-2}\ px^{-1}$). \added{This is approximately the saturation level of the detector.}\explain{Addressing minor comment 7 of the referee.} We median filter the remaining emission over a length scale 30$\times$ the stellar PSF, corresponding to 83.1 pixels.
Finally, we subtract that medianed image from the original data, smoothed to the seeing to improve S/N.
Following \citet{Pellegrini2012ApJ...755...40P}, we classify the M33 \ion{H}{2}\ regions into 5 classes: (0) indeterminate, (1) optically thick, (2) blister, (3) optically thin, and (4) shocked, based on the ionization structure in the halos of individual regions. Objects are considered optically thick if [\ion{O}{2}]\ and/or [\ion{S}{2}]\ dominates over at least two-thirds of the circumference in projection. Blister and optically thin \ion{H}{2}\ regions are those with the low-ionization species dominating over one-third to two-thirds; and less than one-third of the circumference, respectively.
Shocked nebulae show an ionization structure that is unlikely to have resulted from photoionization alone, with a pocket of low ionization species on the interior surrounded by [\ion{O}{3}]\ on the outside. Objects are categorised as ``indeterminate" due to poor S/N, incomplete data, or abnormal ionization morphology.
Our classifications are given in Table~\ref{tab:object-list} for our sample of 108 objects, and are based on consideration of both [\ion{O}{3}]/[\ion{S}{2}]\ and [\ion{O}{3}]/[\ion{O}{2}]\ ratio maps. The object locations in the disc of M33 are shown in Figure \ref{fig:finder-chart}. We present examples of categories 1--4 in Figures~\ref{fig:bclmp-668} to \ref{fig:bclmp-667}. These figures show the [\ion{O}{3}]/[\ion{S}{2}]\ and [\ion{O}{3}]/[\ion{O}{2}]\ ratio map for these representative objects.
Similar images for all the objects in our sample are provided in the interactive version of Figure~\ref{fig:metal-chart} below.
\begin{figure*}
\fig{M33_Ha_finder.png}{0.5\textwidth}{}
\caption{Continuum-subtracted {\rm H$\alpha$}\ image with the sample \ion{H}{2}\space regions from Table \ref{tab:object-list} marked.
\label{fig:finder-chart}}
\end{figure*}
Figure \ref{fig:bclmp-668} shows ratio maps for BCLMP 668, an optically thick nebula. As evidenced by Figure \ref{fig:bclmp-668}(b) and (c),
this object shows a classic, Str\"omgren sphere structure, with the low-ionization envelope completely surrounding the highly ionized core.
The emission from [\ion{O}{3}] \space is relatively low throughout.
\begin{figure*}
\gridline{\fig{BCLMP_668_halpb.png}{0.3\textwidth}{(a)}
\fig{BCLMP_668_o3o2b.png}{0.3\textwidth}{(b)}
\fig{BCLMP_668_o3s2b.png}{0.3\textwidth}{(c)}
}
\caption{BCLMP 668 in smoothed, unsharp-masked (a) continuum subtracted {\rm H$\alpha$}, (b) [\ion{O}{3}]/[\ion{O}{2}], and (c) [\ion{O}{3}]/[\ion{S}{2}]\space ratio maps.
\added{Color bar units are in image counts.}
This object is classified as optically thick, and displays a shell-like structure of high vs low ionization zones.}
\label{fig:bclmp-668}
\end{figure*}
\begin{figure*}
\gridline{\fig{BCLMP_650_halpb.png}{0.3\textwidth}{(a)}
\fig{BCLMP_650_o3o2b.png}{0.3\textwidth}{(b)}
\fig{BCLMP_650_o3s2b.png}{0.3\textwidth}{(c)}
}
\caption{Same as Figure~\ref{fig:bclmp-668} for BCLMP 650. This object's optical depth is classified as blister, with highly ionized gas subtending between one-third and two-thirds of the object's circumference.}
\label{fig:bclmp-650}
\end{figure*}
Figure \ref{fig:bclmp-650} shows ratio maps for BCLMP 650, a blister object. In Figure \ref{fig:bclmp-650}(b) and (c), we see that the envelope of the low-ionization species [\ion{O}{2}]\space and [\ion{S}{2}]\space extend about halfway around the \ion{H}{2}\space region in projection. There is a break to the west (upper portion in image), where [\ion{O}{3}]\space dominates, allowing the escape of ionizing radiation.
\begin{figure*}
\gridline{\fig{IC_132_halpb.png}{0.3\textwidth}{(a)}
\fig{IC_132_o3o2b.png}{0.3\textwidth}{(b)}
\fig{IC_132_o3s2b.png}{0.3\textwidth}{(c)}
}
\caption{Same as Figure~\ref{fig:bclmp-668} for IC 132. Highly ionized [\ion{O}{3}]\space is seen all around the halo, indicating the escape of ionizing radiation.}
\label{fig:ic-132}
\end{figure*}
IC 132 (Figure \ref{fig:ic-132}) is an optically thin \ion{H}{2}\space region that is extremely bright in [\ion{O}{3}]. Figure \ref{fig:ic-132}(b) shows that the ratio of [\ion{O}{3}]\wavelength{5007}/[\ion{O}{2}]\wavelength{3727} is $>1$ over the entire visible extent of the object.
The ionization structure in Figure \ref{fig:ic-132}(b) is similar to that found by \citet{LopezHernandez2013MNRAS.430..472L} for this object. Comparing Figure \ref{fig:ic-132}(c) with other objects demonstrates how the [\ion{O}{3}]\wavelength{5007}/[\ion{S}{2}]\wavelength{6724} \space morphology changes dramatically for optically thick vs thin regions.
\begin{figure*}
\gridline{\fig{BCLMP_667_halpb.png}{0.3\textwidth}{(a)}
\fig{BCLMP_667_o3o2b.png}{0.3\textwidth}{(b)}
\fig{BCLMP_667_o3s2b.png}{0.3\textwidth}{(c)}
}
\caption{Same as Figure~\ref{fig:bclmp-668} for BCLMP 667. This object is classified as shocked, and the ratio maps reveal the inverted ionization profile.}
\label{fig:bclmp-667}
\end{figure*}
The morphology of BCLMP 667 (Figure \ref{fig:bclmp-667})
shows the lower-ionization species dominating along the inside curve of this object, which is the opposite of what is expected for photoionization from a source driving the shell-like nebular morphology. We therefore classify this as a shock-ionized object.
\subsection{Metallicity dependence of IPM}\label{subsec:metal-dependence}
Figure \ref{fig:metal-chart}(a) shows [\ion{O}{3}]/[\ion{O}{2}]\ excitation vs $12+\log(\rm O/H)$, with optical depth class shown by the symbol colors.
We have used data from \citet{Lin2017ApJ...842...97L} for the values of $12 + \log\rm (O/H)$ for the \ion{H}{2}\space regions in our sample (Table \ref{tab:object-list}). As expected, Figure \ref{fig:metal-chart}(a) shows a strong anticorrelation between optical depth and [\ion{O}{3}]/[\ion{O}{2}]\ ratio. However, there is also a clear trend with metallicity, and there are no objects classified as optically thin (as opposed to blister) having $12 + \log\rm (O/H)> 8.25$.
The strength of the [\ion{O}{3}]\wavelength{5007} line exhibits a strong, non-linear relationship with the O abundance: it gets stronger with increasing metallicity in the metal-poor regime, but at higher metallicities, the greater abundance of metals cools the nebula, rendering it unable to collisionally excite [\ion{O}{3}]\space in the visible-wavelength transitions. Thus, fine-structure lines in the infrared dominate the [\ion{O}{3}]\ emission at higher $12+\log(\rm O/H)$.
Bright-line abundance indices such as $R_{23}\equiv \rm ($[\ion{O}{2}]$\lambda3727 + \rm $[\ion{O}{3}]$\lambda\lambda4959,5007)/\rm H\beta$ and O3N2 $\equiv \log((\rm$ [\ion{O}{3}]$\lambda5007/\rm H\beta)/($[\ion{N}{2}]$\lambda6583/\rm H\alpha$)) are based on this principle, and are thus maximised in the interval $12 + \log\rm (O/H) = 8.0 \text{ to } 8.5$, dropping off steeply at higher values \citep[e.g.,][]{Yin2007A&A...462..535Y, Kewley2002ApJS..142...35K}. For our sample,
\citet{Lin2017ApJ...842...97L} derive the values of $12+\log(\rm O/H)$ using the bright-line indices O3N2, N2 $\equiv \log(\rm [NII]6583/H\alpha)$
\citep[e.g.,][]{Marino2013A&A...559A.114M} as well as direct modelling of the electron temperature $T_{e}$, with a majority derived using methods reliant on the [\ion{O}{3}]\wavelength{5007} emission line. We caution that \ion{H}{2}\ structural evolution effects can shift strong line ratios like N2 by more than an order of magnitude \citep{Pellegrini2020aMNRAS.496..339P}; however, the dominant trends with metallicity are well established.
Since we utilise the [\ion{O}{3}]/[\ion{O}{2}]\ and [\ion{O}{3}]/[\ion{S}{2}]\ ratios as a tracer for degree of ionization, the method of IPM used here is therefore also sensitive to the metallicity. The optical [\ion{O}{3}]\ lines are weak at higher metallicity even though the O$^{++}$ ion may still be prevalent, and therefore the efficacy of IPM is reduced in this regime. Thus, some of the optically thick objects may be mis-classifications on account of [\ion{O}{3}]/[\ion{S}{2}]\ and [\ion{O}{3}]/[\ion{O}{2}]\ being reduced at higher abundances.
As seen in the LMC and SMC, the most luminous \ion{H}{2}\ regions are also the most likely to be optically thin, including blister objects \citep{Pellegrini2012ApJ...755...40P}. This trend is also seen in our sample in Figure~\ref{fig:thinfraction}, which shows the frequency of optically thin and blister \ion{H}{2}\ regions as a function of {\rm H$\alpha$}\ luminosity for objects having $12+\log(\rm O/H) < 8.4$.
In Figure~\ref{fig:metal-chart}(b), we see that the most optically thin objects are tightly clustered at the lowest metallicities. Luminous \ion{H}{2}\ regions with $L(\rm H\alpha) \geq 10^{38}\ \rm erg\ s^{-1}$ at moderate metallicities of $12+\log(\rm O/H)=8.3$ or 8.4 are mostly classified as optically thick, while the opposite is true at lower metallicity. This further supports the likelihood that some of the higher-metallicity, high-luminosity objects classified as optically thick are actually optically thin.
On the other hand, a real trend in increased frequencies of optically thin nebulae must also exist for metal-poor environments. Dust content decreases with metallicity, thereby decreasing the opacity to the Lyman continuum. Also, larger star clusters generating luminous \ion{H}{2}\ regions tend to be more prevalent at lower metallicity, increasing the likelihood of powering objects with the earliest O stars, and enhancing the likelihood of optically thin \ion{H}{2}\ regions. Metal-poor OB atmospheres also tend to be hotter than at solar metallicity \citep[e.g.,][]{MaederMeynet2001A&A...373..555M, MartinsPalacios2021A&A...645A..67M}, driving higher nebular ionization parameters. Thus, without detailed modeling of the individual objects, it is impossible to clarify the locus of optically thin objects in Figure~\ref{fig:metal-chart}, but the IPM classifications in Table~\ref{tab:object-list} likely significantly underestimate the frequency of optically thin nebulae.
\begin{figure*}
\begin{interactive}{js}{metal_chart.zip}
\fig{metal_chart.png}{0.4\textwidth}{(a)}
\fig{metal_luminosity.png}{0.4\textwidth}{(b)}
\end{interactive}
\caption{[\ion{O}{3}]$\lambda5007/$[\ion{O}{2}]$\lambda3727$ (panel (a), left) and {\rm H$\alpha$}\ luminosity (panel (b), right) vs oxygen abundance. Symbol type and color show optical depth classifications.
Objects classified as optically thin and blister have higher [\ion{O}{3}]/[\ion{O}{2}]\ values, which is a function of metallicity.
\added{Median measurement errors for [\ion{O}{3}]/[\ion{O}{2}], $\log L$({\rm H$\alpha$}), and $12+\log(\rm O/H)$ are 0.06 dex, 0.018 dex, and 0.004 dex. Systematic errors are on the order of 20\% for [\ion{O}{3}]/[\ion{O}{2}]\ and $L$({\rm H$\alpha$}), and 0.18 dex for $12+\log(\rm O/H)$.}
An interactive version of panel (b) will be made available in the online version. Users can: \\
i) Filter the objects by $\log$([\ion{O}{3}]/[\ion{O}{2}]) by means of a slider. \\
ii) Select an object by clicking on the symbols or in a drop-down list to view the corresponding ratio maps used for classification as shown in Figures \ref{fig:bclmp-668} -- \ref{fig:bclmp-667}. Both monochrome and colour images are available. \\
iii) Select the different optical depth categories in the legend to plot only objects in the selected categories. \\
iv) Hover with a mouse pointer on the symbols to view the ID, metallicity and [\ion{O}{3}]/[\ion{O}{2}]\ value for each object.}
\label{fig:metal-chart}
\end{figure*}
The classifications are also dependent on the spatial resolution and depth of the ratio maps. Comparing Figure~\ref{fig:thinfraction} with the results of \citet{Pellegrini2012ApJ...755...40P} for the LMC and SMC, the frequency of optically thin objects is much lower, about a factor of 2 at $L(\rm H\alpha)\ \sim10^{38}\ \rm erg\ s^{-1}$. There is no reason to believe that the ISM properties of M33 are substantially different than in these galaxies, and so most likely our classifications are affected by the lower spatial resolution of the M33 imaging data (22 pc in the smoothed images) relative to the imaging of the Magellanic Clouds (1.4 pc).
\added{Given the qualitative similarity of the results from these two studies, the systematic errors generated by the resolution effects do not substantively change the observed trend in Figure~\ref{fig:thinfraction}. However, we caution that absolute interpretation of optical depths is much less reliable.}
\explain{Addressing referee comment 2.2}
\begin{figure*}
\fig{thin_fraction.png}{0.5\textwidth}{}
\caption{Fraction of optically thin \ion{H}{2}\space regions as a function of luminosity for $12+\log(\rm O/H) < 8.4$. Luminosities of the sample objects range from $\log L(\rm H\alpha) = 36.7$ to 39.7, thus the frequencies in bins with $\log L(\rm H\alpha) < 37.5 = 0.$
There are no objects in our sample that have $\log L(\rm H\alpha)$ in the range 38.8 to 39.2. }
\label{fig:thinfraction}
\end{figure*}
We therefore conclude that IPM based on [\ion{O}{3}]/[\ion{O}{2}]\ or [\ion{O}{3}]/[\ion{S}{2}]\ is useful only at lower metallicities, $12 + \log (O/H) \lesssim 8.4$, and it is also dependent on spatial resolution. Other diagnostic lines can extend the use of IPM. For example, \citet{Zastrow2013ApJ...779...76Z} have similarly used [\ion{S}{3}]$\lambda 9069/\rm$[\ion{S}{2}]$\lambda6724$ ratio maps to identify Lyman continuum escape. In general, the underlying principle remains the same: to reveal the nebular ionization structure by differentiating the high vs low ionization zones.
\subsection{A kpc-sized patch of elevated [\ion{O}{3}]}
In general, the [\ion{O}{3}]\wavelength{5007}/[\ion{O}{2}]\wavelength{3727}\space ratio in the diffuse ISM is low, $< 0.4$, and largely invariant. However, in a region bounded approximately by R.A. $\sim$ 1h 33m 40s to 1h 34m 20s and decl. $\sim$ 30\degr 53\arcmin\ to 30\degr 59\arcmin, we observe a large-scale, bi-lobed patch of diffuse emission where the [\ion{O}{3}]\wavelength{5007}/[\ion{O}{2}]\wavelength{3727}\space ratio is $\sim 0.6$, significantly greater than the background (Figure \ref{fig:hii-patch}). This patch corresponds to an area $\sim 9\arcmin\times3\arcmin.5$, implying a structure with dimensions on the order of $\sim$1 kpc for the M33 distance of 840 kpc \citep{Freedman1991ApJ...372..455F}. The integrated [\ion{O}{3}]\wavelength{5007} luminosity of the diffuse patch, excluding other \ion{H}{2}\space regions in the vicinity, is $\sim(4.9\pm 1.5)\times10^{38}$ ergs/s. Such a large scale region of elevated ionization is unusual and difficult to explain.
There is an optically thin \ion{H}{2}\space region, BCLMP 637, at the centre of this patch of excited gas. BCLMP 637 is one of the most highly ionized objects in our sample, with an [\ion{O}{3}]\wavelength{5007}/[\ion{O}{2}]\wavelength{3727}\space ratio of $\sim4$. Its ionization is apparently dominated by [NM2011] J013350.71+305636.7 \citep{NM2011ApJ...733..123N}, a WN3 star. We evaluate whether this WN3 star is responsible for photoionizing the large, excited patch in what follows.
From aperture photometry, we find that the observed [\ion{O}{3}]\wavelength{5007} luminosity of BCLMP 637 is $\sim(1.38\pm 0.08)\times10^{38}$ erg/s, about $4\times$ less than that of the large patch.
The {\rm H$\alpha$}\ luminosity of BCLMP 637 is $(6.75\pm 0.62) \times10^{37}\ \rm erg\ s^{-1}$, corresponding to $Q(\rm H_0) \sim 5\times 10^{49}\ \rm s^{-1}$. We compare this to the PoWR WNE stellar models at LMC metallicity by \citet{Todt2015A&A...579A..75T} and find that this value is roughly an order of magnitude higher than that predicted by the brightest models. In particular, the LMC WNE PoWR models 10-17, 09-16, and 13-21 predict $M_V$ in the range --4.8 to --5.0 for an early type WR star, which agrees with $M_V$ of --4.9 for [NM2011] J013350.71+305636.7 \citep{NM2011ApJ...733..123N}. The ionizing photon flux predicted by these models is in the range $1.1-1.3\times10^{49}$ photons/s, which is much less than the $Q(\rm H_0)$ required to ionize even BCLMP 637. Thus, additional ionizing OB stars are likely present in the nebula. However, there is no further evidence suggesting an unusual stellar population in this object that can be responsible for also ionizing the large, extended patch of elevated ionization.
Thus we also consider candidate high-mass X-ray binaries (HMXBs) within the patch from the X-ray survey of M33 by \citet{Pietsch2004A&A...426...11P}.
We examine two sources: [PMH2004] 192 and [PMH2004] 229. Extrapolating the reported X-ray fluxes in the 0.2--4.5 keV band from \citet{Pietsch2004A&A...426...11P} with typical HMXB power law indices of 1.5-2.5 results in ionizing photon emission rates $Q(H_{0})$ on the order of $10^{46}\ \rm s^{-1}$. Again, this value of $Q(H_{0})$ is orders of magnitude below the required $10^{50}\ \rm s^{-1}$ to explain the origin of the elevated excitation.
Interestingly, \citet{Bigiel2010ApJ...725.1159B} report the existence of unusually hot and bright giant molecular clouds (GMCs) adjacent to this high ionization patch, with a lower inferred CO-to-H$_2$ conversion factor. These GMCs are located between the two lobes, slightly south of center (Figure \ref{fig:hii-patch}). This suggests that the elevated ionization is indeed real and physically associated with M33, and that the source responsible for exciting the diffuse [\ion{O}{3}]\space is also heating up the GMCs. The identity and nature of this source remains unknown.
\begin{figure*}
\fig{o3_o2_power.png}{0.7\textwidth}{}
\caption{Unsmoothed [\ion{O}{3}]/[\ion{O}{2}]\ ratio map. The kpc-sized patch is marked. The location of unusually bright GMCs reported by \citet{Bigiel2010ApJ...725.1159B} is marked with a red X.}
\label{fig:hii-patch}
\end{figure*}
\section{Conclusion} \label{sec:conclusion}
To summarise, we have used narrow-band imaging of M33 to generate ratio maps in [\ion{O}{3}]\wavelength{5007}/[\ion{S}{2}]\wavelength{6724} \space and [\ion{O}{3}]\wavelength{5007}/[\ion{O}{2}]\wavelength{3727} \space to explore the limits of ionization-parameter mapping as a probe of \ion{H}{2}\space region optical depth to ionizing UV radiation. We employ a revised empirical method for continuum subtraction building on methods by \citet{Keenan2017ApJ...848...12K} and \citet{Hong2014PASP..126...79H}. This method uses the pixel histogram distribution for diffuse emission after filtering out bright, resolved emission, and exploits the dispersion around the mode.
We show that, due to the metallicity dependence of the [\ion{O}{3}]\wavelength{5007} emission line, the [\ion{O}{3}]\wavelength{5007}/[\ion{S}{2}]\wavelength{6724} \space and [\ion{O}{3}]\wavelength{5007}/[\ion{O}{2}]\wavelength{3727} \space ratios can only be effective as optical depth diagnostics in the low metallicity regime ($12 + \log\left(O/H\right) \lesssim 8.4$), which is roughly $\lesssim 0.5Z_{\odot}$. Most likely, we are unable to identify a number of optically thin \ion{H}{2} \space regions at higher metallicities due to the weakness of [\ion{O}{3}]\wavelength{5007} emission in this regime. Other emission lines should be used to trace higher ionization species in these conditions.
We report the presence of a peculiar large scale ($\gtrsim 1$ kpc) structure in northern M33 that is excited in [\ion{O}{3}]\wavelength{5007} and conspicuously absent in other bands. The known WR star and HMXBs in the vicinity of this patch cannot provide the required radiation to account for its ionization. Further observations are needed to understand its origin.
\acknowledgments
We thank Rogier Windhorst for use of the continuum filters and Anne Jaskot for assistance with the observing run.
\added{We are also grateful to the anonymous referee for helpful comments.}
This work was supported by NSF AST-1210285, NASA HST-GO-15088, and the University of Michigan.
Data processing was performed using the open-source Python libraries AstroPy, \citep{astropy:2013,astropy:2018}, Photutils \citep{photutilsBradley_2019_2533376}, NumPy \citep{numpy2020NumPy-Array} and SciPy \citep{scipy2020SciPy-NMeth}. Animations of the continuum subtraction process (Figures \ref{fig:video}, \ref{fig:haro-11-histogram}, \ref{fig:eso338-histogram} and \ref{fig:mrk71-histogram}) were prepared by generating individual frames in Matplotlib \citep{matplotlibHunter:2007} and then combined using FFmpeg \citep{ffmpegTomar2006}. Static and interactive versions of Figures \ref{fig:metal-chart} and \ref{fig:thinfraction} were created using the data visualisation libraries Altair \citep{altairVanderPlas2018} and Vega \citep{vegaSatyanarayan2017}.
\vspace{5mm}
\facilities{Mayall(MOSAIC-1), HST(STIS)}
\clearpage
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BOOL GetCurrentProject(LPTSTR lpszProject)
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HKEY hSectionKey = NULL;
HKEY hAppKey = NULL;
HKEY hSoftKey = NULL;
HKEY hCompanyKey = NULL;
BOOL bRet = FALSE;
if (RegOpenKeyEx(HKEY_CURRENT_USER, _T("software"), 0, KEY_WRITE|KEY_READ,
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DWORD dw;
if (RegCreateKeyEx(hSoftKey, _T("SuperCx"), 0, REG_NONE,
REG_OPTION_NON_VOLATILE, KEY_WRITE|KEY_READ, NULL,
&hCompanyKey, &dw) == ERROR_SUCCESS)
{
if (RegCreateKeyEx(hCompanyKey, _T("ProjectSetup"), 0, REG_NONE,
REG_OPTION_NON_VOLATILE, KEY_WRITE|KEY_READ, NULL,
&hAppKey, &dw) == ERROR_SUCCESS)
{
if (RegCreateKeyEx(hAppKey, _T("General"), 0, REG_NONE,
REG_OPTION_NON_VOLATILE, KEY_WRITE|KEY_READ, NULL,
&hSectionKey, &dw) == ERROR_SUCCESS)
{
DWORD dwType, dwCount;
LONG lResult = RegQueryValueEx(hSectionKey, _T("CurrentProject"), NULL, &dwType,
NULL, &dwCount);
if (lResult == ERROR_SUCCESS)
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ATLASSERT(dwType == REG_SZ);
ATLASSERT(dwCount < 128);
lResult = RegQueryValueEx(hSectionKey, _T("CurrentProject"), NULL, &dwType,
(LPBYTE)lpszProject, &dwCount);
if (lResult == ERROR_SUCCESS)
bRet = TRUE;
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if (hSectionKey)
RegCloseKey(hSectionKey);
if (hAppKey)
RegCloseKey(hAppKey);
if (hSoftKey != NULL)
RegCloseKey(hSoftKey);
if (hCompanyKey != NULL)
RegCloseKey(hCompanyKey);
return bRet;
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BOOL GetPictureFolder(LPTSTR lpszFolder)
{
TCHAR szProject[4096];
if (!GetCurrentProject(szProject))
return FALSE;
TCHAR szT[4096];
DWORD dw = ::GetPrivateProfileString(_T("Path"), _T("Picture"),
NULL, szT, 4096, szProject);
ATLASSERT(dw < 4096);
LPTSTR lpszFind = strstr(szT, _T("<¹¤³Ì·¾¶>"));
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LPTSTR lpszTemp = (LPTSTR)szProject;
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{
// remember last directory/drive separator
if (*lpsz == '\\' || *lpsz == '/' || *lpsz == ':')
lpszTemp = (LPTSTR)_tcsinc(lpsz);
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lstrcpyn(lpszFolder, szProject, lpszTemp - szProject);
lstrcat(lpszFolder, szT + 10);
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else
{
lstrcat(lpszFolder, szT);
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return TRUE;
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BOOL GetRecipeFolder(LPTSTR lpszFolder)
{
TCHAR szProject[4096];
if (!GetCurrentProject(szProject))
return FALSE;
TCHAR szT[4096];
DWORD dw = ::GetPrivateProfileString(_T("Path"), _T("Recipe"),
NULL, szT, 4096, szProject);
ATLASSERT(dw < 4096);
LPTSTR lpszFind = strstr(szT, _T("<¹¤³Ì·¾¶>"));
if (lpszFind != NULL)
{
// always capture the complete file name including extension (if present)
LPTSTR lpszTemp = (LPTSTR)szProject;
for (LPCTSTR lpsz = szProject; *lpsz != '\0'; lpsz = _tcsinc(lpsz))
{
// remember last directory/drive separator
if (*lpsz == '\\' || *lpsz == '/' || *lpsz == ':')
lpszTemp = (LPTSTR)_tcsinc(lpsz);
}
lstrcpyn(lpszFolder, szProject, lpszTemp - szProject);
lstrcat(lpszFolder, szT + 10);
}
else
{
lstrcat(lpszFolder, szT);
}
return TRUE;
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Take the Franklin Road exit and go straight ahead (North) on Milwaukee about two miles to Ustick and turn left. Follow Ustick for about 1 1/2 miles and look for the Boise Cycle, LLC Sign on the left side of Ustick. Enter and go to the back of the drive.
Take the Cole Road exit and go straight ahead (North) on Cole about two miles to Ustick and turn left. Follow Ustick for about 2 1/2 miles and look for the Boise Cycle, LLC Sign on the left side of Ustick. Enter and go to the back of the drive.
Saturday 10 a.m. to 4 p.m. ~ Closed Sunday & Monday ~ Tuesday thru Friday 9:30 a.m. to 6:30 p.m.
Closed Sunday & Monday ~ Tuesday thru Friday 10 a.m. to 6 p.m. ~ Saturday 10 a.m. to 2 p.m.
Copyright © 2004-2016 All rights reserved. | {
"redpajama_set_name": "RedPajamaC4"
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{"url":"https:\/\/siebencorgie.rs\/gallery\/feldversuch\/","text":"Feldversuch\n\nSigned distance field based synthesizer\n\nSiebencorgie published on\n5 min, 835 words\n\nIdea and motivation\n\nIn the winter of 2021 I wrote my bachelors thesis named Signed distance field based sound modeling for intuitive audio creation interfaces.\n\nThe main motivation stems from the idea that wavetable editing can be made fun. And maybe a little bit from the existence of Nako \ud83d\ude2c. This is a small video of me editing a wavetable in Vital.\n\nAs you can see at 1:00 there is a 3D representation of whatever I build. So the Idea was: \"What if I could model this surface?\"\n\nAfter some research the main points of interest where identified (actually most of the work for the thesis ^^).\n\n1. How to create the sound (time domain \/ frequency domain)?\n2. How does modeling work related to sound creation?\n3. How could a graphical interface look like for such a tool?\n4. ... And some less important questions...\n\nObviously there are multiple questions spawning from those. It was planned to use hand tracking for the input, since this makes a lot of sense here. I couldn't implement it fast enough however, so I had to default back to good old mouse+keyboard input.\n\nAll the theoretical stuff isn't discussed here, but maybe you can find the thesis by its name, or send me a message if you are interested.\n\nTechnical stuff\n\n3D Rendering\n\nThe easy stuff first: 3D signed distance field rendering. Since I already had Nako at this point it makes sense to use it here as well. It was a good real world test to check what works and what doesn't. One aspect of Nako that directly influenced the development of the synth is its time-line like way of adding primitives to each other. The way Nako combines those is by taking a string of operations. Something like Take this sphere, remove this cube, then add this plane etc.. While this is a technical limitation in reality (I wrote about it here) this actually works quite well. The order can be expressed easily in the interface and the result is predictable.\n\nApart from that everything was already prepared by Nako and its companion crates.\n\nUser interface\n\nA perk of using Nako's renderer was that I got free 2D (also SDF based) rendering for free. Including features like caching already rendered and unchanged layers between frames. This lets the GPU work on the 3D rendering almost exclusively. While there are already some 2D rendering components prepared (in the nako_std crate), like text rendering and some simple buttons, I ended up hard coding most of the interface elements specifically for the synth. I noticed that handling the compositor by hand is not really practical for bigger projects like this.\n\nSound creation\n\nThe whole purpose of creating the 3D surface is to retrieve some kind of signal. This is currently done by stepping along a plane and checking the height of the surface at the given location. The resulting set of points is the base signal that is either played back as is (time domain), or transformed into the time domain from the frequency domain using iFFT.\n\nWhile the time domain is straight forward the frequency domain wasn't as easy to implement. Mostly because I didn't want to give up on the multi-voice feature of the synth. At the moment whenever the model changes a time-domain base signal is created via iFFT. This signal is then pitched to the correct offset for each voice's key. Therefore the amount of iFFT transformations keeps constant and pitching the voices is mostly a matter of reading the base signal more or less fast. Obviously this is not the correct way. This pitching strategy introduces artefacts from resampling, and there is no interaction between the voices frequencies. However doing it the right way requires a lot more engineering which wasn't practical at the time.\n\nTakeaways\n\nNako and graphics\n\nThe biggest problem of the synth is currently its performance. It needs a reasonably powerful graphics card to even start. For instance my notebook's Intel HD 4000 won't even render one frame. Nako also scales with model complexity. Therefore creating highly detailed models is currently not practical. This, paired with the already mentioned Operation Stream nature is what led me to start Algae.\n\nThe 2D UI which is also based on Nako is quite tedious to use. Which brings me to the conclusion that it might be time to overcome not invented here syndrome and start using toolkits like EGUI. UI Toolkit seem to be hard, at least even my third attempt isn't as good as I hoped.\n\nClosing\n\nAfter getting my bachelors for the thesis and the program I got the chance to evolve the project and submit it to a conference under the title Creative Sound Modeling with Signed Distance Fields at the Mensch und Computer conference (MuC). Below is a small demonstration video of the final version.","date":"2022-12-09 23:21:38","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\": 0.3826630115509033, \"perplexity\": 1313.064720417177}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446711552.8\/warc\/CC-MAIN-20221209213503-20221210003503-00738.warc.gz\"}"} | null | null |
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THE
TRUTH
ABOUT
LOVE
JULIE ANNE WACK
Copyright © 2016 by Julie Anne Wack.
Library of Congress Control Number: 2016919529
ISBN: Hardcover 978-1-5245-6382-0
Softcover 978-1-5245-6381-3
eBook 978-1-5245-6380-6
All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the copyright owner.
This is a work of fiction. Names, characters, places and incidents either are the product of the author's imagination or are used fictitiously, and any resemblance to any actual persons, living or dead, events, or locales is entirely coincidental.
Any people depicted in stock imagery provided by Thinkstock are models, and such images are being used for illustrative purposes only.
Certain stock imagery © Thinkstock.
Rev. date: 12/12/2016
Xlibris
1-888-795-4274
www.Xlibris.com
750990
Contents
New York City – April 1886
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
New York City – April 1886
"I don't think this pigeon likes my hat. He's staring at it."
"We're probably standing on the ledge where he rests when he isn't flying. Can we go in now, Tru? I'm starting to get cold and my fingers are getting numb. Besides, I'm afraid we'll be late returning to work and get into trouble."
"Don't worry, Clarice, it's safe to climb back in through the window now. I peeked between the curtains, and the room is empty. We have to hurry, though, in case Mr. Jarvis returns. I'll go first. You take the hatbox and hold it until I climb in—and don't drop it. I don't want to lose it after all this." Clarice took the precious hatbox from her friend and watched as the slender young woman hoisted herself over the windowsill and back into the room from which they'd fled what seemed like an eternity ago.
"I'm in. Hurry, Clarice, hand me the hatbox then climb through the window. That stupid bird can have his ledge. I still say he didn't like my hat."
"The hat is a bit strange."
Gertrude Kueshner gasped at the sound of a deep male voice behind her. She swung around to find herself staring up into eyes the color of shiny gold coins. For one moment, she could have sworn those eyes were laughing at her, but a second glance revealed a face that looked anything but amused.
The tall, handsome man staring down at Tru was indeed trying not to laugh out loud, so he straightened his shoulders in an attempt to look more imposing. But the corners of his mouth had begun to twitch, and despite his best efforts to control himself, he burst out laughing.
Gertrude Kueshner was having a bad afternoon. When she saw where the stranger's gaze was fixed, she realized that the hat perched atop her head was the source of the stranger's laughter. Tru had had all she could take for one day, so she straightened her spine and jutted out her jaw. "How dare you! I'll have you know this hat is as good as any of the latest ones worn by the most fashionable ladies in Paris. Just ask Clarice. You obviously have no taste in fashion."
"Well, I do apologize for my failure to recognize the quality of your bonnet. The cherries on the brim probably looked appetizing to the poor pigeon, and that's why he stared. You're lucky he didn't attack it. Now speaking of your friend, perhaps we should assist her off the ledge."
Tru turned toward the window where her companion was still perched precariously on the ledge. "Clarice, I'm so sorry. I'll help you back in." But before Tru could move, the tall stranger had reached the window and, putting his hands around Clarice's waist, lifted her over the windowsill and back inside.
Once her friend was back within the confines of four walls, Tru turned to face the man, who now stood quietly studying the two young women. At that moment Tru decided it was better to go on the offensive. "Who, may I ask, are you? And what are you doing in the Jarvis's suite?"
"I'm Alexander Marshall of the New York City Police. And who might you two ladies be?"
"Police! You can't be a policeman! You don't have a moustache, your hair is all wild and curly, and you're not wearing a uniform!"
"Well, the lack of a uniform can be easily explained, since I'm a detective." The man was again fighting the impulse to laugh. "And though, I grant you, many of New York's Finest do favor facial hair, it isn't a requirement. I prefer the feel of a clean face. As for the hair, you would have to blame my dear departed mother for this unruly mop."
Tru stared at her feet for a moment as she tried to determine how much trouble she had gotten herself into this time. The man was definitely large enough to be a policeman, and he looked imposing enough to frighten criminals. Besides being tall, he had broad shoulders and strong-looking arms. But when he had laughed, he had looked less intimidating. After Tru finished surveying the stranger, her eyes returned to his face. The man's mouth was now set in a forbidding frown, making Tru wonder if she had only imagined his earlier laughter.
Alexander Marshall returned the young woman's scrutiny with a practiced gaze. He had been surprised when he entered the room to find two such pretty young women instead of the more-hardened types that he usually dealt with in his line of work. A quick search through his memory produced no images of these two women from among the criminal types in the rogue's gallery the police department kept. But he could not rule out that they were newly recruited to a life of crime after a short time in the city. The burglary ring he was currently after was known to use young and pretty women to obtain wax impressions of keys from unsuspecting household servants.
Alex found himself focusing on the obvious leader of the pair. Her eyes were a brilliant blue, startling him at first glance as they peered out from a face that managed to be both elegant and impish. The outrageous concoction that had been the object of his amusement sat atop a shiny pile of light-brown curls. She wore a dress of bright yellow, and the image of a canary poised to take flight entered his mind unbidden.
"Did you capture them, Detective?"
Alex turned to see an anxious face peering around the corner of the entrance to the suite. It was the day manager who had summoned the police. "I'm questioning them now. I'll get to the bottom of this situation, I assure you."
"The Everett is a fine establishment." The man's Adam's apple bobbed up and down as he wrung his hands. "We don't want any trouble for our tenants."
"Just relax and let me proceed." Alex fixed the nervous clerk with his best authoritative stare. "I told you everything is fine. Now, if you'll excuse me, I'd liked to finish questioning the young ladies."
"Ladies indeed!" With those parting words, the man withdrew from the room, closing the door with a disapproving click.
"Now, where were we? Ah, yes. You were about to explain what you were doing in this hotel suite that required you to climb out the window and spend time visiting a pigeon."
"Now see here—what was your name again?" Tru met the detective's gaze straight on.
"Detective Marshall."
"See here, Mr. Marshall, am I to understand that odious man sent for the police because he thought we were criminals? How dare he!"
Once more, the image of a canary flashed through Alex's mind as he fought back another burst of laughter. Right now, the young woman's feathers were definitely ruffled. Either she was a very good actress or a truly outraged innocent.
"Detective, we are here on completely legitimate business, I assure you." Tru glanced back at Clarice, hoping her friend would have sense enough to remain quiet and let her do the explaining. "We came here to see Mrs. Jarvis. We work at Hearns, and we're on our lunch hour." Tru prayed that would be enough of an explanation for the detective.
"Well, that's possible, I suppose. Except Mrs. Jarvis seems to be nowhere in sight. Or were you looking for her out on the ledge and just became engrossed in your conversation with the pigeon?"
"Oh, you don't have to be sarcastic!" Tru resisted stomping her foot in frustration. "I'm telling you the truth."
"I want to believe you, but you'll have to tell the whole story, not just part of it. Continue."
"I told you we work at Hearns—you know, the store across from Macy's."
"You're salesgirls. So why were you here to see Mrs. Jarvis?"
"We prefer salesladies, if you don't mind. And we were here to show her a hat."
"That silly thing on top of your head?"
"No, this hat in this box!" Tru thrusts the hatbox she clutched tightly in her hands toward the skeptical detective. "I made it. Mrs. Jarvis was in the store earlier today looking for something for a special event. She didn't find anything to her liking, and I thought she might like my hat. I had brought it to the store to show Clarice. I want to open my own millinery shop some day." Tru thought of all the hopes and dreams that had gone into the bonnet so lovingly packed in the bright-yellow hatbox.
"I'd better have a look, miss." The detective reached out to gently pry the hatbox loose from the young woman's hands. "A hatbox can hold things other than a hat."
"Oh, and I suppose I have burglary tools hidden inside. Feels a bit light for that, don't you think?"
Alex did feel a little foolish as he prepared to lift the lid and peer into a box that barely weighed anything. But it was his job to be sure, and the criminals he had dealt with in the past had taught him to trust no one. Not even a pretty young woman with guileless blue eyes. When he parted the tissue and pulled out the delicate object nestled inside, he was dismayed to hear a chuckle escape his lips.
The hat was immediately snatched away from the detective. "And just what are you laughing at?" An indignant Tru checked to make sure the detective's large hands hadn't damaged her creation.
"Was I laughing? I don't think so. I'm sure it's the height of fashion to have a dove perched on top of your head. You should show this one to the pigeon. I'm betting he'd really appreciate it. So this hat is the reason you're here. But that doesn't tell me why you were on the ledge."
This time, Tru did stomp her foot. "Oh, you! Very well, if you must have all the embarrassing details before you let us go, so be it. We knocked on the door, but nobody answered. Then the door opened, and the maid emerged dressed to go out. She was a very disagreeable woman. She said Mrs. Jarvis was not in, so we should go away. I tried to leave the hat with her, but she said she was just about to run some important errands and didn't have time for such nonsense. Then she left."
"Why didn't you leave the box outside the door?"
Tru stared at Alex, horrified at the very thought. "And risk someone stealing my hat! Do you know how much work goes into one of these hats? Not to mention the cost of the materials."
"Forgive my ignorance. Now, you were saying... ? The maid left . . . ?"
"We were about to go when I noticed, in her haste, the maid hadn't finished latching the door. I—well, I didn't want to waste the trip on our lunch hour, so—"
"Which is almost over, and we'll be in even bigger trouble if we're not back soon." For the first time, the young woman cowering behind the skirts of her friend spoke up, drawing Alex's attention. A skinny redhead with big green eyes and freckles splashed across her face, the poor girl looked as if she had expended her entire supply of courage with that one timid outburst.
"Oh, Clarice, I promise it will be all right—that is, if this insufferable man lets us go soon."
"As soon as you finish your story, miss."
"Well, as I told you, I couldn't bear the thought of wasting our lunch hour. So I wiggled the door handle a little, and the door opened. I know I should have just closed it and gone away, but it seemed like a sign. I thought if I left the hat on the table with a note for Mrs. Jarvis and she saw the hat, she would love it."
"Did you think Mrs. Jarvis would be happy to have you enter her home without permission?"
Tru blushed at the thought of her rudeness. "I didn't think of that at the time. Besides,"—Tru straightened her shoulders—"you're a man. You can't possibly understand how crucial it is to wear just the right hat to an occasion. That was part of the problem. Just as we were writing the note, we heard a man's voice coming down the hall toward the suite. We knew if a man came in and saw us, he wouldn't understand the importance of the hat, so we decided to hide out on the ledge."
"You were the one who decided that all by yourself." Clarice's small voice piped up emphatically.
Tru frowned in her friend's direction. "All right! I decided."
"Then what happened?" Alex hid a smile at the first sign of a crack in the friends' united front.
"What do you think happened? We climbed out through the window." Tru frowned at the memory. "It was a little more difficult than I imagined. Anyway, I peeked back through the curtains and saw a man and a woman enter the suite. The man was Mr. Jarvis. Unfortunately, the woman was not Mrs. Jarvis, nor did she appear to be a lady of refinement. I had no idea who she was. Well, at that point, I was certain Mr. Jarvis would not be happy to have us enter the suite by way of the window and introduce ourselves. He's a rather disagreeable man anyway. I've seen him at Hearns with Mrs. Jarvis once or twice. He always complains about how much she spends, and he has horrible taste. Well, I was afraid if he caught us, he would inform the manager or, worse, forbid Mrs. Jarvis from shopping at Hearns. She spends a lot of money in the store, and Clarice and I would surely lose our jobs over it."
Alex's sympathies were aroused by the woman's story. But, more importantly, his detective's instincts were inclined to believe this was a case of bad judgment rather than criminal intent. Most criminals he knew would be ashamed to make up such a silly story. "So what happened next?"
"Well, Mr. Jarvis and the woman disappeared into another room. I was afraid we'd be out on that ledge all day. But it was only a few minutes, and they were back where I could see them."
Alex almost choked as he tried to keep from laughing at the puzzled look on the naïve young woman's face. "And they left right after that?"
Tru nodded. "Yes. Mr. Jarvis gave the woman some money, and they left."
Alex suppressed a smile. "Well, miss, let's see that note you say you were writing." Tru hurriedly searched her reticule until she found the crumpled piece of paper and handed it to the detective. "Looks like an unfinished note to Mrs. Jarvis just as you said. All right, I believe you. I'm going to let you go this time, but I'd advise you from now on not to hide on the side of a building if you don't want to attract any notice. I'll need your names before you go. I'll have to check out your story about working at Hearns. If you're lying to me, believe me, I'll find you. I'm very good at what I do."
"By all means, check out our story." Tru never doubted for a minute that the tall policeman was quite capable of finding them no matter where they might be in this vast city. "I work in accessories, and Clarice works in piece goods. Clarice's last name is Lennox and mine is Kueshner. But please don't tell anyone at Hearns what happened."
"If your story checks out, I won't. I can be very discreet. Now, Miss Kueshner, I believe you have a first name."
The young woman with the fiery red hair stepped out from behind her friend and seemed to regain some of her poise with escape close at hand. In a breathless voice, she offered Alex the last piece of information he needed to end the interview. "Her name is Gertrude—Gertrude Kueshner. But no one calls her that. Everyone calls her Tru." The redhead was immediately rewarded with a withering look from her companion.
"Tru? That's a strange name." Alex thought to himself that a strange name suited such a strange creature, even one as appealing as the young woman now lovingly repacking her precious hat. "You ladies are free to go. I will talk to the day manager and persuade him this was all a misunderstanding. But I advise you not to make any more unauthorized entrances into other people's homes."
"Certainly not!" Tru gave the cynical policeman what she hoped was a look of cool disdain and hustled her friend in the direction of the door. She was anxious to go before the detective changed his mind. The whole trip had been for naught, and she would have to wait for another opportunity to impress one of Hearns' wealthy customers with one of her creations. Tru stifled a sigh of disappointment.
"Oh, Miss Kueshner."
Tru turned back. This time, there was no mistaking the broad grin that split the detective's handsome face. Tru's heart made a strange lurching movement in her chest as she froze with her hand on the doorknob.
"It really is a lovely hat. Someone will want it someday."
Tru choked back an unexpected lump in her throat as she turned without a word to follow Clarice out the door and down the hall.
***********************************************************
"I really should dismiss you both. Not just dock your pay. You were five minutes late returning to your departments from lunch."
"Mr. Hartley, we were really only trying to be good Hearns employees."
"Oh, really, Miss Kueshner, and how do you do that by breaking the rules?"
"Well, Mrs. Jarvis was very disappointed when she failed to find just the right hat this morning. She said she would have to try Macy's this afternoon. We wouldn't want that. So after she left, I remembered a new shipment had just arrived from Paris, and I went looking in it. When I found a hat I thought she would like, I decided to take it to her myself on my lunch hour." Tru pushed down the faint stirrings of guilt she felt rising in the pit of her stomach. After all, her hat was as beautiful as any she had seen in that shipment from Paris. "So, you see, Mr. Hartley, Clarice and I were really only making a special delivery. Something the store does all the time for its good customers."
"Well, I suppose if you made a sale, it somewhat lessens the offense. We wouldn't want Mrs. Jarvis to get her hat from Macy's."
Tru could have sworn the manager's lips made a smacking noise at the thought of a sale snatched from under the nose of their giant rival across the street. "Well, as it turned out, she wasn't home. No one was, not even her maid. So we couldn't leave the hat. But I will show it to her the next chance I get."
"Humph! All this nonsense and you didn't even make a sale. If you weren't such a good salesgirl, I'd have let you go a long time ago. Why the customers like you is beyond me! But Mrs. Vanderhollen was overheard to remark only today that she wouldn't have spent nearly as much if you had not helped her decide on several items. At Hearns, we always give the customers what they want. As to Miss Lennox, I know she would not have participated in such an escapade without your urging. Her behavior is usually properly restrained. Speaking of restraint, Miss Kueshner, perhaps tomorrow you could wear a color that is a little less, shall we say, eye-catching. Brown is always a good choice. You can both leave now, but I will tolerate no more tardiness. And don't think I won't dock your pay."
The two young women watched the slight man glide away to take final stock of his kingdom as the great store prepared to go to sleep for the night. "Honestly, Tru, if his moustache gets any longer, it will pull his chin down to his chest."
"That's if he had a chest." Tru laughed for the first time since their adventure had gone wrong earlier in the day. "Let's go home. My feet are killing me."
The chatter of female voices filled the air as a horde of salesladies, released from the requirement to restrain their conversations, shared tales of their day's encounters with demanding department managers and unreasonable customers. Tru and Clarice joined the throng moving in the directions of the employees' cloakroom, the last stop before the women all scattered for their homes in the far corners of the big city.
"You know, Clarice, no matter how tired I am, I never get over how magnificent the store looks. It's hard to believe there are so many beautiful things for people to buy all in one place." Tru's eyes surveyed the endless counters of merchandise.
"The new store does look nice. I love the new electric lights. But I never forget how tired my feet are at the end of the day, especially on sales days, which seems to be every other day since the store moved here."
The store had only recently moved to Fourteenth Street across from its giant rival—Macy's. But already the competition for customers between Hearns and Macy's had begun to intensify. The elevated railroad station at the corner of Sixth Avenue and Fourteenth Street discharged a steady stream of customers making their way to the doors of the giant stores.
"Well, I'm just happy to work here. This way I can keep up with all the latest fashions, though Lord & Taylor is more elegant. The most fashionable younger women shop there." Tru often strolled along the section of Broadway between Fourteenth Street and Twenty-Third Street known as "the ladies mile." It was the place for the wealthy socialites of New York to be seen and where someone like Tru could observe all that was sophisticated and up to the minute in the great city. At those times, Tru's mind was full of dreams about someday making beautiful hats that would be worn by New York's grand ladies of society.
"You know, Tru, there was only one way Mr. Hartley could have known about Mrs. Vanderhollen's remarks." Clarice's voice dragged Tru back from her daydreaming. "It must have been Mr. Kenton. He overhears everything as floorwalker. Betsy certainly wouldn't have told him. She'd never do anything to make you look good in Mr. Hartley's eyes."
"You're right about that." At that moment, Tru spotted the woman who was her immediate superior and who was intent on making Tru's life as miserable as possible at every opportunity. The manager for Tru's department was a well-endowed young woman who always looked slightly frumpy no matter how hard she tried to appear well turned out. "Look at her over there trying to impress Mr. Hartley. I'll never understand how she got to be manager. She's completely lacking in taste or style."
"I'm sure she's jealous of you and that's why she's so mean." Clarice's mind couldn't grasp why anyone would dislike her pretty and vivacious friend. "Oh, look, Tru, there is Mr. Kenton now."
Tru spotted the elegant floorwalker amid a cluster of chattering women and headed in his direction. "Mr. Kenton, may I have a word with you?"
"Of course, Miss Kueshner, what might I do for you?"
The slender man bent slightly at the waist in just a hint of a bow. A relatively recent immigrant to America, the dapper man's voice was pleasantly accented, and his courtly manners thrilled the female shoppers whom he assisted all day as part of his job. His elegant air had smoothed many a ruffled feather for Hearns wealthier customers.
"Mr. Hartley said someone told him of overhearing Mrs. Vanderhollen speak well of me today. I wanted to offer my thanks if it was you." Tru smiled at the man as he inclined his head to acknowledge Tru's gratitude.
"It was simply fair to pass along such remarks. It benefits the store if the customers are pleased with the service they receive."
"Well, thank you again." Tru didn't mention that the fortuitous timing of the floorwalker's remarks might have saved her job.
"If you'll both excuse me now, I have some errands I must run before I return to my home. Miss Kueshner, Miss Lennox, enjoy your evening." With another slight bow, the floorwalker moved off in the direction of the large front doors.
"I swear, Tru, you almost expect him to click his heels when he bows like that. Do you know where he's from? I've never heard anyone say." The bubbly redhead grabbed the cape she had worn this morning to ward off the spring chill and waited impatiently for Tru to retrieve her own hat and dolman from the coat room.
"No, I've never heard anything but gossip." Tru and her friend emerged from the store onto Fourteenth Street and headed in the direction of home. "I don't think Mr. Hartley cares where he came from, only that he's an excellent floorwalker. All the female customers seem to enjoy his courtly manners. He has just the right mix of diffidence and charm."
"Speaking of charm, Tru, how are those two charming brothers of yours?"
"Oh, Carl and John are just fine."
Clarice's small pointed elbow darted out to poke Tru sharply in the ribs. "You know who I mean."
"Oh, you're referring to Walter and Ernest." Tru couldn't resist teasing her friend. She had known full well that Clarice meant her younger twin brothers. A strapping pair of boys who nobody but their family could tell apart, they had been the object of Clarice's adoration since she had first met them. The problem was Clarice had never been able to decide which twin she preferred. So she simply pursued both with equal fervor.
"They're fine, Clarice, but I don't know if they'll be coming home from Rutgers this weekend. They wrote that some of the other football players want to spend the next couple of weekends in spring practice."
Tru chuckled at the crestfallen look on her friend's face. But before she could cheer Clarice up by telling her the boys might still come home for a brief visit, Tru heard her name being called. She turned around and saw a beautiful young woman running to catch up with Tru and her friend.
"Tru, could you give John a message for me?"
Tru nodded at the new arrival as the trio paused on the sidewalk. Sophy Klienst was a fellow employee at Hearns. Sophy had begun seeing Tru's older brother after they had met when John came one day to the store to visit his sister. Tru was not at all certain she approved of the growing closeness between her brother and the exotic beauty. Sophy was a moody person who sometimes struck Tru as being deeply unhappy. There was no doubt, however, that the woman's striking looks had an impact on men, even her serious older brother. Many men dragged to the store by their wives would slyly watch Sophy out of the corners of their eyes, their attention drawn by the woman's lustrous black hair, violet eyes, and voluptuous figure.
"What do you want me to tell him?" Tru pasted a smile on her face and hoped it looked real.
"I can't see him tonight as we planned. My mother wants me to help with some extra baking."
Tru remembered that Sophy's mother was a widow who baked pastries to earn money on which to live. Sophy and her brother had come to this country when they were adolescents, and they had worked ever since to help their mother survive. Sophy's husky voice still carried a slight accent unlike Tru, who had been born after her parents' arrival in America.
"I'll tell him when I see him, but I don't know when he'll be home from the paper. He might go to your house straight from the shop."
The beautiful woman cast a puzzled look at the sister of the man she loved. "Isn't it Tuesday? Aren't you going to the paper now?"
"Oh my goodness, I totally forgot—so much happened today. Papa will be angry if I'm late. Why didn't you remind me, Clarice?"
Tru's friend shrugged. "I guess I forgot what day it was because of all the excitement, just as you did."
Tru didn't hear her friend's apology because she was racing along Fourteenth Street in the direction of her father's print shop as fast as her legs would carry her.
***********************************************************
"Papa, I'm sorry I'm late. I forgot what day it was." Tru leaned against her father's battered old desk, trying to catch her breath.
"And how, mein klein nachtish, can one forget the day of the week? If you were not always thinking of frivolous things such as clothes and useless finery, you would have room in your head for important things."
Tru tried not to feel hurt by her father's words. This was nothing new between them. Still she had to fight to keep resentment from creeping into her voice. "Papa, I'm always here on the days that you need me."
Tru's father rose from his desk and went to pick up a stack of newspapers from a nearby table. Tall and impressive with a barrel chest and a silver moustache, which was in startling contrast to the dark hair that was still abundant on his head, Tru's father was a handsome and vigorous man who wore his sixty years well. Tru could never remember a time when she had not been in awe of the patriarch who so easily commanded the love and respect of his family.
"Liebling, you come three days a week to help distribute the paper, but you are here only in body. Your mind is always with that oversized store and those silly people who have nothing better to do all day than to spend good money on foolishness. You could do so much more with your life."
"Papa, leave her be. Tru has loved pretty things since she was little and dressed her dolls in those silly clothes she made for them."
"I do not need you to explain my daughter to me." Johann Kueshner frowned at his oldest son, who had emerged from the printing room wiping the ink from his hands as he peered sympathetically at his sister from behind his glasses. "When one is a child, such things are charming and easily overlooked. But when we grow up, we must recognize those things that are truly important just as you and Carl have done. You both work at setting the type and putting together our paper so we can reach people with our message. That is dedicating your life to a worthwhile cause."
"It is not our paper—it is your paper, Papa." Tru's oldest brother carefully finished cleaning his hands as he spoke. Then he took the stack of papers from his father's outstretched arms.
"Why would you say such a thing? I have worked very hard since coming to this country when you were but a wee child sitting on your mother's lap so that our family can point with pride to this paper—this paper that tries to do some good in this world. Not like those big papers with their lurid headlines and endless advertising for fancy stores. Fancy stores that drive small shopkeepers out of business."
Tru tried to think of a way to prevent her father and brother from beginning one of their endless arguments over the way her father's newspaper was run. But she knew from past experience that nothing she said or did would stop them. Ever since she could remember, she had both loved and hated this tiny two-room shop that smelled of printer's ink and her father's tobacco.
"Those papers you are so contemptuous of actually report the news, Father. Not like this paper with its constant editorials and political pieces."
Tru watched helplessly as her beloved father and brother faced off against each other. Her father had puffed out his chest and was starting to grow flushed with anger, while her brother stood squarely in front of her father, betraying his own agitated state by repeatedly raking his fingers through his blond hair. Tru knew from experience that neither would give an inch.
"Those opinions you seem to think so little of are expressed in the hope of furthering the cause of the working people of this country."
"Father, those working people you want to reach are reading those other papers. Not a paper that is printed only three times a week and carries only political news."
Tru turned hopefully at the sound of a footfall outside the door and was rewarded with the sight of another brother. Carl Kueshner entered the room and, seeing the look on Tru's face, quickly sized up the situation. Carl was the peacemaker of the family and the only one besides his mother who could intervene between the two bullheaded men in front of him. "Papa, Mr. Schneider is outside at the vegetable stand. Didn't you want to talk to him about the meeting next week at the union hall?"
Carl's melodious voice seemed to cut through the tension in the room immediately. Tru reflected on how much his voice resembled the beautiful music his long fingers could coax from a piano so effortlessly.
"Yes, I do want to speak to him before we print the notice about the meeting."
"Well, you better hurry and go out to talk with him. Tru and I will help John get the rest of the papers ready."
"Very well." Tru's father moved toward the door, thankful for anything that would restore peace with his children. "I will return shortly, and we will go and sell the papers. Then we will all go home to one of Mama's wonderful suppers."
It was with relief that Tru heard the door close behind her father. She joined her brothers, who were already busy folding the stacks of papers and stuffing them into cloth bags to be carried out onto the streets in a few minutes. Tru tried not to think about how tired her feet were or how much she loathed standing on corners, trying to sell the small paper her father had lovingly printed for almost thirty years—the newspaper that had made life so hard for all of them.
Tru watched her beloved brothers as they worked and thought of all that her father's political idealism had cost them through the years. The two brothers looked alike in many ways. Both had blond hair; though behind his glasses, John's eyes were a piercing blue, while Carl's were a dreamy hazel. But their faces were stamped with the same chiseled features and high cheekbones.
Inside they were very different, yet they both had sacrificed their own dreams to carry out their duties as sons. John loved the newspaper business, but he wanted to work on one of the big city papers. He had been offered a job as a reporter on the New York World a couple of years earlier, but he had turned it down. He knew his father could not write and print the small paper by himself.
Carl did not even belong in the small newspaper office. He was a gifted musician whose talent for the piano was being wasted every day he spent in the small shop helping John set up the odd printing jobs that paid most of the cost for continuing with their father's dream. Tru stared down bitterly at the masthead of the paper she was folding. "Der Wahrheit," it proudly proclaimed—"The Truth." If only her dear papa could be persuaded to see a truth other than his own.
"So, little cabbage, what is that silly thing made of straw and fruit that is piled on top of your head? A new creation you stayed up all night to make?" Carl could hardly keep a straight face as he spoke to his sister.
"No, no, Carl. Don't you see? She must have had an accident and fallen into Mrs. Murphy's fruit and vegetable cart. Did you hurt yourself, little cabbage, or just Mrs. Murphy's produce?"
Both men laughed out loud as Tru stomped her foot hard on the floor. "Don't start, you two, or I'll hit you both over the head with these bags of newspapers. Just see if I don't." Being the only girl with four brothers had its benefits, but the teasing she had to endure wasn't one of them. "And, John, you had better behave, or I won't give you the message I have for you."
"What message? From Sophy? I promise, Tru, I won't tease you the rest of the day. What did she say?"
Tru regretted taunting John about Sophy's message as soon as his expression turned anxious. "I'm sorry, John. I should have told you as soon as I got here. Sophy can't meet you tonight. She has to help her mother bake." Tru worried as a cloud fell over her brother's face. John's feelings for Sophy were obviously growing stronger every day.
"She's had to work with her mother in the evenings a lot these last few weeks. I offered to come and help them, but Sophy said a man would just get in the way and upset her mother. I've never met her mother. I don't know why she would be upset. I wish Sophy understood that I would be glad to do anything to be with her."
Tru exchanged a look with Carl, who shared Tru's reservations about John's choice. But neither of them had said a word to their older brother. They knew how stubborn John could be once his passions were engaged. They could only hope that John was not in over his head when he had waited so long before falling in love. Both Tru and Carl were glad their own hearts were not under assault by that most capricious of emotions.
"Come, children, we must hurry." Tru's father burst back into the shop. "It is evening. Everyone rushes home from their work. Now is when we must sell our papers."
Tru and her brothers shouldered the heavy bags and followed their father out onto the street as they had since they were children. John and their father would deliver the papers to their subscribers, while Carl took a position outside Tammany Hall to sell some of the extras. Tru would walk the short distance from her father's shop on East Ninth Street to near Cooper Union. There she would take her place on the corner and try to turn the newspapers in her bag into the money her family needed to survive. Tru readjusted the weight of the heavy bag where it dug into her shoulders and hurried to catch up with her father and brothers.
Chapter 2
"All right, shorty, what have you got for me?"
"Now, sir, what does a fancy headquarters detective want from the likes of me?"
Alex let his smile fade a little as he slowly tightened his hold on the man's filthy vest, twisting the cloth as he squeezed the slight man between his powerful frame and the brick wall of the alley. "Now, shorty, lad, you don't want to disappoint me, do you? You promised me you'd get the information I need about the next little party at the pit."
"What for?" The small man's voice started out defiant but ended in a whimper. "Sergeant Mahoney at the precinct don't care if we have our little bit of fun and games. Why should you? I could get in big trouble if I told yous anything."
The pasty-faced criminal's eyes darted from Alex to the other detective standing quietly nearby. Then the same beady eyes traveled back to the face of the man holding him captive before his gaze slipped downward to stare at the dirt of the alley floor. Alex kept smiling at the man he held up against the wall, but there was something chilling in the big detective's eyes that made the petty thief squirm, an unusual occurrence in this dreary section of the city where a criminal almost always felt safe from the law. Most of the patrolmen in the Fourth Ward were willing to overlook anything but the most egregious of crimes. To do more was almost impossible in this area that contained some of the worst slums in the city.
Out of patience, Alex shoved the man hard against the bricks, deciding the time for subtle persuasion had passed. Alex broadened his grin as he glanced over his shoulder at his partner. "Well, Detective, do you want to have some fun? I'm tired of this pitiful excuse of a man wasting our time. Besides, he smells bad, and you know how I hate foul odors. I say it's time to finish our business and get out of this stinking hole."
Detective Richard Shaw shrugged, feigning indifference as to the fate of the man still dangling from his partner's grip. "Sure, Alex, if he won't talk, we might as well get something out of a trip downtown. You want to take the first swing while I look around for something to hit him with? I've no interest in bruising my knuckles."
"No, I went first last time. I'll hold him and let you take first crack. Besides, he's pretty scrawny, I don't think we need anything but our fists to cave in his head. Do you?" Alex secured the now sniveling man firmly against the wall with his left forearm, pressing it high across the thief's throat, threatening to cut off the man's wind. Alex could feel the petty criminal trembling against his arm.
"All right, yous win." A thin whisper broke the silence. "I'll tell yous what yous need to know." With relief, the nervous man felt a slight lessening of the pressure on his windpipe.
Alex looked back at his partner and winked. "That's better. I don't like to be disappointed. So talk, and we'll be on our merry way. As much as I'm enjoying your sparkling conversation, you've offended my nose long enough."
"Next time the dogs go after the rats is tonight 'round 'bout midnight. Big John's best dog be working the pit, so Big John'll be there." The frightened man made one last attempt at bluster. "But yous'll never take down Big John at the pit. Not with all his chums around."
Alex grinned. "Who says we're going to take down Big John? Now, remember, no one knows you talked to us, and it had better stay that way. If anyone finds out beforehand that we know about tonight's little party, your dear sainted mother will soon be crying in her whiskey over her lost son."
"What kind of flowers does your mother like? Just in case you forget our little warning and we have to take care of you." Alex's fellow detective spoke in a soft, unhurried voice. "The big guy and I like to do things up right. Does she like roses or maybe daisies?"
Alex almost laughed out loud as his friend put the finishing touches on their game. Richard Shaw had the face of a choirboy, which made his words all the more effective. When Alex's partner had tried to convince his ward boss to recommend him for a patrolman's job when he tried to join the ranks of New York's Finest, he'd had to show he was tough enough by beating up a local gang member and dumping him on the ward boss's doorstep to prove he was worthy of the boss's patronage.
"My friend's right. If we didn't get you, which is highly unlikely, we'd just spread the word to certain people that you ratted them out." As Alex spoke, he slowly let go of the thief. "Just keep that in mind."
Alex watched the scrawny criminal scurry away into the shadows. He and his partner needed to get busy making their plans for the night's work. "Let's get out of here." Richard Shaw lit up a cigar, and, after making sure no one was around to see them, the two men emerged from the alley. Alex took a nice deep breath of his partner's fragrant tobacco. "You know, Richard, I think that may have been the worst-smelling alley in this whole city."
"You say that about every alley."
The two men started northeast toward Mulberry Street and police headquarters. Alex tried not to think about how tired he was and what a long night lay ahead of them. "See, Richard, why I prefer uptown crime? The crime is as bad in the Tenderloin District, but the place smells decidedly better."
"Well, I don't know. Some of that cheap perfume those ladies of the evening wear smells pretty awful."
"Well, in my book, it beats urine and human waste any time. Anyway, we'd better hurry. Chief Byrnes just might want to be informed we're going after Big John tonight."
Richard Shaw glanced over at his companion. "Chief Byrnes won't be at the station tonight. The mayor's throwing a big party. Byrnes never misses a chance to rub elbows with the mayor." Richard often marveled at his friend's strange combination of street smarts and political naïveté. Of course, that was probably why Alex Marshall had remained an honest copper in a system so full of corruption—that and his father's death in the Draft Riots of '63 when Alex was nine years old. Richard almost envied his friend's loss, since it had helped make Alex one of the most determined and single-minded detectives in the city.
"Byrnes will still want a report. The organized pickpocket ring we've been trying to catch for months has really got him riled. They're getting bolder. Today a bunch of high tones were robbed, one right after the other, near Union Square. Rumor has it one of the victims was Joseph Pulitzer himself. Byrnes definitely doesn't want to give Pulitzer's paper any more ammunition to hurl at him."
"Was that why you were at the Everett today?" Richard stepped carefully around a drunk lying on the sidewalk, grateful it wasn't his job to deal with the hundreds of common drunks arrested every day.
"No. A patrolman telegraphed from the Eighteenth Precinct. He had been summoned by the manager at the Everett about some unusual intruders. The patrolman thought it might be linked to that hotel burglary ring we've been after." Alex smiled broadly, remembering his encounter at the hotel earlier in the day.
"I gather it wasn't our gang of thieves. Come on, what's the story? I'm curious about that grin on your face."
Alex didn't reply as an image popped into his head of a brightly dressed and very ruffled young woman with eyes the color of sapphires. When he recalled how she had stomped her foot at him, he laughed out loud.
"All right, that's it. Out with it." Richard's voice had grown indignant as he waited for an answer.
"Some other time, Richard, but remind me tomorrow I have to go to Hearns."
"Why Hearns? You always shop at Brooks Brothers. What's at Hearns?"
"The end of the story about the Everett. I'd have gone to Hearns this afternoon, but this little side trip downtown kept me from it. After I go to Hearns tomorrow and check on something, I promise I'll tell you the whole story."
Richard knew by his friend's tone that there was no use pressing him. Once Alex made up his mind, all the money in New York City wouldn't have been enough to persuade him to change it. So the two friends finished their trip uptown in silence.
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"Mama, that smells heavenly. I'm so hungry." Tru leaned over the table and breathed deeply of the wonderful aroma coming from her mother's sauerbraten.
"Mein liebling, you are always hungry like a small bear when you come home. Now hurry and carry the platter to the table. Your papa and brothers are starving also, I'm sure. Never in my memory does my family not eat my cooking."
Tru bent over to kiss the small woman's cheek. "It's your own fault, Mama. You are too good a cook."
"Ach, such nonsense. You must have smelled the kirschenmichel we have for dessert to flatter me so."
Tru's mother patted her hand as they left the cozy kitchen, heading to the dining room table. Tru placed the steaming platter down among the other bowls of food and took her seat beside Carl, while her mother sat at the foot of the table opposite Tru's father. John was already seated at his usual place across from Tru and Carl. The two empty chairs beside John testified to the absence of Tru's youngest brothers away at college.
"And how did the sale of the papers go today?" For almost thirty years, Martha Kueshner had asked her husband the same question three times a week.
Tru marveled that through all the years, not once had a hint of resentment crept into her mother's voice no matter what hardships the remarkable woman had endured, so complete was her support of her husband's dream. Sometimes Tru wished her mother had gotten angry. Even now when the combined hard work and wages of the whole family ensured plenty of food in the pantry, Tru still remembered when she was tiny and they had lived in their first small home near the stench of the slaughterhouses. There they often had only soup and her mother's wonderful bread for their evening meal. But they had never gone hungry. Her mother had taken in laundry and mending, while the savings she and her husband had brought with them from their homeland dwindled under the weight of Johan Kueshner's dream. Tru earliest memories were of helping her mother work on the pretty dresses that were always for someone else.
"Tru had a new hat on today when she came to the shop." John helped himself to a generous portion of his mother's potato dumplings. "It looked as if she'd fallen into a fruit cart, didn't it, Carl?" Carl could only nod his head, since his mouth was full of food.
"You boys have no appreciation for your sister's gift. Don't mind them, daughter, their opinions don't matter. Some grand lady will want one soon. I'm sure of it."
Tru flashed her mother a grateful smile. No matter how much she loved her brothers, she did get tired of her hats being the object of so many of their jokes. "As a matter of fact, Mama, I have sold a hat." Tru had been waiting for just the right time to tell her family the news. "I met a lady while I was selling papers two weeks ago. She was very nice, and she was dressed beautifully. She stopped because she admired the hat I was wearing, and we chatted about fashion a bit,"—Tru glanced hurriedly at her father out of the corner of her eye—"after she bought a paper, of course. She was so nice that I told her about my dream of one day owning my own millinery shop. She said she could tell I had wonderful taste and asked me to make a hat for her. I finally finished it, and I'm taking it tonight to the address she gave me. It's near Union Square."
Tru's father laid his fork back down onto his plate and frowned in his daughter's direction. "It is very late by the time you finish helping your mother with the dishes. I do not want you abroad at such an hour."
"But, Papa, this is very important." Tru tried not to sound angry. She had hoped her father would not stand in her way this evening.
"Johann, I can do the dishes alone tonight. This is a great thing that she has sold her first hat."
Tru smiled thankfully at her mother. Tru was excited because someone with great style would be seen abroad in one of her hats. Also she hoped to take some of the money from the sale and buy material to make a very special bonnet for her mother's upcoming birthday.
"Martha, you work hard all day. It's not much to ask to have the child's help."
John's fork clanked down onto his plate as he tried to hold in his irritation. "Papa, Tru is not a child. She is a grown woman. This is her dream. You above all should understand that. I will help Mama with the dishes tonight."
"Thank you, John." Tru was touched by her brother's offer. "Really, Papa, the countess does not live all that far away."
"Countess! You make a hat for a woman that still calls herself 'countess' in this land where all are free and equal? Why should my daughter serve the privileged classes anyway? This is not what we came to America for—this is not a worthy dream to have! How many times must we discuss this?"
"She mentioned that she was a widow, Papa. It is not her fault that her husband was a count in the old country. It is not as if she commanded me to make the hat for her. It was a request, and she will pay. We can always use the extra money." Tru tried to keep her voice respectful even as she felt a tide of rising anger in the face of her father's opposition. Then Tru decided to try a different approach. She got up and went to stand by her father's chair, bending down to kiss him on the cheek. "Please, Papa, understand how important this is to me. It is only a little hat. I promise it cannot be used to overthrow the American democracy."
Johann Kueshner sighed. He knew he would lose the argument, but he decided to put up one last attempt at resistance. "It is too dark for you to go alone. It is your safety I am considering."
"I will walk her to the place where she needs to go." Carl's soft voice put a final end to the disagreement. "She will be safe with me."
"Very well. My children all go their own way without listening to their papa's advice." The older man shook his head. "Some things about this America are not as good as the old country. Children in the old country obey their elders."
Tru and Carl exchanged glances, and Carl winked at his sister. Once their father began to grow nostalgic for the ways of his homeland, they knew they were past the hardest part. Johann Kueshner could be a stern man, but he loved his children very much. Soon, the room was filled again with the sounds of the laughter and chatter of the family's evening meal.
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"How close are we, Tru, to this woman's house?"
"We're almost there, Carl. Are you sorry you came?" Tru glanced anxiously at her brother's face, afraid of what she might see there. In her excitement, she had forgotten that the path to the countess's house would take them by Union Square with its many music shops and along Fourteenth Street, which led right past Steinway Hall. It was as a small boy that Carl had heard the great Anton Rubenstein play there and afterward had dreamed of one day doing likewise. A dream that now seemed lost to him.
"It's all right, Tru. I enjoy looking at the pianos." Carl stopped in front of the window to stare at the beautiful instruments on display in the Steinway showroom that was located between Union Square and the Academy of Music. When he finally spoke, his words seemed more for himself than for Tru. "Perhaps there is still a chance that someday I will play a Steinway on the concert stage." Carl straightened his shoulders. "If not, I will enjoy playing on Mama's piano for the family. Now we had better hurry. The hour grows late."
The two young people turned the corner from Fourteenth Street onto Lexington and stopped in front of a small but stately home that bore the same number as the paper clutched in Tru's hand. Tru knocked on the door, her stomach in knots. After a short wait, the door was thrown open by a cheerful-looking woman dressed in a maid's cap and apron.
"Yes?"
"We have business with the countess Von Ott. My name is Gertrude Kueshner, and I'm delivering a hat the countess ordered."
"Well, well, come in then. The countess is in the music room. I'll take you to her."
Tru began to relax a little under the warmth of the maid's cheery smile. As Tru and her brother moved down the long hall, she became aware of the sound of a violin being played behind the large double doors that marked the end of the hall. The maid knocked on those doors, and, immediately, the music ceased.
"Countess, there is a young woman here named Gertrude Kueshner who says she has a delivery for you."
"Of course, have her come in." A rich, throaty voice answered the maid, who opened the large doors and hustled them into the room before hurrying on her way.
Tru entered, grateful for the comforting presence of her brother. Then she became lost in admiring the beautiful room in which she found herself. The large salon was dominated by a magnificent, elaborately carved piano of mahogany inlaid with tortoiseshell. The glorious instrument stood on a raised dais by a large window that filled one wall at the end of the room. The interior of the room overflowed with lovely paintings and objets d'art, and it glowed with the rich color of the red velvet upholstery of the furnishings. The walls were covered with gold brocade wall covering and paneling of the same mahogany as the piano. There was a fireplace burning brightly in the wall across from the doorway where Tru and her brother waited nervously. A music stand and a small table holding a violin and bow were placed in front of the fireplace, and there the countess Von Ott stood, wearing a welcoming smile.
Carl waited quietly beside his sister for the countess to speak. His eyes were riveted on the lovely vision the countess made bathed in the light from the fire. A mature woman, her blond hair was pulled back from her delicate features into a knot at the base of her long, graceful neck. Her face was framed by the lace that trimmed the high collar of a pale-pink silk dress that revealed the womanly curves of her body. The warm smile on her face caused a few wrinkles to form around her eyes, telling the world that the petite countess was a few years past the first bloom of youth.
"Welcome to my home, Miss Kueshner. I am so glad you have come. Do sit down, please... both of you." The countess gestured gracefully toward a nearby settee before seating herself in a chair near the table upon which her violin rested.
Carl sat silently observing the older woman as he waited for his sister to begin her business with the countess. He smelled the delicate scent of primroses as the countess settled herself amid the softly rustling folds of her skirt. He watched as her right hand came to rest on her nearby violin, unconsciously caressing the instrument as one might caress a lover. Carl feared he was being rude and hoped the countess didn't notice his rapt attention. Yet he could not take his eyes off the woman his sister had come to see.
Maria Von Ott tried not to betray the strange unease she was feeling in the presence of the attractive young man who had seated himself next to the young woman she had met on the street. Maria had taken an instant liking to the stylish young lady when she had observed her valiantly trying to sell her small newspapers to an indifferent throng of passersby. When the woman had told her of her dream of one day opening her own millinery shop, an impulse to help had seized the older woman, and she had commissioned a hat. Now she hoped she had not made a mistake.
"I've brought the hat, Countess. Would you like to see it?" Tru began to untie the ribbons holding the box shut. To her dismay, her hands were trembling so that she was having a hard time undoing the bow. Carl, seeing his sister's distress, reached over and undid the ribbon without saying a word—a sweet gesture that did not go unnoticed by the countess. Tru gave her brother a grateful smile and lifted the small hat from the tissue. Her heart was pounding so that she almost didn't hear the countess's cry of delight.
"It's wonderful, Miss Kueshner! Simply delightful! I adore it." The countess swept the frothy creation of pink velvet and white egret feathers from Tru's hand and went to stand in front of a nearby mirror as she tied the bow under her chin.
"It is perfection. And you were clever enough not to make it so big as to overwhelm me, since I am, shall we say, rather diminutive." The countess laughed, delighted with her new purchase and the vindication of her instincts about the young hatmaker. "I knew you would make for me a beautiful hat. Thank you. I only wish I went out more into society so that I needed many more hats for you to make." The countess returned to her seat and carefully placed the hat back into its nest of tissue. "Now, please do me the honor of having tea with me before you leave?"
"That would be lovely, if it's no trouble." Tru was weak with relief.
"Of course, it's no trouble. I am anxious to get to know you better." The countess rang a small silver bell, and the same friendly face that had greeted Tru and her brother earlier appeared in the doorway of the music room.
"Yes, madam?"
"Rose, bring us some tea and some of cook's wonderful scones, please."
"Right away, madam." The smiling maid bustled away to return in a remarkably short time with a tray full of tea and scones. She set the tray down in front of Tru and Carl and hurried back out of the room, closing the doors behind her.
"Please, you must introduce me to this young man. In all the excitement, I'm afraid I quite forgot my manners. Is this your beau, Miss Kueshner?" The countess tried to sound normal, but the intense scrutiny of the slender young man was causing her usually unruffled demeanor to slip just a bit.
Tru laughed at the idea of Carl being her beau. "Carl is most certainly not my beau. I am sorry that I did not make the proper introductions. Countess Von Ott, may I present my brother Carl Kueshner."
As soon as the introduction was done, Carl put down his tea, and his eyes locked onto the countess's warm brown ones. Much to Tru's amazement, her shy brother rose from his chair and bent over the countess's hand in a very European bow. "Countess, it is a pleasure to meet you." Carl's eyes held the older woman's for another moment before the countess broke their gaze and looked away. Carl resumed his seat, and his eyes turned in the direction of the beautiful musical instrument at the other end of the room. "May I ask, Countess, if you play the piano?"
"Yes, though the violin is my first love. Do you play, Mr. Kueshner? Please feel free to try the piano if you'd like. It is a Steinway especially built for me and given to me as gift by my late husband when he knew his health was failing." The countess watched as the young man got up and approached the piano. She could sense the wonder he felt for the instrument by the way his hands moved lovingly over the lid as he opened it to reveal the keyboard. "I often have musical nights for my friends. Perhaps you would like to come?"
Even as she spoke, Maria wondered what had possessed her to invite this disturbing young man back to her home. But the next minute, the familiar strains of Beethoven's Impassionata burst into the room as the man's fingers flew over the keys. He played brilliantly, and Maria was soon lost in the music. She was brought abruptly back to earth when Tru's brother stopped playing. The countess sensed this man's passion for music was the equal to hers, and, again, a strange uneasiness washed over her.
"Carl is an excellent pianist, but he doesn't have such a wonderful instrument to play on at home. I'm sure he'd love to come to your musical evenings, Countess." Tru knew what it had cost her brother to stop playing and return to his seat.
"You must come also, Miss Kueshner. And please do not call me countess anymore. My name is Maria. It is many years since it mattered that I was married to a count. In truth, my dear Fritz was not unduly impressed with being a count. He was a socialist. That is why we came here in '71. We were fleeing the antisocialist laws that Bismarck had passed in Prussia. Fortunately for me, Fritz did appreciate the beauty and comfort his wealth could provide. He left me financially independent when he died much too young not long after we arrived in this great country."
"I will be happy to accompany Carl when he comes." The countess's openness and warmth touched Tru. "But I must warn you, I am not a bit musical."
Carl's laughter erupted at his sister's remark. "I can attest to that. We were finally forced to reduce her role in the family's musical evenings to simply turning the pages of the music book for me as I play."
The sound of the pleasant male laughter reverberating in the room was surprisingly moving to the refined widow, and she felt her heart skip a beat. Her sense of uneasiness returned to dampen her enjoyment of these young people's company.
"Oh, you! All of you boys are so proud that you inherited Mama's voice." Tru favored her brother with a mock frown.
"You have more brothers?" The countess was charmed at the thought of such a large and close-knit family.
"I have three more brothers. The oldest, John, lives at home as Carl and I do. The twins, Walter and Ernst, go to Rutgers and are home on weekends. Between my brothers and my papa, I would be all alone in a world of men if it weren't for my mama."
The countess smiled. By the tone of her voice, it did not sound as if the young woman found that male world all that unpleasant. "My dear husband and I were never able to have children." A sad wistfulness had crept into the voice of the older woman. "Your mother is very lucky to have such a delightful family. Perhaps one day I can meet the rest of the Kueshner household."
"I'm sure my mother and father would be delighted to meet you, Countess." Tru smiled and hoped she wasn't lying as she imagined what her father might say upon being introduced to the countess.
"Please do remember to call me Maria."
"Maria." Tru was having a hard time calling the countess by her given name.
"And may I call you something besides Miss Kueshner? Your given name is Gertrude, is it not?"
"Yes, but my brothers have always called me Tru, as do my friends. I really don't like to be called Gertrude."
"Tru. How unusual, but I can see that it suits you. I hope we can be friends, Tru. You're a talented young woman. You must hold tight to your dreams. I am sure you will succeed one day soon."
Tru felt like crying out of gratitude for the older woman's words of encouragement, but, instead, she stood up, knowing the hour was late. She did not want to overstay her welcome on this first visit; besides, their father would be worried. Tru would not put it past Johann Kueshner to come searching for them and end up knocking on the countess's door.
"We really must be on our way, Coun... Maria. My family will worry if we are too late. Thank you again for the tea."
Carl stood up beside his sister, preparing to go. He was full of a strange regret he did not understand. He only knew he very much wanted to see this lovely woman again and hear her play the violin.
"I'm sorry that you must leave so soon. But I would not want to be the cause of distress for your family." The countess stood to show her guests to the door herself. When they reached the entry, Maria Von Ott surprised herself again by placing a brief kiss on Tru's cheek. She felt close to this spirited young woman despite the shortness of their acquaintance. "The next musicale is tomorrow evening. Please do come? I would like it very much if you were here. I know you will enjoy it. A number of very talented musicians from the Academy of Music will be playing for us." The countess spoke to Tru, but she was only too aware of her new friend's brother standing quietly by the door.
"We will both come. Thank you so much for being kind enough to invite us." Tru turned to go but was stopped short by the countess's sudden exclamation.
"Oh my! I almost forgot to pay you for my wonderful hat."
The abject dismay on the other woman's face made Tru smile. "You can pay me tomorrow evening. I can wait. Now, good night, Maria, and thank you again."
Tru and Carl left the doorway and walked slowly down the sidewalk, hearing the door close softly behind them. When they reached the street, they quickened their pace, both of them filled with anticipation at the thought of the next evening's promised pleasure.
Behind the closed door of her home in the stillness of the empty foyer, Maria Von Ott leaned back against the elaborately carved door and felt a surprising mixture of expectation and unease sweep over her at the thought of the coming evening. Then she fled down the hall to the solitude of the music room and the safety of her music.
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"I hate nights like this."
Richard Shaw grinned at his partner. "I don't know why you say that. It's my favorite part of the job, seeing you covered with dirt and smelling like the back room at the Rusty Anchor. It makes all the danger and late nights worthwhile."
"Tie it off, Shaw. You don't smell so sweet yourself." Alex looked down at his filthy disguise, knowing he'd never get the best of his friend on this subject. Richard didn't mind the times they had to get rigged out like the scum they were after. Richard's nose was almost dead as far as Alex could tell. "Let's get on with it. Maybe if we're lucky, we can be in bed by morning."
The two men sauntered down the alley that led to the rat pit that was their destination. Already the detectives could hear the din of male voices raised to a fevered pitch by the promise of the blood sport to come. Alex and Richard strolled up quietly to the back of the crowd. They exchanged a quick glance and then began to move carefully in opposite directions around the tightly packed men. Eventually, they arrived at the front of the crowd where they could easily scan the faces revealed in the glow of the lanterns illuminating the pit. They were looking for one man in particular... Big John. The man they suspected of leading the worrisome gang of pickpockets who had grown too bold for their own good. It was the job of the two detectives this night to capture the elusive thug and bring him to headquarters. In the morning, their boss would have a little informal chat with the big thief and use all means available to persuade him to keep his band of pickpockets' activities below Fourteenth Street.
They soon spotted Big John across from where they stood pretending to wait for the sport to begin. He was about to throw a scrawny brown dog of indeterminate breed into a pit of squealing rats that were being poked from outside the enclosure by several men with long sticks. The prodding was having its desired effect, and by the time the dog was dropped into the pit, the rats were in a frenzy. The plucky dog attacked the rats with a vengeance. Soon, his muzzle was covered with blood and the pit strewn with furry corpses.
Alex gave his partner a look, and Richard nodded his head slightly. They both knew they'd have to wait until Big John left the pit area before they could hope to trap him. If they were lucky, the large man's scruffy dog would win big tonight and his owner would get rip-roaring drunk to celebrate.
Again and again, the pit was filled with new victims. Other dogs took their turn at the excited rodents, but Big John's dog, though not much to look at, was definitely the most enthusiastic and efficient killer this night. The more his dog won, the more whiskey the ugly giant of a man poured down his throat. Finally, Big John had cleaned out the pockets of all who dared bet against him. He collected his winnings and prepared to leave. Alex watched the man stagger away with two companions then signaled Richard it was time to follow their quarry.
Careful not to get too close, the two detectives followed their prey at a distance, looking for an opportunity to capture the man they were after. They waited, hoping one or both of Big John's companions would leave. The detectives' job would be a lot harder if they had to take on other thugs in addition to their target. But as time passed and the three men remained together, Alex and his partner knew they would have to attack soon or lose this opportunity.
Suddenly, the unexpected happened, and Alex and his partner got lucky. A quarrel seemed to break out among the three men weaving along ahead of the detectives. As the two detectives watched from a distance, Big John and one of his friends began to drag the third erstwhile companion into the nearest alley. Soon, the sound of a vicious beating reached the two detectives' ears.
"What now?" Richard whispered to Alex as he joined his partner at the entrance to the alley.
"Get closer and take a look." Alex led the way silently into the dark, shadowy alley. He could see Big John and the second man had the third man down on the ground, kicking him repeatedly with their heavy boots. The two lowlifes were enjoying hurting the man on the ground so much they didn't hear the two detectives approach or so much as glance in the direction of the street to see if they were being observed.
"I don't like men who kick another man when he's down. How about we even the odds a little bit?" Alex murmured to his partner and waited for Richard to nod in agreement.
The two detectives pulled out the nightsticks they had hidden under their coats and moved quietly forward. The drunken thugs were so engrossed in finishing off their victim they never spotted the two men creeping slowly toward them. Before the thugs could mount a defense, Alex and Richard each landed a hard blow to the side of the heads of their targets. The smaller attacker fell to his knees immediately, and Richard quickly laid him out with another blow. Big John's head, however, proved of sterner stuff. It took several well-aimed blows from Alex's club before the larger man lay motionless in the dirt.
"Well, let's get Big John to headquarters. The other one can lay here till he sleeps it off. We'll let Big John sober up behind bars until the captain has at him in the morning." Alex spoke quickly and then went to look at the third man whose beating they had interrupted. The man moaned loudly as he attempted to get up out of the dirt on his own. Alex reached down to help the injured man to his feet and dusted off his clothing.
"Thanks, mates." The grateful man flashed Alex a toothless grin. "Do I know you two fellas? I dinna recall ever laying eyes on you before, but I've only a wee memory for faces anyway. Dinna matter, I owe you. Them two woulda done me in if you hadn't come a long."
"Think nothing of it, friend. Now if you're all right, we'll be on our way." Alex rejoined his partner, who was bending down to put an arm under one of Big John's shoulders as they prepared to drag the large man to the nearest precinct where they would load him into a police wagon for the trip to headquarters.
"Of course, it'd really take more than a few kicks to do old Angus in." The old drunk was feeling braver now that the danger had past. "What would you be about doing with Big John?"
Since the man seemed more curious than concerned, Alex guessed he had no intention of trying to help his former companion. "We're detectives. We're taking him to jail for theft. Do you have a problem with that?" Richard spoke up before Alex had a chance.
"Sure, and I've no problem seeing his ugly carcass rotting in jail. And, sure, and it's a thief he be. Dinna he steal the wee plucky dog from me in the first place. That was what the fight were about."
"The dog is yours?" Alex watched as the ragtag man bent down and picked up the little dog, which promptly began to lick the dirt from the old man's cheek.
"Aye. He came with me all the way from Scotland. Big John stole him from me one night while I lay drunk. I dinna have any way to get him to return the wee dog. So I waited for a chance to steal the wee fella back. I acted as if I dinna care that the great oaf had him. But I was too far in me cups tonight and made the mistake of saying the dog were really mine. Big John got suspicious, and that was when he and the other blackhearted scum started in to beating me."
The little dog seemed perfectly content in the old man's arms, so Alex had no reason to doubt his story. "Well, the dog is yours now. We can't take him, and he'd just be out here on the street. Good luck to you. We have to go before Big John comes to and gives us more trouble. But if I were you and I wanted to keep my dog, I'd move out of this neighborhood before Big John gets out of jail and returns."
"Well, I'll remember that, and thanks again, mates. I owe you one." The battered Scotsman doffed his cap and prepared to amble off in the direction of the nearest saloon clutching his canine companion.
Alex smiled and nodded his head as he and Richard began the task of getting their prisoner to jail. "Just take good care of that little dog, old fella. He's hell on rats."
Chapter 3
"Mrs. Vanderhollen, you might find a parasol that better matches that dress if you look at these others." Tru held out several more parasols for her exasperating customer to inspect.
The elderly woman looked down at the frilly white parasol in her hand then, after a moment's hesitation, tossed it aside with a dismissive gesture. "At second glance, I don't believe I care for it as much as I thought. You may show me the ones you picked out. Perhaps something else will prove more to my liking."
Tru spent another hour trying to steer the society matron to a suitable choice. Finally, she managed to persuade her to purchase an elegant brown silk parasol with black braiding and fringe that went beautifully with the wealthy customer's somber brown day dress with black braiding around the hem and on the bustle. When the sale was complete and Mrs. Vanderhollen gone, Tru glanced over at her friend Clarice, who stood in the piece goods and trim department on the other side of the vast store. The counter in front Tru's friend was ten deep with impatient women. Clarice looked worn out, though it was not yet noon. Hearns and Macy's were having a price war on lace, and, currently, Hearns' price was lower.
Tru knew that in a few days' time, it would be her turn to deal with the hordes of women that appeared whenever there was a sale at Hearns. A shipment of silk gloves newly arrived from Paris even now rested in Hearns' storeroom, waiting to go on sale at the beginning of the new workweek. Tru sighed, anticipating how her feet would feel next week when the long work days were through.
"Now why is such a lovely young lady sighing?"
Tru swung around at the sound of the cultured male voice behind her, dismayed at being caught off guard. She was even more dismayed when she saw who stood at her counter leaning on an expensive walking stick with an ornate gold handle. Randall Bently was a friend of the Hearn family and one of the few men who made a habit of regularly shopping on his own in the vast emporium.
A tall, aesthetic-looking man with pale blond hair that was almost white, many of the salesladies considered the dapper gentleman to be handsome. Some of the women would discreetly try to attract his attention when he made his almost daily forays into the store. Tru, however, had a different opinion of the wealthy socialite. She disliked Randall Bently's pale-green eyes that constantly seemed to shift from side to side as he spoke. And she detested his spoiled, petulant mannerisms.
Perhaps precisely because of Tru's indifference, the man seemed to spend more time in her department than anywhere else in the store. His unwelcome attentions not only repulsed Tru but also were an added source of friction between Tru and her nemesis Betsy Stafford. If Randall Bently had not always made a point of making a purchase whenever he lingered beside her counter, Tru was certain Betsy Stafford would have complained to Mr. Hartley. But since the wealthy man spent large sums of money in Tru's department, there was little the spiteful manager could do.
"You didn't answer my question. What is so distressing the charming Miss Kueshner this day?"
Tru pasted a fake smile on her face and reluctantly turned her full attention to the man now striking a pose of studied casualness as he leaned on his walking stick. Everything about the man was phony, and Tru prayed he would leave soon. She could already sense Betsy Stafford's eyes boring into her from where she stood nearby talking to Mr. Hartley.
"Nothing important, I assure you, Mr. Bently. Now what may I do for you today?"
Randall Bently leaned over the counter and favored the young salesgirl with his most charming smile. "Well, you could meet me at the south entrance to the park tonight and take a walk with me. It is beginning to get warmer when the sun goes down. It would please me a great deal if you would gift me with the pleasure of your company."
"Oh, I'm sure an important man such as yourself has better things to do in the evenings than to share his much sought-after company with a poor salesgirl. Is there something in particular you are looking for today? I have some lovely new opera gloves just arrived from Paris." Tru picked out a pair of gloves from the case to show the wealthy man.
Randall Bently concealed his irritation as he reached out to examine the gloves, taking hold of one of Tru's hands in the process. The man attempted to prolong the contact, but Tru quickly withdrew her hand from his grasp, trying not to let him see the revulsion she felt at the feel of his touch.
The socialite allowed a fleeting expression of anger to cross his face at Tru's reaction but recovered quickly. "Yes, these are very fine gloves, Miss Kueshner, very soft to the touch. Although I can think of something even softer that I would enjoy touching."
Tru was outraged at the man's brazen insinuation, but she controlled her temper for the sake of her job. "Why, yes, Mr. Bently, I do have a pair of softer gloves now that you mention it." Tru reached in the case and brought out the most expensive gloves in the store, smiling sweetly as she laid them on the counter.
Momentarily nonplused, the habitual Casanova regained his air of confidence and pretended to compare the two pairs of gloves. "Why, Miss Kueshner, you are right as always. Your taste is as exquisite as you are. I'll take the pair you chose."
Tru carefully wrapped the obscenely expensive gloves and quickly completed the purchase, counting the seconds until she could be rid of Randall Bently. "Here you are, Mr. Bently. I'm sure whoever is fortunate enough to receive this gift will be very impressed by your generosity." As she turned to hand Randall Bently his newly wrapped purchase, she could see Betsy Stafford starting to move across the store in their direction.
"I can be most generous to the people I call 'friend', Miss Kueshner. Perhaps one day you will allow me to demonstrate just how generous I can be. Good day." Bowing slightly in Tru's direction, Randall Bently sauntered off, having recovered his air of aloof charm.
"Don't think you'll ever land a man like that, Gertrude Kueshner." Tru's immediate superior was now standing across the counter from Tru and lost no time in launching into an attack. "He's only after one thing from the likes of you, Miss Kueshner. I'm warning you, he'd have you and toss you aside the next day. And as long as I am being honest, you had better warn that brother of yours to watch his step also. He is being played for a fool. Others in the store are not so quick to turn down Mr. Bently's favors. Perhaps some others think like you that Randall Bently can help them advance in the store. But that will never happen as long as I am here."
"What are you talking about? Which one of my brothers?" Tru was puzzled and upset by the angry woman's reference to her brothers.
"Never mind, you'll find out soon enough." With one last nasty look, the buyer marched away in the direction of the store warehouse to check on the latest shipments from Paris.
Tru felt her hands shaking at the memory of both encounters she had just endured. But a customer was approaching, and Tru returned to her job. Her mind, though, was still distracted as she puzzled over the buyer's strange warning. Carl and John had been in the store on several different occasions, and the man-hungry buyer had spotted them and demanded Tru introduce her to them both.
"Young woman, I said I'll take this one."
Tru jumped at the sound of an angry voice and realized she had let her mind wander completely away from her work. "I'm so sorry, madam. That's an excellent choice. I'll have it wrapped in a second." Tru hurried to send the money and the bill of sale flying through the pneumatic tubes that crisscrossed the store's ceiling. Soon, the huffy customer was on her way, and Tru was left standing in her now empty department with a mind full of questions. Tru thought of her brother John, and a strange sense of dread came over her. She raised her head, and her eyes scanned the store until her gaze came to rest on the dark-haired beauty her brother John had come to love. Sophy Klienst was standing in her own department behind rows of exotic-looking perfume bottles. She wore her usual troubled expression.
As Tru stared thoughtfully at the beautiful salesgirl, she was surprised to see Randall Bently approach Sophy. Tru thought she saw the young woman stiffen as the sophisticated man began to speak to her. Then to Tru's surprise, Sophy appeared to be angry with Bently. Tru couldn't hear a word they were saying, but she could have sworn they were arguing. Sophy shook her head as if refusing some request, and Randall Bently grasped Sophy elbow and then quickly dropped it. At that point, the argument seemed to end. The slender man spoke a few more words to an obviously distraught Sophy Klienst before moving away toward the front entrance of the store.
Tru looked away before Sophy could notice she had been staring. What had transpired between the Sophy and Randall Bently? It must have been business, but what store business would have brought about such an emotional exchange? Fortunately, a new customer approached, dragging Tru's mind back to the job at hand and away from her painful speculations. Then several more customers appeared in rapid succession, and Tru forgot her earlier worries. When next there was a lull in Tru's busy afternoon, her mind returned to Betsy Stafford's obscure warning, and she was once again preoccupied with worry about her brother.
"Tru... Tru, you really are lost in thought today."
Tru turned her head in the direction of the voice calling her name and found Sophy Klienst standing next to Tru's counter. "I'm sorry, Sophy. What can I do for you?" Tru tried to smile, but her heart wasn't in it.
"I'm on an errand for my department manager, so I haven't long. I need you to take a message to John for me if you would please be so kind."
Tru had an uneasy feeling her brother was about to be disappointed again by the beautiful Miss Klienst. "Today is not a paper day, Sophy. I don't see him until we all sit down for our evening meal together."
"That is all right. We were not to meet until after the evening meal. Please tell him I cannot see him tonight. I must help my mother again." Sophy Klienst's eyes refused to meet Tru's as she gave Tru the message for John Kueshner.
"You know, Sophy, John would be glad to help you or at least keep you company. He looks forward to seeing you."
"No, no... I mean... please, Tru, just give John my message."
Sophy seemed near tears, and Tru regretted pushing her about the matter. Tru was trying to think of something to say to smooth over the awkward moment when suddenly there erupted a loud disturbance near the store's entrance. Both women turned in the direction of the noise in time to see a pathetically thin young woman clutching a shawl around her shoulders as Mr. Kenton and Mr. Hartley each took hold of an arm to escort her out of the store. As the pitiful creature struggled with the men, trying to free herself from their grasp, Tru saw the woman was pregnant. In the next moment, Tru recognized her as a former salesgirl who had been fired only a couple of months earlier.
"Leave me be! I have to talk to Mr. Hearn about getting my job back!" The desperate woman's voice could be heard throughout the store. "Please, all I want is to work. I need money for food, or I'm afraid my baby will die inside of me! I'm begging you, for the love of god, have mercy! Please help me!"
A crowd of customers stood frozen in shocked silence, listening to the pitiful tirade of the young woman who was being firmly removed from the store by the two icy men holding her arms. The poor creature's pleas had fallen on deaf ears, and soon her cries faded as she was escorted off the premises and down the street away from the building. In a few minutes, the customers had returned to their shopping, and the normal air of pleasant elegance had been restored, albeit accompanied by a low buzz of curious speculation.
"Poor thing. I remember when she worked here only a short while ago." Tru gazed sadly at the doors through which the fallen woman had been banished. "I wonder how she came to such a state?"
Sophy Klienst stared at Tru, her face ashen. A strange bitterness distorted her usually lovely features. "How can you be such a naïve fool? It is always the woman who suffers. Believe me, wealthy men never accept responsibility for their indiscretions. They just toss the woman aside like so much garbage." Visibly upset, the angry salesgirl rushed back to her department, leaving an astonished Tru to stare after her.
Tru's mind was racing as she tried to distract herself by needlessly rearranging the merchandise in one of the cases. Why had Sophy responded so intensely to the incident they had just witnessed? And did the anger and bitterness Tru had seen on her face mean future pain for John? Tru's heart sank remembering the message she would have to deliver to her brother.
"From canary to wren." A pleasant male voice sounded from behind Tru, causing her to jump in surprise. This day was turning into a disaster. A thoroughly exasperated Tru whirled around and, as she had feared, found herself staring up into an oddly familiar pair of laughing eyes.
"Pardon me?" Tru recognized the curly hair and the handsome face the instant her eyes met the man's cheerful gaze. "Oh, it's you."
"Of course, it's me. I told you I would stop by the store to check out your story." Alex's smile grew wider at the look of pique that appeared on the face that was every bit as lovely as he remembered.
"Well, you have. And now you know I didn't lie, so please leave before you get me into trouble."
"An angry wren." Alex crossed his legs and leaned against the counter just to let the exasperated young woman know he would not be driven away anytime soon. "First, a ruffled canary and now an angry wren. What other birds do you imitate?"
"What are you talking about?" Tru nearly stomped her foot for a second time in the detective's presence. Stomping her foot was a bad habit from her childhood that she tried very hard to control. Usually, only her brothers could still provoke her to the point of stomping. However, this man seemed to provoke her even faster than her brothers. "What bird imitations are you talking about, or have you just taken leave of your senses?"
"Yesterday you were wearing a yellow frock, and when you got angry, you made me think of an angry canary. Today you're dressed in brown and remind me of an irate wren."
"Oh, that's just wonderful! Yesterday you accuse me of being a thief while all the while I reminded you of a canary. Today I'm not even a canary but just a plain old wren. Do you have any more clever insults in your bag of tricks?" Tru didn't know why, but it bothered her that the detective thought so little of her appearance.
"You misunderstand. I love birds, and I am particularly fond of wrens. They are such busy little birds." Alex's voice had softened as he spoke, and Tru found herself waiting breathlessly for him to continue. "They're adorable really, and they have such a cheerful song to sing."
Tru threw back her head and laughed. It was the first time he had heard her laugh, and he thought it was a lovely sound.
"Your comparison has finally come to naught. I can't sing a note, Mr... . ? I forget your name."
"Marshall. Alex Marshall. But you see, Miss Kueshner, there are many ways to sing. For instance, the sound of your laughter just now reminded me of some of the most delightful melodies I've ever heard."
Startled, Tru looked up into the detective's smiling face, and she did strangely feel like singing. Her gaze held his briefly, and she marveled. She had never before seen eyes such a beautiful color.
"Miss Kueshner, are you taking care of this man's needs?" The store manager's shrill whine shattered Tru's pleasant reverie.
"Mr. Hartley. Yes, of course, I'm just..." Tru stammered for a moment, caught off guard by the sudden presence of her superior.
"The young lady was being most helpful. She was about to show me..." Alex's eyes searched the shelves until they fell on an object that interested him ". . . that fan over there. The one with the birds painted on it." Tru got the fan down and placed it in front of Alex, who picked it up and studied it.
"Good. I'm glad she's being of service. That is one of our many lovely imported fans. I'm sure your wife will love it." The store manager favored the detective with one of his ingratiating smiles.
"Oh, it's not for my wife. It's for my mistress." Alex couldn't resist trying to deflate the pompous little man. "My wife doesn't care for such things. She's too plain to do it justice anyway."
Tru watched in amazement as Alex kept a perfectly straight face while the embarrassed store manager sputtered as he tried to think of a reply. "Well, we at Hearns are always glad to be of assistance in whatever way we can. Carry on, Miss Kueshner. Good day." Wilfred Hartley almost ran in his haste to get away from Tru and her indiscreet customer.
As soon as the manager was safely out of earshot, Tru turned to Alex. "I can't believe you did that, but thank you for not telling him why you are really here."
"But, Miss Kueshner, I am here to buy that fan, although not for my wife, as I'm not married." Alex smiled down at Tru's upturned face. She was so pretty and spirited she made him want to laugh with sheer delight even when she was angry with him and stomping her foot.
"Mr. Hartley's gone now, so you don't have to buy the fan if you don't have anyone to give it to." Tru was puzzled at all the mixed-up sensations she was experiencing. This man obviously had little in common with her, and she felt in a constant state of turmoil when he was near. Yet she was sorry that their encounter would soon be over.
"Don't argue with me. Just wrap it up. How much do I owe you?" Alex used all his powers of deception to keep from betraying his dismay at the price Tru quoted him. He would have to cut back on his cigars for a month to make up for buying the fan, but he would sooner cut off his arm than let Tru down.
Tru took the detective's money and sent it on its way. Then she wrapped his purchase, being careful to tie the small package with an extra special bow. When she was done, she handed the wrapped fan to him, gasping in surprise when he promptly handed it back.
"For you." Alex laughed at the wave of emotions playing across Tru's expressive face. He knew she was torn between proper manners which required her to refuse a gift from a man who was a complete stranger and her obvious love of beautiful things.
"I can't accept this." Tru looked down at the pretty package, longing to possess the exquisite fan inside that was far too expensive for her to afford on her salary.
"Miss Kueshner, I refuse to take it back, and I'm a very stubborn man. I want you to have it to remember the detective who found you on a ledge with the pigeons." Alex's voice grew soft. "And because I'm truly sorry I laughed at your hat." Then he stood and waited for her reply. He was surprised at how important it was to him that his gift be accepted.
Tru raised her head. "I suppose I should keep it... after all your insults." A wistful smile spread across her face, softening her words. "Thank you. I'll enjoy it always."
"I'm glad. Well, good-bye, little bird." Alex Marshall doffed his hat, leaving Tru to stare after him as he walked out of the store, whistling as he went.
***********************************************************
Tru clutched her new fan in one hand as she waited for Carl to knock on the countess's door. Tru certainly hoped this evening proved more sedate than had the rest of her day. She experienced a feeling of uneasy excitement whenever she recalled her brief encounter earlier in the day with the persistent detective.
Then there were the lingering questions about Sophy Klienst that would not stop whirling around in her brain. John had been as disappointed as Tru had feared he would be when she gave her brother Sophy's message. Both she and Carl had tried to persuade John to go to Sophy's house anyway, but he refused to go against Sophy's wishes, worried he would upset her. Instead, he planned on spending the evening in his room, writing in his journal.
Tru's troubled mind was brought back to the present as the door to the countess's home flew open before the sound of Carl's second knock had faded. The same cheerful face that had greeted them the evening before now bid them welcome again. The maid stepped back, gesturing for them to enter the brightly lit home. After taking Tru's dolman from her shoulders, the maid led Carl and Tru down the long hall emblazoned with gaslights for the occasion.
On the way toward the music room, they passed two massive tables each holding a large silver candelabra of flickering candles and huge vases of hothouse flowers. The hall was perfumed with the flowers' delicate scent. The sound of people laughing and talking flowed out to surround them as the maid threw open the doors to the brightly lit music room.
"You've arrived! I'm delighted you came, Tru, and, of course, you also, Mr. Kueshner." A vision of blonde loveliness floated toward Tru and Carl. The countess Von Ott was dressed in a cream-colored gown of silk whose short sleeves and low décolletage emphasized her lovely skin and curvaceous figure.
Tru felt decidedly dowdy in her light muslin gown, though it was one of her favorites. She had copied it after one she had seen in the window of Lord & Taylor. Tru had saved up to buy some beautiful apple-green material with delicate white flowers embroidered all over and sewn it into the same stylish lines she had admired in the store window. But the gown didn't seem quite as fashionable as it had earlier when Tru had examined herself in the mirror in her room.
"You look beautiful, Tru." The petite countess planted a kiss on Tru's cheek. "Today's fashions look especially well on a slender figure like yours." The countess linked arms with the younger woman and drew her into the crowded room. "Come meet some of my guests. I'm sure you'll enjoy them. You also, Mr. Kueshner. There are a number of very fine musicians here this evening."
Carl followed his sister and the countess, his gaze drawn to the graceful sway of the countess's bustle. Carl knew it was wrong, but he was terribly fascinated by the delicate beauty of the older woman who had taken a liking to his sister. The same heavenly scent of primroses he'd smelled the evening before wafted back at him every time the countess Von Ott took a step.
"Professor Baer, I'd like you to meet the young people I was telling you about." The countess stopped in front of a small, round man in a rumpled suit. "This is Gertrude Kueshner and her brother Carl. Tru, Mr. Kueshner, this is Professor Baer. He will accompany me later on the piano."
The courtly academic bowed at the waist, placing a light kiss on Tru's hand before turning his attention to Carl. He regarded Tru's brother intently from behind his spectacles for a few moments then smiled and reached out to grasp Carl's hand in a hearty handshake. "Maria tells me you have a deep love of music and that you have a great talent. You must come to the academy and play for me someday soon."
"Oh, I doubt you would be interested in my playing. The countess is just being kind." Carl fought the rush of excitement that flooded over him at the thought of being able to play for such a man.
"Nonsense, my boy. There is one thing you should know—Maria Von Ott is a very serious musician in her own right. She never indulges in meaningless encouragement. If she saw something in you that impressed her, then I am willing to listen. I hold her opinion in the highest regard. You will see for yourself later when she plays."
"Professor, do not raise the boy's expectations so high I cannot possibly meet them." Maria smiled as she spoke, but she was genuinely nervous at the thought of playing this evening in front of these two young people.
"Maria, we will be enchanted as always. Now I must take my leave. There are such delicious pastries to be had at the refreshment table. I must pay homage to Maria's cook by sampling them all." The rotund professor then proceeded to take his leave.
"Great artists need nourishment—or so he tells me. If the greatness of the artist were measured by what they ate, my dear friend would truly be a genius." Maria laughed and was rewarded with the first genuine smile of the evening from the serious-looking Mr. Kueshner. A feeling of warmth swept over Maria as her eyes met the young man's amused gaze. Right on the heels of her delight at making Carl Kueshner smile, she felt a strong urge to run and hide.
"If you will excuse me, I see some new guests I must welcome. Please feel free to enjoy the refreshments." A suddenly nervous-sounding countess gestured toward a large table crowded with trays of food. "Cook truly is wonderful with pastries, and if you don't hurry, the dear professor may have eaten them all before you can sample any. The music will start shortly, so make yourselves comfortable until then."
Carl watched the countess sweep away across the room, puzzled by his need to follow her every movement. She paused at the doorway to greet a large, rather austere woman in a severe black dress who was leaning heavily on the arm of a young man. The man seemed vaguely familiar to Carl, who continued to watch the group, hoping to recall where he had met the man now engaged in conversation with the countess. Suddenly, he remembered exactly where he had seen that face before.
"Don't you want to eat anything, Carl?" Tru put her hand on her brother's arm to gain his attention. "Who are you watching so intently?" Tru followed her brother's gaze to where it rested on the two people who were still deep in conversation with Maria Von Ott. "Do you know them, Carl?" Something about her brother's expression had begun to make Tru uneasy.
"I know the man."
"Well, for heaven's sake, Carl, who is he?" Tru's unease was turning to anger in the face of her brother's continued silence.
"It's Sophy's brother, Otto Klienst. I met him at the print shop once when he escorted Sophy there. John introduced us."
"Are you certain that you recognize him? He does not look like Sophy." Tru stared at the man who stood next to the countess. The man was short and stocky with light-brown hair and washed-out brown eyes with none of the vivid coloring of Sophy Klienst. His features were nondescript in contrast to the eye-catching beauty of his sister.
"The more I study him, the more certain I am. I talked to him for several minutes that afternoon."
"Who is the woman with him?" Tru feared the answer to her whispered question.
"I am not sure I want to know." Carl sensed his sister's trepidation and shared it. They both knew if the woman with Otto Klienst was his mother, then that meant Sophy had lied to John.
"What should we do?" Tru continued to watch the group by the door. "They appear to know the countess quite well. I wonder how?"
"We must find out if that is Sophy's mother. Perhaps we will learn more about the countess in the process. Come, little sister, let's introduce ourselves before the music starts and the opportunity is lost." Carl took hold of Tru's elbow, guiding her across the room to where the countess stood talking to her new guests.
"Pardon the intrusion, Countess, but I believe I know this gentleman." Carl spoke politely, watching the other man's face for any sign of recognition. After a few moments, he was rewarded with a nod of agreement.
"Ah, yes. You are Carl Kueshner. We met at your father's paper. A good piece of work, your father's paper, although he does not go far enough in his views."
Tru puzzled momentarily at what Sophy's brother meant by that remark. But she was more interested in finding out who the sour-looking woman was who would not let go of Otto Klienst's arm.
Carl smiled and turned to introduce Tru, hoping to force an introduction to Otto Klienst's companion. "This is my sister, Gertrude Kueshner."
"Who are these people, Otto?" An imperious voice, whose strength belied the woman's seeming need for constant support, demanded an explanation from the young man at her side.
"The young woman works with Sophy, Mother."
Tru's heart sank with the confirmation that the woman her brother loved had lied to him. Tru thought of the other similar message she had carried the day before to John and wondered if that, too, had been a deception.
"So you work at that place also." The older woman's dark eyes seemed to bore through Tru. "And do you have problems with unwanted advances from men as my Sophy does?"
Tru was startled at the question. Out of the corner of her eye, she could see the countess begin to look uneasy. "Not often, Mrs. Klienst. But then Sophy is a very beautiful woman. However, I'm sure most of the male customers are polite to her. We really don't have many men who like to shop by themselves. They usually are accompanying their wives."
"Humph!" The older woman was clearly not comforted by Tru's words.
"Mother, we must take a seat now. The music will start soon." Otto Klienst appeared anxious to remove his mother from her present company. He bowed toward the countess before guiding the older woman away. "We will talk later, Countess. Thank you for inviting us. You are kind as always."
Tru watched the strange pair as they found two seats near the music dais. Sophy's brother was certainly solicitous of his unpleasant parent.
"Mrs. Klienst can be somewhat eccentric, but she has had a difficult life." The countess's soft voice broke into Tru's thoughts. Tru turned and was surprised to see a look of concern on Maria Von Ott's face.
"You know her well?" Carl felt compelled to ask what might be considered a prying question after so short an acquaintance.
The countess seemed far away for a moment. Then she returned to the present. "We traveled on the same boat to America in '71." The countess looked as if she wanted to say more, but, instead, she gestured toward the chairs arranged in rows facing the platform holding the Steinway. "It is time for the music to start. Please take a seat. I hope you enjoy our little offering this evening." With those words, the countess hurried away to find Professor Baer, leaving Carl and Tru to take their places among the other guests gathering in anticipation of the evening's entertainment.
Tru sat beside Carl and tried not to squirm. She was too upset by the recent revelations to relax. She kept thinking of Sophy's words earlier in the day, hoping to remember something that could explain the apparent lie. But she remembered the message exactly, and there was no mistaking Sophy Klienst's words.
"Carl?" Tru leaned over and whispered to her brother as the professor and the countess mounted the dais.
"What?" Carl was intent on watching the beautiful woman on the dais as she removed her long gloves in preparation for her performance.
"Don't you think it's strange that Otto Klienst introduced me as someone who works with Sophy and not as John's sister?"
Carl glanced at Tru before turning his eyes back toward the piano and the woman who was about to pick up her violin. "Yes, Tru, I took note of that. It is all very strange, and I don't know what we must do about it. But for now, let us listen to the music, please?"
Tru studied Carl's profile as her brother stared at the two people preparing to play. Carl was right. They could do nothing now, and she knew how much her brother had been looking forward to this evening. She would just have to try to relax and forget for a short while the problem of what to do about John.
The countess smiled at her guests from the platform and glanced down at the music lying open upon the stand in front of her. "I will play for you now Brahm's Violin Sonata no. 1 in G major with Professor Baer's kind accompaniment."
As the opening strains of the violin piece began to fill the room, Carl Kueshner sat transfixed, his eyes riveted on the woman pouring out her soul through the music she made on the instrument cradled under her chin. The countess's small hands alternately beguiled and demanded notes of perfect clarity from the violin that seemed to become a part of her as she played. Carl listened, enthralled, to the music pouring forth from the violin. It sounded to him like the musing of a melancholy heart. The music made him feel his loneliness and his longing for someone to share his dreams. When the music ended, Carl sat very still, too moved to join in the enthusiastic applause.
The rest of the program was a blur to Carl. There were two pianists who performed Chopin and Mendelssohn, but Carl was not impressed. The musicians were young and lacked the emotional depth the countess had brought to her music. Several times during the remaining performances, Carl's gaze strayed to the exquisite profile of the nearby countess, who sat listening to the young musicians. Once, as if sensing Carl's scrutiny, she turned her head, and, for a moment, their eyes locked, but the lovely widow quickly looked away. For the rest of the program, she was careful not to let her attention wander from the dais.
As soon as the musical part of the evening was over, Tru grabbed her brother's arm. "Carl, please, can we leave? I cannot enjoy myself when I am so concerned about John. We must talk, you and I, and decide what to do."
"I will find the maid to get your wrap. Wait here, and when I return, we will say our good-byes to the countess." Carl left the music room in search of the maid. He did not want to disturb his hostess until they were ready to make their early departure. As he stepped out into the hall, he glanced toward the entry and saw a strange sight. Otto Klienst was standing by the door leading to the outside, arguing with a bedraggled man who definitely looked out of place in the grand foyer. The man wore a long unkempt beard and was gesturing wildly at Sophy Klienst's brother as if demanding entrance to the stately home. Carl could not make out what they were saying, but he did hear snatches of the stranger's voice. Enough to know that the man was speaking heavily accented English.
Not wanting to appear to be eavesdropping, Carl walked quickly in the opposite direction until he found the maid, who promptly fetched Tru's dolman. As Carl prepared to return to the music room, he glanced at the entry and saw the two men were gone. He entered the music room and hurried to his sister's side.
"Carl, thank goodness. I am so worried I will go crazy if we do not leave right now." Tru stood and allowed her brother to help her into her wrap.
Carl decided now was not the time to tell Tru about what he had seen in the hall. It was probably nothing, and Tru's imagination was already in a frenzy over why Sophy Klienst had been lying to their brother.
"I see the countess over there. Come, Tru, let's convey our thanks and leave." Carl tucked Tru's hand into the crook of his arm, and the two young people made their way to their hostess's side.
"You are leaving? So soon? Did you not enjoy yourselves?"
"Oh, no, it was wonderful. But we told our mother that we would return by a certain hour. We must not worry her. It was a beautiful evening." Tru hated telling her new friend a lie, but it could not be helped.
"And you, Mr. Kueshner, how did you like the music?" The countess felt flustered as she looked up into the soft hazel eyes of the man who had spent much of the evening staring at her.
Carl bowed low over the countess's hand, briefly touching his lips to her fingertips. "It was the most important night of my life. When I leave this room, I will take your music with me wherever I go."
Maria Von Ott looked into the ardent eyes of Carl Kueshner and knew his words were not idle flattery. "Perhaps you would like to come to the house to practice on the Steinway, Mr. Kueshner?" Maria was dismayed to hear her own voice issuing the invitation even as something deep inside her was warning her against ever seeing this young man again.
"I will not disturb you if I come to play in the evenings?" Carl could barely conceal his excitement.
"No. I give lessons only in the daylight hours. Except for my musicales, my evenings are very quiet. Although sometimes I go to the academy at night to play with Professor Baer and my friends. I will tell the maid to let you in the music room whenever you come even if I'm not here."
"Thank you again, Countess. But now we must go." Tru hated to drag her brother away early, but she felt a deep sense of dread building inside her. She was certain one of her beloved brothers was destined for heartache unless somehow she could prevent it.
"Oh, no, dear child!" The countess's dismayed cry startled Tru. "In the excitement of the evening, I forgot again to pay you for my lovely hat. I will send my maid to your home with the payment. Give me your address so I will be sure to take care of it."
"It's all right. I can wait another few days."
Maria Von Ott shook her head, taking Tru's hands in hers. "No, no. You must always put a high value on yourself. Never forget that. If you do not value who you are and what you do, no one else will." The countess summoned her maid for pen and paper. She watched as Tru wrote down her address, handing the paper to her maid for safekeeping before saying her final good-byes. "Good night, my young friend. I look forward to seeing you soon."
Maria watched her two guests as they left the crowded music room and marveled at how much she did look forward to seeing them again, especially the young man with the soulful eyes that made her feel as nervous as a schoolgirl.
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"That was delightful. Worth every cent I paid." Randall Bently rolled off the young woman beneath him and left the bed to light a cigar. He enjoyed a cigar after pleasuring himself with a young and well-formed female, and Sophy Klienst was both. Although in truth, he preferred the winsomely elegant Miss Kueshner's looks. Ah, well, perhaps he would be able soon to get Miss Kueshner into the bed now occupied by Miss Klienst. Randall took a long, leisurely puff on the expensive cigar he held and wondered if he had time yet this evening to avail himself of the salesgirl once more.
Sophy Klienst concentrated very hard on not losing the contents of her stomach. Every time she remembered the feel of Randall Bently's flesh pressed against hers, she felt sick. She wanted to scream and run from the room to scrub herself clean. But she knew she would never be clean again—never clean enough to be with her beloved John. Sophy feared John was lost to her forever. How could she ever marry such a sweet, trusting man, knowing what she had done?
Soon, it would be over. It had to be. They would close the trap on the smug socialite, and Sophy would never again have to couple with this man she detested. Sophy hoped it would all be worth the sacrifice she had made. Tears filled her eyes, and she glanced quickly in the direction of the loathsome man still puffing away on his cigar. She could not let him see her moment of weakness. To her relief, he was too busy admiring himself in the mirror to take note of her tears. Maybe tonight she would be lucky, and he would be satisfied with one time only. But to Sophy's regret, the self-absorbed dandy turned around and began eyeing her in that way she knew only too well.
"Incidentally, Miss Klienst, I don't want you ever again to argue with me as you did this afternoon about the time or the frequency of our little visits. When I want the pleasure of your body, I expect you here. If you want my help and influence in advancing at the store, you must be completely at my disposal." Randall Bently climbed back into the bed as he spoke and began stroking the flesh of the woman huddled there. "I expect a full return for my investment. In that spirit, I believe we have time for another romp before I must leave."
Sophy clutched the sheets to keep from fleeing. Sometimes she feared she'd go mad at the sight of this man's body sliding on top of hers. She loathed the pasty white flesh concealed beneath the man's elegant clothes. But the only way they could think to get enough money together was by tricking the man who even now was violating her. Sophy had a fleeting moment of panic, wondering if she had been a fool to sacrifice so much. Then she closed her eyes and prayed it would be over soon.
Chapter 4
Tru thought the day would never end as she glanced again at the watch pinned to her bodice. She had to talk to Sophy before they went their separate ways for the evening.
"Is there something the matter, Miss Kueshner?"
Tru jumped as a familiar voice pulled her back from her musings. "No, of course not, Mr. Kenton. Why would you ask?" Tru looked up to find the floorwalker near Tru's counter with a concerned look on his face.
"You seem unusually agitated. I observed you glancing at your timepiece quite often during the last few minutes. Is there a problem?"
Tru managed a smile for the man who had always been friendly to her in his own reserved way. Tru was grateful that Mr. Kenton seemed to like her. Considering her problems with her buyer, it was comforting to know that at least one person in the giant store was on her side. "It's nothing, Mr. Kenton, really. It's just been a slow day. I find I hate days like this more than I hate busy ones."
The cool veneer of the floorwalker warmed just a bit as a hint of a smile played around his mouth. "Too quiet for you? Tomorrow is the big monthly sale. You had better savor the peace, as it is only temporary. The morning will bring you all the activity you desire."
Tru watched the floorwalker move away to help an elderly customer who needed her packages transported to a waiting carriage. Sighing, Tru wished once more that the day were at an end. But her day would not be over until she had spoken with the woman she was afraid was about to break her brother's heart.
Tru recalled the walk home from the musicale the evening before. She and Carl had spent the entire time talking over what to do with their suspicions about Sophy Klienst. Finally, they had agreed not to tell John. They both hoped that they were mistaken and that there was a simple reason for the apparent discrepancies with Sophy's message.
Carl had suggested that Tru speak to Sophy and try to find out what that explanation might be. Carl would try to talk to John at the print shop to learn just how serious their older brother was about the possibly duplicitous Miss Klienst. If John were not yet deeply in love with the strange beauty, he would not be shattered if she were misleading him. Both Tru and Carl clung to that hope. Neither of them wished to interfere in John's life unless it proved absolutely necessary.
With Tru's troubled thoughts continuing to churn in her mind, the long tense afternoon drew slowly to a close. A last-minute shopper insisted on Tru showing her every fan in the department and then left without making a purchase. By the time the irritating woman had gone, Tru was one of the last to close down her department.
Tru hurried through the nearly empty store, afraid she would miss Sophy. As Tru neared the cloakroom, she was relieved to spot Sophy Klienst just getting ready to leave. "Sophy, would you wait a minute?" As Tru drew close to the other woman, she noticed Sophy did not look particularly happy to see her.
"Aren't you in a hurry to get to the print shop today?" Sophy seemed to be suffering from a chill despite the warmth in the building. She clutched her wrap tightly around her.
"I'll hurry in a few minutes. I need to talk to you." Tru observed the nervous woman closely. "You know a most peculiar thing happened. Carl and I saw you brother last night."
Sophy's turned ashen at Tru's words. Tru knew immediately that the woman was desperately afraid of what was to come next. "He was at a musicale with your mother." Tru waited a moment for her words to sink in. "Sophy, I thought you told my brother you could not meet him last night because you were spending the evening helping your mother bake?"
Sophy's face went blank for a moment before she stammered. "I'm sure I did not say it in quite that way, Tru."
"Yes, Sophy, I remember it very clearly. Those were your exact words."
"You did not tell this to John?" Sophy's voice carried an undertone of panic as she pleaded with Tru. "You must not. Believe me, you would upset him for no reason. There is a simple explanation."
"I have not said anything yet. First, I wanted to hear from you why you would lie to my brother." Tru tried to look Sophy in the eyes, but the woman would not meet her gaze.
Sophy hesitated a long moment before answering. "It was not a lie. My mother received the invitation to attend the musicale after I talked to you. I stayed home and did the baking so my mother could go and enjoy herself."
That sounded reasonable, but every aspect of Sophy Klienst's demeanor told Tru that it was a lie. Nothing more would be gained by furthering the conversation. Tru could easily ask the countess to find out if the invitation was a last-minute one. But in her heart, Tru knew that would only confirm what she already knew to be the truth. Sophy was lying to John.
"I'm sorry, Sophy, that I questioned your honesty. I am very protective of my brother. Will you be seeing John tonight?"
A complete transformation swept over Sophy Klienst's face. The look of unhappy wariness was replaced with one of joyful anticipation. Tru could almost believe this woman loved her brother, but then why the lies?
"Yes, my brother is escorting me to the beer garden so I can meet John. I am anxious to see him. I have missed him very much."
Tru was more puzzled than before, but she was late now to help at the print shop. She had to hurry, or she would have to endure another lecture from her papa. "I have to leave, Sophy. I'll tell John you are looking forward to seeing him this evening."
"Please do tell him that because I am. You cannot begin to know how much it means to me to be free to spend time with your brother tonight. Well, good-bye." Sophy hurried away as if she had just escaped a horrible calamity.
Tru quickened her footsteps. She didn't have time now to try to figure out the mystery that was Sophy Klienst. She had to hurry to help her papa and her brothers. She started toward the large front doors, but just as she was about to step out onto the street, a hand reached out and seized her by the arm, holding her tight.
"The two of you don't fool me, you know, whispering together." Betsy Stafford dug her fingers into Tru's flesh. "But it isn't going to work for either of you. I'm going to get promoted, not you or that brainless hussy. All the shameless flaunting of your so-called charms at Mr. Hartley and the important male customers won't get either of you advanced ahead of me. I won't let you."
"You are mad. I don't know what you're talking about." Tru wretched her arm free. It took all Tru's self-control to keep from slapping the vicious woman. "I was talking to Sophy about my brother. Not that it's any of your business. I'm late, and I am walking out that door. Touch me again, and I'll forget I'm a lady." Tru could see the hate in Betsy Stafford's eyes as she turned to leave. Tru had never had anyone's hatred directed at her before, and her hands were shaking as she reached for the door.
Tru started through the double doors, but before she could complete her escape, her enemy hurled one last threat at her departing back. "No amount of influence will help you if I decide to ruin you and Sophy Klienst. Just remember that."
Tru didn't acknowledge the buyer's threatening words but, instead, hurried to where she saw Clarice waiting for her on the street. By the time Tru had reached her friend, she had managed to calm down a bit. The sight of Clarice's friendly smile helped Tru put the nightmare confrontation out of her mind and think about making it to the print shop on time.
"What took you so long? I was going to leave, but I decided to wait, since I have an errand to run in the direction of the print shop." Clarice glanced at Tru's face as she hurried to keep up with her friend. "Did something happen, Tru? You look upset."
"It's nothing I want to discuss. Don't worry." Tru forced herself to slow down. "Betsy Stafford was just being vicious again. I'm just going to ignore her. I've more pressing things to worry about."
"Such as?"
"My brothers, for one. I wish they'd all get married so I could quit fretting about them."
"I'd be glad to marry one for you."
Tru laughed at the excited look on her friend's face at the thought of marrying one of Tru's brothers. "Which one do you want to take off my hands?"
"Either Ernst or Walter will do."
"That's just the problem, Clarice. You have to decide on one or the other, or neither of them will ever take you seriously. They can't both propose to you."
"I know. But I just can't decide. They're both so handsome and wonderfully big." Clarice's heartfelt sigh was loud enough to draw the attention of several passersby.
Tru laughed in spite of her bad mood. "Well, Clarice, they will be home tomorrow. They decided to hold practice next weekend. They will be coming on the last afternoon ferry. Why don't you come to dinner? Maybe you can start trying to decide which one you really want. You simply can't marry both. It's just not done."
Clarice wrinkled her nose at Tru, stopping just short of sticking out her tongue. "I know that. But sometimes I wish I could. Thanks for the invitation, though. Are you sure your mother won't mind?"
"Not Mama. She always cooks enough for a small army when the twins come home. And she loves having a full table."
"Well, I'll be there." Clarice stopped in front of a small meat market. "Here's where I have to do my errand. I'll see you tomorrow."
Tru waved good-bye to her friend and quickened her steps as she drew closer to the print shop. She was in such a hurry she almost missed sight of the ragtag boy signaling her from a nearby corner.
"Hey! If it ain't the pretty lady. You're late as always."
Tru turned to look in the direction of the familiar voice. She smiled when she caught sight of a scrawny youngster of about nine with a bag full of newspapers almost as big as he was hanging from his shoulder.
"Tommy, I missed you Tuesday. Where were you?" Tru glanced down at the small undernourished boy who now walked at her side.
"Ah, my ma was sick. I had to look after her and the baby. Cost me half day's wages to get someone to take over my corner. Then the stupid monkey was thinking he'd be keeping all the day's profits and cutting me out altogether. I set him straight in a hurry."
Tru glanced at the boy's determined face. Life was tough for her young friend. He was too small to be fighting such battles, but there were hundreds more just like him all over the city.
"So, pretty lady, you going to try to sell that pathetic rag of yours again today?"
"So, Tommy, are you going to try to sell that lurid rag of yours today?"
"You bet! Best newsie in New York is what I am. Too bad your father and brothers can't afford to hire me. I'd increase your circulation in a week. You couldn't print enough copies of your paper."
"Good thing we can't afford you then. The boys work hard enough printing the number of copies that they do. Well, here's the print shop. Why don't you come in and say hello before you head out to your corner again?" Tru issued the same invitation she issued every Tuesday, Thursday, and Saturday. Then she waited as the full-time newsie and part-time pickpocket pretended to be considering whether to enter the office of the small paper that could hardly be thought of as a rival to the World News, the paper that was stuffed in the bag slung over the boy's shoulder.
"Well, I guess Mr. Pulitzer won't mind if I stop to say hello. I can give your brothers some tips on how to run a real newspaper."
Tru hid a smile. She never doubted her small friend would accept her invitation. Tru's father always brought extra food from home three days a week to give the hungry youngster when he stopped by. Tommy never missed one of the free offerings. Tru walked up the steps and into the office followed by the tough little survivor her family had befriended.
"Why, if it isn't Mr. Pulitzer's right-hand man." John smiled and called out a cheerful greeting from where he and Carl stood folding newspapers. "Come to help us distribute Der Wahrheit?"
"Not until you make it into a real newspaper, doc." Tommy called all the men in Tru's family by that title. It was his way of showing respect for the men who treated him with more kindness than his own drunken lout of a father ever had before the man up and drowned after an all-night binge.
"You know, Tommy, I think my father has some leftover strudel in his desk drawer. It'll just go to waste. Why don't you do him a favor and eat it?" Carl winked at Tru as he spoke. They had gone through this same ritual for almost a year and a half. "Let Tru get it for you."
Tru removed the strudel along with several thick slices of ham and bread from where her father kept them and laid them out on top of the desk. Soon, the hungry boy was gulping down the food along with the coffee that he insisted was the only liquid he could stomach. As ravenous as the boy was, though, most of the food would be saved and wrapped carefully in newspaper to be taken home and given to his mother and his baby sister.
"Where's the old doc?" Tommy glanced around the office. He missed seeing the gruff old man, who was always so kind to him under all the bluster.
"Yes. Where is Papa?" Tru had joined her brothers at the table to help fold and stuff the newspapers into the waiting bags.
John shook his head. "He's at another political meeting of the Socialist Union. He wants to cover it on the front page on Saturday." John didn't even bother to hide his disgust. "Along with, of course, a book review of the latest political diatribe to arrive from Europe."
"You know, doc, what you need is a good, juicy murder on the front page of your newspaper. Now, me, I got three murders on the front page of mine today. Yes, sir, these babies will practically sell themselves. Almost a waste of my talents! I got my call all worked out . . . Paper! Read all about it... ! Lady of the evening kills customer, then offs herself! . . . Both found naked in a pool of blood! Money in the bank, doc"
Tru didn't know whether to laugh or cry as she watched the worldly-wise youngster tuck his precious food parcels into his news bag, preparing to go. But she did know that everything the boy said was the truth about what sold newspapers.
"You know, doc, I like you. So I'll give you a tip. There's been a murder not very far from here. Going to be big news as soon as the body's discovered because it's in a little alley near the Academy of Music. High-tone place to be having a stiff turn up. People pay more attention than they do when a body shows up down where I live."
"How do you know this?" John sounded horrified and excited at the same time as he knelt and put both his hands on the newsboy's thin shoulders.
"I was taking a short cut through the alley on the way to pick up my papers to sell. I always try to get there first to make sure the papers I get don't have any sections missing. Get a reputation for selling papers with pages missing and you lose customers. I spotted the body hidden behind some trash when I had to take a leak. No one had seen it yet. The alley isn't used much, and the body was covered with garbage."
"Why didn't you tell a policeman?" Tru was appalled at the thought of the small boy finding a dead body. Even though she knew in her heart he'd probably seen worse in his short life.
"Me and New York's Finest ain't exactly on speakin' terms. Besides, if you're late, the other newsies take extra papers, and you end up short-changed. That means you lose money for the day. Can't sell papers you don't have. And I can't afford to lose money. The stiff don't care when he gets found. He's just as dead." The newsboy hoisted his bag onto his shoulder and headed in the direction of the door. Before leaving, he paused and turned back toward Tru and her brothers. "The corpse has probably been spotted by now, but maybe not. Make an interesting story for a good reporter. I'd cover it myself, but I got to sell these newspapers. Well, see you around, pretty lady. Let me know if you get that story, doc."
The door clicked shut behind the young would-be reporter as John leaped toward Carl and grabbed the papers from his brother's hands. "Come on, Carl. Drop that stuff and let's see what's going on."
"Are you out of your mind, John! Papa will go crazy as a loon if we leave and don't get the papers ready on time."
"Carl's right, John. What are you thinking?" Tru stared in disbelief at her near-frantic brother. Tru had never seen the usually steady John so worked up.
"I'm thinking I want to act like a real reporter just once in my life. Tru can keep folding the papers. We'll get back in time to distribute them. This is real news, and it's not very far from here."
"What if Tommy was wrong? Maybe it's not a murder... just someone sleeping off a drunk." Carl tried to reason with his brother, but he knew by the look on John's face, it was no use.
"We'll never know if we don't look. Now, come on and hurry before every reporter in town is there."
"All right, I'm coming. But Papa will kill us. Then there will be two more murders to report." Carl removed his apron covered with printer's ink and grabbed his coat as he followed his older brother, who was already out the door. Just before Carl closed the door behind him, he stopped and turned around. "Don't worry. It's probably nothing."
Tru stared after her brothers. She certainly hoped he was right. No one in her family had ever been near a murder and she certainly didn't want them to be near one now. Anyway, Tru knew that no matter how John might argue with their father, Johann Kueshner would never allow such a story in his beloved paper. Her brothers were on a fool's errand.
***********************************************************
"He was murdered all right." Alex knelt beside the body, continuing to examine the area around the dead man. "His throat has been cut."
Alex stared at the large pool of blood beneath the slain man's head. The blood had seeped deep into the dust of the alley. The puddle of blood made when the life had flowed out of the stranger was a rusty color and drying around the edges. The only blood that was still damp was that which was directly under the dead man's throat where a jagged red line indicated the path of the knife that had nearly separated the head from the man's torso.
"Looks like sometime late last night or early this morning." Alex glanced up to see if his fellow detective agreed.
"About right." Richard Shaw watched his partner search through the man's coat pockets. "Anything to say who he was or why he was in this part of town?"
"No. Nothing that says who he was. I did find one scrap of paper in his pocket with an address on it. It looks to me by the street and the number to be in that area of the Bowery that has a lot of seedy boarding places and cheap flophouses. Probably new to the city, else he wouldn't need to write the address down."
"Makes sense. He doesn't look too prosperous by the state of his wardrobe." Richard studied the corpse lying on its side in the blood and the dirt.
"No, but he did have a few dollars in an old money clip, and it was still in his pocket when I checked. That seems to add to the notion it wasn't a robbery." Alex nudged the body over on its back and examined the dried blood caked in the dead man's beard. "I'd say he was surprised from the rear. Whoever killed him could have grabbed him by the beard from behind and jerked his head back. One good swipe would have done the job if whoever surprised him was strong enough. The poor guy's not too big. Probably not strong enough to break away."
"So we're thinking the killer would have to be fairly tall and solid built to get the job done." Richard Shaw looked out at the street just beyond the alley's entrance and saw a crowd of curious locals had gathered and were inching closer for a better look. "We should start to question the folks from the neighborhood, but I'd bet a week's pay no one here knows him or would admit it if they did."
"What fun would our job be if it was easy? Let's get started talking to the onlookers who are here. Always a chance someone saw something strange or unusual going on." Alex stood up having gotten all the information he could from the corpse. "I already sent someone to the nearest precinct to fetch a wagon. We'll take the body down to headquarters and get that photographer we use for the rogues' gallery to take a picture of him. We might find someone down in the area of the address that was in his pocket that'll recognize him."
The two detectives prepared to exit the alley after getting a patrolman to keep a close eye on the corpse. As Alex turned to leave, his experienced eyes made one last survey of the body and spotted something he had missed. He bent down and pried open the dead man's fingers, extracting a small fragment of paper.
"Did you find something?" Alex's partner took the paper from his hand. "Doesn't look like anything useful is written on it."
"Maybe not, but he was clutching this when he died, and it looks as if the killer took the time to rip it from his hand before leaving the alley." Alex carefully placed the scrap of paper in his pocket. "It could be important. We'll look more closely at it later at headquarters. Meanwhile, let's get started talking to people."
"Sure, before they wander off." The two detectives moved toward the cluster of curious people being held at bay by two precinct cops. Richard Shaw glanced around. "I'll take the folks on the left over by those barrels."
"Suits me. I'll take those folks grouped together in the center of the alley for a start." Alex headed toward the crowd of onlookers, straining their necks to get a glimpse of the dead man.
From where they stood at the edge of the crowd, Carl and John Kueshner watched the two detectives as the men began to question people one by one. "Carl, we need to get a little closer so we can get a better look at the guy's face."
John began to inch deeper into the alley, trying to keep behind other onlookers so as not to draw attention to himself. Carl followed reluctantly, certain he would not enjoy staring down at a corpse. But John Kueshner was determined to get a closer look at the only crime scene he had ever been to so far in his career. When the two brothers had gotten as close as they were going to without attracting attention, they angled themselves so as to get the best-possible look at the body. By straining their necks, they were just able to make out the dead man's face.
Carl choked back a gasp then prayed his reaction would go unnoticed by his brother. But John had heard the slight intake of Carl's breath. Turning at the sound, John was surprised to see shock and what John thought might be a look of recognition on his younger brother's face. "Are you that upset at the sight of a corpse, or do you know this man, Carl?" As he spoke, John looked over his brother's shoulder and saw the tall, curly-haired detective's gaze move in their direction. "Get back from the body, Carl. I think one of the detectives has noticed our interest. I doubt the detectives want two reporters snooping around, and, anyway, we can't afford to spend time talking to the police, since we are already late with the papers."
Carl was grateful his brother couldn't question him further. The two young men tried to appear casual as they retreated to the back of the crowd that had grown steadily as the minutes had passed. Carl had to think how to answer John's question without telling him the truth. Carl did recognize the dead man. It was the same man he had seen at the musicale just the evening before arguing with Otto Klienst.
Just as Carl Kueshner was remembering the angry confrontation between Otto Klienst and the dead man, he felt his brother's elbow strike him sharply in the ribs. "Carl,"—John was whispering so as not to be overheard by anyone nearby—"I could swear that's Sophy's brother, Otto, over there near the corner of that building." Carl turned to look in the direction his brother was now inclining his head. "I wonder what he's doing here."
Carl shrugged his shoulders, all the while fighting a sinking feeling in the pit of his stomach. All he could think about was the image of the lovely countess the evening before deeply engaged in conversation with Otto Klienst and his strange mother. What possible connection could the countess have with those two people or with the man now lying dead in the alley behind them? Carl made the quick decision to share what he knew with no one until he had time to think about what he had seen. When Carl looked again in Otto Klienst's direction, the man had disappeared.
"Carl, what do you know about the man in the alley?" John peered intently into Carl's face.
"I thought I had seen him before, but I was mistaken." Carl hated keeping the truth from his brother. "I think it was just the shock of seeing someone who had been murdered. I really think we ought to return to the shop now. Tru will be worried, and Papa will be angry if the papers are late getting out." Carl started to head away from the crowd of onlookers and in the direction of the print shop. To his relief, John fell into step beside him.
Alex Marshall looked over and noticed the two young men he had seen earlier staring at the corpse were now walking hurriedly away down the street. Still trapped in the middle of listening to the ramblings of an elderly woman who had nothing to contribute to his investigation, Alex was having a hard time keeping a polite smile on his face as he watched the two blond men he had wanted to question disappear around a corner.
Finally able to break free from the elderly woman who insisted a rival fruit seller was capable of committing the murder, since he regularly undercut the older woman's prices, Alex's thoughts returned to the two young men he had seen looking so intently at the corpse. It might have been just morbid curiosity. Still, Alex couldn't shake the feeling he had seen a look of stunned recognition on one of the men's faces. He needed to question the two men, but, first, he would have to find out their identities and where they lived.
Alex walked over to the area of the crowd where the two men had been standing. He spotted an elderly gentleman he thought he remembered being close to them. "Excuse me, sir, did you notice two young men who resembled each other standing near you just a few minutes ago?" Alex smiled the broad grin he used to put people at ease. "Might have been brothers."
"Sure thing, young fella. You mean those two blondish men. I don't miss much, you know. Pretty sharp old bird, I am."
Alex's grin deepened. "I'll bet you are. Did you ever see them before? Any idea who they are?"
"Yep, seen them plenty of times. Their father's one of those political loonies from somewheres in Europe."
"They're immigrants?" Alex remembered the long beard and severe old-fashioned clothes of the victim and was even more certain he needed to find the two young men and question them.
"Yep, but not that new, they've been 'round fer a while, I think. But I heard tell the old man came from somewheres over in that there Europe. Could be, I heard, they might be German? Everyone 'round these parts knows he prints some crazy newspaper full of crazy ideas. Trying to get folks stirred up, you know. I seen them boys out selling it on street corners sometimes with that pretty sister of theirs. Got some foreign name the paper does."
"Whereabouts do they print this newspaper? Do you know?"
"Coupl'a miles toward Second Avenue, maybe? Then I'm thinkin' you turn right, and it's a little piece further. Seems I recall a sign hanging on a shop near Second and Ninth with a foreign name written on it that looked kinda German-like. Those crazy foreigners do something? They kill this guy? Wouldn't surprise me none. Always spouting off about unions and such. Union men can't be trusted - none of them!"
"Far as I know, they haven't done a thing. Just want to ask them a few questions like everyone else. You didn't happen to know the dead man, did you?"
"No, siree, bob. Never seen him before around these streets. Kinda looks like a lot of them foreigners that keep comin' over here, bringing their crazy ideas. Don't need any more crazy ideas. Ain't that so, young fella?"
"You might be right. Well, thanks for your help. I'd better be going. Need to get the body out of the alley. You don't happen to remember the name of that newspaper, do you?"
"Nah. First part's short, I think. Dir or De or some such nonsense. Thought I overheard someone say once that the name meant 'the truth' . . . the truth, now don't that take the cake. Crazy foreigners."
Alex smiled and thanked the man again before heading over to where Richard Shaw stood talking to a precinct officer, making arrangements to have the body carried by wagon up to headquarters. Alex would take one more look, hoping to find more of the torn paper the dead man had been clutching. Then he'd let Richard go alone to see that the corpse got photographed while he went looking for the two young men who had slipped quietly away before he could question them. If they did indeed print a newspaper called the Truth, maybe they would be happy to help Alex solve this murder. But, somehow, he doubted it.
***********************************************************
Tru paced back and forth in the small office. The newspapers had long since been folded and packed in the bags but, there was still no sign of her brothers. She had just reached the decision to leave by herself to start delivering the papers when the door opened. With a sigh of relief, she saw Carl and John enter the shop.
"It's about time you got here."
"I'm sorry, but when we got there, the police were already investigating." John's voice still carried a trace of the excitement he had felt being at a real crime scene. "We had to be careful and figure out a way to get close enough to observe everything we could without being obvious."
"I knew you shouldn't go near such a horrible place." Tru was shocked that John seemed energized by what he saw. She turned to Carl, who, in contrast, looked pale and drained from the experience. "Carl, what's wrong?"
"Nothing! It just wasn't pleasant seeing a man lying in the dirt with his throat slit open." Carl picked up a bag of papers and headed in the direction of the door.
"My god! His throat slit. It wasn't someone we know?"
"Of course not! How could I know someone who was murdered!" Carl shouted at his sister, who looked startled at the sound of her gentle brother yelling in her direction. "Look, Tru, I'm sorry if I sounded angry, but we are very late now delivering the papers, and I am worried about Papa's reaction if he returns from his meeting and the papers are not gone. The murder has nothing to do with us." Carl was halfway out the door.
"Thank goodness for that." Tru grabbed her bag and followed her two brothers.
Once the three young people were outside the print shop, John stopped long enough to take stock of the situation. "Carl, you come with me to deliver papers to the subscribers. It is most important that all their papers are delivered on time. Tru, I hate to ask, but would you go to where Carl usually stands near Tammany Hall? It's a busier corner, and you'll be able to get rid of more papers in a shorter time."
Tru hated standing on Fourteenth Street. She was more afraid of seeing some of her customers there than in her usual spot. But John was right. "All right, let's hurry and go before Papa returns from his meeting and sees how late we are." Tru was as anxious as her brothers to avoid a disapproving lecture. She quickly shouldered her heavy bag and started off at a brisk pace in the direction of Fourteenth Street.
Tru traveled up Second Avenue and turned onto Fourteenth Street heading west. As she crossed Third Avenue, she caught sight of a familiar small figure standing in front of Tammany Hall; at the same time, a colorful cry reached her ears.
"Get your World News here! Prostitute murders customer and mutilates body. Just two cents to find out what parts she cut off. Details in the World News!"
Tru stopped short of her friend's territory and began to quietly ask passersby if they would be interested in buying her small newspaper. When Tommy saw who it was trying to sell papers close to his spot, he waved and walked over to join Tru.
"Pretty lady, what are you doing here? I know your brother works this corner, but it ain't no place for you—too many other newsies. They can be tough on those they think are invading their turf, 'specially if the newcomer doesn't look strong enough to put up a fight."
"I can take care of myself, Tommy. But I hope I'm not hurting your sales."
The small boy laughed and opened his bag to show Tru its empty interior. "Just sold the last one. I told you I never have trouble on a day with a gory headline. Here, let me help you."
Tru watched in amazement as the pint-size whirlwind grabbed several papers and began accosting the people hurrying by. Within minutes, he had sold more papers than Tru usually sold in an hour. Mostly, Tru suspected, because people wanted to be rid of the persistent urchin.
After a few minutes of Tommy's dynamic salesmanship, Tru noticed some other boys with bags on their shoulders were beginning to look in the direction of Tru and her temporary helper. Before Tru could inform Tommy of the unfriendly scrutiny, several of the toughest- and biggest-looking boys approached them.
"What'ya doing selling that rag, small fry? That ain't your paper." A very large boy grabbed Tommy by the arm. "Ain't there enough of us tryin' to sell on this corner?"
"Ah, this paper ain't going to hurt your sales." Tommy yanked his arm away from the bigger boy. "You're all done for the day anyways. Your bags are empty. Get lost. Mr. Hearst got his for the day."
"Mine ain't empty. Neither is Jake's." A fierce-looking lad whose nose looked as if it had already been broken a couple of times in his short life poked the smaller boy in the chest. "Besides, we don't want any paper ain't true American sold this close to the hall." When Tru stepped up to chase the boys away, one of the bullies grabbed her bag and threw it to the sidewalk, scattering her papers all over the ground.
"Besides, ain't no females should be allowed to sell papers. Ain't right." The boy who had stolen Tru's bag grinned, revealing several empty spaces where teeth should have been.
"Let go of me, you young twit! Those are my family's papers, and you are a bunch of cowards trying to push around a woman and a small boy." Tru's temper was beginning to get the best of her. She tried to squirm loose, ignoring her sense of dignity and the curious stares of the people trying to slip past the strange scene. "I'm warning you, I'm about to lose my temper."
"Then I'd advise you boys to let the lady go. I've seen her temper in action, and it's not a pretty sight."
A tingle ran down Tru's spine. There was something oddly familiar about the pleasant male voice coming from behind her. Even without being able to see him, Tru had an uneasy feeling she knew who had come to her rescue.
"All right, boys, you've had your fun." This time, the voice was sharper. "I'd advise you to do as I say and let go of the lady and her friend."
The hands holding Tru's arms pinned to her side dropped. Tru brushed off her dress and then slowly turned around. As luck would have it, her rescuer did indeed wear a now familiar grin. "I could have handled this myself, you know." Tru's chin inched upward. "I just had to break free was all."
Alex laughed. Then he turned and fixed Tru's assailants with a hard stare. "Now off with you boys. And don't let me catch you bothering the lady or her friend again. Understand?"
The group of boys did indeed understand. They gave one last look at the large man, grabbed their bags from the ground, and ran away down the street. Tru was almost sorry to see them go, leaving her standing uncomfortably on the sidewalk with the amused detective.
Tommy's eyes followed the departing newsies, wishing he, too, could run, since his street sense told him the tall man who now stood smiling down at the pretty lady was none other than one of New York's Finest. Being careful not to draw attention to himself, Tommy slipped quietly out of sight behind Tru's skirts.
"So there seems to be no end to your talents, Miss Kueshner... hatmaker, ledge-sitter, saleslady, and now newsboy. I'm mightily impressed." The mock seriousness of the detective's face was harder to take than his customary grin. To her dismay, Tru found herself wrinkling up her nose and making a face that would make a ten-year-old proud.
The detective merely grinned at her response to his words. "Aren't you even going to say thank-you for all my help?"
Before Tru could think of an appropriate retort, another familiar voice reached her ears. "Well, if my daughter does not say thank you, I do. Thank you, sir, for coming to the aid of my precious daughter. I am grateful."
"Papa, what are you doing here?" Tru whirled around to see the puzzled face of her father.
"I am done at my meeting, so I am heading to the print shop. As I come around the corner, I see my daughter accosted by young thugs. Thank goodness this gentleman helped you. And you do not thank him? This I do not understand. Where are your manners, liebling?"
"I would have, Papa, in another minute." Tru could not look at the detective, knowing he was enjoying her predicament immensely.
"Sir, I must thank you properly." Johann Kueshner studied the tall man closely. "And I know my wife will also want to convey her gratitude for your assistance to our daughter and our young friend. Would you be so kind as to accept my invitation to supper tomorrow evening?"
"Oh, Papa, I don't think the busy detec... I mean, this gentleman has far more important things to do than to come to our house for supper, don't you?" Tru tried to catch Alex Marshall's attention, but, for once, he wouldn't look at her.
Alex tried hard to keep from laughing at Tru's desperate attempt to get him to turn down the invitation. "I'd like that very much, sir. I'm sure Mrs. Kueshner is a very good cook." Alex didn't realize the mistake he'd made until he saw a look of dismay wash over Tru's face.
"I'm confused, sir. Do you know my daughter already?"
"No, no. Not really." Alex thought fast. "I met her at Hearns once when I was shopping for a fan. I was completely at a loss as to how to pick out a fan, and your daughter was most helpful, so I asked her name so I could tell the store manager of her first-rate service. And I have a very good head for remembering names and faces." Alex caught Tru's look of relief out of the corner of his eye. "Let me introduce myself, sir. My name is Alex Marshall."
"I am Johann Kueshner." Tru's father offered his hand in a firm handshake. "It is my family's paper my daughter was selling with the help of our young friend—which reminds me." Tru's father looked around for the small newsboy, finally spotting him standing quietly behind Tru. "You have not said a word, Tommy. Are you hurt?"
"I'm fine, doc. But I gotta get going home. I gotta look after me ma and baby sister. They ain't exactly in the pink yet." Tommy decided now was a good time to make his escape. "See you on Saturday." The newsboy took off running down Fourteenth Street, happy to be out of the presence of the large detective.
"This is the address of our home"—Tru's father finished writing on a piece of paper he had pulled from his pocket and handed it to Alex—"and the time we will expect you. Come, daughter, we must return to the print shop to await your brothers. Good day, Mr. Marshall. We will see you tomorrow evening. Auf wiedersehen." Tru's father started off in the direction of the print shop.
"I'm coming, Papa. You go ahead. I'll catch up. I just want to say thank-you to Mr. Marshall myself." Tru waited until her father was out of hearing distance and then turned, snarling, in the direction of Alex. "Well, this is just a fine fix. Do you think you can manage to share a meal with my family without letting something slip about how we really met?"
Alex smiled serenely back at Tru. "Now, Miss Kueshner, you forget I'm a detective. I'm used to keeping secrets. Don't think another thing of it."
Tru snorted by way of reply. She might have been reassured if not for the devilish look on his face. After all, he had been quick with a lie to cover his slip of the tongue. So with one backward glance of derision, Tru took off marching down the street, wondering what disaster lay ahead of her the next evening. Her mind filled with scenes of the horror that would result when the playful detective sat down for an entire evening with her family. A loud groan escaped her lips, startling a priest who happened to be passing by at just that moment.
Alex's smiled faded as he stood staring after the retreating figure of the young woman who was becoming stuck in his head like an unwanted tune. He stooped down to pick up a forgotten paper crumbled at his feet. He did not read German, but he had no doubt what the banner at the top of the page said. The two blond men he was searching for must be Tru's brothers.
Alex frowned. He was torn between pleasure at the stroke of luck that had dropped the two men he sought into his lap and uneasiness over their relationship with the exasperating creature he was beginning to think about more than was wise. Tomorrow he would try to find out the reason these particular young men had been so interested in this afternoon's murder. Alex just hoped the truth didn't come at too high a price.
Chapter 5
"So who is this young man of yours, liebling?"
"Mama, he is not my young man. He is a stranger who simply helped Tommy and I when those rude newsboys scattered my papers. And I'm sure I would have been able to chase them off myself in another minute or two. I don't know why Papa insisted on inviting him to dinner." Tru began stirring the batter more vigorously than was necessary as she helped her mother prepare a strudel for this evening's company.
"My dear daughter, you are getting flour all over yourself."
It had been a long, hard day at Hearns. A sale had been going on, and flocks of customers had traipsed through the front doors all day. Tru had been too tired when she arrived home to go upstairs and change her wilted outfit. The nice neat knot that had contained her abundant curls at the beginning of the day had long since fallen down to one side of her face. The heat of the busy kitchen didn't help matters, and Tru brushed a drooping curl from her forehead, leaving a streak of flour in its wake.
"Don't you think you should change, darling? Your friend will be here soon. Your father mentioned he was a rather good-looking man. You don't want to resemble a collapsed strudel when he comes."
"Mama, he's not supposed to be here for another hour. The boys will be home soon, and they can entertain him when he comes. Besides, how many times must I say he is Papa's guest, not mine? I am certainly not interested in impressing Mr. Marshall."
Tru reached over to take a bite of dough from a nearby bowl, but her mother quickly snatched the bowl away. "You leave my kitchen now, child. If you help any more, there will be no dessert for our guests. Anyway, everything is almost done. You are sure that Clarice is coming too?" As she talked, Martha Kueshner flew around her kitchen, putting the last touches on the meal she was preparing.
"Mama, you know she wouldn't miss a chance to be near the twins." Tru laughed as she removed her apron, preparing to leave the warm kitchen that smelled of veal roast and onion tarts.
"Ach, that is good. She is a dear little thing. I hope someday one of those thickheaded sons of mine will open their eyes and see what a nice little wife she would make. Now run along upstairs. If your brothers come home and see you like this, there will be no end to their teasing." No sooner had those words escaped Martha Kueshner's mouth than there was a knock at the front door.
"I'll get it, Mama. It is probably Clarice coming early to make sure every one of her hairs is in place before the twins arrive." Tru hurried to the door and threw it open, eager to greet her friend. Instead, she found herself staring up into the amused face of Alex Marshall.
"Little bird, your beak is open." Alex reached out and gently pushed Tru's chin up until her lips met in a frown. "Aren't you thrilled to see me?"
"You're early! That's rude! But I should have expected it from you." Tru remained squarely in the doorway, showing no signs of moving to allow the grinning detective to enter her home.
"Your father said six thirty. It's written down on this paper he gave me, so I can prove it."
"Oh, Papa knows better." Tru stomped her foot. "He should have said later. The boys and he aren't even home yet." Tru stepped aside reluctantly and gestured for Alex to enter. "Mama is in the kitchen. Won't you come into the parlor?"
Alex tried not to laugh. Tru's words carried all the enthusiasm of a child being forced to take a tonic. "How can I refuse such a gracious invitation?"
Tru led Alex into a small front parlor. The room was warm and inviting in a plain and simple way. The sun's light was just beginning to fade, and as it filtered through the lace curtains hanging on the windows, it made delicate patterns on the polished wood floor. A well-used piano stood against the opposite wall with a number of chairs arranged nearby. By the arrangement of the furniture, Alex could tell the piano was the center of activity in the room. Alex watched as Tru moved hurriedly around the room, lighting the gas lamps. When she finished her task and came toward him, he took note of her disheveled appearance.
"Were you trying to bake yourself into a pie?" Alex smiled as he removed a handkerchief from his breast pocket and used it to gently wipe the flour from her face. "Whatever is baking in the kitchen, it smells wonderful. Definitely worth a little flour on the face, I'm sure."
"I'm not baking anything. Mama is the one doing the cooking. I'm a terrible cook." Tru thrust her chin defiantly in the air. "I'm a terrible housekeeper too."
"That's all right. I'm sure you are useful in other ways." Alex almost choked when he realized how his words might sound to the inexperienced young woman standing in front of him. He returned his handkerchief to his pocket and hoped the light was too dim for Tru to notice the uncharacteristic color flooding his face. He shifted uncomfortably, looking for something to change the subject. "That's an interesting old trunk."
"Yes." Tru sat on the trunk and patted it lovingly. "Mama put the trunk in the parlor. She said she knew it was a strange place for a trunk, but she wanted to see it every day to remind her of their old home in Baden and of the journey they took to come to this new country. She was sitting on this trunk as the ship approached New York harbor, holding my brother Carl, who was just a baby."
"Why did they come here?" Alex hoped that knowing more about this obviously close knit family would give him a better feel for the young men he planned to discreetly question later in the evening. On the long walk to Tru's home, Alex's mind had been in turmoil over the coming dinner. The early spring night was warmer than usual for the time of year, and he had longed to just enjoy the beautiful evening and the dinner that lay ahead. Unfortunately, this evening was not for pleasure. It was a possible means to try to solve a murder.
As he had drawn near to the address written down in Johann Kueshner's strong, bold hand, Alex had found himself in a neighborhood of modest but pleasant row houses. The neat and tidy homes spoke of occupants who had been in this country long enough to begin to prosper through hard work and determination unlike the unfortunate people Alex usually dealt with in his line of work.
Alex's mind was drawn back to the present when Tru picked up a daguerreotype of a distinguished-looking man and began to speak in a soft voice while studying the picture. "Papa was a young man in his twenties during the European Revolution of 1848. There was no German Empire then, just a loose confederation of states. My grandfather was a professor who believed passionately in the ideas that caused many in Europe to rise up to fight for change. He taught Papa to believe in those ideas too. But the different groups in the different countries were not united enough to prevail, and, within a year, the revolution collapsed. After that, it was very hard for my grandfather to get a job at a university, and his health failed. My papa watched his father sicken and die, and he came to believe the Old World could not change enough to give the people the power over their lives that they deserved. After he met Mama, all he wanted was to go to America. When Carl was born in 1856, they finally had enough money to make my father's dream come true, and they sailed to New York with John, who was only five, and Carl, who was barely six months old. The trip was very hard on Mama, but Papa said that here, new ideas could grow and ordinary men with dreams could make a difference."
"And who plays the piano?" Alex found he wanted to know everything he could about Tru and her family and not just because of his suspicions about her brothers' presence at his crime scene. "Do you play?"
Tru laughed, and the sound made Alex heart skip a beat. "My heavens, no. My mother studied music in the old country. She met my father at university. She also studied painting."
"Your mother is a woman of many talents just like her daughter."
"Thank you, Mr. Marshall. I am most flattered that you appreciate my talents and my daughter."
Alex turned around to face a handsome older woman who was just removing an apron from around her waist. Alex bowed in her direction. "Only a fool fails to appreciate beauty when it's placed in his path."
"And you are most definitely no fool, Mr. Marshall." Martha Kueshner smiled as she assessed the tall young man standing by her flustered daughter.
"Mama, this is Papa's guest." Tru failed to catch the grin that appeared on Alex's face at her carefully worded introduction. "Mr. Marshall, this is my mother, Martha Kueshner."
"We are proud to have you as a guest in our home, Mr. Marshall. I am grateful for your help to my daughter. There are some young men in New York who do not remember their manners these days. My Gertrude is fortunate you came along when you did." Tru's mother had liked the tall man with the warm smile as soon as she had set eyes on him.
"It was nothing really, Mrs. Kueshner." Alex kept a straight face, but his eyes sparkled with repressed laughter. "I'm sure your daughter would have been able to free herself in another moment or two. At least that's what she told me right after the incident."
Martha Kueshner held back a laugh of her own when her daughter snorted at Alex Marshall's gentle barb. But at that moment, the three people in the parlor were distracted by a loud commotion as the front door burst open, letting in a tidal wave of boisterous men. They swept into the parlor, filling the room with the sound of male laughter.
Alex stood aside quietly observing the scene as two young men with powerful builds and identical faces grabbed Martha Kueshner and lifted her off her feet, planting a kiss on each cheek. "Mama, look what we found on the doorstep—a mouse." One of the good-looking twins reached back behind the other one and pulled a small redhead forward, thrusting her into the center of the room. Alex immediately recognized the young woman as Tru's partner in crime from the hotel. Then through the door came another pair of young men immediately followed by Johann Kueshner himself. Alex recognized the second pair as the same two men who had been at his crime scene. As Alex studied the people in the room, the thought crossed his mind that the men now crowding the small parlor were a formidable group.
"Clarice, it is so nice to see you." Tru's mother had extracted herself from the embrace of her two youngest sons and hugged the little redhead warmly.
"Thank you for having me, Mrs. Kueshner." Clarice had barely gotten the words from her mouth when one of the twins pulled her into a bear hug and swung her around, endangering several nearby lamps.
"Why would Mama care if you came to dinner, mouse? You don't eat enough to feed a bird."
"Walter, put Clarice down. Thank goodness someone eats less than you and Ernst do. When you are both home, I am in constant danger of running out of food." The entire family laughed at the absurd notion that Martha Kueshner would ever be caught short of food. Even Tru laughed, although until that moment, she had been uncharacteristically quiet, wishing to avoid notice.
Alex was also trying to melt into the background for a while longer to continue in his role as observer. He took note of the obvious closeness that existed between these family members. Alex also sensed an uneasiness in the two older brothers as he waited quietly to be introduced to all of the young Kueshner men. He dreaded having to mix business with pleasure during what could otherwise have been just a very enjoyable chance to get to know the spirited Miss Kueshner better. But Tru's two older brothers had behaved suspiciously at his crime scene, and he had to look at them as possible witnesses or worse in his recent murder.
Just then, Johann Kueshner noticed Alex standing behind Tru and stepped forward to take charge of the pleasant chaos in the room. "Boys!" At the sound of their father's voice, the four young men immediately gave the older man their complete attention. "I have invited someone to dine with us this evening. He rendered your sister a service yesterday by rescuing her from some ill-mannered thugs. This is Mr. Alexander Marshall."
A slight gasp escaped Clarice's lips as she recognized the detective, but a lethal glance from Tru silenced her. Tru had wanted to warn Clarice about the detective's presence this evening, but the store had been so busy she had never had a chance, especially when Clarice rushed out at the end of the workday without even a good-bye to primp before dinner.
Tru's brothers remained silent as they assessed Alex for several moments. Alex felt as if he were facing down one of the notorious Bowery gangs as he waited for a greeting from the men. Finally, Alex decided to take matters into his own hands, flashing a grin at the imposing wall of male flesh. "The only service I rendered was to the thugs, who made the mistake of accosting your sister. She probably would have injured them severely in another minute or two."
The ice broke as the twins laughed and stepped forward to pump Alex's arm up and down. Tru's other brothers, however, remained in the background, though they finally acknowledged Alex's presence with a smile and a nod. Both John and Carl recognized the detective from the alley the day before and wondered if his visit this evening was a coincidence. Or had he managed to find out their identities and took the opportunity of their sister's assault to be in a position to question them?
Alex Marshall decided at that moment to use a strategy of limited honesty. Extracting himself from one of the twins' hearty handshakes, he turned to the remaining brothers. "We've seen each other before, I believe, but we've never met." Alex held out his hand to John. "I'm Alex Marshall. I saw you and your brother at the murder scene yesterday."
John accepted the proffered hand and shook it. "Yes, Carl and I saw you and the gentleman with you when we arrived in the alley."
"What murder? What are you boys talking about?" Martha Kueshner was visibly upset at the mention of her sons and a violent crime in the same sentence. "What would you boys and Mr. Marshall be doing at a murder? Who was murdered?"
"Mama, calm down. Everything is all right." Tru felt helpless to keep her brothers out of the trouble she feared they were in, especially when she saw the anger and disappointment spreading across her father's face. She could only hope that the presence of guests in the home would help contain her father's anger and shorten the lecture sure to come.
"I'm a detective in the New York Police, Mrs. Kueshner. That's why I was there." Alex smiled at the worried woman.
"I am surprised, young man, but it does explain why you were able to get rid of those young ruffians so quickly." Johann Kueshner turned his frown in the direction of his oldest two sons. "But that does not explain what my boys were doing there. Why were you at a murder scene?"
"Tommy told us about a body that he had seen in an alley yesterday morning." John met his father's gaze squarely. "I know Tommy is just a young boy, but he is serious about the newspaper business, so I believed him, and I made the decision to investigate. I thought there might be a story in it."
"For our paper! That would not be likely!" Johann Kueshner's frown deepened. "I'm surprised at you both."
"We were just curious." As he spoke, Carl was very aware of the detective standing just a few feet away, listening intently to his and John's explanation.
"I thought for once we might act like real reporters." John felt a stab of guilt as soon as he spoke. He knew his mother would be hurt if the evening turned sour in front of their guests.
"We do not print stories of the terrible brutality men are capable of. It only encourages the violence that plagues this city." Johan Kueshner appeared ready to continue his lecture. But before he could utter another word, a thin voice piped up from between the Kueshner twins.
"All this talk of dead people is very unpleasant." Clarice stared up adoringly at one twin and then the other. "I would much rather listen to stories about the football team at Rutgers. Football is so very fascinating."
Tru burst out laughing. Clarice knew nothing about football and expressed no interest in learning anything about it except when in the presence of the twins. But Tru was glad that Clarice's words had managed to derail the argument between her father and her brothers.
"All right, mouse, we'll show you a new play we've thought up to try at spring practice next week." One twin picked the delighted Clarice up by her waist and positioned her in front of him, while the other twin crouched down and prepared to charge the eager redhead.
"Nein, nein. Not in my parlor." Martha Kueshner placed herself between Clarice and her would-be tackler, pointing firmly in the direction of the dining room. "You will stop this nonsense immediately and take your places at the dinner table. We must now eat, or my supper will be ruined." To Clarice's dismay, the twins meekly acquiesced to their mother's wishes, and the opportunity to be swept up into the arms of one of the twins was over.
"Better luck next time, Clarice," Tru whispered to her friend as they started to leave the parlor. "Do you think you could be any more obvious?"
"Me! Look who's talking. What's the good-looking detective doing here? How did you arrange to have him 'rescue' you?" Before Tru could issue an indignant denial, Clarice was hurried away by the twins, who happily escorted their tiny worshiper to her seat at the table and arranged themselves on either side of her.
"She looks like a rowboat between two steamships," Alex murmured close to Tru's ear as he took her arm before she could protest. He walked her to a seat at the table and then, much to her dismay, seated himself beside her.
Once seated at the dining table, the tense exchange between father and sons seemed to be forgotten as the conversation took on a pleasant tone of bantering between the Kueshner siblings, who were obviously close for all their teasing. The meal served by Mrs. Kueshner was a hearty one and required all the attention of the people at the table, which led to some long, comfortable stretches of silence as onion tarts, a veal roast, and cabbage dumplings were consumed and pronounced delicious by Martha Kueshner's appreciative audience.
While he ate, Alex watched the close-knit family as they exchanged views on a variety of subjects. It was clear they had a lively interest in the world around them and that their father encouraged that interest. It was also clear that Johann Kueshner was a man of unbending principles. The idealism of the man was something that Alex could admire, but he wondered how far a man like that would go for his ideals.
Both John and Carl joined in the chatter around the table, yet Alex sensed that they were more subdued than they would normally have been. As the meal progressed, Alex found he was particularly interested in Carl Kueshner. He was quieter than his siblings. But when he did speak, Alex was impressed with his intelligence and sense of humor. There also seemed an extra closeness between this particular brother and Tru. John was obviously the most serious of the Kueshner's children, but Alex sensed a restlessness and slight air of disappointment that was absent from the other brothers. It didn't take long for the detective to realize that the two Kueshner twins were interested solely in football and teasing Tru's redheaded friend, as one would expect from young men in college. Much of the laughter at the table was caused by their constant attempts to make Clarice either giggle or blush. Martha Kueshner was quiet during the meal, but it was clear that the reason for her silence was her profound enjoyment of her family and having them all together in one place.
The wonderful meal and pleasant company made Alex even sorrier that he still remained suspicious of Carl and John Kueshner. He would have preferred to end the evening with complete confidence that they had nothing to do with the murder of the unknown stranger. But something in the men's manner told him he needed to investigate further. Alex was also intrigued with the story of how the Kueshner brothers had come to know about the body. He would have to find a way to talk to the young newsboy who had been the source of their information.
"Tru, I almost forgot." All eyes at the table turned toward Martha Kueshner as she broke her silence. "A woman came here today with money to pay you for a hat."
"Oh, that must have been the countess's maid. The countess said she would send one of her servants over with payment." Tru was anxious to avoid her brothers' traditional teasing whenever the subject of her hats came up, so she hoped her mother would not pursue the subject.
"No, Tru, it was not the maid. It was the countess herself." Tru's mother waited for everyone's surprise to subside before continuing her story. "I was baking this afternoon when a knock came. When I answered the door, there was this woman dressed in very fashionable clothes standing on the stoop. You can imagine my surprise. But I invited her in, and we had a pleasant conversation."
Tru's father stopped eating and slammed his fork onto the table. "Humph! What do you have to discuss with a woman such as that! Send her servants around with payment, indeed."
Tru frowned at her father's words, certain he was about to launch into one of his political tirades. But before Tru could try to stop him, her mother continued.
"You are wrong, Johann. She was very charming and did not put on airs. She seemed sweet and intelligent. We discussed fashion for some time. She is very interested in Tru's dream of opening a millinery shop. She is quite lovely, and it will not hurt to have such a woman wear one of Tru's hats out in public. She is also a musician, and we spoke about music. She is interested in Carl's music. She told me she invited Carl to use her piano and to accompany her to the Academy of Music. Is that right, Carl?"
All eyes turned in Carl's direction as everyone waited for his reply. The slender man shifted uncomfortably, loathed to discuss any aspect of the beautiful countess and her offer. His mind was full of troubling questions about Maria Von Ott at the same time that he was more intrigued by the countess than he had ever been by any woman. "The countess was most gracious to Tru and myself. She has a Steinway that I would enjoy playing, I'm sure."
Alex sensed the strong current of emotion underlying Carl's simple words and wondered what thoughts were hidden behind his shuttered expression. Something about the countess was troubling Carl, that much was clear.
"And why should Carl need to play on this woman's piano when he has a perfectly good one here?" Johann Kueshner's disapproving gaze fixed on his second son. "Carl needs no person's charity."
"Perhaps it is not charity he needs but her interest and encouragement." John's deliberate words reverberated around the table like a drumroll.
Alex watched the exchange, surprised to realize that as close and loving as these people were, there were conflicts that simmered beneath the surface. His own experience with family had been different. After his father's death, there had been only he and his mother. His mother had been a frail woman, and Alex had taken on the role of man of the house despite his tender years. He had tried hard to take the place of his father, but his mother never recovered from the loss of her husband. She died a few years later when Alex was thirteen. The complex feelings that both bind and separate a large family were outside his experience. Alex wasn't sure he could be comfortable with so large and emotional a group of people.
"At least she is willing to wear one of the noodle's monstrosities. That takes courage."
Alex fought his laughter as Tru threw a look that was hot enough to burn flesh across the table at one of the twins. Alex still could not tell the twins apart, so he couldn't be certain which man was in imminent danger of losing his life.
"Little brother, I will hit you over the head with Mama's best frying pan if you call me noodle again tonight."
"Oh, Tru, Walter was just being funny. Weren't you, Walter?" Clarice actually batted her eyes at the big man beside her, bringing forth the first genuine smile of the evening on Carl's face.
"Clarice, if Walter or Ernst said the moon was made of cottage cheese, you'd agree." Carl laughed fondly at his sister's friend. "By the way, are you sure that it is Walter you're defending?"
"Thank you very much, Carl Kueshner, but neither of them would ever say anything that silly. And I think I know which one is Walter and which is Ernst." Clarice tried to sound confident, but everyone at the table could see the doubt beginning to form as she looked from one twin to the other. When the two men sat there giving the pathetic creature no clue as to which was which, the entire table burst out laughing.
The rest of the meal passed pleasantly. Alex knew he would have to find a way to learn more of Carl Kueshner's knowledge, if any, of the murder in the alley, but, for now, he would bide his time. He wanted more information, such as the dead man's identity, before he really questioned Tru's brothers. By the time the meal was almost over, Alex had formulated a plan as to how to proceed with his investigation. And part of that plan was questioning Tru's small friend, the young newsboy.
Remembering the young newsboy's manner yesterday when Alex had come to his and Tru's rescue, Alex doubted the lad would talk willingly to one of New York's Finest. It might be necessary to gain the boy's trust through his friendship with Tru and her family. Alex glanced at the young woman beside him, who looked as delectable as the dessert her mother was serving at that very moment. He hoped he would not have to use Tru to get close to her young friend because Alex was sure Tru would not take kindly to being used that way.
"Mama, you made Grossmader's special bread pudding." Tru tried to smile as her mother nodded and set the dessert down in front of her. But Tru suspected she wouldn't enjoy the treat as much as she usually did. She had been so nervous all evening sitting next to the surprisingly restrained Alex Marshall that everything she ate had been tasteless.
"Mama, will you forgive me if I do not stay any longer?" John had stood up before his mother could put a plate in front of him. "I have to meet Sophy. Her brother is escorting her to the Brunswick Café so that I can spend time with her."
"Your mother went to all this trouble, and we have guests. You were not raised to display such rudeness." Tru's father's disapproval of his eldest son's behavior was plain for everyone at the table to see.
"Johann, our son works hard and has little time to spend with his young woman. I'm sure Mr. Marshall and Clarice will understand." Martha Kueshner's gentle voice pleaded their son's case to her husband.
"Your mother is too indulgent, but I will respect her wishes." Tru's father seldom refused his beloved wife. "But next time, we would like Sophy to come to dinner. I think it is time your mother and I meet this young woman who causes you to forget your manners."
"I want you to meet her, Papa. But Sophy is shy, and her mother relies on her a great deal. I will try to get her to come soon, I promise. It was a pleasure meeting you, Mr. Marshall, and thank you for helping my sister. Now if you will excuse me."
Tru's oldest brother left the dining room and was soon heard shutting the front door behind him. Alex watched as a look of concern passed between Tru and her brother Carl. Something was amiss, but it wasn't any of Alex's business, so he put it out of his mind as the family prepared to retire to the parlor.
At his mother's request, Carl sat down at the piano and began to play for the family and their guests. Alex soon recognized the impressive level of skill that sent Carl's fingers flying over the keys of the old instrument. He wondered why Carl stayed working at the print shop in what was clearly a waste of his talent. Anyone with ears could tell that music was his passion.
After Carl finished his first piece for his appreciative audience, Clarice stood up from the sofa where she had managed to squeeze herself between Tru's twin brothers. "Mrs. Kueshner, I have to go. I promised my mother I wouldn't be too late." Clarice looked anything but happy at the idea of leaving.
"Of course, dear girl." Tru's mother winked at her daughter across the room. "One of the boys will accompany you. Which one of the twins would you like to escort you home?"
Clarice's elfin face lit up like the electric lights at Hearns at the thought of having one of the twins all to herself for the walk home. But after a moment of reflection, her look of delight faded to one of indecision. The two men did nothing to help, standing silently by, while Tru and her mother watched with growing impatience.
Finally, Clarice turned a woebegone face toward Tru's mother. "Mrs. Kueshner, can't they both walk me home?"
"Certainly, dear." Martha Kueshner stifled a sigh. "I'm sure they'll both be delighted."
Her world restored, Clarice left the house wearing a smile as she floated between the two men who seemed content to share the adoration of the irresolute pixie. Tru secretly thought her brothers found safety in numbers when it came to her friend. When the room was settled again, Tru's mother requested her son play several old folk tunes from their homeland. Alex enjoyed hearing the simple melodies and was surprised when he glanced at his father's pocket watch to find that he had been listening to Carl Kueshner play for over an hour. Good manners required him to make his good-byes.
"I'm afraid it is time for me to leave." Alex stood up, surprised at how sorry he was to have to end the evening. He had been uncertain how he would respond to Tru's large family and their somewhat radical ideas. But the time spent in their company had increased his respect for them.
"We are glad you could join us, Mr. Marshall. You are welcome in our home anytime." Tru's father stood up and held out his hand in a firm handshake
Then Martha Kueshner surprised herself and Alex by giving the detective a small kiss on the cheek before turning to her strangely quiet daughter. "See Mr. Marshall to the door, please, liebling."
"Of course, Mama." Tru smoothed her gown as she rose from her chair. All night she had thought she wanted to be rid of the overbearing detective, but, suddenly, that idea had lost some of its appeal, and she didn't know why.
She tried to calm herself as she led Alex Marshall out of the parlor and toward the front door, but her heart refused to stop pounding. Behind them, the strains of the piano had once more resumed. When they reached the door, Tru hastened to say good-bye but was startled into silence when Alex took her hand and pulled her onto the stoop, reaching around her to shut the front door behind them.
Tru began to protest, but the sharp words died in her throat. The soft spring air felt wonderful as it cooled her flushed cheeks. She looked skyward and found the heavens filled with stars.
"It's too beautiful an evening not to linger over our good-byes." Alex followed Tru's gaze upward and gently pulled Tru down the steps and further from her front door. "The stars are sparkling tonight. But, you know, I do believe they pale when compared to the fire in your eyes when you get good and mad, noodle." Tru's head snapped up as she prepared to do battle. But Alex only laughed, placing his fingertips gently over her lips before she could utter one word. "See you proved me right. No star could match the brilliance of your eyes at this moment." Alex let his fingers slowly stray from Tru's lips, brushing her skin softly until he cradled her cheek in the palm of his big hand, while Tru stared up at him. "Do you know what you remind me of tonight?"
"If it's another kind of bird, please don't bother to tell me." Tru managed to whisper her weak protest even as Alex's eyes kept her from thinking too clearly.
"No, not a bird." Alex looked down at Tru's light-green dress, and his smile widened. "You remind me of a meadow in springtime."
"Oh, wonderful! Now I remind you of grass." Tru began to turn her head away, but Alex captured her chin and gently tilted her head until he could once more look into her eyes.
"I didn't say grass. I said a spring meadow. I find spring meadows especially intoxicating." Alex leaned closer to smell her scent, while his free hand reached up to stroke her hair. "Your hair smells of lilacs. I love lilacs." Alex brought his hand down to cup her neck, while one finger lightly traced her features. "You face is like a beautiful wildflower atop a delicate stem." He moved nearer until there was no space between their bodies.
Tru stood mesmerized as Alex bent closer until his lips whispered against hers. A warm languor had invaded her body as she fell under the spell of his soft words. She felt as if she could have melted into a puddle right on the doorstep of her parents' home. For a moment, she thought her heart had seized to beat as she leaned forward, waiting for Alex's kiss.
"Saying good night to Mr. Marshall, noodle?" A deep male voice caused Tru and Alex to jump apart as they turned and saw Tru's youngest brothers grinning at them from the sidewalk leading up to the house.
"I was just about to." Tru tried not to sound disappointed. "Why are you back so soon?" Tru wanted nothing more at that moment but to have her mother's heaviest frying pan in her hand.
"We had plenty of time to see Clarice home and get back. You must have lost track of time, sister. Maybe you were just mesmerized by Carl's playing." One of the twins grinned as he poked his brother in the ribs with his elbow.
"You boys go on in. I'll be in shortly. I think Mama has some of Grossmader's bread pudding left over. You wouldn't want Carl to get it all."
"That's all right, we're stuffed. We'll just wait for you, Gertrude." The two big men grinned broadly at their sister, who recognized defeat.
"Good night, Mr. Marshall." Tru moved away from Alex and started toward the steps of her home.
"Yes, good evening, Miss Kueshner." Alex moved reluctantly past Tru's brothers, who looked so pleased with themselves that Alex had to fight the urge to punch them both.
"Nice to meet you, Mr. Marshall." The twins continued to grin as they took a position on each side of their sister. "Come again when we're home. We'll be glad to keep you and the noodle company." Tru cast one last lingering look over her shoulder as her brothers each grabbed an elbow and propelled her into the house.
As the door closed behind them, Alex was once more aware of the music floating out of the window that had been opened to let in the pleasant spring breeze. He stood on the sidewalk for a moment, listening to the melody that seemed to express the same longing he was feeling deep in the pit of his stomach. As he stared up at the window, a slender shadow was silhouetted in the light. Alex's longing grew as he enjoyed one more glimpse of his adorable hatmaker—his hatmaker! Alex shook his head. What was he thinking?
Alex forced himself to start walking down the street. He should be glad that those two gorillas she called brothers had interrupted them. Gertrude Kueshner's older brothers might very well hold the key to a murder, and it was his job to find out, no matter what it took. Besides, everything about the prickly saleslady and her large family made any romantic notions he might have far too complicated to pursue. Alex picked up his pace as he fought the urge to turn around one more time and look at the house and the stoop where he had come so close to losing his heart forever with a kiss that he knew should never happen.
***********************************************************
"Sophy, marry me now."
"I can't, John. Not now. Mama relies on Otto and me so much. I can't abandon her."
"But you wouldn't be abandoning her. I can help you care for your mother." John reached across the table in the crowded café and captured Sophy's hands in his. "Let me talk to your mother. I will show her I can support you and also provide for her."
"How, John? You do not make much at your papa's newspaper, and he cannot afford to give you more." Sophy couldn't bear to look into John's eyes and see the love there.
"I will quit working for Papa. I can get a job on one of the big newspapers. I have a friend at the World who has told me he could get me a job there if I was looking for work." John tried to keep his feelings of guilt hidden as he pleaded his case to the beautiful woman he loved. In the past, he had never let his feelings for any woman become too strong because he felt his duty to his family came first. But he was thirty-five, and this might be his only chance at happiness with a woman he loved.
"You would do that for me, John?" Sophy's voice was barely a whisper. She was overcome with tenderness as she brought John's ink-stained fingers to her lips to kiss them. Her eyes were full of tears as she laid her cheek against his work-roughened hand. "If you could only know what your love means to me."
"Then marry me, and we will find a way to work things out, I promise."
Sophy searched desperately for the rights words to avoid hurting this man she truly loved. All the while knowing that whatever else her next words might be, they would be a lie. "Give me a little time, John. Let me talk to Otto about Mama in my own way. Perhaps between Otto and myself we can make a plan that Mama will accept. Give me a month, John, and we can set a date to be together. But you must promise to let me do this my own way. You must not speak to Otto or Mama." Sophy Klienst almost choked on the rest of her lie. "I give you my word I will be free to marry you by the end of the month."
"All right, Sophy, I promise. But remember I've waited a long time to find the love I want to keep for the rest of my life. That love is you."
Sophy bowed her head afraid John would read the truth in her eyes; their cause was hopeless. In another month's time, everything would be done. She would be in South America, and John would be lost to her forever. She only hoped that his love would not turn to hate and that he would find it in his heart to forgive her just a little. That was the only hope she had left.
Chapter 6
"Do ya for a penny, mister. I ain't got none of them nasty diseases, I swear."
Alex almost gagged as a disgusting odor made its way to his nostrils. He looked down at the filthy woman whose hand was like a claw wrapped around his ankle. Alex couldn't begin to determine the woman's age through all the dirt and grime on her face, but her eyes spoke of years of suffering on the city's nastier streets.
"Not today." The detective bent down and gently removed her hand from his leg. "But here..." Alex reached into his pocket and gave the woman the few coins he found there. Then he hurried to catch up with his partner, who had kept on going while Alex was being held captive by the pitiful woman.
"What's gotten into you, Alex? We'd go broke if we gave money to every down-and-out person roaming these streets. Besides, you know she'll just use the money to get drunk."
Alex knew Richard was probably right. They'd come across at least a dozen such pathetic creatures as they'd searched the neighborhoods near the wharves looking for the address on the paper Alex had taken from the corpse. "Maybe getting drunk is a blessing for the likes of her." Alex didn't know why he felt different this night. He only knew he had begun to regret the cynicism he had cultivated over his years as one of New York's Finest. The thought of the warmth he'd felt around the Kueshner family's dinner table flashed through his mind.
Richard Shaw's partner had been in a strange mood for the last week. But Richard wasn't sure why, as Alex hadn't confided in him as he usually did. "Why are you so determined to identify that body anyway? It's not like the fellow was the only man murdered in New York last week. With no apparent friends or connections in the city, I doubt the superintendent will be in too much of a hurry to find his killer. We may never find out who he was."
"Maybe not, but I've got a feeling about him, and I trust my feelings—at least when it comes to police work." Alex was sorry he couldn't confide in his friend. But he was not ready to tell Richard that this time he had a personal reason for wanting to solve a case. Almost every night for the last week, his dreams had been filled with the kiss that never happened. Only in his dreams, his lips did meet the lips of the delightful Miss. Kueshner, and the desires aroused by the dreams kept him tossing and turning all night.
"I know you're a good detective, and part of the reason you're good is that you're stubborn as a mule. But even you aren't a magician. This town's full of strangers coming and going, and, unfortunately, some of them meet with violent ends."
Alex kept silent, restlessly running over in his mind what little evidence he had. There was one thing for certain: he had to find out how involved Tru's brothers were in this crime. "Look, Richard, I don't want to give up yet. There are some really seedy flophouses a few blocks from here that we haven't taken a look at. Let's check them next."
The two men walked in silence for the next block. If he didn't know better, Richard would have thought his friend was having woman trouble. But Alex never thought of anything except being a detective.
"You remember, Alex, when we were assigned to stake out the brothel owned by that rich madam who's friends with Captain Byrnes. That crazy customer was beating some of her ladies. The one who kept coming back in different disguises."
"Now what made you think of that?"
"Seems I recall a certain madam taking quite a fancy to you, and if I remember correctly, after we caught the guy who was beating her girls, she was grateful enough to offer some of her services for free. Her exact words were she'd be glad to personally thank you in her private parlor. She was mighty disappointed when you declined her invitation. Since she was that smitten, maybe if you'd played your cards right, you might have become a kept man. You could have retired from the police force and been sitting pretty."
"Now what on earth put that idea in that thing you call a brain?" Alex glanced at his partner to see if he was joking, but his friend's face was serious.
"Well, for one thing, she knows everything there is to know about pleasing a man. With your sour disposition lately, that would make my life easier. Also she is rumored to be one of the wealthiest women in New York. Maybe if you married her, you could pay me the money you owe me from our last poker game."
"Seems to me I remember you being the one who owes me."
"Well, one thing is for sure, that madam is razor sharp and has seen it all. Men like us, in our profession, are better off with women who understand the world we operate in and don't look at everything through rose-colored glasses." Richard Shaw shot a quick glance at his partner. "Women like that—the kind you only marry—can make a lot of trouble for a man who does the job we do."
Richard's words brought the image of the feisty Miss. Kueshner flooding back into Alex's thoughts. Maybe his friend was right. Maybe Tru was too tender and sheltered a creature for a man like himself. Still, Alex thought, no woman had ever stuck in his mind like his little bird.
"Say, Alex, you never told me what happened with that young woman on the ledge. You said you were going to her house to have dinner so you could size up her brothers. Anything ever come of that?"
Alex kept his eyes straight ahead. "I'm still working on it."
"Anything I should know?" Richard sent a puzzled glance toward his longtime friend. "You wouldn't be keeping something from me now, would you?" Alex's silence stirred Richard's suspicions even more. "You never told me what this young woman looks like. Is she pretty?"
"Most people would say so. But she's a bit strange." Alex quickened his pace, trying to outdistance his friend's curiosity, but he failed.
"Didn't you say her family prints a socialist leaning newspaper? Whole family must be strange. Must have been hard having supper with them? People like that don't exactly fit with a member of New York's Finest."
"They're good people." Alex's face was unusually grim as he turned to meet Richard's concerned look. "They're intelligent, hardworking, and kind. The father's just a free thinker in his political views. I enjoyed my evening with them very much."
Richard stopped so they could take stock of how close they were getting to their destination. "Look, Alex, I'm just offering a little advice for your own good. It might get awkward if this investigation leads back to those men we saw at the crime scene if you become friends with this young woman and her family."
"Have you ever known me to put anything ahead of my job?" Alex didn't even wait for his friend's reply before starting off again. "That address should be around here somewhere. I think the problem has been that being a stranger to town, the victim got the numbers in the address turned around, and we didn't realize it. That's why we didn't find it. We'll ask around this block. There are some flophouses with single beds to rent mixed in with the other tenements."
Richard followed Alex down a short alley that ended in a small, dirty courtyard bounded on three sides by the rear of some run-down tenements. At every floor level, a tangle of clotheslines crisscrossed the space above their heads. The man-made web held a parade of ragged garments, waving in the fetid air like pathetic flags of surrender. Beneath this strange canopy, several dirty-faced children played in the mud, while, nearby, a tired-looking young woman watched them as she sat on a stoop peeling potatoes.
"Might as well start with that woman over there. Maybe she knows something about where a man might rent a cot. That would save some time and some shoe leather." Alex took off in the direction of the stoop as the woman watched his approach with a look of suspicion on her face. It was clear that she knew the two men to be policemen and that in this neighborhood, that usually meant trouble for the residents.
"Good day, ma'am." Alex tipped his hat, hoping he'd get lucky and the woman would speak English. "Sorry to bother you, but do you know if there are any single beds to be let in any of the tenements nearby?"
Alex's question was met with a blank stare. But after a few long moments, the woman answered, her voice bearing only a trace of an accent. "That one there has some beds for hire by the day or week." The slight gesture toward the most dilapidated of the structures seemed to use up all the woman's energy. She fell silent as she turned her eyes back to the pile of potatoes in her lap. Alex knew they'd get no more help from this quarter.
"You heard the lady. Let's try that one." Alex led the way across the dirt courtyard, entering the rear of the tenement. A ramshackle set of stairs led up several floors and down one. Alex and Richard heard voices coming from below, so they headed down to the lower level. Alex opened a door and entered a room filled with beds jammed one right up against the other.
Because it was afternoon, most of the cots were empty. Yet on some of them, men sat, staring toward the door through which the detectives had appeared. A game of chance seemed to be going on in a corner, but the participants stopped immediately at the sight of two strangers in their midst. The entire room smelled of unwashed bodies, liquor, and hopelessness.
"We'll just start showing the photograph around." Alex took the photo out of his pocket as he and his partner began the slow process of questioning the men scattered around the room. It only took a short while for it to become clear that the two detectives were not going to get very far questioning the inhabitants of the room. Several of the men did not speak English, and those that did eyed the detectives with ill-concealed fear and hostility. No one would admit to knowing the man whose lifeless eyes stared up at them from the picture Alex held in his hand. Finally, even Alex had to admit defeat.
"We might as well go." Alex's frustration was palpable. "I have a strong hunch we're in the right place. But none of these men are going to help us."
Richard nodded. He knew his friend well enough to know that the fewer words spoken now, the better. The two men headed out of the dark, oppressive room and up the short flight of stairs.
"Hey, young fella! . . . Remember old Angus?" A scruffy-looking old man stood in the doorway, blocking the detectives' path out. Alex stared, not recognizing him, until a sharp bark from inside the man's coat provided a clue. Soon, a small furry muzzle poked out from the stranger's worn vest.
"I remember the dog. Is he still hell on rats?" Alex felt the fluttering in his stomach that always signaled to him that something important was about to happen. Maybe this old man was the luck they needed.
"Aye. But the wee dog's a mite too old for the pit. It's time he retired. So I've decided to head out west to the Mississippi. There's no Big John out there, and it seems like a good place for an old sailor to settle down. I'll be leavin' in a few days' time. What brings you laddies to these parts of town?"
"Do you sleep here at night?" Alex waited anxiously for the old man's next words.
"Aye. Moved in here after me fight with Big John and his thug 'cause it's a wee bit further away from where that thievin' bully be hanging about. Dirty place, but it's cheap, and they generally leave ya be."
Alex reached into his pocket to pull out the photograph. "We're looking for the identity of a man we found murdered up near Union Square. He had an address in his pocket that could have been this place. Would you look at this photo of his corpse and tell us if you recognize him?"
"Sure, laddie, be glad to. I owe ya a favor. Mind ya, I dinna see as well as I used to."
The old man took the photograph in his hand and studied it for some time, bringing it close to his eyes and then holding it back away. He took so long before speaking that Alex found it hard to stand still and keep a smile fixed on his face.
"Yeah, he was here, but it was a while back. Haven't seen him for about a week." The old sailor handed the photograph back to Alex.
"That's how long ago he was found murdered." Richard was beginning to share Alex's hope that the old man might actually be able to help him. "Did he stay here long?"
"Nay. He was here only three or four nights. If I remember right, though, he paid for a week. I wondered what might'a become of him. But in this city..." The old man shrugged his shoulders.
"Did he ever tell you his name or where he came from?" Alex held his breath, hoping their luck would hold out.
The old man shook his head. "He weren't much of a talker. I dinna recall him ever saying but a few words to anyone."
Alex and Richard exchanged a glance; their hope of finding out who the victim had been in life was fading. The old man paused as if deciding whether to say more. A strange look was on his face as he scratched his dog gently behind one ear. Finally, his eyes locked on Alex, and he continued. "I owe ya a debt for helping me get the wee dog back. So I'll be truthful with ya. The man had a gripsack with him when he came to rent the bed. After he dinna come back one night, I took the bag and kept it. I dinna mean to steal anything, but it seemed a shame to be lettin' someone else be having it."
Alex quickly reassured the old sailor. "We aren't going to accuse you of stealing. The man's dead anyway. We just want to find out who he was. Do you still have the gripsack?"
"Aye." The Scotsman gestured for the two detectives to follow him back out of the building.
They trailed the old man and his dog to an alley where an ancient prostitute answered when the old man knocked at her door. After a friendly kiss "hello," the well-used woman disappeared into her small room and returned with the bag, which she handed over to the sailor. Another brief kiss good-bye and a quick attempt by the woman to offer her wares to the two younger men, and Alex and Richard were alone once more with the Scotsman.
"Let's have a look inside." Richard placed the bag on a nearby barrel, opening it as Alex watched intently.
"I hate to disappoint ya, laddies, but there be little in it. I took the best thing." The Scotsman looked down at his feet. "He had a good pair of boots inside. They be well made and only pinch a little. You won't be needing them back, will ya?"
Alex reassured the old man his new boots were safe as Richard pulled an extra set of clothes from the gripsack. Then two newspapers and a book followed and were laid on top of the clothes piled on the barrel.
"That be it, laddies. I told you it weren't much."
Alex picked the book up first. "It's in German." He opened the book carefully, and, to his surprise, tucked between the pages, he found United States currency.
"Well, I'll be." The old sailor stared while the stack of greenbacks grew as Alex emptied the book of its treasure. "I dinna open the book because I can't read. Never regretted it until today."
"There's $300 here." Alex finished counting the money. "Apparently, our corpse wasn't as poor as he appeared." Alex picked up the book again, still frustrated at finding no name to go with his body. There were no more notes contained between the pages, but on the inside of the cover, he found an inscription. "There's a name written in an inscription in the front of the book. But the inscription's written in German."
Alex handed the book to his partner to examine and picked up the newspapers, which were also in German. But on one masthead, the name of the city of Chicago jumped out at Alex. The other paper bore a name Alex recognized immediately: Der Wahrheit. The detective's heart sank at his discovery of yet another link between his victim and Tru' family.
"Thanks, old fella. This helps." Alex picked up the gripsack and replaced the extra clothes. "Take the satchel and the clothes. I'm sure you can use them on your trip west." Alex peeled off a few banknotes from the stack he held. "Take a few greenbacks too. I'm sure our corpse won't mind."
"Much thanks to ya, laddie, I'm mighty grateful." The old man took the money and the satchel. "Well, we probably won't be seeing each other again, so take care, laddies." The Scotsman started to leave the alley, but Alex called out to him after he had taken a few steps.
"By the way, what's the dog's name?"
The Scotsman smiled a toothless grin, bending over to gather the small animal in his arms. "Why, his name is Wee Dog."
Alex laughed as the former rat killer responded to the sound of his name by wagging his tail and licking the face of his master. "Well then, so long, Wee Dog."
Both detectives watched until the two vagabonds disappeared around the corner. Then they turned their attention to the barrel upon which rested the book, the papers, and the stack of money, a pathetically few clues by which to solve a murder but more than they had at the start of the day.
"Well, Alex, what now?"
"Now I visit someone who can read German."
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"Carl, what are you doing here?"
"We ran out of ink for a printing job we're working on, so Papa sent me to the suppliers to purchase some more. I stopped here on my way because I need to make a special purchase of my own—a gift." Carl glanced behind Tru toward the shelves holding so many beautiful things. "I need your advice. I don't know what to get."
"Mama's birthday is still two weeks away. There is plenty of time to decide on a gift."
Carl didn't correct his sister, although his conscience pinched a bit at his omission. He had debated as he walked along Fourteenth Street under growing clouds whether to seek his sister's advice. But he had never bought a gift for a beautiful woman before, and he was afraid of making a mistake. "I want this gift to be special."
Tru looked at her brother, puzzled at how intense he seemed. "Why are you worried? You know Mama will like anything you pick."
"Just help me, please. I was thinking of buying a pair of gloves or a set of handkerchiefs."
"We have some very fine silk handkerchiefs that just arrived from London." Tru turned to remove a box from a nearby shelf, placing it on the counter in front of her brother. "The workmanship is extraordinary. They are edged with the most delicate French lace and embroidered with exquisite little primroses."
"Primroses?" Carl recalled the intoxicating fragrance that greeted him each evening when he entered the countess's music room. "That might be perfect. You do think handkerchiefs are a good choice? Better than gloves?"
"I think Mama would get more use out of the handkerchiefs. And these are so beautiful any woman would be happy to receive them. Feel how fine the silk is." Tru rubbed one lovely silk square against her brother's cheek. "See what I mean?"
Carl knew the handkerchiefs were the right choice as soon as Tru laid one against his face. They felt as smooth and soft as he imagined the countess's skin would feel if he dared to touch her. "I'll take them, Tru. Wrap them in an especially nice package, would you?"
"You know, Carl, Mama would be satisfied with a less expensive gift." Tru knew the price of the handkerchiefs was several weeks' wages for Carl. "But if these are what you want, I'll wrap them."
Carl didn't say anything, but the determined look in his eyes told Tru what she needed to know. She took his money and sent it on its way while she began to wrap his purchase.
As Carl waited for Tru to finish wrapping his gift, his eyes strayed to the perfume counter and the woman his brother John loved. "Have you noticed anything else strange going on with Sophy?"
Tru didn't reply to her brother's query. She decided it was best not to mention Randall Bently and his mysterious visit with Sophy.
"I don't trust her." Carl continued to watch as Sophy Klienst waited on a man who was obviously interested in more than perfume. "John told me he has asked her to marry him, but she still will not take him to meet her mother. I cannot believe our brother is being so foolish."
"I don't know what to think, Carl. I do believe she loves John. But I agree with you that something is not right."
"Look at her, Tru. That man cannot take his eyes off of her. Perhaps we have the reason right before our eyes that she deceives John. Perhaps she simply cannot resist the attention of the wealthy men who come here and stare at her and try to touch her. After all, what woman would choose a poor man over a rich one?"
Tru finished with Carl's package and studied her brother's profile, surprised at the bitterness she detected in his voice. Carl had never before expressed jealousy for those with an excess of wealth. She laid the package on the counter. "Some women care more for what a man is than for what he has. You know that, Carl."
"I'm not sure what I know these days. I have to go. John and Papa cannot finish until I return with the ink. Tru, promise me you'll not say anything about the gift."
"Of course, I would not want to ruin Mama's surprise." Tru was startled when Carl turned abruptly and left without saying another word. She watched him as he disappeared through the doors, puzzling at her brother's unusual behavior. When he was gone, Tru looked down at the counter and discovered that in his agitated state, Carl had left behind the gift he had been so anxious to purchase.
Tru picked up the package and put it into the pocket of her store apron. It was not a day she helped deliver papers, so normally she would go straight home. She wondered whether to wait until she got home that night to give it to Carl or to make a special trip to the print shop after work to return it to him. He had been so insistent on secrecy that Tru thought perhaps it was best not to wait. Surely, Carl would be worried about the gift.
As she considered what to do, a loud crack of thunder sounded from outside the store, promising a rainy afternoon. That meant the store would not be as busy as usual. Already, Tru could see customers leaving hurriedly to find waiting carriages to take them safely home. Another crack of thunder sent Tru's mind wondering back to another rainy afternoon a few days before when she had sat warm and dry in her family's cozy kitchen watching her mother bake pies.
"Mama, you are so good at baking. But don't you ever get tired of cooking for us every day?"
"Nein, liebling. When you do anything, even the lowliest task, for people you love, it is not a burden. You will find that out someday."
Tru's eyes followed the rolling pin as her mother sent it gliding over the dough that would soon line the pie plates resting on the table. She always loved rainy days when she could sit and talk with her mother. But today her mind was distracted, as it had been ever since the memorable evening when she had almost been kissed.
"Gertrude, dear, is there something bothering you? You have not been yourself the last few days. It couldn't have anything to do with the handsome detective, could it?"
"Mama! Why would you think such a thing? Alexander Marshall is a complete stranger. Besides, he is not the kind of man I could ever care for." Tru's voice rose in volume during her heated protest.
"And why is that, liebling?"
"Mama, he is a policeman. You know Papa says the police in New York all answer to Tammany Hall. And Papa says Tammany Hall keeps the people from bettering themselves by making them think they are being helped when all the while the politicians are lining their own pockets. Last month, the police put down the strike by the streetcar workers. They are often used against the trade unions."
"I doubt Mr. Marshall was involved in stopping the strike, since he is a detective. I know the police are sometimes on the opposite side from your papa's political views, but you know your papa would be first to say that a man must be judged by who he is, not just how he earns his bread."
"He isn't like John and Carl, Mama. They are gentle and, mostly, soft-spoken, and they love family above everything. Mr. Marshall is more forceful somehow, and I don't think he believes much in the goodness of people."
"I think Mr. Marshall has kind eyes and a warm smile." Tru's mama looked up from the pies she was filling. "I myself like him. But what I think does not matter. It is what you feel that counts."
Tru sat very still, trying to decide what it was she did feel. Her mind had been in turmoil ever since the night on the front steps when Alex Marshall had turned her world upside down for a few brief moments.
"I don't know, Mama. I'm confused. I've always cared only for my family and my dream of having my own millinery shop. I never felt the need for anything else."
"Tru, you and your brothers have been the best children we could have asked for. You have stayed and helped us build a better life far longer than most children do. But, someday, I hope you all will have your own families and homes. I have never wanted my children to go through their lives without feeling the deep love that I feel for your father. If such loves comes your way, you must embrace it because it makes everything that you are or that you want more complete. And, believe me, if you find such love, your father and I will welcome the man who makes you feel that way with open arms no matter what work he does."
"Miss Kueshner!"
Tru jumped as a sharp voice called her name. The spell her memories had held her in was broken. She turned to see Sophy's Klienst's mother standing on the other side of her counter. "Mrs. Klienst, how nice to see you again." Tru stammered a greeting while trying to recover from the surprise of finding Sophy's widowed mother glowering at her. "Is there something I can do for you? Are you looking for Sophy? She is over that way." Tru pointed in the direction of the perfume counter. As she did, she noticed that Sophy was busy with a customer and had not yet seen her mother.
"I wanted to talk to you, Miss Kueshner."
"What can I do for you, Mrs. Klienst?" Tru wasn't surprised to observe that the somber woman was standing ramrod straight without any help. Tru had suspected the older woman leaned on her son more as a means of control than out of a genuine need for support.
"I noticed as I came in that your brother was here. He seemed to be staring at my Sophy. He is not smitten with my daughter, is he? Because that would not meet with my approval."
Tru could feel her temper rising. "Carl wasn't staring at Sophy. He was waiting for a package to be wrapped. He barely knows her." For an instant, Tru wondered if she should mention John, but she decided it would not be prudent given the woman's strange behavior. "Carl met her only once when she was with her brother."
"That is good. My Sophy does not need more men hovering around her. She is far too beautiful for her own good. Such beauty is a curse. I know that well. Do not think me only a foolish old woman, Miss Kueshner. I am wise in the ways of the world."
Tru stared, at a loss as to how to reply to such an odd statement. "I am sure that you are, Mrs. Klienst." Tru sighed with relief when she realized that Sophy had spotted her mother and was waving at Mrs. Klienst to join her.
"I have to leave you now, Miss Kueshner. Remember what I said. A lovely girl must always be on the lookout for those who would take advantage of her. I watch over my Sophy very carefully."
"I'll remember that, Mrs. Klienst." Tru watched the older woman walk across the store toward her daughter as another loud crack of thunder reminded Tru she would need her umbrella for her trip to the print shop.
***********************************************************
"What took you so long?" John tried not to snap at his brother, who had just entered the print shop soaking wet. The day was not going well, and Carl's tardiness would make it even harder.
"You try dodging raindrops all the way along Fourteenth Street, and then you can complain to me." Carl shook the rain from his hair as he shed his wet coat. "The supplier was busy. Where is Papa?"
John took the package containing the ink and headed back into the print room without saying a word. Carl followed, waiting for an answer.
"Papa went to a meeting called by the trade unionists to decide what to do about the recent arrests of people for picketing. The five men who were convicted for picketing outside the music hall were just sentenced to rather long prison terms." John worked setting type as he spoke. "Papa wants to print a whole new editorial tomorrow about the New York court's ruling. I have to finish this advertising flyer for Mr. Cohn and then reset the entire front page for tomorrow's paper." John's voice betrayed his resentment. "It will take me well into the night, so I won't be able to see Sophy again this evening."
"Maybe Sophy would have cancelled anyway. She has been doing that a lot lately." Carl began to help his brother turn out the flyers.
"Why do you say a thing like that?" John looked up from his task to frown at Carl. "I don't need to hear such things from you. You are my brother."
"Maybe that is why you should hear them from me. Did you ever think that perhaps Sophy Klienst is playing you for a fool?" Both men straightened and stood glaring at each other.
"You know, Carl, it may not be me who is in danger of making a fool of himself over a woman."
"What does that mean?"
"Do you think I haven't noticed that you spend every night at the house of the countess Von Ott? Mama and Papa may be naïve enough to believe that the only attraction is the countess's piano, but I wonder. What does the countess look like, Carl? Tru says she is very beautiful."
"I have no intention of discussing the countess with you. But I'm warning you for your own good, John. You should ask the woman you love if there is someone else. I only say this because I don't want to see you hurt more than you already have been." Carl hated giving voice to his suspicions, but he was afraid his brother was about to be betrayed.
"I will not listen to your ugly accusations, Carl. Just remember my warning. The countess Von Ott is a rich, worldly woman. What possible interest would a woman like that have in someone of your youth and class? You are the one being a fool if you harbor any dreams in that direction."
The brothers' voices had grown steadily louder as they argued. They failed to hear the sound of the door opening in the outer office until a discreet cough drew their attention to the tall detective standing in the doorway, observing them.
"Mr. Marshall, what are you doing here?" John tried to sound cordial as he attempted to calm himself, while Carl stood silent, eyeing the intruder warily.
"I have a book in German, and I need to know what it is about. Does either of you gentlemen read German?" Alex smiled at the two men, hoping to put them at their ease. He had overheard them arguing as soon as he had entered the print shop.
"Carl and I both read some German. And Papa, of course, reads it fluently. What is it you need translated?" John wiped the printer's ink from his hands before carefully accepting the slim volume the detective handed him. "Well, the title translates to 'Power to the Weakest.' This appears to be a political treatise." John tried to hide his surprise as he read the name of the author. "It is written by a man named Fredrick Von Ott." John glanced at his brother, who looked shaken at the sound of the familiar name.
"Let me see that, John." Carl took the book and began thumbing through it. It was indeed a book of socialist theory and Carl could not believe the author was anyone other than the countess's late husband.
Alex's ears had perked up at the name of the author. According to the argument that he had overheard, Carl was involved with a woman with the same last name. Another coincidence linking Tru's brothers with the murder victim. Except Alex didn't believe in coincidences, much to his regret.
"Carl, John, where are you?" A soft feminine voice called from the front of the small shop, followed shortly by the appearance in the doorway of the very woman who at that moment was foremost in Alex's mind.
"Oh. No. Not you." Tru was devastated to see Alex Marshall standing next to her brothers just when she looked a soggy mess.
Alex felt a familiar twitch at the corners of his mouth. He was trying hard not to laugh as Tru stood in the entrance to the room looking like a drowned baby bird. "You are usually supposed to take an umbrella along for a stroll in the rain. See, I brought mine." Alex's held up his neatly closed umbrella for Tru's inspection.
"Thank you for your words of wisdom." Tru shot a withering look in Alex's direction then pointedly addressed her explanation for her sorry state to her brothers. "The wind caught my umbrella when I was at the corner of Fourteenth Street and Second Avenue right when a cloudburst started. The stupid umbrella is probably halfway across Union Square by now."
Alex reached out and plucked the sodden mess of feathers that had once been a hat from atop Tru's head. Then he handed her a large sparkling-clean handkerchief in exchange.
"My hat!" Tru watched in dismay as Alex squeezed a small puddle of water out of the feathers and onto the floor.
"Maybe they'll fluff out when they dry."
"Humph" was Tru's only answer to Alex's remark as she wiped her face with the linen square that smelled disturbingly of the detective's cologne.
"What are you doing here anyway, Tru? It's not a paper day." John resumed working on the flyers he needed to get done.
"I'm on an errand for Carl." Tru grimaced at Carl, who looked properly sheepish as his sister withdrew a small package from her reticule and handed it to him. He quickly slipped it into his pocket.
"Thanks, Tru... Mr. Marshall, here is your book. Is that all you need?" Carl started to hand the thin volume back to the detective, but Alex stopped him.
"No. There is one more thing. There is an inscription inside. I need to know what it says."
Carl opened the book and read the writing inside. "The inscription reads, 'To my dear friend Heinrich Schueller. May we dedicate our lives to advancing the ideas contained in this book.' It is signed—'Fritz.'" Carl handed the book back to Alex.
"Thank you. That helps me a great deal."
Carl felt compelled to ask Alex a question even though he feared the answer. "Why is the book important?"
"I still have not identified that corpse you saw in the alley. This book was among his belongings. I am hoping that the name in the inscription is the name of my victim."
"That still will not tell you why he was murdered or by whom." Carl could not rid himself of the memory of seeing the victim at the countess's house just a short time before he was murdered.
"No. But it's a beginning. There was also a paper in the man's gripsack. Have any of you heard of this paper?" Alex pulled the folded newspaper from his coat pocket and spread it out on a nearby table.
"Of course, that is Der Arbieter-Zeiting." A deep voice rumbled from the doorway as Johann Kueshner hurried in shaking his umbrella as he crossed the room to peer down at the newspaper. "What are you doing with a socialist paper from Chicago, Mr. Marshall?" At that moment, Tru's father spotted his nearly drowned daughter. "Child, you are soaking wet. Where is your umbrella?"
"It's a long story, Papa." Tru kissed her father on the cheek. "I will walk home under yours if that's all right, Papa?"
Tru's father chuckled. "Your mama would have my skin if I let you get any wetter." The newspaper publisher turned his attention back to the detective. "Now, Mr. Marshall, you did not answer my question. Are you developing an interest in all things German?"
"That's it exactly, sir."
"Well, why don't you come to the beer garden tomorrow night and see firsthand how we Germans relax and enjoy ourselves?"
"Papa, that is not a good..." Tru voice trailed off, knowing her objection would fall on deaf ears.
"I'd like that, Mr. Kueshner."
Tru felt as if a hundred birds had been left loose inside her stomach. The thought of another evening in the company of Alex Marshall was very disturbing, and Tru didn't know why. She didn't even like the tall detective, who seemed to always be mocking her. So why did the thought of being near him once more leave her feeling as if she couldn't breathe?
"Good, Mr. Marshall. We all go to Schemer's Beer Garden at the corner of Sixth Avenue and Thirteenth Street. We will see you there at eight o'clock."
Alex gathered up his newspaper and tucked it in his pocket along with the book. "I'll be there. And thank you for the invitation." As he turned to go, a familiar fluttering filled his stomach. But this time, Alex wasn't sure what important thing was about to happen. Was he about to get a step closer to solving his murder or a step closer to being hopelessly in love?
***********************************************************
The countess Von Ott tried to calm herself as she stared in the mirror. How could she be such a hopeless fool at her age? Every night for the past two weeks, she had told herself she would leave her house and not be there when Carl Kueshner arrived. And every night she stayed. Sometimes they talked for hours. Other nights they played together the music of the great composers they both loved. And, always, she listened, enthralled, when Carl's passionate playing filled the room. She felt alive and excited once more, and it terrified her.
"Countess, good evening." Carl stood in the doorway of the music room, watching the woman he had come to love.
"Mr. Kueshner, do come in." Maria turned to welcome the young man who had given her heart new life. "You're early."
"I'm sorry. But I couldn't wait until later." Carl wanted to tell her that he had yearned so for the sight of her that he had left the print shop early and come straight to her side. But he was afraid what she might say. "I've brought a gift for you. I wanted to thank you for all you've done for me by letting me come here."
"You needn't have done that, Mr. Kueshner."
Carl crossed the room and stood as close to the countess as he dared. He handed the small package to her, suddenly terrified his gift would be inadequate.
Maria opened the package slowly and stared down at the exquisite handkerchiefs. "How lovely!"
"I know it is a poor gift to give you after all you've done..." Carl studied the countess's face, afraid she did not really like his offering and was just being kind.
"Oh, no. I love them." Maria lifted one delicate square of silk and lace and touched it to her cheek. "It so soft it feels like a feather against my skin."
"Nothing could be as soft as your beautiful skin." Carl couldn't stop himself. He reached out with a trembling hand and caressed her cheek.
"You mustn't, Carl." Maria's voice was barely a whisper.
"I'm so sorry, Countess, if I've insulted you." Carl drew his arm back, afraid he would be banished from the only place where he felt truly alive.
"No." Maria reached out to take Carl's hand in her own. "You could never insult me with your touch." Unable to resist her own impulse, she brought the hand she was holding to her lips, kissing it tenderly.
A faint hope stirred in his heart. "Maria..." Carl slowly gathered the countess into his arms, overwhelmed by the feelings he had fought to hide.
"No, Carl, we can't." Maria pushed against Carl's chest, and, immediately, he released her.
"I have offended you. Please forgive me, Countess. I will go at once." Carl turned to leave, but a small hand on his arm stopped him.
"You don't understand. I don't want you to go, but I'm afraid of what will happen if you stay. Please let us continue as we have been. I don't want to lose your friendship."
"Of course. I don't want to lose yours either." Carl couldn't bear the look of unhappiness in Maria's eyes.
An uneasy silence fell over the room as the two people tried to reclaim their composure. They had come so close to the precipice, but they had pulled back in time. Carl sat down at the piano that had become as familiar to him as his own face in the mirror, and he began to pour out his frustrations and longings in his music. The room filled with the strains of Beethoven's Moonlight Sonata as he had never played it before.
Maria sat close by, watching every expression on Carl's face as he wove his musical spell. She concentrated on memorizing how he looked as he played against the time when he would grow tired of spending his evenings with a middle-aged widow and no longer come to fill her nights with excitement.
Carl played on, growing more agitated as his fingers flew through each passage of music. He couldn't rid himself of the memory of how Maria had felt in his arms. Yet even as his mind filled with the image of the countess's face looking up at him as he held her to his chest, the memory of a lifeless body lying in an alley pushed its way into his thoughts. Carl desperately wanted to know why the dead man in the alley had been at the countess's home the night of the musicale. At that thought, the music stopped as Carl slammed his hands down on the keyboard. He suddenly realized none of that mattered. Maria could never harm anyone, and he could no longer contain his love for her.
"What is it? Why did you stop?" Maria sat very still. Carl had reached some kind of resolution. She could see it in his eyes. Her heart was pounding in her chest as she sat, fearing he was leaving her forever.
Carl rose from the bench and went to where Maria sat, clasping her hands tightly together. He pulled her up gently into his arms, but when his lips touched hers, he lost all self- control as his kiss took on a desperate hunger. The hands that had played the piano so passionately just moments before now moved hungrily over her body, exploring every curve. Minutes passed before Carl could bring himself to loosen his hold on the woman he loved.
Maria laid her head against his chest and felt his heart racing like hers. "What do we do now?"
"I don't know. I only know I love you. I can't pretend otherwise anymore."
"Carl, I'm older than you." Maria tried to pull away, but now that he had crossed the line, Carl refused to let her leave his embrace. "I've been married. You need to find someone who is young and innocent. Someone who will give you children."
"No!" Carl's voice rose in anger. "I could look the world over and never find a woman who touches my soul as you do." Carl captured Maria's face with his long slender fingers, gently forcing her to look into his eyes. "Of what possible importance can our ages be? Unless you don't want me? I know I am inexperienced and I have nothing to offer you."
"Don't say that." Maria placed her hand over Carl's lips. "You are more wonderful than I could ever have dreamed. I live only for the times we are together. No one has ever made me as happy as you have."
Carl could hardly contain his joy at Maria's words. "Then what can be wrong?"
Maria forced herself to step out of the circle of Carl's arms so she could say what she had to say. She sat down near the fireplace as she suddenly felt cold all over. Her eyes stared down at the polished wood of the floor. She didn't dare to look up at Carl, but she sensed when he began to pace back and forth in front of her.
"Within this room, there is nothing wrong or disgraceful about what we feel. But outside this room, outside in the world, our love would be seen as shameful." Maria eyes darted up at the sound of an object hitting something. Carl stood in the middle of the room flushed with anger, his fists clenched as he stared at the book he'd hurled at the piano he loved.
"Never will I accept that!" Carl came to Maria and fell to his knees before her, burying his head in her lap.
"Darling, think what your family would say." Maria raised her hand to stroke his hair. "Even Tru would be shocked, and I couldn't bear to have your wonderful family hate me." Maria could no longer contain her tears, which began to fall onto Carl's cheek.
"Don't cry, dearest." Carl stood up and lifted Maria out of the chair and into his arms. He sat back down in the chair, cradling the woman he loved like a child. Maria let herself be comforted by his warmth and curled up against his chest. Carl tightened his embrace and began to sing softly to her in German. Exhausted by her emotions, Maria fell asleep.
Carl felt Maria relax in his arms. He stopped singing and rested his cheek on her soft, fragrant hair. "I won't give you up now. I can't. Not even for my family."
Chapter 7
Alex spotted a striking-looking woman emerging from the office of the superintendent of police and immediately recognized the madam who'd been the subject of his partner's long-winded conversation the day before as the two detectives had searched for clues to the identity of their unknown corpse. Alex hoped he could avoid her. However, she spotted him before he could leave the room and headed his way.
"What brings you to our proud establishment? Not having any more trouble with violent customers I hope?" Alex flashed the woman his usual grin.
"No, I'm just visiting an old friend." The colorfully dressed madam winked at the handsome detective. "Thank you again for your help with the problem we had a while back. I certainly wish you had allowed me to show my gratitude personally. Are you sure you wouldn't like to come for a visit soon? You look rather stressed. I'm sure I could help fix that."
Alex shuffled his feet, trying to think of a way not to hurt the madam's feelings. "I've been awfully busy with another case."
The large woman adjusted the rather spectacular hat sitting on top of her head. She knew that the detective was making up an excuse so as not to offend her. "You're looking all spruced up this evening. Are you going somewhere special?"
Alex's grin widened. "You might say that. I'm on my way to meet someone at a German beer garden as soon as I finish my business here."
"Anyone in particular?"
"Only the most exasperating woman I've ever regretted knowing. All in the name of solving a case, mind you."
The madam's laughter drew all eyes in the room toward her and Alex. "That does sound bad, Detective. Well, I'd say you look very well turned out for an evening with this woman you say you regret knowing. But where are the flowers?"
"What flowers?" Alex was completely puzzled at the question.
"The flowers you should be taking to this troublesome woman."
Alex had never thought of flowers as he had hurried home to get ready for the evening. "What should I get?"
The madam was obviously enjoying the flustered look on the face of the detective who had turned down her favors. "What does this creature look like?" She watched as Alex's face turned thoughtful. Having seen her share of men in love, she was quick to recognize that Alex was completely smitten by his mystery woman despite his earlier words.
"She's tall. Willowy, I guess you'd call her. She has bright blue eyes and light-brown hair the color of honey. She has an adorable little chin that she sticks up in the air whenever she's mad." Alex stopped, embarrassed at his inability to do justice to describing the delightful Miss Kueshner.
"Violets, definitely violets." The madam snapped open her parasol and sailed toward the nearest door. "And don't forget them." With those parting words, she left police headquarters, walking right past Alex's partner, who had heard the tail end of the exchange.
"Violets for whom?" Richard was afraid he already knew the answer to his question.
"Never mind. You don't want to know."
"Not Miss Kueshner?" Alex started to turn away from his old friend, but his partner grabbed his arm. "Look, Alex, I really hate to keep saying this to you, but don't you think you're making a mistake? You are bound to move up in the detective division. Byrnes has his eye on you. Even if there wasn't a possibility her brothers are involved in a murder, her family's politics would definitely be a problem."
"They're not a problem for me, and it's no one else's business." Alex's words were defiant, but his mind was torn. He wasn't certain what he thought about Johann Kueshner's radical ideas, but he did know he liked Tru's warm and loving family. And he wouldn't let himself consider the possibility that her brothers were caught up in a murder; at least not tonight.
Richard recognized the stubborn look in his friend's eyes. He knew it was best to say no more. "Did you decide what to do about the name inside the book?"
"I just sent a telegram to the Chicago Police asking them if they have any information on a Heinrich Schueller. Given the newspaper in his possession came from Chicago, it seemed a good place to start. Anyway, it's our best bet. If he was an active socialist, they may know of him."
"Let's hope the Chicago Police don't take forever to get back to us." Richard watched Alex pull the ornate timepiece that had belonged to his father out of his pocket.
"I've got to go. I'm late. Check the telegraph for me a couple of times, will you, Richard?"
"Of course." Richard Shaw was hoping a speedy reply from Chicago might give them some answers and maybe save his friend from making a grave mistake. "Be glad to."
"Thanks. It's time I was on my way." Alex closed his father's timepiece, returning it to his pocket. "I've got to make sure I find some violets."
***********************************************************
John hated himself for what he was about to do. But there was no other way he could lay to rest the uncertainty that had been eating away at him ever since his fight with Carl. He had been angry with his brother, but last night in bed, he had finally faced the truth. He himself had been growing increasingly suspicious of Sophy's strange behavior over the last few weeks. So he lied to his father and brother, making up an excuse this afternoon to leave the print shop early to follow Sophy when she left Hearns. He had asked her a few days earlier to join him and his family at the beer garden on this spring evening. At first, she had said yes. But just as Carl had predicted, she had stopped at the print shop yesterday to tell John she would not be able to come. That was the final push John needed to send him over the edge into a dark downward spiral of doubt and pain.
Now he stood across the street from Hearns, hidden in a doorway, staring at the women exiting the giant store. He watched as Tru and Clarice came through the doors and hurried west along Fourteenth Street. He knew they were hurrying home to get ready for an evening spent relaxing at the beer garden. Shortly after watching his sister and her friend leave, John spotted Sophy as she left the building alone, turning east toward Union Square, heading in the opposite direction from her home.
John slipped from his hiding place and followed behind Sophy at a distance, careful to stay hidden in the midst of the late afternoon crowd filling the sidewalks. Once or twice, he thought he had lost sight of her, but he was always able to find her dark hair again among the throng. When she reached Union Square, she stopped and looked around as if making sure no one was looking her way. John ducked into a nearby alley, observing her as she approached the Morton Hotel at the south end of the square.
A tall man in expensive clothes emerged from a nearby hansom cab and, after signaling the driver to wait, approached Sophy and took her arm to lead her through the grand doors of the exclusive hotel. The familiar way the man touched Sophy left little doubt as to what the relationship was between the dark-eyed beauty and the wealthy stranger. John fought a wave of nausea that threatened to overwhelm him as he left his hiding place and followed the two people into the hotel.
As he pushed through the hotel's ornate doors, he caught a glimpse of his quarry stepping into an otherwise empty elevator. As the elevator rose, he hurried across the plush lobby, watching until he saw the elevator stop at the third floor where he knew the elevator's only two passengers must be getting out. In John's mind's eye, he saw Sophy enter one of the elegant rooms, alone with a man who was obviously far richer than John could ever dream of becoming.
John turned his back on the elevator and surveyed the lobby until he spotted a place where he could position himself unobtrusively to watch for the return of the woman he had thought to make his wife. As he settled in to wait, he almost choked on the bile that rose in his throat as he imagined his beloved at that very moment in another man's embrace.
***********************************************************
"Three hundred dollars! Tonight!" Sophy Klienst felt as if she had been set free from a prison. At least she would no longer need to give herself to this man she loathed. This time had been the very last time.
"What makes you think I will give you such a sum of money?" Randall Bently wore an amused look as he eyed the naked young woman sitting in the large bed propped up by pillows and clutching a bedspread to her breasts.
"Because if you don't, I will go to Mr. Hearns and tell him how you have made his store your private hunting grounds. Not all men are evil like you. I've heard Mr. Hearns is a very moral man who goes to church every Sunday. I doubt he'll be happy to hear how you've abused his friendship and your families' friendship to prey upon young women for immoral purposes." Sophy watched the confident smile begin to fade from the socialite's face. "And if that is not enough, I'll go to your mother and tell her about our liaison. I could hint to her about certain other women you've used and discarded. Just because others have been too scared and intimidated by you to go to your family does not mean I am. I believe your mother would be interested to know you have gotten at least one of Hearns female employees with child. You see how much easier it is to give me the money so you can continue to enjoy your pleasures uninterrupted."
"What makes you think I can get that much money on short notice?"
"Don't play me for a fool. You spend more than that in a week at Hearns. I'm not being greedy. But I want the money tonight." Sophy watched as Bently finished dressing, trying to hide her shaking hand behind the coverlet she clutched for strength.
"You are right, Miss Klienst. That is but a paltry amount to me. Still it does gall one to be blackmailed, especially since I thought our little trysts were mutually pleasurable."
Sophy almost gagged at the memory of what she had let this primping, vile excuse of a man do to her body. "Never pleasurable to me, I assure you. Every time you touched me, it made me sick."
"Stop right there, Miss Klienst. I may tolerate being blackmailed, but I will not tolerate being insulted by a poor, pathetic creature such as you." Randall Bently adjusted his neckpiece and took one last look at himself in the mirror. "You really have no taste at all, young woman. Now how am I to get this money to you this evening?"
Sophy hid the wave of relief that washed over her. "Meet me in the alley behind Hearns in three hours when it will be dark. That is time enough for you to get the money and return. I want to be finished with you. After tonight, never approach me again."
"Let me enlighten you, my dear, by pointing out that there are many young women at Hearns who would welcome my attentions."
"Yes, well, just think of this as a way for you to continue using Hearns as your own private harem."
"That's true. I'll get the money and meet you at the appropriate time and place." The dapper man paused at the door before turning and sending his eyes traveling over the tops of Sophy's breasts, which were visible above the bedclothes. "I'd say it's been a pleasure, dear girl, but you were definitely not worth the price." With those parting words, Randall Bently left the room.
The shattered woman climbed out of bed, fighting back a wave of nausea. She began to slowly dress to leave. Inside, she felt as dead as her hopes for a life with the man she loved, a kind and gentle man whom she had betrayed beyond any possibility of forgiveness.
***********************************************************
Alex looked at the small nosegay of violets he held in his hand and then around the crowded beer garden. Even a seasoned detective might have a problem finding someone in the noisy sea of people crammed into the room enjoying their brief hours of leisure from an almost endless grind of work.
The evening was pleasantly warm for the first day of May, and the doors that formed the side of the building facing Sixth Avenue had been thrown open, and tables spilled out onto the sidewalk. A small band played a sprightly polka at one end of the room where a patch of floor was kept free for dancing. Even though it was early in the evening, a large number of couples were flying around the small dance floor in time to the lively music. Waiters dressed in gaily striped aprons snaked in and out of the crowd, hoisting large trays full of beer steins over their heads.
"Mr. Marshall."
Alex followed the sound of his name being called until he spotted Johann Kueshner, whose hearty voice reached Alex's ears over the din of the crowd. Alex made his way to the table where Tru's parents sat alone in the middle of a cluster of empty chairs.
"Mr. Marshall, how nice to see you again." Tru's mother smiled at Alex and gestured for him to take a seat.
"You look very lovely tonight, Mrs. Kueshner."
"Why, thank you, Mr. Marshall. Gertrude made this dress for me. She is quite talented at making clothes as well as hats."
"Well, the wearer does the dress credit. Where is your talented daughter?" Alex looked around but didn't see Tru anywhere.
Johann Kueshner took a glass of beer from a nearby waiter and placed it in front of Alex. "She is where any self-respecting young German would be... dancing the polka. Have a glass of good German beer and relax, Mr. Marshall. She will be here shortly."
Alex looked over the dance floor, trying to locate Tru among the swirling dancers. It didn't take long for him to spot her dancing on the arm of one of the twins. Just then, the music stopped, and the floor emptied of couples who returned to the tables to refresh themselves before the music commenced again. Alex watched Tru approach feeling as if the sun had emerged from behind the clouds. He felt suddenly lightheaded; or was it his heart that was always lighter when he was near Tru? She looked particularly beautiful at this moment, her face flushed from dancing and her eyes full of fun. Alex took a quick swallow of beer, trying not to stare so much that her family noticed.
Tru was finding that she was having trouble catching her breath, and it had nothing to do with the dance she had just finished. Her heart had started to pound the moment she had spotted the handsome detective sitting with her parents. When she reached the table, she was strangely speechless. Fortunately, her twin brothers were never at a loss for words, sparing her the necessity of talking.
"Look, noodle, it's your friend the detective. How are you, Detective? No more murders to investigate, we hope?"
Alex smiled at Tru's brother. Right behind Tru, Alex spotted Clarice hanging on to the other twin's arm. Still not able to tell the twins apart, he refrained from calling either one by name.
"Walter, please stop calling your sister by your silly pet names this evening." Tru's mother frowned in the general direction of her irrepressible son.
Alex stood, pulling out a chair for Tru next to him. The twins took their usual positions on either side of Clarice. Alex took note of which side of Clarice the one called Walter had sat. If the two men didn't move for the rest of the evening, he might be able to keep them straight. "I thought you might like these." Alex spoke in a soft voice, hoping her brothers were looking the other way. He had never before tried to impress a woman with almost her entire family looking on.
"They're wonderful. Thank you." Tru tucked the flowers into the waistband of her dress. "They look lovely with my dress."
The violets did go well with the dress Tru wore, and Alex said a silent thank-you to the savvy madam for her advice. "You look like a bluebird tonight." Alex bent close to whisper in Tru's ear. "A most delightful bluebird."
"How was your meeting today, Papa?" Tru was desperate to distract herself from the nearness of the detective. The look in his eyes was making her feel warm all over.
"It went well, my dear daughter, I believe the trade unionists are going to organize an independent labor party to run a candidate for mayor this fall. It is about time that Tammany Hall had competition for the votes of the people of this city. Perhaps that way we can see some real changes that benefit the poor and working class." Tru's father looked in Alex's direction. "I hope that doesn't offend you, young man. But it is true. Tammany Hall had grown too corrupt to be a real friend to the workers and the poor."
Alex's thoughtful gaze met that of the fiery newspaper owner. "I've seen a great deal of the corruption you speak of firsthand, and I can't say that I like it. But don't underestimate the deals it takes to keep a city such as ours running. There's much that is dark and ugly that has to be controlled so that decent families such as yours can live in peace."
"Perhaps, Mr. Marshall, but a less cynical view would say that it's better to appeal to what is good in people to change things for the better."
"I hope you're right, sir." Alex smiled as he sipped his beer. "Now if you'll permit me one small request? I'd like it if you all started calling me by my given name—Alex. Where is the rest of your family this evening, Mrs. Kueshner?"
Tru's mother frowned. "I really don't know, Alex. Both John and Carl were to join us as usual. I was certain John said he would bring his young woman with him. They both must have gotten held up at the print shop after Papa left. Perhaps there was an unexpected print job. The boys never turn down an order that would bring in more money."
Tru felt a sense of unease wash over her. "John was to bring Sophy Klienst? Are you sure?"
"Yes, liebling. He seemed most anxious for us to meet her at last."
Alex observed the change of expression that swept over Tru's face at hearing her mother's words. Tru suddenly looked very worried. What was there about this Sophy Klienst that was so disturbing to Tru and her brother?
***********************************************************
Sophy went ashen at the sight of John Kueshner standing a few feet away from her in the alley. The stricken look on his face told Sophy that her worst nightmare had become reality. Somehow, John had learned of her duplicity. Nothing else could explain the look of agony she saw in his gentle eyes, which shone bright with unshed tears behind his glasses.
"John, what are you doing here? You shouldn't be here." Sophy was desperate to get him away from the alley before Randall Bently arrived. She couldn't have him see the exchange of money. That would only make things worse for the man she loved.
"What am I doing? I am following the woman I love like a pathetic fool, stripped of all dignity and pride. Following the woman I asked to be my wife to watch her in the arms of another man." John stared across the small space separating him from the beautiful woman whose face he could barely make out in the dark shadows. But even in the dim light, he could see the guilt in her eyes that was the final proof of the truth from which he could no longer hide.
"Why, Sophy?" John took a step closer. He could have reached out and touched her, but the thought of ever touching her again made him sick to his stomach.
The revulsion she could see on John's face was like a knife slicing into Sophy's heart. It was more than she could bear that she had lost all the regard he had once felt for her. "John, you don't understand. You think you know what's going on, but you don't. I had to do what I did for others. But I never lied to you about my feelings for you. I love you."
John staggered back as if he'd been slapped. "Don't make it worse. You cheapen that word by saying it to me. I've heard enough lies from your mouth."
Suddenly, John's suffering was more than Sophy could endure. She threw herself at him, wrapping her arms around his neck and hanging on in desperation. "I do love you, only you."
John reached up to wrench Sophy's arms from around his neck, but she hung on, driven by one last futile hope she could reach him and erase the wrong she'd done. The feel of her soft, warm body against his was the final mockery of all his hopes and plans. Driven beyond all reason, he reached up to grab Sophy's hair, twisting it cruelly as he ground his mouth against hers in a kiss meant to inflict pain. Then sickened by his actions, John used all his strength to fling Sophy out of his arms. She fell to the alley floor, sobbing. John stared down at the crumbled figure lying in the dirt and then turned away and ran as fast as he could from the sight of the woman who had betrayed his every dream.
Sophy remained on the ground for several minutes and then slowly crawled to her knees. She had to gain control of herself before Randall Bently came to deliver the money. After all that she had lost, the money was all that mattered. As she pushed herself up off the ground, her hand touched an object lying in the dirt beside her—John's glasses. They must have fallen off as he pushed her away. Sophy picked them up and held them cradled in her hands. Tears streamed down her cheeks. They were all she would ever be able to touch of the man she had once dreamed of holding for a lifetime.
***********************************************************
Carl hurried to finish cleaning the press. He had sent his father home early to get ready to go to the beer garden later that evening. His father had been reluctant to leave with John gone, but Carl had insisted. What Carl couldn't tell his father was that he wanted to work late so he could avoid joining the rest of his family at the beer garden. There was only one place he wanted to go after he finished his tasks, and that was to be with his beloved Maria. Carl wiped his hands on a rag, which he then sent sailing across the room as his frustration overwhelmed him. There had to be some way for him to make Maria see that they should be together. A loud racket outside the door to the newspaper office startled Carl out of his private thoughts. He left the printing room to investigate. Just as he entered the front office, the door came crashing open, and in stumbled John.
"John, are you hurt?" Carl hurried across the room in time to save his brother from falling to the floor. "Where are your glasses?" As Carl struggled to hold on to his brother's slumping body, he became aware of the fumes that clung to his brother's clothes. "John, have you been drinking?" Even as Carl asked the question, he spotted a bottle of whiskey clutched in his brother's hand.
"Sophy, . . . why? . . . Why?" John's eyes wouldn't focus as his head rolled back against his brother's shoulder. "I loved you . . ."
Carl watched in dismay as his brother doubled over and emptied the contents of his stomach on the floor. Carl didn't need to hear any more of his brother's tortured murmurings to know that somehow John had found proof of Sophy Klienst's treachery.
"John, listen to me. It's Carl." The Kueshner men were not heavy drinkers, enjoying only an occasional glass of beer at the beer garden. John drank least of all, so whatever alcohol he had consumed this night had left him in a complete stupor. Carl could get no response from him that made sense.
"A whore... nothing but a whore." Tears streamed from John's eyes as he sagged against his brother. Carl finally managed to lower him into the nearest chair. Whatever John had seen, it had sent him into his own private hell. Carl suddenly felt consumed by a red-hot anger at the woman who had betrayed his brother. It threatened to overwhelm him. But he had no time to think of that. Instead, he had to think how he could keep the rest of the family from seeing John in this state.
"It'll be all right, John. I'll take care of you." Carl knelt beside John's chair to keep him from falling to the ground and held his brother in his arms.
"Why? . . . Sophy... why?" John's voice trailed off as he slipped into unconsciousness.
***********************************************************
The only sound in the large room was the rustle of Maria Von Ott's skirts as she paced back and forth in front of the fireplace. Carl was very late. Perhaps he was not coming. Perhaps she had repulsed him the evening before when she had succumbed to her hidden desires and allowed herself to fall into his embrace. She had meant what she had said when she told him their feelings could not be known outside this room. But she had not meant to drive him away. She was sick at the thought of never seeing him again.
"Madam."
"What is it, Rose?" Maria tried not to betray the emotions churning inside her as she turned to face her maid, who stood in the doorway wearing a troubled look.
"Mr. Kueshner is here. But he has a man with him who appears to be ill. He asked me to fetch you."
Maria flew past the startled woman and down the hall. Just inside the door stood Carl holding up a man whose resemblance to her beloved could make him no one else but one of Carl's brothers. "What is the matter, Mr. Kueshner?" Maria forced herself to slow down and appear calm.
"I'm sorry, Countess, to bother you, but your home was the closest place I could think to bring my brother. I hate to impose on your kindness yet again, but could I beg you to give him a bed for the night?"
"Of course, that is no trouble at all. Rose, fetch the coachman and have him help Mr. Kueshner take his brother upstairs. Put him in the blue bedroom. Do you want me to send for a doctor for your brother?" Maria watched the maid leave to get help and then turned anxious eyes toward Carl.
"No doctor, Maria. I'll explain when we are truly alone."
"Don't worry, darling. I'll wait for you in the music room. Come to me as soon as you get your brother comfortably settled." Maria left Carl as soon as the maid and coachman returned. Their appearance made Maria realize that she had spoken aloud the endearment she used only in the privacy of her thoughts.
After what seemed like an eternity, Carl appeared in the doorway to the music room. He quickly closed the doors behind him, locking them to be sure they would have no intrusions. He crossed the room and stood in front of Maria, pulling her into his arms. "I've wanted to do this every minute that has passed since I left you last night."
Maria kissed Carl's lips with a hunger that she didn't think she was still capable of, but she stopped herself before she was completely swept up in the joy of being in his arms once more. She touched his cheek, worried about what had forced him to bring his brother to her house late at night. "Did you get your brother settled comfortably?"
Carl let go of Maria and began to pace the room that had become his haven. "Yes. You don't know how grateful I am that you let him stay here for the night. I couldn't take him home like this. My parents would never understand. They would be terribly hurt."
Maria watched anxiously as Carl continued to move restlessly around the room. Finally, he sat down at the piano and began to play some of the folk melodies that always comforted him. He looked up from the piano as Maria came to sit beside him while he played.
"What happened to your brother tonight, Carl?"
Carl's hands came crashing down on the keys. "A woman betrayed him. He waited a long time before giving his heart to someone, and she betrayed him. I could kill her."
Maria could barely believe her gentle musician was saying these things, but she could see in Carl's eyes how deeply he was feeling his brother's anguish. Desperate to ease his pain, she began to play and sing softly, "You are repose and gentle peace. You are yearning and its fulfillment."
After a few bars, Carl joined his baritone to Maria's soprano, and together they sang one of their favorite lieder by Schubert, "Banish other sorrows from this breast. Let my heart be full of your joy."
Maria's voice faded as Carl gazed into her eyes. In his beautiful baritone, Carl finished the song alone. "This vision of mine, illumined only by your radiance—fill it entirely." Carl's mouth came down to capture Maria's when the last notes of the song ended. There was a long silence as Carl lingered over their kiss. When they at last parted, he reached out to touch Maria's cheek gently. "Your love could fill me with its entirety for all of our days, Maria, if only you'd let it."
Maria didn't say anything. Instead, she took Carl by the hand and led him from the piano to a soft rug in front of the fireplace. She lowered herself onto the rug, pulling Carl down beside her. "Let's not talk anymore of the future. My life has taught me not to trust the future. But I have decided that this moment belongs to us, and I will surrender to it totally to have the beautiful memory of our love being fulfilled." Carl started to speak, but Maria stopped his words with her lips. When she finished kissing him, she pulled his head gently into her lap. "Just rest here in my arms."
Carl buried his face in the soft folds of Maria's silk skirt and smelled the fragrance that would forever remind him of this beautiful woman and the love she had aroused in him for the first time in his life. Unable to stop himself, he reached up and pulled Maria down until she lay beside him. Then he rolled over until he was sprawled on top of her, his legs entangled in her skirts. He seized her mouth and kissed her with all the pent-up desire he had fought for so long. He could feel her melting under him. Her body was growing as soft and pliant as her lips, which were now devouring his with a hunger equal to his own.
"Carl, I can't stand it." Maria stared up into Carl's eyes, all restraint gone. "I have to be one with you. Please take me."
Carl couldn't believe his ears. He was wild with excitement at the thought of possessing this woman whom he loved so deeply and yearned for almost to the point of madness.
"Have you ever been with a woman, darling?" Maria studied the handsome young face hovering above her.
"No." Carl was as afraid as he was eager, but he would let nothing stop what they were about to do, not even his own inexperience.
Maria guided Carl's trembling hands to the ribbon ties on her bodice, letting him know she wanted him to begin to undress her. Carl slowly began to reveal the white chemise under her dress and the smooth silken skin of her shoulders. Maria reached up to help push the bodice of her gown down from her shoulders until the gown was gone entirely and she lay in only her chemise. With his desire growing beyond his control, Carl hurriedly removed her undergarments and began running his hands over every inch of the soft skin now exposed to his touch.
His lips soon followed where his hands had been, and he felt his lovely countess start to tremble as he grew bolder. He cupped her breasts and covered her face and neck with his fervent kisses. Watching the feverish yearning growing in Maria's eyes finally gave Carl confidence that he could satisfy this woman he adored. He tore off his clothing and gave himself over completely to the joy of discovering every secret place that had been hidden from him.
Maria felt his desire becoming more intense and began to arch herself toward him, making small whimpering sounds as her hands explored the beautiful young body that had been hidden under his clothes. She kissed every inch of the smooth skin of his broad chest. She marveled at the feel of his strong arms under her fingertips. It had been so long she had forgotten the way the hard muscles of a man played under their warm flesh. Her eyes devoured him. She knew she was being wanton, but her need for him was too strong for her to control.
Never in his wildest dreams had he imagined the ecstasy he was feeling as the woman he loved offered herself to him with no reservations. "I love you, Maria, more than life itself. If you only knew how the very first sight of you made my heart ache from your beauty."
The world outside seized to exist as the two lovers gave in completely to their passion. They murmured desperate words of love to each other as they climbed to that moment when nothing else is real except the union of a man and a woman, body and soul, and then they reached the instant in time when two people tumble over the edge into paradise together.
When at last their hearts quit racing and they lay quiet in each other's arms, Carl pulled Maria even tighter against the curve of his body, whispering into her hair, "I love you. There is nothing the world could do to make me change my feelings for you. Please trust that my love will last forever."
Maria smiled sadly at the absolute certainty of youth. "Let's just love each other as long as we can, my darling. Don't underestimate the power the world has to separate us from those we love."
Carl only tightened his hold on Maria as he watched her drift off to sleep in his arms. Tomorrow he would try to find a solution to his problem and that of his brother. But for now, his heart was full of the music of love, and he refused to stop listening to its melody.
***********************************************************
"Would you care to dance, Miss Kueshner?"
Tru glanced up at Alex and wondered again for the hundredth time this evening what was the power this man had to turn her whole world upside down with just one look at his crooked smile. All night he had traded opinions with her father and cheerfully borne the ribbing of the twins. He had set out to charm her mother and flatter Clarice and, in general, make himself agreeable to her whole family. Alex's cheerfulness had helped distract her parents from their growing concern over the absence of their two oldest sons. But, most of all, he had made Tru feel that it might be possible to trust her heart to the big detective with eyes like a golden cat.
"I don't blame you for hesitating." Alex extended his hand out to Tru. "But I give you my solemn promise not to step all over those lovely shoes. I've had a word with the bandleader, and I believe this is our waltz."
Tru put her hand in Alex's, trying to hide how it trembled. She followed him onto the dance floor, where she stood very still as he put one arm around her waist and his other hand closed gently over her own. Then he pulled her so tightly against him that she could feel the heat and hardness of his body through his clothes. Tru thought she should pull back from him, since they were closer than was strictly proper. But when Alex smiled down into her eyes, she forgot everything else as he began to whirl her around the room in time to the music.
The strains of the Emperor's Waltz washed over Tru. The beautiful music and the feel of Alex's arms around her waist as they spun around the floor overwhelmed her with sheer delight. Never had she felt so carefree and happy. She threw her head back and laughed with the sheer joy of being alive.
The sound of Tru's laughter made Alex's heart skip a beat. He tightened his hold around her waist, losing himself in the feel of her slender body moving as one with his own as he guided her around the room. The vision in his arms glanced up at him, and Alex knew he could lose himself forever in the depths of her beautiful blue eyes. At that moment, he wanted this waltz to never end as he circled the floor of what now seemed like an enchanted palace instead of an ordinary beer garden.
Just when Tru could have sworn they had sprouted wings and were soaring above the other couples in a place where only music and laughter and love could exist, the music stopped. They clung to each other just a moment longer than they should have before reluctantly drawing apart.
"I swear you have wings, little bird."
Tru smiled at Alex, wanting to prolong the magic they had just experienced. "You make it easy to fly."
Alex wanted nothing more in that moment than to take hold of his luscious lady in blue, kissing that sweet mouth that was tempting him beyond all power to resist until it moaned and parted under his assault. But a certain powerful stirring reminded him of the consequences of such a rash act carried out in full view of Tru's parents and her very large and overprotective brothers. When they reached the table, they were surprised to find only Tru's parents there.
"Where are the twins and Clarice?" Tru knew her mother was worried by the look on her face.
"The boys have taken Clarice home. It was time for her to return to her parents. Your father and I are going home also. I'm sure that Carl and John have a good reason for not joining us tonight, but I am a bit concerned that there was a problem."
Johann Kueshner put a protective arm around his wife. "No matter that they are grown men, you worry about them as if they were still in short pants." Despite his light words, the look on Tru's father's face betrayed his own worry. "So you come home with us, daughter?"
"Sir, would you allow me to escort Tru home?" Alex was not quite ready to end this special evening.
"Of course. The hour is not that late. Stay and enjoy yourselves a little bit longer. If you aren't safe with one of New York's Finest, then with whom?" Tru's papa gave her a kiss on the cheek, and after kissing her mother good-bye, Tru's parents left arm in arm. Alone with Alex, Tru was suddenly overcome with shyness and sat for a few minutes, unable to say a word or even look at Alex.
"Why so quiet, noodle? You usually have plenty to say." Alex laughed as Tru's head snapped up and her fiery eyes met his. "I thought that would bring a response. Come, let's leave and walk a bit along the avenue. The stars are out, and the air is warm and fragrant. I've had my fill of the smell of stale beer and cigars."
Tru allowed Alex to drape her evening cape over her shoulders and maneuver her out of the crowded beer garden. Once they were on the street, Alex was careful not to touch Tru. But as they strolled down Sixth Avenue, his whole body ached with the need to take her in his arms as he watched her lovely profile glowing in the moonlight.
"Look, Alex, an empty store." Tru stopped to stare into the deserted storefront. "Someday I'll have a millinery shop in a place just like this."
"One where pigeons can come and buy hats."
Tru looked up into Alex's eyes, but she saw only warmth there. Somehow, she knew now that this special man would never mock her dreams. "Only pigeons with better taste than a certain detective I know."
Alex's grin faded as his hand reached out to caress Tru's cheek. "This detective has enough good taste to appreciate the loveliness standing before him now." Alex whispered his words as his lips slowly descended to capture Tru's. Tru had never experienced anything like the complete surrender of her mouth under the onslaught of Alex's kiss. As he pulled her to him and molded her body against his, her lips parted of their own volition. She could feel herself growing light-headed as Alex gently explored her mouth with his soft and tender kiss.
After a few long moments of enjoying the taste of Tru's lips, Alex pulled back to look into her eyes, which seemed to reflect every star overhead. "You are so beautiful." He captured her face between his big hands and slowly kissed her forehead and her eyelids and the tip of her nose before returning once more to worship her mouth.
Tru reached up to wrap her arms around Alex, feeling the iron muscles beneath his coat. When he finally loosened his hold on her, she clung to him, breathless, and pressed her lips against his neck, smelling the intoxicating aroma of his skin.
"Excuse me, but this is a public street. You two had best be getting along." Alex and Tru jumped apart, staring, dismayed, at the uniformed policeman who stood under the nearest gaslight twirling his nightstick and frowning in their direction.
"Certainly, Officer, whatever you say." Alex put his arm around Tru and hurried her away from the scandalized policeman. The rest of the walk home passed in blissful silence until at last they stood on the steps of Tru's house.
"I should have thanked that fellow back there." Alex raised Tru's hands to his lips and slowly kissed each small fingertip before turning them over to place a kiss on each palm.
"Why would that be?" Tru was finding it hard to speak with Alex's lips making her skin tingle everywhere they touched. She was anything but grateful for the interruption that had left her feeling in need of something she didn't quite understand.
"If he hadn't happened along, I would have forgotten myself completely." Alex pulled Tru into the circle of his arms.
"Would that have been so bad?" Tru cuddled closer against Alex's chest until no light could have passed between them. She wanted him to kiss her again as she shyly reached inside his coat and ran her hands over his broad shoulders and down the front of his crisp white shirt.
"Very bad for me." Alex captured her hands and held them against his chest. Her innocent exploration was driving him wild. "A man can only resist for so long."
"Then why resist?" Tru ran her finger along Alex's lower lip before bringing her mouth up to meet his once more.
When Alex could speak again, he forced himself to step back from Tru. "Because when I finally make you my own, I don't want anything to stand between us."
"What could stand between us now?" Tru looked into Alex's eyes and saw a strange veil drop over them.
"I hope nothing, little bird. I hope nothing at all."
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Sophy glanced down at the money she had just placed in her reticule and then at the glasses that she held in one hand. Slowly she drew the ribbons closed of the pouch containing the sum total of everything she had sacrificed for, trying to believe it had all been worth it.
The alley was dark and deserted. Randall Bently had departed, leaving her alone again. Suddenly, Sophy was nervous. She had managed to pull herself together before her blackmail victim arrived, but the sense that someone was watching her from the shadows had been with her since John had run from the alley too repulsed by her actions to remain in her presence. She would be glad when her brother arrived to take the money and escort her home.
Sophy tried not to think about the look on John's face as he stood staring down at her where she lay in the dirt at his feet. But the memory persisted despite her efforts to keep it at bay. Her eyes filled with tears. She leaned her head against the hard, rough wall of the store, and while she stood sobbing, a hand reached quietly out of the darkness to end her unhappiness forever. A sharp stabbing pain in the back and a surprised gasp that was not much more than a sigh, and Sophy Klienst lay in the dirt again. But this time, she would never get up.
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Chapter 8
Alex stood looking down at the corpse. A knock on his door in the early morning hours had awakened him out of a sound sleep and sent him hurrying to the alley behind Hearns. A dead body had been found in the wee hours of the morning by a wagon driver picking up garbage at the rear of the department store. The manager of Hearns was summoned and identified the corpse as Sophy Klienst, a name with which Alex, unfortunately, was all too familiar. Sophy Klienst was the name of the woman John was supposed to bring to the beer garden last night. The name that whenever it was mentioned around Tru or her brother Carl caused them to look worried.
Richard knelt beside the corpse and stared at the bloodstain that covered the back of what once had been a pretty green-checked dress. "A single stab wound through the back and into the heart. But no sign of the weapon that was used." Richard gazed up at his partner, who remained strangely silent. "Whoever did this was very angry or very strong. That wound appears to be the result of a single forceful blow." Richard rose to his feet and waited for Alex to say something.
"Do you, gentlemen, need me anymore? I'd like to leave if you don't mind. This has all been most unsettling. But life must go on, and I have things to do, since it is my only day off."
Alex turned toward the manager, who stood nearby trying not to look at the body lying crumbled in the dust. The man's skin was pale with a slightly green hue as he pressed a pristine handkerchief to his nose. Alex knew the man was trying to block out the smell of death, a smell that was nothing new to Alex after years as a New York detective.
"Just a couple of more questions, Mr. Hartley. We appreciate you coming here on a Sunday to help us identify the body." Alex's words were polite, but his face told the nervous manager that he could leave only when the detectives were done with him. "I'm sure you appreciate the gravity of the situation."
"Of course, Detectives, of course." Wilfred Hartley shuffled his feet. He was actually only thinking about his hot breakfast growing cold on the table at home. His housekeeper had just begun to serve him his morning coffee precisely at nine o'clock as always when a burly policeman banged on his door and dragged him away from his morning routine and all because some silly salesgirl had gone and gotten herself murdered behind Hearns. What a bad piece of luck for the store. They might actually have to postpone tomorrow's sale on spring gowns.
"Was there any reason for her to be here last night that would have had anything to do with the store or her job?"
Wilfred Hartley looked positively apoplectic at the thought. "Of course not! My goodness, we would hardly have our salesgirls work over after the store closes. Think of how much extra money the store would be paying out." The Hearn family would not be pleased if a murder were connected to the giant emporium, since it might be bad for business. "When the store closes, the sales force goes home. I'm sure this has nothing whatsoever to do with Hearns. We only hire respectable young ladies."
"She did work at Hearns, and the body was found behind Hearns." Alex raised one eyebrow. "It is logical to think there may be some connection."
Little beads of sweat were beginning to appear on the store manager's forehead. Wilfred Hartley raised his handkerchief to dab them away. "I can't imagine what it would be."
"It's our job to find that out." Alex paused for a moment, fixing the jittery man with a hard stare. "Did Miss Klienst have any problems with anyone at the store that you know of? Anyone she was fighting with?"
"We have no problems like that at Hearns." The store manager's offended sniff would have amused Alex under different circumstances. "Our people all get along. We are just one big happy family at Hearns."
Alex thought about the darker side of human nature that he witnessed every day. Put a group of people together and apply pressure, and there were always problems—some leading to violent conclusions.
"And before you ask, Detective, I know absolutely nothing about Miss Klienst's private life. I don't make a habit of becoming personally involved with those who work under me."
Alex was hardly surprised by that statement. He looked in the direction of his partner. "Anything you want to ask, Mr. Hartley?"
Richard looked up from the body he had been studying. "Can't think of anything right now."
"In that case, you are free to go, Mr. Hartley. But we might have more questions for you later." Alex watched as the store manager almost ran from the alley in his relief at being allowed to leave. As Alex watched the man scurry away, he fought a wave of disappointment. The manager had provided them with scant information, and Alex had desperately wanted the officious little man to supply him with another suspect besides the one whose name kept floating to the front of Alex's mind.
"The sergeant said the woman's body is exactly as he found her." Richard studied Alex's face. Something was bothering his friend. "He said the victim's reticule was lying beside her. All that was in it were a few coins and a handkerchief. We can talk to him again if you think it'll help."
Alex ignored his partner's remarks as he walked slowly around the body of Sophy Klienst. Her face lay sideways in the dirt, yet just a glance at her profile was enough to show Alex the woman had been a striking beauty. Alex could well imagine that such a woman would inspire strong feelings in many men, perhaps even in the loyal and cautious John Kueshner. Alex leaned down to study the ground around the corpse for footprints, but the dirt in the alley was packed down from the many horse-drawn wagons that used the back entrance to Hearns to deliver goods.
"I've found something."
Alex looked over toward where Richard was gently moving one of the victim's arms from under her face. Clutched in her hand was a pair of spectacles like the ones Alex had seen Tru's brother John wear.
"What do you make of these? They're definitely not hers." Richard examined the glasses he held in his hand. "Too big. They look like they fit a man's face."
Alex reached out to take the newfound evidence from his friend. There was nothing left for him to do now but go to John Kueshner's home and confront the man. All the evidence was pointing in the direction of Tru's oldest brother, and Alex could no longer ignore that fact. The memory of the prior evening and the feel of Tru in his arms threatened to overwhelm Alex. For one brief moment, he fought the impulse to crush the glasses and throw the pieces as far away as he could. But that was not possible. Even if Richard hadn't already seen them, Alex knew the detective in him was too strong to ever walk away and leave a case unsolved no matter where the trail led.
The fear of losing Tru nearly brought the big detective to his knees. He knew that it was a very real possibility that the woman who had lodged herself in his heart would turn from him the instant he threatened one of her beloved brothers. Alex straightened his shoulders. He had done difficult things before in the line of duty, and he wouldn't shrink from this. And if Tru's brother was innocent, as Alex wanted to believe, then he would find a way to prove it.
The sound of the police wagon arriving to remove the body brought an end to Alex's painful thoughts. "Come on, Richard. As soon as we help load the body, we have somewhere to go. We'll have the sergeant who found her give her family the bad news."
Richard was concerned. His longtime friend wore an almost desperate look as Richard watched Alex clasp and unclasp the spectacles found in the victim's hand. "Where are we going anyway? Do you already know something about this woman?"
"I'll explain on the way. For now, let's get this poor creature out of this alley. No one deserves to lie in the dirt when they're dead and gone."
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Tru flew to the door at the sound of the first knock. The whole house was on pins and needles because neither Carl nor John had come home last night. When she threw open the door, despite her worry, she was delighted to see Alex Marshall standing on the stoop.
Alex stared at Tru's upturned face and thought how beautiful she looked. Her loveliness always struck him anew each time he saw her after an absence. Alex wondered briefly what it would be like to see that adorable face the first thing each morning for the rest of his life. The need to take her in his arms almost overwhelmed him. He nearly reached out to pull her close, but a discreet cough behind him brought Alex back to reality with a jolt.
"May we come in, Miss Kueshner?"
Tru was surprised by Alex's formality, but she gladly stepped aside to allow him to enter. As he did, she realized that Alex was not alone. A young man with a boyish face followed him into the house. Tru studied Alex's companion for an instant and was struck by the contrast between the man's youthful features and a certain tired worldliness she thought she saw in his eyes.
"This is Richard Shaw, a fellow detective. Richard, this is Miss Gertrude Kueshner."
Richard nodded in response to the introduction. Gertrude Kueshner was indeed lovely. Richard knew Alex well enough to read how hard his longtime friend had fallen for the lithesome beauty with the amazing blue eyes. But after hearing what Alex had to tell him on the way to the Kueshner home, Richard was afraid his friend had no choice but to view John Kueshner as the primary suspect in their latest murder.
"Tru, who is at the door?" Martha Kueshner came down the hall and stopped at the sight of the two men standing with her daughter. "Alex, it is you. It is good to see you. But to what do we owe this pleasure?" Tru's mother smiled at Alex, awaiting an introduction to his companion.
"Mrs. Kueshner, this is my partner, Richard Shaw."
"Would you, gentlemen, like some tea or coffee? I have a fresh cherry strudel I just took out of the oven."
Tru smiled. Her mother always baked whenever she was very worried or very happy. Unfortunately, this morning, it was worry that drove her mother into the kitchen before the sun was even up.
"Thank you for the offer, Mrs. Kueshner, but I'm afraid Detective Shaw and I are here on official police business." Rarely had Alex felt worse in his life than he did at this moment.
Martha Kueshner's hand covered her heart as fear swept over her. "Has something happened to my boys? Mein Gott ein himmel, please tell me they are not dead." Tru put her arm around her mother to support her while fighting her own wave of fear.
"No, no." Alex hastened to reassure Tru's mother. "We have no such news."
Tru shot an indignant glance toward Alex. "Well, you might have made that clear from the start. What is wrong with you, Mr. Marshall, to frighten her so?"
Alex winced at Tru's angry words. This was going badly, and it would only get worse.
Richard Shaw stepped forward, trying to spare Alex a little of this ordeal. "We need to see your son John, Mrs. Kueshner. May we speak to him, please?"
"My son is not here. Why do you wish to speak to him?" Tru's mother knew something was terribly wrong by the look on Alex Marshall's face. The usually twinkling eyes of the detective were clouded over as he stood rigidly by as his partner spoke. At the sound of heavy footsteps coming from the direction of the kitchen, Tru and her mother turned, relieved to have been joined by Johann Kueshner and his twin sons. The three somber-looking men joined the group by the door.
"Perhaps we should move to the parlor to continue this discussion." Tru's father led the way into the homey room and stood by the window, occasionally looking out toward the street, watching for his missing boys. His wife joined him at his side. Their children stood nearby, a sturdy twin taking a stance on each side of Tru. All three siblings glared in the direction of the two detectives.
Alex couldn't help but admire the strong bond that caused this family to present a united front at what they sensed was a threat to one of their own. Tru's chin was thrust out in defiance as if she already knew that a heartbreaking betrayal was imminent.
"We would like to speak to John, sir." Alex repeated his partner's earlier request.
"John isn't here right now, Detective. He is with his brother." Johann Kueshner had already been to the print shop this morning looking for his sons. But there had been no sign of them.
"Did they leave this morning for any place in particular?" Alex could see how worried Tru's father was about something.
Tru's mother turned pleading eyes toward the detectives as her husband's arm tightened around her. "They never came home last night. They have never before stayed out all night. We are most worried about them. Perhaps you can help us to find them?"
Tru winced at her mother's entreaty, while her own gaze never left Alex's face. "I don't think the detectives are here to help us find Carl and John, Mama." Tru looked into Alex's eyes as she prayed her world wasn't about to be shattered. "Why do you want to talk to my brother, Mr. Marshall?"
Alex's eyes met Tru's, and they both knew what he said next might erect a wall between them that would put an end to the promise of love that had begun only a few short hours before. "Sophy Klienst was murdered last night. Stabbed in the heart." Alex's words exploded through the homey room like a curse shouted out in the middle of a cathedral.
"And why do you want to see my son?" Johann Kueshner's sharp mind perceived the threat to his offspring despite his shock at such horrible news.
"Unfortunately, we have evidence that points to your son John as a suspect in this crime." Alex hated the sound of his ugly words even as he spoke them.
A storm of protest erupted at his statement as every Kueshner in the room let the two detectives know that such a thing was not possible. Alex knew he would feel terrible over what he was doing to Tru, but he hadn't known how much it would bother him to see the hurt and anger on the faces of the rest of Tru's family. Martha Kueshner's next words were like a knife to his own heart.
"How could you believe a child of mine would be capable of doing such a thing? You sat at our table and broke bread with us. You know this is not who we are." Tru's mother could see how hard this was on the detective. "Surely, you can't believe such a thing is possible."
"I'm sorry, Mrs. Kueshner, more than you can possibly know. "Alex glanced at Tru, who was watching him now with something close to hatred in her eyes. "But there is strong evidence pointing at John. If he can account for where he was last night, perhaps we will not have to take him in, but we must talk to him."
The sound of the front door opening did nothing to relieve the tension that crackled in the air like heat lightning. All heads turned to watch as the two missing members of the Kueshner family started to walk past the parlor.
"We are all in here, boys." Johann Kueshner called out to his sons. A small part of him wanted to shout at them to turn and run away as fast as they could. But he knew running was never the answer. His son was innocent, and that would be quickly established. But the sight of his eldest sons as they entered the parlor sent a ripple of unease through him. Something had happened last night. John looked ashen and drawn. Carl's concern for his brother was evident by the way he helped his visibly shaken brother through the doorway.
Tru stifled a cry when she saw her brothers. One look at John's face, and she knew he had been through some kind of ordeal. Only one thing would have brought John to such a state. Before she died, Sophy Klienst had betrayed John somehow. But no matter what she had done, Tru was certain John would never harm the woman he had loved so deeply.
"Liebling, what is the matter?" Tru's mother flew to her son's side. Together, she and Carl led her eldest to a nearby chair where she stood behind John trying to comfort him. A silence fell over the room as all eyes turned toward the intruders in the family's midst, waiting for the detectives to speak.
Alex was disturbed by John's appearance. It was impossible not to conclude by his state of dishevelment that the man had been in some kind of trouble. Alex could only hope there was a simple explanation that would relieve him of the obligation to arrest the obviously distraught man.
"John, my partner and I need to ask you a few questions."
"About what?" Carl answered Alex, while John remained silent, sitting with his head buried in his hands.
"The murder of Sophy Klienst."
John's head shot up as he stared, unbelieving, at Alex. Tru was furious at Alex's obvious attempt to trick her brother into some kind of revelation. If that was his game, then he must see how shocked John was at the news.
Alex was glad that John appeared totally surprised. It added to Alex's own conviction that this man was not a murderer. But the detective in him knew it was not enough to keep John from being taken to jail unless he could explain where he was all night and what had happened to him.
"Sophy's dead?" John barely seemed aware of the people around him. "I didn't want that. No matter what she had done, she didn't deserve to die alone in an alley."
Alex's heart fell. Only if he had been there could Tru's brother have known Sophy Klienst was murdered in an alley. When his partner caught his eye, Alex knew Richard had not missed the damning statement. "Where were you last night, John?"
"I'm not sure." Confusion clouded John's eyes. "I don't remember much of the night after I started drinking."
Tru's parents exchanged worried glances. Their eldest son had never been drunk in his life. It would take something terrible to cause this studious man to drink himself senseless.
Alex turned his attention from the figure slumped in the chair to Tru's other brother. "Were you with John all night, Carl?"
"Yes." Carl flinched under the intense stare of the two detectives. "Most of the night. He came to the print shop, where I was working late."
Richard Shaw could see the battle waging in Carl Kueshner. "Was he already drunk when he came to the shop?"
Carl hesitated before reluctantly answering. "Yes."
Richard glanced at Alex before he asked his next question. It was not looking good for Tru's oldest brother. "Where were you both all night?"
"We slept at the print shop."
Alex saw the startled look that Johann Kueshner sent his son. Carl was lying. Alex could see it on the older man's face. But even a man as honest as the fiery printer would hesitate to deliver one of his children into the hands of the law. Alex couldn't blame Tru's father for remaining silent. Already, Tru's parents looked as if they had aged ten years in a few short hours. There was one last question Alex had to ask. If the answer was what Alex feared, he would have no choice but to take John away from his family.
"John, where are your spectacles?"
The dazed man felt slowly in all his pockets but came up empty-handed. "I can't seem to find them."
Alex squared his shoulders under his partner's sympathetic eyes. "John Kueshner, I must take you to police headquarters, as you are under suspicion in the murder of Sophy Klienst." Alex tried to block out the sound of Martha Kueshner sobbing as she clung to her oldest son.
Johann Kueshner stepped in front of his youngest sons when they appeared ready to attack the two policemen. "No, boys. We will straighten this out. The law will protect your brother because he is innocent. This is America, not the old country." The older man watched as the two detectives stepped forward to take hold of his son. "Go with them, John, but have no fear. We will make sure you are not gone long."
Tru thought her heart was breaking as she watched the man she had begun to love pull her brother up from his chair and start to push him gently down the hall and out the door toward the waiting police wagon. The detective with Alex took a pair of handcuffs from his pocket, but a slight shake of Alex's head and he put them back. Tru supposed she should be grateful to Alex for sparing her parents the sight of their eldest child being led away in handcuffs, but at this moment, all she could feel was the pain of Alex's betrayal.
Tru watched in stunned silence as the three men reached the door. But when she heard the door open, she was seized by an impulse and ran down the hall as fast as she could. Alex turned at the sound of her footsteps and stood frozen on the stoop as his partner hurried to place her brother in the police wagon.
"How could you do this, Alex?" Tru's voice was eerily soft as tears streamed down her face. But her eyes flashed fire at the man in whose arms she had waltzed in what now seemed like a dream from a different lifetime.
Alex looked into the face he had come to love, his eyes begging her to believe in him and the possibility of their love. "I can't do anything else, Tru. You must know that." Then he turned away from the sight of Tru's anguished face and almost ran to join his partner.
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Tru couldn't stop thinking about her brother spending the night in jail all alone. She stood by her counter, staring unseeingly across the vast store, oblivious to the people walking by her. Yesterday had been the worst day she could remember her family ever going through. Her mother had tried, but she couldn't hide the fact that she wept the entire day. Tru had never before seen her father look so worried, his usual air of cheerful confidence gone. The twins had stomped around the house like two caged lions. It had taken the combined efforts of both Tru's parents to persuade the unhappy young men to return to college on the last ferry that evening.
Carl had shut himself up in his room right after his brother was led away. He would speak to no one, not even Tru, who had stood outside his door, begging him to let her in. Whatever Carl knew about what had sent John into a state of despair, he was not ready to share it with his sister.
"The delightful Miss Kueshner lost in the clouds. May I join you up there?"
Tru nearly jumped out of her skin at the sound of the unwelcome voice. Randall Bently was the last person Tru wanted to speak to today. "Mr. Bently, may I help you?" Tru kept her tone polite, but a smile was beyond her this morning. Even if Mr. Hartley had been standing right next to her, Tru couldn't have managed her usual solicitous manner.
"You seem out of sorts, Miss Kueshner. Perhaps I could remedy that. Join me for lunch. We could dine nearby. Or, better still, allow me the privilege of dining with you this evening at the Delmonico's. Have you ever dined there?"
"No." Tru's curt answer should have cut short the man's flirtation. But the socialite seemed particularly persistent this morning.
"Well, we should remedy that. Delmonico's would be the gayer for having someone as lovely as you grace it with your presence."
Tru struggled with her temper and lost. "Mr. Bently, I'm hardly in the mood for frivolity. Perhaps you haven't heard, but Sophy Klienst was murdered last night."
"That lovely creature who sold perfume?" The socialite glanced in the direction of the perfume counter, where a plump blonde woman now stood waiting on customers. "What an incredible waste. Do the police have any idea who would commit such a terrible crime?"
Tru hesitated before answering, but she knew the papers that afternoon would be full of the story of her brother's arrest. She stuck her chin in the air and looked Randall Bently straight in the eyes. "Right now, the police are holding the wrong man for the crime. They've arrested my brother, who is completely innocent."
The wealthy man looked momentarily startled at Tru's outburst, and then he recovered his usual aplomb. But before he placed the bored smile that was his customary expression back on his face, Tru could have sworn she saw a look of relief in the man's pale eyes.
"What a shame. But I'm sure they will release your brother soon." Tru could scarcely believe her ears when Bently continued. "All the more reason you should allow me to take you to dine this evening. I would dedicate myself to cheering you up."
"Mr. Bentley, I hardly think a dinner at Delmonico's with you will help me forget my brother's situation. Now I don't wish to be rude, but I have other customers, so unless you want to make a purchase, please excuse me."
Tru turned her back on Bently and was relieved to find she did indeed have a customer at her counter. As she waited on the elderly woman, she could sense Randall Bently's eyes boring into her back. Finally, Tru heard the man's footsteps moving away from her counter. When she finished her sale, her thoughts automatically returned to her brother.
Tru had heard Carl and her father arguing this morning before the men left for the print shop. Her father had accused Carl of lying about where he and John had spent the night. Johann Kueshner was an imposing figure when he was angry, but Carl had stood his ground, refusing to answer his father's questions. Tru couldn't understand why Carl was holding back information if it would help John, since she knew Carl would stop at nothing to help his older brother prove his innocence.
While she stood trying to think of a way to get Carl to confide in her, Tru heard footsteps approaching once more. But when she turned around, instead of a customer, Betsy Stafford stood across the counter looking unusually cheerful. "Miss Kueshner, did you hear the tragic news about poor Miss Klienst?"
The words were the proper ones for the occasion, but the look on the woman's face left no doubt she was glad Sophy Klienst was dead. Tru was shocked at the woman's callous attitude.
"Oh, but you know that already, don't you, Miss Kueshner, since your brother killed her? How terrible for you. Let me know if I can help in anyway."
Tru fought the impulse to leap across the counter and strangle Betsy Stafford to wipe the smirk from her face. Instead, she watched in disbelief as the woman walked away. There had been no mistaking the gloating look the buyer shot Tru as she turned to leave. Tru had known Betsy Stafford disliked her, but she had never realized that the buyer's feelings reached that level of hatred. A shiver ran up Tru's back as a sense of unease washed over her.
Deeply disturbed by the unpleasant encounter, Tru eyes followed Betsy Stafford as she crossed the store heading straight for Randall Bently, who was standing near the men's clothing department. As Tru watched in surprise, the buyer grabbed hold of the wealthy man's arm with an iron grip. Tru was too far away to hear what they were saying, but there was no mistaking the tension in Bently's body as he tried to shake loose of the buyer's possessive hold.
Betsy Stafford, however, was not to be denied as she kept a tight grip on Randall Bently. Something very strange was going on. Normally, a mere employee of Hearns would never have risked offending one of the store's wealthy customers, especially not one who was also a close personal friend of the store's owner. Yet there Betsy Stafford stood holding Randall Bently prisoner when he clearly wanted to leave. After several minutes of what appeared to be an intense conversation with Betsey Stafford doing most of the talking, the buyer finally let go of the man's arm and moved to another part of the store. By the expression on her face, she was pleased with the outcome of her confrontation with Bently. By contrast, the dapper millionaire walked away looking angry and distracted. He was so upset that as he headed straight for the front doors of the store, he nearly knocked over a well-dressed young woman who was very visibly with child.
As Tru watched the embarrassed man apologizing to the young mother-to-be, she suddenly remembered another young woman, heavy with child, who had been in the store only a week or so earlier. Tru vividly recalled the poor creature's pleas for help as she was escorted out of the presence of the customers. And Tru remembered Sophy Klienst's bitter words: "Wealthy men never accept responsibility for their indiscretions. They just toss the woman aside."
Tru stared at the figure of the departing Randall Bently as an idea formed in her mind and stayed there until it became a conviction. Why hadn't she realized before? Randall Bently must be the wealthy man Sophy had been referring to. Tru remembered when she had seen Sophy and Bently deep in conversation and the disgusted, frightened look on Sophy's face when Bently finally left her. There had to be a personal connection between the late salesgirl and the wealthy socialite. Could that connection have something to do with Sophy's murder?
Tru looked down and realized she was gripping the counter so hard her knuckles were white. John didn't kill Sophy, but someone had. What if that someone was Randall Bently? Tru could believe the unctuous man capable of almost anything. But why would he risk being caught and losing his place in New York society? Unless to not kill her meant putting himself at greater risk for losing everything he held dear. Tru knew she was grasping at straws, but her need to help her brother made her desperate.
At that point, Tru decided what she had to do next. She would search until she found the poor creature that had been tossed out onto the street. If Tru could found out that Bently was responsible for the woman's plight, then, perhaps, she could confront him with that fact and learn more about what Sophy had known about the millionaire's dalliances. If she could show the police that someone else besides John had a reason to murder Sophy, then perhaps they would let her brother go.
Tru bit her lip to keep from crying. She had to do something, especially since it was the man she thought she was falling in love with who had arrested him in the first place. The remembrance of the few precious kisses they'd shared swept over Tru, and she closed her eyes to shut out the pain. The truth was she had trusted the wrong man, and she was paying the price.
"Tru, I have to talk to you."
Tru's heart stopped as her eyes snapped open to see Alex standing on the other side of her counter. He looked awful as if he hadn't slept since she had last seen him taking her brother away. A part of her wanted to reach out to him, but, instead, she stiffened her spine.
"You look terrible. Have you been up all hours trying to force my brother to confess to a crime he didn't commit? I've heard that is how New York's Finest solve many of their cases." Tru watched her harsh words hit home. Alex reached out to try to touch her hand, which lay on the counter between them, but Tru pulled it away and stood looking defiantly into his eyes.
"You must know better than that, Tru. I'll do everything in my power to help John."
"You're the reason he's sitting alone in that horrible place to begin with." Tru fought back her tears. Never would she give Alex the satisfaction of seeing her cry. "You shouldn't have arrested him in the first place. How could you believe my brother could do such a thing? And I let you trick me into thinking you cared for me. I was a complete fool. What did you think—that I was fair game because I was a salesgirl and my parents were immigrants?"
"I do care for you." Alex had never felt as helpless as he did now in the face of Tru's anger. How could he make her trust him again? "I'm a detective. It's all I've ever wanted to be since my father died. I have to do my duty, or I'm betraying everything I believe in and that my father died for." Alex looked deep into Tru's eyes, hoping to find a glimmer of understanding, but found nothing but anger and worry. "I promise you I will help John. Please, can't you trust me?"
"My family doesn't need your help. We will find the real killer. Now go away and leave me alone."
"Tru, please listen to . . ."
"Is this man bothering you, Miss Kueshner?" The soft voice of the unflappable Mr. Kenton interrupted Alex's pleas. The floorwalker stood at the counter watching Alex Marshall through hooded eyes. "If you wish to make a purchase, sir, the young lady will be glad to assist you. But we don't allow gentlemen to bother the salesladies."
Tru struggled to calm herself, ignoring the entreaty in Alex's eyes. "I believe the gentleman is through with his transaction. I can't help him anymore, so all business between us is done."
Alex knew he would have to give up for now. His only hope of getting a second chance was to prove her brother was innocent. He prayed his instincts were right and that John Kueshner was not responsible for this horrible crime. Alex took one last look at the face that haunted his every waking moment, and then he headed out of the store. His best move now was to question John once more and see if he would talk now and if he remembered anything that would give Alex a clue where to look next.
Tru's eyes lingered on Alex's departing figure for a few brief moments before turning her attention to the man standing silently nearby. "Thank you, Mr. Kenton."
The floorwalker almost smiled as he brushed aside Tru's thanks. "It was of no consequence, Miss Kueshner. I have heard of your family's troubles. May I offer you my good wishes for a favorable outcome for your brother? If you need any time off from the store, I will be glad to speak to Mr. Hartley on your behalf."
"Thank you, Mr. Kenton, but I will try to maintain my schedule." Tru hesitated then decided to ask the question that was burning in her mind. "Mr. Kenton, do you remember the young woman who came to the store about a week ago? She was in the family way and begged to see Mr. Hearn?"
"Of course. Why do you ask?"
"I feel sorry for her and want to take some clothes and food to her. Do you know her name or her address?"
"I remember her name. It was Ginny Midfield. A rather flighty young woman as I recall. Not one of our better employees. I'm afraid, however, that I don't know where she lives. But I can give you the name of a young woman who is still employed with us whom I used to see talking with Miss Midfield on breaks. She might be able to help you."
Tru smiled her first real smile of the day. "Thank you again, Mr. Kenton. I appreciate your help."
"Think nothing of it, Miss Kueshner. Now if you'll excuse me, I must return to my duties." The floorwalker bowed at the waist and was gone, leaving Tru alone to think of her plan. She would find Ginny Midfield and, perhaps, the person responsible for Sophy's death. But no matter how hard she tried not to, her thoughts gave way to the memory of two amber-colored eyes pleading for her trust.
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"John, you have to trust me enough to tell me the truth about the night Sophy was murdered. I can't help unless you do."
Tru's brother looked up from where he sat slumped on a hard cot and watched Alex Marshall pacing back and forth in the tiny cell. He wanted to say something if it would help catch Sophy's killer. But he didn't seem to have the energy to make a sound come out of his throat. So he lowered his head and resumed staring at the floor. He felt too tired and drained to even take his next breath. How could loving someone lead to such pain? He had never thought to find himself in a situation such as this. Knowing he had brought humiliation and grief to his family was weighing so heavily on him at that moment he didn't even feel he deserved to defend himself.
Alex stopped and stared down at the bowed head of a man who seemed completely beaten down. Alex could see how John was suffering, but there was no time to let Tru's brother wallow in his grief. The detective sat down on the cot and laid a heavy hand on John's knee.
"John, I know you loved Sophy Klienst, and I believe you would never harm her. But no one else will believe that in the face of the evidence against you. Unfortunately, those in charge around here prefer their cases neat, clean, and solved as quickly as possible. You are the easy answer." Alex hoped he was getting through to Tru's brother. "Sometimes it can get a little rough in here when they start questioning people. I've spoken to a few friends, and that is why no one besides me has been in to see you yet. But they will eventually. Talk to me now and give me a chance to find out who really killed Sophy. You want her murderer found, don't you?"
Alex had finally struck the right chord with the grieving man. John eyes met Alex's for the first time since the detective had entered the cell. "I didn't kill her. And I want the bastard who did to get caught."
"That's better. Now let's try to make sense of this whole thing." Alex stood and resumed pacing. "Were you with Sophy the night of the murder? How did you know she was in the alley behind Hearns? Did you see her there?"
"I'd rather not talk about it." John still could not bear to speak of Sophy's betrayal, and he loathed the idea of anyone else knowing what Sophy had done. Better that it died with her.
"You can't afford to hold anything back. Believe me, I wish I didn't have to ask about your private affairs. But I have to know everything to catch the real killer."
John sat in silence for several minutes, and Alex was beginning to fear he would have to start all over to convince the man to help himself. But, finally, John broke the silence, speaking in a voice that was barely above a whisper.
"I followed her that day." John's words were halting, and Alex watched as tears fell silently from the man's eyes onto his clasped hands. "She had begun to lie to me and break appointments. I tried to keep faith with her, but when Carl confronted me about her possible duplicity, I had to admit to myself something was wrong." John looked up briefly. "I'm not proud of what I did—but I'd never been in love before I met Sophy. I was desperate." John lowered his head onto his hands, shaking his head violently. "She said she loved me. I don't understand why she did it."
"What did she do, John?"
"She met another man after work that day and went with him to a room at the Morton Hotel. I followed her to the hotel after she left Hearns." John could hardly bear to say the ugly words aloud. "I waited in the lobby until they were done. The man left first. Sophy left by herself a short while later. I followed her to the alley behind Hearns and confronted her with what I'd seen." John jumped up and began pounding his fists against the nearest wall as Alex waited impatiently for his rage to run its course. "She didn't deny it. I could see in her face that it was true—she'd given herself to another man. Yet even then she swore she loved me. She told me I would never understand but that she had to do it for the others or some such nonsense." John sank to the cot, exhausted by the emotions pummeling him.
Alex was trying to contain his excitement. Someone else had been with Sophy Klienst that night—someone who might prove to be a new suspect. "John, did you know who the man was who met Sophy?"
"No. I never got close enough to see his face. I didn't want Sophy to know I was following her."
"Then give me a general description. Tall, short, fat—whatever."
"He was tall and slender. And he was wealthy, I think. His clothes appeared to be quite expensive, and he carried a fancy-looking walking stick. Also he was in a hired cab, but he could afford to have the driver wait for him outside the hotel the whole time."
"Good, John, good. This helps me. Now tell me—in the alley—did you hit Sophy or struggle with her in any way?" Alex waited for an answer as John avoided meeting the detective's eyes. "I'm not blaming you if you did. You must have been beside yourself."
"I didn't hit her. I could never hit her. But when she put her arms around my shoulders to cling to me, insisting she loved me, I pushed her away, and she fell and landed in the dirt." The memory of Sophy lying in the dirt where his shove had sent her was seared in John's mind. "I couldn't bear to have her touch me and hear her say she loved me when I knew the truth. When I saw her lying in the dirt, I had to run away. That's when I started to drink."
"If you scuffled with her, that might explain how your glasses fell off and were in the alley. Where did you go when you started drinking? What saloon?"
"I only remember the first bar very near the alley. After that, I was drunk and just wondered from one saloon to another. It's all very vague. I don't remember anything else until I woke up this morning."
"In the print shop?"
John felt a momentary unease at lying to Alex, but this was not his secret to reveal. He couldn't betray his brother by telling anyone about the countess and Carl. Besides, it still didn't provide him with an alibi for earlier in the evening, so he would be giving up Carl's secret for nothing. "Yes, in the print shop."
Alex's gut told him that John was still holding something back. Still, he had enough to start looking for the other man who had been with Sophy Klienst the night she died. The wealthy man she was with would have had to hire a hansom cab late Saturday afternoon. Alex knew nearly all the drivers in New York who hired out, and he would question every one of them until he learned the identity of the passenger who had met Sophy at the Morton Hotel. Anyone rich enough to have a hansom cab wait for him outside the Morton while he bed Sophy Klienst would stick in a driver's memory.
"All right, John, that's enough for now. I'm going to look for the man you saw with Sophy. Don't despair. I'll find out who did this, I promise." Alex called for the guard and was soon heading toward Union Square. He would find out who killed Sophy Klienst. Then he would go to Tru and make her give him another chance.
Chapter 9
Tru kicked a poor unsuspecting stone that had the misfortune to find itself in her path. The hat Tru wore, made of blue violets atop a lavender straw bonnet with deep-purple ribbon, had slipped to one side as she puffed along Chrystie Street with steam coming out her ears. She was so angry she didn't even hear the small voice calling to her until a hand reached up to tug her reticule.
"Pretty lady, what's wrong? I've been calling to you for a block." It was too early in the day for Tommy to be hawking the afternoon edition of the paper, but over his shoulder was slung the shoe shine kit he sometimes used to make extra money.
"I'm sorry, Tommy. I guess I was so upset I just didn't hear you." Tru managed a weak smile for her young friend.
"So what's got you riled? Old doc ain't on the warpath about your hats again? Speaking of which—that's a peach of a one you got on today. I don't know what you call them flowers, but I like them. My ma does too. I stole her a bunch from old lady Britey's flower cart once." When Tru didn't respond to his story with the usual lecture about the evils of stealing, Tommy knew something was wrong.
"Pretty lady, 'fess up. Something is wrong. What is it? We ain't exactly headed in an uptown direction here. Who do you know lives on Canal Street?" The young newsboy kept a sharp eye out as they turned a corner and headed down one of the roughest streets in the Lower East Side.
"It's all right, Tommy. I have to find someone. It's very important." Tru began looking for the address the people had given her when she had stopped at Ginny Midfield's home. Or what had been her home until her contemptible family had kicked her out with no place to go. Her parents had disowned her because she was in the family way with no man to care for her. Tru suspected that was because they didn't want another mouth to feed more than the disgrace they claimed she had heaped on them. Tru had to use all her powers of persuasion to get them to even give her an address where the poor girl might be staying.
"This isn't a very nice neighborhood, pretty lady. If you don't know the place, you could look a long time and not find it. You're on your lunch hour, aren't you?" Tru nodded her head. "Then I'd better help you, or you'll be all day."
Tru knew her young friend was right, so, reluctantly, she gave him the address she'd fought so hard to get. Tommy looked at the address and then turned quickly, leading Tru from the dingy street into an even dingier alley. After passing through several more garbage-strewn alleys, they finally arrived at what appeared to be the back of a church surrounded on all sides by some of the worse tenements in the city.
"The Sisters of Our Lady of Compassion take in unwed mothers and sick children. If whoever you're looking for is in here, they must be in a bad way." Tommy took Tru's hand and led her up to the door. "They're pretty nice here and darn good at nursing. Ma and I brought the baby here once when she near died of the croup." Tru's young friend pounded loudly on the door, and soon a sturdy-looking nun answered the summons.
"May I help you?" The woman's stern face broke into a cheery smile when she noticed who it was who had knocked. "Why, Tommy lad, what brings you here? Your sister isn't ill again, I hope."
"Nah, Sister Mary Rose, she's been just peachy since you got her better. I brought my friend." Tommy reached out for Tru's hand. "She's looking for someone who might be staying here. It's real important." Tru smiled gratefully at the young boy still holding her hand.
"And who might that be, child?" The sister's face was friendly as she turned her attention from the young boy to the woman with him.
"I'm looking for a young woman who used to work with me at Hearns. Her name is Ginny Midfield. She's in the family way."
The sister's expression became somber at the mention of the name of the woman Tru sought. "Yes, Miss Midfield is here. But she is not very well."
Tru hated to bother the young woman if she wasn't well, but she had no alternative. "Please, may I see her? It is very important. I promise not to cause her distress. I'd like to help her if I can."
"I fear it is too late for any help but God's." The nun hesitated but finally gestured for Tru and Tommy to follow her. Sister Mary Rose quickly sent Tommy scurrying to the kitchen for a bite of something to eat then guided Tru down another hall. Tru fell in step with the nun, who barely made a sound as she glided through the corridors of the old church.
Finally, they stopped at a closed door. With her hand on the doorknob, the nun turned to look at Tru. "This is the room for unwed mothers. I must warn you, Miss Midfield is very sick. She contracted pneumonia before coming to us. She is extremely weak."
"Will she be able to have the baby?"
"At first, we were afraid she would not live long enough to give birth to the child. But she has an amazing will, and since it is nearly her time, we have grown hopeful the child, at least, will be spared, by the grace of God. Well, now that you are fully prepared, let us go in."
Tru entered a large room with rows of beds on two sides. The faces that turned to stare at the new person in their midst displayed a variety of emotions. Some seemed lost in sadness, some seemed resigned to their circumstances, and a few seemed angry. All the women, however, responded to the presence of Sister Mary Rose with grateful smiles and warm greetings as the nun led Tru toward the far end of the room where a bed stood somewhat apart from the others.
"I'll leave you alone for a few minutes but no more. Miss Midfield tires easily." The nun bent over the pathetically small lump huddled under the covers and laid a gentle hand on the young woman's shoulder. "There is someone here to see you, Ginny. She says she worked with you at Hearns. I'll go and leave you alone to talk, but I'll be back shortly to bathe you."
Sister Mary Rose straightened and headed toward the door of the room, stopping along the way to soothe and encourage her young patients. The small figure in the bed showed no sign of having heard the nun. Ginny Midfield remained curled up, facing a nearby window. She did not turn to look in Tru's direction, leaving Tru at a loss as to what to say. She hated to add to the poor creature suffering, but she had no choice, since John's future was in the balance.
Tru walked around the bed until she could see Ginny Midfield's face. She was shocked at how pale and drawn the young woman appeared. An image of a once happy Ginny laughing as she walked past Tru's counter suddenly popped into Tru's mind. The person in Tru's memory bore no resemblance to this pathetic woman who was so thin that the stomach in which she carried her child jutted out like an obscenity from her skeletal frame.
"Ginny, it's Gertrude Kueshner. Do you remember me from the store? I'm sorry to find you are ill. I didn't know about your troubles until the day you came to Hearns looking for help."
Tru watched as two small tears rolled out of the poor woman's eyes and down her cheek, but Ginny said nothing in response to Tru's words. "I don't want to cause you pain, but I desperately need your help. Sophy Klienst has been murdered. Do you remember Sophy? She worked at the perfume counter." Tru stopped and waited for some sign from the woman that she had heard and understood Tru's words. She was rewarded with a slight nod from the frail woman lying in the bed.
Relieved to know that at least Ginny Midfield was listening and understood what she was saying, Tru continued. "My brother John has been arrested for her murder. He was in love with Sophy, and she betrayed him. But I swear to you he would never have harmed her. He is a good, kind, gentle man. I need your help to find out if someone else might have a reason to murder her." Tru watched as the sick woman's hands tightened like claws around the bedclothes on top of her. Something Tru had said had struck a chord, giving Tru hope that Ginny might actually know something.
"When you came to the store that day, Sophy said something very strange about rich men never paying for their indiscretions. And later I saw Randall Bently engaged in a heated argument with her." Tru was stopped abruptly as the sick woman reached out a trembling hand and grabbed Tru's skirt.
Ginny Midfield's started to speak in a voice barely above a whisper. Tru knelt by the bed and bent close, desperate to catch any word the poor woman might utter. All the sick woman could manage were the words "Bently—evil."
"Ginny, I'm sorry, but I must ask. Is Randall Bently the father of your child?" The eyes that looked up at Tru blazed with hatred for a brief moment as the former salesgirl nodded yes. "Did Sophy Klienst know this?"
The pregnant woman gave another weak nod as she reached up to clasp Tru's hand. "I told her..." Ginny's words were interrupted as she tightened her hold on Tru, fighting for the strength to continue. "She knew... but she still... was taken in..." Exhausted by her effort to speak, the woman's eyes fluttered shut.
Tru sat beside the bed for several minutes, bathing Ginny Midfield's face with cool water she had found nearby. Finally, Ginny's eyes opened again, and she stared at Tru, who searched for words of comfort. "I'll be back in a few days, Ginny, and I'll bring some things for the baby, I promise."
The frail woman appeared comforted by Tru's promise and closed her eyes once more, drifting off into sleep. Just then, Sister Mary Rose appeared to check on her patient. Tru stood to go but felt a compulsion to do more. "Sister, would you send for me when the baby's born? I'd like to be here for her."
"That is good of you. It could be any day now. If you'll leave an address where we can reach you, we will send word to you when her time comes. Tommy is in the kitchen beguiling cookies from Sister Ruth. When you leave the ward, turn right and go to the end of the hall. And perhaps you should hurry, or the little scoundrel will leave no cookies for the patients." With a quick smile of good-bye, the sister returned to her duties as Tru left to find her young guide.
As soon as she collected Tommy, she would return to the store. She had to figure out some way to get Randall Bently alone to confront him with what she knew. And it had to be soon. Tru wasn't sure how long her parents could bear the thought of one of their children in jail.
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"I just overheard the strangest conversation, Tru. Tru!" Clarice waited impatiently for a response. But Tru was so preoccupied she wasn't even aware of the excited redhead standing next to the counter until her friend reached out and touched her arm. "Really, Tru, you'll get in trouble with Mr. Hartley. What if I'd been a customer?"
"Well, you're not!" Tru saw the look of concern on her friend's face and instantly regretted her ill-tempered words. "I'm sorry. I'm just so worried about John. It's all I can think about."
"I know. I feel terrible too."
"What was it you were saying?" Tru forced herself to take her mind off her family's troubles. "And why aren't you at your counter?"
"That's what I have to tell you." Clarice lowered her voice to a whisper. "I was sent to the stockroom by my buyer to get some more Viennese lace. You know how insistent she is about having miles and miles of everything on hand. As if there was that much call for Viennese lace."
Tru stifled a sigh, hoping this was not another of Clarice's long-winded tales. "So what was so important you had to tell me now?"
"I was looking for the lace on a back shelf. You know how they always put the lace at the back of the room. Anyway, I was in the back trying to reach up and get the silly stuff when all of a sudden I heard voices." Clarice couldn't resist pausing to add drama to her revelation. "It was Betsy Stafford, and she was with Randall Bently."
"Randall Bently was in the storeroom?" Tru could hardly contain the wave of excitement that swept over her. "Customers aren't supposed to go in there. What were they saying?"
"I didn't hear everything. After all, I'm not an eavesdropper." Clarice tried to sound indignant and failed. "I did listen long enough to hear most of what they said. It was very strange. I mean, if it was some kind of romantic tryst, that might explain it. But I peeked around the corner to be certain it was them, and I can tell you Randall Bently didn't look the least romantic. In fact, he looked positively apoplectic." Clarice chuckled out loud at the memory of the sour look on the socialite's face. But one glance at Tru's worried expression, and she hastened to finish her story. "All I know is that I overheard them make plans to meet this evening."
"This evening?" Tru's heart began to beat faster. "Did you hear where and when?"
"That's the strangest part. Betsy insisted they meet in Central Park by the pond."
"By the pond?"
"Yes. She told him that at seven thirty tonight, he had to come through the Scholars' Gate and meet her on the west side of the pond. Wouldn't you just love to know what that's about?"
Tru didn't respond to Clarice's last question because she had every intention of finding out exactly what the clandestine meeting between the spiteful buyer and immoral socialite was all about. Whatever it was, Tru had a feeling it had something to do with the murder of Sophy Klienst.
Clarice read the determined look on her friend's face and immediately regretted telling Tru about the planned rendezvous. "Look here, Tru, you're not going to do anything foolish, are you?"
Tru stared at Clarice, but her mind was already plotting how to sneak out of the house so as to be at the pond before seven thirty. She would have to pretend to have a headache and stay in her room to sneak out while the rest of the family was at the dinner table.
"Tru, do you hear me? If I thought you were going to do something foolhardy, I would have to tell your father."
"No, Clarice, you mustn't." Tru pleaded with her friend as she reached over the counter to grab Clarice's hand.
"Promise me you'll tell no one else what you just told me. You want John free, don't you?"
"Of course, I do. How could you even ask that? But what does John's being in jail have to do with Mr. Bently and Betsy Stafford?"
Tru met Clarice's puzzled gaze. "I'm not sure, but I have a strong suspicion it has a lot to do with it. Please trust me? Promise me you'll do as I ask."
Clarice shifted uncomfortably under Tru's intense scrutiny, but, finally, Tru's friend nodded in reluctant agreement. "Thank you, Clarice. You really are a good friend. And if this helps free John, it will be in large part because of you."
That made Clarice feel a little better, so she managed a smile even though she still felt uneasy. "I want to be of help in any way I can. You know I love your family."
"You've been a big help. Now back to your counter and forget we had this conversation."
Tru watched her friend return to her own department. This might be the chance Tru had been praying for—the chance to confront Randall Bently about his dalliances with young women who worked at the store and see if he had anything to do with Sophy's death. If she could find the real killer and bring John back home to the family where he belonged, she could tell Alex Marshall exactly what she thought of him and be rid of him forever. Tru was surprised when tears unexpectedly welled up in her eyes. She impatiently brushed them away. Yes, sir, she would be very relieved to never have to see the arrogant detective again.
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Alex hurried toward the Kueshner house. He was not in the best humor after spending the better part of the day questioning hack drivers. When Alex finally found a driver who remembered taking a tall wealthy man to the Morton Hotel on the right day and time, the driver informed Alex that he never inquired after his passengers' names and wouldn't tell the detective if he did know. At least the driver confirmed that part of John's story to Alex's satisfaction, but that was nowhere near enough to free Tru's brother from jail.
After checking in at headquarters to see if there had been a response yet from the Chicago Police to the telegram he had sent, Alex decided his next step would have to be talking with Tru and Carl to see if they might know who the tall man with the distinctive walking stick might be. Alex shook his head in frustration. He was certainly not looking forward to facing Tru's anger again before he had found the real killer and set her brother free.
The detective's gaze traveled ahead of him to the door of the Kueshner house. To his surprise, he saw a familiar figure emerge from the house and head briskly in the opposite direction away from him. Alex almost called out to Tru, but something about her manner stopped him. Tru was up to something. He could tell by the furtive way she closed the door and the haste in her footsteps. When his stomach began to flutter, that was all Alex needed to let him know Tru was headed for trouble.
Careful to trail her at a safe distance, Alex kept his eyes riveted on Tru's slender figure as she headed toward Sixth Avenue. When she reached Sixth Avenue, she turned uptown with Alex right on her tail. After setting a blistering pace for several blocks, confirming Alex's hunch that she was not on a casual stroll, Tru stopped and checked the timepiece pinned to the waist of her dress.
Alex ducked into a doorway so as not to be seen and watched as Tru stood and waited several minutes until an uptown streetcar halted and Tru hopped on board. Cursing under his breath, Alex dashed from his hiding place, searching frantically until he spotted a hansom cab parked nearby and ran toward it. Coming to a breathless stop beside the cab, Alex looked up at the surprised driver.
"I need to hire your cab! We have to hurry! I want you to follow that streetcar that just went by."
"Now what would you be wanting with that streetcar if you're riding in me cab?"
"I haven't time to explain." Alex was already halfway into the cab. "I'm a New York City detective, and I'd advise you to do what I say and make it snappy."
"Sure thing, governor. Aloysius McCafferty is at your service. Hang on."
The cab driver, having decided this might be a grand adventure, urged his horse off in speedy pursuit of the streetcar that was by now a considerable distance ahead. Alex felt a momentary fear for his life as the cab bolted around several private carriages that had the misfortune to be traveling in a sedate manner up the same avenue as the now grimly determined Aloysius McCafferty. But in mercifully short time, the hack had pulled to within a comfortable distance behind the streetcar and settled down to follow at a saner pace.
Alex watched out the window as the streetcar stopped several times to discharge passengers on its journey uptown. But Tru was never among them. Finally, the streetcar reached the end of its route at the southern edge of Central Park. Alex ordered the hansom cab to stop a safe distance away from where Tru got off the streetcar.
"This where you want out, governor?"
Alex was already on the street, reaching up to pay the driver. "Thanks for moving so quickly." Alex gave the driver a little extra for his valiant effort and then hastened to follow Tru on foot.
"Think nothing of it. If you ever need a hansom cab again, just remember old Aloysius McCafferty, the best and safest driver in New York."
Alex didn't hear the cab driver as he was running to keep Tru in sight as she headed east toward the Scholars' Gate into the park. Alex followed behind as Tru entered the park and moved along the path to the pond. As she neared the pond, she slowed down and stepped off the path, crouching down as if to avoid detection. Alex did likewise, staying just far enough away so as not to arouse Tru's suspicions. The early spring evening was almost dark now, which helped conceal the two people silently circling the edge of the water.
Suddenly, Tru stopped dead in her tracks as two voices floated across the pond toward the side where Alex and Tru were hiding. Alex could tell it was a man and a woman, but he could not quite make out what they were saying. Apparently, Tru couldn't either because she started to move closer toward the voices. Alex realized whoever these people were, they were the reason for Tru's strange nighttime excursion. Alex crept along silently after Tru as the voices grew clearer. Finally, Tru stopped, but Alex kept on for a little distance, wanting to be close in case Tru was in trouble.
"Now, look here, Miss Stafford, I don't know why I even agreed to meet you like this. It's like some silly scene out of a penny dreadful."
"It's not the first time you've met secretly with a woman. I'd say your whole life reads like a cheap novel."
Alex couldn't help smiling to himself as he listened to the exchange. Whoever this Miss Stafford was, she was obviously not afraid of her male companion.
"I must protest the use of the word 'cheap,' Miss Stafford. Now can we get on with this? What do you want from me?"
"You know what I want, Bently. I want you to use your influence to help me advance at Hearns. Just as you were going to do for that little tart Sophy Klienst." A note of seduction crept into the woman's voice. "You know I always wondered why you didn't approach me as you did Sophy."
"I'm afraid, my dear, that is as plain as your face. You aren't my type. Even scoundrels such as I have some standards."
Alex continued to listen as an unmistakable note of venom crept into the woman's voice.
"For that last remark, the price for my silence will also include a rather generous amount of cash, which I'm sure you can afford."
"What makes you think I'll do anything for you?" The man was trying to sound confident, but Alex heard the fear in his voice.
"Because I know you were in that alley with Sophy Klienst the night she died. And if you don't do as I say, I'll go to the police with what I know."
Just as the woman finished making her threat, a loud splash, followed by an unmistakably feminine squeal, shattered the silence. Alex heard startled exclamations coming from the two people who had thought they were alone in the park.
"New York Police! Don't move! Wait right there until I come to talk to you!" Alex shouted commands even as he waded into the pond to rescue Tru. The darkness made the task harder, but Alex followed the sound of splashing and sputtering until he found her, whereby he promptly scooped her up in his arms and carried her, dripping all the way, to the bank of the pond, where he gently set her down.
Tru was beside herself with anger and humiliation. Just as she had tried to get close enough to surprise the two conspirators, she had gotten too near the edge of the pond, and her foot had slipped. Then the last person in the world she wanted to see had pulled her from where she sat, ignominiously, on her posterior in cold water.
Alex tried not to laugh as he reached back into the pond to retrieve a straw bonnet that could only belong to the woman trying to look unflustered as she attempted to squeeze some of the water from her clothes. "You certainly are hard on hats. I think this one is done for." Alex finally let a slight chuckle escape as Tru grabbed her hat from him and sent a withering glance in his direction.
"Don't be so smart, Alex Marshall, you're a bit wet yourself." Tru was pleased to see the detective's clothes were soaked where she had been pressed against him. "Now go do your job and arrest Randall Bently before you let him get away."
Satisfied that Tru was not hurt, Alex moved quickly in the direction of the two people, who stood staring into the shadowy foliage, trying to make out exactly who it was that had followed them. They were less than happy to see a tall, imposing figure emerge from the shrubs at the edge of the pond followed closely behind by the wet and bedraggled Gertrude Kueshner.
"Who are you?" Randall Bently used his most patrician voice to try to intimidate the large man standing before him.
"I'm Alexander Marshall, and I'm a New York City detective." Alex's voice had taken on the hard tone he used with criminals. Tru had never heard him sound this way before, and even she was a bit quailed at his manner. "And a more appropriate question is who are you?"
"Randall Bently of the Fifth Avenue Bentlys. What could you possibly want with me, Officer? I wasn't aware it was against the law for a man and a woman to stroll together in Central Park." The condescending tone had the opposite effect from the one the socialite had intended.
"Drop the act, Bently. First of all, I don't care who you are. Second, I overheard enough of your conversation to know that this little meeting had nothing to do with romance." Alex edged closer just to let the other man feel the full effect of his considerable height. "You were in the alley with Sophy Klienst the night she was murdered. That's reason enough for me to take you to headquarters to question you as a suspect."
"If the lovely Miss Kueshner had not chosen that precise moment to take a quick dip in the pond, you would have heard me confirm that I was in the alley that night but deny that I had anything to do with Miss Klienst's death."
"I can vouch for that, Detective." Betsy Stafford stepped closer to Randall Bently's side, linking her arm with that of the millionaire. "I was going to get what I wanted by threatening to place Bently in the alley that night—that much is true. But what I wasn't going to tell the police unless he gave me what I wanted was that I also followed him when he left. And when I followed him, Sophy Klienst was still standing in the alley alone and alive."
"How nice of you to come to my rescue, Miss Stafford." Randall Bently tried unsuccessfully to free himself from the buyer.
"Think nothing of it, Bently. I expect you to be very grateful for my help."
"Maybe he went back." Tru's voice contained a note of desperation.
"Sorry to disappoint you, but he didn't go back. I followed him to a brothel on Forty-Second Street." Betsy Stafford shot a contemptuous look in the direction of her intended blackmail victim. "Not one of the nicer, refined ones either."
"What can I say?" The wealthy man lifted one elegantly clad shoulder in a casual shrug. "I'm somewhat insatiable in my appetites."
"And the women at the brothel will confirm that besides Miss Stafford?" Alex fought the dark mood descending over him as he watched John Kueshner's chance for early freedom slip away.
"I'm certain they will, as I believe I paid them very well that night. Though after a few brandies, I can't be sure of anything except that I bedded several of them."
Alex wanted to wipe the smirk off the man's face with a blow from his fist, but he restrained himself. There was no use prolonging the interview. Bently had a strong alibi. Alex could see out of the corner of his eye that Tru was very upset by this turn of events. She had begun to shiver in her wet clothes, and Alex was growing concerned for her health.
"Very well, Bently, you and Miss Stafford may go. But don't think I won't still be keeping an eye on you both."
"I'm certain you will, Officer. That's what I pay you for, isn't it?" Bently finally managed to extract himself from Betsy Stafford's clutches and started to leave, but Tru blocked his path.
"Maybe you didn't kill Sophy, but you did help destroy her and broke my brother's heart in the process. And Sophy Klienst isn't the only woman you have hurt. Ginny Midfield is probably dying because she is carrying your child."
"My dear Miss Kueshner, I have no idea what you are talking about. But even if I did, I accept no responsibility for anything that may have happened to either of those two pathetic women."
Tru looked into the smug face of Randall Bently, and something deep inside snapped. Her foot reached out to kick him in the groin while she slapped him hard enough to wipe the smirk off his face. As he doubled over in pain, Tru hit him on top of the head with her reticule. She would have continued to pummel him, but two arms reached around her waist and hauled her a safe distance away as she struggled to break free.
"Leave me alone! He deserves it!"
"I know he does. But you'll only get yourself in trouble." Alex stopped and lifted Tru into his arms. He carried her squirming and fighting out of the park as the groans of her victim faded behind them.
"Put me down, Alex Marshall! Right now!"
"Not until we're away from here and I can be sure that you can't get into any more trouble." Alex tightened his hold on Tru and continued along the path away from the pond.
"Humph!" Tru started to protest once more, but, in truth, she was exhausted. She couldn't sleep from worrying about her brother. It felt comforting to be carried after her long walk into the park. So she kept silent, and, soon, Alex felt her head relax against his chest. When he looked down, he saw that she was asleep.
"Taking the world on by yourself is hard work, little bird." Alex stopped and stared down at the woman sleeping in his arms, and he realized he loved her as he had never loved anyone in his life. The thought frightened him and thrilled him at the same time. What if he lost her because he couldn't find the real killer and set her brother free?
Alex pushed all such thoughts from his mind as he resumed his journey toward the street outside the Scholars' Gate. When he reached Fifth Avenue, he managed to signal a hansom cab. Ignoring the curious stares of the driver, he climbed in, still holding the sleeping woman.
"Where to, mister?"
Alex thought a moment. He didn't think he should take Tru home in this state. Her parents would be upset if they saw her like this. He would have to take her to his house. Even sleeping, he could feel that her shivering had gotten worse. Alex was afraid she'd take sick. He had to get her warm and dry, and then he would think of a plan.
Alex gave the driver his address on Twenty-Fourth Street and settled back against the cushion. He bent to kiss the top of her head, where it lolled against his shoulder. Tru stirred slightly in his embrace, and one slender arm came up to encircle his neck as she pressed herself closer to him.
A little voice at the back of his head began to whisper that perhaps taking Tru to his house was not such a good idea, but Alex silenced it. This was one time he wouldn't listen to his instincts. After all, they couldn't be right all the time.
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Tru blinked as she woke up. She had an uneasy feeling she was not in her bed at home. Then she remembered the night's events, and her heart fell as she realized her plan to free John had failed.
"Feeling better?"
Tru turned her head in the direction of the familiar voice and found Alex sitting beside the bed, sprawled in a chair with his long legs stretched out in front of him. Tru glanced quickly from the man in the chair to the warm cover under which she rested, afraid to look underneath.
"You couldn't expect me to leave you in those wet clothes to catch pneumonia. I had my housekeeper remove them and tuck you in before she left for the night. I had to do a lot of talking to convince her your virtue was safe with me and it was all right for her to go home." Alex grinned at Tru's obvious discomfort. "It is hard to find a good, honest housekeeper. Mrs. Simms is as respectable as they come, so would I risk losing her services to seduce you?"
Tru couldn't tell if Alex was teasing her, but she knew his promise not to seduce her caused her to experience mixed emotions—a fact she tried to ignore. "But what about my parents? They'll be worried to death, and they have enough worries right now."
Alex hated to see the sadness return to Tru's eyes. "I sent the driver of the hansom cab that brought us here over to your home with a message for your parents. I'm afraid I lied a bit. I told the driver to say you were visiting Clarice and he took you to Clarice's home and you sent him to tell them not to worry, as you were going to spend the night. I hope you'll forgive me."
Tru was having a hard time trying not to look into Alex's eyes, which were pleading with her to understand. "I know you mean well. But how can I stay here with just the two of us in the house? It's certainly not proper." Tru ignored the feeling in the pit of her stomach that told her she wanted to stay more than she had ever wanted anything in her life. "And I never lie to my parents, so how can I face them tomorrow?" Tru noticed Alex's dubious expression and amended her statement. "Well, sometimes I may not tell them everything right away."
Alex rose from his chair and began to pace the room. "Including your little excursion tonight? What were you thinking? Did you think that Bently would just confess to you if you confronted him? I've been a policeman for a long time, and I can assure you very few criminals willingly confess their way into prison."
Tru watched Alex moving restlessly around the room and couldn't think of anything to say in her defense. She felt herself growing sleepy again, knowing she was safe in his big, warm bed.
"Tru, are you listening to me?" Alex stopped his pacing to stand by the bed, looking down at Tru, whose hair was spread out in a soft cloud around her face. Her eyes were half closed as she was falling asleep again. Alex's looked down at the lovely, exasperating creature curled up in his bed and knew he was doomed to love her forever. His mind was whirling from trying to figure out how he could manage to remove the obstacles standing in the way of his finally possessing her.
Tru sensed Alex's closeness, and her eyes fluttered open to find Alex standing beside the bed, looking at her with an expression that made her heart race. "You can't expect me to listen to you when you're scolding me like a child now, can you?" Tru made a face at Alex and pulled the covers up to her neck under the intensity of his stare. "You sound like an elderly aunt."
"So I sound to you like an elderly aunt, do I? Perhaps I should correct that impression." Alex whispered his words as his eyes locked with Tru's. His big hand reached out to caress her face before moving to tangle his fingers in her hair. Mesmerized, Tru's breath caught as she watched Alex slowly sink to his knees at the side of the bed.
The memory of another night and a very special kiss flooded Tru's mind. Her lips parted as her eyes stared at Alex's mouth hovering inches away from hers. She knew his lips were soft and his kiss gentle. Yet his kiss also promised something beyond tenderness—something she had yet to experience, something that would change her forever and consume her with its intensity. Tru felt a yearning lighting a fire deep in the core of her being but was she ready to yield to that need? She forced herself to tear her gaze away from Alex's mouth as she raised her hands, creating a barrier between them.
Alex stopped when he sensed Tru pulling back. He cupped Tru face with his powerful hands and gently caressed her cheekbone with his thumb. "I love you, Tru. You must know that by now. I have never felt this way before. I never even dreamed I could feel this way until I met you. To me, you are all the happiness and light that exists in this world." His words caused a tear to begin a slow journey down her cheek. Alex caught it on his finger and brushed it away, kissing her cheek where the tear had been. Tru's arms came up around his neck, and Alex clasped her close to his heart as if he would never let her go again.
Tru moved her head slightly, and their lips met. The kiss that started slowly grew into a hunger like nothing Tru could have imagined. But, suddenly, Alex tore himself away from her, breaking the spell. The sight of her lips parted and wet from their kiss threatened to draw him back, but the eager, trusting look in her eyes strengthened his will.
"Tru, I want you so much at this moment that if the city were crumbling around us, I wouldn't notice. But I don't want to take advantage of this situation. I couldn't bear to hurt you. Whatever we feel for each other, there are obstacles in our way. I want to love you the rest of my life, make no mistake about that. But I can't promise you that I will be able to free John."
Alex saw Tru flinch at his words as she drew away from his embrace. Alex's heart sank. Part of him wished he could take back the words he had just spoken. But there had to be truth between himself and this woman he loved. "How will you feel about me if the worse happens? How would your parents and brothers feel? Could they accept me into the family then? You know you could never be happy if you were estranged from them. And have you thought about what it would mean to be married to a detective? We live very different lives. I'm only saying this because I want you to be sure you could be happy with me. Because if you ever gave me the joy of consenting to marry me, I would never let you out of my life again." Alex reached out to play with the soft curls clinging to Tru's neck. Just touching her made him feel as if flames were traveling through his arms and setting his whole body on fire.
Alex waited anxiously for Tru to say something as the silence between them lengthened. Tru's face betrayed nothing of her thoughts as she lay on the bed near him. She remained close enough for him to touch, yet he feared she was moving out of his reach forever. Finally, she raised her eyes and took his hand and brought it to her lips. She kissed his hand and then pressed it against her cheek tenderly.
Alex watched as Tru caressed his hand. He still had no clue what she would say when she finally answered his questions. But his mind was filled with one thought—that, at any moment, his control would slip and he would make Tru his own in the only way any man ever felt made a woman truly his.
Tru reached out to trace Alex's lips with her fingertip. "My mother told me not long ago that when love finally came my way, I should embrace it, not reject it, because it is a great gift, and I usually listen to her advice. Tomorrow you will be a detective first, and I will fight to set my brother free. But for this moment, there is just you and I and our desires. Kiss me now and remember that whatever happens, it is my choice."
Her words had barely finished when Alex wrapped Tru in his arms. Stretching out beside her on the bed, Alex began to kiss her with a hunger that he knew would last as long as he lived. Tru found herself carried away into a new world full of sensations such as she had never known. Soon, her lips were returning Alex's kisses with the same intense need.
Alex had to use all his self-control to bank his passion. He wanted Tru so desperately he could hardly bear the ache he felt inside, but he needed to proceed slowly. He pushed aside the coverlet until the vision he had dreamed about every night for weeks was at last revealed. She was even more beautiful than in his dreams, and little by little, his hands explored her body. He kissed every inch of her soft, sweet-smelling skin until she moaned out loud. When she began to desperately press herself against him, he knew she wanted him as much as he wanted her. He removed his shirt and felt the warmth of Tru's hands sliding over his back and chest as if she were memorizing the feel of him with her fingertips.
She had grown up surrounded by men, but they were her brothers. She had never thought that a grown man's body could be beautiful until she saw Alex lying beside her. The feel of his broad shoulders and strong arms made her frantic to explore every inch of his body. Her fingers trembled as she tried to unbutton his pants, but when his hands covered hers to help her, the warmth of his touch gave her courage. Her eyes traveled to Alex's face as together they finished removing his clothes. Tru ran her hands over Alex's hard muscled thighs as she looked into his eyes. The love and need she saw there made him seem vulnerable in a way she could never have imagined. Seeing the depths of his longing for her banished any lingering doubts she might have. Her heart was so full of love for him tears formed in her eyes. Her body felt as if someone had started a fire inside her as the need to be one with Alex became so overwhelming the throbbing between her legs turned into an ache that seemed to reach to her soul.
As Alex continued to worship Tru's mouth with his kisses, his large hands began to ease her toward him; Tru's eyes closed, and her head fell back. Tru heard herself moan from a need she didn't really understand but that she knew would only be satisfied when she was fully possessed by this man. Alex eased himself slowly into her until he felt the barrier of her virginity. Gently, he moved in and out, watching Tru grow ever more excited until he gave a strong thrust and sank all the way inside her. Tru whimpered with relief and joy at the exquisite feel of him filling her and becoming one with her. Tru reached up to pull him closer, whispering words of love to him over and over.
Their lovemaking grew in intensity as they were carried away by a passion neither had ever thought possible. Tru's excitement mounted until her body screamed out for release. Just when Tru thought she might faint, the world exploded inside her, and she heard Alex's cry of fulfillment mixing with hers as he collapsed into her outstretched arms. Almost immediately, Alex moved to spare her his weight, but Tru held him tight against her. She knew that she would feel empty when he withdrew from her, and she wanted a few more moments to be one with him. There might never be another night like this for them no matter how much they longed to be together forever.
Finally, Alex rolled on his side, keeping Tru close in his embrace as he kissed her eyelids and lips tenderly. "Tru..." Alex paused, and when at last he spoke, his voice was thick with emotion. "Words are inadequate to tell you how I feel, but like every other fool who's lost his heart completely, I find myself using words to express what can't be expressed. Nothing I might say could ever tell you how much you mean to me." Alex brought Tru's hand to his lips and kissed it reverently. "Just believe that I'll love you always no matter what happens."
Tru looked at Alex, moved by the love she saw in his eyes. She reached out to caress his face. "Words don't matter. Not now. Just hold me."
"I want us to be together always." Alex's arms tightened around Tru as if he already felt events trying to take her from him. "I want a lifetime of nights like this and mornings waking up and feeling you beside me in bed. If I could see your lovely face at the start of each day, I know nothing I saw the rest of the day could make me forget that love does exist in this world."
"I want that too. But we don't know what these next days will bring. My parents are brokenhearted over John. I can't ask them to ignore their pain and accept our love just yet. When John is free, we can be together." Tru kissed Alex tenderly. "Now it's late, and we need to sleep. I must go early to see Clarice and warn her about our lie."
Alex knew Tru was right. He had to find the real killer if anything was to be as he dreamed. But he hated the uneasy feeling gnawing at his stomach. Alex felt Tru stir and looked down to find her already asleep in his arms. He pulled her tighter to him and buried his face in her neck. Her skin smelled of lilacs, and soon he fell asleep with that lovely scent wrapping itself around him.
Chapter 10
Tru couldn't keep her attention fixed on helping the woman standing on the other side of the counter. She kept thinking of the evening before when she had let love sweep her up in its seductive grasp and deliver her body and soul to a certain detective. No matter how hard she tried, she could think of nothing else but his kisses and his naked body lying next to hers. Those memories caused her face to be an embarrassing shade of pink the entire morning. Several customers had even asked her why she was blushing.
By afternoon, Tru had managed to pull herself together and wait on customers without once letting her mind wander to the thought of how Alex's body had felt as she ran her hands over every inch of him. She finished wrapping a package of gloves for the last customer and glanced toward where her best friend stood measuring lace. The tiny redhead looked worried, and Tru knew she was to blame for that fact. When Tru had intercepted Clarice outside her house this morning and explained about Alex's deception, Clarice had been very upset. But to her credit, she was more worried about Tru than any trouble that might land on her doorstep. No amount of reassurance on Tru's part had been able to calm Clarice's fears for her friend's future happiness. But Clarice had given her word she would keep her friend's secret. With no customers at the moment, there wasn't anything to keep Tru's thoughts from returning to Alex and how wonderful it had been making love with him. Tru felt her cheeks growing warm again.
"I know of no pink birds in the state of New York. They are much too exotic. But that particular shade of pink does remind me of a rose bush my mother tried to grow once outside our door."
The sound of Alex's voice behind her sent a strange tingling sensation through her. Tru looked up to see Alex's smile shining down on her, and she felt her heart leap. "Did the rose bush grow?"
Alex's grin broadened. "Yes, but we had a devil of a time picking roses from the bush because there were so many thorns." At that remark, Tru gave a very unladylike snort, which sent Alex into waves of laughter, heedless of the curious stares of nearby customers.
"Be quiet, Alex, before I get in trouble. Mr. Hartley already gave me a strange look this morning when I showed up wearing the same dress as yesterday."
Alex managed to control his laughter, lowering his voice while pretending to examine a pile of gloves lying on the counter. "If Hartley comes, it will probably cost me another month's salary to buy a pair of gloves." Slowly, Alex's smile faded as he gazed into Tru's eyes. "But it would be worth it. You're worth any price a man might have to pay to have you for his own."
Alex and Tru stared at each other, recalling exactly how they had felt when they had become one. Moments passed as they remained oblivious to the people around them. Then, suddenly, a small but insistent voice broke through their reverie.
"Pretty lady!"
Tru was surprised to look around and find Tommy standing only a foot or two away. He was so short thatTru could only see the boy's eyes over the top of the counter, and the look in those eyes told her something was very wrong.
"Tommy, what are you doing here? Is anything wrong with your mother or the baby?" Tru hurried around the counter to kneel beside her small friend and place her hands on his thin shoulders.
"No, they're all right. It's that lady you were visiting the other day. Sister Mary Rose sent me to tell you her baby's coming." The young boy blurted out his message as he glanced uneasily toward the man who stood close by watching the exchange. Tommy paused to catch his breath, trying to ignore the big detective.
"Did you run all the way?" Tru could see Tommy was nervous at Alex's presence. "Well, rest a minute while I try to think of a way to get Mr. Hartley to let me leave the store."
"Is there something wrong here, Miss Kueshner?" Almost as if Tru had conjured him up with her thoughts, the store manager appeared, looking at Tommy as if someone had let a stray dog into his precious store. "Young man, do you have business here? We don't allow begging inside the store."
"He's with me." Alex's firm voice drew the disapproving manager's attention away from the shabbily dressed newsboy. "I'm Detective Marshall. We met when you came to help identify Sophy Klienst."
"Ah, yes. I recognize you, Detective." Wilfred Hartley's manner quickly changed to one of nervous pandering. "What can I do for you? I'd hoped all your questions were answered satisfactorily the other day. The store was not involved in Miss Klienst's unfortunate demise."
"Perhaps not, but I need Miss Kueshner to come with me to answer a few questions. It's very important."
"Well... I... I... that is a most unusual request. I don't see how—"
"Mr. Hartley, the circumstances are most unusual. I'd appreciate your willing help, but I can insist if necessary."
"I suppose we can make do for a short while." The unhappy man frowned in Tru's direction. "But you will return Miss Kueshner quickly to her post?"
"That depends." Alex's answer was called out over his shoulder as he hurried Tru and her small friend toward the front doors. All three of them breathed a sigh of relief when they stepped out onto Sixth Avenue.
"Thank you." Tru smiled up at Alex, wishing she could throw her arms around his neck and kiss him to show her gratitude.
"Thanks for me too." Tommy cautiously looked up at the big detective and stuck out his hand. Alex grinned, reaching out to shake the hand the small boy had offered.
"Think nothing of it. Maybe you can do me a favor someday. Now we had better hurry. Babies don't wait for anyone once they decide to make their entrance into this world. I've had to help deliver a couple, so I know."
"You're going with us?" Tru couldn't hide her relief at the thought of having Alex with her through this troubling situation.
"I might be able to help." Alex looked around and signaled for a hansom cab. "And considering the circumstances, I'll treat us all to a ride downtown. It'll be faster, and I imagine this young lad could use a rest."
Soon, Alex was seated across from the woman he loved and the ragtag newsboy, who was obviously enjoying his first ride in a hansom cab. Tru's face was somber, betraying the worry she was obviously feeling for the young woman about to give birth. Alex was glad he could be of help. But he had to admit to himself it was also a fortuitous opportunity for him to get closer and earn the trust of the newsboy now scrambling up to sit beside the driver. Alex still believed he knew more about the man's corpse in the alley than he had been willing to share so far. If Alex could get him to open up, perhaps he could solve at least one of his murders or maybe both of them. Alex's instincts had begun to whisper to him that these murders were related in a way he hadn't figured out yet. But he had to solve the puzzle soon so he could marry Tru and they could begin their life together. Last night had left him certain that he could not go on living without his little bird.
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"I'm sorry, Miss Kueshner. Ginny is dead."
Tru wanted to weep for the young woman who had been so cruelly used and abandoned, but her outrage at the man who had used her was turning her sadness into anger. "What about the baby?"
Sister Mary Rose motioned for Tru and Alex to follow her into a nearby room. A tiny wail could be heard coming from one of the cradles arranged around the spotless nursery. "Here she is." The nun stopped and beamed down into the nearest cradle. "A lovely little girl. And she's perfectly healthy, thank the Good Lord."
Tru hurried to take a look, marveling at the new life demanding their attention. Little hands and feet pumped fiercely as the tiny face puckered up, ready to let out another demanding wail. Tru sensed Alex had come to stand beside her and glanced up to see him peering curiously at the little creature lying there.
"Would you like to hold her?" Sister Mary Rose scooped up the baby without waiting for an answer, but instead of handing the tiny bundle to Tru, she placed the baby in Alex's arms before he could protest.
Tru tried unsuccessfully to hide a slight smile as Alex froze with the baby in his arms. He looked as if he wished he were facing a whole gang of criminals rather than holding this one tiny creature. Sister Mary Rose started briskly toward the door. "I'll be right back. I've arranged for a wet nurse from the neighborhood. She should have arrived in the kitchen by now."
Alex opened his mouth in protest, but he was too late. The door had already closed before he could get a word out. Alex looked around frantically for Tru and found her laughing. "I don't suppose you'll take pity on me and take the little tike off my hands. I'd much rather go down to the Four Corners at midnight than stand here worrying about dropping this thing."
"This thing, as you call it, is a baby." Tru reached out to relieve Alex of his burden. "So much for the courage of New York's Finest. You have been unmasked as a fraud, Alex Marshall." Tru began to rock the baby gently in her arms.
"You look much better holding her anyway." Alex leaned over to kiss Tru on the forehead. "In fact, you look very beautiful with a baby in your arms." Alex gingerly reached out to touch one of the baby's pink cheeks. "She's so tiny and helpless. It's rather overwhelming. I've never thought about having children. Have you?"
"No. I've always been too wrapped up in my dream of owning my own millinery shop. Maybe someday." Tru gazed at the little girl who would never know her real mother, and a fresh wave of sadness washed over her. One tiny fist had made its way to the baby's mouth, and the baby began to suck noisily. "I hope the good sister hurries back because I'm not sure how long this little one can wait."
"That problem is solved." Sister Mary Rose bustled back into the nursery followed by a young woman whose plump figure and ample bosoms promised immediate relief for the infant Tru held.
After taking the baby from Tru and situating the wet nurse and the child in a nearby room, Sister Mary Rose returned to where Alex and Tru waited. They both were wondering what lay in store for the helpless infant.
"Where will the baby go?" Tru felt compelled to ask the question that was at that moment uppermost in her mind.
"I don't suppose you know who the baby's father is? Or that he would want the child if you did know?" When Tru nodded her head no, a small sigh escaped the nun's lips. "That is almost always the case when the young women end up here." Sister Mary Rose met Tru's worried gaze. "If there are no relatives who want the little one, she will go to an orphanage near here that we also run."
Alex placed a comforting hand on Tru's shoulder as Tru remembered her promise to the poor woman who had fought so hard to bring this little girl into the world. Ginny's parents had made it perfectly clear that they had washed their hands of their daughter and the baby she was carrying, and from the looks of their home, they could barely afford to feed themselves. Tru could still hear the cruel words of Randall Bently as he had callously dismissed Ginny and her plight. It might be possible to convince the Randall Bently's parents of their obligation to the child, but they had obviously not done a good job raising their own son. Tru thought of the little girl growing up where she wasn't wanted and perhaps would be looked down upon, and her mind and heart rebelled at the idea. Every child deserved a home like the one she had known, one where they would be wanted and loved. Suddenly, Tru's face lit up.
"Sister, I may know of someone who might want to give this child a home." Tru prayed she was right. "She is wealthy and could provide the baby with a wonderful life." Tru grew more excited the longer she thought about her idea.
"Tru, what are you thinking?" Alex was worried at the look of determination on Tru's face as she talked. "It is no small matter to adopt a child. You can't just impulsively give the baby to someone."
"Alex, I know what I'm doing. You don't know this woman. She is kind and compassionate, and she told me once that she deeply regretted never having children of her own. Sister, please let me tell her about the baby and see if she would give it a home."
The usually cautious nun hesitated. She liked this young woman who had tried to keep her promise to the baby's mother. There was little chance of the baby finding a family to adopt her at an orphanage already crowded with little ones who had been abandoned in one of the poorest sections of the city. Sister Mary Rose wanted this little girl to find a home and family after all her mother had gone through to bring her into this world. "Very well, Miss Kueshner. You may tell this woman about the infant and the circumstances of her birth. But if the woman rejects the idea, you must give up on this plan and leave the child's future care to me."
Alex had been standing silent as Tru begged the nun for the chance to find a home for the baby. As he watched, an idea came to him that might help Tru's plan become a reality. "Sister, how long will it be before the baby can be taken out into the world?"
"She is in good health as I said. Normally, of course, the child would stay here only until the mother was recovered enough from the birthing to leave. Though, as I told you, many of these young women choose to leave their babies, as they cannot care for them when the father will not admit parentage. Why do you ask?"
"There is a photographer who takes pictures of criminals for us at the jail." Alex hurried to finish telling of his plan, as he saw the skeptical looks on the faces of the two women standing in front of him. "If I fetched him here to take a picture of the baby, we could take the picture today to the woman Tru is thinking of asking to adopt the child. That might help settle the matter one way or another more quickly." Alex wanted Tru's mind at rest, knowing her stubbornness would not allow her to give up on this scheme. He needed to concentrate on finding a murderer if there was ever going to be a future for him and the woman he would never be able to let go.
"Very well, Detective." The tall nun could tell by the look on the younger woman's face that she would haunt the orphanage until she was satisfied as to the future well-being of the infant now sleeping in the nearby basket. "Take your picture and see what comes of it."
Tru finally spoke after listening to Alex's plan, and her eyes told him he had been right to aid in Tru's mission. "Thank you, Alex. Do you think you can get the photographer here right away?"
"I'll head uptown until I can find a hansom cab to hail and bring the photographer back straight away. Will you stay here? And what of the boy?" Alex thought again of his need to get the young newsboy to trust him.
"I'll stay here until you return. If you take Tommy with you, then you can help him get to his corner to sell his newspapers." Tru considered the delight her young friend would feel at a second ride sitting beside the driver of a cab.
Sister Mary Rose smiled at the thought of the young newsie peppering a cabbie with a million questions. "You'll find the lad in the kitchen with cook, Detective. He ran to the kitchen as soon as he entered the building, as he knows only too well that cook cannot resist stuffing him with goodies."
Alex wasted no time gathering up the boy from the kitchen and urging the lad to walk faster until they reached a street where the hansom cabs were waiting for fares. After dropping Tommy off near Tammany Hall, Alex hurried to fetch the photographer. He could waste no time solving this immediate problem so he could resume his quest to free Tru's brother and capture Tru's love forever.
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Maria sat very still listening to Carl's tumultuous playing. She was worried. Carl was always gentle and loving with her, but something was bothering him besides the plight of his brother—something that stood between Maria and this man she loved.
"Darling?"
Maria was startled in her chair, suddenly aware that the room had grown silent while she was lost in her troubled thoughts. She looked up to find Carl sitting at the piano staring at her.
"You look so worried, Maria. What is it you're thinking?" Carl stood up and, without waiting for an answer, walked over to pull Maria up into his arms. When his lips captured hers, his kiss had a desperate quality that frightened her.
The small woman pulled away and reached up to touch Carl's cheek. "Something is wrong. I know it. Are you tired of me and afraid to tell me?"
"Never that." Carl's hands tightened on Maria's shoulders as he pulled her back roughly into his embrace. "I'll love you forever, I swear to you."
Maria wanted to be satisfied with his answer, but the look on Carl's face told her there was a secret he was keeping from her. "What is it, Carl? Please be honest with me. I am frightened I will lose you."
"You'll never lose me unless you send me away." Carl lifted Maria's hand to his lips before turning away to prowl the large room. He finally stopped at the piano and began idly fingering the keys. Maria stood quietly waiting for him to resolve whatever battle was raging inside him.
"Maria, . . . the night of your party... the first night we met, I saw—"
A sharp knock on the door startled the two unhappy people inside the music room. "Yes. What is it?" The countess's impatient voice responded to the interruption.
"It's Rose, ma'am."
"What do you want, Rose?" Maria made no move to open the door to the music room. She wanted to put an end to this intrusion as quickly as possible so she could hear what Carl had been about to say.
"I'm sorry, ma'am, but Miss Kueshner is here, and she insists on seeing you right away."
Maria glanced at Carl, who looked disconcerted at the news his sister was in the house. But Maria could not refuse to see her friend without raising her suspicions. "Very well, Rose. Please send Miss Kueshner in." Maria hastily unlocked the door and composed herself. After a few moments, the maid reappeared, opening the door to let in the countess's unexpected visitor.
"I'm so sorry, Maria, to disturb you, but it is very important." Gertrude Kueshner's words tumbled out as soon as she entered. But she stopped abruptly at the sight of her brother standing by the piano.
Carl and Maria looked at Tru and knew the younger woman was surprised and puzzled to see her brother in the countess's music room at this time of the day. Then it was Maria and Carl's turn to be surprised at the sight of the tall man entering the room immediately behind Tru. Carl was startled to see Alex Marshall, and the detective's presence added to his sense of unease.
"Carl, what are you doing here? It is afternoon. Why aren't you at the shop?" Tru was so taken aback at seeing her brother that she momentarily forgot why she had come.
"I'm not bound to the shop hand and foot, Tru." Carl saw the hurt look on his sister's face and realized he had been too sharp. He tried to calm himself. "Papa sent me on an errand, and I stopped to give the countess some new music I bought. That's all. I was just getting ready to go."
Carl's explanation made sense, and so Tru's mind returned to the pressing reason for her visit. Alex, however, did not buy Carl's explanation. The detective had been very curious to meet the countess Von Ott when Tru had revealed their actual destination on the trip uptown. This was the woman John had warned his brother about that day when Alex walked in on the two brothers arguing—the woman whose late husband's book had been in the possession of Alex's unknown murder victim, a victim who, as it turned out, was found not far from the countess's home.
Alex studied the petite woman closely. He was surprised to see that she was several years older than Tru's brother. But she was also very beautiful, with a fragile quality that would make a man, especially one like Carl Kueshner, want to care for her and protect her. Alex's sharp eyes observed the looks exchanged between the two people who claimed to be just friends, and Alex sensed their bond was far more intimate than mere friendship. Newly in love himself, Alex was sure he looked at Tru the same way Carl was looking at the countess—as if he wanted to kiss her and possess her for the rest of her life. But Alex could tell that Tru was oblivious to her brother's true feelings.
"Countess—I mean, Maria—I have an idea to put to you. I hope I have not made a mistake believing that you might want to—that is . . " Tru was suddenly full of trepidation over her impulsive act. But it might be the only chance the unfortunate infant sleeping peacefully back in the care of the nuns would have for a good home.
"What is it, Tru?" The countess waited impatiently for her young friend to explain what was so urgent for her to discuss that she visited unannounced.
Tru took a photograph from her reticule, careful not to bend it. She moved closer to where the countess and her brother stood and held out the photograph for the countess to take.
"What a sweet baby!" Maria gazed longingly at the photograph as the old pain she thought might have finally been banished forever returned, and she felt tears begin to form in her eyes.
"It's a girl." Tru watched the countess's reaction, and hope grew in her heart.
"She's beautiful. But why are you showing me this photograph, Tru? Whose is she? Surely, you must know that the sight of such a beautiful baby would cause me to revisit my pain at being unable to have a child of my own."
Carl joined Maria and peered down at the infant. "She is a beautiful baby, but what has she to do with you or with the countess?" Carl was angry at his sister for making the woman he loved cry. And he was finding it hard to hide his frustration at not being able to comfort Maria by taking her in his arms and making love to her until she forgot anyone else existed in the world except the two of them.
"Her mother died giving birth to her. I knew her mother from the store." Tru watched Maria's face intently as she told her story. "Her mother was not married, and the father does not want her. No one wants her, not even her mother's family, so she will have to go to an orphanage unless I can find a home for her."
"Oh, but that is so very sad. Surely, something else can be done?" Maria looked genuinely distressed at the thought of such a future for the tiny creature whose photograph she still clutched in her hand.
The moment of truth had come. Tru swallowed hard. "I'd hoped that perhaps you might adopt her?"
"I? But... I..." Maria stopped, her thoughts racing. She stared at the baby in the picture for a long time before raising her eyes and looking at the three people now waiting for her to speak. "I had given up hope of ever being a mother. I am too old—I won't know what to do."
"The look in your eyes tells me the desire to be a mother is still strong in you, Maria." Tru knew she needed to press her case now not just for this abandoned infant but for her friend who clearly wanted to say yes. "As for your being too old, that is nonsense. Isn't it, Carl?"
Carl locked eyes with Maria, and he seemed to forget momentarily that they were not alone in the room. "It's absolute nonsense. You are young and vital and, more important, full of love to give. It would be a crime to waste the love that others need so desperately."
Carl's words were almost too much for the countess. She fought back tears that threatened to betray how much she loved the young man standing beside her. "I do have the servants to help. Rose has children. Perhaps she can teach me how to care for this precious little one." The small woman paused for a brief moment and then seemed to reach a decision. "I will take the child for a while and see how it works out." Tru's face broke into a wide grin. "But I will not make a final decision until I'm absolutely sure I can give this precious little girl the love and care she deserves."
Tru reached out to hug her friend, undaunted by Maria's cautious words. "I know that you will decide to keep her. You will be a wonderful mother. I am so grateful to you." Tru wiped tears from her eyes and smiled at Alex, who still looked worried. Whatever was troubling him, Tru would not let it bother her at this happy moment. "I have the name and address of the hospital where the baby is being cared for, and Sister Mary Rose has already found a wet nurse who I am sure will be glad to come here to feed the little one. After you go to get the baby and bring her home, you can arrange for one of your men to fetch the wet nurse every day."
"Thank you, Tru." The countess was already ringing for her maid. As soon as the woman appeared and recovered from the shock of finding her mistress was set on adding an infant to the household, she was given orders to go immediately and buy everything a baby might need. The butler was quickly dispatched to arrange for the carriage to be brought around to convey everyone back downtown to fetch the baby. The countess was prepared within the space of an afternoon to turn her household upside down for the new life she would now accept into her keeping.
As Carl watched his beloved making arrangements to get the baby, he thought that he had never seen Maria so alive and happy except when she was in his arms or playing her music. He was full of joy for her. Although in the back of his mind lurked the memory of the conversation his sister had interrupted.
When the carriage arrived and everyone set off to rescue the newly born child from an uncertain future, Tru felt relieved that she had been able to keep her promise to help Ginny Midfield's baby in whatever way she could, and she knew that the poor woman would be happy to know her daughter would be loved and cared for as she could never had hoped to do herself.
Alex sat quietly by Tru's side in the carriage as it headed down to the tenements where many stories such as Ginny Midfield's were repeated every day. He seldom got to see a happy ending, but his frown deepened as he thought about the woman sitting across from him by Carl Kueshner's side. The countess appeared to be a lovely and charming woman, and Alex wanted to be happy for the homeless child. But he had an uneasy feeling in the pit of his stomach. He hoped for Tru's sake that nothing was wrong. But the links between the countess and the man found in the alley with his throat cut seemed to be mounting. There was something else bothering the detective; Alex was certain from the look in Tru's brother's eyes that he was a man so in love that he would do anything for the woman who was the object of his passion—perhaps even murder.
Chapter 11
Richard Shaw paced impatiently back and forth in the crowded headquarters, waiting for Alex, as he had been most of the day. The telegram he held crumbled in his hand spelled disaster for his friend. Richard was so lost in his own thoughts that when Alex finally arrived, he failed to notice him until his friend was standing right next to him.
"Richard, did you get an answer from Chicago?" Alex had maneuvered his way across the crowded room until he reached Richard's side. "This place is even busier than usual. What's going on?"
"Plenty. Haven't you seen a newspaper yet today?" Richard turned his back on Alex and walked toward a far corner of the room that was relatively private, knowing his friend would follow.
"No. I had some personal matters to take care of." Alex was surprised when Richard did not try to find out what those personal matters were. Something must really be wrong for his usually curious friend to forgo an opportunity to pry. "Out with it, Richard. What's in that telegram that has you so worked up? And what does any of this have to do with today's news?"
Alex's fellow detective cleared his throat, unable to put his unpleasant task off any longer. "There's been trouble in Chicago. Monday, two strikers at the McCormick Reapers Works were shot and killed by police in a scuffle."
Alex interrupted his friend. "So some police in Chicago got carried away trying to break a strike. That makes them stupid, but what's that got to do with our murder?"
"If you let me finish, I'll tell you. Some socialists and anarchists organized a meeting last night in Haymarket Square to protests the deaths. When the police tried to break up the meeting, a bomb went off, killing a policeman. The police shot into the crowd, and a riot broke out. Several more people were killed or injured."
Alex was stunned. His mind raced back to the terrible day his father died during the draft riots. He remembered hearing the howls of the maddened mob as hordes of people ran mindlessly through neighborhoods, breaking into houses and setting fires everywhere. He could still recall the smell of burning flesh as the mob of working class folks turned their anger over the draft laws toward the freed blacks living in New York and began to beat and hang young Negro men, burning their bodies as a final act of outrage.
"That still doesn't tell me what's in that telegram." Alex forced his mind to return to the present. "Richard, what is it you don't want to tell me?"
"Alex, they arrested eight anarchists for starting the thing. Most of those arrested were German. The Chicago Police began searching the homes of every anarchist and socialist leader in Chicago. They have found papers that, along with what we told them, have helped them confirm our murder victim was a known anarchist from Chicago. Furthermore, when they completed the search of his room, they found evidence that he was involved in plotting some future act of violence with anarchists in our own city."
"What kind of evidence?"
"They don't know for certain. The Chicago Police found letters arranging for Schueller to come to New York to participate in a plot, which isn't spelled out in the letters. In addition, the police found a train schedule with Johann Kueshner and his sons' names written on it along with the address of their print shop. The train schedule had one train circled—a train that arrived here two days before we found Heinrich Schueller's body."
"That doesn't prove Johann Kueshner and his sons are involved in any plot."
"No, Alex, by itself, it wouldn't. But all the letters are signed 'Carl Kueshner.'"
Alex snatched the telegram from his friend's hand. It was as bad as Richard had said. The Chicago Police were advising their counterparts in New York of their findings as a warning. They also wanted New York Police to bring Johann Kueshner and his sons in for questioning to see if there is any connection to the Chicago incident.
"The Chicago Police are overreacting. This proves nothing." Alex tossed the telegram on a nearby desk. "They'll be looking for plots in church sewing circles next."
"Maybe, but both Superintendent Murry and Inspector Byrnes think the evidence from Chicago is strong, and they are worried. They've ordered all known anarchists picked up. They've already taken John from his cell to question him. They want Johann and Carl Kueshner brought in for questioning immediately. I've talked to them, Alex, and they are prepared to arrest at least Carl Kueshner on the evidence they already have."
"How did John look when they were done with him?"
Richard laid a comforting hand on Alex's shoulder. "Not bad, just a few bruises. They didn't get too rough this time."
"So who's supposed to pick up the Kueshner men?" Alex was afraid he already knew which detectives would be ordered to do the job.
Richard shrugged. "Who do you think? But if you want, I'll go alone to try to keep you out of it."
Alex knew if Inspector Byrnes found out Richard had gone alone, they'd both get in trouble, and Alex didn't want his partner's career damaged for his sake. "No. I'll go. Maybe I can make it easier on Tru's father and brother when we bring them back here."
"Whatever you think best." Richard was not very hopeful that his partner would be able to do much to aid the two men caught up in the hysteria following the riot in Chicago. But he'd do what he could to help his friend. "We'd better go. It's not going to get any easier the longer we wait."
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Carl almost cursed as he ruined another line of type for the handbill he was preparing. He had never been as good as John at setting type, and, anyway, his mind was far from the task at hand. He was counting the minutes until he could leave and return to Maria's side.
"Carl, what are you about? That is the third mistake you've made in half an hour. It is not that large a handbill. Your brother would have had the type set long ago."
Carl looked up into his father's disapproving face. "That is right because I'm not as good at this as John."
Johann Kueshner's voice sharpened at the mention of his eldest son. "John will be back soon. But in the meantime, you will try to do the job right. Excuses are the refuge of lazy men. A man must take pride in the work he does."
Carl bit his tongue to keep from yelling that this was not the work he was meant to do. However, before any more angry words could be exchanged, the sound of the front door opening reached their ears followed by the unexpected appearance in the printing room of Alex Marshall and his fellow detective.
"Mr. Marshall." Johann Kueshner's frosty voice chilled Alex to the bone. "Have you realized your mistake in arresting my son and come to tell us you are setting him free?"
"Unfortunately, sir, we can't do that yet." Long moments passed as Alex searched for a way to tell the proud immigrant why he had come to the print shop.
"Papa, Papa."
Alex's heart sank as the sound of Tru's voice preceded her into the small printing room. The last thing he wanted was for Tru to be here when he took her father and brother to the jail to be questioned and possibly locked up."
"Papa, I raced over from the store to see if you needed me." Tru was breathless from running through the streets, and as she entered the back room, she was startled to find Alex and his friend standing there, looking grim. "Mr. Marshall, what are you doing here?"
"My question exactly, liebling. So, Mr. Marshall, you will state your business and leave. We have much work to do and fewer hands to do it."
"Sir, I'm afraid I must escort you and your son to police headquarters for questioning by order of the superintendent of police." Alex heard Tru's sharp cry of protest as he kept his eyes fixed on Tru's father, afraid to look in the direction of the woman he loved.
"Why are you doing this? There must be a mistake." Tru was beside herself, fighting back tears.
"We think it will only be for a short time." Alex prayed he was right and that Tru's father and brother would be let go quickly after those in authority realized their mistake. "They're talking to many people in the wake of the riot in Chicago."
"What riot in Chicago?" Johann Kueshner's puzzled query was echoed by his two children.
"You print a newspaper. Don't you read any?" Richard Shaw was amazed at the bewildered looks of the three people staring at him.
"We are shorthanded with John gone. Papa and I have been in this room since well before dawn." There was no mistaking the bitterness in Carl Kueshner's voice.
"You still have not told us what you are talking about." Tru's gaze was fixed on Alex.
"There was a riot last night at a protest arranged by some anarchists in Chicago." Richard Shaw answered for his partner. "A bomb exploded, killing a policeman."
"What does that have to do with Papa? He is not an anarchist. He does not even belong to any socialist organizations. There is something more you are not saying. Or do they arrest people in this city now for sympathizing with the working man?" Tru's gaze turned to Alex, and her face was grim as she waited for his answer.
"There were some letters found in the house of Heinrich Schueller that implicated your father and brothers in an anarchist's plot that is supposed to occur here in New York. The letters were signed 'Carl Kueshner.'"
"What letters? I never signed any such letters!"
"Who is Heinrich Schueller?" Tru's voice rose in anger. "We know of no one by that name."
"Heinrich Schueller is the name of the man we found murdered in the alley near the Academy of Music." Alex watched Carl, looking for a response, and thought he saw a shadow drop over the younger man's face.
"Alex, we really need to get back to headquarters." At this moment, Richard Shaw could truly say he hated his job. "I'm sorry, Mr. Kueshner, but, believe me, Alex and I have no choice in this matter. Hopefully, the sooner our bosses hear your answers to our questions, the sooner you can return to your home and family."
Tru's mind reeled from the blow she'd just been dealt. Alex had betrayed her trust again. But this time, there was no way she could forgive Alex's part in the pain being inflicted on her family.
"Richard, will you take Mr. Kueshner and his son to headquarters? I need to talk with Miss Kueshner." Alex was certain his fellow detective would have no trouble with the two newspapermen. "Mr. Kueshner, if you will please go with Richard, I will do everything in my power to make your time at headquarters as brief as possible."
Johann Kueshner turned angry eyes toward Alex. "We will go because we have done nothing wrong. But I do not expect or accept any help from you." Johann turned to his daughter, who ran into his arms, burying her head on his shoulder.
"Do not cry, mein klein nachtish. Your brother and I will be home soon." Tru's father took out a large handkerchief and wiped the tears Tru could no longer contain. "Now go home and tell your mother what has happened. But be careful not to alarm her. Will you do that, liebling?"
"Of course, Papa." Tru kissed her father's cheek. "Mama and I will be all right. You will be home soon, I know."
"That is my darling girl. And tell Mama I love her."
"Come with me, sir." Richard Shaw followed his prisoners out the door of the print shop, leaving the two unhappy lovers behind.
"All right, Alex, what do you think you can say to me that will make me understand what you have just done?"
"Tru, I was ordered to bring your father and brother in." Alex tried to get closer to Tru, but she moved away to stand on the other side of the printing press.
"And you obey such orders? You were right when you said we come from different worlds. In my world, a man does not obey an order that is so clearly wrong."
"Darling—" Alex started to move around the printing press, but Tru's angry voice stopped him in his tracks.
"Don't come near me, Alexander Marshall. I will never let you touch me again."
"But I love you, and you love me."
"It was a terrible mistake—loving you. One I mean to correct by forcing you out of my heart and mind." Tru knew even as she let loose her defiant words how difficult it would be to forget this man whose very smile had come to mean everything to her.
"Tru, how can you say such a thing after the night we spent together? I want to marry you. You must know by now how much I love you."
"What happened between us was a lie. There can be no love without trust, and I cannot trust you. If you loved me, you would not have hurt my family like this a second time."
"Tru, I love you so much that, for the first time in my life, I have failed to do my job as I should." Alex kept his eyes riveted on Tru, praying she would listen to him.
"What are you talking about?"
"Carl. I've had evidence for some time that pointed to him as the murderer of Heinrich Schueller."
"What evidence? You are lying! There can be no evidence. Carl would never do such an awful thing!"
"Carl would not easily do such a thing. But what if he were persuaded it was necessary to do something that would normally be against his nature?"
"What circumstances could possibly persuade Carl to kill someone?"
"Love! His love for the countess Von Ott."
Tru took a step backward as if Alex had hit her and clung to the printing press, afraid her legs would give out. "They are just close friends! Besides, the countess is a wonderful and kind person. Why would she wish anyone to be murdered?"
"I don't know yet. But I do know that your brother is desperately in love with her."
"You're insane, Alex Marshall. The countess is older than my brother and very wealthy. She is just helping him with his music."
"She is more than his benefactress. You are just too naïve to see the truth about the feelings she and Carl share." Alex could see that Tru was searching her mind, desperately trying to recall the last time she had seen Carl with the countess. "Tru, I watched them when we visited unexpectedly. They are lovers. I'm sure of it. Her late husband was an influential socialist, Tru. We found a book he had written and signed among Heinrich Schueller's belongings. And we know for certain that Schueller was an anarchist."
"I don't believe you! You are lying! Besides, even if that were the truth, which it isn't, why take my father to jail? You say you had evidence, which you didn't act upon because of your love for me." Tru's voice grew angrier with each word she spoke. "If you didn't act, it was because your evidence was inadequate to arrest Carl before now. You only pretended to fall in love with me to get close to my family. Since you seem to think my family is a nest of murderous vipers, why not complete the job and arrest Mama and I? Then you will be rid of all of us." Tru walked toward the door to the print shop. She stopped, and when she turned back to look at Alex, her face was stone cold. "You must leave now."
Alex stood still, unable to move. "Tru, if only you'd listen."
"I've listened enough to your lies." At that moment, Tru felt such pain that she wanted to double over and drop to the floor. But, instead, she straightened her back. "You are a detective through and through. There is no room in your life for things such as love and trust. You believe only the worst in people."
"That is not the way it is. And if it was that way, you changed that. You are everything to me that is beautiful in this world." Alex made one last desperate plea. "If you could only find it in your heart to trust me a little longer and not give up on our love, I promise to make things right." Even as Alex spoke the words, he could hear how hollow they sounded.
Tru couldn't bear to be near Alex any longer. She knew she was about to collapse. "Get out, Alex, right now. I'm begging you. And don't come back."
Alex hesitated a moment longer, staring at the stiff, unyielding woman he had only recently held cradled in his arms. Then he forced himself to walk past her and out of the print shop.
As soon as she heard the click of the front door, Tru collapsed to the floor and lay sobbing on the worn wooden planks. Love was too painful to bear when you gave your love to the wrong person. She would never let that happen again.
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"Rose, what is it?"
The countess Von Ott looked up from the cradle she was gently rocking to find Gertrude Kueshner standing in the doorway to the music room instead of her maid.
"Tru, how delightful! I'm so glad you've come to visit us. My darling baby is just about to take her nap." The older woman glowed with happiness as she looked down at the tiny infant now peacefully asleep in the ornate cradle. When Tru failed to respond to her greeting, Maria glanced up and realized that Tru was staring at her, a strange expression on her face.
"What is it, Tru? Has something happened to Carl?" The petite woman hurried to her friend's side, anxiously reaching out to touch the younger woman's arm. But Tru shook off the countess's hand and remained silent, watching every expression that crossed the older woman's face.
When she finally spoke, Tru's voice was tight with emotion. "Why do you ask about Carl only?"
"Because he is my particular friend, of course, since we share a love of music." There was an undercurrent in the room that Maria did not understand, and she was growing uneasy. She reached into her pocket and pulled out a handkerchief, patting her temples with the cool silk.
Tru stared at the expensive handkerchief, recognizing it immediately. It was one of the ones Carl had bought on his mysterious trip to Hearns. "So Alex is right." Suddenly, the odd way her brother had acted that day made perfect sense.
"What are you saying?" Maria moved to stand by the grand piano, nervously running her fingers over the keys.
"Don't lie, Maria, not now." Tru stood frozen, occupying the exact same spot she had since entering the room. "Carl's been taken to police headquarters because of you."
"My god, Tru, no! What has happened?"
"My father and Carl have been accused of taking part in some kind of anarchist plot." Tru watched Maria sink down on the piano bench as if her legs could no longer hold her. "And Carl is suspected of murdering a man whose body was found behind the Academy of Music near here. The man's name was Heinrich Schueller." Tru's heart sank as the countess's shocked expression told Tru this was not the first time Maria Von Ott had heard that name. "Detective Alex Marshall thinks it is possible that Carl may have killed him because of his love for you. I don't believe that Carl could murder anyone, but I am starting to believe that you and he are lovers."
Maria raised her eyes and met Tru's accusatory look squarely for the first time since the painful encounter had begun. "Yes, Carl loves me. And I love him. But Carl would never harm anyone for any reason. And I certainly have no reason to want Heinrich Schueller dead. Your Mr. Marshall is wrong about that." Maria stood up and walked to the cradle as if seeking comfort from the sight of the child sleeping within. Moments passed before her voice broke the tense silence filling the room.
"Tru, I am sorry you don't approve of my love for your brother or of his for me. I was afraid that would be your reaction and that of your family. That is why I fought so hard against my feelings." The lovely woman's face seemed to age as she spoke. "But I can't deny my feelings. I love Carl. And, by some miracle, he returns that love. I never thought I could love again as I love him." Maria stepped closer to Tru, her eyes begging for understanding. "I wish you could find a way to be happy for us."
"How can I possibly be happy for you? Even if you hadn't caused my brother to lie to his own family, you are the reason he will probably be jailed for murder."
"How can I be responsible for what has happened when I told you I have no reason to wish harm to anyone?"
Tru watched Maria's face as she spoke and had to admit the woman looked as puzzled as she said she was. But she had to be a good actress to hide her love for Carl from the world as she had. "Was your husband a socialist or not?"
Maria sat down by the cradle and began to rock it gently. "Yes, Fritz was a socialist. I've already told you that. Why does it necessarily follow, then, that I would want this man Heinrich Schueller dead?"
"I don't know. He had a signed copy of your husband's book on socialism in his possession. Maybe he had knowledge that would threaten your life or your wealth in some way? Did you know the dead man or not? Tell me the truth."
"Yes, I knew Heinrich. He and my husband were very close once in Prussia, but they parted ways when Heinrich became an anarchist. My dear Fritz wanted no part of the anarchists. He was not a believer in violence. My husband was a good man who wanted all people to have the same chances in life. The fact that some considered this idea a radical one did not make him a murderer. I could tell you many stories of his acts of kindness, but I will not. I do ask you to believe that the love I feel for your brother would never allow me to harm him in any way. And I never meant to hurt your family. I told Carl we could never love openly for that very reason."
"You never meant to hurt my family!" Tru could no longer control her feelings. "You've seduced my brother, and because of you, he is implicated in a murder. I wish I'd never met you!"
A loud knock on the door froze both women in their places. The countess was the first to regain her composure. "What is it?"
"It's Rose, ma'am. Mrs. Klienst and her son are here to see you. I told her you were busy, but she insists. She says it is most urgent."
Tru was surprised when Maria told her maid to let the new visitors in. What strange relationship existed between these three people that Maria Von Ott would see them now? The image of the countess deep in conversation with Otto Klienst the night of the musicale popped into Tru's head as something began to tug at her memory.
"Tru, we must continue our talk later. I love Carl. You must believe me." Maria turned beseeching eyes toward Tru as she awaited her new guests, but Tru still doubted her sincerity.
"If you do love him, that love has caused him nothing but trouble."
The door opened at that precise moment to admit Otto Klienst and his mother. Before he noticed Tru's presence, the young man approached the countess, bowing before her and kissing her hand. "You look lovely today, Countess. My mother and I are most grateful for your help in securing the money we need."
"Otto, be quiet. We are not alone." Sophy's mother had spotted Tru standing silently nearby. "We will talk to the countess later about the money."
The severely dressed woman approached Tru slowly. Her black mourning colors seemed eminently suited to her unpleasant disposition. A chilling smile appeared on her face. "So, Miss Kueshner, one of your brothers was in love with my beautiful Sophy—I just had the wrong brother." Tru found herself shivering under the woman's unrelenting stare. "And because of his love for her, my Sophy is dead. I warned you, young woman, that love is dangerous."
"My brother didn't kill Sophy, and I will prove it to you and to the police."
Otto Klienst hurried to his mother's side, taking the older woman's arm and guiding her away. "Mama, we are here on business with the countess. It will only bring you more pain to speak of Sophy. Come sit down and rest."
The strange woman seemed to deflate as she sank into a nearby chair with her son's help. Tru looked from the peculiar couple to the countess, who at that moment appeared to be near collapse herself under the emotional distress of the afternoon's events. Tru felt a brief moment of pity for the beautiful woman, but it was quickly banished as the memory that had been playing at the edges of Tru's mind came sharply into focus. Tru heard again Otto Klienst's ardent words and remembered the intense look on his face as he admonished Carl and Tru that their father's paper did not go far enough in its political views. If Der Wahrheit was not radical enough to satisfy Sophy's brother, did that make him an anarchist? Tru now wanted desperately to get out of the countess's house to think.
"By all means, conduct your business with the countess. I'm finished here." Tru's eyes bore into Maria. "Whatever that business is, I'm sure it is more important than my brothers' lives."
Tru started toward the door as the unhappy countess cried out to her not to leave. But Tru didn't stop until she was once more on the street. As she hurried home to her mother, Tru's mind went round and round. What was the connection between the countess Von Ott and the Klienst family? Tru was willing to bet it had something to do with the politics of anarchy despite the countess's fervent denials.
Chapter 12
Alex was exhausted. He had been up all night, sitting by the window in his bedroom, trying to find a solution to his murder cases—one that did not involve Tru's brothers. By dawn, his head was spinning, and as the sun began to paint the early morning sky pink, he was forced to admit to himself that he was no closer to freeing the Kueshner men or to winning back the woman he loved. He had finally fallen into an uneasy sleep just as he should have been rising to go to work.
Now, Alex hurried toward the Kueshner's print shop on a beautiful May afternoon that mocked his dark mood. He had gone to Hearns first, but Tru was not at her counter. Her freckled-face friend had glared at him when he asked her where Tru was, but, finally, she had relented and told him Tru was at the family business. Alex set off in that direction, knowing that he was the last person on earth that Tru would want to see. But he had to talk to her. More and more he felt in his gut that the two murder cases involving her family had to be related. Though on the surface, Sophy Klienst's murder seemed a crime of passion, while the killing of Heinrich Schueller appeared to be politically connected in some way. Perhaps Tru could help him find the connection. She might know more than she realized if only he could prod her memory.
Alex entered the small outer office of Der Wahrheit and found it empty. From the back room, he heard the sound of the press running, so he headed in that direction. He stopped in the doorway to the printing room, surprised by the sight that greeted him. Tru, covered with ink smudges from head to foot, was struggling with the large press that was spitting out copies of the family's newspaper. Tru's mother stood on the opposite end of the press, catching the newspapers and carrying them over to a nearby table where their small friend Tommy was folding and packing the papers into cloth bags.
"That's it, Mama. We did it. They're all printed." Tru shut down the press before noticing Alex standing in the doorway. "What are you doing here? You are not welcome."
"I have to talk to you." Alex shifted uncomfortably as Tru's mother swiveled in his direction. "Tru, it's important. I'm beginning to believe the two murders your brothers are accused of are linked somehow. I thought maybe you might know something without realizing it. Something that would help me figure out the link."
"You mean another link besides the fact that in both cases, you have arrested the wrong man?" Tru turned her back on Alex, trying to hide the tears that had welled up in her eyes. She grabbed a nearby rag and wiped away the tears while removing the ink smudges from her face.
"Mr. Marshall,"—Martha Kueshner stood calmly, wiping her hands as she spoke—"we are aware of no ties between these crimes. We have absolutely no knowledge of Heinrich Schueller. And as to poor Sophy Klienst, we have already told you all we know. Gertrude and I have spoken to a lawyer who sometimes uses our printing services. He has agreed to try to help us. He has advised us not to speak with the police any further until he has a chance to talk to Johann and the boys. So you see you must leave now."
"You heard the ladies. Maybe you'd better scram."
Alex looked down at the small boy, burdened now with two heavy bags of newspapers, who had taken a defiant stance directly in front of the large detective. Alex had to admire the loyalty that spurred the lad to confront the man towering over him.
"I'm not here to hurt anyone, son. I'm trying to help just like you." Alex smiled at the boy, but Tommy refused to respond to the detective's friendly overture.
"Tommy, you had better get going." Tru placed a protective hand on the young boy's shoulder. "We really need to sell those papers today."
"Sure thing, pretty lady. The best man in all of New York is on the job. They'll be gone in an hour." Tommy glared one more time at the detective before hurrying out of the print shop and onto the street.
Tru shook her head and smiled as she watched her young helper leave, marveling at all the spunk contained in one small body. She and her mother had been very touched when Tommy had shown up at the newspaper office offering to help. She doubted they could have finished on time without him. But Tru's brief smile faded as she turned her attention back to Alex.
"You heard Tommy. It is time for you to leave. We have nothing more to say to you, so please go. You are not wanted here."
Alex searched Tru's face for some sign of the love they had shared and found none. Knowing how stubborn Tru was, Alex reluctantly left the newspaper office, emerging back out into the warm air while trying to decide what he should do next. He started down the street and was surprised to spot a crowd of boys in the next block fighting. At second glance, Alex saw that a group of young thugs were actually beating on a small figure lying on the ground at their feet. As Alex moved nearer to the struggle, he realized that the figure covering his head to escape the storm of blows raining down on him was none other than Tru's small protector. Alex started to run toward Tommy and his tormentors. It felt to him as if it took forever to reach the side of the small boy, who was being cursed with every blow.
"Get the little rat! Beat him bloody!"
"What's he think he's doing selling that rag on our streets! Helping them lousy murderers!"
"Stop! Stop right now!" Alex shouted his command as he waded through the mob of attackers, grabbing the bullies left and right and tossing them onto the sidewalk. "I'm a New York policeman. Stop, do you hear me, or you'll all be in jail." Alex had reached Tommy by now and leaned down to help the newsboy to his feet before throwing a protective arm over his shoulders. The last few members of the gang of boys backed off under the heat of Alex's stare.
"Ah, we weren't doing anythin' but our patriotic duty like. Ain't that so, fellas?" One of the larger boys was trying to sound defiant as he faced Alex, but when he looked around, he realized that most of his friends had deserted him, melting back into the crowd that had gathered to watch the curious scene.
"Enough of your crap! Get out of here, and if I catch any of you thugs doing anything like this again, I'll have the whole New York Police Department out looking for you, and when we find you, I'll see you get thrown in jail and forgotten for a few days." Alex watched as the last of the curious passersby resumed their daily activities, and the street returned to normal. A sharp tug on his coat turned his attention downward.
"Thanks. I owe you one."
Alex almost smiled at the solemn expression on the boy's face, but he realized how important it was to the tough little survivor that he acknowledge his debt to the detective. So Alex held out his hand for Tommy to shake. "Think nothing of it. Are you all right?" The bruises on the newsboy's face were beginning to turn purple.
"I'm just fine as frog hair. Can't live in my neighborhood without having a fight or two. If there had been a few less of them, I'd held my own." Tommy squirmed as he lowered his head to stare at the dirt beneath his feet. Alex waited patiently for the boy to reveal whatever was on his mind.
"Look, doc, I might'a been wrong about you. Maybe you ain't such a bad sort. It's just that the pretty lady and her family have been mighty good to me."
Alex patted Tommy on the back and then started to pick up the papers scattered around them. For the next few minutes, nothing was said while the man and boy worked together to return the papers to the bags where they belonged. When the bags were full again, Alex slung them over his shoulders.
"Where do we go to sell these?"
"You want to help?" Tommy tried but couldn't hide his surprise at the offer.
"I'd do anything to help the pretty lady." Alex's steady eyes met the younger ones staring up at him. "You see, son, I love her."
Tommy thought for a minute. He believed the detective. There was something about the look on his face that Tommy guessed was the way older folks looked when they were in love. Tommy had never seen that look on his father's face, but he had seen the older Mr. Kueshner look at his wife that way. Tommy made up his mind in that moment to trust the big detective.
"Look, I got something I've been keeping to myself for a while."
Alex felt a wave of excitement wash over him. He tried to remain calm so he wouldn't scare the boy. "What might that be?"
Tommy resumed walking uptown as Alex fell into step beside him. "I never meant no harm. I didn't figure it was important."
A few more steps passed in silence as Alex waited impatiently for the boy to continue. Finally, Alex decided to take a chance and nudge the boy's story along. "Look, Tommy, whatever it is you've done, I promise I won't tell a soul. If this has something to do with the day you stumbled on the body in the alley, I'm only concerned in proving someone other than Carl Kueshner committed the murder."
Tommy stopped, and Alex watched as the newsboy searched through the pockets of his shabby coat. Finally, he pulled out a small piece of paper with one corner missing. Alex recognized his missing clue and almost snatched the paper from the boy's hand, but he managed to stop himself and wait for the boy to hand it over to him.
"Here." Tommy held out the piece of paper, and Alex's fingers closed quickly around it. "I was in a hurry the day I stumbled on the body. But I did start to search the stiff for money." Tommy's defiant eyes met the detective's. "My ma and sister need things, and the poor sucker wasn't gonna need it no more. I grabbed the paper out of his hand, thinking it was money, and then reached down to dig into his pockets. But that's when I heard a noise nearby. I didn't have no time for someone asking me questions, so I had to run before I could finish going through his pockets. I didn't even realize I'd stuffed the paper into my coat until I got home that night. I don't know why, but I just stuffed it back in and left it there." Tommy looked up at Alex. "Are you going to put me in jail?"
"No. All I'm interested in is what's written on this paper." Alex smoothed out the note and stared.
"I looked at it once, but since I don't read so well, I couldn't make sense of it. Mean anything to you, doc?"
"Not yet. But it's better than the nothing I had before."
Tommy pointed to the paper. "This kinda looks like a map?
"You're right about that being a map of some kind." Alex continued to study the crumpled paper. Days of being crammed into the small boy's coat had blurred the ink a little, but Alex could still make out the map and a name—Jacob Werner. The map looked vaguely familiar, but the name meant nothing to the detective. Still Alex was determined to use all his skills as a detective to find out who Jacob Werner was and the significance of the map. He had to prove the men of Tru's family innocent of all crimes or he would never be able to recapture her trust.
"Come on, Tommy. Let's get these newspapers sold so I can get to headquarters and figure this thing out." The odd-looking pair started off at once, heading for Fourteenth Street near Tammany Hall. "And, Tommy, let's just keep this our secret until I can talk to Mr. Kueshner and his sons. I wouldn't want Miss Kueshner to do anything rash."
The newsboy smiled at the detective. "I know just what you mean, doc. The pretty lady can be one mighty stubborn woman."
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Tru tried to hide her frustration. She had spent an hour with her last customer. Finally, she had managed to satisfy the woman. But the usual feeling of accomplishment Tru experienced when finishing a difficult sale was missing today.
"Tru?" Gertrude Kueshner turned at the sound of her name and found her friend Clarice frowning at her from across the counter. "I've been watching you all morning, but this is the first chance I've had to get away to talk with you. You look terrible. Has something else happened?"
Tru didn't think she should tell Clarice about the disturbing scene between herself and the countess Von Ott two evenings earlier; at least not until she knew what the countess's part was in all of this. "Nothing new is wrong. But Papa and the boys are still in jail, and I'm beginning to fear they will never get out."
"Oh, Tru, you mustn't talk like that." Clarice reached for her friend's hand. "Everything will work out, you'll see." Clarice was shocked to hear a note of despair creep into her friend's voice. Clarice had never known Tru to give up on anything. But then Tru had never before been in love with a man who betrayed her and broke her heart. Clarice wished she could get her hands around the good-looking detective's throat.
"I know it will be all right, Clarice." Tru looked down to where her friend was gently patting her hand and was sorry she couldn't confide in her. "I just forgot for a while. Thanks for reminding me." Tru forced a smile. "Now, you better go back to your counter before you get in trouble with old pie-face Hartley."
Clarice giggled and gave her friend one last reassuring squeeze before returning to her post. Tru watched her go then sighed and bent her head, trying to hide the tears that threatened to come by straightening out her display case. She had meant what she said to Clarice. Everything would be all right because her father and brothers were innocent. But it was up to her to somehow hurry the process along.
"Tru?"
For the second time in only a few minutes, Tru heard her name spoken. Tru knew, without glancing up, who was interrupting her thoughts this time, and it was not someone she wanted to talk to in the middle of a crowded store.
"Countess, why are you here?" Tru looked at the woman now standing beside her counter. It was hard to hate someone who looked so miserable. Maria Von Ott was a mess. Her clothes were rumbled and looked as if they had been slept in. Her eyes were puffy and red. It appeared as if she had been crying ever since Tru had left her two days before.
"Tru, I cannot bear for you to hate me so. I have laid awake the last two nights until the wee hours of the morning, struggling with myself. I finally decided I must tell you the truth even though I will be betraying a secret that is not mine to share. Perhaps then you will understand and not hate me quite so much." Maria paused to regain her composure before continuing. "You must believe that I truly love Carl. I want to help him in any way I can. I have to see him soon, or I will lose my mind thinking of him in that awful place." Maria's eyes implored Tru for a chance to be heard.
"What is this secret you want to share with me?"
The countess fiddled nervously with her reticule, steeling herself to tell her story. "You are right to suspect that my relationship with the Kliensts is not a casual one." Maria looked square into Tru's eyes. "But it is not a political one either. Clara Klienst was the housekeeper years ago on my husband's estate in Prussia when I came there as a very young bride." Maria dropped her gaze to the handkerchief she had taken from her bag. It was one of the ones Carl had given her, and she held it tightly in her hands as if to draw strength from it. "I know it is hard to believe now, but Mrs. Klienst was beautiful in her youth. She had a happy life. She was married to the man who was in charge of my husband's stables, and they had one son—Otto."
"Is there trouble here, Miss Kueshner?" Tru nearly jumped out of her skin as Mr. Hartley appeared out of nowhere.
Maria's cultured voice responded to the interruption before Tru could speak. "Certainly not, sir." Tru almost laughed when the countess managed to put on an imperious air despite her bedraggled appearance. "I'm simply taking my time before making a decision. Does Hearns make a habit of hurrying their patrons into hasty choices?"
Wilfred Hartley swallowed hard and then poured out a stream of reassurances before scurrying away. Tru's amusement at the manager's discomfort was quickly forgotten, however, as she turned her attention back to the nervous countess.
"As I was saying, Clara Klienst had a good life. Unfortunately, my husband's younger brother returned to the family estate after a long absence. He had been traveling the world and had not even returned for our wedding, so I had never met him. But my husband had told me that his brother was a man of little character, and he was right. Heinz proceeded to destroy Clara Klienst's life after he returned."
Tru could see how upset Maria was recalling whatever had occurred years ago. But if it shed light on any of the strange events of recent weeks, Tru could not let Maria stop until everything she knew had been revealed.
"How did this man destroy her life?"
"My brother-in-law was as immoral as my husband was good. He was a man of insatiable appetites, and he believed his status in the world guaranteed that no one could deny him his pleasures no matter how base or cruel. Clara Klienst caught his eye, and he decided he had to have her. When she spurned his advances, he attacked her late one night when no one was near to hear her cries for help."
Tru's heart went out to the woman whose looks had been used as the excuse for a man to behave like an animal. That explained Mrs. Klienst's strange remark about the dangers of being beautiful. "What happened?"
"My husband found out, and, of course, he banished his brother from the family estate. But not before my brother-in-law had managed to brag to Clara's poor husband about what had happened that awful night." Maria sighed, recalling the terrible scene between her gentle husband and his arrogant brother when her Fritz had come upon his vile sibling taunting Rudolph Klienst with the details of what he had done. Even after all these years, the memory made her shudder.
"Did Otto know what had happened to his mother?"
"Thankfully, he was too young to realize what had happened. But his father could not forget or bear to go on with a living reminder of his humiliation. Clara Klienst became pregnant from the vicious attack. Rudolph Klienst hung himself in the stables the day the baby was born." Maria studied Tru, trying to gauge her response. "My husband felt a great sense of responsibility for the tragedy his brother had brought down on the Klienst family. That is why he insisted that Clara Klienst and her children travel to America when he finally decided we must leave Prussia. When we arrived in New York, he offered Clara a job with us, but she refused. While he was alive, he gave the Klienst family money whenever they were in financial need. When he died, I felt it was my sacred duty to carry on with his wishes. So you see, Tru, it had nothing to do with politics."
Tru understood now why Otto and Sophy Klienst had born so little resemblance to each other. And why the Klienst family seemed to have a hold on Maria. It was a tragic story. Tru didn't pull her hand away when Maria reached out to grab it.
"Please, Tru, say you believe that I would never deliberately cause your brother harm. I could not tell anyone Clara's secret. My husband felt very protective of them after all they had been through, especially Otto. He was such a dear little boy then. The entire household tried to keep the truth from him. Even our butler Jacob made sure that Otto never overheard any whispers about his mother or his sister. After the boy's father killed himself, Jacob took Otto under his wing and tried to be a kind of father to him."
Tru remained silent for a few minutes. She did believe Maria Von Ott, but she was not ready yet to accept the idea of a love affair between her brother and the older woman.
Maria could sense the doubts that remained in Tru's heart. "Please, Tru, please understand. My husband's politics would never influence me to want someone killed." Maria reached over the counter to grasp Tru by the arm, but before she could plead her case any further, she saw something over Tru's shoulder that caused her to gasp.
"What is it, Maria?" Tru turned around, following Maria's gaze, trying to see what the countess was staring at across the large room. Tru studied the faces of everyone within the countess's range of sight and saw nothing and no one who appeared out of place. "What are you staring at?"
"At a ghost." Maria looked back at Tru. "I could swear that I just saw Jacob Werner, the butler I was telling you about. But he is dead. We were sent word from Prussia several years ago that Jacob had died."
"Then you must be mistaken, Maria." Tru turned once more in the direction of Maria's gaze. "Where is the man you say resembles your old butler?"
"There, standing next to that strange little man who interrupted us earlier."
Tru was baffled. The only person standing next to Mr. Hartley was Joseph Kenton. "Maria that is the store floorwalker. His name is Joseph Kenton."
"But he looks so much like Jacob." Maria straightened up, her expression turning from one of shock to one of determination. "I'm going over there and talk to him."
Tru watched as the countess Von Ott walked briskly in the direction of the store manager and the floorwalker she believed to be her butler, a man who was supposed to be dead. As she got closer, both men noticed her approach. The manager slapped on the standard smile he wore when dealing with customers, and Joseph Kenton turned and walked away to assist an elderly customer with her packages. To Tru, the floorwalker did not look as if he recognized the beautiful woman heading in his direction, nor did he look startled by the countess's sudden appearance. He merely looked as if he had seen a customer in need of his help and was doing his job. Before Maria could reach him, he was escorting the elderly lady out to her waiting carriage. Maria stared after him for a brief moment and then returned to Tru.
"Do you still think that man was your old butler?" Tru thought the countess appeared to have recovered from her surprise. "He didn't seem startled or flustered when you approached him."
"On the contrary, I am sure I saw a look of recognition on his face right before he turned to walk away." The countess stared thoughtfully at the door through which Joseph Kenton had disappeared. "I'm more certain than I was before."
Tru was impressed by the countess's certainty. "But wouldn't Sophy Klienst have recognized the floorwalker as your old butler?"
"Sophy was barely walking when we came to America. She probably did not even remember him. Anyway, Jacob was closer to Otto than to Sophy." Maria tried to recall more memories from the past. "I don't remember Jacob even being near Sophy. He spent all his spare time teaching Otto to box and shoot. I always assumed he felt he had to teach Otto manly things, since the boy's father wasn't around to do it."
Tru was trying to sort out in her head all the new information that she had learned from Maria. Maria still seemed certain Joseph Kenton and Jacob Werner were one and the same. If that was true, it seemed too big a coincidence not to be looked into by someone. And the more Tru thought about it, the more determined she was that she would be that someone.
"Tru, what are you planning?" Maria Von Ott had grown concerned watching the expression on Tru's face. She had not known the young hatmaker long, but it had been long enough to know that Tru could be very stubborn and she was totally devoted to her family. Maria doubted there was any risk Tru would not take to help her brothers.
"I'm choosing to believe you." Tru was on pins and needles now, wanting her workday to be over. "So I am going to find out exactly who Joseph Kenton is and why he is pretending not to know you."
"I'm not certain that is wise, Tru. Jacob always seemed to be a good man. But if this is Jacob, where he has been and what he has been doing are a mystery. Perhaps there is a good reason he wants to hide his identity. Else, why would he change his name and want people to believe he was dead? I am afraid for you."
"And I am afraid for my brothers and my father and my mother, who cries for her husband every night. If you say Jacob was a good man, then why are you afraid for me?"
"Jacob was very strong willed. That was why he could run our household with such efficiency. Fritz often said it was a good thing Jacob worked for us and not against us because the man possessed a will of iron."
Tru didn't reply. She was already forming a plan to follow the floorwalker. Whatever the truth was about the man, she was determined to find out. And whether his name was Jacob Werner or Joseph Kenton, he wasn't the only strong-willed person working at Hearns.
"Tru,"—Maria's agitated voice brought Tru back to the present—"don't do anything foolish."
"What is more foolish than failing to act if it might help get my father and brothers out of jail?"
Maria blanched at the mention of Carl and the awful place he was in. "I want Carl out of there too. But couldn't you ask the detective who was with you the other afternoon to help you?"
"No! Alexander Marshall didn't believe in my family before—why should he believe in them now? I have to do this on my own." Tru reached out to squeeze Maria's hand. "I promise I'll be careful. I'll only try to find out where he lives. Now go home and wait until I come to you."
Maria gathered her things and took her leave of the obstinate young hatmaker. But when she walked out of Hearns, Maria Von Ott headed in the direction of the police headquarters. She was rather hardheaded herself, and, right now, her instincts were warning her that her friend would get in trouble if she tangled with Jacob Werner alone. Maria would find help for Tru whether she wanted it or not.
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"I have to know what you've been hiding. I can't help you unless you tell me the truth about what happened the night Sophy was murdered." Alex glared at the two men seated in front of him. Now was not the time to go easy. He seized his restless pacing and stopped in front of Carl Kueshner, who sat slumped over, his head in his hands. "I know you are in love with the countess Von Ott. If you don't tell me the truth, I will bring your father in here and ask these same questions in front of him. Do you want that?" Alex knew he had made his point when both men shifted uneasily on the cot where they sat, and Carl Kueshner's head snapped up, his eyes meeting Alex's for the first time during the whole interview.
"Maria and I have done nothing wrong!" Carl glanced around the dreary cell, recalling the last time he had stood in the music room of Maria's home and held her in his arms. Would he ever be free to return to her? "We love each other. That is not a crime."
"You were with the countess the night Sophy died, weren't you?"
Carl glanced at his still silent brother. "I took John there after he came to the print shop where I was working late. I didn't want to take him home because of the condition he was in, and I couldn't think of anywhere to go but to Maria's."
"So you both spent the night at the countess's house?"
"Yes, but there was no reason to tell anyone. It wouldn't have helped John anyway."
"Carl's right. I agreed with him the next morning that it was best not to tell anyone." John broke his silence for the first time since the interview had begun. "I was completely drunk by the time I arrived at the print shop. Hours had passed since I left Sophy in that alley, and I couldn't remember anything after that except that she was alive when I left." John's voice rose in defiance. "I didn't kill her! But Carl couldn't prove that. Why risk bringing shame down on the countess and my brother? Besides, my parents would have been terribly hurt if they had learned about Carl's true feelings for Maria Von Ott in that way."
Alex found himself admiring again the bonds that held this strong family together. But the blind loyalty that had led these two men to conceal the truth for so long had also kept Alex from making headway in his investigation. The big detective returned to pacing the floor. "All my instincts are telling me that the two murders are related. And it's my guess the connection runs through the countess. If we find that link, we find who's responsible for these crimes."
A cold hand closed around Carl's heart at the thought of his beautiful Maria in an awful place such as the cell he and his brother now occupied. "Are you going to arrest Maria? She is not capable of being involved in such acts, I tell you!"
Alex stopped and leaned against the cell bars. He stood lost in thought for a few moments before responding to Carl's urgent inquiry. "I don't have enough evidence to arrest the countess." Alex watched as Carl Kueshner relaxed a little. Then Alex asked the question that was uppermost in his mind. "Do either of you recognize the name Jacob Werner?" Both men looked puzzled. "Did the countess ever mention that name? It could be very important."
"Stop it! Maria had nothing to do with this, I keep telling you!" Carl bolted up from the cot where he had been sitting to stand against the bars of his cell, his fingers gripping the bars so tightly his knuckles were white. "She is too kind and gentle to be connected to anything as ugly as these crimes. You are looking in the wrong place for answers."
Alex jerked Carl around to face him. "Are you trying to convince yourself or me?"
Tru's brother couldn't answer the detective. How could he go on denying that there were questions in his own mind about what he had seen the night of Maria's musicale?"
Alex's hands dropped from Carl's shoulders. "Is there something you're still not telling me, Carl?"
Carl walked back toward his brother and sat down beside him. "The night of the musicale, Otto Klienst and his mother came to Maria's house. She seemed uneasy in their company. I saw the three of them engaged in what appeared to be a rather intense conversation. Later that evening, I happened to see Otto Klienst in the hall arguing violently with the man found dead in the alley."
John stared at his brother in amazement. "Why have you never told me this?"
"At first, I kept it from you because Tru and I were worried about the lie we knew Sophy had told you about her reasons for not meeting you that night." Carl shook his head. "But after I recognized the body in the alley as being the man I saw at the musicale, I was terribly confused. I wanted to find out from Maria why that man had been at her home before I said anything to anyone." Carl looked into his brother's eyes, hoping to find understanding. "I think, John, that I knew somewhere in my heart I was already falling in love with Maria."
Before Alex could ask another question, a discreet cough sounded from behind him, drawing his attention to the other side of the bars where Richard Shaw now stood, quietly observing the scene. "Sorry to interrupt, but there is someone here to see you, Alex. She says it is most urgent she talk with you."
"Can't someone else talk to her? Tell whoever it is I'm too busy."
"I think you had better see her yourself, Alex. She's very beautiful, and she says she's a countess."
Chapter 13
Tru was careful to stay far behind Jacob Werner as he traveled southward along Sixth Avenue. She had been trailing the tall, slender man through the streets of New York ever since he had departed from Hearns. After Maria's strange revelations the day before, Tru had gone home and carefully thought out how she would follow the man she had known as Joseph Kenton when he left work the next day. Now she was determined to see her plan through.
When she had arrived for work this morning, Tru had been afraid the suspicions that now filled her mind would show on her face, putting the floorwalker on guard. But the man avoided her all day, adding to Tru's growing conviction that the countess had been right in her identification of her former butler.
Now Jacob Werner seemed to be nearing his destination as he slowed his brisk pace, forcing Tru to slow hers. She found a large tree and ducked behind it as she watched the impeccably dressed man approach a run-down house huddled on the fringes of a seedy neighborhood in the Lower East Side. As she stayed a safe distance away, she observed Jacob Werner head for the front door of the house, which was separated from its nearest neighbor by a narrow alley. To her surprise, the floorwalker veered away from the path leading up to the entrance of the house, moving instead down a dark, uninviting walkway that ran parallel to the alley. He seemed to be headed toward the rear of the building.
Tru waited until she was certain Werner was far enough ahead of her. Then she darted into the alley and watched Werner from behind the cover of the bushes that separated the alley from the path running beside the house. She was careful to stay in the shadows as she waited in the growing twilight.
When Jacob Werner came to the end of the path, Tru observed him enter a shed in one corner of the untidy yard. As she watched, a lantern flickered to life in one of the windows of the tiny building. Tru left the alley and moved quietly along the path by the house. She stayed in the shadows made by the bushes until she reached the shed and then ducked around to the back of the run-down building. There a small window in the rear wall revealed a glow from within. Tru crept forward, trying to catch a glimpse of the man inside. The shed was small enough that the window was not very high. Tru stood on her tiptoes and dug her nails into the half-rotted wood of the windowsills to help her keep her balance as she stared through the grimy glass.
It took a few moments for her eyes to adjust to the meager light, which was casting strange silhouettes around the crowded space. Boxes were piled up everywhere, and discarded junk littered the floor, but Tru soon spotted her quarry slumped in a chair with his back toward her. Jacob Werner sat at a small wooden table placed along the wall opposite Tru's vantage point. He appeared to be writing in a ledger of some kind. The lantern he had lit upon entering the shack was hanging on a peg some distance away by the door.
Tru stood observing Werner for what seemed like an eternity, growing increasingly uncomfortable as her toes began to cramp and her fingers stiffened from clinging to the sill. Her arms began to feel as if they would fall off, and she wondered if she could keep up her vigil much longer. Finally, just as Tru had decided to leave the way she had come, the floorwalker closed his book and put down his pen. He rubbed his eyes then reached down into a nearby box. Tru watched with renewed interest as the man took out what appeared to be several thick candles with long wicks and began to bundle them together.
Tru had been staring at this strange activity for several minutes when it dawned on her what Jacob Werner was doing. It was dynamite he was bundling together. Tru stifled a gasp. No wonder he was keeping the lantern away from the table. What was the floorwalker from Hearns doing with boxes of dynamite? Tru's heart had begun to pound so loudly she was afraid Werner could hear it. Tru had no choice now. She had to get into that shed. She had to find out what Maria's ex-butler intended to do with all those bundles of dynamite he was preparing.
Fighting back the desire to cry as she thought of her warm home and favorite chair, Tru turned her full attention back to the scene playing out in front of her. Long minutes passed, and the moon grew brighter while Tru watched the pile of deadly bundles grow steadily. Just when Tru became certain her toes were giving out, Werner stopped and stood up, stretching his long arms over his head. Tru waited breathlessly, praying the man would not sit back down. To her relief, he moved in the direction of the only door, removing the lantern from its hook.
Tru abandoned her post underneath the window. She moved to a corner of the building where she could peer around and observe the path leading away from the shed to the rear of the nearby house. As she watched, she heard the door of the shed open and close and the sound of a key turning in a lock. Then Jacob Werner came into view, heading away from her up the narrow stone walkway. In the moonlight, Tru could clearly see Werner as he opened the door to the house and disappeared inside. Soon, a light went on inside the rear of the house. Then more lights blazed on as the man apparently was moving from room to room, lighting the oil lamps inside the ramshackle structure.
Tru watched for some time, but the floorwalker did not reemerge, and, gradually, the lights in the house went out until Tru could make out only one dim light at the front of the house. Finally, she felt safe returning to the window at the rear of the shed, preparing to find a way inside. She couldn't chance trying to enter by the door, since the moonlight was bright and the door was clearly visible from the house. Besides, picking locks was not a skill she had ever learned, so her only option was to try to climb in through the back window.
Tru looked around for anything that she might be able to stand on to climb inside. After a brief search, she noticed an old doghouse half hidden in a nearby tangle of weeds. She tried to move the doghouse and was relieved to discover that it was smaller and lighter than she had initially thought. She dug in her heels and half pushed, half rolled the doghouse to a position under the window. Before climbing up on the roof of the doghouse, Tru checked to make sure the lights were still out in the house where Jacob Werner was hopefully sound asleep. Then praying the wood was not too rotten to hold her weight, Tru hoisted herself up on top. She carefully pushed on the window, which gave way but not without a loud creak. The noise sounded to Tru like an explosion. She froze, expecting to feel the floorwalker's hands grabbing at her at any second. But nothing happened, and, at last, Tru heaved a sigh of relief. She resumed her task, and, gradually, Tru was able to open the window so that there was enough room for her to crawl through.
With only minor regrets at the thought of what it would do to one of her favorite frocks, Tru stood on her toes and hoisted herself halfway through the open window. Hanging half in and half out, she struggled to wiggle her hips through, wincing at the sound of her skirt ripping as it caught on an old rusty nail. Grateful her skirt had been ripped and not her skin, Tru tried to twist around so she could drop feet first onto the floor of the shed. But before she could figure out how to accomplish that maneuver, she found herself falling headfirst onto the shed's dirty floor. She landed with a thud, but after lying in a heap for a moment, she realized that because of the short distance from the window to the floor, no bones were broken. She scrambled to her feet and took a deep breath before moving further into the shed.
Tru brushed off her skirt, trying not to think about what else might be lurking behind the shadows that filled the dark room. When she had landed on the floor, a cloud of dust had been raised that still whirled around her head, making her cough. She took a few steps away from the window, fighting back a scream when she stepped into a tangle of spiderwebs that wrapped themselves around her face. Tru brushed them off then turned her attention to the task at hand.
It took a moment for Tru's eyes to adjust to the shadowy interior. The moon was now her ally as she struggled to navigate from the window to the table where Jacob Werner had sat writing only a short time before. Though the light from the moon was dimmer inside the shed, there was enough of its brilliant glow left to allow Tru to move across the floor with only an occasional injury to her limbs.
Eventually, she reached the table, feeling along its rough surface until her fingers encountered the leather-bound journal she had observed earlier. She picked up the book, returning to stand close to the window where the moon's glow was strongest. She didn't dare remove the entire volume from the shed for fear Werner would notice. But maybe she could remove the last few pages he had been writing on without arousing his suspicions. Those pages might contain information that would provide her with a clue as to what the elegant floorwalker needed with a shed full of dynamite. If it was his habit to write in the journal at the end of each day, that would give Tru twenty-four hours before he would discover some of the journal's pages were missing. Anyway, it was a chance she had to take.
Tru held the book up to the moonlight, flipping hurriedly through its leaves. It was definitely a daily record of some kind. There wasn't enough illumination for Tru to actually read any of the entries, but she could tell when the writing stopped. She counted back several pages from where the bold scrawl seized, carefully ripping those pages free. Then she retraced her steps across the room, returning the journal to the exact place on the table where she had found it. All she could hope was that the pages she had dared to take would hold some clues that would help her solve the murders and set her brothers and father free. Folding the precious papers, she stuffed them into the pocket of her battered skirt, preparing to leave the way she had entered.
She moved one of the boxes that had been stacked away in a corner to a position under the window and hoped that the sheer number and haphazard placement of the boxes would keep Jacob Werner from noting one was out of place the next time he entered the shed. Climbing up on the box, she grabbed the windowsill and attempted to pull herself up and out of the window. It was harder than she thought it would be, and only desperation gave her enough strength to finally manage to pull the upper part of her torso onto the windowsill. There she hung for what seemed like an eternity, half in and half out of the shed. Pushing with all her might against the outside wall while thrusting her hips forward inch by inch, Tru finally managed to drag herself out of the window. Then she promptly fell hard headfirst to the ground, cracking her shoulder against the doghouse on the way down.
Tru lay on the ground for several minutes, catching her breath before she struggled to her feet, wincing in pain. She reached up to shut the window, rubbing her shoulder as she did. Then she moved around to the side of the shed where she could see the house, checking for any sign of movement, but there was none. Staying in the shadows, she quickly fled the yard, disappearing into the dark alley. When she reemerged out onto the street, she realized she had been holding her breath ever since she had emerged from the back of the shed. She took a big gulp of air then hurried in the direction of home, her fingers curled tightly around the pages hidden deep within her pockets.
***********************************************************
"Where have you been!"
Tru's heart leaped to her throat as a nearby shadow split in two, with one half becoming a New York City detective, while the other half stayed a large bush. To her dismay, Alex Marshall stood looming between Tru and the welcoming lights of her home, and by the frown on his handsome face, he was not a happy man.
"I don't see as how that is any of your business." Tru thrust her hands into the pockets of her skirt, closing her fingers around the pages inside. She had to get away from Alex to the privacy of her own room where she could examine the writing on those pages for any clues that might help free the men of her family. Tru made a move to go around Alex, but he reached out to grab her elbow.
"Let go of me, or I'll scream, 'Bloody murder!'"
"Don't try any of your tricks, little bird. My patience is at an all-time low. I just might be tempted to pluck your tail feathers. And, believe me, it would be painful." Alex tightened his hold on this infuriating woman he loved, pulling her closer as she squirmed.
"Stop that wiggling right now and listen to me."
"Never!" A pointed but slightly soggy boot began a swift journey in the direction of Alex's shin, but he anticipated Tru's move and avoided the impact.
"You horrible little minx, why can't you ever trust me and just do as I ask?" A long and anxious wait outside Tru's house had shortened Alex's fuse. When she hadn't arrived home from work on time, his imagination had driven him wild with all manner of terrifying images. Most of them involved seeing Tru lying dead in one of the more-dangerous neighborhoods in New York.
"How dare you call me horrible! You've done nothing but hurt my family! I hate you!"
"Do you now? Well, let's see just how much you hate me."
Before Tru realized what was about to happen, Alex's arms stole around her waist, and his lips crushed hers in a punishing kiss. Once again, she felt a familiar warmth sweep over her as he deepened the kiss. Her heart began to pound, and, for one long moment, Tru forgot everything as her body responded to Alex. But in the next instant, all the painful memories of Alex's past betrayals returned.
"No, Alex, I won't let you trick me again." Tru pushed against Alex's chest, but she couldn't get free. "Let go of me! This isn't doing either of us any good."
"Yes, it is." Alex abruptly released Tru and stepped back, causing her to nearly fall to the sidewalk. As she righted herself, her eyes blazed up at Alex. But instead of letting go of a stream of angry words, she gasped when she saw the papers that had been buried deep in her pocket were now clutched tight in the detective's fist.
"Why... you... you... you only kissed me to get those! Give them here! They don't belong to you." Tru tried to calm down, not wanting Alex to realize the papers he now held were important. "Give them back. They're of no use to you. They're just worthless papers from work."
"Tru, stop insulting my intelligence. I knew something important was up the minute I spotted you coming toward me." Alex reached out to finger the ripped and dirty skirt Tru was wearing. "Since when do you get in such a state selling gloves to little old ladies? And let me give you a little professional advice. If you have something in your pockets you're trying to hide, don't draw attention to it by reaching for it." Alex's eyes softened as he studied Tru's crestfallen face. "But as for the kiss. I'll always kiss you for only one reason—because I love you."
A most unladylike snort greeted Alex's last remark, and he almost laughed. But now was not the time to provoke Tru further. He decided instead to try the element of surprise in an effort to get her to listen to reason. "I know all about Jacob Werner, Tru. The countess paid me a visit at police headquarters."
Alex's words had the desired effect. Tru stopped glaring at him, and her eyes met his. "What do you know?"
"I know that Maria Von Ott is certain the floorwalker at Hearns is her butler from years ago in Prussia, but he avoided her at Hearns, and so she can't swear to it yet. The countess told me everything she told you. And it makes no sense that Jacob Werner wouldn't have kept in touch with a boy he had treated like a son. Nor do I believe that it is a coincidence that the Klienst family and Jacob Werner ended up in the same city so close to one another. It's not much to pin our hopes on, but my gut tells me that Jacob Werner holds the key to both our murders, and there is a good chance the countess is right about your floorwalker's identity." Alex studied Tru's face, looking for signs she might be willing now to open up to him. "What we need to find out is why your floorwalker didn't want to be recognized."
"Those papers you're holding in your hand just might tell us that very thing." Tru looked up into Alex's eyes and made a decision. She would have to trust this man who had hurt her so badly in the past, since it seemed she had no other choice.
"Where did you get these papers, Tru?"
Tru's gaze shifted to the sidewalk. "I followed Werner to his home after work today."
"You what!" Alex's face turned white and then a bright-red color. "Why on earth would you do a stupid thing like that?" Alex's hands reached out to grab Tru by the shoulders. "You know this man at the least is a liar and maybe worse." Alex's couldn't resist giving Tru a small shake. "You could have been killed!"
"What was I supposed to do—stand by and watch my family suffer and do nothing because I was afraid!" Tru stomped hard on Alex's foot so he would let her go. "Just because I need you to help me capture Jacob Werner doesn't mean I'll tolerate you pawing me, Alex Marshall. Now we'd best get back to the task at hand and look at these papers, don't you think?"
Alex didn't know whether to laugh or to cry. He'd been completely unmanned at the thought of Tru in grave danger. But he had to admire her spirit. "So you followed Werner home. What then?"
"I watched him wrapping dynamite sticks into bundles."
Alex's stomach gave a lurch. The idea of the damage a large quantity of explosives could do in a city like New York with its streets teeming with humanity was enough to fix his mind firmly on one track—Jacob Werner had to be found and brought in for questioning. "Tru, how did you get those papers, and what's on them?"
"I climbed through the window into the shed where the dynamite was kept."
Alex struggled to keep from losing control of his emotions a second time. After all, Tru was standing safely in front of him despite her crazy stunt. "Where was Werner while you were executing this little trick?"
"He had gone inside his house." Tru wrinkled her nose at Alex. "You don't think I'm stupid enough to go in the shed when Werner is in there?"
"Heavens no! You always show such good sense and restraint. So how did you get the papers?"
Tru decided to ignore Alex's sarcastic tone. "They were in the shed on a table. I saw him writing in a journal before he prepared the dynamite bundles. I was hoping if I took just the last few pages, he wouldn't miss them, and I might find out something about what he's up to. But it was too dark to really read them in the shed, and I have to admit I was anxious to leave before I got caught."
Alex nodded in understanding. "So let's read them now."
Tru and Alex moved under the light of the nearest lamppost and carefully unfolded the journal pages. The sentences were scrawled haphazardly across the paper as if the writer were in the throes of some intense emotion as he poured out his thoughts. It was sometimes difficult to make out individual words, but Alex and Tru continued to study the handwriting until they were finally able to decipher what Jacob Werner had written.
"I saw the bitch again today after all these years!" Tru's voice reflected her shock as she continued to read out loud from the pages. "How I hated having to wait on that silly woman and her weakling of a husband. They never suspected how much I wanted to strangle them in their beds. How easy it would be to kill her now!"
"He certainly fooled Maria and her husband. She told me that he was a good man who served her and her husband well." Tru looked up at Alex. "How could someone hate that much and keep it concealed?"
"You and your family are so open you can't conceive of the deceit people are capable of, but I've seen how people lie and cheat time after time." Alex smiled gently at Tru trying to soften his words. This was her first real encounter with the dark side of human nature. Something Alex had seen almost every day of his life, starting when he was a boy and his father was murdered by an angry mob. "If a person carries hate in his heart for a long time, it festers and grows until there is nothing left for it to do but explode." Alex thought of the dynamite lying somewhere in a shed, waiting for a human hand to unleash its destructive power. "I'd say that Werner has just such an explosion in mind. Read the rest."
Tru glanced down through the page. It contained more of the same such rantings. But on the next page, the name of Sophy Klienst jumped out at her. Tru's voice trembled as she read the next words out loud. "Our poor beloved Sophy. They killed her! Society and its cruelty! Always destroying the innocent to perpetuate its corruption. Probably one of those rich parasites that liked to leer at her because of her beauty. Pigs! All of them!"
Tru stared at the words she had just read. "That can't be right. Jacob Werner had to have killed Sophy." Tru fought back tears. She felt Alex put a comforting arm around her shoulders, but she moved away from him. Her brother would still be Alex's prime suspect in Sophy's murder.
"Tru, don't give up. I admit that passage seems to indicate that Werner didn't kill Sophy. But there has to be a connection. I know it. And my money, for certain, is on Werner as the killer of Heinrich Schueller."
Tru was grateful for Alex's attempt to comfort her, but nothing in her world would be right unless all her family were free and together again. She turned her attention back to the pages. She hurried through several passages of diatribes against the wealthy and the unfairness of society in general as Alex peered over her shoulder. They were both beginning to despair of ever finding a concrete clue about Jacob Werner's plans when they reached the last page. Alex got so excited he tore the page from Tru's hand and began reading aloud.
"No more time! Thanks to that bitch, the countess. I think she recognized me, but I can't be sure. If she did recognize me, she'll be back with questions about where I've been all these years and why I'm using a different name. We have to move now!"
"There seems to be someone else involved." Alex glanced down at Tru, who looked puzzled at first. Then a light came on in her eyes. Alex knew she had the same thought he had. And their suspicions were confirmed when Alex read aloud the final passage written by this madman.
"'I sent word to Otto tonight. We must act quickly! We will begin our onslaught on our enemies so that we may bring about a glorious new order! We will blow up their pathetic symbol tomorrow night to send a message. With Otto at my side, we will begin to plan our next attacks on their weak and corrupt leaders. My heart is full at the thought of the triumph that awaits us.'"
The writing stopped with those words. Tru glanced at Alex and knew he shared her frustration. They knew more about the crazed workings of Jacob Werner's mind and the type of crime he planned but not enough details to stop him.
"What now, Alex? What symbol could he be talking about?"
"I don't know. But we do know that he intends to move on his plan because his hand was forced when the countess saw him." Alex's frown deepened. "Jacob Werner is a very clever man. He's on alert. He says in the journal they will put the plan in motion tomorrow night, and I believe he'll stick to that timetable. The countess may not have a clue as to his real feelings and intentions, but Werner knows sheer curiosity will eventually bring Maria Von Ott back to Hearns looking for him.
"Whatever he plans, it has to involve the dynamite?" Tru had never seen Alex look so worried. "I know where he lives. What if I tell you and we go there, get the dynamite, and you arrest him?"
"First, there is no 'we.' From this moment forward, you are out of the detective business. As to the dynamite, I can't be sure he won't move it after being seen by the countess. And we don't know how often he writes in that journal. He could find out the pages are missing sooner than we hope."
"Well, he didn't see me following him." Tru cast a defiant look in Alex's direction. "I was very careful. All the lights in the house were off when I left. I'm certain he was asleep. Would he go to sleep if he thought he was being followed?"
"I'm sure you performed like a seasoned New York detective. But this isn't just about whether he sensed he was being followed. I doubt you'd be standing here if he had." Alex forced himself not to dwell on what could have happened to Tru if that had been the case. "He's a fanatic, and fanatics love to spout their theories. Fanatics are the hardest criminals to catch because there is little logic behind their actions. I have to be careful. If I move too fast before I have enough evidence against him, he might go free and escape into hiding."
"But we have the pages from his journal. Isn't that evidence enough?"
"Evidence that was taken without proper procedures and by a woman whose father and brothers are implicated in two murders and an anarchist plot. We have to prove your brothers are innocent as well as prove Jacob Werner guilty." Alex rubbed his weary eyes. He'd gotten little sleep for the last few nights and would get even less tonight. "And I'm sure as fanatical as Werner is, he would never break under questioning, so we wouldn't know if anyone else is involved in the plot besides him and Otto Klienst. I think I have a better chance of getting the evidence I need if I follow them somehow. If I follow them to the target, I can capture both of them red-handed."
"What will that do?"
"Besides stopping the crime, whatever it is, I can get hold of the whole journal before Werner has a chance to destroy it. Whatever plot Werner has devised, I'm positive it is outlined in the journal. That would prove your father and brother Carl are not involved in a terrorist act. I'm also willing to bet that there is evidence in that journal that Werner was working with Heinrich Schueller and something went wrong, causing Werner to kill Schueller. That would prove none of the men in your family murdered Schueller."
"But what about John and Sophy Klienst's murder?"
"We have to take this one step at a time. If we stop whatever it is that Werner and Klienst are planning for tomorrow night, then we get them both in jail. Werner won't break under questioning, but Otto Klienst might. If we can get Klienst to talk and recover Werner's journal, at the very least, we would be giving the police another credible suspect for Sophy Klienst's murder. Do you think there is a chance Werner will be at the store tomorrow?"
"We are having one of the biggest sales of the year. Normally, I'd say, there is no way that Joseph Kenton, the floorwalker, would be absent. He's never missed a day of work as long as I've been there. But I don't know about Jacob Werner, the terrorist."
"Werner wrapped dynamite, wrote in his journal, and then went to sleep. That would seem to mean that though he has to move faster than he planned, he doesn't think the countess will affect his plan yet. After all, she only recognized him as her faithful butler. She is ignorant of his real feelings. If he goes to work tomorrow, as is his routine, then I will follow Otto Klienst until he rendezvouses with Werner and then follow the two of them to their target."
"Can't you get your partner to help or some other policemen?"
"Richard is staying at headquarters to protect your father and brothers from any particularly rough interrogations that might occur at the hands of some of the more-overzealous members of New York's Finest. This anarchist plot has the whole police department on edge. There is no telling how far some of the men might go to get a confession—true or not. I also don't want to let anyone else in on this because not all my colleagues are as honest as I am. They have been known to be indiscreet when it comes to the press. Many of them are not above talking to the reporters from Mr. Hearst's paper if promised remuneration or a career-helping mention in the paper. No. I need to solve this and then present it to my boss with all my evidence in order."
"But what if we lose him?"
"We are not going to lose anyone! How many times do I have to say it? You are not 'helping' anymore." Alex's frustration threatened to boil over. "I will follow Otto Klienst until he joins up with Jacob Werner and then follow them both to their target—alone!"
"Not without me you won't!" Tru's eyes shot fire at the big detective. "I will not sit around and wait to see if you can catch Jacob Werner. You wouldn't even know that they planned anything if it wasn't for me, Alex Marshall!"
Alex knew Tru was right about that, but the thought of her anywhere near such a madman froze the blood in his veins. "Absolutely not, Gertrude Kueshner! You will not put yourself in harm's way ever again while I'm around."
Tru could tell Alex meant it, but she fully intended to ignore him. "Don't worry about me, I can take care of myself."
Alex frantically searched for an argument that once and for all would dissuade Tru from interfering, since she wouldn't listen to any pleas about her own safety. "Tru, if Jacob Werner does come to the store tomorrow to maintain the cover of his usual routine, don't you think your absence might alarm him? After all you, were standing with the countess when she recognized him. We don't want him to bolt before we can prove what he's up to."
Tru thought for a moment and had to admit that Alex's argument made sense. "I'll think about what you said." Tru turned her back on Alex and started toward her home. As she walked away, she glanced back over her shoulder to have the final word. "But I still think you need my help. So far, you've done nothing without me but put the wrong men in jail."
Alex stared at Tru's back as she calmly walked away. Then he headed in the direction of the nearest saloon. He needed a good stiff drink.
***********************************************************
Tru glanced at her watch. The day had been a very long one. The customers had come in droves and been more demanding than usual. And all day, Tru had to make sure she acted completely normal around Jacob Werner. The floorwalker had come to work as always just as Tru thought he would. If Tru hadn't known that he planned an act of terrorism for that very evening, she would never have suspected the dignified man helping old ladies with their packages to be capable of any such thing.
But now the time had arrived for Tru to put her own plan into action. She had stayed awake last night until she thought of a way that she could help Alex Marshall capture the two terrorists whether he wanted her to or not. Finally, an idea had come to her.
Tru reached up as if to remove something from one of the shelves above her. As she reached up, her hand deliberately bumped a decorative vase that sat atop the shelf. She made sure the vase fell in her direction. Then she slumped to the floor, landing near pieces of the vase that were now scattered everywhere. All eyes in the store turned to look in the direction of the loud crash. Immediately, a very nervous Mr. Hartley identified the source of the disruption and rushed toward Tru's department.
"What on earth have you done, Miss Kueshner?"
"I'm so sorry, Mr. Hartley. I've had a terrible headache all day, and it's made me somewhat dizzy. I just didn't want to ask to go home early with the store so busy." Tru held her head as if she was in a great deal of pain and let out a discreet moan, trying not to overdo it.
"Well, I certainly wished you had! You've broken a very valuable vase!" The store's manager was more concerned about the broken vase than Tru's head. He stared down at the shards of broken pottery and frowned. "It will take a while to clean this mess up. I do believe you also caused one of the countertops to crack. I hope you know this damage will come out of your next paycheck."
Tru put her hand to her head again and pretended to be weak in the knees, moaning loudly as she grabbed the nearby counter for support. That was the final straw for the harried manager. "You must go home at once, Miss Kueshner, before you cause any more disruption. For one thing, your moans are upsetting the customers. Please get your wrap and go. I'll see you on Monday when I'll let you know how much all this is going to cost you."
Tru breathed an inward sigh of relief and staggered in the direction of the cloakroom. She avoided looking in the direction of Clarice's department, knowing her friend would be bewildered by her actions. Once she was safely outside the store, she straightened up and started to walk away at a brisk pace. She planned on going to Jacob Werner's house and taking up a hiding place where she could wait until Werner himself came home after the store closed. She knew he would have to get the dynamite he needed and then wait until it was nearly dark before setting off to rendezvous with Otto Klienst. Tru planned on following Werner until he met with Klienst. Since Alex was following Klienst, by the time Alex knew what Tru was up to, it would be too late to send her home without the risk of losing his prey.
***********************************************************
"And where do you think you're going?"
Tru jumped out of her skin at the sound of a familiar voice. "I thought you were following Otto Klienst?" Tru was indignant. "You can't change a plan like that. It's not fair."
"You want to talk about fair..." Alex Marshall was about as mad as he ever got, and that could be pretty mad. "You promised me you'd behave and stay out of danger."
"I did no such thing! You ordered me to stay away, but I never said I would. Why should I listen to you? And you never answered my question. What are you doing here? You're supposed to be following Otto Klienst."
Alex tried to rein in his temper. He didn't have time to continue arguing with Tru. "Otto Klienst hasn't left his house all day. I started to get worried about you and whether you'd stay away as I asked. The more I thought about it, the more I knew that would never happen. I figured you'd try something sneaky, so I hurried over here, thinking I could prevent you from getting yourself hurt or worse. Excuse me for wanting the woman I love to stay alive."
Tru tried not to look into Alex's eyes, knowing her actions had hurt him. "Alex Marshall, you know I can't give up trying to prove Papa and the boys innocent. And if you really loved me, you'd understand that." Tru did look up then and was surprised as the impulse seized her to reach up and caress his face, which looked haggard and worn. She longed for a glimpse of the impish grin that made her heart jump.
"Well, no matter what I want, I know now beyond a doubt you'll do what you believe you must no matter what I say. So all I can do now is try to figure out how to stop Klienst and Werner and protect you at the same time." Alex rubbed his forehead in frustration. "That means making an adjustment in the plans."
"I tricked Mr. Hartley into sending me home early so I could go to Werner's house and wait for him to come home. Then I was going to follow him. Why not let me do that? I would meet up with you when Werner and Klienst rendezvous."
Alex tried to remain calm in the face of Tru's stubbornness. "No! You are staying with me! Otto Klienst's home is very near here, so we will go back and resume watch over him. It is getting close to the time that Hearns closes. I'm not letting you out of my sight again, and if both of us tried to follow Werner, he would certainly spot us. It's better if we follow Klienst to the rendezvous and from there follow both of them to whatever their target turns out to be."
"Why can't you trust me to follow Werner?"
"For one thing, you were very lucky not to get caught following Werner yesterday. For another thing, you've proved to me I can't trust your judgment because your family is involved and you're desperate to get them out of jail." Alex looked deep into Tru's eyes, trying to make her see he was right and that he only had her safety at heart. He knew his own heart would be broken if anything were to happen to this woman that he had come to love so passionately. "Tru, your emotions will cause you to make a mistake—believe me, I've seen it happen many times. This is too important to let you wreck things. People could get killed. From now on, you will do everything I tell you, exactly how and when I tell you. If you don't, I swear I'll tie you up and stash you someplace safe until this is all over. Understand?" Tru managed a weak nod. The look on Alex's face was grim. She knew he meant business.
"Good. So we are leaving right now to go back to watch Otto Klienst's house. We have to hurry. It is still light, so he won't make a move yet. But as soon as it's twilight, they will begin whatever it is they have planned for the city of New York."
Reluctantly, Tru fell in step beside Alex, trying to keep pace with his long strides. They hurried away from the busy street in front of Hearns and, after a series of twists and turns, arrived at a more modest neighborhood of homes. Moments later, they arrived at the street where Otto Klienst and his mother lived in a small but neat row house. Alex looked around for a hiding place from which to observe Otto Klienst's movements. He needed to find a different spot than he had used earlier, since now there were two people trying to conceal themselves. He noticed a shed backed up against a fence behind a dilapidated house. The hiding place was directly across the street from the Klienst home. Dragging Tru behind him, they moved carefully along the fence that ran behind the block of row houses and squeezed into the small space.
"Is it necessary for us to wait in this tiny place?" Tru was trying not to breathe too deeply because when she did, her chest rubbed against Alex's back. "I can't see over you in this position, so what use am I? Why can't I find another hole to hide in? You can still keep an eye on me that way."
Alex took a long whiff of Tru's scent. This was the best-smelling surveillance he had ever been on. "Not a chance. You stay put. We already had this discussion." Alex moved back an inch or two, trying to get as close to Tru as he could. If he was going to spend time in a cramped, dirty space, he might as well enjoy it. "You do exactly as I say. I'm not letting you two feet away from me."
"Well, how about two inches? I'm suffocating." Tru was having trouble catching her breath, but she was afraid it wasn't just because they were tightly packed together.
Alex finally relented after more pleading by Tru. He maneuvered around so Tru was able to get in front of him. Then they both settled in to watch the house across the street. The sky began to turn inky; soon it would be dark. The night was cloudy, blocking the moon. Lights began to appear in various houses around the neighborhood, including the one Alex and Tru stood watching.
As the minutes passed by slowly, Tru found it increasingly hard to maintain a stony silence. She was getting more nervous having to be near Alex for so long. Several times she almost forgot and laid her head back against his broad chest. She would have loved to have his arms around her to ward off the chill from the damp night air. Tru glanced over her shoulder and noticed Alex had taken a small piece of paper from his pocket. The paper was torn at one corner.
"What are you looking at?"
"Something that was found on Heinrich Schueller's body. I always thought there was more to learn from it. There's a map drawn on it that I'm sure is someplace in or near New York City. But the map's poorly drawn and somewhat blurred, and I've never been able to figure out what it's a map of."
"Let me look." Tru twisted around to get a better view of the paper in Alex's hands. The light was growing dim, but she could just make out the drawing on the paper. "I know what this is. It's a map of Bedloe's Island. My father did a whole big article on the island and the preparation of the platform to hold the new statue from France. Then he did another article when the steamer landed with the crates containing the pieces of the statue. He said it was very moving to look around at all the boxes piled everywhere and realize that they would soon be assembled into a giant sculpture that would be a symbol of freedom for all the world to see."
"Symbol!" Two voices gasped out the same word in unison.
"That's it!" Alex couldn't contain himself. He put his arms around Tru and kissed her. Tru resisted for a brief moment then returned his kiss. Only the sound of a wagon approaching on the street brought them back to reality. When their lips parted, Tru was horrified that she had responded to Alex yet again.
"Alex Marshall, that kiss meant noth—" Tru never finished her protest because a large hand was quickly clamped over her mouth. Tru almost bit down on Alex's palm where she thought it'd hurt the most, but, at that same instance, Alex whirled her around to face the street. There she caught sight of a wagonful of bushel baskets sadly in need of repair. Perched on the wagon seat was Jacob Werner.
"There he is." Alex's voice was tight with worry. He had to stop this man for his own sake as well as the sake of New York City.
"Alex, that wagon is one that I saw near the shed at his home yesterday. And those bushel baskets were scattered around the yard."
"He has to transport the dynamite somehow. This means he's planning a big explosion if that wagon is full of dynamite."
"Look." Tru nudged Alex to look in the direction of the house, and they both watched as the front door opened, and Otto Kleinst hurried to join his mentor on the wagon seat. As they continued to watch, the two men headed east a short distance and then turned the wagon south.
"They have to be headed for a pier near Battery Park. Come on, we've got to hurry or we'll lose them."
Alex took off at a brisk trot in the direction the wagon had taken. As Tru started after him, she glanced back at the home they had just spent hours watching and thought she caught a glimpse of a dark shadow peering from one of the windows. But Tru had no time to wonder whether she and Alex had been seen because she was rapidly being left behind in Alex's wake. Tru picked up her skirts and her pace. She had almost caught up with Alex as they rounded a corner, and Alex veered in the direction of a nearby hansom cab.
"Hurry, Tru." Alex glanced back to make sure Tru was with him, then he called to the cab driver. "New York Police. I need you to follow that wagon that passed by here a few minutes ago."
"Fancy meeting you again, governor, especially when I'm so far from my usual bailiwick, so to speak." The driver turned and grinned down at Alex. "Aloysius McCafferty at your service again, sir."
Alex stared at the driver for a brief moment, at a loss for words, then he climbed hurriedly into the cab. "Then you remember the routine. Only this time it's imperative that we not be seen following the wagon. Think you can do that?" Tru was standing now by the steps of the cab, and Alex reached down and lifted her up into the seat beside him.
"Sure thing, governor. Is this the little lady we had the pleasure of following the last time, if I might be so bold as to inquire?"
"Yes, yes! Now hurry up!"
"Pleasure to make your acquaintance, miss, and if I might be permitted to say, you are definitely worth following."
"Why, thank you—I guess."
"Stop talking and starting driving. Tru, sit still and don't make any noise. Now move this bloody cab!"
"Sure thing, Aloysius McCafferty will never let you down." With a sudden lurch, the cab took off, sending Tru and Alex flying back against the seat. By the time they had managed to right themselves, the cab was following the wagonful of explosives at a discreet distance. Alex had to admit that Mr. McCafferty was as good as his word.
Tru looked at Alex's worried face and wanted to offer some encouragement. "At least we know where they're headed."
"Yes, but they must have a boat tied up at one of the piers to take them and the dynamite out to the island. There are dozens of piers they could use. We have to follow them to the right one, or we won't stop them before they're out on the water."
"If we don't catch them at the pier, then what?"
"Then we take a little rowing trip. How are your muscles, Miss Kueshner?"
Tru managed an offended sniff. "As good as yours are, Mr. Marshall."
Chapter 14
As the oars sliced through the water, Tru fought back a groan. She swore she couldn't row another foot. Thank goodness for all the hours spent at Rutgers with her brothers, rowing on the river, else she surely would have disgraced herself in front of Alex. Tru glanced to the front of the boat where Alex sat, a large dark shadow whose powerful arms kept the boat moving on a steady course over the inky waters.
The moon had stayed behind the clouds as they had traveled through the streets of the Lower West Side, following the innocent-looking wagon with its deadly cargo. Aloysius McCafferty had proven a genius at keeping his cab far enough behind their prey so as to be inconspicuous. To make sure they were not noticed, he had even pulled off into alleys on several occasions, waiting patiently for a few minutes before resuming the chase. But even the clever cabbie had been unable to avoid all of the wagons traveling to the warehouses that lined the piers to unload the cargo that would be loaded onto ships the next morning.
Finally, Alex had whispered the order to the cab driver to stop. The wagon they had been following could be seen in the far distance near one of the piers. But the shadowy figures of Jacob Werner and his accomplice were nowhere to be seen. Alex's heart sank as his gaze continued to search for any sign of his prey. Despite the driver's best efforts, they had arrived too late to stop the boat and the deadly cargo headed for Bedloe's Island.
Alex thanked the cab driver and paid him what must have been a princely sum by the size of the smile on the driver's face. "Ever need me again, just look me up. I kinda like this police stuff. Beats squiring around stuck-up swells who never tip enough. Maybe that Pinkerton Detective Agency needs some new blood."
"Maybe they do." Alex shook the cabbie's hand. "If you contact them, I promise to give you a ringing endorsement. And don't breathe a word of this to anyone tonight."
"Please, you insult my professional abilities. Wouldn't dream of blathering, governor." With that, the cab driver disappeared into the fog that now masked the streets near the piers.
Alex listened carefully until he thought he heard the sound of oars moving through the water, and then he went in search of a rowboat. When he found one, he motioned for Tru to climb in. Placing his fingers to his lips to remind Tru they must be completely silent, he handed her a pair of oars, and they began their own journey out onto the dark water.
Ahead of them, they could just make out the faint splashes that marked the other boat's progress toward the island along with the muffled conversation of the boat's two occupants. Alex was relieved to hear the men talking. It meant they were unaware that they had company in the fog.
Eventually, Tru and Alex heard the sound of waves lapping on a shore and the other boat hitting land. Alex stopped their boat immediately while it was still out in the channel and sat leaning on his oars. Tru wasn't sure what he was doing until she saw the faint glow of lanterns through the mist. As soon as Alex saw exactly where the two men had beached their boat, he began to steer silently away from that spot in a direction up shore from where Werner and his partner were unloading the explosives within a tight circle of light.
When Alex and Tru had traveled about a hundred yards away from where the other boat rested, Alex stopped their boat, letting it drift slightly. As their eyes grew used to the darkness of the island, shapes began to emerge. In the near distance, the great pedestal built to receive the grand statue from France appeared massive, solid, and eternal. All around the pedestal, there were crates piled everywhere.
Alex leaned back to whisper in Tru's ear. "They can't touch the pedestal. They don't have enough dynamite to put a dent in it. But they can blow up the crates containing the parts of the statue."
Tru nodded in agreement and whispered back. "What should we do?"
Alex was silent for several minutes, leaving Tru to think of the danger that lay before them. Then she felt the boat moving again. Alex had climbed out of the boat into the shallow water and was pulling it silently to shore. When he came to a place where the rocks jutted out into the water, he pulled the boat up onto land and hid it behind the rocks. Then he reached in to lift Tru out of the boat and onto shore. When Tru was on solid ground, Alex pushed her down behind the rocks, indicating with a forceful gesture that he wanted her to stay put. Then he left Tru and crawled along the shoreline in the general direction of the two men now busy carrying their explosives inland toward the crates.
Tru sat fuming. How dare he order her to stay put? In the glow of the distant lanterns, Tru could see Jacob Werner and Otto Klienst as they began to place their deadly bundles of dynamite among the crates holding pieces of the statue that was to stand as a beacon to all who came to this great city. As she watched, she could make out the dark shadow that was Alex creeping along the shoreline, closing the distance between himself and the two men bent on destruction. Suddenly, Alex stood up, and Tru saw the gleam of a gun in his hand.
"New York Police! You're under arrest. Let me see your hands at all times and step away from the dynamite."
Tru almost jumped up and ran to Alex's side. But before she could move from behind her hiding place, she watched in horror as Jacob Werner suddenly heaved the lantern in his hand in the direction of Alex, hitting the detective in the head and knocking him to his knees. The gun went flying across the sand to land near the water. Tru froze then crouched lower down into the safety of the shadows where she'd been hiding as the two fanatics jumped on a dazed Alex and wrestled him to the ground. Jacob Werner gave a length of rope to his partner and ordered him to tie Alex's hands behind his back.
"Stupid instrument of a corrupt system. Did you really think you could stop Otto and I from having our revenge?" Jacob Werner kicked the helpless detective in the stomach as hard as he could. "We are too smart for you."
Alex lay still for a moment, recovering his wits after the blow to his head. Then he struggled into an upright position despite the pain in his gut. "I wonder, Otto, if Heinrich Schueller or your sister, Sophy, would think Werner's brilliant—that's if they were alive to have an opinion? Was your sister's life worth your cause? You can't believe John Kueshner would kill the woman he loved. What would Jacob Werner be willing to do for his cause?"
"Leave my sister out of this, swine!" Otto Klienst fell on Alex and began beating the detective with his fists, trying to drive Alex into the dirt.
Jacob Werner rushed to stop his partner. "No, Otto. He baits us. We cannot stop our work now. We will deal with him after we have planted the dynamite."
"Then what will we do with him?" The younger man glared at the detective as he allowed himself to be pulled off Alex.
"We will simply tie him to one of the largest crates and let him blow up with his precious symbol of their false freedom built on the backs of the poor." The tall man who had once served the wealthy and privileged with great style laughed. "I believe the French said the name of the statue is Liberty Enlightening the World. The detective will be the first witness to the real enlightenment when the skies around this island are ablaze with our message."
"And what twisted message is that? That it is good to murder innocent young women?" Alex took one look at Otto Klienst's expression and decided to twist the knife some more, hoping to make the younger man lose control. "Or was Sophy still innocent? Is that why you killed her? Because she had disgraced you, Otto? Was it too late? Had Sophy sacrificed her virtue so you could have revenge against the man who ravaged your mother? Did you hate your sister because of who her father was?"
Tru heard Otto Klienst's tortured scream as he tried to break free of Jacob Werner's grip to get to Alex. Tru almost ran to help, but she knew that without a weapon, she was useless. She had to get closer and figure out a plan. So while the two anarchists were distracted, she crawled quickly to the shelter of a pile of crates between her hiding place and where Jacob Werner struggled to keep his young accomplice under control.
"Otto, no! You must stop this. You are letting him make a fool of you. Sophy is gone. The only revenge you have against any of them for what they've done to your family is in destroying the institutions of their privileged world. This is only the beginning. We will destroy other institutions they cherish and create anarchy just as I promised you."
As Werner attempted to calm his enraged companion, Alex scooted a few feet further away from the two men. The rope Otto had used to bind him had been tied hastily and loosened as Alex struggled on the ground with his younger captor. Alex began to work the rope back and forth, trying to loosen it some more. He only hoped Tru had enough good sense to stay put and let him get out of the situation on his own.
"What makes the two of you think you can create even a small ripple in the life of this city?" Alex wanted to keep Jacob Werner talking, and he had the feeling the man was more than willing to oblige.
"You who serve the wealthy and powerful are all the same. Ask Otto about the authorities in Prussia who turned a blind eye when his mother was raped by the count's brother."
Alex continued to work the rope down onto his palms as he watched Otto Klienst's face distort into an ugly mask. It was clear that Jacob Werner had a powerful hold on the younger man.
"They let the count's brother get away with raping my mother and taunting my father until he could no longer bear to live. And Count Von Ott simply sent his brother away to continue his life of debauchery somewhere else." Otto fingered the fuse of the dynamite he held in his hand. "I was too young to realize why my mother cried every night. And I would never have learned the truth if Jacob had not come to America and sought me out to tell me of this injustice. I would never have known that Sophy was the product of a hideous violation."
"Is that why you killed her?" Alex could feel the rope loosening with each twist.
"Why do you keep saying that? I did not kill my sister!"
"No, you just made a whore of her."
"Shut up, or I will stuff this dynamite down your throat and light it now." Otto Klienst looked as if he were going to explode himself. "Sophy was helping our cause. How do you think she felt when she found out the truth about her birth?"
"The way Werner wanted her to feel. Can't you see he used her just like he's using you?" Alex was almost free, but he knew his odds against the two men would be improved if the younger one was not fully in control of his emotions. "Are you sure he wasn't the one who killed your sister?"
"Don't pay any attention to him, Otto. I killed no one but that stupid weakling Heinrich Schueller."
Tru listened to the storm of angry words from her hiding place while she looked around for something she could use as a weapon. So far, she had come up empty-handed. Time was growing short. Even as he responded to Alex's accusations, Jacob Werner was placing the last of his deadly bundles around a large crate. Tru wasn't sure what Alex was trying to do by taunting the two terrorists, but she hoped he kept Werner occupied for a few more minutes until she could come up with a plan of her own. Just then, she spotted a hammer resting near the platform not too far away. A worker in a hurry to get home at the end of a long workday must have left it. Tru started to crawl on her belly toward the potential weapon.
"So you did kill Schueller." Alex scanned the darkness, trying to find Tru. He had to make sure she was safe where he'd left her before he made his move. But he couldn't see her shadow near the rocks where she was supposed to stay.
"Yes. We needed more money than Sophy would be able to extort out of Bentley to pay for our escape to South America, where we will go to avoid capture until we can plan our next attack. I remembered Schueller from his visits to the count before their political paths diverged. I had heard he was now in Chicago and active in the anarchist movement. I wrote to him, telling him I planned an act to further our cause here in New York and could he help us. But he refused to wire the money. Instead, he insisted on coming here himself." Werner's face twisted with contempt. "He knew the count had come to live in New York. He didn't know the count had died, and he thought his old friend had changed his tune and was involved in the plot."
"An impression I'm sure he got from you." Alex had worked his hands free and was holding the rope at his back so neither of his two captors could tell he was no longer bound.
"Perhaps, but the fool went to the countess's house one night before he had even talked with me. He gave his name to the maid. Fortunately, Otto was in the entranceway and not the countess. He recognized Schueller's name from our correspondence. He intercepted the fool and persuaded him to meet later in the alley nearby. Only I was there, with Otto, hiding in the shadows. When the bastard found out the count was dead and not in on our plans, he balked at giving us the money, so I attacked him from behind and cut his throat."
"But in the end, you were the fool. He didn't have the money on him, and you didn't know where to find it." As Alex spoke, he caught sight of Tru crawling in the direction of the platform. What she was up to he had no idea, but he cursed the fact that he had thought she'd listen to him and stay put. He should have tied her up as he had threatened. He couldn't signal her to go back to where she had been hiding because both anarchists stood directly between Alex and Tru. The only thing he could do was to keep the two men's attention until he could attack and pray that Tru didn't try anything on her own.
"Why did you use Carl Kueshner's name on your correspondence?" With his hands completely free now, Alex just needed the right moment to spring.
"You are certainly full of questions, my young friend."
"Just my curious nature. I am, after all, a detective."
"Well, since you have so little time left to be a detective, I will solve the puzzle for you before you die. I will answer this one more question before Otto and I tie you up to a crate and explode you to bits. I was aware of Johann Kueshner's quaint little paper with its silly naïve sentiments. The family was convenient to use in case the letters I wrote fell into the hands of the police in Chicago. And when the older son began to moon over Sophy, it was even more perfect to use the family as scapegoats."
"So you murdered Sophy and set John Kueshner up to take the blame? That makes sense, doesn't it, Otto?"
"No!" Jacob Werner's eyes left Alex and looked toward his partner. "Don't listen to him, Otto. He is only trying to confuse you. For the last time, I did not kill our beloved Sophy. Enough of this talk! Otto, start to light the fuses on the smaller crates while I drag the curious detective to his death."
Jacob Werner watched as his partner struck a match to a long torch. Then the older man started toward Alex. But as he reached out to grab the detective by his shoulders, Alex's arms snapped forward, flipping the madman over his head. He heard the air escaping from the man's body as Werner landed hard on his back. Alex scrambled to his feet and turned to face an onrushing Otto Klienst when a blur of color flashed in the corner of his eye, distracting him. Otto's fist struck Alex hard on the jaw, and the detective fell to his knees. As he went down, Alex saw Tru hit Otto Klienst over the head with what looked like a hammer. Klienst collapsed on the sand, out cold.
But before Alex could get up, Jacob Werner regained his feet and grabbed Tru around her waist and neck, pulling her back in the direction of the massive platform towering above them. Alex staggered after them, but Werner was already dragging Tru up the stairs to the first level of the star-shaped platform.
Tru kicked and screamed but was unable to break the hold her captor had on her. When Werner had succeeded in dragging her up the stairs, he headed toward an entrance that was designed to take visitors into the large two-story interior of the base on which the statue would rest. Inside it was pitch black, and it took a while for Tru's eyes to adjust to the darkness. When they did, Tru saw another set of stairs rising to the second story of the platform, and she realized that Werner was dragging her toward those stairs.
"Let go of me, you monster. I could kill you for what you've done to my family."
"I'm sorry for that, Miss Kueshner. I actually liked you. You were such a rebellious employee, a thorn in the side of that odious little man Hartley."
"Then just let me go. I can't hurt you."
"I don't think Otto would say that. Besides, I have a plan to stop your beloved detective from capturing me, and it involves you." Werner's voice took on an unpleasant note. "You didn't think I missed the looks between you and the detective. It might be romantic for the two of you to die together, don't you think?"
Tru stopped talking and listened for the sound of Alex's footsteps. She prayed Alex was not far behind. Her heart raced even faster as she thought of the danger Alex was in trying to rescue her. If she could only break free on her own and get away from Werner, he couldn't use her to harm Alex.
Alex could hear footsteps moving away from him and up the stairs. He reached the bottom step and began his own ascent, moving as quietly as humanly possible. When he drew near the end of the staircase, he slowed down, not knowing from which direction danger would come. The shadows at the top could easily conceal Jacob Werner, and Alex didn't want to do anything that might endanger Tru. He could see arched openings in the platform's thick walls. With the interior darkness relieved a bit, he was able to make out two forms silhouetted against one of the openings.
"Don't come any closer, Detective, or you'll lose your little salesgirl forever."
"Let her go, Werner." Alex could see now that Tru and her captor stood right next to the edge of one of the openings. He tried to keep his heartbeat steady and not give way to panic at the thought of what Werner might be planning.
"I don't think that will be possible, Detective. If I push her through the opening and she falls to the concrete below, you will be distracted enough, I think, for me to make my escape. It is important that I live to continue my work."
Alex watched in horror as the madman forced Tru to lean backward over the edge of the opening. "Your life is worth nothing, Werner, if you hurt her because I won't stop until I have hunted you down and ripped your heart from your chest."
"Very lovely sentiment, Detective. And I'm sure you think you mean it. But all women are the same. Just ask my father. He worked himself to death on the farm he rented from a nobleman in Prussia. He had to work that hard because my cow of a mother kept popping out more mouths to feed. So, you see, women are really a curse. You should thank me for saving you from this one."
Tru's back felt as if it was breaking as her captor put increasing pressure on her shoulders, forcing her ever more off balance. If she turned her head slightly, she could see the concrete and stones on the ground far below. But she couldn't raise her head or turn it at an angle where she could manage to see how far Alex was from where Werner held her prisoner.
"That's a sad story, Werner. But Tru had nothing to do with your father's death. She's never done you any harm. She's innocent."
"There's no such thing as an innocent person. There are only those willing to die for a great cause and those who care more about their own comfort. Your beloved salesgirl spends her days selling luxurious goods to wealthy people while others starve. Do you still call her innocent?"
"And do you think you make a difference in the lives of the poor by killing people in their name?"
"Enough! Your beloved is about to fall to her death. Say your one last good-bye."
"No!" Alex screamed as he tried to leap across the space separating him from Werner and Tru. He stretched his full length in an attempt to catch one of Tru's legs to prevent her falling and landed hard on the concrete floor, short of the opening where Tru had been struggling with Jacob Werner. He looked up, his eyes searching desperately for sight of Tru, expecting to see the woman he loved plunging to her death.
Instead, he saw Tru turn her head and bite Werner's hand holding onto her right shoulder. She bit down as hard as she could. Her would-be killer screamed in pain and let go with his injured hand. Now with only one hand holding her captive, Tru used her body and her free arm to give one mighty heave in an attempt to push Jacob Werner off her. Werner was off balance, and to Tru's horror, her attempt to escape caused the crazed anarchist to pitch sideways and fall through the opening, plunging to his death. The second after she heard the sickening thud of Werner's body hitting the concrete platform, she felt Alex's arms around her. In shock, Tru tried to look over the edge, thinking the whole thing had to be a terrible dream, but Alex pulled her down beside him on the floor.
"It's all right, Tru." Alex rocked Tru in his arms, kissing the top of her head as he whispered words of comfort. "He can't hurt you or anyone ever again."
Tru knew that Werner was no longer a threat, but she couldn't stop shaking. She was so glad to be held in Alex's arms she wanted to stay there forever and not think about what they had just been through.
"I love you, Tru. I'll do anything to keep you with me. You're the only thing that matters to me. If you want, I'll quit the force today to make you happy." Alex whispered his promise as he kissed Tru on her eyelids, her cheeks, her forehead, covering her face with kisses, trying to assure himself she was really safe in his arms. Then he captured her mouth for a very long kiss, and it was a while before Tru could say anything.
"Not today you won't. You have to get my family free from jail first, Alexander Marshall."
Alex threw back his head and laughed as he helped Tru to her feet. "I'm sure that can be arranged now."
"What about John? Werner never admitted to killing Sophy." Tru began to climb carefully down the dark stairs followed close behind by Alex, who would not let go of her elbow.
"That's true, but this whole business of Jacob Werner and his crazy plot cast enough doubt on John's guilt that I'm sure I can get him set free. We did just stop a terrorist from blowing up a gift from a friendly foreign power. That should count for something." They had finally reached the bottom of the stairs, emerging back out into the night. "Besides, we have Otto Klienst. He may know something that will help us find the real killer of his sister. We had better go collect him."
"Oh my goodness, what if he's not still there?"
"As hard as you hit him, believe me, he's still there."
Alex and Tru walked to where Otto Klienst still lay unconscious. Alex bent down to make sure the man was alive and was able to find a pulse. "Remind me never to give you a hammer for a gift."
Tru felt awful. "I didn't mean to hit him that hard."
"It's good you did, or else we would have had to deal with both of them at the same time. Give me a hand, and we'll tie him up and put him in the boat."
"I don't think that will be acceptable."
Tru nearly jumped out of her skin as a figure dressed totally in black emerged from the shadows. "Mrs. Klienst!" At the same moment, Tru noticed another boat was bobbing alongside the one the would-be terrorists had used to reach the island.
"You will not put my son into your filthy jail. I am here to stop you."
For the first time, Alex and Tru noticed the gun gleaming in Mrs. Klienst's hand. It was the one Alex had lost in his fight with Jacob Werner. If it had landed in the water, it might not work, but Alex wasn't sure he wanted to take that chance. He'd had his fill of risks tonight. "Mrs. Klienst, we won't harm your son. I'm only taking him to jail to talk with the captain and ask him to explain more about Jacob Werner's plot."
"Do you think me a complete fool, Mr. Marshall? I know you must arrest Otto for his part in the crimes he committed with Jacob." The strange woman bent down to caress her child, all the while she kept the gun turned on Alex and Tru. "He was just trying to avenge me and to protect his sister from the same fate as mine." The strange widow stood up straight and looked at Tru. "But it was too late. Men like your brother and that rich bastard at the store had already taken her innocence. That is why I had to protect her myself."
Alex and Tru stared, not understanding the meaning of the woman's words. But before they could say anything, she reached into her apron and brought out a long and very sharp-looking pair of scissors. "I have always been a good seamstress. These are my best pair of sheers. I made all of Sophy's little dresses with them. She looked like an angel when I dressed her up. Then I used them to make her a real angel never to be soiled again by a man's hand. Now I must help Otto escape from the corruption of this world. Despite his failure to avenge me and protect his sister, he was a loving son."
Before Tru or Alex realized what she intended, the troubled woman pointed the gun she held at her son's head and fired. To Alex surprise and horror, the gun worked. Then in the next moment, Mrs. Klienst plunged the deadly looking scissors into her own heart and fell to the ground. Alex and Tru rushed to her side in time to see a smile appear on the woman's face before she breathed her last.
Tru stood up and buried her head into Alex's chest. They both stood holding each other for a very long time. Then slowly Alex pulled Tru away from the terrible scene of violent waste toward the boat that would carry them back across the water to the city and the life that awaited them.
***********************************************************
Four weeks later.
Tru wasn't sure what the commotion was, but she could hear Mr. Hartley's shrill voice all the way from where he stood at the store entrance. She turned in the direction of the sound and was surprised to see the store manager trying to prevent Alex and his fellow detective, Richard Shaw, from approaching her counter.
"I tried to stop them, Miss Kueshner, but they wouldn't let me." Tru could have sworn Mr. Hartley was about to cry.
"What's going on?" Tru looked at Alex. She couldn't understand the grim look on the face of the man she loved. They had decided to get married in just a few short weeks, despite the occasional differences of opinion they continued to have. Tru had accepted Alex's job, so at least that was not something for them to argue about anymore. Tru smiled thinking of the fun they had disagreeing and then making up.
"We've come to place you under arrest." Richard Shaw's boyish face looked deadly serious.
"You're joking? What on earth for?" Tru cast a puzzled look at her handsome fiancé—a look that also contained more than a little threat in it.
"You'll find out in a short while." Alex took out a pair of handcuffs from his pocket and put them around Tru's wrists.
"Wait a minute, Alexander Marshall. What is this all about?" Tru pulled on the handcuffs, but they were already fastened. "Alex, I'll never forgive you for this. Our wedding is off." Tru's face turned scarlet as each of the detectives took her by an elbow and began to escort her out of the store through a sea of curious faces. As Tru passed her best friend's counter, she looked for Clarice, hoping for some help. But the tiny woman was nowhere to be seen.
Neither Alex nor Richard said another word as they hailed a hansom cab. The sound of Tru's protests could be heard up and down Sixth Avenue. But when they got in the hack, instead of heading south in the direction of the jail, the horses headed north on Sixth Avenue and then turned east toward Union Square.
"Where are you taking me?" Tru was starting to get more upset as Alex maintained his stern expression.
"To a precinct jail."
That curt answer was all Tru got out of Alex as the cab turned onto Broadway and headed north until it stopped in the middle of a block of ladies' shops near Twenty-First Street. Tru didn't remember any precinct headquarters here, so she was surprised when Alex got out of the cab and reached in to help her down. Then the two men guided her toward a small shop with a pretty canopied front.
"Hey, pretty lady, what kinda trouble you in this time?"
Tru was startled to see her young friend, his shoe shine kit slung over his shoulder, lounging outside the door through which Alex now led her.
"What do you know about this, young man?" Tommy merely shrugged his shoulders as he entered the building behind the two detectives. Once inside, Tru was surprised to see her whole family lined up, facing the doorway.
Her mother and father were grinning from ear to ear and hugging each other. Carl stood with his arm around his new wife, who looked as if she wanted to cry despite the baby cooing in her arms. John even looked happy over whatever was going on as he stood with the mysteriously disappearing Clarice. He was still mourning Sophy, but Clarice had managed to get a laugh or two out of him on several occasions, which she seemed to have made her pet project of late. Tru couldn't understand why the sight of her in handcuffs was not upsetting her family or even provoking jokes from the oddly silent twins, who just stood looking at her with ridiculous grins on their faces. This was all too much for Tru.
"What is going on here? If someone doesn't tell me soon, I swear I'll tear these things off even if my hands come with them."
Alex, knowing by her tone that his bride-to-be meant what she said, started to remove the handcuffs as Tru's mother spoke up at last. "We will let your future husband tell you about his surprise."
"He's not my future husband anymore."
"Maybe you had better wait to decide that, Tru, until you hear what he has to tell you." Maria smiled knowingly at her new sister-in-law.
"You'll love it, Tru. He's going to—" Clarice never finished her sentence as John's hand came up to gently cover her mouth.
"Quiet, mouse. Let Alex be the one to tell her."
Tru looked at Alex, who now wore the grin he usually sported when he was near Tru. "I have a friend who owns a lot of property, and she owes me a favor. So she's letting me rent this place for a tiny pittance. Your mother and Maria helped me fix it up, and your brothers built all the shelves. Do you like it?"
Tru looked around at the charming room, actually taking note of it for the first time. It was filled with beautiful wood shelves, and the walls were covered with very pretty flowered wallpaper, and lace curtains adorned the windows. It really was a delight to behold, but whatever did Alex want with such a place? "It's lovely, but what are you going to do with it?" Tru was more bewildered than ever.
"Why, give it to you for a millinery shop. You'd like that, wouldn't you?"
Tru couldn't speak for several minutes as tears welled up in her eyes. She looked at her parents, who were beaming with love for her, and then at the rest of her smiling family and friends. In that instant, Tru realized she had never been happier or more excited in her life. But she couldn't find the words to express her feelings.
Alex was getting worried when Tru didn't say anything. "It's all right that I arranged this on my own, isn't it? I wanted it to be a surprise. Your parents thought it was a good idea." But his fears were put to rest when she finally looked up at him through a veil of damp lashes and threw her arms around his neck so tightly he nearly choked.
"I love you more than ever because you did this for me, Alex. It means you understand my dream and that you believe in me. No woman can ask for more from the man she loves or from her family. I love you all." Tru kissed Alex tenderly and for a very long time until two impatient coughs broke the spell. Tru turned and beamed at her twin brothers, who obviously could no longer refrain from teasing her. "Don't worry, boys, I haven't completely lost my senses. I was just about to tell him never to do anything like this again without talking to me first."
"That doesn't bother us, noodle. We know you'll keep the detective in line. We're just anxious for you to get to work. The sooner you start selling hats, the sooner this place will be full of pretty girls."
At that, the sound of hardy laughter filled the room.
THE END
| {
"redpajama_set_name": "RedPajamaBook"
} | 5,953 |
Q: Combine a .txt's name, and second interior string to create a dictionary in Python I've written a batch script that is deployed to our network via Chocolatey and FOG that acquires the serial number of the machine and then ejects it via .txt in a file bearing the name of the PC that the serial number belongs to:
net use y: \\192.168.4.104\shared
wmic bio get serialnumber > Y:\IT\SSoftware\Serial_Numbers\%COMPUTERNAME%.txt
net use y: /delete
The folder Serial_Numbers is subsequently filled with .txts bearing the names of every computer on Campus. With this in mind I'd like to write a Python script to go through and grab every .txt name and second interior string to form a dictionary, where you can call for the PC's name, and have the serial number returned.
I'm aware as to how I'd create the dictionary, and call from it, but I'm having troubles figuring out how to properly grab the .txt's name and second interior string, any help would be greatly appreciated.
Format of .txt documents:
SerialNumber
#############
A: You can use os.listdir to list the directory files nad list comprehension to filter them.
Use glob to list the files in your directory.
You can simply read the first line and stop using the file while populating the dictionary and you're done:
import glob
d = {}
# loop over '.txt' files only
for filename in glob.glob('/path_to_Serial_Numbers_folder/*.txt'):
with open(filename, 'r') as f:
file_name_no_extension = '.'.join(filename.split('.')[:-1])
d[file_name_no_extension] = f.readline().strip()
print d
A: import glob
data = {}
for fnm in glob.glob('*.txt'):
data[fnm[:-4]] = open(fnm).readlines()[1].strip()
or, more succinctly
import glob
data = {f[:-4]:open(f).readlines()[1].strip() for f in glob.glob('*.txt')}
In the dictionary comprehension above,
*
*f[:-4] is the filename except the last four characters (i.e., ".txt"),
*open(f).readlines()[1].strip() is the second line of the file
object and eventually
*f is an element of the list of filenames returned by glob.glob().
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,498 |
Sehembre Hutavi Sobekhotep ali Sobekhotep I., v večini virov Amenemhet Sobekhotep, v starejših virih Sobekhotep II., je bil faraon Trinajste egipčanske dinastije, ki je vladala v drugem vmesnem obdobju Egipta. Vladal je najmanj tri leta okoli 1800 pr. n. št. Njegov kronološki položaj je precej sporen, ker se kot Sobekhotep I. šteje za prvega vladarja Trinajste dinastije, kot Sobekhotep II. pa za dvajsetega vladarja te dinastije. Trenutno prevladuje mnenje, da je bil ustanovitelj dinastije. Leta 2013 so v Abidosu odkrili domnevno njegovo grobnico, vendar domneva še ni dokončno potrjena.
Dokazi
Sehemre Hutavi Sobekhotep je dobro dokazan v primarnih virih. Prvič je omenjen na Kahunskem papirusu IV, ki je zdaj v Petriejevem muzeju egipčanske arheologije v Londonu (UC32166). Papirus je popis gospodinjstva svečenika-lektorja v prvem letu faraonovega vladanja. Na papirusu je omenjeno tudi rojstvo sina svečenika-lektorja v 40. letu vladanja faraona, ki bi lahko bil samo Amenemhet III. Zapis potrjuje domnevo, da je Sobekhotep I. vladal kmalu za Amenemhetom III.
Razen tega zapisa je znanih tudi več arhitekturnih elementov s Sobekhotepovim titularijem: fragment Hebsedove kapele iz Medamuda, tri preklade vrat iz Deir el-Baharija in Medamuda, arhitrav iz Luksorja in podboj iz Mehanmuda, ki je zdaj v Louvreu. Iz Semne in Kumne v Nubiji so trije zapisi o vodostaju Nila, ki jih pripisujejo Sobekhotepu I. Najkasnejši je iz četrtega leta njegovega vladanja, kar pomeni, da je vladal najmanj tri cela leta. Med manjše artefakte, ki omenjajo Sobekhotepa, spadajo valjast pečatnik iz Gebeleina, rezilo tesle, kipec iz Kerme in koralde iz fajanse, ki so zdaj v Petriejevem muzeju (UC 13202).
Domnevna grobnica
Med izkopavanji v Abidosu leta 2013, ki jih je vodil Josef W. Wegner z Univerze Pensilvanija, so odkrili grobnico faraona z imenom Sobekhotep. V več poročilih po letu 2014 naj bi to bil Sobekhotep I. Kasnejše raziskave so pokazale, da je grobnica verjetno pripadala Sobekhotepu IV.
Kronološki položaj
Egiptologi niso povsem soglasni o položaju Sobekhotepa I. znotraj Trinajste dinastije. Prestolno ime Sehembre Hutavire je na Torinskem seznamu kraljev ime 19. vladarja Trinajste dinastije. Zapisi o vodostajih Nila z njegovo podobo na papirusu, odkritem v El-Lahunu, kažejo, da so morda iz zgodnje Trinajste dinastije. Na obeh dokumentih so omenjeni samo vladarji iz pozne Dvanajste in zgodnje Trinajste dinastije.
Na Torinskem seznamu kraljev je Hutavire omenjen samo kot prvi faraon iz Trinajste dinastije. Egiptolog Kim Ryholt domneva, da je pisar zamenjal ime Sehemre Hutavi s Hutavire, kot se je imenoval faraon Vegaf. Prepoznavanje Sehemre Hutavija na drugih napisih je težavno, ker so se tako imenovali najmanj trije faraoni: Sehemre Hutavi Sobekhotep, Sehemre Hutavi Pantjeni in Sehemre Hutavi Habau.
Iz imena Amenemhet Sobekhotep so egiptologi sklepali, da je bil sin Amenemheta IV., predzadnjega faraona Dvanajste dinastije. Njegovo ime se namreč lahko bere tudi "Amenemhetov sin Sobekhotep". Sobekhotep bi zato lahko bil brat Sehemkare Sonbefa, drugega vladarja Trinajste dinastije. Drugi egiptologi ime Amenemhet Sobekhotep berejo kot dvojno ime. Dvojna imena so bila v Dvanajsti in Trinajsti dinastiji kar pogosta.
Sklici
Faraoni Trinajste egipčanske dinastije
Vladarji v 19. stoletju pr. n. št.
Vladarji v 18. stoletju pr. n. št. | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,720 |
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"redpajama_set_name": "RedPajamaC4"
} | 4,934 |
A Inscrição de Gálio, também chamada de Inscrição de Gálio em Delfos e Inscrição de Delfos, é a coleção de nove fragmentos de uma carta escrita pelo imperador romano Cláudio (ca. 54 d.C.) e descoberta em 1885 no Templo de Apolo em Delfos, na Grécia. Ela menciona o procônsul Lúcio Júnio Gálio Aneano (Gálio) e se tornou um importante marco para o desenvolvimento da cronologia da vida do apóstolo Paulo de Tarso (Vide Julgamento de Paulo em Corinto).
Ligações externas
Koiné
Gálio | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,127 |
\section{Introduction}
The speed at which fields (e.g. scalar, vector and gravitational) propagate is a very subtle question. According to the special relativity, energy (and mass) can not travel with speed greater than the speed of light, but there is nothing restricting the speed of auxiliary fields like potentials. Potentials describe interactions, and interactions are mediated by virtual particles, which are off-shell and do not have any a priori preferred speed.
For example, one is tempted to assume that the field of the non-moving source is frozen and does not propagate, until the source/charge moves and the field re-arranges its distribution. This would effectively mean that a static field is infinitely rigid and propagates with infinite speed. This may make sense classically, however, quantum mechanically interactions are fluctuations in space induced by virtual particles. Therefore, the situation is dynamical. To find the speed at which some interaction propagates, one has to calculate explicitly the effects of retardation in the Green's function of the field that mediates that interaction.
It is well known that the QED vacuum structure can affect the propagation of light even in flat space. The so-called Scharnhorst effect is a phenomenon in which light signals travel faster in between the two closely spaced conducting plates, than outside of the plates \cite{Scharnhorst:1998ix}. The reason is the Casimir effect, i.e. the vacuum polarization effect is suppressed in between the plates, so the photon loses less time propagating in between the plates than outside. This gives a hint that a massless particles do not always propagate at the speed of light in vacuum.
There is even more counter-intuitive example in curved space-time. Namely, Drummond and Hathrell demonstrated in \cite{Drummond:1979pp} that vacuum polarization is sensitive to the curvature of spacetime. For example, for a photon propagating in a curved space, vacuum polarization can induce
a modification of the wave equation in such a way so that in some cases photons travel at speeds greater
than unity. The effect seems to be dispersive, and the phase velocity approaches the speed of light at high frequencies. Since the high-frequency limit of the phase velocity determines causality, it seems like causality is preserved in case. Extensive discussion of this effect can be found in \cite{Hollowood:2010xh,Hollowood:2010bd,Hollowood:2008kq,Hollowood:2007ku,Hollowood:2007kt}.
These examples imply that propagation of quantum fields in curved space-time is a very subtle question, with many potential surprises.
The simplest and perhaps the most instructive case to study will be the case of the scalar field. The reason is that the scalar field potential is not gauge dependent. The only freedom we have is to add an extra constant, i.e. $\psi \rightarrow \psi +$const, which in turn has no dynamical effect and can be fixed by setting $\psi=0$ at infinity. The best way to find out the propagation velocity is perhaps to study the Green's function of a field. Once the Green's function is found, we can analyze the retarded potential for a given field and infer the speed at which the signal propagates from a point to a point. However, the difficulty of finding the general Green's function for a field in a curved space-time makes this approach very difficult. Fortunately, the full space Green's function is not absolutely necessary to study the propagation phenomena. A case in which an observer observes a modulated source will be sufficient to study. Therefore, we will consider a massless scalar field potential for a stationary (non-moving) but time-dependent source. Our result shows that the phase velocity of the retarded potentials is position dependent, and may easily be faster than the speed of light. In the case of the Schwarzschild space, this phase velocity at the horizon can even be infinitely greater than the speed of light at the horizon. Though our solution in the linear ansatz is analytic, our analysis of the general form of the source is numerical. We therefore do not have an analytic form for a complete Green's function for an arbitrary source. However, the fact that the phase velocity of the scalar field varies locally is important. Among the other things it implies that gravity must affect the path of the massless scalar field, which for example should lead to the gravitational lensing effect for the massless scalar field.
We would like to emphasis at the very beginning that throughout the paper we will use the term ``signal" in a loose sense. We will call any change in the field a ``signal". While a non-moving source does not emit any real particles, the phase of the field will change, and the speed of that change (phase velocity) we will call the speed of the signal. While this is not a real signal or information in a strict sense (i.e. transmitted by the group velocity) it will have some important consequences.
\section{Retarded Green's function for a massless scalar field in a curved space-time}
As a referent point, we first show the retarded Green's function for a massless scalar field in the flat space-time.
The speed at which the signal propagates through space can be read off the retarded solution. Consider a point particle in Minkowski space carrying a massless-scalar-field charge at the origin. Let the magnitude of its charge increase (or decrease) in time as $g(t)$. The equation of motion is
\begin{equation}
\label{motion1}
\partial_t^2 \psi -\partial_x^2\psi-\partial_y^2\psi-\partial_z^2\psi= 4\pi \delta (\vec{x})g(t)
\end{equation}
The solution for the function $\psi$ is
\begin{equation} \label{flat}
\psi=\frac{g(t-|\vec{r}|)}{|\vec{r}|} ,
\end{equation}
where $\vec{r}=(x,y,z)$. The scalar field potential falls off with distance in flat space as $1/r$. From the numerator, we see that the signal travels from a point to a point with the speed of light $c$, i.e., if we increase the magnitude of the charge at the origin, the potential at a point $\vec{r}$ will be affected after time $t=|\vec{r}|$. So, it takes some time for a signal to propagate even if the source is static. This is best understood in terms of virtual particles. A static source emits the sea of virtual particles which modify the space around it. Result in Eq.~(\ref{flat}) implies that, in flat space virtual (just like real massless) particles propagate with the speed of light, at least as long as they are in vacuum.
\section{Green's function for a massless scalar field in a curved space-time}
\label{sec:gf}
There are very few examples of exact Green functions in curved space-times \cite{Poisson:2003nc,Frolov:2012jj,Frolov:2003mc,Chu:2011ip,Chu:2012kz,Wiseman:2000rm,Burko:2002ge}. The reason is that it is notoriously difficult to find an exact solution without any approximations \cite{Bird:1975we,Ehlers:1976ji}. However, we will demonstrate that the particular case with a great degree of symmetry, i.e. a charge located at the center of a spherical symmetric curved space, is directly solvable. We will then use the explicit solution to discuss the speed at which scalar field potentials propagate in such space-times.
We fix again a point scalar charge at the origin of a spherically symmetric space. Fixing the charge at the origin rather than at an arbitrary point in space will provide the required symmetry and greatly facilitate the problem. We let its magnitude change as $g(t)$. The geometry of the space-time can be written as
\begin{equation}
d\tau ^2 =g_{tt}dt^2+g_{rr} dr^2-r^2 \left(d\theta^2 +\sin^2\theta d\phi^2 \right)
\end{equation}
The equation that we will try to solve is
\begin{equation}
\label{motion2}
D^tD_t\psi +D^rD_r\psi+D^\theta D_\theta\psi+D^\phi D_\phi\psi= \delta (r)\frac{g(t)}{\sqrt{h}}
\end{equation}
where $h=-g_{tt}g_{rr}r^4$.
We note here that our definition of the scalar charge slightly differs from the definition in Eq.~2.2 in \cite{Wiseman:2000rm}. The difference is the time component of the four-velocity $u^t$, which in our static case (charge is not moving) is just a constant and can be absorbed in $g(t)$. Further, strictly speaking, we are dealing with geometries without horizons, so we will not discuss the no-hair theorems.
We will first try to find the time-independent solution, which will correspond to a static scalar charge of constant magnitude. In that case, Eq.~(\ref{motion2}) reduces to
\begin{equation}
\frac{1}{\sqrt{h}}\partial_r \Big( g^{rr}\sqrt{h}\partial_r \psi\Big)=\delta (r)\frac{g(t)}{\sqrt{h}}
\end{equation}
Without loss of generality, we can set $g(t) = 1$. The solution can be found by applying volume integration over the element $\sqrt{h}dtd^3 x$. The static solution $\psi_s$ is
\begin{equation}
\label{state1}
\psi_s(r)=\int_r^{\infty} \sqrt{\frac{-g_{rr}(R)}{g_{tt}(R)}}\frac{1}{R^2} dR
\end{equation}
It is easy to verify that this is a solution by substituting Eq.~(\ref{state1}) back into Eq.~(\ref{motion2}).
Since this solution depends on both $g_{tt}$ and $g_{rr}$, it is different from the result from the flat space, but it will reduce to the flat space solution at large radius $r$.
We will now try to find the time-dependent solution which corresponds to $g(t)$.
Since we a priory expect massless particle to propagate with the speed of light, we may expect this solution to have the following form
\begin{equation}
\psi (t,r)= g\left( t-\int_0^{r} \sqrt{\frac{-g_{rr}(R)}{g_{tt}(R)}}dR \right)\psi_s(r) ,
\end{equation}
where the expression in parentheses on the right-hand side is the argument of the function $g$. This form is a straightforward curved space generalization of the flat space solution given by Eq.~(\ref{flat}).
The term $\sqrt{\frac{-g_{tt}}{g_{rr}}}$ is just the coordinate speed of light in the radial direction (obtained from $d\tau =0$). Though this appears to be a reasonable guess, this form is a solution only if $g_{rr}=-g_{tt}$. However, this requirement brings us back to the flat space. We will therefore generalize the form of the solution allowing for the possibility that the propagation speed is not the speed of light. We now try a more general form
\begin{equation}
\psi (t,r) = g\left( t-\int_0^{r} \frac{1}{v(R)}dR\right)\psi_s(r)
\end{equation}
where $v(r)$ is the coordinate speed at which the signal of the retarded potential travels (not necessarily the speed of light). This $v(r)$ must asymptotically go into the speed of light at large $r$ where the space-time becomes flat. We do not expect this form to always generate a solution.
However, if $g$ is a linear function of its argument, one can find the suitable solution. Therefore we consider the linear form of $g$
\begin{equation}\label{la}
g\left( t-\int_0^{r} \frac{1}{v(R)}dR\right)=A \left( t-\int_0^{r} \frac{1}{v(R)}dR\right)+B
\end{equation}
where $A$ and $B$ are two constants. If we plug this ansatz into Eq.~(\ref{motion2}), we find a condition under which the solution is valid
\begin{equation} \label{v}
v=\sqrt{\frac{-g_{tt}}{g_{rr}}}r^2 \psi_s^2= r^2 \psi_s^2c_l
\end{equation}
Here, $c_l=\sqrt{\frac{-g_{tt}}{g_{rr}}}$ is the coordinate speed of light. Since $v$ is also a coordinate velocity of propagation, it will be different for different observers and it will change from point to point. But, it is clear that the retarded signal propagates at a speed that is different from the speed of light for a given observer.
To make this more clear, we consider the Schwarzschild space-time, i.e.
\begin{equation}
g_{tt}=-g_{rr}^{-1}=1-\frac{2m}{r}
\end{equation}
We plug this condition into Eq.~(\ref{state1}), and the static solution becomes
\begin{equation}
\psi_s(r)=\int_r^{\infty} \frac{1}{1-\frac{2m}{R}} \frac{1}{R^2} dR=\frac{-\ln(1-\frac{2m}{r})}{2m} \neq \frac{1}{r}
\end{equation}
We see that the static scalar field potential does not fall off as $1/r$ which was the case in flat space. However, in the limit of $r \gg 2m$, we recover the usual $1/r$ behavior.
From Eq.~(\ref{v}), the coordinate propagation speed of the retarded potential is
\begin{equation} \label{vphase}
v= r^2 \psi_s^2c_l>c_l
\end{equation}
which is not equal to the coordinate speed of light $c_l$, and in fact is always greater than $c_l$ (for this particular example of the Schwarzschild space-time).
Our calculations will be strictly valid as long as our space-time is not strictly a black hole, but the conclusions will be valid even when we are only slightly outside the horizon.
In the extreme limit, exactly at the horizon, $r=2m$, the propagation speed, $v$, becomes infinitely faster than the speed of light at that point.
In the context of the Schwarzschild black hole, the coordinate speed of light, $c_l$, vanishes at the horizon, and any signal sent from the horizon gets infinitely redshifted. Thus, in the standard description it remains unclear how information about the black hole charge which is presumably imprinted at the horizon can be communicated to the region around the black hole.
[If the scalar charge is conserved, then formation of the black hole can not violate this conservation \cite{Stojkovic:2005zq}.]
Let's check what happens when the propagation speed of the retarded potential is taken into account. We can calculate the time, $\Delta t$, for a signal to propagate from the horizon to some finite distance $R$
\begin{eqnarray}
\Delta t &=&\int_{2m}^R \frac{1}{v} dr\nonumber\\
&=&\int_{2m}^R \frac{1}{1-\frac{2m}{r}} \ \frac{dr}{\left[r\ln(1-\frac{2m}{r})\right]^2} \nonumber\\
&= &\frac{-2m}{\ln(1-2m/R)}
\end{eqnarray}
This time is finite and therefore the potential has no problem to propagate from the horizon outside. Thus, a charged particle (at least with the scalar charge) can keep communicating its potential to the region outside the black hole. We mention again that our results are not strictly applicable to the black hole case, but we can always consider a shell whose radius is just slightly outside its own Schwarzschild radius and preserve the qualitative conclusions drawn here. In the next section we will reveal that the propagating velocity, $v$, is the phase velocity. Thus, this velocity refers to the change of phase, and it is not a group velocity. The casual light cone for real particles remains the same, the Green's function does not have support outside the light cone, and causality is preserved.
\section{General case}
The discussion of the time dependent source so far was based on a solution found in the particular ansatz of Eq.~(\ref{la}). We will now try to analyze the general form of $g(t)$ (not only the linear ansatz that we used). We will first decompose the source $g(t)$ into different frequency modes as
\begin{equation}
g(t)=\int \tilde{g}(\omega)\exp(i\omega t) d\omega
\end{equation}
The wave number of each frequency mode is $\omega/v_\omega (r)$, where $v_\omega$ is the phase velocity of that mode. Then, the scalar potential can be written as
\begin{equation}
\psi(t,r) = \int \bar{g}(\omega)\exp\left[i\omega (t-\int \frac{1}{v_\omega} dr)\right]f_\omega(r) d\omega
\end{equation}
where $f_\omega$ is the amplitude of the mode labeled by the frequency $\omega$. $\bar{g}(\omega)$ and $\tilde{g}$ can be found by matching the boundary condition at $r=0$.
By plugging the above equation into equation (\ref{motion2}), we find that $f_\omega$ and $v_\omega$ must satisfy
\begin{eqnarray}
\label{wave4}
&&-\frac{\omega^2}{g_{tt}}f_\omega -\frac{\omega^2}{v_\omega^2 g_{rr}}f_\omega +\frac{1}{\sqrt{h}}\partial_r \Big( \frac{\sqrt{h}}{g_{rr}}\partial_r f_\omega\Big)\nonumber\\
&&-i\omega\Big(\frac{\partial_r f_\omega}{v_\omega g_{rr}}+\frac{1}{\sqrt{h}}\partial_r (\frac{\sqrt{h}}{g_{rr}v_\omega}f_\omega)\Big)=0
\end{eqnarray}
except at $r=0$.
Since both imaginary and real parts must vanish independently, the above equation can be rewritten as two equations
\begin{eqnarray}
\label{wave5}
&&-\frac{\omega^2}{g_{tt}}f_\omega -\frac{\omega^2}{v_\omega^2 g_{rr}}f_\omega +\frac{1}{\sqrt{h}}\partial_r \Big( \frac{\sqrt{h}}{g_{rr}}\partial_r f_\omega\Big)=0\\
\label{wave-v}
&&\frac{\partial_r f_\omega}{v_\omega g_{rr}}+\frac{1}{\sqrt{h}}\partial_r \Big(\frac{\sqrt{h}}{g_{rr}v_\omega}f_\omega\Big)=0
\end{eqnarray}
Eq.~(\ref{wave-v}) can be easily solved by integrating with respect to $r$.
\begin{equation} \label{eq}
f_\omega^2\frac{\sqrt{h}}{g_{rr}v_\omega}= {\rm constant}
\end{equation}
In the limit of $r\rightarrow \infty$, the space becomes flat, which implies $v_\omega \rightarrow 1$ and $f_\omega \rightarrow 1/r$, as it should.
Eq.~(\ref{eq}) can be rewritten as
\begin{equation}
\label{velocity1}
v_\omega=P_\omega^2\sqrt{\frac{-g_{tt}}{g_{rr}}}
\end{equation}
where $P_\omega \equiv f_\omega r$.
If we substitute this relation into Eq.~(\ref{wave5}), and replace $f_\omega r$ with $P_\omega$ we get
\begin{eqnarray} \label{veleq}
&&\partial_r^2 P_\omega+\frac{1}{2}\partial_r\ln \left(\frac{-g_{tt}}{g_{rr}}\right)\partial_rP_\omega-\frac{1}{2}\partial_r\ln \left(\frac{-g_{tt}}{g_{rr}}\right)\frac{P_\omega}{r}\nonumber\\
\label{wave-6}
&&-\frac{\omega^2 g_{rr}}{g_{tt}}\left(P_\omega -\frac{1}{ P_\omega^3}\right) =0
\end{eqnarray}
The zero mode, $\omega =0$, solution to this equation is exactly $f_\omega =\psi_s$, where $\psi_s$ is the time-independent $g=$const solution given in Eq.~(\ref{state1}). Moreover, the phase velocity $v_\omega$ in this case is the same as the propagation velocity given by Eq.~(\ref{v}) in the ansatz solution we found. This then reveals the meaning of the parameter $v$ in Sec. \ref{sec:gf}.
In the high frequency limit, $\omega \rightarrow \infty$, the last term in Eq.~(\ref{veleq}) dominates. In order to satisfy the equation it has to vanish, thus requiring $P_\omega =1$. Eq.~(\ref{velocity1}) then implies that $v_\omega $ becomes the speed of light $c_l$. It is this feature that ensures causality.
For the $\omega \neq 0$ modes, we will again use the spherically symmetric Schwarzschild geometry. The boundary conditions are
\begin{eqnarray}
\mbox {$r\rightarrow \infty$, $P_\omega =1$ }\\
\mbox {$r\rightarrow \infty$, $\partial_r P_\omega =0$ }\\
\mbox {$\frac{-g_{tt}}{g_{rr}}=(1-1/r)^2$}
\end{eqnarray}
\begin{figure}[h]
\centering
\includegraphics[width=3.2in]{velocity}
\caption{This figure shows $P_\omega(r)$ for three different values of $\omega$, i.e. $\omega=10^{-1}$, $\omega=10^{-2}$ and $\omega =10^{-3}$. We see that $P_\omega$ grows as it is approaching the origin, higher $\omega$ modes increase slower than lower $\omega$ modes, and $P_\omega \geq 1$ everywhere. This behavior implies that the phase velocity $v_\omega$ is always superluminal (for this case of the Schwarzschild geometry), and that lower $\omega$ modes propagate faster than the higher $\omega$ modes.}
\label{velocity}
\end{figure}
In Fig.~\ref{velocity}, we show $P_\omega$ as a function of $r$ for several values of the frequency $\omega$. We can see that $P_\omega$ grows as it is approaching the origin. It is also apparent that higher $\omega$ modes increase slower than lower $\omega$ modes. We do not plot $P_\omega$ near the horizon, because the singularity will cause numerical instabilities. Since $P_\omega \geq 1$ for all values of $r$, the phase velocity defined by Eq.~(\ref{velocity1})
\begin{equation}
v_\omega=P_\omega^2\sqrt{\frac{-g_{tt}}{g_{rr}}}\geq c_l
\end{equation}
is greater than the speed of light everywhere.
\section{Common features with other examples with superluminal phase velocity}
In this section we discuss some other known examples where the phase velocity is superluminal, which may have have something in common with our results.
Perhaps the best known example is that of a massive Klein-Gordon field in flat space-time. The dispersion relation is simply $\omega^2=k^2+m^2$.
The phase velocity
\be
v_p \equiv \omega/k
\ee
is always greater than unity as long as $m \neq 0$. Moreover, superluminality is most pronounced for low frequencies $\omega$, while for large frequencies we have $\omega \approx k$, i.e. $v_p \approx 1$. However, the group velocity
\be
v_g \equiv d\omega/dk
\ee
is always less than unity. Comparing this result with the results we obtained in curved space, we might conclude that the curvature of space induces an effective mass to the massless scalar field, making it formally equivalent to the massive Klein-Gordon field with superluminal phase velocity.
The other, less known example is a massless scalar field in a $(5+1)$-dimensional flat space-time. The wave equation is
\begin{equation}
\partial_t^2 \psi -\partial_{x_1}^2\psi-\partial_{x_2}^2\psi-\partial_{x_3}^2\psi-\partial_{x_4}^2\psi-\partial_{x_5}^2\psi= \delta (\vec{x})\delta(t)
\end{equation}
The general solution can be found in most mathematical physics textbooks (e.g. \cite{Hassani}) or papers \cite{Cardoso:2002pa}. The Green's function for this case is
\begin{equation}
G^{5+1}(t,r)=-\frac{1}{8\pi^2}\Big(\frac{\delta'(t-r)}{r^2}+\frac{\delta(t-r)}{r^3}\Big)
\end{equation}
where, $r=\sqrt{x_1^2+x_2^2+x_3^2+x_4^2+x_5^2}$. If one considers the following concrete source
\begin{equation}
f(t,\vec{x}) =\sin(\omega t)\delta(\vec{x})
\end{equation}
then the wave function can be easily found as
\begin{eqnarray}
\psi&=&\int f(t',\vec{x'}) G^{5+1}(t-t',\vec{x}-\vec{x'})dt' d\vec{x'}\\
&=&-\frac{1}{8\pi^2}\Big( \frac{\omega \cos(\omega (t-r))}{r^2}+\frac{\sin(\omega (t-r))}{r^3} \Big)
\end{eqnarray}
Since this form includes two trigonometric functions, it is hard to see how the phase changes. We will then combine the two terms into a single trigonometric function.
\begin{eqnarray}
\psi&=&S(\omega,r)\sin(\omega(t-R))\\
S(\omega,r)&=&-\frac{1}{8\pi^2}\frac{\sqrt{1+r^2\omega^2}}{r^3}\\
R&=&r-\phi(r)/\omega\\
\phi(r)&=&\sin^{-1}\Big(\frac{r\omega}{\sqrt{1+r^2\omega^2}}\Big)
\end{eqnarray}
This form is similar to the form we studied in the last section. The phase velocity in this case is
\begin{equation}
v_p =\frac{1}{\partial_r R}=1+\frac{1}{r^2\omega^2}
\end{equation}
We can see that the phase velocity is infinite at the origin (for fixed $\omega$) but equal to the speed of light at $r \rightarrow \infty$. In this case the solution is created by two waves with different phases. Since their amplitudes decay in different ways, their combination makes the total phase velocity change with location, and in fact makes it infinitely faster than the speed of light at some locations. These are the features which are shared with our solution for the massless scalar field in a curved space. Moreover, for a fixed finite $r$, superluminality is again most pronounced for small $\omega$, while for large frequencies we have $v_p \approx 1$.
\section{Conclusions}
In this paper we analyzed the question of the speed at which potentials propagate in curved space-time. While finding an answer is easy in flat space, it becomes highly non-trivial in curved space-time. The difficulties range from finding an exact solution for the Green's function to choosing the right definition of the propagation speed. To avoid gauge and other ambiguities we considered the massless scalar potential. We located the scalar charge whose magnitude was changing in time at the origin in a spherically symmetric space-time, and found the solution for different frequency modes for this configuration. A non-moving particle does not emit real scalar field quanta, but what is changing in the system is the phase of the field. We found that the phase velocity is not constant but changes from point to point. Moreover, in the specific case of the Schwarzschild geometry, it is always greater than the coordinate speed of light at any given point. In an extreme limit, exactly at the horizon, the phase velocity becomes infinity faster than the speed of light at that point (which is actually vanishing). In fact, this feature is required if a black hole is going to communicate "information" about its potential which is presumably located at the horizon to the outer world.
It is important to note that the phase velocities, $v_\omega$, for different frequency modes (labeled by the frequency $\omega$) are different for each mode, and in general they are different from the local speed of light. Also, the amplitudes of different frequency mode's ($f_\omega$ in the text), have different $r$ dependence. These two facts make the curved space case quite different from the flat space, and explain why it was impossible to find a uniform propagation mode like in flat space (see Eq.~(\ref{flat}).
Since the propagation is dispersive, the high-frequency limit of the phase velocity will determine causality. Since the phase velocity approaches the speed of light at high frequencies, causality is preserved in our case.
The cases of the electromagnetic and gravitational potentials are more complicated because of the non-zero spin. However, they are also massless fields and will perhaps have some similar properties. In particular, we expect the retarded electromagnetic and gravitational potential from a non-moving source to propagate at a speed different from the speed of light. In other words, the average velocities of virtual photons and gravitons should not be the same as for real photons and gravitons in curved space-times.
A related question can be asked in the context of gravitational lensing. If the retarded gravitational potential of a static source travels with a finite speed (not necessarily the speed of light), it must experience the effect of the gravitational lensing, just as the light does. This would imply that the gravitational lensing effect on gravitons should be able to amplify or reduce the strength of gravity from a given static source \cite{Nemiroff:2005az}. In \cite{Wucknitz}, several examples were constructed to emphasis that the gravitational lensing could affect real gravitons, but could not lens any static gravitational field potential (though the static potential could be affected to some extent). However, the sources used in these examples were infinite planes, and not point sources, so the conclusions are perhaps not general.
If our conclusions for the scalar field hold for gravitons as well, then the static gravitational potential could propagate at any finite speed (except in the extreme case of the black hole horizon where it should be infinite) depending on the curved background. Since this speed is finite and position dependent, the effect of gravitational lensing of gravity should exist, though the magnitude of the effect should be different from the gravitational lensing of the light because of the different speed of propagation.
It is interesting that one of the possible explanations of the Pioneer anomaly \cite{Nieto:2003rq} is the focusing of gravity at around $25$AU \cite{Nemiroff:2005az}, exactly where the Pioneer anomaly arises. This could be a hint that gravity is bent nearby our Sun \cite{Anania:2005dy}, of course if the real explanation is not something more conventional, like the thermal radiation pressure \cite{Turyshev:2011yi,Bertolami:2008qb}.
At the end we would like to compare our findings with the existing similar results in the literature, e.g. \cite{Scharnhorst:1998ix,Drummond:1979pp}.
In \cite{Scharnhorst:1998ix}, using the Casimir effect, the authors showed that vacuum polarization effects may lead to superluminal propagation of photons in between the plates (since the vacuum polarization effects are suppressed there). While this is a flat space result, it is indicative that superluminality may arise in completely physical setups.
In \cite{Drummond:1979pp} it was argued that the quantum corrections in curved space-time are able to introduce tidal gravitational forces on the photons which in general alter the characteristics of propagation, so that in some cases photons travel at speeds greater than unity. In that case it is actually the low-frequency limit of the phase velocity that is superluminal. This indicates that propagation of quantum fields in curved space-time is a very non-trivial problem, and surprising results may be derived. It should be noted that superluminality does not always lead to paradoxes, since in both of the above mentioned cases it is impossible to send signals backward in time. While work presented in \cite{Scharnhorst:1998ix,Drummond:1979pp} is perturbative, our analysis is exact since it based on the exact solution of the Green's function in curved space-time.
It is interesting that our analysis also indicate that lower frequency modes propagate faster than high frequency modes, in good agreement with \cite{Drummond:1979pp}.
Finally, we emphasis again that we found only the phase velocity to be superluminal. If the group velocity is not superluminal, then the Green's function does not have support outside the light cone, and causality is preserved. Strictly speaking, a second order linear wave equation can not have a "wavefront" propagating faster than the speed of light. However, this statement does not affect velocity of an individual frequency component of the phase. It is only when one takes into account all the frequencies (where the higher frequencies give the dominant contribution) that he has to obey that statement.
\begin{acknowledgments}
This work was partially supported by the US National Science Foundation, under Grants No. PHY-0914893, PHY-1066278, and by Shanghai Institutions of Higher Learning, the Science and Technology Commission of Shanghai Municipality (11DZ2260700). We thank Y.Z. Chu, J. Wang, V. Frolov and A. Zelnikov for very useful discussions.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 971 |
Sodium thiopental production
Legislation and litigation
Infusion pump system firmware vulnerability disclosures
Hospira
Defunct American healthcare firm owned by Pfizer
Traded as
NYSE: HSP
May 3, 2004 (spun off from Abbott Laboratories)
Lake Forest, Illinois, United States
F. Michael (Mike) Ball (CEO); John C. Staley, Chairman of the Board of Directors
Generic acute-care and oncology injectables, integrated infusion therapy, medication management systems
$4.5 billion (2014)
Approximately 19,000
www.hospira.com
Hospira was an American global pharmaceutical and medical device company with headquarters in Lake Forest, Illinois. It had approximately 19,000 employees.[1] Before its acquisition by Pfizer, Hospira was the world's largest producer of generic injectable pharmaceuticals,[2] manufacturing generic acute-care and oncology injectables, as well as integrated infusion therapy and medication management systems. Hospira's products are used by hospitals and alternate site providers, such as clinics, home healthcare providers and long-term care facilities.[3] It was formerly the hospital products division of Abbott Laboratories. On September 3, 2015, Hospira was acquired by Pfizer, who subsequently sold off the medical devices portion of Hospira to ICU Medical.
Worldwide sales in 2014 were approximately $4.5 billion.[4] Current results are now part of Pfizer's consolidated statements.
Hospira corporate headquarters in Lake Forest, Illinois
In January 2004, Abbott announced it was spinning off its hospital products division.[5]
Hospira's name was picked by employee vote. The name is derived from the words hospital, spirit, inspire and the Latin word "spero," which means "hope."[3] Hospira became an independent company on May 3, 2004, with 14,000 employees, 14 manufacturing sites and an estimated $2.5 billion in annual sales.[6]
In 2007, Hospira purchased Mayne Pharma Ltd., an Australian-based specialty injectable pharmaceuticals company, for $2.1 billion.[7]
In 2009, Hospira acquired the biotechnology business from Pliva-Croatia, the generic injectable pharmaceuticals business of Orchid Chemicals & Pharmaceuticals Ltd., a leading Indian pharmaceuticals company, for approximately $400 million,[8] and TheraDoc, a clinical informatics company that develops hospital surveillance systems, in 2009.[9] In 2010, Hospira acquired Javelin Pharmaceuticals, Inc., maker of post-operative pain management drug Dyloject, for approximately $145 million.[10]
In 2011, Hospira's board chose Mike Ball, formerly president of Allergan, as Hospira's new CEO. Ball became CEO in March 2011.[11] Hospira named John Staley its non-executive chairman with the retirement of former executive chairman Christopher Begley in January 2012. Begley had announced his retirement as Hospira's chief executive in August 2010, but had remained as executive chairman.[12]
In 2015, Pfizer signed an agreement to acquire Hospira.[13] The roughly $17 billion acquisition was completed in September, 2015.[14] A year later Pfizer sold the medical devices portion of Hospira to ICU Medical for roughly $900 million in cash, stock, and other consideration.[15][16][17]
Sodium thiopental is an anesthetic discovered by Abbott Laboratories in the 1930s.[18] Hospira manufactured the drug after splitting off from Abbott under the brand name Pentothal. The WHO considers it an essential drug. However, it is also used as part of the lethal injection protocol in many US states.[19] Though Hospira has supplied these states with the drug, it has said, "we do not support the use of any of our products in capital punishment procedures."[20]
Wikinews has related news:
Final US manufacturer ceases production of lethal injection drug; executions delayed
In January 2011, the company announced that it would stop producing sodium thiopental.[21] Hospira had recently moved production of the drug from a plant in North Carolina to a plant in Liscate, Italy.[21] However, the Italian government would only allow Hospira to manufacture it if they could guarantee it wouldn't be used in capital punishment.[22] The Italian constitution bans the use of capital punishment.[23] Company officials determined there was no way it could prevent sodium thiopental from being used in executions, and did not want to expose their employees to liability.[24][23][25]
Oxaliplatin: In August 2009, Hospira introduced a generic version of Sanofi-Aventis SA's (SNY) colon-cancer drug known generically as oxaliplatin and by the brand name Eloxatin, in the United States. In April 2010, Hospira announced a legal settlement with Sanofi-Aventis. Under the settlement terms, Hospira agreed to stop selling oxaliplatin injection in the United States by June 30, 2010, and can relaunch the product in the United States on Aug. 9, 2012.[26]
Biosimilars: In 2010, the U.S. Congress passed legislation that would allow the marketing of biosimilar drugs in the United States. The legislation would allow 12 years of data exclusivity for brand-name biologics. Some consumer groups, like AARP, oppose this provision, saying it would cause lack of access to the promise of such drugs.[27]
Hospira's competitors in specialty injectable pharmaceuticals include Fresenius AG, Baxter International Inc., Bedford Laboratories, Mylan, Sandoz, Teva Pharmaceuticals as well as divisions of several multinational pharmaceutical companies. Its competitors in medication management systems include Baxter, B. Braun Melsungen AG, CareFusion and Fresenius Medical Care AG.[28]
In 2014-2015 two security researchers independently identified what were described as severe defects in Hospira's PCA system firmware, the software controlling various of their drug infusion equipment (CVE-2015-3459[29] and further advisory ICSA-15-125-01B[30]). Numerous remote exploit vulnerabilities were found, in what was believed to be the first FDA safety advisory of its kind.[31] This was followed in July 2015 by a second FDA recommendation that hospitals discontinue use of the affected pumps entirely.[32] The devices, extent of their flaws, and implications, were widely discussed.[33][34][35]
^ "Hospira - Investor Relations - Shareholder FAQ".
^ "US-based Hospira to buy Orchid Chemicals' injectables biz for $400 mn". The Economic Times. 16 December 2009.
^ a b "About Hospira".
^ "HSP Key Statistics". Yahoo.com.
^ Higginbotham, Stacey (25 January 2004). "Abbott Labs to spin off unit". Austin Business Journal.
^ "Hospira Begins Trading As Part of the S&P 500". Wall Street Journal. 2004-05-04. ISSN 0099-9660. Retrieved 2020-04-23.
^ "Hospira close to purchase of Mayne". Chicago Tribune. 19 January 2007. Retrieved 2020-04-23.
^ "US-based Hospira to Buy Orchid Chemical's Injectables Biz For $400 Mn". The Economic Times. 16 December 2009.
^ "Hospira Acquires Theradoc, Enhances Medication Safety and Infection Management Offerings". Infection Control Today. 2 December 2009.
^ "Hospira To Close $145M Javelin Deal This Week". BusinessWeek. 29 June 2010. Archived from the original on 26 October 2012.
^ "Hospira Names Allergan's Michael Ball as CEO". Daily Herald. 8 March 2011.
^ "Archived copy". Archived from the original on 2012-07-07. Retrieved 2012-04-11. CS1 maint: archived copy as title (link)
^ "Pfizer to buy Hospira to boost biosimilar pipeline". 5 February 2015.
^ "Pfizer completes $17-billion Hospira acquisition". The Pharma Letter. 4 September 2015. Retrieved 28 April 2017.
^ Pringle, Sarah (6 January 2017). "ICU Medical Wins Big Price Cut to Buy Pfizer's Hospira Unit". TheStreet.com. Retrieved 28 April 2017.
^ Jamerson, Joshua (6 October 2016). "ICU Medical To Buy Pfizer Unit in $1 Billion Deal". The Wall Street Journal. Retrieved 28 April 2017.
^ "ICU Medical Completes the Acquisition of Hospira Infusion Systems from Pfizer". ICU Medical. ICU Medical. Retrieved 28 April 2017.
^ Thatcher, Virginia S. (1953). "Chapter 7: Illegal or Legal?". History of Anesthesia with Emphasis on the Nurse Specialist. J.B. Lippincott. Archived from the original (PDF) on 2011-05-01.
^ Allen, Nick (27 September 2010). "US executions on hold due to lethal injection drug shortage". London: The Telegraph.
^ Welsh-Huggins, Andrew (27 September 2010). "Shortage of drug holds up some U.S. executions". NBC News. AP.
^ a b McGreal, Chris (2011-01-23). "Lethal injection drug production ends in the US". The Guardian. ISSN 0261-3077. Retrieved 2020-04-23.
^ "EU puts squeeze on drug supplies for U.S. executions". Reuters. 2011-12-20. Retrieved 2020-04-23.
^ a b Koppel, Nathan (January 22, 2011). "Drug Halt Hinders Executions in the U.S." The Wall Street Journal.
^ Eckholm, Erik; Zezima, Katie (21 January 2011). "States Face Shortage of Key Lethal Injection Drug". New York Times.
^ "Hospira - Investor Relations - Press Release". corporate-ir.net.
^ "Sanofi-Aventis Settles Additional Eloxatin Suits". MarketWatch. 6 April 2010.
^ "Home - AARP Online Community". aarp.org.
^ "Hsp". CNN.
^ "NVD - Detail". nist.gov.
^ "Hospira LifeCare PCA Infusion System Vulnerabilities (Update B) - ICS-CERT". us-cert.gov.
^ "Billy (BK) Rios". xs-sniper.com.
^ "Symbiq Infusion System by Hospira: FDA Safety Communication - Cybersecurity Vulnerabilities". fda.gov.
^ "Researcher: Drug Pump the 'Least Secure IP Device I've Ever Seen' - The Security Ledger". The Security Ledger. 5 May 2015.
^ "Serious Security Flaws Found in Hospira LifeCare Drug Pumps - SecurityWeek.Com". securityweek.com.
^ Andrea Peterson (3 August 2015). "Connected medical devices: The Internet of things-that-could-kill-you". Washington Post.
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,060 |
De Nationale Numismatische Collectie (NNC) is de grootste collectie in Nederland van papiergeld, munten, penningen en daaraan gerelateerde objecten zoals hierbij aan niet-westerse betaalmiddelen, productiemiddelen en een uitgebreide bibliotheek. In totaal gaat het om circa 400.000 objecten. De Nederlandsche Bank (DNB) beheert de NNC.
Tot het begin van de 21e eeuw had Nederland drie numismatische collecties van een grote omvang:
Het Rijksmuseum Het Koninklijk Penningkabinet (KPK) beheerde een bijzondere collectie munten en penningen. Sinds de jaren 60 van de twintigste eeuw maakte ook papiergeld deel uit van de collectie. Daarnaast bezat het KPK een belangrijke numismatische bibliotheek.
Het Nederlands Muntmuseum (HNM) beheerde de collectie van de voormalige Koninklijke Nederlandse Munt. Hierbij lag het zwaartepunt op producten en productiemiddelen van het bedrijf en de Utrechtse muntslag.
De Nederlandsche Bank bezat een nagenoeg complete collectie Nederlands papiergeld aangevuld met een bijzondere collectie Hollandse munten, buitenlandse biljetten en munten.
In 2004 zijn deze collecties in beheer gekomen van de Stichting Het Geld- en Bankmuseum. Toen het Geldmuseum in 2013 de deuren sloot, is het beheer van de numismatische collecties overgedragen aan De Nederlandsche Bank. De collectie gesneden stenen is overgegaan naar het Rijksmuseum van Oudheden.
Externe link
De Nationale Numismatische Collectie - dnb.nl
Verzameling | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,343 |
National security expert Dr. Sebastian Gorka said that over President Obama's eight years, he has created "global chaos", culminating in allowing his ambassador to the UN to abstain from a motion condemning Israeli settlements.
Israeli Prime Minister Benjamin Netanyahu hammered the Obama administration for failing to record a veto against a United Nations resolution condemning West Bank and East Jerusalem settlements, accusing the president of initiating the motion.
Mike Huckabee criticized the Obama administration for the United States' abstention before the United Nations during a vote condemning Israeli settlement expansion in the West Bank.
While briefing On the Record about the mass shooting at a Tel Aviv open-air market, an Israeli police spokesperson said that Palestinians have been observed celebrating in the wake of the attack.
Some grim news reported this afternoon on Happening Now as the bodies of three kidnapped Israeli teenagers were found in a shallow grave near the West Bank town of Hebron. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,033 |
Shape Magazine for $4.50 Per Year!
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This sale is good today only. Grab your subscription here! | {
"redpajama_set_name": "RedPajamaC4"
} | 9,809 |
The coolest needlepoint accessory certain to make your dog the envy of the neighborhood!
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Made with Pearl Cotton Thread, 18-Count Mono Canvas, Full-Grain Leather, Solid Brass Buckle.
Size Small: Neck Size 13″-16″.
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Size Large: Neck Size 19″-22″. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,119 |
<meta>
<info author="xXMADEXx" type="script" version="1.0" name="NG Health packs" />
<file src="pack.png" />
<script src="pack_c.lua" type="client" />
<script src="pack_s.lua" type="server" />
<export function="useHealthPack" type="client" />
<export function="useHealthPack" type="server" />
<oop>true</oop>
<min_mta_version server="1.3.4-0.00000" client="1.3.4-0.00000"></min_mta_version>
</meta>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 14 |
==============
``vcd.writer``
==============
.. automodule:: vcd.writer
.. autoclass:: VCDPhaseError
:members:
.. autoclass:: VCDWriter
:members:
.. autoclass:: Variable
:members:
.. autoclass:: ScalarVariable
:members:
.. autoclass:: RealVariable
:members:
.. autoclass:: VectorVariable
:members:
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,840 |
// Copyright (c) Microsoft Corporation
// All rights reserved.
//
// Licensed under the Apache License, Version 2.0 (the "License"); you may not
// use this file except in compliance with the License. You may obtain a copy
// of the License at http://www.apache.org/licenses/LICENSE-2.0
//
// THIS CODE IS PROVIDED *AS IS* BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
// KIND, EITHER EXPRESS OR IMPLIED, INCLUDING WITHOUT LIMITATION ANY IMPLIED
// WARRANTIES OR CONDITIONS OF TITLE, FITNESS FOR A PARTICULAR PURPOSE,
// MERCHANTABLITY OR NON-INFRINGEMENT.
//
// See the Apache Version 2.0 License for specific language governing
// permissions and limitations under the License.
namespace Microsoft.WindowsAzure.Management.HDInsight.ClusterProvisioning.LocationFinder
{
using System.Threading;
using Microsoft.WindowsAzure.Management.HDInsight;
/// <summary>
/// Factory interface to create a Location finder client.
/// </summary>
internal interface ILocationFinderClientFactory
{
/// <summary>
/// Creates a Location finder client instance.
/// </summary>
/// <param name="credentials">Credentials containing user subscription id.</param>
/// <param name="context">
/// A context instance that can be used to cancel the task.
/// </param>
/// <param name="ignoreSslErrors">
/// Specifies that server side SSL errors should be ignored.
/// </param>
/// <returns>A Location finder client instance.</returns>
ILocationFinderClient Create(IHDInsightSubscriptionCredentials credentials, IAbstractionContext context, bool ignoreSslErrors);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,668 |
\section{Introduction}
The transport of a polymer through a nanopore plays a critical role in
numerous biological processes, such as DNA and RNA translocation across
nuclear pores, protein transport through membrane channels, and virus
injection. For a polymer threading through a nanopore, loss of available
configurations due to the geometric constriction leads to an effective
entropic barrier. Kasianowicz \textit{et al.}~\cite{Kasianowicz}
demonstrated that an electric field can drive single-stranded DNA and RNA
molecules through the water-filled $\alpha$-hemolysin channel and that the
passage of each molecule is signaled by a blockade in the channel current.
These observations can directly be used to characterize the polymer length.
Due to various potential technological applications~\cite{Kasianowicz,Meller03},
such as rapid DNA sequencing, gene therapy and controlled drug delivery,
the polymer translocation has become a subject of intensive experimental
~\cite{Akeson,Meller00,Meller01,Meller02,Henrickson,Sauer,Krasilnikov,Storm}
and theoretical
~\cite{Storm,Sung,Muthukumar99,Lubensky,Metzler,Ambj3,Chuang,Kantor,Milchev,
Luo1,Luo2,Luo3,Huopaniemi1,Tian,Matysiak} studies.
As to translocation, one of the basic questions concerns the
dependence of the translocation time $\tau$ on the system parameters
such as the polymer chain length
$N$~\cite{Meller01,Meller02,Storm,Sung,Muthukumar99,
Lubensky,Chuang,Kantor,Milchev,Luo1,Luo2,Luo3,Huopaniemi1,Tian,Matysiak},
sequence and secondary
structure~\cite{Akeson,Meller00,Meller02,Luo3}, pore length $L$ and
pore width $W$~\cite{Luo1}, driving force
$F$~\cite{Meller01,Meller02,Henrickson,Sauer,Kantor,Luo2,Huopaniemi1,Tian,Matysiak},
and polymer-pore
interaction~\cite{Meller00,Meller02,Krasilnikov,Lubensky,Tian}.
In a recent experiment,~\cite{Meller00,Meller02} striking
differences were found for the translocation time distribution of
polydeoxyadenylic acid (poly(dA)$_{100}$) and polydeoxycytidylic
acid (poly(dC)$_{100}$) DNA molecules. The origin of the different
behavior was attributed to stronger attractive interaction of
poly(dA) with the pore. Also, recently Krasilnikov \textit{et
al}.~\cite{Krasilnikov} have investigated the dynamics of single
poly (ethylene glycol) (PEG) molecules in the $\alpha$-hemolysin
channel in the limit of a strong attractive polymer-pore attraction.
The result for the residence time in the channel shows a novel
non-monotonic behavior as a function of the molecular weight.
On the theoretical front, not only the quantitative but also the
qualitative picture of the polymer-nanopore interactions is still
elusive. Based on a Smoluchowski equation with a phenomenological
microscopic potential to describe the polymer-pore interactions,
Lubensky and Nelson~\cite{Lubensky} captured the main ingredients of
the translocation process. However, when comparing with experiments,
their model is not sufficient. Numerically, Tian and
Smith~\cite{Tian} found that attraction facilitates the
translocation process by shortening the translocation time, which
contradicts experimental findings~\cite{Meller00,Meller02}.
To this end, in this letter we use Langevin dynamics (LD) to
investigate the influence of polymer-pore interactions on
translocation. In our numerical simulations, the polymer chains are
modeled as bead-spring chains of Lennard-Jones (LJ) particles with
the Finite Extension Nonlinear Elastic (FENE) potential. Excluded
volume interaction between monomers is modeled by a short range
repulsive LJ potential: $U_{LJ} (r)=4\epsilon
[{(\frac{\sigma}{r})}^{12}-{(\frac{\sigma} {r})}^6]+\epsilon$ for
$r\le 2^{1/6}\sigma$ and 0 for $r>2^{1/6}\sigma$. Here, $\sigma$ is
the diameter of a monomer, and $\epsilon$ is the depth of the
potential. The connectivity between neighboring monomers is modeled
as a FENE spring with $U_{FENE}
(r)=-\frac{1}{2}kR_0^2\ln(1-r^2/R_0^2)$, where $r$ is the distance
between consecutive monomers, $k$ is the spring constant and $R_0$
is the maximum allowed separation between connected monomers.
\begin{figure}
\includegraphics*[width=0.39\textwidth]{Fig1.eps}
\caption{ A schematic representation of the system. The pore length
$L=5 $ and the pore width $W=3$ ( See text for units)
}
\label{Fig1}
\end{figure}
We consider a 2D geometry as shown in Fig. \ref{Fig1}, where the
wall in the $y$ direction is described as stationary particles
within a distance $\sigma$ from each other. The pore of length $L$
and width $W$ in the center of the wall is composed of stationary
black particles.
Between all monomer-wall particle pairs, there exist the same short
range repulsive LJ interaction as described above.
The pore-monomer interaction is modeled by a LJ potential with
a cutoff of $2.5\sigma$ and interaction strength $\epsilon_{pm}$.
This interaction can be either attractive or repulsive depending on
the position of the monomer from the pore particles.
We have also performed numerical calculation for the case of a pure
short range repulsive LJ potential for the pore-monomer interaction.
As expected, the results for the long range LJ pore-monomer interaction
approaches that for the pure repulsive pore-monomer interaction in the
limit $\epsilon_{pm}\rightarrow 0$.
In the Langevin dynamics simulation, each monomer is subjected to
conservative, frictional, and random forces, respectively,
with~\cite{Allen} $m{\bf \ddot
{r}}_i =-{\bf \nabla}({U}_{LJ}+{U}_{FENE})+{\bf F}_{\textrm{ext}}
-\xi {\bf v}_i + {\bf F}_i^R$, where $m$ is the monomer's mass,
$\xi$ is the friction coefficient, ${\bf v}_i$ is the monomer's
velocity, and ${\bf F}_i^R$ is the random force which satisfies the
fluctuation-dissipation theorem.
The external force is expressed as ${\bf F}_{\textrm{ext}}=F\hat{x}$,
where $F$ is the external force strength exerted on the monomers in the pore,
and $\hat{x}$ is a unit vector in the direction along the pore axis.
In the present work, we use the LJ parameters $\epsilon$ and
$\sigma$ and the monomer mass $m$ to fix the energy, length and
mass scales respectively. Time scale is then given by
$t_{LJ}=(m\sigma^2/\epsilon)^{1/2}$ The dimensionless parameters in
our simulations are $R_0=2$, $k/m=7$, $k_{B}T=1.2$, and $\xi/m=0.7$.
For the pore, we set $L=5$ unless otherwise stated. A choice of
$W=3$ ensures that the polymer encounters an attractive force inside
the pore. We have checked that a choice of $W=4$ yields similar
results. The driving force $F$ is set between $0.5$ and $2.0$,
which correspond to the range of voltages used in the
experiments~\cite{Kasianowicz,Meller01}.
The Langevin equation is integrated in time by a method described by
Ermak and Buckholtz~\cite{Ermak} in 2D.
Initially, the first monomer of the chain is placed in the entrance of
the pore, while the remaining monomers are under thermal collisions described
by the Langevin thermostat to obtain an equilibrium configuration.
The translocation time is defined as
the time interval between the entrance of the first segment into the
pore and the exit of the last segment. Typically, we average our
data over 2000 independent runs.
The translocation probability, $P_{trans}$, is calculated as the
fraction of runs leading to successful translocation at given
conditions. Fig. 2(a) shows $P_{trans}$ as a function of
$\epsilon_{pm}$ for $N=128$ under different driving forces.
Specifically, the numerical results clearly show two different
regimes. With increasing $\epsilon_{pm}$, $P_{trans}$ increases
rapidly first, and then slowly approaches saturation at larger
$\epsilon_{pm}$.
It is known that attractive interaction with the
channel can facilitate the translocation of metabolite molecule
across cellular and organelle membranes~\cite{Berez}. Here, our
results show that polymer translocation through the attractive
nanopore shares the same character.
\begin{figure}
\includegraphics*[width=0.39\textwidth]{Fig2.eps}
\caption{
(a) The translocation probability as a function of the attractive
strength for different driving forces.
(b) The distribution of translocation time for different attractive
strengths under the driving force $F=0.5$. The chain length $N=128$.
The data point at $\epsilon_{pm} =0$ corresponds to a pure repulsive
pore-monomer interaction
}
\label{Fig2}
\end{figure}
Experimentally, Meller \textit{et al.}~\cite{Meller00,Meller02} have
investigated the translocation of homepolynucleotides of different
bases: poly(dA)$_{100}$ and poly(dC)$_{100}$. The translocation time
distributions in both cases are well approximated by fast-growing
Gaussian for translocation time lower than the most probable value
$\tau_p$ and falling exponentials for translocation time larger than
$\tau_p$. The decay time scale for poly(dA) is found to be much
longer than poly(dC), by a factor of $\sim 7$. There exists also a
large difference between the value of $\tau_p$ for the two,
corresponding to 1.2 $\mu s$/base for poly(dC), and 3.3 $\mu s$/base
for poly(dA). These differences have been attributed to the base
specific nucleotide-pore interactions, with the adenines having a
stronger attractive interaction with the pore as compared with
cytosines.
In our numerical results, we have found that indeed for
$\epsilon_{pm}$=2 and 3, the attractive potential has a marked
impact on the shape of the histogram of the translocation time as
shown in Fig. 2(b). The shape of the histogram changes from a nearly
Gaussian below the most probable value to a long exponential tail.
The value of $\tau_p$ as well as the characteristic decaying
time scale increases with $\epsilon_{pm}$.
These findings are in excellent agreement with the experimental
observation of Meller \textit{et al.}~\cite{Meller00,Meller02}, and
provide further support that the base specific interaction with the
pore plays a pivotal role in the translocation dynamics of single-stranded
DNA and RNA molecules.
\begin{figure}
\includegraphics*[width=0.39\textwidth]{Fig3.eps}
\caption{ Translocation time as a function of the attractive
strength for different driving forces. The chain length $N=128$.
Here, $\epsilon_{pm} =0$ corresponds to a pure repulsive pore-monomer
interaction}
\label{Fig3}
\end{figure}
Fig. 3 shows the calculated $\tau F$ as a function of
$\epsilon_{pm}$ for $N=128$ under different driving forces.
Initially, $\tau$ increases very slowly with increasing
$\epsilon_{pm}$. Then, it crosses over to a different regime and
increases sharply to the asymptotic behavior $\tau F \sim
e^{L/k_BT}$. The crossover threshold value of $\epsilon_{pm}$
increases with increasing $F$. Surprisingly, previous numerical
work~\cite{Tian} failed to capture the essential feature that the
translocation time increases with increasing attractive base-pore
interaction.
An important element of our analysis is the fact that the
translocation time can be written as $\tau \sim \tau_1 + \tau_2 +
\tau_3$, where $\tau_1$, $\tau_2$ and $\tau_3$ correspond to initial
filling of the pore, transfer of the DNA from the \textit{cis} side
to the \textit{trans} side, and finally the emptying of the pore,
respectively. In the presence of the attractive pore-monomer
interaction and driving force across the pore, $\tau_1 <<
\tau_2,\;\tau_3$, while $\tau_2$ increases monotonically with $N$.
For strong attraction and intermediate values of $N$, $\tau$ is
determined mainly by $\tau_3$ related to the emptying of the pore.
This process involves a free energy difference of $\Delta
\widetilde{F}=L(\epsilon_{pm}-F\sigma/2-f(N))$ between the final and the
initial state. The term $f(N)$ here accounts for the entropic
driving force which should kick in at larger values of $N$ and
eventually saturate for very long polymers. For the region of weak
attraction below the threshold, $\Delta \widetilde{F}<0$ and the
translocation time depends weakly on $\epsilon_{pm}$. Above the
threshold when $\Delta \widetilde{F}>0$, the process is activated
with a barrier $\sim \Delta \widetilde{F}$ and increases rapidly
with increasing strength of attraction $\epsilon_{pm}$. This
accounts for the observed crossover behavior of $\tau$ as a function
of $\epsilon_{pm}$. In the weak attraction non-activated region, the
overall $\tau$ is determined mainly by $\tau_2$ and its dependence
on the driving force scales as $F^{-1}$ which comes from the
velocity dependence on $F$. However, once one enters the activated
region, the force $F$ also influences the activation barrier besides
affecting the prefactor and $\tau$ drops off with increasing $F$
much faster than the simple $F^{-1}$ behavior.
For $F=0.5$ and a pure repulsive pore-monomer interaction, we have
shown in our earlier work~\cite{Luo2,Huopaniemi1} that $\tau\sim
N^{2\nu}$ for relatively short chains and crosses over to $\tau\sim
N^{1+\nu}$ for longer chains as shown in Fig. 4,
where the Flory exponent $\nu=0.75$ in 2D~\cite{de Gennes},
and the crossover length $N_c \sim 200$. The scaling behavior for
attractive interaction strength $\epsilon_{pm}= 1$ is very similar
to the pure repulsive case. For $\epsilon_{pm}= 2$, we found that
$N_c \sim 310$. For stronger attractive strength $\epsilon_{pm}=3$,
only $\tau \sim N^{2\nu}$ is observed for the
$N$ values studied under $F=1$ and $F=2$, with no indication of
crossover behavior as shown in the insert of Fig. 4.
\begin{figure}
\includegraphics*[width=0.39\textwidth]{Fig4.eps}
\caption{
Translocation time as a function of the chain length for
$\epsilon_{pm}=0$ and $\epsilon_{pm}=1$ with $F=0.5$.
The insert shows results for $\epsilon_{pm}=3$ with $F=1.0$ and $F=2.0$.
}
\label{Fig4}
\end{figure}
Under a strong attractive force with $\epsilon_{pm}=3$ and a weak
driving force $F=0.5$, the translocation time $\tau$ has a
qualitatively different dependence on $N$ as compared with the pure
repulsive or weak attractive pore interaction. It has a novel
non-monotonic behavior with a rapid increase to a maximum at $N \sim
14$, followed by a decrease for $14<N<32$and an increases again for
$N>32$ as shown in Fig. 5(a). This can be understood by considering
the different N dependence of $\tau_1$, $\tau_2$ and $\tau_3$ in the
strong attraction limit. For small and intermediate values of $N$,
$\tau$ is dominated by $\tau_3$. Here the entropic factor $f(N)$ in
the barrier for $\tau_3$ fights against the simple power law
increase in the prefactor accounting for the number of monomers
needed to cross the pore. This leads to an initial increase of
$\tau$ to a maximum value followed by a subsequent decrease.
Eventually, for larger $N$, the $\tau_2$ process (which approaches
$N^{2\nu}$ asymptotically) takes over, leading to the increase of
$\tau$ with increasing $N$ again.
For this case, we found that there is about $20\%$ of the total
translocation processes in which the polymer enters and reexits the
\textit{cis} side of the pore. It is useful to define an additional
residence time $\tau_r$ as the weighted sum of the translocation
time and the return time which corresponds to the experimentally
measured blockage time. For the case with no external driving
force, the translocation probability is very small and the residence
time is almost all due to return events. We have calculated the
residence time $\tau_r$ for $F=0$ and $\epsilon_{pm}=3$ and the
result is shown in Fig. 5(b). The $N$ dependence here is again
non-monotonic similar to the translocation time for $F=0.5$ except
for the absence of the eventual increase at the large $N$ limit, due
to the absence of the $\tau_2$ contribution for the return process.
Our numerical result of $\tau_r$ for $F=0$ is in good agreement with
recent experimental data of Krasilnikov \textit{et
al}.~\cite{Krasilnikov} in which the residence time of neutral PEG
molecule in $\alpha$-Hemolysin pore was measured.
\begin{figure}
\includegraphics*[width=0.39\textwidth]{Fig5.eps}
\caption{
(a) Translocation time $\tau$ as a function of the chain length for
$\epsilon_{pm}=3$ and $F=0.5$.
(b) Residence time $\tau_r$ as a function of the chain length for
$\epsilon_{pm}=3$ and $F=0$.}
\label{Fig5}
\end{figure}
To summarize, we have investigated the influence of attractive
polymer-pore interactions on the translocation dynamics via
numerical simulation studies of a simple course grained model. Our
results are in good agreement with recent experimental data for
driven translocation of poly(dA)and poly(dC) molecules, and for the
blockage time study of poly(ethylene glycol) molecule through $\alpha$-Hemolysin pore.
They
clearly demonstrate the important role of polymer-pore interaction
factor in the translocation dynamics.
\begin{acknowledgments}
This work has been supported in part by The Academy of Finland
through its Center of Excellence (COMP) and TransPoly Consortium grants.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,822 |
.中国 е интернет домейн от първо ниво за Китай. Това е китайското име на домейн за Китай.
Източници
中国
Интернет в Китай
Китайски език | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,813 |
repo="$1"
if [ -z "$repo" ]
then
repo="pypi"
fi
make cleanall || exit 1
python2 setup.py sdist bdist_wheel || exit 1
python3 setup.py sdist bdist_wheel || exit 1
twine upload -r "$repo" dist/* || exit 1
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,195 |
Q: Get Length of split in java There is a string and I split it by whitespace and I want to get the last one.
String test = "Hey Hi Hello"; // Defining "test" String
test = test.split(" ")[2]; // Now "test" is "Hello"
System.out.print(test); // prints Hello
I should do this to get the last word of "test" String. But I know the length of the string. What should I do if I don't know the length of the String? What should I write between [ ] to get the last word?
For example when I get a data from an web page and I don't now the value is what.
A: test.split returns an array of Strings. Just save it somewhere instead of using it immediately and check its length.
String test = "Hey Hi Hello";
String[] words = test.split(" ");
test = words[words.length - 1];
System.out.print(test);
A: String[] temp = test.split(" "); //this will split whole string into array using white space.
test = temp[temp.length-1]; //gets the element at the last index of temp array
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,779 |
{"url":"https:\/\/psyteachr.github.io\/msc-data-skills\/joins.html","text":"# Chapter 6 Data Relations\n\n## 6.1 Learning Objectives\n\n1. Be able to use the 4 mutating join verbs: (video)\n2. Be able to use the 2 filtering join verbs: (video)\n3. Be able to use the 2 binding join verbs: (video)\n4. Be able to use the 3 set operations: (video)\n\n## 6.3 Setup\n\n# libraries needed\nlibrary(tidyverse)\n\n## 6.4 Data\n\nFirst, we\u2019ll create two small data tables.\n\nsubject has id, gender and age for subjects 1-5. Age and gender are missing for subject 3.\n\nsubject <- tibble(\nid = 1:5,\ngender = c(\"m\", \"m\", NA, \"nb\", \"f\"),\nage = c(19, 22, NA, 19, 18)\n)\nid gender age\n1 m 19\n2 m 22\n3 NA NA\n4 nb 19\n5 f 18\n\nexp has subject id and the score from an experiment. Some subjects are missing, some completed twice, and some are not in the subject table.\n\nexp <- tibble(\nid = c(2, 3, 4, 4, 5, 5, 6, 6, 7),\nscore = c(10, 18, 21, 23, 9, 11, 11, 12, 3)\n)\nid score\n2 10\n3 18\n4 21\n4 23\n5 9\n5 11\n6 11\n6 12\n7 3\n\n## 6.5 Mutating Joins\n\nMutating joins act like the mutate() function in that they add new columns to one table based on values in another table.\n\nAll the mutating joins have this basic syntax:\n\n****_join(x, y, by = NULL, suffix = c(\".x\", \".y\")\n\n\u2022 x = the first (left) table\n\u2022 y = the second (right) table\n\u2022 by = what columns to match on. If you leave this blank, it will match on all columns with the same names in the two tables.\n\u2022 suffix = if columns have the same name in the two tables, but you aren\u2019t joining by them, they get a suffix to make them unambiguous. This defaults to \u201c.x\u201d and \u201c.y,\u201d but you can change it to something more meaningful.\n\nYou can leave out the by argument if you\u2019re matching on all of the columns with the same name, but it\u2019s good practice to always specify it so your code is robust to changes in the loaded data.\n\n### 6.5.1 left_join()\n\nA left_join keeps all the data from the first (left) table and joins anything that matches from the second (right) table. If the right table has more than one match for a row in the right table, there will be more than one row in the joined table (see ids 4 and 5).\n\nleft_join(subject, exp, by = \"id\")\nid gender age score\n1 m 19 NA\n2 m 22 10\n3 NA NA 18\n4 nb 19 21\n4 nb 19 23\n5 f 18 9\n5 f 18 11\n\nThe order of tables is swapped here, so the result is all rows from the exp table joined to any matching rows from the subject table.\n\nleft_join(exp, subject, by = \"id\")\nid score gender age\n2 10 m 22\n3 18 NA NA\n4 21 nb 19\n4 23 nb 19\n5 9 f 18\n5 11 f 18\n6 11 NA NA\n6 12 NA NA\n7 3 NA NA\n\n### 6.5.2 right_join()\n\nA right_join keeps all the data from the second (right) table and joins anything that matches from the first (left) table.\n\nright_join(subject, exp, by = \"id\")\nid gender age score\n2 m 22 10\n3 NA NA 18\n4 nb 19 21\n4 nb 19 23\n5 f 18 9\n5 f 18 11\n6 NA NA 11\n6 NA NA 12\n7 NA NA 3\n\nThis table has the same information as left_join(exp, subject, by = \"id\"), but the columns are in a different order (left table, then right table).\n\n### 6.5.3 inner_join()\n\nAn inner_join returns all the rows that have a match in the other table.\n\ninner_join(subject, exp, by = \"id\")\nid gender age score\n2 m 22 10\n3 NA NA 18\n4 nb 19 21\n4 nb 19 23\n5 f 18 9\n5 f 18 11\n\n### 6.5.4 full_join()\n\nA full_join lets you join up rows in two tables while keeping all of the information from both tables. If a row doesn\u2019t have a match in the other table, the other table\u2019s column values are set to NA.\n\nfull_join(subject, exp, by = \"id\")\nid gender age score\n1 m 19 NA\n2 m 22 10\n3 NA NA 18\n4 nb 19 21\n4 nb 19 23\n5 f 18 9\n5 f 18 11\n6 NA NA 11\n6 NA NA 12\n7 NA NA 3\n\n## 6.6 Filtering Joins\n\nFiltering joins act like the filter() function in that they remove rows from the data in one table based on the values in another table. The result of a filtering join will only contain rows from the left table and have the same number or fewer rows than the left table.\n\n### 6.6.1 semi_join()\n\nA semi_join returns all rows from the left table where there are matching values in the right table, keeping just columns from the left table.\n\nsemi_join(subject, exp, by = \"id\")\nid gender age\n2 m 22\n3 NA NA\n4 nb 19\n5 f 18\n\nUnlike an inner join, a semi join will never duplicate the rows in the left table if there is more than one matching row in the right table.\n\nOrder matters in a semi join.\n\nsemi_join(exp, subject, by = \"id\")\nid score\n2 10\n3 18\n4 21\n4 23\n5 9\n5 11\n\n### 6.6.2 anti_join()\n\nAn anti_join return all rows from the left table where there are not matching values in the right table, keeping just columns from the left table.\n\nanti_join(subject, exp, by = \"id\")\nid gender age\n1 m 19\n\nOrder matters in an anti join.\n\nanti_join(exp, subject, by = \"id\")\nid score\n6 11\n6 12\n7 3\n\n## 6.7 Binding Joins\n\nBinding joins bind one table to another by adding their rows or columns together.\n\n### 6.7.1 bind_rows()\n\nYou can combine the rows of two tables with bind_rows.\n\nHere we\u2019ll add subject data for subjects 6-9 and bind that to the original subject table.\n\nnew_subjects <- tibble(\nid = 6:9,\ngender = c(\"nb\", \"m\", \"f\", \"f\"),\nage = c(19, 16, 20, 19)\n)\n\nbind_rows(subject, new_subjects)\nid gender age\n1 m 19\n2 m 22\n3 NA NA\n4 nb 19\n5 f 18\n6 nb 19\n7 m 16\n8 f 20\n9 f 19\n\nThe columns just have to have the same names, they don\u2019t have to be in the same order. Any columns that differ between the two tables will just have NA values for entries from the other table.\n\nIf a row is duplicated between the two tables (like id 5 below), the row will also be duplicated in the resulting table. If your tables have the exact same columns, you can use union() (see below) to avoid duplicates.\n\nnew_subjects <- tibble(\nid = 5:9,\nage = c(18, 19, 16, 20, 19),\ngender = c(\"f\", \"nb\", \"m\", \"f\", \"f\"),\nnew = c(1,2,3,4,5)\n)\n\nbind_rows(subject, new_subjects)\nid gender age new\n1 m 19 NA\n2 m 22 NA\n3 NA NA NA\n4 nb 19 NA\n5 f 18 NA\n5 f 18 1\n6 nb 19 2\n7 m 16 3\n8 f 20 4\n9 f 19 5\n\n### 6.7.2 bind_cols()\n\nYou can merge two tables with the same number of rows using bind_cols. This is only useful if the two tables have their rows in the exact same order. The only advantage over a left join is when the tables don\u2019t have any IDs to join by and you have to rely solely on their order.\n\nnew_info <- tibble(\ncolour = c(\"red\", \"orange\", \"yellow\", \"green\", \"blue\")\n)\n\nbind_cols(subject, new_info)\nid gender age colour\n1 m 19 red\n2 m 22 orange\n3 NA NA yellow\n4 nb 19 green\n5 f 18 blue\n\n## 6.8 Set Operations\n\nSet operations compare two tables and return rows that match (intersect), are in either table (union), or are in one table but not the other (setdiff).\n\n### 6.8.1 intersect()\n\nintersect() returns all rows in two tables that match exactly. The columns don\u2019t have to be in the same order.\n\nnew_subjects <- tibble(\nid = seq(4, 9),\nage = c(19, 18, 19, 16, 20, 19),\ngender = c(\"f\", \"f\", \"m\", \"m\", \"f\", \"f\")\n)\n\nintersect(subject, new_subjects)\nid gender age\n5 f 18\n\nIf you\u2019ve forgotten to load dplyr or the tidyverse, base R also has an intersect() function. The error message can be confusing and looks something like this:\n\nbase::intersect(subject, new_subjects)\n## Error: Must subset rows with a valid subscript vector.\n## \u2139 Logical subscripts must match the size of the indexed input.\n## x Input has size 6 but subscript !duplicated(x, fromLast = fromLast, ...) has size 0.\n\n### 6.8.2 union()\n\nunion() returns all the rows from both tables, removing duplicate rows.\n\nunion(subject, new_subjects)\nid gender age\n1 m 19\n2 m 22\n3 NA NA\n4 nb 19\n5 f 18\n4 f 19\n6 m 19\n7 m 16\n8 f 20\n9 f 19\n\nIf you\u2019ve forgotten to load dplyr or the tidyverse, base R also has a union() function. You usually won\u2019t get an error message, but the output won\u2019t be what you expect.\n\nbase::union(subject, new_subjects)\n## [[1]]\n## [1] 1 2 3 4 5\n##\n## [[2]]\n## [1] \"m\" \"m\" NA \"nb\" \"f\"\n##\n## [[3]]\n## [1] 19 22 NA 19 18\n##\n## [[4]]\n## [1] 4 5 6 7 8 9\n##\n## [[5]]\n## [1] 19 18 19 16 20 19\n##\n## [[6]]\n## [1] \"f\" \"f\" \"m\" \"m\" \"f\" \"f\"\n\n### 6.8.3 setdiff()\n\nsetdiff returns rows that are in the first table, but not in the second table.\n\nsetdiff(subject, new_subjects)\nid gender age\n1 m 19\n2 m 22\n3 NA NA\n4 nb 19\n\nOrder matters for setdiff.\n\nsetdiff(new_subjects, subject)\nid age gender\n4 19 f\n6 19 m\n7 16 m\n8 20 f\n9 19 f\n\nIf you\u2019ve forgotten to load dplyr or the tidyverse, base R also has a setdiff() function. You usually won\u2019t get an error message, but the output might not be what you expect because the base R setdiff() expects columns to be in the same order, so id 5 here registers as different between the two tables.\n\nbase::setdiff(subject, new_subjects)\nid gender age\n1 m 19\n2 m 22\n3 NA NA\n4 nb 19\n5 f 18\n\n## 6.9 Glossary\n\nterm definition\nbase r The set of R functions that come with a basic installation of R, before you add external packages\nbinding joins Joins that bind one table to another by adding their rows or columns together.\nfiltering joins Joins that act like the dplyr::filter() function in that they remove rows from the data in one table based on the values in another table.\nmutating joins Joins that act like the dplyr::mutate() function in that they add new columns to one table based on values in another table.\nset operations Functions that compare two tables and return rows that match (intersect), are in either table (union), or are in one table but not the other (setdiff).\n\n## 6.10 Exercises\n\nDownload the exercises. See the answers only after you\u2019ve attempted all the questions.\n\n# run this to access the exercise\ndataskills::exercise(6)\n\n# run this to access the answers\ndataskills::exercise(6, answers = TRUE)","date":"2022-07-03 09:15:25","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\": 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\": 0.22518569231033325, \"perplexity\": 4653.984614181857}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656104215805.66\/warc\/CC-MAIN-20220703073750-20220703103750-00371.warc.gz\"}"} | null | null |
Minister of Foreign Affairs (New Zealand)
The Minister of Foreign Affairs is a senior member of the Government of New Zealand heading the Ministry of Foreign Affairs and Trade and responsible for relations with foreign countries.
Coat of Arms of New Zealand
Flag of New Zealand
since 26 October 2017
Ministry of Foreign Affairs and Trade
Cabinet of New Zealand
Prime Minister of New Zealand
Governor-General of New Zealand
At Her Majesty's pleasure
$288,900[1]
www.beehive.govt.nz
The current Minister of Foreign Affairs is Winston Peters, who has held the position since 2017.
Responsibilities and powersEdit
The Minister of Foreign Affairs is responsible for overseeing New Zealand's relations with foreign countries and the promotion of New Zealand's interests abroad.[2] The Minister is in charge of the Ministry of Foreign Affairs and Trade, including New Zealand's diplomatic staff. The office is often considered to be one of the more distinguished ministerial posts, and has at times been counted as the most senior role below that of the Prime Minister. In terms of actual political power, however, the Minister of Foreign Affairs is not as prominent as in countries such as Australia, Canada, the United Kingdom and the United States, with the Minister of Finance being considerably more influential.
Historically, the Minister of Foreign Affairs has been a member of Cabinet, with the exception of the Rt Hon. Winston Peters between 2005 and 2008. This situation came about as the result of coalition negotiations in which it was agreed that New Zealand First would take a senior ministerial portfolio but would not join Cabinet.
The first New Zealand foreign minister was James Allen, appointed to the post of "Minister of External Affairs" by William Massey in 1919. Before this time, there was no dedicated ministerial portfolio for foreign relations. A Department of External Affairs was created in 1919 but its functions were limited to administering New Zealand's Island Territories in the Pacific; namely the Cook Islands, Niue, Tokelau, and the League of Nations Mandate of Samoa.[3] In 1943, a new Department of External Affairs was created to conduct the country's external relations. The older department was then renamed the Department of Island Territories and a separate portfolio called the Minister of Island Territories was subsequently created.[4]
From 1943, the Minister of External Affairs became the main ministerial portfolio for conducting New Zealand's external relations.[5] Like its similarly named Australian and Canadian counterparts, the portfolio was called "External Affairs" rather than "Foreign Affairs" in deference of the British Government's responsibility for conducting foreign policy on behalf of the British Empire and later the Commonwealth of Nations.[6] The title was changed to "Minister of Foreign Affairs" in 1970 after the Department was renamed the Ministry of Foreign Affairs. The title became "Minister of Foreign Affairs and Trade" following the abandonment of the short-lived "Minister of External Relations and Trade" title, created in September 1988 when the Ministry of Foreign Affairs absorbed the Trade functions of the old Department of Trade and Industry. In 2005 responsibility for trade was split into a separate portfolio, with the title reverting to "Minister of Foreign Affairs".
Historically it has been common for Prime Ministers to take on the role of Foreign Minister themselves, particularly if they have an interest in the field. Several New Zealand Prime Ministers including Peter Fraser, Walter Nash, Keith Holyoake, and David Lange held the External Affairs portfolio.[5] The most recent Prime Minister to do this is Helen Clark in 2008 as Acting Minister, and prior to her was Mike Moore, in 1990. Thirteen Prime Ministers have served as Foreign Minister for all or part of their terms.
New Zealand has had 27 foreign ministers (regardless of exact title). The longest-serving was Keith Holyoake, who held the post for the duration of his 11-year premiership. The second longest-serving, and the longest-serving who was not also Prime Minister, was Don McKinnon, who became Commonwealth Secretary-General.
List of Ministers of Foreign AffairsEdit
Reform United Labour National NZ First
24 November 1919 28 April 1920 Massey
Ernest Lee
17 May 1920 13 January 1923
Francis Bell
7 June 1923 18 January 1926
William Nosworthy
24 May 1926 24 August 1928
Gordon Coates
25 August 1928 10 December 1928
Joseph Ward
10 December 1928 28 May 1930 Ward
George Forbes
28 May 1930 6 December 1935 Forbes
Michael Joseph Savage
6 December 1935 27 March 1940 Savage
Frank Langstone
1 April 1940 21 December 1942 Fraser
Peter Fraser
7 July 1943 13 December 1949
Frederick Doidge
13 December 1949 19 September 1951 Holland
19 September 1951 26 November 1954
Tom Macdonald
26 November 1954 12 December 1957
Holyoake
Walter Nash
12 December 1957 12 December 1960 Nash
Keith Holyoake
12 December 1960 8 December 1972 Holyoake
Norman Kirk
8 December 1972 31 August 1974 Kirk
Bill Rowling
6 September 1974 12 December 1975 Rowling
Brian Talboys
12 December 1975 11 December 1981 Muldoon
Warren Cooper
11 December 1981 26 July 1984
26 July 1984 24 August 1987 Lange
Russell Marshall
24 August 1987 9 February 1990
Mike Moore
9 February 1990 2 November 1990
Don McKinnon
2 November 1990 10 December 1999 Bolger
Phil Goff
10 December 1999 19 October 2005 Clark
Winston Peters
19 October 2005 29 August 2008
Acting Minister
29 August 2008 19 November 2008
Murray McCully
19 November 2008 2 May 2017 Key
Gerry Brownlee
2 May 2017 26 October 2017
26 October 2017 Incumbent Ardern
^ https://www.parliament.nz/media/3151/parliamentary-salaries-and-allowances-determination-2016.pdf
^ "Ministerial Portfolio: Foreign Affairs". The Department of the Prime Minister and Cabinet. Retrieved 29 December 2017.
^ "External Affairs Bill", in New Zealand Parliamentary Debates, Vol. 185 (3 October–5 November 1919), p.337.
^ Malcolm Templeton, An Eye, an Ear, and a Voice: 50 years in New Zealand's External Relations, 1943-1993, p.1.
^ a b Malcolm Templeton, ed., An Eye, An Ear, And a Voice, pp.1-2.
^ Alan Watt, "The Department of Foreign Affairs," in The Times Survey of Foreign Ministries of the World,Department of External Affairs (1921–70) ed. Zara Steiner (London: Times Books Limited, 1982), p.35; James Eary, "The Department of External Affairs," in The Times Survey of Foreign Ministries of the World, p.96.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Minister_of_Foreign_Affairs_(New_Zealand)&oldid=902004336" | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,694 |
\section{Introduction}
Let us consider the wave equation in $\mathbb R^{n} \times \mathbb R$:
\begin{equation}\label{wveq}\begin{cases}
\partial_t^2 u -\Delta u=0, \\
u(x,0)=f,~\,\partial_tu(x,0)=g.
\end{cases}
\end{equation}
The space-time estimate for the solution of \eqref{wveq} which is called
\emph{Strichartz estimate} has been proven to be an important tool
in studies of various problems. (See \cite{st, pe, km, ls, MS, kt}.)
It is well-known that the estimate
\begin{equation}\label{stri}
\|u\|_{L_t^q(\mathbb R,\, L_x^r(\mathbb R^{n}))}\lesssim\|f\|_{\dot H^s} +
\|g\|_{\dot H^{s-1}}
\end{equation}
holds for $s\ge 0, 2 \le q , r <\infty$
which satisfy
\[
\frac1q+\frac{n}r=\frac{n}2-s,\,\, \frac1q+\frac{n-1}{2r} \le
\frac{n-1}4.\]
Here $\dot H^s$ is the homogeneous $L^2$ Sobolev space of order $s$. See \cite{FW} for the estimates with $r=\infty$.
It was Strichartz \cite{st} who first proved the estimate \eqref{stri} when $q=r$. This was later extended to mixed norm estimates by Pecher \cite{pe}. (Also see \cite{GV}.)
The endpoint cases $(r,q) = (2(n-1)/(n-3),2)$ except $n=3$ were obtained by Keel--Tao \cite{kt}. Klainerman and Machedon \cite{km} showed the failure of \eqref{stri} when $(n,r,q)=(3,\infty,2)$.
\
In this note we consider a generalization of \eqref{stri}
by replacing the Lebesgue measure with general measure $\mu$.
More precisely, we study the estimate
\begin{eqnarray}\label{fstrichartz}
\|u\|_{L^q(d\mu)}\lesssim\|f\|_{H^s} + \|g\|_{H^{s-1}}.
\end{eqnarray}
Here we denote by $H^s(\mathbb R^{n})$
the inhomogeneous $L^2$ Sobolev space of order $s$, which is the space of all tempered distributions $f$ such that $(1 + |\cdot|^2)^{\frac s2} \widehat f \in L^2(\mathbb R^n)$, equipped with the norm
\[
\|f\|_{H^s(\mathbb R^n) } = \| (1 + |\cdot|^2)^{\frac s2} \widehat f \|_{L^2(\mathbb R^n)}.
\]
This kind of estimates was studied in connection with problems in geometric measure theory, precisely, the sphere packing problem (see \cite{Mi, w, ob, ob2}).
Throughout this paper, the measure $\mu$ is assumed to be a nonnegative Borel
regular measure with compact support in $\mathbb R^{n+1}$.
Let us denote by $\mathfrak M(\mathbb R^{n+1})$ the space of such measures.
In addition we impose uniform growth condition on $\mu$ as follows.
\begin{defn} Let $\alpha\in (0, n+1]$. For $\mu \in \mathfrak M(\mathbb R^{n+1})$, we say that $\mu$ is $\alpha$-dimensional if there exists a constant $C_\mu$, independent of $x$ and $\rho$, such that
\begin{equation}\label{def-measure}
\mu(B(x,\rho))\le C_\mu \rho^\alpha \quad\textrm{ for all } x\in \mathbb R^{n+1},\, \rho>0.
\end{equation}
Here $B(x,\rho)$ denotes the open ball of radius $\rho$ centered at $x$.
Also we define
\[
\langle \mu\rangle_\alpha = \sup_{x\in \mathbb R^{n+1},\,\rho>0} \rho^{-\alpha}\mu(B(x,\rho)).
\]
\end{defn}
For $1\le q\le \infty$
let us set
\begin{equation}\label{necessary}
s(\alpha,q,n)=\begin{cases}
\max \{ \frac{n}2-\frac{\alpha}q ,\,
\frac{n+1}4
\}, \,\, \,\, &\text{ if }{0}<
\alpha \le 1, \\
\max \{ \frac{n}2-\frac{\alpha}q ,\,
\frac{n+1}4 +\frac{1-\alpha}{2q},\,~~~~~~
\frac{n+2}{4}-\frac{\alpha}{4} \}, \,\, \,\, &\text{ if } {1}<
\alpha \le n, \\
\max \{ \frac{n}2-\frac{\alpha}q ,\,
\frac{n+1}4+\frac{n+1-2\alpha}{2q},\,\frac{n+1}2-\frac{\alpha}{2}
\},\, &\text{ if } n< \alpha \le n+1.
\end{cases}
\end{equation}
When $n=2$
Wolff \cite{w} showed that \eqref{fstrichartz}
holds for $\alpha$-dimensional measure $\mu$ if $s>\max(\frac34, 1-\frac{\alpha}4, 1-\frac{\alpha}q)$, $\alpha \in (1,3)$.
Erdo$\breve{\textrm g}$an \cite{er3} improved Wolff's result so that
\eqref{fstrichartz} holds for $s>s(\alpha,q,2)$, $\alpha\in(1,3)$
and also showed that \eqref{fstrichartz} generally
fails if $s<s(\alpha,q,2)$.
When $n\ge3$,
Oberlin \cite{ob} obtained \eqref{fstrichartz} for $
\alpha\in (1,n+1)$ provided that $q<\alpha$ and $s>\frac{n-1}2$.
\
It is plausible to conjecture that \eqref{fstrichartz} holds if $s> s(\alpha, q,n)$ (see Proposition \ref{prop:necessary}) but like other open problems of similar nature complete resolution seems out of reach at this moment. However, for $n=3$ and $\alpha \in[1,4]$, we obtain the sharp estimate by the following theorem and Proposition \ref{prop:necessary}.
\begin{theorem}\label{wave}
Let $n=3$.
Also let $\mu$ be an $\alpha$-dimensional measure.
Suppose that $u$ is a solution to the equation \eqref{wveq}.
Then \eqref{fstrichartz} holds with
\begin{equation}\label{scon3}
s>
\begin{cases}
s(\alpha,q,3), \, \text{ if }\, 2 \le q \le \infty,\\
s(\alpha,2,3), \, \text{ if }\, 1 \le q \le 2.
\end{cases}
\end{equation}
Furthermore, the implicit constant in \eqref{fstrichartz} does not depend on particular choice of $\mu$ as long as $\langle\mu\rangle_\alpha$ is uniformly bounded.
\end{theorem}
For $\delta >0$ and $R \gg 1$, let us define the truncated cone $\Gamma_R$ by
\[\Gamma_R=\{\xi=(\xi',\xi_{n+1})\in \mathbb R^{n+1}: |\xi'|=\xi_{n+1},
R\le \xi_{n+1}\le 2R \}\] and its $\delta$-neighborhood by
\[\Gamma_R(\delta)=\{\xi\in \mathbb R^{n+1}: \text{dist} (\xi, \Gamma_R)<\delta \}.\]
Note that the space-time Fourier transform of $u$ is supported in
the forward and backward light cones. By reflection in frequency spaces, Littlewood-Paley decomposition, and Plancherel theorem,
Theorem \ref{wave} is a consequence of the following.
(For details, see the end of Section \ref{sec:rescaling}.)
\begin{theorem}\label{thecone}
Let $n=3$ and $\mu$ be an $\alpha$-dimensional measure supported in $\overline{B(0,1)}$.
Then, for $f$ supported in $\Gamma_R(1)$ there exists a constant $C>0$ such that
\begin{equation}
\label{frac} \|\widehat f\|_{L^q(d\mu)}\le C {\langle\mu\rangle_\alpha^{\frac1q }}\, R^s\|f\|_2
\end{equation}
holds if $s$ satisfies \eqref{scon3}.
\end{theorem}
\begin{rmk}
If $n=3$ and $ \alpha \in [1, 4]$, \eqref{frac} is sharp because $s(\alpha,q,3)=s(\alpha,2,3)$ when $1 \le q \le2$.
If $0 < \alpha <1$, we have $s(\alpha,2,3) > s(\alpha,q,3)$ for $1\le q \le 2$.
\end{rmk}
In general, $s(\alpha,q,n)$, $n \ge 3$ provides a lower bound on $s$ for \eqref{frac} to hold.
\begin{proposition}\label{prop:necessary}
Suppose that for any $\alpha$-dimensional measure $\mu$, there is a constant $C>0$ such that \eqref{frac} holds whenever $f$ is supported in $\Gamma_R(1)\subset \mathbb R^{n+1}$.
Then
$ s \ge s(\alpha,q, n) $ for $1\le q \le \infty$.
\end{proposition}
In order to show this, we construct $\alpha$-dimensional measures and functions for which $\eqref{frac}$ fails if $s<s(\alpha,q, n) $. For $\alpha\in (1,n+1]$, we modify the examples in \cite{er3} and for $\alpha\in (0,1]$ we use the construction in \cite{HL}. (See Section \ref{sec:sharpness} for details.) For each $\alpha$ some of the conditions appearing in \eqref{necessary} are natural in view of dilation
and rescaling structure of the estimate \eqref{frac}. To be precise, for the case $n <\alpha \le n+1$ in \eqref{necessary} the condition $s(\alpha,q,n)= \frac{n+1}4 + \frac{n+1-2\alpha}{2q}$
may be understood as a homogeneity condition and $s(\alpha,q,n)=\frac n2 - \frac \alpha q$ is related to the Knapp type example.
A similar statement also applies to the case $1<\alpha\le n$.
\smallskip
In contrast with the Lebesgue measure, there is no obvious scaling structure for a general $\alpha$-dimensional
measure. But, as is to be seen in what follows, if we assume uniform bound which only depends on $\langle\mu\rangle_\alpha$, the measure $\mu$ can be handled as if it is homogeneous of degree $\alpha$
with respect to isotropic dilation. This observation plays an important role throughout the paper. Similar idea was used in \cite{HL} to obtain
restriction estimate for the curve with respect to general measures other than the Lebesgue measure.
\smallskip
The estimate \eqref{frac} is closely related to Fourier restriction problem. We make use of the bilinear approach which was extensively utilized to tackle the restriction problem. Some aspects of our paper are similar to those of \cite{er3}. However, unlike \cite{er3} we use the induction on scale argument in more direct way without relying on weighted inequality and this enables us to expose underlying structure more clearly. Also our argument can be used to recover the results in \cite{er3}. It is natural to expect that multilinear restriction estimates \cite{BCT} and its recent development (see \cite{BG, G}) can be used for further improvement of the estimate \eqref{fstrichartz}. However, it does not seem likely that these estimates would give the sharp estimate such as our result in Theorem \ref{wave}.
\smallskip
In this paper we will prove \eqref{frac} for $n\ge3$ (see Theorem \ref{thecone1}) while our results are sharp only for $n=3$.
(The sharp estimates for $n=2$ and $1\le\alpha\le3$ was previously obtained by Erdo\u{g}an \cite{er3}.)
When $n \ge 4$, the necessary condition $s \ge s(\alpha,q, n)$ (except the equality case) is sufficient only for $n< \alpha \le n+1$ or for large $q$ if $\alpha \le n $.
\subsubsection*{Average decay estimate}
It is well-known that \eqref{frac} implies the associated average decay estimate.
In fact the decay rate is determined by $s(=s(\alpha,2,n))$ of the estimate \eqref{frac}.
Let $I_\alpha(\mu)$ be an $\alpha$-dimensional energy of $\mu$ which is given by
$I_\alpha(\mu)=\iint |x-y|^{-\alpha} d\mu(x)d\mu(y)$.
If $\mu$ is a positive Borel measure supported in the unit ball and satisfies $I_\alpha(\mu)=1$, there exists a constant $C_\alpha > 0$ such that
\begin{equation}\label{energy}
\int_{\Gamma_1}
|\widehat \mu(R\xi)|^2 d\sigma(\xi) \le C_{\alpha}
R^{-n + 2 s(\alpha,2, 3)+\varepsilon}
\end{equation}
for $R>1$ and any $\varepsilon >0$.
This can be shown by the argument in \cite{w2} which makes use of Lemma 1.5 and duality.
(See Section 2 in \cite{w2} for details.)
The average decay estimate over the sphere has been studied in
connection with Falconer's distance set conjecture. (See \cite{Matt1}, \cite{b1},
\cite{w2}, \cite{er2}, \cite{er1}, \cite{LR} and references therein.)
\subsubsection*{The sphere packing problem}
Let $S(x,r)$ be a sphere in $\mathbb R^{n}$ with center $x$ and radius $r > 0$. We denote the Hausdorff dimension by $dim_{H}$ and the $d$-dimensional Lebesgue measure by $|\cdot|_{d}$. Theorem \ref{thecone} immediately implies the following.
\begin{corollary}\label{corollary}
Let $E \subset \mathbb R^{n}$ and $P
$ be a Borel set in $\mathbb R^{n+1}$ with $dim_H(P) > 1$.
Assume that $|S(x,r) \cap E|_{n-1} > 0$ for any $(x,r) \in P$.
Then $|E|_n >0$.
\end{corollary}
Wolff \cite{w} proved that {Corollary \ref{corollary}} is valid when $n=2$. When $n \ge 3$, Oberlin \cite{ob} showed that the statement holds by obtaining estimate for the spherical average.
\subsubsection*{Organization of the paper}
In Section \ref{sec:bilinear} we prove a bilinear version of \eqref{frac},
which is obtained by an adaptation of the induction on scale argument. In Section \ref{sec:rescaling}
proofs of Theorem \ref{thecone} and Theorem \ref{wave} are given.
In Section \ref{sec:sharpness}, we discuss the necessary conditions in Proposition \ref{prop:necessary}.
%
\section{Bilinear estimates}\label{sec:bilinear}
In this section we prove a bilinear version of the estimate \eqref{frac} which is closely related to bilinear restriction estimate for the cone (\cite{tvv, tv1, w3, tao-bilinear, le}).
Under an additional transversality condition, bilinear estimate gives formally better estimate than linear one by weakening Kakeya compression phenomena.
\begin{defn} For a function $f$ which is supported away from the origin we define the angular support $\textit{$\mathcal A$supp$\!$ } f$ by
\[ \textit{$\mathcal A$supp$\!$ } f=\Big\{ \frac \xi{|\xi|} : \xi\in \text{supp$\!$ } f \Big \}. \]
\end{defn}
The following may be regarded as a generalization of bilinear restriction estimate for the cone in \cite{w3}.
\begin{theorem}\label{bifrac}
Let $R\gg 1$ and let $\mu$ be an $\alpha$-dimensional
measure supported in $\overline{B(0,1)}$. Suppose that $f$ and $g$
are supported in $\Gamma_R(1)$ and
\begin{equation}\label{angular-separation}
\text{dist}( \textit{$\mathcal A$supp$\!$ } f, \textit{$\mathcal A$supp$\!$ } g)\ge \frac1{100}.
\end{equation}
For $2\le
q\le\infty$, there is a constant {$C = C(\beta,n)$}
such that
\begin{equation}\label{bi}
\left(\int |\widehat f\,\, \widehat g|^{\frac q2} d\mu\right)^\frac2q\le
CR^{2\beta} {\langle\mu\rangle_\alpha^{\frac2q}}\|f\|_2 \|g\|_2
\end{equation}
for any $\beta > \beta(\alpha,q):= \max\{ \frac{n}2-\frac{\alpha} q, \,\frac{3n+1-2\alpha}{8} \} $.
\end{theorem}
Since $|\widehat f|\lesssim R^{\frac n2} \|f\|_2$ and $|\widehat g|\lesssim R^{\frac n2} \|g\|_2$, \eqref{bi} trivially holds with $2\beta\ge n$.
It is easy to verify that the condition $\beta \ge \frac{n}2-\frac{\alpha} q$ is necessary by adopting the examples which are used to show
the necessity of the condition \eqref{scaling}.
Another necessary condition for \eqref{bi} with $\alpha >2$ is
\begin{equation}\label{bilinear-condition}
\beta\ge \frac{n-1}4-\frac{\alpha-2}{2q}.
\end{equation}
To see this, we consider the squashed
cap function (see \cite{tvv}) and a suitable measure. Precisely,
let us set $d\mu=\psi(x)|x''|^{-n-1+\alpha}
dx_1dx_2dx''$, $x=(x_1, x_2, x'')$ for a smooth cutoff function $\psi$.
Then $\mu$ is an $\alpha$-dimensional measure if $n+1\ge
\alpha>2$.
Considering a pair of characteristic functions $f, g$ supported in $\Gamma_R(1)$ with large
angular separation and dimensions $1\times 1\times
\overset{\,\,\,\,\,(n-1)\text{ times}}{\sqrt R \times \cdots \times \sqrt R}$ it is easy to show \eqref{bilinear-condition}.
\subsubsection*{Localization to a smaller cube} Improvement due to localization is important for the induction on scale argument.
In the following lemma we make it precise how localization to smaller cubes affects the estimate \eqref{bi}.
\begin{lem}\label{localization} Let $\beta\ge 0$, $q\ge2$.
Suppose that \eqref{bi} is valid for any $\alpha$-dimensional measure $\mu$ supported in $\overline{B(0,1)}$, and for any $f$,
$g$ supported in $\Gamma_R(A)$ for some $A \sim 1$ satisfying the condition \eqref{angular-separation}. Then, for $x_0 \in \mathbb R^{n+1}$ and $\rho\le (10A)^{-1}$,
\begin{align}\label{local1}
\left(\int_{B(x_0,\rho)} |\widehat f\,\, \widehat g|^{\frac q2} d\mu\right)^\frac2q\le
C {\langle\mu\rangle_\alpha}
^{\frac2q}R^{2\beta}\rho^{\frac{2\alpha} q+2\beta-n} \|f\|_2 \|g\|_2.
\end{align}
\end{lem}
\begin{proof}
By translation we may
assume $x_0=0$ without loss of generality.
Since the map $\phi \mapsto \rho^{-\alpha} \int \phi(\rho^{-1} x) d\mu(x)$ is a positive linear functional on $C_c(\mathbb R^{n+1})$, by the Riesz representation theorem there exists a Radon measure $\mu_\rho$ such that
\[\int \phi(y) d\mu_\rho(y)= \rho^{-\alpha}\int \phi(\rho^{-1} x) \chi_{B(0,\rho)} (x)d\mu(x)
\]
for any continuous function $\phi$.
Then it is obvious that
$\mu_\rho$ is supported in $ \overline{B(0,1)}$ and $\langle\mu_\rho\rangle_\alpha \le
\langle\mu\rangle_\alpha$.
Let us set $f_\rho=\rho^{-n-1} f(\cdot/\rho)$, $g_\rho=\rho^{-n-1} g(\cdot/\rho)$. Then,
it follows that
\begin{align}\label{local2}
\int_{B(0,\rho)} |\widehat f\, \widehat g |^{\frac q2} d
\mu
=\rho^\alpha \int_{{B(0,1)}} |\widehat f_\rho\,\widehat g_\rho|^{\frac q2}
d \mu_\rho .
\end{align}
We note that $f_\rho$ and $ g_\rho$ are supported in the set $\Gamma_{R\rho}(A \rho).$ Since the measure $ \mu_\rho$ is supported in $\overline{B(0,1)}$,
we may put a harmless smooth function $\eta$ in the integral which
satisfies $\widehat \eta\sim 1$ on $\overline{B(0,1)}$ and $\text{supp$\!$ } \eta \subset
B(0,\frac12)$.
Then we see
\begin{align} \label{local3}
\begin{aligned}
\int |\widehat f_\rho(x)\widehat g_\rho(x)|^{\frac q2} d \mu_\rho
(x)&\sim \int |\widehat\eta ^2 \widehat f_\rho(x)\widehat
g_\rho(x)|^{\frac q2} d \mu_\rho (x)
\\&=\int | \widehat {\eta\ast
f_\rho}(x)\widehat{ \eta\ast g_\rho}(x)|^{\frac q2} d \mu_\rho (x).
\end{aligned} \end{align}
Note
that $\eta\ast f_\rho$ and $\eta\ast g_\rho$ are
supported in $\Gamma_{R\rho}(1)$.
Hence, from \eqref{bi} we have
\[
\Big(\int | \widehat {\eta\ast
f_\rho}(x)\widehat{ \eta\ast g_\rho}(x)|^{\frac q2} d \mu_\rho
(x)\Big)^\frac2q \le C \langle \mu\rangle_{\alpha}^{\frac2q}(R\rho)^{2\beta} \|\eta\ast f_\rho\|_2\|\eta\ast
g_\rho\|_2.
\]
Since $f_\rho$ and $g_\rho$ are supported in
$\Gamma_{R\rho }( A\rho)$,
$\|\eta\ast f_\rho\|_2\le
C\rho^{-n/2}\|f\|_2$\footnote{This follows from H\"older inequality and easy estimates $ \|\eta\ast f_\rho\|_1\lesssim
\|f\|_1$, $\|\eta\ast f_\rho\|_\infty \lesssim \rho^{-n}
\|f\|_\infty$. } and $\|\eta\ast g_\rho\|_2\le C\rho^{-n/2}
\|g\|_2$.
Therefore \eqref{local1} follows by \eqref{local2} and \eqref{local3}.
\end{proof}
\subsubsection*{Decomposition of $f$ and $g$} For the proof of Theorem \ref{bifrac} we adapt the induction on scale argument which has been used to prove sharp bilinear restriction estimate \cite{w3, tao-bilinear, le}. The following is actually a rescaled version of \cite[Lemma 3.5] {w3} (also see \cite{ tao-bilinear, le} for a simpler proof based on wave packet decomposition, especially the proof of Theorem 1.3 in \cite{le}).
\begin{lemma}\label{wavepacket}
Let $R \gg 1$ and $0<\delta\le 1$. Let $\{\mathbf
q\} $ be a collection of cubes of sidelength $\sim R^{-\delta}$ which partition the cube
$[-1,1]^{n+1}$. Suppose $f$ and $g$ are supported in $\Gamma_R(1)$ and satisfy \eqref{angular-separation}. Then for each $\mathbf
q$, $f$ and $g$ can be decomposed such that
\[ f=f_{\mathbf q}+ f_{\overset{\not\sim}{\mathbf q}} \quad \text{and} \quad g=g_{\mathbf q}+ g_{\overset{\not\sim}{\mathbf q}} \]
with $f_{\mathbf q}$,
$f_{\overset{\not\sim}{\mathbf q}}$, $g_{\mathbf q}$,
$g_{\overset{\not\sim}{\mathbf q}} $ supported in $\Gamma_R(C_d)$ for some $C_d\sim 1$
and, for $0<\epsilon\ll \delta$,
\begin{align}
\label{l2sumq}
&\sum_{\mathbf q} \|f_{\mathbf q}\|_2^2\le CR^\epsilon
\|f\|_2^2,\qquad \sum_{\mathbf q} \|g_{\mathbf q}\|_2^2\le
CR^\epsilon \|g\|_2^2,
\\
\label{bil2} \|\widehat {f_{\mathbf
q}}\widehat{g_{\overset{\not\sim}{\mathbf q}}}\|&_{L^2(\mathbf q)},
\quad \|\widehat{f_{\overset{\not\sim}{\mathbf q}}}\widehat
{g_{\mathbf q}}\|_{L^2(\mathbf q)}, \quad
\|\widehat{f_{\overset{\not\sim}{\mathbf
q}}}\widehat{g_{\overset{\not\sim}{\mathbf q}}}\|_{L^2(\mathbf q)}
\le C R^{\frac{n-1}4+c\delta+\epsilon} \|f\|_2\|g\|_2
\end{align}
with $c$ independent of $\delta$ and $\epsilon$.
\end{lemma}
\begin{proof} Let $F$ and $G$ be functions supported in $\Gamma_1(R^{-1})$.
Then, by using wave packet decomposition\footnote{It should be noted that we are working with functions instead of extension operators. Nevertheless, the argument in \cite{le} works without extra difficulty.} (\cite{tao-bilinear, le}), on $B(0, R)$ we may write \[\widehat F = \sum_{w_1}c_{w_1}p_{w_1} \quad \text{and} \quad \widehat G = \sum_{w_2}c_{w_2}p_{w_2},\]
where $c_{w_i}$ and $p_{w_i}$ satisfy the following:
\begin{align*}
&(1)\,\, \big(\sum_{w_1 \in W_1}|c_{w_1}|^2\big)^{1/2} \le C\|F\|_2,\,\,\big(\sum_{w_2 \in W_2}|c_{w_2}|^2\big)^{1/2} \le C \|G\|_2; \\
&(2)\,\,\, \text{If}\,\, w_i\in W_i, \, \text{supp$\!$ } \widehat {p_{w_i}} \subset \{\xi : \xi = v_i + O(R^{-1/2})\};\\
&(3)\,\,\, \text{If} \,\, \text{dist} (x,T_{w_i}) \ge R^{1/2 + \delta}, \,\, |p_{w_i}(x)|\le CR^{-100n}; \\ &(4) \,\,\,
\text{For any subset}\,\, S \subset W_{i},\, \|\sum_{w_i \in S}p_{w_i}\|_2^2 \le C \# S.
\end{align*}
Here $W_i$ is the set of all pairs $w_i = (y_i,v_i)$ with $y_i \in R^{1/2}\mathbb Z^n$ and $v_i \in R^{-1/2}\mathbb Z^n \cap \{ v \in \mathbb R^n: 2^{-1} \le |v| \le 2^2\} )$ and $T_{w_i}$ is defined by
\[T_{w_i} =\big \{(x',x_{n+1}) \in \mathbb R^n \times \mathbb R : |x_{n+1}| \le R, \big|x'-\big(y_i+x_{n+1}\frac{v_i}{|v_i|}\big)\big| \le R^{1/2}\big\}\] for each $w_i \in W_i$, $i=1,2$.
Partition $[-R,R]^{n+1}$ into $R^{1-\delta}$-cubes $\{R\mathbf q\}$ which are essentially disjoint. Then, \[\|\widehat F\,\widehat{G}\|_{L^q(B(0,R))} \le \sum_{\mathbf q}\|\widehat F\,\widehat{G}\|_{L^q( R\mathbf q)}.\]
Now we use the relation $\sim$ between $w_i$ and $R^{1-\delta}$ cube $R\mathbf q$ which was introduced in \cite{tao-bilinear} (also see \cite{le}). For $\mathbf q$ fixed, we say $w_i\sim \mathbf q$ if $w_i\sim R\mathbf q$ and, otherwise, we say $w_i \not\sim \mathbf q$.
We keep the same notation $\sim$ since it does not cause any ambiguity.
We now set \[\widehat{F_{\mathbf q}} = \sum_{w_1 \sim \mathbf q} c_{w_1} p_{w_1},\quad \widehat{F_{\overset{\not\sim}{\mathbf q}}} = \sum_{w_1\not\sim \mathbf q} c_{w_1} p_{w_1}, \quad
\widehat{G_{\mathbf q}} = \sum_{w_2 \sim \mathbf q} c_{w_2} p_{w_2}\quad \widehat{G_{\overset{\not\sim}{\mathbf q}}}
= \sum_{w \not\sim \mathbf q} c_{w_2} p_{w_2}.\]
Then, by repeating the argument in \cite{tao-bilinear, le} it is not difficult to see that
\begin{align*}
&\sum_{\mathbf q} \|\widehat {F_{\mathbf q}}\|_2^2\lesssim R^\epsilon
\|F\|_2^2, \quad \sum_{\mathbf q} \|\widehat {G_{\mathbf q}}\|_2^2\lesssim
R^\epsilon \|G\|_2^2;
\\
\|\widehat {F_{\mathbf
q}}\widehat{G_{\overset{\not\sim}{\mathbf q}}}\|_{L^2(R\mathbf q)},
&
\quad \|\widehat{F_{\overset{\not\sim}{\mathbf q}}}\widehat
{G_{\mathbf q}}\|_{L^2( R\mathbf q)}, \quad
\|\widehat{F_{\overset{\not\sim}{\mathbf
q}}}\widehat{G_{\overset{\not\sim}{\mathbf q}}}\|_{L^2( R\mathbf q)}
\lesssim R^{-\frac{n+3}4+c\delta+\epsilon} \|F\|_2\|G\|_2.
\end{align*}
Since we are dealing with functions $F$ and $G$ which are supported in $\Gamma_1(R^{-1})$ instead of being supported on the surfaces,
we obtain an extra power $R^{-1}$ compared to the extension operator (\textit{cf.} \cite{tao-bilinear, le}).
Since $f, g$ are supported in $\Gamma_R(1)$ and satisfy \eqref{angular-separation}, we apply the above to \[F = R^{\frac{n+1}{2}}f(R\cdot) \quad \text{and}\quad G = R^{ \frac{n+1}{2}}g(R\cdot). \]
Then, the change of variable $x\to x/R $ gives the desired estimates \eqref{l2sumq} and \eqref{bil2}.
\end{proof}
To prove Theorem \ref{bifrac} we make use of the following simple lemma.
\begin{lem}\label{fractal} Let $R\gg 1$ and $\mu$ be an $\alpha$-dimensional
measure supported in $\overline{B(0,1)}$.
Set $\phi_R=R^{n+1}\phi(R\cdot)$ for a Schwartz function $\phi$. Then, for any $r \ge 1$ and any Schwartz function $\phi$,
there is a constant $C$, independent of $\mu$, such that
\begin{equation}\label{convolution}
\|\phi_R\ast d\mu\|_{r}\le C\langle\mu\rangle_\alpha R^{(n+1-\alpha)(1-\frac1r)}.
\end{equation}
\end{lem}
\begin{proof} By rapid decay of $\phi$ it follows that
$\phi_{R}\le C_N R^{n+1}\sum_{j\ge 0} 2^{-Nj}\chi_{B(0,2^{j}R^{-1})}$ for all $N >0$.
{Hence, recalling the definition of $\langle\mu\rangle_\alpha$ in Definition 1.1} and choosing a sufficiently large $N$, we have
\begin{align*}
\phi_R\ast d\mu(x)&\le C_N R^{n+1}\sum_{j\ge 0}
2^{-Nj}\mu(B(x,2^{j}R^{-1}))\\
&\le C_N \langle\mu\rangle_\alpha R^{n+1-\alpha}\sum_{j\ge 0} 2^{(\alpha-N)j}
\le C_N \langle\mu\rangle_\alpha R^{n+1-\alpha}.
\end{align*}
This yields \eqref{convolution} with $r=\infty$.
Since $\mu$ is supported in $\overline{B(0,1)}$, we also have $\|\mu\|\le 2^\alpha\langle\mu\rangle_\alpha$ where $ \|\mu\| = \sup \{ |\int f(x) d\mu(x) | : |f|\le 1,\, f \in C(\mathbb R^{n+1}) \}$.
So, from Young inequality $\|\phi_R\ast d\mu\|_1\le \|\phi_R\|_1 \|\mu\|\lesssim \langle\mu\rangle_\alpha$.
Hence, \eqref{convolution} follows from $\| \phi_R \ast d\mu\|_r^r \le \|\phi_R \ast d\mu \|_\infty^{r-1} \|\phi_R \ast d\mu\|_1$ for any $r\ge 1$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{bifrac}]
It is sufficient to show the case $2 \le q \le 4$.
Extension to $q\ge 4$ can be obtained by interpolation with
the {trivial estimate}
\
\| \widehat f\,\,\widehat g\|_{L^\infty {(d\mu)}} \lesssim R^{n} \| f \|_{L^2} \| g \|_{L^2},
\
which is valid since $\widehat f$ and $\widehat g$ are continuous.
In what follows we prove the implication from \eqref{bi} to \eqref{imply}.
We assume that \eqref{bi} holds for some $\beta>0$ and $f,g$ supported in $\Gamma_R(1)$. As is mentioned before, this is true with a large $\beta>0$. Since $\mu$ is supported in $\overline{B(0,1)}$, we have
\[\Big(\int |\widehat f\,\,\widehat g|^{q/2}
d\mu \Big)^\frac2q
\le \sum_{\mathbf q} \Big( \int_{\mathbf q} | \widehat
f\,\,\widehat g|^{q/2} d\mu \Big)^\frac2q.\]
Using the decomposition in Lemma \ref{wavepacket}, we have
\begin{equation} \label{qube}
\begin{aligned}
\Big(\int |\widehat f\,\,\widehat g|^{q/2}
d\mu \Big)^\frac2q
&
\le \sum_{\mathbf q} \Bigg[\Big( \int_{\mathbf q} | \widehat
f_{\mathbf q}\widehat g_{\mathbf q}|^{q/2} d\mu \Big)^\frac2q +
\Big( \int_{\mathbf q} | \widehat {f_{\mathbf
q}}\widehat{g_{\overset{\not\sim}{\mathbf q}}}|^{q/2}
d\mu\Big)^\frac2q
\\
&\qquad +
\Big(
\int_{\mathbf q} | \widehat{f_{\overset{\not\sim}{\mathbf
q}}}\widehat {g_{\mathbf q}}|^{q/2} d\mu \Big)^\frac2q+
\Big(
\int_{\mathbf q} | \widehat{f_{\overset{\not\sim}{\mathbf
q}}}\widehat{g_{\overset{\not\sim}{\mathbf q}}}|^{q/2} d\mu\Big)^\frac2q\Bigg].
\end{aligned}\end{equation}
By Lemma \ref{localization}, it follows that
\[\Big( \int_{\mathbf q} | \widehat
f_{\mathbf q}\widehat g_{\mathbf q}|^{q/2} d\mu \Big)^\frac2q \le
C \langle\mu\rangle_{\alpha}^{\frac 2q} R^{2\beta}R^{\delta(n-\frac{2\alpha}q-2\beta)}\|f_{\mathbf q}\|_2
\|g_{\mathbf q}\|_2.\]
Schwarz inequality and \eqref{l2sumq}
give
\begin{equation}\label{local}
\sum_{\mathbf q} \Big( \int_{\mathbf q} | \widehat
f_{\mathbf q}\widehat g_{\mathbf q}|^{q/2} d\mu \Big)^\frac2q \lesssim C \langle\mu\rangle_{\alpha}^{\frac 2q} R^{2\beta+\epsilon}R^{\delta(n-\frac{2\alpha}q-2\beta)}\|f\|_2
\|g\|_2.
\end{equation}
For the other terms in \eqref{qube}
we use \eqref{bil2}. Let $\eta$ be a Schwartz function such that $\widehat \eta(\xi)=1$ if
$|\xi|\le 10C$ and $\text{supp$\!$ }\widehat \eta\subset B(0,20C)$ and set $\eta_R=R^{n+1} \eta(R\cdot)$. Since $f_{\mathbf
q}\ast g_{\overset{\not\sim}{\mathbf q}}$ is supported in a ball of
radius $10R$,
we have $ |\widehat {f_{\mathbf
q}}\widehat{g_{\overset{\not\sim}{\mathbf q}}} | = |(\widehat
{f_{\mathbf q}}\widehat{g_{\overset{\not\sim}{\mathbf q}}} )\ast \eta_R |
\lesssim ( |\widehat{f_{\mathbf q}}\widehat{g_{\overset{\not\sim}{\mathbf q}}} |^{q/2} \ast \eta_R )^{2/q}$.
By \eqref{convolution}, we thus obtain
\begin{align*}
\Big( \int_{\mathbf q} | \eta_R \ast \big(\widehat {f_{\mathbf
q}}\widehat{g_{\overset{\not\sim}{\mathbf q}}}\big)|^{q/2}
d\mu \Big)^\frac2q & \le C\Big( \int_{\mathbf q} | \big(\widehat
{f_{\mathbf q}}\widehat{g_{\overset{\not\sim}{\mathbf
q}}}\big)|^{q/2} |\eta_R| \ast d\mu \Big)^\frac2q
\\
&\le C
\begin{cases}
\|\widehat {f_{\mathbf
q}}\widehat{g_{\overset{\not\sim}{\mathbf q}}}\|_{L^2(\mathbf q)}
\| |\eta_R| \ast d\mu\|_\infty^\frac12& \text { if } q=4\\
\|\widehat {f_{\mathbf
q}}\widehat{g_{\overset{\not\sim}{\mathbf q}}}\|_{L^2(\mathbf q)} \|
|\eta_R| \ast d\mu\|_2& \text { if } q=2.
\end{cases}
\end{align*}
By \eqref{bil2} it follows that for $2\le q\le 4$,
\begin{align*}
\Big( \int_{\mathbf q} \big| \widehat {f_{\mathbf
q}}\widehat{g_{\overset{\not\sim}{\mathbf q}}}\big|^{q/2}
d\mu \Big)^\frac2q &\le C\|\widehat {f_{\mathbf
q}}\widehat{g_{\overset{\not\sim}{\mathbf q}}}\|_{L^2(\mathbf q)} \langle\mu\rangle_\alpha^{\frac2q}
R^{\frac{n+1-\alpha}2}
\\
&\le C \langle\mu\rangle_\alpha^{\frac2q} R^{\frac{3n+1-2\alpha}4+c\delta+\epsilon} \|f\|_2\|g\|_2.
\end{align*}
The terms
$\widehat{f_{\overset{\not\sim}{\mathbf q}}}\widehat {g_{\mathbf
q}}$ and $\widehat{f_{\overset{\not\sim}{\mathbf
q}}}\widehat{g_{\overset{\not\sim}{\mathbf q}}}$
can be treated similarly. Since
$ \#\{\mathbf q\}\le CR^{(n+1)\delta}$,
combining the estimates for these terms, we get
\begin{align*}
\sum_{\mathbf q} \Big( \int_{\mathbf q} | \widehat {f_{\mathbf
q}}\widehat{g_{\overset{\not\sim}{\mathbf q}}}|^{q/2}
d\mu\Big)^\frac2q &+
\Big(\int_{\mathbf q} | \widehat{f_{\overset{\not\sim}{\mathbf
q}}}\widehat {g_{\mathbf q}}|^{q/2} d\mu \Big)^\frac2q+
\Big(
\int_{\mathbf q} | \widehat{f_{\overset{\not\sim}{\mathbf
q}}}\widehat{g_{\overset{\not\sim}{\mathbf q}}}|^{q/2} d\mu\Big)^\frac2q
\\ &\le
C \langle\mu\rangle_\alpha^{\frac2q} R^{\frac{3n+1-2\alpha}4+{\tilde c \delta}+\epsilon}
\|f\|_2\|g\|_2.
\end{align*}
Here $\tilde c = c +(n+1)$.
By this and \eqref{local} it follows from \eqref{qube} that for any $\epsilon>0$,
\begin{align}\label{imply}
\Big(\int |\widehat f\,\,\widehat g|^{q/2} d\mu \Big)^\frac2q
&\le C \langle\mu\rangle_\alpha^{\frac2q} R^{\max \left\{ 2\beta + \delta\left( n-\frac{2\alpha}q-2\beta \right),\, \frac{3n+1-2\alpha}4 + \tilde c \delta \right\}+\epsilon} \|f\|_2\|g\|_2.
\end{align}
Hence we have shown that $\eqref{bi}$ implies \eqref{imply}.
If we have \eqref{bi} for $\beta= \beta_i$, then we see that \eqref{bi} holds with
\[
\beta = \frac{3n +1 -2\alpha}{8} + \tilde c \left( \frac{8\beta_i - 3n-1+2\alpha}{8(\tilde c - n +2\alpha/ q +2\beta_i)} \right) := \beta_{i+1}
\]
by choosing $\delta = \frac{8\beta - 3n-1+2\alpha}{4(\tilde c - n +2\alpha/ q +2\beta)}$.
Iterating this implication we obtain a sequence $\{\beta_i\}_{i=0}^\infty$.
Note that the sequence $\{\beta_i\}_{i=0}^\infty$ is strictly decreasing as long as
$- n +2\alpha/ q +2\beta_i > 0$ i.e. $\beta_i > n/2 - \alpha/q$.
Since $\beta_i$ converges to $(3n+1 -2\alpha)/8$,
we conclude that \eqref{bi} holds for $2\le q\le 4$ and for any $\beta > \max\{ \frac{n}2-\frac{\alpha} q, \,\frac{3n+1-2\alpha}{8} \}$.
This completes the proof.
\end{proof}
\section{Proof of Theorem \ref{thecone}: Rescaling}\label{sec:rescaling}
In this section we prove the following theorem by deducing linear estimate from the bilinear one \eqref{bi}.
\begin{theorem}\label{thecone1}
Let $n\ge3$ and $\mu$ be an $\alpha$-dimensional measure supported in $\overline{B(0,1)}$.
Then, for $f$ supported in $\Gamma_R(1)$ there exists a constant $C>0$ such that \eqref{frac} holds for $s$ satisfying
\begin{align*}
s > \widetilde s (\alpha,q,d ) :=
\begin{cases}
\, \max \{\frac {n}2-\frac\alpha q, \frac{n+1}4,\frac{3n+1}8-\frac\alpha 4\}, &\text{if } 0<\alpha\le1,\\
\, \max\{\frac {n}2-\frac\alpha q, \frac{n+1}4+\frac{1-\alpha}{2q},\frac{3n+1}8-\frac\alpha 4\}, &\text{if } 1 <\alpha\le n,\\
\, \max \{ \frac {n}2-\frac\alpha q, \frac{n+1}4+\frac{n+1-2\alpha}{2q},\frac{n+1-\alpha}{2} \},
& \text{if } n <\alpha \le n+1.
\end{cases}
\end{align*}
\end{theorem}
For $n=3$, this immediately implies Theorem \ref{thecone}.
\begin{proof}[Proof of Theorem \ref{thecone}]
It suffices to check that $\widetilde s (\alpha,q,3) = s(\alpha,q,3)$ for $ 0 <\alpha \le 3$.
If $0<\alpha\le 1$, it is easy to check
$\widetilde s (\alpha, q, 3) = n/2 - \alpha/q = s(\alpha,q,3)$ for $q \ge 2$.
If $1<\alpha\le 3$, we see that $\widetilde s(\alpha,q,3) = s(\alpha,q,3)$ because $(3n+1-2\alpha)/8 = (n+2-\alpha)/4$ when $n=3$.
This completes the proof.
\end{proof}
To prove Theorem \ref{thecone1}, we begin with a Whitney type decomposition to exploit bilinear estimate. (See \cite{tvv,w3}.)
\subsubsection*{Whitney type decomposition of $\mathbb S^{n-1}\times \mathbb S^{n-1}$}
For each $j\ge 1$, let us dyadically divide the sphere
$\mathbb S^{n-1}$ into $O(2^{(n-1)j})$
caps ${\theta^j_k}$ of diameter $\sim 2^{-j}$.
We will write $\theta^j_k\approx \theta^j_{k^\prime}$ to mean that
$\theta^j_k$ and $\theta^j_{k^\prime}$ are not adjacent but have
adjacent parent caps of diameter $\sim$ $2^{1-j}$.
Then, by Whitney decomposition of $\mathbb S^{n-1}\times \mathbb S^{n-1}$ away from its diagonal $D = \{ (x,x) : x \in \mathbb S^{n-1} \}$
we have
\[
\mathbb S^{n-1}\times \mathbb S^{n-1}\setminus D
=\bigcup{}_{ j\ge 1 } \bigcup{}_{(k,k^\prime):\theta^j_k\approx
\theta^j_{k^\prime}}\theta^j_k\times \theta^j_{k^\prime}.
\]
Let $D(R^{-1/2})$ be a $O(R^{-1/2})$ neighborhood of $D$.
Then $\mathbb S^{n-1}\times \mathbb S^{n-1}\setminus D(R^{-1/2})$ can be covered by $\theta^j_k\times \theta^j_{k^\prime}$ such that $\theta^j_k\approx \theta^j_{k^\prime}$ and $1\le j \le \log R^{1/2}$.
Also $D(R^{-1/2})$ is covered by a union of disjoint cubes $\theta^{j_\circ}_k\times \theta^{j_\circ}_{k}$ of sidelength $O(R^{-1/2})$. So, $2^{-j_\circ}\sim 1/{\sqrt R}$.
\
For $f$ supported away from the origin we set
\[f_k^j(\xi,\tau)=\chi_{\theta_k^j}({\xi}/{\tau}) f(\xi,\tau), \quad (\xi,\tau)\in \mathbb R^n\times\mathbb R.\]
Then by the above decomposition we have
\begin{equation}\label{decomp}
| \widehat f |^2 \le \sum_{j=1}^{\log R^{1/2}}\sum_{(k,k^\prime):\theta^j_k\approx \theta^j_{k^\prime}}
|\widehat {f_k^j}\widehat {f_{k'}^j}| + \sum_{k} |\widehat{f_k^{j_\circ}}|^2.
\end{equation}
Note that the diameters of $\textit{$\mathcal A$supp$\!$ } f_k^j$ and $\textit{$\mathcal A$supp$\!$ } f_{k'}^j$ are $O(2^{-j})$. In order to handle the first sum we need the following.
\begin{lem}\label{rescaling}
Let $1\le j \le \log(R^{1/2})$.
Suppose $f$ and $g$ are supported in $\Gamma_R(1)$ and the diameters of $\textit{$\mathcal A$supp$\!$ } f$ and $\textit{$\mathcal A$supp$\!$ } g\le 2^{-j-2}$. If
$
\text{dist} (\textit{$\mathcal A$supp$\!$ } f, \textit{$\mathcal A$supp$\!$ } g) \ge 2^{-j},
$
there exists a constant $C>0$ such that
\begin{equation}\label{sep}
\| \widehat {f}\widehat {g}\|_{L^{\frac q2}(d\mu)}\le
C \langle \mu\rangle_\alpha^{\frac 2 q} \, R^{2\beta}\, 2^{\gamma j} \,\|f\|_2 \|g\|_2,
\end{equation}
where
$\beta > \beta(\alpha,q)$ and
\[\gamma=\gamma(\alpha,\beta, n)=\begin{cases}
-4\beta+ 2(n+1-2\alpha)/q+n+1, &\text{ if } n< \alpha \le n+1,\\
-4\beta + 2(1-\alpha)/q +n+1, &\text{ if } 1\le \alpha \le n,\\
-4\beta+n+1, &\text{ if } 0<\alpha\le 1.
\end{cases}\]
\end{lem}
\begin{proof
By rotation we may assume that
$\textit{$\mathcal A$supp$\!$ } f$ and $\textit{$\mathcal A$supp$\!$ } g$ are contained in
$O(2^{-j})$ neighborhood of $ (0,\dots, 0,\frac 1{\sqrt 2}, \frac1{\sqrt 2}) \in \mathbb R^{n+1}$.
More precisely, we
use the null coordinates
\[\xi''=(\xi_1,\xi_2, \dots, \xi_{n-1}), \, \sigma=\frac{\xi_{n+1}-\xi_n}{\sqrt 2},\, \tau=\frac{\xi_{n+1}+\xi_n}{\sqrt 2}.\]
Then the cone $\Gamma_R$ can be written as $|\xi''|^2=2\sigma\tau$. So
\[\Gamma_{R}(1)\subset\{(\xi'',\sigma,\tau): |\xi''|^2=2\sigma\tau+ O(1),\, \tau\sim R \}.\]
Additionally, after a slight adjustment we may assume that $\text{supp$\!$ } f \subset V_1$ and $\text{supp$\!$ } g \subset V_2$,
where
\[
V_i = \{(\xi'',\sigma,\tau): |\xi''|^2=2\sigma\tau+O(1),\, \tau\sim R,\,
|\xi''/\tau +(-1)^i 2^{-j}e_1|\le 2^{-j-2}\}, \,\, i=1,2.
\]
Now, using an anisotropic transformation $T_j : (\xi'',\sigma,\tau) \mapsto (2^{j}\xi'',\sigma,2^{2j}\tau)$,
we have
\begin{align*}
\| \widehat {f}\widehat {g}\|_{L^{q/2}( d\mu)}=\| \widehat {f_j}\widehat
{g_j}\|_{L^{q/2}( d\mu_j)},
\end{align*}
where $f_j=|\det T_j| \, f \circ T_j = 2^{(n+1)j} f \circ T_j $, $g_j=|\det T_j| \, g\circ T_j $,
and
\[ \mu_j(\phi)=\int \phi\circ T_j \, d\mu.\]
So, we see
\[\text{supp$\!$ } f_j
\subset\Big\{(\xi'',\sigma,\tau):
|\xi''|^2=2\sigma\tau + O(2^{-2j}),\, \tau\sim 2^{-2j}R,\,
|\xi''/\tau - e_1|\le 2^{-2} \Big\}
\]
and
\[\text{supp$\!$ } g_j
\subset\Big\{(\xi'',\sigma,\tau):
|\xi''|^2=2\sigma\tau + O(2^{-2j}),\, \tau\sim 2^{-2j}R,\,
|\xi''/\tau + e_1|\le 2^{-2} \Big\}.
\]
Clearly, $\mu_j$ is supported in $\overline{B(0,1)}$ because $j\ge 1$, and note that $f$ and $g$ are supported in $\Gamma_{2^{-2j}R}(2^{-2j})$ and satisfy the separation condition \eqref{angular-separation} in Theorem \ref{bifrac}, by which we have that, for $ \beta > \beta(\alpha,q)$,
\begin{equation}\label{scaled}
\| \widehat {f_j}\widehat {g_j}\|_{L^{q/2}( d\mu_j)} \le C \langle \mu_j\rangle_\alpha^{\frac 2q} (2^{-2j}R)^{2\beta} \| f_j\|_2 \| g_j\|_2.
\end{equation}
We now estimate $\langle \mu_j\rangle_{\alpha}$. For any $( x,\rho) \in \mathbb R^{n+1}\times \mathbb R$, we have
\begin{align*}
\mu_j( B(x,\rho))&=\int_{T_j^{-1} B(x,\rho)} d\mu
\lesssim \int_{\mathcal R} d\mu,
\end{align*}
where $\mathcal R$ is a rectangle of dimensions $ 2^{-j}\rho \times2^{-j}\rho\times\cdots \times2^{-j}\rho\times \rho\times 2^{-2j}\rho$.
There are three different ways of decomposing $\mathcal R$ into cubes, namely, cubes of side length $2^{-2j}\rho$, $2^{-j}\rho$, and $\rho$, respectively. Considering these three cases and
using \eqref{def-measure} for each case, we obtain
\begin{align*}
\mu_j(B(x,\rho))
&\lesssim \langle\mu\rangle_\alpha \,\min\{ 2^{(n+1-2\alpha)j}\rho^\alpha, \,2^{(1-\alpha)j}\rho^\alpha,\, \rho^\alpha \} \\
&= \langle\mu\rangle_\alpha \times \begin{cases}
\, 2^{(n+1-2\alpha)j}\rho^\alpha, &\text{ if } n< \alpha \le n+1,\\
\, 2^{(1-\alpha)j}\rho^\alpha, &\text{ if } 1<\alpha\le n,\\
\, \rho^\alpha, &\text{ if } 0< \alpha \le 1.
\end{cases}
\end{align*}
Hence, $\langle \mu_j\rangle_\alpha \lesssim \,\min\{ 2^{(n+1-2\alpha)j} , \,2^{(1-\alpha)j} ,\, 1 \} \times\langle\mu\rangle_\alpha$.
Combining this with \eqref{scaled}, $\|f_j\|_2 = 2^{j(n+1)/2}\|f\|_2$ and $\|g_j\|_2 = 2^{j(n+1)/2}\|g\|_2$ yields \eqref{sep}.
\end{proof}
To handle the second sum in \eqref{decomp}, which is easier, we use the following.
\begin{lem}\label{diagonal}
Suppose that $f$ is supported in $\Gamma_R(1)$ and the diameter of $\textit{$\mathcal A$supp$\!$ } f$ is $O(R^{-1/2})$.
For $q \ge 2$, there exists a constant $C>0$ such that
\begin{equation}\label{diagonaleq}
\| \widehat {f}\|_{L^{q}(d\mu)}\le
C \langle \mu\rangle_\alpha^{\frac 1 q} \, R^{\beta_\circ(\alpha,q)}\, \|f\|_2,
\end{equation}
where
\[\beta_\circ(\alpha,q) = \begin{cases}
(n+1)/4+(n+1-2\alpha)/2q, &\text{ if } n< \alpha \le n+1 ,\\
(n+1)/4+(1-\alpha)/2q, &\text{ if } 1\le \alpha \le n,\\
(n+1)/4, &\text{ if } 0<\alpha\le 1.
\end{cases}\]
\end{lem}
\begin{proof}
Note that $f$ is supported in a rectangle $\mathcal R$ of dimensions
$C \overset{n-1 \text{ times } }{R^{1/2} \times \cdots \times CR^{1/2 } }\times C\times CR$ for a positive constant $C$.
We consider $T_R$ which takes the unit cube $Q$ onto $\mathcal R$.
Then we have
\[
\widehat f(x) = R^{\frac{n+1}{2}} \int e^{-i (T_R x) \cdot y} f_R (y) dy,
\]
where $f_R := f\circ T_R $ is supported in the unit cube $Q$.
As before, let $\mu_R$ be the measure given by
$
\mu_R (\phi) = \int \phi\circ T_R \, d\mu
$ for continuous function $\phi$.
Since $f_R$ is supported in $Q$, $|\widehat{f_R}| \le C(|\widehat{f_R}|^q\ast |\eta_{Q}|)^{\frac1q}$ for any Schwartz function $\eta_Q$ satisfying $\eta_Q=1$ on $Q$.
Using Hausdorff-Young inequality, Lemma \ref{fractal}, and H\"older inequality we have, for $q \ge 2$,
\begin{align*}
&\qquad\qquad \| \widehat f \,\|_{L^q(d\mu)} \lesssim R^{\frac{n+1}{2}} \| |\widehat{\eta_Q}|\ast d \mu_R\|_\infty^{\frac 1q} \, \| \widehat{f_R} \|_{L^q(\mathbb R^{n+1})}
\\&\lesssim R^{\frac{n+1}{2}} \langle \mu_R \rangle_\alpha^{\frac 1q} |\text{supp$\!$ } f_R|^{\frac12-\frac1q} \, \| {f_R} \|_{L^2(\mathbb R^{n+1})}
\lesssim R^{\frac{n+1}{4}} \langle \mu_R \rangle_\alpha^{\frac 1q} \, \| f \|_{L^2(\mathbb R^{n+1})}.
\end{align*}
%
As in the proof of Theorem \ref{rescaling} (this corresponds to the case $2^j\sim \sqrt R$), it is easy to see
\begin{align*}
\langle \mu_R \rangle_\alpha^{\frac1q}
\lesssim \langle\mu\rangle_\alpha^{\frac1q} \times \begin{cases}
\, R^{\frac{n+1-2\alpha}{2q}}, &\text{ if } n< \alpha \le n+1,\\
\, R^{\frac{1-\alpha}{2q}}, &\text{ if } 1<\alpha\le n,\\
\, 1, &\text{ if } 0< \alpha \le 1.
\end{cases}
\end{align*}
Therefore, combining these two estimates gives \eqref{diagonaleq}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thecone1}]
By \eqref{decomp},
we have
\[
\| \widehat f \|_{L^{q}(d\mu)}^2 \le \sum_{j=1}^{\log R^{1/2}}\sum_{(k,k^\prime):\theta^j_k\approx \theta^j_{k^\prime}}
\|\widehat {f_k^j}\widehat {f_{k'}^j}\|_{L^{q/2}(d\mu)} + \sum_{k} \|\widehat{f_k^{j_\circ}}\|_{L^{q}(d\mu)}^2 =: I + II.
\]
By Lemma \ref{rescaling}, for any $\epsilon >0$ and $R>1$, we have
\begin{align*}
I &\lesssim \langle \mu \rangle_\alpha^{\frac 2q} R^{2\beta} \sum_{j=1}^{\log R^{1/2}} 2^{\gamma j} \sum_{(k,k^\prime):\theta^j_k\approx \theta^j_{k^\prime}} \| f_k^j\|_2 \| f_{k'}^j \|_2.
\end{align*}
Hence, by Schwarz inequality, we get
\begin{align*}
I \lesssim \langle \mu \rangle_\alpha^{\frac 2q} R^{2\beta} \times
\begin{cases} \log R^{1/2} \| f\|_2^2 \\
R^{\gamma /2} \log R^{1/2} \| f\|_2^2 \end{cases}
\lesssim \langle \mu \rangle_\alpha^{\frac 2q} \times
\begin{cases} R^{2\beta(\alpha,q) + \varepsilon} \| f\|_2^2 , &\text{ if} \, \gamma \le 0, \\
R^{2 \beta_\circ(\alpha,q) + \varepsilon} \| f\|_2^2 , &\text{ if} \, \gamma > 0. \end{cases}
\end{align*}
From Lemma \ref{diagonal}, we also see
\begin{align*}
II \lesssim \langle \mu\rangle_\alpha^{\frac2q} R^{2\beta_\circ(\alpha,q)} \sum_k \|f_k^{j_\circ}\|_2^2 \lesssim \langle \mu\rangle_\alpha^{\frac2q} R^{2\beta_\circ(\alpha,q)} \|f\|_2^2.
\end{align*}
Combining these estimates, we obtain that
for $ q \ge 2$, and for any $\epsilon>0$ and $R>1$,
\begin{equation}\label{ndimension}
\| \widehat f\|_{L^{q}(d\mu)}\lesssim
\langle\mu\rangle_\alpha^{\frac 1q}\,R^{\widetilde s(\alpha,q,n) +\epsilon}\|f\|_2,
\end{equation}
where $\widetilde{s}(\alpha,q,n)=\max \{\beta(\alpha,q),\beta_\circ(\alpha,q)\}$.
Note that
\[
\widetilde{s}(\alpha,q,n) =
\begin{cases}
\, \max \{\frac {n}2-\frac\alpha q, \frac{n+1}4,\frac{3n+1}8-\frac\alpha 4\}, &\text{if } 0<\alpha\le1,\\
\, \max\{\frac {n}2-\frac\alpha q, \frac{n+1}4+\frac{1-\alpha}{2q},\frac{3n+1}8-\frac\alpha 4\}, &\text{if } 1 <\alpha\le n,\\
\, \max \{ \frac {n}2-\frac\alpha q, \frac{n+1}4+\frac{n+1-2\alpha}{2q},\frac{3n+1}8-\frac\alpha 4 \},
& \text{if } n <\alpha \le n+1.
\end{cases}
\]
If $3\le n <\alpha$, we can improve $\widetilde{s}(\alpha,q,n)$ slightly.
In fact, we use Plancherel theorem and Lemma \ref{fractal} so that, for $q \le 2$, we have
\begin{align*}
\|\widehat f\|_{L^q(d\mu)} \le \mu(\mathbb R^{n+1})^{\frac1q -\frac12} \|\widehat f\|_{L^2(d\mu)} \lesssim \|\widehat f\|_{L^2} \| |\phi_R| \ast d\mu\|_\infty^{\frac12} \le R^{\frac{n+1-\alpha}2}\|f\|_{L^2}.
\end{align*}
Since $\frac{ 3n+1}8-\frac\alpha 4 > \frac{n+1-\alpha}2$ if $ 3 \le n <\alpha \le n+1$, it follows that, for $n <\alpha\le n+1$
\[
\widetilde{s}(\alpha,q,n) = \max \left\{ \frac {n}2-\frac\alpha q, \frac{n+1}4+\frac{n+1-2\alpha}{2q},\frac{n+1-\alpha}{2} \right\}.
\]
This completes the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{wave}.]
We now show that if \eqref{frac} holds for some $s=s_0$, then \eqref{fstrichartz} holds for $s>s_0$. Since $\mu$ has compact support, by finite decomposition we may assume that $\mu$ is supported in $\overline{B(0,1)}$. Recall that the solution $u(x,t)$ is given by
\begin{equation}\label{solution}
u(x,t) = \frac{1}{(2\pi)^{n}}\int_{\mathbb R^n} e^{i x\cdot \xi} \cos (t |\xi|) \widehat f(\xi) d\xi
+ \frac{1}{(2\pi)^{n}} \int_{\mathbb R^n} e^{i x\cdot \xi} \sin (t |\xi|) \frac{\widehat g(\xi)}{|\xi|} d\xi.
\end{equation}
Let $P_j$ be the projection operator $P_j$ for $j\ge1$ given by $
\widehat{ P_j f}(\xi) = \beta(| \xi | / 2^j) \widehat f(\xi),
$
where $\beta$ is a $C_0^\infty(\mathbb R)$ function which is supported in $[1/2, 2]$ and $\sum_{j\in \mathbb Z} \beta (|\xi|/2^j)= 1$ for $\xi\neq0$.
Also we define $P_{\le 0} f$ such that $\widehat{P_{\le 0}f} = \beta_0 \widehat f$, where $\beta_0$ is a $C_0^{\infty}(\mathbb R)$ function such that $\beta_0(|\xi|) = 1- \sum_{j \ge 1} \beta(|\xi| / 2^j)$.
We write
\[u(x,t)=P_{\le 0}( u(\cdot,t))(x) +\sum_{j\ge 1} P_{j}( u(\cdot,t))(x).\]
By Cauchy-Schwarz inequality and Plancherel theorem we have
$|P_{\le 0}( u(\cdot,t))(x)| \lesssim \|f\|_2 +\|g\|_2.$
Since $\mu$ is supported in $\overline{B(0,1)}$, it follows that
\[\| P_{\le 0}( u(\cdot,t)) \|_{L^q(d\mu)} \lesssim \|f\|_2 +\|g\|_2. \]
So, in order to show \eqref{fstrichartz} for $s>s_0$ it is sufficient to show that \eqref{frac} with $s=s_0$ implies
\[\| P_{j}( u(\cdot,t)) \|_{L^q(d\mu)} \lesssim 2^{s_0j} \|f\|_2 + 2^{(s_0-1)j}\|g\|_2.\]
By \eqref{solution} and time reversal symmetry this in turn follows from
\begin{align}\label{decomp2}
\| e^{i t \sqrt{-\Delta}} P_j h \|_{L^q(d\mu)} \le 2^{s_0j} \|h\|_2 .
\end{align}
Since $\mu$ is supported in $\overline{B(0,1)}$, as before, using the smooth function $\eta$ satisfying $ \eta\sim 1$ on $\overline{B(0,1)}$ and $\text{supp$\!$ } \widehat\eta \subset
B(0,\frac12)$, we have
\begin{align*}
\| e^{i t \sqrt{-\Delta}} P_j h \|_{L^q(d\mu)}\sim \| \eta\, e^{i t \sqrt{-\Delta}} P_j h \|_{L^q(d\mu)}.
\end{align*}
Note that the space time Fourier transform of $\eta\, e^{i t \sqrt{-\Delta}} P_j f (x)$ is supported in $\Gamma_{2^j}(1)$. Using \eqref{frac} with $s=s_0$ and Plancherel theorem gives
\begin{align*}
\| e^{i t \sqrt{-\Delta}} P_j h \|_{L^q(d\mu)}\lesssim 2^{s_0j} \|\eta\, e^{i t \sqrt{-\Delta}} P_j h\|_{2} \lesssim 2^{s_0j} \|h\|_2 .
\end{align*}
Hence we get \eqref{decomp} and complete proof.
\end{proof}
\section{Proof of Proposition \ref{prop:necessary}}\label{sec:sharpness}
Now, we obtain lower bounds on $s$ for which \eqref{frac} may hold. This is done by constructing suitable functions and measures.
\
Firstly we show that if \eqref{frac} holds, then
\begin{equation} \label{scaling}
s\ge \frac{n}2-\frac{\alpha}q.
\end{equation}
Let $\mu$ be the measure given by
$d\mu(x)=\chi_{\overline{B(0,1)}}(x)|x|^{\alpha-n-1}dx$ and
$f=\chi_{\Gamma_R(1)}$. Then, $\mu$ is obviously $\alpha$-dimensional and $|\widehat{f}(x)|\gtrsim R^{n}$ if $|x|\le cR^{-1}$
with sufficiently small $c>0.$ Hence \eqref{frac} implies $
R^{n}R^{-\alpha/q}\le CR^sR^\frac {n}2$. So, letting $R \rightarrow \infty$ gives \eqref{scaling}.
\
We now show the second condition:
\begin{align} \label{second_1} s \ge
\begin{cases}
\frac{n+1}{4},\, &\text{ if }\, 0 < \alpha \le 1,
\\
\frac{n+1}4 +\frac{1-\alpha}{2q},\, &\text{ if }\,1 < \alpha \le n, \\
\frac{n+1}4+ \frac{n+1-2\alpha}{2q},\, &\text{ if }\, n < \alpha \le n+1.
\end{cases}
\end{align}
Let us set
\begin{equation}\label{P}
P=\Big\{(\xi_1,\xi'',\xi_{n+1}):
|\xi_1-\xi_{n+1}|\le \frac1{100},\, |{\xi''}|\le \frac{\sqrt{R}}{100},\,
\frac54 R\le |\xi_1+\xi_{n+1}|\le \frac 32 R \Big \}.
\end{equation}
Here $\xi''\in \mathbb R^{n-1}$. Then $P$ is contained in $\Gamma_R(1)$.
Let $x=(x_1, x'', x_{n+1})$ be the dual variables of $(\eta_1,\xi'',
\eta_{n+1})$, where $ \eta _1=(\xi_1+\xi_{n+1})/\sqrt 2$ and $\eta_{n+1}=(\xi_1-\xi_{n+1})/\sqrt 2$.
For a given $\alpha$ let $\ell$ be a positive integer satisfying $\ell-1 <\alpha \le \ell$.
We consider a measure $\mu$ defined by
\[
d \mu (x ) = \prod_{i=1}^{n+1-\ell} d\delta(x_i) |x_{n-\ell+2}|^{\alpha - \ell} d x_{n-\ell+2} d x_{n-\ell+3} \cdots d x_{n+1},
\]
where $\delta$ is the 1-dimensional delta measure and $x = (x_1,\dots,x_{n+1})\in\mathbb R^{n+1}$.
If $n <\alpha\le n+1$, we set
\[
d\mu(x) = |x_1|^{\alpha - n-1} dx_1 \cdots dx_{n+1}.
\]
Then it is easy to see that $\mu$ is $\alpha$-dimensional. In fact, considering the delta measure, $\mu(B(x_0,\rho)) \le C \rho^{\alpha -\ell +1} \cdot \rho^{\ell -1} = C \rho^\alpha$ for any $\rho >0$ and $x_0\in\mathbb R^{n+1}$.
Let $f = \chi_P$. Then we have $\| f \|_2 \lesssim R^{(n+1)/4}$.
{We denote by} $P^*$ the {dual rectangle} of $P$ of which dimensions are $C R^{-1} \times \overset{(n-1) \text{ times}}{C R^{-1/2}\times\cdots \times C R^{-1/2}} \times C $ for some constant $C$.
{It follows that $|\widehat f(x) | \gtrsim R^{1+(n-1)/2}$ on a rectangle of which size is comparable to $P^*$.}
{Hence, \eqref{frac} gives
\[
R^{1+\frac{n-1}2} \mu(P^*)^{\frac1q} \lesssim R^{s + \frac{n+1}4}.
\]
Letting $R \rightarrow \infty$, we obtain \eqref{second_1} because
\[
\mu(P^*) \approx
\begin{cases}
\, 1,\,&\text{ if }\, 0<\alpha \le 1,\\
\, R^{-\frac 12(\alpha - \ell +1) }\times R^{-\frac12(\ell-2)},\,&\text{ if }\, 1< \alpha \le n,\\
\, R^{-(\alpha -n)}\times R^{-\frac{n-1}{2}},\,&\text{ if }\, n< \alpha\le n+1.
\end{cases}
\]}
\
{Finally we show that \eqref{frac} implies
\begin{align} \label{third_1} s\ge
\begin{cases}
\,\frac{n+2}{4}-\frac{\alpha}{4},&\text{ if }\, 1< \alpha \le n,
\\
\, \frac{n+1}2-\frac{\alpha}{2},&\text{ if }\, n < \alpha \le n+1.
\end{cases}
\end{align}}
{The condition \eqref{third_1} can be obtained by} an adaptation of the example in \cite{er3} which was based on
the one due to Wolff \cite{w2}.
First we show \eqref{third_1} for $1 <\alpha \le n$.
Let $\phi_P$ be a Schwartz function supported in $P$, where $P$ is given by \eqref{P}.
Let $N$ be an integer such that $ R^{\frac{\alpha-1}{2}}\sim N$, and let $v_1, \dots, v_N$ be the lattice points which are contained in $B_{n-1}(0,1)$ and separated by distance $\sim R^{-\frac{\alpha-1}{2(n-1)}}$.
Now we consider a Schwartz function $F$ supported in $\Gamma_R(1)$, which is given by
\[ F(\eta_1,\xi'', \eta_{n+1})= N^{-\frac12} \sum_{k=1}^N \phi_P(\xi)e^{iv_k\cdot\xi''}.\]
Here again, $\eta_1$ and $\eta_{n+1}$ are the coordinates defined as in the above.
Note that $\widehat F$ is a sum of translations of $\widehat{\phi_P}$, i.e. $\widehat F = N^{-1/2} \sum_{k=1}^N \widehat{\phi_P}(x_1, x'' - v_k, x_{n+1})$.
Since $1 < \alpha \le n$, we see $R^{-\frac{\alpha-1}{2(n-1)}}\ge R^{-\frac12}$, which implies that $P^* + v_k$'s are almost disjoint.
By rapid decay of {$\widehat{\phi_P}$} outside of $P^*$ we see that $|\widehat
F|\gtrsim N^{-\frac12} R^\frac{n+1}2 \sim R^{\frac{2n+3-\alpha}4}$ on $S
:= \bigcup_{k=1}^N(P^*+ v_k)$. Consider the measure $ d\mu
=R^{\frac{n+2-\alpha}2} \chi_{S}\,dx, $ which is an
$\alpha$-dimensional measure with $\langle\mu\rangle_\alpha\lesssim 1$ when
$1< \alpha\le n$.
In fact, for $R^{-1} \le \rho < R^{-1/2}$ and $x_\circ \in \mathbb R^{n+1}$ it is easy to see
\begin{align*}
\mu(B(x_\circ,\rho)) \le R^{\frac{n+2-\alpha}2} |S\cap B(x_\circ,\rho)|
\le R^{\frac{n+2-\alpha}2} R^{-1} \rho^{n}
\le \rho^{-(n-\alpha)}\rho^{n} = \rho^\alpha.
\end{align*}
The other cases $\rho<R^{-1}$, $R^{-1/2}<\rho \le 1$, and $\rho >1$ can be treated similarly.
Note that $ \int_S d\mu\sim 1$ and $\|F\|_2^2 \le N^{-1} \sum_{k=1}^N \|\phi_P\|_2^2 \le C R^{\frac{n+1}4}$.
Hence \eqref{frac} implies $R^{\frac{n+2-\alpha}2}\lesssim R^s$,
which gives \eqref{third_1}.
Now we proceed to show \eqref{third_1} for $\alpha> n$.
Similarly as before, let $u_1, \dots, u_M$ be the lattice points
which are contained in $B_{n-1}(0,1)$ and separated by about
$R^{-\frac{2\alpha-n-1}{2(n+1)}}\gg R^{-\frac12}$ so that $M\sim
R^{\frac{(n-1)(2\alpha-n-1)}{2(n+1)}}$. Under the same assumption, let $w_1, \dots, w_L$ be
the lattice points which are contained in $(-1/100,1/100)$ and
separated by $R^{-\frac{2\alpha-n-1}{n+1}}$ such that $L\sim
R^{\frac{2\alpha-n-1}{n+1}} $. We set
\[ G(\eta_1, \xi'', \eta_{n+1})= (ML)^{-\frac12} \sum_{k=1}^M\sum_{j=1}^L
\phi_P(\xi)e^{i(u_k\cdot\xi''+w_j \eta_{n+1})}.\]
Hence it follows that
$|\widehat G|\gtrsim (ML)^{-\frac12}
R^\frac{n+1}2=R^{\frac{3n+3-2\alpha}4}$ on $T = \bigcup_{k=1}^M
\bigcup_{j=1}^L (P^*+ u_k+ w_j)$. We now consider an $\alpha$-dimensional measure
$d\mu=R^{n+1-\alpha} \chi_{T}\, dx$.
Noting $ \int_T d\mu\sim 1$ and $\|G\|_2\sim R^{\frac{n+1}4}$, we get the second condition in \eqref{third_1} by letting $R\to \infty$.
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,720 |
Isabella Nielsen (* 21. August 1995) ist eine dänische Badmintonspielerin.
Karriere
Isabella Nielsen stand beim Danish Junior Cup 2011, 2012 und 2013 auf dem Siegerpodest. 2011 siegte sie bei den Israel Juniors. Bei den Greece International 2013 wurde sie Dritte ebenso wie bei den Dutch International 2014 und den Czech International 2014. Im letztgenannten Jahr wurde sie ebenfalls Dritte bei den nationalen Titelkämpfen.
Weblinks
Badmintonspieler (Dänemark)
Däne
Geboren 1995
Frau | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,732 |
James Christian (Chris) Bollwage is the mayor of Elizabeth, New Jersey, the state's fourth-largest city. A lifelong resident of Elizabeth, he was elected into his first term in 1992, and reelected in November 1996, 2000, 2004, 2008, 2012, 2016 and 2020. He has been the president of the New Jersey State League of Municipalities.
He is a 1972 graduate of St. Mary of the Assumption High School. He graduated from Kean University with a bachelor's degree in 1981 and with a master's degree in 1989. Kean University gave Bollwage an honorary degree in 2002. Bollwage is an adjunct professor in the Public Administration Department at Kean University.
Bollwage's vision of the future of Elizabeth included "Go-Green" initiatives, collegiate corridor concept, remodeling of Midtown train station, and expansion of economic development, recreation, housing and transportation. He also unveiled a foreclosure assistance program, a Healthy Elizabeth initiative and camera surveillance network.
See also
List of longest-serving mayors in the United States
References
Living people
New Jersey city council members
New Jersey Democrats
Mayors of Elizabeth, New Jersey
1954 births
Kean University alumni
Kean University faculty
21st-century American politicians | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,128 |
\section*{Abstract}
\begin{quote}
{\footnotesize We classify the harmonic morphisms with one-dimensional fibres
(1) from real-analytic conformally-flat Riemannian manifolds of dimension at least four,
and (2) between conformally-flat Riemannian manifolds of dimensions at least
three.}
\end{quote}
\section*{Introduction}
\indent
Harmonic morphisms between Riemannian manifolds are maps which pull-back (local)
harmonic functions to harmonic functions. By a basic result, a map is a harmonic morphisms
if and only if it is a harmonic map which is horizontally weakly conformal (see \cite{BaiWoo2}\,).\\
\indent
There are, now, several classification results for harmonic morphisms with one-dimensional fibres.
In \cite{Bry}\,, it was proved that there are precisely two types of such harmonic morphisms
from Riemannian manifolds, with constant curvature, of dimension at least four.
This result was generalized, in \cite{PanWoo-d}\,, to Einstein manifolds of dimension
at least five; in dimension four, the situation is different, there appears
a third type of harmonic morphism \cite{Pan-4to3} (see \cite{PanWoo-exm}\,).
Also, in \cite{PanWoo-sd}\,, are classified the `twistorial' harmonic morphisms
with one-dimensional fibres from self-dual four-manifolds.\\
\indent
In this paper, we classify the harmonic morphisms with
one-dimensional fibres from conformally-flat Riemannian manifolds
of dimension at least four. We prove that there are just two types
of such harmonic morphisms, one of which (the `Killing type'), also,
appears in the above mentioned results, whilst the second type is an extension
of the `warped product type', involved in \cite{Bry}\,, \cite{Pan-4to3}\,,
\cite{PanWoo-d} and \cite{PanWoo-sd}\,.\\
\indent
The main result is given in Section \ref{section:main} (Theorem \ref{thm:cf1d}\,) after
a brief review of harmonic morphisms, and conformally-flat Riemannian manifolds, given
in Sections \ref{section:harmorphs} and \ref{section:conf-flat}\,, respectively.\\
\indent
In Section \ref{section:main}\,, we also classify the harmonic morphisms with one-dimensional
fibres between conformally-flat Riemannian manifolds (Corollary \ref{cor:cf1d}\,).
It follows that \emph{the Hopf polynomial map $\mathbb{R}^4\to\mathbb{R}^3$\,,
$(z_1,z_2)\mapsto(|z_1|^2-|z_2|^2,2z_1\overline{z_2})$\,, is,
up to local conformal diffeomorphisms, the only harmonic morphism with one-dimensional fibres
and nonintegrable horizontal distribution between conformally-flat Riemannian manifolds,
of dimensions at least three} (Corollary \ref{cor:cf1dHopf}\,).\\
\indent
I am grateful to John~C.~Wood for useful comments.
\section{Harmonic morphisms with one-dimensional fibres} \label{section:harmorphs}
\indent
In this section we recall a few facts on harmonic morphisms with one-dimensional fibres.\\
\indent
Unless otherwise stated, all the manifolds are assumed to be connected and smooth
and all the maps are assumed to be smooth.
\begin{defn}
A \emph{harmonic morphism (between Riemannian manifolds)} is a map $\varphi:(M^m,g)\to(N^n,h)$
such that if $U$ is an open set of $N$, with $\varphi^{-1}(U)\neq\emptyset$\,, and
$f$ is a harmonic function on $(U,h|_U)$ then $f\circ\varphi$ is a harmonic function
on $(\varphi^{-1}(U),g|_{\varphi^{-1}(U)})$\,.
\end{defn}
\begin{defn}
A map between Riemannian manifolds $\varphi:(M^m,g)\to(N^n,h)$ is \emph{horizontally
weakly conformal} if, for any $x\in M$, either $\dif\!\varphi_x=0$ or,
for any $X,Y\in({\rm ker}\dif\!\varphi_x)^{\perp}$, we have
$h(\dif\!\varphi(X),\dif\!\varphi(Y))=\lambda(x)^2g(X,Y)$ for some positive number $\lambda(x)$\,.\\
\indent
The function $\lambda$\,, extended to be zero over the set of points $x\in M$ where $\dif\!\varphi_x=0$\,,
is the \emph{dilation} of $\varphi$\,. (Note that the dilation $\lambda$ is continuous on $M^m$ whilst
the \emph{square dilation} $\lambda^2$ is smooth on $M^m$.)
\end{defn}
\indent
Let $\varphi:(M^m,g)\to(N^n,h)$ be a horizontally conformal submersion; denote by $\lambda$ its dilation.
Then $\lambda=1$ if and only if $\varphi$ is a \emph{Riemannian submersion}.\\
\indent
If $m=n$ then a surjective map $\varphi:(M^m,g)\to(N^m,h)$ is a horizontally conformal submersion
if and only if it is a \emph{conformal diffeomorphism}. The dilation of a conformal diffeomorphism $\varphi$
is called the \emph{conformality factor} of $\varphi$\,.\\
\indent
The study of harmonic morphisms is based on the following result of B.~Fuglede and T.~Ishihara
(see \cite{BaiWoo2}\,).
\begin{thm} \label{thm:FugIsh}
A map is a harmonic morphism if and only if it is a harmonic map which is horizontally
weakly conformal.
\end{thm}
\indent
As usual, if $\varphi:(M^m,g)\to N^n$ is a submersion we denote by $\mathscr{V}={\rm ker}\dif\!\varphi$
the \emph{vertical distribution} and by $\H=\mathscr{V}^{\perp}$ the \emph{horizontal distribution}.\\
\indent
For horizontally conformal submersions with one-dimensional fibres the condition
of harmonicity can be expressed as follows.
\begin{prop}[\,\cite{Bry}\,, see \cite{Pan}\,, \cite{BaiWoo2}\,] \label{prop:Bryant}
Let $\varphi:(M^{n+1},g)\to(N^n,h)$ be a horizontally conformal submersion with one-dimensional
fibres; $n\geq3$\,. Let $\lambda$ be the dilation of $\varphi$ and let $V$ be the vertical vector field
(well-defined up to sign) such that $g(V,V)=\lambda^{2n-4}$.\\
\indent
The following assertions are equivalent.\\
\indent
\quad{\rm (i)} $\varphi:(M^{n+1},g)\to(N^n,h)$ is a harmonic morphism.\\
\indent
\quad{\rm (ii)} $[V,X]=0$ for any basic (horizontal) vector field $X$\,.\\
\indent
Furthermore, if {\rm (i)} or {\rm (ii)} holds then $\Omega=\dif\!\theta$ is basic,
where $\theta$ is the vertical dual of $V$, characterised by $\theta(V)=1$ and
$\theta|_{\H}=0$\,.
\end{prop}
\indent
Let $\varphi:(M^{n+1})\to(N^n,h)$ be a harmonic morphism with one-dimensional fibres.
With the same notations as in Proposition \ref{prop:Bryant}\,, the vector field $V$
is called the \emph{fundamental vector field}. It is easy to prove that $\Omega=0$
if and only if $\H$ is integrable; also, we have $g=\lambda^{-2}\varphi^*(h)+\lambda^{2n-4}\theta^2$.
It follows that $V$ is a Killing vector field if and only if $\mathscr{V}$ is a Riemannian foliation;
equivalently, $V(\lambda)=0$ (\,\cite{Bry}\,, see \cite{Pan}\,, \cite{BaiWoo2}\,).
If $V$ is Killing then $\varphi$ is called of \emph{Killing type}.
\begin{lem}[\,\cite{Pan-thesis}\,, cf.\ \cite{BaiWoo2}\,] \label{lem:curv}
Let $\varphi:(M^{n+1},g)\to(N^n,h)$ be a harmonic morphism. Let $\lambda$ be its dilation
and let $V$ be the fundamental vector field of $\varphi$\,; we shall denote by
$\sigma=\log\lambda$\,.\\
\indent
Then we have the following relations for the curvature tensors
$R^M$ and $R^N$ of $(M^{n+1},g)$ and $(N^n,h)$\,, respectively:
\begin{equation} \label{e:xvyv} \begin{split}
R^M(X,V,Y,V)=&-\frac{1}{2}(n-2)e^{(2n-4)\sigma}(\Lie_{\H(\grad_{h}\sigma)}h)(X,Y)\\
&-(n-2)e^{(2n-4)\sigma}\{nX(\sigma)Y(\sigma)-|\H(\grad_{h}\sigma)|_{h}^{2}\;h(X,Y)\}\\
&+e^{-2\sigma}\{V(V(\sigma))-(n-1)V(\sigma)^{2}\}h(X,Y)\\
&+\frac{1}{4}\,e^{(4n-6)\sigma}\;h(i_{X}\Omega,i_{Y}\Omega)\:,
\end{split} \end{equation}
\begin{equation} \label{e:xyzv} \begin{split}
R^M(X,&Y,Z,V)=-\frac{1}{2}e^{(2n-4)\sigma}(^{h}\nabla\Omega)(X,Y,Z)\\
&+\frac{1}{2}(n-1)e^{(2n-4)\sigma}\{X(\sigma)\Omega(Y,Z)+Y(\sigma)\Omega(Z,X)-2Z(\sigma)\Omega(X,Y)\}\\
&-e^{-2\sigma}\{X(V(\sigma))-(n-2)X(\sigma)V(\sigma)\}h(Y,Z)\\
&+e^{-2\sigma}\{Y(V(\sigma))-(n-2)Y(\sigma)V(\sigma)\}h(X,Z)\\
&+\frac{1}{2}e^{(2n-4)\sigma}\{\Omega(X,\grad_{h}\sigma)h(Y,Z)-\Omega(Y,\grad_{h}\sigma)h(X,Z)\}\:,
\end{split} \end{equation}
\begin{equation} \label{e:xyzh} \begin{split}
&R^M(X,Y,Z,H)=e^{-2\sigma}\varphi^*(R^N)(X,Y,Z,H)\\
&-\frac14\,e^{(2n-4)\sigma}\{\Omega(H,X)\Omega(Y,Z)+\Omega(H,Y)\Omega(Z,X)-2\Omega(H,Z)\Omega(X,Y)\}\\
&-\frac12\,e^{-2\sigma}V(\sigma)\,\{-\Omega(Y,H)h(X,Z)+\Omega(X,H)h(Y,Z)-\Omega(X,Z)h(Y,H)+\Omega(Y,Z)h(X,H)\}\\
&-e^{-2\sigma}\{X(\sigma)H(\sigma)h(Y,Z)-X(\sigma)Z(\sigma)h(Y,H)-Y(\sigma)H(\sigma)h(X,Z)+Y(\sigma)Z(\sigma)h(X,H)\}\\
&+e^{-2\sigma}\{h(X,Z)h(^h\nabla_Y(\H(\grad_h\sigma)),H)-h(Y,Z)h(^h\nabla_X(\H(\grad_h\sigma)),H)\\
&+h(Y,H)h(^h\nabla_X(\H(\grad_h\sigma)),Z)-h(X,H)h(^h\nabla_Y(\H(\grad_h\sigma)),Z)\}\\
&-e^{-2\sigma}\{h(X,Z)h(Y,H)-h(X,H)h(Y,Z)\}\{e^{(-2n+2)\sigma}\,V(\sigma)^2+|\H(\grad_h\sigma|_h^2\}\:,
\end{split} \end{equation}
where $X,Y,Z,H$ are horizontal and $^{h}\nabla$ denotes the Levi-Civita connection of $(M,h)$.
\end{lem}
\begin{rem}
See \cite{BaiWoo2} and the references therein for more information on harmonic morphisms
between Riemannian manifolds and, in particular, for the notion of $p$-harmonic morphism.
Also, see \cite{LouPan,LouPan-II,Pan-Deva} for harmonic morphisms in the more general
setting of Weyl geometry.
\end{rem}
\section{Conformally-flat Riemannian manifolds} \label{section:conf-flat}
\indent
Firstly, we recall (see \cite{Laf-Weyl}\,) the definition of the Weyl tensor of a Riemannian
manifold.\\
\indent
Let $(M^m,g)$ be a Riemannian manifold. For $h$ and $k$ sections of $\odot^2(T^*M)$
(that is, $h$ and $k$ are symmetric covariant tensor fields
of degree two on $M^m$), we shall denote by $h\owedge k$ the section of
$\odot^2(\Lambda^2(T^*M))$ defined by
\begin{equation*}
\begin{split}
(h\owedge k)(T,X,Y,Z)=&h(T,Y)k(X,Z)+h(X,Z)k(T,Y)\\
&-h(T,Z)k(X,Y)-h(X,Y)k(T,Z)\;,
\end{split}
\end{equation*}
for any $T,X,Y,Z\in TM$.\\
\indent
If $S$ is a (1,3)-tensor field on $(M,g)$ then we shall denote by the same symbol $S$
the (0,4)-tensor field defined by $S(T,X,Y,Z)=-g(S(T,X,Y),Z)$\,, for any $T,X,Y,Z\in TM$.\\
\indent
The \emph{Weyl (curvature) tensor} of $(M^m,g)$ is the (1,3)-tensor field $W$ characterised
by the following two conditions:\\
\indent
\quad1) $\trace(X\mapsto W(X,Y)Z)=0$\,, for any $Y,Z\in TM$,\\
\indent
\quad2) $R=g\owedge r+W$ for some (necessarily unique) section $r$ of $\odot^2(T^*M)$\,,
where $R$ is the curvature tensor of $(M^m,g)$\,.\\
\indent
The Weyl tensor is conformally invariant; that is, if we denote by $W^g$ the Weyl tensor
of $(M^m,g)$ then $W^{\lambda^2g}=W^g$, for any positive function $\lambda$ on $M^m$.\\
\indent
The Riemannian manifold $(M^m,g)$ is called \emph{(locally) conformally-flat} if
for each point of $M^m$ there exists an open neighbourhood $U$ and a conformal diffeomorphism
$\varphi$ from $U$ onto some open set of $\mathbb{R}^m$ (endowed with its canonical Riemannian metric);
the local coordinates on $U$ induced by $\varphi$ are called \emph{flat}.\\
\indent
{}From Liouville's theorem on local conformal diffeomorphisms between Euclidean spaces (see \cite{BaiWoo2}\,)\,,
it follows easily that if $(M^m,g)$ is conformally-flat then $M^m$ is real-analytic in flat
local coordinates $(m\geq2)$\,.\\
\indent
The following theorem is due to H.~Weyl (see \cite{Laf-Weyl}\,).
\begin{thm} \label{thm:Weyl}
A Riemannian manifold, of dimension at least four, is conformally-flat if and only if its
Weyl tensor is zero.
\end{thm}
\noindent
(See \cite{Laf-Weyl} for the case when the dimension is less than four.)\\
\indent
We do not imagine that the following result is new.
\begin{prop} \label{prop:Wxyxy}
Let $(M^m,g)$ be a Riemannian manifold, $m\geq4$\,. The following assertions are equivalent.\\
\indent
{\rm (i)} $(M^m,g)$ is conformally-flat.\\
\indent
{\rm (ii)} $R(X,Y,X,Y)=0$ for any $X,Y\in TM$ spanning an isotropic space on $(M,g)$\,,
where $R$ is the curvature tensor of $(M,g)$\,, and $TM$ now denotes the complexified
tangent bundle.
\end{prop}
\begin{proof}
Clearly, assertion (ii) is equivalent to $W(X,Y,X,Y)=0$ for any $X,Y\in TM$ spanning an
isotropic space on $(M,g)$\,, where $W$ is the Weyl tensor of $(M,g)$\,. Therefore,
by Theorem \ref{thm:Weyl}\,, we have (i)$\Longrightarrow$(ii)\,.\\
\indent
Suppose that (ii) holds and let $(X_1,\ldots,X_m)$ be an orthonormal frame on $(M^m,g)$\,.
Then for any distinct $i,j,k,l\in\{1,\ldots,m\}$ we have
$$W(X_i\pm{\rm i}X_j,X_k+{\rm i}X_l,X_i\pm{\rm i}X_j,X_k+{\rm i}X_l)=0\;.$$
This is equivalent to the following two relations
\begin{equation} \label{e:Wxyxy1}
W(X_i,X_k+{\rm i}X_l,X_i,X_k+{\rm i}X_l)=W(X_j,X_k+{\rm i}X_l,X_j,X_k+{\rm i}X_l)\;,
\end{equation}
\begin{equation} \label{e:Wxyxy1'}
W(X_i,X_k+{\rm i}X_l,X_j,X_k+{\rm i}X_l)=0\;.
\end{equation}
\indent
Also, by applying condition (2) of the definition of the Weyl tensor, we obtain
\begin{equation} \label{e:Wxyxy2}
\sum_{r=1}^mW(X_r,X_k+{\rm i}X_l,X_r,X_k+{\rm i}X_l)=0\;.
\end{equation}
\indent
{}From \eqref{e:Wxyxy1} and \eqref{e:Wxyxy2}\,, it follows that
$W(X_j,X_k+{\rm i}X_l,X_j,X_k+{\rm i}X_l)=0$ and, hence, $W_{jkjk}=W_{jljl}$\,,
for any distinct $j,k,l\in\{1,\ldots,m\}$\,. Therefore, for any distinct
$i,j\in\{1,\ldots,m\}$\,, we have
$$(m-1)W_{ijij}=\sum_{r=1}^mW_{irir}=0\;.$$
\indent
{}From \eqref{e:Wxyxy1'} we obtain that, for any distinct $i,j,k,l\in\{1,\ldots,m\}$\,,
we have
\begin{equation} \label{e:Wxyxy3}
\begin{split}
W_{ikjk}&=W_{iljl}\;,\\
W_{ikjl}&=-W_{iljk}\;.
\end{split}
\end{equation}
\indent
The first relation of \eqref{e:Wxyxy3} implies $W_{ikjk}=0$\,, whilst from the
second relation of \eqref{e:Wxyxy3} and the algebraic Bianchi identity
it follows quickly that $W_{ijkl}=0$\,, for any distinct $i,j,k,l\in\{1,\ldots,m\}$\,.\\
\indent
Thus, if (ii) holds then $W=0$ which, by Theorem \ref{thm:Weyl}\,, is equivalent to (i)\,.
\end{proof}
\indent
If $\H$ is a distribution on a Riemannian manifold $(M^m,g)$ we shall denote by $I^{\H}$
the integrability tensor of $\H$, which is the $\mathscr{V}$-valued horizontal two-form on $M^m$ defined by
$I^{\H}\!(X,Y)=-\mathscr{V}[X,Y]$\,, for any horizontal vector fields $X$ and $Y$, where $\mathscr{V}=\H^{\perp}$.\\
\indent
Next, we prove the following:
\begin{prop} \label{prop:isoI}
Let $\varphi:(M^m,g)\to(N^n,h)$ be a horizontally conformal submersion between
conformally-flat Riemannian manifolds.\\
\indent
Then $g\bigl(I^{\H}\!(X,Y),I^{\H}\!(X,Y)\bigr)=0$\,, for any horizontal vectors $X$ and $Y$
spanning an isotropic space on $(M^m,g)$\,.
\end{prop}
\begin{proof}
As both the hypothesis and the conclusion are conformally-invariant, we may suppose
that $\varphi:(M^m,g)\to(N^n,h)$ is a Riemannian submersion. Then the proof
follows easily from Proposition \ref{prop:Wxyxy} and the following
well-known relation of B.~O'Neill (see \cite{BaiWoo2}\,):
\begin{equation*}
R^M(X,Y,X,Y)=\varphi^*(R^N)(X,Y,X,Y)-\frac34\,g(\mathscr{V}[X,Y],\mathscr{V}[X,Y]\bigr)\;,
\end{equation*}
for any horizontal vector fields $X$ and $Y$.
\end{proof}
\begin{cor} \label{cor:isoI}
Any horizontally conformal submersion, with fibres of dimension at most two,
between conformally-flat Riemannian manifolds has integrable horizontal distribution,
if the codomain has dimension at least four.
\end{cor}
\begin{proof}
Let $\varphi:(M^m,g)\to(N^n,h)$ be a horizontally conformal submersion between
conformally-flat Riemannian manifolds, $m\geq n\geq4$\,.\\
\indent
Let $x\in M$ and let $E\subseteq T_xM$ be an oriented four-dimensional subspace.
{}From Proposition \ref{prop:isoI}\,, it follows that $I^{\H}_x:\Lambda^2_+E\to\mathscr{V}_x$
is conformal, where $\Lambda^2_+E$ is the space of self-dual bivectors on $(E,g|_E)$\,.
As $\Lambda^2_+E$ is three-dimensional, we obtain that either $I^{\H}_x=0$
or $\dim(\mathscr{V}_x)\geq3$\,.
\end{proof}
\indent
We end this section with an application of Corollary \ref{cor:isoI}\,.\\
\indent
An \emph{almost CR-structure}, on a manifold $M^m$, is a section $J$ of
${\rm End}(\H)$ such that $J^2=-\,{\rm Id}_{\H}$\,, where $\H$ is some distribution
on $M^m$. Obviously, $J$ is determined by its eigenbundle corresponding to $-{\rm i}$
(or ${\rm i}$)\,. Furthermore, a subbundle $\mathscr{F}$ of the complexified tangent bundle
of $M^m$ is the eigenbundle corresponding to $-{\rm i}$ of an almost CR-structure
on $M^m$ if and only if $\mathscr{F}\cap\overline{\mathscr{F}}=\{0\}$\,.\\
\indent
Let $J$ be an almost CR-structure on $M^m$ and let $\mathscr{F}$ be its eigenbundle
corresponding to $-{\rm i}$\,; $J$ is called \emph{integrable} if for any $X,\,Y\in\Gamma(\mathscr{F})$
we have $[X,Y]\in\Gamma(\mathscr{F})$\,. A \emph{CR-structure} is an integrable almost CR-structure
(see \cite{LouPan-II}\,). If $M^m$ is endowed with a Riemannian metric $g$ then
$\mathscr{F}$ is isotropic, with respect to $g$\,, if and only if $J$ is orthogonal,
with respect to (the Riemannian metric induced on $\H$ by) $g$\,.\\
\indent
For example, any oriented two-dimensional distribution $\mathscr{V}$, on a Riemannian manifold
$(M^m,g)$\,, determines two orthogonal CR-structures on $(M^m,g)$\,; at each point
$x\in M$, these are given by the rotations of angles $\pm\,\pi/2$ on $\mathscr{V}_x$
(cf.\ \cite{Woo-4d}\,).\\
\indent
Let $\varphi:(M^{n+2},g)\to(N^n,h)$ and $\psi:(N^n,h)\to(P^2,k)$ be
horizontally conformal submersions, $n\geq2$\,. Let $\mathscr{V}={\rm ker}\dif\!\varphi$\,,
$\H=\mathscr{V}^{\perp}$ and let $\mathscr{K}\subseteq\H$ be the horizontal lift of
$({\rm ker}\dif\!\psi)^{\perp}$\,. Assume $\mathscr{V}$ and $P^2$ oriented and orient
$\mathscr{K}$ such that the isomorphism $\mathscr{K}=(\psi\circ\varphi)^*(TP)$
to be orientation preserving.\\
\indent
Then the positive/negative orthogonal CR-structures determined by $\mathscr{V}$ and the
positive orthogonal CR-structure determined by $\mathscr{K}$ sum up to give orthogonal
almost CR-structures $J_{\pm}^{\varphi,\psi}$ on $(M^{n+2},g)$\,. Obviously,
if we endow $(P^2,k)$ with its positive Hermitian structure $J^P$ then
$\psi\circ\varphi:(M^{n+2},J_{\pm}^{\varphi,\psi})\to(P^2,J^P)$ is holomorphic;
that is, the differential of $\psi\circ\varphi$ intertwines $J_{\pm}^{\varphi,\psi}$
and $J^P$.\\
\indent
We call $J_{\pm}^{\varphi,\psi}$ the \emph{positive/negative almost CR-structures associated
to $\varphi$ and $\psi$}\,.
\begin{prop} \label{prop:n-Wood}
Let $\varphi:(M^{n+2},g)\to(N^n,h)$ be an $n$-harmonic morphism from
a Riemannian manifold of constant curvature to a conformally-flat Riemannian manifold,
and let $\psi:(N^n,h)\to(P^2,k)$ be a horizontally conformal submersion
which is real-analytic in flat local coordinates, $(n\geq4)$\,; assume
$\mathscr{V}\,(={\rm ker}\dif\!\varphi)$ and $P^2$ oriented.
Denote by $J_{\pm}^{\varphi,\psi}$ the almost CR-structures associated to $\varphi$ and $\psi$\,.\\
\indent
Then either $J_+^{\varphi,\psi}$ or $J_-^{\varphi,\psi}$ is integrable and parallel along the
fibres of $\varphi$\,.
\end{prop}
\begin{proof}
As the $n$-Laplacian on $n$-dimensional riemannian manifolds is conformally invariant,
we may suppose $(N^n,h)$ real-analytic, in flat local coordinates. Therefore, also,
$\varphi$ is real-analytic.\\
\indent
Note that $\varphi$ has minimal fibres \cite{Lou-p}\,. Also, by
Corollary \ref{cor:isoI}\,, the distribution $\H$ is integrable.\\
\indent
Let $\mathscr{F}_{\pm}$ be the eigenbundles of $J_{\pm}^{\varphi,\psi}$ corresponding to $-{\rm i}$\,.
Let $Y$ be a basic vector field which locally generates $\mathscr{F}_+\cap\mathscr{F}_-$\,.
Then $Y$ is isotropic. Moreover, from the fact that $\varphi$ and $\psi$ are
horizontally conformal, it follows that $\nabla_YY$ is proportional to $Y$,
where $\nabla$ is the Levi-Civita connection of $(M^{n+2},g)$\,.\\
\indent
There exists an isotropic vertical vector field $U$ such that $\mathscr{F}_+$ and $\mathscr{F}_-$
are, locally, generated by $\{U,Y\}$ and $\{\overline{U},Y\}$\,, respectively;
we may suppose that $g(U,\overline{U})=1$\,. As $Y$ is basic, $[U,Y]$ and $[\overline{U},Y]$
are vertical. Thus, $\mathscr{F}_+$ and $\mathscr{F}_-$ are integrable if and only if $g([U,Y],U)=0$
and $g([\overline{U},Y],\overline{U})=0$\,, respectively.\\
\indent
As $(M^{n+2},g)$ has constant curvature, $R^M(U,Y,Y,\overline{U})=0$\,. On the other hand,
a straightforward calculation shows that (cf.\ \cite{LouPan}\,)
\begin{equation} \label{e:n-Wood}
R^M(U,Y,Y,\overline{U})=g([U,Y],U)g([\overline{U},Y],\overline{U})\;.
\end{equation}
\indent
The proof follows.
\end{proof}
\begin{rem}
1) If $n=2$ then the conclusion of Proposition \ref{prop:n-Wood} holds
under the assumption that $(M^4,g)$ is Einstein \cite{Woo-4d}
(see \cite{LouPan} for a generalization of this result to Einstein--Weyl spaces).\\
\indent
2) Proposition \ref{prop:n-Wood}\,, also, holds under the assumption that
$\varphi$ is a real-analytic horizontally conformal
submersions such that the mean curvature of $\mathscr{V}$ takes values
in $(\mathscr{V}\oplus\mathscr{K})^{\perp}$\,. Also, note that, in the proof, we have not use
the fact that $\dif\!\varphi(\mathscr{K})^{\perp}\,\bigl(={\rm ker}\dif\!\psi\bigr)$ is integrable.
\end{rem}
\section{The main result} \label{section:main}
\indent
This section is devoted to the following result and its consequences.
\begin{thm} \label{thm:cf1d}
Let $\varphi:(M^{n+1},g)\to(N^n,h)$ be a harmonic morphism between Riemannian manifolds,
$n\geq3$\,; denote by $\lambda$ the dilation of $\varphi$\,.\\
\indent
If $(M^{n+1},g)$ is real-analytic and conformally-flat then \emph{either}\\
\indent
\quad{\rm (i)} $\varphi$ is of Killing type, \emph{or}\\
\indent
\quad{\rm (ii)} the horizontal distribution of $\varphi$ is integrable and its
leaves endowed with the metrics induced by $\lambda^{-2n+4}g$ have constant curvature.
\end{thm}
\begin{proof}
\indent
By a result of \cite{PanWoo-d}\,, at least away of the critical points (which may
occur only if $n=3$\,, see \cite{BaiWoo2}\,), we have
$\varphi:(M^{n+1},g)\to(N^n,h)$ real-analytic.\\
\indent
As the dimension of the intersection of (the complexification of) $\H$
with any isotropic two-dimensional space, on $(M^{n+1},g)$\,, is at least $1$\,,
Proposition \ref{prop:Wxyxy} implies that $(M^{n+1},g)$ is
conformally-flat if and only if, for any $U\in\Gamma(\mathscr{V})$ and $X,Y\in\Gamma(\H)$
with $g(U,U)=g(X,X)$\,, $g(X,Y)=0$\,, $g(Y,Y)=0$\,, we have
$R^M(U\pm{\rm i}X,Y,U\pm{\rm i}X,Y)=0$\,; equivalently,
\begin{equation} \label{e:cf1d1}
\begin{split}
R^M(U,Y,U,Y)&=R^M(X,Y,X,Y)\\
R^M(U,Y,X,Y)&=0\;.
\end{split}
\end{equation}
{}From \eqref{e:xyzv}\,, it follows quickly that the second relation of \eqref{e:cf1d1}
is equivalent to
\begin{equation} \label{e:cf1d1second}
(^h\nabla_Y\Omega)(X,Y)+3(n-1)Y(\sigma)\Omega(X,Y)=0\;.
\end{equation}
Thus, by assuming $X$ and $Y$ basic and using Proposition \ref{prop:Bryant}\,,
we obtain
\begin{equation} \label{e:cf1d2}
Y(V(\sigma))\Omega(X,Y)=0\;,
\end{equation}
where $V$ is the fundamental vector field of $\varphi$\,.\\
\indent
Next, we shall use the first relation of \eqref{e:cf1d1}\,. For this, we assume
$X$ and $Y$ basic with $g(X,X)=e^{-2\sigma}$ (equivalently, $h(X,X)=1$\,), and
$U=e^{-(n-1)\sigma}V$ (so that, $g(U,U)=g(X,X)$\,). Thus, the first relation
of \eqref{e:cf1d1} becomes $$e^{-(2n-2)\sigma}R^M(V,Y,V,Y)=R^M(X,Y,X,Y)$$
which, by applying \eqref{e:xvyv} and \eqref{e:xyzh}\,, is equivalent to
\begin{equation} \label{e:cf1d3}
\begin{split}
R^N(X,Y,X,Y)=-(n-1)h(^h&\nabla_Y(\H(\grad_h\sigma)),Y)-(n-1)^2Y(\sigma)^2\\
&+\frac14\,e^{(2n-2)\sigma}\bigl\{h(i_Y\Omega,i_Y\Omega)+3\Omega(X,Y)^2\bigr\}\;,
\end{split}
\end{equation}
where we have denoted by the samy symbol $R^N$ and its pull-back by $\varphi$ to $M^{n+1}$.\\
\indent
We may assume that $Y$ is the horizontal lift of an isotropic geodesic (local)
vector field on (the complexification of) $(N^n,h)$\,; equivalently, $^h\nabla_YY=0$\,.
Then \eqref{e:cf1d3} becomes
\begin{equation} \label{e:cf1d3'}
\begin{split}
R^N(X,Y,X,Y)=-(n-1)&Y(Y(\sigma))-(n-1)^2Y(\sigma)^2\\
&+\frac14\,e^{(2n-2)\sigma}\bigl\{h(i_Y\Omega,i_Y\Omega)+3\Omega(X,Y)^2\bigr\}\;.
\end{split}
\end{equation}
\indent
As $R^N(X,Y,X,Y)$ is basic, from \eqref{e:cf1d2} and \eqref{e:cf1d3'} it easily follows
that either $\Omega=0$ or
\begin{equation} \label{e:cf1d4}
V(\sigma)\bigl\{h(i_Y\Omega,i_Y\Omega)+3\Omega(X,Y)^2\bigr\}=0\;.
\end{equation}
\indent
Now, from $\Omega\neq0$ it follows that there exist $Y\in\H$ isotropic
and $X\in Y^{\perp}\cap\H$ such that the second factor of the left hand side of
\eqref{e:cf1d4} is not zero. Thus, we have proved that either $\Omega=0$ (equivalently,
$\H$ is integrable) or $V(\sigma)=0$ (equivalently, $\varphi$ is of Killing type).\\
\indent
Next, we study the case $\Omega=0$\,. Then \eqref{e:cf1d1second} (and hence, also, the
second relation of \eqref{e:cf1d1}\,) is automatically satisfied, whilst \eqref{e:cf1d3}
is equivalent to
\begin{equation} \label{e:cf1d5}
R^N(X,Y,X,Y)=\,^h\nabla(\dif^{\H}\!u)(Y,Y)-(\dif^{\H}\!u)(Y)^2\;,
\end{equation}
where $u=-(n-1)\sigma$ and, recall that, $X$ and $Y$ are basic with $h(X,X)=1$\,, $h(X,Y)=0$
and $h(Y,Y)=0$\,.\\
\indent
Let $h_1=e^{2u}h|_{\H}=e^{(-2n+4)\sigma}g|_{\H}$\,.\\
\indent
We have proved that, if $\H$ is integrable, \eqref{e:cf1d1} is equivalent
to the fact that the curvature tensor $R^P$ of any leaf $P$ of $\H$\,, endowed with the
metric induced by $h_1$\,, satisfies $R^P(X,Y,X,Y)=0$\,.\\
\indent
It follows that if $\H$ is integrable then $h_1$ induces a conformally-flat Einstein metric
on each leaf of $\H$\,; equivalently, $h_1$ induces a metric of constant curvature
on each leaf of $\H$\,. The proof is complete.
\end{proof}
\begin{exm}
Let $(N^n,h)$ be $\mathbb{R}^n$, endowed with the canonical metric, and let
$$M^{n+1}=\bigl\{(t,x)\in\mathbb{R}\times\mathbb{R}^n\,\big{|}\,|tx|<1\bigr\}\;,$$
where $|\cdot|$ denotes the Euclidean norm on $\mathbb{R}^n$.\\
\indent
Define $\lambda:M^{n+1}\to(0,\infty)$ by $\lambda(t,x)=(1-|tx|^2)^{\frac{1}{n-1}}$, $(t,x)\in M^{n+1}$,
and let $g=\lambda^{-2}h+\lambda^{2n-4}\dif\!t^2$.\\
\indent
Then $\varphi:(M^{n+1},g)\to(N^n,h)$\,, $(t,x)\mapsto x$\,, is a harmonic morphism which
satisfies assertion (ii) of Theorem \ref{thm:cf1d}\,; in particular, $(M^{n+1},g)$
is conformally-flat, $(n\geq3)$\,. Furthermore,
$\varphi$ is neither of Killing type nor its fibres are geodesics.
\end{exm}
\begin{rem}
If $n=3$ then Theorem \ref{thm:cf1d} holds, also, in the complex-analytic category.
Indeed, the only point in the proof
of Theorem \ref{thm:cf1d} where it is essential for $\varphi$
to be `real' is when we deduce from $\Omega\neq0$ that there exist $Y\in\H$
isotropic and $X\in Y^{\perp}\cap\H$ such that the second factor of the left
hand side of \eqref{e:cf1d4} is not zero. But, if $n=3$ and $h(X,X)=1$ then
$$h(i_Y\Omega,i_Y\Omega)+3\Omega(X,Y)^2=4\Omega(X,Y)^2\;,$$
which, also, in the complex-analytic category, is not zero, for suitable
choices of $X$ and $Y$, if $\Omega\neq0$\,.
\end{rem}
\indent
Next, we discuss the case when both the domain and codomain, of a harmonic
morphism with one-dimensional fibres, are conformally-flat; the notations
are as in Section 1.
\begin{cor} \label{cor:cf1d}
Let $\varphi:(M^{n+1},g)\to(N^n,h)$ be a submersive harmonic morphism
with connected one-dimensional fibres, $(n\geq3)$\,.\\
\indent
The following assertions are equivalent.\\
\indent
\quad{\rm (i)} $(M^{n+1},g)$ and $(N^n,h)$ are conformally-flat.\\
\indent
\quad{\rm (ii)} One of the following assertions holds:\\
\indent
\quad\quad{\rm (iia)} $\varphi$ is of Killing type, $n=3$\,, and, up to a
homothety, $\Omega$ is the volume form of a Riemannian foliation by
geodesic surfaces, of sectional curvature $1$\,, on $(N^3,\lambda^{-4}h)$\,.\\
\indent
\quad\quad{\rm (iib)} The horizontal distribution of $\varphi$ is integrable and its
leaves endowed with the metrics induced by $\lambda^{-2n+4}g$ have constant curvature.
\end{cor}
\begin{proof}
If $n\geq4$ this follows from Corollary \ref{cor:isoI} and the proof
of Theorem \ref{thm:cf1d}\,.\\
\indent
Assume $n=3$\,. Then by the proof of Theorem \ref{thm:cf1d}\,, if $(M^4,g)$ is conformally-flat,
on each connected component of a dense open subset of $M^4$,
either $\varphi$ is of Killing type or (iib) holds.\\
\indent
If $\varphi$ is of Killing type then there exist Weyl connections $D_{\pm}$
on $(N^3,[h])$ such that $\varphi:(M^4,[g])\to(N^3,[h],D_{\pm})$ is $\pm$twistorial,
in the sense of \cite[Example 4.8]{LouPan-II} (cf.\ \cite{PanWoo-sd}\,). Furthermore,
$(M^4,g)$ is conformally-flat if and only if both $D_{\pm}$ are Einstein--Weyl
(see \cite{PanWoo-sd} and the references therein).
Also, if $(N^3,h)$ is conformally-flat then $D_{\pm}$ are Einstein--Weyl
if and only if, locally, $D_{\pm}$ are the Levi-Civita connections of
constant curvature representatives $h_{\pm}$ of $[h]$
(see \cite{Cal-F}\,).\\
\indent
We claim that if $n=3$ and $\varphi$ is of Killing type then, with the same notations
as above, the following assertions are equivalent:\\
\indent
\quad(a) Up to a homothety, $\Omega$ is the volume form of a Riemannian foliation
by geodesic surfaces, of sectional curvature $1$\,, on $(N^3,\lambda^{-4}h)$\,.\\
\indent
\quad(a$'$) $D_{\pm}$ are, locally, the Levi-Civita connections of constant curvature
representatives $h_{\pm}$ of $[h]$\,, and $D_+\neq D_-$\,.\\
\indent
Indeed, if $\varphi$ is of Killing type then, by replacing $g$ and $h$ with $\lambda^{-2}g$
and $\lambda^{-4}h$\,, respectively, we may suppose that $\varphi$ is a Riemannian submersion
with geodesic fibres. Then the Lee forms $\alpha_{\pm}$ of $D_{\pm}$\,, with respect to $h$\,,
are given by $\alpha_{\pm}=\pm*_h\Omega$ (see \cite{LouPan-II}\,, \cite{PanWoo-sd}\,),
where $*_h$ is the Hodge $*$-operator of $h$\,,
with respect to some local orientation, and we have denoted by the same symbol $\Omega$ and
the two-form on $N^3$ whose pull-back by $\varphi$ is $\Omega$\,. Hence, if (a$'$) holds then
$h_{\pm}=e^{\pm2u}h$ where $u$ is characterised by $\dif\!u=*_h\Omega$\,;
in particular, $u$ is a harmonic (local) function on $(N^3,h)$\,.\\
\indent
It follows that (a$'$) is equivalent to the following assertion:\\
\indent
\quad(a$''$) Locally, there exists a nonconstant function $u$ on $N^3$ such that
$$\dif\!u=*_h\Omega\,,\;(\nabla^h\!\dif\!u)(Y,Y)=0\,,\;{\rm Ric}^h(Y,Y)=-\dif\!u(Y)^2\,,$$
for any isotropic vector $Y$ on $(N^3,h)$\,, where $\nabla^h$
is the Levi-Civita connection of $(N^3,h)$ and ${\rm Ric}^h$ is the Ricci tensor
of $(N^3,h)$\,.\\
\indent
Now, by applying, for example, Lemma \ref{lem:curv}\,, we obtain that assertion (a$''$)
is equivalent to the following:\\
\indent
\quad(a$'''$) Locally, there exists a nonconstant function $u$ on $N^3$ such that
$$\dif\!u=*_h\Omega\,,\;\nabla^h\!\dif\!u=0$$
and the level surfaces of $u$ have sectional curvature equal to $|\!\dif\!u|^2$.\\
\indent
The proof of (a)$\iff$(a$'$) follows.\\
\indent
We have thus proved that (ii)$\Longrightarrow$(i)\,, and if (i) holds then, also, (ii) holds
on each connected component of a dense open subset
of $M^4$.\\
\indent
To complete the prof of (i)$\Longrightarrow$(ii)\,, define a connection $\nabla$ on $\H$ by
$$\nabla_EX=\H\nabla^{\lambda^{-2}g}_{\H E}X+\H[\mathscr{V} E,X]$$
for any vector field $E$ and horizontal vector field $X$\,, where
$\nabla^{\lambda^{-2}g}$ is the Levi-Civita connection of $(M^4,\lambda^{-2}g)$\,.\\
\indent
If we assume (i) then, from the fact that (ii) holds on each connected component of a dense open subset
of $M^4$, it follows quickly that $\nabla\Omega=0$\,. Therefore either $\Omega$ is nowhere zero or $\Omega=0$ on $M^4$.
The proof is complete.
\end{proof}
\begin{exm} \label{exm:Hopfmap}
Let $\pi:\mathbb{R}^4\to\mathbb{R}^3$ be the \emph{Hopf polynomial map} defined by
$\pi(z_1,z_2)=(|z_1|^2-|z_2|^2,2z_1\overline{z_2})$\,, for any $(z_1,z_2)\in\mathbb{R}^4\,(=\mathbb{C}\,^{\!2})$\,.\\
\indent
Then $\pi|_{\mathbb{R}^4\setminus\{0\}}$ satisfies assertion (iia) of Corollary \ref{cor:cf1d}\,.
\end{exm}
\indent
We end with the following consequence of Corollary \ref{cor:cf1d}\,.
\begin{cor} \label{cor:cf1dHopf}
The Hopf polynomial map $\pi:\mathbb{R}^4\to\mathbb{R}^3$ is, up to local conformal diffeomorphisms with basic conformality factors,
the only harmonic morphism with one-dimensional fibres and nonintegrable horizontal distribution between
conformally-flat Riemannian manifolds, of dimensions at least three.
\end{cor}
\begin{proof}
This follows from the fact that any harmonic morphism which satisfies assertion (iia) of
Corollary \ref{cor:cf1d} is, locally, the Hopf polynomial map $\pi:\mathbb{R}^4\to\mathbb{R}^3$, up to conformal
changes with basic factor.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
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Older and Wiser
Privacy Confidentiality Security Data Protection Policy
Psychotherapy Counselling
NLP Therapy- What is it?
NLP Pre-suppositions
Gender and Sexual Diversity (GSD)
Life-Goals Blog (Ideas and Suggestions)
Workplace Sexual Harassment Can Cause Depression
Author: Ann Ingham
Categories: counselling
A new study looking at the impact of sexual harassment in the workplace has found that when someone is harassed by a colleague or boss the impact on mental health is worse than when the harassment comes from a customer or client.
The Independent reported on the Danish study, which surveyed more than 7,600 people working for over 1,000 different companies. It noted that over one per cent suffered sexual harassment from a colleague, with 2.4 per cent experiencing the same treatment from someone else they dealt with through their job.
Researchers used the Major Depression Inventory (MDI) to assess the impact of sexual harassment on mental health. By filling in a questionnaire each person is given a score, where 20 indicates mild depression and 30 or more is serious depression.
If a person experienced harassment from a client or customer, their score on this scale increased by 2.05 points. However, when the harassment came from a colleague, the score increased by 4.5 points.
Dr Ida Madsen, from the National Research Centre for the Working Environment in Denmark, said that they were surprised by the results, as this is not a distinction they've seen in previous studies.
"This is important as some workplaces, for example in person-related work, may have an attitude that dealing with sexual harassment by clients or customers is 'part of the job'," she stated.
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Amy Wilson Carmichael (* 16. Dezember 1867 in Millisle, County Down, Nordirland; † 18. Januar 1951) war eine Indien-Missionarin und Autorin. Sie eröffnete ein Waisenhaus und gründete eine Missionsstation in Dohnavur. 55 Jahre lang arbeitete sie in Indien ohne Heimaturlaub und schrieb viele Bücher über ihre missionarische Arbeit.
Frühes Leben
Amy Wilson Carmichael wurde in dem kleinen Dorf Millisle, County Downin Nordirland, geboren, als Älteste von sieben Geschwistern. Ihre Eltern waren David und Catherine Carmichael, gläubige Presbyterianer. Aus ihrer Kindheit wurde erzählt, dass sie oft betete, Jesus möge ihr blaue statt ihrer braunen Augen geben. Sie war sehr enttäuscht, als sie keine blauen Augen bekam. Als Erwachsene jedoch erkannte sie, dass sie viel besser von den Indern akzeptiert wurde, weil sie auch deren Augenfarbe hatte, und dass es ihr mit blauen Augen viel schwerer geworden wäre, dort Fuß zu fassen. Amys Vater starb, als sie 18 war.
Carmichael war Gründerin der Welcome Evangelical Church in Belfast. Die Geschichte der Gemeinde begann damit, dass sie Mitte der 1880er Jahre eine Sonntagmorgen-Klasse für "Shawlies" sammelte; das waren Arbeitermädchen, die Schals statt Hüte trugen (Mill girls). Sie trafen sich im Gemeindesaal der Presbyterianischen Rosemary-Street-Kirche. Diese Arbeit erwies sich als sehr erfolgreich. Die Zusammenkünfte der "Shawlies" wuchsen so sehr, dass Amy schließlich eine Halle mit Platz für 500 Personen benötigte. Genau zu dieser Zeit sah sie eine Anzeige in der Zeitschrift "The Christian", in der eine Halle aus einer Eisenkonstruktion angeboten wurde, die 500 Personen fassen konnte und 500 £ kostete. Eine Spende in Höhe von 500 £ von Miss Kate Mitchell und dazu die Spende eines Grundstücks von einem der Mühlenbesitzer ermöglichte die Errichtung der ersten Welcome Hall an der Ecke Cambrai-Street / Heather Street in Belfast im Jahr 1887. Amy arbeitete in der Welcome Church, bis sie 1889 ein Ruf aus Manchester erreichte. Sie wurde gebeten, auch dort unter den Mill Girls zu arbeiten, bevor sie schließlich in die Missionsarbeit ging.
In vielerlei Hinsicht war sie für diese Arbeit ungeeignet. Sie litt unter Neuralgie, einer Nervenerkrankung, wodurch sie ständig Schwächegefühle und Schmerzen hatte. Oft war sie für Wochen bettlägerig. Bei der Keswick Convention 1887 hörte sie Hudson Taylor, den Gründer der China-Inland-Mission, über missionarisches Leben sprechen. Bald darauf war sie sich ihrer Berufung in die Missionsarbeit sicher. Sie bewarb sich bei der China-Inland-Mission und bereitete sich dann in London auf die Mission vor. Dort traf sie die Schriftstellerin und China-Missionarin Mary Geraldine Guinness, die sie ermutigte, in die Mission zu gehen. Sie war zwar bereit, nach Asien in ein Missionsgebiet zu reisen, wurde aber wegen ihres Gesundheitszustandes für untauglich befunden. Sie verschob ihre Missionars-Laufbahn bei der China-Inland-Mission und beschloss, der Church Missionary Society beizutreten.
Arbeit in Indien
Zunächst reiste Carmichael für 15 Monate nach Japan, aber nach einer kurzen Dienstzeit in Lanka fand sie ihre lebenslange Berufung in Indien, für die sie von der Church of England Zenana Mission beauftragt wurde. Sie erlebte Tempelprostitution und Kinderhandel: Es gab Tempel-Kinder, junge Mädchen die oft mit nur fünf Jahren in den Tempel als Prostituierte gezwungen wurden, um Geld für die Hindu-Priester zu verdienen (Devadasi, eigentlich Tempel-Tänzerinnen), und als Opfer der Eltern für die Götter. Carmichael beschäftigte sich mit ihnen und konnte einige aus der Zwangsprostitution retten. Die von ihr gegründete Organisation Dohnavur Fellowship wurde ein Refugium für über tausend Kinder. Dohnavur liegt in Tamil Nadu, 30 Meilen von der Südspitze Indiens entfernt.
In dem Bemühen, die indische Kultur zu respektieren, trugen Mitglieder der Organisation indische Kleidung und den Kindern wurden indische Namen gegeben. Auch Carmichael trug indische Kleidung und färbte ihre Haut mit Kaffeesatz, um sie dunkler zu machen. So reiste sie oft weite Strecken, um manchmal nur ein Kind zu retten.
Während ihrer Tätigkeit in Indien erhielt Amy einen Brief von einer jungen Frau, die erwog, selbst in die Mission zu gehen. Diese fragte Amy: "Was macht missionarisches Leben aus?" Amy schrieb einfach zurück: "Missionarisches Leben ist einfach eine Möglichkeit zu sterben."
Carmichael war auch eine überaus produktive Autorin; sie veröffentlichte 35 Bücher, u. a.: Mission Work in Southern India (1903), His Thoughts Said... His Father Said (1951), If (1953), Edges of His Ways (1955) and God's Missionary (1957).
Letzte Zeit und Vermächtnis
Im Jahr 1931 war Carmichael nach einem Sturz ans Bett gefesselt. An der Verletzung und deren Folgen hatte sie bis zu ihrem Tod zu leiden. Sie starb in Indien 1951 im Alter von 83 Jahren; und sie bat darum, keinen Grabstein zu bekommen, sondern eine Vogeltränke mit der Aufschrift "Amma" – in Tamil "Mutter".
Ihr Beispiel als Missionarin inspirierte z. B. Jim Elliot und dessen Frau Elisabeth.
Werke (Auswahl)
From Sunrise Land: Letters from Japan. Marshall, 1895
Things as they are; mission work in southern India. Morgan & Scott, London 1905
Overweights of Joy. 1906
Lotus Buds. Morgan & Scott, London 1912
Walker of Tinnevelly. Morgan & Scott, London 1916 (Biografie von Thomas Walker)
Ragland, pioneer. S.P.C.K. Depository, Madras 1922 (Biografie von Thomas Gajetan Ragland)
Ploughed Under: The Story of a Little Lover. Society for Promoting Christian Knowledge (SPCK), 1934
Candles in the Dark. Christian Literature Crusade, 1982
Rose from Brier. Christian Literature Crusade, 1972
Mimosa: A True Story. CLC Publications, 2005
If. Christian Literature Crusade, 1999
Gold Cord. Christian Literature Crusade, 1957
Edges of His Ways. Christian Literature Crusade, Fort Washington 1955
Mountain Breezes: The Collected Poems of Amy Carmichael. Christian Literature Crusade, 1999
Whispers of His Power. CLC Publications, 1993
Thou Givest They Gather. CLC Publications, 1970
Kohila: The Shaping of an Indian Nurse. CLC Publications, July 2002
Biographien
Elisabeth Elliot: A Chance to Die: the Life and Legacy of Amy Carmichael. Fleming H. Revell Company, Old Tappan, NJ 1987, ISBN 0-8007-1535-7.
Sam Wellman: Amy Carmichael: A Life Abandoned to God. Barbour Publishing, Uhrichville, Ohio 1998, ISBN 1-57748-364-2.
Derick Bingham: The Wild-Bird Child: A Life of Amy Carmichael. Ambassador-Emerald International, 2004, ISBN 1-84030-144-9
Frank Houghton: Amy Carmichael. Die Geschichte einer großen Frau. R. Brockhaus Verlag, Wuppertal 1990, ISBN 3-417-20830-0.
Hildegard Horie: Denn Götter weinen nicht – Amy Carmichael und ihre Tempelkinder. Esras.net, Niederbüren, 2017 (2. Aufl.) ISBN 978-3-905899-92-4.
Evangelischer Missionar
Autor
Brite
Geboren 1867
Gestorben 1951
Frau | {
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{"url":"https:\/\/pesquisa.bvsalud.org\/portal\/?lang=pt&q=au:%22Dong,%20Xu%22","text":"Mostrar: 20 | 50 | 100\nResultados 1 - 20 de 510\nFiltrar\nMais filtros\n\nAssunto principal\nTipo de estudo\nIdioma\nIntervalo de ano de publica\u00e7\u00e3o\n1.\nBlood ; 2020 May 08.\nArtigo em Ingl\u00eas | MEDLINE | ID: mdl-32384137\n\nRESUMO\n\nErythropoiesis is a complex multistage process that involves differentiation of early erythroid progenitors to enucleated mature red blood cells, in which lineage-specific transcription factors play essential roles. Erythroid Kr\u00fcppel-like factor (EKLF\/KLF1) is a pleiotropic erythroid transcription factor that is required for the proper maturation of the erythroid cells, whose expression and activation are tightly controlled in a temporal and differentiation stage-specific manner. Here, we uncover a novel role of G protein pathway suppressor 2 (GPS2), a subunit of the NCoR\/SMRT nuclear receptor corepressor complex, in erythrocyte differentiation. Our study demonstrates that knockdown of GPS2 significantly suppresses erythroid differentiation of human CD34+ cells cultured in vitro and xenotransplanted in NOD\/SCID\/IL2R\u00ce\u00b3null (NSG) mice. Moreover, global deletion of GPS2 in mice causes impaired erythropoiesis in the fetal liver and leads to severe anemia. Flow cytometric analysis and Wright-Giemsa staining show a defective differentiation at late stages of erythropoiesis in Gps2-\/- embryos. Mechanistically, GPS2 interacts with EKLF and prevents proteasome-mediated degradation of EKLF, thereby increases EKLF stability and transcriptional activity. Moreover, we identify the amino acids 191-230 region in EKLF protein, responsible for GPS2 binding, that is highly conserved in mammals and essential for EKLF protein stability. Collectively, our study uncovers a previously unknown role of GPS2 as a post-translational regulator which enhances the stability of EKLF protein and thereby promotes erythroid differentiation.\n\n2.\nLife Sci ; 253: 117675, 2020 May 01.\nArtigo em Ingl\u00eas | MEDLINE | ID: mdl-32360621\n\nRESUMO\n\nAIMS: Gliomas are responsible for the majority of deaths from primary brain tumours. Sevoflurane showed inhibition effects on the tumor progression in vitro. However, whether sevoflurane could affect the stemness of glioma stem cells (GSCs) and the potential molecular mechanism have not been well elucidated. MAIN METHODS: Effects of sevoflurane on cell viability, proliferation and invasion ability of glioma cells as well as tumor growth in vivo were assessed. Sphere formation assay was performed to evaluate the effect of sevoflurane on the stemness of GSCs. Effects of sevoflurane on mitochondrial function was evaluated by intracellular\/mitochondrial reactive oxygen species (ROS) level and mitochondrial membrane potential. Expression levels of proliferation-related proteins, stemness markers and proteins in CaMKII\/JNK cascade were measured by Western blot. KEY FINDINGS: Sevoflurane inhibited the viability, proliferation and invasion ability of glioma cells (U87MG and U373MG). Western blot showed that sevoflurane decreased the expression levels of proliferation and invasion-related proteins. Sphere formation ability of GSCs, expression levels of stemness markers and mitochondrial function were significantly suppressed by sevoflurane. Moreover, sevoflurane treatment significantly increased the Ca2+ concentration and stimulated phosphorylation of CaMKII, JNK and IRS1. Ca2+ chelator BAPTA-AM combined with sevoflurane synergistically inhibited colony forming ability and the expression levels of proliferation-related proteins and stemness markers. In addition, the in vivo study further confirmed that sevoflurane inhibited tumor growth via Ca2+-dependent CaMKII\/JNK cascade. SIGNIFICANCE: The present study demonstrated that sevoflurane inhibited glioma tumorigenesis and modulated the cancer stem cell-like properties and mitochondrial membrane potential via activation of Ca2+-dependent CaMKII\/JNK cascade.\n\n3.\nArtigo em Ingl\u00eas | MEDLINE | ID: mdl-32333194\n\nRESUMO\n\nTransaminase responsible for alienating prochiral ketone compound is applicable to asymmetric synthesis of herbicide L-phosphinothricin (L-PPT). In this work, the covalent immobilization of recombinant transaminase from Citrobacter koseri (CkTA) was investigated on different epoxy resins. Using optimum ES-105 support, a higher immobilized activity was obtained via optimizing immobilization process in terms of enzyme loading, coupling time and initial PLP concentration. Crucially, due to blocking unreacted epoxy groups on support surface with amino acids, the reaction temperature of blocked immobilized biocatalyst was enhanced from 37 to 57\u00a0\u00b0C. Its thermostability at 57\u00a0\u00b0C was also found to be superior to that of free CkTA. The Km value was shifted from 36.75\u00a0mM of free CkTA to 39.87\u00a0mM of blocked immobilized biocatalyst, demonstrating that the affinity of enzyme to the substrate has not been apparently altered. Accordingly, the biocatalyst performed the consecutive synthesis of L-PPT for 11 cycles (yields>91%) with retaining more than 91.13% of the initial activity. The seemingly the highest reusability demonstrates this biocatalyst has prospective for reducing the costs of consecutive synthesis of L-PPT with high conversion.\n\n4.\nJ Phys Condens Matter ; 32(33): 335803, 2020 Apr 15.\nArtigo em Ingl\u00eas | MEDLINE | ID: mdl-32294629\n\nRESUMO\n\nMultiferroic materials endowed with both dielectric and magnetic orders, are ideal candidates for a wide range of applications. In this work, we reported two phase transitions of MnI2 at 3.45 K and 4 K by systemically measuring the magnetic-field and temperature-dependent magnetization of the MnI2 thin flakes. Furthermore, we observed similar temperature and field-dependent behaviours for the magnetic susceptibility of MnI2 and electronic capacitance of the Ag\/MnI2\/Ag devices below 3.5 K. Considering the related theory work, we discussed the relationship between the antiferromagnetic and ferroelectric orders in MnI2. Our work reveals the in-plane magnetic and electric properties of MnI2 materials, which might be helpful for the further investigation and application of MnI2 multiferroics in the future.\n\n5.\nPhotodiagnosis Photodyn Ther ; : 101761, 2020 Apr 10.\nArtigo em Ingl\u00eas | MEDLINE | ID: mdl-32283311\n\nRESUMO\n\nXeroderma pigmentosum (XP) is a rare autosomal recessive dermatosis that is often complicated by multiple skin tumours at exposed locations, which are difficult to treat. We report a case of a 12-year-old girl with XP treated with oral retinoic acid and photodynamic therapy (PDT) with good clinical results. She had an 8-year history of multiple skin lesions that first appeared on her nasal dorsum, but gradually increased in size and spread to her entire face, neck, and upper limbs. Notably, the lesions became evidently aggravated after sun exposure. When she was 6 years old, sesame-seed-sized papules and plaques appeared, which were fragile and irregular in shape and would self-rupture, accompanied with slight itchiness and bloody exudate. Examination revealed multiple basal cell carcinomas. The tumours were treated with local carbon dioxide laser therapy combined with PDT. On the follow-up visit 2 months after the surgery, most of the skin lesions on her face had subsided. In cases of multiple tumours, PDT can be the treatment method of choice because it is less invasive, has less side effects, and does not damage the surrounding normal tissues.\n\n6.\nTrends Cancer ; 2020 Apr 11.\nArtigo em Ingl\u00eas | MEDLINE | ID: mdl-32291236\n\nRESUMO\n\nSperm-associated Antigen 5 (SPAG5) is a mitotic spindle protein. Recent studies have found that it is overexpressed in many human cancers and functions as an oncogene. Here, we summarize the current underlying mechanisms for its oncogenic roles in regulating cellular behaviors of cancer cells and discuss the possibility of targeting SPAG5 for cancer treatment.\n\n7.\nZhongguo Zhen Jiu ; 40(3): 337-41, 2020 Mar 12.\nArtigo em Chin\u00eas | MEDLINE | ID: mdl-32270653\n\nRESUMO\n\nThere is no criteria of placebo acupuncture method and no suitable method for all kinds of acupuncture research currently. In this paper, the methods and theories of placebo acupuncture were collected in recent 10 years at home and abroad. The analysis was conducted in the aspects of the premise of placebo acupuncture design, the common methods and their advantages and disadvantages, the application of various placebo acupuncture methods and the controversy on placebo acupuncture. It is required to further improve the design of placebo acupuncture control, explore the key questions of it and specify the criteria of its method so as to lay the foundation for the establishment of scientific and rational placebo acupuncture control in acupuncture research.\n\n8.\nJ Clin Nurs ; 2020 Apr 11.\nArtigo em Ingl\u00eas | MEDLINE | ID: mdl-32279372\n\nRESUMO\n\n9.\nAm J Orthod Dentofacial Orthop ; 157(3): 329-339, 2020 Mar.\nArtigo em Ingl\u00eas | MEDLINE | ID: mdl-32115111\n\nRESUMO\n\nINTRODUCTION: The purpose of this study was to comparatively evaluate the effects of Twin-block (TB) appliance and sagittal-guidance Twin-block (SGTB) appliance on alveolar bone around mandibular incisors in growing patients with Class II Division 1 malocclusion, using cone-beam computed tomography. METHODS: The sample consisted of 25 growing patients with Class II Division 1 malocclusion (14 boys and 11 girls, mean age 11.92\u00a0\u00b1\u00a01.62\u00a0years) and was randomly distributed into the TB group (n\u00a0=\u00a013) and the SGTB group (n\u00a0=\u00a012). The treatment duration was 11.56\u00a0\u00b1\u00a01.73\u00a0months. Pretreatment (T1) and posttreatment (T2) cone-beam computed tomography scans were taken in both groups. Height, thickness at apex level, and volume of the alveolar bone around mandibular left central incisors were measured respectively on labial and lingual side, using Mimics software (version 19.0; Materialise, Leuven, Belgium). Based on the stable structures, 3-dimensional (3D) registrations of T1 and T2 models were taken to measure the sagittal displacement of incisors. Intragroup comparisons were evaluated by paired-samples t tests and Wilcoxon tests. Independent-samples t tests and Mann-Whitney U tests were used for intergroup comparisons. RESULTS: In both groups, alveolar bone height and volume on the labial side of the incisors significantly decreased after treatment (P\u00a0<0.05). Lingual alveolar bone height, lingual and total alveolar bone volume, labial, lingual and total alveolar bone thickness showed no significant difference between T1 and T2 (P\u00a0>0.05). In both groups the incisors tipped labially and drifted to the labial side. Compared with the TB group, less labial alveolar bone loss, less incisor proclination and crown edge drift were found in the SGTB group (P\u00a0<0.05). CONCLUSIONS: Labial alveolar bone loss around mandibular incisors was observed after both types of appliances treatment in growing patients with Class II Division 1 malocclusion. Less labial alveolar bone loss, less incisor proclination, and crown edge drift were found in the SGTB group than in the TB group during treatment.\n\nAssuntos\nPerda do Osso Alveolar , M\u00e1 Oclus\u00e3o de Angle Classe II , Aparelhos Ortod\u00f4nticos , Adolescente , Cefalometria , Crian\u00e7a , Tomografia Computadorizada de Feixe C\u00f4nico , Feminino , Humanos , Incisivo , Masculino , M\u00e1 Oclus\u00e3o de Angle Classe II\/diagn\u00f3stico por imagem , M\u00e1 Oclus\u00e3o de Angle Classe II\/terapia , Mand\u00edbula , Coroa do Dente\n10.\nIran J Kidney Dis ; 14(2): 107-118, 2020 Mar.\nArtigo em Ingl\u00eas | MEDLINE | ID: mdl-32165595\n\nRESUMO\n\nINTRODUCTION: Previous studies have shown that TGF-\u00df1\/Smad3 signaling promotes renal fibrosis by inhibiting miR-29. To date, only few studies have reportedon circulating microRNAs in IgA nephropathy (IgAN). However, the plasma expression of miR-29a and its role in patients with IgAN remains unclear. In this study, we attempted to elucidate whether plasma miR-29a expression can be used as a biomarker for monitoring disease states. METHODS: For this study, 15 healthy subjects, 36 patients with untreated renal biopsy-proven IgAN, and 79 patients with IgAN, who were under treatment for a period of 1 year on an average, all of whom had similar age and gender distributions, were included. The plasma expression of miR-29a in each group was explored by real-time PCR, and the relationship between miR-29a expression and clinical, pathological, and prognostic indicators of IgAN was further evaluated. RESULTS: Relative plasma expression of miR-29a in patients with IgAN was significantly lower than that in healthy controls (P < .001), and these changes in plasma miR-29a could be suppressed by treatment (P < .05). Plasma miR-29a was positively correlated with eGFR and negatively correlated with proteinuria and serum creatinine, irrespective of whether or not the patients with IgAN accepted treatment (P < .05). Plasma miR-29a level was negatively correlated with primary pathological parameters such as crescent formation, Lee's and Oxford classification (P < .05). Kaplan-Meier analysis revealed that patients with high plasma expression of miR-29a had better renal function and better response to treatment compared to those with low expression (P < .05). CONCLUSION: Plasma miR-29a could be considered as a biological marker that reflects renal damage and function, to predict the progression of IgAN.\n\n11.\nACS Sens ; 5(4): 994-1001, 2020 04 24.\nArtigo em Ingl\u00eas | MEDLINE | ID: mdl-32174111\n\nRESUMO\n\nAlthough volatile organic compound samples can be detected by gas nanosensors in adsorption principles, extreme concentrations of target gases imply the excessive adsorption, which would lead to a long recovery time and even a shortened lifetime. Herein, we report the observations of the ionization current sensing behavior on the volatile organic compounds in an ionization gas sensor with silicon-based nanostructures. The micro ionization gas sensor consists of a pair of silicon microneedle array electrodes covered by nanolayer structures and a microdischarge gas gap. The dynamic response behaviors of the sensors to the exposure of ethanol, acetone, and 2-chloroethyl ethyl sulfide have been carefully scrutinized. The sensor exhibits sound performances to the high-concentration volatile organic compounds with a fast-recovery property and could generate effective responses well at 36 V, namely, the safety operation voltages. It could be well understood by the Jesse effect where small proportion of impurities in gases could lead to an intensive increase in the overall ionization probability. Besides, the reproducibility, recovery time, sensitivity, and selectivity properties have been systematically characterized.\n\n12.\nInt J Biol Macromol ; 154: 878-887, 2020 Jul 01.\nArtigo em Ingl\u00eas | MEDLINE | ID: mdl-32173428\n\nRESUMO\n\nA series of agar\/\u03ba-carrageenan mixed hydrogels with different mass ratios were prepared, and their physicochemical properties, gelling behavior and drug release performance were determined and analyzed. The results showed that the gel strength, the gelling temperature and the gel melting temperature decreased with the increase of \u03ba-carrageenan, while the apparent viscosities increased. Optical rotation and differential scanning calorimetry (DSC) indicated that there did exist intermolecular interactions between agar and \u03ba-carrageenan, and the detailed gelling mechanism of the mixed hydrogels was proposed, which was different from that of the previous research. Besides, agar\/\u03ba-carrageenan mixed hydrogels were used as carriers for the delivery of metformin hydrochloride (MET). The results showed that the drug loading efficiency and the sustained release capacity of agar hydrogels could be enhanced by the addition of \u03ba-carrageenan, and the release profile was mainly dominated by the electrostatic interaction between the MET and the polysaccharides. These results indicated that \u03ba-carrageenan had the potential to improve the physicochemical properties and drug release performance of agar hydrogel.\n\n13.\nJ Int Med Res ; 48(3): 300060519889458, 2020 03.\nArtigo em Ingl\u00eas | MEDLINE | ID: mdl-32216522\n14.\nSci Adv ; 6(12): eaaz3367, 2020 03.\nArtigo em Ingl\u00eas | MEDLINE | ID: mdl-32206724\n\nRESUMO\n\nMammalian transient receptor potential (TRP) channels are major components of Ca2+ signaling pathways and control a diversity of physiological functions. Here, we report a specific role for TRPC1 in the entry of herpes simplex virus type 1 (HSV-1) into cells. HSV-1-induced Ca2+ release and entry were dependent on Orai1, STIM1, and TRPC1. Inhibition of Ca2+ entry or knockdown of these proteins attenuated viral entry and infection. HSV-1 glycoprotein D interacted with the third ectodomain of TRPC1, and this interaction facilitated viral entry. Knockout of TRPC1 attenuated HSV-1-induced ocular abnormality and morbidity in vivo in TRPC1-\/- mice. There was a strong correlation between HSV-1 infection and plasma membrane localization of TRPC1 in epithelial cells within oral lesions in buccal biopsies from HSV-1-infected patients. Together, our findings demonstrate a critical role for TRPC1 in HSV-1 infection and suggest the channel as a potential target for anti-HSV therapy.\n\n15.\nOpt Lett ; 45(4): 1009-1012, 2020 Feb 15.\nArtigo em Ingl\u00eas | MEDLINE | ID: mdl-32058528\n\nRESUMO\n\nWe experimentally and numerically demonstrated the generation of a (3, 1) vector signal by a single Mach-Zehnder modulator (MZM) without pre-coding. The MZM is driven by the (3, 1) modulated signal after photoelectric conversion by the \"square law\" of a photodetector. Although the phase changes, the corresponding constellation distribution is consistent with that of the regular signal. Our proposed scheme effectively avoids the pre-coding process with a simple architecture. The bit-error-ratio (BER) results indicate that the (3, 1) signal has a better BER performance than the pre-coded quadrature phase shift keying vector signal, and both are below ${3.8}\\times {{10}^{ - 3}}$3.8\u00d710-3 after 25 km optical fiber transmission.\n\n16.\nFood Funct ; 11(2): 1754-1763, 2020 Feb 26.\nArtigo em Ingl\u00eas | MEDLINE | ID: mdl-32043502\n\nRESUMO\n\nCarvacryl acetate (CA) is a semisynthetic monoterpenic ester obtained from essential oils, and it exerts an antioxidation effect. The purpose of our study was to investigate whether CA could provide neuroprotection against oxidative stress caused by cerebral ischemia-reperfusion injury (CIRI) and elucidate the underlying mechanism. Middle cerebral artery occlusion (MCAO)-induced damage was established in Sprague Dawley (SD) rats and PC12 cells were exposed to hydrogen peroxide (H2O2) to imitate oxidative stress damage. TTC, HE and Nissl staining were used to observe the pathological morphology of lesions. The contents of ROS and MDA, and the activity of SOD were measured to reflect the level of oxidative stress. In addition, the TUNEL method was used to assess injuries in vitro, and the expression of Nrf2 was determined by immunohistochemical staining and western blot analysis. Importantly, we constructed and validated Nrf2 knockdown PC12 cells to confirm the key role of Nrf2 in the neuroprotective effect of CA against oxidative stress injuries. CA alleviated CIRI in rats with MCAO, as shown by brain tissue pathophysiology. The contents of ROS and MDA were reduced, and the SOD activity was augmented by the simultaneous promotion of Nrf2 expression. In addition, the H2O2-induced injury in Nrf2-knockdown PC12 cells was more serious than it was in control cells, and CA-mediated neuroprotection was exclusively inhibited by the knock down of Nrf2 in PC12 cells. In conclusion, it is shown here that CA has the effect of relieving cerebral ischemia reperfusion-induced oxidative stress injury via the Nrf2 signalling pathway.\n\n17.\nMedicine (Baltimore) ; 99(3): e18800, 2020 Jan.\nArtigo em Ingl\u00eas | MEDLINE | ID: mdl-32011484\n\nRESUMO\n\nRATIONALE: Low-grade malignant fibrous myxoid sarcoma (LGFMS) is a malignant tumor that originates from soft tissues and has specific clinical and histopathological characteristics. Paravertebral LGFMS is rarely reported. PATIENT CONCERNS: A 60-year-old woman had pain in the lower back and right anterior thigh for more than 3 years. DIAGNOSIS: Paravertebral LGFMS. INTERVENTIONS: Tumor resection, vertebral canal decompression and pedicle screw fixation. OUTCOMES: The tumor was excised, and the vertebral arch was fixed with pedicle screws at the root. Chemoradiotherapy was not performed. Her postoperative visual analogue scale (VAS) score decreased from 7 points at admission to 2 points at follow-up. The patient was discharged at postoperative day 13, and no recurrence was observed at the 6-month follow-up. LESSONS: Although LGFMS is rare, it should be considered in differential diagnosis of other soft tissue tumors to avoid misdiagnosis and inappropriate treatment.\n\nAssuntos\nSarcoma\/diagn\u00f3stico , Sarcoma\/cirurgia , Neoplasias de Tecidos Moles\/diagn\u00f3stico , Neoplasias de Tecidos Moles\/cirurgia , Diagn\u00f3stico Diferencial , Feminino , Humanos , Pessoa de Meia-Idade , Sarcoma\/complica\u00e7\u00f5es , Neoplasias de Tecidos Moles\/complica\u00e7\u00f5es , Compress\u00e3o da Medula Espinal\/diagn\u00f3stico , Compress\u00e3o da Medula Espinal\/etiologia , Compress\u00e3o da Medula Espinal\/cirurgia , Coluna Vertebral\n18.\nInt J Biol Macromol ; 148: 777-784, 2020 Apr 01.\nArtigo em Ingl\u00eas | MEDLINE | ID: mdl-31978475\n\nRESUMO\n\nThe wounds of diabetic patients are difficult to heal, which could lead to a limb amputation or even death. The experiment aims to develop a new type of nanoparticles that could accelerate wound healing. Epigallocatechin gallate, ascorbic acid, gelatin and chitosan nanoparticles (EV NPS) were prepared by ion cross-linking method, and their properties were studied. The optimal formula ratio of EV NPS is Vc:EGCG:Gel:CS\u00a0=\u00a00.2:3:1:1. Transmission electron microscope (TEM) images show that it is a roughly uniform spherical nanoparticle with a diameter of 200\u00a0nm. ICR mice were intraperitoneally injected with streptozocin (STZ) to establish diabetic mice. Full-thickness excisional wounds were established on the back of mice. The results showed that EV NPS can promote wound healing in diabetic mice, and the mechanism may be through increasing collagen accumulation, promoting angiogenesis and reducing the infiltration of inflammatory cells. EV NPS may have potential application values for wound healing in diabetic mice.\n\n19.\nJ Nurs Manag ; 28(3): 559-566, 2020 Apr.\nArtigo em Ingl\u00eas | MEDLINE | ID: mdl-31954085\n\nRESUMO\n\nAIM: To analyse the structural associations among job characteristics, organizational justice, work engagement and nursing care quality in Chinese nurses. BACKGROUND: Nursing care quality helps ensure patient safety, which are core concerns. The explicit relationships among the study's variables from a management perspective can help hospital managers to implement effective strategies to improve nursing care quality. METHODS: This cross-sectional study was conducted to investigate the relationships among the variables in 1,615 nurses in eight Chinese tertiary hospitals. Structural equation modelling was used to test a proposed model of these relationships. RESULTS: The scores of job characteristics, organizational justice, work engagement and nursing care quality were 3.55\u00a0\u00b1\u00a00.41, 3.84\u00a0\u00b1\u00a00.77, 4.67\u00a0\u00b1\u00a01.30 and 3.42\u00a0\u00b1\u00a00.70. Job characteristics and organizational justice had direct effects on nursing care quality. Work engagement mediated the relationship of nursing care quality with job characteristics and organizational justice. The final model explained 24% of nursing care quality. CONCLUSION: The results provide a better understanding of the associations between the study's variables. Perceived job characteristics and organizational justice can improve nursing care quality through work engagement. IMPLICATIONS FOR NURSING MANAGEMENT: Reconfiguring work design to strengthen nurses' positive perceptions of job characteristics and organizational justice can enhance nursing care quality.\n\n20.\nACS Appl Mater Interfaces ; 12(5): 6460-6470, 2020 Feb 05.\nArtigo em Ingl\u00eas | MEDLINE | ID: mdl-31942793\n\nRESUMO\n\nLight-driven actuators that directly convert light into mechanical work have attracted significant attention due to their wireless advantage and ability to be easily controlled. However, a fundamental impediment to their application is that the continuous motion of light-driven flexible actuators usually requires a periodically switching light source or the coordination of other additional hardware. Here, for the first time, continuous flapping-wing motion under sunlight is realized through the utilization of a simple nanocrystalline metal polymer bilayer structure without the coordination of additional hardware. The light-driven performance can be controlled by adjusting the grain size of the upper nanocrystalline metallic layer or selecting metals with different thermodynamic parameters. The achieved highest frequency of flapping-wing motion is 4.49 Hz, which exceeds the frequency of real butterfly wings, thus informing the further development of sunlight-driven bionic flying animal robotics without external energy consumption. The flapping-wing motion has been used to realize a light-driven whirligig, a light-driven sailboat, and photoelectric energy harvesting. Furthermore, the flexible bilayer actuator features the ability to be driven by light and electricity, low-power actuation, a large deflection, fast actuation speed, long-time stability, strong design ability, and large-area facile fabrication. The bilayer film considered herein represents a simple, general, and effective strategy for preparing photoelectric-driven flexible actuators with target performances and informs the standardization and industrial application of flexible actuators in the future.","date":"2020-05-27 16:30:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.28809595108032227, \"perplexity\": 12568.782643869268}, \"config\": {\"markdown_headings\": false, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590347394756.31\/warc\/CC-MAIN-20200527141855-20200527171855-00258.warc.gz\"}"} | null | null |
\chapter{The Origin of the String Field}
Much of the work in string theory has been done in the
context of first quantization. This means working with
two dimensional field theories, and making use of some
interpretation in order to relate observables in the
two dimensional theory, the world-sheet theory, to observables
in the physical spacetime theory, the target space theory.
This two dimensional field theory, typically a conformal
field theory, is the two dimensional analog of one-dimensional
field theories that describe the classical
mechanics of point particles.
Most of particle physics, however, is not done in first quantization.
Our description of Yang-Mills theories, or gravitation, is
done in the context of relativistic quantum field theory, or second
quantization. String field theory aims to formulate string theory as
a spacetime field theory. It follows that string field theory is the
{\it unconventional} approach to string theory based on the idea of
extending and generalizing the {\it conventional} field theory approach
to particle physics.
One must certainly keep in mind that generalizations of the usual
particle field theory concepts are necessary when one formulates
string field theory. For example, in classical mechanics one writes
an action for the coordinates $x^\mu (\tau )$ of a point
particle, and in the passage to field theory one
defines the field $\phi (x^\mu )$, dropping the proper
time variable $\tau$. For strings the analog is to pass
from $X^\mu (\sigma , \tau )$ to {\it functional} fields
$\Phi (X^\mu (\sigma ))$. This is well known not to work,
the string field must have extra arguments that include the
ghost fields of reparametrization invariance (see [\siegel ]). Only for
such string fields one can write suitable string field
actions. Further surprises were found in writing closed
string field theory (see [\zwiebachl ]), and very likely additional ones are
awaiting us in our way to a complete formulation.
Thus, in the final formulation, string field theory may have
little in common with standard formulations of particle
field theory except for the use of fields as dynamical
variables. We can also hope, since the physics of strings
is different from that of particles, that nonperturbative
string phenomena may be extracted from string field theory
less painfully than is typically the case in particle field theory.
It is very important to emphasize that, in current approaches
to string field theory, we work
in the path integral approach. That is, we do not work with
string field operators that create or destroy strings, but
rather with string fields which are classical $c$-numbers
(and classical anticommuting numbers, for ghost fields or
fermi fields). Quantization is defined by doing path
integrals with the string field action.
Let us consider in some detail the type of first-quantized actions
used to describe string theory. This will allow us to understand the
origin of the string field. One of the simplest actions used to
describe strings in first quantization is the following
$$S \sim \int d^2\xi \, \sqrt{h} h^{ab} \partial_a X^\mu
\partial_b X^\nu \, \eta_{\mu\nu}, \eqn\simplfqa$$
\noindent
where $a,b=1,2$ are indices labeling the coordinates $\xi^a$ in the
two dimensional surface,
$h_{ab}$ is a two dimensional metric, the scalar fields
$X^\mu (\xi)$ give the embedding of the worldsheet in spacetime,
and $\eta_{\mu\nu}$ is the flat Minkowski spacetime metric.
String theory is supposed to be a theory of gravity so one
may ask, where is the graviton, or the metric tensor in the
above action ? It is not there. That is the case because this is a first
quantized action. One finds that this action can be used
in a very indirect way to describe gravitons.
There are two
dimensional actions that provide a somewhat more explicit
framework for the study of the dynamics of the target space
fields. These are the sigma model actions, and typically
read as follows
$$\eqalign{
S' \sim \int d^2\xi \,\bigl(&
\sqrt{h} h^{ab} \partial_a X^\mu \partial_b X^\nu \, G_{\mu\nu}(X)
+\epsilon^{ab} \partial_a X^\mu
\partial_b X^\nu \, B_{\mu\nu}(X)\cr
& + \sqrt{h} R^{(2)}(h) \Phi(X) +
\cdots \bigr). \cr}\eqn\lsimplfqa$$
\noindent
Here we see that the Minkowski metric $\eta_{\mu\nu}$ was replaced by the
arbitrary metric $G_{\mu\nu}(X)$ and extra interactions were
added, including in particular one parametrized by an antisymmetric
object $B_{\mu\nu}(X)$, and another by $\Phi(X)$. In a sigma model action,
the objects $G_{\mu\nu}(X)$, $B_{\mu\nu}(X)$, and $\Phi(X)$, are
prescribed nonlinear functions of the two dimensional field $X(\xi )$.
The above action is therefore that of a nonlinear sigma model.
The object $G_{\mu\nu}(X)$
should play the role of a metric tensor in spacetime, $B_{\mu\nu}(X)$
should correspond to an antisymmetric tensor in spacetime, and $\Phi(X)$
should correspond to a scalar field, a dilaton in spacetime. Now we have the
metric
tensor, but where are Einstein's equations for $G_{\mu\nu}$, or the equations
for the antisymmetric tensor $B_{\mu\nu}$, or the equations for the
dilaton field $\Phi$ ? They are not found directly from the action,
again because this is not a second quantized action. They arise,
however, from a rather remarkable condition. The condition that
the above action be conformal invariant yields Einstein's equations
for $G_{\mu\nu}$ along with equations for the other fields.
While fascinating, it seems very difficult to use this approach for
a complete formulation of string theory. The complete classical equations
for the spacetime fields involve calculations to all loops in the
two dimensional field theory. The method is also very difficult
to use for spacetime fields that are not massless (we did not include
those in the above action) since they correspond to nonrenormalizable
interactions in two dimensions.
Nevertheless we see a general pattern. Each possible two dimensional
interaction, or local operator, is accompanied by a spacetime field
which appears as a coupling ``constant'' multiplying the interaction.
If we think of the string field as a collection of spacetime fields, the
above action suggests that the string field simply encodes the
data of a two dimensional field theory. Given a particular
string field we can associate a particular two dimensional field
theory. It seems very difficult at this stage to make precise
this idea for a variety of technical and conceptual
difficulties. The way we {\it know} how to proceed goes
as follows.
If we have a conformal field theory, as is the case for the first
action we wrote above,\foot{We must include, however, the reparametrization
ghosts.} conformal invariance provides us the
complete and explicit list of all possible local operators $\Phi_s$ of
the two dimensional conformal theory. For each of these
operators we associate a spacetime field $\psi_s$. The state space
${\cal H}_{\hbox{CFT}}$ of a conformal field is the space of states created by
letting all
possible local operators act on the vacuum. Such states are denoted
by $\ket{\Phi_s}$. For example, to the familiar graviton vertex operator
$\partial X^\mu\overline\partial X^\nu e^{ipX}$ we must
associate the graviton field $h_{\mu\nu}(p)$
$$\partial X^\mu\overline\partial X^\nu e^{ipX}\leftrightarrow h_{\mu\nu}(p).
\eqn\assvof$$
\noindent
We are then naturally led to assemble the string field as a general
vector in the state space ${\cal H}_{\hbox{CFT}}$
$$\ket{\Psi} = \sum_s \ket{\Phi_s}\, \psi_s \,\, .\eqn\setupsf$$
Here each {\it target space field} $\psi_s$ is the component
of the vector $\ket{\Psi}$ along the basis vector $\ket{\Phi_s}$.
The target space fields are in general complex numbers, and may be
Grassmann even or odd. The string field action $S(\Psi )$ is a function
from ${\cal H}_{\hbox{CFT}}$ to the real numbers.
Loosely speaking, the string field does indeed appear to encode the data of
a two dimensional field theory. If $S_c$ denotes the action
for the conformal field theory that is being used to define
the string field, then associated to the string field in
Eqn.\setupsf\ we may attempt to define the two dimensional action
$$S_c+ \sum_s \, \psi^s \cdot \int d^2 \xi\, \Phi_s(\xi ,\bar \xi),
\eqn\actdef$$
using the components of the string field (the spacetime fields)
to weight the various interactions.\foot{I am ignoring ghost insertions that
are necessary to go from the string field to the world sheet interactions.}
Again, due to technical difficulties it is difficult
to define precisely the meaning of the above expression, unless
the operators $\Phi_s$ are primary fields of dimension (1,1).
Therefore, at present, the precise formulation of closed string
field theory does not attempt to define the string action as
a function in the space of two dimensional theories, but rather
as a function in the state space of a given conformal field theory.
Therefore, in order to write a string field theory, we must first choose
a conformal field theory. A conformal field theory defines a consistent
spacetime background for string propagation. This means that we are only
able to write string field theory once we pick a background that must
correspond to a classical solution of the theory we are aiming to write.
An analogy is useful to understand the situation.
In Einstein's gravity the dynamical variable is the metric tensor
$g_{\mu\nu}(x)$ on some manifold $M$. The Einstein action, given a metric
$g_{\mu\nu}$ gives us a number, this action is therefore a function on the
space ${\cal G}$ of metrics on $M$. There are some special metrics on
${\cal G}$, the Ricci flat ones. They solve the classical Einstein's equations
and therefore define consistent spacetime backgrounds.
We can study physics around any Ricci flat background $\hat g_{\mu\nu}$ by
expanding the metric tensor as $g_{\mu\nu} = \hat g_{\mu\nu} + h_{\mu\nu}$,
where $h_{\mu\nu}$ represents a fluctuation. The gravity action becomes a
function of the field $h_{\mu\nu}$. While this action for $h_{\mu\nu}$ does
contain all of the physics of gravity, the action depends explicitly of
the background metric $\hat g_{\mu\nu}$ in a complicated way. In string
theory the analog of a given metric $g_{\mu\nu}$ is a two dimensional
field theory, and the analog of the space ${\cal G}$ of metrics on $M$
is the space of two dimensional field theories (it is not clear how to think
of a two-dimensional field theory as a structure in some space, thus
there is no compelling analog for $M$). Corresponding to the Ricci-flat
metrics we have the conformal field theories. The string field $\ket{\Psi}$
we have discussed above
corresponds to the fluctuation field $h_{\mu\nu}$. Indeed the string field
action has background dependence; it uses, for example, the BRST operator of
the
conformal field theory. This necessity to fix a conformal field theory to get
started writing a string field action is usually referred to as the issue
of background independence of string field theory. It is certainly the
central question facing string field theory. A background independent
string field theory would most likely be the formulation of string theory we
are looking for.
The problem of setting up a background independent string field theory
is exactly analogous as that of reconstructing Einstein's theory if
we only knew the expansion of the Einstein lagrangian around flat space.
Of course, the Einstein-Hilbert action was written before the flat space
expansion was known, but as a problem of principle, the issue of reconstructing
the gravity action from the interactions of a spin two field $h_{\mu\nu}$
has received considerable attention. Our current problem, and in this case
we do not know the answer in advance, is to find the action from which
our string field action $S(\Psi )$ arises by expansion around a classical
solution. We need to understand the geometrical meaning of the series
of self-interactions of the string field $\Psi$.
There are various ideas that are probably going to be helpful in the
near future. They will be discussed in \S4 . Some of them center around the
Batalin-Vilkovisky (BV) theory
of quantization. The string field action $S(\Psi )$ actually satisfies
a second order, nonlinear partial differential equation called the BV
master equation. A generalization of the concept of a Lie algebra, a
structure called ``homotopy Lie algebra'' seems to be at the basis of
classical closed string field theory. At the quantum level, the algebraic
structure becomes what one could call a ``quantum homotopy Lie algebra''.
Understanding
these algebraic structures seems essential. At a geometrical level it
is probably necessary, at least in the short run, to develop more of the
geometry in the space of two dimensional field theories. Here again BV
seems to be of help; it is reasonable to assume that this theory space
is actually a (super) symplectic manifold. Finally, geometry of two
dimensional Riemann surfaces, or geometry on the space of Riemann surfaces
(moduli spaces) plays a crucial role in writing string field theory.
I would expect this geometry to tie in nicely with theory space geometry
in the future formulations of string field theory.
\chapter{String Diagrams from an Extremal Problem}
If string theory is to be formulated in a natural way as a field
theory there has to exist a natural way to generate the sum over
all surfaces (necessary in the computation of any string amplitude)
with the use of a propagator and a set of vertices. For a covariant
string field theory, the vertices should be symmetric under the
exchage of the various strings. It was widely believed that no
natural way would be found to generate all Riemann surfaces once and
only once. Indeed, experience suggested that any simple choice of
vertices and propagator would either miss surfaces, produce some surfaces
more than once (typically an infinite number of times), or both.
I would like to show here how one can build
all the different Riemann surfaces efficiently by using
vertices and propagators. I will prove explicitly two basic results
and then explain why they guarantee that all Riemann surfaces can
be produced without missing any surface and without overcounting any
surface. In order to achieve this we use string diagrams. For original
references the reader may consult [\saadizwiebach--\wolfzwiebach ].
A string
diagram is the analog of a particle field theory Feynman diagram. One speaks
of the string diagram corresponding to a given Riemann surface. The
string diagram is nothing else than the Riemann surface equipped with some
extra structure. This structure can be a analytic one-form (abelian
differential), an analytic two-form (quadratic differential), or typically,
a (conformal) metric. It is the extra structure on the Riemann surface
that should tell us how the surface is built in terms of vertices and
propagators. What we have found in the last few years is that, for
covariant string field theory, a metric seems to be the right
structure to put on a surface. That metric,
moreover, is very special, it is the solution to an extremal problem.
Not only for closed string field theory these metrics are relevant.
Extremal metrics actually define the string diagrams of open string
theory and of open-closed string field theory. Thus actually {\it all} string
diagrams for
covariant string field theory are defined by extremal metrics.
Let us first explain what is a (conformal) metric on a Riemann surface (see
[\strebel ]).
A two dimensional Riemannian manifold, that is, a two dimensional
manifold with a Riemannian metric $g_{\alpha\beta}$, actually defines
a fixed Riemann surface ${\cal R}$, since a metric defines a conformal structure.
The metric can be put locally in the conformal form $dl^2 = \rho^2(dx^2+dy^2)$,
with $z=x+iy$ the local analytic coordinate, and, with $\rho (x,y)$ the
positive semidefinite Weyl factor. $\rho$ is said to be a conformal metric
(metric, for short) on the Riemann surface ${\cal R}$. In the extremal problem we
wish to consider the complex structure will be kept fixed, and we will choose
a (conformal) metric satisfying some extremal property. Varying the Riemannian
metric would mean that we are also changing the conformal structure -this
we do not want to do.
A (conformal) metric $\rho$ on a Riemann surface ${\cal R}$ defines
a length element $dl= \rho |dz|$, and an area element
$\rho^2 dx \wedge dy$. Under analytic changes of coodinates
the metric must transform as $\rho (z) |dz| = \widetilde \rho (w) |dw|$.
This transformation guarantees the invariance of the length of a curve
and the invariance of the area under analytic mappings.
In the standard minimal area problem one has a Riemann surface ${\cal R}$, and
one chooses a set $\Gamma_i$ of free\foot{Basepoint free.} nontrivial homotopy
classes of closed
curves. Let $\widehat\gamma_i$ be representative curves for the chosen
homotopy classes. Associated to each homotopy class one chooses a constant
$A_i\geq 0$. One then asks for the metric $\rho$ of least possible
area on ${\cal R}$ under the condition that for any curve $\gamma$ freely
homotopic to $\gamma_i$ ($\gamma \sim \gamma_i$) the length of $\gamma$
on the metric $\rho$ must be greater than or equal to $A_i$:
$$\int_{\gamma \sim \gamma_i}\rho\, |dz| \geq A_i \, . \eqn\admcond$$
This requirement must hold for all $i$ in the set.
A metric is called {\it admissible} if it satisfies the length
conditions \admcond . Thus the minimal area problem asks for the
admissible metric of least possible area.
A useful property of the
space of admissible metrics is that it is a convex space; if
$\rho_0$ and $\rho_1$ are admissible metrics then
$$\rho_t = (1-t) \rho_0 + t \rho_1, \eqn\convexspace$$
is an admissible metric for all $t \in [0,1]$, as follows directly from
\admcond . Let ${\cal A} (\rho )$ denote the area of ${\cal R}$ with the metric $\rho$.
\section{Two Useful Results}
The area functional ${\cal A}$ has a very nice property: it is
strictly convex. This means that for the above one-parameter
family of metrics, with $\rho_0 \not= \rho_1$, one has
$${\cal A} (\rho_t ) < (1-t) {\cal A} (\rho_0 ) + t {\cal A} (\rho_1 ),\eqn\strictcon$$
for $t \in (0,1)$. In words, the area is lower than the linear interpolation
of the areas at the endpoints. The inequality is strict since we do not include
the endpoints $t=0$, and $t=1$, corresponding to $\rho_0$ and $\rho_1$.
It simply means that the area, as a function of $t$, is a convex function.
This relation is simple to derive, so we will do so now. By definition
$$\eqalign{{\cal A} (\rho_t ) = \int \rho_t^2 \,\hbox{dxdy} &= \int
((1-t) \rho_0 + t \rho_1)^2 \,\hbox{dxdy}\cr
&= (1-t)^2 {\cal A}(\rho_0)+ t^2 {\cal A}(\rho_1 )+
2t\,(1-t)\int \rho_0\rho_1\,\hbox{dxdy} .\cr}\eqn\bproo$$
Since $\rho_0 \not= \rho_1$ (by assumption) Schwarz's inequality implies that
$$\int \rho_0\rho_1\,\hbox{dxdy}< \Bigl( \int \rho_0^2\,\hbox{dxdy}
\int \rho_1^2\,\hbox{dxdy}\Bigr)^{1/2} = \bigl( {\cal A} (\rho_0) {\cal A} (\rho_1)
\bigr)^{1/2} .\eqn\schwarzi$$
Using this inequality for the last term in \bproo\ we find
$$\sqrt{{\cal A} (\rho_t )} < (1-t) \, \sqrt{{\cal A} (\rho_0 )} + t\, \sqrt{{\cal A} (\rho_1 )},
\eqn\sqiscon$$
which shows that $f(t) = \sqrt{{\cal A} (\rho_t)}$ is convex. But, if an everywhere
positive function is convex, its square is also convex. This implies the
desired result, namely, the area functional is convex.
An immediate consequence
is the uniqueness of the minimal area metric (if it exists).
This is proven as follows. Assume there are two different metrics
$\rho_0$ and $\rho_1$ on the surface ${\cal R}$, both admissible and
both having the (same) least possible area ${\cal A}$. It follows that
the admissible metric $\rho_t$ (for any
$t\in (0,1)$ ) must satisfy
${\cal A} (\rho_t) < (1-t) {\cal A} + t{\cal A} = {\cal A}$, in contradiction with the assumption
that $A$ was the least possible value for the area.
The uniqueness of the minimal area metric is fundamental for our purposes,
as will be explained later.
There is another result, which is also standard, easy to derive, and
very powerful. Consider the rectangular region in the complex plane
bounded by the points $0,a,ib$, and $a+ib$, where $a$ and $b$ are
real numbers greater than zero. On this region of the complex plane
we would like to find the metric of least possible area such that any
curve starting anywhere on the left vertical segment $[0,ib\, ]$ and ending
anywhere on the right vertical segment $[a,a+ib\, ]$ should be longer than
or equal to $a$. This problem is the canonical (and simplest) minimal
area problem. We are looking for a metric $dl^2 = \rho^2 (\hbox{dx}^2
+ \hbox{dy}^2)$ with least possible area ${\cal A} (\rho ) = \int_0^b \hbox{dy}
\int_0^a \hbox{dx}\,\rho^2$. It is clear that the metric $\rho(x,y) =1$
is admissible, indeed, any curve crossing the rectangle from left to right
must be longer than or equal to the length of a strictly horizontal line
($y$ constant). The horizontal lines are precisely of length $a$. The
area of this metric is $ab$. It follows that for the metric $\hat \rho$
of least possible area we must have ${\cal A} (\hat \rho ) \leq ab$. We will
show now that actually $\rho = 1$ is the metric of least possible area.
To this end consider an admissible metric $\rho$. Since the length condition
must be satisfied for a horizontal curve with fixed $y$
$$\int_0^a \rho (x,y)\, \hbox{dx} \geq a, \quad \forall y \, ,\,\quad\to\quad
\int_0^b \hbox{dy} \int_0^a \rho (x,y)\, \hbox{dx} \geq ab \, .\eqn\ffcond$$
The Schwarz inequality
$$\int f^2 \hbox{dxdy}\int g^2 \hbox{dxdy}
\geq \Bigl( \int f\, g \, \hbox{dxdy} \Bigr)^2 ,\eqn\sineq$$
with the choice $f= \rho$, and $g=1$, gives us
$${\cal A} (\rho ) = \int \rho^2 \hbox{dxdy}
\geq \, {1\over ab} \, \Bigl( \int \rho\, \hbox{dxdy} \Bigr)^2 .\eqn\neem$$
All integrals here are over the rectangular region. Using \ffcond\ we
immediately see that for any admissible metric $\rho$ we have that
${\cal A} (\rho) \geq ab$. Thus the least possible area is indeed $(ab)$. Since
that was the area for the metric $\rho(x,y) = 1$, the uniqueness of the
minimal area metric implies that the minimal area metric is indeed
$\rho (x,y ) =1$. This is what we wanted to show.
We can get a further result as a simple consequence of the above.
Identify the left edge and the right edge of the rectangle via the
relation $iy \equiv a+iy$ , for $0\leq y\leq b$. We then get an annulus, or
a cylinder.
It is clear from the above derivation that the metric of least possible
area under the condition that any closed curve, freely homotopic to
a horizontal closed curve, be longer than or equal to $a$, is still the
flat metric $\rho (x,y) = 1$.
\section{The Minimal Area Problem for Closed String Theory}
We discussed at the beginning of \S2 what is the
general minimal area problem. One chooses a set (finite or infinite)
of homotopy classes of closed curves and imposes length conditions
on all curves belonging to the chosen homotopy classes. This problem
has not yet been solved, in fact, very little is
known. The solution is known for the case when we choose a set of
homotopy classes having representative curves that can be chosen
to be nonintersecting simple closed Jordan curves (no self intersections).
Such a set of representative curves is called, in
the mathematical literature, an admissible set. For any fixed genus
the maximum number of curves in an admissible set is finite --if we
try to add additional curves we must produce intersections. If we
impose length conditions on curves homotopic to those on an admissible set
the minimal area metric is known. It is a metric that arises from a
Jenkins-Strebel quadratic differential [\strebel]. Intuitively, this
means that the surface, with the minimal area metric, is built by gluing
together flat euclidean cylinders of circumferences equal to the length
parameters $A_i$ (see equation \admcond ). If we put length conditions
on curves homotopic to representatives that must intersect, even on
two such representatives, the solution is not known.
The minimal area problem relevant for closed string field theory is
the following. For any
surface ${\cal R}$ (with or without punctures) we are interested in the
metric of least possible area under the condition that
{\it all} nontrivial closed curves on ${\cal R}$ be longer than or
equal to $2\pi$ [\zwiebachma ]. This minimal area problem, in contrast
with the related extremal problems studied earlier
[\strebel ], does not require specifying some homotopy classes of
curves on the Riemann surface on which we impose length conditions.
The length conditions are imposed on {\it all} homotopy classes. This is why
this is a problem defined on moduli space, why the problem is modular
invariant. Whenever we choose particular curves on a surface we are
introducing extra data. For example, on a torus, choosing two curves
without self intersections, and intersecting each other once, amounts to
choosing a particular representative for the torus in Teichmuller space.
Having a well defined problem which does not require extra structure
is crucial for us. Since we have put conditions on all homotopy classes
of curves, and clearly their representatives intersect, this problem does
not fall within the class of problems whose solution is known.
The results established in \S2.1 will now allow us to
prove a result that gives us a fair amount of intuition about the
minimal area problem for closed string theory.
I claim that if a surface is built by gluing
together flat cylinders of circumference $2\pi$, and moreover, no closed
curve on the resulting surface is shorter than $2\pi$, then the surface
has a metric solving the minimal area problem. The argument (which is
truly simple) goes as follows. The cylinders split the surface into
a set of rectangular domains of the type considered in the previous
section, each with $a=2\pi$, and with the vertical edges identified. Since
the core curve of each rectangle (a closed horizontal curve) is nontrivial,
it is necessary that any curve homotopic to a core curve
and contained completely on the corresponding rectangle be longer than or
equal to $2\pi$. Let us call this condition $(\alpha )$. Condition
$(\alpha )$ is necessary, although not sufficient for a metric to be
admissible. Imagine
there is an admissible metric with area lower than that of the
original metric. Then it
would have to have lower area at least on one of the rectangles. But this is
impossible since, on each rectangle, the original flat $\rho =1$ metric has
already the least possible area under condition $(\alpha )$. This proves the
result. We therefore have a simple way to construct many metrics of minimal
area.
We glue flat cylinders watching out that no closed curve is smaller than
$2\pi$.
Let us now explain a very fundamental idea, the idea that shows the
relevance of minimal area metrics. The origin of all the difficulties
in constructing a field theory of closed strings lies on the fact that given
two Riemann surfaces it is, in general, very hard to tell whether or
not they are the same, that is, whether or not there is a conformal
map from one to the other. This means that when we construct
Riemann surfaces using vertices and propagators it is very hard to guarantee
that no Riemann surface is produced more than once. The situation is
very much improved if we put metrics on the surfaces. It is actually
easy to see if two surfaces with metrics are the same. For example, if
we build our surfaces using the cylinders discussed in the paragraph
above two metrics are the same if {\it they look the same}. That is,
they must have the same number of cylinders, and the gluing patterns
must be the same. This is the case because the cylinders determine
very special {\it saturating geodesics} on the surface, length $2\pi$
geodesics that saturate the length conditions. Two surfaces cannot have
the same metric if their patterns of saturating geodesics are not the same.
When we construct
our surfaces with metrics, our problem is making sure that for any
two such surfaces $({\cal R}_1,\rho_1)$, and $({\cal R}_2,\rho_2)$ the underlying
Riemann surfaces ${\cal R}_1$ and ${\cal R}_2$ are not the same (otherwise we have
overcounting). Suppose we can
tell that the metrics are not the same, what does this buy for us?
If the metrics are not the same and the Riemann surfaces are different
that is no problem\foot{Precisely speaking, in this case, the metrics
are different as Riemannian metrics on the underlying two dimensional
manifold, since as conformal metrics they are not defined on the same
Riemann surface and any comparison is meaningless.}. The problem happens
if the Riemann surfaces are the same despite the fact that the metrics
are different.
Here is where the minimal area principle helps; if we can assure that
the metrics are of minimal area, their being different guarantees that
the Riemann surfaces are different! The reason is uniqueness of the
minimal area metric on a given Riemann surface. If the two Riemann surfaces
were the same they could not
have two different metrics solving our extremal problem. As one can
imagine, different Feynman graphs in string theory correspond to
different length of cylinders and different gluing patterns. Therefore
different Feynman graphs produce different metrics, and if we guarantee
that they are all of minimal area, we are free of the problem of overcounting.
I have explained above one simple criterion to tell whether a metric is
of minimal area (the surface is built with cylinders). This criterion is
sufficient but not necessary, in fact
not all minimal area metrics are of this type.
In the next section I will sketch an argument why the rules of
sewing, which allow us to build complicated surfaces starting from
vertices and propagators, are compatible with minimal area.
A very interesting problem, an extremal problem for metrics on
Riemannian manifolds, has been investigated by Gromov [\gromov ] and
Calabi [\calabi ]. Their problem for the case of two dimensional surfaces
is a particular case of our minimal area problem. In their case they
consider a fixed two dimensional manifold $M$ and the space of
Riemannian metrics on $M$ such that all nontrivial closed curves are
longer than or equal to $2\pi$. Then they ask for the Riemannian metric
of least possible area. Since they can change the complex structure of
the surface, it seems clear that their ``extremal isosystolic'' metrics
should correspond, at every genus, to a minimal area metric on the Riemann
surface whose extremal metric has the least possible area.
They can prove existence of the
extremal isosystolic metric. Uniqueness is not clear, it could be
that a finite number of different Riemann surfaces have extremal metrics
whose area is the same, and at the same time lowest among all
the areas of the extremal metrics on all other Riemann surfaces.
\section{Why Minimal Area Metrics Work}
Let us now sketch the logic that shows that minimal area metrics
solve our problem of generating all Riemann surfaces once and only once.
Here I will not be completely self-contained, but hope to give
a reasonably clear idea of how things fit together. For complete details
the reader should consult the original papers.
A metric solving the minimal area problem is expected to
give rise to closed geodesics of length $2\pi$ that foliate
the surface completely. If there is a point through which there is no
saturating geodesic the
metric could be lowered at this point without destroying
admissability.
We can show that such curves must foliate the surface since
they cannot intersect whenever they are of the same homotopy
type. In fact, generically, two saturating geodesics can at most
intersect in one point.
For the case of Riemann spheres with punctures,
since any Jordan closed curve must cut the surface into two separate
pieces, two saturating geodesics that cross they must do so
at least in two points. Since this cannot happen,
the surface will be foliated by bands of geodesics that do not
intersect. This is precisely what happens with Jenkins-Strebel (JS)
quadratic differentials, the horizontal trajectories of the
quadratic differential are the saturating geodesics. The surface
is then foliated by bands of geodesics, the so-called ring-domains
of the quadratic differential. The horizontal trajectories
can intersect in the critical graph of the quadratic differential,
but this graph is of zero measure on the surface. Thus all minimal
area metrics on Riemann spheres (that define the classical closed
string field theory) arise from JS quadratic differentials. They
are now known explicitly and can be described in terms of polyhedra.
In higher genus Riemann
surfaces one can have saturating bands of geodesics
that cross. Thus higher genus minimal area metrics
do not always arise from JS quadratic differentials. One concrete
example showing crossing of foliations was given
in [\wolfzwiebach ].
In a punctured Riemann surface, the minimal area metric must have
all closed curves homotopic to the punctures satisfying the length
conditions. This actually allows one to show that
a minimal area metric is isometric to a
semiinfinite cylinder of circumference $2\pi$ for some neighborhood
of each puncture. This semiinfinite cylinder must end somewhere on
the surface; let ${\cal C}_0$ denote the boundary
curve of the semiinfinite cylinder. Let ${\cal C}_l$ denote the saturating geodesic
in the cylinder a distance $l$ away from ${\cal C}_0$.
The minimal area metrics satisfy an amputation property; if we amputate
the semiinfinite cylinder along ${\cal C}_l$
for $l >0$, the resulting surface still has a minimal area metric.
This property allows us to show that the plumbing
of surfaces with minimal area metrics gives a surface with a
minimal area metric [\zwiebachqcs ,\wolfzwiebach]. The basic idea is
simple, given two surfaces to be sewn together (or a single surface
with two legs to be sewn), one first amputates the semiinfinite cylinders
corresponding to the relevant legs. Given the amputation property the
resulting surfaces with boundaries have minimal area metrics. If we glue
together the open boundaries, and, by doing so we do not create closed
curves that are smaller than $2\pi$, (this is the reason for stubs, as
we will see), then the resulting surface inherits a minimal area metric.
This follows because any candidate metric with less area would have to
have less area in at least one of the surfaces that were glued, but this
is impossible given that the amputated surfaces have minimal area metrics.
The reader interested in the complete detailed argument should consult
[\wolfzwiebach ].
We define the height $h$ of a foliation (a band of geodesics making up
an annulus) to be the shortest distance, along the annulus, between the two
boundary components. It can be shown that a foliation
with height $h$ greater than $2\pi$ is isometric to a flat cylinder
of circumference $2\pi$ and height $h$. This suggests that on a string
diagram we can identify propagators with the cylinders that have
heights greater than or equal to $2\pi$. One can prove a cutting
theorem (closely related to amputation); if we cut open a foliation
of height greater than $2\pi$ on a metric of minimal area, the resulting
(cut) surface still has a minimal area metric.
When building the string
field theory we must choose vertices, we call ${\cal V}_{g,n}$ the vertex
that must be introduced at genus $g$ for processes with $n$
punctures. This vertex corresponds to the surfaces that are
missed by the Feynman rules using lower dimensional vertices. There is
a simple criterion to tell whether or not a surface ${\cal R}$ belongs to the
vertex:
\noindent
$\underline{\hbox{The String Vertex}\,\, {\cal V}_{g,n}}.\,$ ${\cal R} \in {\cal V}_{g,n}$ if
and only if the heights of all internal foliations are less than or equal to
$2\pi$. If ${\cal R}$ is in ${\cal V}_{g,n}$ we define the coordinate curves to be
the ${\cal C}_\pi$ curves around each puncture.
\noindent
In the above, internal foliation refers to a foliation
that does not correspond to one of the semiinfinite cylinders.
We have placed
coordinate curves leaving ``stubs'' of length $\pi$ for each
puncture. The coordinate curves define the amputated vertices, that
is the vertices as surfaces with holes ready to be joined to each
other with propagators. The short cylinder of length $\pi$ left
from each semiinfinite cylinder is necessary in order to make sure that the
plumbing procedure produces admissible metrics; if we had
no stubs the plumbing of two holes on a single surface,
with a propagator of small length, could introduce curves
shorter than $2\pi$, and the resulting surface would not inherit a minimal
area metric. The stubs guarantee that
upon plumbing no curve shorter than $2\pi$ is generated. Then plumbing
is guaranteed to preserve the minimal area property.
\noindent
Two things must be checked. First, no surface in ${\cal V}$ must be produced by
sewing. Indeed, due to the stubs of length
$\pi$, sewing must create foliations
of height greater than to $2\pi$, and, by definition, the surface
cannot belong in ${\cal V}$.
The second point that must be checked is that all surfaces which
are not in ${\cal V}$ are produced by sewing, and produced only once.
The first part is clear because
for any surface which is not in ${\cal V}$ there is at least one foliation whose
height is greater than $2\pi$. By the cutting theorem we can cut all such
foliations open and obtain a surface(s) with a minimal area metric(s).
Restoring the semiinfinite cylinders around all boundaries one obtains
the Riemann surface(s) whose plumbing must give us the original surface.
Therefore no surface is missed. Finally, no surface could have been produced
more than once, since
different Feynman diagrams correspond to different metrics, which
by uniqueness of the minimal area metric, cannot correspond to the
same Riemann surface.
There are some important open questions left about the extremal
metrics solving the minimal area problem at higher genus. We do not
have a mathematical proof of the existance of such metrics. While I
am not concerned about the possibility that the metrics might not
exist,\foot{At higher genus a fraction of each moduli space is made of Riemann
surfaces whose minimal area metrics arise from quadratic differentials.
We also know now of explicit examples of minimal area metrics that
do not arise from quadratic differentials.} a proof of existance is likely to
be helpful in understanding
further properties of the extremal metrics. We would like to know
how the metrics look in general, and what type of curvature singularities
they have. It would also be interesting to know how to parametrize the
general metrics of minimal area. Explicit knowledge of the minimal area
metrics is likely to be useful in future calculations using string field
theory.
\chapter{Batalin-Vilkovisky Structures}
Batalin-Vilkovisky (BV) quantization is the complete solution
of the problem of quantizing a general gauge theory (having no
second class constraints). In the usual applications one starts
with a classical gauge invariant theory with an action $S_0 (\phi^i)$ defined
on a set of fields $\phi^i$. As a first step one extends the set of fields into
a larger set comprised of fields $\psi^s$, and antifields $\psi^*_i$. One then
finds a master action $S_M (\psi_i , \psi^*_i)$ which reduces to the original
action $S_0$ upon setting the antifields to zero, and that solves the classical
master equation $\{ S_M , S_M \} = 0$. The antibracket appearing on
the left hand side in the analog of the Poisson bracket between coordinates
and momenta in classical mechanics. Here it is defined using the pairing
between fields and antifields. To define a quantum theory, however, one must
make sure that the master action actually satisfies the complete master
equation $\hbar\Delta S_M + {1\over 2}\{ S_M , S_M \} = 0$. This typically
requires extra work, since $\Delta S_M$ need not vanish in general.
The final solution $S_M (\hbar )$ is the quantum master action.
BV quantization is extremely efficient in closed string field theory
because, once we have a consistent set of string vertices ${\cal V}_{g,n}$, we
can write directly the full quantum master action $S_M(\hbar )$ solving
the quantum master equation [\zwiebachqcs ]. The full spectrum of fields and
antifields
also arises naturally from the conformal field theory. In string field
theory there is no need to go through the steps listed in the above
paragraph.
In the present section I will explain in detail why the string field
can be broken naturally into fields and antifields. This happens because
in the state space ${\cal H}_{\hbox{CFT}}$ one can introduce a symplectic structure.
In understanding this point we will have to set up the kinetic term
of closed string field theory.
Batalin-Vilkovisky quantization in covariant open string field theory was
developed by Thorn [\thorn ,\thornpr ] and Bochicchio [\bochicchio ].
Hata found BV quantization relevant to the study of the unitarity of
the HIKKO (light-cone-style) closed string field theory [\hata ]
\section{Symplectic Vector Spaces}
In ordinary symplectic geometry, a symplectic vector space is a real
vector space $V$ equipped with a nondegenerate bilinear
two form $\omega$ (taking $V\otimes V$ to $R$). The antisymmetry property of
the form implies that given two
vectors $X, Y \in V$, $\omega (X, Y) = - \omega (Y , X)$. The nondegeneracy
property requires that whenever $\omega (X, Y) = 0,\, \forall Y$, this
must imply that $X=0$. Using a basis we can write explicitly
$\omega (X,Y) = \omega_{IJ}X^I Y^J$, and nondegeneracy implies that the matrix
$\omega_{IJ}$ is invertible. We let $\omega^{IJ}$ denote the inverse
matrix. A symplectic vector space must be even dimensional,
and one can always find a symplectic basis $(X_1,\cdots ,X_n \, ; Y^1,\cdots ,
Y^n)$ such that $\omega (X_i , Y^j) = \delta_i^j$, and, $\omega (X_i , X_j ) =
\omega (Y^i , Y^j ) = 0$. This is how the symplectic structure can be used
to provide a pairing between basis vectors.
For the cases of vector spaces whose vectors can be either even or odd
objects, as is the case for the conformal field theories relevant to us,
we must consider the super case. We then have a super vector space
with an odd, bilinear, nondegenerate two form $\omega$. The form being
odd means that $\omega (A,B)$, for vectors $A$ and
$B$ of definite statistics, is nonvanishing only if $A$ and $B$ have
opposite statistics. Bilinearity and
nondegeneracy have exactly the same meaning as in the
commuting case. Finally, the exchange symmetry of a (super) two form
requires that
$$\omega (A, B) = -(-)^{AB} \, \omega ( B,A) ,\eqn\exchpro$$
where $A,B$ appearing in the exponent refer to the Grassmanality of the
object (0[mod 2] for even objects, and, 1[mod 2] for odd objects). For even
vectors this
gives the expected minus sign. A (super) vector space equipped with
such (super) symplectic structure, as in the bosonic case, admits a
symplectic basis, which determines a pairing of basis vectors.
We want to exhibit explicitly this pairing for ${\cal H}_{\hbox{CFT}}$.
I want to show why \exchpro\ is compatible with the
exchange properties of the antibracket. The antibracket is defined
on a (super)symplectic manifold, that is, a manifold equipped with a
odd nondegenerate {\it closed} two form $\omega$. At every point on the
manifold the tangent space is a symplectic vector space. Given two
functions ${\cal A}$ and ${\cal B}$ on the manifold, the antibracket is defined
as follows
$$\{ {\cal A} , {\cal B} \} = \omega ( A , B), \eqn\antbder$$
where $A,B$ are the Hamiltonian vector fields associated to the functions
${\cal A} ,{\cal B}$. Given the standard relation $i_{{}_A} \omega = -\hbox{d}{\cal A}$, between
functions and their corresponding Hamiltonian vectors, we note that they have
opposite
statistics. Therefore
$$\{ {\cal A} , {\cal B} \} = \omega ( A , B)= -(-)^{AB}\, \omega ( B , A)
= -(-)^{({\cal A}+1)({\cal B}+1)}\, \{ B , A \} ,\eqn\antbder$$
which is the correct exchange property of the antibracket.
\section{Ghost Conformal Field Theory}
The conformal field theories relevant for string theory are
those with total central charge zero. These conformal theories must
include the conformal field theory of the reparametrization ghosts,
having central charge $(-26)$, together with some other conformal theories
adding up to a central charge of $(+26)$. We need to know the
basics of this ever present ghost conformal field theory. We
consider the conformal field theory formulated in the $z$-plane
with $z= \exp (\tau + i\sigma )$.
We have ghost fields $c(z)$ and $ \overline c ({\overline z} )$
of dimensions $(-1,0)$ and $(0,-1)$ respectively, and
antighost fields $b(z)$ and $\overline b ({\overline z})$ of dimensions
$(2,0)$ and $(0,2)$ respectively:
$$c(z) = \sum_n {c_n\over z^{n-1}},\quad
\overline {c}({\overline z} ) = \sum_n {\bar{c}_n\over\bar{z}^{n-1}},\eqn\first$$
$$b(z ) = \sum_n {b_n\over z^{n+2}},\quad
\overline {b}({\overline z} ) = \sum_n {\bar{b}_n\over \bar{z}^{n+2}}.\eqn\second$$
The stress tensor corresponding to this conformal theory is given by
$$T_g(z) = -2b(z)\cdot \partial c(z) -\partial b(z) \cdot c(z),\eqn\ghset$$
with a similar relation for the antiholomorphic piece. The basic operator
product expansion is
$$b(z)c(w) \sim {1\over z-w} . \eqn\opebc$$
The modes satisfy the anticommutation relations
$$\{b_n , c_m\} = \{ \bar{b}_n , \bar{c}_m \} = \delta_{m+n,0},\eqn\anticom$$
with all other anticommutators equal to zero.
It is convenient to define new zero modes by linear combinations
of the old ones
$$c_0^\pm={1\over 2}( c_0 \pm \bar{c}_0),
\quad b_0^\pm = b_0\pm\bar{b}_0.\eqn\newzm$$
One defines an in-vacuum $\ket{0} \in {\cal H}_{\hbox{CFT}}$, corresponding to $z=0$,
with no operator inserted there, and an out-vacuum $\bra{0} \in
{\cal H}^*_{\hbox{CFT}}$, corresponding to $z=\infty$, with no operator inserted
there.
It follows from the regularity of the conformal fields $c$ and $\bar c$ at
$z=0$, and $z=\infty$, that the oscillators $(c_{-1},\bar c_{-1}, c_0^+, c_0^-,
c_1, \bar c_1)$ do not annihilate the in-vacuum nor the out-vacuum. This
requires that the basic overlap be of the form
$$\bra{0} c_{-1}\bar{c}_{-1}c_0^+ c_0^- c_1 \bar{c}_1
\ket{0} \sim 1 \, . \eqn\begininner$$
We define the first quantized ghost number operator $G$ by
$$ G = 3+ \biggl[ {1\over 2} (c_0 b_0 - b_0c_0) +
\sum_{n=1}^\infty (c_{-n}b_n -b_{-n}c_n) + \hbox{a.h.} \biggr] .
\eqn\ghostdef$$
The reader should verify that $G\ket{0} = 0$. This operator satisfies
the following commutation relations
$$[G,c_n ] = c_n, \quad [G, b_n] = -b_n, \quad [G, Q] = Q, \eqn\ghostcomm$$
and analogous relations for the antiholomorphic objects.
\section{Symplectic Structure on ${{\cal H}'}_{\hbox{cft}}$}
Once the ghost conformal field theory is combined with a matter
conformal field theory, the total stress tensor is the sum of the matter stress
$(T_m(z),\overline T_m(\overline z))$, and the ghost stress tensor
$(T_g(z),\overline T_g(\overline z))$. It has central charge zero
and dimension two
$$T(z) = \sum_n {L_n \over z^{n+2}}, \quad
\overline T(z) = \sum_n {\overline L_n \over \overline z^{n+2}}.\eqn\oevo$$
The BRST operator of the combined conformal theory is given by
$$Q = \int {dz\over 2\pi i} c(z) \bigl( T_m(z)+{1\over 2}T_g(z)\bigr)
+\int {d\overline z\over 2\pi i} \overline c(\overline z) \bigl( \overline T_m(\overline z)+ {1\over 2}
\overline T_g(\overline z)\bigr),\eqn\brstdef$$
where the operators in the integrand are normal ordered. We can verify that
$$\{ Q , b(z) \} = T_m(z)+T_g(z) = T(z),\quad
\{ Q , \overline b(z) \} = \overline T_m(\overline z)+\overline T_g(\overline z) = \overline T(\overline z).
\eqn\frbb$$
\subsection{Reflector State}
Let $\ket{\Phi_i} \in {\cal H}_{\hbox{CFT}}$ be a basis for states, and
$\bra{\Phi^i} \in {\cal H}^*_{\hbox{CFT}}$ be the dual basis:
$\bra{\Phi^j}\Phi_i\rangle =
\delta_i^j$. In any conformal theory there is an association of surfaces
to states. Consider the two punctured sphere with uniformizing coordinate
$z$, with a puncture at $z=0$, and local coordinate $z_1=z$ at that point,
and a puncture at $z=\infty$, with local coordinate $z_2 = 1/z$ at that
point. The state $\bra{R_{12}} \in {\cal H}^*_{\hbox{CFT}}\otimes {\cal H}^*_{\hbox{CFT}}$
corresponding to this surface is defined via the relation
$$\bra{R_{12}}\Phi_i\rangle_{(1)}\ket{\Phi_j}_{(2)} \equiv \langle
\Phi_i\, (z_1=0)\, \Phi_j\, (z_2=0)\rangle \equiv G_{ij},\eqn\metrdef$$
namely, the components of the state $\bra{R_{12}}$ are given by correlators
on the above two punctured sphere. The local coordinates we have chosen
are necessary to be able to define the correlators of local operators
which are not dimension zero primaries. The metric $G_{ij}$ must be
nondegenerate.
We also have the useful relations
$$\eqalign{
{}&\bra{R_{12}} (c_n^{(1)} + c_{-n}^{(2)} ) = 0,\cr
{}&\bra{R_{12}} (b_{n}^{(1)} - b_{-n}^{(2)}) = 0,\cr
{}&\bra{R_{12}} (Q^{(1)} + Q^{(2)}) = 0, \cr
{}&\bra{R_{12}} (G^{(1)} + G^{(2)} -6) = 0.
\cr}\eqn\qonref$$
The first two identities, which hold for all $n$, can be obtained by
expressing the oscillators in terms of contour integrals of the ghosts
(or antighost) conformal fields, and using the definition \metrdef . The
last two identities also follow from contour deformation, both the
BRST charge and the ghost charge are contour integrals of holomorphic
currents. The last identity also follows directly from the first
two ones, and Eqn.\ghostdef ; it implies that a nonvanishing
overlap $\bra{R_{12}}A\rangle_{(1)}
\ket{B}_{(2)} \not= 0$ requires that the ghost numbers of $A$ and $B$
add up to six: $G(A) + G (B) = 6$.
\subsection{A Kinetic Term for Closed String Fields}
There is no unique symplectic structure that can be introduced
in ${\cal H}_{\hbox{CFT}}$. We need to understand which is the physically
relevant one. The answer emerges clearly, as I will show now, from
an analysis of equations of motion for closed string fields.
It was long felt that a satisfactory linearized closed string field equation
would be
$Q \ket{\Psi} = 0$, possibly supplemented with a ghost number constraint
for the string field. This would have been the analog of the open string
linearized field equation. It is very clear now that this could not
be the correct answer. It is interesting that the difficulty and its
resolution can be understood by attempting to write a kinetic term
for string fields that would give $Q\ket{\Psi}=0$ as an equation of motion.
Due to the nondegeneracy of the reflector an obvious choice for
kinetic term would seem to be
$$S_0^2\, \sim \,\bra{R_{12}} \Psi \rangle_{(1)} \, Q^{(2)}\,
\ket{\Psi}_{(2)}\,\quad ?\eqn\nonum$$
The problem now is ghost number. Since the vacuum was assigned
ghost number zero, and we only have operators of integer ghost number,
the ghost number of the string field should be an interger. Moreover,
the ghost numbers of $\ket{\Psi}$ and that of $Q\ket{\Psi}$ must add up
to six. This is clearly impossible, as it would require a string field
of ghost number $5/2$. Therefore the above candidate action does not
work. The fact that closed string vertex operators are usually of
the form $c(z) \overline c (\overline z) V(z,\overline z)$, with $V$ a matter operator,
suggests that the string field ought to be of ghost number $+2$. In
that case the kinetic operator $Q$ of ghost number $+1$ should be
replaced by an operator of ghost number $+2$.
A clue emerges when we note that we can restrict ourselves
to work with the subspace of the state space consisting of states that are
annihilated by $L_0 -\overline L_0$. We do not lose anything because any physical
state (a state annihilated by $Q$) which is not annihilated by $L_0 -\overline L_0$
is actually trivial (can be written as $Q$ acting on something). Indeed,
let $h-\overline h$ denote the eigenvalue of $L_0-\overline L_0$ on $\ket{\Psi}$, we then
have
$$\ket{\Psi} = (L_0-\overline L_0) \cdot {1\over h-\overline h} \,\ket{\Psi}=
\{ Q, b_0-\overline b_0 \} \cdot {1\over h-\overline h}\, \ket{\Psi}
= Q \cdot { b_0-\overline b_0 \over h-\overline h}\, \ket{\Psi}, \eqn\trivin$$
showing that indeed, such states are trivial. As a subsidiary
condition $L_0-\overline L_0 =0$ is not a strong condition, even with this
condition the string field still can be off the mass shell. The same
would not be true for the condition $L_0 + \overline L_0 =0$,
such a condition would impose a familiar field equation.
Our ability to impose an $L_0-\overline L_0=0$ condition suggests that we could
impose a further condition on the off shell string field; we could demand that
$(b_0 - \overline b_0) \ket{\Psi} = 0$. This condition cannot be justified
as we justified the $(L_0 -\overline L_0) = 0$ condition. We {\it cannot} prove that
a physical state which is not annihilated by $b_0-\overline b_0$ must be
trivial. Therefore this condition has a nontrivial effect on the definition
of physical states.
We may reasonably expect not to get in trouble
since on physical states $Q\ket{\Psi} = 0$, the condition
$(b_0 -\overline b_0 ) \ket{\Psi} =0$ does not lead to further constraints, as
we have already required that $L_0 -\overline L_0 = \{ Q, b_0-\overline b_0 \} $ annihilate
all states. The cohomology of $Q$ on the space of states annihilated
by $b_0-\overline b_0$ is called semirelative cohomology.
All of the pieces of the puzzle are now in place. If we impose
a $b_0^-(=b_0-\overline b_0)$ condition on all states, the overlap
$\bra{R_{12}} A\rangle \ket {B}$ would always vanish. This is
not hard to see. A state annihilated by $b_0^-$ can always be written
as $b_0^-$ acting on something ($b_0^- \ket{\alpha} = 0 \,\to
\ket{\alpha} = b_0^- c_0^- \ket{\alpha}$ since no state can be simultaneously
annihilated by $b_0^-$ and $c_0^-$). Since $b_0^-$ acting on the reflector
gives us another $b_0^-$ (Eqn.\qonref ) the relation $b_0^-b_0^- =0$
implies that we always get zero.
The solution is clear, a nondegenerate inner product must include
a $c_0^-$. We are therefore led to define
$$\langle A , B \rangle \equiv \bra{R_{12}}~A\rangle_{(1)} \, c_0^{-(2)}
\, \ket{B}_{(2)}.\eqn\inpdef$$
This bilinear inner product has the following exchange property
$$\langle A , B \rangle = (-)^{(A+1)(B+1)} \langle B , A \rangle,\eqn\skd$$
which follows from the symmetry of the reflector $\bra{R_{12}}$ under the
exchange of state spaces (see [\zwiebachl ] for details).
Restricting ourselves to ${{\cal H}'}_{\hbox{CFT}}$, the subspace of
${\cal H}_{\hbox{CFT}}$ where all states are annihilated by $b_0^-$ and
$L_0^-$, we have that
\noindent
\subsection{Claim:~} $\langle \,\, , \,\, \rangle$ is nondegenerate on
${{\cal H}'}_{\hbox{CFT}}$.
Here is a proof. Assume $\langle A , B \rangle = 0$ , for all $\ket{A}
\in {{\cal H}'}_{\hbox{CFT}}$. It follows from
$$\langle A , B \rangle = \bra{R_{12}}~A\rangle_{(1)} \, c_0^{-(2)}
\, \ket{B}_{(2)}=0, \eqn\vanisheq$$
that the inner product actually vanishes for all $\ket{A}\in
{{\cal H}}_{\hbox{CFT}}$.
This is the case because any state that is not annihilated by $b_0^-$ can
be written as $c_0^-$ acting on a state. The $c_0^-$ acting on the
reflector gives us another $c_0^-$ which kills the $c_0^-\ket{B}$ state.
The nondegeneracy of the metric arising from $\bra{R_{12}}$ therefore
implies that $c_0^-\ket{B} =0$. Multiplying by $b_0^-$ we find that
$\ket{B}=0$, thus establishing the claim.
Another basic property of this inner product is that, on ${{\cal H}'}_{\hbox{CFT}}$
the BRST operator satisfies
$$\langle QA , B \rangle = (-)^A \langle A , QB \rangle . \eqn\brstinp$$
This is readily proven:
$$\eqalign{\langle QA , B \rangle &=
\bra{R_{12}}Q^{(1)}\ket{ A}_{(1)} \, c_0^{-(2)} \ket{B}_{(2)}\cr
&= (-)^{A+1} \bra{R_{12}}A\rangle_{(1)} \, Q^{(2)}c_0^{-(2)} \ket{B}_{(2)}\cr
&= (-)^{A+1} \bra{R_{12}}A\rangle_{(1)}(b_0^-c_0^-+c_0^-b_0^-)^{(2)}
\, Q^{(2)}c_0^{-(2)} \ket{B}_{(2)}\cr
&= (-)^{A+1} \bra{R_{12}}A\rangle_{(1)}\, c_0^{-(2)}b_0^{-(2)}
\, Q^{(2)}c_0^{-(2)} \ket{B}_{(2)},\cr}\eqn\gthere$$
where we introduced $1=\{b_0^-,c_0^-\} $, and in the next step the
$b_0^-c_0^-$ term was seen to vanish upon taking $b_0^-$ into the reflector.
Since the anticommutator of $b_0^-$ with $Q$ gives $L_0^-$, which
kills $c_0^-\ket{B}$, we find
$$\eqalign{\langle QA , B \rangle
&= (-)^A \bra{R_{12}}A\rangle_{(1)}c_0^{-(2)}
\, Q^{(2)}b_0^{-(2)}c_0^{-(2)} \ket{B}_{(2)},\cr
&= (-)^A \bra{R_{12}}A\rangle_{(1)}c_0^{-(2)}
\, Q^{(2)} \ket{B}_{(2)},\cr
&=(-)^A \langle A , QB \rangle , \cr} \eqn\brstxnp$$
as we wanted to show.
All the above makes it clear that the correct kinetic term
for closed string field theory is
$$S_0^2= {1\over 2} \, \langle \, \Psi\, , \, Q \Psi \, \rangle \,\, ,\,\,
\quad \ket{\Psi} \in {{\cal H}'}_{\hbox{CFT}}.\eqn\kincsft$$
The equation of motion, which follows upon variation, use of the
nondegeneracy of the inner product, and Eqn.\brstinp , is $Q\ket{\Psi} =0$.
This action is gauge invariant under $\delta \ket{\Psi} = Q \ket{\Lambda}$,
with $\ket{\Lambda} \in {{\cal H}'}_{\hbox{CFT}}$. For the classical string
field theory one must restrict the sum over states in $\ket{\Psi}$ to
states of ghost number $(+2)$. It turns out that the kinetic term of the
master action is given by the same formula, the only difference being that
the string field satisfies no ghost number condition.
\subsection{Symplectic Structure in ${{\cal H}'}_{\hbox{CFT}}$.}
The physically relevant bilinear nondegenerate two-form we must choose is
therefore
$$\omega (A, B) \equiv (-)^A \, \langle\, A \, , \, B \, \rangle
.\eqn\symhcft$$
Indeed this is a bilinear, nondegenerate pairing. The sign factor in front
was introduced to get the correct exchange property
$$\eqalign{
\omega (A, B) &= (-)^A \, \langle\, A \, , \, B \, \rangle \cr
&= (-)^{A+(A+1)(B+1)} \langle\, B \, , \, A \rangle \cr
&= (-)^{A+(A+1)(B+1)+B }\, \, \omega ( B \, , \, A \, ) \cr
&= -(-)^{AB}\, \, \omega ( B \, , \, A \, ) ,\cr}\eqn\symhcft$$
as we wanted to show (Eqn.\exchpro )
\subsection{BV structure in Spacetime.}
We can now use the pairing of states in ${{\cal H}'}_{\hbox{CFT}}$ to pair
up the spacetime fields, which are nothing else but the expansion
coefficients of the string field $\ket{\Psi}$. In all generality
we may choose a basis where we pair up the states
$$\Bigl\{\, \ket{\Phi_1}, \ket{\Phi_2}, \cdots \, \Bigr\} \, \,\,
; \, \,\, \Bigl\{\,
\ket{\widetilde \Phi^1},\ket{\widetilde \Phi^2}\cdots \,\, \Bigr\} \eqn\pairu$$
with the condition
$$\omega \, ( \, \ket{\Phi_i} \, , \, \ket{\widetilde \Phi^j} \, ) = \delta _i^j .
\eqn\fgws$$
Let us denote the first group of states by $\ket{\Phi_s}$. It is
convenient to have those fields be of ghost number less than or
equal to $+2$. It then follows that the states $\ket{\widetilde\Phi^s}$ are
all of ghost number greater than or equal to $+3$ (with our inner
product with $c_0^-$, the nonvanishing condition demands that the ghost
numbers should add up to five). Then the expansion of the string field
reads
$$ \ket{\Psi} = \sum_s \Bigl( \, \ket{\Phi_s} \, \psi^s + \ket{\widetilde\Phi^s}
\, \psi^*_s\, \Bigr) .\eqn\strfld$$
The spacetime fields $\psi^s$, and spacetime antifields $\psi_s^*$ are paired.
This is how spacetime fields
and antifields arise. The string field is defined to be Grassmann even.
Since the statistics of $\ket{\Phi_s}$ and $\ket{\widetilde\Phi^s}$ are opposite,
the statistics of $\psi^s$ and that of $\psi^*_s$ must also be opposite.
We define the spacetime ghost number $g^t$ of a spacetime field to be equal
to $2-G$, with $G$ the ghost number of the corresponding
first quantized state. Then, we readily find that
$g^t (\psi^s) + g^t (\psi^*_s) =
4- (G_s + (5-G_s)) = -1$. These are standard properties of the BV
pairing. The reader may wish to verify that one can define explicitly
the tilde states as $\ket{\widetilde\Phi^j} = b_0^-\, G^{ji}\,\ket{\Phi_i}$.
In some sense this is the beginning of all the algebraic work in setting up
closed string field theory. I have tried in the above to be very
explicit about all of the basic and fundamental points. The discussion
of the construction of the complete string field action still requires
further work. In the next section I will give a brief discussion of
some of the remaining issues and of recent developments.
\chapter{Recent Developments}
We discussed in \S2 and \S3 how an extremal problem in Riemann surface
theory allows us to find string vertices, and how we go about setting
up the Batalin-Vilkovisky field-antifield structure of string field
theory. In order to write the string field theory explicitly one needs
to develop the differential geometry on an extended space
$\widehat{\cal P}_{g,n}$ fibered over the space ${\cal M}_{g,n}$ of Riemann surfaces
with $n$ punctures and genus $g$ (for details consult
[\alvarez ,\nelson ,\zwiebachl ,\hatazwiebach ]). The fiber over a surface ${\cal R}$
corresponds to all possible choices of local coordinates, up to a phase, at
every puncture of ${\cal R}$. The string vertex ${\cal V}_{g,n}$ is properly thought as
a subspace of $\widehat{\cal P}_{g,n}$. One can define differential forms
$\Omega^{(k)}_{\bf \Psi}$ of degree $k$ on
$\widehat{\cal P}_{g,n}$. The forms are labeled by ${\bf \Psi}$, which denotes the $n$
states in ${{\cal H}'}_{\hbox{CFT}}$
to be inserted on the punctures of the
surface. These forms satisfy very nice relations that tie up the differential
geometry of $\widehat{\cal P}_{g,n}$ and the algebraic structures in the conformal theory.
For example, the role of the exterior derivative $d$ is played by the BRST
operator: $d\Omega^{(k)}_{\bf \Psi} \sim \Omega^{(k+1)}_{Q {\bf \Psi}}$.
The Lie derivative along some vector field $U$ in $\widehat{\cal P}_{g,n}$ is represented
by a stress energy insertion: ${\cal L}_{{}_U} \Omega_{\bf \Psi} = \Omega_{\bf
T(u) \Psi}$, with ${\bf u}$ a `Schiffer' vector on the surface representing
the vector $U$. Finally, the role of the contraction operator on forms is
played by an antighost insertion: $i_{{}_U} \Omega^{(k+1)}_{\bf \Psi} =
\Omega^{(k)}_{\bf b(u) \Psi}$. In addition to providing conceptual
understanding, such relations simplify the verification that the string
action satisfies the master equation.
The understanding of the geometry of BV quantization has improved considerably
thanks to the work of A. Schwarz [\schwarz ]. His work allows a covariant
description, in the sense of symplectic geometry, of the BV formalism.
The BV master equation, as originally formulated reads
$$\hbar \Delta S +{1\over 2} \{ S,S\} =0 , \eqn\bveqnshort$$
where the antibracket $\{ \cdot , \cdot \}$ and the $\Delta$
operator are given by
$$\{ G , H \} = {\partial_r G \over \partial \psi^s}
{\partial_l H \over \partial \psi_s^*}-
{\partial_r G \over \partial \psi_s^*}
{\partial_l H \over \partial \psi^s}, \quad
\Delta =(-)^{\phi_s}{\partial_l \over \partial \psi^s}
{\partial_l \over \partial \psi_s^*}.\eqn\defbracket$$
While the antibracket has a covariant expression using the symplectic
two form of the manifold: $\{ G, H\} \sim \partial_I G \, \omega^{IJ}
\partial_J H$, the second order differential operator
$\Delta$ is manifestly noncovariant.
The solution [\schwarz ] was to interpret $\Delta$ as an operator which
acting on functions gave the divergence of the corresponding Hamiltonian
vector field. In order to define a divergence, however, one must introduce
a volume element $d\mu = \rho \prod_I dz^I$ on the symplectic
manifold. Consequently the delta operator
$\Delta_\rho$ acquires a $\rho$ dependence. The main result of [\schwarz ]
is that $\rho$ must be chosen so that $\Delta_\rho^2 =0$. If this is
satisfied one can prove the existence of a `Darboux-Schwarz' system of
coordinates where the symplectic form $\omega$ becomes a constant and
$\rho =1$. Such system of coordinates seems necessary to carry out the
standard BV argument for the gauge independence of the observables.
In summary, a BV manifold is the object $(M,\omega ,d\mu )$, with $M$
a supermanifold, $\omega$ an odd, symplectic, closed two-form, and $\mu$ a
volume element that leads to a nilpotent operator $\Delta$.
As we discussed in \S1 there are indications that string field theory
may be thought as a function on the space of two dimensional field theories.
Nobody has been able to understand this space concretely. I believe it is
likely that string field theory will be eventually formulated on some abstract
space closely related to a theory space, or to theory space with some extra
conditions.
As we have seen the relevant state space of conformal theories including
ghosts have a symplectic structure. It seems clear that this state space
corresponds to the tangent space to `theory space'. This suggests that
theory space may be a symplectic manifold. This idea was advocated by
Witten [\witten ] who also proposed a concrete construction for the
case of open string field theory. In this case, theory space is the space
of one dimensional lagrangians defined on the boundary of a disk. Rather
than finding an action $S$ (which would be a function in this theory space)
satisfying the master equation, one looks for
the corresponding Hamiltonian vector field $V_S$ . This vector must be odd and
should satisfy the condition $\{ V_S , V_S\} =0$ where the bracket is the
graded
Lie bracket. This equation, which can also be written as $V^2_S =0$, must hold
everywhere, as it guarantees that the master equation is satisfied everywhere.
At the points where $V_S=0$, we have conformal field theories.
The setup is very attractive, although it seems likely that the explicit
construction needs further work [\liwitten ,\shatashvili ]. In order to
formulate a complete quantum theory we need to obtain the full master
equation. This would mean that theory space should be a BV manifold in the
full sense, that is, an object $(M,\omega ,d\mu)$, with $d\mu$ a consistent
volume element. This volume element has not been found yet. Nevertheless
for a complete formulation the equation we must solve is not $V^2_S=0$ but
rather $V^2_S = -\hbar\, V_{\hbox{div}V_S}$ [\hatazwiebach ].
Hata and the author [\hatazwiebach ] have found the BV approach useful to
understand the
significance of the freedom to choose different sets of string vertices
${\cal V}_{g,n}$ in order
to reproduce the sum over surfaces. More concretely, one wished to know
what is the relation between string field theories that use different
ways of breaking up the sum over surfaces. The answer is simple: two such
string field master
actions differ by terms induced by a field-antifield transformation
canonical with respect to the antibracket. It is possible
to use the differential geometry language described above in order to write
explicitly the generator of the canonical transformation in terms of the
vector field $U$ in $\widehat{\cal P}_{g,n}$
generating the variation of the string vertices ${\cal V}_{g,n}$. Using the
covariant description of gauge fixing one can then show that two theories
using different decompositions of moduli space yield the
same gauge fixed action upon use of different gauge fixing conditions.
During the last year the algebraic structures that arise in closed string
field theory have been brought into the open [\wittenzwiebach ,
\everlinde ,\zwiebachl]. The first object to emerge was a homotopy Lie
algebra.
In some sense homotopy Lie algebras (and their more familiar relatives,
the homotopy associative algebras [\stasheff ]) are the answer to a
longstanding
question. We have long suspected that string theory on-shell amplitudes
have a group theoretical meaning, indeed, special three point functions have
been related to structure constants of Lie algebras. The three point
amplitudes,
together will all the higher $n$-point functions (at the classical level)
define the structure constants of a homotopy Lie algebra~! This structure
can be readily explained. Suppose we have a graded Lie algebra
$\{ T_a , T_b \} = f_{ab}^{~c }\, T_c$. Introduce for each generator
$T_a$ an object $\eta^a$ of opposite statistics. Then consider the
following anticommuting vector field
$$V = \bigl( f_{a_1a_2}^{~~~~b} \,\eta^{a_1}\eta^{a_2} +
f_{a_1a_2a_3}^{~~~~~~b} \,\eta^{a_1}\eta^{a_2}\eta^{a_3} + \cdots \, \bigr)\,
{\partial\over \partial \eta^b} .\eqn\hlathree$$
where the $f$'s with more than three indices are the higher structure constants
of the algebra. The condition $V^2=0$ is then imposed. This condition
gives, at lowest order, the Jacobi identity for the original structure
constants of the graded Lie algebra. At higher orders it gives an infinite
set of quadratic constraints on the structure constants. If the condition
$V^2=0$ is satisfied, we have a homotopy Lie algebra.
The physical interpretation arises when
we show that, for string theory, the structure constants are nothing else
than on-shell
scattering amplitudes, and the quadratic relations are the Ward identities
of the theory. In particular, if we label the physical states (BRST
semirelative cohomology classes) as $\ket{\Phi_a}$, the structure constant
$f_{a_1a_2\cdots a_n}^{~~~~~~~~b}$ is simply given by the correlation
function $\langle\langle \Phi_{a_1} \Phi_{a_2} \cdots \Phi_{a_n}\widetilde\Phi^b
\rangle\rangle$ where the double bracket indicates that one integrates over
the positions of the punctures (this is a string amplitude). While this
homotopy Lie algebra exists on the cohomology of $Q$ in ${{\cal H}'}_{\hbox{CFT}}$,
there is a homotopy Lie algebra defined on the full state space
${{\cal H}'}_{\hbox{CFT}}$.
This is the homotopy Lie algebra that underlies classical closed string
field theory, and it corresponds to a simple modification of the previous
equation
$$V = \bigl(\, f_{a_1}^{~~b} \,\eta^{a_1}+
f_{a_1a_2}^{~~~~b} \,\eta^{a_1}\eta^{a_2} +
f_{a_1a_2a_3}^{~~~~~~b} \,\eta^{a_1}\eta^{a_2}\eta^{a_3}\, + \cdots\, \bigr)
\,
{\partial\over \partial \eta^b} .\eqn\hlatwo$$
Here we have added a structure constant with two indices that can be
thought as the matrix elements of some operator: $f_{a_1}^{~~b} =
(Q)_{a_1}^{~~b}$. This time $V^2=0$ requires, as the lowest
consistency condition, that $Q$ be a nilpotent matrix. The next
consistency condition demands that $Q$
be a derivation of a product defined by the $f_{a_1a_2}^{~~~b}$'s.
The third consistency condition is quite amusing. It expresses the fact
that structure constants $f_{a_1a_2}^{~~~b}$ need
not satisfy anymore the Jacobi identity; the Jacobi identity must only
be satisfied weakly, that is, it differs from zero by terms having to
to with $Q$ acting on the structure constants representing four point
correlators. Therefore the $f_{a_1a_2}^{~~~b}$'s of this homotopy Lie algebra
do not define a Lie algebra. The reader may have guessed what is the
physical relevance of this algebra. This time the index $a$ labels a
general state $\ket{\Phi_a}$ in ${{\cal H}'}_{\hbox{CFT}}$. $Q$ is nothing
else but the BRST operator, and the higher structure constant
$f_{a_1a_2\cdots a_n}^{~~~~~~~~b}$ is given by the off shell correlation
$\langle \Phi_{a_1} \Phi_{a_2} \cdots \Phi_{a_n}\widetilde\Phi^b
\rangle$ integrated over the set ${\cal V}_{0,n+1}$ defining the string vertex
coupling $n+1$ strings at genus zero. Thus the off-shell amplitudes
of string field theory, together with the BRST operator form a homotopy
Lie algebra. I would like to emphasize that closed string field theory
seems to be necessarily nonpolynomial. It was shown by Sonoda and the
author [\sonodazw ], under very general conditions, that no cubic interaction
alone
could give a consistent four point amplitude. This actually means that
for covariant closed string field theory there is no clever choice of
$f_{a_1a_2}^{~~~b}$'s that satisfies a strict Jacobi identity. It would
seem that there is no Lie algebra underlying closed string field theory,
just a homotopy Lie algebra. I believe this point should be investigated
more closely.
Finally, consider the most general homotopy Lie algebra:
$$V = \bigl(\, f^{~b} \,+
f_{a_1}^{~~b} \,\eta^{a_1} +
f_{a_1a_2}^{~~~b} \,\eta^{a_1}\eta^{a_2}+ \cdots \, \bigr)\,
{\partial\over \partial \eta^b} .\eqn\hlaone$$
If we view the lowest structure constant $f^b$ as a vector $F$, and
$f_{a_1}^{~~b}$ as a matrix $Q$, this time the lowest identities
following from $V^2=0$ can be written as
$ QF = 0$, and ${Q}^2 B + F \star B = 0$, with $B$ arbitrary, and
where $\star$ denotes the product defined with the structure constants
$f_{a_1a_2}^{~~~b}$. The reader may note that we have lost the nilpotency
of $Q$! This is the algebraic setup necessary for background independence.
Indeed, only when $f^b=0$ we recover the BRST operator, and therefore,
a CFT. This corresponds to a zero of the vector $V$ at $\eta^a=0$.
We recall that $V=0$ is indeed the equation of motion in the setup
of [\witten ]. The homotopy Lie algebra of \hlaone\ can be constructed
indirectly (and in a background dependent way) by shifting closed string
field theory with a string field which is not a classical solution
[\zwiebachl].
If we wish to consider the quantum theory this all gets generalized.
One can write a master equation for the on-shell action [\everlinde ]
and this structure can be seen to arise from the master equation
for the complete off shell action [\zwiebachl]. If we wish to use the
description using an anticommuting vector field, then,
following [\hatazwiebach ], the equation $V^2=0$ must be turned into
$V^2 = -\hbar\, v_{\hbox{div}V}$ (with $v_f$ denoting the Hamiltonian
vector corresponding to the function $f$). Quantum closed string field
theory defines a concrete realization of this algebra (constructed
in a Darboux-Schwarz frame of fields and antifields, with a density $\rho=1$).
This structure may be called a `quantum homotopy Lie algebra'. The
more desirable name of BV algebra is now being used by mathematicians
to describe a differential graded commutative algebra with
a nilpotent second order operator $\Delta$
[\lianzuckerman ,\getzler ,\penkavaschwarz ,\horava ]. The antibracket can be
defined as the failure of $\Delta $ to be a derivation.
Back to theory space and background independence. The problem of background
independence in the simplest setting is that of proving that closed
string field theories formulated around nearby conformal field theories are
actually equivalent. This is not so simple to prove, but has been achieved to
a large degree through the work of A. Sen [\sen ]. I believe a better
geometrical understanding of his result is very much needed to appreciate
well the issues involved in background independence. A host of problems
arise when one realizes that in comparing string field theories formulated
on state spaces ${\cal H}$ and ${\cal H}'$ corresponding to different conformal field
theories many natural identifications of the two spaces become meaningless
as soon as the theories are not infinitesimally away. Some of these
difficulties, whose origin lies in the fact that the state spaces are
infinite dimensional, were found in [\kugozwiebach ]. Together with
K. Ranganathan and H. Sonoda we have investigated geometry of the vector
bundle whose base manifold is a space of conformal field theories and
whose vector space at each point is the infinite dimensional state space
of the theory [\rangasonoda ]. We gave a characterization of the connections
that can be introduced in this vector bundle, and studied special connections
that allow the construction of a conformal theory using the state space of
another theory a finite distance away. I would expect connections to be
a useful ingredient in future analysis of background independence.
As we have reviewed in this section, from the algebraic viewpoint
homotopy Lie algebras and their generalizations seem to play a prominent
role in string field theory. From the geometrical viewpoint we have
seen the role of geometry on moduli spaces of Riemann surfaces,
geometry of BV quantization, and theory space geometry. I look
forward to see the fascinating relations that are likely to be
uncovered between the different geometries. Such developments
should pave the way to a complete formulation of string theory.
\ack
I am grateful to Bernard Julia for his invitation to lecture at Les Houches.
\singlespace
\refout
\end
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,983 |
A contemporary design style offers a clean look to bring to your kitchen. People love the fresh look that clean lines, minimalist shapes, and cool colors provide. The only problem with this style is it can be uninviting. Adding warmth can help the kitchen feel like a cozy gathering place. Here are a few ideas you can incorporate when designing a warm contemporary kitchen.
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For a free in-home consultation, contact Polar Shades at 702-260-6110. We'll help you find the perfect interior window shades to complement your warm contemporary kitchen. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,574 |
Q: Is there a way to get types for @pixi/app (and other pixi modules) Im currently setting up a project using pixiJs and typescript, i want to use the @pixi/app and other pixi packages so that the final bundle is smaller, my current bundle is 387 KiB (without my own code), using @pixi/app, @pixi/loaders and @pixi/sprite my bundle is 211 KiB which is 176 KiB smaller.
The problem is there is no typing that i could find for @pixi/anyPackage, is there a way to get the typing in a way that is not super hacky or do i have to choose between a big bundle and types ?
A: I've quickly searched around and found this. Does it suit your needs?
However, please note that this package isn't for v5.
Note This repository is not intended for v5.x which ships with its own generated declarations.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,487 |
\section*{Introduction}
Since the seminal work of Jacobson \cite{Jacobson:1995ab} which recovered Einstein Equations from the Clausius Relation applied to a local Rindler horizon, many attempts were made to adapt this idea to a Friedmann-Lemaître-Robertson-Walker spacetime. The first challenge was to find the right horizon on which to work. The local Rindler horizon, where the expansion $\theta$ of null geodesic congruence vanishes, finds its quasi-local equivalent in the cosmological apparent horizon, on which the ingoing expansion $\theta_-$ vanishes \cite{Bak:1999hd}. Then the Unified First Law of Hayward \cite{Hayward:1997jp}, initially designed for black holes, was identified as the relevant relation, along with the tools needed to build it (Misner-Sharp energy \cite{Misner:1964je}, Kodama vector \cite{Kodama:1979vn}). In particular, the associated, dynamical surface gravity was computed. However, no one succeeded in establishing Friedmann Equations from the Unified First Law using this dynamical $\kappa$ (only the reverse way, from gravity to thermodynamics, was achieved \cite{Cai:2006rs}).
Very recently, we presented a clean and perfectly well-defined way of recovering Friedman from the UFL \cite{Binetruy:2014ela}. Nevertheless, later works continued to claim that the dynamical surface gravity was not relevant, and that the static form $\kappa=1/R_A$ was to be used.
In the present work, we intend to clarify the situation. In Section \ref{section_dynamical_setup}, we present the tools we need in our dynamical, spherically symmetric framework. In Section \ref{section_Friedmann_Eq}, we reproduce the computation of our previous work \cite{Binetruy:2014ela}, in a more detailed manner. Section \ref{section_misconceptions} compares the logic of our derivation with the logic of the usual, approximate computation that was up to now found in the literature. This will give us some extra information on the way to project the Unified First Law. Section \ref{section_temperature} shows how to relate our surface gravity to a physical notion of temperature, in the tunneling method for modelling Hawking radiation. In the last section, we show that the sign of our temperature does not depend on the causal nature of the horizon, and that a change of sign occurs only when one departs from description of a past-inner horizon. We conclude that the dynamical surface gravity and associated temperature are perfectly well-defined for our expanding Universe, and pose no problem in recovering the Friedmann Equations from the Unified First Law.
\section{The Unified First Law in spherically symmetric spacetimes}
\label{section_dynamical_setup}
\subsection{Hayward's Unified First Law}
We will work here in the context of spherically symmetric spacetimes:
\begin{equation}
ds^2 = \gamma_{ij}(x)dx^idx^j +R^2(x)d\Omega^2 \ ,
\label{eq_spherical}
\end{equation}
with signature (-,+,+,+) and coordinates $x^0=t$, $x^1=r$, R being a function of $t$ and $r$.
Within this symmetry, a relevant definition for the energy is the Misner-Sharp one :
\begin{equation}
E(R) \equiv \frac{R}{2G}(1-\nabla^a R \nabla_a R ) \ ,
\label{eq_Misner_Sharp}
\end{equation}
which is the total gravitational energy inside a sphere of physical radius $R$ \cite{Misner:1964je}. On the apparent/trapping horizon, $\nabla^a R \nabla_a R=0$, and the Misner-Sharp energy reduces to the Schwarzschild mass.
Another important quantity is the energy-supply covector:
\begin{equation}
\psi_a \equiv T_a^{\phantom{a}b}\nabla_b R + \omega \nabla_a R \ ,
\label{eq_energy_supply}
\end{equation}
with an energy-momentum tensor $T_{ab}$, the trace of which defines:
\begin{equation}
\omega \equiv -\frac{1}{2}T^{ij} \gamma_{ij} \ .
\label{eq_omega}
\end{equation}
Then the Unified First Law of Thermodynamics established by Hayward \cite{Hayward:1997jp} is:
\begin{equation}
\nabla_a E = A \psi_a + \omega \nabla_a V \ ,
\label{eq_Unified_First_Law}
\end{equation}
where we use flat area and volume, $A=4\pi R^2 \ $ and $V=\frac{4}{3}\pi R^3$.
\subsection{Definition of the surface gravity for dynamical spacetimes}
In static spacetimes, one may encode the time-translational invariance into a Killing vector field $\xi^a$, which satisfies:
\begin{equation}
\nabla_a \xi_b + \nabla_b \xi_a=0 \ .
\label{eq_Killing}
\end{equation}
The usual notion of surface gravity $\kappa$ is defined from the Killing:
\begin{equation}
\xi^a(\nabla_a \xi_b - \nabla_b \xi_a)=2\kappa \xi_b \ .
\label{eq_surface_grav_static}
\end{equation}
However, in a dynamical spacetime, one does not have a time-translational Killing field anymore. Still, this notion may be generalized to the notion of Kodama field $K^a$ \cite{Kodama:1979vn}:
\begin{equation}
K^a \equiv \epsilon^{ab}_{\perp} \nabla_b R \ ,
\label{eq_Kodama}
\end{equation}
where $\epsilon^{ab}_{\perp}$ is the (1+1) antisymmetric tensor in the (t,r) plane. Unlike the Killing, the Kodama does not give a direction of invariance in time, but it still provides a preferred time-direction. It therefore does not satisfy to the Killing Equation Eq.\eqref{eq_Killing}, but to a slightly altered form of it:
\begin{equation}
K^a(\nabla_a K_b + \nabla_b K_a)= 8\pi G R \psi_b \ .
\label{eq_Kodama_equation}
\end{equation}
One may then understand the energy-supply as a measure of departure from static symmetry. Both the Misner-Sharp energy and the energy-supply vector may be expressed in terms of the Kodama vector, which we take as a sign of coherency between these quantities.
The defining equation Eq.\eqref{eq_surface_grav_static} for the surface gravity $\kappa$ must also be adapted, since $K^a (\nabla_a K_b - \nabla_b K_a)$ is no longer colinear to $K_b$:
\begin{equation}
K^a(\nabla_a K_b - \nabla_b K_a)=2\kappa \nabla_b R \neq 2\kappa K_b \ .
\label{eq_surface_grav_dyn}
\end{equation}
We thus have a perfectly well-defined notion of surface gravity. It may be expressed as:
\begin{align}
&\kappa = \frac{1}{2\sqrt{-\gamma}}\partial_i[\sqrt{-\gamma}\gamma^{ij}\nabla_j R]
\label{eq_surface_grav1} \\
\text{or } \nonumber \\
&\kappa = G\frac{E}{R^2} - 4\pi \omega R G \ .
\label{eq_surface_grav2}
\end{align}
This is sometimes referred to as the ``Hayward-Kodama'' surface gravity in the literature.
\subsection{Reformulation of the Unified First Law}
Starting from Eq.\eqref{eq_Unified_First_Law}, we may now rewrite the Unified First Law as:
\begin{align}
A\psi_a &= \nabla_a E - \omega \nabla_a V \nonumber \\
&= R \nabla_a \left(\frac{E}{R}\right) + \frac{E}{R} \nabla_a (R) -\frac{\omega}{3}(A\nabla_a R+R\nabla_a A) \nonumber \\
&= R \nabla_a \left(\frac{E}{R}\right) + \frac{E}{8\pi R^2} \nabla_a A -\frac{\omega}{3}\left(\frac{A}{8\pi R}\nabla_a A +R\nabla_a A\right) \nonumber \\
&= R \nabla_a \left(\frac{E}{R}\right) + \nabla_a A \left( \frac{E}{8\pi R^2} -\frac{\omega}{3}\frac{A}{8\pi R} -\frac{\omega R}{3}\right) \nonumber \\
&= R \nabla_a \left(\frac{E}{R}\right) + \nabla_a A \left( \frac{E}{8\pi R^2} -\frac{\omega R}{6} -\frac{\omega R}{3}\right) \nonumber \\
&= R \nabla_a \left(\frac{E}{R}\right) + \frac{1}{8\pi G} \nabla_a A \left( G\frac{E}{R^2} -4\pi \omega R G \right) \nonumber \\
&= R \nabla_a \left(\frac{E}{R}\right) + \frac{\kappa}{8\pi G} \nabla_a A
\label{eq_unified_first_law_reform}
\end{align}
where in the last equality we recognized the surface gravity $\kappa$ from Eq.\eqref{eq_surface_grav2}.
Here one should not get confused: this is the full Unified First Law, not just an independent expression of the energy-supply term. The fact that the Unified First Law is written the other way around should not occult that Eq.\eqref{eq_unified_first_law_reform} is the full law.
\section{Recovering Friedmann Equations from the Unified First Law projected on the Apparent Horizon}
\label{section_Friedmann_Eq}
In this section, we give a detailed explanation of our derivation of the Friedmann Equations. The original computation may be found in \cite{Binetruy:2014ela}.
\subsection{The Unified First Law projected}
Let us now work in a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) Universe:
\begin{equation}
ds^2 = -dt^2 + \frac{a(t)^2}{1-k r^2}dr^2 + R^2 d\Omega^2 \ , \\\ \text{ } \\\ R=a(t)r \ .
\label{eq_FLRW}
\end{equation}
This spacetime has an apparent/trapping horizon satisfying $\nabla^a R \nabla_a R=0$, and so its radius is:
\begin{equation}
R_A = \left( H^2 + \frac{k}{a^2} \right)^{-1/2} \ ,
\label{eq_apparent_radius}
\end{equation}
which is nothing else than the Hubble radius $H^{-1}$ in the flat case ($k=0$). We also give the time-derivative of the apparent horizon radius:
\begin{equation}
\dot{R}_A = -H R_A^3 \left( \dot{H} - \frac{k}{a^2} \right) \ .
\label{eq_apparent_radius_deriv}
\end{equation}
We have shown in Eq.\eqref{eq_unified_first_law_reform} that the Unified First Law writes:
\begin{equation}
A\psi_a = \frac{\kappa}{8\pi G}\nabla_a A + R \nabla_a \left(\frac{E}{R}\right) \ ,
\label{eq_Apsi}
\end{equation}
where the Misner-Sharp energy is, in this cosmological context:
\begin{equation}
E=\frac{R^3}{2G}\left(H^2+\frac{k}{a^2}\right)=\frac{R^3}{2GR_A^2} \ ,
\label{eq_energy}
\end{equation}
so that the last term in Eq.\eqref{eq_Apsi} reads:
\begin{equation}
R \nabla_a \left(\frac{E}{R}\right)= \frac{R}{2G} \nabla_a \left(\frac{R^2}{R_A^2}\right) \ .
\label{eq_vanishing_term}
\end{equation}
It is often stated in the literature that this term vanishes on the horizon. This is completely false, except if one means that it vanishes ``along'' the horizon. The quantity in the covariant derivative is indeed constant when evaluated at $R=R_A$, but this is also true for any other fixed value of $R$. Therefore, if one wants to dismiss this term in order to identify a Clausius relation in $A\psi=\frac{\kappa}{8\pi G}dA$, one should keep in mind that it implies all the following computations hold only along a surface of constant $R$. In other words, in order to get rid of the term in question, one must project the Unified First Law Eq.\eqref{eq_Apsi} on a vector $t^a$ which is tangent to the horizon. Since the apparent horizon is defined as the surface where $\nabla^a R \nabla_a R=0$, this tangent vector is defined by $t^a \nabla_a [\nabla^b R \nabla_b R]=0$. For the FLRW Universe, it yields:
\begin{equation}
t^a = \frac{1}{2}\left( 1 , -\frac{RH}{a}\left(1-\frac{\dot{R}_A}{H R_A}\right) , 0, 0 \right) \ .
\label{eq_tangent_vector}
\end{equation}
The Unified First Law may be identified to the Clausius Relation only once it is projected on this tangent vector $t^a$. It then reads:
\begin{equation}
t^a (A \psi_a) = t^a \left(\frac{\kappa}{8\pi G}\nabla_a A + R \nabla_a \left(\frac{E}{R}\right) \right)= t^a \left(\frac{\kappa}{8\pi G}\nabla_a A \right) \ .
\label{eq_Clausius}
\end{equation}
Let us now detail the computations for each of the three terms in this projected law.
\subsection{The Friedmann Equation}
\label{subsection_Friedmann}
\subsubsection*{The vanishing term}
We here check that the second term on the right-hand side of Eq.\eqref{eq_Clausius} does vanish when projected onto $t^a$:
\begin{align}
t^a \nabla_a \left(\frac{E}{R}\right) &= \frac{t^t}{2G}\partial_t \left( \frac{R^2}{R_A^2}\right) +\frac{t^r}{2G}\partial_r \left( \frac{R^2}{R_A^2}\right) \nonumber \\
&= \frac{1}{4G} r^2 \partial_t \left( \frac{a^2}{R_A^2} \right) - \frac{1}{4G} \frac{RH}{a}\left(1-\frac{\dot{R}_A}{H R_A}\right) \frac{a^2}{R_A^2} \partial_r (r^2) \nonumber \\
&= \frac{1}{4G} r^2 \left( \frac{2a\dot{a}R_A^2 - a^2 2 R_A \dot{R}_A}{R_A^4} \right) - \frac{1}{2G} \frac{R^2 H}{R_A^2}\left(1-\frac{\dot{R}_A}{H R_A}\right) \nonumber \\
&= \frac{1}{2G}\frac{R^2 H}{R_A^2} -\frac{1}{2G} \frac{R^2\dot{R}_A}{R_A^3} - \frac{1}{2G} \frac{R^2 H}{R_A^2} + \frac{1}{2G} \frac{R^2\dot{R}_A}{R_A^3} \nonumber \\
&= 0 \ .
\label{eq_vanishing_term}
\end{align}
Note that this is true for any radius $R$, not just for $R=R_A$.
\subsubsection*{The energy-supply term}
Now let us compute the non-vanishing terms in Eq.\eqref{eq_Clausius}. Assuming the energy-momentum of a perfect fluid in our FLRW Universe, one gets the energy-supply covector from Eq.\eqref{eq_energy_supply} :
\begin{equation}
\psi_a = \frac{1}{2}(\rho +p) \left( -RH , a , 0, 0\right) \ .
\label{eq_energy_supply_FLRW}
\end{equation}
Its projection is then:
\begin{align}
t^a (A \psi_a) &= - \frac{1}{4}ARH(\rho +p) - \frac{a}{4}(\rho +p)\frac{ARH}{a}\left(1-\frac{\dot{R}_A}{H R_A}\right)\nonumber \\
&= - \frac{1}{2}ARH(\rho +p) + (\rho +p)\frac{AR}{4}\frac{\dot{R}_A}{R_A}\nonumber \\
&= - 2\pi HR^3(\rho +p) \left(1- \frac{\dot{R}_A}{2HR_A} \right) \ .
\label{eq_psi_projected}
\end{align}
\subsubsection*{The area term}
The right-hand side of Eq.\eqref{eq_Clausius} reads:
\begin{align}
t^a \left(\frac{\kappa}{8\pi G}\nabla_a A \right) &= \frac{\kappa}{4G} \left( \partial_t (R^2) -\frac{RH}{a}\left(1-\frac{\dot{R}_A}{H R_A}\right) \partial_r(R^2) \right) \nonumber \\
&= \frac{\kappa}{4G} \left( 2HR^2 - 2HR^2\left(1-\frac{\dot{R}_A}{H R_A}\right) \right) \nonumber \\
&= \frac{\kappa R^2}{2G}\frac{\dot{R}_A}{R_A} \ .
\label{eq_dA_projected}
\end{align}
In a FLRW spacetime, the surface gravity Eq.\eqref{eq_surface_grav1} is:
\begin{align}
\kappa &= -\frac{R}{2} \left( 2H^2 +\dot{H} +\frac{k}{a^2}\right) \nonumber \\
&= -\frac{R}{2} \left( \frac{2}{R_A^2} -\frac{\dot{R}_A}{H R_A^3}\right) \nonumber \\
&= -\frac{R}{R_A^2} \left( 1 -\frac{\dot{R}_A}{2HR_A}\right) \ ,
\label{eq_surfgravFLRW}
\end{align}
so that Eq.\eqref{eq_dA_projected} also writes:
\begin{equation}
t^a \left(\frac{\kappa}{8\pi G}\nabla_a A \right)= \frac{\kappa R^2}{2G}\frac{\dot{R}_A}{R_A} = -\frac{R^3 }{2G}\frac{\dot{R}_A}{R_A^3}\left( 1 -\frac{\dot{R}_A}{2HR_A}\right) \ .
\label{eq_dA_projected_final}
\end{equation}
\subsubsection*{The Friedmann Equation}
Finally, using the results of our projections Eqs. \eqref{eq_vanishing_term}, \eqref{eq_psi_projected} and \eqref{eq_dA_projected_final} into the full projected Unified First Law Eq.\eqref{eq_Clausius}, we get:
\begin{align}
t^a (A \psi_a) &= t^a(\frac{\kappa}{8\pi G}\nabla_a A ) \nonumber \\
- 2\pi HR^3(\rho +p) \left(1- \frac{\dot{R}_A}{2HR_A} \right) &= -\frac{R^3 }{2G}\frac{\dot{R}_A}{R_A^3}\left( 1 -\frac{\dot{R}_A}{2HR_A}\right) \nonumber \\
2\pi H(\rho +p) &= \frac{1}{2G}\frac{\dot{R}_A}{R_A^3} \nonumber \\
2\pi H(\rho +p) &= -\frac{1}{2G} H\left( \dot{H}-\frac{k}{a^2} \right)\nonumber \\
-4\pi G(\rho +p) &= \dot{H}-\frac{k}{a^2} \ .
\label{eq_Friedmann_eq}
\end{align}
This is indeed the second Friedmann Equation.
Note that up until now, we have not made any mention of the notions of temperature or entropy. We \emph{do not need} to assume any such relations as $T=\frac{\kappa}{2\pi}$ or $S=\frac{A}{4G}$ to derive the Friedmann Equation from the Unified First Law. The introduction of temperature and entropy should come afterwards, when one tries to identify the projected Unified First Law, $t^a (A \psi_a) = t^a(\frac{\kappa}{8\pi G}\nabla_a A )$, with a Clausius relation of the form $\delta Q = TdS$. However, one should be extremely careful with the \emph{physical interpretation} of these quantities. Defining a new quantity $T$ as $T=\frac{\kappa}{2\pi}$ does not mean it is a temperature!
We will postpone to Section \ref{section_temperature} the interpretation of our surface gravity $\kappa$ in terms of temperature. But first we deem it useful to present the common way of recovering Friedmann Equations from thermodynamics, the conceptual differences it has with the above derivation, and the extra-information we can get from it.
\section{The original derivation and a new way of projecting the Unified First Law}
\label{section_misconceptions}
\subsection{Comparison with the original derivation}
The original computations were going from Thermodynamics to Friedmann Equations \cite{Cai:2005ra}, and then the other way around \cite{Cai:2006rs}. Many other works followed these seminal computations, and applied them to several theories of gravity, but with no major variation of the line of reasoning. The logic of all these derivations is quite different from ours, as well as the needed assumptions. The original derivation \cite{Cai:2005ra} starts from an infinitesimal time variation of the Unified First Law expressed in the $(t,R,\theta,\phi)$ coordinates instead of the previous $(t,r,\theta,\phi)$. Then one invokes the first law of thermodynamics $-dE=TdS$, with a temperature $T$ assumed to have the usual (but static!) form $T=1/2\pi R_A$ and an entropy $S$ that is assumed to scale as the area of the apparent horizon. (Note once again that the derivation detailed in Section \ref{section_Friedmann_Eq} is completely free of such assumptions). One still has to evaluate this equation on the apparent horizon to finally get the Friedmann Equation (which makes it valid only at the horizon).
Therefore the reasoning goes as follows: on the one hand, the time variation of the Unified First Law gives an expression for $dE$. On the other hand, one must call for the usual first law of thermodynamics (the Clausius Relation) to express $dE$ in another way. Equating both expressions for $dE$ yields the Friedmann Equation. This is drastically different from the reasoning in the computation of Section \ref{section_Friedmann_Eq}, which only uses the Unified First Law and makes no assumption on temperature and entropy. However it is true that our computation, though it does not use the Clausius Relation, relies on another hypothesis: that the energy takes the Misner-Sharp form Eq.\eqref{eq_energy}. We see this hypothesis as better motivated than those on temperature and entropy. Hayward has justified the use of the Misner-Sharp energy in spherical symmetry \cite{Hayward:1994bu}, and this definition is widely used by the community. The logic of the two different derivations can be sketched as follows:
\begin{itemize}
\item our computation: Unified First Law + Misner-Sharp = Friedmann (which may also be interpreted \emph{a posteriori} as a Clausius Relation, with temperature and entropy),
\item common computation: Unified First Law + Clausius Relation (with temperature and entropy)= Friedmann.
\end{itemize}
Thus, one can wonder whether the two situations are equivalent: the common derivation may take for hypothesis the conclusions of our derivation, and vice-versa. If it were the case, then the common computation should recover the Misner-Sharp energy as a conclusion. However, integrating the infinitesimal $dE=-TdS=-\frac{\dot{R}_A}{G}dt$, one gets:
\begin{equation}
E = -\frac{R_A}{G} \ ,
\label{eq_E_Cai}
\end{equation}
which is a strange form for the energy, and not the Misner-Sharp one. We take this as a hint that the original computation holds only as an approximation. Indeed, researchers of the field who tried to justify the form of the temperature $T=1/2\pi R_A$, started from the full Kodama-Hayward surface gravity Eq.\eqref{eq_surfgravFLRW} evaluated on the horizon, and then argued that the process should be adiabatic and the time derivative $\dot{R}_A$ negligible. In this sense, $T=1/2\pi R_A$ can only be an approximate temperature at best. Section \ref{section_Friedmann_Eq} does not use this approximation.
Therefore the exact derivation of Friedmann Equations from thermodynamics of the horizon was not achieved, until a rigorous computation was provided in \cite{Binetruy:2014ela}. However, the Cai-Kim derivation \cite{Cai:2005ra} turns out to be very instructive when performed in a projective manner: it yields another vector field on which to project the UFL in order to recover Friedmann Equations.
\subsection{Projection of the Unified First Law on $\partial_t$}
\label{section_cleaner_Cai}
Indeed, when considering only a time variation in the Unified First Law, one is doing nothing else than projecting onto a vector $t'^a$ which is just $\partial_t$:
\begin{equation}
t'^a = (1,0,0,0)_{(t,R,\theta,\phi)} \ .
\label{eq_time_vector}
\end{equation}
The Unified First Law reads, still in the $(t,R,\theta,\phi)$ coordinates:
\begin{align}
A t'^a \psi_a &= R t'^a \nabla_a \left(\frac{E}{R}\right) + \frac{\kappa}{8\pi G} t'^a \nabla_a A \nonumber \\
-AHR(\rho +p) &= R \partial_t \left(\frac{R^2}{2GR_A^2}\right) + \frac{\kappa}{8\pi G} 8\pi R \nabla_t R \nonumber \\
-4\pi HR^3(\rho +p) &= \frac{R^3}{2G} \partial_t \left( H^2 +\frac{k}{a^2}\right) + 0 \nonumber \\
-4\pi HR^3(\rho +p) &= \frac{HR^3}{G} \left( \dot{H} -\frac{k}{a^2}\right)
\label{eq_Cai_neat}
\end{align}
The Friedmann Equation is recovered straightforwardly, once again without any assumptions on entropy or temperature (though assuming a Misner-Sharp energy). We have just found another vector field on which to project the Unified First Law! An interesting point is that this new vector $t'^a$ turns out to be colinear to the Kodama vector, which reads in the $(t,R,\theta,\phi)$ coordinates:
\begin{equation}
K^a = \sqrt{1-k\frac{R^2}{a^2}} \times \left( 1,0,0,0 \right) = \sqrt{1-k\frac{R^2}{a^2}} \times t'^a \ .
\end{equation}
Thus projecting on the Kodama also yields the Friedmann Equations! This is not so surprising, as $K^a$ is not \emph{any} vector: it indeed gives a preferred time-direction.
A puzzling difference arises when one compares this new projection to that of Section \ref{section_Friedmann_Eq}. There it is shown that, when projecting on $t^a$, the vanishing term and the term yielding $\left( \dot{H} -\frac{k}{a^2}\right)$ are the $\nabla_a \left(\frac{E}{R}\right)$ term and the $(\frac{\kappa}{8\pi G}\nabla_a A) $ term respectively. It is exactly the contrary for projection on $K^a$. How is one to identify a Clausius relation $dE=TdS$ in the projected Unified First Law $A K^a \psi_a = R K^a \nabla_a \left(\frac{E}{R}\right)$? What are the temperature and entropy now? It is not possible to identify the surface gravity $\kappa$ of Eq.\eqref{eq_surfgravFLRW} in the previous equation. Moreover, although it was identified in the first projection (on $t^a$), $\kappa$ appeared on both sides of the Unified First Law, and was thus of no use for the establishment of the Friedmann law. This seems to point at the Clausius relation as not being fundamental in the derivation of Friedmann Equations. In the end, the original derivation, written in this projective manner, is the one that does not even need to express the surface gravity $\kappa$ and that demands no assumption on the temperature.
We summarise the situation in Table \ref{table_projection}.
\begin{table}
\begin{tabular}{|l|c|c|c|}
\hline
projection & $A\psi_a$ & $\frac{\kappa}{8\pi G}\nabla_a A$ & $R\nabla_a \left(\frac{E}{R}\right)$ \\
\hline
on tangent field $t^a$ & $2\pi \kappa H R^2 R_A^2(\rho +p)$ & $\frac{\kappa}{2G} \frac{R^2 \dot{R}_A}{R_A}$ & 0\\
\hline
on Kodama field $K^a$ & $-4\pi HR^3(\rho +p)$ & 0 & $-\frac{R^3}{G R_A^3}\dot{R}_A$\\
\hline
\end{tabular}
\caption{Projection of the UFL on two different vectors, term by term.}
\label{table_projection}
\end{table}
In the previous sections, we have established that the dynamical surface gravity Eq.\eqref{eq_surface_grav1} poses no problem in the derivation of Friedmann Equations from the Unified First Law. In a spherically symmetric, dynamical framework, it is the relevant quantity to use, along with the Kodama vector and Misner-Sharp energy. In the next section, we will try to see whether this surface gravity can be used to define a notion of temperature.
\section{From Surface Gravity to Temperature}
\label{section_temperature}
In order to understand whether the surface gravity $\kappa_{\cal H}$ may be related to a notion of temperature, let us use the tunneling approach by Parikh and Wilczek \cite{Parikh:1999mf}, in the Hamilton-Jacobi formulation (see \cite{Hayward:2008jq} for the black hole case).
\begin{center}
\begin{itemize}
\item Past Horizon (FLRW) $\rightarrow$ Retarded Eddington-Finkelstein coordinates:
\begin{equation}
ds^2 = -e^{2\Psi}Cdx_-^2 - 2e^{\Psi}dx_-dR + R^2d\Omega^2 \ .
\end{equation}
with $x_-=\eta-\chi$ (conformal time and comoving distance).
In order to give an idea of what the functions $\Psi$ and $C$ are, we give their expression for a FLRW Universe:
\begin{equation}
e^{\Psi}=\frac{a}{\sqrt{1-kr^2}+RH} \ ,
\end{equation}
\begin{equation}
e^{2\Psi}C=a^2\frac{\sqrt{1-kr^2}-RH}{\sqrt{1-kr^2}+RH} \ ,
\end{equation}
\begin{equation}
C=1-kr^2-R^2H^2 \ ,
\label{eq_def_C_cosmo}
\end{equation}
but we do not need to specialize to FLRW now, and so we go back to the general case.
\item Kodama vector: $K^a=(e^{-\Psi} ; 0 ; 0 ; 0) \ .$
\item Surface gravity: $\kappa_{\cal H} = \frac{1}{2} \left. \partial_R C\right|_{\cal H} \leqslant 0 \ .$
\item BKW approximation of tunneling probability for a massless scalar field $\phi=\phi_0 \exp(i\mathcal{I})$:
\begin{equation}
\Gamma \propto exp\left(-2\frac{Im\mathcal{I}}{\hbar}\right) \ .
\end{equation}
\item Equation of Motion is Hamilton-Jacobi Equation:
\begin{equation}
g^{ab} \nabla_a \mathcal{I} \nabla_b \mathcal{I}= 0 \ .
\end{equation}
\item Action: $\mathcal{I}=\int\partial_{x_-}(\mathcal{I})dx_- + \int\partial_{R}(\mathcal{I})dR\ .$
\item Kodama energy\footnote{This is a generalization of the Killing energy. Not to be confused with $\omega$ in the Unified First Law.}: $\omega=-K^a\nabla_a \mathcal{I} = -e^{-\Psi}\partial_{x_-}(\mathcal{I}) \ .$
\item Wave number: $k=\partial_{R} \mathcal{I} \ .$
\item EoM: $k(Ck+2\omega) = 0 \ .$
\\$\rightarrow$ $k=0$ (outgoing solution).
\\$\rightarrow$ $k=-2\omega/C$ (ingoing).
\item Ingoing solution has a pole and contributes to imaginary part of the action:
\begin{align}
Im\mathcal{I} =& Im \int \partial_R (\mathcal{I}) dR \nonumber = Im \int k dR \\ \nonumber
=& Im \int -\frac{2\omega}{C}dR \\ \nonumber
=& Im \int -\frac{\omega}{\kappa_{\cal H}(R-R_A)}dR \ , \nonumber
\end{align}
since near the horizon, $C \approx \partial_R C (R-R_A)= 2\kappa_{\cal H}(R-R_A)$.
Using Feynman's $i\epsilon$-prescription, we circumvent the real pole from below:
\begin{align}
Im &\int -\frac{\omega}{\kappa_{\cal H}(R-R_A-i\epsilon)}dR \nonumber \\ \nonumber
&= Im \int -\frac{\omega(R-R_A+i\epsilon)}{\kappa_{\cal H}(R-R_A-i\epsilon)(R-R_A+i\epsilon)}dR \\ \nonumber
&= \int -\frac{\omega\epsilon}{\kappa_{\cal H}(R-R_A-i\epsilon)(R-R_A+i\epsilon)}dR \\ \nonumber
&= 2i\pi \times \lim_{R\rightarrow R_A+i\epsilon} \left( -\frac{\omega\epsilon(R-R_A-i\epsilon)}{\kappa_{\cal H}(R-R_A-i\epsilon)(R-R_A+i\epsilon)} \right) \\ \nonumber
&=-\frac{2i\pi \epsilon \omega}{2i \epsilon \kappa_{\cal H}} \\ \nonumber
&=-\frac{\pi\omega}{\kappa_{\cal H}}\ ,
\end{align}
where we have used the residue theorem to compute the integral.
\item The tunneling probability takes a thermal form:
\begin{equation}
\Gamma \propto exp\left(-2Im\mathcal{I}\right) \propto exp(-\omega/T) \Leftrightarrow T= \frac{\omega}{2 Im \mathcal{I}} \ ,
\end{equation}
\item with temperature:
\begin{equation}
T=-\frac{\kappa_{\cal H}}{2\pi} \geqslant 0 \ .
\label{eq_temperature_sign}
\end{equation}
\end{itemize}
\end{center}
We indeed get a positive temperature for the thermal spectrum of ingoing modes, \emph{i.e.} of particles tunneling from beyond the horizon to the interior (towards the central observer), as was originally suggested in \cite{Gibbons:1977mu}. The inner character of the horizon for an expanding Universe yields a negative surface gravity, while its past nature imposes a minus sign in the relation between $T$ and $\kappa_{\cal H}$. This combining sign effect is explained in details in \cite{Helou}.
\section{Sign of surface gravity and causal nature of the horizon}
\label{section_causal_nature}
In a FLRW Universe, we have seen that the surface gravity can be expressed as in Eq.\eqref{eq_surfgravFLRW}. Using Friedmann Equations, one can also express it in terms of a parameter of state $w$:
\begin{align}
\kappa &= -\frac{R}{2} \left( 2H^2 +\dot{H} +\frac{k}{a^2} \right) \nonumber \\
&= -\frac{R}{2} \left( 2\left(H^2+\frac{k}{a^2}\right) +\dot{H} -\frac{k}{a^2} \right) \nonumber \\
&= -\frac{R}{2} \left( \frac{16\pi G}{3}\rho -4\pi G(\rho+p) \right) \nonumber \\
&= -\frac{R}{2} \left( \frac{4\pi G}{3}\rho -4\pi G w\rho \right)\nonumber \\
&= - 2\pi G R \rho \left( \frac{1}{3} - w \right) \ .
\label{eq_surfgrav_state_param}
\end{align}
From the above expression for the surface gravity, one can see that it changes sign depending on the energy budget of the FLRW Universe: $\kappa$ is negative for $w<1/3$, positive for $w>1/3$. However, the standard model of cosmology, or $\Lambda$-CDM, describes the history of our Universe by successive eras: inflation, when $w\sim -1$, radiation-domination, $w=1/3$, matter-domination, $w=0$, and the current phase of cosmic acceleration or ``dark energy'', $w\sim -1$. All these phases have $w\leqslant 1/3$, and thus $\kappa$ negative or null. But the signature of the horizon is not constrained: it can be timelike, null or spacelike (see the four cases to the right of Figure \ref{fig_causal_nature_horizon} \footnote{In this figure we have used the Bousso wedge convention, which represents, out of the four null directions of the lightcones, only those along which geodesic congruences are contracting ($\theta<0$). For a Minkowski-like region, these directions are the future and past inner ones (Bousso wedge: $>$). For the trapped region of an expanding cosmology, they are the two past directions (Bousso wedge: $\wedge$).}).
\begin{center}
\begin{figure}[h]
\centering
\includegraphics[width=15cm,angle=0]{trapping_horizon_signature.png}
\caption{Causal nature of the horizon and the sign of $\kappa \ .$}
\label{fig_causal_nature_horizon}
\end{figure}
\end{center}
The last case to the right of Figure \ref{fig_causal_nature_horizon} represents a spacelike past-inner horizon, with $w<-1$, which could represent a phase of even more accelerated expansion, driven by a hypothetical ``phantom energy''. The previous case to the left is a null past-inner horizon for $w=-1$, and is the usual half of the Penrose diagram for de Sitter Universe. The previous case shows a timelike past-inner horizon for a matter-dominated Universe with $w=0$. The previous drawing (second from the left) is a null past-inner horizon for a radiation-dominated Universe, $w=1/3$. All four cases are past-inner horizons (the last one being degenerate), with $\kappa\leqslant 0$, though the signature can be timelike/lightlike/spacelike.
Only when we tilt the horizon further still do we get a positive $\kappa$ (see the left case on Figure \ref{fig_causal_nature_horizon}). That is only because we have crossed the limit between past-inner horizons ($\mathcal{L}_+ \theta_->0$) and past-outer ones ($\mathcal{L}_+ \theta_-<0$) \footnote{Here the +/- indices refer to outgoing/ingoing future null directions respectively.}. We are no longer describing the same system, but rather something resembling a white-hole \footnote{Nevertheless, past-outer configurations could be used to describe some special expanding cosmologies, for example a stiff-matter cosmology, for $w=1$.}.
Therefore the sign of $\kappa$ does not depend on the signature of the horizon. It is solely dictated by the inner/outer nature of the apparent horizon. The sign of the temperature, however, is a result of both the sign of $\kappa$ and the sign entering in the definition of $T$, as in Eq.\eqref{eq_temperature_sign}. For further details on this combining sign effect, see \cite{Helou}.
Note that this is all exactly similar to the black hole case: a future-outer horizon can be either timelike, null, or spacelike, its surface gravity $\kappa$ will still be of constant, positive sign (the equivalent of the three cases to the right of Figure \ref{fig_causal_nature_horizon}). Of course, if one pushes the horizon further still, the surface gravity will change sign, but that is just because the configuration will have become future-\emph{inner}. We will not be describing a black hole anymore.
A practical way of knowing the causal nature of a horizon is the sign of parameter $\alpha$ \cite{Dreyer:2002mx}:
\begin{equation}
\alpha = \frac{ \mathcal{L}_- \theta_- }{ \mathcal{L}_+ \theta_- } \ .
\label{eq_alpha}
\end{equation}
$\alpha$ is negative/null/positive for timelike/lightlike/spacelike horizons respectively. This is true for both inner \emph{and} outer past-horizons.
\section{Conclusions}
We have seen that the sign of the Hayward-Kodama surface gravity \emph{does} depend on the inner/outer character of the trapping horizon. However, the sign of the temperature depends on both the inner/outer \emph{and} the past/future character of the horizon. This is clearly established in \cite{Helou}. The statement is valid for black holes, white holes, expanding cosmologies as well as contracting ones \footnote{It has been stated in \cite{Tian:2014sca} that in the case of a contracting cosmology ($H<0$), one will have to deal with an outer horizon. However in the language of Hayward that we use here, a contracting Universe has a future-inner trapping horizon.}
We have shown in Section \ref{section_dynamical_setup} that the Hayward-Kodama surface gravity is perfectly well-defined in a dynamical context, using the Kodama field (the generalization of the static Killing field). Researchers of the field who have used this definition of surface gravity, have unfortunately let go of the time-dependent part, and approximated $\kappa$ to the usual, static surface gravity $1/2\pi R_A \ .$ Later on, the full Hayward-Kodama surface gravity was rehabilitated \cite{Cai:2006rs}, in a computation going from gravity to thermodynamics. A full, exact computation establishing Friedmann Equations from the Unified First Law, using this dynamical $\kappa$, was nevertheless still needed. It was recently provided in \cite{Binetruy:2014ela}. This derivation was reproduced here in Section \ref{section_Friedmann_Eq}.
Moreover, the link of our surface gravity to a temperature was also established in \cite{Binetruy:2014ela}, and detailed in this present work, Section \ref{section_temperature}. This quantity $T$ turns out to be positive, and interpretable as the temperature of a thermal spectrum for particles tunneling from outside the horizon towards the central observer. Here, no approximation, truncation, or convenient absolute value have been applied on the temperature. We hope that the present work, added to our previous article \cite{Binetruy:2014ela}, will rehabilitate the Hayward-Kodama surface gravity as the relevant quantity to use in spherically-symmetric dynamical spacetimes.
\textbf{Acknowledgements:} I thank Florian Gautier and Jean-Philippe Bruneton for useful discussions. I thank Pierre Binétruy for comments on the manuscript.
\newpage
\section*{Appendix A}
It is also possible to get the Friedmann Equations from each components of the Unified First Law expressed in the $(t,R,\theta,\phi)$-coordinates. Here again, we use the Misner-Sharp energy.
The $R$-component of the Unified First Law reads:
\begin{align}
\nabla_R{E} = A\psi_R + \omega \nabla_R V &= A\frac{\rho+p}{2} + \frac{\rho-p}{2}\nabla_R V \nonumber \\ \frac{3}{2G}R^2(H^2+k/a^2) &= 4\pi R^2 \rho \ ,
\label{eq_unified_first_law_R}
\end{align}
which immediately yields the First Friedmann Equation:
\begin{equation}
H^2+\frac{k}{a^2} = \frac{8\pi G}{3} \rho \ .
\label{eq_first_friedmann_eq_from_unified_law}
\end{equation}
Now for the $t$-component:
\begin{align}
\nabla_t{E}= A\psi_t + \omega \nabla_t V &= -HRA(\rho+p) + \frac{\rho-p}{2}\nabla_t V \nonumber \\
-\frac{R^3 \dot{R}_A}{G R_A^3} &= -4\pi HR^3(\rho+p) \ ,
\label{eq_unified_first_law_t}
\end{align}
which yields the Second Equation:
\begin{equation}
\dot{H}-\frac{k}{a^2} = -4\pi G (\rho+p) \ .
\label{eq_second_friedmann_from_unified_law2}
\end{equation}
This is nothing else than the computation of Section \ref{section_cleaner_Cai} (projecting on Kodama amounts to selecting the $t$-component in these coordinates).
Note here that this is once again valid for all $R$, not only on the apparent horizon $R=R_A$.
\underline{Nota Bene:}
\begin{itemize}
\item The UFL in the form of Eq.\eqref{eq_Unified_First_Law}, written in the $(t,R)$ coordinates, yields the First and Second Friedmann Equations directly from its $R$ and $t$-components respectively.
\item The UFL in the form of Eq.\eqref{eq_Unified_First_Law}, written in the $(t,r)$ coordinates, yields the First Friedmann Equation directly from its $r$-component. The $t$-component supplemented with the First Friedmann Equation yields the Second Friedmann Equation.
\item The UFL in the form of Eq.\eqref{eq_unified_first_law_reform}, written in the $(t,R)$ coordinates, yields the Second Friedmann Equation directly from its $t$-component. The $R$-component gives nothing...
\item The UFL in the form of Eq.\eqref{eq_unified_first_law_reform}, written in the $(t,r)$ coordinates, yields the Second Friedmann Equation directly from its $t$-component. The $r$-component gives nothing...
\end{itemize}
The last two remarks are understandable: Eq.\eqref{eq_Unified_First_Law} is not symmetric in $p$ and $\rho$, which allows us to single out an expression for $\rho$ alone. On the other hand, Eq.\eqref{eq_unified_first_law_reform} is symmetric in $p$ and $\rho$, and cannot recover the First Friedmann Equation.
Let us interpret the first remark: the $R$-component of Eq.\eqref{eq_Unified_First_Law} gives the First Friedmann Equation, which links the energy density $\rho$ to the size of the apparent horizon $R_A$. Nowadays, this is dominated by dark energy, $\Omega_{\Lambda}=0.7$, which we take as vacuum energy. An explanation to this has been provided in \cite{Binetruy:2012kx}, in terms of collapse of the quantum wave-function on the most probable state (with highest entropy). Now the $t$-component of Eq.\eqref{eq_Unified_First_Law} gives the Second Friedmann Equation, which links the sum $(\rho+p)$ to the time variation of the apparent horizon $\dot{R}_A$. This, on the contrary, receives no contribution from dark energy: $(\rho_{\Lambda}+p_{\Lambda})=0$. Matter, and radiation, are the contributing components here. Hence, as already noticed in \cite{Binetruy:2014ela}, vacuum energy dictates the position in $R$ of the apparent/trapping horizon, while matter gives its variation in $t$.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,108 |
# **Contents**
1. Cover
2. Dedication
3. Title Page
4. Copyright
5. Introduction
1. Mind and Geist
2. Elementary particles and conscious organisms
3. The decade of the brain
4. Can the mind be free in a brain scan?
5. The self as a USB stick
6. Neuromania and Darwinitis – the example of Fargo
7. Mind – brain – ideology
8. The cartography of self-interpretation
9. Notes
6. 1 What is at Stake in the Philosophy of Mind?
1. Mind in the universe?
2. In the spirit of Hegel
3. The historical animal on the social stage
4. Why not everything, but at least something, is teleological
5. Notes
7. 2 Consciousness
1. I see something that you do not see!
2. Neuronal thunderstorms and the arena of consciousness
3. Buddha, the snake and the bat – again
4. Surfing on the wave of neuro-Kantianism
5. Nothing is beyond our experience – or is it?
6. Faith, love, hope – are they all just illusions?
7. An altruist is lodged in every ego
8. Davidson's dog and Derrida's cat
9. Tasty consciousness
10. The intelligence of the robot vacuum cleaner
11. Strange Days – the noise of consciousness
12. What Mary still doesn't know
13. The discovery of the universe in a monastery
14. Sensations are not subtitles to a Chinese movie
15. God's-eye view
16. Notes
8. 3 Self-Consciousness
1. How history can expand our consciousness
2. Monads in the mill
3. Bio is not always better than techno
4. How the clown attempted to get rid of omnipotence Self-consciousness in a circle
5. Notes
9. 4 Who or What is This Thing We Call the Self?
1. The reality of illusions
2. Puberty-reductionism and the toilet theory
3. Self is god
4. Fichte: the almost forgotten grandmaster of the self
5. The three pillars of the science of knowledge
6. In the human being nature opens her eyes and sees that she exists
7. "Let Daddy take care of this": Freud and Stromberg
8. Drives meet hard facts
9. Oedipus and the milk carton
10. Notes
10. 5 Freedom
1. Can I will not to will what I will?
2. The self is not a one-armed bandit
3. Why cause and reason are not the same thing and what that has to do with tomato sauce
4. Friendly smites meanie and defeats metaphysical pessimism
5. Human dignity is inviolable
6. On the same level as God or nature?
7. PS: There are no savages
8. Man is not a face drawn in the sand
9. Notes
11. Index
12. End User License Agreement
## Guide
1. Cover
2. Table of Contents
3. Begin Reading
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_For Marisa Lux
Become who you are!_
# I am Not a Brain
## _Philosophy of Mind for the Twenty-First Century_
Markus Gabriel
Translated by Christopher Turner
polity
First published in German as _Ich ist nicht Gehirn. Philosophie des Geistes für das 21. Jahrhundert_ © by Ullstein Buchverlage GmbH, Berlin. Published in 2015 by Ullstein Verlag.
This English edition © Polity Press, 2017
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All rights reserved. Except for the quotation of short passages for the purpose of criticism and review, no part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher.
ISBN-13: 978-1-5095-1478-6
A catalogue record for this book is available from the British Library.
Library of Congress Cataloging-in-Publication Data
Names: Gabriel, Markus, 1980- author.
Title: I am not a brain : philosophy of mind for the 21st century / Markus Gabriel.
Description: Malden, MA : Polity, 2017. | Includes bibliographical references and index.
Identifiers: LCCN 2017010106 (print) | LCCN 2017030527 (ebook) | ISBN 9781509514779 (Mobi) | ISBN 9781509514786 (Epub) | ISBN 9781509514755 (hardback)
Subjects: LCSH: Philosophy of mind. | Consciousness.
Classification: LCC BD418.3 (ebook) | LCC BD418.3 .G33 2017 (print) | DOC 128/.2--dc23
LC record available at <https://lccn.loc.gov/2017010106>
The publisher has used its best endeavours to ensure that the URLs for external websites referred to in this book are correct and active at the time of going to press. However, the publisher has no responsibility for the websites and can make no guarantee that a site will remain live or that the content is or will remain appropriate.
Every effort has been made to trace all copyright holders, but if any have been inadvertently overlooked the publisher will be pleased to include any necessary credits in any subsequent reprint or edition.
For further information on Polity, visit our website: politybooks.com
... as we remind ourselves that no age has been more skillful than our own in producing myths of the understanding, an age that produces myths and at the same time wants to eradicate all myths.
Søren Kierkegaard, _The Concept of Anxiety_
# **Introduction**
We are awake and thus conscious; we have thoughts, feelings, worries and hopes. We speak with each other, found states, choose parties, conduct research, produce artworks, fall in love, deceive ourselves and are able to know the truth. In short: we humans are minded animals. Thanks to neuroscience we know, to some extent, which areas of the brain are active when someone shows us a picture, for instance, or prompts us to think of something in particular. We also know something about the neurochemistry of emotional states and disorders. But does the neurochemistry of our brain ultimately guide our entire conscious mental life and relations? Is our conscious self only our brain's user interface, so to speak, which in reality does not contribute at all to our behavior but only accompanies what actually happens, as if it were an unimportant spectator? Is our conscious life thus only a stage upon which a show is performed, in which we cannot really – that is, freely and consciously – intervene?
Nothing is more obvious than the fact that we are minded animals who lead a conscious life. And yet, this most evident fact about ourselves gives rise to countless puzzles. Philosophy has occupied itself with these puzzles for millennia. The branch of philosophy that is concerned with human beings as minded animals, these days, is called _philosophy of mind_. It is more relevant today than ever before, as consciousness and the mind in general are at the center of a whole variety of questions for which we currently have nothing that even comes close to a full explanation in terms of our best natural sciences.
Many consider the nature of consciousness to be one of the last great unsolved puzzles. Why, anyway, should the light turn on, so to speak, in some product of nature? And how is the electrical storming of neurons in our skull connected to our consciousness? Questions such as these are treated in subfields of the philosophy of mind, such as the _philosophy of consciousness_ , and in _neurophilosophy_.
Thus it is here a question of our very selves. I first present a few of the main thoughts of the philosophy of mind with reference to central concepts such as consciousness, self-consciousness and self. In the wider public and in various disciplines outside of philosophy, there is much talk about these concepts, mostly without awareness of the philosophical background, which leads to confusion. Hence, to start out with, I discuss this background with as few philosophical assumptions as possible.
My sketch of the philosophical background of many of the conceptual building blocks of our self-understanding as minded animals forms the foundation of the second main goal of this book: the defense of our freedom (our free will) against the common idea that someone or something deprives us of our freedom unbeknown to us – whether it be God, the universe, nature, the brain or society. We are free through and through, precisely because we are minded animals of a particular kind. The particularity of our mind consists in the fact that we constantly work out a historically shifting and culturally varying account of what exactly it takes to be the kind of minded animal we are. Whereas some believe that we have an immortal soul which accounts for the various mental processes we constantly experience, at the other extreme end of possibilities, many are happy to accept that all their mental processes are ultimately identical to brain states.
In this book, I argue that the truth indeed lies between these untenable extremes (and their more sophisticated versions spelled out by contemporary philosophers and mind scientists of various stripes). Here, the main idea is that what we call "the mind" is really the capacity to produce an open-ended list of self-descriptions which have consequences for how humans act. Humans act in light of their self-understanding, in light of what they believe to be constitutive of a human being. For instance, if you believe that your capacity to act morally presupposes that your soul has an immortal nature, you will live differently from someone who (like me) is convinced that they will live only once and this is the source for our ethical claims. Given that nothing would matter to us at all if we weren't conscious, sentient creatures with beliefs about what that very fact means, anything which matters to us, anything of any importance hinges on our self-conception as minded. However, there is a vast plurality of such self-conceptions spread out through human history as we know it from the writings and cultural artifacts of our ancestors as well as from the cultural variation across humanity as we find it on our planet right now.
It will turn out that we are neither pure genetic copying machines, in which a brain is implanted giving rise to the illusion of consciousness, nor angels who have strayed into a body but, in fact, the free-minded animals whom we have considered ourselves to be for millennia, animals who also stand up for their political freedoms. Yet, this fact about ourselves is obscured if we ignore the variation built into the capacity to conceive of oneself as minded.
## _**Mind and**_ **Geist**
Let me give you an example of the variation we need to explore. Remarkably, there is no real equivalent of the English word "mind" in German. Likewise, the German word _Geist_ , which plays a similar role as mind in English, cannot be translated in a literal way – that is, without further explanation. Here are more elements in a list of terms that have a whole variety of meanings even within English and which cannot easily be translated into every language: consciousness, the self, awareness, intuition, belief, cognition, thinking, opinion. Let us call the vocabulary in which our self-conception as minded animals is couched our _mentalistic vocabulary_. This vocabulary has different rules in different languages, and even within a given natural language, such as English or German, there is really a variety of usages tied to specific local ways of understanding ourselves as minded. Depending on the kind of vocabulary to which you explicitly and implicitly subscribe as privileged over alternatives, you will, for example, think of your mind as extended into your computer, as locked in your brain, as spread out over the entire cosmos, or as connected to a divine spiritual realm in principle inaccessible to any modern scientific investigation. However, in order to make sense of this very variation, we have to assume that there is a core concept defining human self-understanding as minded. For simplicity's sake, my choice for the central term here is "mind."
I am aware that this might be more misleading than in the German context in which I started to work out my views about the topic at hand. For, in the German philosophical tradition, we rather speak of _Geist_ as the relevant invariant. However, the notion behind the somewhat mysterious term _Geist_ can be summarized in roughly the following way: what it is to be a minded ( _geistig_ ) animal is to conceive of oneself in such a variety of ways. Human beings essentially respond to the question of what it means for them to be at the center of their own lives in different ways. What does not vary is the capacity, nay, the necessity, to respond differently to this question. Our response to the question of what it means to be a human minded animal in part shapes what it is to be such an animal. We turn ourselves into the creatures we are in each case by developing a mentalistic vocabulary.
Let me give you another example. We all believe that there are pressing moral issues having to do with, say, abortion, economic equality, warfare, love, education, etc. At the same time, we all have beliefs about why we are even open to these issues. Again, here are two possible extremes. A social Darwinist will believe that morality is nothing but a question of altruistic cooperation, a behavioral pattern whose existence can be explained in terms of evolutionary biology/psychology. By contrast, a Christian fundamentalist maintains that morality is a divine challenge to humans, whose nature is corrupted by sin. This has consequences for their actions and for their answers to pressing moral questions, as we all know. In both cases, the divergence of opinion results from the way in which the social Darwinist and the Christian fundamentalist think about their mental lives. I suggest that the social Darwinist will turn herself into the kind of altruistically inclined animal she takes herself to be, whereas the Christian fundamentalist will literally have a different mindset, one in which her mind will be shaped in light of her conception of the divine. The Christian fundamentalist might be much more obsessed with the idea that God is watching and judging her every deed (and even her intimate thoughts), which will give rise to thought processes utterly absent in the social Darwinist, and vice versa.
In my view, there is no neutral ground to settle the issue, no pure standpoint from which we can start to answer the question of what human mindedness really is. For human mindedness actually exists only in the plurality of self-conceptions. If we strip it from its differentiation into a plurality of self-conceptions, we wind up with an almost empty, formal core: the capacity to create self-conceptions.
However, this formal core matters a lot, as I will also argue in what follows that this formal core necessarily gives rise to an irreducible variety of self-conceptions, and that this fact about us as minded animals is also at the center of morality in general.
## _**Elementary particles and conscious organisms**_
A major challenge of our times is the attempt to come up with a scientific image of the human being. We would like to obtain objective knowledge finally of who or what the human being really is. However, the human mind still stands in the way of achieving a purely scientific story, since the knowledge obtainable only from a subjective point of view has eluded scientific research to this point. To address this problem, future neuroscience is decreed as the final science of the human mind.
How plausible is the assumption that neuroscience can finally give us a fully objective, scientific understanding of the human mind? Until recently, one would hardly have thought that a neurologist or a neurobiologist, for example, should be the specialist for the human mind. Can we really trust neuroscience in general, or brain science in particular, to provide us with the relevant information about ourselves? Do they hold the key to the secret which has haunted philosophy and humanity ever since the Delphic oracle told us to know ourselves?
To what extent should we align our image of the human being with technological progress? In order to address central questions such as this in a sensible manner, we should scrutinize _concepts of our self-portrait_ such as consciousness, mind, self, thinking or freedom more carefully than we are accustomed to do in an everyday sense. Only then can we figure out where we are being led up the garden path, if someone were to claim, for instance, that there is really no such thing as free will or that the human mind (consciousness) is a kind of surface tension of the brain, as Francis Crick and Christof Koch at some point supposed: synchronized neural firing in the 40-Hertz range – a conjecture which they have since qualified.1
In contrast to the mainstream of contemporary philosophy of mind, the proposal introduced in this book is an antinaturalistic one. **Naturalism**2 proceeds on the basis that everything which really exists can ultimately be investigated in a purely naturalscientific manner, be it by physics alone or by the ensemble of the natural sciences. In addition, naturalism at the same time typically assumes that **materialism** is correct – i.e., the thesis that it is only material objects that really exist, only things which belong to a reality that is exhaustively composed of matter/energy. But what then is the status of consciousness, which until now has not been explained scientifically – and in the case of which it cannot even be foreseen how this is supposed to be at all possible?
Remarkably, the German word for the humanities is _Geisteswissenschaften_ – that is, the sciences which deal with the mind in the sense of the invariant structure which is constituted via historically and culturally shifting self-conceptions. The point of dividing academic labor into the natural sciences and the mind sciences corresponds to the fact that it is hard to see how the Federal Republic of Germany, the worlds of Houellebecq's novels, dreams about the deceased, thoughts and feelings in general, as well as the number π could really turn out to be material objects. Do they not exist, perhaps, or are they not real? Naturalists attempt to establish precisely the latter claim by clearing up the impression that there are immaterial realities, which according to them is deceptive. I will have more to say about this in what is to come.
As previously mentioned, I adopt the stance of **antinaturalism** , according to which not everything which exists can be investigated by the natural sciences. I thus contend that there are immaterial realities which I consider essential for any accessible insight of sound human understanding. When I consider someone a friend, and consequently have corresponding feelings for him and adjust my behavior accordingly, I do not suppose that the friendship between him and me is a material thing. I also do not consider myself to be only a material thing, although I would obviously not be who I am if I had no proper body, which in turn I could not have if the laws of nature of our universe had been very different or if biological evolution had taken another course. Antinaturalism does, therefore, not deny the obvious fact that there are necessary material conditions for the existence of many immaterial realities. The immaterial does not belong to another world. Rather, both the material and the immaterial can be parts of objects and processes such as the Brexit, Mahler's symphonies, or my wish to finish this sentence while I am writing it.
The question of whether naturalism or antinaturalism is ultimately right is not merely important for the academic discipline of philosophy or simply for the academic division of labor between the natural sciences and the humanities, say. It concerns all of us, insofar as we are humans – that is, insofar as who we are in part depends on our self-conception. The philosophical question of naturalism vs. antinaturalism also plays an important historical role in our era of religious revival, since religion is quite rightly considered to be the bastion of the immaterial. If one overhastily ignores immaterial realities, one ends up not even being able to understand the rational core of the phenomenon of religion, since one views it from the start as a kind of superstition or cognitively empty ghost story. There are shortcomings in the idea that we could understand human subjectivity by way of scientific, technological and economic advances and bring them under our control by means of such an understanding. If we simply ignore this and pretend that naturalism in the form of futuristic science will solve all existential issues by showing that the mind is identical to the brain and that therefore nothing immaterial really exists, we will achieve the opposite of the process of enlightenment. For religion will simply retreat to the stance of irrational faith assigned to it by a misguided foundation of modernity. Modernity is ill-advised to define itself in terms of an all-encompassing science yet to come. In other words, as the case of scientology proves: we should not base our overall conception of who we are and how absolutely everything hangs together on science fiction. Science fiction is not science, even though it often enables actual science to make progress. But so does religion.
Already in the last century thinkers from different orientations pointed out the limitations of a misguided Enlightenment project based on the notion that we can extend scientific rationality to all spheres of human existence. For instance, the first generation of critical theorists, most notably Theodor W. Adorno (1903–1969) and Max Horkheimer (1895–1973) in their influential book _Dialectic of Enlightenment_ , went so far as to claim that modernity was ultimately a misfortune that had to end in totalitarianism. I disagree. But I do believe that modernity will remain deficient for as long as it props up the fundamental materialist conviction that there are ultimately only particles dispersed in an enormous worldcontainer structured according to natural laws, in which only after billions of years organisms emerged for the first time, of which by now quite a few are conscious – which then poses the riddle as to how to fit the obviously immaterial reality of the mind into the assumption that it should not exist according to the materialist's lights. We will never understand the human mind in this way! Arguably, this very insight led the ancient Greeks to the invention of philosophy, which at least holds for Plato and Aristotle, who both resisted naturalism with arguments still valid today.
To reclaim the standpoint of an antinaturalist philosophy of mind, we must give up the idea that we have to choose between a scientific and a religious image of the world, since both are fundamentally mistaken. There are today a group of critics of religion, poorly informed both historically and theologically, who are gathered together under the name of a "New Atheism," among whom are counted prominent thinkers such as Sam Harris (b. 1967), Richard Dawkins (b. 1941), Michel Onfray (b. 1959) and Daniel Dennett (b. 1942). These thinkers believe that it is necessary to choose between religion – that is, superstition – and science – that is, clinical, unvarnished truth. I have already rebutted at length the idea that our modern democratic societies have to stage a constitutive conflict of world images in _Why the World Does Not Exist_. My thesis there was that, in any case, there are no coherent world pictures, and that religion is no more identical to superstition than science is to enlightenment.3 Both science and religion fail insofar as they are supposed to provide us with complete world pictures, ways of making sense of absolutely everything that exists, or reality as a whole if you like. There simply is no such thing as reality as a whole.
Yet, even if I am right about this (as I still believe after several waves of criticism and polemics against my no-world-view), it still leaves open the question to be dealt with here of how to conceive of the relation between the mind and its non-mental natural environment. It is now a matter of developing an antinaturalist perspective vis-à-vis ourselves as conscious living beings, a perspective happy to join in the great traditions of self-knowledge that were developed in the history of ideas – and not just in the West. These traditions will not disappear because a small technological and economic elite profit from the progress of natural science and now believe that they must drive out ostensible and real religious superstitions and, along with them, expel mind from the human sciences. Truth is not limited to natural science; one also finds it in the social and human sciences, in art, in religion, and under the most mundane circumstances, such as when one notices that someone is sad without theorizing about her mental state in any more complicated, scientific manner.
## _**The decade of the brain**_
The recent history of the idea that neuroscience is the leading discipline for our research into the self is noteworthy and telling. Usually, this political background story is not mentioned in the set-up of the metaphysical riddle of finding a place for mind in nature. In 1989, the United States Congress decided to begin a decade of research into the brain. On July 17, 1990, the then president, George H. W. Bush (that is, Bush senior), officially proclaimed the "Decade of the Brain."4 Bush's proclamation ends solemnly and grandiosely – as is customary in this genre: "Now, therefore, I, George Bush, President of the United States of America, do hereby proclaim the decade beginning January 1, 1990, as the Decade of the Brain. I call upon all public officials and the people of the United States to observe that decade with appropriate programs, ceremonies, and activities." A decade later, a similar initiative in Germany, with the title "Dekade des menschlichen Gehirns" ["The Decade of the Human Brain"] was launched at the University of Bonn under the auspices of the governor of North Rhine-Westphalia, Wolfgang Clement.
It is irritating that the press release for this very initiative began with a statement that is not acceptable as it stands: "As recently as ten years ago the idea that it could ever be possible to see the brain as it thinks was pure speculation."5 This statement implies that it is _now_ supposed to be possible "to see the brain as it thinks,"6 which, however, looked at more closely, is quite an astounding assertion, since it is ultimately a preposterous idea that one could _see_ an act of thinking. Acts of thinking are not visible. One can at best see areas of the brain (or images thereof) which one might consider to be necessary prerequisites for acts of thinking. Is the expression "to see the brain as it thinks" supposed to mean that one can literally _see_ how the brain processes thoughts? Does that mean that now one no longer merely _has_ or _understands_ thoughts but can also _see_ them in a single glance? Or is it just associated with the modest claim of seeing the brain at work, without already implying as a consequence that one can somehow literally see or even read thoughts?
As far as I know, George Bush senior is not a brain scientist (let alone a philosopher), which means that his declaration of a decade of the brain can at best be a political gesture performed in order to funnel more government resources into brain research. But what would need to happen for one to be able to "see" the brain as it thinks?
Neuroimaging techniques such as functional magnetic resonance imaging, to which the German declaration alludes with its claim, constitute progress in medicine. Unlike earlier attempts to understand the living brain, they are not invasive. Thus, we can visualize the living brain with computer-generated models (but not directly!) without serious medical interventions in the actual organ. However, in this case, medical progress is associated with a further promise, the promise _to make thinking visible_. And this promise cannot be honored. In the strict sense, it is quite absurd. That is to say, if one understands by "thinking" the conscious having of thoughts, much more is involved than brain processes that one could make visible by means of neuroimaging techniques. To be sure, one can make brain processes visible in a certain sense, but not thinking.
The two decades of the brain, which officially came to an end on December 31, 2010, were not intended to be restricted to medical progress but offered hope for progress in self-knowledge. In this context, neuroscience has for a while been charged with the task of serving as the lead discipline for the human being's research into itself, since it is believed that human thinking, consciousness, the self, indeed our mind as such can be located in and identified with a spatio-temporally observable thing: the brain or central nervous system. This idea, which I criticize in this book and would like to refute, I call for brevity's sake _neurocentrism_. With the rise of other superpowers, **Eurocentrism** – that is, the old colonialist view of Europe's cultural superiority over the rest of the world – is no longer taken seriously. One must now fight against a new ideological monster, neurocentrism, which is no less a misguided fantasy of omnipotence (and, incidentally, not very scientific either). While Eurocentrism mistakenly thought that human thinking at its peak was bound to a continent (Europe) or a cardinal point (the West), neurocentrism now locates human thinking in the brain. This spurs the hope that, in this way, thinking can be better examined, by mapping it, as Barack Obama's more recent initiative of a "Brain Activity Map" suggests. As if the brain was the continent of thinking such that we could literally draw a map with the structure of neuroimaging models on which we could locate, for instance, the (false) thought that naturalism is true.
The basic idea of **neurocentrism** is that to be a minded animal consists in nothing more than the presence of a suitable brain. In a nutshell, neurocentrism thus preaches: _the self is a brain_. To understand the meaning of "self," "consciousness," "will," "freedom" or "mind," neurocentrism advises us against turning to philosophy, religion or common sense and instead recommends investigating the brain with neuroscientific methods – preferably paired with evolutionary biology. I deny this and thus arrive at the critical main thesis of this book: _the self is not a brain_.
In what follows, I will take a look at some core concepts of our self-understanding as minded, such as consciousness, self-consciousness and the self. I will do so from an antinaturalistic perspective. In this context, I will sketch some of the absurd consequences and extreme views that came out of naturalism, such as **epiphenomenalism** – that is, the view that the mind does not causally interfere with anything which really happens. Along with taking stock of some conceptual bits and pieces of our mentalistic vocabulary, I will scrutinize the troublesome issue of free will. Are we free at all, or are there good reasons to doubt this and to conceive ourselves to be biological machines that are driven by the hunger for life and that really strive for nothing other than passing on our genes? I believe that we are in fact free and that this is connected to the fact that we are minded animals of the sort which necessarily work out conceptions of what this means in light of which people become who they are. We are creative thinkers, thinkers of thoughts which in turn change the thinkers. It really matters how you think of yourself, a fact well known to psychologists and to human subjects in general, as we constantly during our conscious life engage with the realm of thoughts about ourselves and others.
Mainstream philosophy of mind for quite a while has sought to provide a theoretical basis for neurocentrism. This seemed necessary given that neurocentrism cannot yet claim to be based on empirical results, as neuroscience is infinitely far away from having solved even "minor" problems, such as finding a physical/neural correlate for consciousness, not to mention finding a location in the brain which correlates with insight into some complicated quantum-mechanical truth or the concept of justice. It has participated, sometimes even enthusiastically, in the decade of the brain. Yet, in the course of the unfolding of mainstream philosophy of mind it has become apparent to many that it is anything but obvious that the self is a brain.
Many reflections and key results of the last two centuries of the philosophy of mind already speak against the basic idea of neurocentrism. I will thus also refer to long-dead thinkers, since in philosophy one should almost never assume that someone was wrong simply because they lived in the past. Plato's philosophy of mind loses nothing through the circumstance that it emerged in ancient Athens – and thus, incidentally, in the context of an advanced culture to which we owe some of the most profound insights concerning ourselves. Of course, Plato was wrong if he believed that we had an immortal soul somehow invisibly governing the actions of the human body. However, if you actually carefully read Plato, let alone Aristotle, it is far from clear that he believed in what the British philosopher Gilbert Ryle famously derided as "the ghost in the machine."
Homer, Sophocles, Shakespeare, Tom Stoppard or Elfriede Jelinek can teach us more about ourselves than neuroscience. Neuroscience attends to our brain or central nervous system and its mode of functioning. Without the brain there can be no mind. The brain is a necessary precondition for human mindedness. Without it we would not lead a conscious life but simply be dead and gone. But it is not identical with our conscious life. A necessary condition for human mindedness is nowhere near a sufficient condition. Having legs is a necessary condition for riding my bicycle. But it is not a sufficient one, since I have to master the art of riding a bicycle and must be present in the same place as my bicycle, and so forth. To believe that we completely understand our mind as soon as the brain is understood would be as though we believed that we would completely understand bicycle riding as soon as our knees are understood.
Let us call the idea that we are our brains the **crude identity thesis**. A major weakness of the crude identity thesis is that it immediately threatens to encapsulate us within our skull as minded, thinking, perceiving creatures. It becomes all too tempting to associate the thesis with the view that our entire mental life could be or even is a kind of illusion or hallucination. I have already criticized this thesis in _Why the World Does Not Exist_ , under the heading of **constructivism**.7 This view, coupled with neurocentrism, supposes that our mental faculties can be entirely identified with regions of the brain whose function consists in constructing mental images of reality. We cannot disengage ourselves from these images in order to compare them with reality itself. Rainer Maria Rilke gets to the heart of this idea in his famous poem "The Panther": "To him, there seem to be a thousand bars / and back behind those thousand bars no world."8
A recent German radio series called _Philosophie im Hirnscan_ [Philosophy in the Brain Scan] asked the question whether it is "not the human mind but rather the brain" that "governs decisions." "Free will is demonstrably an illusion."9 What is worse, it claimed that neuroscience finally provides support for a thesis allegedly held by Immanuel Kant, namely that we cannot know the world as it is in itself.10 The German philosopher of consciousness Thomas Metzinger (b. 1958), a prime representative of neurocentrism, is reported to claim that
> Philosophy and neuroscience agree: perception does not reveal the world, but a model of the world. A tiny snippet, highly processed, adjusted to the needs of the organism. Even space and time, as well as cause and effect, are created by the brain. Nevertheless, there is a reality, of course. It cannot be directly experienced, but it can be isolated by considering it from various vantage points.11
Fortunately, many philosophers who work in epistemology and the theory of perception would not accept this statement today. The theory that we do not directly experience reality but can only isolate it by considering it from various vantage points proves on closer inspection to be incoherent. For one thing, it presupposes that one can directly experience a model of the world, as the quotation indicates. If one had to isolate this model itself indirectly by considering reality from various vantage points on one's own, one would not even know that, on the one hand, there exists a model and, on the other, a world of which we construct internal models. To know that one constructs or even has to construct a model of reality implies knowing something about reality outright, such as that we can only know anything about it indirectly for some reason or other. Thus one does not always need models and is not trapped within them, so to speak. And why then, pray tell, does the model not belong to reality? Why should, for instance, my thought that it is raining in London, which indeed I do not first need to isolate by considering reality from various vantage points in order to have it, not belong to reality? If my thought that it is raining in London right now does not belong to reality (as I can know it without first having to construct a model of mental reality), then where exactly does it take place? With this in mind, new realism in philosophy claims that our thoughts are no less real than what we are thinking about, and thus that we can know reality directly and need not make do with mere models.12
My own view, **New Realism** , is a version of the idea that we can actually grasp reality as it is in itself by way of our mental faculties. We are not stuck in our brains and affected by an external world only via our nerve endings such that our mental life is basically a useful illusion, an interface or computational platform with a basic evolutionary survival value.
After the failures of the loudmouthed exaggerated promises of the decades of the brain, the _Süddeutsche Zeitung_ , one of the biggest German newspapers, pithily reported:"The human being remains indecipherable."13 It is thus time for a reconsideration of who or what the human mind really is. Against this background, in this book I sketch the outlines of a philosophy of mind for the twenty-first century.
Many people are interested in philosophical questions, in particular those pertaining to their own minds. Human beings care about what it means for them to be minded, suffering, enjoying, thinking and dreaming animals. Readers who are interested in philosophy but do not spend the whole day going through philosophical literature often have the reasonable impression that one can only understand a philosophical work if one has read countless other books first. In contrast, the present work should be accessible without such assumptions, insofar as it also provides information about the relevant basic ideas lurking in the background. Unfortunately, many generally accessible books about the human mind in our time either simply assume a naturalistic framework or are driven by the equally misguided idea that we have immortal souls on top of our bodies. My own stance is thoroughly antinaturalistic in that I do not believe that nature or the universe (or any other domain of inquiry or objects) is the sole domain there is. We do not have to fit everything into a single frame. This is an age-old metaphysical illusion. However, there is a further question concerning the structure of human mindedness, as the human mind certainly has neurobiological preconditions (no mind without a suitable brain) but also goes beyond these conditions by having a genuinely immaterial side which we will explore.
## _**Can the mind be free in a brain scan?**_
My overall goal is the defense of a concept of _the mind's freedom_. This includes the fact that we can deceive ourselves and be irrational. But it also includes the fact that we are able to discover many truths. Like any other science or discipline, philosophy formulates theories, gives reasons for them, appeals to facts that are supposed to be acceptable without further ado, and so forth. A theory consists of propositions – claims that can be true or false. No one is infallible, certainly not in the area of self-knowledge. Sophocles portrayed this harshly in _Oedipus the King_ , but hopefully things will not unfold so tragically here.
My main targets in this book, neurocentrism and its pioneering precursors – the scientific image of the world, structuralism and poststructuralism – are all _philosophical_ theories. Sometimes it seems as if the empirical findings of brain research entail that the self and the brain are identical. Advocates of what I critically call "neurocentrism" act as though they could appeal to scientific discoveries that should not be doubted by reasonable modern citizens and thus to alleged facts recognized by experts. Yet, with its sweeping assumptions, neurocentrism formulates genuinely philosophical claims, which here means claims that one cannot delegate to some other branch of learning. Science itself does not solve philosophy's problems unaided by philosophy's interpretation of its results. Neurocentrism is ultimately just bad philosophy trying to immunize itself against philosophical critique by claiming to be justified not by philosophy, but by actual neuroscientific discoveries. Notice, though, that no neuroscientific discovery, no fact about our neurobiological equipment without further interpretation, entails anything about human mindedness.
In particular, for its interpretation of neuroscientific knowledge, neurocentrism brings to bear _philosophical_ concepts such as consciousness, cognition, representation, thinking, self, mind, free will, and so forth. Our understanding of those concepts by means of which we describe ourselves as minded animals is stamped by a millennia-long intellectual, cultural and linguistic history. There is no possibility for us simply to sidestep this fact and take, as it were, a neutral or fresh look at the human mind, as if from nowhere or from a God's-eye point of view. Our ways of thinking about ourselves as thinkers are mediated by the language we speak and by the manifold cultural assumptions which govern our self-conception, as well as by a huge range of affective predispositions. Our self-conception as minded always also reflects our value system and our personal experience with mindedness. It has developed in complex ways, in the tension between our understanding of nature, literature, legal systems, values of justice, our arts, religions, socio-historical and personal experience. There just is no way to describe these developments in the language of neuroscience that would be superior or even equal to the vocabulary already at hand. Disciplines such as neurotheology or neuroaesthetics are "terrifying theory-golems," as Thomas E. Schmidt sharply puts it in an article on new realism for _Die Zeit_.14 If a discipline only gains legitimacy by being able to observe the brain while the brain engages with a given topic or object, we would ultimately need a neuro-neuroscience. Whether we then would also still have to come up with a neuro-neuro-neuroscience, time will tell...
There is also a suspicious political motivation associated with neurocentrism. Is it really an accident that the decade of the brain was proclaimed by George H. W. Bush shortly after the fall of the Berlin Wall in 1989 and thus with the end of the Cold War looming? Is this just a matter of political support for medical research? Does the idea of being able to watch the brain – and thereby the citizen – while thinking not also imply a new possibility for controlling social surveillance (and the military-industrial complex)? It has long been well known that possibilities for controlling consumers are expected from a better understanding of the brain: think of neuro-economics, another theory-golem out there.
As the German neuroscientist Felix Hasler (b. 1965) plausibly argued in his book _Neuromythology_ , the decade of the brain also goes along with various lobbying efforts. By now, more students at American research universities take psychotropic drugs than smoke cigarettes.15 The higher resolution and more detailed knowledge of our images of the brain promise to contribute to social transformation in the context of what the German sociologist Christoph Kucklick aptly summarizes as a "control-revolution." He observes that we live in a "granular society," where we are no longer merely exploited but are also put in a position to interpret ourselves as objects of medical knowledge.16 The crude identity thesis corresponds to the fantasy that our self, our entire human mind, turns out to be a physical object among others, no longer hidden from view.
The question, of who or what the ominous self really is, is thus revealed to be significant not merely for the discipline of philosophy but rather in political terms as well, and it concerns each one of us on an everyday level. Just think about the popular idea that love can be defined as a specific "neurococktail" or our bonding behavior be reduced to patterns trained in prehistoric times in which our evolutionary ancestors acquired now hard-wired circuits of chemical flows. In my opinion, these attempts are really relief fantasies, as they defer responsibility to an in itself irresponsible and nonpsychological machinery which runs the show of our lives behind our backs. It is quite burdensome to be free and to thus figure that others are free, too. It would be nice if we were relieved of decisions and if our life played out like a serenely beautiful film in our mind's eye. As the American philosopher Stanley Cavell puts it: "Nothing is more human than the wish to deny one's humanity."17
I reject this wish, and in this book I argue for the idea that the concept of mind be brought into connection with a concept of freedom, as it is used in the political context. Freedom is not merely a very abstract word that we defend without knowing what we actually mean by it. It is not merely the freedom guaranteed by a market-based economy, the freedom of choice of consumers confronted with different products. On closer inspection, it turns out that human freedom is grounded in the fact that we are minded animals who cannot be completely understood in terms of any natural-scientific model or any combination of them, be it present or futuristic. Natural science will never figure us out, not just because the brain is too complex (which might be a sufficient ground to be skeptical about the big claims of neurocentrism), but also because the human mind is an open-ended process of creation of self-conceptions of itself. The core of the human mind, the capacity to create said images, is itself empty. Without the variation in self-images, no one is at home in our minds, as it were. We really exist in thought about ourselves, which does not mean that we are infallible or illusions.
And thus we come to the heart of the matter. We exist precisely in the process of reflecting on ourselves. This is the lot of our form of life. We are not merely conscious of many things in our environment, and we do not merely have conscious sensations and experiences (including feelings); rather, we even have consciousness of consciousness. In philosophy we call this _self-consciousness_ , which has little to do with the everyday sense of a self-conscious person. Self-consciousness is consciousness of consciousness; it is the kind of state you are in right now as I instruct you to think about your own thought processes, to relate mentally to your own minds.
The concept of mental freedom that I develop in this book is connected to the so-called existentialism of Jean-Paul Sartre (1905–1980). In his philosophical and literary works, Sartre sketched an image of freedom whose origins lie in antiquity and whose traces are to be found in the French Enlightenment, in Immanuel Kant, in German Idealism (Johann Gottlieb Fichte, Friedrich Wilhelm Joseph Schelling, Georg Wilhelm Friedrich Hegel), Karl Marx, Søren Kierkegaard, Friedrich Nietzsche, Sigmund Freud and beyond. In contemporary philosophy this tradition is carried on in the United States primarily on behalf of Kant and Hegel, although Kierkegaard and Nietzsche are also assigned a role. So far, Albert Camus and Sartre have played a minor role in the revival of the existentialist tradition in the philosophy of mind.
I mention these names only in order to remind us of an important strand of human thought about ourselves. The philosophical tenet I take over from that tradition I call **neo-existentialism** , which claims that human beings are free insofar as we must form an image of ourselves in order to be individuals. To have a human mind is to be in a position in which one constantly creates self-conceptions in light of which we exist. We project self-portraits of ourselves, who we are and who we want to be, as well as who we should be, and via our self-portraits we relate to norms, values, laws, institutions, and rules of various kinds. We have to interpret ourselves in order to have any idea of what to do. We thus inevitably develop values as reference points for our behavior.
Another decisive factor at this point is that we often project false and distorted self-images onto social reality and even make them politically effective. The human being is that being which forms an idea of how it is included in realities that go far beyond it. Hence, we project visions of society, images of the world, and even metaphysical belief-systems that are supposed to make everything which exists part of a gigantic panorama. As far as we know, we are the only animals who do this. In my view, however, it does not diminish other animals or elevate us morally in any sense over the rest of the animal kingdom which would justify the current destruction of the flora and fauna of our planet, the only home to creatures who are able to orient their actions in light of a conception of a reality that goes beyond them. It is not that we human beings should be triumphantly intoxicated on our freedom and should now, as masters of the planet, propose a toast to a successful **Anthropocene** , as the epoch of human terrestrial dominance is called today.
Many of the main findings arrived at by the philosophy of mind in the twentieth and twenty-first century to this point are still relatively little known to a wider public. One reason for this certainly lies in the fact that both the methods and the arguments that are employed in contemporary philosophy typically rest on complicated assumptions and are carried out in a quite refined specialized language. In this respect, philosophy is of course also a specialized discipline like psychology, botany, astrophysics, French studies or statistical social research. That is a good thing. Philosophy often works out detailed thought patterns about particular matters in a technical language out of sheer intellectual curiosity. This is an important training and discipline.
However, philosophy has the additional and almost equally important task of what Kant called "enlightenment," which means that philosophy also plays a role in the public sphere. Given that all human beings in full possession of their mental powers constantly construct self-images which play out in social and political contexts, philosophy can teach everybody something about the very structure of that activity. Precisely because we are nowhere close to infallible with respect to our mindedness, we often create misguided self-conceptions and even support them financially, such as the erroneous but widespread conception that we are identical with our brains or that consciousness and free will are illusions.
Kant explicitly distinguishes between a "scholastic concept" [ _Schulbegriff_ ] and a "cosmopolitan concept" [ _Weltbegriff_ ]18 of philosophy. By this he meant that philosophers do not only exchange rigorous, logically proficient arguments and on this basis develop a specialized language. That is the "scholastic concept" of academic philosophy. Beyond this, in Kant's view, we are obligated to provide the public with as extensive an insight as possible into the consequences of our reflections for our image of the human being. That is the "cosmopolitan concept" of philosophy. The two concepts go hand in hand, such that they can reciprocally critique each other. This corresponds to the fundamental idea of Kantian enlightenment – a role that philosophy already played in ancient Greece. The word "politics" itself, together with the set of concepts which still structure our overall relation to the public sphere, the polity, derives from ancient Greek philosophy. The idea made prominent by Plato's Socrates is that philosophy, among other things, serves the function of critically investigating our self-conception as minded, rational animals. For many central concepts, including justice, morality, freedom, friendship, and so forth, have been forged in light of our capacity to create an image of ourselves as a guide to human action. Again, we can only act as human beings in light of an implicit or explicit account of what it is to be the kind of minded creature we happen to be. Given that I believe that neurocentrism is a distorted self-conception of the human mind, full of mistakes and based on bad philosophy, it is time to attack in the name of enlightenment.
## _**The self as a USB stick**_
The _Human Brain Project_ , which was endowed with more than a billion euros by the European Commission, has come under heavy criticism. Originally, its aim was to consolidate current knowledge of the human brain by producing computer-based models and simulations of it. The harmless idea was to accumulate all the knowledge there is about the brain, to store it and to make it readily available for future research.
Corresponding to the idea of medical progress based on knowledge acquisition are wildly exaggerated ideas of the capability of artificial intelligence which permeate the _zeitgeist_. In films such as Spike Jonze's _Her_ , Luc Besson's _Lucy_ , Wally Pfister's _Transcendence_ , Neill Blomkamp's _Chappie_ or Alex Garland's _Ex Machina_ , mind, brain and computer are confused so as to give rise to the illusion that we could somehow (and soon) upload our mental profiles on non-biological hardware in order to overcome the limitations of biologically realized mental life. In order to convince us that there is no harm in such fantasies, in _Her_ we are shown a protagonist who falls in love with his apparently highly intelligent software, "who" happens to develop a personality with existential problems so that the program decides to break up with its human lover. In _Transcendence_ , the protagonist becomes immortal and omnipotent by uploading his self onto a computer platform in order to disseminate himself on the internet. In _Lucy_ , the female protagonist, after she is able consciously to control 100 percent of her brain activity under the influence of a new drug from Asia, succeeds in transferring herself onto a USB stick. She becomes immortal by transforming herself into a pure mass of data on a data carrier. This is a rampantly materialist form of pseudo-Buddhist fantasy represented by the idea that such a mind/ brain-changing drug has to come from Asia, of course...
Along with the imaginary relief that we get in wishing to identify our self with the grey matter in our skull, our wish for immortality and invulnerability also plays a decisive role in the current world picture. The internet is presented as a platform of imperishability, onto which one can someday upload one's mind purified of the body in order to surf through infinite binary space forever as an information ghost.
Somewhat more soberly, the scientists of the _Human Brain Project_ anticipate medical progress through a better understanding of the brain. At the same time, this project also proclaimed on its homepage, under the tab "Vision," that an artificial neuroscience based on computer models, which no longer has to work with actual brains (which among other things resolves ethical problems), "has the potential to reveal the detailed mechanisms leading from genes to cells and circuits, and ultimately to cognition and behaviour – the biology that makes us human."19
Scientific and technological progress are welcome, without question. It would be irrational to condemn actual progress merely on the ground that science over the last couple of hundred years has led to problems such as nuclear bombs, global warming, and ever more pervasive surveillance of citizens through data collection. We can only counter the problems caused by modern humanity with further progress directed by enhanced value systems based on the idea of an ethically acceptable sustainability of human life on earth. Whether we resolve them or whether humanity will obliterate itself sometime in the foreseeable future cannot be predicted. This depends on whether we will even recognize the problems and identify them adequately. It is in our hands to implement our insights into the threatened survival conditions of human animals. We underestimate many difficulties, such as, for instance, the overproduction of plastic or the devastating air pollution in China, which already afflicts hundreds of millions. Other problems are hardly understood at all at this point, such as, for example, the complex socioeconomic situation in the Middle East. These problems cannot be dealt with by outsourcing human self-conceptions to objective representations, maps and models of the brain. It is not only that we do not have enough time left to wait for neuroscience to translate everything we know about the human mind into a neurochemically respectable vocabulary. Rather, there is no need to do this, as we are already equipped to deal with the problems, but underfinanced and understaffed in the relevant fields of inquiry.
We certainly do not want to return to the Stone Age, indeed not even to technological conditions in the nineteenth century. A lamenting critique of modernity does not lead anywhere except for those who long for the end of civilization, which more likely than not reflects their own fears of civilization, their "discontent with civilization" (Freud).20 It is already almost unthinkable for us _digital natives_ to write someone a letter via snail mail. If there were no email, how would we organize our workplace? As always, technological progress is accompanied by technophobia and the potential for ideological exploitation, which govern the debates over the digital revolution and the misuse, as well as the monitoring, of data on the internet.
General technological progress is evidently not that bad. On the contrary: I am glad that I can write emails and be electronically connected with my friends around the globe. Furthermore, I am glad that I no longer have to go to a video store in order to rent films. I am glad that I can order pizza, book my vacation online, and find information ahead of time concerning hotels, beaches or art exhibitions. Our technologically sophisticated and scientifically respectable progressive civilization is not in itself a "context of delusion," as pessimistic cultural critics following the philosopher Adorno suppose.
Yet, as always, there are delusions today. The major delusion is that scientific and technological process unaided by cultural, philosophical, ethical, religious and artistic reflection could by itself lead to an enhanced understanding of the human mind in light of which we could improve on our decision-making or courses of action.
Admittedly, our present-day techno-scientific progress has its dark sides, problems of our own making on an undreamt-of scale: cyberwars, ecological degradation, overpopulation, drones, cybermobbing, terrorist attacks prepared in social networks, nuclear weapons, attention deficit disorders, and so forth. Nevertheless, remarkable regressions in the domain of self-knowledge are to be noted, which are this book's concern. When it comes to such regressions, we are dealing with ideology, thus with a certain kind of illusion that proliferates for as long as no one revolts against it. _Ideology critique_ is one of the main functions of philosophy in society, a responsibility that one should not avoid.
## _**Neuromania and Darwinitis – the example of**_ **Fargo**
The British medical physician and clinical neuroscientist Raymond Tallis (b. 1946) coined the terms "neuromania" and "Darwinitis," by which he understands humanity's current misrepresentation of itself.21 **Neuromania** consists in the belief that one can know oneself by learning ever more about one's central nervous system, especially concerning the workings of the brain. **Darwinitis** complements this view with the dimension of our deep biological past in an attempt to make us believe that typical present-day human behavior is to be better understood, or perhaps only explainable at all, if we reconstruct its adaptive advantage in the struggle for survival amid the tumult of species on our planet. Neurocentrism is the combination of neuromania and Darwinitis, and thus the idea that we can understand ourselves as minded animals only if we investigate the brain while considering its evolutionary prehistory.
A wonderfully ironic example of Darwinitis can be found in an episode of the brilliant television series _Fargo_. The psychopathic killer Lorne Malvo, skillfully played by Billy Bob Thornton, is temporarily arrested by a policeman who recognized him. However, Malvo had already previously come up with the ingenious idea of presenting himself as a priest on a church website, so that he is quickly released, since the police all too credulously take a supposed church's website on the internet for the real thing. When Malvo heads out of the police station, the above-mentioned policeman, who knows his true identity, asks him how he can reconcile his behavior – being a killer who uses a fake identity as a priest – with his human conscience. Malvo replies by asking him why human beings can visually distinguish so many shades of the color green. The policeman is at a loss but later asks his fiancée about it. She answers him as follows. Our refined color palette for green stems from the fact that, back in the time when we were hunter-gatherers, we had to recognize our natural enemies as well as our prey in bushes and dense forests. Natural selection thus brought about our specific color vision, which generally speaking is hard to deny. Without natural selection and the color spectrum consciously available to us due to our biological equipment, our species would probably not exist.
Yet, Malvo does not merely evoke a biological fact. With his response, he actually wants to communicate that he is a hunter and to justify it. He would like to _justify_ his behavior by pointing out that we are descended from hunters and killers and hence that his killing represents a kind of natural necessity. He thus advocates a crass form of social Darwinism and consequently a philosophical position. **Social Darwinism** advocates the thesis that every kind of interpersonal behavior between human beings can be understood, explained and justified according to parameters implemented in the survival of biological species that can be investigated by evolutionary biology. What we do is driven by behavioral patterns ultimately rooted in hard-wired facts about our biological make-up and not in ethical reflection which floats free from our biology.
Although _Darwinism_ , of course, first emerged in the second half of the nineteenth century, certain basic ideas of _social Darwinism_ are much older. The ancient Greeks already discussed it – for instance, Plato in one of his main works, in the first book of _The Republic_. The ancient philosopher Thrasymachus appears there and defines justice as "nothing else than the advantage of the stronger."22 In the first half of the nineteenth century, Arthur Schopenhauer began describing specifically human behavior in a proto-social-Darwinistic manner. For example, he explained romantic love as well as many other social facts and processes in general on the basis of human sexuality as courtship behavior in the biological sense of the term, which in his case was accompanied by a marked general misanthropy and misogyny in particular. Schopenhauer – to put it lightly – had problems dealing with the opposite sex.
One finds explanations like this everywhere in popular culture and science today. In particular, biological categories are employed to explain the basic structures of human behavior in relationships that we are all involved in on an everyday level. We indeed wish that it might finally be revealed that the human being is "only" an animal, too, and in any case we do not want to be so naïve as to believe that we have completely dropped out of the animal realm. Perhaps, from a bad conscience toward the other animals (whom we gawk at in zoos and happily grill on warm, summer evenings with a bottle of beer in one hand), we wish to pretend that the human being is no exception in the animal realm but rather, simply by coincidence, also a living being with a very peculiar kind of mind – and, by the way, most likely the only animal worried about its position in the animal realm in light of the very idea of such a realm. This is one of the reasons why, as far as I know, we are the only animals that rip other animals to shreds with machines, pack their meat into their intestines, turn this mess into a sausage, and then enter into a dialogue over the most skillful preparations for roasting them. In its sophistication, this goes way beyond any kind of cruelty we find in the animal kingdom, and it serves the function of creating a culture of meat consumption which makes it look as if meat was not really meat, given that sausages are highly artificial products obscuring their origins.
## _**Mind – brain – ideology**_
One of the main theses of this book is that the issues that have been touched on up to this point, of delegating our self-knowledge to new scientific disciplines, are ideological and thus misguided fantasies. What I criticize here under the heading of **ideology** is a system of ideas and knowledge claims in the realm of self-knowledge, which misconstrues the products of the mind's freedom as natural, biological processes. Seen in this light, it is no wonder that the contemporary ideology of neurocentrism has particularly tried to dismiss the concept of human freedom. It is supposedly for the best that there are no products of human freedom at all.
The goal would be achieved if one were able to trace Heinrich von Kleist's _Amphitryon_ , Gioachino Rossini's _Petite messe solennelle_ , the hip-hop of the 1990s or the architecture of the Empire State Building as somewhat complex variants of the ludic drive prevalent in the animal kingdom. It is widely acknowledged that our sciences are still infinitely far from achieving this goal. Thus we find, toward the end of the much discussed German "Manifesto: Eleven Leading Neuroscientists on the Present and Future of Brain Research," which appeared in the popular science journal _Gehirn und Geist_ [Brain and Mind]:
> Even if we should someday eventually bring to light all of the neural processes that underlie human sympathy, being in love, or moral responsibility, the uniqueness of this "inner perspective" is still preserved. Even one of Bach's fugues is no less fascinating when one has precisely understood how it was composed. Brain research will have to clearly distinguish what it can say and what lies beyond its domain, as musicology – to stick to this example – has a few things to say about Bach's fugue but must remain silent when it is a matter of explaining its singular beauty.23
As a representative example of neurocentrism – that is, the thesis that self = brain – one can point to the book by the Dutch brain scientist Dick Swaab (b. 1944) entitled _We Are Our Brains: A Neurobiography of the Brain, from the Womb to Alzheimer's_.24 Right at the beginning of the introduction we read that:
> Everything we think, do, and refrain from doing is determined by the brain. The construction of this fantastic machine determines our potential, our limitations, and our characters; _we are our brains_. Brain research is no longer confined to looking for the cause of brain disorders; it also seeks to establish why we are as we are. It is a quest to find ourselves.25
This quote nicely illustrates how, according to neurocentrists, brain research is no longer supposed to be research only into the mode of functioning of an organ. It now wishes, or at least Dick Swaab wishes, to embark on the "quest to find ourselves." Without a healthy brain, admittedly, we would not exist. We could not think, be aware or live consciously. However, without many more arguments, it does not follow that we are identical to our brain.
An initial distinction which helps to clarify why we should resist the scientifically premature and philosophically deluded claim that we are identical to our brain even if we cannot exist without it is the distinction between _necessary_ and _sufficient conditions_. For example, it is necessary that I have a bagel and jelly in order to be able to spread jelly on a bagel. But it is by no means sufficient. If the jelly is in the fridge and the bagel is in China, I do not easily achieve the state of having bagel with jelly smeared on top of it. For that to happen, I must properly blend jelly and bagel, for instance with butter, and all of us – me, jelly, bagel and butter – have to be in the same place. The necessary material conditions for there being an event of bagel consumption do not suffice for this event to happen.
We are in a similar situation when it comes to the brain. One reason why we are not identical to our brain consists simply in the fact that we do not first of all have a body that is composed only of neurons but have many additional organs that consist of other kinds of cells. Furthermore, we would not even be close to being what we are if we did not exist in social interaction with other human beings. We would have no language and would indeed not even be capable of surviving, since human beings are anything but born solipsists, as we cannot have a normal kind of consciousness without communicating with others.
Many cultural facts simply cannot be explained by observing a brain. At the very least, one must take into consideration a variety of brains that are found in mature, healthy human organisms. This makes the subject matter of neuroscientific observation forever too complex, since, because of its individual character and plasticity, even a single brain cannot be anywhere near to being completely described. Good luck to the attempt to investigate, for instance, the sociocultural structure of a current Chinese metropolis or even a small town in the Black Forest with the methods of neurobiology! Not only is such a project completely utopian, but it is also superfluous, since we are in possession of far better means. Our methods of describing and explaining cultural facts stem from the long history of the acquisition of self-knowledge, which besides philosophy includes, of course, literature, music, art, sociology, psychology, the colorful bouquet of the human sciences, religions, and so forth.
For more than a hundred years, the philosophy of mind has been centrally concerned with the relation between mind and brain, a question that became acute in early modern philosophy, especially with René Descartes (1596–1650). Descartes famously formulated a (pretty crude) way of looking at the problem: for he asked how an immaterial (meaning: not material) thing, such as a thought or a series of thoughts, could ever interact with the material world. If it could, it could not really be immaterial, as only matter causally interferes with matter. If it could not, our actions seemed to be utterly mysterious, as we often act by forming a plan in the realm of thoughts about the near future and then carry it out in the world of bodies by moving our limbs accordingly.
However, the mind–brain problem harks back to antiquity, to the question already formulated in ancient Greece, namely, how our body relates in general to our mind or our soul. This has created the general **mind–body problem** , of which our more recent _mind–brain_ problem is a variant. The most general formulation of the problem runs as follows. How can conscious subjective mental experience take place at all in an unconscious, cold, purely objective universe that unfolds according to natural laws? How could our subjective conscious states possibly fit into a larger scheme of things in the universe, which is not governed at all by the kinds of psychological or logical laws constitutive of our self-encounter as thinking beings? Suns, moons, fermions and galaxy clusters, dark matter and CO2 do not have or support an inner life. How come that the brain does? The prominent Australian philosopher of consciousness David Chalmers (b. 1966) has called this the **hard problem of consciousness**. In other words, how does our apparently provincial perspective on the universe fit into the natural order of the universe which exceeds our attempt to conceive it?
At this point, it is very tempting in our current cultural climate to come up with a simplistic solution, one which basically just shrugs its shoulders and flatfootedly denies that there is a problem. For instance, one could suppose that the conscious self is generated by the brain. One could perhaps understand this production as a side-effect of an adaptive advantage in the struggle for survival of species and individuals. Consciousness would then simply exist because certain kinds of brains prevailed in evolutionary terms, namely those generating consciousness. This certainly looks like some kind of explanation. But if our conscious self is _generated_ by the brain, it cannot simply and self-evidently be _identical_ to what generated it. If A is generated by B, then A and B are in any event not strictly identical. Hence, either the self is _generated_ by the brain (perhaps as an illusion that is useful for survival or as our organism's user interface) or it is _identical_ with the brain. This is where theorizing begins, as one cannot obtain clarity and coherence in this matter by confusing production and identity, which Sam Harris (b. 1966) absolutely fails to do in his book _Free Will_.26 Harris there actually claims that the self is generated by the brain and hence is not free, while he also denies the existence of an independent mental level. However, how is it possible that the brain creates something with which it is identical? Traditionally, this model, the model of a _causa sui_ , a cause of itself, was reserved to God, and it did not help theology's coherence either... The problem is that, if there is really only a brain and not in addition to it a mental level (let alone a soul), then, if the brain produces consciousness, it has to produce a brain. Of course, this invites the immediate response that consciousness is identical only to a part of the brain, but again this does not help, as it is not the case that the non-conscious parts of the brain literally produce the conscious parts of the brain – at least there is no known mechanism that instantiates this part–whole relation.
To deny human freedom on the basis of the claim that we are identical to our brain will never work, even if our brain unconsciously makes decisions for us. For one thing, on this model we are precisely still free, since the brain for its part is not supposed to depend on the unconscious decisions of another system. If my brain controls me, but I am my brain, then my brain controls itself, or I control myself. Thus freedom is not imperiled but rather elucidated. If the brain is a self-determining system where the non-conscious parts bring about explicit and conscious decision-making, this does not undermine our freedom but is, rather, an account of it. One of the many sources of confusion in Sam Harris's wishful thinking that he can eliminate his own free will by theorizing about it in an incoherent manner is that he really provides us with a (bad) model of free will and not with a way of undermining it. Also, why would free will require that I consciously create a conscious decision to do something? This view immediately runs into a vicious infinite regress, as I would have to pile up infinitely many conscious decisions in order to act in a conscious manner! I would have consciously to produce my consciousness of my consciousness,... , to act. This will not work. But it has nothing to do with the brain.
## _**The cartography of self-interpretation**_
The critique of neurocentrism is nevertheless only one objective of this book. At the same time, I would like to map the intellectual landscape of our self-knowledge by presenting some central basic concepts of the philosophy of mind. How are concepts such as consciousness, self-consciousness, self, perception and thinking connected and how, anyway, do they become part of our vocabulary?
In what follows, I will hence also be concerned with positive self-knowledge, and thus with the question of who we really are. The **main positive thesis** of the neo-existentialism sketched out here is the claim that the human mind engenders an open multiplicity of capacities all of which involve the mind, because the mind creates an image of itself by way of these self-interpretations. _The human mind makes an image of itself and thereby engenders a multiplicity of mental realities_. This process has a structure that is historically open, and which cannot be conceived only in the language of neurobiology. Neurobiology will only ever be able to account for some necessary conditions of human mindedness, even if it informs us that there are restrictions on human action of which we might not have been aware before. The fact that no complete form of self-knowledge of the human mind can ever be achieved via neurobiology alone is grounded in the fact that the human mind is not purely a biological phenomenon.
Our capacity for developing false world pictures and misguided self-conceptions is intertwined with the phenomenon of ideology. What is crucial is that even false self-conceptions express something, as they present those who hold them as taking certain things to be true and accurately stated even though they are not. We are in constant dialogue with others about our self-image and general self-conception precisely because we are constantly negotiating and testing new modules of self-understanding. Some turn out to be delusional. Yet, those suffering from self-delusional models are afflicted by the associated beliefs. It is constitutive of the human mind that it can change according to its beliefs about itself, which includes the false ones. If I mistakenly believe that I am a great dancer and lead my life in light of this illusion regardless of my experience (which should teach me a lesson here), this changes my status as an agent.
The spectrum of the production of mental realities extends from a profound understanding of ourselves in art, religion and science (which includes the human, social, technological and natural sciences) all the way to the quite various forms of illusion: ideology, self-deception, hallucination, mental illnesses, and so forth. We have at our disposal, among other things, consciousness, self-consciousness, thinking, a self, a body, an unconscious, etc. The human mind is irreducibly multifarious and ever-changing. What remains is the core invariant of self-production.
The human mind does not have a reality that is independent from its self-images, such that one could simply compare this independent reality of the mind with its self-images. It exists only in such a way that it makes self-images. It thus always becomes what it makes of itself. It has a history for precisely this reason, the history of mind or _Geistesgeschichte_ , as we say in my neck of the woods.
A simple way to illustrate this thought is by the following contrast. As a matter of fact, I hardly know anything about trees. However, I know that elms are trees. But I could certainly not tell an Anhui elm from a Nikko elm without doing some kind of research (at least Google). Now imagine I see a tree somewhere and say to myself or to others that this is a Nikko elm. It might not even be an elm or an Anhui elm. Be that as it may, the tree is what it is regardless of my beliefs about it. It does not turn into an Anhui elm if I believe it to be one. And it does not change its status at all relative to my beliefs, which simply do not matter for the tree's being what it is. By contrast, if I am self-deluded and believe that I am a great tango dancer, I might start behaving in light of these false beliefs which sustain my self-delusion. Maybe I start traveling to Buenos Aires, where no one wants to dance with me, which I explain as a lack of recognition of my true tango genius, etc. In this case, my belief about myself, my capacities, my style, and so forth, changes me from someone with an accurate self-image into a deluded person. My beliefs about myself (including my beliefs about my beliefs) affect me profoundly and constantly. A mind is just not an elm, as you can tell from this little philosophical exercise.
Our epoch of the history of mind, modernity, has produced neurocentrism, which seems to be in harmony with a great basic motif of this epoch: to achieve enlightenment (the realization of a particular value system) through science. However, what we have increasingly forgotten in the recent history of this valuable epoch is the fact that it can founder and run aground. We need more modernity, not less. We currently lack a sufficiently widespread insight into the constitutive historicity of our self-images, which has been a central theme of philosophy for the last two hundred years and which ought not to be erased from our current concept of mind, which threatens to lose touch with its historical reality in the context of its wish to deny its own mindedness by replacing it with a silicon, plastic or neural counterpart.
## **Notes**
1. Francis Crick and Christof Koch, "Towards a Neurobiological Theory of Consciousness," _Seminars in the Neurosciences_ 2 (1990), pp. 263–75. 2. Concepts in bold print play a central role and for that reason, if possible, are introduced in the form of a definition or gloss. They can be kept track of with reference to the index of concepts. 3. See Markus Gabriel, _Why the World Does Not Exist_ (Cambridge: Polity, 2015). 4. George H. W. Bush, "Presidential Proclamation 6158," _Project on the Decade of the Brain_ , July 17, 1990, www.loc.gov/loc/brain/proclaim.html. See also Felix Hasler, _Neuromythologie: Eine Streitschrift gegen die Deutungsmacht der Hirnforschung_ (Bielefeld: Transcript, 2013). 5. Brigitte Stahl-Busse, "Dekade des menschlichen Gehirns," _idw – Informationsdienst Wissenschaft_ , November 5, 1999, <https://idw-online.de/pages/de/news15426>. 6. Ibid. 7. Gabriel, _Why the World Does Not Exist_ , pp. 38–44. 8. Rainer Maria Rilke, _The Selected Poetry of Rainer Maria Rilke_ , ed. and trans. Stephen Mitchell (New York: Vintage, 1989), p. 25. 9. Martin Hubert, "Teil 1: Des Menschen freier Wille," Philosophie im Hirnscan, April 4, 2014, www.deutschlandfunk.de/philosophie-im-hirnscan-manuskript-teil-1-des-menschen.740.de.html?dram:article_id=283145 trans. MG]. 10. Volkart Wildermuth, "Die Welt, wie sie scheint," Philosophie im Hirnscan, May 29, 2014,[www.deutschlandfunk.de/philosophieim-hirnscan-manuskript-die-welt-wie.740.de.html?dram:arrticle_id=287724 trans. MG]. 11. Ibid. 12. See at length the contribution in Markus Gabriel, "Wir haben Zugang zu den Dingen an sich," _Gehirn und Geist_ 3 (2014), pp. 42ff. 13. Christian Weber, "Der Mensch bleibt unlesbar," _Süddeutsche Zeitung_ , October 18–19, 2014 [trans. MG]. 14. Thomas Schmidt, "Die Wirklichkeit ist anders!," _Die Zeit_ , April 3, 2014, [www.zeit.de/2014/15/neuer-realismus trans MG]. See also the other contributions in the series on new realism in the issues of _Die Zeit_ from April 16 to July 16, 2014. There is an overview of the state of the debate in Gabriel (ed.) _Der neue Realismus_ (Berlin: Suhrkamp, 2014). 15. Hasler, _Neuromythologie: Eine Streitschrift gegen die Deutungsmacht der Hirnforschung_ , pp. 159ff. 16.Christoph Kucklick, _Die granulare Gesellschaft: Wie das Digitale unsere Wirklichkeit auflöst_ (Berlin: Ullstein, 2014), p. 11. 17. Stanley Cavell, _The Claim of Reason: Wittgenstein, Skepticism, Morality, and Tragedy_ (Oxford: Oxford University Press, 1999), p. 109. 18. Immanuel Kant, _Critique of Pure Reason_ , trans. Paul Guyer and Allen W. Wood (Cambridge: Cambridge University Press, 1998), p. 695 [A840/B868]. 19. European Commission, 2013; website no longer active. 20. _Das Unbehagen in der Kultur_ is the original German title of Freud's _Civilization and its Discontents_. The German literally means "Discontent in Civilization" [Trans.]. 21. Raymond Tallis, _Aping Mankind: Neuromania, Darwinitis and the Misrepresentation of Humanity_ (Abingdon and New York: Routledge, 2011). 22. Plato, _The Republic_ , trans. Paul Shorey (Cambridge, MA: Loeb, 1930), 338c. 23. Christian E. Elger et al., "Das Manifest: Elf führende Wissenschaftler über Gegenwart und Zukunft der Hirnforschung," _Gehirn und Geist_ 6 (2004), pp. 31–7; [www.spektrum.de/thema/das-manifest/852357. 24. D. F. Swaab, _We Are Our Brains: A Neurobiography of the Brain, from the Womb to Alzheimer's_ , trans. Jane Hedley-Prôle (New York: Spiegel & Grau, 2014). 25. Ibid., p. 3. 26. Sam Harris, _Free Will_ (New York: Free Press, 2012).
# 1
**What is at Stake in the Philosophy of Mind?**
At first glance, nothing seems more obvious than that mind is at stake in the philosophy of mind. But I have already argued that the mind is not a thing, a natural kind out there in the universe, whose nature we need to study in some natural scientific way. In the last century, in particular, a new approach emerged to the fact that we are creatures who are aware of their surroundings by perceiving them, who have feelings, thoughts and dreams, etc. This new approach is called "philosophy of mind"1 and was presented in paradigmatic form in Bertrand Russell's influential book _The Analysis of Mind_.2 What is called _Philosophie des Geistes_ in German, even today by many authors in the German-speaking world, stems from the discipline of the "philosophy of mind" in the English-speaking world. However, in German, _Bewusstseinsphilosophie_ [philosophy of consciousness] would be a more accurate translation of "philosophy of mind," which I would like to use in order to be able to distinguish this new orientation from previous currents of thought. Philosophers from Hegel to Habermas suggested a distinction between philosophy of consciousness and philosophy which deals with mind in the sense of _Geist_. This distinction is obscured by the decision to come up with a unified category, the mind, and think of it along the lines of consciousness. Consciousness is, roughly speaking, a subjective process that, for all we know, is somehow connected with the fact that we have a suitable brain. In order to be able to sketch the outlines for my contribution to the revival of self-conceptions in terms of _Geist_ , I will distinguish philosophy of consciousness and philosophy of mind. _Philosophy of consciousness_ , then, corresponds to the mainstream discipline called "philosophy of mind" in the English-speaking world, and _philosophy of mind_ , as in the title of this book, refers to an investigation into human mindedness, into the invariant capacity to produce self-conceptions and its differentiation into conceptual modules, such as consciousness, self-consciousness, thought, representation, will, etc.
What is problematically new in philosophy of consciousness is not so much to be found in its content but, rather, consists in the fact that the philosophy of consciousness typically assigns the philosophy of mind in general the task of seeking the answer to a specific question: What marks something out as a mental state or a mental event? For, if we ever want to make progress on the mind–brain problem, it seems a good move to carve out the concept of a mark of the mental so that we know what to relate to what. Notice that the concept of a brain is also more complicated than is presented by standard discussions. Usually, we speak of a "normal" healthy adult brain, for which we have a map on which we find the neural correlates of specific mental functions, such as the visual cortex with its subregions. However, this brain is really a model of a brain and not the kind of thing everybody has within their skulls. Brain science can only ever work out a model of the brain on the basis of very limited samples, and it tells us that brains have plasticity – that is, they can be highly individual and even change the function of some areas in order to replace other areas, etc. Yet, for the rest of this book I will simply play along and assume that we know how to individuate a brain. Hence, I will not try to attack neuroscience on that front and simply grant the concept of the brain.
This leaves us with the task of finding a "mark of the mental," as the saying goes, something which helps us to distinguish mind and brain first on a conceptual level in order to find out how they hang together in the natural world (if they do!). The mark accepted by many turns out to be consciousness, which is why the philosophy of mind has been concentrated one-sidedly on a single capacity of the human mind: consciousness.
The question referred to, concerning the mark of the mental, arises against the backdrop of the modern presumption that much of what we once may have considered in terms of mind turned out to be purely natural. Here once more the modern struggle against superstition comes to the fore. While it may once have been believed that the heavenly bodies move in regular paths and constellations in order to transmit the messages of the gods, in modernity we have finally realized that the universe contains no such messages for us. It is meaningless in that it does not contain messages and is not driven by any mind. The regular movements of the heavenly bodies can be explained mechanistically, and neither intention nor mind of any other kind is behind such an explanation. I have no intention to deny this.
Yet, according to this view, the mind was progressively banished from the universe or nature until it was resurrected in the shape of philosophy of consciousness. Mind has turned into consciousness, which, in turn, is supposed to rest on essentially subpersonal, non-conscious processes encapsulated in our brain. Some associate this progressive banishment with **secularization** , and thus with the disappearance of religion in favor of that which is not religious, above all scientific explanations, something supposedly characteristic of modernity. However, the question here is whether we even have criteria for something that can be considered as mental, and under what conditions a religious and a scientific explanation are really incompatible.
## _**Mind in the universe?**_
The first contrast that is prominent in the modern line of reasoning is that of nature and mind. In this vein, Russell ventures the claim that we should not make use of precisely this contrast, since otherwise the dualism feared and despised by nearly everyone will force itself upon us. This **dualism** is the thesis that the universe consists of two different kinds of objects or events: mental and natural. Most philosophers of consciousness today consider it untenable, because one must then assume that mental events would somehow have to impinge on the mechanism of the conservation and transformation of energy that belongs to purely natural processes. According to many contemporary scientists (but not according to Isaac Newton himself!), the laws of nature that teach us how the conservation and transformation of energy function tell us nothing about there being a mind that impinges causally on what happens. On the contrary, everything which takes place in nature or the universe apparently can be explained without recourse to a mind, since the laws of nature teach us that nothing can impinge causally on something without a transformation of energy/matter. Sure, this assumes that mind itself is not something material. If it were, by this logic it could easily impinge on what happens causally. Mind-matter has not yet been discovered, or so it seems. Thus one prefers to seek mind in the brain, because without the latter we would in fact have no conscious inner life, and therefore no consciousness, which the philosophy of consciousness ultimately considers to be the mark of the mental.
It is easy to imagine a perspective from which the mind-matter problem appears quite striking. Imagine Yonca would like to drink coffee. Accordingly, she goes to the kitchen and turns on the coffee machine. From the perspective of physics, we have no reason to assume that somewhere in Yonca's body a vital force, a soul or a mind is diffused and guides her body into the kitchen. Were something like this the case, its interaction with the body would have been proven long ago, since such a mind can impinge on the material-energetic reality studied by physics only if it leaves material-energetic traces according to the laws of nature. This means that energy would have been put to use, which can be measured. From this perspective, it seems most natural that we have to look for Yonca's apparent wish to drink coffee somewhere in the energetic meshwork of nature. Since, however, no soul is to be found there, but at most a brain, the question of how brain and mind are related to one another starts to look to make sense. We know from the description of the scenario that Yonca wants to drink coffee, and we know from the point of view of physics that this cannot mean that an immaterial soul interacts with her body without leaving any material-energetic trace. In this context, one usually invokes **the principle of the causal closure of nature** , which involves the claim that purely natural processes can never be interfered with by purely mental processes. Nothing which does not leave any material-energetic footprints can interfere causally with processes which require a material-energetic grounding in order to take place. This causes a problem if one wishes to distinguish mind from nature by the fact that the former is understood to be a non-material substance, a purely mental bearer of thoughts, and the latter is understood as the closed realm of the causal world, the universe or nature.
Within this framework, the American philosopher of consciousness Jaegwon Kim (b. 1934) asked somewhat derisively whether an almost immaterial mind, so to speak, could not still be somehow causally connected to our body. But then the question arises as to how the mind manages to accompany the body at great velocity. For example, how does the mind accelerate when an astronaut is sent into outer space? Can one measure it physically, or how does one conceive it? Could our body outrun our mind if only it were quick enough? Or is the mind fastened to the body somewhere, perhaps in the pineal gland in the brain, as Descartes, the greatgrandfather of the philosophy of consciousness, supposed?
The very idea of a causal closure of reality relies on the notion that there is a purely natural reality, an objective realm occupied by processes and objects such that everything in that domain can be apprehended, described and explained with scientific precision and objectivity. Let us call the realm of physical reality thus conceived **the universe**.
So far, so good. But what about psychology? Does it not examine something like the human mind with the aid of experiments, and thus also with scientific precision and objectivity? Yet, if this is the case, then the mind must belong to the universe. The contrast between nature and the mind that leads to a mind-matter problem quickly collapses. Notice that dualism looms large as soon as one supposes that there is a problem at all with the embedding of mind in nature. Hence, to be a dualist, one does not need to believe that there is a secret, non-material source of energy (the mind, the soul) which prowls around within our skull.
It is indeed correct that we do not find mind in the universe. But it does not follow from this that the mind does not exist! This only follows if we have an image of the universe as the only realm of what exists, as the one and only true reality. Yet such an image of the world is not scientifically or physically verifiable but has to remain a pure article of faith. At best, one can argue for it philosophically.
One prominent strategy at this point is to try to explain mind away altogether, which is called **theory reductionism** , in order simply to eliminate the problem we have outlined. This thesis has the name it does because we are supposed to reduce every theory that avails itself of mental processes into a theory in which the word "mind" no longer occurs. For instance, **behaviorism** attempted to translate all statements about mental processes into statements about observable, ultimately physical, measurable behavior, and flourished in the early days of reductive views. To be in pain meant only to exhibit pain behavior. The assumption that there are mental processes at all came to seem like a remnant of premodern superstition – a side-effect of the misguided attempt to find mind in the universe while considering the latter to be the one and only true reality.
## _**In the spirit of Hegel**_
The search for a mark of the mental is largely occupied with the question of how mental states and events can have a place in a purely natural universe. But, in framing the question this way, it is assumed that we should accept a standard or paradigmatic concept of reality: physical reality. In presupposing this, we already oppose mental reality to physical reality by definition. The question then becomes how such a rift can be overcome or resolved.
Let us give this whole way of posing the question a name: **naturalistic metaphysics**. **Metaphysics** is the theory of reality as a whole, also called "the world," "the universe," "reality (full stop)" or "the cosmos." Metaphysics deals with absolutely everything which exists, with the most encompassing totality of them all. If the world is identified with nature, this means that everything which really exists must be natural. The sense of "naturalism" in "naturalistic metaphysics" is that **naturalism** claims that absolutely everything which exists is natural in the sense of being an object of the best (natural-)scientific theory. We should note that this metaphysics is neither a finding confirmed by research in some particular scientific discipline – or even by all the sciences together – nor a presupposition of research in physics, chemistry, biology or any other branch of natural science. Rather, it is a philosophical theory about how the world as a whole is constituted.
In _Why the World Does Not Exist_ , I argued against all metaphysical world pictures which hold that there is a single reality, regardless of whether this is equated with the universe or with a giant hallucination in some god's mind. The assumptions driving a good deal of discussions in mainstream philosophy of mind are thus at least not without an alternative. If it turns out that we simply do not have to fit absolutely everything which exists into a single frame, there is room for a total reconfiguration of the very problems of the philosophy of mind, a reconfiguration I highly recommend given the manifold problems of naturalistic metaphysics.
The first problem is that naturalistic metaphysics relies on a concept of nature that has completely run its course and is even obsolete. That there is a universe, which at best can be completely explained by a unified physics in the sense of a "theory of everything," appears at present to be extremely utopian from the standpoint of science itself. Naturalistic metaphysics emerged in times when it seemed as though first Newton and then Newton + Einstein would suffice to provide a fundamentally complete account of the universe in the language of mathematics. Since the advent of quantum physics, that no longer sounds very plausible. The candidate that, these days, stands for a unified physics, namely string theory with its many varieties, does not appear to be experimentally verifiable at all. Currently, the very notion of absolutely everything which is physical cannot be pinned down by attaching it to physics. Physics is not unified, and it does not have an account of absolutely everything which is physical. At its borders, there are all sorts of speculations and open questions, from the nature of dark matter and dark energy to the hypotheses of a plurality of universes which might in turn be part of a multiverse landscape we will never be able to explore scientifically. In short, we are no longer even close to being able to specify how one could actually investigate the universe as a whole by means of experimentally grounded science.
The second problem of mainstream naturalistic metaphysics is that it does not ever concern itself with the human mind, but at most with consciousness. Thus it excludes the greatest part of the philosophical tradition that does not deem mind to be a subjective phenomenon of the kind that belongs to conscious experience. For example, in German one speaks of the "spirit of the age" [ _Zeitgeist_ ]. Hegel introduces the term "objective spirit" [ _objektiver Geist_ ], by which he means, for example, that a traffic sign is spiritual in the sense that it is the expression of an intention publicly to determine recognized rules for human behavior. Our social reality is defined by a huge array of artifacts which embody meaning so that our everyday experience is one where we constantly encounter objective manifestations of mind. In the philosophical tradition, human mindedness and language are intimately connected. But is language, for instance, purely subjective? Even texts that are handed down from the past can convey to us an impression of a _zeitgeist_ that no longer exists but which nevertheless was once a reality. The German philosopher Wolfram Hogrebe (b. 1945) expresses this concisely in an aphorism: "Spirit is outside, but breaks through inside."3 Our subjective mind is formatted in an encounter not with brute physical objects but with embodied meaning and social interaction. We never merely face an objective physical reality which we take in perceptually. The idea that the human mind in its infant stages, as it were, looks at the physical world and tries to make sense of it, is completely mythical, a modern myth based on the denial of the fact that our first encounter with reality is an encounter with people who interact with an environment consisting both of physical objects and artifacts.
In the nineteenth century in Germany, the concept of mind [ _Geist_ ] led to the adoption of the expression _Geisteswissenschaften_ as a designation for the humanities. This led to a bad opposition between the disciplines dealing with the human mind and the disciplines dealing with nature, the natural sciences. In the twentieth century, so-called hermeneutics (from the ancient Greek _hermēneia_ = understanding) postulated that the humanities investigate only that which one can understand, while the natural sciences do not _understand_ but rather wish to _explain_. Along these lines, the great Heidelberg philosopher Hans-Georg Gadamer (1900–2002) wrote that language qua "being that can be understood"4 is the object of the humanities.
The division of sciences into two broad categories over the decades happened to create the notion that ultimately only the natural sciences study the world, as they deal with things that are the way they are regardless of how we take them be, whereas the humanities deal with "softer" topics. But this climate assumes that there is only one true reality, the tough and blind world of physical forces into which we poor creatures are thrown so that our conscious life consists in coming up with illusions which help us to cover up the real.
At the same time, we have also witnessed a revival of serious philosophical interest in the tradition sometimes called German Idealism – i.e., the great philosophical systems that began with Kant and were further developed in the first half of the nineteenth century. This goes hand in hand with a revaluation of the humanities, including philosophy. Hegel's philosophy, which is fundamentally an attempt to give an account of what he calls _Geist_ , surprisingly occupies center stage once more. As a matter of fact, he provides us with nowhere near exhausted insights into the relation of "nature" and "spirit." In particular, he proposed a quite plausible version of the idea that the human mind consists in the formation of its own image and of its place in a reality which goes far beyond it.
Hegel puts this as follows: "Spirit is only what it makes of itself; it is the activity of producing itself, grasping itself."5 He thus takes up a few of the fundamental ideas of Kant and Fichte, both of whom are also relevant for contemporary ethics and practical philosophy, as you will see.
The philosophy of mind should include the tradition of philosophy from Plato to Sartre and beyond. Philosophy of consciousness should be a part of the larger enterprise to understand human mindedness and its relation to non-human reality, which includes the minds of other animals as well as purely physical reality. In any event, we have to reject the starting point of mainstream philosophy of consciousness – that is, first and foremost that we have to grope for a response to the misleading question of how the subjective phenomenon of our interior mental life fits into an anonymous, blind, unconscious nature without purpose, whose regularities can be described scientifically in the language of the natural sciences. When it comes to developing a philosophy of mind for the twenty-first century, it is first necessary to break through this dogma which makes us blind as to who we really are.
## _**The historical animal on the social stage**_
Hegel's basic idea, that spirit first forms itself by way of self-images, also implies that spirit cannot be a thing among things. One does not come upon it as one comes upon mountain ridges, lakes or algae. The philosophy that immediately followed German Idealism drew upon this fundamental Kantian–Hegelian insight. In this context, Karl Marx believed precisely that the human being to this point had constructed false ideological self-images which must be overcome – admittedly, on the level of political economy and not merely by way of a correction of the images in philosophy, or natural science for that matter. Marx identifies false self-images in the ideological superstructure of art, religion, philosophy and science, a method which presupposes a specific form of the basic existentialist idea that the human being is a historical animal. It should be noted that even Marx did not suppose that the mind belonged to nature in the sense of the universe. This is why in a famous dictum he speaks of a "naturalism of the human being," which at the same time is supposed to be a "humanism of nature."6 Marx's idea behind this is that the human mind is a piece of nature insofar as we produce artifacts in which human being and nature are united on the basis of social practices and particularly on the basis of labor. Labor, on Marx's account, is blind with respect to its material conditions, which is why it generates mistaken representations of these, which in turn have consequences for the formation of social reality. This feedback loop is the motor of history. In contrast to Hegel, Marx thus believes that the mind is essentially engendered through materially enacted labor, thus through the transformation of the natural environment with which it is sensuously engaged and not primarily in the form of art, religion and philosophy. But even Marx does not consider the human mind to be a subjective phenomenon that is played out within our skull and which could be investigated by psychological experiments, for instance, or even with a brain scanner – had he known of such a thing. Yet, he relegates adequate self-knowledge of the human mind to the discipline of political economy, for which he was trying to provide a philosophically sound foundation.
One of the central thoughts of the mid-nineteenth-century philosophy of mind up to the so-called existentialism of the twentieth century was that the human being is free in that it takes up its own role on the social stage. This thought attained particular prominence in Sartre's novels and plays, but also of course in the literary works of Camus. **Existentialism** is the view that the human being initially discovers itself as simply existing and must continually respond to this situation, and it is precisely this that distinguishes it from all other living beings. "Existence precedes essence."7 This means that we are surprised by our own existence, by the fact that we find ourselves in this world. This fact as such does not give us the merest clue as to the meaning of it all, which is why Sartre takes the human mind to be a response to this predicament.
The basic idea behind this thought was indeed already the guiding theme of Kant's ethics. Its existentialist undertones also play a crucial role in contemporary philosophy – for instance, in the influential work of Christine Korsgaard (b. 1952), Stanley Cavell (b. 1926), Robert B. Pippin (b. 1948), Judith Butler (b. 1956), Jonathan Lear (b. 1948) and Sebastian Rödl (b. 1967), to mention but a few contemporary philosophers working in that tradition.
The fundamental thought of this neo-existentialism has the advantage of actually being easily understandable. In order to examine it more closely, let us imagine an everyday scenario: we are browsing through the supermarket looking for cucumbers, in the course of which we see someone who moves toward the shelves of bread, picks out a number of packages of bread and is examining them. We now first ask ourselves what appears to be the simplest question of all: What is he doing there? The obvious answer is that he is looking at a bunch of different packages of bread and examining them.
So far, so good. Yet is this actually a suitable description? How many human beings do you know who go to the supermarket in order to take hold of packages of bread and examine them? That would actually be a strange thing to do. It seems more likely to be the kind of thing one would do if one was perhaps drugged or otherwise suffering from an abnormality. But, then, what is going on? What is the man doing?
Now, here is another explanation: he went to the supermarket to buy bread and, because he bought moldy bread in the recent past, he is now examining the bread more closely. The fact that he is examining packages of bread is thus situated in a larger context which bestows meaning on his action in such a way that he does not fall into any specifically abnormal category.
In general, in order to understand what anyone is doing, we refer to a specific structure: someone is doing something _in order_ to achieve something. In sociology and philosophy, this is called an _action_. An action is thus always an activity that is oriented toward goals and is accompanied by motives. Actions have a meaning that one must understand in order to know which, if any, takes place before we can set out further to describe, understand or explain it.
Neo-existentialism takes up Sartre's idea – which, as I said, has an influential precursor in Kant – that a person's action is really understandable only when we understand their project, as Sartre puts it. This project consists in the person's not only carrying out individual, idiosyncratic actions but also choosing between different courses of action within the context of a general idea of how they are meaningfully shaping their life.
## _**Why not everything, but at least something, is teleological**_
This apparently completely trivial observation has quite far-reaching consequences. We are talking here about _teleological action explanation_. **Teleology** (from the Greek _to telos_ = goal, purpose) is the doctrine of goal-directed processes. A **teleological action explanation** postulates that someone does something because she is pursuing a goal. It would be completely absurd to claim seriously that teleological action explanations are always objectively false and inadequate. Yet neurocentrism virtually forces just this presumption on us, which is why the very existence of successful teleological action explanation limits the pernicious influence of neurocentric thinking.
Let us for once assume that we are our brain. It follows that the man in the supermarket also is his brain. Thus we would always have to attribute his actions to his brain. His brain looks at bread and so forth (indeed it is certainly not his fingernails that do so, and his eyes are connected with his brain). Yet in what language does one best objectively describe brain processes?
It is a widespread assumption that modern science is so successful in making predictions and understanding nature in general because it gets by without teleology. On such a view, no sort of intention lies behind natural processes, and even the laws of nature are normally conceived of not as purposeful frameworks but as brute facts. There is no further reason why certain brute facts are as they are and, hence, no intention can lie behind them. According to this understanding of the sciences, one should not assume that anything happens in nature _in order for_ something else to happen. Rather, purely natural processes are interpreted according to laws of nature by means of which we explain in general terms which event brings about which subsequent event.
The wish not to buy moldy bread, on this view, is not a scientific matter. For it to be a scientific matter, one would first have to translate it into an appropriate language, a language that dispenses with the teleological structure of action explanation. Then it would be a matter of the chemical composition of what we call mold and of the biological fitness of the organism, which is increased to a certain extent by avoiding mold. It would be a matter of photons hitting sensory receptors. It would be a matter of the complex processing of stimuli on the retina, and so forth. But it would not be a matter of the man doing something because he is pursuing a goal. Brains do not pursue any goals in this sense; they fulfill certain functions of processing information in a complex organism, which is situated in a natural environment to which it can react when necessary. Brains have no goals, only people do, who are much more than their brain and much more even than their present, past and future bodies.
One of the repeatedly emphasized breakthroughs of modern science consists in the fact that it formulates laws of nature without teleology. We can see what this means when we recall the glaring contrast between Isaac Newton and Aristotle in this regard, as it is usually represented. Aristotle postulates, among other things, that there are five elements: earth, water, fire, air and the ether, a substance which he introduces in order to account for the movement of the heavenly bodies. Our expression "quintessence" stems from this Aristotelian element, which literally means "the fifth element," ether, which for Aristotle borders on the divine. According to Aristotle, all of the elements have a natural place toward which they strive: fire and air move upward, earth downward, and so forth. In addition, Aristotle assumes in general that everything which occurs in nature has an ideal or perfect form after which it strives. This was the basic thought of his teleology. Nature is ordered and understandable for us, because we know in advance that there is a perfect form of the given natural species to discover. The perfect form makes any natural phenomenon intelligible.
In contrast to this, the modern idea has it that we can discover laws of nature that can be expressed mathematically without invoking anything which has the shape of an intention or a goal-directed activity. Accordingly, nature is understandable for us not because it fulfills our expectations of perfection, but because we are able to recognize relations and formulate them precisely by means of experiments and mathematical equations. Newton discovered gravity in this way, and in this context the Aristotelian doctrine of the five elements can be considered to be refuted once and for all.
Over the course of centuries, science has undermined our tendency to conceive of natural events in terms of the kinds of activities we understand on the model of human agency. Let it be noted in passing that this is not actually true of Newton's understanding of physics, as the context in which he works out laws of nature is highly theological. Be that as it may, modern physics teaches us that nature does not in fact generally function in the way one might think if one investigates it with the naked eye. What we see and perceive with our other senses more or less directly are objects on a specific level of physical reality. If we take only these objects into consideration, as Aristotle does, we do not get very far in explaining nature. Nature is fundamentally different from what meets the eye. Teleological action explanations hold true only on a limited everyday spectrum, in which, for example, people bump into us. In light of these considerations it seems perfectly legitimate to wonder how the reality of human action looks from the standpoint of natural science, a standpoint not accessible to the naked eye and one which dispenses with teleological explanations on all levels. This thought first spurred the attempt to seek a science of human behavior based on a scientific psychology.
The realization that nature, in a scientifically precise manner, is scaled entirely differently than it appears to us at first sight has resulted in massive disillusionment. Without this disillusionment we would never have been able actually to figure out (and not merely conjecture in a speculative way) the fact that we are situated in a possibly infinite universe and perhaps even in only one universe among infinitely many. We would have no telephones, no internet, no trains, indeed, we would not even have electricity in our homes, which was also mastered only after the surprising unifying discovery in the nineteenth century that electricity and magnetism are fundamentally the same.
The view that there is one all-encompassing reality laid out there in front of our eyes, as it were, underlies the ancient Greek conception of the cosmos. In _Why the World Does Not Exist_ I argued that this conception, the very idea of metaphysics as a theory of absolutely everything which exists, has to be dismissed. It is as misguided as the Aristotelian doctrine of the five elements or the medieval theory of the four humors, according to which health and illness derive from different humor flows in our bodies.
Brain research can have a disillusioning effect. It should have relieved humanity of the superstitious notion that a soul dwells somewhere within the skull or that a mind is burrowing around in our brain or trapped by it. The British philosopher Gilbert Ryle (1900–1976) nicely summed up this whole syndrome of nonsense in his book _The Concept of Mind_ , with his famous phrase of the "ghost in the machine."8
The evolutionary biologist and proud atheist Richard Dawkins explains this kind of superstitious belief in souls with his claim that there is a "native dualism" of body and soul and a "native teleology" entrenched in us human beings: "The idea that there is a _me_ perched somewhere behind my eyes and capable, at least in fiction, of migrating into somebody else's head, is deeply ingrained in me and in every other human being."9 Furthermore, "childish teleology" "assigns purpose to everything,"10 a misperception in which Dawkins recognizes one of the origins of religion, which for him, as the famous title of his book already states, is merely a _God Delusion_.
Dawkins is right insofar as it is in fact not a self that is situated behind our eyes but, rather, a brain. The overall idea of an animating psychic substance is refuted by modern biology, since it has become clear that we can understand the chemistry of life without recourse to a vital force. And why should there be another vital force alongside of the chemistry of life (in the sense of biological life)? Such ideas are rendered obsolete by modern science, an insight that is quite important for our image of the self. My attack on neurocentrism and naturalistic metaphysics is not an attack on the more modest view according to which we have to take natural science seriously. Of course, the heavenly bodies are not driven around ethereal spheres by some kind of world soul. But none of this means that mind and brain are identical or that everything there is can be studied and explained by the natural sciences. This is just another form of superstitious overextension of one model of explanation over the entirety of the cosmos, a modern form of mythology.
Scientifically advanced brain research in fact contravenes ancient assumptions that stem from a teleological conception of the world and which are invalid within the framework of modern science's methodologically guided knowledge interest. The internal conduction of signals by neurons, for instance, goes through ion channels and hydrophobic fatty acid chains. In order to form a comprehensive picture of the working of the brain, information concerning countless other chemical details has to be gathered, and one must familiarize oneself with the function of different areas of the brain as well. This is the everyday business of neuroscience and brain research, which is as complex as it gets.
From the standpoint of this research orientation, one is not likely to come up with any plausible teleological action explanations. To say that certain neurons fire _in order to_ communicate information to other neurons in a distant area of the brain is a misleading metaphorical expression that is nevertheless sometimes used in brain research, heedless of its teleological implications. Things do not happen in the organism as portrayed to our children in the classic French cartoon series _Il était une fois... la vie_ [Once Upon a Time... Life]. The processes in the brain involving individual cells and neural networks do not happen because they are pursuing certain goals but only because they perform a function.
A _biological function_ in an organism is not a _goal_ in the sense of teleological action explanation. The description of the functional architecture of the brain in terms of the neurobiological mechanisms unearthed by brain research is not an insight into the goals and intentional actions of an actor. In the brain, no one is at home running the show. Indeed, no one is sitting in a cell and pulling a lever so that something can be diffused through a cell membrane. Synapses are not sluices that someone opens and closes. It is characteristic of this level of description to disregard ideas of origination, goal and action, since they lead to absurd (albeit amusing) picture thinking.
However, the fact that no one is at home in the functional biochemical architecture of the brain does not after all entail that no one ever carries out a plan such as buying a plane ticket, or that in reality brains and not people buy plane tickets without any purpose whatsoever. Let us begin with a scenario and then interpose another one in the middle of it, as in a film. In the first scene we see a man who is taking bread from a shelf. The second scene suddenly shows neurons firing in a specific area of the brain or another physical process within the organism – a change of perspective that is often found in movies – think of _Lucy_ , for instance, or Shane Carruth's _Upstream Color_. With a catchy phrase by the German philosopher Peter Bieri (b. 1944), we can summarize the effect of such changes in point of view through a sudden cross-cutting: "All of a sudden no one who is doing anything is there anymore. All that's left is the scene of an occurrence."11
Finally, we should give our man a name: Larry. Larry wants to buy bread. He is doing what he is doing because he is pursuing this very goal. But, someone might want to object, is Larry's setting of a goal in reality not ultimately a biological function? His experience of setting such a goal for himself might turn out to be a kind of illusion, an explanation that comes to mind because we simply do not yet know enough about evolution and about Larry's brain. Maybe teleological action explanation is shorthand for the real thing – that is, an explanation of certain events in the language of natural sciences, which, however, would be much less handy than our present account.
The thesis that all our teleological action explanations rest on illusions was already advocated as a response to existentialism in early postwar philosophy. And it also has a previous history in which Marx and some sociologists connected to him play a decisive role. **Structuralism** supposes that human goals are a kind of illusion engendered by the fact that individuals are determined by systemic structures. It attributes the superficial impression of freedom to institutions or anonymous power structures that, behind the backs of the participants in a given social field, see to it that we relate to each other in a certain way. I consider structuralism to be false, as well as the corrections added by the poststructuralism that succeeded it, which today still has many adherents. Neurocentrism is ultimately just a rewriting of structuralism and poststructuralism into a "scientific" key. Yet, if there are anonymous structures that guide us, we ourselves already belong to these structures and guide ourselves. In the chapter on freedom we will see that self-determination is unavoidable, however many there are who would like to believe that some kind of structure sustains us and frees us from the anxiety-inducing thought that we are forced to be free by our own nature as agents.
## **Notes**
1. In English in the original. 2. Bertrand Russell, _The Analysis of Mind_ (London: Routledge, 1995). 3. Wolfram Hogrebe, _Riskante Lebensnähe: Die szenische Existenz des Menschen_ (Berlin: Akademie, 2009), p. 17. 4. _Truth and Method_ (London: Continuum, 2004), p. 470. 5. _The Philosophy of History_ (Kitchener: Batoche, 2001), p. 90; translation modified. 6. Karl Marx and Friedrich Engels, _The Marx–Engels Reader_ , ed. Robert Tucker (New York: W. W. Norton, 1978), p. 85; translation modified. 7. Jean-Paul Sartre, _Essays in Existentialism_ , ed. Wade Baskin (Secaucus: Citadel Press, 1965), p. 34. 8. Gilbert Ryle, _The Concept of Mind_ (London: Routledge, 2009), pp. 5ff. 9. Richard Dawkins, _The God Delusion_ (Boston: Houghton Mifflin, 2008), p. 180. 10. Ibid., pp. 180–1. 11. Peter Bieri, _Handwerk der Freiheit_ (Munich: Carl Hanser, 2001), p. 32.
# 2
**Consciousness**
Let us begin our tour through our self-conception with the fundamental concept of consciousness, the notion at the center of the philosophy of consciousness. The advantage and disadvantage of consciousness is that we are all familiar with it as an everyday phenomenon. As soon as we wake up from dreamless sleep, but also while we dream, we are conscious. Nothing could matter to us, nothing would be present to us as thinkers, believers or perceivers, if we were not consciously experiencing something. Consciousness is at the very heart of our self-conception and, despite the familiarity of the phenomena of consciousness, the concept of consciousness immediately poses many difficulties. Soon one must face up to the fact that consciousness is a really strange phenomenon after all.
Just ask yourself the question as to what happens when you become conscious of the fact that you exist as an animal with a human mind on this particular planet? Following a good old idea of the philosopher Friedrich Wilhelm Joseph Schelling (1775–1854), the astonishment of consciousness is today still nicely expressed with the beautiful metaphor that, in the human being, nature opens her eyes, as it were. Against this backdrop, for instance, the American philosopher Thomas Nagel (b. 1937), in his book _Mind and Cosmos_ , maintains that present-day science will never be able to address this question appropriately.1 For Nagel, the idea of a cosmos gradually becoming aware of itself in conscious creatures belonging to it cannot be accounted for in terms of our contemporary materialistic world-view. Let us speak here of **the cosmological riddle of consciousness**. It derives from the impression that consciousness adds something fundamental to nature precisely because, in consciousness, nature somehow begins to become aware of itself via one of its products.
The cosmological riddle of consciousness should not be equated with what Chalmers has called _the hard problem of consciousness_ , which I have already addressed above (p. 26). Notice that Chalmers's problem was already formulated in particularly clear terms by Gottfried Wilhelm Leibniz (1646–1716). The hard problem consists in the fact that it seems to be a complete mystery how material things (such as neurons or even all the areas of the brain together) can have an inner life at all. Why do we experience the firing of neurons in any specific way? Chalmers considers this problem a hard one because he supposes that we cannot resolve it with the scientific methods developed to this point, which have a quarrel with consciousness – a claim with which many, admittedly, do not concur.
A crucial point on which much hinges in his reflections is the thought experiment of the _philosophical zombie._ A **philosophical zombie** is an exact physical replica of a human being – let us say, of Chalmers himself. Furthermore, this exact replica behaves exactly as he does – with one key difference, namely that it does not have a consciousness, and thus no one is home in this organism, as one might aptly put it. Chalmers considers philosophical zombies in purely logical terms and takes them to be metaphysically possible, though many others do not. In my view, all that matters is that philosophical zombies are not biologically possible, as everything which functions in the way in which Chalmers's brain does and accordingly has the kind of neurochemistry we find within his skull will be conscious. The question as to whether philosophical zombies contain a hidden contradiction cannot be resolved, either scientifically or philosophically, as science deals not with possible worlds but with our natural environment, and philosophical thought experiments about the structure of nature really have cognitive value only if we can cash them out with respect to nature, as we find it. Nature, as we find it, does not contain philosophical zombies.
Usually, both the cosmological riddle and the hard problem are invoked as evidence for the notion that the philosophy of consciousness is a particularly pressing field. It seems as if the natural sciences were about to provide us with the solution to almost all problems concerning the universe with the exception of consciousness. Consciousness is the last remaining mystery, or so the story goes if we accept that there is a cosmological riddle or a hard problem.
In order to get a sense of what is at stake, let us take a step back and proceed gradually. What is consciousness and how do we know anything about it?
We wake up in the morning and slowly come to consciousness. In the course of doing so, we sometimes recall our dreams, and thus the fact that we were also somehow conscious in our dream state. As we go about our daily business, various things, processes and persons continually come into focus and receive our attention: the coffee machine, the toothbrush, the commute to work, our boss. We feel somewhat tired in the morning, refreshed after a shower, angry when we feel that we are treated unfairly, delighted when someone delivers good news to us. We experience all of this consciously, while we do not consciously experience the growing of our fingernails or hair, for instance. Some processes that concern us are experienced, they are consciously available; other processes constitutively remain in the unconscious background, such as the growing of fingernails and digestion.
Most of what happens to us every day and a good deal of what we do is processed on an unconscious level, even when it comes to activities of which we can easily become conscious. We open the car door without noticing how exactly this is done. We move our fingers – for instance, to type out these lines. We are not conscious for the most part of the vast majority of processes within our organism: the digestion of a glass of water, the circulation of blood flow to the liver, or the firing of countless neurons that enable us to move our fingers, to be angry with someone or to recognize this page in front of our eyes as an expression of intentionally formed thoughts. Only a few of the impressions processed by our organism pass through the _narrowness of consciousness_ , as psychologists called this around the turn of the twentieth century. Considered in terms of the whole organism, conscious processes, experiences, are only the tip of an iceberg that protrudes from the deep ocean of purely natural, unconscious processes.
We have thus already come to know a key feature of consciousness: we are familiar with consciousness; the fact that we find ourselves in the state of consciousness is quite well known to us. Philosophers commonly nail this point down by speaking of an _internal perspective_ or the _standpoint of the first person_ – that is, the standpoint of the self. The **standpoint of the first person** consists in the fact that we are familiar with consciousness precisely and only because each of us is conscious. Consciousness is available from within, and it can certainly only be experienced from within. Knowing that one is conscious is first-hand knowledge, if anything is.
Consciousness thus lies quite close to us, as long as we are conscious. For this reason, it is natural to highlight this by speaking of "my" or "your" consciousness. Let us call this the **possession condition** , which means that consciousness is always someone's consciousness. I have my consciousness, you have your consciousness. I cannot be directly familiar with your consciousness any more than you can be with mine, because I would have to become you, or you would have to become me, for either of us to be directly familiar with each other's consciousness. As the philosopher Michael Patrick Lynch sums up this point in a discussion of the right to privacy in the digital age: "part of what makes your individual mind _your_ mind is that you have a degree of privileged access to your mental states."2
Consciousness has other features which seem to make it very special among the phenomena we encounter in our life. In a sense, one can only get to know one's consciousness from within, as it were. You have to _be_ conscious in order to be acquainted with that fact. The object I know about when I know that I am conscious – i.e., myself as a conscious subject – is at the same time the subject which knows this. One can only become conscious through one's own consciousness – so it seems, at any rate. We can describe this idea as the _privacy of consciousness_ – that is, the assumption that each person can attain immediate consciousness only of his or her own consciousness.
Of course, I can be conscious, in a certain sense, of the fact that Martha has a stomach ache, that she is thus consciously experiencing a stomach ache. Even then I still do not experience her stomach ache, regardless of how vividly I imagine it – and regardless of how much I suffer in thinking of Martha's stomach ache. Suffering in imagining someone else's pain is not feeling someone else's pain. The two pains are still numerically different. Hence it stands to reason that consciousness of other consciousness, of other minds, is only _indirectly_ accessible to us, while we are _directly_ present to ourselves in our own consciousness, so long as it exists.
Considered from the standpoint of the first person, it very quickly seems as if we are stuck in our consciousness, so to speak, each of us in our own one. In this context, the influential American philosopher of consciousness Daniel Dennett speaks of the _Cartesian theater_ , by which he means, as the name suggests, that this idea ultimately dates back to Descartes.
Strictly speaking, the implied historical claim is incorrect. In reality René Descartes sets out from the special status of consciousness precisely in order to argue that we cannot be encapsulated in our consciousness. That is the real goal of the reflections in his famous _Meditations on First Philosophy_ , in which, proceeding from the mind's capacity for self-knowledge, he concludes that there are objective truths that are independent of and knowable by the human mind. But over the last hundred years it has become typical in the philosophy of consciousness in the English-speaking world to present Descartes as a bogeyman. I leave it open to what extent this strangely reflects a cultural divide or even geopolitical circumstances separating Great Britain from its nearest continent. As a matter of fact, the idea of the Cartesian theater which Dennett rightly attacks is much more widespread in the history of English philosophy, as it is basically the definitive idea of so-called British empiricism – that is, roughly the philosophies of John Locke, David Hume and George Berkeley – but thereby hangs a tale.
What Dennett means by the "Cartesian theater" and what he justly criticizes can be easily understood. Hitherto we approached consciousness by way of the internal perspective. From that vantage point it seems as if there is a stage on which appears everything of which we become conscious: our boss, our anger, our delight, our visual field, the sometimes annoying ambient noise in the background, and so forth. I am the only one directly aware or conscious of that stage, whereas you are aware of your own stage. If one approaches the issue in this way, the question immediately arises as to who is observing what is playing out on the stage. Who is witnessing the spectacle? And the obvious answer to this question is: I am or you are, insofar as you are also an "I" – that is, someone who illuminates his or her own mental inner space attentively and observes what is happening there. It thus seems obvious to look for whoever is controlling consciousness from within. In this way we are led to the much criticized idea of a homunculus – that is, a little person in our mind or head who observes the subjective, mental state.
## _**I see something that you do not see!**_
The word "homunculus" stems from Latin and means "little man." The **homunculus fallacy** consists in imagining that our consciousness is a purely private stage on which something proceeds that a self observes and that cannot ever be observed by anyone else from an outside perspective. The private stage, the Cartesian theater, is in principle sealed off from anyone but the self forced to observe and control it. The idea of the homunculus is by no means a recent invention. It is rather ancient and extends over millennia of the mind's self-examination. Recently, neuroscientists and philosophers have regularly pointed out that in reality there is no ego or self. What they mean (or at any rate what they should mean by such a claim) is that there is no homunculus. At the very least, neuroscience has not been able to identify any brain area as the neural correlate of consciousness – that is, as the area which is always active whenever consciousness takes place. There are various hypotheses which try to explain the problem away by identifying consciousness with a different kind of structure realized by the brain – that is, not with one specific area. For instance, the global workspace model propounded by the neuroscientists Bernard Baars, Stanislas Dehaene and Jean-Pierre Changeux is roughly the idea that consciousness is more a firing-pattern which connects different areas of the brain than the activity of one area or a bunch of areas. In this case, consciousness could not be identified with any specific subregion of the brain. Yet, the real incoherence of the homunculus cannot be revealed in this way, as even the global workspace model is immediately open to a homuncularistic interpretation. In addition to any such model, the idea has to be ruled out that, whatever corresponds to consciousness in the brain or the brain's activity, it cannot have the form of a homunculus, of an observer of an internal scene. It is important to bear in mind that the idea of a tiny spectator who is sitting in our head was rejected in philosophy a long time ago, for instance, in Plato's _Theaetetus_ and Aristotle's _On the Soul_. Dennett is right in calling attention to the fact that belief in the homunculus is still widespread even where the Cartesian theater has been officially rejected. He calls this position "Cartesian materialism." You do not have to believe in an immaterial soul in order to be entangled in the homunculus fallacy. The fallacy does not go away whatever neuroscience tells us about the brain, as any neuroscientific discovery concerning the neural correlates of the conscious mind will be open to various interpretations.
In his wonderful polemical book _Dreams of a Spirit-Seer_ , Immanuel Kant argues not merely against the superstitious belief in spirits of his age but also against a belief that is widespread up to the present day, namely the belief "that my thinking Ego [ _Ich_ ] is in a place which differs from the places of other parts of that body which belongs to me."3 Kant adds that "no experience" teaches me "to shut up my ego into a microscopically small place in my brain."4 He derisively compares this to the idea that the human soul is located in an "indescribably small... abode" in the brain and perceives things there
> as does the spider in the centre of its web. The nerves of the brain push or shake it, and cause thereby that not this immediate impression, but the one which is made upon quite remote parts of the body, is represented as an object which is present outside of the brain. From this seat it moves the ropes and levers of the whole machinery, causing arbitrary movements at will. Such propositions can be proved only very superficially or not at all, and as the nature of the soul is, indeed, not well enough known, they can be just as weakly combatted.5
Kant's spider in the center of its web is the homunculus, and it is easy to see how the idea of such a spider has a materialistic equivalent such as various possible interpretations of the global workspace model and certainly the brute identification of consciousness with the brain. In our era, neuroscience has been accompanied by new avatars of the homunculus and the spider in the web. The homunculus fallacy is correlated with the idea that we can never have direct access to the external world but can only construct mental images that originate in the brain and have little or nothing to do with things "out there" – an idea that Kant likewise already attacks in _Dreams of a Spirit-Seer_. Such an idea assumes that these mental images are seen by someone and are somehow taken to be representations of the external world. Thus one conjures the homunculus, even if perhaps in the scientificsounding form of a certain area of the brain or even the entire brain. As the German philosopher Geert Keil (b. 1963) remarks in a similar context: "In the neurosciences, a few authors now view the homunculus-hypothesis as a legitimate research program, namely as the search for an area of the brain that carries out operations of central control and integration."6
It can be made clear with a simple example that, in the case of the homunculus and the Cartesian theater, we are dealing with an obvious error – even if it is to some extent very well hidden. Let us suppose that I am looking right at a statue of the Buddha. According to the homunculus model, the statue of the Buddha makes its appearance in my Cartesian theater in the form of a consciously processed mental representation directly available to me but not to you. One could connect this to the fact that I see the statue from a certain perspective. It seems distorted to me in terms of perspective, depending on my vantage point, and I know that it must appear differently in your Cartesian theater, since you must stand somewhere else in order to see the statue (I am standing here already). _I see something that you do not see_.
The statue thus appears in my Cartesian theater, and I see it – that is, my self sees it – there. But now the question immediately arises as to how I actually know that there is a statue of the Buddha which appears onstage in my theater, on the one hand, and in your theater, on the other hand, and yet at the same time also still appears differently to each of us. What do I know about how exactly the statue appears to you? Are you perhaps an expert on such statues and do you thus experience certain details in a completely different way than I do? Do you even experience that thing as a Buddha statue? It seems as if you could not if you did not know anything about Buddha statues.
One can further complicate the issue by placing a dog, a cat or a goldfish in a bowl in front of the statue of the Buddha. Does a statue of the Buddha now appear on the mental stage of these animals? Can a statue of the Buddha appear to something that has no conception of the Buddha or even of statues at all?
One is caught in the many tentacles of the homunculus fallacy if one concludes on the basis of this thought experiment that one could only know that a Buddha statue is standing there by becoming conscious of a mental representation which might or might not be about a Buddha statue or which is of a Buddha statue for me and of something else for the cat. If this implies that the statue appears on my internal stage, then I will not be conscious of the same statue as you are and I will not be conscious of the things a dog, a cat or a goldfish are when put in front of what I perceive as a Buddha statue. Hence, the question: What, if anything, is out there? And what does "out there" mean here? If it refers to reality without anyone present, without anyone looking at it in such a way that she perceives it as being a certain way, is it even any particular way?
If we take dolphins or bats into consideration, things become even weirder. Thomas Nagel wrote one of the most influential philosophical essays of the last fifty years on this topic, entitled "What is it Like to be a Bat?"7 In this, Nagel invites us to imagine what it would be like to receive sonar signals from our environment and live the life of a bat. But it is simply not possible to shift ourselves into this perspective. This is also true for the underwater perceptions of dolphins, for the sensitivity of snakes to temperature, or for the spectacular capacity of bees to perceive properties of light in the sky which turn the sky for them into a very detailed map by which they can travel incredible distances and still find their way back to their hive. Let us thus place what a few of us human beings consider to be a statue of the Buddha in a pit full of snakes and bats (a scene familiar to us from the _Indiana Jones_ films). No statue of the Buddha will appear in the consciousness of snakes and bats. What is it that appears to them? Something evidently does, which we can conclude from their perception-based behavior with respect to the object we experience as a Buddha statue. We do not really know, since we have no access to such an alien consciousness from an internal perspective.
A very nice illustration of the pitfalls of the homunculus idea can be found in _Doctor Who_ , one of the most successful TV series of all time, which has been on the air for more than fifty years. In the most recent season 8 (2014), episode 2 bears the title "Into the Dalek." Daleks are murderous cyborgs whose only aim is to annihilate all lifeforms other than Daleks. They consist of a neurobiological structure (a brain) which is embedded in a tank-like war machine. Their most commonly used saying is "Exterminate!" They are pure killers. Originally they were mutant neural networks set into war machines controlled by their neural activity. Daleks are thus something akin to what neurocentrism today considers us human beings to be: brains embedded in non-conscious machines functional only in the struggle for survival between species and fundamentally, if not exclusively, driven by their own egoism.
The Daleks are the main adversaries of the protagonist of the series, the Doctor, whom one could interpret as a philosophically trained critic of ideology. As "Into the Dalek" unfolds, the Doctor is shrunken into a nano-sized version of himself and climbs into a Dalek that had surprisingly began to engage in moral deliberations, which is extremely uncharacteristic for Daleks. This is why the humans wonder what is going on, as they do not trust the fact that an actual ethical conversion has taken hold of a Dalek. The Doctor is transported onto a spaceship by the name of "Aristotle," whose crew captured this remarkable Dalek. Incidentally, it is interesting in this context that Aristotle, with his book _On the Soul_ , can be viewed as the originator of the homunculus fallacy.8 Even though he ultimately overcomes the fallacy, he is one of the first to point it out. Perhaps the name of the spaceship on which the homunculus scene occurs alludes to this.
In any case, the Doctor climbs into the Dalek's skull. We then see that the Dalek brain has an eye in front of which a film is playing that consists partly of memory images and partly of images of the external world that it receives through the outstretched lens of the Dalek's body machine, which each Dalek operates as a kind of eye in addition to its internal eye. A brain is situated in the Dalek with an eye installed that watches a movie theater hidden from the view of others. This immediately raises the question of whether there is an even smaller Dalek in the Dalek's brain, and so forth, which is one of the problems with the homunculus fallacy. One simply needs too many homunculi (infinitely many) to solve this self-created problem. We cannot solve the problem as to how an observer observes the scenes of which we are conscious by introducing an observer who in turn can be observed. In this way, we create the Dalek problem of always having to bring in another smaller Dalek, who watches a Dalek one level up watching a Dalek one level up, and so on _ad infinitum._
## _**Neuronal thunderstorms and the arena of consciousness**_
Of course, one can simply insist on the fact that the brain or some region of the brain is in fact a homunculus. Yet, this brute assumption does not explain anything but simply takes a myth at face value. Another option, which is only slightly better, can be introduced with recourse to the issue of the Buddha statue. The problem with the idea that our consciousness is a private stage, exclusively accessible and intimately familiar to each individual, consists in the fact that, on this model, it is hard to see how one could ever make sense of our perceptions of things belonging to the external world. If anything, perception is a form of consciousness. If consciousness always shields us from reality or from things "out there," this would also be true of perception. Yet, this turns us into Daleks; it pushes perception entirely into our head so that it begins to look reasonable that we never know what is really "out there," since all we are acquainted with in perception are glimpses of the consciousness movie played in the brain's internal home theater. But, then, how could we so much as know that things "out there" are even approximately as they appear on our private screen? Indeed, how could we know that there are any things "out there" at all?
The only reason that seems to speak in favor of this explanatory model is that, in a sense, each of us really does see something different when we stand before a Buddha statue, whereas other animals do not even get to see one. In addition, some other animals have entirely different sensory modalities and sensations we cannot even imagine. This almost immediately leads to the position of **neuroconstructivism** – that is, to the view which denies that we directly perceive reality and the things in it and holds instead that we can only ever perceive mental images or representations that the brain has constructed on the basis of sense-impressions of which we never become aware as such. The sensory registration takes place below the threshold of conscious experience but is turned into conscious experience via non-conscious neural channels of information-processing which result in mental images, but not in direct perceptions of the external world.
All of this can be made to sound very scientific. But this should not mislead us. Neuroconstructivism must also suppose that there is something "out there" which is portrayed in my theater of consciousness as a Buddha statue which shows up differently in the mental realm of a snake. Thus, according to neuroconstructivism, there had better be something that appears to be a Buddha statue to me and appears to be X to a snake and to be Y to a bat (X and Y are placeholders here for something that we human beings can never fully picture to ourselves). This raises the perfectly legitimate question: What's "out there" then? The neuroconstructivist replies: "A physical object or an enormous mass of physical objects are out there: some kind of electromagnetic swirl, which we upload through _our_ sensory receptors and which other animals simply upload through _their_ sensory receptors onto their mental user interface, their consciousness." Here, you can fill in your favorite metaphysical picture of physical reality out there which is supposed to explain how physical states carrying information can affect our nerve endings.
But does such an obvious seeming and scientifically informed response provide much help at this point? No it does not! The entire thought experiment simply amounts to the claim that we can only ever attain consciousness of physical objects (including electrons, photons, electromagnetic fields) because they make an impression on our private mental screen. Indeed, to this point, we have not been provided with any reason for the claim that we can somehow be directly conscious of anything, that we can perceive things "out there" at all. Here, the strategy of the neuroconstructivist is usually to be content with the thought that we cannot perceive electrons, photons, etc., but that we must infer their existence by means of methods, experiments and models. We _infer_ that the objects underlying our perception of a Buddha statue as well as the X and Y of bats and snakes are a certain way.
Yet what about our consciousness of the experiments, the measurements, the instruments and our theoretical understanding of electrons? Where does any of this come from? If it is impossible consciously to take note of anything without uploading it onto our purely private mental user interface, then this also holds for everything which we can ever know about things "out there," in the external world. But then we do not even perceive measurement instruments or brains, hands, etc. Thus, we do not get through to things by means of our scientific experiments, as we do not get through to our scientific experiments.
If consciousness is literally nothing but a neurochemical process within our skull, through which a theater in the head is set in motion, which might or might not depict the external world to some degree, how do we even know that there really is an "out there" that our mental images are connected to in any relevant way whatsoever? At this point, we cannot backpedal to the claim that we discovered any of this via scientific experiment, since we would have conducted these in our internal "consciousness movie," too. Yet, an imaginary experiment is not an experiment. Ludwig Wittgenstein (1889–1951) formulated a series of fitting metaphors for this problem in §265 of his _Philosophical Investigations_ : "(As if someone were to buy several copies of the morning paper to assure himself that what it said was true.) Looking up a table in the imagination is no more looking up a table than the image of the result of an imagined experiment is the result of an experiment."9 It would amount to a contradiction to believe on the one hand that we can only ever attain consciousness of anything by projecting it onto our own screen, and on the other hand to proceed as though we know this because we attained consciousness of something that does and cannot appear on our own screen. The neuroconstructivist suffers from a split mind: by trying to withdraw himself into his most private realm of conscious experience he first loses touch with reality, which he smuggles back into the picture by secretly leaving the private home of his consciousness in order to conduct experiments in the real world...
Contrary to the overly complicated – and fundamentally misguided – theory construction of the neuroconstructivist, it is most natural to suppose that I see a red cup when I turn my head to the left because there actually is a red cup over there to the left and actually a coat hanger over there to the right. Cups and coat hangers are not just lying around in our consciousness; rather, they are in living rooms or clothing stores. There is also no room for all the things I perceive on an everyday basis in my brain. Where in my brain should I fit my office in which I am working right now? And don't tell me my office is in my mind, as my mind is not the kind of place where you could store all those books, the wood of my floors and the pictures on my wall.
## _**Buddha, the snake and the bat – again**_
Sometimes, neuroconstructivists invoke the epistemology and view of the mind you find in Immanuel Kant. However, Kant was much more consistent than neuroconstructivism. In particular, Kant saw through the contradiction in which neuroconstructivism is continually entangled. In one of his major works, the _Critique of Pure Reason_ , he is concerned with demonstrating, among other things, that the identification of the bearer of our thought processes with any kind of thing (be it an immaterial soul or the brain) is a fallacy that imposes itself on us in the realm of self-consciousness and self-knowledge, a fallacy which he is eager to expose. Kant chose the term "paralogism" (from Greek for fallacy) for this specific fallacy. According to Kant, **paralogism** is the misguided belief that the bearer of the capacity for thought must be a thing that one can find somewhere in the external world: an immaterial soul that is difficult to detect or just a property or activity of the brain, or even a cluster of brain regions actively creating a neural symphony that cannot be precisely localized. A paralogism identifies the thinking subject with an object in the external world.
Kant is especially aware of the fact that we cannot even in principle _scientifically_ solve the problem of the Buddha statue, which appears to one human being in a certain way but to other human beings or other animals in a completely different way. If consciousness were really a private user interface upon which constructed mental images appear, science alone would be of no help to us, since it would only describe what appears in the theater of consciousness. It would just pile one layer of appearances onto another one. The kind of objectivity on the basis of which any science proceeds would not be possible at all. If we are all trapped in our experiential bubbles, we cannot even succeed in inferring that there are such things or phenomena as electromagnetic fields by interpreting data we receive from our measuring instruments, as we could never perceive these instruments as they are in themselves.
How does Kant approach the problem of the Buddha statue? He starts with the unproblematic assumption that a Buddha statue appears _to us_. He then can add that an X appears to the snake, a Y to the bat. He could call the Buddha statue, the X and the Y "appearances" and write a list of appearances:
Appearance 1 | the Buddha statue (human representation)
---|---
Appearance 2 | X (snake's representation)
Appearance 3 | Y (bat's representation).
Roughly speaking, his term for what is there independently of appearances and which we can only ever hope to access via inferences on the basis of what is immediately available to us is the "thing-in-itself." Kant's absolutely central thesis, without which his entire edifice of thought collapses, and which he very consistently argues for, is that we can neither perceive nor know the thing-in-itself or things-in-themselves. **Kant's main thesis** , that we can perceive and know only appearances but not things-in-themselves, is called **transcendental idealism**. Thus Kant writes:
> No subtle reflection is required to make the following remark, but rather one can assume that the commonest understanding might make it, even if in its own way, through an obscure distinction of the power of judgment that it calls feeling: that all representations that come to us without our choice (like those of sense) give us objects to cognize only as they affect us, so that what they might be in themselves remains unknown to us; hence that as regards this species of representations, even with the most strenuous attention and distinctness that the understanding might add to them, we can attain merely to the cognition of _appearances_ , never to _things in themselves_.10
One can illustrate Kant's thought here as follows: imagine that your only active sensory modality as your contact with the external world was touch. Indeed, it is already sufficient for the thought experiment to close your eyes and attempt to concentrate only on your sense of touch. Now imagine that someone were to touch you gently on the top of your hand with a certain object (let's say a matchbox). On the basis of your tactile sensation alone, it is very unlikely that you could tell the touch of a matchbox from the touch of any other object with a similar kind of structure. The tactile and the visual registration of a matchbox hang together for you only if you have seen matchboxes or have received information about them from people who had already seen matchboxes. If no one had ever seen matchboxes, no one could come to know that there are matchboxes on the basis of tactile sensory information alone. Whatever theories you would formulate about the thing "out there" – the thing-in-itself – on the basis of your sense of touch – the appearances – your descriptions would still be quite different than what you would give if you could see the thing "out there." Kant extends this line of thought to all of our senses and believes in particular that space and time are forms of our intuition that quite possibly have nothing to do with how things are in themselves. For him, space and time are, as it were, ways in which reality looks for the eye, not ways for reality to be.
## _**Surfing on the wave of neuro-Kantianism**_
Of course, there is plenty of room for disagreement with Kant's transcendental idealism, and I, for one, do not believe that it is true. What is important here is that Kant would certainly have protested had he known how popular it is in our time to attribute to him the thesis that we only construct our world mentally, and that this is because our nervous system processes certain stimuli internally in the way it does. This is precisely a paralogism, an identification of the functions of thought with objects "out there" (such as brains or brain regions). Kant would have had countless objections to precisely this thesis, although it became fashionable in nineteenth-century physiology to construct neuroconstructivism in the wake of Kant. Thus, for instance, the famous German physiologist and physicist Hermann von Helmholtz (1821–1894) wrote in 1855:
> That the mode of our perceptions is conditioned by the nature of our senses just as much as it is by external things is quite evidently brought to light by the aforementioned facts, and is of the utmost significance for the theory of our cognitive capacity. The very thing that, in recent times, sensory physiology has established through experience, Kant already sought to do earlier for the representations of the human mind in general by explaining the share that the particular innate laws of the mind, the mind's organization, as it were, have in our representations.11
The bizarre view that Kant paved the way for neuroconstructivism has been a topos since the nineteenth century and has been extended into present-day textbooks. The Nobel prizewinning neuropsychiatrist Eric Kandel (b. 1929) writes in his standard text _Principles of Neural Science_ :
> Thus we _receive_ electromagnetic waves of different frequencies, but we _perceive_ them as the colors red, blue and green. We receive pressure waves from objects vibrating at different frequencies, but we hear sounds, words, and music. We encounter chemical compounds floating in the air or water, but we experience them as smells and tastes.
>
> Colors, tones, smells and tastes are mental creations constructed by the brain out of sensory experience. They do not exist as such, outside the brain... Our perceptions are not direct records of the world around us. Rather, they are constructed internally according to constraints imposed by the architecture of the nervous system and its functional abilities. The philosopher Immanuel Kant referred to these inherent brain properties as _a priori_ knowledge. In Kant's view the mind was not the passive receiver of sense impressions envisaged by empiricists. Rather the human mind was built to conform with certain preexisting conditions, such as space, time, and causality. The existence of these ideals was _independent_ of any physical stimuli coming from beyond the body.12
Kant advocates virtually nothing of what Helmholtz and Kandel freely associate with him. He believes neither that we receive electromagnetic waves which we perceive as colors nor that space and time are pre-existing conditions, as Kandel writes. Kant distinguishes between forms of intuition (space and time) and categories (which include causality) and denies the very claim that we can know how things are in themselves through the application of categories to spatio-temporal sensations. Furthermore, Kant does not consider perceptions to be mental constructions that have anything to do with our nervous system. He argues, rather, that in principle we cannot know who or what is actually thinking inside of us. For this reason, he shies away from the identification of thinking with anything beyond the realm of thinking – again: be it with an immaterial soul or a material brain.
The thesis that the self can be identified with the brain is ruled out by Kant just as much as the thesis that we have an immaterial soul that thinks inside of us. It is reasonable to suppose that Kandel has not read Kant and that, if he has, in any case he did not understand much. From the standpoint of epistemology and the philosophical theory of perception, most of what he writes about the nature of perception and its relation to the perceived environment is confused and false.
Be that as it may, there is a line of interpretation which began shortly after Kant's death and attempts to locate Kant's term "the 'I think'" in the brain, a tradition which includes thinkers such as Schopenhauer but also in due course scientists such as Helmholtz. Kandel accepts in an uncritical way a topos of the erroneous interpretation of Kant's theory of knowledge and carries it to an extreme. He reclaims Kant for himself – whatever the motivation for this appropriation might be. What Kandel describes as a philosophical option – leaving aside the question of what Kant really thought – tends not to be advocated by contemporary epistemologists in this form, since it is connected to a whole series of quite obvious fallacies to which Kant had attempted to find an alternative.
The claim that brains construct distinct user interfaces, so that we tap into the existence of an external world only in a hopelessly mediated and distorting way, but never directly, rests on a fallacy or an error that cannot be remedied by any kind of scientific progress. If this were indeed the case, how could we even know that we only ever have mediated and distorted access to an external reality? Assessing the veridicality – i.e., the correctness or degree of distortion – of our perceptual representation of external reality presupposes an independent grasp of that reality, a grasp simply assumed but unargued for in the view according to which we perceive reality not as it is in itself, but only its shadows, as it were, constructed by and in our brains.
If consciousness boiled down to a relation between a homunculus in the mind and the show on its private stage, the appeal to science would never get us out of the quandary. The sciences could then describe only what plays out on the private stage of the individual scientist. The claim that in reality there is only a swarm of particles dancing where I believe myself to be seeing a Buddha statue, a swarm of particles that in itself has neither the color nor the form of a Buddha in any way, would be only a further representation of what might be there in itself, a random guess in the dark. We could in principle formulate only relatively wild speculative hypotheses about the external world if our entire experience were only ever a mental construct that presents images on a private, internal stage located within our skull.
## _**Nothing is beyond our experience – or is it?**_
That all of this presents an enormous problem for the scientific image of the world today is relatively easy to recognize in a typical sentiment regularly expressed in this context. This view has severe effects on the common conceptions of the relation between consciousness and the brain. I am thinking here of _empiricism_. **Empiricism** (from the Greek _empeiria_ = experience) is the thesis that the only source of our knowledge is experience, which in this context specifically means sense experience. This claim, which is ancient in terms of the history of philosophy, and which recurs repeatedly over millennia, is today defended in a particularly aggressive manner by people such as the popular American physicist Lawrence Krauss (b. 1954). Krauss emphasizes again and again that there is only a single scientific method. This method supposedly consists in the fact that our critical thinking can be guided by experience. He seems to understand "experience" as a combination of sense experience, measurement, and a theoretical edifice constructed from these two elements. In more precise terms, he believes that experience with reality ensures that one makes reasonable predictions and can then prove these predictions by means of experiments. We already adapt this method in everyday life, and for him modern science is a way of perfecting it.13
Krauss bases his campaign against the world's religions on this point, a campaign that he is currently conducting together with Richard Dawkins. Both understand "religion" as merely irrational superstition that clings to obviously false theses concerning nature's mode of functioning – a dangerous superstition which stands in the way of any advance in knowledge. As a philosopher, I am of course sympathetic to the struggle against superstition. But one does not fight very effectively against superstition on the basis of a very poorly established philosophical position which on closer inspection turns out to be itself a form of superstition.
Dawkins reduces religion to "baseless and arbitrary beliefs and injunctions, handed down the generations."14 In addition, he assumes that all religions are concerned with God, and that this word is a way "to denote a supernatural creator that is 'appropriate for us to worship.'"15 Yet from where does Dawkins receive the supposed insight that each religion is concerned with God and that "God" means what he thinks it does? How does he justify this knowledge claim?
What Dawkins overlooks is the fact that the world's religions (whether they accept a God, like Judaism, Christianity and Islam, or whether they are polytheistic, like Hinduism, or ultimately atheistic, as are some versions of Buddhism) emerged in an era when the modern conception of nature did not yet exist. How is the author of the Book of Genesis in the Bible supposed to arrive at the thought that God is a super _natural_ creator when he never had the concept of nature to begin with, which Dawkins anachronistically projects into the past? The anachronistic contrasting of a scientific and a supernatural explanation of natural occurrences pervades the historically uninformed neo-atheism that is advanced by Dawkins.
His simple-minded model of the critique of religion has admittedly already been around for centuries. However, from a perspective that is both historically informed and critical of actual textual sources – a perspective that has long been adopted by academic theology and religious studies – it is simply an unfounded and arbitrary belief that religion generally is what Dawkins imagines it to be. His account of religion does not stand the test of religious studies, theology, sociology or the philosophy of religion. It is not scientifically backed up by any discipline based on actual empirical and conceptual engagement with the phenomena grouped under the heading of "religion." What he says about religion is, thus, unscientific by any respectable standard. Since he takes his critique of religion to be the prototype of the rules for the enlightenment of humanity and its liberation from supposedly childish superstition, his critique of religion fulfills his own definition of religious superstition, even if he does not believe in God but instead replaces the latter with nature. At this point it is important to emphasize that nothing which we know about nature speaks in favor or against a specific metaphysics, be it a metaphysics according to which there is a God or one which denies this. On no minimally sophisticated account of the monotheistic divine can it be located within the realm of nature insofar as it is studied by the natural sciences. That is the point of thinking of the divine as supernatural! It simply does not belong to the natural order. Yet, this is something God or the divine shares with the conceptual setup even Dawkins has to rely on: the concepts, inferential patterns (logical rules) and truths he believes to have grasped and employed are not themselves on the same level of natural phenomena as, say, digestion or gene mutation.
The question of how we conceive of human knowledge acquisition has many far-reaching consequences and does not concern merely philosophical epistemology. Rampant empiricism – i.e., the thesis that all knowledge derives exclusively from the source of experience – means trouble. If all knowledge stemmed from experience, and we could hence never really know anything definitively – since experience could always still correct us – how could we know, for example, that one should not torture children or that political equality should be a goal of democratic politics? If empiricism were correct, how would we be supposed to know that 1 + 2 = 3, since it is hard to see how this could be easily revised by experience? How could we know on the basis of experience that we know everything only on the basis of experience?
Rampant empiricism breaks down in the face of simple questions. If _all_ knowledge really stems from the source of sense experience, what are we to make of the knowledge concerning this supposed fact? Do we know from sense experience that all knowledge stems from sense experience? One would then have to accept that experience can teach us wrongly even in regard to this claim. In principle, one would have to be able to learn through experience that we cannot learn anything through experience... How would this work? What kind of empirical discovery would tell us that not everything we know is through empirical discovery?
In order to delve a little bit deeper into the incoherence of rampant empiricism, we might consider another simple line of thought. Krauss and many other scientists in their philosophical moments say that all knowledge is based on experience, or is "evidence-based," as one also calls it. They point this out because they would like to draw our attention to the fact that we can err. We are susceptible to error, which is called **fallibility**. Empiricism's recommendation derives from its modest-sounding commitment to our fallibility and to its attempt to explain why we are susceptible to error, namely because we receive data from the external world that we must interpret and order in a theoretical way. If we ever wish to know anything on this basis we must eventually claim that we have arrived at our goal. We then formulate a knowledge claim. Yet we can also be deceived and corrected by experience. For that reason, one must actually be open to revisions. Empirical, experience- or evidence-based knowledge claims are defeasible; they come with the idea that they could be retracted or corrected.
Yet how could the claim to know that all knowledge stems from sense experience be corrected by sense experience? The simple answer is: it can't! The statement that all knowledge stems from sense experience is for this reason not a scientific hypothesis that can be proven or disproven. This is because, no matter how well a scientific hypothesis is established and embedded in a theory, it must remain conceivable that pieces of evidence can emerge that would reveal the hypothesis to be false. Therefore, rampant empiricism is not itself a scientific hypothesis.
To be sure, there are subtle varieties of empiricism that were articulated in the last century under the heading of "logical empiricism" in particularly impressive form by Rudolf Carnap (1891–1970) and Willard van Orman Quine (1908–2000). Thus, for instance, Carnap claims that there are truths that one cannot grasp through sense experience, while Quine corrects him to the effect that _all_ knowledge is always a combination of theoretical elements and experiential data. According to Quine, there is neither pure empirical knowledge nor pure _a priori_ knowledge – that is, knowledge not derived from sense experience at all. Yet all of this ultimately makes the situation philosophically subtle and complicated in such a way that Carnap and Quine are in any case not of the opinion that scientific knowledge might be a description of processes in our theater of consciousness.
However, the situation becomes even more uncomfortable for the kind of crude empiricism that Krauss propagates. In public debates, philosophers repeatedly ask him whether there is mathematical knowledge, thus for instance knowledge of the fact that 1 + 2 = 3. Since Krauss is a theoretical physicist who works with mathematical equations in his everyday research, one would hardly expect him to contest the fact that he possesses mathematical knowledge which is incomparably more complex than multiplication tables from elementary school. Krauss's answer to this question is typical. He suspects that he has been lured into a trap but persists in maintaining that he certainly does need sense experience to attain mathematical knowledge: he has to read somewhere that 1 + 2 = 3, for instance, or become informed about mathematical axioms and how mathematical symbols function.
Yet this solution has been arrived at by cheating. If I know through sense experience that there are still two bread rolls in the breadbasket (I see them lying there), this is possible because the bread rolls emit light that hits my photoreceptors. I can attain knowledge of bread rolls from sense experience because bread rolls are the kinds of things that can enter into an appropriate causal relationship with my sensory receptors. But the numbers 1, 2 and 3, as well as mathematical symbols, are not identical with signs that have been written down. When I write the number 1 down three times, thus – 1, 1, 1 – I repeat the exact same number three times, namely the number 1. There are not nine different number ones on this page, but simply nine different cases in which I have written the same number down. If I see two bread rolls lying in the breadbasket, there are actually two different things – the one bread roll and the other bread roll. In contrast, numbers can neither be seen nor measured – they can only be visualized by means of signs, which is not the same thing. The visualization or vocalization of a mathematical truth is not itself a mathematical truth, nor is the fact that at some point in my conscious life I learned a mathematical truth in some way or another. To know that 1 + 2 = 3 is not to know as such that someone taught me this or that I saw something on a piece of paper.
By the way, one can also neither see nor measure the difference and identity of things. In this context, in philosophy one speaks of the _**a priori**_ (Latin _a priori_ = in advance, independent of all experience). This means that we can have sophisticated empirical knowledge only because we employ theoretical concepts – such as cause, law of nature, identity, object, thing, consciousness – that are certainly connected to our experience, but which cannot simply be picked up by sense experience. No science is possible without some _a priori_ assumptions. No scientifically respectable body of knowledge will be entirely empirical or grounded in sense experience.
Things become still more uncomfortable for the crude empiricism represented here when we return from this somewhat abstract consideration to consciousness. There may indeed be different degrees of consciousness, from complete alertness to the feeling of pain all the way to the dim consciousness of being lost in daydreams. There are also fleeting transitions such as nodding off, in many ways dangerous or simply just annoying (if one sleeps through one's train stop or has to stay awake through a boring lecture, which is part of everyday academic life). Here, the empiricist idea that we know through sense experience that we are conscious bears pathological features, since it inherently includes the possibility of a misinterpretation and could thus at any time reveal the fact that we are not really conscious. If my way of knowing that I am conscious right now is a kind of hypothesis fumbling on the basis of sense experience, there could be a way for me to be wrong about this. But being conscious and wondering whether one might not be conscious borders on a symptom of mental illness. Fortunately it is not the case that at any time in my conscious life I might fear undergoing the experience of not being conscious at all. It seems as if, at the very least, one could not be deceived regarding the question as to whether one is actually conscious. This insight is encapsulated in Descartes' most famous statement: "I think, therefore I am," the cogito. The **Cartesian cogito** amounts to the claim that, as long as we are conscious, we cannot be deceived about the fact that we are conscious, while we can certainly be deceived about what consciousness consists in, which Descartes does not deny.
Krauss sometimes discusses this topic with the previously mentioned philosopher of consciousness Daniel Dennett and even admits that consciousness is a difficult problem. At the same time, however, he believes (as does Dennett) that future neuroscientists will eventually find a solution. But how, one wonders, are we supposed to be informed by sense experience, and thus in an empirical manner, that we are conscious? Could it be revealed that this is false? Could experiments persuade you that you are not conscious right now?
In the nineteenth century, the French neurologist Jules Cotard (1840–1889) discovered an illness that is named after him, the Cotard syndrome, in which afflicted patients profess that they are dead and do not exist. A strict empiricist would have to assume that patients suffering from Cotard syndrome are perhaps right and really are dead. They would then be zombies and _World War Z_ could soon break out. Yet of course they are not dead, and even the idea that they might be is odd. Patients who suffer from Cotard syndrome cannot be right, because they obviously still have a living body and are conscious. They suffer from a neurological disorder that requires a treatment to allow them to return once more to a healthy consciousness, which then helps them to realize that they have been conscious and alive all along.
As we can see, it is a ridiculous idea to suppose that it could one day come to light that we have no consciousness at all. Whoever sincerely claims that he is not conscious is examined with good reason so that it can be discovered what illness he is suffering from or whether he is an ingeniously constructed android. Since such androids do not exist at present, all that remains is the presumption that one cannot deny that one is conscious without at the same time being conscious. As I have already emphasized, this does not imply that one learns about the nature or the necessary biochemical preconditions for consciousness through introspection. At any rate, no one, not even Descartes, claimed that this was the case! Even though we are infallible in believing that we are conscious when we are, this does not mean that we are also infallible with respect to what consciousness is. However, the self-intimating aspect of consciousness at least rules out a number of verdicts, such as the ultimately pathological denial of the very existence of consciousness.
Of course, we can have false ideas of what it means to be conscious. In this regard, we are susceptible to error. It is simply a different matter to know _that one is conscious_ and to know _what consciousness is_. There are indeed scientific disciplines designed to give us a better grasp of consciousness, including philosophy. In any event, this should never mislead us into believing that no one was ever conscious or that we are just not conscious, since these "findings" would clearly imply that – to put it mildly – something has gone wrong in the process of theory construction.
## _**Faith, love, hope – are they all just illusions?**_
Our present culture is very fond of the fear that we are in fact not really conscious at all. The fear of losing one's own consciousness while fully conscious is expressed in zombie films and television series such as _The Walking Dead_ or in the classic movie _Awakenings_ from 1990. The latter is the film adaptation of a book by the same title written by the famous neurologist Oliver Sacks (1933–2015). The book documents a real case that transpired in New York. Sacks succeeded in waking up, as it were, some victims of the epidemic _Encephalitis lethargica_ from their nearly complete unconsciousness. Unfortunately, this worked for only a short period of time, which is portrayed as the tragic story it is in the movie (with Robin Williams and Robert de Niro in the starring roles).
Surprisingly, there are philosophers who believe that we are not conscious, because they believe that the term "consciousness," strictly speaking, does not really refer to anything. I am thinking here of the position of so-called _eliminative materialism_ , which is advocated with especial verve by the neurophilosophers Patricia Churchland (b. 1943) and Paul Churchland (b. 1942), both of whom taught at the University of California in San Diego until recently. **Eliminative materialism** claims that our mental states are on the whole illusory, because in reality there are only material states and processes in the universe, such as the information-processing states in the brain studied by cognitive science and neurobiology.
In a well-known essay Paul Churchland sets out to eliminate mental states as we believe we know them. Roughly, he proceeds as follows. First he points out that there is such a thing as an _everyday psychology_ , which he dubs "folk psychology." We talk of human beings as conscious, say that they have opinions, that they are rational or irrational, unconsciously repress feelings, have good intentions, and so forth. We all have our own unique views of consciousness and continually imagine how other human beings perceive us and feel about us and the things surrounding us. At this stage of the argument Churchland assumes that folk psychology is an empirical theory like any other. This already betrays an empiricist assumption in that he views our account of consciousness as an empirical theory.
On his view, we attribute mental states to ourselves and others because we have a certain theory of such states that we derive from our experience. In contemporary psychology and cognitive science this is called a "theory of mind": the capacity to form assumptions about other minds and thus about the feelings, intentions, hopes and beliefs of others. According to Churchland, we have constructed folk psychology on the empirically shaky ground of somewhat naïve theorizing about the mental states of others. In particular, he maintains that folk psychology has made no progress over the millennia (an unargued assumption which happens to be false).
To sum up, Churchland begins with the twofold assumption that our everyday psychology is a folk psychology, which he regards as an empirical theory. Here, his ulterior motive is to suggest to the reader not only that this theory could possibly be false, but that it is indeed almost certainly false. He explicitly compares it with the geocentric image of the world, which has also been exposed as false. The theory that the sun rises can be characterized as false because the earth, as is well known, revolves around the sun and rotates around its own axis, which, so the story goes, makes it seem as though the sun were moving while the earth stands still, as a naïve observer might not notice the movement of the earth (which is why it took so long for us even to accept the de-centering thought that we are flying around in the universe). Strictly speaking, the sun rises no more than the earth is situated at the center of the solar system. In reality, from this vantage point there are thus no sunrises; rather, there is only a kind of illusion that tricks us into believing in sunrises. The empirical theory that the sun rises each morning has simply been proven to be false. Yet the everyday illusion remains when we stand on a firmly anchored flat surface and see something we refer to as a sunrise. We do not see the earth moving in the same way in which we see the sun moving, even if we know that the earth is moving too.
In his next step, Churchland claims that folk psychology assumes that there are propositional attitudes. Note that this is a typical philosophical technical term and thus, strictly speaking, is not expected to be a component of folk psychology, which already causes trouble for Churchland's account. A **propositional attitude** is a mental attitude that one entertains toward a state of affairs: for instance, one can _fear_ that the Syrian civil war will spread; one can _hope_ that there is some bread left; one can _believe_ that some people eat cats; one can _know_ that one has hands – and so forth. Fear, hope, belief and knowledge are propositional attitudes. In the philosophy of language, one calls a **proposition** the content of an utterance, which can be true or false and which is linguistically formed with "that" clauses. A person can have different attitudes toward the same proposition. Let us consider an example:
... that the civil war rages in Syria.
One can know this, believe this, be worried about it, be deeply saddened by it, or ignore it. Thus it is possible to relate to this proposition in different ways. In fact, many philosophers claim that the human mind is in essence the capacity to have propositional attitudes, which in my view is a good starting point.
However, Churchland doubts that we have propositional attitudes. And thus he immediately walks right into the trap of a contradiction that the American philosopher Lynne Rudder Baker (b. 1944) quite fittingly called _cognitive suicide_.16 This can easily be seen if we ask ourselves what Churchland denies. He denies:
... that there are propositional attitudes.
One of his reasons for wanting to deny the existence of propositional attitudes is that it is hard to fit them into a materialistic conception of reality according to which it consists only of material states and processes which can be scientifically investigated. Churchland would like to do away with the alleged superstition that not everything there is turns out to be an object of natural scientific investigation. The contradiction consists in the fact that Churchland thus adopts the propositional attitude of doubt toward the proposition that there are propositional attitudes. If there are no propositional attitudes, then Churchland cannot even _believe_ that there are none! And he cannot know this either.
At the end of his essay "Eliminative Materialism and the Propositional Attitudes," Churchland seeks refuge in sciencefiction fantasies that are designed to conceal this contradiction. He envisions a future in which we could somehow wire our brains together and recommends that we do everything possible to make this amazing future happen rather sooner than later:
> Think what this might do for hockey teams, and ballet companies, and research teams! If the entire population were thus fitted out, spoken language of any kind might well disappear completely, a victim of the "why crawl when you can fly?" principle. Libraries become filled not with books, but with long recordings of exemplary bouts of neural activity.17
The essay culminates in Californian science-fiction literature. Dreamers, seers and bigshots from Silicon Valley would perhaps be in agreement with the idea of abolishing libraries in favor of neural activation patterns that in the near or rather distant future can be downloaded into the brain. Yet such visions only gloss over the incoherence of a conception that would have us believe that our assumption that we actually have beliefs and thoughts is ultimately an antiquated superstition. Cognitive suicide only works at the cost of actual suicide.
In distinction to her husband, Patricia Churchland works predominantly on eliminating the idea that philosophy could discover anything whatsoever about consciousness without help from the neurosciences, a project that has been given the name of "neurophilosophy."18 To ponder concepts such as "propositional attitude" and their embeddedness in other concepts such as perception, knowledge, belief, and so forth, and to analyze them firmly with arguments, is thus not supposed to further the state of knowledge, although the philosophy of mind has done this for millennia.
But could we be conscious without having propositional attitudes? Consciousness is obviously a quite complicated affair. In particular, it is not self-evidently certain which phenomena, processes and states we associate with consciousness. But it is certain that we cannot even imagine what it would be like to be conscious without having propositional attitudes. For instance, let us consider an everyday case, such as sitting on a train and listening to music. We look around us, observe other passengers and, while doing so, pay attention to the music, but sometimes we forget about it and our thoughts wander from one topic to another. Such everyday situations have been described in literature by means of the well-known narrative technique of the _stream of consciousness_. This was also very humorously front and center in the brilliant British comedy series _Peep Show_ , in which we can hear the protagonists' inner monologue and hence get a voyeuristic look into their inner lives, into their consciousness.
With the help of our imagination, let us thus climb aboard a typical stream of consciousness while on the train. As soon as we begin to describe what we consciously experience in this situation we will ascribe propositional attitudes to ourselves: we pay attention to the music, look around the train and spot an individual who seems interesting. We try not to stare at that person, since there are many reasons to look around at our fellow commuters discreetly, although these reasons can mostly be traced back to the fact that we are not sure which propositional attitudes the interesting person there in the front might have if we were to stare at them for too long. The whole situation would no longer make sense at all; strictly speaking, it could not happen if there were actually no propositional attitudes. Human consciousness, as we are familiar with it from our own experience, would not exist without hopes, convictions, opinions, doubts, intentions, and so forth. Sociologists speak here of communication as a form of **double contingency** , which in simple terms means that human beings always develop their beliefs by comparing them to the beliefs which they ascribe to other human beings, and vice versa.
This quite obvious insight is eagerly denied, however, within neurocentrism's framework of interpretation. That is already remarkable in itself. This poses a decisive question: Why would anyone wish to deny that there really are hopes, convictions, opinions, doubts and intentions? What lies concealed behind the illusion that a great part of our consciousness is an illusion?
Accepting the claim that propositional attitudes, and thus an essential aspect of our consciousness, are an illusory affair, a kind of folklore that is supposed to be eliminated by the amazing, evidence-based future technologies based on futuristic neuroscientific discoveries, is a philosophical thesis. If one probes more closely, one soon realizes that it rests on untenable assumptions. These assumptions are a matter of not one, but a whole series of errors, many of which we still have to identify and, well, eliminate. It is better to eliminate the errors of eliminative materialism than to eliminate consciousness.
Yet these errors have in the meanwhile come to shape the everyday self-images of many human beings in economically advanced industrial societies, since they are considered to be progressive and scientific. We should all like to be progressive and scientific if the alternative is superstition, ignorance and manipulation. But the problem is that the errors of neurocentrism are precisely superstition, ignorance and manipulation, though these facts are to some extent well concealed. The errors become evident only when one takes the time to question the image of the world that stands behind them, all the way to its underlying assumptions. That holds true for any superstition which prevails at a given time. Unreason is always disguised, or else it would not be as effective as it unfortunately is. How is it possible to gloss over the quite obviously absurd claim that we do not have any propositional attitudes?
Various strategies are employed to this end. A common strategy that is not of much philosophical interest simply appeals to scientific facts. This strategy maintains that the accumulation of comprehensive knowledge of the human brain and of the neurobiology of human cognition will eventually lead to a realization of the fact that we do not have any propositional attitudes. Of course, for the most part this is not expressed very clearly or distinctly. Instead, for instance, there is research into what causes human beings to fall in love, to vote, or to act morally, in the most general sense of acting in a minimally altruistic manner. The normal answer to the question of why someone acts altruistically does not appeal to scientific findings, however. For thousands of years the question has been posed as to why human beings have regard for other human beings and live in communities instead of roaming the forests as solitary loners. For a long time now, we have heard myriad answers to questions such as that concerning the origin of morality. Almost the entire history of thinking about these topics, from Plato to Nietzsche and beyond, is ignored in favor of the naïve and much refused assumption that moral behavior is tantamount to altruistic behavior. However, the actual history of philosophy has a lot to teach us in the area of self-knowledge, which has certainly not been made obsolete by modern science. Ignoring history does not make it go away.
## _**An altruist is lodged in every ego**_
An answer to the question why human beings are not simply merciless egotistical predators – although they can act this way! – is that they have the capacity to recognize that other human beings should be respected. For other human beings lead a conscious life, too. To lead a conscious life means to experience oneself as the subjective center of events, to experience oneself as a self or an "I." To modify slightly a proposal by Daniel Dennett: to be someone, to experience oneself as a self, is to be the rationalizing "center of narrative gravity." What often goes unnoticed is the remarkable feature that at the center of being is our potential of being able to understand that there are other such centers apart from our own. To be a conscious self is immediately to be in contact with the fact that we are not the only centers. Anyone who is really a practical egotist and tries to get rid of the presence of other minds at the core of her very being turns out to suffer from one of the manifold psychological disorders associated with an exaggerated sense of self.
As I am conscious right now of being conscious, I can easily remind myself that at this very moment there is a child in Shanghai for whom nothing is more important than a certain toy, which in contrast would not be of interest to me. It is a characteristic feature of conscious human life to know that others are consciously alive, too, and that different things matter to them just as different things matter to us at different stages on our life's way (and even at different moments in a single day). Without the capacity to recognize that different things can matter to different people or to ourselves over the course of a single day (or even a single hour), we would indeed be no one at all.
Thomas Nagel rightly considers this insight into the inevitable presence of otherness at the rational core of our centers of narrative gravity to be the basis of **ethics**. By _ethics_ , I understand the systematic reflection on the foundation of the principles for our action in light of the fact that we are capable of good and evil (of morally recommended and morally prohibited action). In his books _The Possibility of Altruism_ and _The View from Nowhere_ , Nagel distinguishes two categories: the subjective and the objective. The "subjective" is his name for our thinking that is bound to a conscious standpoint. Insofar as we are conscious, we experience everything from a subjective standpoint. This holds true not only for our perspectival sensorily based perceptions but also for all of our beliefs, insofar as these too are embedded in a personal network of beliefs that we do not survey in one fell swoop. It is for this reason that we can entertain contradictory beliefs, because we do not have an overview of everything that we actually believe and assume.
A simple demonstration: I am absolutely positive that almost any reader of these lines (at least anyone who has read from the beginning up to this sentence) agrees that more than seven people in India have seen a tree more than once in their life. Furthermore, any reader believes that more than eight people in New York City have noticed the Empire State Building. Until this moment, you have probably never explicitly formulated this belief. And for the same reason we can also change our beliefs if we realize that some of our beliefs are actually incompatible with others that we also hold. Our beliefs do not continually flutter around in our conscious interior life; they are not like birds in an aviary, as Plato put it in his dialogue _Theaetetus_. This means that our belief-system is not itself an occurring thought or conscious stream but, rather, something which consists largely of a logical network of presuppositions, inferences we can spontaneously draw in order to travel through the logical space of our beliefs, as well as contradictory assumptions isolated from each other precisely because we did consider all of them together as we formed them. In other words, most of our beliefs are unconscious and non-objectively entertained. To hold a belief is not to be conscious of that belief. Even though we can become conscious of some of our beliefs, we are never in a position to survey them completely. This is why our rationality is subjective to the extent to which it can operate only on a limited data set we consciously process.
In contrast, "the objective" consists in the realization that we are part of a context which to a large extent is completely independent of what we suppose about it. Regardless of how much power we ascribe to our thoughts and our linguistically coded theoretical constructs, we all know that most facts are simply what they are no matter what opinions we have and no matter what perspectives we adopt. Reality and our belief system do not simply match each other. On the contrary, given that there are many belief systems apart from my own, I can know with absolute certainty that my belief system is nothing like an adequate representation of reality as a whole. To know anything about our beliefs is to know that we do not know everything. Hence there is an ideal of objectivity that consists in describing a reality which is abstracted from ourselves – from our interests and our perspectival consciousness. Admittedly, we do not always attain this ideal, and in the interpersonal realm it is perhaps even unattainable in principle, for which reason it is part and parcel of ethics to realize that our contexts for action are never ideally constructed. The ethical dimension for evaluating our actions is formed from this weighing up of subjective and objective perspectives on facts.
Every consciousness must recognize the fact that other consciousnesses exist independently of one's own. Even if one wishes to be a completely ruthless egotist, as Lorne Malvo from the brilliant _Fargo_ series does, one must acknowledge these facts, since otherwise one cannot successfully pursue one's own goals. Human beings can only be egotistical if they understand that others are conscious, too. Altruism, a mindset characterized by regard for others, which initially seems to be contrary to egoism, is based on the potential for realizing that others are conscious, too. If I have even one reason for not using someone merely as a means for my egotistical ends, then the reason for this is precisely that the other has conscious experience and unconscious beliefs, too: he or she feels pain, hopes for something, wishes for and avoids things, and believes certain things. People and other animals are not merely tools like screwdrivers. In order to be egotistical in a meaningful, goal-oriented way, we must have taken the standpoints of others into consideration, since otherwise we cannot understand their motives or use them for our purposes. There would be no egoism without this decentering. The potential for altruism is thus based on it. Altruism is made possible by egoism. Whether one likes it or not, others are conscious, too, and pursue goals, and thus already fundamentally deserve ethically relevant consideration. The egotist recognizes this but acts according to her wish to disrespect that very fact by taking it into consideration in her strategic plan to make use of others whatever their intentions. One should not, however, conclude from this that we only act ethically when we are somehow like Mother Teresa and sacrifice our life for others. Egoism is not evil per se, nor is altruism good per se. They are both aspects of ethical action that reciprocally condition each other. If we reproach someone for egoism, we really mean that this person is pursuing a poor balance of egoism and altruism – for instance, by merely pretending to do something for others while their own interests secretly prevail. However, the problem here is one of fraudulent intent rather than egoism. Since each of us only lives once, we have a right and an obligation consciously to lead this life as our life. We are entitled to egoism, but this does not mean that egomania is the right mental profile.
In light of this, it is ethically convenient simply to deny the consciousness of others. However, it does not work. If one denies the consciousness of others (including of other conscious animals), one avoids recognizing the fact that others are living beings able to feel pain who can also suffer from the cruel actions and injustice that human beings perpetrate on each other. When one succeeds in eliminating the standpoint of the subjective and replacing it with a purely objective standpoint, such as the standpoint of a neurobiology of the human mind that is supposedly attainable someday in either the near or the distant future, one is thus freed of the burden of being bothered by the claims of human freedom. The irony in this is that one can regard the fantasy of a completely objective description of reality, in which no subjective experience occurs any longer, as very egotistical. All ethical claims, which rest on the fact that there is conscious life which harbors propositional attitudes, are thus removed from one's image of the world. If we array ourselves against consciousness and instead conceive of ourselves as neurocomputers, it makes it easier for us to ignore the fact that actually we are not neurocomputers. It is a relief to envision one's own freedom in this way and then ultimately delegate it to our neurochemistry. Yet all of this is just a kind of self-blinding. It is an ethical stance which sells itself as purely objective, non-ethical scientific insight.
The Churchlands are only too happy to gloss over this. This takes the form of complaisant affirmations such as that family life and canoeing are important, or that they still like going into nature as though into a museum and how they marvel at the indigenous peoples of Canada, and so forth.19 Whether one sympathizes with such declarations or not, it remains the case that human society is structured by linguistically coded propositional attitudes. Such attitudes are not annoying vestiges of a pre-technological era, and they will not be changed by any neurobiological discovery. No possible medical or technological advancement of human nature will ever make it the case that we can get rid of "folk psychology," let alone of propositional attitudes. What is more, ethics requires that we do not even desire to overcome the ethical condition of human agency. We ought to be ethical.
The subjective and the objective are interrelated. Without subjective standpoints we would not have any beliefs, or any networks of beliefs, which distinguish us as persons from one another and make our different life projects possible. It is a condition of possibility of our ethical life that we regularly disconnect from ourselves and can visualize how others perceive us – which we constantly do. Even scientific objectivity is attained only by our conscious striving for it, which of course does not imply that it is not really there. To claim that everything is subjective or that everything is objective is equally absurd.
## _**Davidson's dog and Derrida's cat**_
At this point, at the very latest, some among you will have asked how things stand with other animal species besides the human being. For surely they, too, have consciousness, while up to this point I have been speaking almost exclusively of human beings only. In no way would I want to deny that other animal species have consciousness, too, and from this formulation you can infer without further ado that I evidently consider the human being to be an animal among other animal species. Despite the fact that it is as hard as it gets to find a precise cut-off point in the animal kingdom which would separate conscious from non-conscious life, I assume that lots of animals belonging to different branches of evolution are conscious. Some life forms might turn out be automata, mechanical data machines of self-replication in open systems. Yet this does not apply across the board, and this is all that matters at this point in the argument.
"Human being," or to express it better in this context, _Homo sapiens_ , is among other things a name for a specific kind of animal. Carl Linnaeus (1707–1778), who in his _Systema naturae_ introduced the species name _Homo sapiens_ , cites as the human being's characteristic feature the famous ancient precept "Know thyself!" ( _nosce te ipsum_ )20 The capacity for self-knowledge presupposed by this precept is, according to Linnaeus, precisely what makes us " _sapiens_ ," a living being that is capable of wisdom.
This notion has an important history. In his famous defense speech before the Athenian jury, which Plato composed as _The Apology of Socrates_ , Socrates recounts that the Delphic Oracle called him the wisest of human beings.21 The Oracle calls for universal self-knowledge. Its precept "know thyself" was a great theme of ancient Greek literature and philosophy. Wisdom is based on self-knowledge, for which reason Linnaeus defines the human being as _Homo sapiens_ , since the Latin verb _sapere_ means "to be knowledgeable." However, Linnaeus' classification applies not to every kind of knowing but, in particular, to the specifically human form of self-knowledge that makes wisdom possible. The possibility of wisdom is anchored in the fact that human beings live their life in accordance with a symbolic representation of what that means. Ever since Socrates and Linnaeus, the human being, as we know it in modernity, has been explicitly defined via its capacity for wisdom. Philosophy is notoriously the activity we engage in when we reflect upon that very capacity, as philosophy is literally the love of wisdom. We distinguish ourselves from other human beings, from inanimate nature and from other animals by ascribing different mental states, the absence of mental states or different kinds of mental states to these entities respectively.
Other animal species live a conscious life as well. We even ascribe propositional attitudes to them. One could not imagine any documentation of animals that would not describe how a lion lurks in ambush or how a herd of gazelles runs away in terror because they have sensed the lion. Of course, our practice of ascribing propositional attitudes in us to the animal realm around us is in part problematic, since we ascribe certain attitudes to animal species that we have come to prefer for historical reasons and deny that other animal species have these attitudes. Most of us consider our domestic animals to be friendlier than poisonous snakes in the Amazon jungle, and almost anyone holds that dolphins are nicer than sharks (which can be dangerous when confronted with orcas, which happen to be much more dangerous to humans than sharks). Yet we also do this in the case of human animals when we ascribe only to our enemies motives such as envy, ruthless egoism or other flaws, for instance, while in contrast we ascribe goodwill and commendable altruism to our friends (or those whom we consider to be friends). Which form or what degree of biological organization must be present for one justifiably to ascribe consciousness to an organism is in fact not yet clear. This remains an important and open question, since there are of course (neuro)biological bases as well as necessary preconditions for consciousness of which we do not yet know enough to be able to say precisely which living beings are really conscious. In fact, to discover this is an important task for neurobiology in the future, as well as to spell out the consequences of future discoveries in this field for ethics and philosophy in general.
In any event, one should never get carried away in denying consciousness to all other animal species, although of course this is quite convenient if one wants to ease one's conscience concerning the slaughterhouses and other torture chambers for animals that are dedicated to medical progress for humans.
However, there is an interesting line of thought that problematizes the ascription of propositional attitudes to the animal realm beyond us. It was pioneered by the American philosopher Donald Davidson (1917–2003), to whom we are indebted for important contributions to the philosophy of mind. Unfortunately, his writings range from being partly cryptic to completely incomprehensible, although fortunately his basic ideas have often been reconstructed, more or less intelligibly, by other authors.
Davidson, at any rate, is (in)famous for the fact that he denies our form of consciousness to other animal species, since he considers our form of consciousness to be linguistically structured through and through. His line of thought runs roughly as follows.22 Assume that you come home and your dog is already standing at the front door wagging its tail and yelping. Ordinarily, we interpret this as the dog's delight in the fact that we are home. As the comedian Jerry Seinfeld once said, all of this boosts our self-confidence. A dog, according to Seinfeld, is an animal that is always impressed all over again at the fact that we manage to go away and then return home, and yet somehow also see to it that it is fed. The dog has no idea how we do this and continually looks up to us, impressed – an admiration that, in contrast, we encounter in our children only until they understand how to care for themselves. Then the whole enchantment of the supposedly mysterious power of adults goes up in smoke and it is revealed that they, too, are normal people. The greatest episode of this insight is called "puberty," a process that to my knowledge has never been attested to in the case of domestic animals...
However, an ethically relevant respect for other animal species does not stem from the fact that they have particularly subtle networks of propositional attitudes. Otherwise, we would never have a reason to respect infants, small children or developmentally disabled adults in ethically relevant ways. Thus, even if some Davidsonian argument successfully established that other animals have no propositional attitudes (which is extremely doubtful), this would not entail that they are mere objects not worthy of ethical attention, as the same argument would apply to our own children too.
Davidson invites us to imagine what is really going on in the dog. When we say that our dog _is delighted that we are coming home_ , this is much more problematic than we initially imagine. This becomes clearer when we imagine that both our seventeen-year-old daughter and our dog are at home when we arrive. Let us assume that our daughter is also delighted to see us. She thus knows what it means for someone to come home: one must unlock the door, one must have been somewhere else, in a city or in the country, for example, one must have gotten around in a vehicle or on foot, and so forth. In addition, our daughter understands the process in a certain way: she is delighted that we are coming home, because we promised her something. She knows that we are coming home, because it is the end of the workday, and so forth.
Davidson thus focuses on the "fine-grainedness" of our beliefs. We are not simply delighted when someone comes home, but, rather, our delight belongs to a network of changing attitudes. In contrast, it is to some extent plausible to believe that the dog is always simply delighted – unless it is so ill and feeble that it does not wake up when we arrive. The dog's delight over the fact that we are coming home perhaps never actually has to do with the explicit content of coming home. What we call "coming home" is something that the dog does not know at all, as it is questionable that the dog believes that he is at home and that someone is coming home in anything like the way we believe this.
Davidson concludes from this that, in any case, the dog is by no means delighted by the fact that we are arriving home in the same way that our daughter is. Let us think it through a little more deeply and ask whether a dog can be delighted that grandma is calling on the phone. The answer to this is surely no, since a dog has no idea what a telephone call is. At best, a dog is delighted by the ringing of a telephone, insofar as it resembles the sound of a doorbell. Dogs are not delighted by telephone calls from grandmothers, in the same way that they do not hope that Scarlett Johansson's next film will be good.
These simple reflections on the consciousness of other animal species, incidentally, are by no means superfluous even though they might seem to be self-evident to many readers. Geert Keil points out that the principle of guilt which prevails in our criminal justice system ("No punishment without guilt") does not belong "to the earliest achievements of Western legal history. In rural France it was common into the seventeenth century to apprehend and to punish animals in court because of alleged crimes."23 The world of human experience is continually in contact with the rest of the animal realm. We hunt, kill and breed animals. We are stung by mosquitoes, observed by seagulls, dwelt in by microorganisms, and so forth. We naturally ascribe propositional attitudes to other kinds of animals with good reason, as we live in the same realm with them all the time (our bodies happen to be complex breeding grounds of many animals we do not notice in our everyday lives unless we are doctors or biologists, say). Our own body is indeed more like a zoo than a closed system controlled by a conscious ego running the show. That much has been established by biology over the last hundred and fifty years. Our body is not just one organism but an entire ecosystem.
Indeed, Davidson rightly points out how easily we fall prey to false ideas of the inner life of other animals because we describe their behavior in a language that is tailored to our form of life. In our association with other animal species, we tend toward **anthropomorphism** – that is, toward the projection of our form of life onto other animal species. Our beliefs are fine-grained and differentiated in a way that reflects millennia of cultural and intellectual history. Thus the human mind, as we know it, has a history. We share some episodes of this history with other animal species, especially with those kinds of animals that we have explicitly integrated as domesticated ones into our form of life.
Yet a much greater divide persists between us and our animal friends than one imagines in our at least somewhat partially animal-friendly moments. The French philosopher Jacques Derrida (1930–2004) makes this clear in reference to a situation he found himself in one morning as he stood naked in front of his cat and met her gaze. At the time, he reacted as though another person had seen him naked, until it became clear to him that the cat did not even recognize that he was naked, in any case not in the sense in which he spontaneously felt exposed in front of her.24
Our feeling of shame belongs to our form of life; it is embedded in our human ideas of how we should live together. To be sure, cats and other animals live among us, but they share our form of life only in a very limited way. Of course, the reverse also holds, as we share the cat's form of life only in a very limited way. Our vocabulary, designed to describe mental processes in our fellow human beings, is certainly not able to describe in an adequate and fine-grained enough way what it is like when cats slink through a city park at night and hunt mice. We cannot even imagine this, since for us there would simply be no sense in hunting mice in a city park at night and then carrying them home between our teeth.
## _**Tasty consciousness**_
Many other living beings definitely have components of consciousness that remain for the most part unintelligible to us, insofar as we simply do not know what it is like or how it feels to undergo certain experiences. That also holds true even for members of one's own species. When I was a child, I always asked myself how it was possible for other people to like different dishes than I did. How could it be that someone did not always consider spaghetti with tomato sauce and a glass of lemonade to be delicious but instead preferred foie gras, for instance, and a glass of champagne?
If something tastes good to one, it is hard to really imagine what it would be like not to enjoy the very same-tasting thing. For that to happen, one must have already had the peculiar experience that one's own taste changes over the course of life and that one can of course also deliberately cultivate it. For the most part, we do not experience taste as a neutral fact that we then further evaluate as to whether we assess it as good or bad. Taste is experienced as good or bad; it is not a neutral fact in the world, as it were. Something always tastes more or less good to us. But how do we know what it is like not to enjoy something that we do enjoy or, conversely, to enjoy something that just appears repulsive to us?
Recent philosophy of consciousness has coined the term _qualia_ in this context. **Qualia** (from the Latin _qualis_ = qualitatively constituted in some way; singular: _quale_ ) are the contents of conscious, purely subjective experiences. They are ways in which we experience things. Examples of qualia are impressions of color and sensations of taste or sensitivity to heat, for instance.
We have thus already learned of two facets that are associated with consciousness. We can distinguish them conceptually as follows.
1. The first facet consists in the fact that we can be _conscious of something_. "Intentionality" stems from the Latin verb _intendere_ , which means to extend or stretch outward. **Intentional consciousness** consists in our reference to something typically different from consciousness. A prime example of intentional consciousness is conscious perception. When I perceive a tree in front of me, my consciousness "stretches out" to the tree and is, thereby, of the tree. We are able to focus consciously on something and think about it. We extend our consciousness, as it were, by directing our attention to it. With a different emphasis, this has recently also been characterized by the philosopher Ned Block as _access consciousness_ – that is, consciousness that has an attentive access to states of information.
2. The second facet is associated with our internal perspective. In this context, one speaks of **phenomenal consciousness** – that is, of our subjective, conscious experience. The contents of this experience are the already mentioned qualia.
The concept of consciousness, which incidentally has many more additional aspects (and is for that reason probably a whole meshwork of concepts), inclines us to conflate intentional and phenomenal consciousness. For instance, one might suppose that we are able intentionally to focus on our percepts. I can easily focus on my experience of taste while drinking a good red wine, thus I must be able to focus consciously on this experience. One thereby turns the taste of red wine into a private object – that is, an object that only the person who experiences the taste can observe, as if my taste of wine was a secret object, hidden from your view. How do I know whether the wine really tastes to others exactly as it does to me? Certainly, they may tell me that it tastes good to them, too. Perhaps they agree with me when I emphasize the note of vanilla. But if it makes sense that one is not able to have experiences of taste without at the same time also evaluating them, then one can also experience the vanilla note of a red wine differently, even if different people publicly agree that a vanilla note is present. Are my and your vanilla notes even comparable in any way? How could I ever be in a position to know how vanilla or red wine taste to you?
Perhaps you are acquainted with a similar phenomenon, the **problem of the inverted spectrum**. Ned Block succinctly summarizes the problem as follows. Could it not be the case that "the things we both call red look to you the way the things we both call green look to me?"25
This line of thought poses a particularly intricate riddle if one pursues it a little further. Experiences of color are no more neutral than those of taste. Colors please or displease us, too. We do not simply just notice them and then evaluate them, but we already experience them with a view toward evaluation. We sense them immediately as cold or warm, agreeable or bothersome.
Furthermore, we also learn to see colors we might not have consciously noticed and can become better at distinguishing them. If one now adds to the account the fact that our visual field is colored through and through, it stands to reason that in fact the entire reality perceived through our sense of sight perhaps appears completely different to me than it does to you. Even if we agree that blue dice are lying there in front of us, my experience of blue is perhaps so fundamentally different from yours that we are not really speaking about the same blue dice after all. How do we know we are not basically constantly talking past each other even in the most mundane situations, say, where we search for the red pair of keys in the living room?
At this point, everything hinges on what is meant when one speaks of the vanilla note of a red wine or of the blue of a pair of dice. Does one aim at an _objective_ common property of red wine or of dice, or at our individual _subjective_ sensations? In the first case, one would have to have an intentional consciousness of these properties. In the second case, one would have to have a phenomenal consciousness that consists in experiencing qualia. One can pin this difference down in the following way: intentional consciousness _is related_ to properties of commonly present objects, while qualia are ways in which we _experience_ something. To relate to something and to experience something is not the same thing. Hence, we now have a conceptual distinction in our hands, the distinction between _intentional_ and _phenomenal consciousness_ , which can help us to avoid many confusions widespread in our contemporary scientific and everyday culture.
## _**The intelligence of the robot vacuum cleaner**_
All of this now perhaps seems almost so self-evident that you are asking yourself why it is worth talking about. However, there are many reasons for our overlooking the difference between intentional and phenomenal consciousness all too easily in important contexts. Let us take the example of artificial intelligence and thus the question of whether computers, smartphones and robots can think intelligently.
The topic of artificial intelligence has occupied the philosophy of consciousness since early modernity – it did not first become virulent with the emergence of the computer. Descartes already posed the question of how he could know that people are passing in front of his window and not merely robots (or, as one said in those days, automata): "... I see the men themselves... yet do I see any more than hats and coats which could conceal automata? I _judge_ that they are men. And so something which I thought I was seeing with my eyes is in fact grasped solely by the faculty of judgment which is in my mind."26 What if there were perfectly constructed humanoid robots which we could not distinguish from humans with the naked eye, such as in the movie and HBO series _Westworld_? If one thinks here again of _Doctor Who_ , all kinds of scenarios can be imagined in which this could be relevant. Of course, one can also think of _Blade Runner_ , zombie films, the Swedish television series _Real Humans_ – or one among the countless products of literary fiction that treat this topic. One could also include classical works of literature such as those by E. T. A. Hoffmann and Heinrich von Kleist. Would we not attribute consciousness without further ado to humanoid robots, especially if they could have an intelligent conversation with us? It is even reported that many people over the course of time got the impression that their model of a robot vacuum cleaner such as the iRobot Roomba possesses its own conscious will – indeed, that it has a personality.27
Let us now pose the question as to whether we would actually consider humanoid robots to be conscious if they had only intentional consciousness but entirely lacked phenomenal consciousness. Of course, this already presupposes that they could have intentional consciousness, which I will provisionally accept for the following line of thought. In this case, they might well provide all kinds of accurate reports by processing information given to them on the basis of reality-tracking devices such as photosensors. Perhaps through an internal analysis of the chemical composition of the aforementioned red wine such a robot would even be able correctly to express the fact that the wine reveals a note of vanilla. In our era of digital revolution, it is particularly easy to imagine robots with an improved version of intentional consciousness – a consciousness upgrade – for example, being equipped with greater cognitive processing power in certain areas. Standard commercially available computers have for a long time played better chess or Nine Men's Morris than most human beings. Yet would all of this actually justify us in characterizing robots as "conscious"?
I do not believe so. Let us suppose that the left cheek of an otherwise perfectly humanoid robot (a hubot from the series _Real Humans_ or a robot from Spielberg's _Artificial Intelligence_ ) suddenly fell off and it then had a similar appearance to the robot in the first episode of the eighth season of _Doctor Who_. We would be looking at a rusty primitive mechanism in its interior. It would then naturally seem reasonable to suppose that the robot does not have any phenomenal consciousness: it does not have experiences; it does not feel in any way. It is just a robot that is operated by means of a rather crude mechanism. If it now tells us that it sees blue dice, it could still utter a true statement or give an accurate account of there being blue dice. But, since it has no experience of blue at all, something essential would be missing from its perception, namely consciousness. And even if there are cases of unconscious perception (also in humans, such as cases of blindsight), they are not the standard cases for which we designed our mentalistic vocabulary. In any case, a purely intentional machine would not be conscious in the way that we are. But could the robot have phenomenal consciousness despite that? How do we know for sure that it does not?
At this point, it is important to recall that consciousness, as we know it, simply has necessary biological preconditions – which I would not wish to call into question at any point. A rather crude mechanism which does not consist of the relevant organic material that would need to be assumed as a precondition for phenomenal consciousness (lacking neurons and other kinds of cells) could not sustain any phenomenal consciousness for biological reasons. In fact, our phenomenal consciousness emerged over the course of evolution, which does not merely mean – and it is important to bear this in mind – that the human mind is on the whole an evolutionary phenomenon. Neurobiology and cognitive science do not completely cover the investigation of the mind, since they have for their focus only some of the necessary conditions for the presence of consciousness and other mental phenomena (such as the relevant wetware and crude cognitive computational power).
When we speak about colors, tastes or other qualia, we speak about something that one has to experience to be able really to know what one is talking about. It is not enough to be able to give accurate reports about such things. The relevant phenomenal finegrainedness is simply lacking in any artificially constructed robot, regardless of how sophisticated the robot's vocabulary might be when it analyzes how a red wine tastes. A sommelier machine is just not a sommelier, even if a sommelier machine might sometimes be better at its job than an actual sommelier.28
Human consciousness is not a set of general rules that we can translate completely into algorithms or model in some other way. Computers are much better at processing information with the aid of algorithms than we ourselves are. What distinguishes us in this area is that we are not like Mr Spock from _Star Trek_ and we do not approach everything logically and unemotionally. In part, we have irrational feelings, which include not only much maligned forms of anxiety but also our capacity to fall in love or enjoy drinking wine. The human being is neither merely an _animal rationale_ nor simply an _animal irrationalis_ but, rather, a being which creates self-images that can prove to be illusory and celebrates and cultivates them in concert with others, changing them if they turn out to be harmful. For this reason, we even have the right to pursue cultured absurdity, irony and illusions as well, so long as no one is harmed who for their part would like to pursue their own illusions.
A simple example of this is our everyday relation to material goods. We purchase all sorts of things, and we typically have the illusory impression of finding precisely the right thing that we sought, which satisfies our need. As soon as we own a thing or have consumed it, we again desire something else. This quite insatiable desire for material goods has been critically put to the test again and again, from Buddha to Marx and all the way to psychoanalysis and contemporary behavorial economics, with its discovery of the manifold biases underlying our exchange of goods. One must see through desire as a source of illusions. However, we cannot give up these illusions entirely, since we only come to be anyone at all through them. To live is to suffer from illusions; it is not a way of calculating our survival chances by becoming smarter. An utterly uninterested intentional consciousness not worried by illusions simply would not exist in the way that we do. It is no accident that for thousands of years we have been surrounded by beautiful things onto which we project our consciousness, even though we all know that it is an illusion, for instance, that gold has a value in itself that is greater than the value of peanuts.
The term "consciousness" refers to a combination of intentional and phenomenal elements. Phenomenal consciousness in part is grounded in facts discoverable with recourse to the theory of evolution. If one were to think that pure intentionality – that is, the fact that some system regularly gives accurate reports – could count as consciousness independently of phenomenal consciousness, this would be as though one believed that water molecules could also consist only of hydrogen and not, for instance, of the right combination of hydrogen and oxygen. Pure water molecules just are H2O molecules. H alone is far from being water.
## **Strange Days** _**– the noise of consciousness**_
It turns out that there is something wrong with the idea of dividing consciousness as we know it into two halves. Or, rather, it is perfectly fine to offer a conceptual analysis of the term "consciousness" in different aspects, but this should not mislead us into thinking that we actually have a hold on what a truncated form of consciousness, such as a purely intentional artificial intelligence, would look like. At the very least, it is highly questionable to assume that the form of representation we ascribe to a vacuum cleaner deserves the title of consciousness of any sort.
Let us now take up a thought experiment that starts from the other side and imagine what it would be like to possess only phenomenal but no intentional consciousness. In Kathryn Bigelow's science-fiction thriller _Strange Days_ there is a particularly good illustration of what it would be like to be reduced to one's phenomenal consciousness. The movie transports us to a dystopian future that plays out – as is typical – in Los Angeles, the city of film itself. In the futuristic vision of _Strange Days_ there are virtual reality engines with which one can record a human being's phenomenal consciousness and then experience what it is like to be them, like a multimodal, multidimensional movie. A VR device is strapped to one's head, and one then lives through exactly what the person experienced during the recording process.
Now, the arc of suspense revolves around a recording which circulates in LA's underground and on which one can see that the police murdered a black hip-hop star, a fact which could trigger a revolt, for which reason the murderers (who happen to be white police officers) attempt to get hold of the neuromovie. In the course of doing so, they employ a horrible method: the device with which the consciousness films are recorded and played back can also be used to fry a brain, so to speak. Those afflicted in such a manner end up in an irreversable condition, suffering from a completely chaotic flickering of data – their experience is forever like the flickering screen of an old television in the days when broadcasting closed down for the night. It is impossible for their phenomenal consciousness to stretch out to any real pattern external to the flickering they experience, and hence no one can get through to them anymore. In one scene this experience is shown from a firstperson perspective, which leaves a quite horrifying impression.
Some forms of psychotic experience, other pathological mental states or the state of intoxication triggered by psychoactive drugs maybe come close to such a jumble of data. But we can also just try to imagine our consciousness being dissolved into a pure, temporal impressionism in which we no longer perceive any objects but experience only flowing colors and other impressions – as if one were standing too close to a late Monet or a pointillist painting.
The great Scottish philosopher David Hume (1711–1776) distinguished between _impressions_ and _ideas_ in order to classify forms of internal experience. On his understanding, an _impression_ was a jumble of data, while an _idea_ emerges from a more stable organization of one's jumble of data. Ideas are objectifications out of impressions; they help us discern real patterns (information) in the data with which we are confronted as sensuous creatures.
Impressionist painting ultimately traces back to Hume's distinction. It illustrates ideas on the basis of impressions, for which reason one must move to an appropriate distance from the painting so that the objects it presents emerge from the jumble of color. If, sitting at my writing desk, I quickly turn my head a little to the left and try to avoid too much focus on specific objects, I see blurred red, blue, black, flecks of light, and so forth. But as soon as I fix my gaze on the fleeting impressions I immediately recognize books, a glass of water, pens, and other things. The perfection of the presentation of our impressions is certainly a great theme of modern painting. Hume's helpful distinction corresponds to our distinction of phenomenal consciousness (impressions) and intentional consciousness (ideas).
If our consciousness were only a stream of experience similar to the state of intoxication, a pure jumble of data, we could no longer communicate with each other. We would not be able to distinguish our impressions from what causes them and to which we can intentionally refer. Our waking consciousness is not ordinarily impressionistic in such a way that upon closer inspection everything looks blurred. If things were always like that, we would no longer have impressions of anything. We would merge with our impressions and would no longer be present as consciously acting agents capable of making up our minds, so to speak. If we had just impressions, but no ideas, impressions would not even be classifiable by us – for instance, as an impression of red or one of pain. A pure jumble of data, without any intentional order, cannot be observed or consciously experienced by anyone, since everyone who observes must also classify. One could not register pure data without any order; strictly speaking, such data does not exist at all. It is not the case that our consciousness consists of two layers, one pure data layer of sensory registration and a second one of explicit, reflective recognition or objectification. The level described by cognitive science or vision science, for instance, does not tell us anything about how our consciousness works. It rather gives us a model of how non-conscious processes of information-processing can be accounted for in a scientific theory. Whatever the outcomes of such an investigation, they will always only unearth necessary preconditions for consciousness but never reveal the whole thing. This is a sense in which all natural sciences of the mind are and will forever remain reductionist in outlook, as they have no room for conscious experience. Actual conscious experience is not governed by any idealized laws. This does not mean that we are not able to work out scientific theories of aspects of the mind. However, these aspects of the mind are not "out there" in reality, causally interacting with each other as so many levels of physical organization. This is just a misguided projection of criteria of successful theorizing onto an external, natural world.
Phenomenal and intentional consciousness thus work together in actual consciousness. If one separates them from each other, one is left with a fundamentally unintelligible image of consciousness – as though our consciousness were either a vacuum-cleaning robot or an unfortunate hippie who cannot come down from a LSD trip. The realization that neither describes the normal case lies behind a much cited and rightly celebrated formulation of Kant's, which we can simply call **Kant's insight** : "Thoughts without content are empty, intuitions without concepts are blind."29 What is potentially misleading in this formulation, however, is that Kant links intentionality to concepts and phenomenal consciousness to intuitions. In turn, this makes it seem all too plausible somehow to privilege reflectively competent users of concepts such as us humans and to presume that other kinds of animals have a lesser form of consciousness or no consciousness at all, as they do not possess an explicit grasp of concepts. As, for instance, the philosopher Tyler Burge has shown in detail in his major book _Origins of Objectivity_ , one must guard against this. Of course, other kinds of animals and human infants have concepts too, even if they are not linguistically represented by them and are thus characterized by a different degree of fine-grainedness. In order to be able to have concepts and to classify impressions, one is not required also to be able to classify these concepts linguistically. To be sure, we human beings can do this – on the basis of our language we are quite successful in classifying reality. Yet it does not follow from this that it is only us who are able to classify, while all the other animals, on the contrary, could be reduced to being hungry impression machines, as it were.
According to Kant, our thoughts would have no content if we did not have impressions (what he calls "sensation" [ _Empfindung_ ]). If we had only impressions but no stabilizing concepts, our impressions would be blind intuitions, since we would no longer be able to cognize anything, to discern real patterns of experience. A pure jumble of data does not even consist of green spots in one's field of vision; it consists only of pure ineffable experiences. As soon as one sees green spots, one already has intentional consciousness of something, namely of green spots.
Kant also describes the state of pure, jumbled data that is psychotically blurred in similar terms to what we witnessed in _Strange Days_. As in the movie, such consciousness would be "less than a dream."30 For even in dreams we still dream of something. In our dreams, the sequence of representations is indeed occasionally quite odd, as is well known; one jumps from one sequence to another and has quite indefinite sensations. Also, our sensory modalities are reduced in dreams, as we usually do not have the impression of tasting or feeling anything (think of the dream of an orgasm, which is not typically accompanied by the real deal). However, one does not dream merely in tastes, tones and colors. Dreams are not pure symphonies of sensation but forms of conscious experience.
Let us call the thesis that consciousness is exclusively intentional **consciousness rationalism**. The corresponding other extreme which reduces consciousness to its phenomenal aspect is **consciousness empiricism**. Both mistakes repeatedly arise up to the present day in the sciences that are concerned with consciousness. They form the basis for different varieties of neurocentrism.
While the one takes us ultimately to be purely rational quasicomputers (biologically implemented artificial intelligence, as it were), the other views us as experience and desire machines. This is the postmodern version of the ancient Christian doctrine that the human being wavers between God and animal. The mistake of the ancient as well as the postmodern doctrine consists in the fact that these two extremes – the purely rational God and the purely emotional desiring animal – simply do not exist and at any rate have nothing to do with the reality of human consciousness as we know it, even though they are characteristic of the human mind's fantasies about itself.
A representative example of consciousness rationalism is an approach advocated by Daniel Dennett. In his highly regarded book _The Intentional Stance_ , he spells out his (in)famous **thesis of the intentional stance** : this thesis amounts to the claim that every system which we can successfully describe as intentional is intentional. For him, then, intentionality is in the eye of the beholder, but not out there. Of course, it is not merely arbitrary to describe a system in intentional jargon. Yet, the reality underlying the phenomena itself for Dennett constitutively differs from the illusory potential of the intentional stance. Let us again consider Davidson's dog. We can predict that the dog will bark and wag its tail when it hears Davidson coming home. We can accordingly express ourselves as follows: the dog is delighted because it knows that Davison is coming home. We thus ascribe propositional attitudes to it. But we could also carry out a scientific investigation and merely describe how the sound waves affect the dog's sensory receptors, which trigger further processes. In this descriptive account, which Dennett calls the _physical stance_ , we do not employ any intentional vocabulary. For Dennett, the physical account is ultimately better off when it comes to the description of reality as it is in itself, whereas the intentional stance is pragmatically useful and probably forever indispensable, as it helps us to navigate the otherwise overly complex social world of human intentional behavior.
Dennett argues that everything which we can successfully describe as intentional is in fact intentional. Accordingly, even computer programs can be intentional, since we can say that our computer beat us in chess or that our smartphone is "smart" and thus intelligent, because it keeps certain things in mind for us and is much better at solving certain problems than we are. Dennett thereby supports a crass version of consciousness rationalism, and thus it is also no wonder that he attempts to deny that there are qualia. This leads him to **qualia eliminativism** , the thesis that in reality there are no qualia, which results in the following conception of consciousness: "Conscious human minds are more-or-less serial virtual machines implemented – inefficiently – on the parallel hardware that evolution has provided for us."31 For Dennett, whether or not we are robots is not really relevant. Physically we are indeed like a vacuum-cleaning robot, because we then detach ourselves from the fact that we have intentional and phenomenal consciousness and take only physically describable processes into account. For Dennett, we are agents who reciprocally describe each other in the intentional stance, virtual programs who exist only because and for as long as we reciprocally describe each other in the intentional stance. Consciousness is then a kind of useful fiction, since the ascription of consciousness helps us to predict behavior correctly to some extent. This fiction would vanish if we had sufficient physical or neuroscientific knowledge at our disposal at all times, so that we no longer had to envision other human beings as intentional systems.
The absurdity of this position is exposed if we realize that it amounts to the claim that no one would ever have been conscious had no one described or experienced themselves as conscious, and the real question is how it is possible for Dennett to arrive at this idea. Obviously, this model is oriented by science fiction rather than by consciousness. Many positions in contemporary philosophy of consciousness generally suffer from scientism. **Scientism** is the claim that only knowledge which is arrived at by natural scientific research and accepted by a group of scientific experts counts as genuine knowledge. Since we have only a relatively small degree of scientific knowledge concerning the neurobiological foundations of consciousness, scientistic philosophers and scientists alike tend to compensate for their ignorance by considering science fiction to be an anticipation of future scientific knowledge. We know comparatively little about the neurobiology of consciousness, because research on the brain is still a relatively new science whose subject matter is almost infinitely complex by human standards. What complicates matters is that we currently have no idea as to whether there is extraterrestrial conscious life and, if there is, what kind of neurobiological foundations such beings might have. In addition, the brain is not just a network of neurons, as there are other cells there as well as chemical processes which do not function in a digital manner according to a binary code. Here every comparison to less complex systems we fully understand fails us entirely, so that it is precisely these areas that are a welcome hunting ground for science-fiction philosophy.
We know much more about the human mind and human consciousness than about their biological foundations. This sometimes bothers natural scientists and their philosophical PR spokespeople, because they might have to admit that the ancient Greek tragedians, Augustine, Hildegard von Bingen, the Buddha, Solomon, Lao-tzu, Sappho or Shakespeare already knew more about consciousness and the mind than most of our contemporaries (which happens to be true...).
But then one can no longer speak in a contemptuous tone of folk psychology and treat it at best as a useful virtual machine – that is, as a somewhat useful fiction. The self-knowledge of the human mind has long been much further advanced than the best scientific study of the neurobiological foundations of consciousness. To the extent to which scientists suffering from scientism intend to make a claim on advancement of knowledge in all areas, this fact has long been a thorn in their sides. Hence, the mind's actually progressive history of self-understanding since the dawn of language and writing is either simply suppressed or reinterpreted, in short, as a continuation of biological evolution by other means, as is propagated by Richard Dawkins and repeated by Dennett. However, what a genetic mutation or an environmental adaptation in the realm of nature, which includes us too, is supposed to have to do with Shakespeare or differential calculus is not immediately apparent.
## _**What Mary still doesn't know**_
Once again, the underlying problem can be illustrated with the help of an easily understandable and much discussed thought experiment that goes back to the Australian philosopher Frank Cameron Jackson (b. 1943).32 Jackson imagines a scientist by the name of Mary. She lives in the future and knows everything that can be known about nature with the methods of the ensemble of the best natural sciences. Maybe in Mary's future the attempt to reduce all other natural sciences to physics will be successful. The theory that expects something like this to occur is called **physicalism** , which claims that all knowledge of nature is physical knowledge.
One way of supporting physicalism is to rely on the assumption that everything that exists in nature consists of elementary particles whose behavior is accounted for by natural laws. This can be characterized as **microfundamentalism**. Since, according to this thesis, neurons or DNA consist of elementary particles as well, one might believe that in the future all biological (and thus also neurobiological) facts could be expressed in the language of physics. Of course, at the moment physicalism is at best a wildly speculative hypothesis. In any case, upon closer examination, physicalism is a quite unscientific presumption, and it boggles the mind as to how this presumption could be scientifically proven. It is certainly not a consequence of physics as we know it and, for instance, it is incompatible with many interpretations of quantum mechanics. In any event, we have nothing even close to scientific evidence for microfundamentalism. Rather, the very existence of different levels of complexity in various physical and biological systems (just to mention those broad categories) clearly speaks against it on the level of an adequate description of the universe itself.
Be that as it may, the idea we pursue is that Mary knows everything that there is to know about nature, and she expresses this exclusively in the language of physics (in mathematical equations). However, Mary is not entirely perfect because she suffers from total color blindness. She sees only shades of gray – that is, degrees of lightness and darkness. She lives in a world that is completely black and white. Incidentally, a case in which this happened to a painter, of all people, has been documented and studied.33 Other cases have been documented as well. Mary has thus never seen colors. But she knows that people speak of experiences involving colors and use words such as "green" and "red." She obviously knows, in addition, that these people call something "green" when their sensory receptors process information after being activated by electromagnetic waves in the range of 497 to 530 nm.
At this point, Jackson offers a litmus test for physicalism: Does Mary know everything or is her knowledge lacking in some way? His answer is that in fact Mary does not know everything. To be sure, she has every conceivable bit of physical information at her disposal, but nevertheless there are some things she does not know – for instance, what the green of the Colombian jungle looks like. She does indeed know that the Colombian jungle is overwhelmingly green (which she can figure out from measurements). But in knowing this she knows nothing about my experience of green during my first landing in Colombia, except that my rods and cones were activated while I had intact visual brain areas. If she were to have an operation that subsequently enabled her to see colors (which, given her omniscience regarding physics, should pose no problem to her!), she would come to know something about nature that she did not know before, to wit, what the green of the Colombian jungle looks like. Jackson concludes from this that physicalism is false. This is called the **knowledge argument** , since it is supposed to prove that not everything about nature can be _known_ in terms of physics, from which it follows that physicalism is wrong.
Many objections have been raised against this. In particular, the physicalist could indeed say that what is at stake is real knowledge about nature, which for different reasons just would not include knowledge about color experience based on having it. Yet, regardless of the question as to whether it effectively refutes physicalism, Jackson's knowledge argument has revealed an essential aspect of phenomenal consciousness. Let us stick here to the example of colors, whose nature is still widely debated in contemporary philosophy. One can even claim that modernity is a quarrel over colors, because since early modern physics the tendency has been to think that the universe is colorless, that colors are only a kind of illusory symphony in our organism that is played on the keyboard of our nerve fibers by electromagnetic waves.
In any case, we know that whenever we have an experience of color a physical event has occurred: photons must have hit our sensory receptors (whatever this might mean). In the case of _visible_ light, of course, a spectrum of electromagnetic radiation is experienced by us as being a certain way. Infrared and ultraviolet radiation are perceived by us not as colors but as heat (if, for instance, one is sweating in an infrared sauna), so that the entire spectrum of physical light is not visible.
But how do we know that visible light is a spectrum of electromagnetic radiation? And why do we distinguish this spectrum from other spectra? To put it quite simply: because we have experiences of color. Had we never perceived color, the thought would never have entered our heads that certain parts of nature can be described in such a way that visible light is a natural phenomenon. Our access to the physical realm is always inexorably mediated by our grasp of objects available to us in a phenomenal format.
Let us call this line of thought, in distinction to the knowledge argument, **the indispensability thesis**. This thesis claims that our subjective standpoint remains unaffected however the very best science progresses in the future. Our subjective standpoint is simply a condition for our access to the ideal of absolute objectivity. This certainly does not mean that we cannot achieve absolute objectivity in the sense of a grasp of how things are and would have been had we never been around to notice. It just means that our ways of coming to know these kinds of things presupposes that our thoughts be anchored in our conscious experience. We have to sense colors and tastes in order to be able to arrive at the thought that they represent excerpts from a physical reality, which has in itself different structures than those which characterize the experiential sphere of colors and tastes. Our theories are always irreducibly established on the basis of data that we subjectively experience as phenomena. In any case, one always has first to master the instruments to gather data on what is not directly accessible to our senses. To this end, one must use the senses as well as the vocabulary that we employ to orient ourselves in the "lifeworld," as Edmund Husserl (1859–1938) called the shared experienced world in his famous late piece from 1936 _The Crisis of European Sciences and Transcendental Phenomenology_. We cannot escape from the lifeworld – but why would this even be desirable in the first place? The fact that our thought about the human mind relies inexorably on the continued existence of subjective standpoints fortunately does not make it impossible for us to grasp how things are in mind-independent reality. On the contrary, the fact that we are capable of working out theories of natural necessary conditions for certain events to take place is as much a fact about human consciousness as is its subjectivity. Subjectivity does not stand between us and the world.
## _**The discovery of the universe in a monastery**_
Our classification of nature into visible and invisible processes depends on the fact that we have experientially grounded knowledge, without which we would not classify nature in this way at all. Physical classifications for this reason are a long way from being independent of the human subject, as is often pretended, and are only in part objective. The language of physics trivially reflects our knowledge interests, as there are many physical facts no one cares about. Let me just repeat that this does not mean that physics is not objective or that we do not know anything about physical reality as it is in itself.
It should be pointed out, though, that the contemporary physicalist image of the world is firmly built upon the well-established hypothesis that, within our cosmic horizon, and thus within nature as observable to us, nothing in space-time moves faster than light. As one of Einstein's famous discoveries shows, in this respect the speed of light is the absolute limit for all movement. Nothing within space-time is diffused faster than those electromagnetic waves which we call "light." This has been tested, and many revolutionary discoveries of modern physics follow from this hypothesis.
So far, so good. Yet one might wonder why living beings whose survival depends to a large extent on a sense of sight that functions halfway decently have developed a physics in which everything hinges on space-time and the diffusion of light. Our physics takes its starting point from our position on our planet and attempts from that vantage point to reveal the universe on both a large and a small scale. Crucial advances were made by instruments such as the microscope and the telescope: the microscope led to evidence that matter is composed of smaller units, and thus to the discovery of a micro-world, about which ancient philosophers had already speculated. In more recent times, the telescope led to the Big Bang theory. Edwin Hubble (1889–1953), after whom the Hubble telescope is named, first discovered in the 1920s that there were additional galaxies besides the Milky Way. He is considered as one of the discoverers of the expanding universe as well.
Incidentally, the fact that the Big Bang theory, as well as the theory of the expanding universe – aside from an anticipation in Immanuel Kant's work on the universe (in his _Universal Natural History and Theory of Heaven_ of 1755) – can be traced back to the Belgian priest and theologian Georges Lemaître (1894–1966), who had already formulated both theories some years before Hubble, is quite conveniently ignored by many popular physicalists. Remarkably, Einstein initially rejected Lemaître's Big Bang theory because it struck him as too strongly influenced by the Christian doctrine of creation, while the Catholic Church accepted Lemaître into the Pontifical Academy of Sciences for his discoveries. In one word, the Catholic Church accepted the Big Bang theory before the scientific community did.
This does not fit in so well with the historical misrepresentation attempted by some ideologues of science today, according to which modern science, above all physics, is supposed to have taken up the battle with religious superstition. Time and again one reads that Galileo or even Giordano Bruno – who in the sixteenth century already taught that the universe is infinitely large – were persecuted by the Church and concludes from this that theology and the Church stood in the way of progress as they stuck to their superstitions. But one then fails to note the fact that Isaac Newton considered the universe to be a sensorium of God (whatever exactly that might mean), that Bruno was and remained a Dominican who did not believe that his insight into the infinity of the universe would lead to atheism, and that Newton happened to be a creationist who calculated the age of the universe on the basis of the Bible. Newton was neither a physicalist nor any other kind of materialist. One fails to note the fact that the Big Bang theory was discovered by a monk, that Kant was deeply religious, and so forth. Error and superstition, accordingly, do not necessarily result from the fact that someone is religious, and these are by no means found only among religious people. Religion and science are not distinguished as easily from one another as are superstition and reason.
If one would like to view this historical misrepresentation in its pure form, one should watch the first episode of the remake of the popular documentary series _Cosmos_. The original version by Carl Sagan (1934–1996), which ran in the 1980s, made a large contribution in the United States to the dissemination of the scientific image of the world. The remake, whose first season ran in 2014 as _Cosmos: A Spacetime Odyssey_ , was hosted by Neil deGrasse Tyson (b. 1958). In the first episode, every conceivable error concerning Giordano Bruno, the Catholic Church and medieval "Italy" is on display, illustrated with the help of the Hollywood machinery of Seth MacFarlane (the creator of _Family Guy_ ). The remake is basically pure propaganda shot through with scientific fact (as most propaganda is!), and one might ask who is really supposed to profit from such historical nonsense. Bruno is presented as a stereotypical hero of science, who travels through medieval Europe and is persecuted by the powers of darkness. This image of the Middle Ages, which is prominent in the United States (think of _Game of Thrones_ ), and by which the United States continues to distinguish itself from Europe (or its fantasies of Europe), is about as accurate as Terry Gilliam's _Monty Python and the Holy Grail_ (1975) or _Jabberwocky_ (1977).
Let us return to consciousness. The crucial realization is that our knowledge of our own experience, thus our subjective knowledge, cannot be eliminated, improved or ignored by employing the language of physics. Like any other language, one understands this language only to the extent to which the technical vocabulary in which it is conveyed is understood. And this vocabulary includes words such as "color" or "time." Our understanding of the meaning of these words is naturally enriched by our discovery of which processes in the universe are connected to experiences of color or to our awareness of time. Yet it by no means follows from this that we could disregard our experiences of color or our awareness of time. Physical time and the time we experience are simply not the same, even though they are related to each other. However, the relation between physical and experiential time is not itself an object of scientific investigation. Even if psychology can reveal something about the experience of time, there is still a gap between Proust and empirical psychology which in principle cannot be closed.
That time passes is not something that can be physically understood. It has been a mystery of physics up to the present day just how there can be an "arrow of time" directed from the past into the future. In popular accounts, one sometimes reads that this is supposed to be explained by **entropy** – in simple terms, by the law that disorder always increases in physical systems. An ice cube has a specific order. If one puts it into lukewarm water, which is disordered relative to the ice cube's order, in time it takes on the disorder of its surroundings and melts. Sometimes physicists even let themselves get carried away and explain the always increasing disorder of a nursery as a case of entropy. My explanation of an untidy nursery is better, however: nurseries typically go from order to disorder because children do not respect our need for order. Children throw toys around, which is not a problem of physics – and thus not a case of entropy – but at most an issue for pedagogy, psychology or sociology.
Be that as it may, if one wishes to explain the orientation of time (the arrow of time) by means of entropy, one fails on a conceptual level, as one hides the notion of time passing in the notion of entropy. If entropy means that disorder is followed by order, then this presupposes that time passes and does not explain it. Recourse to entropy is only a pseudo-explanation which obscures the fact that one does not understand our awareness of time. On the other hand, it is of course not a good idea to say that everything is literally happening at the same time or that we will never understand how an irreversible timeline oriented from the past to the future could possibly exist. Even if the physical discoveries concerning time – in particular, relativity theory – are indeed spectacular breakthroughs with major impact on our understanding of our awareness of time, they cannot replace that very awareness or fundamentally explain it. How physical time and our awareness of time are connected is still a mystery that must be addressed by the philosophy of time.
## _**Sensations are not the subtitles to a Chinese movie**_
Let us go back to Jackson's knowledge argument. In my view, Jackson misjudges the scope of his thought experiment. He believes that he has refuted physicalism, but at the same time he sacrifices his hard-won insight on the altar of scientism, which is the real problem. In particular, Jackson additionally believes that qualia do not have any causal power in the universe. That now sounds somewhat overblown and once more obscures how erroneous it is. Hence we must proceed a bit further into his argumentative presuppositions.
**Epiphenomenalism** generally speaking is the thesis that mental states and processes as a whole have no causal effects on processes in the universe. Epiphenomenalists consider mental states to be pure concomitant features. Epiphenomenalism thus admittedly accepts that there are mental states and processes (at least). But it denies that they causally impact natural occurrences (a shame, since this spoils everything). The American philosopher John Searle (b. 1932), who manifestly rejects epiphenomenalism, brought things to a head with the following ironical description: "The brain is just a total mechanical hunk of junk, like a car engine only wetter, and... it functions by absolutely straightforward mechanical connections."34 Just picture in some detail what this way of thinking amounts to. If you are thirsty in the middle of the summer and your tongue is almost stuck to the roof of your mouth, you will try everything in your power to get your hands on a cold drink. You are conscious of an already unbearable thirst; your tongue and your throat feel dry. Now you make your way to the nearest kiosk. While doing so, you imagine how something like a bottle of Coke, a can of iced tea, or just a glass of cold water will be emptied in a few gulps. You stand before the refrigerated shelf and reach, anything but timidly, for a bottle of orange juice, say, which you had not even thought of before, but which evokes memories of a beautiful summer evening last year. One could even narrate this short story, which is not very demanding in literary terms, as an inner monologue, in which you as the first-person narrator chronologically describe your mental representations. One agony thus follows the next: thirst, joyful anticipation, the odd feeling that the individual mental representations of the beverage evoke in you, and finally the cold drink that runs down your throat.
Epiphenomenalism claims that none of this has anything to do with what actually happens in reality. In reality, what takes place is that your organism is deployed into internal states of information, which ultimately traces back to the fact that certain patterns of neurons are firing. This firing of neurons triggers all of the impressions that manifest as qualia to you while in the mode of consciousness. Yet regardless of how it feels, your organism moves to the kiosk, guided only by the laws of nature. Photons are deflected by the kiosk (otherwise it would be invisible) that are again picked up by your organism and, mediated by processes that appear to be unbelievably complicated, lead to the fact that you are drawn to the kiosk.
The actual causal story could never really be told in full detail, since it contains a virtual infinity of information: all of the elementary particles of which you, the kiosk and your surroundings consist, and because of which your organism moves. Basically, on this line of thinking, the whole universe can be traced back to the fact that a chain of events takes place, which is supposed to lead in a strict causal sense (and thus, above all, without alternative and of necessity) from one cause to an effect that is in turn another cause of another effect, and so on. Thus qualia come about as mere sideeffects which do not contribute anything to what really happens. Qualia are not supposed to be causes that lead to effects in their train; rather, they are only an incidental rush of sensations. Just as the English subtitles to a Chinese movie do not contribute anything to the movie's plot (indeed, they do not belong to the world of the movie but are just there for us because we do not understand Chinese well enough), so qualia simply run alongside what really happens.
Infected by a common case of Darwinitis, Jackson wonders how the existence of epiphenomenal qualia can be made compatible with the theory of evolution. If qualia do not contribute anything to the survival of the organism, why are they around? Why is there consciousness at all, if things could have taken place without it? Were microbes or bacteria not enough? Would it not be enough for survival, much better or at least as good, if we human beings had no feelings of pain, or if the organism dealt with everything unconsciously – not merely digestion or the growth of fingernails, but simply everything? At this point, Jackson surmises that qualia "are a byproduct of certain brain processes that are highly conducive to survival."35
His other example for such evolutionary byproducts speaks volumes. Polar bears are weighed down by an extremely heavy coat. One could ask what evolutionary advantage is entailed by such a heavy coat, as it apparently makes survival harder (carry that around!). It quickly becomes apparent that a heavy coat is hardly an advantage, since their weight slows one down (although polar bears actually are alarmingly fast, up to 30 km/h!). In any case, one must somehow compensate for such a heavy coat (with strong muscles, for instance). However, according to Jackson, the fact that the polar bear's coat is heavy is simply a byproduct of the fact that the coat keeps the animal warm. This is the case for many qualities that emerged in the course of evolution. One must choose the right aspects in the description of a phenotype if one is to carry out a successful zoological study. What Jackson thus claims is that qualia should not play any real role in a zoological study. They are more akin to the heaviness of the coat than to the fact that such a coat keeps the creature warm.
Epiphenomenalism here is in line with the tradition of early modern thought. Descartes above all is (in)famous for his thesis that animals are fundamentally automatons, and Julien Offray de La Mettrie (1709–1751) developed a corresponding image of the human being in his book _Man a Machine_. Jackson slyly suggests that a large part of our behavior is not guided at all by the fact that we have qualitative experiences, a few of which we avoid (usually, pain, at least aside from S&M play and self-destructive behavior) and others at which we aim (usually, the satisfaction of desires and inclinations of all kinds). However, these qualia must never be seen to interfere with the mechanism of what happens, otherwise one would no longer be an epiphenomenalist.
Behind this whole debate over epiphenomenalism lies the belief that the universe is causally closed and does not actually contain any qualia. In the background is the idea that everything which happens in nature is determined by the laws of nature. Each event in space-time follows upon another event in space-time according to the laws of nature, to which there are no exceptions. This thesis is known as **determinism** and is examined in detail in the chapter on freedom that concludes this book.
At this point, I expect that some readers would like to object to such an image of causation and the laws of nature being modified, if not superseded, by quantum physics (or more sophisticated philosophical theories of causation, for that matter). Yet whatever kind of causality holds at the quantum level is irrelevant to our investigation to the extent to which it remains unclear how consciousness and the quantum level are related. To be sure, there is speculation over the fact that processes occur in so-called microtubules (certain protein structures that are found in cells) that can be described in terms of quantum mechanics, through which consciousness might be generated. For instance, a version of this has been proposed by the famous British mathematician Sir Roger Penrose (b. 1931). But, strictly speaking, the retreat to genuine or putative chance and indeterminacy in physically describable nature does not contribute anything to the understanding of the mind. There is no such thing as a quantum theory of consciousness, as the kind of object actually characterized by quantum theory is precisely not conscious mental activity or intentional thought but, rather, anonymous processes subject to laws of nature which differ from classical mechanical laws – that's all.
The idea that we are situated in an enormous container, in which lumps of matter that are invisible to the naked eye move to and fro according to inviolable natural laws that can be mathematically formulated, is indeed for many reasons a kind of fairy tale. Let us give this idea a name, too: **the fairy tale of the container**. This fairy tale draws on a world-view that rests on numerous unsupported claims. Again, at best it is a philosophical interpretation of physics, but never an actual result of physics per se.
Contemporary theoretical physics invites all manner of speculation. In particular, with regard to the various hypotheses of the multiverse, the question arises as to whether there are universal laws of nature that prevail everywhere in the space-time continuum, or whether there might not be universes which cannot even be conceived of in terms of space-time as we know it (or believe to know it). For an overview of the often wildly metaphysical conjectures at the frontiers of physical knowledge, I recommend Brian Greene's _The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos_.36
In my earlier book _Why the World Does Not Exist_ I argued for the claim that it is absurd, in any case, to believe that there is an all-encompassing world that includes everything which really exists. There simply is no such thing as an all-encompassing singular reality. To believe in reality as a whole, or in the world, is the remainder of an ancient idea that arose among us human beings a couple of thousand years ago as a side-effect of the misguided impression that our habitat (earth) is somehow enclosed by a sphere. This impression has still not completely fizzled out. In our time, it takes the shape of the notion that the (physical) universe is all there is and that the universe is some kind of total sum or matter/energy spread out in a certain way according to laws of nature forged in a ridiculously small amount of time right after the Big Bang (fill in the details with your favorite overall cosmology).
I leave all of this aside, since in any case I do not think that wandering conceptually into the depths of the cosmos is particularly helpful if we want to understand whether our thirst on a hot summer afternoon offers a genuine contribution to what happens on earth. However, it is important to bear in mind that our overall conception of reality as a whole (if any) has immediate consequences for our conception of ourselves as minded creatures within that whole.
A large part of human life (which includes ethics and politics) can only be understood when we take the reality of qualia into account. If I want to explain, for example, why Hans went to the kiosk, I will not be able to avoid mentioning that he was thirsty. Surely there might be numerous other reasons why he went to the kiosk: he wanted to meet Joe, who works there; he hoped that Josephine, with whom he is secretly in love, would be there; he wanted to buy a magazine. No external description of his behavior, let alone a description of the distribution of mass/matter/energy in some region of the universe, will be a better account of why Hans does what he does than the account which rightly mentions his motivations. And Hans would have no motivations to do anything whatsoever if he were not an animal of a specific kind endowed with sensibilities which provide him with a constantly changing flow of qualia.
The idea that the universe is nothing but an "interlocking mechanism,"37 as Keil has fittingly called it, and that our qualia or even our consciousness as a whole contribute nothing in causal terms to the cosmic works, in reality just provides an imaginary relief. It is an attempt to avoid confronting the complexity of human action. If it makes no difference whether Hans is on his way to get a soda, or to see Joe or Josephine, one has already taken the easy way out of human action explanation. The situation is continually complicated by our qualia, and indeed for everyone involved.
What if Hans, for instance, is a closet homosexual, since he has the misfortune of coming from a corner of a city that is homophobic and does not want to admit that he really likes Joe rather than Josephine? A psychoanalyst would be delighted to investigate this and would perhaps surmise that it is no accident that Hans is convinced that he likes Josephine, since her name is similar to Joe, the name of the real object of his affection.
The image of the world as a universe that is completely structured according to laws of nature, a view often called determinism, is an unproven hypothesis in almost every regard. The claim that everything which happens does so because there are laws of nature does not follow from the fact that there are laws of nature. When humans speculate, they quickly tend to get lost in metaphysical ideas about reality as a whole. The idea that there is just one world which may or may not be identical with the (physical) universe is metaphysical poppycock, no better than the idea that we live on a planet which is encompassed by a cluster of spheres which revolve around us according to the wishes of the gods. When faced with the infinite, events in everyday life seem overwhelmingly negligible. Yet this impression is only an effect of our perspective, and it does not give us the truth about whether human freedom or even human consciousness is an illusion. In a nutshell, we should not get lost in our ideas of the vast, maybe infinite regions of the universe hitherto undiscovered and feel small and unimportant compared to such a conception. The impression that our innermost mental states contribute nothing to what is going on in causal terms is really a form of conceptual or philosophical mild form of depression.
## _**God's-eye view**_
In the absence of further elucidation, the question of how the human mind or human consciousness in general actually fits in with nature or the universe is not well formulated. On closer inspection, it is about as meaningful as the question as to whether the moon is a square root. All kinds of things can be imagined in regards to it, yet one should not be fooled by one's own imagination. The question concerning the relation of nature and mind ordinarily draws on far too many assumptions. For the most part, it proceeds from an image of the world and of the human being that can be exposed as providing imaginary relief.
In modernity, one finds, on the one hand, the representation of a universe without mind, which one then, on the other hand, complements with mind. The mind, as it were, illuminates blind nature without anyone knowing how this works. As a result, the universe, or nature, feels like a "cold place to call home," as Wolfram Hogrebe puts it.38 That is, the naturalistic world-view still describes one's own abode, one's home, but robs this place of all its magic, which gives the impression that things have a firm, entrenched structure, albeit one which does not respect our craving for meaning. One's home is retained, but it is now as cold as the apparently black and sepulchral space outside our atmosphere.
The human mind produces images of itself. It thus transports us from the internal perspective of our everyday experience to the bird's-eye view. This is also the origin of the ancient idea that God is a kind of all-seeing eye, which in real historical terms leads to Google's image of the human being, something that is compellingly presented in Dave Eggers's novel _The Circle_. One major problem with God, indeed, was that one did not easily measure up to his omniscience. Google is a bit more helpful, even though it serves the same type of imaginary function of giving a determinate place to everything and of imagining that someone must know where and how things are.
Currently, there is a new ideology on the horizon attached to the notion of the infosphere. It sometimes seems as though many of us would like nothing more than actually to be perpetually observed and controlled, as this corresponds to the age-old fantasy of a God who takes away our freedom and forces us to obey his laws. But, for the most part, an actual God is no longer welcome today in our modern scientifically oriented society, since he is traditionally described as supremely free and a rather fearsome guy (one does not want to run into him in the desert, as the poor prophets do). With an actual God, freedom would again be in play – at least for him.
In ideological terms, human freedom is a potentially disruptive factor: if I am free to choose my own search engine, Google's profits might decrease. The tendency to form monopolies is constitutive of present-day market conditions. To be sure, we all want to have a choice (one does not want to see only sausages and pickles on the supermarket shelves, as in good old socialism). From a business perspective, it would be best for all consumers always to choose only a single product and hence not really to have a choice at all. Hence the fantasy to make humanity computable and to restrict freedom in favor of clearly marked paths. Humanity still quarrels over who is adopting the real bird's-eye view, over whom or what is considered its God: science, technology, progress, Google or, for the more classically minded, a personal actual God. In all these cases, the frame of mind, the fundamental fantasy, remains the same – that is, the fantasy of a God's- or bird's-eye view which knows that everything is always already in order and guarantees that nothing goes unnoticed.
## **Notes**
1. Thomas Nagel, _Mind and Cosmos: Why the Materialist Neo-Darwinian Conception of Nature is Almost Certainly False_ (Oxford: Oxford University Press, 2012). See my review of this, " _Da schlug die Natur die Augen auf_ ," in the _Frankfurter Allgemeine Zeitung_ of October 7, 2013. 2. Michael Patrick Lynch: _The Internet of Us: Knowing More and Understanding Less in the Age of Big Data_ (New York and London: Liveright, 2016). 3. Immanuel Kant, _Dreams of a Spirit-Seer_ , trans. Emanuel F. Goerwitz (London: Swan Sonnenschein, 1900), p. 49. 4. Ibid. 5. Ibid., p. 51; translation modified. 6. Geert Keil, _Willensfreiheit_ (Berlin: De Gruyter, 2012), p. 208. Keil understands the homunculus fallacy somewhat differently than I do. He writes that a homunculus is "a human-like entity that is posited in recent philosophy of mind, which is either explicitly or implicitly called upon to explain the human mind's way of functioning" (ibid.). 7. Thomas Nagel, "What is it Like to be a Bat?," _Philosophical Review_ 83 (1974), pp. 435–50. 8. Aristotle, _On the Soul; Parva Naturalia; On Breath_ (Cambridge, MA: Harvard University Press, 1957), p. 73 (413a). 9.Ludwig Wittgenstein, _Philosophical Investigations_ , trans. G. E. M. Anscombe (Oxford: Blackwell, 1958), 94e. 10.Immanuel Kant, _Groundwork for the Metaphysics of Morals_ , ed. and trans. Allen Wood (New Haven, CT: Yale University Press, 2002), p. 67 (Ak 4:450–451). 11. Hermann von Helmholtz, "Über das Sehen," in _Abhandlungen zur Philosophie und Geometrie_ (Cuxhaven: Traude Junghans, 1987), p. 21. 12. Eric R. Kandel, James H. Schwarz and Thomas M. Jessell, _Principles of Neural Science_ (4th edn, New York: McGraw-Hill, 2000), p. 412. 13. See, for example, the following statement by Krauss: "the ultimate arbiter of truth is experiment, not the comfort one derives from one's a priori beliefs, nor the beauty or elegance one ascribes to one's theoretical models." Krauss, _A Universe from Nothing: Why There is Something rather Than Nothing_ (New York: Free Press, 2012), p. xvi. 14. Richard Dawkins, _The God Delusion_ (Boston: Houghton Mifflin, 2008), p. 177. 15. Ibid., p. 13. 16. Lynne Rudder Baker, "Cognitive Suicide," in Robert H. Grimm et al. (eds), _Contents of Thought_ (Tucson: University of Arizona Press, 1988), pp. 1–18; <http://people.umass.edu/lrb/files/bak88cogS.pdf>. 17. Paul Churchland, "Eliminative Materialism and the Propositional Attitudes," _Journal of Philosophy_ 78/2 (1981), p. 88. 18. In her book of the same name: Patricia Churchland, _Neurophilosophy_ (Cambridge, MA: MIT Press, 1989). 19. See, for instance, the interview with both in Susan Blackmore, _Conversations on Consciousness: What the Best Minds Think about the Brain, Free Will, and What it Means to be Human_ (Oxford: Oxford University Press, 2006), pp. 50–67. 20. _Caroli Linnaei Systema naturae: A Photographic Facsimile of the First Volume of the Tenth Edition, 1758_. London: British Museum, 1939. 21. Plato, _Euthyphro. Apology. Crito. Phaedo. Phaedrus_ (Cambridge, MA: Harvard University Press, 1982), p. 81 (20e). 22. Donald Davidson, "Rational Animals," _Dialectica_ 36 (1982), pp. 317–27. 23. Geert Keil, _Willensfreiheit_ (Berlin: De Gruyter 2012), p. 159. 24. Jacques Derrida, _The Animal That Therefore I Am_ (New York: Fordham University Press, 2008), p. 3. 25. Blackmore, _Conversations on Consciousness_ , p. 27. 26. René Descartes, _Meditations on First Philosophy_ (Cambridge: Cambridge University Press, 1986), p. 21. 27. Christoph Kucklick, _Die granulare Gesellschaft: Wie das Digitale unsere Wirklichkeit auflöst_ (Berlin: Ullstein, 2014), p. 67. 28. Ibid., p. 90. 29. Kant, _Critique of Pure Reason_ , trans. Paul Guyer and Allen W. Wood (Cambridge: Cambridge University Press, 1998), pp. 193–4 [A51/B75]. 30. Ibid., p. 235 [A112]. 31. Daniel Dennett, _Consciousness Explained_ (New York: Back Bay Books, 1991), p. 218. 32. Frank Cameron Jackson, "Epiphenomenal Qualia," _Philosophical Quarterly_ 32/127 (1982), pp. 127–36. See also "What Mary Didn't Know," _Journal of Philosophy_ 83/5 (1986), pp. 291–5. 33. Kandel et al., _Principles of Neural Science_ , pp. 586–7. 34. Blackmore, _Conversations on Consciousness_ , p. 206. 35. Jackson, "Epiphenomenal Qualia," p. 134. 36. Brian Greene, _The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos_ (New York: Vintage, 2011). 37. Keil, _Willensfreiheit_ , p. 41. 38. Wolfram Hogrebe, _Riskante Lebensnähe: Die szenische Existenz des Menschen_ (Berlin: Akademie, 2009), p. 40.
# 3
**Self-Consciousness**
Having dealt with some crucial aspects of consciousness naturally leads us to our next major topic: self-consciousness. In examining consciousness, we have adopted an odd perspective. We were not only conscious of certain kinds of things in our environment, as usual, but we also consciously reflected on consciousness. Most of us are not normally worried about this every day, and nobody spends her entire conscious life dedicated to the fact that she has a conscious life – with the possible exception of the truly enlightened, if such there be. Yet, it is desirable to spend some time reflecting on consciousness, at least in a context in which we are looking for a mark of the mental and deem consciousness special in that regard. Tyler Burge, a leading American philosopher of mind, nicely sums up the relevance of self-consciousness thus: "The claim that one has a conception of mind-independent entities _as_ mind-independent entails that one has a concept of mind. An ability to hold that physical entities are independent of one's mind, and everyone else's mind, requires a capacity for self-consciousness."1 Thus, to know that some things are real, and that other things are products of our minds, presupposes a grasp of the concept of consciousness and, therefore, self-consciousness. Self-consciousness is a central feature of our lives without which we would constantly confuse perception and hallucination, "inner" mental states and objects "out there."
One of the most remarkable facts of consciousness is that one cannot really deny that one is conscious while being conscious. Admittedly, we have already seen how some have attempted to do precisely this. This insight into the irreducibility of consciousness even lurks behind what is probably the most famous sentence of modern philosophy, namely Descartes' infinitely cited "I think, therefore I am." In his _Meditations on First Philosophy_ , he expresses this thought as follows:
> Thinking? At last I have discovered it – thought; this alone is inseparable from me. I am, I exist – that is certain. But for how long? For as long as I am thinking. For it could be that were I totally to cease from thinking, I should totally cease to exist. At present I am not admitting anything except what is necessarily true. I am, then, in the strict sense only a thing that thinks; that is, I am a mind, or intelligence, or intellect, or reason – words whose meaning I have been ignorant of until now. But for all that I am a thing which is real and which truly exists. But what kind of thing? As I have just said – a thinking thing.2
One easily falls prey to a conceptual confusion on the basis of this citation. The confusion consists in mistaking the fact that, while one cannot be deceived about one's being conscious, one can be very much deceived by one's opinions about consciousness. Consciousness itself is not generally or in all cases the best guide to the nature of consciousness. For instance, my conscious experience of colors brings with it many riddles that cannot be solved simply by observing my consciousness by staring at colors. As long as one is conscious it cannot be doubted _that_ one is conscious, from which it does not follow that one thus has indubitable knowledge of the nature of consciousness. It is actually quite difficult to distinguish the contribution of consciousness in conscious experience from the contribution of mind-independent reality, which is why there is philosophy.
Hence, for centuries the following reasonable objection has been raised again and again against the Cartesian cogito: Descartes reifies consciousness; he thus makes it into a thing or a substance, the thinking substance ( _res cogitans_ ). This objection, which was formulated in especially clear terms by Kant, then by Fichte and, later, above all by Husserl and the philosophical movement known as phenomenology which followed him, draws our attention to the fact that Descartes conflates the certainty that we cannot be deceived about whether we are conscious with the claim that we thus also have insight into a particularly spectacular thing in the universe: consciousness. To be more precise, the objection runs as follows. Descartes conflates an _epistemological_ insight, an insight that characterizes our form of cognition (our infallibility with regard to our being conscious) with a _metaphysical_ insight into the structure of the universe.
Yet this fallacy secretly forms the basis even for neurocentrism. Neurocentrism departs from the assumption that there is a thing, consciousness, whose properties have to be more precisely investigated so that we might eventually answer the question of how this thing relates to material things in the universe, above all to our brain. Neurocentrism's response is that consciousness is not something in addition to the brain-thing, and for this reason neurocentrism identifies consciousness with it. It turns consciousness into a brain-thing. It thereby fares no better than Descartes' identification of consciousness with a consciousness-thing.
The question of how the brain and consciousness are related to each other was incidentally also already formulated by Descartes. He thought that contact between immaterial consciousness ( _mens_ ) or soul ( _animus_ ) takes place in the pineal gland (a part of the vertebrate brain involved in the production of melatonin in the sleep–wake cycle).
Following Descartes, two major options concerning the mind–brain relationship were offered. **Dualism** claims that, along with brain-things, there is also a consciousness-thing in the universe, while **monism** denies this. **Neuromonism** claims that the consciousness-thing is identical either to the entire brain or to some areas of the brain and their activities. However, both positions presuppose that consciousness is a thing in the universe, which is the crucial mistake.
Self-consciousness is consciousness of consciousness. We could not be concerned with this fact without at the same time being conscious: we are conscious while we concern ourselves with consciousness. Self-consciousness is a special state that deserves some scrutiny, as it promises to be a key element for our understanding of human consciousness.
Despite its special status, self-consciousness is a much more mundane phenomenon than one might initially suppose. We continually take a stance toward our own mental states, toward our thoughts, without explicitly reflecting on the fact that we are reflecting on our consciousness. Our stream of consciousness does not simply run like subtitles beneath our actions, as though we were sometimes additionally focusing our mind's eye on the processes in our consciousness.3 Since Kant, many philosophers of the past and present have pointed out that our consciousness would thus oddly be made into something unconscious. For this reason, to the present day the Kantian view that there is actually no consciousness without self-consciousness is widespread, as consciousness and self-consciousness seem to be somehow inextricably interwoven. In any event, if consciousness were only accidentally related to self-consciousness, it would be hard to account for the fact that we constantly experience consciousness itself in experiencing anything through consciousness. This is why it is not hard to focus on consciousness, as it is already present and available for further reflection. It makes no sense to assume that consciousness is like any old object out there in the world, as it is the only object which makes itself known to us as knowers without mediation by consciousness. Consciousness is available to consciousness without a further consciousness mediating between consciousness and consciousness. If this were not the case, we could never achieve consciousness of consciousness. This would destroy the entire discipline of a philosophy of mind or any other knowledge of ourselves as minded creatures.
Let us make things a bit more concrete. We continually evaluate the thoughts and impressions that we consciously experience. They do not suddenly appear in the obscure field of our consciousness and then vanish again, but they are immediately consciously experienced by us in such a way that they bear a stamp. For instance, it occurs to us that we still have to write an email to a friend. As this occurs to us, we feel immediately ashamed that we have not yet replied. Perhaps it thus dawns on us that we really do not wish to reply to her at all, since her last email had strange undertones. In any case, our friendship is on shaky ground and we would prefer not to reply. The consciousness of the fact that we have not written a reply, which can suddenly arise in the midst of an entirely normal situation, implies that we go around evaluating our thoughts – we are ashamed – without being able actively and consciously to control them. For its part, shame comes in many shapes and sizes, if you will. There are degrees of shame, which shrewd observers of the inner lives of human beings have illuminated in countless novels. Our conscious experience is always at the same time an experience of our conscious experience in light of certain aesthetical, ethical or purely biological values. If you become aware of a lion running toward you, this perceptual experience should trigger flight behavior. The conscious experience of a lion is judged in a certain way if it provokes the right response in you. This is self-consciousness in action.
Of course, we can be delighted or annoyed by a thought or a sense-impression. We experience a beautiful sunrise and are delighted by it. Perhaps it is connected to memories of other sunrises. We enjoy the taste of a brioche. We are annoyed by the watered-down coffee we have purchased at the train station. We clench our fist in our trouser pocket when considering that our favorite sports team is about to lose again, and so forth.
A large part of our emotional world, thus, has the form of self-consciousness. This does not mean that we explore our inner world from an internal vantage point; rather, it means that intentional and phenomenal consciousness are fundamentally intertwined. As mentally more or less healthy human beings, we do not have any states of consciousness without their already appearing to us to be evaluated in a certain way. On this basis, it becomes clear why our moral values are connected to our world of emotions, since we first become capable of morality when we are conscious that others are conscious too and experience this structure as a system of values that is to be perpetually calibrated anew.
## **How history can expand our consciousness**
The modern history of research into self-consciousness reaches a major peak in Marcel Proust's epoch-making masterpiece _In Search of Lost Time_. Literature, perhaps paradigmatically narrative literature and poetry, has for millennia contributed to our achievement of a better understanding of consciousness. In the course of the mind's history, humanity has become more conscious, if you will, insofar as for thousands of years we have no longer merely had attitudes toward our natural environment and vague attitudes toward ourselves and others. Rather, at the latest since the advanced cultures in Mesopotamia, a subtle language has developed that enables us not merely to explore the fine-grainedness of our consciousness but to deepen it. It is hardly an accident that the scientific discipline of psychology, which, to the present day, has continually vacillated between the natural sciences and the humanities, emerged in the second half of the nineteenth century in the wake of the literary ascendancy of self-exploration. Nietzsche, for instance, was able at this time to be simultaneously one of the best writers in the German language (as both a prose stylist and a lyrical poet) and a critical psychologist of our moral attitudes, just as Freud around the same period achieved breakthroughs in psychoanalysis and developed a profound understanding of mythology, art and literature, to which even someone like Eric Kandel has recently called attention.4 Freud too was a first-rate writer, which is certainly a major part of his cultural impact beyond academia.
The history of the mind is thus among other things a history of the expansion and transformation of consciousness. Consciousness is not a thing that was always already precisely exactly what it is. We do not discover it in nature as we do neutrons, chestnuts and rainstorms. "Consciousness" is a concept that belongs to our self-portrait – that is, the word "consciousness" does not simply pick a natural kind or object in nature. Consciousness is connected to natural phenomena (no human consciousness without a functioning brain), but it is neither identical nor in any other meaningful way reducible to them.
We describe ourselves as conscious living beings with minds and thus produce a certain image of ourselves. Consciousness is not a reality that exists completely independently of how we form a concept of it. This makes it different from a crater on the moon, for example. Of course, other living beings, or even children who have not yet been culturally socialized, also have states of consciousness, such as impressions of color, auditory impressions and impressions of taste. However, whether one can identify Yves Klein blue on first sight in a museum, and knows how to bring a framework of allusions to the history of art to bear on it or not, makes a crucial difference even on the level of conscious perception. This cultural and historical potential for classifying sense-impressions fundamentally changes our consciousness. There is a gradual and progressive formation of consciousness – again a great theme in literature – which was treated in the genre of the _Bildungsroman_ , from which arose Hegel's _Phenomenology of Spirit_. Hegel's great work examines the formation of consciousness in a philosophical way by investigating different forms or shapes of consciousness that have become differentiated over the course of millennia. His basic idea is precisely that consciousness itself can change according to our conception of it. This is why there can be such a thing as psychotherapy where you can learn to change your consciousness by changing your concept of it. If you realize, for instance, that certain negative thoughts come to your mind only because you belittle the role played by your conscious experience of social situations, you are in a position literally to change your consciousness for the better. The concept of consciousness is connected to that of self-consciousness, which for its part can have a history in which new facets that were not there before are cultivated.
The findings of paleoanthropology, which are popular today and supposedly obtained in a strictly scientific manner, often invite people to make big claims about everything that our ancestors in caves experienced and how morality and civilization are supposed to have gradually developed in the savannah, making it seem as though one might have a glimpse into the inner life of our prehistoric ancestors. For instance, such a glimpse might be obtained by forming ideas of how tool use, burial rites and inner life are related. But even tool use has a history and is embedded within the history of the mind. There is no overall relation between tool use and the mind, as there is no stable entity called "the mind." What we share with our minded ancestors is only the capacity to create self-images and not necessarily the same self-images. As a matter of fact, without written documents or oral evidence we cannot know anything about the minds of our ancestors. Visual imagery is insufficient as long as one knows nothing about the function of the images, for which reason the impressive cave paintings of our ancestors will forever remain a fundamental mystery to us, since we simply do not have any reliable evidence that would enable us to figure out what was really going on. Any substantial hypothesis (be it a reading of cave painting as art, maps or religion) is unsubstantiated, albeit amusing, speculation.
Depending on how one envisions the manner in which a subtle inner life arose in preliterate epochs, it can be surprising to discover what level the first literary works of advanced Mesopotamian cultures reached – for instance, the _Epic of Gilgamesh_ or the lyric poetry (in the form of temple hymns) of the Assyrian high-priestess En-Hedu-Anna, who wrote in the third millennium BC (in the area of present-day Iraq, the cradle of advanced culture) and who was possibly the first person (at least, as far as I know) to sign her name to a literary text.
The first written evidence of advanced cultures certainly did not fall from heaven but, rather, suggests that it was preceded by a long prehistory of self-conscious inner life, which is accompanied by complex self-depictions that were orally transmitted and refined over a long span of time – as with the Homeric epics the _Iliad_ and the _Odyssey_ , or the much longer Indian epics _Mahabharata_ and _Bhagavad Gita_. One should also not overestimate our "Western" subjectivity and capacity for self-depiction and self-knowledge, just because we happen to live at this moment and are proud of the way in which our technological achievements have been arranged.
Unfortunately, there is a widespread tendency to look anachronistically for contemporary forms of self-consciousness in the scenes depicted in cave paintings, although it must have been clear, at the latest since sociology emerged in the nineteenth century, that there is a history of self-consciousness. This idea was first formulated quite clearly by the post-Kantian philosophers Fichte, Schelling and Hegel, the so-called German Idealists. It forms the core of Hegel's _Phenomenology of Spirit_ , in which he wants to establish that the human mind, _Geist_ , considered as a whole, has a history that cannot be understood if one conceives of it as a continuation of biological relations to the external, natural world by means of consciousness.
Unfortunately this history – like every other real history – does not simply proceed in such a way that we always make further progress in ways that leave a trace. That literary works or languages fade into oblivion is enough to cause self-consciousness to regress to stages that have long been rendered obsolete. A reason for this lies simply in human freedom, which likewise consists in being able to forget, overlook or repress something.
However, it is crucial that we are not simply conscious but that we also have a consciousness of consciousness, even in everyday contexts. We do not know this merely because we take up attitudes to our own beliefs, wishes, feelings or sense-impressions, attitudes that we cultivate and develop. We can become wine connoisseurs and gourmets. We can learn to recognize Yves Klein blue at first sight. We can become judo masters. We can classify a brief sequence of chords as typical of Beethoven or Nirvana. We can familiarize ourselves with our own patterns of reaction, and so forth. In addition to all of this, we are immediately conscious of the fact that others are conscious. Consciousness of the fact that others are conscious is just as primordial a fact as consciousness of ourselves. To borrow a term from Martin Heidegger: consciousness and consciousness of the consciousness of others are equi-primordial [ _gleichursprünglich_ ]. Self-consciousness is not only directed toward ourselves as individuals but is already structurally connected to the consciousness of others. The judgments belonging to our consciously experienced attitudes (such as being delighted or ashamed about something) are oriented to our expectation that others take up attitudes to our attitudes. From the outset, we experience ourselves in social contexts and do not first need to find our way into them by leaving our own private Cartesian theater behind. Each of us experiences this primordial sociality of consciousness, for instance, in the well-known phenomenon of being only too happy to share wonderful experiences (such as a trip abroad or a visit to a museum), since only in this way can one expect confirmation of the structure of these experiences. We cultivate our consciousness in light of the criterion of the consciousness of others, a criterion that has already been inscribed into our inner life the moment that we are able to put anything at all into language. This is actually quite obvious, insofar as we must have learned the rules of our language from others, which already imposes a discipline on our individual consciousness so that we are no longer able arbitrarily to let our thoughts wander (like a small child who calls everything that pleases it and is to its liking "mama" or everything that moves "car").
Our consciousness of the fact that others are conscious was recognized as primary by Kant and then recognized even more explicitly as the basis for human society by Fichte and Hegel after him. Marx and the founding fathers of sociology and psychology have contributed to this idea in their own respective ways. Contemporary feminist theories, in particular Judith Butler's gender performativity theory, are also based on this idea, and Butler quite explicitly invokes Hegel.
In times of neurocentrism, the capacity to attain consciousness of the fact that others are conscious is called a "theory of mind," which is a misleading expression, since the word "theory" is here used in an improper way. Since the 1990s, it has also been believed that the biological basis for this phenomenon has been discovered. So-called mirror neurons are accordingly supposed to see to it that we reproduce, as it were, the behavior and feelings of others through the firing of neurons and the internal simulation of the conscious attitudes of others. One feels a shadow of pain, so to speak, for a person writhing on the ground, because certain neurons emerged and lasted in the course of evolution (in other words, random genetic mutation, not history, is supposed to explain the vagaries of self-consciousness).
I am certainly not in the business of denying that there are mirror neurons, nor do I even want to deny that, without them, we would not have been able to attain either self-consciousness or the consciousness that others are conscious. No more do I deny that important biological foundations of human mental states have ultimately emerged through random genetic mutation – that is, not in a goal-directed or planned way – and were then selected due to environmental pressure. That there are conscious living beings on our planet, and perhaps elsewhere, too, is a matter of chance, when viewed from the perspective of the universe. That is, there are no particular reasons for it. However, from the perspective of the universe there is no particular reason for anything whatsoever; in the universe, there simply is what there is.
All this means, though, is that we are able to form an image of the universe and its sheer infinite expanses, in which we, with our consciousness and the needs and values associated with it, play no constitutive role. The universe, understood in this way, is of course not concerned about us. For all we know, the processes that result in the emergence of species, which we call "evolution," are not consciously or intentionally guided. There is no deeper biological reason for the fact that beings like us exist beyond the fact that animals like us are simply here now and the fact that our biological emergence was possible, because certain functional structures were selected in time immemorial and were passed on through reproduction. Our existence as animals and its continuation depend on our adaptation to an extremely fragile natural environment, an adaptation that can come to an end at any time as a result of a natural or humanly created catastrophe. The universe, our planet or nature will not see to it that we survive when the composition of our atmosphere changes (whether because of man-made global warming, a meteor crashing into earth, or a volcanic eruption) in such a way that we can no longer breathe. Yet, none of this speaks in favor of an identification of mind and brain.
Insofar as human beings are prone to misjudge their own position as minded, historical animals because they do not know much about the universe, the progress of our discoveries in the field of neurobiology is obviously conducive to progress in self-knowledge. Whether one believes that the sun shines so that we may see, or that the fact that we are conscious and self-conscious is a kind of symphony of the mind which is arranged for the glory of God, who literally views the performance "down here" from a distance, is in fact quite beside the point. Yet it would be a misunderstanding if one thought that all of our recently acquired neurobiological knowledge would give us a better understanding of self-consciousness than what was at the disposal of En-Hedu-Anna, for instance, or Homer and Sophocles, Nagarjuna, Jane Austen, Augustine, Hildegard von Bingen, George Sand, Hegel, Bettina von Arnim or Marx. Knowing more about the neurobiological conditions of the mind contributes to self-understanding. However, the biological perspective also obscures the historical dimension of self-consciousness.
Why reducing the mind to its neurobiological conditions is prone to misunderstandings is revealed by a now already familiar line of thought. Indeed, we have seen that we know how impressions of color are associated with electromagnetic waves only because we have impressions of color and learn to distinguish them ( _the indispensability thesis_ ). Impressions of color do not disappear with our increasing knowledge of electromagnetic waves. Our experience always remains its own dimension, without which we would have no access at all to the scientific facts and processes that are associated with it.
## _**Monads in the mill**_
The indispensability of subjective experience becomes clearer with a helpful distinction from John Searle. In his book _The Rediscovery of Mind_ , he refers to the fact that our experience is **ontologically subjective** – i.e., that consciousness can be described from an internal perspective in which we also experience colors, feelings and so forth. He distinguishes **ontologically objective facts** – i.e., facts that one can describe in such a way that no consciousness is involved in their existence – from what is ontologically subjective. For example, if one describes a chemical reaction, one thus describes an ontologically objective fact. Consciousness, in contradistinction to chemical reactions, is ontologically subjective; it would not take place without anyone noticing that it takes place.
At this point, Searle adds a decisive insight by referring to the fact that consciousness is nevertheless **epistemically objective**. This means that claims to truth can be formulated about what consciousness is, which in turn means that one can get consciousness right or wrong. The fact that consciousness is ontologically subjective, hence, does not imply that we cannot be wrong about it. It means simply that the facts of consciousness are subjective, not that our beliefs about it are not truth-apt.
Yet, when it comes to consciousness, that which is formulated in terms of truth claims is and remains irreducibly ontologically subjective. The confusion to which many who want to deny consciousness fall victim, according to Searle, can be attributed to the fact that they do not understand that something can be epistemically objective and ontologically subjective at the same time. There is simply no reason to consider our consciousness only from the ontologically objective perspective.
One source of the contemporary, neurocentric alienation of self-consciousness from itself can be identified as follows. We have learned that many phenomena which we experience every day prove to be entirely different than we had previously thought. As we obtain more precise knowledge of the universe, it begins to look like a strange place. Let us once again take the example of sunrises. As a matter of fact, when one thinks that one sees the upper edge of the sun on the horizon, this is actually a perception not of the sun but of a chemically induced reflection of it, comparable to a Fata Morgana, which is made possible by the chemical composition of the atmosphere. Besides, the sun does not rise in the literal sense of an upward movement. Therefore, in a certain sense there are no sunrises. At any rate, some express themselves in such a fashion, and something similar holds true for the horizon or for the blue sky, two effects that are not ontologically objective. There is no blue sphere above our heads screening us off from the cosmos – at least, not in the sense in which one might interpret the phenomenon of a blue sky before knowing anything about the atmosphere. The ontologically objective facts about celestial bodies and our atmosphere make the situation seem as though events such as sunrises turn out to be a mere semblance, a kind of illusion.
But this distinction between objective reality and subjective appearance does not apply to the case of consciousness. Consciousness cannot be an illusion in this way. Rather, consciousness is the source of illusions as well as of perceptions, and so on. Without consciousness, we would neither deceive ourselves nor ever have true beliefs. Consciousness is simply the standpoint from which something appears to us, and we cannot dispense with this standpoint, since then no one would actually be there any more to be able still to recognize anything.
We would not even know what we should be looking for in the brain, for instance, or in the traces of our species' biological past, if we were not already familiar with what it means to be conscious or self-conscious. Our fallible acquaintance with consciousness – i.e., self-consciousness – is an indispensable requisite for any further inquiry into the nature of consciousness. This is what the famous German philosopher Gottfried Wilhelm Leibniz had in mind with his **metaphor of the mill** , which has been a hot topic up to the present day. It is actually one of the earliest modern formulations of the so-called **hard problem of consciousness.** In §17 of his short text _Monadology_ (1714), he claims that one could not explain our subjective impressions " _in terms of mechanical reasons_ " – that is, "through shapes and motions" that are at work in our organism.5 To be sure, Leibniz could not yet know much about the precise structure of the physiology of our brain. Nevertheless, the misleading metaphor of a mill can easily be replaced by any other kind of machine (be it a network of neurons or a silicon-based computer).
Leibniz imagines being able to climb into a thought machine, thus into the brain. He compares this with going into a mill and claims: "Assuming that, when inspecting its interior, we will only find parts that push one another, and we will never find anything to explain a perception."6 From this he draws the quite unjustified conclusion that the human mind does not have any internal complexity at all, but is just a monad (from the ancient Greek _monas_ = a unit). The famous Nobel prizewinning physicist Erwin Schrödinger (1887–1961) goes further in _Mind and Matter_ , and in opposition to "Leibniz's fearful doctrine of monads"7 even believes that "... in truth there is only _one_ mind. That is the doctrine of the Upanishads. And not only of the Upanishads. The mystically experienced union with God regularly entails this attitude unless it is opposed by strong existing prejudices; and this means that it is less easily accepted in the West than in the East."8 Admittedly, one can hardly take this thesis seriously, either scientifically or philosophically, without extensive justification. Schrödinger himself probably also suspected this, which is why he immunized himself from critique by "the West." However, locating a thought in the West or the East has no philosophical purchase power. Where does the mysterious East in which strong prejudices are less common begin? Am I more spiritual than anyone in London or New York because I live east of them? Might I be a Far-Eastern thinker for Californians or for Indians, should they ever travel east on their way to meet me? Is there more consciousness of unity in Germany than in France, because Germany lies to the east of France? Were the Californian hippies so "spiritual" because they resided so far to the west that they almost made it to the Far East?
Schrödinger, like many before and after him, unfortunately falls prey to **Euro-Hinduism** , a view that spread out in the wake of German Romanticism. It is the idea that there is a greater and truer mystical consciousness of unity in India – as though it were harder for Indian citizens to understand that their neighbors had consciousnesses of their own! The pseudo-mystical idea to which Schrödinger adheres (which of course originates in what he would call the "West") is in his case a consequence of the fact that he is not able to solve the hard problem of consciousness, which he formulates in neuro-constructivist terms right at the beginning of his book:
> The world is a construct of our sensations, perceptions, memories. It is convenient to regard it as existing objectively on its own. But it certainly does not become manifest by its mere existence. Its becoming manifest is conditional on very special goings-on in very special parts of this very world, namely on certain events that happen in a brain. That is an inordinately peculiar kind of implication.9
This is, in fact, "an inordinately peculiar kind of implication," though not in itself but in the eye of the misguided beholder Erwin Schrödinger, who works out an incoherent account of the nature of consciousness: if (1) the world is a construct of our sensations and (2) the world includes the brain, then the brain is also a construct of our sensations. But how then is the brain supposed to form the basis for these constructions if it is thus something that is not itself constructed from its sensations? How is such an impossible miracle supposed to happen?
Leibniz and Schrödinger both overlook the fact that, when something does not function in a purely mechanical way, like clockwork, we are still a long way from being able to conclude that it does not possess any internal complexity. One cannot oppose mind and matter as one does the simple and the complex. Here it is enough, as Brigitte Falkenburg (b. 1953), a German philosopher and physicist, emphasizes, to point out that mental phenomena are structured in a different way than physical phenomena, which is why the true statements that one can utter respectively concerning both cannot be consolidated into a single theoretical discipline. Mind and matter are both complex in their own way. For this reason, Falkenburg adheres, within new parameters, to Leibniz's thesis that the mental and the corporeal are fundamentally measured according to different standards, that they are thus incommensurable.10
It is not my concern to champion Leibniz over Schrödinger or Falkenburg over both. I consider Leibniz's theory of consciousness to be false, since he goes so far as to think that everything which exists fundamentally has a kind of consciousness. This position is known as **panpsychism** , which, it should be noted, has had many adherents up to the present day who defend it with impressive precision. Among others, Chalmers comes close to such a position. He goes so far as to entertain the notion that, wherever information-processing is going on, experience might also be present. For this reason, he considers it to be an "open question" as to whether a thermostat could be conscious (even if its consciousness has to be fairly dull).11
The real open question at this point is probably just how much must have gone wrong if one no longer knows whether thermostats, photoelectric sensors, lightbulbs and rainstorms, contrary to our expectations, might somehow be conscious. Obviously, such things are not conscious, and any other claims can only be the result of false premises or muddled conceptions.
The metaphor of the mill nevertheless contains a glimmer of truth. What it reveals is that the description of mechanical or, rather, quasi-mechanical (chemical) processes that run their course while we are conscious or self-conscious, and which must be somehow connected with consciousness and self-consciousness, follows different principles than the description of consciousness and self-consciousness. If I am speaking about ion channels and neural networks, this will presuppose completely different considerations than if I am speaking about self-consciousness. Self-consciousness is embedded in social contexts. In the case of brain processes, however, there are no social contexts; rather, an internal processing of information takes place, if one may even put it in these terms. A variety of people participate in genuine social interaction who never turn up in my brain. It is not as though I absorb them into myself. That is, consciousness which is socially interactive and directed outward cannot be captured by concentrating on the description of processes in the brain. Once again, in the words of Wolfram Hogrebe's aphorism: "Spirit is outside but breaks through inside."12
## _**Bio is not always better than techno**_
The thesis that some contents of our consciousness are located outside of our skull is called **externalism**. One important version of externalism is **social externalism** , which amounts to the claim that some contents of our consciousness exist only as a result of our contact with another consciousness. The most radical version of this thesis is **social interactionism** , which claims that we would not be conscious at all without such contact, an idea that harks back to George Herbert Mead (1863–1931) and was recently restated by the renowned psychologist German Wolfgang Prinz (b. 1942), in his book _Open Minds: The Social Making of Agency and Intentionality_.13
A founding father of externalism, the recently deceased American philosopher Hilary Putnam (1926–2016), was one of the most important postwar philosophers. He is well known for being at least an indirect influence on the _Matrix_ trilogy of films. In the first chapter of his book _Reason, Truth and History_ , entitled "Brains in a Vat," he sketches out a dystopian vision of the possibility that we are all just brains which are electrically stimulated, so that we are led to believe in the illusion of a reality independent of ourselves.14 This idea is taken up and put into action by the _Matrix_ films. Putnam is also one of the first to have suggested that the brain be envisioned as a computer, on which consciousness or the mind in general runs like a piece of software. This thesis is also known as **machine state functionalism**.
Putnam originally introduced externalism into the philosophy of language and then went on to employ it fruitfully in the philosophy of consciousness. One of the central questions of the philosophy of language is how it is possible for our words to mean anything at all. The things that we name and the states of affairs of which we speak do not cry out to be named and carved up in the way we refer to them in our natural languages. This is why different natural languages can exist. It has been pointed out that many claims about the human mind and its relationship to the natural environment – which for the most part is characterized as an "external world" – are in part based on quite confused claims about the meaning of our words. Putnam has repeatedly posed the question in his books as to which implicit or explicit theory of the meaning of linguistic expressions guides our reflections on ourselves as minded animals. This illuminating approach has led to far-reaching insights, particularly to the fact that almost no one who has worked with such insights for a while says to themselves "I am a brain."
Putnam uses the now overused example of the meaning of the term "water," a consideration which led him to externalism. Let us pose what appears to be the easiest question of all: What does the term "water" mean? The answer, as easy as it is superficial, is that "water" means water. "Water" refers to water. But this is by no means self-evident. That is to say, if we were living in the Matrix, if I were thus identical to my brain, which is not in Germany at all but somewhere on a space station and has been electrically stimulated there from the beginning of time, the issue with water would pose a genuine problem. If that were the case, I would never have seen real water. Every time I had had the opportunity to think about water, the machines inputted thoughts of water into me, and when, perhaps, I thought that I was a child who had been taught the meaning of "water," that was all just a radical illusion. I was never in contact with water, and maybe in reality there is no water (who knows what my brain on that space station out there in the real world outside of the Matrix is made of!).
As a brain in the Matrix, everything that I know about "water" I know only from my own imagination, from my hallucinations that the machines I am plugged into actually produce. This sounds horrific, and one quickly sees that in such a scenario we could never really be sure of anything, not even of the meaning of our most simple words. All of our seeming knowledge would consist of illusory impressions that the machines feed us – perhaps only in order to keep our brains intact. A brain that has beautiful dreams can potentially be better preserved than a brain which also knows that it will be forever misused as just a source of energy by machines, and in the midst of such thoughts would probably gradually sink into a lasting depression before finally drowning in its sorrows.
All of this sounds dreadful. It sounds even more dreadful when it becomes clear that neurocentrism would have us believe, in all seriousness, that in fact _we_ are essentially brains in a vat. According to neurocentrism, we are indeed brains that are enclosed within our skulls. Evolution is supposed to guide our thoughts, since everything we consider to be true only seems plausible because the bio-machines to which we are connected pursue certain egotistical interests: they want to pass on their DNA and aside from that simply live on without further purpose. Neurocentrism accordingly teaches that we really are brains in a vat, which are guided by alien processes and machines – by evolution, genes, neurotransmitters, and so forth. Thus it is as if we are dealing with a bio-variant of the _Matrix_. However, bio is not always better than techno.
Of course, Putnam does not stop there. Rather, he imagines a scenario that helps him to illustrate why we cannot be brains in a vat. Imagine that you see ants crawling in front of you on the beach and executing a remarkable march back and forth. You then notice that the ants have apparently sketched a portrait of Winston Churchill in the sand. Question: Are the ants familiar with Winston Churchill and have they made a picture of him in the sand? This would be just as inconceivable as the proverbial ape who hits the keys on a typewriter at random and as a result produces a manuscript of _Macbeth_. The ants are unfamiliar with Winston Churchill and the ape is unacquainted with _Macbeth_. Nothing against ants and apes, but they are simply no more educated within our lifeworld than we fully understand theirs. What do I know of what ants experience or what they think if they communicate with each other via pheromones – that is, by way of scents? Apes are closer to us in evolutionary terms, which is why we do not hesitate to characterize some of them as humanoid – that is, similar to humans. Nevertheless, this does not mean that we can imagine what their lifeworld is like, how they experience their reality.
Ants which are crawling in the sand are simply in no position whatsoever to sketch a portrait of Winston Churchill. It would be a miracle if they knew that Churchill had even lived. How would they supposedly have obtained this information? Who would have explained this to them and how could they ever understand it?
At this point, Putnam connects his two lines of thought together. If we were brains in a vat, we would know as little about water as the ants do about Churchill and the ape about _Macbeth_. Brains in a vat are simply missing the right kind of contact with water, which consists in the fact that one has at some point actually touched, seen and drunk water. Similarly, the ants are missing the right kind of contact with Churchill, which consists in the fact that one has had history lessons on him, that one has seen him on YouTube, in TV documentaries or in books, for example.
Generally speaking, the following is the case: the poor brains in a vat know nothing of anything that goes beyond the realm of their imagination. This means that they never have a language that even remotely resembles ours. All of their words refer only to hallucinated episodes and objects. They know neither water nor land, neither friends nor elevators, nay, not even their own fingers! We only imagine that we could be such brains in a vat because we ignore the fact that our language does not continually report what is going on inside of us but speaks about the common environment in which we are situated.15 This thesis is the **basic idea of externalism**.
To believe that the simple words which we first learn at a very young age could not have any relation to the external world but are, rather, just labels we attach to our mental imagery is again to invoke the homunculus. This would be like believing that children are effectively stuck inside their minds.
Putting aside the sinister scenario of brains in a vat for a moment, the idea that we have an interiority that no one else can access tickles our fancy. We would be perfectly happy if our whole life played out on a private stage that would not exist without us. This would give us much more control over our lives than we actually have. It would mean that the entirety of colorful reality comes to an end with our demise, as if every death were a true apocalypse in which everything hangs in the balance. But we are not that important.
## _**How the clown attempted to get rid of omnipotence**_
The claim that we are brains in a vat or brains in a skull (which amounts to the same thing) is a narcissistic fantasy. Sigmund Freud (1856–1939) investigated the fundamental structures of this fantasy in a wonderful chapter of his not so wonderful book _Totem and Taboo_ entitled "Animism, Magic and the Omnipotence of Thoughts." Indirectly referring to Freud, Putnam himself calls the notion that the ants might after all be sketching a portrait of Churchill a "magical theory" of linguistic meaning.16
Freud reminds us of a phenomenon with which we are all familiar: sometimes we think of a person, for instance, and then suddenly meet this person on a streetcar or receive an unexpected phone call from her. In general, we more or less have the experience of an unexpected meaningfulness in common. Despite its utter randomness and complexity, social life seems to have an underlying thread. In moments in which we give in to this fantasy, it is as though the chance occurrences on our planet were somehow tailored for us: when the sun suddenly and unexpectedly shines on the day of a long-awaited reunion or when it begins to thunder at the moment of a friend's death. Such experiences can of course be quite profound and leave us with the impression that it is as if there is a purpose concealed behind what are only apparently chance occurrences, that this whole show (as in _The Truman Show_ ) had something to do with us. If one were to carry the impression that we are at the center of a show too far, one would wind up with superstitious beliefs or, much worse, psychosis.
Freud thinks that humanity has lived for long stretches as if chance occurrences were always an expression of psychical connections, a view he calls **animism**. Today, animism is entertained even by philosophers and scientists in the form of panpsychism, or the **hypothesis of panspermia**. The former, as has already been described above, holds that everything is ultimately animate, that our mind is only a complex variant of a basic mental force that is just as real as the strong nuclear force. The latter holds that perhaps life on our planet emerged because it was floating around through the universe from primordial times in embryonic form and finally settled here.
Admittedly, Freud's own world-view does not get us much further. In particular, he claims that we are quite primitive machines that are continually beset by a stream of desires, which he calls **libido**. For him, our mental life consists essentially in organizing our pleasurable sensations in some way. According to Freud, mental life is fundamentally the attempt to achieve pleasurable feelings. On his view, it is simply the case that we are desiremachines that emerged through evolution who fool ourselves into thinking that there are purposive structures which go beyond the regulation of our libido. Hence he rejects the idea of a homunculus but goes so far in the process that he immediately suspects that we are never really a "master in... [our] own house."17 If one looks a little more closely, Freud thus secretly adheres to the homunculus but turns him into a stupid puppet.
Yet he does not even stop there but shamelessly indulges in a complete exaggeration. Because, like many of his contemporaries – Nietzsche, above all – he saw through the fantasy of the omnipotence of thought as an illusion, he rashly concluded that our conscious thoughts, our "ego," as he calls the level of conscious thought, is just a complete idiot: "The ego here plays the ludicrous role of the clown in the circus, who, by his gestures, tries to convince the audience that every change in the circus ring happens as a result of his orders. But only the youngest in the audience are taken in."18 A few things get quite mixed up here. If the ego is only supposed to be the plaything of anonymous forces, who is playing the clown? Someone is considered to be a clown precisely because others are more intelligent. It cannot be the case that someone is simply a clown insofar as he or she is actually an ego. The clown would not be a clown then and the adults would not be smarter. And who is the audience?
Freud's belief that the ego is a clown is typical of the denial or denigration of the ego or the self: like so many others, he conflates the realization that there is no homunculus with the supposed realization that there is no ego or no self – or, if you will, that both are ultimately just more or less useful fictions. But, in doing so, he overlooks the really obvious option that the ego or the self is just not a homunculus and that virtually no one who has spoken of an ego or a self in the long history of the philosophy of mind has spoken of it in such terms!
For this reason, one should draw on Freud in another way that is more fruitful. One can follow his observation that our thoughts do not have any omnipotence or magic, without concluding from this that our thoughts have no power at all and are only waves of illusion that break on an immense, wild ocean. We can call the thesis that the mind is somehow everything and already disseminated across the universe – the omnipotence of thought – **joyful animism**. Such animism is an expression of a complete overestimation of the self, which is something we can learn from Freud. The converse claim, that we are basically clowns who are witnessing a circus show that in reality has nothing to do with us, is in contrast a complete underestimation of the self. I would like to call this – recalling Hegel's apt expression – an **unhappy consciousness**.
At this point, one can see that there is a connection between the technical philosophical term "self-consciousness" and the everyday usage of the word in the sense of lacking "self-assurance." In fact, our conceptions of consciousness, and thus of our consciousness of consciousness, are linked to the value we ascribe to consciousness. Often, if not always (as Freud believes), our own feelings of self-assurance are for this reason reflected in our theorizing, when it is a matter of figuring out what the human mind is.
There is a whole series of problems that come along with self-consciousness. These problems arise as a result of the fact that it is difficult to form an exact picture of how one can genuinely have consciousness of something, on the one hand, and have consciousness of consciousness, on the other. How are consciousness and self-consciousness related?
For a start, ask yourself the following question: Can you be in a conscious state of any sort without being able to give an account of it when asked? It seems as though one always consciously experiences one's own conscious impressions and could give an account of them in some way. Such a diagnosis has led to the establishment of an entire branch of theory that is today called HOT – that is, "higher-order theory."
On closer examination, we are once again dealing here with old wine in new bottles. These ideas were spelled out for the first time in Kant's _Critique of Pure Reason_ and were then developed further by those who came after him, primarily by Fichte, a virtuoso of the self [ _Ich_ ] (who even has "ich" embedded in his last name).19 Kant and Fichte considered the crucial distinguishing feature of our consciousness to lie in the fact that we can accompany any single conscious state with a higher-level consciousness of our consciousness. For example, I am tasting vanilla right now because I have eaten vanilla yogurt, and I can tell you this. In the same way, I see ever more letters appearing on my computer screen, feel how my fingers just move effortlessly over the keyboard, sense the contact with my desk chair and can give an account of all of these things. We can also intensify our impressions and manipulate them to a certain degree by moving to the level of self-consciousness – that is, by occupying ourselves with our own consciousness.
However, self-consciousness can also be frightening, which is quite vividly depicted in the work of the German poet Heinrich von Kleist, who was very exercised by Kant's philosophy, even having what is termed a Kant crisis which contributed to his actual suicide. Kleist wrote a wonderful short text _On the Marionette Theater_ , in which he investigates the uncanny phenomenon of being unsettled by attaining a consciousness of our consciousness. In this text, a scene occurs in which two male characters are in a bathhouse and one of them (a sixteen-year-old boy) lifts his foot and thereby gestures in a manner that reminds both of a famous motif in the history of sculpture, the "Boy with Thorn." The "Boy with Thorn" is a statue of a youth who is pulling out a thorn that has pricked his foot (Freud would be delighted that a thorn should be pulled out!).
The many homoerotic and borderline pedophilic connotations of this scene are unmistakable (above all, if one thinks of Thomas Mann's _Death in Venice_ , for instance, where Gustav von Aschenbach also compares his beloved Tadzio to "Boy with Thorn"). Be that as it may, the crucial point that is emphasized by Kleist arises from the fact that the older observer of the scene irritates the youth by laughing at him and pretending not to have noticed the unwanted, natural grace of his gesture at all:
> My friend was reminded of this statue when after our swim he placed his foot on the footstool to dry it and at the same time glanced into a large mirror; he smiled and told me what a discovery he had made. And indeed I had made the same observation at the same moment; but whether it was that I wanted to test the security of his natural grace, or whether I wanted to challenge his vanity, I laughed and replied that he was imagining things. He blushed and lifted his foot a second time to show me; as one could have easily predicted, the attempt failed.20
A similar point is made in a brilliant episode of the American sitcom _Seinfeld_ , to be found in the episode "The Chaperone" (sixth season, first episode). Elaine learns a painful lesson about grace in a job interview:
> LANDIS: Of course, Jackie O. was a great lady. Those are going to be some tough shoes to fill. Everyone loved her. She had such ... grace.
>
> ELAINE (gushing): Yes! Grace!
>
> LANDIS: Not many people have grace.
>
> ELAINE: Well, you know, grace is a tough one. I like to think I have a little grace ... not as much as Jackie –
>
> LANDIS: You can't have "a little grace." You either have grace, or you ... don't.
>
> ELAINE: O.K., fine, I have ... no grace.
>
> LANDIS: Grace isn't something you can pick up at the market.
>
> ELAINE (fed up): Alright, alright, look – I don't have grace, I don't want grace ... I don't even say grace, O.K.?
>
> LANDIS: Thank you for coming in.
>
> ELAINE: Yeah, yeah, right.
If we focus our consciousness on our consciousness, our consciousness is transformed. Whether we achieve grace depends on the degree to which we are able to lose control over our gestures and let them happen without conscious access.
Concealed behind this phenomenon lies a quite general problem that was examined closely not only by Kant but also by Fichte and Hegel after him, influencing various Romantic circles. This combination of philosophy and literature ultimately led to the great achievements of psychology in the discovery of the unconscious.
The general problem identified by all three consists in the fact that one cannot really maintain that consciousness is always accompanied by self-consciousness, as if all of our conscious states were always also simultaneously monitored by a higher, evaluating authority. The question could then immediately arise as to whether this authority is itself conscious or not. If it is conscious, there must in turn be an even higher authority that monitors it. Thus, a problematic infinite regress emerges: for any consciousness another consciousness is needed for the former to be conscious, and so on to infinity.
Now of course one could attempt the following evasive maneuver: Why shouldn't the higher authority (self-consciousness) simply be able to monitor itself? It would then have the twofold task of monitoring consciousness (or evaluating it) and also monitoring or evaluating itself in the bargain.
However, this does not work, which one can illustrate with a simple case. Try sometime to play a strategic board game – Nine Men's Morris, let us say, or chess – against yourself. Such an attempt will fail, as is well known. We cannot really play Nine Men's Morris or chess against ourselves, since it would not then make any sense to devise a complicated plan, because we could not keep it a secret from our "opponent." Something like this is the case with self-consciousness monitoring itself. Here there is a vicious circle, a circle in which an attempt is made to engender self-consciousness through an impossible form of self-monitoring. If we try to generate consciousness and self-consciousness in one act of self-monitoring, we are stuck in either a regress or a circle.
## _**Self-consciousness in a circle**_
We are familiar with these structures not merely from philosophical theories of self-consciousness but also from everyday situations. Let us assume that we want to go on a diet right after a lazy holiday of overindulging in food and lying on the beach. We now experience being in a restaurant where we could order a tempting pizza. We have a choice between the extremely unsatisfying salad, which is consistent with our plans to be on a diet, and the tempting pizza. At this point, self-consciousness comes into play and begins to evaluate our consciousness, as with the well-known image of a little angel on one shoulder and a little devil on the other. The all but clearly noticed, almost even unconscious to and fro runs its course. Eventually, a decision is made and the little devil and little angel disappear for the moment.
Whatever one makes of this, it is usually not the case that there is someone else standing behind the little devil and little angel, evaluating them in turn. And even if this were the case, if another voice were thus added and disparaged the little devil, for instance, there would not be yet another voice behind this voice. The chain comes to an end somewhere, usually with a little devil and a little angel. Thus we hear an inner voice full stop, without another commentary following it again.
The issue can be grasped even more succinctly. If consciousness only ever exists in such a way that there exists an accompanying self-consciousness that is distinct from consciousness, the question arises as to whether self-consciousness is conscious, too. Since one has assumed that consciousness could by definition always be accompanied by self-consciousness, this also holds true for self-consciousness. But then immediately there are infinitely many levels: consciousness of consciousness of consciousness, and so on. This is the infinite regress _par excellence_ – that is, an infinite regress: in order to be conscious, one must be conscious of being conscious; in order to be conscious of being conscious, one must be conscious of being conscious of being conscious; and so on.
This theory does not add up. Hence, it was traditionally assumed (especially in Kant's and Fichte's influential theories of self-consciousness) that the chain terminates in sheer self-consciousness. Yet the latter was then assumed to be able to observe itself. How else would we know that self-consciousness exists at all? This is known as the **problem of circular reasoning** : self-consciousness knows about itself through itself; it is intimately acquainted with itself. Yet, this comes suspiciously close to a brute assertion which covers up the problem that we have to stop somewhere but do not really know where.
Kant clearly recognized this problem of circular reasoning: If we are self-conscious, we must already be conscious of being conscious. We cannot conjure this up from nothing and hence cannot explain it from a perspective in which we act as though we did not yet know what we are talking about. He expresses this in a somewhat complicated manner thus:
> Through this I, or He, or It (the thing), which thinks, nothing further is represented than a transcendental subject of thoughts = x, which is recognized only through the thoughts that are its predicates, and about which, in abstraction, we can never have even the least concept; because of which we therefore turn in a constant circle, since we must always already avail ourselves of the representation of it at all times in order to judge anything about it.21
Kant here speaks of the "I, or He, or It" that is supposed to think and leaves open what this actually amounts to. He gives the following reason for this. Our representation of consciousness crucially depends on how we represent the bearer of thoughts. Who or what actually has thoughts? Who or what thinks inside of us? Kant's answer is that we can never really know this, since we must already form a representation of the bearer of thoughts before we can delve deeper into its nature. According to Kant, one cannot get behind one's own representations to find a reality of consciousness that is hidden there, whether this be an immortal soul or the brain.
Those who followed Kant were not satisfied with this conclusion. We still want to know who or what is actually the bearer of thoughts. Are _we_ thinkers of thoughts or are _they_ thinking in us? Georg Christoph Lichtenberg (1742–1799), a contemporary of Kant's, considered something like this when he wrote: "We are conscious of certain ideas that do not depend on us; we believe that others, at least, do depend on us. Where is the line drawn? We know only the existence of our sensations, representations, and thoughts. _It thinks_ , we should say, just as one says, _it lightnings_."22 At this point, Fichte and Hegel had considered a strategy for resolving the problem, the strategy of **social interactionism** , with which we have already briefly familiarized ourselves. This strategy solves the problem by proceeding from the fact that there must be more than one consciousness in order for self-consciousness to exist. Fichte and Hegel maintained that we understand consciousness and self-consciousness because we perceive the consciousness of others, which then installs their self-consciousness into us, as it were, through interaction.
The basic idea is perhaps familiar to you from other contexts (pedagogy or psychology). Especially widespread is the idea that the "voice of conscience" – that is, the evaluating authority in us – develops through education. Our parents and other teachers are accordingly supposed to have installed a conscience in us. Like conscience, on this model self-consciousness would then be a social artifact.
Wolfgang Prinz has recently elaborated this line of thought in his book _Open Minds_ , where he argues for the thesis that "self can emerge" only "from others."23 Prinz goes so far as to claim systematically that none of us could have intentions without having first learned from the intentions of others: "Agency and intention, I contend, are initially perceived and understood to be operating in others, and it is only through practices of social mirroring that individuals come to apply these notions to themselves and to implement related control mechanisms for their own actions."24 To my knowledge, this idea was first systematically elaborated by Fichte, who formulated it as follows: "The human being... becomes a human being only among human beings... _if there are to be human beings at all, there must be more than one_."25 Fichte calls this structure _recognition_. According to him, at some point we are summoned to be a self. This summons is implemented as a steering mechanism, and in this way we attain self-consciousness.
At first sight, all of that sounds plausible; it allows us to make sense of how we internalize ideas of _who_ and _how_ we ought to be. Indeed, it is certainly true that we incorporate values and patterns of conduct from our social surroundings. But can we really apply this model as an account of self-consciousness? Are we self-conscious, self-aware of our own consciousness, because someone teaches us to become conscious?
Social interactionism goes too far. How is it supposed to be possible for me to recognize another person as self-conscious if I am not already, for my part, conscious and thus implicitly self-conscious? I cannot first attain consciousness, which is linked to self-consciousness, as a result of learning that others are conscious, as they summon me to be conscious. It would never be possible to upload consciousness, so to speak, to a completely unformed mind. One resolves neither the problem of the infinite regress nor the problem of circular reasoning by increasing the number of self-consciousnesses.
Hegel, who critiqued Fichte's attempt at a solution to the problem at hand in his notoriously difficult book _Phenomenology of Spirit_ , was probably the first to have recognized this. Hegel's point is that we do not achieve a better understanding of self-consciousness by claiming that it exists in the plural. If we have a problem with something, it is usually no solution to multiply that thing!
Furthermore, one can ask social interactionism the following: Why can we not successfully convey self-consciousness to a stone? Probably because it does not even have _the potential for self-consciousness_. The potential to be a responsive person must already be present when one is summoned. What one learns through social interaction, in fact, are steering mechanisms and complex ideas of courses of action, roles, and much more. What one cannot learn in this way, however, is how to be conscious and self-conscious as such. Social interactionism fails at this level and, therefore, explains the existence neither of self-consciousness nor of consciousness.
But what now? Does self-consciousness perhaps not exist at all, since every attempt to render it intelligible founders on fundamental problems? We have seen that one cannot explain consciousness by means of consciousness itself. One winds up in inescapable situations – that is, in aporia – if self-consciousness is understood to be a kind of higher-level consciousness.
Consciousness is not a skeleton key for the human mind. The concept(s) of consciousness provide us with only a limited model of the human mind. This can already be seen from the fact that one runs into trouble having to explain consciousness by means of consciousness, which triggers the problems of self-consciousness already outlined above. If we wish to understand ourselves as the minded animals we are, it is not enough to notice that we are conscious. For this reason, neither the philosophy of consciousness alone nor an upgraded version enhanced by research in psychology and neuroscience is the philosopher's stone in the realm of self-knowledge.
We find ourselves in quest of ourselves. If one wants to achieve self-knowledge, it stands to reason to ask first what the self that one would like to know could be. We have now become acquainted with two candidates: consciousness and self-consciousness. Both concepts fail as candidates for answering the question of who or what we really are. They are limited explanatory elements in our self-portrait.
## **Notes**
1. Tyler Burge, _Origins of Objectivity_ (Oxford: Clarendon Press, 2010), p. 157. 2. René Descartes, _Meditations on First Philosophy_ (Cambridge: Cambridge University Press, 1986), p. 18. 3. This model of self-consciousness as an inner mind's eye with which we observe our mental processes has been refuted numerous times in twentieth-century philosophy, by thinkers as diverse as Martin Heidegger, Ludwig Wittgenstein, Gilbert Ryle, Jacques Derrida, the so-called Heidelberg School and Ernst Tugendhat (b. 1930). An excellent and clear overview of this discussion can be found in Tugendhat's classic _Self-Consciousness and Self-Determination_ , trans. Paul Stern (Cambridge, MA: MIT Press, 1986). 4. Eric Kandel, _The Age of Insight: The Quest to Understand the Unconscious in Art, Mind, and Brain, from Vienna 1900 to the Present_ (New York: Random House, 2012). 5. Gottfried Wilhelm Leibniz, _Philosophical Essays_ , ed. and trans. Roger Ariew and Daniel Garber (Indianapolis: Hackett, 1989), p. 215. 6. Ibid. 7. Erwin Schrödinger, _What is Life?_ (Cambridge: Cambridge University Press, 1967), p. 129. 8. Ibid. 9. Ibid., p. 93. 10. Brigitte Falkenburg, _Mythos Determinismus: Wieviel erklärt uns die Hirnforschung?_ (Berlin: Springer, 2012), pp. 354ff. 11. David Chalmers, "What is it Like to be a Thermostat?" in _The Conscious Mind_ (Oxford: Oxford University Press, 1996). 12. Wolfram Hogrebe, _Riskante Lebensnähe: Die szenische Existenz des Menschen_ (Berlin: Akademie, 2009), p. 17. 13. Wolfgang Prinz, _Open Minds: The Social Making of Agency and Intentionality_ (Cambridge, MA: MIT Press, 2012). 14. Hilary Putnam, _Reason, Truth and History_ (Cambridge: Cambridge University Press, 1981), pp. 1–21. 15. Unfortunately, Putnam's argument, on close examination, does not suffice to strictly prove that we are not brains in a vat. For more details, see Gabriel, _Antike und Moderne Skepsis_ (Hamburg: Junius, 2008), pp. 93–109. 16. Sigmund Freud, _Totem and Taboo_ , trans. James Strachey (London: Routledge, 1950), pp. 87–115; Putnam, _Reason, Truth and History_ , p. 3. 17. Sigmund Freud, _Introductory Lectures on Psycho-Analysis (Part III)_ (London: Hogarth Press, 1953), p. 285. 18. Sigmund Freud, _On the History of the Psycho-Analytic Movement: Papers on Metapsychology and Other Works_ (London: Hogarth Press, 1957), p. 53. 19. In reference to this, see the useful overview of the major arguments from Fichte all the way to higher-order theory in Manfred Frank's _Präreflexives Selbstbewusstsein: Vier Vorlesungen_ (Stuttgart: Reclam, 2015). 20. Heinrich von Kleist, "On the Marionette Theater," trans. Thomas G. Neumiller, _Drama Review_ 16/3 (1972); trans. modified. 21. Kant, _Critique of Pure Reason_ , trans. Paul Guyer and Allen W. Wood (Cambridge: Cambridge University Press, 1998), p. 414 [A346/B404]. 22. Georg Christoph Lichtenberg, _Philosophical Writings_ , ed. and trans. Steven Tester (Albany: SUNY Press, 2012), p. 5; translation modified. 23. Prinz, _Open Minds_ , p. 63. 24. Ibid., p. xvi. 25. Johann Gottlieb Fichte, _Foundations of Natural Right According to the Principles of the Wissenschaftslehre_ , ed. Frederick Neuhouser, trans. Michael Baur (Cambridge: Cambridge University Press, 2000), p. 37 (§3).
# 4
**Who or What is This Thing We Call the Self?**
"The self" is a rather ominous concept that is used today in a vague manner as a name for the control center of thinking, feeling and willing. Neurocentrics often argue on behalf of the claim that there is no self, since there is no evidence of it in the brain. At the same time, some of them, such as the German philosopher of consciousness Thomas Metzinger (b. 1958), maintain that the self is a kind of simulation that the brain produces, a "transparent model of the self," as Metzinger puts it, which gives us the impression that we are looking through an "ego-tunnel" positioned behind our eyes, as it were, out into a directly accessible reality.1
In any case, a contradiction emerges: on the one hand, the self is not supposed to exist – which is a sensational thesis allegedly supported by brain research and the theory of evolution – but, on the other hand it certainly does exist, namely as an ego-tunnel, though not as a permanent thing. Metzinger thinks that we have a multiplicity of different self-models at our disposal, a structure that has been developed by organisms via selection in evolution's icy struggle for survival.
Now, that may well be. In any case, there can be no doubt that at some point we must reckon with an explanation from the bottom up, and thus with the claim that complex neurobiological structures such as the human brain emerged over the course of millions of years of evolution. Metzinger believes that the capacity of organisms to create self-models is "a weapon that emerged in the course of the cognitive arms race."2 Yet, on closer inspection, he does not achieve his denial of the existence of a self, as this would lead him to a contradiction: on the one hand to claim that there is no self, and on the other hand to give an account of a self that supposedly does not exist. The only "advantage" of this account is that it can easily be embellished with technological and martial metaphors.
> I like to look at the human self-model as a neurocomputational weapon, a certain data structure that the brain can activate from time to time, such as when you have to wake up in the morning and integrate your sensory perceptions with your motor behaviour. The ego machine just turns on its phenomenal self, and that is the moment when _you_ come to.3
It is entirely correct that the self is not an object among other objects. It does not exist within the order of objects which includes rats, cats and mattresses. Whoever thinks it does deceives herself. However, one cannot rid oneself of the self by affirming that we are really no one at all on the basis of metaphors indebted to Darwinist and martial influences.4 If the self were the organism's conscious user interface, which is experienced as a supervisory control center, a simulation through which a tunnel-like self-model emerges (what an odd account of the self!), it would exist after all.
Simulations exist no less than drops of water. The effect of a spectacular discovery that there is no self, supposedly supported by the latest research on the brain, arises from first fueling the expectation that the self has to be an object which one can look for within the skull, so that one can then come up with the discovery that no self is to be found there. But why should one expect that the self should be a spatio-temporal thing? This expectation already presupposes modern naturalism, and thus the notion that everything which exists can be scientifically researched. Despite assurances to the contrary, the concept of the self is retained, but now in the form of a simulation. The real question, however, is: What is this ominous self, really, such that one can even ask to what extent it is a simulation? And to what extent can research on the brain answer this question?
Furthermore, how does Metzinger know that he is tricked into believing in an ego-tunnel? It is as if someone described on the telephone how it feels to sit in a flight simulator. Such a description, in any case, is not a scientific perspective on inner experience. Thus it may well be that some philosophers of consciousness and some neuroscientists experience themselves as an ego-tunnel that is positioned behind their eyes. And I agree with them that this is a kind of illusion which occurs with the involvement of their brains (which holds true for every illusion that is experienced). The question as to whether such accounts of the theater of consciousness are conceptually coherent, however, cannot be delegated to brain research. The real venue here would be philosophy, which developed the concept of the self.
## _**The reality of illusions**_
It has long been suspected that, in the case of the self, we could be dealing with an illusion, a position prominently advocated by no lesser figures than Buddha, David Hume and Friedrich Nietzsche. Within this tradition, Hume speaks of a _bundle theory of the self_ , which is distinguished from a _substance theory_. The **bundle theory of the self** claims that we find ourselves in many states of consciousness (i.e., states of thinking, feeling and willing), but that the self is no more than the mere sum or aggregate of these states. Accordingly, the self constantly changes depending on what states are present and is nothing which exists continually. The self is thus a bundle.
The **substance theory of the self** , in contrast, holds that the self is an entity which has the experience of thinking, feeling or volition. According to the substance theory, the self is the bearer of mental states, he or she to whom something becomes manifest. It is distinct from all of its states, insofar as it can have them. A substance is the bearer of properties; it has properties. If the self is a substance, it is for example distinct from its thoughts and sensations because it is the very entity that has these thoughts and sensations. According to this theory, the self does not exhaust itself in these states but, rather, is situated at an evaluative distance from them.
The pros and cons of both theories have been pondered for thousands of years. Research on the brain is not of much help in this matter, which is concerned with the conceptual question of whether the self can be coherently conceived of as a bundle. To identify the self with the brain amounts to a form of substance theory (other substance theories identify the self with the soul or with the whole body). If one considers the self to be a simulation that the brain produces in a specific area or by synchronizing many areas, it still remains a substance, but one that is identical with parts of the brain rather than with the whole brain. Hence, if anything, research on the brain imposes a substance theory on us rather than a bundle theory.
An influential branch of contemporary philosophy of consciousness draws on so-called phenomenology, a philosophical orientation that started in the nineteenth century and was definitively elaborated by the German mathematician and philosopher Edmund Husserl. **Phenomenology** (from the Greek _to phainomenon_ = what appears) is concerned with different forms of seeming, appearance, illusion, and so forth, as well as their counterparts evidence, intuition, etc. Phenomenology deals with how reality appears to us. From its inception, the self played a central role in it, insofar as the self can be conceived as something to which something else appears. My computer screen appears to me, as do a variety of other objects and subjectively experienced impressions which are all connected in one field of consciousness, which is experienced as unified over spatio-temporal distances.
Many contemporary philosophers of consciousness are affiliated with phenomenology, even if they are at the same time naturalists – something that differentiates them from the great phenomenologists of the past such as Franz Brentano, Husserl, Heidegger, Sartre and Maurice Merleau-Ponty. In one of his influential books, John Searle has built on an important basic idea of phenomenology, which he sums up very accurately as follows: "The marvellous thing about consciousness is that if you have the illusion that you're conscious then you are conscious. See, the normal appearance/reality distinction doesn't work in quite the same way for consciousness as it does for other phenomena."5 Searle of course concedes that we can consciously deceive ourselves about all sorts of things: for example, we can believe at first sight that something is a hedgehog when it is really a cactus, or briefly think that something tastes sweet but then quickly learn better. In all of this, however, we do not deceive ourselves about the fact that we are undergoing a conscious experience. As was mentioned above, many are convinced that this is what lies behind Descartes' famous statement "I think, therefore I am," though considered in historical terms it is problematic to equate what Descartes calls "thinking" ( _cogitare_ ) with our later concept of the self. But thereby hangs a tale.
What is crucial is that consciousness is another case that displays the typical structure of the mind in general, namely that, when it comes to the mind or spirit, even an illusion is a kind of reality. If I experience a Fata Morgana, there is of course no water where I believe to be seeing water. Nevertheless, I experience water and may perhaps rush thirstily to the water's apparent location. Even if consciousness is on the whole an illusory structure that our body, as a genetic copying machine programmed purely for self-preservation, produces completely unconsciously, this illusion still exists. Furthermore, it would be the decisive factor for us conscious living beings. As Searle, once again, puts it: "Consciousness is our life... So what's special about consciousness is that as far as human life is concerned it is pretty much the precondition of everything important."6
## _**Puberty-reductionism and the toilet theory**_
Neurocentrism's typical strategy of claiming the territory of selfhood and self-knowledge typically consists in "naturalizing" the self – that is, incorporating it into a subject area of phenomena that can be scientifically explained and understood. In this way, the self loses its mysterious character. The **naturalization of a phenomenon** is the attempt to treat a phenomenon that can apparently not be investigated scientifically as something that, contrary to its appearance, can be scientifically articulated and researched. Here, "natural" accordingly means something like "potential object of natural scientific research," which already raises innumerable questions, since it is anything but clear what allows something to be scientifically investigated.
A slightly different concept for this explanatory procedure is _reductionism_. **Reductionism** in this context is the reduction of a concept that does not seem to be established in a scientific framework to a scientific equivalent. For a reduction to be meaningfully made, there must be a _reason for it_. In a given subject area, discoveries need to have been made that seem to entail that we can achieve a better explanation of a phenomenon only by reducing some concepts by which we characterize it into the language of science.
Accordingly, **neuroreductionism** is the reduction of a phenomenon that, at first glance, is not subject to neuroscientific explanation to a discourse capable of couching the phenomenon in relevant neurobiological terms. Puberty is a simple everyday example of such a process that could be cited here, as it offers a reason for the reduction. On the one hand, we are familiar with puberty as continuous sequence of behavioral changes: rebelliousness, fighting against parental authority, the withdrawal into a private sphere, changes in the pitch of a boy's voice, and so forth. On the other hand, all of this has to do with the concomitant phenomenon of hormonal changes associated with sexual maturation. So far, so good. A reductive explanation at this point tells us that the system of behavioral changes can be reduced to processes that can be neuroscientifically explained. If one wishes actually to explain adequately what is going on in puberty, one has to take facts about hormones and their effects on the human brain into consideration. The reductionist view claims that, in the case of puberty, one need not take anything more into consideration. Rebelliousness or criticism of authority such as parents and teachers in this perspective is nothing but a byproduct, a concomitant phenomenon of hormonal changes, and thus nothing with which one should be concerned if one is interested in the objective facts of the matter. The behavioral phenomena typical of puberty can accordingly be reduced to hormonal changes, with the result that one can leave aside changes of behavior when offering an explanation of the phenomena.
At this point, it is important to take into account a distinction that is often overlooked in public debates on these issues. There are at least two kinds of reductionism. **Ontology** (from the ancient Greek _to on_ = being and _logos_ = rational discourse or theory) is concerned with the question of what it means for something to exist. Correspondingly, there is an **ontological reductionism** , which claims that a phenomenon that does not appear to be natural exists only as a phenomenon, as an appearance, behind which lies a natural process and nothing else. An example of this would be the reduction of pure water to H20: viewed objectively, pure water is nothing but H20. One can, at least so the reductionist believes, ontologically reduce pure water to H20. Philosophers often cite as an example the reduction of perceived temperature to the average kinetic energy of particles, which allows us to bring thermodynamics to bear instead of sensations of heat. In this way, temperature can be reduced to properties of a particle system, with the result that one can leave our sensations out of the picture, in a truly ontologically objective manner. **Theory reductionism** , in contrast, is more modest and claims only that a phenomenon that can be described with non-scientific concepts can be more adequately described with scientific concepts.
If one now applies this to puberty, it is already no longer clear which form of reductionism is even in question. Does one wish to say that puberty can be described by taking hormonal changes into consideration, or does one wish to say that puberty is nothing but hormonal changes?
Neuroreductionism arrives on the scene with a typical nineteenth-century style cockiness by attempting an ontological reduction and a theory reduction at the same time. And that is precisely the problem. In the nineteenth century, neuroreductionism was advocated in its original form by Karl Vogt (1817–1895), a Swiss-German scientist who, in the middle of the century, denied the freedom of the human will in the manner that has become customary today. Vogt was notorious in his time for declaring that "Thoughts stand in the same relation to the brain as the gall does to the liver or urine does to the kidneys."7 If one ontologically reduces puberty according to such a model, then the moodiness of a pubescent youth is nothing but hormonal fluctuation. But what is that supposed to tell us about our association with each other? Can we really deal with the behavior of our children as if it was literally nothing but hormonal fluctuation? And what about other complex forms of behavior which we classify as crime?
In his _Phenomenology of Spirit_ , Hegel, in his inimitable way, anticipated and dismissed with sharp irony theses such as Vogt's. In a famous passage, he writes:
> Brain fibres and the like, when regarded as the being of Spirit, are no more than a merely hypothetical reality existing only in one's head, not the true reality which has an _outer_ existence, and which can be felt and seen; when they exist _out there_ , when they are seen, they are dead objects, and then no longer pass for the being of Spirit... the Notion underlying this idea is that Reason takes itself to be _all thinghood_ , even _purely objective_ thinghood itself; but it is this only _in the Notion_ , or only the Notion is the truth of this idea; and the purer the Notion itself is, the sillier an idea it becomes when its content is in the form, not of the Notion, but of picture-thinking... The _depth_ which Spirit brings forth from within – but only as far as its picture-thinking consciousness where it lets it remain – and the _ignorance_ of this consciousness about what it really is saying, are the same conjunction of the high and the low which, in the living being, Nature naively expresses when it combines the organ of its highest fulfilment, the organ of generation, with the organ of pissing. The infinite judgment, _qua_ infinite, would be the fulfilment of life that comprehends itself; the consciousness of the infinite judgment that remains at the level of picture-thinking behaves as pissing.8
Vogt's toilet theory of thinking, as we can now call it following Hegel, originally intended to point out to us that our thinking is not a divine spark but, rather, a natural organic process. Yet Vogt is by no means only a coarse critic of religious or theological conceptions of the soul (as was, for instance, Karl Marx) against which he marshals the brain. Vogt and other neurocentrists of his time, such as Ludwig Büchner (1824–1899, brother of the famous writer Georg Büchner), who was very much read and discussed at that time, were, incidentally, severe racists and misogynists – as were, unfortunately, the overwhelming majority of scientists and thinkers at this time, not merely the neurocentrists of the day. In his bestseller _Force and Matter: Empirico-Philosophical Studies, Intelligibly Rendered_ (1855), Büchner explains (supposed) differences in behavior between the "Caucasian race" and the "other varieties inferior to" them with reference to the physiology of the brain, since according to this kind of reasoning a difference in behavior must always be reduced to a function of the brain: "The natives of Australia, in whom the superior parts of the brain are almost wanting, possess neither great intellectual capacity nor any sense of art or moral worth. All attempts of the English to civilize them have hitherto failed."9 The following argument of Büchner's, which is still popular today in a somewhat better concealed form, is not much more pleasant: "It is known that the female sex is intellectually inferior to the male sex. Peacock found that the average weight of the male brain considerably exceeds that of the female."10 Were these remarks not all too pathetic nonsense, they would almost border on irony, when Büchner, on the same page no less, supports his claim by citing a "Dr _Geist_ " (Dr _Spirit_!) from a "hospital" who is also supposed to have proven it.
One of the major weaknesses of ontological neuroreductionism lies in the fact that it identifies social and historically variable (institutional) channels of behavior in natural terms. This is also known as **essentialism** (from the Latin _essentia_ = essence) – i.e., the assumption of an immutable essence for which one is not responsible, but which one also cannot change. One cannot fundamentally change processes of the brain with only persuasive coaxing any less than one can suppress hormonal changes that way. If there were a female brain, one could do nothing about it through processes of social emancipation. Women would then be determined to behave in certain ways by their specifically grown brains. I hope none of my readers believes such crap (Trump voters and racist Brexiteers probably did not read up to this point). Given the weaknesses of all-out reductionism, it would be more advisable for neurocentrism to adopt the more modest strategy of theory reductionism. Certainly puberty can be better understood if one also explains it neuroscientifically. However, theory reductionism is not a spectacular insight and it also cannot promise to replace our traditionally developed modes of accounting for human action by the fancy neurospeak of the future.
## _**Self is god**_
All of this now in fact leads us to the self, the small god upon the earth. There is much talk of the self these days. There are questions of to what extent egoism is supposed to be justified, of self-employment, and much more. The I or ego can be found in everyday application from iTunes to iRobots. Neurocentrists vacillate between identifying the self with the brain and denying its existence. Yet who or what is this self after all?
At this point, it is advisable to look at the history of mind or spirit. One of the first who substantivized the personal pronoun "I" [ _Ich_ ] in the German language to "the self" [ _das Ich_ ] or "this self" was a medieval philosopher who is usually referred to as Meister Eckhart (1260–1328). "Meister" (as translation of _magister_ ) in this case basically means professor of philosophy and theology. Meister Eckhart was accused of heresy in Cologne by the Inquisition and died during his trial in Avignon, where his main doctrines were posthumously condemned as heretical. It is precisely his concept of the self that can be cited as a reason for the fact that the ecclesiastical authorities of his time were not particularly pleased with him. Later this concept of the self was seen as the opening move of the Enlightenment, which can thus be found as early as the Middle Ages (as is so much of what is considered to be modern – that is, neither ancient nor medieval). In a famous sermon – mind you, not incriminated by the Inquisition – Meister Eckhart said:
> _Therefore I pray God that he may quit me of God_ , for my unconditioned being is above God and all distinctions, insofar as we conceive God as the origin of creatures... If I had not been, there would have been no God: I am thus the cause of the fact that "God" exists; If I had not been, God would not be "God"... in bursting forth I discover that God and I are One.11
This is a much more radical move than cheap contemporary atheism which merely denies the existence of God. For Meister Eckhart points out that those who speak of God actually create an image of themselves. His name for this structure – which indeed is the structure of a _Geist_ or a mind – is the self, the I.
Meister Eckhart developed a theory of "the agent of the soul" and hypothesized a "spark" of "the soul" in us.12 This is connected to his theory of the self. On closer inspection, his idea is quite radical, and it rests on a simple line of thought. Suppose I pick up a coffee mug. I am now looking at my coffee mug. There thus exists a relation between me and the coffee mug, the relation of seeing. To see something means to perceive something by means of our senses. I thus perceive the coffee mug. Now let us pose the question: Am I identical to this perception or am I to be found somewhere in this perception in such a way that I can grasp my identity with an object given to me in perception? I am not the coffee mug; that much is clear. But do I come into existence by being that which perceives the coffee mug? This also does not add up, since in any case I cannot strictly be identical with that which perceives the coffee mug because I, the same thinker, can perceive many other things too. If I exist at all, I am thus someone who can perceive coffee mugs and at the same time someone who can perceive other things too. I am still myself when I perceive the lamp.
Thus I can perceive all kinds of things, and as long as I live it is an open question what I will perceive in the future. Let us assume that it is possible to perceive myself as I am perceiving a coffee mug. If self = brain, this is quite possible. I would then perceive myself in a functional magnetic resonance imaging (fMRI), for instance. To be sure, neuroscientists to this point have told us that this is not yet feasible, because the self has yet to be found. It is unclear how multiple streams of information can be incorporated into a unity at all at the level of our perception, which is known as the **binding problem**. Not to mention the question of how all the processes that are relevant for perception could be bound together in such a way that a self could be perceived in them. But let us assume for the time being that this will be solved in the future. Then I could see my self flickering in the fMRI. Yet, this self would still not be the self that we are looking for. The self perceived in the fMRI would be no more strictly identical to the perceiving self than in the case of my perception of the coffee mug. The very reason why I cannot be identical to a coffee mug I perceive counts against identifying myself with any object I can perceive by means of an fMRI! According to Meister Eckhart, then, we thinkers of thoughts are categorically distinct from any object we could ever perceive or think of.
With this basic line of thought, which I have here extended to the topic of the brain, Meister Eckhart discovered the topic of the radical non-objectivity of the self. This idea lies concealed behind the modern concept of _autonomy_ – that is, our giving laws to ourselves (from the ancient Greek _autos_ = self and _nomos_ = law). What Meister Eckhart is saying amounts to the claim that the self cannot be strictly identical with any object that it perceives. The self only falls under laws it gives to itself; it cannot be strictly determined by the objects it perceives. Meister Eckhart proceeds so far on this basis as to consider the self "godlike," since the distinction between objects and the divine mind, in his view, is one of monotheism's fundamental insights.
If one subscribes to this line of thought and cuts God out of the picture, one quite quickly arrives at the idea that there is an all-knowing self that strives to perceive reality as a whole and to unlock its secrets. This profoundly theological model forms the basis for the early modern understanding of science, a fact which today is too often pushed aside into oblivion. The self does not merely want to be godlike, it wants to be God and hence strikes him from its world picture in order to take his place. This is how science started occupying the very position of religion. This does not mean that science is just another religion. However, it does mean that science currently occupies the position of religion, which is ultimately just as bad.
Actually to cut God out of the picture and not still remain secretly in the clutches of theology presupposes that one has developed another concept of the self, a genuinely modern one. Otherwise we will be stuck with the fantasy of a creation liberated from God. The paradox is that the early modern scientific image of the world is actually taken from Judeo-Christian-Islamic monotheism. The difference is that nature is radically emptied of God, and in the place of the divine spectator of the world-spectacle there now sits the thinking, critically examining self.
This corresponds to Max Weber's thesis of the "disenchantment of the world," which for him is by no means a specifically modern phenomenon, but which in his eyes begins with the prophets of the Old Testament, who, in the name of the newly revealed desert God, turn against the mages of neighboring polytheistic religions and in this way disenchant nature. All enchantment lies in God's hands. Nature turns into a desert, into the absence of God from his creation.
The best illustration of the theological fantasies of the scientific image of the world is found in the already mentioned new edition of the American television documentary series _Cosmos_. The narrator, the astrophysicist Neil deGrasse Tyson, presents himself as capable of swiftly moving through space-time at will in his spaceship (his vessel of imagination) so as to inform the audience about the Big Bang, quite distant galaxies, gravity, and so forth. His spaceship can also shrink down at will, doing so in order to make the universe visible at different scales of magnitude. He suggestively names it "the Ship of the Imagination," though in fact, on close examination, it reveals more about the imagination of Tyson than it does about the history of science. Granted, the scientists and their history are lucidly illustrated by means of the animated illustrations of Seth MacFarlane (the creator of the cartoon series _Family Guy_ ). In many regards, the series is a paradigmatic example of successful science education for the broad public and, thus, welcome in our dark times of widespread superstition and ignorance (a very serious problem for the US these days as well as for the rest of the world). We could use more of such programs on our television, but perhaps with a little less ideology.
The critical point can be grasped especially well from a scene in the first season that is concerned with the age of the planet earth. The narrator heads to the rim of the Grand Canyon with the Ship of the Imagination. In order to be able to show the audience the hidden strata where fossils can be found, he raises his hands – like Moses or a small god – whereupon the strata are parted from each other with an Old Testament-like roar and hidden treasure is uncovered. The show's production here suggests, as at many other points, that the narrator is pretty much like God, who can be present at will in all places and at all times of creation, as well as on all scales. The narrator even works miracles, since it would be a miracle if someone could stretch apart the strata of the Grand Canyon at will, or (as in another episode) increase the gravitational force in Manhattan from a g-force of 1 all the way up to a g-force of 100 and can then reduce it again to the same degree, so that water hydrants are crushed by their own mass and furniture movers are flying through the streets. The Ship of the Imagination is an example of the fantasy of the omnipotence of thought.
The thought that lies concealed behind the self often plays a role where one would least suspect it at first sight. In particular, the self comes to fruition against the background of the modern conception of absolute objectivity. **Absolute objectivity** is a vantage point from which the universe can be observed as if no intelligent observers with their species-specific conditions for knowledge were present. Absolute objectivity tries to break free from the obvious fact that no one knows anything except from a subjective standpoint. Knowledge in light of this counts as more objective the less it is marked by the fact that it is acquired from a subjective standpoint.
Thomas Nagel, whom we have already come across a number of times, has summed this up with his famous metaphor and rejection of the "view from nowhere." In _Why the World Does Not Exist_ , I spoke in this connection of the "world without spectators."13 Many believe that science should strive for absolute objectivity. Ideal knowledge would then consist in developing a world-view in which our standpoint would no longer play any role.
Neither Nagel nor I deny that we in fact succeed in dissociating from our standpoints to some extent and that we accordingly achieve objective knowledge. As already discussed above (p. 76), Nagel even considers our capacity to dissociate from ourselves as a foundation for ethics. As a matter of fact, if I want to restrict claims to ownership and to share resources with other people, I must be able to understand that objects are not simply out there so as to belong to me. Without the capacity to empathize with others and to desist from doing whatever one wants or evaluating a situation however one likes, one could never understand how any kind of ethically or politically relevant decision would be possible. At most, one could suppose that the egoistical struggle for survival of citizens or other members of communities engenders the illusion that one can renounce standpoints, just as we saw in the account of egoism above. Our capacity to examine critically the relation of interests that belongs to our beliefs is connected to our capacity for theoretical abstraction. Traditionally, this capacity is known as _reason_.
To be sure, no one can adopt the paradoxical standpoint of standpointlessness, since it contains a contradiction in itself. Hence, this standpoint can remain at best an ultimately incoherent ideal that we strive for but, in principle, can never attain. One must become conscious of this fact, since otherwise one takes on God's theological role without even noticing it.
## _**Fichte: the almost forgotten grandmaster of the self**_
The topic of the self or the I is at the center of the thought of the German philosopher Johann Gottlieb Fichte (1762–1814). Fichte was made famous by Kant, who anonymously published Fichte's first work, _Attempt at a Critique of All Revelation_ , with the help of his publisher, so that readers at the time believed that it was a new book by Kant himself. In the wake of this event, in 1794 Fichte obtained his first position as professor in Jena, which he eventually had to resign due to a harsh dispute over his supposed atheism. Remarkably, Fichte was a thorn in the side of Goethe, who was then the minister responsible for the University of Jena. Goethe did not think much of the idea that the self is a small god on earth, since he was instead of the opinion that the divine can be found in nature rather than in us. He thus indirectly supported the demise of Fichte in Jena.
We can already conclude from the title of his first work that it was no accident that Fichte was considered to be close to atheism. Fichte, incidentally, later became the founding rector of the Friedrich Wilhelm University, which is known today as the Humboldt University of Berlin, and was socially and politically active throughout his lifetime. As a prodigy from a poor part of the Eastern German countryside, he was from a young age promoted by a nobleman who was impressed by the young man's stunning foreign-language skills. In contemporary philosophy, Fichte still plays a role, as he was the first philosopher who established a conceptual connection between the autonomy of the self and its recognition by others. He thus also became the originator of social interactionism, discussed above (p. 124), on the basis of which he founded his philosophy of law and the state.
At any rate, Fichte is particularly well known for the fact that he took up the question of who or what the self really is. In this context, he developed his philosophical program, which he called "the science of knowledge." His basic thought is easy to understand, and throughout his career Fichte never stopped wondering why so many people apparently did not want to understand him. Fichte's writings, at first sight, are difficult or even incomprehensible, simply because he attempted to avoid unnecessary conceptual ballast or complicated debts to the history of philosophy, as he actually wanted everyone who was interested to be able to understand him. He invented a new language for philosophy based on ordinary German in order to undermine traditional authorities and to claim more territory for the radical freedom of thought. However, one must carry out a bit of translation work to reconstruct his ideas for our time.
In order to do this, let us depart from the **basic thought of the science of knowledge** : There are distinct domains of knowledge whose fundamental features we are taught in elementary school: mathematics, geography, English, physical education, and so forth. What we learn about in the course of studying these subjects are not only matters such as multiplication tables, spelling and the names of capitals. We also learn how to learn something. School is not just about collecting information; it is about collecting information about how to collect information in the future, how to become a responsible knower. Accordingly, one might ask what the common denominator for all domains of knowledge is, since all of them must have a common form that links them to one another. There seems to be an overall structure of learning something, of coming to know something. This is why Fichte departs from the assumption that all domains of knowledge are connected despite the fact that they respectively convey distinct substantive contents. Despite its distinct contents, according to Fichte, knowledge must nevertheless have a unified form.
And this is precisely Fichte's Archimedean point. His science of knowledge examines how the form and content of knowledge are connected. His basic question is actually as old as Plato's philosophy. Even today it is still lodged in our word for mathematics, since this word is derived from the ancient Greek verb _manthano_ , which means "to learn." For Plato, mathematics is a matter of learning how to learn something, by gradually being led to an insight that can be taught to all human beings. He illustrates this in his dialogue _Meno_ , in which an uneducated slave is taught fundamental geometrical insights in a step-by-step manner. Hence, class distinctions according to Plato and Fichte play no role when it comes to our coming to know something. As knowers or rational animals we do not differ with respect to our rationality. Fichte thus aims, as Plato did before him, to demonstrate to us human beings that we are all endowed with reason. This means that we can learn from others, because we share the capacity to know something with other human beings. This is **rationalist universalism** , which is a basic assumption of the Enlightenment.
Furthermore, Fichte supposes that the form of all knowledge stems from the fact that we have the capacity to understand. Thus, even though we aim at absolute objectivity, we must still be able to understand what we find out. That is, even the limiting case of absolute objectivity remains related to us. Against this backdrop, Fichte speaks not merely of the self or "I" [ _Ich_ ] but of the "we" [ _wir_ ], which Hegel took up in a much cited formulation: "'I' that is 'We' and 'We' that is 'I'."14 What _I_ know is something _we_ can know. I can tell you what I know and you thereby come to know it, and vice versa. Knowledge is essentially shareable. Even in the limiting case of absolute objectivity, there is still a distinction between the self or "I" and what is not the self, between the knower and what is known.
In this context, Fichte introduces the notion of an absolute I. This just means that the self is released from or categorically distinct from what is known (the Latin term _absolutum_ means nothing more than "that which is detached"). On this basis, Fichte develops three fundamental propositions, which become the supporting pillars of his philosophy of the self. These should be kept in mind when one speaks of the self today, since Fichte's basic ideas have had a lasting influence all the way to Freud and Sartre, which still resonates in our contemporary psychological vocabulary. If one disregards this history, it is all too easy to fall prey to thinking that "the self" is something so utterly familiar that we can ask whether it is an illusion or not. Fichte had to defend himself against various misunderstandings of his view throughout his lifetime and as a reaction to this came to the conclusion that "The majority of men could sooner be brought to believe themselves a piece of lava in the moon than to take themselves for a _self_."15 As neurocentrism proves, in this respect nothing has changed since that time.
## _**The three pillars of the science of knowledge**_
Let us very briefly go over the three tenets of the science of knowledge. They will help us to finally come to terms with the self (and therefore with ourselves).
The **first tenet of the science of knowledge** is: "I = I [ _Ich = Ich_ ]."16 Of course, that sounds trivial, but it is not. One might think that this proposition is just an instance of the claim that everything is identical to itself, that everything is what it is. Then the proposition would be a **tautology**. But consider the following:
The round square = the round square;
or
The current king of France = the current king of France.
If the round square or the current king of France are identical to themselves, then apparently there is a difference between self-identical things that exist and those that do not. There are no round squares and there is no longer a current king of France (Nicolas Sarkozy was an exception...). The problem does not arise for the self or "I," however, since the term "I" means nothing more than the knower. We want to develop a science of knowledge and thus think about how it is possible that different domains of knowledge are connected. "I" is the name for the fact that there is a knower. One cannot deny this. _In particular, if for some reason one might feel tempted to deny that anyone knows anything, one could not even know that there is any reason to believe this!_ That is Fichte's scaled-down version of Descartes' cogito. "I think, therefore I am" becomes, in Fichte, "I = I" or "I am I." The first tenet thus guarantees that in the realm of knowledge there is at least one thing that is identical to itself: the "I" or the knower. If I know that it is raining right now in London, and then I also know that 2 + 2 = 4, I am not split into two separate beings: one who knows that it is raining, and another who knows that 2 + 2 = 4. I can be identical to myself as knower on various occasions and with respect to different contents of knowledge.
The **second tenet of the science of knowledge** is (slightly simplified): "self ≠ not-self." The second tenet expresses precisely the by now familiar idea of absolute objectivity. It tells us that we have a concept of a reality beyond ourselves. Stones, meadows, neutrinos, galaxies, mountains, and so forth, can be subsumed under the concept of a not-self. They fall within the category of objects that do not think. Fichte also calls this category "nature," which immediately led Goethe among others to raise objections, since he did not think that the self ought to be excluded from nature. Yet Fichte here has been historically triumphant, since with the second tenet he got to the bottom of the idea of nature as not-self, the idea of absolute objectivity. Nevertheless, in so doing he was not of the opinion that nature simply is what it is; rather, he thought that the concept of nature as a totality of everything that belongs to the not-self is a product of the self's abstraction. The concept of nature plays a role in our theories of reality, as it classifies some phenomena we encounter as natural and others as social or mental.
To be sure, it was Fichte's intention to save the self from nature in this way. He was interested in telling us apart from merely natural objects, to draw a principled distinction between thinkers and things. As a consequence, he completely purged nature of traces of the self, so that subsequently it was no longer evident how the self could ever belong to nature again.
This dilemma has been a concern to the present day, to the extent that, by "nature," we understand everything which we investigate from the standpoint of absolute objectivity. By definition, this excludes our subjective standpoints. The self became in this way radically opposed to nature, so that the temptation immediately arose in Fichte's wake likewise to cut it out of the picture entirely. Neurocentrism would like to explain the self away by attempting to translate everything that has the form of the self into the language of neurochemistry or evolutionary psychology. Neurocentrism is a violent way of integrating the self into nature – that is, into a domain which in principle cannot serve as a home for the self. It, thus, is a form of radical alienation of the self from its world.
Fichte struggled mightily against this – and with good reason. The conception of absolute objectivity disregards the fact that it forms nature into a unified concept. Let us follow a simple line of thought. What do protons, bosons, photons and neurons actually have in common? It is not only the fact that they all end with "ons" (otherwise we would have to add lemons to the standard model of particle physics). In addition to such particles, molecules and galaxies, the Big Bang, gravity, bacteria, tardigrades, supernovae and space-time belong to nature. Thus, again: What do all these objects, laws or facts actually have in common, such that we can actually recognize that they belong to the same realm?
And here Fichte replies: they have in common the fact that they play a role in an account from the standpoint of absolute objectivity. But that means that all of these objects, laws and facts are connected within the framework of a theory. Strictly speaking, there has been no such theory up to this point; there is no unified theory of nature that is recognized as some particular science. Physics has never achieved this. But why are we sure that there is even a unified realm of nature that we can investigate in the mode of absolute objectivity? Where do we derive our concept of nature from if not from nature itself?
The key to answering this question lies in the fact that the standpoint of absolute objectivity cannot itself be investigated at all from the standpoint of absolute objectivity. The idea of absolute objectivity is a product of abstraction that arises from the fact that we put ourselves to one side while investigating. In so doing, we do not disappear but keep ourselves out of the picture that we are forming of the actual situation we are in as thinkers of thoughts and builders of theories.
More recently, Nagel and Searle have reformulated precisely these thoughts wholly in keeping with Fichte's intention. They both refer to the fact that our ideal of objectivity is formulated from a standpoint that cannot itself be absolutely objective. The distinction between our subjective standpoint, the self, and the domain of objects under investigation from the objective standpoint, nature or the not-self, cannot be derived from the objective standpoint alone. These theories were not forged first in laboratories but in social relations, which Nagel and Searle do not foreground, but which Fichte before Marx recognized.
Notice that it does not follow from all this that absolute objectivity does not exist – i.e., that neutrinos are "socially constructed." Neutrinos actually do exist, and their hypothesis and ultimate discovery forms a fascinating chapter of the last century's history of science. However, what does follow is that it is impossible for absolute objectivity to exist independently of a context in which subjectivity plays a role. An all-encompassing, purely objective world picture, in which the self does not feature, is radically incomplete and impossible. In the first place, all-encompassing world-views are incoherent anyhow – which was the theme of my earlier book _Why the World Does Not Exist_. Yet we do not have to rehearse the arguments developed there in detail. All we need to do here is simply realize that our claims about what belongs to nature as a whole are made from a standpoint. This standpoint investigates things and states of affairs in the mode of absolute objectivity, in which we are trained by modern scientific methods. Yet these methods and thus the framework of absolute objectivity itself simply cannot be investigated by using these methods on it. There is no natural science of natural science, which is why until now there have indeed been the most absurd neurodisciplines – neuro-German studies, neurosociology or neurotheology. But there is no neuro-neuroscience, which would then have to be trumped in turn by a neuro-neuro-neuroscience. At some point or other we will always encounter ourselves as the originators of the subjective standpoint we inhabit even when engaging in dissociating from it and observing natural phenomena, as if we were not present.
With this in mind, we can now turn to the **third tenet of the science of knowledge**. One must let this dissolve on the tongue, as it were, before I can explain it in a way that is understandable for those of us native to the twenty-first century: " _In the self I oppose a divisible not-self to the divisible self_."17 There are three protagonists in this tenet:
1. the self
2. the divisible self
3. the divisible not-self.
The tenet sounds odd, but it is easy to reconstruct. In distinction from the divisible self, "the self" is the general circumstance that you share with me: the circumstance that we can know something. Here it is important to keep in view an important difference between _knowledge_ and _representation_ , which is very easily blurred today.
From Plato up to contemporary epistemology – which among other things is intensely concerned with the question of what knowledge is – one speaks of the **standard definition of knowledge**. This runs as follows: _knowledge is justified true belief_. Behind this lies the following idea, which was for the first time developed by Plato in his dialogue _Theaetetus_ , the foundational text of epistemology. Ask yourself for a start whether someone can know something that is false. Can I know that Angela Merkel has seventeen index fingers? How am I then supposed to know that she only has two (at least, to this point)? This is known as a **truth condition**. It entails that one can only know something that is true. What one knows is a fact, something that is true.
Next question: Can you know something that you do not believe at all? Suppose I were to tell you that I know that 2 + 2 = 4. Would you answer back: Do you really believe that? And what if I were then to let you know that I am by no means convinced of it. That would be strange. Hence there has been a discussion lasting millennia about the relation and difference of knowledge, certainty and belief. One cannot know anything that one does not consider to be true with a high degree of certainty. If I know something, I should be willing to bet on it – at least under the right circumstances, in which I have no reason to question or restrict my own knowledge.
Now perhaps you will want to point out to me, however, that we are almost never absolutely certain. We often get it wrong even though we are certain that we are right. Likewise, one can persuade people that they know something which they really do not know at all – which is a condition of possibility of ideology. We often make others and ourselves believe that we know certain things we do not know, as well as that we do not know certain things we actually do know (think about the incredibly stupid US debates about evolution vs. creation or debates about climate change; of course, animal species are not created by a very powerful dude from heaven from mud, and, of course, there is man-made climate change).
The third condition for knowledge, the **justification condition** , has it that one cannot know anything which one cannot defend with good reasons as soon as legitimate doubts are raised. If I say that I know exactly where Angela Merkel is right now, and you question this claim, I can point out to you that I just saw her on live television giving a speech in Berlin or that an acquaintance of mine just called me because she saw the chancellor shopping with her bodyguards. To know something presupposes that one can provide good reasons for what one considers to be true. It also means, at the very least, to consider something that is true to be true on the basis of sound reasoning or some other kind of legitimate evidence, along with quite firm belief. In this sense, the testimony of our senses typically serves as a good reason to believe something and to claim knowledge.
Knowledge is essentially shareable. If I call my wife and ask her whether our dog is sleeping in the living room, she can take a look and confirm this. My wife then knows through simple observation (which is a good reason for her knowledge) that the dog is sleeping in the living room. Knowledge is shareable and communicable. Fichte refers to this aspect of knowledge as "the divisible" or "shareable self." If I tell you what I know, you can thereby come to know it yourself. We can share knowledge.
What I cannot share with my wife, however, is her _representation_ of the scene. If she goes into the living room and sees our dog there, she sees her from a particular perspective; she has particular feelings about the dog, perceives particular objects that I would perhaps not pay attention to at all, because she has other background assumptions and experiences, many of which are unconsciously processed. Our representations are embedded in a background, as Searle calls it, thus in a prevailingly unconscious concurrent repertoire of capacities and assumptions.18 Representations are not shareable, they are private.
By a **representation** here, we can understand the psychological episode that occurs when one processes sensory impressions or calls to mind impressions that have been processed with the aid of our imagination. The concept of representation, in turn, introduces difficulties, which can be avoided if one understands representations as those pieces of information that are accessible to an individual on the basis of his or her specific situation considered as a whole, which includes the fact that the individual is located in a specific place at a specific time. I do not know exactly what my wife is representing and imagining as she shares her knowledge of the dog's whereabouts with me, and vice versa. Even if one knows a person very well, one cannot step into their world of imagination in this sense and perceive it by viewing it from the inside – as occurs, for instance, in _Being John Malkovich_. No one except John Malkovich can be John Malkovich. However, John Malkovich can easily share his knowledge with us by communicating it to us. In order to do this, we do not have to step into his mind and merge with him in an impossible manner.
Hence one can communicate representations but cannot literally share them. In contrast, one can both communicate and share knowledge. That simply follows from the concept of knowledge, which I would here like to summarize once more: I can know the same thing that someone else knows if both of us recognize the same state of affairs as being true on the basis of having good reasons for our recognition of the fact. We then find ourselves in the same state of knowledge. But I cannot have exactly the same representation as another person, since I would then have to be that other person.
Fichte's fundamental idea is to reject the notion that we can always only know something because we have representations that arise in us as a result of the stimulation of our nerve endings or any other kind of information-processing. Ideas structurally similar to the neurocentrism of our day circulated at his time and were employed to call our knowledge into question. Fichte points out that these arguments conflate knowledge and representation. His ultimate aim is the defense of enlightenment, knowledge and facts against the dark forces of anti-modernism, which are once more on the rise in our allegedly "post-truth" or "post-factual" era. It is simply an ideological lie that we live in a post-truth era, a lie designed to convince the people that they do not know what they actually know.
Knowledge that can be communicated and shared is universal. "The self" is Fichte's name for the universal dimension of knowledge. It is the universal knowledge subject. "The divisible self," in contrast, is Fichte's name for the fact that many thinkers can know the same thing. "The divisible not-self" is everything which one can know in the mode of absolute objectivity. You and I know the same thing (the same divisible not-self) if we know the speed of light or the mass of certain elementary particles.
One could draw on Fichte to justify contemporary scientific culture by enriching the latter with the self rather than letting it conflate the self with the brain, as it currently does – and thus with something that belongs to the category of the divisible not-self (nature) rather than to the adequate category of a thinker. Even so, one would then be on the cusp of the beginning of the nineteenth century, regressing behind neurocentrism. Yet philosophy did not end back then; rather, it was just getting started on the right track.
## _**In the human being nature opens her eyes and sees that she exists**_
We now know what the self is: it is the subject of universal knowledge. To be a self means to know something and to be able to communicate it. In no way does it mean to be alone with oneself or to dwell like a homunculus in the brain. That said, it is already clear: the self is not a brain.
However, Fichte's philosophy of the self runs into problems as soon as one poses the question of how the self is actually related to nature. In Fichte's lifetime, Schelling, the major thinker of German Romanticism, formulated an all-decisive objection, the aftermath of which, incidentally, made existentialism, Marxism and modern depth psychology possible. As if that were not enough, Schelling's student Johannes Müller (1801–1858) ranks alongside Charles Darwin (if not actually on the same level) as one of the most important biologists of the nineteenth century. His formulation of the **law of specific nerve energies** is still found in neuroscience textbooks up to the present day. This law states that the objective structure of an external stimulus does not determine a sense-impression; rather, the stimulated sense organ (in other words, the respective nerve cells) is responsible for the modality in which a stimulus is processed as a perception (i.e., seeing, hearing, tasting). Considerations of this kind imposed themselves upon Müller, since he takes Schelling's objection against Fichte seriously, an objection which ultimately led to so-called _Naturphilosophie_.
_**Naturphilosophie**_ poses the question of how nature must be constituted if at some point in the course of its development beings could arise who are capable of conceiving of nature's development. Reflections of this kind are today familiar under the heading of the _anthropic principle_. The **anthropic principle** (from the ancient Greek _ho anthropos_ = human being) in general is the view that the observable universe is manifestly suited to the development of living beings who observe this universe. Indeed, we are such beings who have developed within the universe. This is expressed metaphorically in the claim that nature wakes up, as it were, in the human being and attains consciousness of herself, originally a Romantic metaphor, which in our times Nagel took up in his most recent book, _Mind and Cosmos_ , explicitly borrowing it from Schelling.19
In fact, it is remarkable that we are able to understand nature at all. This can be considered simply amazing, since in any case there is no evidence that nature was literally waiting around so that living beings with minds to figure her out would develop and begin to decipher the laws of nature. That there are living beings with minds does not appear to be necessary, at any rate. That is, we can easily imagine an alternative course of evolution in which living beings like us, who attempt to conceive of nature and their own place in it, would never have emerged at all. If our own existence could not have happened, if we are contingent, there is the worry that we might not be in any position whatsoever to figure out nature. However, we only discovered that we are contingent through figuring out something about nature (in particular, evolutionary theory as the only correct explanatory framework of the origin of the plurality of species on our planet). Hence, there are limitations on contingency, as our existence cannot bring with it the utter unintelligibility of nature. Otherwise, we would be in a position to wonder how much we might ever know about nature. Therefore, the very fact that we are able to work out a scientifically respectable sense of our own contingency proves that nature is at least in part intelligible, understandable and knowable by us. This is the basic structure of Schelling's overall philosophical project and his _Naturphilosophie_ in particular.
As a matter of fact, it is misleading if one speaks at this point of the fact that evolution has spawned living beings endowed with minds. For "evolution" is a name for processes in which species emerge. The term does not designate an overall process, let alone a direction in which animals evolve. There are many independent trees of the evolution of species, and none follows any specific direction. We account for these processes by means of the theory of evolution. It is not the case that evolution is a "blind watchmaker," as Dawkins put it in his book of the same name.20 Evolution is neither a watchmaker nor any other kind of maker; it does nothing at all, because it is neither a subject nor any other kind of person with blind intentions but our collective term for complex processes constitutive of the emergence of new species, which can be better explained with the aid of the theory of evolution than with any other available alternative.
Admittedly, one often reads that evolution does this or that or has led to this or that. That is absurd, since evolution leads to nothing at all. At best, genetic mutations caused by cosmic radiation, as well as other entirely natural processes that take place in the division of cells, lead to the transformation of phenotypes – that is, to the transformation of the external appearance of living beings of a particular species. If their environment has also been somehow transformed – all the way to great catastrophic upheavals that have affected the earth again and again – living beings with mutated genes will survive and reproduce under certain conditions, and so on. The basic ideas of Darwinism are well known.
The term "evolution" unfortunately suggests that the processes accounted for by the theory of evolution are connected as if there were still a kind of intention, even if a blind one, such as the survival of individuals and species. Evolution is unwittingly imagined as a process that is intentionally carried out, for instance as a struggle for survival. The crucial advance of evolutionary biology in comparison to older attempts at explaining the origin of species lies in the fact that evolutionary biology gets by without assuming any intentions. It has become clear from the progress of biology and massive leaps forward such as evidence for DNA and its decoding in the last century that one can dispense with blind watchmakers too. The theory of evolution has no need for such metaphors, which only accommodate the need to replace the old God full of intentions with at least an ersatz God: the blind watchmaker or evolution.
Schelling's _Naturphilosophie_ , in contrast to Fichte's philosophy of the self, insists that living beings with minds, which identify as selves and in this way can attain knowledge about nature, do not fall from heaven. We are not angels bolted together with the bodies of apes or pure spirits imprisoned in animal bodies. But that only means that there are necessary biological conditions for the fact that we can identify ourselves as selves.
Within the framework of the self-investigation of the human mind, nature is thus discovered under the banner of the self and becomes a great theme of the nineteenth century, because it troubles the self, so to speak. _Naturphilosophie_ and what succeeded it in the nineteenth century, including Marxism and psychoanalysis, objected to the idea of a complete autonomy of the self by pointing out that there are mental illnesses, and that the self is in general only able to develop a dimension of knowledge when it is not disturbed by its natural conditions or when it does not allow itself to be disrupted by them.
## _**"Let Daddy take care of this": Freud and**_ **Stromberg**
Against the backdrop of Fichte's philosophy and German Romanticism, Freud developed his influential concept of the self or the ego. He elaborates this with particular clarity in his text _The Ego and the Id_ of 1923, in which he enriches his master distinction of the conscious and the unconscious with his equally famous conception of an ego, a superego and an id. In this context, Freud, unlike Fichte, understands the self or "the ego" no longer as a universal dimension of knowledge but, rather, as a facet of our psychical life. "We have formed the idea that in each individual there is a coherent organization of mental processes; and we call this his _ego_."21 The self becomes a subject for psychology, a process whose groundwork was laid by Schopenhauer and Nietzsche, who heavily influenced Freud's way of thinking about the self.
Neurocentrists routinely point out in their favor that Freud was actually a kind of neuroscientist, since in fact he began early on to seek physiological causes of mental illnesses. Freud's _Project for a Scientific Psychology_ of 1895, a text in which he developed a biologically grounded "neuron theory," as he himself called it, is often cited in support of this claim. Since we now know that a majority of the processes that run their course in the brain are not actually consciously experienced, even as they make our conscious life possible, some believe that a scientifically secure basis for psychoanalysis or something close enough has been discovered.
However, Freud actually discovered psychoanalysis because he understood that there are structures of our mental life that manifest in the ways in which we _describe_ ourselves and our attitudes to others. In this vein, he founded psychotherapy – the talking cure – which consists in describing our attitudes to ourselves and others and questioning in what ways these attitudes are experienced as painful and how they can be transformed. An intervention into organic processes – for instance, by taking antidepressants – is thus necessary in some cases but by no means in all. There is a link between our self-descriptions and the quality of our conscious life. Whether this link is regulated by neuro-cocktails in _all_ cases and whether biochemical regularities, which are subject to the release of neurotransmitters, are correlated in a law-like way with the regularities that emerge in the self-description of our conscious experience is _de facto_ not known by anyone at this time. It is and will forever remain a purely speculative question as to what specific biochemical basis is supposed to correspond to Napoleon's ambition to conquer Europe – insofar as the question is even ever formulated.
Psychoanalysis has regularly been criticized for not being scientific enough. The success of any such critique depends on what one means by "science." In any case, psychoanalysis up to the present day is recognized as a legitimate form of therapy. Freud and, in his wake, the French psychoanalyst and philosopher Jacques Lacan (1901–1981), in particular, among other things critically evaluated the ultimately superstitious belief that there is an expert panel watching over humanity composed of scientists, who worship an absolutely objective goddess named "science."22 Psychoanalysis is often associated with a critique of a false ideal of science that believes absolute objectivity can be achieved. That does not make it popular with everyone; it also lies behind the resistance summed up in the slogan that it is not scientific enough.
Since Freud and psychoanalysis are frequently cited by neurocentrism and sometimes honored as precursors, it is worthwhile looking a little more closely for a moment at what Freud understands by the self or ego and whether he wanted to identify it with the brain. What is "the self" or ego in Freud and which of his insights are relevant for our context?
Freud's seminal distinction consists in dividing the psychical into the conscious and the unconscious, which he called "the fundamental premise of psycho-analysis."23 Accordingly, consciousness does not exhaust our mental life (the psychical). This distinction has remained intact to the present day, and there is only contention over how exactly one should understand it. According to Freud, it is crucial that the unconscious does not consist of purely organic processes but already belongs to the psychical. Of course, neurobiological processes ensure that protons, which affect my sensory receptors, can be registered as pieces of information. These processes are unconscious in the sense that I do not notice them as such but only experience their results: my conscious impressions. If I look at my hand, I do not simultaneously see the neurobiological processes that must have taken place in the background.
Yet these processes do not belong to the category of the unconscious in Freud's sense. For him, what is unconscious can be made accessible by giving up the resistance one has built up because one censors the particular ideas and wishes that one has. One need not immediately think here of repressed sexual fantasies or the (in)famous Oedipus complex. It suffices to consider entirely everyday experiences.
Let us consider the everyday experience of boarding an airport bus that travels from the terminal to the airplane. One boards and sees an open place to sit. While heading for the seat, one sees other people whom one has perhaps already observed at the airport. One takes an interest in some people (in whatever manner you like), though is not interested in others, "and the act of perception itself tells us nothing of the reason why a thing is or is not perceived."24 Hence one constructs a proper theory that makes it possible to guide one's own behavior. It would be nice to see the apparently helpless older woman in the seat that one has selected. It is probably not a good idea to stand next to the man acting like a jerk, who spends the whole time talking on the phone to his business partner so that it is hard to avoid the impression that he wants to show everyone how important he is, and so forth. In all of these cases, different thoughts go through one's head that are not very pleasant and which one quickly puts aside, so to speak. According to Freud, such thoughts are subconscious messages that ensure that certain things attract our attention, that we are interested in certain people and have a disposition to evaluate them that has somehow become second nature.
Let us suppose that someone – let us call him Donald – is particularly irritated by tourists in shorts who swarm onto the plane. He could perhaps justify his irritation if questioned by complaining that people are being quite pushy, and he might even get worked up over the fact that someone in those kinds of shorts who is obviously going on vacation is now pushing their way past everyone else. While he complains to himself or to a fellow passenger, it suddenly occurs to him that in reality he himself would love to go on vacation and is instead flying on business to Palma de Mallorca. Furthermore, perhaps at some point he had the experience of someone not appreciating his legs and hence would never himself travel in shorts or even go out in public wearing shorts. Yet this flash of insight, so to speak, is at best a brief glimpse into the unconscious and is replaced by a bad attitude, apparently well founded, toward pushy travelers. Donald pushes back, as he is not in a position to face his unconscious fantasy and wish structure which actually tells him that he himself would like to be the tourist in shorts.
The basic idea of psychoanalysis is that everyday speech in which we justify ourselves, in which we articulate our attitudes to ourselves and to others, is always anchored in experiences, many of which we believe we have completely forgotten. However, they actually exert a decisive influence on us precisely because we have developed a resistance against them. The unconscious emerges through the development of resistance against ideas that we were once conscious of in our life. This is how we construct our personality or our "character," as Freud sometimes calls it with reservations.25 "The ego" or the self in this context is Freud's term for the internal discourse of rational justification, the very level at which we believe that our attitudes toward others are justified and reasonable rather than merely personal, emotional and irrational. The ego serves the function of making us believe that we are fundamentally objectively justified in feeling the way we do about others and ourselves. It makes us look rational in our own eyes.
This can be illustrated with reference to a well-known phenomenon. Consider a workplace, for instance – an open-plan office, let us say. So that we are imagining roughly the same thing, think of _Stromberg_ , the British or American versions of _The Office_ , or the French _Le Bureau_ , which are all variations of the British prototype originally created by Ricky Gervais. These TV series are concerned with the psychodynamics of the workplace.
The German open-plan office from _Stromberg_ belongs to an insurance company named Capitol. At least since _Stromberg – the Movie_ it is clear that "Capitol" is a name for German capitalism. The German social market economy appeals to the need for security, whereas the American model of economic exchange addresses us as potential consumers and business people interested in pure liberty. Capitalism in the German version is experienced as a contribution to security. This is, of course, a fantasy and not itself an economic fact. There just is no specific relationship between the production and exchange of goods and political emancipation or social progress and stability. Any such relation is negotiated between a network of subsystems of society that are in principle too complex in order for anyone to give an overall account of what is going on. This is one of the reasons why there is such a thing as ideology, a fantasy supplement that makes our lives as members of structures no one fully understands more bearable. Unfortunately, this can easily be exploited. A TV series such as _Stromberg_ expresses unconscious ideas. Indeed, the unconscious is continually erupting in the series, which is part of the comedy – something that reaches its peak in the movie when the executive board meets with prostitutes for wild orgies under the pretext of a party for management. In the movie we see a brutal outburst of sexist and repressive authoritarian behavior among the management of Capitol insurance, something which unfortunately nowadays can even result in your being the president of a powerful country.
In any social situation, such as the one depicted in _The Office_ , every individual projects a kind of psychical map onto the social space. Everyone in the network enters the personal interpretation space of the others in terms of specific evaluations. If one listens to their evaluations, every single person will describe the social network of the workplace in a different way. Besides the _legally regulated order_ – which determines the positions in the company – there is always a _psychical order_ , which partly overlaps with the legal order. This becomes all the more transparent as we gradually realize that there is no purely legal order that is utterly independent of our psychical orders and that the two are often connected (one should think here of the discussions of bullying, stress, sexual harassment, sexism, red tape, and so on – in short, one should think of business psychology).
The omnipresence of the psychical order is illustrated in _The Office_ and _Stromberg_ by the fact that short individual interviews with the various parties are conducted over and over again. The individuals who are interviewed speak in front of the camera that follows everyday life at the office, since a documentary is being produced about it. Those involved describe the social order in light of their evaluations, in the course of which it subtly becomes clear that their evaluations are far from rational. Participation in the game of giving and asking for reasons is not itself a rational business; it is always grounded in emotional experience and, thereby, in the unconscious of the members of a given social system.
The self or ego in the Freudian model boils down to a level of description for our social interaction. At this level, our descriptions appear to be justified and supported with good reason. This corresponds to Freud's understanding of the psychological function of science. He succinctly writes at the very beginning of his essay "Instincts and Their Vicissitudes" of 1915:
> We have often heard it maintained that sciences should be built up on clear and sharply defined basic concepts. In actual fact no science, not even the most exact, begins with such definitions. The true beginning of scientific activity consists rather in describing phenomena and then in proceeding to group, classify and correlate them. Even at the stage of description it is not possible to avoid applying certain abstract ideas to the material in hand, ideas derived from somewhere or other but certainly not from the new observations alone.26
## _**Drives meet hard facts**_
In contemporary philosophy, there are different accounts of the relation between our rationality (from the Latin _ratio_ = reason) and the structure of what Freud calls "the ego" and what we have been referring to as the self. Neo-pragmatism holds that to be a self is to be rational, which is in conflict with Freud's view that the ego is actually a part of the id – that is, that its primary function is to create the misleading impression (the illusion) that our drives are justified. The neo-pragmatist American philosopher Robert Brandom (b. 1950) speaks here of "the game of giving and asking for reasons," and he has made a case for conceiving of the self as the general name for our fellow players in this game.27 To be a self means to be accessible at the level of description on which we support and justify our attitudes to ourselves and to others with reasons.
Unfortunately, despite his valid insight that rationalism cannot be the whole truth about the self, Freud himself was often inclined to treat the ego or self like a homunculus that in his view enters the scene because perceptions arise in us through mere sensory stimulation. "It is like a demonstration of the theorem that all knowledge has its origin in external perception."28 To support this thesis he draws on "cerebral anatomy"29 and considers the ego to be a kind of anatomically verifiable surface or interface in the neo-cortex. How confusing things thus become is shown by the following fundamentally indecipherable passage, in which Freud explicitly defines the ego as a homunculus:
> The ego is first and foremost a bodily ego; it is not merely a surface entity, but is itself the projection of a surface. If we wish to find an anatomical analogy for it we can best identify it with the "cortical homunculus" of the anatomists, which stands on its head in the cortex, sticks up its heels, faces backwards and, as we know, has its speech-area on the left-hand side.30
Here Freud brazenly reifies the ego. However, this can be avoided with the help of Brandom's suggestion that the self is to be understood as a social function, as the level of description of social interaction in the light of our justificatory speech. One thus already treads a fine path between bundle theory and substance theory, since the level of description belonging to the self is neither a bundle nor a bearer of mental states. If the self is not an entity but, rather, a capacity to participate in a certain practice, it ceases to make sense to try to locate it and identify it with some brain area (or any number of brain areas working together in whatever spectacular way).
However, Brandom is not bothered by the fact that the self is not merely a neutral player among others. He does not focus on the fact that every self describes the whole game from a standpoint that others do not immediately and self-evidently share. The reason for this is not only that each of us just has different theories with different experiential bases, which could then be aligned with one another in the game of giving and asking for reasons. Rather, one can learn precisely from Freud that the ego or self becomes individual because we have experiences that are evaluated by others and which we then acquire as habits.
In order to account for this, Freud introduces the superego and the id alongside the ego. The superego, which he also characterizes as the ego-ideal or the ideal ego, plays a constitutive role in our self-description as an ego, in our considering certain modes of conduct and attitudes to be acceptable or even imperative. The superego stipulates what we should accept in general as good reason, which allows the ego to transform an emotional attitude toward others into something that appears to be justified. To recognize that we find ourselves in certain recurring situations in a certain way due to certain experiences in the past (not only in childhood) is difficult for us because we thus to some extent surrender the conscious control for which the ego feels responsible. The ego as an advocate of good reason is simply not the only factor of mental life, and it does not merely exist alongside other factors. Feeling seems to be something that just somehow comes over us, which is why the task of self-knowledge is to make a practice of how not to become the victim of one's feelings, as it were. Such a practice would not be possible if we were already in full control of any feature of our mental lives, as we could just happily play the game of giving and asking for reasons without bothering with our emotions, habits, etc. But how could we ever be initiated into the game without having been taught to behave in certain ways prior to acquiring the (often ideological) belief that our practices are in play because they are justified?
In the case of the id, his name for the drives, Freud distinguishes between a positive drive (eros) and a death drive. The positive, often sexual drive strives for self-preservation, the death drive for self-destruction. Furthermore, Freud is of the opinion that both of these drives can be biologically verified. He even cites "simpler organisms" in support of his distinction of ego and id, "for it is the inevitable expression of the influence of the external world."31 On this level, he attributes to the ego perceptions that enable our inner life to have contact with reality, while the id potentially resists the recognition of an independent reality by forming an inner world.
What Freud is proposing here can be to a large extent reconstructed without the absurd idea that ego, id and superego can be localized, as if they were literally a matter of three regions of the body. Unfortunately, Freud actually does accept this crude materialistic idea, which is why he calls his theory "topical" (from the ancient Greek _topos_ = place).
We have beliefs and opinions about what is the case independently of our beliefs and opinions. To perceive something means to come into contact with the kinds of facts that we cannot change by means of our perception but can only take in. That already resonates in the popular notion among traditional German philosophers that the etymology of the word "perception" [ _Wahrnehmung_ ] indeed suggests that one takes in [ _nehmen_ ] something true [ _wahr_ ] and thus conceives of something true, a fact. As a matter of fact, the German word for perception etymologically means to become aware of something rather than to believe that something is true. In languages that use a version of the Latin-derived _perceptio_ for perception (as in English or French), the idea is expressed that we are collecting or gathering something in perception. For Freud, the ego is close to the **reality principle** , as he calls it – that is, the fact that we typically come into contact with given states of affairs in whose continued existence or coming about we do not participate.
Facts whose existence is not brought about by us in any manner can be characterized as **hard realities**. We perceive some hard realities simply because they act upon our sensory receptors. But this is not unconditionally true of all facts, of course: the fact that Napoleon crowned himself emperor, at any rate, exists independently of whether anyone living today was involved in making it happen. It is certainly not easy to specify just when something actually exists independently of us. Yet this should not concern us further here. Right now, the only thing that is crucial to note is that, at some level or other, the self encounters facts, some of which are hard realities whose existence must simply be accepted if one is at all interested in truth.
In contrast, the id stands for drives. These are able to exist independently of hard realities. Thanks to our drives, we are in a position actually to change some hard realities: for example, we used to be confronted by a mountain range that made it difficult to get to Italy from Switzerland (and vice versa). So we simply drilled the Gotthard Base Tunnel in order to travel more quickly from one country to another. And why do we want to do that? Because we want to do business with each other, go skiing and visit museums, meet up with friends who live elsewhere, and so forth. The reason for the existence of a tunnel is not that it is rational to have tunnels, but rather that tunnels help us realize our wishes. There is nothing essentially rational in turn about our wishes themselves.
Because we have drives we transform realities and create new ones. Without drives, we would be passive windows into reality that would not belong to this world. Animals are distinguished from many plants by their self-determined motion. We animals experience an impulse toward change of place as a drive. For this reason, Freud supposes that the id is a kind of energy center that is irritated by perceptions and reacts to them. For Freud, the ego is the eyes and ears of the id; he expressly considers it to be a part of the id, which raises difficulties for his distinctions. Ego, superego and id are much more closely connected than one might believe at first sight. The superego, according to Freud, is for its part the "representative of the internal world of the id,"32 which communicates the id's wishes to the ego in censored form. "What has belonged to the lowest part of the mental life of each of us is changed, through the formation of the ideal, into what is highest in the human mind by our scale of values."33 An odd construction.
In any case, it is clear that the very thing which Freud calls a "drive" can be understood as a striving for transformation, in distinction from "perception." Perception does not want to transform anything but, rather, simply accepts things, while the id drives us from one state to the next. It thus only actually obtains some specific form – that is, the form of concrete wishes – because it is articulated by the ego. The ego is a transformation engine which translates drives into more specific wishes by attaching the (ultimately illusory) notion to them that it is inherently rational to realize the wishes, to change reality so as to make what we wish for happen. This is part of why egoism is so hard to overcome and why humanity is so amazing at destroying itself by destroying its natural habitat, planet earth: we are prone to believe that what we want to have and want to do is the right thing to have and the right thing to do – a pure illusion, as thinkers as diverse as Buddha, Marx and Freud have rightly pointed out.
Let us choose another example which comes to mind for us inhabitants of wealthy globalized societies. We go into the supermarket and ask ourselves which kind of milk we want to buy. We allow the color and form of the packaging to stimulate our imagination (Mmm, that tastes just like milk at breakfast in Bavaria, and I will feel the way I did that time I was at Lake Starnberg). The very moment we pose ourselves the question as to which kind of milk we want to buy, calculations begin that put the energies of the id (Mmm, quite tasty!) in contact with the energies of the superego (Too expensive! Milk makes us fat!). In Freud's somewhat archaically tinged image of the human being, the pre-conscious and unconscious synchronization of the layers of consciousness is rather as follows: "Mmm, I am secretly recalling my mama's breast and my infantile love for my mother," whispers the id. "You are fat and should not be, otherwise you do not deserve mama's love," roars the superego. This, or something close to it, is probably how Freud would have seen it. One need not follow him in this regard.
## _**Oedipus and the milk carton**_
The perception of various milk cartons always occurs in conjunction with evaluations that do not unconditionally have anything to do with actual facts. Our perceptions are embedded in systems of wishes and desires of all sorts and never arise in our life in some pure state that is free of wishes. All thinking always involves an element of wishful thinking. Even when, for instance, a scientist stands in the laboratory and is researching a protein, he or she must dissociate from his or her wishes in order to perceive or discern the bare facts. Yet in order to be able to dissociate from one's wishes, one must want to do precisely that. Indeed, one has already resolved to be a scientist and orders one's drives as expediently as possible in accordance with this decision. Consciously lived existence, about which we communicate in the form of reasons accepted in conversations with each other every day, is in that sense "a specially differentiated part of the id,"34 as we do indeed experience quite a variety of things but never the manner in which the choice is made as to what we are or are not to concentrate on. Hence Freud can characterize the ego as a "poor creature,"35 because he sees it as a surface on which perceptions are displayed on the one hand, and drives on the other, which are immediately filtered and censored by the superego. The ego for Freud is an interface inserted between drives and imperatives, and it deals with this by experiencing its drives as inherently rational and, therefore, justified (at least, generally or typically justified).
In the version of the distinction between perception and drive – between reality and wish – advanced by Freud, at the end of the day unfortunately everything gets so mixed up that, on closer inspection, the structure collapses. How, for instance, is the id supposed to be not merely an undifferentiated single drive, a feeling emotionally hot and bothered, so to speak, that we sense and that urges us on? How can there be many drives and not just a boiling emotional heat? It so happens that Freud understands the ego as part of the id and thus inserts an eye into the latter with which the blunt and stupid id now looks around at realities to which it can adhere, a mechanism Freud calls "cathexis." Lo and behold, the mother's breast is already there, onto which the id can latch – and the often caricatured psychoanalytic explanation of our sexual drive with reference to our deep desire to sleep with our mothers and their future surrogates or our fathers and their future surrogates runs its course in a hundred different ways in literature, film and television.
Yet what does the claim that the ego is a part of the id mean? To formulate the question in another way: If the id is unconscious and its messages or impulses reach the conscious ego only after having been filtered, how can the ego be a part of the id? It would have to be unconscious and instinctual. But then it could no longer be responsible for perceptions. The reality principle collapses into the pleasure principle, and from the science of psychoanalysis out comes Pippi Longstocking:
> Three times three makes four,
> widdle widdle wid,
> and three makes nine,
> I'm making the world,
> widdle widdle wid,
> how I like it.
Freud himself explains: "Psycho-analysis is an instrument to enable the ego to achieve a progressive conquest of the id."36 Yet what is going on in this model? Is the analyst in his or her function as conversational partner who approaches another person a representative of the external world for this person – that is, the ego – or is the analyst the voice of conscience, the superego? Or is the analyst even the id? If the id is already the ego, which indeed is supposed to be a part of it, how is the ego's conquest of the id supposed to proceed at all?
It is evident that a few things are here conflated with each other. Like he did in replying to objections raised by representatives of the philosophy of his time, Freud would now presumably respond that he learned of the relations between ego, superego and id in clinical practice, that he has empirical, scientific ground to believe that his theory is true. Yet this claim, which retreats to supposed experiences that the specialist will have had, cannot be persuasive even according to Freud's own premises. If a contradictory model of the self or ego is presented, one cannot fast-talk one's way out of it by claiming that one has already seen this in psychotherapy or, as is more typical of late, in a brain scan. Surely there are no theoretically far-fetched contradictions to be seen anywhere, neither in the talking cure nor in a brain scanner. They are believed to be there only because one is looking for them.
And yet Freud is on the right track. His reflections must be modernized, however, which in this case means above all freeing them from the erroneous assumption that the self or the ego is a biological entity that is formed by the interaction between organism and natural environment (external world), and that, moreover, a long cultural history has led to the emergence of a superego. These assumptions by Freud led to legendary myth-making – to which we are admittedly indebted for genuine insights and many works of art. Without Freud there would be no surrealism, no films by Alejandro Jodorowsky or Woody Allen, and many other things would be missing. Without him, there would probably not have been a sexual revolution, even if, as a man of the nineteenth century, he still partly clung to its ideas, above all concerning feminine sexuality, but also concerning homosexuality and, well, sexuality as such.
We are indebted to psychoanalysis for decisive advances in the political emancipation of minorities or oppressed groups. The controversy over the fact that we repress sexual wishes, indeed that we all have sexual fantasies, which are not self-evidently to be classified as perversions or to be condemned as crimes, ultimately led to the fact that we no longer consider homosexuality to be a sickness (at least this is still true for many Western societies, as I am writing this) or believe that women are seductresses, akin the serpent in paradise (or even the converse, that they are projection screens for masculine desires and not really capable of desire themselves), who come between us and the voice of God. In general terms, Freud laid the groundwork for our recognition of sexuality as a central component of our very selves.
Different branches split off in the twentieth century from his original version, branches which for their part led to new emancipatory theoretical developments, such as **gender theory** , for example. These new theoretical developments fundamentally proceed from the fact that there are gender roles that cannot be completely explained by examining our organism and determining whether we are male or female. The most prominent advocate of gender theory today, the American philosopher Judith Butler, has referred to the fact that even the search for feminine or masculine elements in a human organism – by classifying hormones as "feminine" or "masculine," for instance – is often already determined by the fact that we bring certain ideas of gender roles to bear with us and project them onto bodies.37 It is crucial that we do not ignore the insights from psychoanalysis or gender theory, despite the fact that they have been overgeneralized by their adherents and misused as an excuse to ignore new scientific evidence.
On closer inspection, Freud's psychoanalysis holds crude assumptions about gender roles in its pseudo-biological details which are, strictly speaking, rather absurd and mostly unfounded. For instance, Freud is never quite certain whether the superego represents the internalized father or the internalized parents. He does not consider the possibility that it could come from the mother, simply because he probably thought that religious and moral ideas were advocated and enforced only by men. Ultimately, his writings swarm with patriarchal assumptions supported by myth-making.
A typical procedure for the avoidance of self-knowledge is currently infected with Darwinitis. Instead of asking who or what the self really is and developing a coherent response to this question, a response that is at the same time conceptually and historically informed about what account of the self all this talk of the self brings along with it, an ultimately unknowable past is invoked. This past must go back far enough in time and be attested to only by a few archeological findings, such as skulls, perhaps also a spearhead, in any case by a cave painting. Then one is able to narrate popular histories of how we originated from this past in order to make it look as if one has thereby achieved a better self-understanding.
This is mythology. The **main political function of mythology** consists in forming an idea of the total social situation of one's own time by imagining a primordial time. The less one really knows about this past, the more inventive one can be. Freud himself, in his wonderfully written book _Moses and Monotheism_ , tells a story according to which we have a conscience and believe that there are objective moral laws, and thus in general something like good and evil, only because a primal horde killed Moses in the desert. Having a bad conscience emerged as a reaction to this slaughter. But how would this have worked and how did they transmit their bad conscience to the next generation? Why did they feel so bad about this particular act of killing and how does Freud know any of this, as there is no written or other record of this extraordinary "fact"?
Freud famously thinks that the essential theme of individuation, the cultivation of a self, lies in the male wish to murder one's father and sleep with one's mother, the much discussed **Oedipus complex** , to which Carl Gustav Jung (1875–1961) added an Electra complex – that is, the equivalent for girls. Freud did not approve of Jung's addition, as he explained in his essay "Female Sexuality" of 1931. These themes are in fact found in many mythological texts from the past, primarily of course in Sophocles' tragedy _Oedipus the King_.
Today there is a widespread tendency to reduce good and evil to value judgments that came to prominence in the course of evolution because they were useful. Something which was conducive to survival counted as "good" and something which endangered the survival of the species _Homo sapiens_ counted as "evil." In particular, there is discussion of why **altruism** emerged – that is, the fact that living beings sacrifice their own well-being or even their lives for other living beings, or take an interest in the well-being of others at all. The presupposition for this question is the assumption that from the outset all living beings are actually egoistic. This is how the opposition egoism–altruism emerges.
Against this backdrop, the evolutionary biologist Richard Dawkins, for instance, in his widely noted book _The Selfish Gene_ , has attempted to explain why we consider our relatives to be morally closer to us than random strangers.38 He thinks that what drives egoism is not the individual (that is, not you or I) but a certain gene which we represent. Since this gene is also present in our relatives, we protect them even if it brings about the sacrifice of our own interests. Thus it becomes understandable why we normally prefer a stranger to die rather than our own children (but how do things then stand with adopted children or friends? – one of the many problems immediately prompted by this line of thought).
And how does one know that any of this is actually the case? If we are asked in the context of our everyday socially arranged, more or less amicable relationships, many of us will quickly answer that we would protect our relatives before protecting random strangers. However, this is not simply a universal truth (moreover, it should be noted that our relatives do not all belong to the same gene pool...). Here is a simple counter-example from the present day. Until the end of August 2014, the town of Amirli, which was overwhelmingly populated by Shia Turkmen, was besieged by jihadist troops from ISIS, who threatened to massacre them. During the siege, an Iraqi helicopter occasionally flew into the town to ensure that food was provided and to fly the sick and wounded out. The local dentist was assigned the task of caring for all of the sick and wounded, while his own family was still stuck in Amirli. What is more, he told the German magazine _Der Spiegel_ that he deliberately did not put his own family on the helicopter and have them flown out, since a panic would otherwise have ensued.39 Things would have ended badly if the doctor had pulled strings to save his family. It is not difficult to find countless examples (but also countless counter-examples) in which someone acts altruistically, according to generally accepted standards, and subordinates his or her own relatives to the common good. Why is genetic egoism supposed to be the rule and altruism an exception that is hard to explain?
The point is that we just do not have any data that sheds light on whether and how some humanoid or human ancestor acted in prehistoric times, which principles they followed and why they did what they did. More or less arbitrary assumptions about ordinary behavior in one's own society (or what is considered to be ordinary behavior in one's own society) are simply projected onto the past and combined with facts from evolutionary biology. In this way, mythology appears to be scientific and one is not directly exposed to public criticism. Here we have a strategy for immunizing oneself that is easy to see through. This is only too willingly concealed behind a perfidious lip-service to science, which on the basis of universal human reason is supposedly opposed to appeals to authority. However, there is a significant gap between natural science and philosophical interpretations of natural science. You simply cannot read philosophical truths off from science without first and foremost resolving the conceptual philosophical issues. This does not rule out the fact that philosophers need to respect natural science and acquire as much knowledge as possible in its manifold fields. However, there is a widespread tendency in our culture (more so in the Anglophone than in the German-speaking world) to outsource philosophical issues from philosophy to natural science, which is a fundamental mistake.
It is simply not the case that overall we tend to be egoistic. But it is also not the case that we generally prefer to care for others and sacrifice ourselves. Precisely therein lies human freedom: we are able to dissociate ourselves from our own standpoint by understanding that others have standpoints, too. This insight is associated with the mind's account of itself as a self. In particular, the self is both something individual (just you or I) and something universal (each of us is a self). However, by now it should be evident that the self cannot be the brain or a gene, let alone a gene pool. Rather, it is the case that without brains of a certain type the dimension of the self would never have been able to develop historically. Brains are a _necessary condition_ for the fact that there are practices in which selves are involved. Yet the discovery of the self takes place within the context of historical processes of self-knowledge.
To this point, I have reconstructed a few cornerstones of this history, which in the West stretches back to Greek philosophy. "The self" is a philosophical concept, and it fulfils a function in the account of ourselves. It belongs to our self-portrait. The point is to pay attention to this account and to ask what assumptions are in play and whether any kind of coherent picture of the self can be constructed from them.
The self was also introduced so that we might understand what it means to be able to act in a good or an evil manner. It does not help to want to get rid of the concepts of good and evil by replacing them with the conceptual pairings of useful and harmful, or altruism and egoism. This only obscures the intention to introduce another vocabulary to account for the self. Yet this other vocabulary then typically adopts elements of its antecedent, and one still speaks of the self – or of the ego, as Freud does, for instance – in order to suggest that it has been only a biological matter all along. The self thus became reified by considering the account of it to be a biological issue.
We can understand ourselves as selves, can experience ourselves as conscious and self-conscious, can know and communicate thoughts. None of this can be sufficiently explained by realizing that we need a specific kind of organism to be able to do this. If we believe that such an explanation is sufficient, we conflate the necessary biological or natural conditions for the fact that we are living beings endowed with a specific mind with elements of our account of ourselves that have arisen historically. This conflation is a basic form of ideology, and behind it lies concealed, in each instance in a new way, the attempt to get rid of freedom and ultimately to become a thing. Neurocentrism is an ideological fantasy of self-objectification.
## **Notes**
1. Thomas Metzinger, _The Ego-Tunnel: The Science of the Mind and the Myth of the Self_ (New York: Basic Books, 2009). 2. Susan Blackmore, _Conversations on Consciousness: What the Best Minds Think about the Brain, Free Will, and What it Means to be Human_ (Oxford: Oxford University Press, 2006), p. 153. 3. Ibid. 4. Thomas Metzinger, _Being No One: The Self-Model Theory of Subjectivity_ (Cambridge, MA: MIT Press, 2003). 5.Blackmore, _Conversations on Consciousness_ , p. 203. 6. Ibid., pp. 198–9. 7. Karl Vogt, "Physiologische Briefe: 12. Brief," in Dieter Wittich (ed.), _Vogt, Moleschott, Büchner: Schriften zum kleinbürgerlichen Materialismus in Deutschland_ (Berlin: Akademie, 1971). 8. Georg Wilhelm Friedrich Hegel, _Phenomenology of Spirit_ , trans. A. V. Miller (Oxford: Oxford University Press, 1977), p. 210 (§346). 9. Ludwig Büchner, _Force and Matter: Empirico-Philosophical Studies, Intelligibly Rendered_ (London: Trübner, 1864), p. 117. 10. Ibid., p. 112. 11. Raymond Bernard Blakney, _Meister Eckhart: A Modern Translation_ (New York: Harper & Row, 1941), Sermon 52, pp. 231–2; translation modified. 12. Ibid., Sermon 48, pp. 220–1. 13. Thomas Nagel, _The View from Nowhere_ (Oxford: Oxford University Press, 1989); Gabriel, _Why the World Does Not Exist_ (Cambridge: Polity, 2015), p. 7. 14. Hegel, _Phenomenology of Spirit_ , p. 110 (§177). 15. Johann Gottlieb Fichte, _The Science of Knowledge_ , ed. and trans. Peter Heath and John Lachs (Cambridge: Cambridge University Press, 1982), p. 162n. 16. Ibid., p. 96. 17. Ibid., p. 110. 18. John Searle, _Intentionality: An Essay in the Philosophy of Mind_ (Cambridge: Cambridge University Press, 1983), pp. 141–59. 19. Nagel, _Mind and Cosmos: Why the Materialist Neo-Darwinian Conception of Nature is Almost Certainly False_ (Oxford: Oxford University Press, 2012). 20. Richard Dawkins, _The Blind Watchmaker: Why the Evidence of Evolution Reveals a Universe without Design_ (New York: W. W. Norton, 1996). 21. Sigmund Freud, _The Ego and the Id and Other Works_ (London: Hogarth Press, 1961), p. 17 (the Standard Edition of the Complete Psychological Works of Sigmund Freud, Vol. 19, 1923–1925). 22. The German word for science, _Wissenschaft_ , is feminine and hence becomes a "goddess," _Göttin_ , when fetishized in this manner Trans.]. 23. Freud, _The Ego and the Id and Other Works_ , p. 13. 24. Ibid., pp. 15–16. 25. Ibid., p. 28. See also Freud's "Character and Anal Eroticism," in _Jensen's "Gradiva" and Other Works_ (London: Hogarth Press, 1959), pp. 167–75 (the Standard Edition of the Complete Psychological Works of Sigmund Freud, Vol. 9, 1906–1908). 26. Freud, "Instincts and Their Vicissitudes," in _On the History of the Psycho-Analytic Movement: Papers on Metapsychology and Other Works_ (London: Hogarth Press, 1957), p. 116 (the Standard Edition of the Complete Psychological Works of Sigmund Freud, Vol. 14, 1914–1916). 27. Robert Brandom, _Articulating Reasons: An Introduction to Inferentialism_ (Cambridge, MA: Harvard University Press, 2000), p. 11. 28. Freud, _The Ego and the Id and Other Works_ , p. 23. 29. Ibid., p. 25. 30. Ibid., p. 26. 31. Ibid., p. 38. 32. Ibid., p. 36. 33. Ibid. 34. Ibid., p. 38. 35. Ibid., p. 56. 36. Ibid. 37. See, for instance, Judith Butler, _Gender Trouble: Feminism and the Subversion of Identity_ (New York: Routledge, 1990). 38. Richard Dawkins, _The Selfish Gene_ (Oxford: Oxford University Press, 1976). 39. Christoph Reuter and Jacob Russell, "Die Vergessenen von Amirli," _Der Spiegel_ , August 25, 2014; [www.spiegel.de/spiegel/print/d-128859935.html.
# 5
**Freedom**
In recent years, the classical debate whether we are free, or, rather, whether we have free will, has assumed a new shape, as some neuroscientists have entered this discussion with supposedly new findings that might speak against free will. Some new discoveries in brain research seemed to suggest for a while that even decisions that we consciously make and which then determine our actions must have been unconsciously prepared in the brain.1 It thus appears as though our decisions do not lie in our hands. The idea that our brain could be directing us was born.
The debate about free will is not new. It has been prevalent for a long time and took a new turn in the nineteenth and the early twentieth century, when the suspicion arose that the human will is defined or determined by the fact that, although we are living beings with minds, we nevertheless belong to the animal kingdom. At that time, it was Darwinism, nascent sociology, psychology and also brain research to which one looked for evidence that the human being did not really have free will.
In fact, we know that there are many factors that influence the decisions we make and the personality we develop. We are not in control of our preferences in the sense that we are able to choose all of them in the same way we are able to choose an appetizer in a restaurant or a brand of sausages in the supermarket. We come to the world with preferences that are in part genetically determined, and over the course of our life we cultivate further preferences in our association with other people and with authorities, without being consciously aware of the mechanisms for selection that in the end result in patterns of conduct.
Such truisms, which have fortunately long since been acknowledged, have certainly shaken the outdated (never uncritically predominant) image of the human being, according to which each of us is an autonomous self to the extent that there is a command center for our life. On this view, we are situated in such a command center and decide with complete freedom, in a frictionless vacuum, who or what we are and what we want to do. This model has rightly been recognized as a variant of the homunculus fallacy. If we had free will only in this sense, if we were the helmsman of such a command center, we could not in fact have free will at all. The idea of a completely autonomous little helmsman within our skull or in the depths of our soul is simply incoherent. But we do not need neuroscience to tell us that, as the incoherence is a conceptual matter: it simply is not possible to be responsible for all your preferences in the sense of having to create them _ex nihilo_. If we had to create ourselves in this way, we would never come into existence in the first place. No one is absolutely autonomous; no one acts in a void. Hence, it cannot be a requirement of actual free will to be absolutely autonomous.
Furthermore, it is a fact that many decisions which we experience as consciously made are unconsciously prepared at the neural level. This seems to speak in favor of the claim that our brain directs us, in which case "we" would then just be a conscious user interface able to have experiences and the brain would be the real central processor. A part of the brain whose activity we do not experience would accordingly somehow direct the very activities of the brain that we consciously experience or in which consciousness arises.
That is roughly the basic structure which, along with many others, the German neurophysiologist Wolf Singer outlined in a much discussed article published in the newspaper _Frankfurter Allgemeine Zeitung_ on January 8, 2004, entitled "No One Can Be Other Than They Are" – an idea that he further elaborates in his essay "Neural Connections Determine Us" as well as in subsequent books.2
Naturally, there is no reason to deny the fact that our organism functions as a biological entity in such a way that many operations of information-processing and decision-making must occur without our noticing them. In a similar vein, the Nobel prizewinner Daniel Kahneman (b. 1934), in his book _Thinking, Fast and Slow_ , refers to how important it is for us as living beings with minds that we can think so quickly, that we make decisions without going through explicit and time-consuming reflections.
All of this has already been common knowledge for a long time in the philosophy of consciousness and is discussed there, for instance, under the heading of the **Dreyfus Model of Skill Acquisition**. As the name suggests, it was developed in particular by the philosopher Hubert Dreyfus (b. 1929), who teaches at Berkeley and who has pointed out that actual _expertise_ is distinguished from mere _competence_ by the fact that an expert just sees right away what action has to be taken in a given situation. To be knowledgeable in an area consists in transforming consciously available information into unconscious processes which structure patterns of behavior.
There has been good research showing that the acquisition of competence in chess does not happen in such a way that the grandmaster can calculate better positions, as it were, and can thus become ever more like a computer. In truth, the concept of intuition plays a large role in the game of chess. Good players see possibilities in a position and choose paths and moves intuitively because they are short on time. Only then do they calculate the paths they have chosen. Which paths and moves are arrived at on the basis of calculations and which are excluded in advance (because they are counter-intuitive) determines the playing ability of a chess player, among other things. One certainly needs to be able to calculate very well and to have a proper visual imagination, but without intuitive insight into the essential structure of a given chess position chess masters would not be able to play as well as they do (any more than we could drive from one exit on the highway to the next without an intuitive overview of the situation, which relies on unconscious information-processing and percepts we do not take notice of).
We are all familiar with such phenomena: when one gets one's first driver's license, one might think that good drivers would continually keep in mind all of the different skills and knowledge of the rules that they have acquired. Yet, the more experience one has of driving a car, the less one explicitly recalls the rules by which one has learned to drive. The driver probably forgets the specific wording of the rules, although they remain second nature. The same holds true for those who know foreign languages. As soon as one speaks a language fluently, one no longer needs explicitly to recall the rules to be applied, which have probably been forgotten. Thus it is generally true that we must make use of an **unconscious background of skills** , as Dreyfus's colleague Searle puts it. Accordingly, even at the high level of playing chess or solving mathematical problems, we are not necessarily conscious of the cognitive skills we are engaged in, nor are we conscious of all the information-processing required in order to carry out even mundane tasks, such as crossing the street.
Yet why should the fact that the many processes we unconsciously call on to make decisions – processes which escape our attention – somehow threaten our freedom or free will? Singer argues as follows: "If it is granted that conscious deliberation on arguments rests on neural processes, then it must be subject to neural determinism in the same way as the unconscious decisions where we concede this happens."3 He assumes that there are "genetically determined... fundamental neural connections"4 and that they determine all our conscious decisions (including his belief that his view is correct). Accordingly, he has not freely decided on his own theory of neural determinism. His brain dictated it to him. Had his theory been true, he would have been lucky to have a brain which communicated the theory to him.
At first glance, it might seem entirely plausible to some that their brain dictates their thoughts to them. On closer inspection, however, it turns out that any view along those lines is quite untenable. Let us have closer look, then! Let us start from the notion of **naïve determinism** , on which Singer's position is based. In a nutshell, naïve determinism is the claim that everything which happens in nature proceeds according to natural laws from which there can be no deviation and which at any moment determines exactly what will happen in the next moment. Just as everything that one throws out of the window falls onto the street – as a result of gravity, a law of nature – so everything in nature occurs inevitably in the way prescribed by the laws of nature. The statement of naïve determinism immediately invites all sorts of objections, since, for instance, some things that one throws out of the window fly upwards (a healthy bird, for instance, or balloons filled with helium). But the basic idea seems to be clear, in any case. The basic idea is that nothing goes against the laws of nature, from which the determinist concludes that everything which happens takes place in accordance with the laws of nature. Since our neural processes belong to nature, they too are simply reeled off according to fixed rules. And thus we would no longer be free, because we would be defined or determined by the fundamental neural connections of our brain. Free will would be an illusion whose advantages and disadvantages for evolution one could continue to discuss.
Many arguments in the history and philosophy of science speak against determinism in this sense, arguments that were recently elaborated in a particularly distinct way by the German philosopher and physicist Brigitte Falkenburg in her lucid book _Mythos Determinismus_ (The Myth of Determinism). Let us take as an example the following simple line of thought: since the time of Galileo's formulation of the law of falling bodies, we know that, in a vacuum, a cannonball and a crystal ball fall with equal velocity.
Thus one can formulate a law of nature on this basis that expresses relationships which can be rendered mathematically precise. Yet, this law of nature says nothing about what actually happens when I throw a feather out of my window. What if a wind blows from below and propels the feather upwards? Furthermore, feathers and cannonballs do not actually fall out of my window with equal velocity, something which holds true only in a vacuum. It quickly becomes evident that laws of nature only hold true without exception when we carry out certain idealizations. Laws of nature do not describe what actually happens at some moment in nature or even what must happen; rather, they describe idealized conditions. Hence, one cannot predict on the basis of knowledge of natural laws alone what will happen in the next moment.
The laws of nature thus do not stipulate, so to speak, what must happen. They are not like rules of a game, which are fixed and which determine precisely what the next move will be. In contrast, this does hold true for movies, since the events that one can repeatedly view by fast-forwarding and rewinding always run according to the exact same pattern. In light of this, let us call naïve determinism the **movie theory of nature**. It claims that nature, in the eyes of an omniscient external observer (if such an observer were even possible), would be like a movie in which the exact same thing always happened every time it is replayed. To be sure, this is a widespread but quite untenable idea of how nature functions, as if it were a very complex machine that always proceeds from one state to the next according to the same principles. Of course, none of this follows from the present state of knowledge in physics. Determinism is metaphysical speculation, at best, and not a hypothesis that has been or even can be proven by physics. As Falkenburg notes: "... determinism can _always be saved_ by _some kind_ of speculative metaphysics. Indeed, this shows only that it is not an empirically testable scientific hypothesis but a sheer case of belief."5 For this reason, the neuroscientist cannot simply appeal to the fact that our neural processes are determined, insofar as this is supposed to mean that they are somehow inserted into the gigantic machinery of nature that changes from one state to the next without our help, our wishes or our volition. Determinism is not a scientifically proven hypothesis but is actually a case of wild philosophical speculation.
In any case, it is certain that there are natural conditions for how we are able to act in general, and we cannot change all of these conditions. Whether we want to or not, we cannot travel anywhere near the speed of light and certainly not faster than that – in any case, not by blasting off and accelerating into the universe in some kind of tin can. Hence recent science-fiction films such as _Interstellar_ must introduce "wormholes" into space-time to keep alive the vain hope that one day we will live in outer space and on other planets in order to escape our earthly conditions and found colonies elsewhere.
It seems hard for many of us to recognize that as a biological species we are stuck on this planet – which is still a wonderful place to be – and that sooner or later human beings will not exist any longer. Sooner, if we do not live in a demonstrably more environmentally sustainable manner in the near future. Later, if we survive until our planet disappears for good along with all human intelligent life, never to be seen again, as our dying sun expands.
Unfortunately, the rejection of naïve determinism as bad metaphysics alone does not suffice to return our free will to us, as there is a more fundamental conceptual threat to our understanding of ourselves as the free agents we happen to be.
## _**Can I will not to will what I will?**_
There is a serious problem with free will which goes beyond the metaphysical question of whether the movie theory of nature is true. The **hard problem of free will** , as I call it, is a paradox – that is, a series of claims all of which intuitively we must accept, but which together entail an untenable conclusion. The positive part of the paradox looks like this:
1. I can do what I want (this is **freedom of action** ).
2. I can influence what I want by forming a will (an intention to do something).
3. To form such a will (an intention to do something) is an action.
4. Therefore, free will exists (this is **freedom of the will** ).
This good news is unfortunately rescinded by the negative part of the paradox.
1. If the will is freely formed, I can also form another will.
2. I must thus not only be able to do what I want but must also be able to choose for myself what I want (to set an agenda for myself).
3. But I cannot choose for myself what I want ("I can do what I will... But I am not able to _will_ it," as Schopenhauer put it).6
4. Consequently, the will is ultimately not freely formed and is thus unfree.
5. Therefore, free will does not exist.
Naturally, my freedom of action is limited since I cannot always do what I want. What is worse, if to do what one wants always presupposes the prior activity of forming a will, we would never be entirely free, because the fact that we have a will itself, of all things, would make us unfree. Schopenhauer is right to point out that I cannot choose my agenda from scratch. What is more, I cannot change my will from scratch either, since I would have to want this, too. The desire to change my will would then be what determines me to carry it out – something I cannot in turn choose on pain of a vicious infinite regress. Therefore, at some point, one must indeed assume that one simply just wants something for no reason whatsoever. However, one would apparently not be free then but, rather, determined by one's own will, which in principle cannot be chosen. If there were such a thing or agent as our will, she, he or it could actually not be free. It would lie in the nature of the foundation of the freedom of action that we are never really free.
In November 2010, I traveled to a philosophy conference through the state of Goa in India. At some point, a truck with an advertisement for a product named Zinzi drove in front of our taxi (no one has paid me yet for my covert product placement here). On the pink advertisement alongside the product's name was written only the memorable sentence "Find any reason." Until I wrote these lines, I figured that it must have something to do with a soft drink. I just found out that Zinzi is an Indian red wine.
The point of this is that, instead of giving us a reason to want precisely _this_ drink, the apparent freedom to allow some kind of arbitrary reason to occur to us is demanded of us so as to justify the ultimately groundless wish to drink this beverage. In short, at some point in the system of our preferences we always run into groundless desires and patterns of preference, a fact the ad for Zinzi ironically exploits. One can often still do what one wants on this basis, but one can no longer want what one wants. Our reasons seem to be pretexts, convincing our will to "Find any reason!"
If neural processes run their course unconsciously and lie behind these patterns of preference (as Singer claims), we seem to have arrived at an improved argumentative version of his position of neural determinism. The groundless will, which we simply have in a given way, only has to be identified with a neural pattern. If you combine neuroscience with Schopenhauer's paradox, you might claim to have found empirical support for the paradox's conclusion that no one is ever really free, or that there is no free will.
In philosophy, there are a variety of attempts to solve the hard problem of free will. What needs to be made clear at the outset is that the problem has very little to do with the brain. Whether my will is identical to neural processes or is formed by unconscious neural processes – which is why I am not able to choose what I want but, rather, have to find it out – plays only a subordinate role in the hard problem. For thousands of years there have been other variants of the hard problem of free will which did not involve our brain.
For instance, for a long time **theological determinism** was central. It taught that, as an omniscient being, God already knew before creation everything that was going to happen. As a human being, then, one apparently cannot change this, since what was going to happen to one was already fixed before birth. Accordingly, as the classical argument runs, we have no free will ( _liberum arbitrium_ ) but an unfree will ( _servum arbitrium_ ). Incidentally, Martin Luther himself belongs to this tradition. In 1525, he composed an elaborate text _On the Bondage of the Will_ ( _De servo arbitrio_ ), in which he unmistakably notes that:
> It is most necessary and most salutary, then, for a Christian to know this also; that God foreknows nothing contingently, but foresees and purposes, and accomplishes every thing, by an unchangeable, eternal, and infallible will. But, by this thunderbolt, Freewill is struck to the earth and completely ground to powder... Hence it irresistibly follows, that all which we do, and all which happens, although it seems to happen mutably and contingently, does in reality happen necessarily and unalterably, insofar as respects the will of God.7
Luther calls this a "'paradox" because it amounts to "publishing to the world that whatsoever is done by us is not done by Freewill, but by mere necessity."8 There are countless attempts, some of which are ridiculous in their hairsplitting, at solving this theological problem. For now, it is crucial only to note that determinism does not necessarily have anything to do with the brain. God can also apparently threaten our freedom or not even allow it to emerge in the first place, as Luther believes.
In contrast to this, **physical determinism** (at least one prominent variety) claims that we are not free because, in truth, only the things and events that particle physics talks about exist, since the entirety of physical reality is composed of elementary particles (which, it should be noted, is a metaphysical claim, not something necessitated by physics). Depending on which laws one postulates to understand the behavior of elementary particles, everything happens according to necessity or at least according to laws of probability – perhaps with a dash of randomness or uncertainty in quantum reality. Yet randomness or uncertainty is nowhere near freedom, as we will soon see. Thus in this model, too, we are unfree, but not so much because the movie theory of nature is true but, rather, because we do not really exist according to this metaphysical interpretation of physics.
**Neural determinism** adds to the hard problem of free will the claim that unconscious neural processes, which run their course in the brain, merely follow basic established neural connections and thus make decisions for us – that is, in place of a consciously experiencing, supposedly autonomous subject – before any supposedly consciously made decision. The brain supposedly decides for us, and indeed it fundamentally does so in such a way that we have no more direct influence on it than we do on our blood sugar level or digestive processes.
All kinds of things accordingly seem to threaten our poor little freedom: God, the physical universe, and even our very own brain. Yet these are only a few examples of the hard problem of free will, which actually does not have much to do with such things. At bottom, the problem is much more troublesome than any machination of the brain, God, and the universe altogether could be. In the case of the hard problem, it is a question of whether the concept of free will is even coherent or whether it is not ultimately as absurd as the concept of the biggest natural number. Closer inspection of our concepts (or elementary mathematics) shows that there cannot be a biggest natural number. Perhaps closer inspection of the concept of free will will also show that this, too, cannot exist at all, since the concept itself is incoherent. In any case, this is the hard problem that really lies concealed behind theological, physical and neural determinism.
The hard problem of free will proceeds from the fact that our decisions are conditioned by something that for the most part we never survey. We are certainly never in a position to survey all necessary conditions for our acts, as we do not know all the almost infinitely many details that are involved in what we do and why we do it. Against this background, Spinoza already almost cynically pointed out that freedom is the consciousness of our actions without the consciousness of the determining grounds for our actions.9 In other words: we consider ourselves to be free because we do not know what exactly is determining us. According to this view, free will is an illusion which arises on account of our ignorance of the determining factors of our actions.
The claim that we really only consider ourselves to be free out of ignorance of the necessary conditions for everything that happens is deeply unsatisfying. Our freedom would then consist only in the fact that we are too stupid to realize that, in reality, we are not free. We are quite justified in expecting more than such a cynical theory of freedom.
## _**The self is not a one-armed bandit**_
Against the backdrop of the hard problem of free will, the very concept of the will was introduced to the history of philosophy. Admittedly, this concept was severely criticized, though it was also accepted by many. Contemporary philosophy mostly replaces it with talk of capabilities and capacities that we can or simply cannot put into practice. Freedom is thus sought in the fact that we have capacities and capabilities. Let us call this approach the **capacity theory of human freedom**. The capacity theory replaces talk of the will by talk of capacities in order to circumnavigate the hard problem of free will.
However, capacity theory, at bottom, is not really very helpful since, rather than solving the paradox of free will, it only defers it. The problem now runs as follows: suppose that I take measures to become a better swimmer. In order to reach this goal, I must decide both for and against certain actions. I must either wake up earlier to go swimming before work or schedule my daily routine in a completely different manner right away. I need to eat less Wiener schnitzel and more salad, and so on. To the extent that, in principle, no obstacles stand in my way, I could decide to become a better swimmer – something which could, however, fail to happen for various reasons. Thus situations arise in my life, after I have made decisions, in which I must choose: schnitzel or salad; sleeping in or getting up when I hear the alarm clock. Nothing prevents me from choosing salad. In this regard, I am free to do what I want.
But when I have chosen the salad, am I then still free not to have chosen the salad? Obviously not: I have indeed chosen the salad, as was already mentioned. I have made my decision. But how does this happen? One possibility is to claim that I have chosen between salad and schnitzel. But, in this case, I have not chosen between my choice of salad and my choice of schnitzel, but between salad and schnitzel. I choose not my choice but an action: to order salad or to order schnitzel. But is my choice then free at all?
If one wants to prevent the paradox at this point from resurfacing, one should absolutely not say that my choice is a further action that is free. For then the question would arise as to whether my action of choosing in turn was free or not. If it was free, there was another choice – the choice to carry out this action of choosing. This was in turn either free or unfree. Once again, we run into a vicious infinite regress: one must choose to choose to choose, and so on. At some point one must choose something without choosing that one chooses it. Sooner or later a decision needs to be made which cannot have a further ground.
But how does this decision come about? Capacity theory, as its name suggests, has it that the decision is the exercise of a capacity. Traditionally, this capacity was identified with the will. On closer inspection, capacity theory ultimately replaces the problems of the will by multiplying the problems with recourse to a bunch of capacities which are supposed to interact. Be that as it may, capacity theory needs to address the following regress: if I exercise my capacity of necessity in the sense that it is not up to me either to exercise it or not, I am unfree. However, if the reasons for the exercise of my capacity are supposed to be up to me, the regress begins. If Fred (the will) is unfree, he does not become freer by renaming himself Freddie (capacities). The hard problem of free will persists.
Naturally, the capacity theorist assumes that it is not the case that there is always a multitude of necessary conditions that compel my capacity to be exercised. The exercise of my capacity is supposed to be free, without my thus being permitted to have chosen ahead of time to exercise my capacity. In exercising my capacity, I need to be free and, hence, unconstrained by prior parameters which would undermine my capacity to act in one way or another.
This gives rise to a version of the **problem of chance** , however. One can illustrate this problem as follows. Imagine that our self functioned like a one-armed bandit in Las Vegas. Assume further (which of course is not quite right) that the reels were not preprogrammed to spin on the one-armed bandits. It would thus actually be a matter of chance whether the three reels, for instance, displayed "cherry, melon, 10" or "cherry, 10, 10." The problem of chance arises in that it is hard to avoid the claim that the decision as to whether we are exercising a capacity or not is like pulling on a one-armed bandit. In this case, our decisions would not be determined, they would not be preprogrammed. In each and every case, determinism would thus be false.
However, in its place the capacity theorist has so far merely put **indeterminism** , which claims that some events occur without either necessary or sufficient conditions compelling them to happen. The problem of chance consists in that our actions, according to capacity theory, are indeed free, but that the decision to exercise a capacity is ultimately left to chance.
A typical move of the capacity theorist at this point is to allow for the notion that we have various capacities that work together and thereby solve the problem of chance. I have the capacity to improve my swimming, to control my eating behavior, to change my sleeping habits. This interplay of various capacities is supposed to minimize chance, since they are all unified in me as an agent, a picture which makes the exercise of my capacities look less random and more specific to me. Yet, this maneuver just obscures the fact that, for every capacity, it is also true that either it is activated by necessary and jointly sufficient conditions or it is not. The problem of chance thus recurs. The more capacities, the more one-armed bandits.
A one-armed bandit is not free but a chance machine. Sure, the counter-argument against the capacity theory just sketched presupposes that there could actually be one-armed bandits that produce genuinely random outcomes – a strict naïve determinist would already doubt whether this is conceivable. But even if there were chance machines and our capacities were exercised like one-armed bandits, this would not make us one jot freer. If we are played like a throw of the dice, so to speak, this does not make us any freer than if we are "compelled" by necessary conditions or the neural connections in our brain to exercise our capacities, a thought that is reflected in a highly refined poetical manner in Stéphane Mallarmé's masterpiece _Un coup de dés jamais n'abolira le hazard_ (A Throw of the Dice Will Never Abolish Chance). If the actual exercise of our capacities were activated by chance on every occasion, this would make us not free but dependent on chance. Whether one is dependent on fate, natural laws or chance plays only a subordinate role in the formulation of the hard problem of free will.
The American philosopher Peter van Inwagen (b. 1942) – incidentally, an avowed Christian thinker – introduced a famous and helpful distinction in his _Essay on Free Will_ of 1983, one of the most influential texts in the literature on our theme.10 On closer examination, I believe his distinction does not stand up to analysis, but it does point us in the right direction. He distinguishes two possible positions in the debate between free will and determinism, one of which he calls **compatibilism**. Compatibilism assumes that freedom and determinism are actually compatible, and so it is perfectly consistent, on the one hand, for there to be necessary and jointly sufficient conditions for everything that happens – conditions that can extend far into the past – and, on the other hand, for us to be free. Van Inwagen calls the other position **incompatibilism**. This position correspondingly claims that we can only be free if determinism is false, since freedom and determinism are incompatible.
In my view, incompatibilism fails to the extent that it cannot solve the problem of chance. At some point in their arguments, incompatibilists always assume that we have capacities – or a will – that are activated so as to trigger an action (or motivate it in some such manner that freedom, as they conceive of it, is not threatened). If the activation of our capacities at any point functions like a one-armed bandit, however, one has failed to salvage freedom, even if it is also true that one has reined in determinism. This is why compatibilism is the safer bet. As the German philosopher Thomas Buchheim (b. 1957) writes: "Ultimately all of our activity must be determined by _some_ factors, otherwise it would be a product of chance and not a person's decision."11
Usually, incompatibilists reproach compatibilists for giving up freedom. There are very detailed points of disagreement here. One can, however, immediately reject the basic reproach: compatibilism maintains not that we are not free but that everything that happens to us is determined, and that determinism and freedom, seriously considered, are compatible.
In my view, the debate really hinges neither on freedom nor on free will but on the world picture or metaphysics that is operative in the background. Almost everything depends on the meaning of "determinism," as some forms of determinism are compatible with free will, whereas others are not. Evidently, if "determinism" referred to the view that non-conscious processes in the universe force us to do whatever we think we do out of our own choice, determinism would be incompatible with freedom. It is not hard to work out a meaning for the term "determinism" such that determinism clashes with freedom. Yet, this does not mean that determinism in the most plausible sense of the term (on which it has a chance to be true) conflicts with freedom.
The crucial first point of my own contribution to this debate is the claim that there is no all-encompassing reality through which a single causal chain runs from the beginning of all time. The claim that there is such a chain is a world picture against which I argued in more detail in _Why the World Does Not Exist_. Since _the_ world that would connect everything together does not exist, there is no way to pose a problem for freedom on such a metaphysical basis. The universe is not an all-encompassing reality that reels off everything which happens on a single timeline, as though time were a chain on which everything that happens hung like pearls. The universe is also not a movie from which nothing and no one can escape. The movie theory of nature is simply false, which is not to deny that many elements in the universe can be understood deterministically. It is, however, not the case that absolutely everything that takes place falls within the domain of natural phenomena governed by strict deterministic laws which are incompatible with free will.
The hard problem of free will, however, does not even presume such a metaphysics of nature. It is not concerned with whether we are only pearls on a chain, which from the very beginning of time are continuously bound to the inexorable anonymous laws of nature. The fear that incompatibilists stoke is the fear that we could be the playthings of an anonymous causal chain. Yet one need not assume this at all in order to formulate a version of determinism. The idea of an anonymous causal chain is a red herring in the question of determinism. Since it presupposes a metaphysically false idea of the universe, every determinist should gladly disregard it because it does not contribute anything to what is at stake.
Let us return to the everyday level and simplify the thought at stake with recourse to an example. Imagine that I find myself at a buffet in a cafeteria during lunchtime and must choose whether I would prefer to have a baloney sandwich or pasta. My freedom of action consists in the fact that I can decide for myself whether to have the sandwich or pasta. Nothing prevents me from making this decision. On the contrary! Imagine now that I take the sandwich; this is no accident. Indeed, I can tell you exactly what led me to such a decision: yesterday I had pasta, the sandwich looks more appealing, I am not a vegetarian, and I would like to get a rise out of my vegetarian companion. Of course, neural processes are also taking place, without which I could not have any thoughts and which are possibly also guided by microorganisms that belong to my gut flora.12 Moreover, these processes are restricted by laws of nature, since my thoughts rely on the existence of time, and no information can be conveyed at a velocity greater than the speed of light. In addition, the sandwich lies on a counter, which among other things is related to the fact that the earth's gravity is 1g. If it were greater, the sandwiches would be flat and I would not be able to lift anything. If it were less, the sandwiches would fly through the cafeteria.
Thus, there are a number of necessary conditions, including at the very least the ones just listed. All of this pertains to my choice of the sandwich. Otherwise I would have perhaps chosen the pasta or something else entirely. In this sense, all the necessary conditions, taken together, are sufficient. If they are all present, nothing is missing and I do indeed grab the sandwich. My freedom cannot lie in the fact that there is a kind of gap in between the conditions. If it did, we would again be at the mercy of a one-armed bandit. Freedom and determinism are thus not incompatible to the extent to which determinism means commitment to the absence of gaps in the set of necessary and jointly sufficient conditions for an event which actually takes place.
The point of this reflection is that not all conditions are causal or even anonymous and blind. It is not as if all conditions of my freedom of action are tangible causes that push me around, so to speak, urging me on and ultimately compelling me to opt for the sandwich. Furthermore, some conditions are just not causal at all.
A simple example: I must pay for the sandwich. It is for sale. But this does not cause me to buy the sandwich, even though it is a condition for my choice. Our economy indicates exchange of goods not governed by deterministic causes as it is built on the free negotiation of the exchange value of goods and services. Indeed, this is one of Marx's most basic insights: the reason why cigarettes cost what they do is not because they have a certain completely objective use value but, rather, because processes of negotiation have taken place that terminate in a given price. Thus, normally, things do not cost exactly what they are worth in themselves, which is why _exchange value_ always differs from _use value_. This can be observed simply by noting that the price of a good can change in the course of time without the good itself changing. It is enough to keep things in storage and wait. Their value (their price) will change soon enough.
Our actions are thus free because they are not exhaustively caused by anonymous blind factors such as causal chains. We can nevertheless in principle completely understand our actions by listing all of the necessary and jointly sufficient conditions that lead to the fact that a given action results. That is no accident. Some of us are bandits but none of us is a one-armed bandit (in the sense of what is to be found in casinos, since there may well be bandits who have had an arm amputated).
## _**Why cause and reason are not the same thing and what that has to do with tomato sauce**_
Leibniz – one of the great thinkers of early modernity who has already been mentioned a number of times – formulated a famous principle: the **principle of sufficient reason**. This principle, which states that nothing happens without a sufficient reason, lies concealed behind the hard problem of free will. Leibniz's basic idea is very simple: if something happens (for instance, the kick-off of a soccer match), there is a series of conditions for why it does so. One can identify these conditions in a list:
* Twenty-two (sufficiently healthy) players are on the field.
* The referee has a whistle.
* The referee absolutely wants the match to begin on time (since last time he got in trouble with FIFA for blowing the whistle too late).
* The playing field consists of an appropriate turf (and not of concrete or ice, for instance).
* The spectators are behaving themselves (they are not throwing cups of beer onto the playing field).
* The earth's gravity is 1g.
* There are agreed-upon rules of soccer.
...
Each item on this list is a necessary condition for the fact that a soccer match can begin with the starting whistle. If one of these conditions were not met, the soccer match would not start. Not one of the conditions is sufficient on its own. If the spectators are behaving themselves, but there are no players on the field, the referee cannot blow the whistle to start the match. And the fact that the earth's gravity is 1g alone has precious little to do with the soccer match.
Leibniz's principle states that an event occurs if and only if its list of conditions is complete. There are no gaps in the tapestry of conditions. Indeed, the match would not begin if the referee did not have a whistle, and so forth. The necessary conditions are only sufficient for the fact that a soccer match can start on the referee's whistle when they are present altogether in the right way.
However, some readers will already want to object here: what happens if the referee suddenly loses the desire to begin the match, calls the whole thing off and heads home? In that case, a different event occurs, namely the referee's trip home. In such a case, however, there is once again a list of conditions for this event, which includes the fact that the referee suddenly no longer had the desire to officiate the match, but likewise includes the fact that neural processes occurred that are connected to his lack of desire. For instance, perhaps he is extremely depressed because for the last few months he has binged on cocaine. In this scenario he would actually be guided by his brain's neurochemistry, which is something that can occur (far be it from me to cast doubt on this!). The fact that the referee no longer wants to officiate matches does not fall from heaven, and even if this were the case – if he thus went home by chance instead of remaining – it would just add a new condition that explains why the football game lacked a starting whistle.
The point of this line of thought is that, whenever an event takes place, necessary conditions (the reasons, causes, conceptual and legal preconditions, and so on) are present for its taking place. Taken altogether they are sufficient for the fact that the event takes place. In my view, **the principle of sufficient reason** accordingly states: _for every event that takes place there is a collection of necessary conditions that are jointly sufficient for the event to take place_.
At first sight, one might think that the principle of sufficient reason limits our freedom or even renders it void. It does indeed state that we do not do anything just so or without any reason. It also rules out the possibility of there being a list of necessary conditions for the occurrence of an event that then activate, as it were, a will that is free of what comes from outside in order for the event actually to take place. The fact that one decides for this or for that does not fall from heaven but, rather, stems from all of the necessary conditions being present, which can include the fact that we would like something specific to happen. One cannot free oneself from the corset of necessary conditions, as it were. If that were to happen, one would not be free but, once again, at best be a one-armed bandit who depends on chance.
The impression that the principle of sufficient reason means that we are not free is deceptive. In order to see why this is so, a first step is to distinguish between hard anonymous causes and reasons. A **hard anonymous cause** is a cause that will have an effect, whether one wants it to or not. For instance, if someone were to strangle me for ten minutes I would be dead, whatever my wishes might be. If I am stabbed in the leg with a knife, it will hurt me, whether I want it to or not. If gravity follows Newton's law, one will be able to observe from the earth at a particular time precisely the same stars and constellations, whether one wants to or not. On many conceptions, laws of nature are supposed to express the actual connections between hard anonymous causes and their effects. Laws of nature are, so to speak, particularly hard, necessary and unyielding.
In contrast to this notion of hard anonymous causes, there is the concept of a reason. It is indeed true that there are many reasons that would never compel me to do something. Let us imagine that Olaf – a longstanding chain smoker – were to have the best possible reason finally to quit smoking. The reason might be that he would live longer, cough less, smell better, have better teeth, etc. It does not follow from his insight into this reason or this set of reasons that he will necessarily quit smoking. He will quit smoking if he wants to, but otherwise he will just go on smoking regardless of any list of reasons we might give him. Of course, we can try to compel him to quit smoking if we threaten him with prison if he carries on smoking. Yet, even then, there is always still a chance that, should he consider a life without cigarettes not to be worthwhile, Olaf will prefer not to be free to give up smoking. Reasons generally do not compel; they merely suggest that following the path of action they recommend might be a good idea. Reasons are proposals and not some kind of spiritual cause. Reasons can become motives for actions, which is not in any obvious way the circumstance with hard anonymous causes. In any case, adhering to the laws of nature is not a typical motif. Unless we are God, we do not usually intend to follow the laws of nature because it would make no sense to attempt not to follow them. To respect a reason for doing something, to incorporate it into one's life, means to accept that we could also not have incorporated it. Accepting a reason as a guiding motif for a course of action categorically differs from being compelled to do something.
This distinction of hard causes and reasons can lead to the paradox of free will. To that end, one must accept that nature is merely the reeling off of hard anonymous causes that lead to effects, which will in turn be hard causes for something else, and so on. In addition one must accept that nature is all there is. Then, there is no place for us as free agents in such a mechanism, since everything that happens at all happens whether one wants it to or not. Any kind of reasonable will that allows itself to be led by reasons seems to be opposed to this kind of mechanism.
Our freedom indeed has no place in the hard anonymous chain of cause and effect of an inexorable nature that is not concerned with us. And, conversely, nature has no place in what many contemporary philosophers call the "space of reasons," precisely because reasons are not supposed to be a case of hard causes. One is supposed to follow them by understanding them and not by being compelled by them, otherwise they would coincide with hard causes. In this sense, the philosopher John McDowell (b. 1942), in his very influential book _Mind and World_ , distinguishes a "space of causes" from a "space of reasons."13
But is that really enough? How do we know whether we are ever guided by reasons? If we comply with a reason, such as a wise bit of advice, or do not comply with it, how does that happen? Does it simply lie in our neurochemistry? If that were the case, we would always be guided by our brain and its mechanism of hard anonymous causes after all. So it seems, at any rate. Hence, many believe that it is not sufficient to lay out a concept of reasons if one cannot show in addition that we are ever really free to follow them without being secretly compelled to follow them.
However, this impression is deceptive. Everything hinges on recognizing that both reasons and hard anonymous causes can guide our behavior at the same time. We do not have to choose between a depiction of the human being, on the one hand, as fully autonomous and, on the other, as being fundamentally part of non-human nature. Assume that on the basis of my neurobiological make-up, tomato sauce tastes better to me than Alfredo sauce. In that case, the relevant neural connections could be cited as hard causes for the fact that tomato sauce tastes better to me than Alfredo sauce. But perhaps today I have a good reason to opt for the Alfredo sauce, since it contains more protein and fat, which I would like to incorporate into my diet. In such a case, this reason would take precedence despite the fact that my overall preference for tomato sauce might turn out to be biologically hardwired.
Let us imagine that I pick the tomato sauce as usual. We can now draw up a list that includes the necessary conditions for this specific, actually occurring event. Hard causes will not be the only things on this list, as it will include some reasons. Here is an excerpt from the list of necessary conditions.
* My neurochemistry predisposes me to tomato sauce (this is expressed by the fact that it tastes better to me).
* My gut microbiota (my gut flora) are inclined to tomato sauce. • I know that I get tomato sauce when I order spaghetti al pomodoro.
* Tomatoes are available today.
* Tomatoes have a certain price.
* I can afford the spaghetti al pomodoro.
...
Some of the entries on the list are not hard anonymous causes. That tomatoes have a certain price surely has to do with the fact that they grow on our planet. But their price (their exchange value) is not itself a natural fact. Prices do not grow like tomatoes; they result from negotiations in complex economic systems. I could not order tomato sauce in a restaurant if there were no economic system that consisted in a set of negotiations driving the exchange value of goods. But such a set does not obey the laws of nature, as negotiations do not follow physical laws. This is one of the many reasons why economics is not at all like a natural science. Too bad, it is still in the grip of the ideological illusion that it can be expressed in a mathematical symbolism supposed to resemble the expression of laws of nature. And thereby hangs a tale ...
In any event, the principle of sufficient reason does not state that everything which happens does so as a result of hard anonymous causes. Since it is simply not the case that in all circumstances all necessary conditions that underlie an event are hard anonymous causes, what happens on our planet cannot be explained entirely by taking only the latter into account. It is simply an error to reduce all events to natural events. This mistake makes it look as if freedom could not find a place in nature. This impression of an incompatibility of freedom and nature derives from a misguided conception of nature and our overall place in it.
For instance, if one explains why a car thief is punished, one will employ all sorts of concepts that are considered necessary conditions for the event of punishment, even though these concepts do not all pick out hard causes. Naturally, this means that it is not the case that there is a single, very long causal chain which determines, directs or governs everything that exists. In other words, neither the Big Bang nor the Big Bang taken together with all laws of nature and the initial conditions of the universe is a cause for today's exchange rate of the US dollar or the election of Donald Trump as US president. Rather, there exist an enormous, completely incalculable number of necessary conditions, and given events happen because of them. _We are free because many of the necessary conditions for our actions are not hard causes. And we are not one-armed bandits because there is no room for chance thanks to the principle of sufficient reason. Everything has a sufficient reason (the collection of necessary conditions that jointly suffice for an occurrence), and nothing happens without reason and/or cause_.
I have claimed above that determinism and freedom are compatible. But have I not ultimately denied that determinism is true? Not necessarily! If determinism were the thesis that there are only hard causes for events and no other kinds of conditions, it would in fact be false. However, one does injustice to determinism by associating it with such an outdated and at any rate false metaphysics.
**Metaphysics** , in general, is concerned with absolutely everything, with absolute totality, the world, the universe, reality as a whole, the cosmos, or whatever else you want to call it. But, as I have argued at length in _Why the World Does Not Exist_ , absolute totality does not exist anyway. Hence, there is no overall metaphysical reason to assume that there is a single enormous causal chain by which everything that ever happens is linked. For this reason, determinism should not offer itself as a metaphysical thesis and attempt to inflate itself into a world picture which weakens its case. Also, qua metaphysical world picture, determinism would be fundamentally unscientific, as one could neither prove nor falsify it by actually observing the universe – or, for that matter, the brain. It would have already been decided in advance that there is a single enormous causal chain, which is neither a presupposition of physics and neuroscience nor anything that follows from empirical, scientific discoveries that have been made up to this point. Causal determinism is simply a myth from the past. At best, it is a philosophical, metaphysical claim.
Traditionally, the principle of sufficient reason is considered to be the epitome of a metaphysical thesis. Does it not state that everything that happens is determined by a sufficient reason? Do I not therefore simply replace one metaphysics with another? Some even think that the principle of sufficient reason tells us not only something about events but literally something about absolutely everything that exists. The principle would then in fact be metaphysical in the problematic sense I reject. For this reason, on my construal, the principle of sufficient reason is restricted; it talks not about absolutely everything but only about events. In my preferred version, it says that no event can occur without the necessary conditions for its occurrence being jointly sufficient.
An important twist that should be noted in this line of thought is the fact that I do not assume that all conditions can ever be reduced to a common category. Not all conditions are (hard causes or) reasons. The list of conditions is open. There are countless types of conditions that we cannot survey in a single theory, a metaphysics. Hence, all we need to safeguard our free will is a categorical distinction that allows us to understand that we have no credible evidence which forces us to identify all conditions of human action with hard anonymous causes.
My resolution of the hard problem of free will is thus a specific form of compatibilism. It claims that determinism in the form of the principle of sufficient reason is true and that nevertheless at the same time we are free, because some conditions of those events which we conceive of as emerging through freedom are properly referred to as free. We are free, because many of the necessary conditions of human action are precisely such that we cannot understand them without ascribing free will to agents. And we have no credible, non-metaphysical evidence which undermines this image of the human being as a free agent.
## _**Friendly smites meanie and defeats metaphysical pessimism**_
An exemplary case can further illustrate the resolution of the hard problem of free will. Romeo sends Juliet a rose to make her happy. We would normally say that Romeo was free to do this – unless we were to learn that he did it under the influence of a drug that limits his freedom or something similar. However, we have no reason to suspect that Romeo was drugged or otherwise manipulated into sending the rose. Here are a few conditions for our classifying this event as an expression of freedom. Let us call this list the **friendly list** :
* Romeo likes Juliet.
* Romeo has the little bit of money needed to buy the rose.
* Romeo knows where to find Juliet before the rose withers.
* Romeo is able to move.
* Romeo wants to send Juliet a rose to make her happy.
* Romeo is someone who is made happy when others are happy, but most of all when Juliet is happy.
...
No element on this list has a compulsory or coercive character that would limit Romeo's freedom. To be sure, the actual list is quite long. A skeptic about freedom would have to deal with the problem of having to argue that no entry on the list requires Romeo's freedom. Accordingly, he would have to replace the friendly list by one that deprives him of freedom. Let us call the alternative list the **meanie list** , because it undermines the expression of freedom with a mean-spirited maneuver.
* Romeo likes Juliet – only because of a genetic disposition.
* Romeo has the little bit of money needed to buy the rose – it is on hand just by chance (it fell off of a roof), while he just happened to be stumbling around the florist's.
* Romeo knows where to find Juliet before the rose withers. His knowledge is purely instinctive: he heads to Juliet's place because his nervous system sniffs out her scent trail, without his even noting this himself.
* Romeo can move – he is blown around by gusts of wind. He is moved about like a piece of paper in the wind.
* Romeo wants to send Juliet a rose to make her happy – but only because a certain hormone is released due to a brain lesion.
* Romeo is someone who is made happy when others are happy, but most of all when Juliet is happy – because he has just gotten over a long clinical depression that did not allow him to feel happy about anything other than someone else being happy, which made him into a compulsive do-gooder.
...
We are all familiar with the fact that we are sometimes uncertain of the motivations of our action, both with respect to ourselves and in our encounters with others. Hence, we look for **action explanations** – that is, explanations which allow us to understand why someone (including ourselves) does something or did something. In this context, we can always suppose either benevolence or ulterior motives. The former lies behind the friendly list: one attributes freedom to someone, which is a charitable interpretation of an event that has every appearance of being a happy one (such as Romeo sending Juliet roses). Meanie lists substitute the appearance of benevolence either with ulterior motives (base motivations) or with explanations that allow us to disburden someone of the imposition of freedom. For example, if I am stumbling around and bump into someone on the street, no one will accuse me of having a base motive. In this respect, I simply was not free.
At this point of the argument, we need to realize that there is no sufficiently justified general suspicion that permits us to replace _all_ motives for human action and _every_ appearance of benevolence or freedom with an all-encompassing meanie list. If we had reason to believe that, in general, human action is never free, that would lead to a form of **metaphysical pessimism** , a philosophy prominently advocated by the grim Arthur Schopenhauer. He thought that there was no room for freedom in the structure of this world and preferred to understand every apparently benevolent action as the bare will to survive or the equally bare, purely biological will to reproduce. As he notes in his "The Metaphysics of Sexual Love," chapter 44 of the second volume of _The World as Will and Representation_ (one should call this volume: The Second Part of the Tragedy): "For all amorousness is rooted in the sexual impulse alone, is in fact absolutely only a more closely determined, specialized, and indeed, in the strictest sense, individualized sexual impulse, however ethereally it may deport itself."14 Schopenhauer goes to great lengths to explain the increasing rate of divorce in modern times in this way, too, for he believes that divorce is merely a consequence of the lie concealed in the miserable world's eternal cycle of rebirth, meaningless in itself. He does not explain divorce as a social phenomenon or as a series of free decisions. For him, it is simply the necessary consequence of falling in love, and love itself is nothing but a side-effect of the imperative to procreate. Schopenhauer's social agenda, which is fairly obvious, incidentally, is a justification of the Indian caste system along with its practice of arranged marriage.15
Accordingly, on this view, Romeo sends Juliet roses only because his motive is to sleep with her. Whatever he might think of tender motives, they play no actual role for Schopenhauer, who was still willing to credit some men (but no women) for sometimes breaking through the fatal cycle of rebirth if they make it to the special status of genius or saint. However, as his awful essay _On Women_ makes unmistakably clear, he allotted to women only the role of "nurses and governesses of our earliest childhood"16 or that of sexual temptress. Women, so we read, are "by nature... meant to obey."17 What happens if one renounces the cycle of the Indian caste system is clear to Schopenhauer: then, as in France, there is a revolution!
> In Hindustan no woman is ever independent, but each is under the guardianship of a father, husband, brother, or son... That widows burn themselves on the corpses of their husbands is of course shocking [a brief flash of insight in Schopenhauer]; but [and thus the flash of insight is immediately extinguished] that they squander on their lovers the fortune which has been acquired by the husband through the incessant hard work of a lifetime, and in the belief that he was working for his children, is also shocking... Was not the evergrowing influence of women in France from the time of Louis XIII responsible for the gradual corruption of the court and government which produced the first revolution, the consequences of this being all the subsequent upheavals?18
Schopenhauer, scorner of freedom, misogynist, and also not very sympathetic in other regards, is at the same time the master of the meanie list, as this passage proves in a paradigmatic way. At a historical remove from him, we recognize that these texts are purely ideological, that they lay claim to a discovery of a nature (for instance, "of women") in order to justify social structures that exist because of freedom and which can accordingly be transformed (abolished!).
Schopenhauer's purely ideological ensnarement should be more or less obvious to most readers of this book. At this point in history, one can only hope that the contemporary scientific and philosophical literature weighed down with Darwinitis and neuromania will in the future be perceived as ideological. Since one should not wait until then and hope for better times, one must intervene today.
There is no reason for an all-encompassing metaphysical pessimism that replaces all of the friendly items on the lists that express our action explanations with mean-spirited ones. Such an imposition is unacceptable, as well as being a form of paranoia supported by pseudoscience that is to be mistrusted in principle, both on its own account and for other reasons, too.
The resolution of the paradox of free will sketched here consists in the claim that the principle of sufficient reason is true and that both physical and neural determinism might be true (we simply do not know if they are). However, these two kinds of determinism only threaten our freedom if all events, including all human actions, belong entirely only to the realm of physics-cum-biology. Among other things, this presupposes the unwarranted assumption that we can always understand and explain our actions better if we phrase our accounts of what we are doing in the language of the natural sciences. This amounts to the thesis of naturalism, which under the scalpel of philosophical analysis turns out to be largely unfounded and certainly overly metaphysical (and unscientific).19 Naturalism is ultimately simply not true, because many events take place only if agents are involved in them, and agents can only be involved in events if they actually exist. Yet, agents cannot be ontologically reduced to epiphenomena of purely natural events. We can do what we want to do. Our actions are mostly free – or, rather, we have no good reason to doubt this. Naturalism in general and neurocentrism in particular are forms of ideologically driven metaphysical paranoia. They are neither scientific findings nor warranted suspicions that human action never has the motivation the agents ascribe to themselves.
However, I do not mean to convey the idea that there is a will which is just as free as our actions are. The claim that such a will exists as a second agent located somewhere in ourselves (either in the brain as a neurobiological homunculus or in some more spiritual layer of ourselves as souls) is a source of much confusion. For this reason, Nietzsche, in his critique of Schopenhauer, considered a properly radical solution by pointing out that _the_ will does not exist, that there is never an interplay of capacities and capabilities which one could characterize as _the_ will. "I laugh at your free will and at your unfree will, too. What you call your will is an illusion to me, there is no will."20 In contrast, Nietzsche argues that "will" is "a false reification."21 Thus he responds to Schopenhauer, who was initially a great model, an "educator," for him.22
This is all the more significant given that Schopenhauer took the concept of the will to extremes. And, indeed, he did this especially in the framework of the debate over human freedom. Schopenhauer produced the most influential version of the notion that we can indeed do what we want to do (freedom of action) but cannot choose what we want as an act of our own volition, a paradox definitive of debates about free will up to the present day. In this context, Schopenhauer claimed that the will determines us, that we are directed or "willed" by the will itself, so to speak, without ever being able to change this heteronomy. The will is our fate as humans because we have a character as respective persons only by our being committed to want this or that. Schopenhauer, like many popular neuroscientists, sees us as victims of ourselves qua brains.
Yet Schopenhauer falls into the very trap into which he had earlier tried to run his opponents. In particular, he accepts that there is a capacity, the will, whose activation is not up to us but without which we would not have any freedom of action. How are we supposed to be able to do what we want to do without wanting anything in the first place? This is a valid question and rightly posed. But it does not follow that there is such a thing as _the_ will. Why should we not simply say that we want, wish for, like, prefer, or choose all kinds of things, without it being the case that this is because "the will," "the wish," "the liking," "the preference" or "the choice" is operative in us, like an alien being lodged behind our consciousness, directing us from there? Errors of this sort are known as **reifications**. Schopenhauer, like Freud and the neurocentric enemies of their own free will, reifies his own vocabulary in which we couch our self-portrait as minded animals. This is a form of avoidable self-alienation.
Let us briefly return to the self. One might think that we bear a self-thing inside ourselves or are a self-thing that dwells behind our eyes (the good old homunculus). One might even seemingly
"provide reasons" for this, for instance, by means of the following fallacy (which Kant calls a "paralogism," as we have seen).
1. We are able to observe things (cats, worms, trees, and so forth).
2. If we observe things, we have thoughts.
3. If we have thoughts, then someone must exist who has these thoughts.
4. Since we are only too happy to say "I" [ _ich_ ], let us call the bearer of thoughts "the self" [ _das Ich_ ].
5. We are able to observe the bearer of thoughts.
Conclusion: Therefore, the self (the bearer of thoughts) is a thing.
The mistake here is the identification of a formal object of our thought (the self) with a thing in the world (a brain, say, or a soul for that matter). It is simply not the case that everything we can think about with true or false thoughts is an object or thing in the natural world. There are some things we can observe and track with thoughts that are not part of the natural world, and the self is precisely one of those "things." Neurocentrism ideologically assumes that we are a natural thing among natural things and that it is only such things that exist. But it is false that only natural things exist, since values, hopes and numbers exist, _inter alia_ , which do not self-evidently qualify as natural things.
The basic error lies in the idea that there is a world out there that, as a metaphor commonly used by contemporary philosophers suggests, has furniture which is completely independent from us, namely the "furniture of reality." The object furnishings of reality are then supposedly tangible things that are pushed around through space and time according to inviolable laws of nature. As corporeal things, we would then only be one thing among all other things. In that case, the existence of consciousness, numbers, values, open possibilities, and so forth, would of course be an enigma. However, consciousness, free will, and so on, are not objectively enigmatic things in the natural world. This seems plausible only if one is in the grip of the misguided metaphysical fantasy of a completely reified universe of which everything and everyone is a part.
If we reify everything, we get caught in contradictions. We then all too quickly have to introduce a self-thing and identify it with a brain-thing in which a will-thing is lodged (or several will-things which are located in different areas of the brain). The next thing we know, we are now supposed to believe that nothing can be free because all things are related to each other according to laws of nature that govern what happens to each and every thing. Yet this building-block metaphysics that conjures up an image of the world in which things bump into each other and sometimes get stuck to each other, and so on, does not of course fit very well with our self-image as freely acting persons among other such persons (nor does it fit with actual physics, by the way). I call this kind of metaphysics **legocentrism** (in honor of the well-known toy). Persons are simply not natural things (they are neither like Playmobil nor like Lego).
## _**Human dignity is inviolable**_
This line of thought has important ethical consequences. Article 1 of the Basic Law for the Federal Republic of Germany famously states: "Human dignity shall be inviolable. To respect and protect it shall be the duty of all state authority." Of course, one can interpret and explain this article in different ways. In any event, it is important to respect the valuable insight that human dignity and human rights are conceptually linked.
What should not be overlooked here is that human dignity is literally inviolable, since it is not a natural thing. One cannot grab hold of human dignity or even perceive it with one's senses as natural things are perceived. It does not belong to us because we are human-things whose brain-things have been installed to grow and flourish within our skulls.
Here Kant's famous **distinction between economic value (price) and dignity** can be of further help:
>... everything has either a price or a dignity. What has a price is such that something else can also be put in its place as its equivalent; by contrast, that which is elevated above all price, and admits of no equivalent, has a dignity. That which refers to universal human inclinations and needs has a _market price_... but that which constitutes the condition under which alone something can be an end in itself does not have merely a relative worth, i.e., a price, but rather an inner worth, i.e., dignity.23
In this passage, Kant points out that our condition as human agents is an end in itself. Here one has to make the reader aware of a common wordplay in German philosophy. The German word for "condition" is _Bedingung_ , which contains the word _Ding_ (= thing). A _Be-Dingung_ turns something into a thing without necessarily being itself a thing. A condition makes a thing into what it is without all conditions necessarily having to be hard causes. According to Kant, human dignity belongs to us because we live in the "realm of ends."24 It is part of our constitution as agents to be conditioned by human dignity. The realm of ends is a system of concepts that we use to make human actions understandable to ourselves. This includes concepts such as friendship, deception, gift, president, copyright, exploitation, free will, alienation, ideology, revolution, reform and history. These concepts are distinct from the ones that we use to make natural processes understandable, processes which automatically run their course without any help from us.
In Kant's time, the dividing line between natural automatic processes and free actions was admittedly in dispute, which received literary treatment in Romanticism. It is sufficient to think of famous automata such as Olimpia from E. T. A. Hoffmann's _The Sandman_. In another well-known passage, from the _Critique of Practical Reason_ , Kant opposes "mechanism of nature" and "freedom" to each other, in the course of which he formulates the suspicion that we could after all be automata. Kant emphasizes the central thought that things which are not subject to the " _mechanism_ of nature" need not "be actual material _machines_."25 In distinction to many contemporary advocates of determinism, Kant explicitly takes issue with the hard problem of free will as formulated by Leibniz and made popular by Schopenhauer.
> One is here taking account only of the necessity of the connection of events in a time series as it develops according to natural law, whether the subject in whom this elapsing occurs is called _automaton materiale_ , inasmuch as the machinery is driven by matter, or, with Leibniz, [ _automaton_ ] _spirituale_ , inasmuch as it is driven by presentations; and if the freedom of our will were none other than the latter [kind] (say, psychological and comparative freedom, not simultaneously transcendental, i.e., absolute, freedom), then it would basically be no better than the freedom of a turnspit, which, once it has been wound up, also performs its motions on its own.26
On the basis of an admittedly somewhat unorthodox reading of Kant, one can claim that, as inhabitants of the realm of ends, we are really free insofar as we can only understand each other if we allow for genuine teleological explanations of action. Let us recall Larry and his brown bread: Larry goes into the supermarket to buy brown bread that is not moldy. Thus, he is not moving around merely in the realm of hard causes. It is not the case that Larry has been blown into the supermarket by hurricane-force winds and has landed in front of shelves of bread. Larry goes to the supermarket of his own accord. His action is an element of a whole system of actions that constitutes Larry's biography.
Actions are distinguished from natural events by the fact that a natural event need not be an element of a teleological system where someone does something in order to achieve an end. For this reason, natural events (such as having to answer the "call of nature") are involuntary. Having to answer the "call of nature" indeed has a function in the organism, but it is not an end toward which someone has decided to aim.
A large part of human civilization consists of the fact that we eliminate natural events from occurring around human bodies or at least make them look more beautiful: we trim our fingernails and cut our hair, we clothe ourselves, have private bathrooms that can be locked, make use of technology with which we improve our natural potential (calculators, trains, and so on). This also includes the fact that we do not live with wild animals but would like to know that they are somewhere else entirely. They are supposed to remain in wilderness areas undisturbed by human beings, or in zoos, and not bother us.
Be that as it may, the idea of a dignified human life is the idea of a life that is able to move in the "realm of ends." If we are sick or terminally ill, natural events take over our lives. However, this is not the normal state of affairs, as neurocentrism would have us believe. We are governed by hard causes and neural connections in our body only when we are sick. Whoever supports environmental conservation and the protection of animals does not do this only because there is an electrical storming of neurons within the skull that drives one against one's will to engage in such actions, or that implants the will, whether one wants it or not, to be used in this way for the community.
Human dignity is inviolable because we are _not only_ organisms and animals of a specific kind but are those particular animals who live in the realm of ends. Please note that I do not mean that other animal species are excluded from this realm. Indeed, domestic animals already live with us in our realm of ends, even if they do not understand it nearly as well as we do. Admittedly, no single person comes even close to knowing everything about the realm of ends in general, because innumerably many people and institutions contribute to it. Hence it sometimes seems to us to be more like a natural force than a manifestation of freedom.
We create soft conditions, so to speak, through which we partially free ourselves from the concatenation of hard anonymous causes. Other animal species do not have a kind of reflexively, philosophically grounded legal system like ours, which resulted from millennia-long reflections on the structure of a just political system, among other things. Such reflections were paradigmatically elaborated at an advanced theoretical level by the ancient
Greeks (primarily Plato and Aristotle, but also tragedians such as Aeschylus and historians such as Thucydides and Herodotus). Jurisprudence and political science have been established on this ancient foundation. They are expressions of free will, because they are constitutive elements in the self-portrait of human agents. Such elements are subject to historical change, a change not governed by laws of nature.
We are not more valuable than other animal species because of these reflexive achievements. As Kant writes, dignity is not a relative value and thus not something that elevates us above the rest of the animal realm. From the fact that human dignity exists, which is perhaps also what establishes our human rights, it does not follow that we may treat other animal species badly just because they are unfamiliar with our realm of ends. Otherwise, our dignity would be reduced to a relative value, one that allows us to look good in relation to other living beings. However, according to Kant, dignity is an inner value that I see substantiated in the fact that our actions are free because not all necessary conditions of our actions are hard causes.
## _**On the same level as God or nature?**_
I have already cited Cavell's insightful dictum that "Nothing is more human than the wish to deny one's humanity."27 Sartre famously advocated a similar thesis, namely, the thesis that a large part of human behavior aims to absolve us of our own freedom. As he puts it, "man fundamentally is the desire to be God."28 I call this **dehumanization from above** : human beings become inhuman through self-divination.
Sartre's basic philosophical idea is not hard to understand. He distinguishes between in-itself and for-itself. Something that is in itself is completely identical to itself without any conceptual mediation turning it into the thing it is. Think of a rock. It has an absolute being. To be sure, it can be destroyed, but it cannot change itself by changing its beliefs about itself – because it has no beliefs. Rocks are an example of the in-itself. Some primitive organisms, too, should undoubtedly be included in the category of the in-itself (we do not know at this time at what "level" in the animal realm the for-itself begins). In contrast to such beings, "human reality," as Sartre calls it, entails that we belong to the for-itself. This is Sartre's version of the already well-known idea that our image of ourselves (even if it is a false image) expresses something essential about us, something in light of which we can change. We are always to some extent what we consider ourselves to be. Even if we are wrong about who or what we are, this error constitutes who we are. If I consider myself to be a good dancer (without being one), I am a person who considers himself to be a good dancer. This says something about me (for example, that I am too vain to admit that I am a bad dancer or that I would love to be a good dancer).
For Sartre, human beings devise the idea of God because God stands for a perfect combination of in-itself and for-itself. God has no false beliefs about himself; he cannot be self-deluded, vain, envious, etc., while he is at the same time also not like a stupid, lifeless rock. Rather, he is imagined to be an exceptionally perfect person (in any case, this is how philosophers imagine their god, who is for that reason called the "God of the philosophers"). In Sartre's view, we are free because there is a gap between our in-itself (our body, our parents, and so forth) and our for-itself. This gap can never be closed, even if we like to develop a variety of strategies to cover it up. In particular, a classical strategy for covering up the gap at the very center of our being is **essentialism** , which in this context is the claim that a human being or a social group is determined by their essence to engage in specific patterns of conduct that are only apparently free. Racist and sexist ideas are essentialist in that sense, as are nationalist ideas that claim Bavarians or Greeks have an essence that manifests itself in their actions: for instance, Bavarians like to drink beer and vote CSU, while Greeks are corrupt, lazy, congenial, radically left-wing or radically right-wing, and so forth. Stereotypical notions such as these are currently part of the portfolio of German ideology, not to mention the ideological regression of the US. Ideology is as widespread as humans, of course. It is not restricted to a specific nation-state or time in history.
Unfortunately, there are countless stereotypes of the essentialist kind, and from the perspective of neo-existentialism they serve only to deny our own freedom by denying it to others. If others are automata, we cannot be blamed for not making any use of our own freedom to treat them like free living beings. Ideology has harmful consequences for others, as it relieves those who cling to their ideology from the moral imperative to change their behavior and their very selves by giving up cherished, but ultimately ungrounded and even evil assumptions about others.
That said, one can combine the existentialist motifs of Cavell and Sartre and identify two dangers: a dehumanization from above and one from below. Dehumanization from above is a threat if we choose God as our ideal self – that is, if we want to become like God (something religion itself in all of its varieties typically prohibits). **Dehumanization from below** is a threat if, infected with Darwinitis, we believe that all human behavior can be completely explained by evolutionary biology.
In present-day society, dehumanization from above is not associated with religion as much as it is with the fantasies of omnipotence of post-humanism and trans-humanism, as well as the idea of a digital revolution, engulfing everything, lying in the hands of the gods of Silicon Valley. There are currents of thought that proceed by claiming that, due to our technology, we human beings have already been cyborgs for a while now. The laptop on which I am composing these lines, according to the advocates of the thesis of the "extended mind," supposedly belongs to me just as much as my liver does. At the extreme, think of scenarios like those from the film _Transcendence_ , which many take to be desirable as they come with the hope that we can be uploaded to an online platform after death, where we will enjoy an eternal life on the exciting internet.
Dehumanization from above stems from "progressive cerebration" – that is, the cephalization of our culture, pithily described by the German poet and medical doctor Gottfried Benn. Recent neuromania is another chapter in the progressive cerebration. Benn opposes a thesis advanced by the Austrian neurologist Constantin von Economo (1876–1931), who was well known for, among other things, having discovered the brain inflammation _encephalitis lethargica_ , which we have already discussed (p. 70). Von Economo hypothesized that the history of mind is understandable via the evolution of the brain. In accordance with this, he claimed that there was a kind of progress of the growing brain that drives cultural progress in its train.
Benn rejects Economo's idea, which he considers along with nihilism to be a symptom of decline of an unsteady modernity, primarily in the first half of the last century. Benn – who sadly opted politically for National Socialism, which he wrongly thought was going to be a political system supportive of the arts – in 1932 asked in his essay "After Nihilism":
> Do we still have the strength, asks the author, to oppose the scientific-deterministic world view with a self that is grounded in creative freedom? Do we still have a strength drawn from the power of traditional Western thought, not from economic chiliasms and political mythologems, to break through the materialistic-mechanical formworld and to design the images of deeper worlds out of an ideality that posits itself as such, and in a measure that sets its own limits?29
In place of materialism and nihilism, Benn posits "constructive intelligence... as an emphatic and conscious principle of far-reaching liberation from materialisms of every stripe."30 Within the framework of progressive cerebration, Benn recognizes in the combination of "Newton Imperator, Darwin Rex"31 an unbridled "intellectualistic exponential increase of becoming"32 – in short: a dehumanization from above.
This dehumanization from above is also operative in **functionalism** , which claims that consciousness or mind is a formal functional structure that can be implemented or realized in various materials – in the age of Silicon Valley, silicon is repeatedly cited as an alternative to our brain tissue. One might think that functionalism is a new thesis that came into play with the arrival of computers. But here Benn is remarkably clear-sighted. In his "Speech to the Academy," he describes the basic structure of neurocentrism, which has not changed to the present day, as follows:
> A new stage of cerebration seems to be around the corner, a more frigid, colder one: to conceive our own existence, history, the universe in only two categories: the concept and the hallucination. From Goethe's time, the _disintegration of reality_ has transgressed every measure, so that even the wader, if he notices it, must plunge into the water: the earth is ruined by pure dynamics and by pure relation. _Functionalism_ , you know, means the time of unbridled movement, inexistent being.33
Just like Goethe and Nietzsche before him, Benn recommends that we reflect on the historical background of modern intellectualization. In Goethe's epic _Faust: The Second Part of the Tragedy_ , a homunculus succeeds in escaping from the phial in which he had been living by smashing it on "the bright throne."34 As the philosopher Thales – who appears on the scene in the "Classical Walpurgis Night" – remarks, the homunculus is "beguiled by Proteus"35 into giving himself over to "Eros... who gave all things beginning"36 and causes the glass in which he is trapped to shatter. Nietzsche will later call this "the Dionysian,"37 which in any case is not of much help, since even the Silicon Valley bandits, to relieve themselves of functionalism, participate in "Burning Man," a festival of intoxicated ecstasy in the Black Rock Desert of Nevada. There, appropriately enough, a very large wooden effigy is set on fire and burned. Post-humanism receives its sacrifice. "Burning Man" is a post-humanist celebration of Silicon Valley's dehumanization from above.
With the homunculus's act of shattering, Goethe alludes to Schelling's _Naturphilosophie_. Schelling radically broke with the idea that the human mind as an intellectual self, as pure consciousness, confronts a nature that is inanimate and without self or consciousness. Schelling and Goethe both thought that Fichte advocated precisely this position – and Fichte, as we have already learned, was a thorn in the side of Goethe, who was politically very influential. As professor in Jena, let us recall, Fichte became involved in the atheism dispute, since he ultimately advocated a form of purely rational religion that viewed God not as a person or an authority beyond human reason but, rather, as the moral order immanent to the world: "That living and active moral order is itself God; we require no other God and can grasp no other. There is no ground in reason from which one can proceed and, by means of an inference from that which is grounded to its ground, assume some separate being as the cause of that which is grounded."38 In complete contrast, Goethe and Schelling (represented by Proteus in _Faust: The Second Part of the Tragedy_ ) object that we should not imagine nature as a mechanical contraption. If we belong to nature at all, we must assume that our mind, including our emotional world, is no stranger to a nature devoid of meaning and morality. For this reason, these two authors mobilize the human being's pre-logical, irrational prehistory against progressive cerebration. In other words, they rightly insist on the fact that there is no intentional consciousness without phenomenal consciousness, that as thinking beings we also feel. The idea that our life is governed by an egotistical calculus for survival is corrected with reference to our capacity for erotic enthusiasm. Dehumanization from above is thus foiled.
As an argument against neurocentrism, Benn accurately points out that "the biological basis of the personality is not the cerebrum, as was assumed in an earlier scientific era, but the whole organism."39 Yet, at the other end of the spectrum, there is all too easily the threat of a dehumanization from below, which today assumes the form of Darwinitis. I am thinking in particular of the attempt to ground human goodness – that is, our moral capacity – in non-human nature by studying primate communities in order to find altruistic social relations there.
Authors such as the ethologist Frans de Waal (b. 1948) are surely right when they point out that a dualism "which pits morality against nature and humanity against other animals" is untenable.40 The idea that brutality exists in the rest of the animal realm, while the human being alone has somehow tamed itself – or has been tamed by God – is not empirically supportable. For this reason, primatology is of course also philosophically relevant. However, it never proves that morality is merely a side-effect of evolution. At best, it shows that other primates also participate in the moral realm of ends, that they also have a mind in a full sense of the term, and maybe even a history. We do not know any of this yet, as we have not deciphered the languages of other animals. In any event, it seems clear that they have no written records and institutions based on traditions – but, again, this does not elevate us above the rest of the animal kingdom in any special sense.
## _**PS: There are no savages**_
In modern political philosophy there has been a prolonged debate over the question of whether "savages" (in other words, the inhabitants of North and South America who were newly discovered by Europeans at the end of the fifteenth century) lived in a natural condition free of states, and indeed of morality. In modern anthropology, it became clear that humanity could not be classified into the supposedly "savage" on the one hand and the "civilized" on the other – one of the reasons being that, from antiquity, this classification has been used in order to justify slavery, which was finally recognized as a moral evil in the nineteenth century. Instead of such classifications, the French sociologist Bruno Latour (b. 1947), in his book _We Have Never Been Modern_ , aptly postulates a "symmetrical anthropology."41
**Symmetrical anthropology** no longer divides humanity into a premodern (the formerly "savage") group and a modern (the formerly "civilized") group. Rather, it proceeds from the fact that we appear just as different in the eyes of every genuine or supposed other as they do in ours. Here a symmetry prevails in which we are no longer conceptually privileged in understanding the human mind because we are more technologically advanced.
After a long course of events that consumed many human lives, it finally became clear that universalism really has no philosophical alternative, even if recent political events can be read as a rebellion against the truth of universalism. **Universalism** is the thesis that all human beings are fundamentally equal, that there are no different species (or, even worse, races) of human beings who are somehow situated at different stages of development and hence have fundamentally different values that are incompatible with the advanced values of modernity.
The realization of the truth of universalism means that a typical justification of political power should no longer function. This justification – which has been extensively used both politically and ideologically – proceeds from the claim that human beings are naturally anarchistic and tend to raid, plunder and instrumentalize each other for egotistical ends.
This ideological representation of the "state of nature," as Thomas Hobbes quite influentially described it in his gloomy, visionary _Leviathan_ , is familiar today primarily in the Hollywood genre of apocalyptic movies. As soon as public order is threated by an alien invasion, by zombies, by a natural catastrophe of unimaginable magnitude or simply by the end of the world, such movies usually depict raids on supermarkets and a state of generalized civil war. The first thing human beings do in that genre, now that the police are no longer watching, is take advantage of their newly won freedom to seize hold of every conceivable good by force. Such scenarios imply that public order protects us from ourselves, so to speak. If public order were to disappear, man would again be "an arrant wolf to man," as Hobbes puts it in his famous formulation.42
In any case, it should no longer be officially acceptable today to justify the state's monopoly on violence by referring to the brutal savages who supposedly lie concealed inside all of us, because this has long been perceived as an ideological construct by advanced sociology and anthropology. Fortunately, our idea of constitutional democracies is still in many places aligned with universalism – that is, it is not justified by claiming that we view constitutional democracy as our contingent system of government, which the state merely uses to justify its retention of power over us evil, brutal citizens. This model would not be compatible with our claims to political freedom. If we all thought that the state and only the state imposes on us codes of conduct such as morality, so that we do not continually threaten each other with murder and mayhem, we would be right not to feel very free. It would be hard to escape the impression that we are a "nation of devils"43 tamed by the equally brutal government.
In contrast, in his famous text _Toward Perpetual Peace_ , Kant recognizes that we are not such devils, even though the justification of a public order would have to function even if we were devils. The assumption that we are evil by nature is only a fiction that we employ to make it understandable to ourselves why institutions are reasonable both for good and for evil human beings.
> The problem of establishing a state, no matter how hard it may sound, is _soluble_ even for a nation of devils (if only they have understanding) and goes like this: "Given a multitude of rational beings all of whom need universal laws for their preservation but each of whom is inclined covertly to exempt himself from them, so to order this multitude and establish their constitution that, although in their private dispositions they strive against one another, these yet so check one another that in their public conduct the result is the same as if they had no such evil dispositions."44
The point is that the state (in other words, its chosen representatives) can proceed neither from the supposed fact that we are a nation of angels nor from the supposed fact that we are a nation of devils. Rather, the state is a matter of guaranteeing political freedom for everyone, whether they are good or evil, or, to put it better, whether they act in a good way or in an evil way. Fortunately, many of us have been more or less able to free ourselves from the idea that there is a human nature which can be determined phenotypically – for instance, as race – or in historicalnationalistic terms (the Germans, the Norwegians, the Chinese) – by linking essences instead to the state of economic development, for instance. We need to recognize fully that our essence as human beings is a function of the historical structure of our mind as _Geist_ – i.e., of our capacity to create images of what it is to be a human being and act in accordance with these images. In this context, it is important to emphasize, with Kant, that the creation of self-images is constrained by moral norms: how we conceive of ourselves as human beings in part determines the moral value of our actions. This is one of the many reasons why ideologies can be dangerous, as they impose distorted representations of the human mind on us and are thereby able to manipulate us into acting against our own interests and those of everybody else.
Certainly, there is a lot to be done to spread the insight into universalism. For example, I fear that, if one were to launch a survey of how a typical German looks, many people would probably believe they have an answer. The point, however, is that "a German" has no appearance, since being German is a normative status. A German is someone who has German citizenship. One need not have a particular kind of appearance for that. Admittedly, citizenship is associated with values, since, in the case of Germany, as it is right now, it is anchored in the framework of a free democratic constitutional order. Since this includes human rights, equality of opportunity, indeed a universal idea of freedom and equality before the law, it follows from all of this that, politically speaking, we have a great chance to rid ourselves of the idea that human beings can be broken up into different subspecies.
However, it remains true that we are nowhere near a universal realization of the universal structure of morality. If human rights are supposed to distinguish us from other animal species, the fact that we can arbitrarily misuse animal species for our own purposes once more brings us back to the idea of the "savage." In this case, one imagines that "wild animals" are a danger to civilization so that one can claim a universal human monopoly of violence against the rest of the animal realm, a thought process underlying the current ecological disasters we are witnessing. The current ecological crisis, among other things, is a function of the dehumanization from above: humans believe they are beyond nature and that God may even tell them to exploit the planet at will (think of the dangerous association of Christian fundamentalism and Republican climate change denial in the US).
Kant offered a well-known **dehumanization argument** that refers to the fact that "violent and cruel treatment of animals is... intimately opposed to a human being's duty to himself," since it thus "dulls" the "shared feeling of their suffering" even toward other human beings.45 Hence, with Kant, we could derive animal rights from the very foundation of human rights: "Even gratitude for the long service of an old horse or dog (just as if they were members of the household) belongs _indirectly_ to a human being's duty _with regard to_ these animals; considered as a _direct_ duty, however, it is always only a duty of the human being _to_ himself."46 We should thus remain human beings and only become better at understanding what moral and political opportunities this entails. To this end, one must criticize the ideologists who, as I have sketched in this book, are associated with a double dehumanization. An important step in overcoming one's ideological ensnarement is to draw the lesson from modernity that there are no savages, indeed that even other animal species are not savages. We have fundamentally distinguished ourselves from them so that we could go around with a glorified self-image. We are different from other animals – that is all. Our dignity does not derive from this difference. It is not given to or implanted in us as nature but, rather, assigned in advance as a task that we are still far from completing. The human being has not yet established a moral and political order in which all human beings are recognized for what they are: minded animals with a history which all share the capacity to lead a life in light of their conceptions of what it is to be a human being.
## _**Man is not a face drawn in the sand**_
In this book, I have sketched the outlines of a philosophy of mind – or, rather, of _Geist_ – for the twenty-first century. In so doing, my intention was to elaborate the concept of spiritual freedom and to advocate it against reductionist and eliminativist programs that would like to persuade us that we have neither minds in any demanding sense nor freedom. The neurocentric opponent represents a widespread form of ideology, whose main intention from epoch to epoch I see in the various attempts of the human being to get rid of itself. This self-alienation of the human mind assumes many forms today, which to some extent revolve around the thoughts of trans-humanism and post-humanism. That is, they revolve around the idea that the era of the human being has come to an end because, heading into the future as cyborgs, we have outgrown our biological nature.
A problem that contributes to the ideology of our time can be linked to the fact that the humanities, which in German are called _Geisteswissenschaften_ – i.e., sciences of _Geist_ – after World War II began radically to dismiss _Geist_. The famous title of a series of lectures that was held at the University of Freiburg encapsulates this: _Die Austreibung des Geistes aus den Geisteswissenschaften_ [The Expulsion of _Geist_ from the Humanities].47 The organizer of this lecture series, the literary theorist Friedrich Kittler, who died in 2011, explained this expulsion in the following manner. According to him, the concepts of a "mind" and of a "human being" were first constructed in early modernity and consolidated by different domains of knowledge. The former spiritual realm [ _Geisterwelt_ ] of superstition was in this way replaced by a single spirit, a _Geist_ , or a human mind. Yet this notion for him is now itself to be overcome as a superstition, because it involves an untenable construct that in any case is about to disappear in favor of new technological orders as well as of an overall new order of things.
Modernity in German is also called the time of the new [ _Neuzeit_ ], because it is the epoch based on the belief that one could start something radically new which changes everything so fundamentally that we will finally arrive at the end time or a state of post-history. The time of the new longs to be the end time, as numerous apocalyptic films, novels and television series make clear. Like good old Christians, many people wish for a decisive final event to occur, a revolution after which nothing will be as it was before, but everything will be better and settled definitively, once and for all. In particular, one might wish that demands for freedom will eventually come to an end.
In a nutshell: modernity fosters imaginary forms of relief. Yet, one must stand up in opposition to this in the name of freedom. True progress consists not in the illusory ideal of overcoming the mind and the human being but in improving the moral and juridical order in light of our insights into who we are as free human agents, without identifying us with things in the natural world (such as a brain, an assortment of genes or a race).
Thus, there is no utopia still to come, no time after historical times that would in principle be better placed for freedom to be advanced than the one in which we already find ourselves. There is no better place, neither in the future, nor in the past, nor on Mars. Neither postmodernity nor post-humanism will better fulfill the demands for freedom than our current era. We have the improvement of humanity in our hands, as individuals and as institutions. No one can take this away; it can only be oppressed.
The expulsion of _Geist_ and the human being from the humanities has an ominous historical background, since it springs not least from Martin Heidegger's _Letter on Humanism_. Heidegger explicitly composed this work to distance himself from existentialism, which he scorned because it was a modern urban, progressive and liberal philosophy. In contrast, as the publication of his so-called _Black Notebooks_ has once more called to mind, he preferred to align himself with National Socialism and an idea of the essence of the German people that was utopian to the point of madness.48 Heidegger exchanged the concept of _Geist_ , which refers to the fact that our freedom consists in working in historically and socially relevant ways on our self-portrait in the light of conceptual and ethical demands, for a concept of essence that is supposed to bind us surreptitiously to our soil. Thus, in a seminar from 1933–4 on _Nature, History and the State_ , he speaks of the supposed fact that "the nature of our German space would definitely be revealed differently" to a "Slavic people... from the way it is revealed to us; to Semitic nomads, it will perhaps never be revealed at all."49
Another source for the supposed overcoming of humanism is the Heidegger-inspired _The Order of Things_ by the influential French sociologist, historian and philosopher Michel Foucault (1926–1984). This book describes how the concept of the human being emerged in the modern life sciences and human sciences and how it has changed over time. From this Foucault draws the absurd conclusion that the human being has existed for only a few hundred years, because he considers the human being to be only a point of intersection in different scientific discourses, and thus a construct, which is a terrible philosophical confusion: "Taking a relatively short chronological sample within a restricted geographical area – European culture since the sixteenth century – one can be certain that man is a recent invention within it."50 Foucault concludes from this that the epoch of the human being could also be brought to a total end – indeed, just like Heidegger and other eschatologists, he even hopes for "some event of which we can at the moment do no more than sense the possibility."51 The book closes with the prospect, indeed with the wager, "that man would be erased, like a face drawn in sand at the edge of the sea."52 I venture to bet against this!
Of course, the human being will disappear someday, simply because – this is a detail of my bet – we will never reach the Star Trek universe via androids and warp-drives and colonize planets that are far enough away from our sun when the latter dies one day, and we have already long since been pulled into the abyss.
Humanity as a whole is not to be preserved for eternity. We will be extinct sooner or later. Humanity is finite, as is each individual, too. We find ourselves in dimensions of unimaginable vastness and will never sufficiently understand the universe in order to be able to provide an exact assessment of our place in it. And yet, in modernity we have much progress to report. We live in an age of knowledge. However, this will lead to further progress only if we stop fooling ourselves into believing that we are on the verge of realizing that we are neither minds nor human beings which are still free to make progress.
Hence an important task for us in our century is to take a new look at our situation as minded animals. We must overcome materialism, which would have us believe that all that exists is what is found in the universe (in the sense of the reality of hard anonymous causes, of matter and energy), and which for that reason desperately seeks a conception of the mind that is able to reduce _Geist_ to consciousness and then reduce consciousness to an electrical storming of neurons. We are citizens of many worlds, we move in the realm of ends. This provides us a series of conditions for freedom.
There is no reason in principle to engage in escapism. But there are many reasons to further social and political progress, because untold numbers of human beings live at present under conditions which make it difficult simply to tell them to live with human dignity. Man is still an arrant wolf to man. This is our real problem, and we cannot get rid of it by all becoming vegetarians in affluent societies, by attending meditation classes, or by identifying mind and brain. Euro-Hinduism is only another evasion, a looking away from the real problems. The human being's greatest enemy is still the human being, and this spoils for many the only prospect which we human beings have: this life that we are living right now.
There is no reason to pin our hopes on a utopian future. We exist here and now, and that is all. The poet Rainer Maria Rilke celebrated this in his _Duino Elegies_ with his concept of our beinghere (which we should oppose to Heidegger's being-there, always in the future).
> Life is glorious here. You girls knew it, even you
> who seem to have gone without it – you who sank under
> in the cities' vilest streets festering like open sewers.
> For there was one hour for each of you, maybe
> less than an hour, some span between two whiles
> that can hardly be measured, when you possessed Being.
> All. Your veins swelled with existence.
> But we forget so easily what our laughing neighbor
> neither confirms nor envies. We want to make it
> visible, even though the most visible joy reveals
> itself to us only when we've transformed it, within.53
Actually, life can be glorious here, but not always and not for everyone. We human beings are to blame if we do not work in common to improve the conditions for freedom, prosperity, health and justice on this planet for everyone. We have no other planet, and we should not seriously count on another life after death in which we can make everything better. For this reason, it is a central task of philosophy to work on an avatar of the human mind that can be led into the field, in the sense of an ideology critique, against the empty promises of a post-human age. I would like to conclude this book with the words of Schelling, who wrote them in a letter to his friend Hegel on February 4, 1795: "The alpha and omega of all philosophy is freedom."54
## **Notes**
1. See with reference to this claim the book _Mind Time: The Temporal Factor in Consciousness_ (Cambridge, MA: Harvard University Press, 2004) by the neuroscientist Benjamin Libet, who has become famous for his experiments. The best critique of the entire experimental setup and of the philosophical theses associated with it can be found in Alfred R. Mele's _Free: Why Science Hasn't Disproved Free Will_ (Oxford: Oxford University Press, 2014). 2. Wolf Singer, "Verschaltungen legen uns fest: wir sollten aufhören, von Freiheit zu sprechen," in Christian Geyer (ed.), _Hirnforschung und Willensfreiheit: Zur Deutung der neuesten Ergebnisse_ (Frankfurt am Main: Suhrkamp, 2004), pp. 30–65. 3. Wolf Singer, "Keiner kann anders, als er ist," _Frankfurter Allgemeine Zeitung_ , January 8, 2004 (www.faz.net/aktuell/feuilleton/hirnforschung-keiner-kann-anders-als-er-ist1147780-p4.html). 4. Ibid. 5. Brigitte Falkenburg, _Mythos Determinismus: Wieviel erklärt uns die Hirnforschung?_ (Berlin and Heidelberg: Springer, 2012), p. 255. 6. Arthur Schopenhauer, _The Two Fundamental Problems of Ethics_ , trans. David E. Cartwright and Edward E. Erdmann (Oxford: Oxford University Press, 2010), p. 69. 7. Martin Luther, _On the Bondage of the Will_ , trans. Edward Thomas Vaughan (London: Hamilton, 1823), pp. 32–3. 8.Ibid., pp. 58–9; translation modified. 9. "Therefore, the idea of their freedom is simply the ignorance of the cause of their actions." Baruch Spinoza, _Complete Works_ (Indianapolis: Hackett, 2002), p. 264. 10. Peter van Inwagen, _An Essay on Free Will_ (New York: Oxford University Press, 1983). 11. Thomas Buchheim, "Wer kann, der kann auch anders," in Christian Geyer (ed.), _Hirnforschung und Willensfreiheit: Zur Deutung der neuesten Experimente_ (Frankfurt am Main: Suhrkamp, 2004), pp. 158–65 (here, p. 162). See also his _Unser Verlangen nach Freiheit: Kein Traum, sondern Drama mit Zukunft_ (Hamburg: Meiner, 2006). 12. I would like to thank the biologist Michael Hoch for an interesting conversation about the notion that one could also actually formulate interesting objections to free will on the basis of our discoveries of microbiota, which play a crucial role in our understanding of eating habits. Thus potential determination by nature actually begins in the stomach. 13. John McDowell, _Mind and World_ (Cambridge, MA: Harvard University Press, 1994). 14. Arthur Schopenhauer, _The World as Will and Representation_ , Vol. 2, trans. E. F. J. Payne (New York: Dover, 1966), p. 533. 15.Ibid., p. 557: Marriages from love are contracted in the interest of the species, not of individuals. It is true that the persons concerned imagine they are advancing their own happiness; but their actual aim is one that is foreign to themselves, since it lies in the production of an individual that is possible only through them. Brought together by this aim, they ought then to get on with each other as well as possible. However, the two persons, brought together by that instinctive delusion that is the essence of passionate love, will in other respects be very often of quite different natures. This comes to light when the delusion vanishes, as it necessarily must. Accordingly, marriages contracted from love prove as a rule unhappy . . . 16. Arthur Schopenhauer, _Parerga and Paralipomena: Short Philosophical Essays_ , Vol. 2, trans. E. F. J. Payne (Oxford: Clarendon Press, 1974), p. 614. 17. Ibid., p. 626. 18. Ibid. 19. To learn more about this critical analysis in detail, I recommend Geert Keil's informative _Kritik des Naturalismus_ (Berlin und New York: De Gruyter, 1993). 20. Friedrich Nietzsche, _Nachgelassene Fragmente 1882–1884_ , Vol. 10 of _Sämtliche Werke: Kritische Studienausgabe in 15 Bänden_ (Munich: Deutscher Taschenbuch, 2009), p. 420. 21. Nietzsche, _Nachgelassene Fragmente 1885–1887_ , Vol. 12, p. 26. 22. Nietzsche, "Schopenhauer as Educator," in _Untimely Meditations_ (Cambridge: Cambridge University Press, 1997), pp. 125–94. 23. Immanuel Kant, _Groundwork for the Metaphysics of Morals_ , ed. and trans. Allen W. Wood (New Haven, CT: Yale University Press, 2002), p. 53. 24. Ibid. 25. Immanuel Kant, _Critique of Practical Reason_ , trans. Werner S. Pluhar (Indianapolis: Hackett, 2002), p. 123. 26. Ibid. 27. Stanley Cavell, _The Claim of Reason: Wittgenstein, Skepticism, Morality, and Tragedy_ (Oxford: Oxford University Press, 1999), p. 109. 28. Jean-Paul Sartre, _Being and Nothingness_ , trans. Hazel E. Barnes (New York: Citadel Press, 1969), p. 566. 29. Gottfried Benn, "After Nihilism," in _The Weimar Republic Sourcebook_ (Berkeley: University of California Press, 1994), p. 380. 30. Ibid. 31. Gottfried Benn, _Gesammelte Werke_ (Wiesbaden: Limes, 1968), p. 197. 32. Ibid. 33. Ibid., p. 433. 34. Johann Wolfgang von Goethe, _Faust: Parts 1 and 2_ , trans. Louis MacNeice (New York: Continuum, 1994), p. 220 (Verse 8472). 35. Ibid. (Verse 8469); translation modified. 36. Ibid. (Verse 8479). 37. For an extensive treatment of this term, see Nietzsche's _The Birth of Tragedy_ (Cambridge: Cambridge University Press, 1999). 38. Johann Gottlieb Fichte, "On the Ground of Our Belief in a Divine World-Governance," in _J. G. Fichte and the Atheism Dispute (1798–1800)_ , trans. Curtis Bowman (Farnham: Ashgate, 2010), pp. 17–29 (here p. 26). 39. Benn, _Gesammelte Werke_ , p. 97. 40. Frans de Waal, _Primates and Philosophers: How Morality Evolved_ (Princeton, NJ: Princeton University Press, 2006), p. 8. 41. Bruno Latour, _We Have Never Been Modern_ (Cambridge, MA: Harvard University Press, 1993). 42. Thomas Hobbes, _De cive, or, The Citizen_ (New York: Appleton-Century-Crofts, 1949), p. 1. 43. Immanuel Kant, "Toward Perpetual Peace," in _Practical Philosophy_ , ed. and trans. Mary J. Gregor (Cambridge: Cambridge University Press, 1996), p. 335. 44. Ibid. 45. Immanuel Kant, _Metaphysics of Morals_ , trans. Mary J. Gregor (Cambridge: Cambridge University Press, 1996), pp. 192–3. 46.Ibid. 47. We lose something of the force of the German here by rendering the title of this lecture series in a manner consistent with the rest of the book, so the reader is here advised to consider the title more literally – "Driving the Mind out of the Sciences of Mind" [Trans.]. 48. See my reviews of the three volumes that have appeared to this point: "Wesentliche Bejahung des Nationalsozialismus," _Die Welt_ , April 7, 2014; "Der Nazi aus dem Hinterhalt," _Die Welt_ , March 8, 2014; and "Wo 'Geschick' waltet, darf keine Schuld sein," _Die Welt_ , March 21, 2015. 49. Martin Heidegger, _Nature, History, State, 1933–1934_ , trans. and ed. Gregory Fried and Richard Polt (London: Bloomsbury, 2013), p. 56. See also the references in Emmanuel Faye's _Heidegger: The Introduction of Nazism into Philosophy in Light of the Unpublished Seminars of 1933–1935_ , trans. Michael B. Smith (New Haven, CT: Yale University Press, 2009), ch. 5. 50. Michel Foucault, _The Order of Things: An Archeology of the Human Sciences_ (London: Routledge, 1989), pp. 421–2. 51.Ibid., p. 422. 52. Ibid. 53. Rainer Maria Rilke, _Duino Elegies and The Sonnets to Orpheus_ , trans. A. Poulin, Jr. (Boston: Houghton Mifflin, 1975), pp. 49–51. 54. Friedrich Wilhelm Joseph Schelling, _Briefe von und an Hegel_ , Vol. 1, ed. Johannes Hofmeister (Hamburg: Meiner, 1952), p. 22.
# **Index**
## **A**
* action
* explaining –
* freedom of –
* philosophical definition of –
* teleology
* voluntary/involuntary –
* Adorno, Theodor W.
* _Dialectic of Enlightenment_ (with Horkheimer)
* Aeschylus
* "After Nihilism" (Benn) –,
* altruism
* Dawkins on –
* egoism andº6, –,
* society and –
* _The Analysis of Mind_ (Russell)
* animals
* anthropomorphism
* as automatons
* consciousness of –
* Davidson's dog
* duality with humans –
* natural actions and –
* rights of
* animism
* Freud and –
* joyful
* anthropic principle, nature and
* Anthropocene
* anthropology, symmetrical
* antinaturalism –
* _The Apology of Socrates_ (Plato)
* Aristotle
* five elements ,
* _On the Soul_ ,
* philosophy of mind
* resists naturalism
* artificial intelligence
* consciousness of robots –
* extended minds
* fictional images of –
* intentionalism
* _Artificial Intelligence_ (film)
* arts/humanities
* division from science
* expansion of consciousness –
* _Geist_ and –
* Indian epic literature
* atheism –
* _Attempt at a Critique of All Revelation_ (Fichte)
* _Awakenings_ (film)
## **B**
* Baars, Bernard
* Bach, Johann Sebastian
* Baker, Lynne Rudder
* behaviorism, reductionism and
* _Being John Malkovich_ (film)
* beliefs, Plato on
* Benn, Gottfried
* "After Nihilism" –,
* Berkeley, George
* Besson, Luc
* _Lucy_ ,
* _Bhagavad Gita_
* Bieri, Peter
* Bigelow, Kathryn
* _Strange Days_
* binding problem
* Block, Ned
* Blomkamp, Neil
* _Chappie_
* Bonn, University of
* "Decade of the Human Brain"
* the brain
* conditions of identity
* dictation of –
* mental states as identical to
* mind states as identical
* research in –
* as self –
* varieties of –
* _Brain and Mind/Gehirn und Geist_ journal
* Brandom, Robert ,
* Brentano, Franz
* Bruno, Giordano ,
* Buchheim, Thomas
* Büchner, Ludwig
* _Force and Matter_ –
* Burge, Tyler
* _Origins of Objectivity_
* relevance of self-consciousness
* Bush, George H. W.
* "Decade of the Brain" –,
* Butler, Judith
* gender theory
* performativity theory
## **C**
* Camus, Albert ,
* capacity theory of human freedom –
* Carnap, Rudolf
* Carruth, Shane
* _Upstream Color_
* Catholic Church, Big Bang theory and ,
* causal closure of nature principle
* causation
* hard anonymous cause –
* metaphysics and
* Cavell, Stanley , , ,
* Chalmers, David
* hard problem of consciousness
* panpsychism
* problem of consciousness
* chance, capacity theory of freedom and
* Changeaux, Jean-Pierre
* _Chappie_ (film)
* Churchland, Patricia
* eliminative materialism , –
* Churchland, Paul
* "Eliminative Materialism and the Propositional Attitudes" –
* _The Circle_ (Eggers)
* circular reasoning
* about self-consciousness –
* cognition
* cognitive suicide
* neuroscience and
* color, impressions of
* compatibilism –
* free will and
* _The Concept of Mind_ (Ryle)
* consciousness
* of animals –
* awareness of others' –
* as brain interface
* Cartesian theater , , –, –,
* changing conception of
* clown ego of –
* cosmological riddle of –
* degrees of –
* double contingency
* eliminative materialism –
* empiricism as phenomenal –
* Freudian unconscious and –
* God's eye view –
* hard problem of , , –
* homunculus fallacy –,
* intentional , –
* inverted spectrum problem –
* Kant's paralogism and –
* Leibniz's hard problem of –
* mind and , –,
* naturalistic metaphysics and
* necessity of self-reflection –
* neuroconstructivism –
* neuroscience and
* of others , –
* phenomenal –, –
* phenomenology and –
* philosophy of
* physical correlate
* possession condition –
* propositional attitudes –
* qualia ,
* at quantum level
* rationalism as intentionalism –
* of robots –
* secularization and –
* self and , , –
* skill acquisition –
* stream of
* time and –
* unconscious bodily functions and
* _see also_ mind; self-consciousness
* constructivism
* contingency, double
* cosmology, consciousness and –
* _Cosmos: A Spacetime Odyssey_ (television)
* deGrasse on medieval Italy –
* ship of imagination –
* Cotard, Jules –
* Crick, Francis
* _The Crisis of European Science and Transcendental Phenomenology_ (Husserl)
* _The Critique of Pure Reason_ (Kant) ,
* crude identity thesis ,
* culture, neuroscience and –
## **D**
* Darwinitis
* avoiding self-knowledge –
* dehumanization from below
* epiphenomenal qualia and
* _Fargo_ example of
* Davidson, Donald
* dog's consciousness –,
* Dawkins, Richard
* the "blind watchmaker"
* critique of religion –
* _The God Delusion_
* scientism –
* _The Selfish Gene_ –
* on teleology
* _Death in Venice_ (Mann)
* deGrasse Tyson, Neil
* _Cosmos_ remake –
* ship of imagination –
* Dehaene, Stanislas
* Dennett, Daniel
* Cartesian theater ,
* consciousness
* the homunculus fallacy
* _The Intentional Stance_ –
* sense of self
* Derrida, Jacques
* nakedness and his cat
* Descartes, René
* animals and
* on automata –,
* conditions for consciousness
* Fichte's "I am I"
* location of mind
* _Meditations on First Philosophy_ –, –
* mind and brain
* self and thinking
* self-consciousness –
* determinism
* awareness of determiners –
* compatiblism/ incompatibilism –
* indeterminism
* movie theory of nature –,
* naïve –
* neural , , –
* physical –
* strict laws of nature ,
* theological ,
* dignity of human beings –
* _Doctor Who_ (television) –,
* _Dreams of a Spirit-Seer_ (Kant) –
* Dreyfus, Hubert –
* dualism
* human–animal –
* mind–nature/body –, ,
* _Duino Elegies_ (Rilke)
## **E**
* Eckhart, Meister –
* Economo, Constantin von
* Eggers, Dave
* _The Circle_
* Einstein, Albert
* naturalistic metaphysics
* space, time and light
* "Eliminative Materialism and the Propositional Attitudes" (Churchland) –
* empiricism
* consciousness –
* definition of
* fallibility and
* Krauss and –
* logical
* scientific method and –
* En-Hedu-Anna
* Enlightenment thought
* mental freedom
* rationalist universalism
* scientific rationality
* entropy, time and –
* _Epic of Gilgamesh_
* epiphenomenalism
* concept of –
* qualia and –, –
* epistemology
* _a priori_ knowledge –
* Descartes' insight
* neuroscience and reality
* objectivity –
* _Essay on Free Will_ (Inwagen) –
* essentialism
* freedom and
* ethics and morality
* biology's detachment from
* universal humanity and
* varied standpoints
* Eurocentrism
* evolution
* _see also_ Darwinitis
* _Ex Machina_ (film)
* existentialism
* concepts of –
* mental freedom and
* teleology and
* _see also_ neo-existentialism
* experience, empiricism and –
* _The Expulsion of_ Geist _from the Humanities_ (lecture series)
* externalism
* Putnam and concepts of –
* social
## **F**
* fairy tale of the container
* Falkenburg, Brigitte
* mental and physical phenomena
* _Mythos Determinismus_ –
* fallibility, empiricism and
* _Fargo_ (television) –,
* _Faust_ (Goethe) –
* Fichte, Johann Gottlieb
* absolute I –
* _Attempt at a Critique of All Revelation_
* category of "nature" –
* _Geist_
* Goethe and
* mental freedom
* objection to Descartes
* pillars of science of knowledge –
* self and knowledge –
* self-consciousness , , ,
* social interactionism
* _Force and Matter_ (Büchner) –
* Foucault, Michel
* _The Order of Things_ –
* _Free Will_ (Harris) ,
* freedom/free will
* action –
* brain and –
* capacity theory of –
* chance and
* classic debates about –
* compatibilism –
* control-revolution
* defense of
* explaining actions –
* God and –
* hard problem of –, ,
* of the mind , –
* neuroscience and , , –
* principle of sufficient reason –
* reification of "will"
* Schelling on
* self-concept and , –
* skill acquisition and –
* subjectivity and
* _see also_ determinism
* Freud, Sigmund
* animism and –
* cathexis
* discontent with civilization
* ego and self –, –,
* _The Ego and the Id_
* ego/consciousness as a clown –
* "Female Sexuality"
* hard realities
* history and literature
* the id and ego –
* "Instincts and Their Vicissitudes"
* libido
* mental freedom
* _Moses and Monotheism_
* Oedipus complex
* politics of mythology
* _Project for a Scientific Psychology_
* the reality principle
* reification
* sexuality and –
* the superego
* _Totem and Taboo_
* functionalism
* dehumanization from above
* machine state –
## **G**
* Gabriel, Markus
* _Why the World Does Not Exist_ , , ,
* Gadamer, Hans-Georg
* Galileo Galilei
* Garland, Alex
* _Ex Machina_
* _Geist_
* for 21st century –
* Eckhart's "I"
* humanity and
* mind and –
* naturalistic metaphysics and ,
* self-consciousness and
* gender
* Freud and
* Schopenhauer on women –
* Germany
* "Decade of the Human Brain"
* human dignity
* modernity –
* global warming
* God
* Dawkins on
* dehumanization from above –
* human freedom and –
* mind and brain
* self as god –
* _see also_ religion
* _The God Delusion_ (Dawkins)
* Goethe, Johann Wolfgang von
* _Faust_ –
* Greek philosophy
* the five elements –
* mind and body problem
* Green, Brian
* _The Hidden Reality_
* guilt, animals and –
## **H**
* Harris, Sam
* _Free Will_ ,
* Hasler, Felix
* _Neuromythology_
* Hegel, Georg W. F.
* dismisses neuroreductionism
* _Geist_
* "I"
* mental freedom
* _Phenomenology of Spirit_ , , , ,
* self-consciousness , ,
* social interactionism
* unhappy consciousness
* Heidegger, Martin
* equi-primordial consciousness
* Foucault and –
* National Socialism and
* phenomenology and
* Helmholtz, Hermann von –
* _Her_ (film) –
* hermeneutics –
* Herodotus
* _The Hidden Reality_ (Green)
* history
* expansion of consciousness –
* Marx on
* Hobbes, Thomas
* _Leviathan_
* Hoffmann, E. T. A.
* _The Sandman_
* Hogrebe, Wolfram ,
* Homer
* homosexuality
* homunculus fallacy –,
* freedom and
* Freud and –, –
* Goethe and
* reification of self
* Horkheimer, Max
* _Dialectic of Enlightenment_ (with Adorno)
* Hubble, Edwin –
* human beings
* action in self-understanding
* animals and –, –
* the Anthropocene
* "civilization" and "savages" –
* dehumanization and –
* dignity of –
* expanding consciousness –
* first-person standpoint –
* goals of
* God and freedom –
* interest in the mind –
* irrational feelings and –
* Linnaeus names "wise" –
* philosophical zombies
* prospects for –, –
* scientific image of
* universalism and –
* _Human Brain Project_
* European Commission project –
* Humboldt (formerly Friedrich Wilhelm) University
* Hume, David
* bundle theory of self
* impressions and ideas
* Husserl, Edmund
* _The Crisis of European Science and Transcendental Phenomenology_
* "life world" –
* objection to Descartes
* phenomenology and –
## **I**
* idealism
* _Geist_ –
* transcendental
* ideas, Hume's definition of
* identity
* crude identity thesis
* ideology
* Marx and self-images
* mind–brain relation
* philosophy and society
* self-conceptions and –
* _Iliad_ (Homer)
* immortality, self and data
* impressions –
* _see also_ perception
* _In Search of Lost Time_ (Proust)
* indispensability thesis
* individuals _see_ human beings
* "Instincts and Their Vicissitudes" (Freud)
* intention
* Dennett and consciousness –
* evolution and
* phenomenal and intentional consciousness –
* _The Intentional Stance_ (Dennett) –
* Inwagen, Peter van
* _Essay on Free Will_ –
* ISIS, altruism under
## **J**
* Jackson, Frank Cameron
* epiphenomenalism –
* knowledge argument and physical ism –
* Jonze, Spike
* _Her_ –
* Jung, Carl G.
* justification condition
## **K**
* Kahneman, Daniel
* _Thinking, Fast and Slow_
* Kandel, Eric
* _Principles of Neural Science_ –
* Kant, Immanuel
* _a priori_ knowledge –
* on circular reasoning –
* _The Critique of Pure Reason_ ,
* dehumanization argument
* _Dreams of a Spirit-Seer_ –
* "enlightenment" –
* German Idealism
* human dignity –
* impressions and consciousness –
* influence on existentialism
* mental freedom
* on mind –
* neuro-Kantianism and –
* neuroscience and
* objection to Descartes
* paralogism –
* self-consciousness , –
* _Towards Perpetual Peace_ –
* transcendental realism
* _Universal Natural History_
* Keil, Geert
* on guilt –
* on the homunculus fallacy
* universe as interlocking mechanism
* Kierkegaard, Søren
* Kim, Jaegwon
* Kittler, Friedrich
* _The Expulsion of Geist from the Humanities_
* Kleist, Heinrich von
* _On the Marionette Theater_ –
* knowledge
* _a priori_
* difference from representation –
* effect on reality
* Fichte's science of , –
* indispensability thesis –
* justification condition
* the knowledge argument –
* rampant empiricism and –
* self-knowledge –
* truth condition
* Koch, Christof
* Korsgaard, Christine
* Krauss, Lawrence
* _a priori_ knowledge and –
* consciousness
* empiricism
* experience and critical thinking –
* Kucklick, Christoph
## **L**
* La Mettrie, Julien Offray de
* _Man a Machine_ –
* Lacan, Jacques
* language, the brain and
* Latour, Bruno
* _We Have Never Been Modern_
* Lear, Jonathan
* legocentrism
* Leibniz, Gottfried Wilhelm
* consciousness and , –
* hard problem of free will
* metaphor of the mill
* principle of sufficient reason –
* Lemaître, Georges
* Big Bang theory
* _Leviathan_ (Hobbes)
* Lichtenberg, Georg Christoph
* Lindgren, Astrid
* _Pippi Longstocking_
* Linnaeus, Carl ,
* literature _see_ arts/humanities
* Locke, John
* _Lucy_ (film) , ,
* ludic drive, arts and
* Luther, Martin
* _On the Bondage of the Will_
* Lynch, Michael Patrick
## **M**
* machine state functionalism –
* _Mahabharata_
* Mallarmé, Stéphane
* "A Throw of the Dice Will Never Abolish Chance"
* _Man a Machine_ (La Mettrie) –
* "Manifesto: Eleven Leading Neuroscientists"
* Mann, Thomas
* _Death in Venice_
* Marx, Karl
* consciousness of others
* exchange and use values
* false self-images
* Fichte and
* mental freedom
* teleology and
* materialism
* concept of
* eliminative –
* _Matrix_ (film series) , ,
* McDowell, John
* _Mind and World_
* MacFarlane, Seth , –
* Mead, George Herbert
* _Meditations on First Philosophy_ (Descartes) –, –
* Meister Eckhart –
* _Meno_ (Plato)
* mental states _see_ consciousness; mind
* Merleau-Ponty, Maurice
* metaphysics
* causal chains and
* definition of
* Descartes' insight
* legocentrism
* nature and
* pessimism –
* "Metaphysics of Sexual Love" (Schopenhauer)
* Metzinger, Thomas
* on philosophy and neuroscience
* self and brain –
* microfundamentalism
* mill, metaphor of the , –
* mind
* causal closure of nature
* conditions of identity
* consciousness and –,
* dualism with nature ,
* freedom of –
* _Geist_ and –,
* location of
* Marx's concept of
* neuroscience and
* philosophy of ,
* propositional attitudes –
* reductionism and
* relevance of philosophy
* secularization and –
* self-concept and
* self-knowledge of –
* theory of
* _Mind and Cosmos_ (Nagel) ,
* _Mind and Matter_ (Schrödinger) –
* _Mind and World_ (McDowell)
* modernity, science and
* monism
* monads and self-consciousness –
* neuromonism
* morality _see_ ethics and morality
* _Moses and Monotheism_ (Freud)
* movie theory of nature –,
* Müller, Johannes –
* mythology
* _Mythos Determinismus_ (Falkenburg) –
## **N**
* Nagel, Thomas
* Fichte's "nature"
* _Mind and Cosmos_ ,
* _The Possibility of Altruism_
* view from nowhere
* _The View from Nowhere_
* "What is it Like to be a Bat?"
* naturalism
* antinaturalism and –
* concept of
* consciousness and –
* definition of
* free will and
* _Geist_ and ,
* Marx and
* naturalistic metaphysics –
* search for self
* nature
* anthropic principle
* causal closure principle
* determinist laws of
* dualism with mind –,
* fairy tale of the container
* Fichte's category of –
* metaphysics and
* self in –
* visible/invisible processes –
* _Naturphilosophie_ (Schelling) –, –
* necessity, principle of sufficient reason and –
* neo-existentialism –
* concepts of
* essentialism
* main positive thesis of
* neurocentrism
* claims of –
* concept of –
* Descartes and –
* metaphysical paranoia
* naturalization of phenomenon
* reductionism and
* self as brain –
* self-consciousness –
* teleology and
* neuroconstructivism –
* Kant's paralogism and –
* neuromania –
* neuromonism
* _Neuromythology_ (Hasler)
* neurophilosophy
* neuroreductionism
* essentialism and
* puberty and –
* neuroscience
* biology and ethics
* brain necessary for mind
* culture and –
* determinism and , , –, –
* eliminative materialism –
* free will and
* the homunculus fallacy –
* image of humans
* knowledge from
* law of specific nerve energies –
* "theory golems"
* thought and neuroimaging –
* New Atheism –
* New Realism –
* Newton, Isaac
* mind and nature
* naturalistic metaphysics
* understanding of physics –
* Nietzsche, Friedrich
* the Dionysian
* mental freedom
* on reification of "will"
* "No One Can Be Other Than They Are" (Singer)
* nuclear weapons
## **O**
* Obama, Barack
* "Brain Activity Map" initiative
* objectivity
* absolute
* definition of –
* _see also_ subjectivity
* _Odyssey_ (Homer)
* Oedipus complex
* _Oedipus the King_ (Sophocles) ,
* _The Office_ (television) –
* _On the Bondage of the Will_ (Luther)
* _On the Marionette Theater_ (Kleist) –
* _On the Soul_ (Aristotle) ,
* _On Women_ (Schopenhauer) –
* Onfray, Michael
* ontology
* ontological reductionism
* subjectivity –
* _Open Minds_ (Prinz) , –
* _The Order of Things_ (Foucault) –
* _Origins of Objectivity_ (Burge)
## **P**
* panpsychism
* panspermia hypothesis
* "The Panther" (Rilke)
* _Peep Show_ (television)
* Penrose, Sir Roger
* perception
* color blindness and knowledge –
* Husserl's "lifeworld" –
* impressions –
* neuro-Kantianism –
* neuroconstructivism –
* neuroscience and reality
* performativity theory
* pessimism, metaphysical –
* Pfister, Wally
* _Transcendence_ ,
* phenomena
* artificial intelligence and –
* consciousness and –
* and intentional consciousness –
* phenomenology
* concern with consciousness –
* _The Phenomenology of Spirit_ (Hegel) , ,
* dismisses neuroreductionism
* self-consciousness
* _Philosophical Investigations_ (Wittgenstein)
* philosophy
* ideology critique and
* philosophical zombies
* propositions and theory
* scholastic _versus_ cosmopolitan –
* science and –
* _Philosophy in the Brain Scan/ Philosphie im Hirnscan_ (radio)
* physical being
* mind–body dualism
* physicalism
* Jackson on –
* microfundamentalism
* physics
* Big Bang theory –
* fairy tale of the container
* theory of everything
* universal laws of nature –
* visible/invisible processes –
* _Pippi Longstocking_ (Lindgren)
* Pippin, Robert B.
* Plato
* _The Apology of Socrates_
* investigating the self
* _Meno_ –
* philosophy of mind
* _The Republic_
* resists naturalism
* _Theaetetus_
* politics
* Greek philosophy and
* universalism –
* _The Possibility of Altruism_ (Nagel)
* poststructuralism, neuroscience and
* _Principles of Neural Science_ (Kandel) –
* Prinz, Wolfgang
* _Open Minds_ , –
* _Project for a Scientific Psychology_ (Freud)
* propositional attitudes
* consciousness and –
* Proust, Marcel
* _In Search of Lost Time_
* psychoanalysis –, –
* psychology, folk –, ,
* puberty, neuroreductionism and –
* Putnam, Hilary
* on animism
* externalism/machine state functionalism –
* _Reason, Truth and History_
## **Q**
* qualia
* definition of
* eliminativism
* epiphenomenalism and –, –
* physicalism and
* Quine, Willard van Orman
* logical empiricism
* racism
* Büchner's reductionism –
* essentialism
## **R**
* _Real Humans_ (television)
* realism
* Kant's transcendental
* New Realism –
* reality
* knowledge and
* naturalistic metaphysics
* neuroconstructivism –
* neuroscience and –
* physical world –
* the universe
* the universe and
* reason and rationalism
* consciousness –
* Freudian ego and
* principle of sufficient reason –,
* rationalist universalism
* _The Rediscovery of Mind_ (Searle)
* reductionism
* explaining away mind
* neurocentrism and
* ontological
* theory ,
* reification of "will" and self –
* religion
* antinaturalism and –
* Dawkins's critique of –
* dehumanization and –
* disenchantment of the world
* Euro-Hinduism
* God and freedom –
* moral standpoint of Christianity
* self as god –
* theological determinism ,
* representation
* difference from knowledge –
* neuroscience and
* _The Republic_ (Plato)
* rights, human –
* Rilke, Rainer Maria
* _Duino Elegies_
* "The Panther"
* Rödl, Sebastian
* Romanticism, German
* automatic _versus_ free actions
* Euro-Hinduism
* Russell, Bertrand
* _The Analysis of Mind_
* mental–natural dualism
* Ryle, Gilbert
* _The Concept of Mind_
* derides "ghost" , –
## **S**
* Sacks, Oliver
* _Awakenings_ –
* Sagan, Carl
* _The Sandman_ (Hoffman)
* Sartre, Jean-Paul ,
* neo-existentialism and
* phenomenology and
* self-divination –
* Schelling, Friedrich W. J.
* on consciousness
* Fichte and
* freedom ,
* _Naturphilosophie_ –,
* self-consciousness
* Schmidt, Thomas E.
* Schopenhauer, Arthur
* free will and free action –
* hard problem of free will
* metaphysical pessimism –
* "Metaphysics of Sexual Love"
* _On Women_ –
* reification of "will"
* social Darwinism and
* _The World as Will and Representation_
* Schrödinger, Erwin
* _Mind and Matter_ –
* science
* _a priori_ knowledge and –
* the five elements
* futuristic
* God and human mind –
* microfundamentalism
* philosophy and –
* _see also_ naturalism
* scientism, definition of
* Searle, John
* on consciousness
* against epiphenomenalism –
* Fichte's "nature"
* _The Rediscovery of Mind_
* unconscious and skills
* _Seinfeld_ (television) ,
* self
* awareness of others –
* as the brain , –
* bundle theory of
* data and immortality
* divination of –
* freedom and –
* Freudian ego and –, –
* as god –
* human self-concept
* mind and brain –
* nature and –, –
* neuroscience and
* Platonic investigation of
* seen as a USB stick –
* self-knowledge , –
* society and egoism –
* standpoint view of –, –
* substance theory of
* self-concept
* from consciousness
* plurality of
* vocabulary of
* self-consciousness
* circular reasoning about –
* definition of
* Descartes' cogito –
* feelings of self-assurance –
* as the higher authority
* history and literature –
* monad approach to –
* panpsychism –
* relevance of
* sensory impressions and –
* subjective and objective –
* _The Selfish Gene_ (Dawkins) –
* sexuality
* biology and
* Freud and –,
* Singer, Wolf
* naïve determinism
* "No One Can Be Other Than They Are"
* skill acquisition, Dreyfus model of –
* social Darwinism
* Darwinitis –
* moral standpoint of
* social interactionism
* externalism and
* Fichte and
* self-consciousness and others –
* society
* egoism and altruism –
* ideology critique
* self-images and –
* Sophocles
* _Oedipus the King_ ,
* soul, belief in
* Spinoza, Baruch –
* _Strange Days_ (film) ,
* _Stromberg_ (television) –
* structuralism, human goals and
* subjectivity
* awareness of others –
* defining "subjective"
* indispensability thesis –
* inverted spectrum problem –
* ontological –
* qualia of
* _see also_ objectivity; self
* sufficiency
* principle of sufficient reason –
* superstition, religion and
* surveillance
* Swaab, Dick
* _We Are Our Brains_
## **T**
* Tallis, Raymond
* teleology
* action
* brain research and –
* _Theaetetus_ (Plato) ,
* thinking
* invisibility of –
* neurocentrism and –
* neuroscience and
* _Thinking, Fast and Slow_ (Kahneman)
* Thornton, Billy Bob
* Thrasymachus
* "A Throw of the Dice Will Never Abolish Chance" (Mallarmé)
* Thucydides
* time, conscious experience of –
* totalitarianism, modernity and
* _Totem and Taboo_ (Freud)
* _Towards Perpetual Peace_ (Kant) –
* _Transcendence_ (film) ,
* transcendental realism
* truth
* in all areas
* condition
* _see also_ knowledge
## **U**
* United States
* "Decade of the Brain" –
* _Universal Natural History and Theory of Heaven_ (Kant)
* universalism
* humans and rights –
* the universe
* Big Bang and physics of –
* as cold home
* enclosed reality notion
* science and mind and –
* _Upstream Color_ (film)
* utopianism –
## **V**
* value
* dignity _versus_ price
* _The View from Nowhere_ (Nagel)
* Vogt, Karl –
## **W**
* Waal, Frans de
* _The Walking Dead_ (television)
* _We Are Our Brains_ (Swaab)
* _We Have Never Been Modern_ (Latour)
* Weber, Max
* _Westworld_ (television)
* "What is it Like to be a Bat?" (Nagel)
* _Why the World Does Not Exist_ (Gabriel)
* freedom and
* objectivity ,
* reality , ,
* Wittgenstein, Ludwig
* _Philosophical Investigations_
* _The World as Will and Representation_ (Schopenhauer)
## **Z**
* zombies –
* philosophical
# **POLITY END USER LICENSE AGREEMENT**
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AGDC '07: Dave Perry's back pages
On the final day of the conference, the veteran designer talks about his humble beginnings, getting burned believing his own hype, and why he doesn't like the Wii.
By Brendan Sinclair on September 10, 2007 at 10:01AM PDT
AUSTIN, Texas--Veteran developer Dave Perry's name is most closely associated with games that came out years ago: MDK, Enter the Matrix, Disney's Aladdin, Earthworm Jim. After leaving Atari in February of 2006, Perry hooked up with the resurrected Acclaim, an upstart developer of free-to-play massively multiplayer online games, and has since been working with a number of Korean MMO development teams on a half-dozen projects.
"Dave, Jamil. Jamil, Dave."
Perry took time out from his development schedule to appear on the last day of the Austin Game Developers Conference and field questions from GDC director Jamil Moledina for a crowd of attendees. The discussion traced Perry's history in games, from teaching himself how to code in Basic from magazines as a boy in Northern Ireland to his current Top Secret community-designed MMO project at Acclaim. The talk was also peppered with interesting trivia, humorous anecdotes, and opinionated (and sometimes self-effacing) jabs along the way.
To show how "ghetto" the early development process was, he described an early attempt at photorealistic graphics. To get digitized images, his team took pictures of themselves performing the various animations, and then cut the backgrounds out. Not thinking to do it in front of black backdrop, the team found itself cutting an old backyard shed out of every picture in order to get the process to work.
To demonstrate the process (and more likely to razz a friend), Perry showed a picture of himself hitting his coworker Neil Young in the face with a toilet plunger. Young is now vice president and general manager of Electronic Arts Los Angeles.
"I left school to do this, and the first job offer I had was 3,500 pounds per year, which is like $5,000, to move to London," Perry said. "And I didn't do any math or ask what it costs to live in London."
Perry called it a sticker shock when he arrived in England's capital, noting that it cost almost his entire salary just to take the train to work each day.
"But that was just the way it was," Perry said. "If you wanted in, you had to do that. But very quickly the salaries started to rise. You do a game, it sells, and they say, 'Please don't leave, here's some money.'"
One of the tricks to getting by for Perry was cranking out a lot of games. He wrote his own engine, which allowed him to churn out titles like the Teenage Mutant Ninja Turtles PC game in a week. At one time, he had 11 games on the charts at once.
That sort of prodigious output was attractive to prospective publishing partners. Virgin Games lured Perry over to the US when they needed a licensed game made for McDonald's, and they needed it on short notice. The game was called Global Gladiators, and eschewed McDonald's trademark stable of characters in favor of Mick and Mack, a couple of generic kids that fit right in with the traditional platformer-shooter gameplay.
"This was a bit of a disaster," Perry said. "We made the game, which we thought was fun, and McDonald's came to have a look at it. They said, 'What is this? What are you thinking? Where are the restaurants in the game? How do I buy burgers and fries?'"
Perry said nobody wanted to buy burgers and fries in a game, and even avoided using Ronald McDonald in the game originally. When the company insisted, he said the team "threw Ronald McDonald waving a flag at some point."
"The game won Game of the Year," Perry said, "so we were all high-fiving and saying, 'To hell with McDonald's!' But it turned out that they were pretty pissed. They went and hired another team and they made a new McDonald's game with restaurants that you can buy burgers and fries in. It was called McDonaldland, and it didn't get Game of the Year."
That experience didn't sour Perry on advertising-based games, as he went on to make Cool Spot based on the 7-Up mascot. Back then, advertising in games was shameless, Perry said, saying the whole game was built on pushing the brand, and noting that Cool Spot even began with the company mascot riding in on a 7-Up bottle.
Despite dealing with other people's intellectual properties, Perry was able to call the shots on a lot of his games from pretty early on.
"It's almost like you're bit of a liability, to be honest," Perry said. "You start to get trusted; you do a few things that sell, and people go, 'whatever you want to do.' It's like if Will Wright said he wanted to do a game about snakes next, everyone would say it sounds like a great idea because the guy's hitting home runs one after another. So snakes are obviously the future."
"I got to the point where I could abuse that a little bit and whatever I would want to do would get funded. I did some pretty wacky stuff. I did a little helicopter game [R/C Stunt Copter for the PlayStation] and things like that which really shouldn't get done. But at the time, no one was saying no."
Perry has dealt with other people's brands on and off throughout his career. Most recently, and perhaps most famously, he and Shiny handled the game tie-ins to The Matrix trilogy. But his two games based on the sci-fi films almost didn't happen. Before the first movie debuted, Perry said he was approached about making a Matrix game by the film's producer Joel Silver (Action Jackson, Road House). Perry said he was called into Silver's office, where he was shown an early demo of the film's signature "bullet-time" technology, starring nothing more than a burning barrel of fire. Unimpressed by the burning barrel and completely engrossed in the development of Shiny's PC game Sacrifice, Perry passed on the project.
"It all made sense, but at the end of the day, after seeing The Matrix, it didn't make sense," Perry said. "We should have put the pause on Sacrifice."
When the filmmakers geared up for a sequel to The Matrix, they again turned to Perry to see if he was interested in handling the game adaptation. Given a second chance, he readily agreed, and work began on Enter the Matrix for the PC, PlayStation 2, GameCube, and Xbox.
The ambitious project was a side story to the films, but one created with the filmmakers' blessing and participation. As part of the production, they shot an hour of new footage for cutscenes that would go into the game, including one scene in which two female characters kiss.
We were working really hard to try and get a [T for Teen] rating, and the [Entertainment Software Rating Board] was like, 'What are you thinking? Lesbian kissing in your game and you want a teen rating? Forget about it.' We actually said as our response that it wasn't two women kissing; it's two computer programs kissing. And wouldn't you believe it, the ESRB accepted that. We did it kind of as a joke, and they took it."
Moving to the current day, Moledina and Perry discussed his latest project, Top Secret. An MMO racing game, Top Secret is being developed by a community of gamers all vying for a "grand prize" that will see publisher Acclaim give one creative, aspiring designer an all-new MMO to direct.
Like all of Acclaim's games, Top Secret will be free to play and supported by in-game ads and microtransactions. That's a business model imported from Korea (like many of Acclaim's games) that Perry thinks has the potential to be a disrupting force in the gaming world. He said it's inevitable that a designer on par with Hideo Kojima (Metal Gear Solid series) or Shigeru Miyamoto (Super Mario, Zelda) will emerge from Korea or China, where these types of games are prevalent. And once that designer does emerge, Perry said nothing would be the same.
Moledina seized on the issue to ask Perry for his thoughts on Nintendo's Wii and its motion-sensing controller.
"That's truly disruptive," Perry said of the Wii. "I got sort of killed in the press recently because I said people are going to put their Wii-motes down when they start to play all the new stuff on PlayStation 3 and Xbox 360. It wasn't dismissive at all. Nintendo has disrupted our industry; they've done a fantastic job of that. My point is the games I personally like to play--Assassin's Creed and Killzone 2--I have a list full of them and none of them are on the Wii."
Mick & Mack as the Global Gladiators
Top Secret (Acclaim Games) | {
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Might it be through grief
at sight of the bush clover,
colored by autumn,
that the stag's cries continue
until the foothills resound?
—from Kokin Wakashū: The First Imperial Anthology of Japanese Poetry, translated by Helen Craig McCullough
THE TALE OF SHIKANOKO LIST OF CHARACTERS
MAIN CHARACTERS
Kumayama no Kazumaru, later known as Shikanoko or Shika
Nishimi no Akihime, the Autumn Princess, Aki
Kuromori no Kiyoyori, the Kuromori lord
Lady Tama, his wife, the Matsutani lady
Masachika, Kiyoyori's younger brother
Hina, sometimes known as Yayoi, his daughter
Tsumaru, his son
Bara or Ibara, Hina's servant
Yoshimori, also Yoshimaru, the Hidden Emperor, Yoshi
Takeyoshi, also Takemaru, son of Shikanoko and Akihime, Take
Lady Tora
Shisoku, the mountain sorcerer
Sesshin, an old wise man
The Prince Abbot
Akuzenji, King of the Mountain, a bandit
Hisoku, Lady Tama's retainer
THE MIBOSHI CLAN
Lord Aritomo, head of the clan, also known as the Minatogura lord
Yukikuni no Takaakira
The Yukikuni lady, his wife
Takauji, their son
Arinori, lord of the Aomizu area, a sea captain
Yamada Keisaku, Masachika's adoptive father
Gensaku, one of Takaakira's retinue
Yasuie, one of Masachika's men
Yasunobu, his brother
THE KAKIZUKI CLAN
Lord Keita, head of the clan
Hosokawa no Masafusa, a kinsman of Kiyoyori
Tsuneto, one of Kiyoyori's warriors
Sadaike, one of Kiyoyori's warriors
Tachiyama no Enryo, one of Kiyoyori's warriors
Hatsu, his wife
Kongyo, Kiyoyori's senior retainer
Haru, his wife
Chikamaru, later Motochika, Chika, his son
Kaze, his daughter
Hironaga, a retainer at Kuromori
Tsunesada, a retainer at Kuromori
Taro, a servant in Kiyoyori's household in Miyako
THE IMPERIAL COURT
The Emperor
Prince Momozono, the Crown Prince
Lady Shinmei'in, his wife, Yoshimori's mother
Daigen, his younger brother, later Emperor
Lady Natsue, Daigen's mother, sister of the Prince Abbot
Yoriie, an attendant
Nishimi no Hidetake, Aki's father, foster father to Yoshimori
Kai, his adopted daughter
AT THE TEMPLE OF RYUSONJI
Gessho, a warrior monk
Eisei, a young monk, later one of the Burnt Twins
AT KUMAYAMA
Shigetomo, Shikanoko's father
Sademasa, his brother, Shikanoko's uncle, now lord of the estate
Nobuto, one of his warriors
Tsunemasa, one of his warriors
Naganori, one of his warriors
Nagatomo, Naganori's son, Shika's childhood friend, later one of the Burnt Twins
AT NISHIMI
Lady Sadako and Lady Masako, Hina's teachers
Saburo, a groom
THE RIVERBANK PEOPLE
Lady Fuji, the mistress of the pleasure boats
Asagao, a musician and entertainer
Yuri, Sen, Sada, and Teru, young girls at the convent
Sarumaru, Saru, an acrobat and monkey trainer
Kinmaru and Monmaru, acrobats and monkey trainers
THE SPIDER TRIBE
Kiku, later Master Kikuta, Lady Tora's oldest son
Mu, her second son
Kuro, her third son
Ima, her fourth son
Ku, her fifth son
Tsunetomo, a warrior, Kiku's retainer
Shida, Mu's wife, a fox woman
Kinpoge, their daughter
Unagi, a merchant in Kitakami
SUPERNATURAL BEINGS
Tadashii, a tengu
Hidari and Migi, guardian spirits of Matsutani
The dragon child
Ban, a flying horse
Gen, a fake wolf
Kon and Zen, werehawks
HORSES
Nyorin, Akuzenji's white stallion, later Shikanoko's
Risu, a bad-tempered brown mare
Tan, their foal
WEAPONS
Jato, Snake Sword
Jinan, Second Son
Ameyumi, Rain Bow
Kodama, Echo
HINA (YAYOI)
The girl could see nothing. Her lungs were bursting. At any moment, she would open her mouth and breathe in the fatal waters of the lake. Snatches of her brief life came to her: her mother's face, her father's last words, her brother's cry for help before he disappeared. She had been one of the few survivors after the massacre in Miyako. Now her life was over, and she and Takemaru, the baby she clutched desperately, would join the dead. Tears formed in her eyes, only to be lost in the ebb and flow of Lake Kasumi.
Then suddenly there were dark shapes next to her, strong arms seized her. She was pulled upward toward the light, miraculously still holding the baby. She retched and coughed, gasping for air, taking great gulps of it into her lungs. Hands reached down from the side of the boat and took Take from her. He was limp and pale, but, as she herself was pulled on board, she heard him scream in ragged, outraged gasps. He was alive.
The boat bucked like a living animal in the strong westerly wind. She saw the ocher-colored sail lowered quickly, dropped on the deck, while the helmsman struggled with the oar at the stern. The men who had plunged into the water to save her were lifted up; they tore their wet clothes off and went naked, laughing. Monkeys screamed and chattered at them, dancing at the end of their cords. The sun in the east was dazzling. A crowd surrounded her. The men who were not naked were all dressed in red. They looked like beings from another world and she was afraid that she had drowned. But women stripped the heavy robes from her with hands that felt real, exclaiming at their fine quality in human voices. She and the baby were wrapped in furs, wolf and bear skins, and a bowl of some warm, strange-smelling liquid was pushed into her hands.
Men hoisted the sail again, the hemp flapping, fighting them, ropes snapping, snaking through the air. The monkeys screamed more loudly. In the confusion, one of the boys approached her, holding the lute. Beneath the howl of the wind, the slap of the waves, it was still playing, but more softly, its mother-of-pearl and gold-inlaid rosewood gleaming in the sun.
"Who are you?" he said quietly. "What are you doing with Genzo?"
Fragments of memories came to her. It is Genzo, the Emperor's lute, Take's mother, Akihime, the Autumn Princess, had said, and she had promised to tell her where the child Emperor was, but she had not. Could this be him standing before her? It must be, the lute revealed him. But she must hide the fact she knew who he was.
She shook her head at him, as though she did not understand, and held out her hands. His eyes narrowed as he thrust the lute at her. She saw his unease, longed to speak to reassure him, but did not dare say anything. How would she address him, for a start? Words of honor and deference rose on her tongue, but then the sailors shouted roughly at him to come and help them. Beside him the other boy was holding a text, made up of pages stitched together.
"Yoshi caught the lute and I caught this," he said, holding it out to her. "It's heavy! How did a girl like you manage to throw it so far?"
She grabbed it from him. She could not explain it, maybe it had sprouted wings and flown. She already knew the Kudzu Vine Treasure Store was enchanted. She tucked it under one arm while she turned her attention to the lute. It gave a sigh, as if it would start playing; she gripped it with her other hand.
More shouts echoed around her. The boys darted from her side and the lute quieted. It retained all its beauty, but it surrendered to her touch and allowed her to play it. It no longer played itself, in that wild irrepressible outburst of joy.
"She is a musician," one of the men who had rescued her exclaimed. "We must take her to Lady Fuji."
The other looked back toward Nishimi, now barely visible over the choppy surface of the lake. "She must be from a noble family. Someone will miss her, someone will come looking for her."
"That was Lord Hidetake's home," the oarsman called. "He is dead."
"Could this be his daughter? The one they call the Autumn Princess?"
"The Autumn Princess would be a grown woman by now," said one of the women, who had already put Take to her breast and was nursing him. "This one is still a girl. How old are you, lady?"
"I turned twelve this year," the girl replied.
"And what do they call you?"
She did not want to say her name. There came into her mind a fragment of memory, a poem. "Yayoi," she said. It meant Spring.
"Is this little man your brother?" the woman asked, stroking Take's black hair tenderly.
She knew she must not tell them that the baby was the Autumn Princess's son. "No, my mother died, a long time ago. He is the child of one of my maids." She went on, improvising, "She died giving birth to him. I like to play with him. I was holding him when I had to run away."
"What were you running from?" They were sympathetic toward her, but their curiosity was becoming tinged with anxiety.
The girl who had named herself Yayoi began to shiver, despite the furs and the warm drink.
"A bad man came," she said, and then regretted sounding so childish. "I was afraid he was going to kill me."
"We should take her back," one of the men suggested.
"Kinmaru," the other man reproved him. "Someone was going to kill her!"
"And that someone, Monmaru, could very well come looking for her and then who will get killed? Us, that's who!"
"Can't turn back against this wind," the helmsman called. "It's impossible."
* * *
It was late in the afternoon by the time they came to the shore near the Rainbow Bridge. The market was almost over. Lanterns were being lit in the streets of Aomizu, on the island of Majima, and along the bridge. As soon as the boat grounded, the acrobats leaped ashore with the monkeys.
"It's not too late to do a trick or two," Kinmaru cried. Monmaru began to beat a small drum and immediately the boys threw themselves into a performance, a circle of somersaults with the monkeys, a high tower with three of the monkeys on top, a wild dance where the animals jumped from man to boy to man. A crowd soon gathered around them. Yayoi realized the audience knew the monkeys by name, calling out to them, Shiro, Tomo, Kemuri, and had their favorites, whom they applauded wildly. She was dazed by the noise, the colorful clothes, the shouts in a dialect she could barely understand. She gripped the lute and the text close to her chest, as though they could shield her from this strange, new world.
"Come," said the woman who had nursed Take—he was now asleep in her arms. "You will stay with us tonight and tomorrow we will ask Lady Fuji what she thinks we should do with you."
Yayoi slept restlessly on a thin mat in a room with three women and a clutch of children—one other young infant and three toddlers. The toddlers slept deeply like kittens. Take woke once screaming, and the other baby was colicky and fretful. Every time Yayoi felt herself dropping into sleep, the baby wailed and she woke in alarm, half-dreaming something had happened to Take, he had slipped from her arms underwater, he'd been stolen by monkeys. She heard the men and boys return later, their exaggerated efforts to keep quiet, their muffled laughter, the monkeys chattering as they were returned to their cages. For a few hours the house fell silent, but she thought she heard a bird call, while it was still dark, before even the roosters had woken, a long, fluting call like an echo from the past.
The women rose at dawn to prepare the morning meal. Yayoi, who had never made a meal in her life, held Take for a while. He was nearly two months old. He looked closely at her face and smiled.
He will never know his mother, she thought, and felt tears pool in her eyes. What would this day bring for them both? She felt sick and faint with fear.
"Don't cry, lady."
"Look how pale she is, white as a spirit."
"You need to be beautiful for Lady Fuji."
The women's voices echoed around her.
"Will Lady Fuji let me keep Takemaru?" she said.
They exchanged looks that she was not meant to see.
"The baby can stay with us."
"Yes, I have plenty of milk for two."
"You cannot look after him, you are still a child yourself."
"Then let me stay with you too!" Yayoi could not hold the tears in.
"This is no place for a young lady like you," Take's foster mother said.
It was cool in the early morning, but by the time Lady Fuji arrived the sun was high in the sky and the air was warm. She came in with a rustle of silk, cherry blossom petals in her hair, the sweet perfume of spring all around her.
The women immediately started to apologize on Yayoi's behalf.
"Her clothes are not yet dry."
"She's been crying, her eyes are red."
"She nearly drowned yesterday; she can't be expected to look her best."
Fuji studied Yayoi carefully, taking her head between her hands and tilting it from side to side. "I can see how she looks. What a beautiful child. Who are you, my dear, and where do you come from?"
Some instinct warned Yayoi that her former life was over and she should never speak of it. She shook her head.
"You can't tell me? Well, that may be for the best. You have a Kakizuki look to you. Are you a survivor of the massacre in the capital?"
Yayoi did not answer, but Fuji smiled as if she had acquiesced.
"Someone hid you at Nishimi, but you were discovered and that is why you ran away?"
This time Yayoi nodded.
"Can you imagine any man wanting to kill something so precious?" Fuji said. "Yet hundreds of women and children were put to death in Miyako last year when the Kakizuki warriors fled, leaving their families behind. I am of a mind to protect this one."
She looked around and saw the lute and the text. "You brought these with you? As well as the baby?" She picked up the lute and studied it with an expressionless face. It had lost its glowing rosewood and its gleaming inlay, yet Yayoi thought the older woman recognized it.
"So what am I to do with you?" Fuji said finally. "Is anyone going to come in pursuit of you?"
"I don't know," Yayoi replied. "Maybe." She held herself rigid, trying not to tremble.
"Someone must have seen you fall in the lake, but did they see you rescued? If anyone is looking for you, they will start their search with our boats, so I think I will take you somewhere you can be safely hidden. We will hold a funeral service for the children who sadly drowned."
Hina drowned and Yayoi was rescued.
"Will Take come with me?"
"How can a girl like you take care of a baby? And that would only draw unwanted attention to you. Take can stay here, the women will look after him. One more baby makes little difference to this troop of children."
She called to the women to bring some clothes, not Yayoi's own robes, which she told the women to cut up for costumes, but old castoffs that smelled of mildew and something sour like vinegar. When she was dressed, they covered her head with a cloth, which concealed her hair and most of her face.
"I must take my things," she said anxiously. "The lute and the text." Clasping them to her chest, she followed Fuji into the rear courtyard of the house, where the boys from the boat were feeding the monkeys and playing with them. A young girl was with them, idly beating a small drum, laughing at the monkeys and teasing the boys when they yawned and rubbed their eyes. Yayoi wanted to stay with them, to be one of them.
She felt the lute stir and quiver and the notes began to trickle from it. She gripped it, willing it to be silent. The girl came to Yoshi's side and took his hand protectively. Yayoi wondered if they had grown up together, if the girl was a princess like Aki.
Fuji shook her head. "It will be safer hidden away too," she said. "Kai, dear, I've told you before not to hang around here with the monkeys. Go back to your own place. I'm sure you have plenty of chores there."
"I wish I could stay here," Kai replied.
"What nonsense! Girls are never acrobats. Be thankful the musicians took you in."
Fuji helped Yayoi into the palanquin that rested on the ground outside the rear gate, the porters, two strong young men, beside it. They both bowed respectfully to Fuji, who gave them directions in a quick, low voice before she climbed in next to Yayoi and let down the bamboo blinds.
She heard the women call, "Goodbye! Goodbye! Take care of yourself."
"Goodbye, Takemaru," Yayoi whispered.
* * *
The lute quieted as the men jogged and the palanquin swayed. The stuffy heat and the motion made Yayoi sleepy and she nodded off several times, dreaming in brief, lucid snatches, then jolting suddenly awake. She could see nothing outside, only had the sensation of moving from light into shade, splashing through water, then going up a steep hill, the palanquin wobbling alarmingly as the men negotiated the steps. Finally, the palanquin was set down. Fuji raised the blind and stepped out.
Yayoi followed her, glad to breathe the cool mountain air. Below her, framed by twisted pine trees, lay Lake Kasumi. She could see smoke rising from the villages around its edge and the tiny sails of boats, gleaming yellow in the sun. Behind her a bell tolled. It must be midday.
"This is a temple for women," Fuji said. "I have sent a few girls here to be looked after, until they are old enough."
Old enough for what? Yayoi wondered, her mind shying away from the answer. She concentrated on what was around her: the vermilion wooden gate, the flowering mountain cherries, the steps that led upward beneath pines that curved over them like a dark tunnel.
Fuji began to climb them swiftly. Yayoi had to trot to keep up with her. The stones were set too high for a child and, by the time they reached the top, her legs ached. Someone must have been told of their arrival, for at the top of the steps a nun was waiting to greet them. Behind her was a garden, with a spring that filled a cistern then overflowed and ran trickling away from them into a large fishpond.
"Our abbess asks that you will take some refreshment with her." She looked at Yayoi with cool, unfriendly eyes. "You have another foundling for us to look after?"
"She is called Yayoi," Fuji said. "I would prefer as few people as possible to know she is here. It will not be for long."
"No," the nun agreed, her eyes appraising Yayoi's height and age. "I suppose she can join the other girls in prayer and study." She turned and began to walk toward a low building at the side of the temple. Its roof was curved at each end in an upward swoop, like wings, as if it would take flight at any moment.
The nun paused and said to Fuji, "Asagao will want to see you. She can be this girl's friend. They are about the same age." She clapped her hands.
A girl came from the building and dropped to her knees before Fuji, who stepped forward to take her hands and lift her to her feet. She looked carefully at her, much as the nun had studied Yayoi. The girl blushed. Yayoi thought her very pretty.
"Lady Fuji," Asagao said. "I am so happy. I missed you so much."
"Sweet child, I have brought someone to be your friend. Please take care of her for me."
"Go with her to the girls' room and show her where everything is," the nun said. "Give me your things. Well, well, what have you brought with you? An old lute and an even older text? The lute will be useful, but you won't need the text here. Don't worry, we will keep it safe for you. When you leave, you may take it with you."
"Reverend Nun, may we walk a little way with you and Lady Fuji?" Asagao pleaded.
She had an enchanting manner and the nun was charmed. "Very well, since it is so long since you have seen your benefactress. Just as far as the fishpond."
Red and white carp swam peacefully in the large stone basin, beneath lotus leaves from which the flower stems were just beginning to emerge.
"See how the red and the white can live together?" Asagao said. "Why is our world so torn by war?"
Fuji smiled. "You are very poetic, my dear. I can see you have been learning well. But it is best not to speak of the red and the white. As far as the Miboshi are concerned, there are now only the white."
"Yet in this pond the white are outnumbered by the red," Asagao said, so quietly only Yayoi heard. She wondered what her story was and how she had ended up under Fuji's protection. The two older women walked on and the girls were left alone.
* * *
Over the next few days she was able to learn more about Asagao and the other girls. Their ages ranged from six to fourteen. The oldest was gentle, rather tall, as slender as a reed, and seemed shy and younger than her age. Her name was Yuri. The next oldest was Asagao, born the year before Yayoi. Then there were two sisters, so close in age they looked like twins, with red cheeks and a stocky plumpness that the meager food at the temple did nothing to diminish. They were ten and nine years old and were called Sada and Sen. The youngest, the six-year-old, was Teru, a thin, wiry little girl who reminded Yayoi of the monkey acrobat children. She wondered if she was of the same family and, if so, why she had been sent away to the temple.
She mentioned this to Asagao one night as they were preparing for bed. The older girls helped the younger ones, combing their hair, hanging their day clothes on the racks. Teru had fallen asleep while Yayoi was smoothing out the wrinkles from her robe. Yuri was at the far end of the room, singing quietly to Sada and Sen, who were already lying curled together. Her voice sounded thin and mournful. The plum rains had begun and everything was damp. The water fell in a steady cascade from the roofs, drowning all other sound. In the dim days, the girls became both febrile and depressed.
"Lady Fuji probably bought her from her family," Asagao said. "Many parents have no choice. Daughters fetch a good price. Everyone wants girls these days."
"Is that what happened to you?" Yayoi was ashamed of asking so directly, but could not control her curiosity.
"My mother was one of Lady Fuji's entertainers," Asagao whispered. "I am not meant to speak of it, but I want to tell you. My father was a Kakizuki warrior. They fell in love, he bought her freedom and took her to his house in Miyako. Women on the boats don't have children—you will find out, I suppose—so I was lucky to be born at all. When the capital fell to the Miboshi, my father did not flee with the Kakizuki, but sent me to Lady Fuji, and killed my mother and himself."
"How horrible, how sad," Yayoi murmured, wondering how Asagao could still grow up so pretty and so charming.
"I think you would find all the women on the boats are the same these days," Asagao said. "They all hide tragic stories of loss and grief beneath the songs and the smiles."
She stroked Yayoi's cheek. "I am sure we will be friends."
At that moment Yayoi wanted nothing more. "Let's be friends forever," she said, seizing Asagao's hand and pressing it.
* * *
The following morning Reverend Nun came into the room where the girls were practicing serving tea and other drinks, though water replaced wine, taking turns to be the male guest and the female entertainer. Playing the role of the men made the two sisters giggle uncontrollably, and Sada, in particular, proved extremely inventive in portraying drunken behavior. Asagao was equally gifted as the entertainer, distracting and calming the guests with songs and dances. They did not have to pretend to be in love with her. Even Reverend Nun watched for a few moments, her face softening. Then she recollected why she had come and said, "Yayoi, our reverend abbess wishes to see you."
This message was obviously shocking to the other girls, who all stopped what they were doing and stared openmouthed. Sada broke off in mid-sentence and began to hiccup for real. Reverend Nun gave her a disapproving look. "Perhaps this role play is becoming a little too realistic. Asagao, put away the bedding. The rest of you can do your dance practice now with Yuri. Come, Yayoi."
The cloisters that linked the buildings around the main hall were flooded and rain poured down on each side. It was exhilarating, like running through a waterfall. Yayoi found she was stamping deliberately in puddles, as if she were a little girl again, playing with her brother, Tsumaru, and Kaze and Chika, the children of Tsumaru's nurse.
"Walk properly," Reverend Nun scolded her when one unexpectedly deep puddle sent water splashing up her legs.
At the end of the cloister stood a small detached residence, not much more than a hut. On the narrow veranda a ginger cat sat with its paws tucked under it, looking morose. The hut was old and weatherbeaten; the bamboo blinds over the doorway hung crookedly and were black with mold. One of the steps was broken and there were several boards missing from the walls and shingles from the roof.
"Did you say the Abbess wanted to see me?" Yayoi said doubtfully.
"Yes—don't ask me why! She has never asked to see any of the girls before. It is most unusual."
"And she lives here?"
"Our abbess is an unworldly woman. She does not concern herself with material things. She chose this hut as her abode when she took over the headship of our community. She agreed to it only if she was permitted to live in this way, as humbly as the poorest peasant. The former abbess was very different, very different. We all miss her."
Yayoi was hoping Reverend Nun would expound more on the former abbess, who sounded interesting, but at that moment a voice called from inside.
"Send the child in."
Yayoi stepped up onto the veranda, avoiding the broken step, and pushed aside the bamboo blind. Gloomy as the day was, it was even darker inside, though one small oil lamp burned in front of a statue that Yayoi recognized, when her eyes adjusted to the dimness, as the horse-headed Kannon. A flowering branch had been placed in front of it and the sweet smell filled the room, mingling with incense, not quite concealing the whiff of dampness and mold.
"Come here. I am told your name is Yayoi." The woman stretched out a pale hand and beckoned to Yayoi to approach. Her head was shaved and her skull gleamed in the light, as if it were carved from ivory. Her features were ordinary—snub nose, wide mouth, small, rather close-set eyes—and her build, though not at all fat, was solid. She wore a simple robe, dyed a deep maroon. Her feet were tucked under her, reminding Yayoi of the cat outside.
Yayoi saw the Kudzu Vine Treasure Store, lying on a shabby cushion beside the Abbess.
The older woman followed her gaze. "You brought this with you. Can you read it?"
"I can read a little," Yayoi said. "But it often seems very difficult."
"I should say it does!" The Abbess laughed, a surprisingly merry note. "Many would call it the most difficult text in the world, if they were lucky enough to get their hands on it. Do you mind telling me how it came into your possession?"
There was something about her that made Yayoi relax, as if the woman were a relative, an old aunt or a grandmother, neither of which Yayoi had ever known. She knelt down on the cushion, moving the text aside, happy to feel its familiar touch beneath her hand.
"An old man gave it to me. I was interested in plants and healing when I was little. I used to brew up potions from dandelion, burdock roots, charcoal, and try to get the dogs and cats to drink them, when they were sick. Master... he, the old man, came upon me one day and asked me seriously about my ingredients and measurements and if I was keeping records of the results. Later he gave me the Kudzu Vine Treasure Store and said I would find many cures in it, but I haven't got to that bit yet." She hesitated for a moment and then said confidingly, "It only lets me read certain parts."
"Oh yes," said the Abbess. "It is a text of great power, but I can see it would be tricky. This old man, can you tell me his name?"
"Master Sesshin," Yayoi said, and immediately wished she had not.
"Don't be afraid," the Abbess said. "Only truth is spoken in this hut. Truth is what I seek: true thought, true sight, true speech. This Master Sesshin, what kind of person was he?"
"He had a lot of books. He lived in my father's house, I don't know why, but for as long as I can remember he was there. Even when my mother was alive, before Lady Tama..." She recalled her stepmother's cruelty and fell silent.
"What is it that Lady Tama did?" the Abbess prompted.
"She had his eyes put out," Yayoi whispered, "and she drove him away, into the Darkwood."
"Poor man," said the Abbess. "And poor Lady Tama, who has added such darkness to her life. Was she your father's second wife?"
"My mother died when I was very young," Yayoi said. "My grandfather took Lady Tama from her husband, my uncle, and made my father marry her."
"Ah, what trouble these old men cause with their attempts to control everything! If only they could foresee the ripples that go on through generations!" The Abbess said nothing more for a few moments but took Yayoi's hand and stroked it gently.
"My husband died," she said finally. "I was still a young woman, and we had one son. I had been married at my father's command. I had not seen my husband previously. But I came to adore him, and he me, I believe. He died in the north. After his death, his brother begged me to marry him and swore he would preserve the estate for my son, but my grief was so great I could not bear to look at either of them, for they both resembled my dead husband. I chose to leave my son in his uncle's care and I renounced the binding ties of love and affection. I wanted to know the truth of this treacherous, cruel world, and why humans have to live lives filled with such deep pain."
"Did you find any answers?" Yayoi asked.
"In a way. We worship the goddess of healing and compassion here, and she has helped me. But I missed my son terribly, and when I was told he had died in the mountains my pain was no less intense than it had been for his father."
A long silence followed.
"What am I to do here?" Yayoi asked, not knowing how to respond to the Abbess's disclosures. She thought of her own uncle, her own mother and father. Why were some forced to die and others permitted to live? Where did the dead go? Did they still see all that took place on earth? How could they watch those they loved and not grieve over them and long to be with them? Why did their spirits not return more often?
"Lady Fuji has asked us to take care of you and teach you all you need to know. We do this for several girls she has sent to us. In return, she pays for the upkeep of our temple, our food, and so on. And she protects us. She has many powerful friends. There are not a few, these days, who are offended at the idea of women running their own affairs. They would like to impose a male priest to keep an eye on us. Times are changing, my dear Yayoi; even in this remote place we can sense it. The Miboshi are warriors, not swayed by gentler pursuits as the Kakizuki were."
"Can I stay here, always?" Yayoi said. She did not want to be reminded of the power struggles in the capital in which her father had died.
The Abbess said gently, "I'm afraid Lady Fuji has other plans for you. We try to give the girls skills, both physical and spiritual, so they may live the best life they can. I see you can read and write, but do you know how to calculate?"
Yayoi shook her head.
"Well, I will teach you that. And you will come to me once a week and we will read your text together."
* * *
"What did she say to you?" Asagao asked jealously. "None of us has ever been sent for. What is she like?"
Yayoi had returned to the girls' room, puzzled by the conversation with the Abbess. Asagao was alone; the other girls were dancing in the exercise hall. Asagao had been told to put away the bedding, after which she was supposed to sweep the floor, but she was still lying on one of the mats, the broom abandoned at her side. Her face was flushed, her sash loosened.
"I am to learn to calculate," Yayoi replied. She did not want to speak about the Kudzu Vine Treasure Store.
"Why? Are they going to marry you to a merchant?" Asagao giggled. "You will be totting up how much rice you have sold and working out the value of the bean harvest. What a waste of a beautiful girl!"
"The Abbess will be giving me lessons herself," Yayoi said.
Asagao pouted. "You are going to be everyone's favorite. I shall be jealous. But what was the Abbess like?"
"She is rather like a cat. In fact she has a cat, a ginger one. She is merry and playful, but you feel she might scratch at any time." Yayoi looked at Asagao sprawled on the mat, saw the translucent white of her skin. "Hadn't you better hurry up? Reverend Nun will be angry if she catches you with the bedding not put away and the floor unswept."
"I have been practicing for my first time." Asagao giggled again. "I can't help myself. It's so much fun. Yuri showed me. You know she is leaving soon? Here, I'll show you. Lie down and we'll pretend I'm your merchant husband."
Yayoi's heart was beating fast, with a kind of terror. She could not put it into words, but she suddenly saw her future. She turned and ran from Asagao, ran from the room, out into the garden. Her eyes were filling with tears. She came to the top of the steps. Where would she go, if she did run away? The choices seemed stark. She could stay where she was, and hand control of her life and her body over to these others, or she could die. By this time sobs were shaking her. She crouched down, her head in her hands. She did not want to die. But she did not want to go where they intended she should either.
She heard someone behind her, and Asagao put her arms around her.
"Don't cry," the other girl soothed her. "Don't cry. I'm sorry I upset you. Our lives may be hard, but they will have pleasures, too. Maybe you are too young to understand now, but one day you will. And we will always be friends, I promise you."
They heard the Reverend Nun calling them.
"I suppose I had better finish the floor," Asagao said.
BARA
A little way from the capital, while they could still smell the smoke from the fires at Ryusonji, the fugitives, Shikanoko and the Burnt Twins, paused in their flight at a remote temple. Eisei insisted they bury the Autumn Princess though Nagatomo thought Shikanoko, numbed and silenced by grief, would have ridden on with her dead body until he too passed away. The temple was neglected and the monks, whom Eisei knew, were reluctant and taciturn, yet Nagatomo thought he would not mind it as a final resting place, against the side of the mountain, looking out over the narrow valley where the flooded fields reflected the bamboo groves and the clouds, and the wind sighed in the cedars. The funeral was hasty, with little ceremony. The lord, as Nagatomo called Shikanoko in his mind, stayed with the horses, watching from a distance.
Nagatomo thought someone else watched, too. In the next few days he was aware a woman was following them. The horses knew she was there; the foal frequently turned with pricked ears and alert eyes, staring back the way they had come, until its mother called in her fretful, anxious way. The lord did not notice. He noticed nothing.
"It's just a coincidence," Eisei said, when Nagatomo mentioned her. "She is on a pilgrimage or going home to her birthplace."
"Traveling alone?" Nagatomo replied. "And who goes into the Darkwood on a pilgrimage?"
The great pilgrim routes all lay to the south. There were no sacred shrines or temples, and no villages, in the huge forest that spread all the way to the Northern Sea. Apart from the occasional hermit, no humans dwelled there, just wild animals, deer, bears, wolves, monkeys, and, it was said, tengu—mountain goblins—as well as huge magic snakes and other supernatural beings.
When they stopped to eat and sleep—though Nagatomo knew the lord did neither—the woman hid herself. She lit no fire; he wondered what she ate, who she was, what she wanted from them.
The rain had lessened to a steady drizzle, but the trees still dripped heavily and the streams and rivers spread out, drowning the path. The fake wolf jumped from rock to rock. It did not like getting wet. The horses waded through water up to their hocks. The lord rode the silver white stallion, Nyorin, and Nagatomo and Eisei doubled up on the mare, Risu, though both preferred to walk. The mare was bad tempered and ill mannered, and bucked and bit, without provocation. The foal was still nursing and the mare stopped dead whenever it demanded the teat.
At night they removed the black silk coverings they both wore and caressed each other's ruined face. It did not matter, then, that no one else would ever look on them with desire again or understand the terror and agony as the mask seared away skin and flesh. They were the Burnt Twins. They had found each other.
Only the lord could wear the mask. Nagatomo knew it had been made for him in a secret ritual by a mountain sorcerer. Usually it was kept in a seven-layered brocade bag to be taken out when the lord walked between the worlds and talked to the dead. But now, on the journey into the Darkwood, he wore it day and night. The polished skull bone, the cinnabar lips and tongue, the antlers, one broken, the black-fringed eye sockets through which glistened the unending tears, transformed him into a different creature.
"He cannot take it off," Nagatomo whispered to Eisei.
"Cannot or will not?"
"It is fused to his face in some way."
"It must be because of what happened at Ryusonji," Eisei said, as if he had been thinking about it over and over. "The dragon child was awakened, my former master destroyed. Finding that overwhelming power, and releasing it, came at a price."
"Has it burned him?" Nagatomo wondered aloud. "As it did us?"
"He does not seem to be in pain," Eisei replied. "Not physical pain," he added, after a long pause.
Mid-afternoon on the fourth or fifth day—he was beginning to lose count; every day was the same: steep gorges, flooding rivers, huge boulders, the wild cries of kites in the day and owls at night, the humid air that made them sweat profusely until just before dawn, when they shivered in their sodden clothes—Nagatomo noticed the woman was no longer following. The foal had been restless, trotting back along the track, almost as if it were trying to attract his attention, making its mother balk and neigh piercingly after it.
The lord was far in front, Gen, the fake wolf, close to the stallion's heels, as always. Eisei pulled on the mare's bridle, yelling at her.
"I'll catch up with you," Nagatomo said, and began to walk back the way they had come. The foal whickered at him. It was uncanny how intelligent it was; often it seemed on the point of speaking in a human voice. It trotted confidently ahead of him.
He told himself he was being a fool, trailing after a horse. As Eisei said, it was just a coincidence; she had not been following them, and even if she had, he should be relieved she no longer was. After Ryusonji the lord was a hunted man, an outlaw. Any one of Aritomo's retainers might be on their trail, hoping to win the Minatogura lord's favor as well as great rewards. Maybe it was not a woman, at all, but a warrior in disguise. Maybe it was a mountain sorcerer or a witch.
But the foal knew her. He was certain of that.
How long was it since he had last been aware of her? He could not be sure. There was no way of knowing exactly what hour it was, with the sun hidden all day behind dense cloud. He was hungry enough for it to be almost evening, but he had been hungry since he woke, and the sparse dried meat and unripe yams had done little to fill his stomach. He walked for what seemed like a long time. The mare's cries grew fainter, and then he could no longer hear her at all, but the foal still trotted forward, stopping at every bend to check if Nagatomo was following.
The woman was sitting on a rock by the track, her head low, her face buried in her arms, her hands bound together in front of her. She did not move at their approach, but when the foal nuzzled her she put out her tied hands and pulled its head close to her. It allowed her to embrace it for a few moments, breathing out heavily. Then it nudged her more insistently. She raised her head slowly and looked at Nagatomo.
Her face was streaked with tears, her eyes and lips swollen with grief. He thought he must be an alarming sight, with the black face covering and his long sword and knife, but she showed no fear. In fact, she looked as if grief had consumed her and left no room for any other emotion.
He started to speak, but at that moment the foal squealed and leaped backward. The woman, as she fell, looked beyond Nagatomo, and he, forewarned by something in her eyes, had drawn his sword in an instant and turned to face his attackers.
One called, "Are you Kumayama no Kazumaru, known as Shikanoko, wanted for murder and rebellion?"
"Come and find out," Nagatomo said. He was assessing them quickly. They had emerged from the forest while he was distracted by the woman. How long had they been following them? Was she one of them, part of the trap? The foal whinnied and horses neighed in reply. The men wore crests of three pine trees on their jackets, and held swords, but they did not appear to have bows.
"It is he," the second man said. "He covers his face to hide the demon mask."
"There should be three horses," the first said, hesitating for one fatal moment, during which Nagatomo flew at him, flicking the man's sword from his hand with a twist of his own and with the returning stroke slicing him through the neck. The blood spurted from the opened artery, and the foal screamed like a human.
The second man, his eyes dark with shock, took a step back, gripping his sword. He was more prepared and, Nagatomo guessed, a better swordsman. He and Nagatomo circled each other, assessing stance, grip, weapon. Nagatomo's sword was longer and heavier. It gave him more reach, but his opponent's lighter blade gave its owner greater speed and flexibility. The other man was fitter, and probably better fed. Nagatomo wondered about him briefly, where he was from, what his name was, what fate had led them to encounter each other in the Darkwood, one evening in summer. Then he thought of nothing, as his enemy thrust at him and he began to fight for his life.
It had started to rain again and the ground was becoming slippery. For a long time they exchanged blows, parrying and ducking, grunting with exertion, now and then uttering cries of hatred. Nagatomo was slowly forcing the other back toward the stream, which was spilling over its banks and flooding onto the track. The water splashed around their ankles, hiding roots and holes, and one of these was his opponent's undoing. His foot slipped into it, he stumbled and dropped his guard.
Nagatomo rushed forward, the point of his sword entering the man's throat and coming out the other side, skewering him. The force of the blow threw the dying man backward into the water, his blood streaking the surface briefly, before becoming lost in the murky current.
Nagatomo put one foot on the man's chest, to pull out his weapon. Bubbles burst from the mouth and the wound. For a moment he thought the sword was stuck, but then it came free. His opponent's mouth under the water went slack and air no longer came from it, though blood did.
He staggered back to the bank, gasping for breath and trembling as the tension ebbed from his limbs. Elation seized him. He was not dead; his attackers were. He saw life and death, side by side, in their raw simplicity.
The foal came docilely to his side and sniffed at the man's legs. They looked foolish, half-covered by water. Nagatomo wanted to laugh; he wanted to embrace the foal. He gave it a thump on its hindquarters and turned to face the woman.
She was on her feet, her eyes fixed on him. He had hardly had time to look at her before. Now he studied her as he walked rapidly toward her, his eyes flicking over the undergrowth behind her in case there were any more men hidden there.
He stopped a few paces from her. She was tall, only a little shorter than he was, and large boned. Her face was tanned dark by the sun, her nose flat, her mouth wide. Her hair was covered by a sedge hat tied down by a scarf, but he guessed it would be as coarse as a horse's mane. It angered him irrationally that even a woman like this would never look on him with love or desire, and for that reason, or maybe because he suspected she had been in league with his attackers, he addressed her roughly.
"So you thought to entice me into an ambush? Your companions are dead. Who are you and why are you following us? Answer me truthfully or I'll send you to join them in the next world."
"They are no companions of mine," she said angrily. "They wear the crest of Matsutani—that means they serve Masachika. I was following the horses, and have been for weeks, ever since they were taken from Nishimi, when Masachika captured the Princess. I waited at Ryusonji. I saw you leave and watched you bury her. Then, when I realized Masachika's men were also on your trail, I stopped. I didn't want to lead them to you. I thought I might distract them while you vanished beyond their reach."
"And did you?" he said, unable to keep contempt from his voice.
"I think they were saving me for later," she said, without emotion. "That's why they tied me up." She held out her hands; he sheathed his sword, took out his knife, and cut the cords. The foal gave a low whinny, and, when her hands were free, she embraced it, as Nagatomo had wanted to earlier.
"Dear Tan," she said. "I never thought I would see you again."
"Tan?" he questioned. They had never given it a name; it was just called the foal.
"It's what my lady called him because when he was born he was as dark as coal. His coat is lightening now just as Saburo said it would." Her eyes filled with tears.
Nagatomo felt a perverse pang of jealousy. She was weeping for someone, in a way no one ever would for him. "So why follow us in the first place? You have not answered me."
"My name is Ibara. I have a favor to ask of you," she said, hesitant. "I am sorry, I know a woman like me should not speak so directly to a great warrior like yourself, but I am beyond caring about all that now."
One of the horses neighed from the grove.
"We should ride on," Nagatomo said. "Wait here while I get their horses. We will talk further as we ride."
The two horses were tethered beneath an oak tree. They laid back their ears at his approach and swung their haunches toward him, as though they would kick him, but the foal came barging through and its presence seemed to calm them. He untied them and led them back to the track, where he stripped the corpses of their clothes and footwear and gathered up the weapons, using one of the tethering cords to tie them into a bundle and strap them behind the saddle. Then he helped the woman mount one of the horses and, still holding its reins, leaped nimbly onto the back of the other.
"They are good horses," he exclaimed.
"Masachika is a rich man now," the woman replied. "The body of the Princess gained him many rewards."
"I imagine the favor you mean to ask is that we should kill him," Nagatomo said.
"Not exactly. I want you to teach me to fight with the sword, so I can kill him myself."
There was a clap of thunder and the rain began to fall more heavily.
It was nearly dusk when they caught up with Eisei, who had taken shelter beneath a rocky overhang, where the stream emerged from between steep cliffs. It offered some protection from the direct rain, but the walls and the boulders on the ground were dank with moisture. Farther back, a kind of low cave extended beneath the cliff, where the ground at least was dry.
"You can sleep here," Nagatomo told the woman, ignoring Eisei's disapproving look.
"Surely the lord..." she began.
"He will not sleep or seek shelter. We will take it in turn to keep watch."
"Where is he?" she said, gazing out at the rainy darkness.
Nagatomo looked at Eisei, who made a small movement with his shoulders and said, "Somewhere. Not far away, I think."
"What is wrong with him?" she said in a hesitant voice.
"He loved the Princess. She died," Eisei said curtly.
"He wants to die, too," the woman said, partly to herself. "I know that feeling." And then, even more quietly, so only Nagatomo heard her, "But we will see Masachika dead first."
* * *
The horses were restless all night, upset by the two new stallions, who, excited by the presence of the mare, challenged Nyorin with loud calls. Bara hardly slept, and when she did, the dead talked to her in muffled voices and wrote messages she could not read. At dawn, she crawled from the cave and went toward the bushes. The man who had rescued her was asleep on his back. The other, the monk, was tending the smoky fire. Both had removed their face coverings and, after one shocked glance, she averted her gaze.
The monk did not look at her but, as she walked past, said, "See if you can find some dry wood."
"I will," she replied.
It had stopped raining and there were patches of blue sky overhead between the pink- and orange-tinged clouds. The foal came up to her eagerly and followed her through the undergrowth. Then it went ahead, while she squatted to relieve herself, turning back, when she stood, to whicker at her. The mare responded in the distance with an anxious neigh.
Bara walked after it. To her right, she could hear the endless babbling of the stream as it rushed over rocks and through pools. On her left, the forest rose in a steep slope, thick with trees she did not recognize, apart from maples. She had grown up in the port city of Akashi and then had worked in the house in Miyako, and the Nishimi palace, on the shore of the lake. Everything here alarmed her—the strange bird calls, the half-seen creatures that slithered away, the darkness between the trees that seemed to stretch away forever, the uncanny dappled circles where the sunlight shot through.
The ground was still sodden, but there were plenty of dead branches on tree trunks that she could reach easily, and she was breaking these off and making a bundle in her left arm when the foal, which had been walking ahead of her, stopped dead and snorted through its nose.
Pushing past it, she in turn halted suddenly. In the path stood the animal that she had noticed following the horses. She had thought it was a dog, but it did not look like any dog she had ever seen. Perhaps it was a wolf. Close up it did not seem quite real. Its eyes were as hard as gemstones and its movements awkward. The idea came to her that she was dreaming; she had fallen asleep, after all; she could almost feel the rocky floor of the cave beneath her. She struggled to wake. The wolf curled its painted lips, showing its carved teeth and its manlike tongue.
"Gen!" a low voice called, as if it were an angel or a demon, making a pronouncement. Shivers ran down her backbone. "Gen!"
The wolf seemed to sigh as it retreated. She went forward slowly, Tan's nose in her back, pushing her on.
She saw the head first and thought she had come upon a stag. The antlers, one broken, were lit up by the morning sun. Then she realized it was a man wearing a mask. She recognized the shape; it matched exactly the pattern of the burns on the other men's faces. It covered three quarters of his face, leaving his chin free. Through the sockets she could see his eyes, so black the iris and the pupil merged.
He had been sitting, his legs folded beneath him, but he stood as she approached. The antlers gave him added height and he seemed, to her, like some spirit of the forest, half man, half deer. He reminded her of the dancers at the summer festivals of her childhood. In Akashi they had danced the heron dance, wearing beaked and feathered headdress. In that garb, the men had become protective, chaste, quite unlike their usual truculent, predatory selves.
She felt no fear now, only pity, for somehow she recognized a grief as deep as her own.
He did not speak to her but addressed the foal.
"Who is this my lord has brought to me?"
She recognized the sword he wore at his hip—it was the sword the Princess had left in the shrine at Nishimi, as an offering to the Lake Goddess. He also bore a rattan bow and a quiver on his back, filled with arrows fletched with black feathers. She could see traces of cobwebs spun between them. It was a long time since they had been disturbed.
The foal nudged her, pushing her forward. She fell to her knees and said, "My name is Bara, but now I call myself Ibara. I was at Nishimi when the Princess came, with the horses and with that sword you wear."
A stillness came over him, like a deer after the first startle. She was afraid he was going to leap away and disappear into the forest, but then a long shudder ran through him and he sank to his knees in front of her.
"With Jato?" he touched its hilt briefly.
"If that is its name. The last time I saw it was before the altar in the shrine."
"Masachika must have taken it. It was made for me, but he had it at Matsutani. I took it back from him."
"You should have killed him then," Ibara said. "Swords return for a purpose."
He did not respond to this but said, "Tell me what happened at Nishimi."
She could see him more clearly now. How young he was! She had formed a picture of an older man, for that was what the word lord suggested to her. But he was not much more than a boy. How had he destroyed the Prince Abbot, in an act of such power the temple at Ryusonji had burned to the ground?
He loved the Princess. She died. His mouth had the same shape as little Take's, and his long fingers, too. Tears welled in her eyes.
"Akihime came to Nishimi, with Risu and Nyorin. Yukikuni no Takaakira employed me to look after Lady Hina. Hina knew the horses, she knew their names."
"Hina?" the lord said wonderingly. "Lord Kiyoyori's daughter?" And the foal came closer, nodding its head.
"I had no idea who Lady Hina was, other than that she was his ward and that he was secretive about her and didn't want anyone to know she was living with him. I guessed he'd saved her life. I would have done the same thing—anyone would. She was enchantingly pretty and so brave. She hid the Princess, and we pretended she had been rescued from the lake. After the baby was born Akihime worked in the kitchen. We thought no one would suspect her of being anything but a servant girl."
"She had a child?" His lips were ashen.
"A son. She wanted him to be called Takeyoshi. Lady Hina often played with him and she was carrying him when Masachika came over the mountains from the west." She halted abruptly. "This is the bit I don't understand. For he came with Saburo."
Tan gave a low whinny.
"Yes, Tan, Saburo, the man who saved your life at birth. He must have told Masachika that the Princess was at Nishimi, but I cannot believe he would betray her. And then Masachika killed him, stabbing him in the back."
"Masachika often works as a spy," the lord said. "Your Saburo would not be the first to be deceived into trusting him."
He laid one hand against the foal's neck. "What happened to Hina?"
Ibara replied, "She jumped into the water with the baby in her arms."
Tears splashed on her arm and hand. The foal was weeping.
"What is this animal?" she cried, half-rising. "How does it understand every word and why is it shedding tears like a human?"
The lord said quietly, "The foal is a vessel for the spirit of Lord Kiyoyori, Hina's father."
"The one who died at the side of the Crown Prince? How can that be possible?"
He looked at her. The bone mask allowed no expression apart from the eyes, but they seemed to open onto a world she did not know existed. She could not hold his gaze.
"Is it like rebirth?" she said after a long silence.
"Not quite. Lord Kiyoyori's spirit refused to cross the river of death. A man who owed him an unthinkable debt took his place. I summoned the lord back. The mare was pregnant. His spirit took over the unborn foal."
It might have been the wild claim of a man driven mad by grief, yet, if she accepted it, so many things made sense—the foal's devotion to Hina, its ability to understand human speech, its tears.
"I don't believe Lady Hina is dead," she said, addressing the foal. "There was a boat. I think they saved her and the baby. Don't grieve for her yet..." And then, deeply uncomfortable, she added, "Lord."
Accepting it was true gave her new hope. "Why can't you summon the Princess back? Or Saburo? Summon him back into whatever shape you like. He died before we even held each other. I cannot stand it." She was twisting her hands together frantically.
"I know," he said, and, for a moment, Ibara felt their deep grief unite them. Then he said bitterly, "Don't think I haven't tried. Night after night, I attempt to walk again between the worlds and summon up the dead. But she is gone. Maybe she is in Paradise, maybe she is reborn, either way she is forever lost to me. Your Saburo must have died even earlier. He also will have crossed the last of the rivers that flow between this world and the next. I was given much power and taught many things, but I lay with her when I should not have done, and though together we destroyed the Prince Abbot, we did not escape punishment. She forfeited her life and I cannot remove the mask. I am condemned to live half animal, half human, belonging to neither world. I will go without food or sleep until I follow her into the realm of the dead. Maybe there I can find forgiveness."
"It is not forgiveness I seek," Ibara said in a low voice. "It is revenge."
The foal gave a sharp neigh of encouragement.
How strange, she thought, I am just a girl from Akashi, a servant, but my desire for revenge is stronger than this boy's, who was born a lord, brought up as a warrior.
"No one is to blame for the Princess's death but myself," he said. "It is on myself that I am taking revenge."
"The baby looks like you. He is your son, isn't he? Don't you want to find him?"
"Better he died in the water," Shikanoko said, "than grow up in this world of sorrow."
MU
Three of the five boys who had been born from cocoons, Mu, Ima, and Ku, had been left at the mountain hut all winter. Once a messenger had come from Shikanoko at Kumayama to check that they were still alive, but after that they heard nothing of him or their other brothers, Kiku and Kuro. At first they did not worry, living day to day without much thought, like animals, but when spring came Mu began to be plagued by restlessness and a sort of anxiety. He took to roaming through the Darkwood and it was there that he saw a foxes' wedding, though at the time he did not know what it was. It was the third month, when showers chased sunshine. For several days he had been away from the hut, sleeping under the stars or in caves when it was too wet, feeling almost like a fox himself. One morning he was plucking young fern shoots, cramming the tender stems into his mouth, in one of the hidden clearings on the lower slopes of Kuroyama, when he heard curious noises, the soft padding of many feet and flute music, so high he could not tell if it was really music or the wind in the pine trees, and drums that might have been rain falling. He quickly climbed an oak tree and hid in the foliage as a procession came into the clearing.
At first, he thought they were people, dressed in colorful clothes, walking upright, playing flutes and drums, but then he saw their pointed ears, their black-tipped snouts, their precise, delicate paws. A male and a female were carried on the shoulders of the largest foxes, who were the size of wolves. Like the music, they hovered between reality and imagination, filling him with an intense longing. He did not think they were aware of him, but, as they passed beneath the oak tree, one young female looked up and smiled in his direction, a smile that seemed to be an invitation into worlds he had not known existed.
The sun shone brilliantly on the short winter grass, only recently liberated from snow, starred with flowers, yellow aconites and celandines, white anemones. The bride and groom were lowered to the ground and stood facing each other. They joined hands—paws, Mu thought—as the flutes played even more sweetly and the drums more loudly. Then the sky darkened, sudden rain joined in the drumming, and, when Mu could see again, they had all disappeared, as if the shower had dissolved them.
When he came home his brothers Kiku and Kuro had returned and were crouched by the fire, silent and miserable. He felt a moment of relief, as if his anxiety had been for them, but why had they come alone and why did they look like that? The youngest brother, Ku, was sitting near them, watching them with a troubled expression on his face, a bunch of puppies, as usual, crawling over him and tumbling around him. The fourth boy, Ima, was tending a pot in which a stew of spring shoots was simmering along with some sort of meat.
Ima scooped broth into wooden bowls and offered them to Kiku and Kuro. Kuro took one and drank without a word, but Kiku refused with a gesture that made Mu's heart sink.
"What's happened?" he said.
"Shikanoko..." Kuro began.
"Don't you dare speak!" Kiku shouted. "It was all your fault!" He hit Kuro over the shoulders so violently the soup flew from the bowl, scalding Kuro's face and hands. Kuro swore, grabbed a smoldering stick from the fire, and thrust it toward Kiku's face.
"Stop it, stop it!" Mu cried. "What happened to Shikanoko? He's not dead?"
"He might as well be," Kiku said angrily. "He has sent us away. He never wants to see us again. It was all Kuro's stupid fault. I told him to leave all his creatures behind. But he had to bring the deadliest one."
"The snake? The snake bit someone?" Mu said.
"Only a woman." Kuro tried to defend himself.
"Only a woman?" Kiku repeated. "The woman we were meant to rescue, the Autumn Princess, the woman Shika loved."
"I don't understand that," Kuro muttered. "I don't know what love means."
"I'm not sure I do either," Kiku admitted.
Mu thought of the fox girl and how her look had transfixed him. Do I love her? he wondered.
"Shika felt something for her," Kiku tried to explain. "An emotion so strong her death destroyed him. He has turned us away, our older brother, our father, the only one who cared for us, who brought us up." He said all this in a bewildered voice as though, for the first time in his life, he himself was feeling some strong emotion. He brushed his hand against his eyes. "What is this? Is it the smoke making my eyes water?"
Tears were staining his cheeks. Mu could not remember ever seeing him cry, not even when the rest of them had wept after Shisoku died. "Where has Shika gone?" he said.
Kiku sniffed. "He rode away into the Darkwood, with Gen, three horses, and two men with burnt faces. He performed an act of great magic and defeated the priest. He raised a dragon child from the lake. You should have seen it, Mu, it was magnificent. Balls of lightning everywhere, a roaring like you've never heard. The priest dissolved in fire."
"Tell Mu about the price Shika paid," Kuro said. "Tell him about the mask."
"The stag mask he uses," Kiku said. "It stuck to his face. It cannot be taken off. Now he is half man and half deer."
"Is that so bad?" Mu asked, wishing he could be half fox.
"It would not matter if he had stayed with us." Kiku gestured toward the fake animals that the old mountain sorcerer Shisoku had created from skins and skeletons. "He would have fitted in perfectly here. Or he could still have been a warlord like he intended. He would have been all the more terrifying. He did not need to send us away. He could have achieved anything he wanted with our help. Look what we have done for him so far! He would never have got the better of that monk, Gessho, or taken his old home back from his uncle."
"It was my bee that killed the uncle," Kuro added proudly.
"And then getting into Ryusonji," Kiku continued. "It's a shame about the Princess—my eyes are doing that strange thing again. Why is your fire so smoky, Ima?—but the Prince Abbot was destroyed. Shika could have done none of those things without us."
At that moment one of the fake wolves gave a long, muffled howl and fell over with a thump. Ku pushed away the pile of dogs that surrounded him and ran to it. The puppies yelped and snarled at it in playful attacks, but it did not move. The other boys stared at it.
"It's dead," Ku said.
"It was never really alive." Kuro moved toward it and knelt beside it, pushing the puppies away. He looked up at Kiku. "Whatever power was holding it together has left it."
Kiku looked wildly around at the other fake animals, making no effort now to control his tears. Mu followed his gaze. He realized what he had not noticed before: Shisoku's creations were winding down, fading in some way. Regret stabbed him. He also felt his eyes water. Why hadn't he looked after them better?
A crow plummeted from the branch it had been perched on and lay broken and silent on the ground, its borrowed feathers scattered by the breeze.
"No!" Kiku sobbed.
"You never liked them much, anyway," Mu said, surprised at his apparent sorrow.
"I hate them," Kiku replied, controlling himself with an effort. "But they are breaking down before I've had a chance to learn how they work, how to make them. How did Shisoku get them to move, to live to the extent they did? How did he and Shika make the mask? What would he have done with the monk's skull that we buried? And the horse's? I need to learn all these things, and now there is no one to teach me."
"What's going to happen to us?" Ima said, suddenly anxious.
Kuro said, "The old man Sesshin..."
Mu started at the name. "He is one of our fathers. The only one still alive, apart from Shika."
"Well, he told Shika to kill us. He called us imps. My snake was meant to bite him!"
"He must know some sorcery," Kiku said.
"He gave all his power away to Shika," Kuro said. "I heard the torturers tell the Abbot Prince."
"Prince Abbot," Kiku corrected him.
"Whatever, he is gone." Kuro stood up. "But wasn't the dragon superb? If only I could learn to summon one up like that."
"Well, you won't now," Kiku retorted. "Because Shika is never going to want to see you again."
They looked wildly at one another. They were all crying now, even Kuro.
What will become of us? Mu thought. There is no one in the world who cares about us.
* * *
Over the next few weeks the boys sulked and squabbled as more of Shisoku's animals ran out of living force and fell to the ground. Mu wanted to burn them; they did not exactly decay like real animals, but they gave out a strange smell; insects began to dwell in the hides and maggots hatched. The corpses heaved with a new movement that nauseated him. But Kiku would not allow it. He studied each one's unique makeup, committing to memory how they were put together, out of which materials.
He went through the hut, looking at, smelling, tasting the contents of all the flasks of potions and jars of incense and ointments that Shisoku had concocted or collected. The sorcerer had kept records in an arcane script, which none of them could read, but Kiku searched out every object of power, every amulet and statue and figurine. He knew their weight and what they were made from, but he did not know how to use them for his own ends. That did not stop him trying everything out, experimenting fearlessly.
Sometimes he raved uncontrollably about visions and deep insights, sometimes he seemed to work magic by accident. Once he threw up so violently and for so long the others thought he was dying.
He gathered the remains of the werehawk, from where they still lay on the roof, and made a necklace from the beak and talons. He dug up the horse's skull. Worms and insects had done their work and the flesh was stripped from the bone. The last remnants fell away when Kiku boiled the skull in an iron pot on the fire.
"You can't make a horse," Mu said. "Even Shisoku never made anything so large."
"I want to make a mask like Shika's," Kiku replied.
Mu had never seen the mask, only knew the seven-layered brocade bag in which it used to be kept, but Kiku had watched Shika wear it and had held it in his hands.
"I carried it," he said, with a note of pride in his voice. "He said he wanted to leave it behind, but I knew he didn't really, so I took it to him."
He described it to Mu: the stag's skull, the antlers, one broken, the half-human, half-animal face with its carved features, smoothly lacquered, and its cinnabar-reddened lips. But he did not know the ritual in which it had been created, months before the boys were born, the blending of the red and white essences of male and female.
When the horse head was reduced to gleaming bone, Kiku tried to shape it, but his chisel often slipped and the resulting skull pan was lopsided and jagged. He made a face mask from wood, carving out eye sockets and a mouth hole, and he and Kuro lacquered it without really knowing the method. The lacquer bubbled and cracked, as if it were diseased, and the result was monstrous, both laughable and sinister. When Kiku put it on, the dogs howled and ran to Ku, and two more fake animals lay down and did not get up again.
"It's useless," Kiku said, taking the mask off and throwing it to the ground. "It's ugly and it has no power."
"It's only your first attempt," Mu said. "Imagine how many times Shisoku had to experiment and practice before he got it right. And he was still making mistakes up to the time he died."
"But he mostly knew what he was doing. He must have had so much knowledge," Kiku said. "Why do I have no one to teach me? Don't you ever feel it? That there is a huge part of our lives missing? Why is there no one like us? Where did they all go?" He sighed, and glanced around the clearing, his eyes falling on the dogs, cowering around his youngest brother. "Maybe the skull has to come from something I kill myself."
"No!" Ku said defiantly.
"A dog is too easy," Mu added. "It is not enough of a challenge for you." He picked up the horse mask and set it on a pole near the hut. "It'll make a good guard."
The mask was not what Kiku had intended, yet it was not a complete failure, and some strange force had attached itself to it. At night they heard hoofbeats and whinnying, and several times, the post seemed to have moved by morning. Ima was fascinated by it. He patted the post and clicked his tongue at it when he went past, and brought offerings of fresh grass and water.
Weeks went by. Shikanoko did not return. Kiku continued his experiments. Kuro set about replacing his collection of poisonous creatures, and managed to capture another sparrow bee.
One morning Mu had gone with Ima to the stream to gather grass and check the fish traps. The boys were always hungry, and although they preferred meat to fish, fish were easier to get and more plentiful. The stream did not flood that spring and, in the deep pools, sweetfish hid in the shadows, while crabs could be found under every rock. Sometimes the traps would catch an eel, which was as rich and tasty as meat. They were both knee-deep in the water when they heard someone approaching. Neither of them had Kiku's acute hearing, but the sounds were unmistakable: a snapped twig, a dislodged stone, and then a quickly muffled gasp as a foot slipped. The two boys were out of the water and into the undergrowth in one movement, as quick as lizards.
A boy and a girl came warily down to the stream. The boy looked familiar, and Mu realized it was the messenger who had been sent by Shikanoko in the winter, the boy called Chika. He was still not very clear about human ages—his own growth, like all his brothers', had been so rapid he had nothing to go by—but he knew Chika was a boy, definitely not yet a man. The girl seemed younger, but maybe not by much. They were both thin, legs scratched and bleeding in several places, barefoot, burned brown by the sun. Yet the boy carried a sword, and the girl a knife, and, Mu thought, they both looked as if they knew how to use them.
The boy knew his way, leading the girl across the stream, helping her jump from boulder to boulder. When they reached the bank, they walked downstream toward the hut. Mu picked up the fish they had already caught, still flapping on the grass stem threaded through their gills, and gestured to Ima to bring the bucket of crabs. They followed the pair, silent and unseen.
The boy halted near the horse skull, hand on sword, and called, "Is anyone there? I am Chikamaru, son of Kongyo, from Kuromori. I am looking for the man known as Shikanoko."
Kiku emerged from the hut, blinking in the sunlight. "We know who you are, Chika. Shikanoko is not here."
Slowly the other boys appeared and surrounded the pair. The girl held her knife out threateningly, but Kiku brushed it aside and stepped close to her, touching her face and her hair, in a gentle way that both astonished and alarmed Mu.
"Kongyo?" Kiku said finally. "He was the man who came with the horse." His eyes flickered to the horse mask on the pole.
Chika said, "That's Ban? My father said he died. He was our last horse. But what have you done to his skull?"
Kiku made a dismissive gesture. "It doesn't matter."
The girl began to cry silently, as if the sight of the horse, once no doubt magnificent and prized, now a hideous replica, had unleashed all her grief.
"I've done that," Kiku told her. "Water has come from my eyes. It will dry up, don't worry."
His face had taken on an intense fixed expression, like a male animal about to mate or kill.
"Cook the fish," Mu said to Ima, to break the uncomfortable silence, and then addressed the boy, Chika. "Sit down, we'll eat something. Are you hungry?"
They both nodded. The girl slumped down, still weeping. Chika said, "Our mother told us to flee. After our father died she was afraid his murderers might seek to kill us, too. I don't know what will happen to her. My sister is still in shock, I think. She hardly speaks and the slightest thing sets off her tears."
"Our mother is dead," Kiku said, sitting down next to the girl. "She died just after we were born."
"Lady Tora?" Chika said.
The boys stared at him. "You knew our mother?" Mu said.
"She came to Matsutani with Akuzenji, the King of the Mountain. Shikanoko was with them, too."
Mu remembered the name, Akuzenji. Shika had told them he was one of their five fathers.
"What does that mean, King of the Mountain?" said Kiku.
"That's what he called himself. He wasn't really a king, he was a bandit. Merchants paid him so they could travel safely along the northern highway. If they didn't pay, he robbed them and usually killed them. He set an ambush for Lord Kiyoyori, whom my father served, but he was captured and the lord beheaded him and all his men, except Shikanoko. Then Lord Kiyoyori fell madly in love with Lady Tora and made her his mistress, even though she was said to be a sorceress."
"One of our fathers took the head of another of them," Kiku murmured. "That would be a skull worth having."
"The bodies were burned and the heads displayed at the borders of the estate," Chika said. "You should have seen it—thirty men separated from their heads in as many minutes. It was brutal. I've been in sieges and battles, but nothing was as horrifying as that day."
"You say you have been in battles," Mu said, "but you are not yet a grown man."
"I still know how to fight with this." Chika tapped the sword that lay beside him on the grass. "I have just escaped from the battle in which my father died."
"Why have you come here?" Ima said from the fire. The sweet smell of grilling fish rose in the air.
"I could think of nowhere else to go. Our father is dead, along with all Lord Kiyoyori's men and their families. We held out for months in the fortress at Kuromori, but after Shikanoko left for the capital, and never came back, Lord Masachika attacked for the second time, took the fortress, and put all the defenders to death. Then he did the same at Kumayama. He holds a huge domain now for the Miboshi. No one is left to oppose him in all the east."
"It sounds very complicated," Kiku said. "You'll have to explain it to us. We need to understand all these things, if we are to live in the world."
"No one understood why Shikanoko disappeared," Chika said. "They felt betrayed and abandoned. At first we thought he must have died, but then we heard that he destroyed the Prince Abbot at Ryusonji. He could have dominated the capital himself, but he rode away, no one knows where."
"Someone died," Kiku said, glancing at Kuro, who sat a little way off, letting a snake slither up his arms and around his neck. "A girl Shika liked."
"Loved," Mu said.
Kiku frowned. "Loved," he repeated, and bent forward to look in the girl's face. She squirmed away and said to her brother, "I don't want to stay here."
"You spoke," he said in delight. "You see, we will be safe here. We can stay, can't we?"
"Of course," Kiku said. "You are welcome, you and your sister. What did you call her?"
"Kaze."
"And your name, Chika—that's like the other lord you mentioned."
"Masachika. I wish it were not. I hate him more than any man alive. It's one of the clan names—Chika, Masa, Kiyo, Yori. Many of us are called some variation of it. Masachika is Lord Kiyoyori's younger brother."
So he is some relation to us, Mu thought. If Kiyoyori is one of our fathers, this Masachika is our uncle. Kaze is Chika's sister. He will be uncle to her children. And if I have children my brothers will be their uncles. He thought of the foxes and the girl he had seen, thought of having children with her. The blood rushed to his face and he trembled. In the following weeks he often went back to the clearing, looking for her, but he did not find her.
* * *
During that time Kiku and Chika had many conversations about the realm and governance of the Eight Islands: the emperor, the nobility, the great families who held roles of state, the warlords and their warriors, the rich merchants who had their own sort of power. Kiku's frustration increased daily, until finally he announced his intention to go out into the strange and enticing world Chika described. "You can't be a sorcerer without someone to teach you," he said. "We can't live in this stinking place forever. We must find some way of having power in the world."
"Maybe we should be bandits," Kuro said. "I would like to be called King of the Mountain."
"King of the Insects, that's what you are," Ku jeered.
"And you are King of the Dogs," Kuro countered.
"Bandits are like crows," Kiku said. "They swoop down and steal, they scavenge. But if you are rich you don't have to scavenge."
"Others steal from you, then," Mu said.
"Then you could be a bandit, in secret, as well as a merchant," Kiku suggested. "Yes, that's what I would like to be."
"Being a warrior sounds very fine," said Ima, who had been entranced by Chika's tales of heroism and sacrifice.
"You can't be a warrior," Chika told him. "You have to be born into a clan."
"Lord Kiyoyori was our father," Ima reminded him. "And so was Shika."
"Well, if Shikanoko had stayed around he might have brought you up as warriors. But he's disappeared and Lord Kiyoyori is dead, and no one's going to believe you're sons of either of them."
"Why not?" Ku asked.
"You don't look like it," Chika replied.
"What do we look like?" said Kiku.
"Not like anyone else, really," the warrior's son said.
The brothers exchanged glances, seeing one another's coppery skin, their sharp bony faces, their unkempt black hair.
"I have no intention of being a warrior," Kiku declared. "They all end up being killed or killing themselves. I am going to be a merchant by day, and a bandit by night. And maybe a spy or an assassin, but only for the highest reward."
Chika laughed. "To be a merchant you have to have something to sell, and ways of either making it or buying it."
Kiku laughed, too, but more loudly. "I have something that I think will get me started. Come into the hut."
They crowded into the hut after him and watched as he pulled a pile of old rags away from the wall. Beneath it lay a large flat stone.
"I'd never been able to move it," Kiku said. "Then, one day, it shifted. Something I did must have unlocked it. Help me lift it—Kuro, you're the strongest."
Together, they raised the stone and slid it aside. A wooden chest had been buried under it. Kiku removed the lid and plunged both hands in, pulling out pearls, golden statues, silver prayer beads, copper coins, jade carvings—all small, light things that could be easily carried. "They must be valuable," he said.
"Where did it come from?" Kuro asked.
"Maybe Akuzenji got the sorcerer to hide it for him?" Mu suggested.
"That's what I think," Kiku said. "Our fathers provided for us, before they knew of our existence. It's touching, isn't it, Chika?"
Chika said nothing, just stared at the treasure.
"What should I deal in, Chika? You know what men buy and sell. What are the things people cannot do without?"
"Wine, I suppose, and the things you make from soybeans: paste, curds, sauce."
"I imagine I'll find out what all that stuff is," Kiku said. "Chika and I will go to... what's the best place?"
"Maybe Kitakami," Chika said. "I've never been there, but it has the reputation of being a city where anyone can make a fortune. It trades with countries on the mainland. It is said to be rough and wild."
"Then we will go to Kitakami."
"I don't mind coming with you," Chika said. "But what about Kaze?"
"Kaze can stay here. I'll come back for her once I've started making my fortune."
"I'm coming with you, too," Kuro announced.
Kiku stared at him for a moment, and then nodded. "Yes, I'm sure I'll need you."
Kuro grinned. "Me and my creatures."
* * *
After the three left, Mu, Ima, and Ku went back to the peaceful life they had been leading before their brothers returned. Yet it was not quite the same, for now they had a girl living with them. She followed Ku and the dogs around and joined Ima in bringing offerings to the skull of her father's horse. She ate what they ate and slept alongside them, outside on fine nights, inside the hut when it rained. It did not rain often—even though the plum rains should have set in—and the days and nights were very hot.
Shikanoko had taught all the boys to use the bow. One day, Kaze took up Mu's bow, went off into the forest, and returned with a hare and two squirrels. They could all move silently, and could take on invisibility, but she was a better shot than any of them. She still did not speak much, but now and then, at the end of the day when they sat around the fire watching the flames grow brighter as night fell, she would sing—lullabies that made the dogs sigh in their sleep, ballads of love and courage that filled the boys with yearning.
They were fascinated and intimidated by her. She ordered them around, as if it were her right to be served. Ku and Ima adored her. Her presence, Mu thought, made them all more gentle, more complex—perhaps more like real people. He remembered Shika's wolf companion, Gen, who had been as artificial as the ones whose remains now littered the clearing, but who had grown more and more real because of its attachment to its master.
He began to pay more attention to the fake animals, tried to revive their strange spark of life.
Even objects need attention, he thought. Even the lifeless need love.
Some nights, he saw green eyes shine in the darkness under the trees and he imagined the foxes had come to listen to the singing. He heard vixens scream at midnight. One day, when he went to the stream to get water, a fox was drinking from one of the pools. It ran into the undergrowth at his approach. He called, without knowing why, "Come back! I won't hurt you!" and a few moments later the leaves rustled and the fox girl stepped out.
They stood and gazed at each other, the stream flowing sluggishly between them. She seemed less like a fox than before and more like a human. Her ears were only slightly pointed and her feet were small and delicate, but they were definitely feet, not paws. Beneath her robe, which was dyed red and tied with a yellow sash, was a hint of a tail, but, at second glance, it was not a tail at all but just the way the robe fell.
"I saw you," he said.
"I know. You were in the oak tree."
"What was it?"
"A wedding," she said gravely. "I'll show you if you like."
He held out his hand. "Come across."
"No, you come to me," she commanded.
He leaped the stream in a bound and found himself so close to her he could smell her faint animal scent. She smiled at him and, reaching up—she was much smaller than he was—kissed him on the mouth.
The effect on him was so surprising he broke away from her, crying out, which made her laugh. Taking his hand, she led him into the bushes and pulled him down, so they were lying side by side. The sunlight dappled the moss with tiny circles. In the distance a warbler was calling. She loosened her robe and guided his hands to her body, then reached under his clothes to caress him.
* * *
She lived in the hut with him as his wife. She combed and braided Kaze's hair—the girl responded as if the fox woman were her mother, climbing on her knee when they sat by the fire, though Kaze was really too big to be cuddled. The dogs growled at her at first, but she rubbed their ears, picked off their fleas, and won them over. She was always merry; everything made her laugh. She liked to dance in the twilight, while Kaze sang. And every night she lay down with Mu and made him happy in a way he had never dreamed of.
Once he said to her, "Why did you choose me?"
"Your mother was one of the Old People."
"I don't know what that means," he said.
"They were here before the horsemen came. They are more like us, both animal and human. Shisoku was one, too. The Old People know many things about other realms that the horse people have never learned."
Mu thought about the mask and the skulls. "Can you teach me those things?"
"I am teaching you already," she said, laughing. "Didn't you notice?"
SHIKANOKO
Throughout the summer the Burnt Twins and Shikanoko kept traveling north, following the tracks of deer and foxes through the tangled forest. Not knowing where else to go, Ibara went with them. They saw no one nor was there any sign that they were being followed.
That first autumn the rain ceased and though the following winter there were heavy snowfalls, the seasonal summer rains failed. They had found shelter in an abandoned building where a solitary hermit had once lived. It stood near a spring from which water flowed constantly, and the former occupant had made a garden, which still existed wild and overgrown. Yams and taro had self-seeded, as well as pumpkins, and there were fruit trees, an apricot and a loquat. They found his bones in the garden, in a tangle of grass and kudzu vine, scattered by scavenging birds and animals. Eisei gathered them up and buried them.
Animals came to drink at the spring, foxes, wolves, and deer. The deer lingered to graze on the sweet grass in the clearing, which, thanks to the spring, flourished all through the hot summers while the rest of the land dried up. Nagatomo trapped hares and rabbits and shot birds—pigeons mostly and the occasional pheasant—but the lord would not allow them to hunt deer. Consequently the deer became more and more tame, allowing him to mingle with them and feeding from his hand.
"That is why he is called Shikanoko," Nagatomo said. "The deer's child."
Ibara had not known his name, though she had heard it once, a long time ago, it seemed, from the mouth of the man with the Matsutani crest whom Nagatomo had killed. In the long months when there was little to do, Nagatomo taught her with that dead man's weapons and now she carried his sword at her hip; she wore his clothes and tied her hair back. She had fallen a little in love with Nagatomo, almost overcome with desire when he held her hips or shoulders to correct her stance, but he never responded. She knew he and the monk were lovers, twinned in some way by their shared suffering. In time she recovered her equilibrium, but she did not lose her desire for revenge. Every day she thought of Masachika and how she was going to kill him.
Shikanoko spent hours in meditation, but he began to eat. Ibara could feel how, little by little, grief gave up its grip on her heart, and she thought she could sense the same in him. When the deer came in the evening he moved among them as if he were following the steps of an ancient dance. When she had first seen him she had recalled the heron dancers. Now he danced with the deer—they all did—and in the dance created the ties that bind Heaven and Earth, humans and animals, the living and the dead.
The foal grew to its full size and its dark coat turned to silver. It looked like its father but had a black mane and tail. Nagatomo introduced it to the saddle and bridle, but it did not need breaking in. It accepted the saddle but did not like to be bridled. However, no one felt comfortable riding it. There were enough horses for the three men, and Ibara took Risu. Tan followed close by her side and she talked to him about Hina, feeling sorry for the man trapped in a horse's body, wondering what good it had done him to be summoned back.
The mare lay down one cold winter night and did not get up again. The death rekindled Shika's grief and he wept bitterly. The two stallions mourned like humans.
Something about Ibara's presence goaded Shikanoko. She was like the thorn she had named herself. She pricked and scratched away at the armor he was building around himself. Perhaps it was that she often talked of Hina to the horse Tan, and then she would mention Tan's twin, his own son, Takeyoshi. So he heard of the baby's birth, and Hina's beauty, her great intelligence, her kindness, and it filled him with a longing to see them both.
It was not magic or sorcery, nor did she do it deliberately, but little by little he awakened and began to emerge from grief. He took out Jato and cleaned and polished it, rebound his bow, Kodama, and carved new arrows, fletching them with white winter plumage.
They were in the northernmost part of the Darkwood, right up near the Snow Country. The snow was heavier than Shika had ever seen. After blizzards they had to dig their way out of the hut. The clearing, the grave, every branch and twig, every boulder and rock was blanketed in white.
For weeks they were confined inside together. On the worst nights they brought the horses inside, too, otherwise they would have been buried under the snow. Shika's intention had been, when winter came, to continue his meditation and fasting in the open, which was really a slow way of killing himself, fading back into the forest, ceasing to live in a world so full of pain. But perversely his body showed signs of wanting to live. It became hungry and demanded food; suddenly it slept again, in the deep, healing sleep of boyhood. His mind awoke, too. His hours of solitary meditation had revealed to him, among other things, how little he knew. He understood nothing about how the world worked, what Aritomo's motives were, why the Miboshi and the Kakizuki fought each other, what it meant to be a warrior, what the nature of revenge, such as Ibara desired, was. He recalled his own upbringing, his father's death, up here in the north; the loss of the bow, Ameyumi; his uncle's brutality. And now he had a son, Takeyoshi, whom he was condemning to the same orphan state.
He saw how impetuous and thoughtless his actions had been, all his life, how he had been used and manipulated by others, for both good and bad, in his need for approval and affection, in his quest to redress his own pain.
On days when the snow fell heavily, there was nothing to do but talk. Eisei had received the usual education of a monk, could read and write, and knew sutras and other holy writings by heart. He had also absorbed, over the years, the songs and ballads that were sung in the outer courtyards at Ryusonji, and he often recited these: long, intricate tales of heroes, warriors, powerful priests, warring clans, child emperors. Nagatomo and Shika had shared the rudimentary education of provincial warriors. They could read and write, and knew the history of their clan and the legends of Kumayama, but they had never learned how to conduct a careful argument or correct a false idea. Ibara could read a little, write in women's script, and calculate. She knew a great deal about her hometown, Akashi, and the way the free port and its merchants operated. Since she did not defer to any of them, dressed like a man, and usually spoke like one, they forgot she was a woman. Three of them were born in the same year. Nagatomo was a year older.
One evening, Ibara said, "Takaakira, the man who employed me to look after Lady Hina, was called the Lord of the Snow Country. I suppose his estates were not far from here?"
"We are alongside them," Nagatomo replied. "They begin on the eastern edge of the Darkwood and extend far to the north."
Shika recalled the day Eisei had told him of Takaakira's death. He had not known then that it was on Hina's account. Now he felt a new interest in the man who, in disobeying Aritomo, had saved Hina's life and paid for it with his own.
"What sort of a man was he?" he asked Ibara.
"In truth, I hardly spoke to him, and then only about Hina. He knew a great deal about all sorts of things; he wrote poetry. There were two women whom he arranged to instruct Hina in history, music, and so on. He talked more to them, about her progress and her timetables. I had no idea a child could absorb so much. It used to worry me sometimes. I had to make sure she went outside from time to time, even though he didn't really approve. He wanted her to keep her pale complexion."
"He had a reputation for courage," Nagatomo said.
"I heard him claim once he was an adept," Eisei added. "There must be some strange esoteric practices in the Snow Country. He believed Yoshimori was the true emperor and he was going to tell Aritomo that on the day he died, and plead for your life and the Princess's. I suppose he never got the chance."
Ibara had a remote expression on her face as if she was dwelling on the past. "He adored her," she said finally. "I've never seen a man so obsessed."
Shika was surprised at the strong emotion that welled up in him, part jealousy, part affront, but also, mingled in, gratitude and relief. Before he met Ibara, he had assumed Hina had died in the massacre of the Kakizuki women and children. Now Ibara was convinced she had not drowned. But where had she gone? What had happened to her?
"Yet she was only a child," he said. "And how old was he?"
"Well over thirty, I would imagine," Ibara replied. "She was about eleven years old. He intended to make her his wife—but he had not touched her," she added, maybe noticing Shika's face.
The shutters rattled as the wind howled against them. The snow fell with the lightest of sounds, like insects swarming.
* * *
On sunny days, when the snow did not fall, they rode out through the forest—no longer the dark wood but gleaming white. The horses plunged through the deep snow, snorting with excitement. Gen was light enough to run over the frozen surface. They took their bows and hunted hares and squirrels. Sometimes they saw wolf tracks. Every now and then, Shika caught sight of his antlered shadow, blue-black on the snowy ground. Each time he felt the shock anew.
Will I ever be rid of it?
As spring approached, the snow fell with rain mixed in. The icicles that clung to the roof began to drip in the sun. The stream melted and the water roared with its new fierce flow. The deer dropped their antlers. Nagatomo and Ibara collected them, polishing them. Fawns were born, and bounded after their mothers on long, delicate legs.
The short summer brought biting flies and heavy humid air. Violent thunderstorms crackled round the mountain peaks. In the autumn, drums sounded from far away, giving a rhythm to their own dances. Another winter passed: the same deep drifts of snow, the same long conversations in the smoky hut. They saw no one else and began to forget they were fugitives. No one ventured so deep into the forest, or so they thought, until one day in early spring when Nagatomo returned from collecting water, saying, "Am I going mad and hearing things, or is someone beating a drum in the distance?"
Once he mentioned it, they all heard it, a dull, monotonous pounding on a solitary drum. It stopped for a while, then started up again. It was the wrong time of year for the drum festivals and, in truth, the playing did not sound skillful.
"Someone practicing?" Ibara suggested.
After the drumming had persisted for a full day and a half, Shika said, "I'm going to see what it is."
The Burnt Twins exchanged a swift look, and said together, "We should go."
"You still act as if it matters whether I live or die," Shika said, with amusement.
"It matters to us," Nagatomo said.
Shika was touched, though he did not show it. "Well, you can come with me. Ibara, do you mind keeping guard here?"
The drumbeat was halting and uncertain, yet there was something compelling about it. The mask responded to it in some way. He felt the rhythm pass through his skull and reverberate within the antlers.
The snowmelt filled the streams and the trees were just beginning to put on their first green sheen. Frogs rejoiced and birds sang, skirmished, mated in their urge to raise young ones before the short summer ended. The horses trotted eagerly, nipping at one another and bucking occasionally for the sheer joy of being alive.
On the edge of the forest, where the huge trees gave way to bamboo groves and then to coppice, the stream widened into a marshy lake. Sedge and susuki reeds grew around it; a snipe took off at their approach with a sudden cry of alarm.
A shaggy northern pony, dun colored with a brown mane and tail, threw up its head from where it had been grazing and whinnied loudly. The drumbeat stopped abruptly. A young boy, about ten years old, got to his feet, took one look at the antlered man on the white horse, and ran away, shouting to the pony to come to him. But he was slowed by the drum and Nagatomo easily caught up with him before he could reach his mount.
He tried to lean over and scoop up boy and drum together, but his horse was spooked by the drum's hollow sound against its neck, and shied, throwing its rider. Nagatomo fell to the ground, still holding the boy, the silk covering slipping from his face.
When he stood, the boy looked up at his ruined features but did not say a word. Nagatomo's horse ran back to the others. Eisei seized its reins. The boy's eyes followed it, he saw Eisei's black-covered face, and then he saw Shika. He tried to wriggle out of Nagatomo's grasp, not to escape but to throw himself facedown on the ground.
"Kamisama, kamisama," he wailed. "Please help me!"
Shika dismounted and told him to sit up. It saddened him to see the fear and shock in the boy's face.
"I am no god," he said, thinking, But I will never be human again.
"Then take off the mask," the boy challenged.
"That I cannot do."
"Then you must be gods or spirits, all of you. I was trying to summon the deer god. I didn't really think I could, but I was so desperate I didn't know what else to do. And you came."
"Is that why you were beating the drum?" Eisei said.
The boy replied, "People have said they've seen a figure, part deer, part man, in the forest. I thought I would try to call him out. I borrowed the drum from the shrine, but it's not as easy to make it speak as I thought it would be."
"You were certainly persistent," Nagatomo remarked. "In what way do you need help?"
"My father died two years ago in Miyako, and now his cousins want to divide his estate among them. They say he was ordered to take his own life because he was a traitor and therefore his lands must be forfeited and I should not inherit them. When my mother opposes them, they say she and I deserve death, too, and we will be killed if we do not submit. But my mother thinks we will be killed even if we do. There is no one to help us. We cannot appeal to Lord Aritomo—my mother suspects he might even be supporting these false claims."
"What was your father's name?" Shika said, though he had already guessed the answer.
"Yukikuni no Takaakira, Lord of the Snow Country. My name is Takauji. I am his only son." He had recovered some of his composure and now studied them, with their ragged clothes and their wild hair and beards, almost insolently. "You are so few. And you look like bandits. Is this how the deer god answers my prayers?"
Shika liked his defiant attitude and found himself inclined to help Takaakira's son. He put his hand on Jato and felt the sword quiver in response, as if it sensed his desire to fight and shared it.
"There is one more of us," he said. "We are not bandits, but you are right, we are few. How many cousins are there, and what forces do they command?"
"There are three of them. Each has about twenty men. There are a few hundred warriors still attached to the estate and they are mostly undecided. They don't think the land should be split up and they are loyal to my father's memory. But I am not yet a man and they are not used to the idea of serving a woman. However, around here everyone worships the deer god and would do whatever he says."
Shika drew the Burnt Twins aside. "What should we do?" he asked quietly.
"People obviously know we are here," Nagatomo said. "Sooner or later they will learn who you are. Either we move on now, farther north, or we take advantage of this new situation."
"It's an opportunity to stamp your mark here," Eisei added. "If you have allies and men in the Snow Country, you will return to Miyako with the east protected."
They both spoke of the future, Shika thought, whereas he still did not consider he had a future. One action would lead to another if he started to reengage with the world.
Tan, who had been following them through the forest at his own speed, trotted up to them, sniffing curiously at Takauji.
"What a fine horse," the boy exclaimed.
"What should I do, Lord Kiyoyori?" Shika said.
Tan pawed the ground and neighed shrilly. Naturally, Lord Kiyoyori wanted to fight.
"We should talk to your mother," Shika said to Takauji. "Where can we meet?"
Takauji pointed to a small shrine hut at the farther side of the lake, half-obscured by willow trees and hazel bushes. It had the high, steep roof of Snow Country buildings. Above its door, antlers had been fastened, some of the largest Shika had ever seen, and newly tanned deer hides were spread over the small veranda. A frayed and knotted rope hung beneath a wooden bell. "I will bring her there at this time tomorrow."
"We will be there," Shika promised.
"Can I touch your antlers?" Takauji asked.
Shika lowered his head. Takauji grasped the unbroken branch and pulled sharply, jerking Shika forward, unbalancing him.
Shika cuffed him. "I told you, the mask does not come off."
"You really are the deer god, aren't you?" Takauji said, with awe.
* * *
When they returned to the forest hut and told Ibara what they had learned, she said, "I would like to do something to help Lord Takaakira's son, and his wife—she has suffered a lot in her life, I think."
"She must have seen very little of him," Eisei remarked.
"Yet he trusted her to run that vast estate in his absence," Ibara said. "She is by no means a helpless woman; it may be some kind of trap. Let me go first tomorrow. I should be sorry to die before Masachika, but apart from that my life is unimportant."
Shika smiled, though he knew no one could see it beneath the mask. "We could argue all night over whose life matters the least."
"I don't believe either of us has yet fulfilled our destiny," she replied in a low voice.
"Then you and I will go to the shrine and meet her together and give Heaven a chance to prove you are right. Nagatomo and Eisei will keep watch outside."
They rose early and were on the edge of the forest, with a clear view of the shrine, just after daybreak. They had hidden the horses farther back among the trees, but Tan had followed them, picking his way carefully among the winter's dead leaves, through which blades of grass and flower stems were emerging. Under rocks the last of the snow still lingered. Gen walked stiff-legged behind Shika, turning his head to favor his right ear, now and then stopping to sniff the air.
Shika settled into a meditation position, Jato on the ground beside him, his bow on his back, while Eisei and Nagatomo checked that the shrine was empty. They then melted back into the forest.
Gen crouched on his haunches, a little in front. After some time, when the sun had burned the mist from the fields, the fake wolf raised his head and gave a faint whine. A few moments later, Shika heard the muffled tread of hoofs in the soft earth; only one horse, he thought.
A crow called: Nagatomo had heard the horse, too. A real crow replied from a tree nearby, making Shika grin.
The three figures came into sight, the woman riding, the boy running alongside. It was the same pony from the day before. It caught Tan's scent and stared toward the forest, whinnying a greeting. The other horses would have neighed back, had they been within earshot, but Tan remained silent.
The woman did not really ride the pony but rather used it as a method of transport. She had no other interest in it. He guessed she did not ride often. It stopped abruptly not far from the shrine and she slid from its back, as though thankful to get off.
The pony looked thankful, too, shook itself vigorously, and began to crop the new grass. The boy ran to the shrine, his knife drawn, and entered cautiously. After a few moments, he came out and beckoned to his mother.
She looked around once, took some offerings from a cloth, and went forward to place them on the steps. The boy pulled the bell rope and the wooden clapper gave out a hollow, eerie sound that made Shika's neck prickle. Gen gave a muffled howl.
Shika picked up his weapons and approached the shrine. Ibara emerged from the dead bracken where she had been concealed, and followed him. At the steps she touched his arm and indicated that she should go first, but at that moment Takauji appeared on the threshold and gestured to them to come in. Bending his head, Shika stepped inside.
For a moment he could see nothing in the gloom. He heard her gasp and could only imagine how the antlered mask had startled her. He made no bow or greeting, but she dropped to her knees, laid her palms flat on the floor, and lowered her head.
"My son told me," she whispered.
"I am not the deer god," he answered. "I am a man under enchantment, a curse, you could say."
"It must make you powerful," she said, more loudly, sitting up and gazing at him frankly. "Yes, I can see it does. Believe me, I know all about power."
He could see, now, the planes of her face, sharp features, pale, northern skin. There was something birdlike about her; she reminded him of a falcon, fierce and swift. Her hands and feet were very small, her wrists slender.
"Are you better with the bow or the sword?" she said, wasting no time.
"I believe I can outfight most men with both," he said, "but probably the bow suits me more."
She said, "I am glad of that. This is my plan. I don't want to plunge the whole of the Snow Country into war, brothers against brothers, fathers against sons, but until those who challenge me are dead, that war cannot be avoided. I am going to invite my husband's cousins to an archery contest, in honor of the deer god, and you will kill them."
She gave a thin-lipped smile. "Of course, there is no reason why you should help me. I don't know who you are or where you have come from."
"There are bonds between us," Shika replied. "My companion was employed by your late husband."
"Were you there when he died?" the lady said, turning her piercing eyes toward Ibara.
"No, lady."
"Are you a woman?" her voice was bitter. "Is that why you found employment with him?"
"No," Ibara said simply, and then, "I worked in his household."
"Looking after Kiyoyori's daughter, I suppose. I heard all about it. What a foolish thing to die for, don't you think? I will never forgive him, but I will never forgive Aritomo either." She was twisting her hands together, and then struck one fist with the other palm, and held her hands firmly so they would not move. "I'm sorry," she said. "It has become a habit, since I had the news of his death. What happened to the girl? I hope she is dead, too. It was an evil day when Takaakira came upon her."
"She drowned while trying to escape," Ibara said, levelly.
"So much the better. But you are a strange one! Dressed as a man, carrying a sword. What are you trying to achieve?"
Ibara gave Shika a look, and stepped to the open door, where she crouched down, staring at the lake.
"I have offended her... him... which should I say? I did not mean to talk about these things. Now I am upset."
Shika could see that her life had made her selfish and angry. He was inclined to leave her to it: let the relatives divide the estate as they desired. But Takauji interested him and he wanted to spare the boy the sort of childhood he himself had had.
She studied him as though divining his thoughts and said, "I suppose Aritomo would be very interested in knowing about you."
"He already knows about me," Shika said. "He has nightmares about me."
"Does he know that you are here, in the Snow Country?"
"By the time he learns that, I will be somewhere else." Her gall in trying to threaten him, in this oblique way, made him laugh.
She had the grace to look uncomfortable. "Well, as I said, I am no friend of the Minatogura lord. He will not hear about you from me. My men deeply resent their lord's death. They are Snow Country people. They know how to keep silent."
She got to her feet and came close to him. For a moment he thought she was going to touch the mask, and he wondered briefly if it would come off under her fingers, but though she looked at it intently, she did not reach out. She lowered her eyes to his legs, as though appraising him, making him aware that her husband had been dead for many months, and even before that she had been more or less abandoned. He felt her hawklike determination. He admired her, but he did not in any way desire her.
* * *
At night, it was still cold, and sometimes there were new falls of snow, but once the equinox had passed, spring came with a rush. In the days before the competition, targets made of bundles of straw, shaped like wild boar, with large painted eyes and real tusks, were set up along the lakeshore. Horses were washed and groomed, their winter coats brushed out, their manes and tails plaited with red. They caught the excitement and neighed wildly, stamping their feet and tossing their heads. From the marshes, nesting birds shrieked in response.
Takauji and his mother were given the place of honor on the steps of the shrine. All day men competed in heats, galloping past the targets and losing their arrows. The boar's eyes were considered the winning shot. Finally there were four rivals left, the three cousins and one of Lady Yukikuni's men, middle-aged, skillful and cunning, with a clever, nimble horse. None of them had achieved the perfect score of three eyes in a row.
As they were preparing to ride off against one another, the lady said in a clear voice, "There is one more competitor, the representative of the deer god himself. Who dares take him on?"
She beckoned to Shika and called, "Come out!"
Nyorin stepped out of the forest, his silver coat gleaming, his long mane decorated with flowers and leaves. A cry of surprise and awe came from the watching crowd. The other competitors fell back as the horse approached the shrine. Shika bowed his antlered head to the lady and her son and took the huge bow from his shoulder. Nyorin's nostrils flared and he uttered a challenging neigh as Shika turned him toward the starting point, breaking into a swift gallop as Shika dropped the reins on his neck and drew from his quiver one of the arrows he had made in the forest.
It was easy for him, far easier than shooting the werehawk from the sky. One after another, three arrows slammed into the boars' eyes.
Nyorin came to a halt and trotted back to the starting line, where he stood snorting in triumph as though saying, Beat that!
Shika was about to dismount and go to the lady when one of the cousins rode toward him, shouting, "Take off that mask and let us see who you really are!"
Several tried to dissuade him, but he had already drawn his sword and was thrusting toward Shika. Nyorin moved like lightning, striking out with his forefeet, giving Shika time to draw Jato.
The lady said in a clear voice, "He has drawn his sword against the deer god. Let him die."
No one saw who loosed the shaft that pierced the man's chest. It came out of the forest. Shika knew it was Nagatomo's. Within moments the other two cousins were dead. Eisei and Ibara, he thought. The straw boars stared with their blind eyes as the blood soaked into mud and sand.
The lady was on her feet, her pale face flushed with triumph. "It is the judgment of the forest itself!" she cried.
People drew back as Shikanoko rode away, afraid his shadow would fall on them.
* * *
The next day there were offerings on the edge of the forest, and every day after that. A week went by before the lady came, riding the dun pony, with Takauji leading it. She had brought Shika a pair of chaps made from wolf skins. The fur was gray and white, the bushy tails still attached.
After she had given them to him she said, "Stay with me as my husband. I know you are a man."
"If you can remove the mask, I will," he replied, not believing she could.
She smiled and reached out immediately, certain she would succeed, but she could not shift it from his face. Tears of disappointment came to her eyes.
"Stay anyway," she begged. "I will tell no one. You may hide out in the forest all summer and I will come to you at night. You have seen how the farmers are already bringing you food. You will lack nothing. I will show you my gratitude."
Shika thanked her, but, as soon as she had left, without even any discussion among them, they prepared the horses, packed up their few possessions, and began to ride to the north.
KIKU
Once they had struck the northern coast highway and turned to the west, the three boys, Chika, Kiku, and Kuro, came across many other travelers: merchants with trains of packhorses; monks carrying stout sticks and begging bowls; officials and their retinues; warriors on horseback, in groups of three or four, offering their swords and their services as bodyguards; beggars, and probably not a few thieves, Chika thought, taking care to keep the treasure well hidden. They had divided it into three and put it into separate bags, though Kiku and Kuro had each taken a strand of pearl prayer beads to string around his neck. Kiku's entangled with the beak and talons of the werehawk.
Chika explained who all these people were, as best he could, adding to the knowledge Kiku and Kuro had acquired in their previous forays into the human world, but their grasp of how everything hung together was still flimsy.
"Is this the mountain where Akuzenji was king?" Kuro asked as they climbed up to the pass. The road was steep, hardly more than a track. Horses stumbled over rocks and slipped on the scree. Even high in the mountains, it was hot. Sweat dripped from men's faces, and the animals' bellies and legs were flecked with white stains.
"I suppose so, and this must have been the most dangerous part of the journey," Chika said. "Any number of men could lie hidden behind the outcrops, and there is nowhere to escape to."
The mountain rose steep and jagged on one side, and, on the other, the valley fell away, a sheer drop of hundreds of feet.
"That's where Akuzenji used to throw the bodies of those who refused to pay him," said a man who had been walking just behind them, leading a small horse laden with baskets. "Only they were not strictly speaking bodies, not until they reached the bottom, that is."
Kuro's eyes brightened with interest.
"Where are you boys walking to?" the stranger went on.
"Kitakami," Chika said.
"All alone? No family?"
Chika said warily, "In Kitakami—we are going to a relative's house."
"What is their name? I know most people in Kitakami."
"I don't remember."
"So where do they live?"
"Near the port," Chika improvised.
The man laughed. "In Kitakami everyone lives near the port. Well, if you find them, tell them Sansaburo from Asano says good day. May your journey be safe!"
He clicked his tongue at the horse and walked past them.
"What did he mean by all that?" Kiku said.
"I think he was just being friendly," Chika replied.
Kiku looked at Kuro, who raised his eyebrows as if he, also, did not understand.
At the top of the pass, they paused to catch their breath. The black cone of Kuroyama rose behind them, and in front lay fold after fold of ranges all the way to the west. In the distance, to the north, Chika could see the glimmer of the sea, and in the south a huge lake—Kasumi, he supposed. It was approaching evening; most travelers had hastened on to find lodging before dark. Only the friendly merchant was still on the road ahead of them.
Kiku, deep in thought, hardly looked at the view but, as they began the descent, said to Kuro, "Let's take that horse."
"All right," Kuro replied agreeably. "Shall I use the sparrow bee?"
"That'll do," said Kiku.
They quickened their pace until they had almost caught up with the man. Kuro took the cover from the wicker cage and the sparrow bee began to buzz angrily. He released the catch and the bee shot out, soaring briefly, then descending to attack. The man called Sansaburo from Asano gave a cry and danced around, waving his arms futilely. The bee stung him on both hands, then on the neck. Within moments he was lying in the dust, clutching at his throat.
"What have you done?" Chika cried in shock.
Kuro had grabbed the lead rope of the startled horse and was trying to catch the bee before it stung the horse, too. When he had succeeded he said, "Let's throw him over the edge."
"No, we'll leave him here," Kiku said.
"You killed him!" Chika said. "He'd done nothing to you!"
"Don't worry. I have a plan. We'll take the horse."
"What's in the baskets?" Kuro said, curious.
They loosened the well-tied knots on one of the baskets and prized open the lid. A faint fishy smell floated out.
"Just old shells?" Kiku said. "What use are they?"
Chika slipped his hand in among the shells and felt their smooth interiors. "It must be mother-of-pearl. It's used as decoration, in inlays and so on."
"Is it valuable?" Kiku asked.
"Very," Chika replied.
"We'll take it to the... who should we hand it over to?"
"What do you mean?" Chika asked.
"Aren't we going to keep it?" Kuro said.
"We must give it back," Kiku declared. "We'll say we found the horse, straying. But first we need to kill a couple more merchants, so people begin to be frightened."
Chika said doubtfully, "We'll be caught."
"I promise you, we will never be caught," Kiku said, with conviction. "Just tell me who we should take the shells to."
"I suppose to whoever ordered them. This man probably took them to the same person, year after year."
"Hmm. I should have asked him that, before he died," Kiku said. "There's so much to learn. Well, you can make inquiries when we get to Kitakami."
They walked a little farther and, when night fell, went off the track into the forest to rest for a while. Kuro sat by the horse, keeping watch, and Chika and Kiku lay down, side by side. Kiku seemed to fall asleep immediately, but Chika was wakeful, and when he did finally sleep he had a nightmare that one of Kuro's snakes was slithering toward him. He tried to run, but his limbs were paralyzed. The snake hissed and flickered its tongue and he knew that at any moment it would bite him and he would die. He woke to find Kiku's arms around him.
"You were screaming. What's wrong?"
"I had a bad dream."
Kiku said, sounding surprised, "It feels nice, holding you like this."
Chika lay without moving, letting the other boy touch him with exploring hands. He felt his body begin to respond to the pleasure. Their limbs entwined, their mouths joined, as they took the first steps on the journey of sex and death that would bind them to each other.
The second merchant was garroted by an invisible Kiku. They took his horse, loaded this time with bamboo scoops and bowls. Kuro looked after the horses while Chika and Kiku fell on each other with the ferocious lust of young males, incited by their seeming power over life and death.
"Can you kill anyone you choose?" Chika asked, as he lay panting and exhausted.
"I don't know yet. I am finding out. It's fun, isn't it? The killing, and then this afterward? I had no idea it was all so much fun. I don't understand why people don't do it all the time."
"Some do," Chika replied. "But it's not so much fun for the people who die."
"If people did not die, there would be no room for new ones. And don't they just get reborn into a new life, anyway?"
They had heard monks and priests expounding the doctrine along the road.
"Can you be killed?" Chika asked, a new fear seizing him.
"I suppose so. Our fathers are mostly dead, and our mother." Kiku did not want to dwell on the subject. "So who shall we kill next? You choose."
The idea seemed suddenly irresistible, yet a few months ago it would have appalled him. His warrior upbringing was being corrupted and tainted by Kiku. He tried not to think of his father's stern teachings. He knew he was enthralled by Kiku, as by no one else in the world. He wanted to please him; he could not resist him.
"I bet you cannot kill a warrior," he challenged.
* * *
He was not much of a warrior, a solitary, grizzled man with only one eye, the empty socket yawning disconcertingly. He sang ballads in a monotonous, melancholy voice outside a lodging place not far from Kitakami. Chika and Kiku watched him while the horses rested and drank and Kuro procured food. Chika found the strains of his voice, in the summer evening, curiously moving. He wondered where he had come from, what had reduced him to this.
He wore a faded green hunting robe and under it a corselet, missing much of its silk lacing; at his hip was a long sword. He did not earn enough for a meal, let alone a room, and began to walk away, his measured gait and the set of his shoulders indicating he was resigned to another night on the road.
The two boys sauntered after him, and Kuro pulled the reluctant horses along behind them. Kiku waved at his brother to fall back out of sight.
"Attract your warrior's attention," he whispered to Chika. "I'll grab him from behind and then you can use your sword."
Chika watched, with all the shock and thrill it always gave him, as Kiku faded into nothingness, and then, hastening his steps, called, "Hey, sir, wait a minute!"
The man turned. The sunset behind him made him appear solid, black and featureless, and for a moment Chika felt afraid, but as he came closer he saw the lines in the face and the gray hair. The remaining eye was clouded, two fingers were missing from the left hand. When the man finally moved, it was stiffly—no doubt he was troubled by old wounds.
"What do you want, boy?"
"I am traveling alone. I thought we could walk together."
The warrior gave him a shrewd look. "What happened to your companions and the horses?"
Chika said, "They have gone their own way. Maybe they stopped for the night."
"You carry a sword; you look like a warrior's son. Were you not taught to speak the truth, at all times?" His tone was forbearing, but the words flicked Chika like a whip.
"You are an expert on the life of a warrior, are you, you beggar?"
The man turned. "Walk by yourself. I am a little fussy in my choice of companions. If you ever learn any manners, you can approach me again."
Chika's hand was on his sword. At the same moment the warrior drew his own, turned as fast as a snake, and struck out.
Kiku gave a shriek, breaking into sight, grasping his arm, blood beginning to drip from it.
"What are you?" the warrior cried. "Forgive me, you should not sneak up on a man like that. It's instinctive, you see. I can't help but react. Now I have cut you!" His eyes went back to Chika's sword. "What? You were intending to kill me? Two shabby boys like you? You must be desperate! I have nothing but these clothes and my sword. I'll die before I hand that over!"
"I'm bleeding," Kiku said.
"It's not fatal," the man assured him. "Unless you are unlucky enough to get wound fever. Wash it well, that's my advice. And now, my young friends, unless the warrior's son wants to fight me, I'll be on my way."
"Wait," Kiku said. His brow was taut with pain and concentration. "What are you planning to do in Kitakami?"
"None of your business, brat."
"You must be hoping to find someone who can feed you, someone you can serve."
The warrior laughed. "And if I am, what's it to you?"
"How would you feel about serving me?"
He laughed again but more bitterly. "The first requirement of anyone I serve is that they be rich!"
"We are rich," Kiku said, pulling out the pearl prayer beads and fingering them. "But we don't know what to do in Kitakami and we are afraid of being cheated. We found these horses, wandering. We want to take them to their rightful owners. Not for any reward, we don't actually need it. Just to do the right thing."
Kiku had been turning paler and paler while he talked, and now he swayed as if he was about to faint.
The warrior sheathed his sword. "Come, let me take a look at that cut."
Chika knew he had a chance of killing him now, while he was unprepared. He saw the exact place in the neck where his blade would open the flesh and the blood vessels within. His hand flexed and clenched, his sword quivered. He was not sure if Kiku's faintness was a ploy to get the man off his guard.
"Put the sword away," the man said. "I may be one-eyed and crippled, but you still wouldn't stand a chance against me. Here," he tore a strip from his underrobe. "Run to the spring and wet this. We'll bind his arm."
"What is your name?" Chika asked after the wound was bound and Kuro and the horses had caught up with them.
"Yamanaka no Tsunetomo. And yours?"
"Kuromori no Motochika." He used his adult name.
"Huh? Everyone at Kuromori was slaughtered. So why are you still alive?"
"That's my business," Chika said, making Tsunetomo laugh.
"That's right, my friend. Some of us are called to die and some to survive at any price. If that's your path, embrace it, without regret or shame."
"Has it been your path?" Chika asked.
"Maybe it has," Tsunetomo said. "At any rate, I am still alive. Now, let's see what those horses are carrying."
It was almost dark, a warm night with no moon, the starlight diffused by the hazy air. Kuro made a fire and unloaded the baskets from the horses' backs. Tsunetomo inspected the contents.
"You found the horses straying, you say?"
Kiku nodded. His eyes were a little brighter in the firelight and his cheeks were flushed, but he no longer seemed faint, nor otherwise affected by the wound.
"What happened to the owners, I wonder?" Tsunetomo's face was expressionless, his voice bland.
"No sign of them," Kiku said.
"Maybe someone murdered them?" Kuro suggested.
"That's what people are saying," said Tsunetomo. "I've already heard one or two rumors—Akuzenji's ghost, or some new bandit chief, or ogres who kill people to eat them. If such rumors continue to spread, more people will become afraid and soon no one will want to travel alone."
"That's good," Kiku said. "We can offer them protection—you and your sword."
"Am I to guard the whole length of the highway?" Tsunetomo laughed.
"You must know other people you can hire to help you."
"As a matter of fact, I do."
"So will you serve me?" Kiku asked.
Tsunetomo stared at him. "I will," he said finally. "You're a strange creature, but there's something about you... keep me in food and shelter, and a little extra for wine, and my sword is yours."
ARITOMO
The fiery death of the Prince Abbot had not only shocked and grieved Lord Aritomo but had also alarmed him. His hold on power was weakened, his authority shaken. With his usual clear-sightedness he knew it would take only one more blow to dislodge him. He strengthened the capital's defenses, while making plans to retreat to Minatogura, preparing boats at Akashi, in case attacks came from Shikanoko in the east and the Kakizuki in the west.
But his enemies did not take advantage of his momentary weakness. Shikanoko vanished into the Darkwood, Lord Keita and his retinue made no move from Rakuhara. Within a few weeks Masachika, whom Aritomo came to depend on even though he could not forgive him for Takaakira's death and some days could hardly bear to look at him, finally captured Kuromori and went on to take Kumayama. The east was once more secure.
Hoping to placate the vengeful spirit of the Prince Abbot, Aritomo gave orders for Ryusonji to be rebuilt, exactly as before, and for the dragon child to be worshipped there, yet the construction progressed slowly. After a series of inexplicable accidents, the carpenters refused to work, saying the place was occupied by ghosts and untethered spirits whom no one could control now that their master was gone.
The Imperial Palace, which had burned to the ground in the Ninpei rebellion, was also being rebuilt. In the meantime the Emperor, Daigen, and his mother were living in a nearby temple. The treasures that had been destroyed were slowly being replaced, but expenses were high and even Aritomo's new taxation system could not produce enough revenue. It was his custom to visit Daigen weekly, to take part in the rituals that bound Heaven and Earth through the sacred person of the Emperor. Daigen had been the Prince Abbot's choice and Aritomo could not fault him. He was intelligent, courteous, and, most important, biddable, seemingly resigned to his role as a figurehead and happy to play it in return for beautiful companions, wine, and poetry. There was no reason for harmony not to be restored, but, as the years passed, the drought worsened; rain hardly fell, the lake shrank and the river dried up.
Aritomo tried to wipe Takaakira's dying words from his mind: Yoshimori is the true emperor. Yet they haunted his dreams and he often woke suddenly in the night hearing a ghostly voice speak them in the empty room.
On one of his visits the Emperor's mother sent a message through a courtier that she wanted to speak to him. He had to obey, yet he went with reluctance, fearing she was going to grumble about their living conditions or demand some new luxury for which he would have to find the money.
Lady Natsue received him alone. He prostrated himself before her, as was required, feeling a twinge of pain in his hips, regretting his sedentary life, longing for a horse beneath him, a hawk above, the brisk air and huge skies of the east.
It was a warm spring day and water trickled through the garden. The room was not unpleasant; it faced south and was elegantly appointed with flowing silk hangings and a few exquisite pieces of lacquered furniture. He could not see what she had to complain about.
"Please sit up, lord," Lady Natsue said.
He dared to look directly at her. She had been the late emperor's second wife, always, it was whispered, jealous of the first, Momozono's mother. Yet surely no one could have surpassed her in beauty. Even now, in apparent middle age, she seemed perfect, still youthful. She spoke at length about the joys of the season and the various flowers and birds of the garden, then told an amusing anecdote about a court lady and a mouse, which His Majesty had turned into a poem. When she fell silent he said, half-irritated, half-charmed, "What can I do for Your Majesty?"
"I need to speak to you about Ryusonji." She gestured that he should come nearer. The tone of her voice changed though it was no less attractive. "My late brother and I were very close," she whispered. "He shared many of his secrets with me. Under his rule Ryusonji became a place of great power. Now he is gone it lies empty; its power leaks from it."
"I am trying to rebuild it," Aritomo replied. "But the work is proving difficult and slow. No one can replace the Prince Abbot."
"I had heard about the problems. Last week I went to see for myself. I wanted to prepare for the anniversary of my brother's death, pay my respects and mourn him. Women, as you know, are not usually permitted to enter into spiritual mysteries, yet my brother recognized that in me dwelled an ancient soul that had acquired great wisdom. He often sought my advice and he promised me that if he died before me he would attempt to reach me from the other world. When I knelt before the half-completed altar I felt him call to me. He wants me to move into Ryusonji. I will be able to ensure that the repairs progress smoothly and the various disruptive spirits are appeased. My son will come with me. It is fitting that the Emperor should be in the spiritual heart of his capital."
"I am not at all sure that it is fitting," Aritomo said, wanting to speak with his customary frankness yet fearing to offend her. "How shall I put it? The events that took place there, the deaths, the dark forces unleashed..."
"I can handle any darkness at Ryusonji. The shadows are a source of power just as much as the light. When the new palace is finished maybe my son can live there. But it is imperative that I move quickly, for someone else is about to take possession of the temple."
"What do you mean?" Ever since he had been told of the details of the confrontation between the Prince Abbot and Shikanoko, he had had nightmares in which a masked half-human figure confronted him in judgment. Now the Empress's words summoned up that image. He feared it was what he would find at Ryusonji. Yet Shikanoko was surely far away, in the Darkwood.
"An old man is there, camped out in one of the cloisters. He plays the lute and sings. I was told he was harmless, wandering in his mind, but his presence seemed offensive so I ordered him to be removed. However, no matter how many times he was thrown out, he always returned. Finally the guards lost patience with him and beat him to death, they thought, but the next day they heard the lute and his voice—he was back in the cloister. Now no one dares approach him. I believe I know what he is doing there. He has obtained the Book of the Future and means to erase my son's name and inscribe that of Yoshimori."
Neither of them spoke for a moment. The trickling of the stream seemed suddenly louder and birds called from the garden.
"Who is he?" Aritomo whispered.
"The monks who survived told me he is Sesshin, once many years ago a fellow student of my brother. He became a great master who gave his power away to the evil man they call Shikanoko."
"Gave his power away?" His skin was crawling. He had heard of Sesshin before, some connection with Matsutani and Masachika. And then he remembered, and the terrible day Takaakira died came back to him.
"So he could pass as a foolish old man," Lady Natsue explained. "But little by little he is gathering knowledge again. He has all the time in the world since he has made himself immortal."
He stared at her in disbelief, wondering if he had misheard.
She repeated the word, "Immortal."
"What is his secret?" Aritomo said hoarsely.
"That interests you, Lord Aritomo?" Her gaze pierced him. "Would you steal it from him? Would you wish never to die?"
"I want more time," he replied. "I don't want to die before I have achieved all I strive for."
"None of us can know the hour of our death," she said, her eyes not leaving his face. "The water from the well at Ryusonji is reputed to prolong life. My brother and I have both drunk from it. I am much older than you think, but I am still as mortal as my brother proved to be."
The wind had risen and leaves rustled from outside, a branch scraping against the roof. A crow called harshly as if it were sitting directly above them. He felt parched, almost feverish. Surely it was hotter than it should be?
"Lord Aritomo," she said. "Are you unwell?"
"No!" he replied, his voice suddenly loud. He was never sick; he denied illness access to his body. Even battle blows glanced off him, hardly leaving a wound. But the idea of an immortal at Ryusonji, slowly rewriting the Book of the Future, had struck deeply inside him. He struggled to regain calm.
"I will inspect Ryusonji myself," he said. "If I consider it suitable you and His Imperial Majesty may move there."
"Let us not waste any more time." Lady Natsue inclined her head graciously.
* * *
When Aritomo returned to his own palace, the one abandoned by Lord Keita when the Kakizuki fled from the capital, he sent for Masachika, who, he knew, had just come back from Minatogura. It was not long before the Matsutani lord was kneeling in front of him, apparently in perfect submission. Aritomo studied him for a few moments. Masachika was undeniably a handsome man, and he had gained great popularity and respect since the discovery and capture of the Autumn Princess, but Aritomo thought he could read his deeper character clearly, seeing how opportunistic and self-serving all his actions and words were. He did not trust his loyalty, yet, though he did not like admitting it, Masachika had made himself indispensable.
First he told Masachika of the Empress's request and asked him to inspect the temple and make all necessary preparations.
"I will come with you. I have not visited Ryusonji myself since the Prince Abbot died. But what news do you bring from the east? I hope you have sorted out your personal life."
Masachika smiled, a little embarrassed. "I finally convinced Keisaku and his daughter that I was never going to marry her. I could have taken her as a second wife, but I did not want to distress Lady Tama, after all she has suffered. I found a suitable husband for the young woman, and released Keisaku from all obligations to me. They will hold Keisaku's estate in vassalage to you, which protects Minatogura from the north. It seemed an acceptable solution all around, provided Lord Aritomo agrees, of course."
"It will be good to have someone loyal in between the port and the Snow Country. I had hoped Takauji would be removed. I cannot trust him not to challenge me sooner or later. But I hear the cousins failed in their efforts to get rid of him?"
"Yes, and they are all dead now. The mother arranged an archery contest. An unknown archer, who she claimed was the deer god, came out of the forest to win it, and the challengers were all killed. She said it was the judgment of the forest. Takauji is, unfortunately, more secure than ever."
When Aritomo made no response Masachika said, "He is the son of the man who betrayed you. You cannot trust him."
"I am fully aware of that," Aritomo snapped, enraged that Masachika should speak so of Takaakira, who had been so superior to him in every way. Yet he knew he was right. Unless he was removed, Takauji would be a continuing threat. "I cannot deal with him now," he said, more calmly. "First we must destroy the Kakizuki. Did you find out the identity of this so-called deer god?"
Masachika said, "All the evidence—the antlered mask, the skill with the bow, the fake wolflike creature—suggests it was Shikanoko."
Aritomo kept his face still, his expression impassive, yet a kind of dread was welling up in him.
Masachika went on. "By the time my men investigated he had disappeared again. The archery contest took place weeks ago. Shikanoko could be anywhere by now. Takauji was extremely hostile and my men were lucky to return alive. Unlike those I sent immediately after the disaster at Ryusonji, who never came back. Remember, we are not dealing with an ordinary fugitive but with a sorcerer."
"Is it his power that makes the rain dry up? How do I combat that? I don't mind facing a thousand men on a battlefield, but this one sorcerer keeps evading me."
"Shikanoko has no men, no army," Masachika said. "All were destroyed at Kumayama. If he had been going to challenge you he would have done it immediately after the death of the Prince Abbot. I don't think he will ever emerge from the Darkwood. If you don't provoke a snake it will not bite you."
"Is he completely alone?"
"He has a few companions, I believe: the ones they call the Burnt Twins, one is a former monk from Ryusonji, the other from Kumayama, and one other whose name and identity I have not been able to discover."
"So some survived from Kumayama?"
"These were already with him. But there are always some survivors. Some hide, some run away, some are left for dead but recover from their wounds."
"I will never eradicate all my enemies," Aritomo said.
Masachika nodded in sympathy. "But we will do our best to control and weaken them. I did find out something else, probably not very important. One of the women left at Kumayama told me. Shikanoko's mother became a nun, after her husband, Shigetomo, died. Apparently she is still alive and is in a convent a little way from Aomizu, on Lake Kasumi."
"What would she know about anything? It must be years since she forsook the world."
"As I said, it's not likely to be important."
"Well, follow it up anyway," Aritomo said. "Arinori is in Aomizu now. He can look into it. There's no need to go yourself. Write a message."
Arinori had served him for years and had been rewarded with Lake Kasumi and the surrounding districts. He was an experienced seaman, ambitious and determined. Aritomo trusted him far more than Masachika, though he had to admit the latter was considerably more intelligent.
* * *
The next day they rode in an ox carriage to Ryusonji. Both had dressed carefully and soberly in formal clothes, each with a small black hat on his head. A large retinue followed on horseback. Aritomo traveled frequently around the capital, inspecting new buildings and repairs, overseeing merchants and craftsmen, keeping an eye out for excesses of luxury and extravagance that would attract new taxes to pay for horses, armor, and weapons.
People dropped to their knees as he went by, but he inspired fear, not love. The city ran smoothly, his officials keeping every section meticulously administered, but neither he nor they could make the rains fall at their appointed time or save the crops when they failed.
The lake at Ryusonji had shrunk; the exposed bed was muddy and foul-smelling. A charred spiral of black across rocks and moss still showed where the burst of flame had scorched the ground and set fire to the buildings. Most of the blackened beams had been removed and new lumber was stacked in the courtyards. There seemed to be some desultory activity, workmen sawing planks and preparing floors, but it was still far from finished.
"The Empress wants to move here as soon as possible," he said to Masachika as they descended from the carriage. "See if there is anywhere suitable for temporary lodging. If she is to be believed, her presence here will speed the completion."
Masachika went to speak to the head carpenter. Aritomo waited in the shade of the cloister, trying to sharpen all his senses, to discern what was really going on at Ryusonji.
The sound of a lute came to him, its mournful, plangent notes turning his spine cold. Masachika came back, saying enthusiastically, "This hall is nearly finished. It could be ready before the end of the month. I will start arranging furnishings and servants."
"They will need many rooms," Aritomo said. "And priests, guards, and so on. What happened to all the priests and monks who were here before?"
"Some died in the fire, I believe. The rest must have run away."
"Well, the Empress will bring her own, no doubt. Consult the steward of the Imperial Household."
Masachika inclined his head. "I will, lord."
If he resented being given this tedious, if prestigious, responsibility, he gave no sign of it. Aritomo knew he could rely on him, that Masachika would complete the task as swiftly and efficiently as he did everything. Yet, no matter how competent Masachika was, Aritomo would never warm to him.
The notes of the lute trickled through the air as if they were summoning him.
"Let us inspect the other courtyards," he said.
The sun beat down on the blindingly white stones, making his head ache. The new moss was an unnaturally brilliant green. The shadows under the cloisters were deep black.
The lute player sat cross-legged, the lute in his lap, his face turned to them as if he had been waiting for them.
Aritomo saw the hollow eye sockets, the shriveled lids.
Masachika exclaimed, "It is Master Sesshin!"
"The one whom your wife had blinded?"
"Yes, it was his eyes that I replaced at Matsutani and so subdued the guardian spirits."
"I remember," Aritomo said coldly. He could not take his eyes off the old man. So this was what immortality looked like! When the physical body could not die, did it simply mean endless pain and suffering? The deformed frailty before him tempted him briefly, savagely. He had often seen how under torture life persisted longer than he would have believed possible, but eventually it was extinguished. The Empress had told him the old man could not be killed and he wanted to put her claim to the test.
"Your lordship should not concern himself with the old lute player." The head carpenter had followed them into the courtyard and stood beside them, regarding Sesshin with an indulgent smile. "He is our talisman, aren't you, grandfather?"
He spoke in a loud voice and Sesshin nodded and smiled with senile glee.
"As long as he is left alone to play and sing, our work progresses. I bring him some food every day, not that he takes more than a mouthful. Since he returned there have been no accidents, no fires. The men say the dragon child must like his songs."
"Does he have any books?" Aritomo said, remembering what the Empress had said.
The workman shook his head. "I don't think so. What use would he have for books since he cannot see?"
Aritomo leaned over Sesshin and said loudly, "Where is the Book of the Future?"
"I will show it to you, one day," Sesshin replied, his voice low and rational. "And you will see whose name is written in it. There is no need to shout. I am not deaf."
Then he took up the lute and began to pluck the strings with his long fingernails.
"That's the way," the head carpenter said approvingly. "Keep playing for us!"
"I will come back and talk to him again," Aritomo said. There was so much he did not understand, it was unsettling him. The feeble old man who was somehow immortal, the Book of the Future: none of it made sense. And it was too hot, the wind was too dry. He longed for the gray skies and constant drizzle of the plum rains. He decided he would make sure the old man stayed here so he knew where he was, and he would come to question him, alone, and discover his secrets.
HINA (YAYOI)
Yayoi grew taller and, in her third spring at the temple, her body began to change. Her breasts swelled and she bled monthly, like all the women, attuned to the cycle of the moon. She had seen first Yuri, and then Asagao, become women in the same way—indeed, Asagao made sure there was not a single detail in the process that Yayoi did not know about—so she was not shocked or frightened as young girls sometimes were. Mostly she accepted womanhood—what else could she do?—but she also grieved for that girl child, single-minded and courageous, now gone forever.
Yuri left the temple one spring, making the journey on her own, the palanquin waiting at the foot of the steps. Sada and Sen, the two sisters, wept for days. Asagao was now the oldest girl. She did not sing them to sleep as Yuri had. She teased them for their red eyes and, when no one else was around, she bullied them. She left the following year. Lady Fuji herself came for her, bringing another young girl, whom she entrusted to Yayoi's care.
"You, yourself, will be ready soon," she said with her familiar, appraising look.
The girls missed Asagao, her high spirits, her teasing ways. Even Sada and Sen wept for her, while Yayoi felt bereft, as though she had lost a limb.
In the sixth month of that year, when they had given up hoping for the plum rains, men came on horseback, demanding to speak to the Abbess. Their leader was obviously a warrior of high rank, though the nuns did not know him. He wore a crest of a sail above waves and the white stars of the Miboshi.
Men's voices, their heavy tread, their sweaty smell, were so unusual, it threw the girls and the nuns, even the ginger cat, into a state of anxiety and agitation. The men spoke politely, yet there was an undertone of menace. The lord made it clear that the women's temple survived only on his sufferance. One word from him and it would all be destroyed.
The Reverend Nun tried to hide the girls, but the men demanded to see every person living in the place. She rapidly tore off the robes they usually wore and made them put on nuns' habits and servants' clothes, rubbed dirt and ash on their faces and hands. They had never seen her so distressed and it alarmed them into silence.
Yayoi was terrified one of the men might be Masachika, her uncle. Would she even recognize the man who had killed Saburo in front of her eyes? Would he know her, after all these years? But all the men seemed strangers to her and, though they studied her intently, she did not think any of them knew her.
They did not lay hands on the girls, but they touched the older women, patting their breasts and feeling between their thighs, and one nun, taller than the others and somewhat masculine in look, was forced to disrobe to prove she was not a man.
Since the men hardly spoke, no one knew who or what they were looking for. They searched every corner of the temple, from the chests that held the sacred sutras to the woodpile stacked on the southern wall.
"Why did they spend so long at the fishpond?" Yayoi asked after they had left.
"People hide underwater," the Reverend Nun replied, "and breathe through hollow reeds—or so I have heard."
She helped the girls clean themselves, her hands shaking. "I suppose we were lucky. None of you has been harmed and nothing has been damaged that can't be mended."
Torn manuscripts, a broken statue, ripped hangings, a shattered ceiling, doors forced out of their tracks—the tall nun, weeping silently, set herself to repairing them.
Yayoi, now the oldest, undertook the task of calming and consoling the other girls. They all, like her, had fears from the past that had been reawakened by the armed men. She took out the lute and forced it to play soothing music, wondering where Yoshi was and if she and Genzo would ever be in his presence again. Several sutras needed recopying; Sada had quite a gift for writing, and she and Yayoi worked on them together, Yayoi reading them aloud.
When they were finished, she took them to the Abbess for her approval, kneeling quietly while the older woman read through them, reciting each syllable under her breath. Yayoi felt that peace was being restored with every precious word. Even the cat was calmed; she could hear its loud rhythmic purring.
The Abbess said suddenly, "They were looking for a man whom they called Shikanoko."
Yayoi felt shock run tingling through her limbs. Her heart thumped.
"I did not understand why they thought such a man, an enemy of Lord Aritomo, a warrior and a sorcerer, would be here at our insignificant temple. They told me they had recently discovered he was my son, and therefore there was every reason to suspect I was hiding him."
Yayoi said nothing, silenced by astonishment.
"My son, whom I have not seen since he was a young child, whom I believed to be dead. Not a day has passed that I have not prayed for him, but all my prayers, it seems, have gone unanswered. They say he has become a monster. He murdered his uncle through dark magic and then destroyed the Prince Abbot and the temple at Ryusonji. Now, these men say, Lord Aritomo believes it is his evil power that has cursed the realm and caused the rain to cease and the rivers to dry up. I told them if I knew where he was, I would have handed him over to them. But I do not know."
She stopped speaking and stood abruptly. The cat, alarmed, ran out of the room.
"And now our temple has come to the attention of Lord Aritomo. He sent Arinori to investigate me. I could tell Lord Arinori was shocked that we had no priests overseeing us. He asked how we ran our affairs and I was forced to reveal our benefactress, Lady Fuji. She will not thank me for that. No doubt she will come under his scrutiny next. They also inquired to whom we paid tax and I had to admit that we pay no tax to anyone." She went to the door and gazed out on the garden. "I cannot believe he has grown up to be so evil, but he had a fierce temper even as a baby. He screamed and bit my breasts. My women said that meant he would be a powerful warrior. But how did he fall into sorcery?"
Yayoi longed to reveal all she knew of Shikanoko, how he had served her father for a while, his kindness to her and her brother, the way he had cared for Sesshin after the old man had been blinded. She remembered the day he had arrived at Matsutani, riding Risu, the bad-tempered brown mare, and how he had been able to shoot down the werehawk that no one else could. But she did not dare open her mouth. Waves of emotion swept over her, so violent she feared she would faint. What is this? What is wrong with me?
"I must renew my prayers," the Abbess was saying. "I will fast for the next week, while I endeavor to restore tranquillity to myself and my temple. Let no one disturb me."
Through the next few weeks, Yayoi became aware that what had been a childish fantasy about Shikanoko—that she would grow up to marry him, cherish him, make him smile, wipe the sorrow from his heart—had become a true adult emotion. The knowledge sustained her. It was a secret as precious as a sutra to be carried in her heart, binding her spirit to his, in the same way the sacred words bound the earthly and the divine.
Lady Fuji came unexpectedly at the beginning of autumn. The typhoons had brought some rain, but less than usual. Summer crops had been sparse, and the shortage of food and the approach of winter had caused unrest among the farmers. It was being harshly suppressed, Lady Fuji told them, and Miboshi warriors were everywhere.
"That is why I need you, Yayoi, my dear," she said, with a sigh. "It is earlier than I planned, but only by six months or so. And look at you, you are ready. Something has awakened you."
The other girls wept bitterly and Yayoi could not prevent tears forming in her own eyes. She went to bid farewell to the Abbess and received back the text, the Kudzu Vine Treasure Store, that Master Sesshin had given her all those years ago.
"Let us read it together one last time," the older woman said. "Really it is an honor that it has dwelled with us all these years. It has blessed us and so have you."
Yayoi let the pages fall open where they willed.
The Abbess was studying her face. "What has it shown you today?"
"It tells of a stone that reveals sickness," Yayoi said, deciphering the faded gold letters on the indigo dyed page. "Here is a picture—but now it is gone again!"
It had given her a tantalizing glimpse: a surface of perfect smoothness, a dark mirror that had allowed her just one brief glance into its depths.
"Ah, I wonder," the Abbess murmured.
"What is it?" Yayoi asked.
"If you were staying here I would tell you, but there is no point now." The Abbess could not hide her distress. "How I will miss you! I deeply regret the path you are being forced to follow."
"I still don't understand the truth of the world and why there is so much pain!" Yayoi could feel tears threatening. She did not want to leave; she wanted to stay and learn more about the mysterious stone, but she knew the peace and seclusion of the temple were no more than an illusion. Ever since the Miboshi warriors had come to search for Shikanoko, none of the women had felt entirely safe.
"I will always pray for you," the Abbess continued.
"And I for you," Yayoi replied.
"Use what you find in the text only for healing. Do not follow my son into sorcery."
Yayoi bowed without speaking. I would follow him even into the realms of hell, she thought.
Fuji had brought a second palanquin with her. When Yayoi had arrived, she had been a child, small for her age. Now that she had reached her adult height she would no longer fit inside a palanquin with the other woman. But she also felt Fuji was glad to keep her distance, that she was not entirely comfortable with whatever plan she had for her. She did not look Yayoi in the eye or embrace her spontaneously as she had when she had come to collect Asagao. Alone in the palanquin, Yayoi had plenty of time to imagine what might be going to happen to her and to dread it.
It was dusk when they arrived at Aomizu and the boats on the dock were bright with red lanterns, the lights reflected in the still waters like a host of fireflies. The moon was a crescent in the sky, waxing toward its ninth-month fullness. Music was playing and Yayoi could hear singing. She thought she recognized Asagao's voice and was slightly comforted. She longed to see Take, and then thought of the boy called Yoshi, and felt the lute in her lap begin to stir.
One of the women whispered to Fuji as they stepped onto the boat. "He is here."
"What, already?" Fuji exclaimed, biting her lip. "We hardly have time to prepare her. Quick, bring some water. Let me wash her feet at least. How impatient these Miboshi are," she muttered as she brushed dust from Yayoi's robes and combed her hair until it fell silky and tangle-free down her back.
"Yayoi," she said, her voice serious and cold. "You are a clever girl, they tell me. You know what is expected of you. Charm this warrior, do whatever he wants, please him in every way. My future, the future of all of us on the boats, depends on it."
She led Yayoi to the stern of the boat where the bamboo blinds surrounded the largest of the separate spaces. She dropped to her knees, raised the blind on one side, and, bowing to the ground, said, "My lord, I have brought the one you requested."
Yayoi found herself on her knees shuffling forward. The blind unrolled behind her with a slight rustle. The lanterns outside threw wavering shadows as the boat rocked slightly on the water of the lake.
A lamp burned in one corner, the scent of oil strong. By its light she saw him sitting cross-legged, a flask of wine and a bowl beside him. It was the lord, Arinori, who had come to the temple.
"Don't be afraid," he said. "Here, drink a little wine, it will relax you."
She took the bowl and sipped, the liquor flowing like fire into her throat and stomach. It did not relax her but had the opposite effect, making her heart pound with fear. Her whole being recoiled from the idea that this stranger, enemy even, should have intimate access to her body. How is it possible, she thought, that men have such power over women? That even Fuji was complicit in this transaction, which probably bought her privileges, which was pleasing to all parties except Yayoi herself, without whom the transaction would never have taken place?
"They could not really turn such a little pearl into a serving girl," he said, coming closer, putting his hand on the back of her neck, feeling her hair, pulling her face to his. His other hand was inside her robe, caressing her breast, and then reaching farther down, forcing her thighs apart.
He had the hard body, the iron muscles, of a warrior. He was nearly twice her size. When he thrust into her it was like being knifed. She could not help crying out from pain and fear. It excited him and she felt the gush of his release, alien in its smell and wetness.
Afterward he was kind to her, in a way. He stroked her hair and called her his little princess. He held the wine bowl close to her lips so she could drink, and kissed the tears from her eyes.
He wanted her to know he was a prize, how lucky she was to have attracted him. He was rich and powerful, he would always take care of her. Lord Aritomo himself had named him, he boasted. They were as close as brothers. That week he came every day, expressing his pleasure with gifts of silk robes, casks of wine, and the fine-quality rice that was otherwise almost impossible to obtain. Fuji was delighted and showered Yayoi with compliments and affection, any remorse she might have felt dispelled by the success of the transaction.
"Lord Arinori has become our protector. It is just as I hoped. But, Yayoi, you look so thin, you must not lose weight. Eat, eat, it's the sweetest rice we've had all year. Don't fret, don't dwell on what might have been. This is your life now and, while it may not be what your parents might have hoped for you or what you would have chosen, it is better than being dead. Enjoy it, strive to please Lord Arinori, and one day he may even buy you from me for his own."
Yayoi could not imagine anything worse. When Arinori's duties took him back to the capital, she began to choose men from the visitors that came aboard at Majima, Kasumiguchi, Kitakami, and the other market towns around the lake. She learned their desires and their foibles, their needs and their strengths. Some she liked more than others, some became almost friends. But not one of them ever knew that, when she embraced him and gave herself to him, in her heart it was not him she was holding but the wild boy who had ridden into her father's lands on a brown mare all those years ago.
MU
The fox girl laughed when Mu asked her name, as she laughed at most of his questions.
"You can call me Shida, if I have to have a name," she said, tickling him with a dried fern. She liked to drape herself in the leaves and flowers of the forest as they came into season, making garlands from strands of berries and the brown fronds of bracken for herself and Kaze, and for the child, when it came.
They had been together for four winters. The child was a girl, as slight and delicate as a fern leaf herself. They called her Kinpoge, after the celandines that starred the forest floor around the time of her birth. She did not seem to have any fox attributes, apart from her animal-like agility and her rapid growth, though in certain lights her thick hair had a russet gleam to it and her eyes were amber.
During those years, Shida taught Mu the bright playful magic of the fox people. It seemed to have no purpose other than to make life amusing. She cast a spell on Ban's skull, to Kaze's utter delight, that sent the horse flying through the air. She summoned up shape shifters, tanuki, cranes, turtles, and snakes, just for the fun of startling Ima or Ku. They never knew if an iron pot was really a pot or a grinning fat-bellied tanuki, or if an old robe, thrown on the ground, might not suddenly sprout wings and launch itself, squawking, into the air.
Her presence seemed to revive many of Shisoku's fake animals. One of the creatures, a cross between a dog and a wolf, which had been Shisoku's water carrier, raised its head from where it had fallen by the stream and Shida ran to help it to its feet, chuckling at its awkward gait.
"You should make a companion for it," she said, when it tried to lift the water vessel on its own and the water spilled out lopsidedly.
Mu had to admit he did not know how to and that all he did not know overwhelmed him. The forest and the mountain were home to thousands of plants, flowers, trees, and grasses, and myriad creatures, insects, birds, and small animals as well as deer, monkeys, foxes, bears, and wolves. There was no one who could teach him their names. Shida did not understand his need to label them. They were all instantly recognizable to her, she did not need words. Mu realized she lived like an animal, in each single moment, observing, feeling, enjoying, but not reflecting or recording. Sometimes he felt himself slipping into the same way of being, and days would pass when he hardly had a single thought. Then he would wake in the night from a bad dream in which a stranger who was at the same time familiar accused him of wasting his life. He would lie there in the dark, hearing the others breathing around him, alarmed and uneasy at what he was leaving undone, yet ignorant of what it might be.
He became preoccupied and silent. Shida accused him of being gloomy and turning into an old man before his time. There were no mirrors to look in; he could not check his appearance, yet he felt she was right. He was aging rapidly. He could see the same thing happening to Ima and Ku. Kaze grew like a human child, but the brothers seemed fated to have lives as short as insects'.
And as pointless, he thought.
Even before Kiku returned, Mu and his fox wife quarreled. Her playful magic no longer enchanted but irritated him. She began to spend time away, with her own people. He missed her with a kind of agony, but was angry with her when she came back.
And then one day Kiku rode into the clearing with Chika and an older man, a warrior with missing fingers and one eye.
They had an air of prosperity about them; their cheeks were fat and their hair sleek. The horses were sturdy, with bright eyes and round haunches. Mu saw the hut and the clearing through his brother's eyes and felt ashamed.
Kiku made no comment, hardly even greeted his brothers, but said as he dismounted, "I have come for the skull."
"Do you remember where we buried it?" Mu said, thinking of the day when its owner, the monk, Gessho, had died.
"Oh, yes!" Kiku said.
"What will you do with it?" Mu asked.
Kiku cast a look at Ban, the horse skull he had tried to infuse with power, which stood on its post, motionless. "I know what I'm doing now. Our father, Akuzenji, had other sorcerers in his service. Tsunetomo took me to one who was familiar with these old matters. He taught me what I have to do. I'm going to try again."
"Ban can fly," Kaze told him. "Shida did it."
Kiku turned his gaze on Shida, who sat by the fire staring at him with frank interest. "You can do magic, Lady Shida?"
"A bit of this, a bit of that," she said carelessly. "Why do you call me Lady?"
"You're a beautiful woman. Are you my brother's wife?"
"No!" she said, laughing, even as Mu said, "Yes!"
Kiku said, "I need a beautiful woman to infuse the skull with power."
"Find your own wife," Mu said.
"I have come for my wife," Kiku replied. "I am going to take Kaze as my wife, sister of my dearest friend. Isn't that right, Chika?"
Chika nodded without speaking. Mu noticed he stayed by Kiku's side, as close as a dog.
Kiku went on, "But the ritual demands something different, some other woman, one more like our mother."
"If you touch her, I will kill you," Mu said.
"Don't be silly," said Shida. "You can't dictate what I can and can't do. I don't belong to you. If he wants me to join him in some magic, where's the harm in that?"
Kiku said to Chika, "Go and dig up the skull. Ima will show you where it is. Then boil a pot of water, Ima. We will clean it tonight, and tomorrow we will start the ritual."
"No!" Mu cried and leaped at his brother, not knowing what he intended to do, driven only by fear and frustration. But his arms were seized by the warrior, who up to this time had not spoken and who moved faster than Mu would have thought possible. He was strong, too, and held Mu with no effort. Mu struggled to use the second self, to turn invisible, but it was so long since he had used either that he was not quick enough.
"What shall I do with him?" the warrior asked.
"I don't know," Kiku said impatiently.
"Do you want me to kill him?"
"No, not really. I don't want him interfering or distracting me."
"He won't distract you if he's dead," the warrior said with a laugh, and tightened his grip on Mu's neck, as if he would break it with his bare hands.
"But it would be an inauspicious start to the rituals," Chika said. "Take him to the other side of the stream, Tsunetomo, and tie him up there."
Tsunetomo picked Mu up and strode across the stream. Then, despite Mu's struggles, he trussed him like a goose, in a kneeling position, his hands tied to his feet behind his back. His knots were expert; there was no way Mu could wriggle out of the ropes. After an hour his joints and muscles had set up a scream of pain that dulled his hearing and his senses to everything else.
As night came, Ku brought water and stayed beside him, helping him to drink, whimpering like a dog.
"Untie me," Mu begged, unable to keep himself from whimpering, too.
"They say they will kill you if I do."
"I would rather be dead, for then I would not feel."
He could hear all night long the hiss and bubble of the water that was boiling the skull clean.
* * *
The rituals lasted for several days, during which time, as far as Mu could tell, Kiku and Shida were alone in the hut, with the skull. Every now and then he heard the others talking, smelled the food Ima and Kaze prepared—he himself refused to eat—and saw the horses come to the stream to drink. The sight of him alarmed them, as if they could not determine what he was, and they gazed at him with huge eyes and pricked ears. Often, a silence descended on the clearing, a sudden hush as if the whole forest held its breath in awe at what was taking place within the hut, the transformations that were occurring. Nothing, no one could help being affected by it. When sounds and voices returned, they were solemn and muted. Under Mu's tormented eyes, the hut seemed to glow with light, transformed into an enchanted palace in a garden of wisteria.
Kiku finally emerged and held the skull aloft. It was lacquered now and gleamed black and red with brilliant green jeweled eyes and cinnabar lips, the teeth inlaid with mother-of-pearl. Mu saw it clearly, because his brother brought it to the side of the stream to wave it in his face.
It was midday. The sun sparkled on the water, on the wet stones, on the mother-of-pearl.
They began to make preparations to leave. Mu heard Kiku's voice.
"Ku and Ima, you will come with me. From now on we must all be together. I have taken everything I need from this place, its treasure, its knowledge."
"What about Mu?" Ima said.
"He can stay here and become like Shisoku," Kiku replied.
"I'm not leaving the animals," Ku said stubbornly.
Kiku turned the skull toward him and went closer. "I am your older brother and you will obey me."
Ku tried to take a step back, but Kiku lowered the skull, grasped his arm, and forced him to stare into his eyes. Mu could not see what happened, but within seconds Ku had slumped to the ground.
"Pick him up and put him on your horse," Kiku ordered Tsunetomo, and the warrior obeyed, screwing up his face in a sort of unwilling admiration.
"Now you," Kiku addressed Ima, but Ima shook his head.
"I need to look after things here. I'll stay with Mu and Kinpoge."
Mu saw Kiku repeat the same process, and stare intently into Ima's eyes. But Ima stared back. Whatever power the skull had given Kiku did not work on him.
Kiku's eyes flashed with anger and for a moment Mu feared Tsunetomo would be ordered to kill them both. The warrior had his hand on his sword, as eager as a hunting dog.
Kiku turned to the horse, which Chika held ready for him. As he mounted, and Chika lifted Kaze up behind him, he said, "Stay, then. You can untie Mu now."
Mu screamed as the blood flowed back into his cramped legs. It was a long time before he could stand. When he finally managed it, on ankles that kept bending the wrong way beneath his weight, the clearing was empty, apart from Ima, and Kinpoge, who flitted around him like some ghostly spirit.
"Where is your mother?" he said to her, and her amber eyes filled with tears.
"She is in the hut," Ima said awkwardly. "I have tried to rouse her but..."
Mu hobbled slowly toward it, seeing it clearly in the afternoon light. The magic had all fled. It looked as dilapidated as usual, the roof sagging, the walls subsiding, revealing no trace of what had happened within it. He slid the door open and stepped in, his eyes adjusting to the dimness.
Shida lay on the ground, half-naked still, her legs apart, her arms above her head. He thought for a moment that she was dead, but when he knelt beside her she stirred and said something he did not understand. He took her in his arms, helpless to avoid his own clumsiness and the pain it caused them both, but even as he held her he knew her shape was changing.
The fox snarled and snapped at him. He embraced all its loveliness for one last time and then he released his grip and the creature ran from the hut and from his life.
He crawled out after her and saw her disappear into the forest, her tail burning as if with foxfire. He collapsed on the threshold.
Kinpoge knelt beside him, her hand like ice on his brow.
Ima brought rags steeped in hot water to bind the joints, and rocks from the fire to lay against his muscles. The heat seemed to enter his entire body and he burned like the fire itself, until it fled from him, like steam evaporating, and a chill followed that made him shake and shiver uncontrollably. When the fever subsided, it left him weak and in despair. At first he did not recognize this new emotion, but then he realized it was grief, and he knew that he had loved her and love had changed him.
* * *
For a long time, he limped like a cripple and had to use a cane. Winter came early, with the first snow in the eleventh month. It drifted waist-deep around the hut and continued to fall for the next two months. If Ima had not been there, Mu and his daughter would have starved, or frozen to death, drifting into their long sleep without noticing. But Ima kept them both alive, going out daily to track hares or rabbits, occasionally bringing down, with an arrow, a squirrel or a pheasant, once even a serow, whose skin made a warm cape for Kinpoge. In some ways, hunting was easier, for against the white snow there was nowhere to hide and neither humans nor animals could conceal their tracks.
Ima kept the fire going, too, coming home with armfuls of dead wood, and cooked meals, roasting the tender joints, stewing the rest. Maybe it was the tempting smell of food, or Ima's tracks clear in the snow, that showed the tengu where they were.
The tengu came in the late afternoon, when a blood-red sun was sinking rapidly behind the western mountains, and it was already freezing. The light from the fire and from the lamps in the hut looked tiny against the great snowy mass that the Darkwood had become.
Kinpoge must have seen him first, lurking in the shadows just beyond the firelight, for she cried out and jumped into Ima's arms. Mu looked into the darkness and saw two red eyes glaring at them. He felt for the knife that Ima had been using to cut up the rabbit that would be their supper. Ima slid Kinpoge off his lap, pushed her behind him, and reached for his bow.
The tengu was dressed in bright blue leggings and a short red jacket. He had a long, beaklike nose, and when he sat down opposite Mu, he made a curious shrugging movement to adjust something feathery between his shoulders, which, at first, Mu thought was a dead bird and then realized was wings. The wings were grayish white and shaggy, almost indistinguishable from the tengu's thick shock of hair. He pulled out a sword and a bow, which he had been carrying on his back between the wings, and laid them down, the sword on his left, the bow on his right.
He gave Mu a long, penetrating look and said, "We would really like to know what's going on. And that rabbit looks good. Give me a piece. I love rabbit."
He reached out to the embers. He had only three fingers and a thumb.
Ima said, "It's not cooked yet."
"I don't mind," the tengu said, and crammed the half-raw rabbit's leg into his wide mouth.
"He's your brother, isn't he?" he said indistinctly.
"Who?" Mu said, thinking for a moment he meant Ima.
"So-called Master Kikuta, who claims to be Akuzenji's son and the new King of the Mountain."
"Kiku?" Mu said, with pain, after a long pause.
"Is that his name? Kiku meaning the flower, or Kiku meaning 'listen'?"
"Listen, I think." Mu had never heard of the flower.
"Well, he writes it like the flower these days, with a fancy crest to go with it, a crest that now appears on the robes of fifty warriors and is stamped on tons of goods going between Kitakami and the east, by road and by sea."
He had said all this while chewing vigorously, blood and grease running down his chin. Now he swallowed and reached for another piece.
"You know more than we do," Mu said. "Why have you come to ask us? We can tell you nothing you don't already know. We have not seen or heard from our brothers since they left in the ninth month. Before that, they were away for years." He pressed his lips together, trying to master the agony of remembering.
The tengu watched him intently over the rabbit bone he was chewing. He bit into it with his powerful teeth and sucked out the marrow noisily.
"And what were you doing all that time?" There was a note of accusation in his voice that Mu did not care for.
"What business is it of yours?"
The tengu hissed through his teeth in annoyance. "If I am informed correctly, you are the son of several powerful men and a sorceress of the Old People. I daresay you have many talents. Yet you are skulking here with the half-dead—and even they are ceasing to live—only kept from death yourself by the efforts of your brother, who may not have your abilities but is a lot more practical than you."
"How can you tell that?" Mu said. "You've only known us for five minutes!"
"I know many things. I am not without some supernatural ability myself," the tengu said smugly. Then he addressed Ima in a kind voice. "This rabbit is delicious. Well caught! Well cooked! In fact, well done, all around."
Ima narrowed his eyes and said nothing. Kinpoge peeked out from behind him.
"Ah, a little child!" the tengu exclaimed. "I love children!"
Not in the same way you love rabbit, I hope, Mu said to himself.
"So, what have you been doing?" the tengu repeated.
"Nothing," Mu admitted. "What should I be doing?" He recalled his nightmares and immediately wanted to defend himself. "I have no one to teach me anything. Those you say are my fathers are either dead or distant—either way, they are no use to me and never have been. So I can do certain things real, ordinary people can't, but, as you pointed out, here in the Darkwood Ima's skills are more useful. I can take on invisibility to surprise my daughter, or use the second self to make her laugh, but even that I don't do often, and when I needed to, seriously, I was too slow. I was tied up for days, and now I am half-crippled."
"Your brother tied you up?"
"Not himself, but on his orders. A warrior who serves him, called Tsunetomo."
"I know Tsunetomo. He leads the band they called the Crippled Army."
"Who are they?" Mu said. "Maybe I should join them."
"Well, it's a possibility," the tengu replied. "But that's some time ahead. They are a bunch of warriors, both Kakizuki and Miboshi, seriously injured in battle, mutilated, scarred, some blind, some without arms, some legless. They became ugly and imperfect and were turned out by their former masters, to starve to death or become bandits. Most of them are thickheads, but one or two among them have picked up some knowledge of this and that. Tsunetomo is not a complete idiot. Now they serve your brother, Master Kikuta. At first he seemed just another ambitious merchant, good at seizing opportunities and ruthless in eliminating his rivals—there are many men like that in Kitakami, but this year something changed." The tengu had been staring into the flames as he spoke. Now he fixed Mu with his glittering eyes and said, "He has acquired some magic object from which he derives extreme power."
Ima looked across at Mu and their eyes locked. Mu raised his eyebrows slightly and Ima made an almost imperceptible movement with his head. The tengu intrigued Mu, and, somewhat to his surprise, he felt he could trust him.
"It is a skull," he said slowly. "The head was taken from the monk Gessho by Shikanoko when he killed him, and after Shisoku was killed in the same fight. Shisoku was the sorcerer who lived here, and made the creatures."
"I know Shisoku," the tengu said impatiently. "Or knew, I should say."
"It was buried for years," Mu went on. "Kiku returned to retrieve it and invest it with power in secret rituals."
"Were these rituals conducted by himself alone?"
"With a woman," Mu forced himself to say. "A fox woman."
"Mu's wife," Ima explained.
Mu steeled himself to meet the tengu's eyes. He felt they saw deep into his heart, even into his soul. They examined him without pity, saw through all the defenses he might erect, all the excuses he might make.
"Your name is Mu?" the tengu said. "Is that Mu written as 'warrior' or Mu written as 'nothing'?"
"I don't know. I have never seen it written."
"Well, when I have finished with you, you will be both. You will be a great warrior, but you will be as nothing, free from all attachments. That is what I am going to teach you."
The tengu spoke with such assurance, Mu could not help laughing. "You speak as though I have no choice in the matter."
"That is correct."
"What is your purpose?" Mu asked.
"I'm not going to tell you." The tengu cackled with sudden brusque laughter. "Not yet, anyway."
The tengu started the next day, waking Mu before it was light. There had been a deep frost in the night and the surface of the snow crackled beneath their feet when they walked outside.
"Since it is winter, we will start with a lesson on how to stay warm," the tengu announced. He surveyed Mu by the light of a flaming branch he had plucked from the fire. "Look at you! There is nothing to you. You are as frail as a dead spider. Don't you eat anything?"
"I eat plenty," Mu said, trying not to shiver.
"I saw you last night, toying with a tiny bit of rabbit, drowning your appetite with that vile twig brew. You should have grabbed that carcass and shoved the whole thing in your mouth. That's what Master Kikuta would do!"
"My daughter and my brother needed to eat, too," Mu said. He had intended his voice to be mild, but it came out whiny. "Not to mention you, Sir Tengu, our honored guest."
The tengu cackled. "Sir Tengu! That's a good one. No one's called me that before."
"Do you have a name?" Mu said.
"You can call me Tadashii, because I am always right. Now, to work. Watch this."
He handed the burning torch to Mu and began to breathe in a rapid rhythm. The snow beneath his feet melted immediately, steaming as he sank through the frozen surface down to the buried grass. Standing next to the tengu, Mu felt the heat radiate from him, making him believe for a moment that winter was over and spring had come.
"Now you do it," Tadashii said.
"Just like that? You aren't going to give me any instructions?"
"It should be second nature for you, just like the other skills that you've neglected. Imitate my breathing and think of the warmth of your own blood. That's all I'm going to tell you."
Tadashii took the smoldering branch back from Mu, waving it in the air so sparks flew from it and it crackled into flame again.
Mu began to breathe in the same rapid way as Tadashii had. He was watching the branch's fiery arc when suddenly he felt its heat inside his belly. His blood began to boil, racing through his veins. The snow steamed around him as he sank through it to the grass beneath. He felt mud under his feet.
"Ha! Ha! Ha!" Tadashii's laughter rang out through the silent forest. "Easy as breathing, isn't it?"
Mu did not reply at once. Within himself something was melting like the snow. He saw a life beyond the great drifts of grief that had all but buried him, a life where warmth and laughter—and power—were all possible again.
"What else am I going to learn?" he said.
"Everything," Tadashii promised. "I am going to teach you everything. It will be very hard work, but fun, too."
* * *
It was hard work such as Mu had never known, but he reveled in it. All that he had felt was empty was now filled. He no longer dreamed of Shida or yearned after the foxlight that flickered in the marsh. He had a new purpose: to meet every challenge Tadashii threw at him and to master it. He stopped caring what the tengu's intentions might be. The training had no end other than itself. By the time summer came he could use his own innate skills flawlessly and he had learned much more: the art of sword and bow; the roots and herbs of the forest that poisoned or cured; the names and properties of trees, plants, animals, insects; how to trap a stoat, whose meat when dried was a source of courage; how to track bears and wolves; how to recognize scorpions, spiders, snakes, and toads, and milk their venom.
His physical strength increased, as Tadashii showed him how to use his muscles and how to build them up. For hours he carried boulders the length of the stream and back again, and while he would never approach the tengu in strength—Tadashii could lift great rocks with one hand—he surpassed most ordinary men, despite his slight build and appearance. He was no longer lame. He kept the stick, but as a weapon. Tadashii forged a sword for him.
Tadashii could not give Mu wings, but he showed him how to leap to great heights, how to swing from treetop to treetop like a monkey, how to stride the crags like a mountain goat. He seemed to have an inexhaustible patience, which he also taught to Mu. Indeed, Mu thought he must have a different sense of time for, though he had spoken with some urgency on their first meeting, now he seemed to be in no hurry, either to solve the problem of Master Kikuta or to leave.
The seasons passed. It was winter again, and then another winter. Often in the long dark nights they played Go or checkers or chess, for the tengu loved all games, but still he gave no indication of what his original purpose might have been.
* * *
One spring night, two years after the tengu arrived, he took Mu by the shoulders and flew with him high above the forest toward the side of the mountain. It was the night of the full moon of the third month and they could see as clearly as if it were day. Mu caught a glimpse of water that was Lake Kasumi, and the river that flowed from it all the way to the capital, and, in the other direction, the Northern Sea.
Tadashii landed on a ledge where rocks had been placed in a circle, dropping Mu gently in the center of them. On each rock perched winged tengu; some, like Tadashii, had beaks, others long red noses. They were all armed with swords and bows. To see so many at once was alarming. Mu had landed on his hands and knees and he now turned this into a deep, reverent bow.
"Welcome," said a number of voices, all low and gruff, like Tadashii's.
"So this is your pupil?" said one long-nosed being.
"It is," Tadashii replied. "His name is Mu, the warrior of nothingness."
"Does he understand the principles of being and non-being, of form and no-form?" the tengu asked.
"That's not as important as being able to play a good game," another tengu interrupted. "Is he going to be a player or a stone on the board?"
"That is not yet decided," Tadashii said. "I am hoping you, my masters, will look on him favorably and instruct him."
"If he is able to learn we will teach him. Let us see if he can survive our lessons." There was a ripple of laughter, as if the tengu did not really believe that was possible.
Tadashii touched Mu's head. "Be strong," he whispered. "I hope we meet again."
Mu shivered slightly. The uncharacteristic affection alarmed him as much as Tadashii's words. There was a faint rush of air against his face, as Tadashii flexed his wings, followed by a greater rush as all the tengu rose into the air, leaving him alone on the mountainside.
Alone, but not alone. Physically the tengu might have departed, but they were still present in some way, observing Mu, as the moon set and the stars wheeled overhead. He settled, cross-legged, in the meditation position Tadashii had taught him, reminding himself he had been tied up for a week, in a far more uncomfortable way, and had survived. The first light of dawn filled the sky and birds began to sing piercingly. The mountain air was cold, heavy with dew. Mu warmed himself, almost without thinking. The hours passed. He was neither hungry nor thirsty. He had no needs and no desires. He knew only the eternal being that includes all life, all death, in which each person exists for a tiny moment and is then absorbed back into the endless void of all and nothing.
He knew he had great powers; he saw they were meaningless. He embraced the nothingness of his name.
He lost all track of time. He seemed to take leave of his body and fly above the land. At first it looked like a Go board and then more like a vast scroll, presenting various scenes to him. He left the Darkwood behind and soared over Lake Kasumi. He saw the lake was shrinking, its marshlands drying out, a rim of half-dried, foul-smelling mud clogging its beaches. He was above a city, to the north of the lake. Could this be Kitakami? He peered down, through buildings that seemed to have no roofs, so he could see straight into them. He saw Kiku, grown older, surrounded by servants and retainers. Mu perceived the network of his business empire, spreading like a spider's web over the land.
Then the wind took him and blew him to the south. He saw a young woman on a pleasure boat, entertaining a man who looked like a merchant, one of Kiku's rivals perhaps, and he looked deep down into the lake, where someone lay hidden, breathing through a reed. To his surprise, he recognized Chika. On the shore red lanterns denoted a festival. A troupe of acrobats were performing with monkeys. Mu noticed a strong, well-built boy, about twelve or thirteen years old, and an older lad, maybe in his twenties, wiry and flexible, a natural performer, with attractive, expressive features, who kept the crowd spellbound. A young girl of the same age beat time on a drum, her eyes fixed on the performer. A large black bird with a sprinkle of gold feathers hovered above him and it suddenly soared upward, as if it sensed Mu's presence. He saw its bright yellow eyes searching, but it did not see him.
Now he was over the capital, floating above mansions and palaces. In a sumptuous room a great lord, his face gray-tinged and gaunt, was retching into a silver bowl. Outside, warriors and noblemen gathered anxiously. Along the riverbank were strewn corpses. Dogs scavenged among them. The river was a thin, dirty trickle. He was above a temple, newly built from gleaming cypress and cedar. Again he could see straight down into the halls and cloisters. An old man sat with a lute on his knees. He seemed to be dozing, but as Mu passed overhead he startled and turned his face upward, listening. Mu saw into the depths of the lake, where a sleeping dragon lay coiled, its scales shimmering dully in the murky water. A swirling motion began, a whirlpool formed, the dragon stretched and flexed. In the main hall of the temple a figure knelt, chanting sutras, before an altar where golden statues and painted deities kept watch. A woman's voice rose and circled around the lutist's song. The notes vibrated and echoed against one another until the friction became unbearable. At the edges of the land, smoke was rising, as when a scroll is first thrown on the fire. Its edges blacken, it begins to contort in the heat, scorch marks appear here and there, finally flames seize hold.
The land is about to burn, Mu realized.
Only the Darkwood seemed untouched. With a sense of gratitude and relief, he turned back to its dark green mass. He and Tadashii had made many journeys through the forest, but now Mu went farther than he ever had before. He was shown a building hidden among the trees, high in the mountains, a small shrine perhaps, or a hermit's retreat. Looking down through the canopy of leaves he saw two silver-gray horses grazing in a clearing. Nearby a wolflike creature kept watch.
Gen!
In the clearing were four figures involved in an intricate dance. One was clearly a woman, though she dressed like a man. Two wore black cloths over their faces, covering everything but their eyes. The fourth wore a mask, a stag's head with one broken antler.
"Shikanoko!" Mu cried.
Shikanoko looked up, the only person to notice Mu's presence. Unlike the bird, he could see him. Mu felt their eyes meet and lock, but before he could speak he sensed he was losing the power of flight. For a moment he thought he would plummet to earth, and he almost blacked out as he rushed through the air, but he calmed himself and summoned his concentration, and found himself with a thump back on his mountain ledge.
Tadashii was sitting waiting for him. "Not a very elegant landing," he said. "But otherwise, my masters have reported you did quite well. You have seen the state of the board. I hope we will soon be ready for my next move."
"Do you think you could explain a little more clearly?" Mu said.
The tengu did not reply.
"I saw many strange things that I don't understand," Mu persisted.
"Meditate on them." Tadashii refused to say anything else.
* * *
Her father was so preoccupied during the years of the tengu's training, he hardly noticed Kinpoge growing up. She turned more and more to Ima, who took care of her, fed her, and taught her how to hunt and to cook. She liked to catch fish, taking them out of the water with her bare hands; she knew where to gather fern heads and burdock, mushrooms and chestnuts. She looked after the fake animals that remained and, particularly, the skull horse, Ban. She gave it grass and water every morning and, in the evenings, rode it through the air. She wove reins for Ban from the green rushes, and tied two cross pieces onto the pole, one as a handhold and one for her feet. In spring and summer she made garlands of flowers and decorated the skull.
Sometimes she wished she had a real horse, but, as Ima pointed out, a real horse could not fly and it would grow old and die, whereas Kiku had, unwittingly, given Ban another kind of life.
Ban responded to her attentions, turned its head to her when she approached, and leaped joyfully into the air when she untethered it.
She did not go far from the hut where she had lived all her life. Mountains surrounded it and she did not think Ban could fly that high. But she often followed the course of the stream that flowed past the hut. After a few miles, it divided into two, one branch continuing to the west, the other turning southward. Once she had gone south, but after a while the land became cultivated and there were too many people around. She knew instinctively that she should keep hidden and that she should never give away the location of the hut.
So, instead, she and Ban explored the west branch, which flowed through a steep, thickly wooded valley. Occasionally she saw movements in the trees and she realized monkeys lived there.
The monkeys fascinated her. She watched the mothers and babies with a kind of hunger—she who had hardly known her own mother. The mothers took such good care of the babies. In the summer they roamed carefree through the forest, leaping from tree to tree like her father, and in winter they gathered around the pools of the hot springs. Kinpoge spied on them through the branches of the leafless trees. Once or twice they noticed the skull horse and shrieked in alarm, as they did when eagles flew overhead.
One day, near sunset, it was maybe her tenth or eleventh spring on earth—like her father she had matured quickly and was nearly an adult—she and Ban were hovering over the thick canopy, hoping to catch sight of the monkeys, when a boy's face popped out through the leaves, staring at her in astonishment. She could not tell his age, but he seemed taller than she was. His eyes were long and narrow, his nose rather sharp, his cheekbones high. The sun's rays shone round him like a halo.
"Hello!" he exclaimed, and then, hurriedly, "Don't be frightened! Don't leave!"
Ban was quivering beneath Kinpoge's hands. She knew she should escape quickly, but then the boy pulled himself a little higher so his feet were planted firmly in the crook of the branches, and stood up. Two monkeys pushed through the leaves. One climbed onto his shoulders, peering at Kinpoge and Ban and chattering excitedly. The other sat beside the boy, holding on to his leg with one hand and scratching his own belly with the other.
"Who are you?" the boy said. "Are you some magic creature? You must be, for you are so small and you are riding a very strange-looking steed. Do you understand human speech?" When she did not reply he persisted. "Do you speak some fairy language?" He began to mime his words with extravagant gestures that made her laugh.
"I understand you," she called across the space between them.
"What's your name? Mine is Takemaru—everyone calls me Take—but that's a child's name. Soon I will take an adult name, for I am nearly grown up."
"Kinpoge," she said.
"Like the flower? That is so beautiful. And it suits you, you are so small and bright! Where do you live? In the treetops?"
"I live with my father and my uncle. A little way upstream. I must go now."
"Come again," he said. "I will look out for you."
"Goodbye!" Kinpoge cried, turning Ban's head to the east.
She did not tell her father or Ima about the encounter. Both had warned her never to let herself be seen, never to talk to anyone. But she could not stop thinking about the boy, Takemaru, and she wanted very much to see him again. The next day she wove a fresh garland of spring flowers for Ban and, in the afternoon, she set out again. She knew she should not fly toward the west, but somehow she could not help it.
The days were lengthening and there were still several hours before nightfall. The sun in the west dazzled her. It made the shiny new green leaves glisten as they danced in the breeze. It was the fourth month and already very hot. There was no sign of rain and even the dew, which usually soaked the forest every night, had dried up. Every tree was familiar to her and she could tell each one was suffering. They had responded to the demands of the season and had put out new leaves, but it had cost them; they were becoming frail, their roots no longer held firmly by the embracing earth.
She guided Ban to the same tree and there was the boy, alone this time. His face lit up when he saw her and he held out his hand. Kinpoge took it and, still holding Ban's bridle so the skull horse could not fly away, stepped nimbly onto the branch.
The tree swayed in the wind, the leaves rustled, the humming of insects rose around them. There was a strong, heady smell of blossoms and catkins. Take held her firmly.
"I'm all right," she said, easing herself from his grasp and sitting down astride the branch. "I won't fall."
"You should be an acrobat," he said, sitting down facing her. "You are so light and adroit. But, I don't know why, girls never are, only boys. Even Kai, who is agile like you, has to be content with playing the drum."
"I don't know what you're talking about," she said. They were so close, their knees almost touched. "What's an acrobat? Is it something to do with the monkeys?"
"In a way. We do tricks with the monkeys. People like to watch us. They give us food, clothes, even coins sometimes. We go to all the markets and every year, at this time, we come to the forest to look for suitable monkeys to train—do you know what train means?"
"I do!" Kinpoge screwed up her face. "My father is being trained by a tengu. It's been going on for years. I hope your monkeys don't take as long!"
"A tengu?" She could tell this interested Take very much. "A real tengu? What is he teaching him?"
"Everything. But mostly how to fight with sword and bow."
"How to kill people?" Take's eyes gleamed.
"I suppose so, though I think it's more about not letting them kill you, as far as I can see. And then there's a lot of meditation and spiritual exercises. My father is often absent for weeks and when he comes back he seems like a different person."
"Different in what way?" Take asked, and then added quietly, "I have never known my father."
"You haven't missed much. Fathers are very tiresome, at least mine is. He has seen and learned things most people don't know about. Well, I can't really say what most people do or don't know, as you are the only real person I've met. But the tengu teaches him secrets and shows him hidden things."
Take sighed. "I'd give anything for that kind of instruction. I feel I should have been born to the way of the sword and the bow. But the acrobats I grew up among follow a different path. They will not kill anything. They eat only fruit and plants."
"Come back with me," Kinpoge said eagerly. "Ima, my uncle, will make you roast hare or a meat stew. And we'll ask my father if he will share the tengu's teaching with you."
* * *
Ima was out in the forest, somewhere. Mu was alone, going through the rigorous exercises he followed every day. The tengu no longer lived at the hut—he had gone away on a mission he did not reveal—but Mu continued to work as if Tadashii still breathed down his neck with his hot peppery breath and clacked his beak in admonishment.
He was inside the hut, in front of the altar that Shisoku had made years before. The tengu had shown him the meaning of all the objects the old hermit had collected: the augury sticks, the reed arrows, the protective carvings made of peachwood, the panels depicting the twelve cardinal points, the twelve-month guardians, the twelve animals of the cycle of the years. He had explained how to use them, and access their power, just as he had explained the curses that lay sealed, with the five poisonous creatures, in their jars—curses that killed an enemy and then controlled his soul.
Mu had grown up among these things and had never appreciated their power, though Kiku had. His brother had known enough to perform rituals in this place, with the fox woman, Shida. After that time, Mu could hardly bear to enter the hut and at one time had wanted to burn it down. Through Tadashii's teaching, he had faced that pain and humiliation and seen them as illusions of heart and mind. The memory no longer touched him.
He did not like to be interrupted or even watched. Usually, Kinpoge and Ima kept out of his way. But now his daughter's voice broke into the clear well his mind had become, sending unwelcome ripples through it. At first he ignored her, wanting to stay in that removed state of concentration that had become the source of knowledge and power for him, but her voice was as sharp and insistent as a crow's.
"Father! Father, where are you? I've brought someone to meet you."
He heard steps right outside and leaped to his feet. He did not want to let any stranger in. Taking on invisibility, he slipped through the doorway. For a moment, unseen, he studied them: the girl, his child, her ragged clothes and unkempt hair, her small face appearing like the pale moon among dark clouds, her bare arms and legs, scratched and scarred. And alongside her the boy, tall, handsome, he supposed, with the face of a young warrior, but wearing strange red clothes, his hair tied in a topknot, his shoulders unexpectedly broad for his age, his arms and legs as muscled as a grown man's. He recognized him, and after a moment remembered he had seen him on his flight over the land. What did that mean? That he and the other acrobat and the girl drummer were all somehow connected to him and Tadashii? Despite this, the sight of him filled Mu with a kind of unreasoning anger. It would be a pleasure to kill him.
He was surprised by the anger. It was a long time since his peace of mind had been disturbed by emotion. He looked at it dispassionately and let it slip away.
Then he said quietly, "Kinpoge. I am here." He let himself be seen.
They both turned at the sound of his voice, the boy with a startled expression on his face, the girl more exasperated.
"Don't play tricks on us, Father!"
"Surely we walked right past... how did we not see him?" the boy whispered in Kinpoge's ear. Mu heard him clearly.
"It's just something he does. I told you he was tiresome." Kinpoge held the skull horse by the woven reins. She gave it a perfunctory pat and thrust its pole into the ground.
"Did the tengu teach him?"
"He's always been able to do it. But he's got better at it, since the tengu came."
It amused Mu to hear Kinpoge's assessment of his progress. He was smiling, as the boy approached him, which must have encouraged him, for he bowed his head and said boldly, "Sir, my name is Takemaru. There's no reason why you should show me any favor, but your daughter has told me about your great skill and, well..." His formal tone deserted him and he dropped to his knees. "I have no one to teach me how to fight with the sword and the bow. Please let me become your pupil."
"What a ridiculous idea," Mu said, neither moving nor bowing. "Go away. Don't come back. Kinpoge, you are not to meet him again."
As the boy raised his head Mu saw the disappointment in his face. Kinpoge said, "Father, please!"
"Don't argue! It's impossible. Now go away." Mu settled himself, cross-legged, and pretended to tend the fire.
"I'd better go," Takemaru said.
"I'll come and see you again." Kinpoge's voice was thin with emotion.
"No, if your father forbids it, you must not," he said seriously. "You must obey your father."
"Quite right," Mu remarked. "Now get going!"
"Goodbye, sir. Goodbye, Lady Kinpoge." He bowed formally and began to walk away, his back straight, his stride proud.
"Wait!" Kinpoge cried. "I'll go with you. I'll show you the way."
"Stay here," Mu ordered.
He sensed the conflict within her between desire and obedience. He saw her struggle and then, suddenly, with no warning, the second self emerged. He had seen it so often in his brothers and, since the tengu's arrival, he had used it himself, with increasingly refined mastery, but he had not really expected his daughter to have the same skills. The shadow Kinpoge began to flit after the boy who walked resolutely on, ignorant of what was happening behind him. The real Kinpoge wavered for a moment. Her eyes began to roll backward. Mu caught her as she fell.
The other Kinpoge faded as the two selves merged. He splashed water on her face and rubbed her wrists. After a moment she opened her eyes.
"What happened?"
"You did one of my tiresome tricks," Mu said.
"Was I invisible?"
"No, there were two of you. You discovered your second self."
"Really? It felt strange."
"You will learn to control it," he said, "and use it when you want to."
"How exciting!" Kinpoge's eyes gleamed. "But I'd really like to be able to become invisible."
"Maybe that will come, too." It made Mu sad. She was almost grown up. What would become of her? Who would marry her? Who would look after her when he and Ima passed on? The tengu had promised he would be free of all attachments, but this one for his daughter obstinately remained.
She lay in his arms, in a way she had not since she was a child. They were still sitting like that when Ima returned carrying a large dead hare. He looked at them but said nothing, then began to build up the fire, which had almost burned out, with exaggerated care.
"Uncle, I used the second self," Kinpoge announced.
"You did? I thought you would sooner or later!"
"You expected it?" Mu said. "I didn't. It took me by surprise."
"You only have to consider who her father is, and her mother, for that matter," Ima replied.
While the fire burned brightly, producing the glowing embers that would roast the hare, Ima skinned the creature and removed the entrails. Kinpoge took the skin down to the creek. Two of the dogs followed her hopefully. Mu could hear the scrape of the knife on the skin. The fur was thick and soft. They sewed the hides together to make blankets for winter.
The meat smelled fragrant as it began to roast. He thought it would bring her back soon; she was always hungry. Then his sharp ears caught another voice. The dogs barked and fled back to the fire.
"Hello, little girl. That smells good. I think I will stay for supper." The tengu stepped out of the shadows and jumped nimbly across the stream.
Kinpoge dropped the skin and the knife, and hugged him. Tadashii picked her up with one hand, set her on his shoulders, and walked toward the fire.
"It's not cooked yet," Ima warned before he could say anything. "Don't touch it!"
"You know I don't mind raw meat," Tadashii replied sulkily.
Kinpoge slipped down to the ground. "Let's play a game while we wait."
"Maybe later," he said. "I need to talk to your father."
"Did you finish cleaning the skin?" Ima said to Kinpoge.
"Nearly," she replied.
"Well, go and finish it. Then string it up where the animals can't reach it." Ima's voice, as always, was kindly but firm. Kinpoge usually obeyed him, without question, whereas she tested her father, arguing with him endlessly. Now she went back to the bank of the stream and picked up the hare skin. From the way she shook it, Mu guessed it was already crawling with ants.
"Let's go inside," the tengu suggested.
Mu looked at Ima before he agreed, but his brother was staring at the hare as it sizzled in the embers, and did not return his gaze.
When they were in the hut, the tengu bowed respectfully in the direction of the altar and sat cross-legged on the floor. Mu sat opposite him, just where he had been a short time ago when Kinpoge had called him.
"Don't feel sorry for him," the tengu said.
"Who?" Mu's thoughts had progressed to the boy, Takemaru.
"Your brother, Ima."
Mu shrugged. "I don't, on the whole. But sometimes it seems a little unfair. We were all born at the same time from the same mother. None of us chose our parents or our circumstances. Yet Kiku, Kuro, and I have talents our brothers don't have."
"Fair, unfair, these words have no meaning for me." Tadashii dismissed the idea with a contemptuous wave of his four-fingered hand. "Your brother Ku is perfectly happy being a servant to Master Kikuta in Kitakami. And Ima has talents you still don't appreciate. He plays a very good game of chess, for example. He is content with his life, isn't he?"
"I don't know. Is he?"
The tengu scowled, as though he was unable to answer this. "That's not what I've come to talk to you about," he said.
Mu raised his eyebrows and remained silent.
"I sent you a pupil," said the tengu. "And you sent him away."
"You sent him?"
"Well, not in so many words. But I intended him to come."
"You could have told me," Mu replied.
"I expect you to discern this sort of thing," Tadashii said, sounding irritated. "Didn't you recognize him and guess who he might be?"
"He told me his name was Takemaru."
"Takeyoshi is his real name. He is the son of Shikanoko and the Autumn Princess. He has come to the Darkwood with the one they call Yoshi, Yoshimori, the true emperor. Neither of them know who they are. They think they are monkey boys and acrobats. We need you to teach Takeyoshi to be a warrior, join forces with your brother, find Shikanoko, and offer him these forces so the Emperor might be restored and Heaven placated. Then we can get back to normal."
Mu recalled the acrobats he had seen, the girl with the drum. "Is that what it's all been for?" he said. He gazed past the tengu's face at the objects of power clustered all around him, some on shelves, hidden in awe of their effect, beneath seven-layered cloths or in boxes within boxes.
"Well, it's part of a plan we came up with. Out of desperation, if you must know."
"Why don't you teach him?" Mu asked.
"I might, in due course. In fact, I must. An injustice was done a long time ago that I am trying to put right. Something that was stolen must be recovered. But it is hard for me to make contact with a fully human person who has no knowledge of the other worlds. You have shown that you can travel between them. You will be my bridge to Takeyoshi. It's your turn to be the teacher. You never know how complete your learning is until you pass it on. You could call it the ultimate stage. And the other parts of your mission could not be achieved by anyone else. You alone can be reconciled with your brother. You alone can find Shikanoko."
"I saw him," Mu said. "That time I flew. He is in the Darkwood, but far to the north."
"Very good!" Tadashii seemed more pleased than Mu had ever known him before, and even though he suspected the tengu was flattering him so Mu would carry out his wishes, he still felt a warm glow from the praise.
"I thought I saw a woman there with him," he said, "and two men."
"We hope the woman is going to kill the man called Masachika," the tengu said. "And the men are known as the Burnt Twins. Shikanoko has to be brought back to this world. If he stays much longer in the Darkwood he will become a deerlike creature, and if he dies his spirit will be that of a stag, possibly a god, but he will not be able either to enter the pure land or to be reborn."
"When I saw him he wore the deer mask and was dancing," Mu said.
"He is close to the edge. Soon he will be beyond saving. Unless the mask is removed by a pure spirit who loves him, he will be lost."
"Is there any such person?" Mu remembered the raw emotion with which Kiku had revealed what happened at Ryusonji. "He loved the Autumn Princess, but she is dead."
"I don't know much about that side of human life," Tadashii said. "I've observed there are certain acts that bring pleasure and produce children—that's all very well, I suppose, but why complicate it?"
Mu said, "There is passion, and jealousy, the desire to possess another, the fear of losing her."
"But you have put all that behind you, haven't you?"
"I suppose so," Mu said. "Living here, I haven't had much choice."
"I'm glad, though, that you seem to know something about it, for you may recognize such a person."
"It could be any one of us," Mu said. "We all loved and respected Shikanoko."
"I think it has to be female," Tadashii said bluntly.
"What about the woman I saw?"
The tengu laughed, in a coarse way. "No, she is not for him."
"Which should I do first?" Mu asked, thinking about the various tasks that lay ahead of him.
"Follow your nose," Tadashii said, tapping his own beak and cackling.
"Should I go after this Takeyoshi and bring him back?" Mu said.
"You have missed the opportunity," the tengu replied. "It will be another year before he returns to the Darkwood. Next time, don't turn him away."
YOSHI
That was one of the best performances we've ever done, Yoshi thought as he walked toward where Kai sat with her drum balanced between her knees. It was a warm, still night—too hot for early spring—but the drought seemed more bearable once darkness came and the brilliant stars appeared in the clear sky. Crowds lately had been harder to please. People had more serious concerns: their crops, their children's health, shortages of food, ever-increasing taxes. It had been a hard winter followed as usual by the most difficult time of year. Spring brought more work but less food and in the warmer weather diseases spread more rapidly. Tonight at the start the crowd had been sullen, even hostile, but by the end they were laughing.
I won them over! he thought with pride, for he knew it was he they stayed to watch. The music and the monkeys caught their attention, but it was his performance that kept them spellbound. Saru had always been popular and had taught Yoshi everything he knew, but now Yoshi's leaps, somersaults, and back flips had more daring and assurance, looked more dangerous, yet never failed. The monkeys were inspired by him, would do anything he asked of them, and watched him with devoted eyes.
He was not sure of his exact age but knew he was at the height of his ability. He had seen the older men age suddenly before they were thirty. Their life was physically hard and the demands they made on their bodies huge. He himself probably did not have many years left, but at the moment he was the shining star, the sun, the moon, of the troupe.
Saru called, "We're going back to the boat to eat."
The villagers had been generous in the end, but they had little to share: a basket with four eggs, a handful of early greens, little cakes of millet and dried seaweed. The acrobats were always hungry. Performing and traveling used up so much energy, and none of them ate meat or fish, having taken vows since childhood to take no life, not animal or insect, and certainly not human.
Yoshi made a gesture to show he'd heard him. No doubt Saru would be annoyed, but Yoshi was going to speak to Kai and make sure she came with them. Nothing else made Saru jealous, but this would. Yoshi knew Saru loved him as much as the monkeys did. They had grown up together and shared everything in life. But Kai had known Yoshi in his other life, which no one else knew about, and which he never talked about. She was as close to him and as essential as one of his own limbs.
He studied her as he went toward her. The torchlight fell on her face, which was flushed with the thrill of performing, and reflected in her shining eyes. A piece of material wound around her head held her hair back and hid her ears, but she pulled it away as he approached and let her hair fall around her. He was still half-drunk with the applause and the excitement and a wave of desire for her swept over him.
Lady Fuji had tried to keep Kai away from the acrobats, telling her she must stay with the family of musicians who had adopted her, but no one could stop Kai doing what she wanted, especially once she turned sixteen and came of age. She scorned any suggestion that she might marry one of the boys she had grown up with, and while she still played on the pleasure boats with the musicians, she joined the acrobats whenever she could and beat her drum for them. She had no fear of water, and she followed them around the lake in her own boat. A fisherman from Aomizu had vanished from this vessel one night. The next day it could be seen bobbing about on the waves. Sometimes it seemed there was a figure in it; sometimes it looked empty. No one dared go near it in case it had become possessed by the spirit who had pulled the unfortunate man into the water and who might drown them, too. But Kai swam out to it and brought it back to shore, had it blessed by the old priest with whom the acrobats worshipped, and from then on navigated it skillfully from shore to shore, followed by flocks of blue-and-white herons with whom she seemed to have a deep connection, calling to them and imitating their harsh cries. Often Yoshi would find himself thinking of her and would soon after hear the splash of the single oar and see her shape outlined against the evening sky and feel his heart expand with sheer joy.
About a year before, on another spring evening, they had become lovers. He had seen her out on the lake and had waited for her after the others had packed up and left to find shelter for the night. She had jumped out of the boat straight into his arms as though the moment had been preordained by the gods. He had pushed her hair back and kissed her tiny, unformed ears. The boat had drifted off into deep water and Kai had broken away from him to swim after it and pull it back to shore. They were both laughing with happiness as she took off her wet clothes and he pulled her close to warm her, feeling all the curves and planes of her body and marveling at how he fitted perfectly against them, then within them.
Now he held her again, remembering that time, recapturing its thrill and ecstasy. She was trembling and her eyes were full of longing as she led him to a deserted part of the beach and they lay down together under the stars.
Afterward as he rested, spent, against her, caressing her silky skin, there was a sharp call and he heard the beat of wings as Kon settled on a rock nearby.
"He never leaves you," Kai said. "I would worry more about you if Kon were not always watching over you."
"He doesn't actually do anything," Yoshi replied, "except irritate me. He hides for a while among other birds, and I think he's gone, but he always reappears again."
"Kon bears witness to who you are," Kai said quietly. "I think I would have forgotten if it weren't for him."
"Better if we all forget it." Yoshi eased himself away from her a little.
"How much do you remember?" They had never spoken about their past, their childhood in the palace, their flight from the burning city. "Do you remember the day the werehawk came? It was after that that everything began to change. It knew you, then. It bowed to you. Was that Kon, or another one?"
"Let's not talk about it," Yoshi muttered.
"We need to." She took his hand and laid it on her belly. "Your child is growing inside me. Can you feel how my body is changing?" She guided his hand to her breasts. "See how they are heavier and fuller. It will be born in the winter of this year."
He wanted to make love to her again, but she hesitated. "If you are the true emperor," she whispered in a tiny voice, "you cannot be kept from the Lotus Throne. Who are we to try to defy the will of Heaven? But then we will be separated. What will become of our child?"
"I am just Yoshimaru, the monkey boy," he whispered back. "We have kept it secret for so long, we will continue to do so. We will be a family. You are my wife. It's what I've always wanted ever since we were children. You must know that."
"I do," she replied. "I remember being told all the time that I would never be suitable for you because of my ears, and I used to cry myself to sleep at the thought of you marrying someone else."
"We had the luck to end up together in a world where these things don't matter," Yoshi said. "If your ears were not misshapen, Fuji would have taken you for her own trade." He kissed first one, then the other. "We should be thankful to them."
"It's only that when Akihime saved your life and we fled from the palace, you said, 'If I am to reign I cannot die now.' Do you remember that?"
"I was just a child," Yoshi said. "I didn't understand anything. After Saru and the others found me in the forest, for a long time I expected Akihime to come back for me. I dreaded it. When I heard she was dead I grieved for her, but then I realized she had died without giving me away, and I was profoundly grateful, but most of all relieved. I've never wanted to be anything else but an acrobat, to be with you, to follow the teachings of the Secret One, and now to have our child. If anyone finds out about me I'll be put to death. The Miboshi have their own emperor, my uncle Daigen. My death would legitimize him."
Kai pulled him close. "Then we will never breathe a word, and we will pray that Heaven continues to ignore us. No one else knows, do they?"
He did not answer her, but he was thinking of the girl pulled from the lake, the girl who had the lute, Genzo, who was now one of Lady Fuji's most popular pleasure women. She had never spoken of his secret to him and she had kept the lute hidden away. But she had known who he was in that moment on the boat, for Genzo had told her. And then there was Shikanoko. Yoshi did not know if he was alive or dead, and had never mentioned him to Kai. But he still relived that moment on the edge of the stream when he had thought he would die, and Shikanoko still strode through his dreams with his antlered mask and drawn sword.
HINA (YAYOI)
Lake Kasumi was drying up. Villages that had been on the water's edge were now half a mile away, and often boats ran aground as the channels became shallow. This was not the only way life had become harder for the riverbank people. For years they had escaped scrutiny, living and working as they did between the worlds, on thresholds, in the spaces between high and low water, which are neither inside nor outside, neither land nor sea. They considered themselves different from ordinary people and therefore not subject to the same laws. Everything they did had a kind of magic to it: they created wares that had not existed before and transformed them into other things by way of exchange and barter, increasingly for coins, which were themselves a numinous creation. They trained animals for entertainment and lived alongside them. They controlled and dispensed the ephemeral ecstasies of music and sex, both inexhaustible, given away freely and constantly renewed, never drying up.
These gifts were not paid for, as such, but were reciprocated with other gifts, silken robes, bolts of cloth, the finest teas and wines, ceramic bowls, carvings, parasols, prayer beads. Eventually, this came to the attention of Lord Aritomo, who could not rest unless every part of his realm was brought under his control. He made a law that all entertainers and traders should be licensed. His own officials would issue permits, in return for a share in the gifts, which suddenly turned, in a quite unmagical way, into taxable produce, providing income for Aritomo's armies, his roads, and his fortifications.
Because Yayoi could not only read and write but also calculate, and because her charm was famous, it fell to her to deal with these officials. Lady Fuji, who had built up a floating empire of pleasure boats, and who was most reluctant to share the results of her good fortune and business skills with anyone, relied on her more and more.
"I wish I could come upon some scheme to stop them bothering us," she said after one difficult encounter, when they had finally managed to make an official forget why he had paid them a visit. "I don't know what I would do without you. Truly, it must have been Heaven that sent you. Wasn't it a miracle how the wind drove the acrobats' boat across the lake to gather you up, you and Takemaru, and then changed to bring you to me?"
They had often talked about that day, twelve or more years earlier, when Yayoi had fled from Nishimi, a baby in her arms, along with the lute and the Kudzu Vine Treasure Store. It was almost like a ritual or a ballad. Yayoi gave the expected response, hardly needing to take her attention away from her calculations, thinking fleetingly of the child she had been, when her name had been Hina.
"You and the gods saved my life that day."
"It was surely our fate, for now you are as close to me as the daughter I never had."
"I will always look after you, as if you were my own mother," Yayoi replied.
"You were not the only young girl I rescued," Fuji mused.
"I know, you have helped many who would otherwise have died."
The girls Yayoi had known at the convent had grown up like her to become the women on the pleasure boats: Asagao, still her dearest friend, Yuri, Sada, Sen, and Teru, and all the others selling their songs and smiles.
"One came like you, with a boy and a younger girl."
Yayoi marked her place with her finger and began to pay attention. Fuji was a woman of many secrets and divulged them only when it served her purpose.
"She said he was her brother. He was about six or seven, a proper little princeling. I remember explaining the sacred and the profane to him. He was afraid of pollution. And now look at him—he has lived with the monkeys for so many years, he has almost become one of them. And the little girl is Kai, the drummer."
"Which boy is that?" Yayoi said striving for calm. "I can hardly tell them apart."
"Yoshimaru. His older sister carried a lute just like yours, though she did not have your talent. I don't believe she could really play at all, yet the lute played itself, the sweetest music you've ever heard. I suppose it was enchanted in some way. Which reminds me, I haven't seen your lute for years. Do you keep it hidden away?"
Yayoi said nothing until Fuji had fallen silent for so long it seemed unnatural not to respond. "What happened to her?" Yayoi asked, though she knew better than Fuji.
"Kai was too ill to travel when the other two left us to go to Rinrakuji. We heard the temple burned down around the same time. Yoshimaru turned up with Sarumaru, and the monkeys, a few months later, but there was no sign of his sister and he's never mentioned her. I often wondered what became of her. She was not as beautiful as you, but she had a sort of wild charm, like a young boy. Her father had laid a condition of purity on her, which I would have respected. It suited her. She was to be a shrine maiden."
Neither of them spoke for a few moments. Yayoi looked out across the lake. The mountain ranges beyond were beginning to turn hazy and mauve as the sun passed over them toward the west. It would have been a perfect spring afternoon were it not for the turbid water and the exposed stretches of mud.
Fuji said, "A few months later I heard of a young girl who rode a white stallion on the roads around the lake, fighting off men, with a sword that had itself become famous. I thought it might be her, but then no more was heard of her. I suppose she is dead now. Like the Autumn Princess."
Yayoi said nothing.
"I have been thinking about her a lot, lately," Fuji said, leaning closer and dropping her voice. "Yoshimaru has become such a fine young man. And you see, dear Yayoi, if he is who I think he is, some very important people might be interested—interested enough to stop persecuting us with their demands for licenses and fees. I see you are astonished. You would never have guessed, would you, that Yoshimaru, our monkey boy, who is rather fond of the little drummer, is the missing emperor?"
"It cannot be true," Yayoi said, though she knew it was, had known ever since the lute, Genzo, had burst into melody in his presence when she had escaped from Nishimi. The ancient lute knew the true emperor. For years she had said nothing, had simply prayed for his safety, as she watched him grow from a child of eight to a young man of about twenty.
Like all the acrobats, he dressed and wore his hair in the style of childhood and still carried his childish name. He and Saru were inseparable, both handsome, lively young men. Take, the baby she had brought from Nishimi, adored them both, having been brought up by them, among the monkeys. And lately she along with everyone else had noticed that Yoshi and Kai were in love.
"Does he know?" she wondered aloud.
Fuji said, "He has never given the slightest sign. He must have forgotten. He was only six years old when we first saw him."
"We should leave things as they are," Yayoi said. "He will have a far happier life here."
"But if he is restored to the throne, maybe the drought will stop and the lake will go back to how it used to be. And we would gain considerable rewards."
"Restored to the throne? You are dreaming if you think that will happen! The Miboshi will put him to death, and probably everyone who knows of his existence!"
"You are always so pessimistic, Yayoi! You always expect the worst!" Fuji turned away, biting a hangnail in exasperation.
Isn't that how my life has turned out? Yayoi thought. My mother passed away when I was a child, my father died at the side of the Crown Prince in the Ninpei rebellion, my little brother was killed by mistake after he had been kidnapped. My own life has been spared only through Fuji's discretion.
Fuji spat out the nail and said in a malicious voice, "It happened when you were away at the temple so I don't believe you ever heard of it, but Lord Aritomo forced his favorite, Yukikuni no Takaakira, to commit suicide."
"I did not know," Yayoi said. "But what has it to do with me?"
"He was accused of harboring a Kakizuki girl, Kiyoyori's daughter in fact, first in Miyako and then at Nishimi." She looked up at Yayoi, her usual charming smile on her face. "Aritomo saw it as unpardonable treachery. Takaakira ripped his belly. They say it took him hours to die. Nobody knows Kiyoyori's daughter survived, except me. And you, of course."
Yayoi had not known he was dead, the man who had saved her life when the capital fell to the Miboshi. He had, undoubtedly, had his own motives, of which she had been vaguely aware as a child; he would have made her his wife, once she was old enough. But he had been kind to her; he had taught her to read and write, and so many other things. Deep grief assailed her and then she turned cold with sudden fear, hearing the threat, knowing that Fuji would not hesitate to sacrifice Yoshi to gain some advantage for herself. And that Yayoi and Take were no more than pawns in Fuji's game. The safety of all three of them depended on Fuji's silence. But how could she be prevented from betraying them?
* * *
Yayoi did not have time to reflect more on this disturbing conversation, for her first guest arrived, and then she was kept busy for the rest of the day. Her last visitor was one of her favorites, a merchant from Kitakami. He was no longer young, but not quite middle-aged, the son of an influential family whose specialty was fermentation—soybean products, rice wine, and so on. Their name was Unagi, or Eel, and they guarded carefully both their secret recipes and the contracts they made with farmers all around Lake Kasumi, in which the promise of beans at harvest was exchanged for tools necessary in the planting season, lengths of cloth for summer weddings and festivals, drums for local temples, cord ropes and bamboo baskets.
He lived up to his family name, Yayoi thought, being intelligent, strong, and enterprising, as well as able to slither out of any unpleasant situation. She enjoyed his company as much as his gifts, and the wholehearted pleasure he took in lovemaking reminded her of grilled eel—rich, tasty, good for the health.
But on this day, though they brought considerable pleasure to each other's body, afterward he seemed unusually preoccupied, almost despondent.
"Something is troubling you?" Yayoi said, and called softly to one of the girls to bring more wine.
"Forgive me, Lady Yayoi. I thought I would leave my troubles on the shore, or at least on the boat my servant brought me over on. It's been a strange spring... but I don't want to burden you with my problems."
"You can talk to me about anything," Yayoi said. "Even if I can't help, voicing these concerns often clarifies the way you see them."
"Maybe you can help, you are the wisest woman I've ever met. You know my family has been in this business for as long as anyone can remember. We've dealt fairly with people, our house was founded on mutual trust, and that's the way we've always run things. This year more than half of our suppliers have said they can't carry on in the usual way. It appears someone is muscling in on contracts we've had for years, taking them over, blackening our name, and deliberately trying to ruin us. They call themselves the Kikuta—they have been around for some years, though no one seems to know where they came from, but now they have become much more aggressive. The head of the family lets people believe he is Akuzenji's son, though all Akuzenji's children were supposedly killed by Kiyoyori's men years ago."
Yayoi said, "I have never heard of them." She remembered clearly the day Akuzenji died, when she had been so afraid her father would have Shikanoko executed, too.
"We have competitors, naturally, always have had, but this family is different. They use intimidation, and don't hesitate to follow through with their threats, to the point of murder. And not only of farmers but of their wives and children, too. No one dares stand up to them. Now they have started on us, demanding we sell our business, our warehouses, our stock, the vats and all our tools, as well as our secrets, to them. If we don't, they say they will destroy everything and eliminate our family. I didn't take them seriously at first, but now I don't know what to do about it. My father isn't well and I'm afraid the anxiety is going to kill him. I hate to buckle under to bullying, but I have to be realistic."
"What can you do?" Yayoi felt a twinge of unease.
"I am trying to come to some agreement with them. After all, there are precedents—we used to pay Akuzenji to ensure safe transport of our goods overland, and we still employ seamen, who many would describe as pirates, to protect our ships at sea. It's to be expected and saves us keeping a small army of bodyguards. But the Kikuta will not discuss or negotiate; they want complete control. Our only weapon is that they cannot match us for quality, yet. My father has always had the highest standards and he refuses to compromise on that. But even if our buyers are loyal to us, we are falling behind in supplying them because we cannot get our raw ingredients."
Yayoi poured more wine. "Do these people seek to control other merchant houses or only yours?"
"We are the first, I believe," Unagi said, draining the bowl. "However, if we go under, they will start to attack the rest. They treat it like a military campaign. They are the Miboshi with their white banners and we are the red-flagged Kakizuki." He smiled wryly. "And we all know what happened to them! I often wonder if we should not pack up and flee to the west, while we still have the chance."
"But do they ally themselves with the Miboshi? Do they have their support or protection?"
"No, that was just a figure of speech. They ally with no one. But sometimes I feel we are in a kind of war and I must prepare weapons and men. Maybe the Kakizuki should not have fled but fought back, and so should I. That's what my sons want." He sighed. "This isn't what I'd meant to discuss with you tonight. I had another suggestion to put to you."
He took her hand and gazed intently at her face. "I wish I could bring you with me to Kitakami. I've dreamed of approaching Lady Fuji with an offer. But would you be willing?"
Yayoi was touched and for a moment deeply tempted. She liked and respected Unagi; she knew he would give her a good life.
"Forgive me," he said. "I shouldn't have brought it up at this time. Let me deal with the Kikuta one way or another and then I will speak to Lady Fuji. At least let me know you will consider it."
"I will," she said. "Thank you. I am very grateful."
He stood up. "I will send you a message. Thinking of you is going to give me courage."
He refused Yayoi's offers of food or music, saying he preferred to return to his lodgings before nightfall. She heard the splash of the oar as his servant sculled the boat away.
* * *
Yayoi washed and changed her clothes. She took out the Kudzu Vine Treasure Store, intending to study it as she often did at night, but her heart was heavy. The way Unagi had, uncharacteristically, spoken of his problems had unsettled her, and her mind was full of thoughts of the dead. Takaakira must have died years ago, but she had not known of it, and the news had awakened many memories of the past. She had heard snatches of information and gossip on the boats and in the markets, but mostly men came to forget the world of intrigue and strife. If Takaakira had died without her knowing, there was every possibility Shikanoko had, too. She was trapped here in Lady Fuji's world; she would never escape, never find out. Unagi's offer to buy her freedom pulled at her. She tried to imagine for a moment what her life would be like, but she could get no further than the love of a good man, maybe a child, and then she heard Asagao's voice from years ago: Are they going to marry you to a merchant? What a waste of a beautiful girl!
She thought how useful she might be to him, since she knew how to write and to calculate. But how far removed it was from her dreams as a child, when she was a warrior's daughter. She wanted to talk it over with Asagao, but it was getting late. We will talk tomorrow, she thought, and turned to the text, trying to calm herself in prayer. Whenever she took out the text, she began by meditating on Sesshin, who had given it to her. She did not know if he was alive or dead; she had heard nothing of him since he had been blinded by her stepmother and turned away from Matsutani. She sat motionless, eyes closed, with one hand on the pages.
She felt them rustle, as if a strong wind had suddenly blown through the boat. She opened her eyes and saw for a moment a page that showed the mirrorlike stone. Her hands curved instinctively as if they would clasp it, but then the page turned and, search as she might, she could not find it again.
"Well, I will not read more tonight," she said, almost addressing it as you wretched book, trying to control her frustration and disappointment. As she sat back, the pages rustled again. She looked down and saw the text had opened at a place it had never showed her before.
An image leaped out at her. It was a mask, carved from a stag's skull, with antlers. She had seen such things at festivals. Men wore them to dance in, becoming animals or birds, bridging the spaces between the worlds. There were living eyes behind the mask. They looked at her with silent appeal.
"Shikanoko!" she whispered.
But, before she could be sure, the text had closed the page and opened another, showing her a second mask, made from a human skull. Its eyes glittered with gemstones, its lips were painted red, black silky hair had been pasted to the bone. It seemed to turn and look in her direction, as if it were seeking her out. She felt its malevolence and its jealous, restless desire. It was not content with its own power, it could not endure anyone else's but sought to claim all power for itself. With all her effort, she folded the text closed, feeling its resistance, and sat shaking with fear.
What did it mean? Was Shikanoko dead? Or trapped in the world of sorcery, where his mother had warned Yayoi not to follow him? She felt tears forming and struggled not to weep aloud. She remembered so clearly the evening when he had come to tell her about Tsumaru's death. And then she could not keep the tears from falling, recalling her little brother, the last time she had seen him alive, before he had been kidnapped. He had wanted to play with Chika and Kaze, but the other two children had been unwell, and she and Tsumaru had gone out alone into the Darkwood. After that she could only remember the strangers, Tsumaru's cry, her helplessness, her aching head.
Someone called softly, "Hina!" A voice and a name from the past, a whisper, almost lost among the lap of the waves against the side of the boat and the intermittent sound of music. Hina, her childhood name, all but forgotten, so long had it been since anyone had used it.
"Hina! Are you awake? I must speak with you."
Wiping her eyes on her sleeve, she hid the Kudzu Vine Treasure Store under a cushion, then lifted the bamboo blind and looked down onto the water. Unagi's narrow skiff was just below, and gripping the side of the pleasure boat to hold it steady was his servant. She had never looked at him closely before, but now, in the light of the lanterns, she recognised him as her childhood companion, the son of Kongyo, one of her father's senior retainers, and of Tsumaru's nurse, Haru.
"Chika? Can it really be you?"
"Can you come down? I need to talk to you."
She pulled a cloth from the rack and wound it around her head and face, then, just as she was, in her nightclothes, climbed over the side and stepped nimbly into the skiff. It rocked and Chika held her to steady her. It was too familiar a touch for a servant and she wondered briefly if she had been wise, trusting a boy she used to play with, now a man, a stranger.
"Don't worry," he said, reading her mind. "I am not going to hurt you or force myself on you. I can't deny I've dreamed you were my wife. I used to imagine we would be married when we were children, playing at being the emperor and the empress. Perhaps we might have been, back then, when we were almost equals. Now I am obliged to work for a merchant and you have ended up a pleasure woman. We have both fallen, but we are further apart than ever."
"The great wheel turns," Yayoi said. "We all rise and fall with it, as we reap the harvest from seeds sown in former lives."
"No," he said. "The harvest we reap is sown by those who wronged us. If neither Heaven nor Earth gives us justice, then we must seek our own revenge."
He helped Yayoi sit in the bow, then took up the single oar at the stern and began to scull. It was a warm evening and the surface of the lake was only slightly ruffled, like twisted silk.
"Unagi is a good man," Yayoi said finally.
"They say he is a good lover," Chika replied.
"That is none of your concern." She heard the bitterness and envy in his voice and pitied him. "How did you come to be in his household, and how did you know me?"
"He talks to me about you—he's not a discreet man, he can't keep his mouth shut—and he mentioned the scroll, the one Master Sesshin gave you that you were always trying to read. I remembered it clearly. Perhaps I was jealous that you should receive such a gift. When I managed to see you for myself, I recognized you." His voice changed slightly, growing more tender. "I had never forgotten you, Hina."
"You should not serve a man you despise," she said, feeling a need to defend Unagi.
"All men despise those they serve," Chika replied, the bitterness returning. "But he is not my true master. I serve him on my master's orders. I will tell you how it all came about. My father died in the battle of Kuromori, and my mother sent my sister and me into the Darkwood. Masachika was searching for anyone who survived, to put them to death. I knew a place where Shikanoko used to live. I took Kaze there."
Yayoi was momentarily deafened by the thump of her own heart. "Was Shikanoko there?"
"No, he has disappeared. People say he is dead, or that he lives the life of a stag, somewhere in the Darkwood." He was silent for a moment, and when he spoke again, his voice was full of contempt. "He ran away. He abandoned us, leaving everyone to die. The only people there were the imps, one of whom I now serve."
"The imps?"
"Lady Tora's children. Do you remember her?"
Yayoi was suddenly cold and nauseated.
"She bewitched your father. It was after she and Shikanoko came to Matsutani that everything started to go wrong. She had five children, all at one time, and they had not one father, but five. One of them was Shikanoko, another Lord Kiyoyori."
"Does that make them brothers to me?"
Chika smiled. "I suppose it does."
While she was absorbing this, he related in a whisper a long account of the brothers, their fathers' names, their magic skills, their use of poisons and venomous creatures, how the Princess died, how they grew as fast as insects and had taken wives, how they had quarreled.
"We returned from Kitakami. Kiku dug up the skull of a man, whom Shikanoko had killed there some time ago, and, with Mu's fox wife, carried out the rituals that have given him such great power."
Yayoi thought, This is what the book was showing me. The image of the skull, its searching eyes, made her tremble again. Yet the book must have shown it to her with a purpose, just as it had shown her the stag mask through which shone Shikanoko's eyes.
Chika said, "That is why the brothers are estranged. Mu has many gifts, but now Kiku's are much greater."
"Kiku? Are you talking about the family called Kikuta?"
"That's the name he gave himself when he became a merchant."
My poor Unagi, you are doomed!
"So you are also under his power," she said. "And your sister?"
"Kaze is his wife," Chika replied. "And I am his closest friend, more of a brother than his own siblings. I would do anything for him. He decided I could be an informant and asked me to seek work with Unagi. It was not difficult. I had learned many things from Kiku and I knew how to make myself useful to the house of the Eel. He has come to trust me."
"You will betray him," she said flatly, thinking, What can I do to prevent that?
"If I were a servant, it could be called betrayal. But I am a warrior. I have years of disdain and insults to redress."
"Good and evil are not defined by status," Yayoi said.
"You have been sheltered from the world for too long. Everything is defined by status now. Do you think Aritomo does not dispense a different justice to his nobles and lords from that which he metes out to commoners?"
Takaakira's status did not save him, Yayoi thought, but all she said was, "I know very little of Lord Aritomo."
"No doubt he would be very interested to know more about you," Chika said, with a flash of malice.
When Yayoi did not respond, he went on, a little awkwardly, "I do not mean to threaten you."
"I think you do. You have been well taught by your master." She had been fortunate to survive for so long among the riverbank people, but now two people in one day had threatened to expose her. I must get away. I must warn Yoshi. But she had no idea how to do either.
Chika said, as though trying to excuse himself, "I was afraid of what Kiku might do to my sister. I had to obey him."
"Why have you come to tell me this?" Yayoi demanded. "What do you want from me?"
He took a deep breath, as though he had finally reached the point of his visit. "Shikanoko possessed a mask, made powerful by the same rituals Kiku used on the skull. After the confrontation with the Prince Abbot, apparently, it became fused to his face. That is why, after the death of the Princess, he fled to the forest, and shuns the company of men and women. I know you are a wise woman, and you have the Kudzu Vine Treasure Store, which must tell you many secrets. Furthermore, my sister had a dream about you, that you put your arms around a stag in the forest, and it turned into a man. I believe you could bring Shikanoko back."
"What are you suggesting? That I go deep into the Darkwood to search for a man who is probably dead, certainly an outlaw?" It was exactly what she longed to do, but surely it was impossible. "You don't understand the circumstances in which I live. I'm not free to come and go as I please."
"You're clever, Hina. You'll find a way. And I'll help you."
Yayoi knew it was unlikely she would be allowed to go anywhere, let alone into the Darkwood on such an illusory mission. She did not trust Chika, suspecting that he, or his master, had other motives to find Shikanoko, and that they would lie to her and try to manipulate her. She remembered the skull's restless searching gaze. But all she could think of was the eyes she had seen behind the mask, their mute appeal, and the dream image of herself, her arms around the stag, her beloved.
* * *
She hardly slept. Whenever she closed her eyes she saw the mask. She had short broken dreams in which her hands curved around the stone and she understood everything. During the night she remembered it was the time of year when a few of the acrobats, Yoshi among them, went into the forest to look for young monkeys. The idea came to Yayoi that she might go with them. She knew she was being foolish, that the Darkwood was vast, that she was familiar with only the tiny southwestern corner of it, but she was impelled by a belief that fate would bring her to him, wherever he was and in whatever form. And, in the Darkwood, she would find a way to warn Yoshi not to return.
For years she had done nothing without Lady Fuji's permission. She tried to plan how best to approach her, but, as she had feared, Fuji's instant reaction was a refusal.
"It is our busiest time of year; the fine weather, the summer festivals, all the extra gifts that will need to be recorded. It is very selfish of you even to think of such a thing. Whatever reason can you have for wanting to traipse through the forest with the monkey boys?"
"I am a little tired," Yayoi said, fanning herself. "I feel jaded. I will be better for a short break from entertaining."
"Well, we will go on a pilgrimage somewhere in the autumn." Fuji was looking at her shrewdly. "There is some other reason, I feel. Are you planning to run away with one of our clients? It's Unagi, isn't it?"
"In truth, Unagi said last night he would like me to go with him, but naturally he would approach you first. I am wondering whether to encourage him or not. Some time away will help me think clearly. And I thought I might call in at the convent. I would like to see the Abbess again."
"Whatever for? You can't go back, Yayoi. If you want to bury your past you must bury all of it. And put all thoughts of Unagi out of your mind. He is not as rich as he once was and he can't afford you. No, it is quite impossible!" She began to fan herself vigorously.
They were sitting in the stern of the boat. It was still early morning, but the sky was already an intense blue and the sun was hot. A shade awning protected them, but Yayoi could feel the sweat gathering on her skin. The water was green and clear. She longed to lower herself into it. She felt a sudden wave of fury that she was not allowed to act as she wished, that she would always be trapped by Fuji, always afraid that the woman would betray her and Yoshi. She pressed her lips together, not daring to let any words escape her, wishing with all her heart that Fuji were dead.
There was a small splash and a ripple of movement. They both looked over the side of the boat. Far below a shadow flickered across the lake floor.
"It is just a water rat," Fuji said. "Come, enough sitting around. We must get ready for the day."
But Yayoi knew the creature underwater was too large to be a rat. She followed the ripple with her eyes and thought she saw a reed moving through the water.
* * *
Fuji died that night. It had been a busy day, with many visitors. Yayoi had entertained three of her special guests and had then played with the musicians until her fingers were stiff and her head ached. She had fallen asleep soon after the moon had risen, and had been woken at dawn, while the moon was still high in the sky, by the shocked cries and wailing of Fuji's maid.
She ran immediately to the lifeless body, slapped her cheeks, rubbed her wrists and ankles, burned incense under her nose, called her name repeatedly, but no breath returned. Fuji, so healthy and lively the night before, had departed on her final journey.
There were no marks on her body, no external wounds. Her mouth smelled faintly sweetish and Yayoi guessed she must have been poisoned, though by whom, or for what reason, no one could fathom.
The boats left at once for Aomizu. They were supposed to be heading for Kitakami, for the twenty-fifth-day market, but that would have to be canceled. The funeral had to be held quickly, because of the intense heat, and the speed of it all somehow increased the shock and disbelief. But Yayoi noticed that, despite their shared grief, the other women and the musicians were wary of her and talked about her when they thought she could not hear.
The following day, Takemaru came to the boat, calling out to her from the shore, addressing her as Older Sister. She knew that he was uncomfortable on the boats, that the pleasures of love both attracted and repelled him. He was at that age, confused by desire and emotion, happiest in the company of boys his own age and the young men whom he admired excessively, yet drawn to girls. Soon, she knew, one of the women would find it entertaining to take him behind the bamboo blinds and initiate him, and then he would probably lose his mind and be insatiable for a couple of years. It amused and saddened her at the same time. She did not expect, now, to have children herself. Take was both younger brother and son to her.
He was a tall, well-built boy—too tall to be an acrobat, the others said, when they wanted to annoy him, but they could not deny that his strength made him useful, as a baseman, in the living towers they created of humans and monkeys. Already, he could take the weight of the older men on his shoulders or on his upturned feet. He was quick-tempered, bold, and determined in nature; if he could not conquer something he practiced obsessively until he could do it perfectly. He loved listening to tales of warriors of old, their battles, their victories and defeats, and often played with a wooden pole as if it were a sword or a spear. The acrobats teased Take for his bloodthirsty and violent games, but Yayoi, who knew his parentage, saw in him Shikanoko's warrior traits as well as Akihime's nobility and courage.
The drummer girl, Kai, was with him. Yayoi had never been close to her. They had almost instinctively stayed away from each other, as though knowing they had overlapping secrets that they did not dare reveal. Because of some slight deformity, Kai had never joined the pleasure women on the boats but had been brought up by the musicians. Yayoi had seen her tiny shell-shaped ears once or twice when the wind blew her hair away from her face. Yet Yoshi had fallen in love with her; they were as good as married. Yayoi could not help feeling a pang of regret and envy.
She took her sandals in her hand, and a parasol to protect her face from the sun's glare, and crossed to the shore. It was a relief to get away from the sobbing women—and from some other oppressive, disturbing feeling, some accusation in their eyes and the way they fell silent at her approach.
Kai greeted her warmly and the three of them walked to the end of the dock.
Take said, "They are saying you killed Lady Fuji."
"How can anyone believe that? Of course I did not!" Yet, Yayoi thought, I wished her dead.
"You were the last person to see her alive," Kai said, "and you know magic arts, fatal ones. They are saying you cast a spell on her because she would not grant your request."
"Where did you hear this?" Yayoi asked.
"Gossip in town," Take replied. "Yoshimaru told us."
"Does Yoshi believe it?"
"No, of course not, and nor do Kai and I. But he thinks you should come away with us, in case some official hears the rumors and decides to act on them."
"If I run away, I will be confirming their suspicions," Yayoi said.
"Older Sister, only you can decide what is best; you are wiser than any of us. We are leaving directly for the forest. I was coming to say goodbye. Get what you want to bring, don't tell anyone, just say you are walking to the crossroads with Kai to bid us farewell."
"You have thought it all out," she whispered.
"Yoshi told me what to say," he admitted.
Yoshi. Fuji had threatened to turn him over to Lord Aritomo, to expose who Yayoi really was, and then had tried to prevent her from leaving. She felt a pang of guilt. Even though she had not killed Fuji, there was no doubt she was going to benefit from her death. Was it a miracle from Heaven, or had it been the creature that was not a water rat, Chika, his mysterious master, or someone else from the Kikuta tribe?
"I will accompany you a little way," she said in a louder voice. "Just wait a moment."
She knew she must act. She would never have another opportunity like this. She had to find Shikanoko, take Yoshimori to him, so that the true heir of the previous emperor could be restored to the throne. And she had to give Shikanoko his son, Take.
She went back to the boat and collected a few things together, the Kudzu Vine Treasure Store among them. She did not dare take too much; she left her writing implements and her clothes. She was just tying the corners of the carrying cloth when a shadow fell against the blind and a voice called quietly, "Yayoi!"
It was Asagao, the only person Yayoi could call a friend, apart from Bara, who had been her maid long ago and whom she remembered vaguely but fondly. She and Asagao were close in age, had slept side by side when they were children, hidden away in the women's temple, had caressed and kissed each other when they had begun to learn about love. They had laughed over the ridiculous men who fell in love with them, shed tears for the charming ones who would wed other ordinary women, nursed each other in sickness, bled every month on the same days.
"Are you leaving?" Asagao said.
"No, I am just walking with Kai as far as the crossroads." It pained Yayoi to lie to her, so she said no more.
"It isn't true, is it, what people are saying?" Asagao was watching her closely.
"No," she said simply.
"But you are not grieving. You have hardly shed a tear. You might not have killed her, but you are not sorry she is dead."
"I am grieving. I just find it hard to express my feelings, you know that." Yayoi strove to keep her voice light and natural.
"Yes, Yayoi, you keep everything hidden, even from your friend," Asagao replied.
Yayoi tied the last knot and lifted the bundle.
Asagao said, "What about the lute? Aren't you taking that?"
"Why should I? I will be back very soon." Yayoi hated leaving Genzo, but she did not dare take it, for it had no guile. It would begin to play in Yoshi's presence and betray him.
"Can I play it?" Asagao asked.
"Of course, but it is not easy."
"I will look after it for you."
Yayoi looked at her and saw she had not convinced her. She took her in her arms and whispered, "Goodbye."
"Don't leave. Why do you have to go? What am I going to do without you?"
"I'm sorry. I can't explain."
Asagao began to weep and Yayoi felt her own eyes moisten in sympathy. She could think of nothing else to say. She knew only that she had to take this chance and leave now, before it was too late. She joined Take and Kai on the shore and walked away from the pleasure boats, which had been her life for so many years.
CHIKA
Chika returned to Kitakami and immediately went to the merchant house, which was now the center of the Kikuta empire, and his home. Kiku was overseeing the sampling of the latest batch of soy bean paste. His eyes lit up when Chika came in, and he handed his ladle over to the nearest servant and took Chika into the back room, which overlooked the port, the estuary, and the vast expanse of the Northern Sea.
"Lady Hina has left?" he asked eagerly.
"Yes, she has gone to the Darkwood with the monkey acrobats."
"Were there any problems?"
"Fuji was going to prevent her, so..."
"So you very cleverly killed her, leaving no trace?"
Chika nodded.
"Kuro has taught you something after all. Come here. You did well."
Around him Chika could hear the sounds of the vats being weighed down with stones, to bring the soybeans to fermentation. The smell was intense, making the days of high summer seem even hotter. From the shop, he could hear his sister's voice, greeting customers, giving orders to other women, scolding the children. It should have annoyed him—after all, she was a warrior's daughter, she should not have ended up a merchant's wife, especially when that merchant was not even fully human. Yet he did not refuse to approach Kiku, and allowed the other man to embrace him, feeling the familiar stab of desire, all the stronger for being tinged with repulsion. No one else, since his father died, had met his need for approval and love, and Kiku still fascinated him, as he had since the first day Chika had spied on the boys at the hermit sorcerer's place in the forest and seen how they could split into two separate selves and fade into invisibility. He had been following the monk Gessho, had watched the fight in which both the monk and the sorcerer died, and had longed to be like those boys, to have such skills.
He had learned everything Kiku and Kuro could teach him, but some things could not be taught. They were innate skills and could not be acquired. His nephews and nieces possessed them in varying degrees. He had watched them develop, as the children grew, envious of them and delighted by them at the same time. There were a lot of children. Kiku had restrained his pleasure in killing to some extent, but not his lust. Chika's sister, Kaze, seemed always to be pregnant, as were the female servants. Kuro was the same, fathering many children, a couple of whom he brought home in a basket, handing them over without explanation to Kaze to bring up. Even Ku had found a wife and started a family.
The brothers liked the children, almost to excess, Chika thought. It surprised him, for in all other matters they showed little gentleness and no sentimentality. The children were precocious, walking at six months, talking before they were a year old, but they matured more slowly than their fathers, due to their mothers' human blood.
"There is no one like us," Kiku often said. "We have to make our own tribe." More and more frequently he referred to the three linked families in that fashion, and soon they were all calling themselves the Tribe.
Sometimes Chika envied them. If his life had not been disrupted by war, he would be married, with children of his own. But who would he marry now? He was no longer a warrior, yet he was not really anything else. Outside the Tribe he had no caste or family, yet he would never be truly one of them. He thought of Hina, as he often did, recalling her beauty and charm. She could have been mine. What would our children be like? The idea that Unagi hoped to take her as his wife enraged and saddened him, as did the realization that Hina loved Shikanoko. He had seen it in her face when he had told her of his sister's dream.
One is a merchant, one an outlaw, yet they both have more chance of winning her than I do, he reflected.
Now Kiku said, still holding him close, "You and your sister have made me more human. You came to me when I needed to learn how to relate to other people. We were the same age. I know that warriors, like the family you came from, feel strong bonds of loyalty. While I am not sure I fully understand that idea, I do feel a bond with you. I will always be grateful for that."
"I owe you everything," Chika replied. "And yes, there is a bond between us."
Kiku said, "I used to watch the fake wolf, the one that attached itself to Shika—did you ever see it?"
"Once, at Matsutani," Chika replied. "And of course during the winter he spent at Kumayama it was always at his heels."
"Affection made it become more real—it grew and changed, in a way the other animals Shisoku created could not. I often wonder what he did, when he made that one, that enabled it to love and so to grow. Do you suppose it has died, as a real wolf would have done by now, or has its artificiality extended its life?"
"With luck, we will find out before too long," Chika said.
Kiku smiled as he released him. "I hope so."
"What do you want me to do now?" Chika said.
"It is the mask that gives Shika such great power. As we know, it cannot now be taken from his face. But like my skull it was created through combining male and female essences. I've learned from Akuzenji's sorcerers that in such circumstances the mask can only be removed by a woman who loves him. You told me before, after Kaze's dream, that Hina might be that woman."
"I'm sure of it now," Chika said, after a moment.
"You are jealous, Chika?" Kiku said with his customary acute perception. "Do you want her for your wife?"
"Maybe I do. Maybe I always have."
"Your family have significant dreams," Kiku said. "What about your father? Didn't he have a dream about Shika, that he straddled the realm, holding power in one hand and the Emperor in the other?"
"My father believed it was a prophecy," Chika said. "But Shikanoko rejected the opportunity to take power when it was offered to him."
Kiku said slowly, "The Princess's death affected him so strongly."
"If my sister died," Chika said, "would you walk away from your little empire, from all you have built up?"
Kiku stared at him, trying to fathom the meaning behind the question. "Probably not," he admitted. "Though I am very fond of her, in the same way I am fond of you. But all that happened years ago. Surely Shika will have recovered from grief by now."
"There are some things we never recover from," Chika replied.
Kiku said, "That is hard for me to understand. The thing is, I really need the mask, with or without Shika." He smiled, with the small gratification of using an exact word whose meaning had never been clear to him till now. "If Hina cannot remove it, we will take it, still attached to his head. He turned us away. 'Let me never set eyes on you again,' he said. Once he is dead, you can have Hina."
"He betrayed many," Chika said, "when he did not return to Kumayama. I'll never forget those who died as a result, and I'll never forgive him. It will be a pleasure to kill him."
Kiku turned pale, and for a moment did not respond. Then he seemed to gather himself together. "It disturbs me to talk of killing him," he said. "I am very confused. Sometimes I hate him, sometimes I feel another kind of emotion. I long to see him again." He struck his chin with his fist two or three times. "It is as if something is driving me to confront him, almost as if the skull wants to challenge the mask. I will have no rest until I hold it in my hands."
After a few moments of silence, Chika said, "So I am to go after Hina and take her to Shikanoko, and once he is released from the mask—what then? Do you want me to kill him or not?"
"I cannot decide," Kiku said. "I must give it more thought. Maybe you should go first to my brothers Ima and Mu. I have been thinking for some time that we five were born together—we should all live together, all five families. Only Mu fully understands Kuro and myself. Tell him I want to see him. I want us to work together."
"You will have to apologize to him and beg his forgiveness," Chika said. "You tied him up and slept with his wife. Most people would consider that a terrible betrayal."
"She was a fox woman," Kiku said, "less human than I am."
"Mu loved her deeply, though," Chika said.
Kiku shifted uncomfortably. "Maybe I envied that, being able to love."
"That's why you have to ask him to forgive you."
"That word again," Kiku said. "What does it mean?"
"That you are sorry you hurt him."
Kiku scowled. "Very well. Tell him I am sorry."
"Deeply sorry."
"Whatever you like," Kiku said, with a flash of impatience. "Whatever it takes."
HINA (YAYOI)
Kai did not turn back at the crossroads but walked on alongside Yoshi, her drum slung across her back. From time to time she brought it forward and sent its dull note reverberating through the trees. She laughed and chattered with Yoshi and Saru. Yayoi remembered how Fuji had told her that Kai and Yoshi had been taken onto the boat at the same time. Now she wondered how much Kai remembered of her previous life, and what she knew of Yoshi. Watching them together, she became aware of a deep understanding between them, the sort that people described as a bond from a former life.
From time to time she caught a glimpse of the bird that always followed Yoshi. Yayoi had long suspected it was a werehawk, like the one that had flown to Matsutani, the day the bandits were captured. Shikanoko had killed it with an arrow when no one else had been able to. It was many years ago, yet she still remembered vividly the creature plummeting to the ground, its blood sizzling, Shikanoko's stance, her father's expression.
That bird had been completely black, apart from its yellow eyes, but this one had a spangling of gold, as though its plumage were changing color from year to year. It did not seem to age or suffer. Yoshi did not care for it; he never fed it or spoke to it, yet he sometimes referred to it by its name: Kon. Kai was kind to it, as she was to all birds. It was obvious to Yayoi that, like Genzo, Kon knew the young acrobat's true identity. Together with Kai, they were all that remained of Yoshi's former life, a link with the past that he did not—or did not want to—remember. She had left the lute with Asagao, but Kon chose where he went and whom he followed.
"Sometimes they try to trap him," Take told her, following her gaze. "Yoshi would love it if he stayed home in a cage. But he is too clever to be caught. I have offered to try to shoot him down, but of course they would not allow him to be killed."
Yayoi knew the open secret that the acrobats all belonged to a sect, a kind of hidden religion, that forbade the taking of any life. There was some divine mother and child they worshipped, which she often thought must be the reason they loved children, and remained in some ways children themselves. She was also aware that they always sought a blessing before going on a journey, and that a priest of the sect lived not far from Aomizu, so she was not altogether surprised when, late in the first day, in silent agreement, Yoshi and Saru took a side track that led away to the north.
They took one monkey with them to entice the wild ones they hoped to capture to replace Yoshi's two companions, Kemuri and Shiro, who had both died the previous winter. Saru's favorite, Tomo, had died the year before. This monkey was a young one, captured in the forest two summers earlier. They called it Noboru. Saru led it by a long red cord, and when it was tired it sat on his shoulders. They also had one packhorse, carrying their provisions and empty baskets for the new monkeys. Toward the end of the day, Yayoi sat on the horse, perched between the baskets. The acrobats were tireless, but it was a long time since she had walked anywhere and her legs were aching. The packhorse plodded, stumbling frequently. She wished she could put on leggings and ride as she had when she was a child, astride, freely.
After more than three hours, when it was almost dark, they came to a tiny village, four or five huts huddled together at the foot of a tall hill, almost a true mountain. The way was overgrown and led through thick groves of trees and clumps of bamboo. Every now and then, Yoshi and Saru removed, and replaced behind them, the brushwood that had been laid across the path.
It must be a love of secrecy for its own sake, Yayoi thought, for she did not believe there was any real danger of attack. Many sects had sprung up in the years of difficulty and famine, as people sought to understand Heaven's hostility and placate it. Some were followers of the Enlightened One who taught a new, austere path, others turned to the old gods of mountain and forest. Unless they caused riots and disturbed the civil peace, they were allowed to flourish, especially if they paid contributions to Lord Aritomo's system of taxation.
The old man came out to greet them, as happy as a father meeting his children again after a long absence. A meal was quickly prepared, taro with millet, flavored with the dried seaweed Saru had brought as a gift, followed by mulberries and loquats, picked from the trees that surrounded the small fields. Before they ate, the old man prayed over the food, speaking a blessing on the visitors and on their journey.
When he came to Yayoi he said quietly, "You have not been here before, but you are welcome. What is your reason for traveling to the Darkwood?"
"I hope to gather herbs of healing," she said. "There are many that can be found nowhere else."
"Use them only for good," he said. "Beware of being led into sorcery. And turn to the Secret One, for he is the source of true healing."
She bowed her head, saying nothing, but she couldn't help glancing at the others. Yoshi's eyes were closed and his face calm and rapt. All this has so much meaning for him, she thought. He believes with all his heart. But the emperor is called upon to carry out rituals that bind Heaven and Earth. How would he be able to do that? Better he remains undiscovered and lives out his life among the acrobats.
She prayed now, to any god that might listen, that Fuji had not had time to report her suspicions of Yoshi before her death, that they would not be pursued, that Yoshi would be able to return at the end of the summer, and then she regretted her presumption in daring to suggest that the powers of Heaven might be turned from their purpose. He was the Emperor. He could not avoid his destiny, or the sacrifices that would be demanded of him.
The sparse food and the turmoil of her thoughts gave her a restless night. For a long time she lay, eyes wide open, alongside the women and their children, dozed eventually, and awoke at dawn. When she went outside Kon was calling quietly from the roof, and Take was standing at the entrance to the path, as if on guard, his staff in his hand.
"Have you been keeping watch all night?" she asked.
"I slept for a couple of hours. Then I dreamed Kon was speaking to me, some urgent message. It woke me up and I came outside. There is some danger, I can feel it."
"Do you think we are being followed?" Yayoi felt her world shrink again, as though she had just escaped from prison, only to be recaptured.
He gave her a measured look, mature beyond his years. "Yoshi and Saru aren't worried. They believe, since they threaten no one, no one will threaten them. But someone could be following us—maybe the authorities investigating Fuji's death, or maybe..."
Lord Arinori, my so-called protector, who, if he is not going to have me executed for murder, might seek to own me completely, or Chika or his master and his brothers.
"What should we do?" she said.
"We are safer here than on the road. We should stay for a few days. I'll see what news I can discover and return by nightfall. Tell the others where I've gone and wait for me here."
She tried to persuade him not to go alone, but he was impatient and would not countenance any opposition to his plan. He set out at once before the others woke, leaving Yayoi to explain where he had gone.
Saru and Yoshi mocked his concerns but were happy to spend at least one more day with their beloved teacher. Take returned in the late afternoon, looking pleased with himself.
"I was right," he whispered to Yayoi. "Someone did come after you—Lord Arinori."
She felt a jolt of fear. If they had stayed on the road, he would have caught up with them.
"It's a good thing I stuck to my decision. He has gone to the temple where you lived for a while. He thinks you would have fled there. We will stay a few more days until he has returned to Aomizu."
She remembered the earlier search at the temple, the destruction, the nuns' terror.
"Don't worry," Take said, seeing her expression. "If he does not find you there, what will he do? He is not going to hurt the nuns."
Yayoi gave her fine robes to the village women, telling them to get rid of them or sell them at the market. She dressed in the dull, shabby clothes of a peasant, and worked alongside the women in the fields, letting the earth stain her hands and the sun darken her skin. The young men cut wood for the winter, helped build a new shed to stack it in. There was always work to do and the villagers were grateful for the extra hands. Many days passed before they were ready to move on. They laughed at the dangers that Take saw everywhere, his mind influenced by the tales and legends of the past that he so loved, their intrigues, betrayals, battles, and uprisings, and teased him until he lost his temper.
The morning they left, they knelt before the old man to say goodbye and receive his blessing.
He smiled when he looked down at Saru. "May you find a friend to replace the one you lost." To Kai he said, "I am glad you are staying here with us. You and your child will be safe here."
Kai smiled, blushing a little, and reached out to touch Yoshi's hand. He grinned, too, but the old man turned to him with a somber face. "What you seek will not be found in the Darkwood. It is not what you think, maybe not even what you desire. I told you once, all paths lead to your destiny."
"Master your anger," he said to Take. "It blinds you to what is real and what is best for you."
To Yayoi, he said, "You may use your real name now. Your old life is finished." And from that moment she called herself Hina again.
* * *
Take hurried them off the track, as soon as possible, and they began to make their way eastward, meeting the road to Shimaura some way south of the crossroads where the highway from Aomizu went on to Rinrakuji. They slept for a short time on the edge of the fields, and were woken in the early morning by two small boys who demanded to see the monkey.
"Wait a few months and we will be back with a whole troupe," Saru promised.
They walked all day, and then took a track that turned off to the east. The two young men and Take had been here many times, but for Hina it was completely new. The forest closed around them. Cicadas shrilled in a constant shower of sound and mosquitoes whined. The air was stifling, the track stony. For a while she rode the horse, but it stumbled often, its straw shoes slipping on the rocky ground, and she felt safer walking.
She thought they would sleep in the open air again, but in the late afternoon she saw they were approaching a derelict hut. Take had gone ahead to scout, and came running back.
"There are people in the hut," he said in a loud whisper.
Hina stopped, Yoshi beside her.
"I don't like this place," Yoshi said. "I've been past it many times, and it always makes me tremble."
"Did something bad happen here?" she asked.
His face closed and she knew she was right, but he would never tell her.
Saru, with the horse and Noboru the monkey, went blithely on, calling out a greeting.
There was a slight noise from inside and a tall woman stepped out, holding a broken plank of wood in both hands, as if it were a club, a look of fear on her face. Her head was shaved, her tattered robe a dull brown color.
Hina thought she recognized her but could not believe it was the same woman. Was it a ghost or an illusion? "Reverend Nun?" she questioned, walking forward.
Astonishment, then anger, replaced fear as the woman lowered the plank. "You are one of Lady Fuji's girls. The one we called Yayoi. What are you doing here? It is on your account that all these disasters came upon us. What have you done?"
"What happened?" Hina said.
The monkey was screaming loudly from Saru's shoulder and showing its teeth. The nun looked at it, and then back at Hina. She swayed slightly. The plank dropped from her hands. She crouched down, her face in her palms, her shoulders heaving.
Hina knelt in front of her, the others waiting a few paces behind; Yoshi and Saru silent and concerned, Take turning constantly, his eyes raking the forest and the track they had come along, as if suspecting a trap. There was a clatter of wings and Kon alighted on the roof. It called in its fluting voice, silencing the birds of the forest. In this hush, a voice came from inside.
"Who is there?"
Hina would never forget that voice. "It is the Abbess," she whispered. The nun nodded, without speaking.
"Shall I go in to her?"
Take rushed forward. "It may be a trap. Let me go."
"There is no one inside but our lady," the nun said, her voice hoarse with tears.
Nevertheless, Take, holding the pole ready, stepped inside. Hina followed him. There was no door—it had warped and fallen years before—and the hut smelled of damp and mildew and of something else, a sweetish, stomach-turning whiff of flesh rotting.
"She's telling the truth," Take said. He moved back to the door as Hina went forward and knelt beside the small figure lying on the ground. She was about to take one of the Abbess's hands, when her eyes adjusted to the gloom and she saw the injuries.
The skin had been seared away. The flesh was raw and swollen. Yellow and black streaks of infection ran up both arms. The fingers were turning dark.
"It is Hina," she said softly. "I used to be called Yayoi. I lived at the temple."
"Yayoi, dear child," the Abbess said. Her voice was calm and clear, despite the fever. "Look at what has become of me! I am dying, but I am glad to see you. Heaven has sent you to me."
"What happened to you?" Hina said. "Who did this to you?"
"I did it to myself, foolish old woman that I am. Lord Arinori came to the temple again. This time he was looking for you. Of course, I did not know where you were, nor had I heard the news of Fuji's death. I could tell him nothing. He became very angry when none of his threats worked on us, and had his men set fire to the building. My little cat—you probably knew her mother—was trapped inside. I tried to save her, but the flames were too fierce. Poor creature, she was the victim of human rage and hatred, and I was punished for my stupid, vain attachment."
"Don't blame yourself," Hina said. "Blame the cruelty of men."
"Men will always be cruel and destructive," the Abbess said. "We live with that as we live with typhoons and earthquakes. I could not reach my cat, but I was able to snatch one object from the flames. Now you are here, I understand it was for you. It is by my side. Can you see it?"
Hina groped around with her hands in the half darkness and came upon what felt like a smooth, rounded stone. Her palms seemed to recognize it and it knew them in return, nestling into them. She lifted it and held it up so the Abbess could see it.
"Is this it?"
"Yes."
Hina peered at it. It gleamed slightly even in the gloom inside the hut. It was reflective, like a mirror. She could almost see her face in it.
"It is a medicine stone," the Abbess said. "I knew it was for you when you came to the temple with the Kudzu Vine Treasure Store—do you still have it?"
"I do," Hina said. "I left almost everything else behind, but the text I brought with me."
A smile flitted over the Abbess's face. "The stone and the text belong together. I should have given it to you then, but you were only a child, and you seemed destined for another kind of life. Now you are here, like a miracle. I can only conclude the stone brought you here so you could be united."
"What is it for?" Hina asked.
"Hold it to my mouth so it catches my breath."
Hina did so and a mist covered the polished surface.
"Now look deeply into it," the Abbess said.
Hina could not help crying out.
"What did you see?"
"I cannot say!"
"Say it," the Abbess commanded her. "I am not afraid. It revealed I am dying, didn't it?"
Hina found she could not put into words what the stone had shown her: the intricate workings of the body, all failing one after another, before the inexorable invasion that was death. Tears formed in her eyes and she wept for the incurable frailty of the human body, its passage from birth and growth to decay and death, through a brief moment of passionate, striving life.
"It will show you the fate of any sick person," the Abbess said. "Whether they will recover or if they should prepare themselves to cross the three-streamed river of death. To most people it will seem like a dull black stone. Only in cases of imminent death does it reveal itself to be a mirror."
Her calmness added to the awe Hina felt for the magical object in her hands. She put it down carefully, leaned over the older woman, and placed her hand on the burning forehead.
"Your hands are so cool," the Abbess said. Her eyes closed and she seemed to sleep for a few moments. Then she said, "Where are you going?"
Hina said, "I am going into the Darkwood to find Shikanoko."
"Shikanoko, the outlaw?"
"Your son. You called him Kazumaru. I don't believe he became a monster, as you feared."
"So you are going in search of him?" the Abbess said wonderingly. "He has been much on my mind, as I lie here, dying. Why are you looking for him? Is it because you love him? But how can that be? You can't have been much more than a child when you knew him, if you knew him at all..." Her speech became more rambling and incoherent and Hina could not follow everything she said. She was afraid the end was near, and was about to call the nun, when the dying woman spoke more clearly. "When you find him, tell him his mother forgives him."
"Maybe you should ask him to forgive you," Hina said. "If you had not left him when he was a child... I am sorry, it is none of my concern." But then she felt strongly that it was her concern, and her anger and pity rushed to the surface. "You abandoned him! That is what made him become a sorcerer."
There was a long silence. She feared the Abbess had stopped breathing and leaned over her to check. The woman raised her head toward her and spoke with surprising force. "You are right. I see it all so clearly now. I thought I was seeking holiness. I so wanted to be good. But in the end I gave my cat more affection than I ever gave my son, and for that I am dying." Her voice was filled with despair and bitterness.
She must not die like this, after a whole life dedicated to the sacred, Hina thought. Take had remained on the threshold while they had been talking. Now Hina turned to him. She had not intended to tell him who his father was until they found Shikanoko—for all her confident words, she could not know what he might have become, what grief and loneliness might have wrought in him. She might never find him; she might find a monster. But she had to let Take meet his grandmother, now fate had brought them so close.
"Take," she called softly, "come here!"
He knelt beside them, his eyes widening in pity as he saw the damaged hands.
"You know her?" he said. "Who is she, poor lady?"
"She is the abbess of the temple where I lived for some years, after you and I were rescued from the lake. And she is your grandmother."
The Abbess's eyelids had closed, but now they flew open and she searched for Take's face. "Who is this boy?" she whispered.
"He is called Takeyoshi. He is Shikanoko's son. His mother was Akihime, the Autumn Princess. He is your grandson."
"Is it true?" the Abbess said, and Take echoed her with the same words, as their eyes locked.
"It is true," Hina said.
Tears flowed from the Abbess's eyes. "I want to touch his face, stroke his hair, but I cannot bear the pain."
Take put his own hand to her face and wiped away the tears with his fingers.
"When you find your father, ask him to forgive me," she said.
She did not speak again. Her face took on a calm and joyful expression. Little by little the smell of sickness abated and was replaced by a fragrance like jasmine.
Hina found her lips repeating one of the sutras, that she had chanted so many times at the temple, that she had read aloud to the Abbess, as her tears fell for the dying woman.
The nun came in and joined in the chanting. The hut seemed to glow with light.
"The Enlightened One is coming for her," the nun whispered. "He will take her straight to Paradise."
The Abbess began to breathe rapidly. Her eyelids fluttered. She seemed to want to speak, or maybe she was praying. Then the quick breaths ceased in one last sigh. Her eyes opened, but they no longer looked on this world.
Kon called piercingly and the monkey, Noboru, screeched in response.
The nun said, "The other nuns went to Rinrakuji, to get help. They will be back soon. I'll stay with her body, but you should not linger here. Rinrakuji is a Miboshi temple now. I don't know what you are supposed to have done, or who you really are, but you don't want to get embroiled in their questions and their procedures."
"What will happen to you?" Hina said.
"They will no doubt find a place for us, washing dishes, sweeping floors. There are many ways a nun can serve."
"But you have had your own temple, free from the control of men! You will find it hard to serve them now."
"It could not last," the nun said in a resigned voice. "All over the country, men are gaining power over women. They are in the ascendant, and will be for years to come. Women are condemned to begin their decline. It is all one, part of the great cycle."
Hina knelt to ask for her blessing and Take imitated her. Then they bowed in farewell to the corpse and left the hut, Hina clasping the stone.
Once outside Take turned to her, his eyes bright with unshed tears.
"Tell me everything."
"I will," she replied, with a swift glance at Yoshi, who was waiting with Saru, both sitting on their haunches. The monkey was on Saru's shoulder, searching his hair for fleas. The horse was cropping grass at the edge of the stream. "But not now. Later, when we are alone, I promise."
"What's happening?" Saru said. "Are we stopping here for the night?"
"What's the matter with you?" Yoshi said to Take. "Is something wrong?"
"A woman died in there," Hina said.
Both young men drew the cross sign in the air.
"Let's get going, then," Saru said with a nervous laugh. "I'm not all that fond of the dead."
"Shouldn't we help bury her?" Yoshi said.
"People will be coming soon," Hina said. "Really, it's best if we leave without delay."
Take seemed about to speak, but Hina shook her head at him. He ran to the stream, surprised the horse with a whack on its rump, jumped from rock to rock, and disappeared into the forest. The horse flung up its head and galloped after him. The others had to follow.
* * *
They walked until well after nightfall, the three-quarters moon of the seventh month lighting their path, and slept briefly on the ground until the forest birds began to call before dawn.
Yoshi and Saru went on ahead, but Take, alongside Hina, walked more and more slowly until they were a long distance behind.
"I thought my father must have been a warrior," he said, when the others were out of earshot. "It would explain so much about me. But what else do you know about my parents?"
She told him all she remembered from her childhood, the day Shikanoko arrived on the brown mare, his unparalleled skill with the bow, how he brought down the Prince Abbot's werehawk and had been able to ride the stallion Nyorin, which no one else could, after the death of its master, Akuzenji.
"He was born at Kumayama, and is the true heir to that estate. It lies a little farther to the east from my father's twin estates of Matsutani and Kuromori."
"Kuromori? The Darkwood?"
"Yes."
Take gestured at the huge forest through which they were walking, the mossy trunks, the twisted roots, the fern-fringed streambeds. "So all this was your father's?"
"If the Darkwood belonged to anyone, it was to him. But we lived on the southwestern corner. All this part is completely wild."
"What happened to your father?" Take asked.
"He died at the side of the Crown Prince, along with your other grandfather, Hidetake, in the Ninpei rebellion."
Take absorbed this silently, glancing at Hina with new concern. She wondered how much he had heard of the legends, rumors, and ballads that had sprung up around Lord Kiyoyori and his son, the dragon child, and what he knew of the struggle between the Miboshi and the Kakizuki.
"Who owns the Darkwood now?" he said. "Weren't you his heir?"
"My uncle, Masachika. He had been sent to join the Miboshi when he was a young man, so he ended up on the side of the victors. He thinks I am dead, and must never find out otherwise. It was he who came to Nishimi and discovered the Princess, your mother, hiding there, not long after you were born. That's when I ran away with you, and the acrobats rescued us."
"Did he kill her?" Hina saw in his face that he was already thinking of revenge.
"Not directly. He had her transported to Miyako and she died there."
"And my father—what is his name?" he said after a long silence.
"Shikanoko. He was always called just that. It means the deer's child."
"Is he still alive?"
"It seems so, for they are searching for him. Unless he died in the Darkwood. But, as I told the Abbess—I don't know if you heard—I am also looking for him."
"My grandmother," he stated. "The first of my family I have ever met, and then she died within moments. I lay awake all night, thinking of her, praying for her soul."
"Yes, I did, too," Hina replied.
They walked on slowly. Yoshi and Saru were out of sight ahead, but from time to time they heard Kon calling and Noboru chattering.
"A little while ago," Hina said, thinking she should explain her reasons more fully, "a man came to visit me. I knew him when we were children. He was the son of my father's senior retainer, and the same age as me. After my father's death he fell on hard times, but was taken in by a man who has become powerful in the north, in Kitakami. This man and his brothers were the children of a woman who came to our house at the same time as Shikanoko. She bewitched my father and he fell in love with her."
She was surprised how hard it was to say this. Her face was burning.
"They are his children? Your brothers?" Take said, puzzled.
"There was some sorcery at work. They were all born at one time, they had several men for their fathers. My father was one, Shikanoko another."
"So they are my brothers, too?"
"In a way, yes." She did not want to tell him everything she had learned from Chika, how the brothers had gone with Shikanoko to Ryusonji and caused the Princess's death. "This man, my childhood friend, Chika, begged me to go and find Shikanoko. There are many forces at work and I don't understand them all, but I believe they are converging, with the purpose of restoring the true emperor to the throne."
"People say this terrible drought and the other disasters are all a punishment for the Miboshi's arrogance in choosing the emperor they wanted," Take said.
"You can say such things here in the forest," Hina said, "but never utter them where anyone else can hear you. Your tongue would be ripped out! But certainly in Heaven's eyes there is something grievously wrong. I feel we are being called to set it right. I don't know what to do, except go into the Darkwood in search of your father."
"So my father knows who and where the true emperor is?"
Hina said nothing, not sure how to answer.
Take was frowning as he persisted, "Or is it that you are going to tell him? Are you the only person who knows?"
"Maybe I am, apart from the gods," Hina said quietly. And Kai, she thought, but she did not voice this.
AUTHOR'S NOTE
The Tale of Shikanoko was partly inspired by the great medieval warrior tales of Japan: The Tale of the Heike, The Taiheiki, the tales of Hōgen and Heiji, the Jōkyūki, and The Tale of the Soga Brothers. I have borrowed descriptions of weapons and clothes from these and am indebted to their English translators Royall Tyler, Helen Craig McCullough, and Thomas J. Cogan.
I would like to thank in particular Randy Schadel, who read early versions of the novels and made many invaluable suggestions.
ALSO BY LIAN HEARN
TALES OF THE OTORI
Across the Nightingale Floor
Grass for His Pillow
Brilliance of the Moon
The Harsh Cry of the Heron
Heaven's Net Is Wide
Blossoms and Shadows
The Storyteller and His Three Daughters
THE TALE OF SHIKANOKO
Emperor of the Eight Islands
Autumn Princess, Dragon Child
A NOTE ABOUT THE AUTHOR
Lian Hearn is the pseudonym of a writer—born in England, educated at Oxford, currently living in Australia—who has had a lifelong interest in Japan, has lived there, and studies Japanese. She is the author of the bestselling series Tales of the Otori. You can sign up for email updates here.
All four volumes of Lian Hearn's
The Tale of Shikanoko will be published in 2016.
EMPEROR OF THE EIGHT ISLANDS
April 2016
AUTUMN PRINCESS, DRAGON CHILD
June 2016
LORD OF THE DARKWOOD
August 2016
THE TENGU'S GAME OF GO
September 2016
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Contents
Title Page
Copyright Notice
Epigraph
The Tale of Shikanoko List of Characters
Map
1. Hina (Yayoi)
2. Bara
3. Mu
4. Shikanoko
5. Kiku
6. Aritomo
7. Hina (Yayoi)
8. Mu
9. Yoshi
10. Hina (Yayoi)
11. Chika
12. Hina (Yayoi)
Author's Note
Also by Lian Hearn
A Note About the Author
Books in the Tale of Shikanoko Series
Copyright
Farrar, Straus and Giroux
18 West 18th Street, New York 10011
Copyright © 2016 by Lian Hearn Associates Pty Ltd.
All rights reserved
Originally published in 2016 by Hachette Australia
Published in the United States by Farrar, Straus and Giroux
First American edition, 2016
Map by K1229 Design
Library of Congress Cataloging-in-Publication Data
Names: Hearn, Lian, author.
Title: Lord of the Darkwood / Lian Hearn.
Description: First American edition.|New York: Farrar, Straus and Giroux, 2016.|Series: The tale of Shikanoko series; book 3
Identifiers: LCCN 2016016694|ISBN 9780374536336 (paperback)|ISBN 9780374715038 (ebook)
Subjects: LCSH: Japan—History—1185–1600—Fiction.|BISAC: FICTION / Literary.|FICTION / Fantasy / Historical.|GSAFD: Fantasy fiction.|Adventure fiction.|Historical fiction.
Classification: LCC PR9619.3.H3725 L67 2016|DDC 823/.914—dc23
LC record available at <https://lccn.loc.gov/2016016694>
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## Contents
1. Title Page
2. Copyright Notice
3. Epigraph
4. The Tale of Shikanoko List of Characters
5. Map
6. 1. Hina (Yayoi)
7. 2. Bara
8. 3. Mu
9. 4. Shikanoko
10. 5. Kiku
11. 6. Aritomo
12. 7. Hina (Yayoi)
13. 8. Mu
14. 9. Yoshi
15. 10. Hina (Yayoi)
16. 11. Chika
17. 12. Hina (Yayoi)
18. Author's Note
19. Also by Lian Hearn
20. A Note About the Author
21. Books in the Tale of Shikanoko Series
22. Newsletter Sign-up
23. Copyright
## Guide
1. Cover
2. Table of Contents
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North Catasauqua is a borough in Northampton County, Pennsylvania. The borough was founded in 1907. The population of North Catasauqua was 2,971 at the 2020 census. The borough is part of the Lehigh Valley metropolitan area, which had a population of 861,899 and was the 68th most populous metropolitan area in the U.S. as of the 2020 census.
In August 2008, borough council officially adopted and christened North Catasauqua as "small Town U.S.A.", with a campaign devised by the North Catasauqua Betterment Committee for the official recognition.
History
20th century
In 1907, the citizens of western Allen Township met at the Northampton Hotel, and signed a petition declaring the formation of the North Catasauqua. This petition was submitted to the county in early June, the final decree of incorporation was made on June 11, and was officially entered into the records on June 17. Shortly after the incorporation was made official, the first borough elections were held, with William H. Thomas becoming the first burgess, an office equivalent to a present-day mayor.
Located between the larger boroughs of Catasauqua in Lehigh County and Northampton in Northampton County, North Catasauqua quickly grew as businesses and residents came into the new borough. Eventually a police department was formed along with the Charotin Fire Department in 1909. A new combined firehouse and municipal building was built in 1911. Prior to this, fire fighting equipment was housed at the Hoffman and Follweiler stables. Church services were held around the borough at various places of worship including St. Lawrence Church (built in 1857), St. Andrew's Church (built in 1903), and Holy Trinity Church, which was built in 1903.
In 1913, North Catasauqua School District razed the four-room Allen Township schoolhouse so that a new one could be built. The new school building housed grades 1-8 and featured, what were at the time, state-of-the-art amenities. After a merger of North Catasauqua and Catasauqua school districts in 1953, the school housed elementary grades 1-6 until June 1972.
On June 14, 1947, the borough celebrated the opening of the North Catasauqua Park and Playground built on of land purchased from Northampton County. At the time, the park included a war memorial, complete with flagpole and monument (erected in 1945); playground equipment; a baseball diamond; and 95 Sycamore trees. The park and playground quickly became a popular spot for play and recreation, as well as band concerts, picnics, the borough Christmas tree, and carnivals. The Willowbrook Golf Course has played an important role in the borough's development. The golf course was originally a private five-hole course built by Colonel James Fuller on the Willowbrook Farm Estate to entertain personal friends and executives visiting the Fuller Company in Catasauqua. In 1932, the golf course opened as a private nine-hole course leased to Dr. Calvin Miller from Mrs. Dorothy Fuller, the Colonel's widow. The golf course was open to the public in 1934 as a way to pay for expenses. The golf course has grown over the years and has become a popular course for local residents and area golfers. Today, the golf course has been closed so more housing can be built in the borough.
In 1975, North Catasauqua borough obtained possession of the building and transferred the municipal offices to the site in 1976. In 1995, extensive renovations took place and a rededication ceremony was held in 1996. The municipal building now houses the borough police department, borough offices, council chambers, a meeting room, fire department administrative headquarters, as well as room for future expansion.
Charotin Fire Department was relocated to the municipal building site in 1980 with the building of a new combined fire station and public works garage. The North Catasauqua Public Works Department moved to its present location at the Main Street complex when the old Charotin fire station and municipal garages were sold in 1986. A public works building was built in 1983 at the Main Street complex. This new building included garages and an office, and allowed all of the public works' equipment to be housed in one building.
21st century
In 2004, an additional building was erected. This new facility allows for increased storage and includes bigger garage bays, offices, a break room, and shower facilities for employees. Additionally, a paved pathway leading down to the D&L Canal towpath and Lehigh River was also created, along with a parking lot, for future public recreation at the river. North Catasauqua borough celebrated the centennial of its incorporation with a year-long centennial celebration in 2007.
Geography
North Catasauqua is located at (40.661896, -75.476015).
According to the U.S. Census Bureau, the borough has a total area of 0.8 square mile (1.9 km2), of which 0.7 square mile (1.9 km2) is land and 0.04 square mile (0.1 km2) (2.67%) is water.
The Lehigh River and Lehigh Canal and towpath provides the western border of the borough.
Transportation
As of 2007, there were of public roads in North Catasauqua, all of which were maintained by the borough.
No numbered highways serve North Catasauqua directly. Main thoroughfares in the borough include Howertown Road and Eugene Street.
Demographics
As of the 2010 census, there were 2,849 people, 1,181 households, and 809 families residing in the borough. The population density was 3,825.5 people per square mile (1,468.2/km2). There were 1,186 housing units at an average density of 1,612.3 per square mile (618.8/km2). The racial makeup of the borough was 95.2% White, 1.2% African American, 0.1% Native American, 0.9% Asian, 1.7% from other races, and 1% from two or more races. Hispanic or Latino of any race were 5.2% of the population.
There were 1,181 households, out of which 25.5% had children under the age of 18 living with them, 54.1% were married couples living together, 10.2% had a female householder with no husband present, and 31.5% were non-families. 27% of all households were made up of individuals, and 11.3% had someone living alone who was 65 years of age or older. The average household size was 2.41 and the average family size was 2.9.
In the borough, the population was spread out, with 23.2% under the age of 18, 7.1% from 18 to 24, 30.3% from 25 to 44, 23.2% from 45 to 64, and 16.1% who were 65 years of age or older. The median age was 39 years. For every 100 females there were 92.6 males. For every 100 females age 18 and over, there were 88.2 males. The median income for a household in the borough was $39,375, and the median income for a family was $47,214. Males had a median income of $35,324 versus $26,250 for females. The per capita income for the borough was $19,424. About 5.2% of families and 6.6% of the population were below the poverty line, including 6.2% of those under age 18 and 5.3% of those age 65 or over.
Notable people
Pat Kelly, former professional baseball player, New York Yankees, St. Louis Cardinals, and Toronto Blue Jays
Public education
The borough is served by the Catasauqua Area School District. Catasauqua High School serves grades 9-12. Catasauqua Middle School serves grades 5-8. Sheckler Elementary School serves grades K-4.
References
1907 establishments in Pennsylvania
Boroughs in Northampton County, Pennsylvania
Boroughs in Pennsylvania
Populated places established in 1731 | {
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Meet the People of Let's Get Real!
National Contact
Name (click for Bio)
Founder and President Dr. Dan Holt LGR@epix.net
Director--Australia Arthur Boucaut-Jones JP anc@internode.on.net
Georgia Ms. Melanie Beaver mcb8461@fc.dekalb.k12.ga.us
Illinois Dr. Carolyn Anderson carand@niles-hs.k12.il.us
Kansas Ms. Margo Hosie mhosie@usd273.k12.ks.us
Minnesota Dr. Bill Keilty bkeilt@splkpark.k12.mn.us
Nebraska Ms. Jennifer Selting Bauer jbauer@westside66.org
New Hampshire Ms. Carol Foley cfoley@Londonderry.org
New Jersey Ms. Jennifer Liss jliss@mansfieldschool.com
Ohio Mr. Jason Holt jasontheholt@hotmail.com
Oregon Mr. Larry Miller LarryM@coos-bay.k12.or.us
Pennsylvania Dr. Colleen Willard-Holt cxw20@psu.edu
Wyoming Ms. Cindy Reardon reardon@ccsd.k12.wy.us
Program Coordinators
PPL Corporation
Ms. Margaret Welker mewelker@pplweb.com
Ms. Criss Kerkendall
Ms. Joni Pfautz
Mr. Larry Campbell calbe@hersheys.com
jpfautz@hersheys.com
Mr. Ray Yoder ryoder@visteon.com
Dr. Dan Holt
Let's Get Real™
Founder and President
Dan G. Holt, Ph.D. (formerly an assistant professor of Educational Psychology at Penn State-Harrisburg) most recently he served as the Midwest Regional Director for the College Gifted Program, and currently works as President of Let's Get Real™ and as an educational consultant.
Dr. Holt holds a Bachelor of Fine Arts degree from the University of Oklahoma, a Masters in Education in Counseling and Psychology from the State University of West Georgia, and obtained his Ph.D. in Educational Psychology, with an emphasis in education of gifted students, from Purdue University. He is the author of Cartoon Thinking, a book that teaches both the thinking processes involved in and the art of cartooning. He has published articles in Gifted Child Today, International Journal of Humor Research, Gifted Education Press Quarterly, the Illinois Association for Gifted Journal, and the Kappan.
He is co-author of the book Applying Multiple Intelligences to Gifted Education: I'm Not Just an IQ Score! and the chapter on Assessment Tools for Counselors in the book Counseling the Gifted and Talented. He is the illustrator of the book Quotes for Kids. Dr. Holt has presented at dozens of state, regional, national, and international conferences and invited workshops.
[FrontPage Save Results Component]
Arthur Boucaut-Jones JP (Bouc)
Principal, Adelaide Nautical College
Director, Let's Get Real-Australia
Bouc has been active in the development of Outdoor and Environmental Education in South Australia for many years. He trained as a Physical Education teacher and taught in both primary and high schools for several years prior to taking a government trainer consultant position to work with teachers taking their students outside of the school property on excursions, camps and educational tours. He then moved on into eco-adventure tourism operating a residential conference centre with coaches and boat cruises into the world heritage Coorong National Park. Education groups of all ages became regular clients of this business. Having disposed of the tourism business, the development of Adelaide Nautical College as a residential international senior high and undergraduate college with the option of school ship experiences for young people has become a new and exciting challenge for Bouc. Bouc is also working to promote LGR in Australia and the South Western Pacific. More details can be found on www.anc.yachting.org.au
Dr. Colleen Willard-Holt
Associate Professor of Education
State Coordinator for Pennsylvania
Dr. Colleen Willard-Holt obtained her Ph.D. in Educational Psychology at Purdue University, West Lafayette, IN with emphases in Qualitative and Quantitative Research Methodologies, Assessment, and Gifted Education in 1993. She received her Masters of Science in Education from The Johns Hopkins University, Baltimore, MD in 1983. She received her Bachelor of Science in Special Education & Elementary Education from the University of Wisconsin in 1979.
Dr. Willard-Holt is currently an Associate Professor of Education at Penn State University in Harrisburg. Her teaching experience includes ten years experience in teaching graduate and undergraduate courses in assessment, learning theories, general teaching methods, professional development, higher order thinking, gifted education, and special education. In addition, she has ten years experience in teaching K-12 gifted education, high school special education, and high school math. Dr. Willard-Holt has published nine refereed journal articles, one refereed monograph, two book chapters, two curriculum handbooks, fifteen non-refereed articles, eight book reviews, and two ERIC documents, all in the areas of gifted education and teacher education. She has made over one hundred presentations to local, regional, state, national and international audiences on the topics of gifted education, assessment, brain-based education, and teacher education.
Melanie C. Beaver, Ed.S.
Teacher of Gifted at DeKalb Elementary
School of the Arts at Hooper Alexander
3414 Memorial Drive
Let's Get Real™ State Coordinator for Georgia
Melanie Beaver is a gifted teacher in the DeKalb County School System. She has taught school for over 8 years. The first 5 years of her career
were spent teaching sixth grade science at E.L. Bouie, Sr. Traditional Theme School. For the last three years, she has taught gifted students in grades K-7. She currently teaches gifted students in grades sixth and seventh at DeKalb Elementary School of the Arts. In addition to educating students inside of the classroom, Ms. Beaver enjoys sponsoring Jr. Beta Club, 4-H Club, The Stock Market Game, and Science Olympiad. Ms. Beaver had be nominated by her peers for Teacher of the Year and she has presented at the Georgia Science Teacher's Association Annual Conference for three years. She was recognize as a Science Teacher of Promise in 1999 during her first year of teaching. During her spare time, she enjoys reading, exercising, traveling, and dining out. Ms. Beaver is currently working on her Ed. D in Educational Leadership at Georgia Southern University.
Carolyn S. Anderson, Ph.D.
Assistant Superintendent - Curriculum and Instruction
Niles Township High Schools
7700 Gross Point Road, Skokie, IL 60077
Let's Get Real™ State Coordinator for Illinois
Margo Hosie
K-12 Gifted Consultant
U.S.D. #273
Beloit, Kansas
Let's Get Real™ State Coordinator for Kansas
Hosie holds a Bachelor of Arts Degree in Education from the University of Nebraska at Kearney and a Master of Science Degree with an emphasis in Gifted Education from Kansas State University. The 2004-2005 school year is her 13th year with Beloit schools.
Cindy Reardon
Teacher of the Gifted
Sage Valley Jr. High
1000 Lakeway Road, Gillette, WY 82718
Let's Get Real™ State Coordinator for Wyoming
Cindy is currently in her third year teaching Gifted and Talented and academic competition classes at a 7-9 jr. high. She coaches and sponsors academic competitions including: Let's Get Real, Destination Imagination, Science Olympiad, Science Fair, History Day, Knowledge Master, Geography Bee, and Young Authors. Prior to her jr. high assignment Cindy taught Gifted and Talented elementary students for ten years. She has a bachelor's degree in elementary education from the University of Wyoming and a Masters degree in Gifted Education from the University of Connecticut. She also teaches a Gifted and Talented class for teachers through the University of Wyoming. During her spare time, Cindy enjoys flower gardening, golf, reading, and snowmobiling.
Jennifer Selting Bauer
EY Coordinator
Westside Community Schools
9801 West Center
jbauer@westside66.org
Let's Get Real™ State Coordinator for Nebraska
Jennifer Selting Bauer currently works for Westside
Community Schools coordinating the
services for students in the Excellence in Youth
Program at Oakdale and Sunset Hills Elementary
Schools. In addition, Jennifer serves on the board
for the NE Association for the Gifted. She holds a
Masters degree in Gifted Education from University of
Connecticut. In her spare time she enjoys spending
time with her family and friends, traveling, and
Ms. Carol Foley
Let's Get Real™ State Coordinator for New Hampshire
Ms. Jennifer Liss
Gifted Education Coordinator & Teacher
Mansfield Township School District
200 Mansfield Rd. East, Columbus, NJ 08022
Let's Get Real™ State Coordinator for New Jersey
Jennifer Liss is the coordinator and teacher of the Gifted Education Programs for Kindergarten through Sixth grade in Mansfield Township School District of Columbus, NJ. Highlights of these programs include technology integration and applications, such as Web Design and Multimedia, community service learning projects, and mathematics applications. Students participate in Destination ImagiNation, Continental Mathematics League, NJAGC Contests, NSTA/Toshiba ExploraVision, Northern Burlington County Gifted Consortium, and John Hopkin's Center for Talented Youth. Jennifer holds both a Bachelor of Arts in Elementary Education and a Master's degree in Education from the University of Florida and specializes in Educational Technology. Outside of school she is working towards an additional graduate degree in Leadership in Technology Integration from Pennsylvania State University.
Jason A. Holt, MBA
Let's Get Real™ State Coordinator for Ohio
Mr. Holt, currently a consultant with American Management Systems Inc., obtained his Masters of Business Administration from the University of Oklahoma in 1999. He received his bachelors of science – Zoology, cum laude in 1997; also from the University of Oklahoma. In 1990, Mr. Holt spent several months in the USSR as a student ambassador. 1998 found Mr. Holt spending time in the Africa and Caribbean (AFCAR) section of Barclays Bank's while working in London, England. Mr. Holt, was an active participant in the Gifted and Talented program at Jenks High School in Jenks, Oklahoma, and has participated in several academic competitions, as well as volunteered both as a tutor and soccer coach.
Mr. Larry Miller
Coos Bay, Oregon
Let's Get Real™ State Coordinator for Oregon
Larry Miller is currently an Alternative Education Teacher for Coos Bay Public Schools. He teaches AIMS, (Alternative Instruction for Middle School) and IPASS, (Interim Program for Academic and Social Skills). These are two programs in the Harding Learning Center, which houses the majority of Coos Bays Public Schools' alternative education. Mr. Miller enjoys the Lets Get Real Competitions with his students along with the Stock Market Game and other project oriented activities that Harding Learning Center focuses on. Prior to this Mr. Miller taught for Jefferson School District 14J for four years. He taught alternative education under Project Connect for 2 years, and taught a mainstream class at Jefferson Middle School for 2 years. His favorite subject is mathematics and he enjoys integrating music throughout subject instruction. In 1999, Mr. Miller received his Masters Degree in Early Childhood/Elementary Education from Rutgers University in Central New Jersey, where he grew up. Larry enjoys snowboarding, swimming, biking, and hiking. He has a fondness for music, (playing guitar, as well as synthesizing and processing electronic sounds), and owns a Deejay company through which he hosts parties such as weddings and proms. He has a Black Lab/Great Dane dog named Bennie who is his best buddy.
Ms. Margaret E. Welker
Education Relations Director
As PPL's Education Relations Director, Meg Welker develops and implements initiatives that align with corporate strategies and that address the needs and interests of community and academic partners. Under the umbrella of Education Relations are five broad strategies: school-to-career/workforce development, strategic alliances, employee volunteerism, energy education and environmental education. Prior to her appointment to this current position in December of 2003, she was the naturalist at PPL's Lake Wallenpaupack since 1996.
Ms. Welker was appointed by Governor Tom Ridge as a member of the Pennsylvania Department of Conservation and Natural Resources Advisory Council in 2000. She also serves on the Education Task Force of the Pennsylvania Biodiversity Partnership, The Pennsylvania Center for Environmental Education's education task force and is PPL's representative to the Northeast PA Environmental Partners.
Ms. Welker received a B.S.degree in Environmental Resource Management from The Pennsylvania State University in 1995. She continues to volunteer as a board member and scholarship committee member with her local Penn State Alumni Chapter. She also enjoys traveling on her motorcycle in her spare time.
Senior Business Systems Analyst - Information Services
Let's Get Real™ Program Coordinator
Joni has worked for The Hershey Company for a total of 20 years in various administrative support and desktop/web publishing positions. Currently, she is a member of The Hershey Company's Information Services department, where she has worked for 8 years as the technical webmaster for The Hershey Company's web sites. Joni also acts as webmaster for www.LGReal.org.
Mr. Larry Campbell
Staff Scientist - Ingredients Research (Retired)
Larry B. Campbell retired from his position as Staff Scientist in The Hershey Company Research and Development Department, Hershey, PA in February, 2002. He has a BS degree in AgBioSci ('63) and an MS degree in Dairy Science ('67) both from The Pennsylvania State University. His career includes four years on the research staff in the Food Science Department at Penn State followed by 32 years in various R&D positions at The Hershey Company. His expertise includes chocolate products, dairy products, syrups and toppings. Larry has been responsible for a number of new products and product reformulations in the U.S., Canadian and International markets. These products include SYMPHONY Milk Chocolate, Hershey's Chocolate Syrup, Brown Cow Syrup (International), Top Scotch Topping (Canada), Hershey's Lite n' Creamy Milk Chocolates (International) and Hershey's Chocolate Milk.
Larry's professional affiliations include the American Association of Candy Technologists, American Dairy Science Association and the Institute of Food Technologists (past chair). He serves on the Board of Directors of the PMCA and International Confectionery Association where he also served as President and Chairman of the Board, and the Chair of the Research, Scholarship and Education Committees. He is currently President of the Penn State College of Agricultural Sciences Alumni Society and also serves on the Penn State Alumni Council. Over the years Larry has held leadership positions in a number of community and church organizations. He has been involved with the United Way of the Capital Region serving as a past Chair of the Business and Industry Division 2 and as a Loaned Executive. He has also served as a member of Governor Ridge's Technology 21 Initiative and participated in the Capital Area Math/Science Alliance. | {
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Humana citogenetika je grana citogenetike se bavi istraživanjem čovjekovih kromosoma. Ona se kao poseban istraživački pravac u okviru humane genetike relativno kasno etablirala. Za to su prije svega bile odgovorne metodičke poteškoće koje su nadiđene tek pedesetih godina prošlog stoljeća. Tek je 1956. godine bilo moguće konačno utvrditi točan broj kromosoma kod čovjeka. Pritom se pokazalo da čovjek ima 46 kromosoma, a ne 48 - koliko imaju stanice čovjekolikih majmuna što je bilo do tada očekivana pretpostavka.
U godinama koje su slijedile znatno su poboljšane metode prikazivanja kromosoma tako da su uskoro mogle biti dijagnosticirane prve numeričke kromoske aberacije (npr. Trisomija 13 - Patauov sindrom, 18 - Edwardsov sindrom, 21 - Downov sindrom). Ipak su prilično dugo ostale naučno nepriznate mnoge strukturne kromosomske aberacije jer je aberacije u okviru jednog kromosoma bilo moguće dokazati samo ako se prilikom aberacije događala značajna promjena dužine kromosoma ili centromera. Tek razvojem posebnih tehnika bojenja početkom 1970-ih godina bilo je moguće u okviru jednog hromozoma učiniti vidljivim različite mustre u vidu traka. Time je bilo moguće uočiti i standardizirati čak i manje strukture u jednom kromosomu.
Novi veliki razvojni skok citogenetika je doživjela osamdesetih godina razvojem tzv. "fluorescenog in situ hibridiziranja (FISH)". U ovom trenutku razvoj se karakterizira prije svega masovnom primjenom tzv. "multikolor FISH". To podrazumijeva istovremenu upotrebu više DNK sondi markiranih različitim fluoroscentnim bojama. Uz spomenutu metodu još je značajno "komparativno genom hibridiziranje (CGH)". Veći metodski napredak uključivanjem molekularno-bioloških tehnika doprinijelo je brojnim novim spoznajama, posebno u citogenetici tumora i omogućilo rutinsku dijagnostiku u medicini.
Genetika | {
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package com.evolveum.midpoint.web.component.wizard.resource.dto;
import com.evolveum.midpoint.xml.ns._public.common.common_3.ResourceObjectTypeDefinitionType;
import javax.xml.namespace.QName;
import java.io.Serializable;
import java.util.ArrayList;
import java.util.List;
/**
* @author shood
* */
public class SchemaHandlingDto implements Serializable{
public static final String F_OBJECT_TYPES = "objectTypeList";
public static final String F_SELECTED = "selected";
public static final String F_SELECTED_OBJECT_CLASS = "selectedObjectClass";
private List<ResourceObjectTypeDefinitionTypeDto> objectTypeList = new ArrayList<>();
private ResourceObjectTypeDefinitionType selected;
private String selectedObjectClass;
private List<QName> objectClassList;
public List<ResourceObjectTypeDefinitionTypeDto> getObjectTypeList() {
return objectTypeList;
}
public void setObjectTypeList(List<ResourceObjectTypeDefinitionTypeDto> objectTypeList) {
this.objectTypeList = objectTypeList;
}
public ResourceObjectTypeDefinitionType getSelected() {
return selected;
}
public void setSelected(ResourceObjectTypeDefinitionType selected) {
this.selected = selected;
if(selected == null){
selectedObjectClass = null;
} else if(selected.getObjectClass() != null){
selectedObjectClass = selected.getObjectClass().getLocalPart();
} else if(selected.getObjectClass() == null){
selectedObjectClass = null;
}
}
public List<QName> getObjectClassList() {
return objectClassList;
}
public void setObjectClassList(List<QName> objectClassList) {
this.objectClassList = objectClassList;
}
public String getSelectedObjectClass() {
return selectedObjectClass;
}
public void setSelectedObjectClass(String selectedObjectClass) {
this.selectedObjectClass = selectedObjectClass;
if(selectedObjectClass == null && selected != null){
selected.setObjectClass(null);
}
if(selected != null){
for(QName q: objectClassList){
if(q.getLocalPart().equals(selectedObjectClass)){
selected.setObjectClass(q);
}
}
}
}
}
| {
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We service you with different technologies!
BEAM International Group is a network of companies in different branches located in Doha - Qatar.
Copyright © 2010 BEAM International Group W.L.L. - All rights reserved. | {
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\section{I.~Introduction}
In general, based on their electronic band structures, solids may be classified as either metal or insulator/semiconductor.
These two different classes may combine in single magnetic crystals, the so-called half-metallic magnets, in which one spin part of the band structure is metallic and the other semiconducting.
Half-metallic magnets have promising uses as spintronics devices because a 100\% spin polarization is expected at the Fermi level.
From first-principle calculations, some of the Heusler alloys have been predicted to possess half-metallic band structures~\cite{Groot1983,Kubler1983,Ishida1995,Picozzi2002}.
Among them, Co$_2$MnSi and Co$_2$MnGe are prototypes that are predicted to exhibit a relatively large minority spin gap, which is preserved as long as they are in the ordered phase.
Later, it was theoretically pointed out that in-gap minority spin states appear when Co antisite defects are created at Mn sites, resulting in much degraded spin polarizations~\cite{Picozzi2004}.
Indeed, Co$_2$MnSi with excess Mn, which prevented Co atoms from occupying Mn sites, had a substantially improved tunneling magnetoresistance ratio~\cite{Ishikawa2009,Liu2012}.
Incorporating Fe atoms at the original Mn sites placed the Fermi level in the center of the minority spin gap and further enhanced the magnetoresistance that relies on the spin polarization by up to 2610\% at 4.2 K~\cite{Moges2016}.
From recent experimental studies, another striking event happens when the Ge site is substituted with Ga.
A huge anomalous Nernst effect takes place~\cite{Sakai2018,Guin2019,Hu2020} caused by a high Berry flux that originates from the bulk band crossings located near the Fermi level ($E_\mathrm{F}$)~\cite{Xiao2006,Xiao2010}.
We thus claim that the locus of $E_\mathrm{F}$ is quite important when engineering the bulk and interface band structures by controlling defects and tuning.
The process of computational material design and experimental confirmation has to be iterated to realize the best materials with extremely high functionality.
To confirm their highly spin-polarized conducting electrons, numerous experiments employing for example point-contact Andreev reflection spectroscopy~\cite{Ritchie2003} and spin-resolved photoelectron spectroscopy~\cite{Fetzer2013,Jourdan2014,Andrieu2016,Guillemard2019_1,Guillemard2019_2} were performed.
However, the three-dimensional nature of this crystal family has prevented us from approaching their bulk band structures mainly because of the surface and interface sensitivities of these techniques.
We note that truly bulk-sensitive hard X-ray photoelectron spectroscopy actually helps in studying the valence band density of states (DOS) of bulk and buried interface, although no momentum-resolved information that is a key to understanding the physical properties has ever been obtained~\cite{Brown1998,Miyamoto2009,Ouardi2011,Kozina2014}.
A recent soft X-ray angle-resolved photoelectron spectroscopy (ARPES) study of Co$_2$MnSi films, each with Al-O$_x$ capping layer, probed mainly the surface and/or interface states, blocking the band structure underneath~\cite{Lidig2019}.
Therefore, to suppress the influence of the results from the surface and/or interface states, it is mandatory to prepare a clean surface by cleaving the crystal under an ultrahigh vacuum.
However, because of the robust 3D nature of its crystal structure [Fig.~\ref{fig:kz}(a)], it is generally difficult to produce a flat surface and consequently multiple steps remain [Fig.~\ref{fig:kz}(b)].
In this \textit{Letter}, we report the direct evidence of the half-metallic band structure of Co$_2$MnGe and the presence of multiple Weyl cones using ARPES with a beam spot size of $10 \times 10~\mu$m.
It enables a single domain with a flat surface to be proved for ARPES\@.
A high quality single crystalline sample was grown by the Bridgeman method (see Sec.~S1 of Supplementary Material for the detailed sample growth conditions).
The composition of the sample was Co: 49.3, Mn: 24.9, and Ge: 25.8 (at.~\%) as determined by energy dispersive X-ray microanalysis (EDX), which is stoichiometric enough to prevent Co anti-site defect and preserve its half-metallicity.
The saturation magnetic moment evaluated from the magnetization curve measured with a SQUID at 5~K was 115.3~emu/g and converted to 4.96~$\mu_\mathrm{B}/\mathrm{f.u.}$, being close to an integer number and consistent with the expected value obtained from the Slater-Pauling rule~\cite{Galanakis2002}.
From differential scanning calorimetry, the Curie temperature of the present specimen was 913~K, being comparable with our previous experimental result~\cite{Okubo2010}.
ARPES measurements were performed at BL25SU at SPring-8.
Circularly polarized synchrotron radiation beam was used to maximize the number of bands that appear in the ARPES results.
The beam spot size was adjusted to $10 \times 10~\mu$m.
The energy and angular resolutions for ARPES were set to $<80$~meV and $<0.2^\circ$, respectively.
A clean $(001)$ surface of Co$_2$MnGe was obtained by cleaving in an ultra-high vacuum (pressure $<2\times 10^{-8}$~Pa).
All measurements were conducted at a temperature of $\sim30$~K.
The analyzer slit was set along the $k_x$-axis [parallel to $[110]$; see Figs.~\ref{fig:kz}(a,~b)].
All first-principles density-functional calculations were performed using the WIEN2k program code~\cite{wien2k}.
We used the spin-polarized generalized gradient approximation~\cite{Perdew1997} (see Sec.~S2 for further calculation details).
The Coulomb interaction $U$ was not considered because previous studies indicated that $U$ plays a minor role in Co$_2$MnGe~\cite{Tsirogiannis2015,Sharma2016}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fig1.pdf}
\caption{
(a) Crystal structure of Co$_2$MnGe.
(b) Experimental setup.
(c-j) ARPES images v.s.~photon energy.
(k,~l) Constant energy surface at (k) $E_\mathrm{B}=0$ eV ($E_\mathrm{F}$) and (l) $E_\mathrm{B}=0.28$~eV in $k_x$-$k_z$ plane from ARPES denoted with high symmetry lines.
Green (red) lines correspond to $k_z=0~(2\pi/a)$ at $k_x=0$.
(m) Calculated constant energy surfaces ($E_\mathrm{B}=0.2$~eV).
(n) BZ and $k_x$-$k_z$ surface (shaded by green color).
Red curves correspond to $k_z$ line of each ARPES images at fixed photon energies.
}
\label{fig:kz}
\end{figure}
From the photon energy ($h\nu$) dependence of the ARPES spectra [Figs.~\ref{fig:kz}(c-j)], we see that the ARPES image evolves with photon energy (also see the Supplemental Movie).
At $h\nu=435$~eV, a band that disperses downwards from its peak at $E_\mathrm{B}=0.1$~eV appears at $k_x=0$ [overlaid with white dashed line in Fig.~\ref{fig:kz}(c)].
The photoelectron intensity from this band weakens at $h\nu=470$~eV [Fig.~\ref{fig:kz}(d)], and another feature emerges at $h\nu=500$~eV [B1 in Fig.~\ref{fig:kz}(e)] and its intensity is maximized at $h\nu=535$~eV [Fig.~\ref{fig:kz}(f)].
Interestingly, the bands cross $E_\mathrm{F}$ (B1, B2) and two bands, each with a downward dispersion, emerge on both sides of the $E_\mathrm{F}$ crossing feature (white dashed lines).
Increasing the photon energy further, the band indicated by the black arrow moves to $k_x=0~\text{\AA}^{-1}$.
Moreover, a steeply dispersing band crosses $E_\mathrm{F}$ [red arrow labeled with B2 in Fig.~\ref{fig:kz}(f)] and moves away from $k_x=0~\text{\AA}^{-1}$ with higher photon energies.
A spectral weight of the band, indicated by the red arrow, is maximized at $h\nu=620$~eV [Fig.~\ref{fig:kz}(i)].
The downward-dispersing band re-emerges above $h\nu=620$~eV, and its intensity is maximized at $h\nu=640$~eV [dashed line in Fig.~\ref{fig:kz}(j)].
We realized that the circular features in the $k_x$-$k_z$ constant energy map at $E_\mathrm{B}=0.28$~eV [Fig.~\ref{fig:kz}(l)] stem from the downward-dispersing bands near the $\Gamma$ point.
Importantly, they are not seen at $E_\mathrm{F}$ [Fig.~\ref{fig:kz}(k)].
This demonstrates that our ARPES measurement tracks the band structure along the out-of-plane momentum ($k_z$) line.
We have determined an inner potential of $V_0=18$~eV from the photon energy dependence of the ARPES spectra with downward dispersions.
The computed $k_x$-$k_z$ constant energy map [Fig.~\ref{fig:kz}(m)] reproduces well the observed features with minority spin character.
\begin{figure*}
\centering
\includegraphics[width=0.8\textwidth]{fig2.pdf}
\caption{
(a,~b) ARPES images measured with a photon energies ($h\nu$s) of 435 and 535~eV [same $k$ cuts with Figs.~\ref{fig:kz}(c,~f), respectively].
(c) Second derivative momentum distribution curve along $E_\mathrm{B}=0$~eV line ($E_\mathrm{F}$) in panel (b).
(d) Calculated band dispersions along $\Gamma$-$K$-$X_\mathrm{2nd}$ line.
Red (blue) color corresponds to majority (minority) spin components.
(e) Calculated band dispersions around $K$ and $X$ point.
(f) Measured ARPES band dispersions along $\Gamma$-$X$ line. [Red dashed line in Fig.~\ref{fig:kxky}(c)]
(g) ARPES image normalized by integrated intensity of momentum distribution curve from panel (f).
(h) Calculated band dispersions along $\Gamma$-$X$ line.
(i) BZ and high symmetry points.
Red line corresponds to $k_z$ line of panel (f).
}
\label{fig:BD}
\end{figure*}
We extracted typical ARPES images acquired at the incident photon energies of 435~eV and 535~eV from Fig.~\ref{fig:kz} along the $K$-$\Gamma$-$K$ and $K_\mathrm{2nd}$-$X$-$K_\mathrm{2nd}$ lines [Figs.~\ref{fig:BD}(a,~b), respectively].
A parabolic band with a downward dispersion dominates near $k_x=0~\text{\AA}^{-1}$ at $h\nu=435$~eV\@.
At $h\nu=535$~eV, a spectral feature near $E_\mathrm{F}$ around $k_x=0~\text{\AA}^{-1}$ is modified considerably.
This feature corresponds to B1 in Fig.~\ref{fig:kz} and consists of two separated bands that are more clearly identified by the second derivative of the momentum distribution curve at $E_\mathrm{F}$ [indicated by inverted triangles in Fig.~\ref{fig:BD}(c)].
By combining this plot with B2 in Fig.~\ref{fig:kz}, we conclude that the three bands cross $E_\mathrm{F}$.
Furthermore, we compared these ARPES images with calculated band dispersions along the $\Gamma$-$K$-$X_\mathrm{2nd}$ line [Fig.~\ref{fig:BD}(d)].
The calculated results have been shifted in energy by $-100$~meV (40~meV) toward higher $E_\mathrm{B}$ in the majority (minority) spin channel to adjust to the ARPES results.
The downward-dispersing ARPES band [Fig.~\ref{fig:BD}(a)] fits very well with the theoretical minority spin band.
A band that disperses upwards and crosses $E_\mathrm{F}$ [Fig.~\ref{fig:BD}(b)] is ascribed to the three computed majority spin bands.
For the experimental band dispersion along the $\Gamma$-$X$ line [Fig.~\ref{fig:BD}(f)], the raw ARPES intensity has been normalized to enhance its visibility using the integrated intensity of the momentum distribution curve [Fig.~\ref{fig:BD}(g)].
From a comparison with the calculated band dispersion along the $\Gamma$-$X$ line [Fig.~\ref{fig:BD}(h)], we claim that the minority and majority spin components contribute respectively to the observed bands that disperse downwards near the $\Gamma$ points and the band that disperses upwards and crosses $E_\mathrm{F}$.
\begin{figure*}
\centering
\includegraphics[width=0.8\textwidth]{fig3.pdf}
\caption{
(a-e) Constant energy surfaces in $k_x$-$k_y$ plane mapped by ARPES with photon energy of 535~eV\@.
(f-j) Characteristic features of constant energy surfaces depicted from panels (a-e).
(k-o) Calculated constant energy surfaces with an energy offset of $-100$~meV (40~meV) toward higher $E_\mathrm{B}$ in a majority (minority) spin channel.
(p) BZ and $k_x$-$k_y$ surface (shaded by green color).
}
\label{fig:kxky}
\end{figure*}
Figures~\ref{fig:kxky}(a-e) shows the experimental constant energy surfaces at $E_\mathrm{B}=0$, 0.1, 0.2, 0.3 and 0.4~eV\@.
The characteristic features in Figs.~\ref{fig:kxky}(a-e) are re-drawn in Figs.~\ref{fig:kxky}(f-j).
The calculated constant energy surfaces are shown in Figs.~\ref{fig:kxky}(k-o).
Constant energy surfaces with four-fold symmetry are observed that correlate with the symmetry of the $(001)$ plane of the crystal.
Four elliptical pockets [red solid line in Fig.~\ref{fig:kxky}(f)] are seen at $E_\mathrm{B}=0$~eV and diminish as $E_\mathrm{B}$ increases.
At $E_\mathrm{B}=0.4$~eV [Figs.~\ref{fig:kxky}(e,~j)], we find four circles that diminish with decreasing $E_\mathrm{B}$ and ultimately disappear at $E_\mathrm{F}$.
In a comparison with the calculated results [Figs.~\ref{fig:kxky}(k-o)], we determined that these circles are ascribable to the minority spin components.
All the other features are well explained by the bands of the majority spin channel.
This means that only the majority spin bands cross $E_\mathrm{F}$ and no minority spin band exists, signifying that a half-metallic band structure is realized in this crystal.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fig4.pdf}
\caption{
(a-e) ARPES images with changing photon energy around $h\nu=535$~eV\@.
Panel (d) correspond to blue dotted box in Fig.~\ref{fig:kz}(g).
(f-j) Second derivative ARPES images from panels (a-e).
(k-o) Calculated band dispersion with an energy offset of $-100$~meV (40~meV) toward higher $E_\mathrm{B}$ in a majority (minority) spin channel.
Momentum path is simulated from kinetic energy of photoelectron and considering $k_z$ broadening of $\delta k_z \sim \pm 0.2~\text{\AA}^{-1}$.
(p-t) Red curves are correspond to each ARPES images and calculated results.
}
\label{fig:cross}
\end{figure}
Finally, we discuss the topological aspects of the band structure of Co$_2$MnGe.
Weyl fermions appear in materials that break inversion symmetry~\cite{Xu2015} or time-reversal symmetry~\cite{Kuroda2017}.
It is expected that the Weyl fermions emerge also in Co$_2$MnGe as a result of time-reversal symmetry breaking as has recently been predicted for some of the Heusler alloys~\cite{Chang2017,Belopolski2019}.
Figures~\ref{fig:cross}(a-e) show the ARPES images along the $K_\mathrm{2nd}$-$X$-$K_\mathrm{2nd}$ line acquired at $h\nu=535$-570~eV\@.
To exclude the effect of a momentum constant background, we differentiated the ARPES intensity twice along the energy axis in Figs.~\ref{fig:cross}(f-j).
We see again a band that disperses upwards and crosses $E_\mathrm{F}$, at $h\nu=535$~eV [Figs.~\ref{fig:cross}(a,~f)].
The inner band exhibits a strong intensity whereas the outer band [small red arrow in Fig.~\ref{fig:cross}(a)] is weak.
Marked by two red arrows A and B, the band dispersing downwards intersects other bands dispersing upwards.
These two crossings are persistently seen in the image taken at $h\nu=550$~eV [Fig.~\ref{fig:cross}(h)].
At $h\nu=560$~eV, the crossing of the bands involving the outer band is observed even more clearly at $E_\mathrm{B}=0.4$~eV\@.
Above $h\nu=570$~eV, all crossing points disappear.
To compare the ARPES result with the theoretical band dispersions more rigorously, we considered real momentum ``paths'' using $k_z=\sqrt{2m_e(h\nu-W+V_0)/\hbar^2-k_x^2}~(W=4.5~\text{eV})$.
It is also necessary to consider $k_z$ broadening effects that stem from the limited probing depth of the photoelectrons ($\delta k_z=\pm0.2~\text{\AA}^{-1}$).
In Figs.~\ref{fig:cross}(k-o), we show the computed band dispersions integrated in the $\delta k_z$ window together with the band dispersions along the real $k$-lines [Figs.~\ref{fig:cross}(p-t)].
We find that the theoretical ARPES images reproduce the persistent band crossings for $h\nu=540$-560~eV and its disappearance above $h\nu=570$~eV, although the crossings of the line dispersions are already broken above 550~eV\@.
Our first-principles calculation predicts two band crossings around the $X$ point [A and B in Fig.~\ref{fig:BD}(e)].
Further theoretical analysis tells us that all of them form nodal-lines [Figs.~S3(a-c)].
We, therefore, conclude that crossing points A and B correspond to the type-II Weyl points produced from the tilted cones.
We note that Co$_2$MnGa exhibits similar crossing points [Figs.~S3(d-f)] and that these crossings may generate a high Berry flux~\cite{Chang2017,Belopolski2019}.
The gigantic anomalous electrical and thermal conductivities that give rise to the anomalous Hall effect and the Nernst effect emerge when the gap opens at the crossing point and $E_\mathrm{F}$ is tuned inside the gap.
Because the crossing points A and B are located above $E_\mathrm{F}$ for Co$_2$MnGa, further carrier tuning is required to improve these anomalous conductivities.
We propose that the substitution of Ge atoms into the Ga sites may improve the anomalous transport properties.
Our study with soft X-ray ARPES confirms that these crossing points are of bulk origin and are maintained even for the end material, Co$_2$MnGe.
In conclusion, we performed ARPES on a full-Heusler-type Co$_2$MnGe bulk crystal utilizing micro-spot-size soft X-ray synchrotron radiation.
No contribution of the minority spin band structures to the Fermi surface was observed.
All the observed Fermi surfaces were reproduced by the calculated results for the majority spin channel.
Moreover, two topological Weyl cones were clearly observed indicating Berry flux sources.
Our findings provide strong evidence of half-metallicity coexisting with multiple Weyl cones of the Co$_2$MnGe alloy.
They also shed light on the currently elusive spin-gapless Heusler semiconductors, for which one spin part is semimetallic and the other semiconducting.
Tuning the highly spin-polarized carriers is possible via an electric field if that becomes feasible.
We finally remark that by choosing appropriate elements the Heusler alloys comprising more than three elements give rise to various types of physical behaviors such as the magneto-caloric effect, thermoelectricity and superconductivity, all of which rely on their band structures.
Micro-spot ARPES with soft X-ray synchrotron radiation beam affords opportunities to realize highly functional materials for such alloys.
\begin{acknowledgements}
This work was financially supported by KAKENHI (Nos.~17H06152, 17H06138, 18H01690, and 18H03683).
The soft X-ray ARPES experiment was performed with the approval of JASRI (Proposal No.~2019A1548).
Micro-ARPES instruments was developed by Photon and Quantum Basic Research Coordinated Development Program from MEXT\@.
T.Y. was financially supported by Grants-in-Aid for JSPS Fellows No.~18J22309.
We sincerely thank T.~Sugawara and I.~Narita in Tohoku University for their help to make the single crystals and to perform the EDX experiment.
\end{acknowledgements}
\section{S\lowercase{ample preparations}}
The mother ingot of the polycrystalline Co$_2$MnGe was fabricated by induction melting in an argon gas atmosphere.
A single crystal with a diameter size of 12~mm and a length of about 30~mm was grown by the Bridgeman method.
The obtained ingot was annealed at 1273~K and slowly cooled down to room temperature.
The crystal orientation was checked using Laue's back-reflection method and the specimen was cut into stripes in the direction parallel to the \verb|<|100\verb|>|.
From energy dispersive X-ray spectroscopy, the sample's composition was evaluated to be Co:~49.3, Mn:~24.9, and Ge:~25.8 (at.~\%).
This is stoichiometric enough to prevent Co anti-site defect and preserve its half-metallicity~[4, 5].
\section{F\lowercase{irst-principle calculation}}
The muffin-tin approximation was used for the potential; the muffin-tin radius $R_\mathrm{MT}$ of each atom was taken to be $R_\mathrm{MT}^\mathrm{Co}=R_\mathrm{MT}^\mathrm{Mn}=2.30$~Bohr, and $R_\mathrm{MT}^\mathrm{Ge}=2.23$~Bohr.
The wave functions were expanded into spherical harmonics with integer $\ell$ ranging up to $\ell_\mathrm{max}=10$ in the muffin-tin spheres and by plane waves in the interstitial region with a cut-off value of $R_\mathrm{MT}^\mathrm{Ge}\cdot K_\mathrm{max}=7$.
The Fourier charge density was expanded up to $G_\mathrm{max}=12~\mathrm{Bohr}^{-1}$.
The $k$ space was divided into a uniform $21\times 21\times 21$ mesh.
These $RK_\mathrm{max}$, $\ell_\mathrm{max}$, $G_\mathrm{max}$ and $k$-points were sufficient to stabilize the shape of the DOS\@.
In this calculation, we set the lattice constants to $a=b=c=5.751~\text{\AA}$ and $\alpha=\beta=\gamma=90^\circ$ (the experimental values reported in Ref.~[26] of the main text), and the L2$_1$ ($Fm\bar{3}m$) phase with atomic positions of Co~$(0.25,0.25,0.25)$, Mn~$(0,0,0)$, and Ge~$(0.5,0.5,0.5)$.
\section{P\lowercase{hoton energy dependence of} ARPES \lowercase{spectra}}
As shown in Supplemental movie, we can see a quasi-continuous change in the ARPES band structure by changing incident photon energy.
This further clarifies that our ARPES measurement with variable incident photon energy in the soft X-ray region can track the band structure in three-dimension.
Here, we describe how the inner potential is determined from $h\nu$ dependence of ARPES spectra.
The vertical component of the wavenumber of the $N^\mathrm{th}$ $\Gamma$ point is expressed as $k_\perp=4\pi N/a$ using the lattice constant ($a$).
This is due to the vertical periodicity of BZ.
On the other hand, to estimate $k_\perp$ from the ARPES measurements, the information on the photoemission final state is necessary.
However, it is difficult to observe or rigorously calculate the band structure of the final state, so we assume that the final state is a free electron.
The parameter characterizing the free electrons is the inner potential ($V_0$).
Using the photon energy ($h\nu$) where the electronic structure at the $\Gamma$ point (in this case a convex parabolic band) is observed by ARPES measurements, there exists pairs of $N,V_0$ yielding $k_\perp=4\pi N/a=\sqrt{2m_e(h\nu-W+V_0)/\hbar^2}$.
Here, if there exists a natural number $N$ such that $V_0$ takes a reasonable value that fits to the realistic band structure.
In this particular case of Co$_2$MnGe, the valence band bottom is located at 12~eV below the Fermi level~[22] and the obtained value of $V_0~(=18~\mathrm{eV})$ is found to be reasonable when the work-function (energy distance between the Fermi level and the vacuum level) is considered.
Therefore, we can say that the inner potential is not only a fitting parameter but also an important physical quantity to determine whether the final state can be assumed as a free electron or not.
\section{C\lowercase{omputational simulation of nodal-lines}}
\begin{figure*}
\centering
\includegraphics[width=\columnwidth]{BD_p.pdf}
\caption{\textbf{Band crossing points on high-symmetry lines.}
\textbf{(a,~c)}~Band crossing points of (a)~Co$_2$MnGe and (c)~Co$_2$MnGa.
\textbf{(b,~d)}~Enlarged view of cross points E and F in panel~(a,~c), respectively.
\textbf{(e)}~Crystal structure of Co$_2$MnGe.
}
\label{fig:s_BD}
\end{figure*}
Figure~\ref{fig:s_BD} shows the theoretical band dispersions along high-symmetry lines of Co$_2$MnGe and Co$_2$MnGa.
We find some band crossing points labeled A to F in the vicinity of the $E_\mathrm{F}$.
Here, Band-1 and Band-2 form crossing points A, E, and F\@.
On the other hand, Band-2 and Band-3 generate crossing points B, C, and D\@.
We confirm that these degenerate points form nodal-lines (NLs) in three-dimensional $k$-space.
We calculated the size of band gap $(\Delta \varepsilon)$ between Band-1(2) and Band-2(3) in the whole BZ to visualize NLs.
Computational calculation gives us finite values of $\Delta \varepsilon$ even if there is a degenerate point.
$\Delta \varepsilon$ converges to zero with infinite BZ-mesh (the number of $k$ points in the BZ).
Figure~\ref{fig:s_BZmesh} shows BZ-mesh dependence of the crossing point A of Co$_2$MnGe.
Band gaps between Band-1 $(\varepsilon_1)$ and Band-2 $(\varepsilon_2)$ are calculated in each $k$ points with $\Delta \varepsilon (k)=|\varepsilon_1(k)-\varepsilon_2(k)|$, and shown in Figs.~\ref{fig:s_BZmesh}(e-h).
For $100\times100\times100$ mesh, the band dispersion is found to be unclear [Fig.~\ref{fig:s_BZmesh}(a)] and it is difficult to find out degenerate points from Fig.~\ref{fig:s_BZmesh}(e).
The minimum value of $\Delta \varepsilon$ gets smaller as the number of BZ-mesh increases.
For the finest mesh in our study ($5000\times5000\times5000$) used in the main NL calculation, we can see a crossing feature in the band dispersion [Fig.~\ref{fig:s_BZmesh}(d)] and the degenerate point is obvious from $\Delta\varepsilon(k)$ [Fig.~\ref{fig:s_BZmesh}(h)].
In the main NL calculation, we plot the $k$ point as a degenerate point that satisfies $\Delta \varepsilon(k) < 0.001$~eV in Fig.~\ref{fig:s_NL}.
\begin{figure*}
\centering
\includegraphics{BZmesh.pdf}
\caption{\textbf{BZ-mesh dependence of the band crossing feature near the point A of Co$_2$MnGe in Fig.~\ref{fig:s_BD}(a).}
\textbf{(a-d)}~Energy eigenvalues of Band-1 and Band-2.
\textbf{(e-h)}~Size of band gap between Band-1 and Band-2.
}
\label{fig:s_BZmesh}
\end{figure*}
Calculated degenerate points are shown in Fig.~\ref{fig:s_NL}.
We realize that all the degenerate points between Band-1(2) and Band-2(3) form one-dimensional lines.
Crossing points A to F in Fig.~\ref{fig:s_BD} are indicated by red arrow in Fig.~\ref{fig:s_NL}.
Our calculated NLs formed by Band-2 and Band-3 of Co$_2$MnGa [Fig.~\ref{fig:s_NL}(f)] reproduce previous theoretical results~\cite{Chang2017}, and we confirmed that Co$_2$MnGe has similar NLs [Fig.~\ref{fig:s_NL}(c)].
Furthermore, the degenerate points between Band-1 and Band-2 form another NL [Figs.~\ref{fig:s_NL}(a,~d)].
From ARPES, we identified crossing points A and B (Fig.~4) for which both of the corresponding NLs are formed between Bands-1 and -2 and between Bands-2 and -3.
\begin{figure*}
\centering
\includegraphics{NL.pdf}
\caption{\textbf{Theoretical simulation of NLs.}
\textbf{(a,~c)}~Degenerate points between (a)~Band-1 and -2 and (c)~Band-2 and -3 of Co$_2$MnGe.
\textbf{(b)}~Enlarged view around $W$ point in panel~(a).
\textbf{(d,~f)}~Degenerate points between (d)~Band-1 and -2 and (f)~Band-2 and -3 of Co$_2$MnGa.
\textbf{(e)}~Enlarged view around $W$ point in panel~(d).
Color correspond to binding energy of degenerate points.
Degenerate points in second BZ are shown in gray.
}
\label{fig:s_NL}
\end{figure*}
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\section{Introduction}
\subsection{Semidefinite programs} \label{subsec:sdp}
In the last decades, particularly since the work of Gr{\"o}tschel, Lov{\'{a}}sz, and Schrijver~\cite{gls:geometric}, \emph{semidefinite programs} (SDPs) have become an important tool for designing efficient optimization and approximation algorithms. SDPs generalize and strengthen the better-known \emph{linear} programs (LPs), but (like LPs) they are still efficiently solvable. The basic form of an SDP is the following:
\begin{align} \label{eq:SDP}
\max \quad &\mbox{\rm Tr}(CX) \\
\text{s.t.}\ \ \ &\mbox{\rm Tr}(A_j X) \leq b_j \quad \text{ for all } j \in [m], \notag \\
&X \succeq 0, \notag
\end{align}
where $[m]:=\{1,\ldots,m\}$.
The input to the problem consists of Hermitian $n\times n$ matrices $C,A_1,\ldots,A_m$ and reals $b_1,\ldots,b_m$. For normalization purposes we assume $\nrm{C},\nrm{A_j} \leq 1$.
The number of constraints is~$m$ (we do not count the standard $X\succeq 0$ constraint for this). The variable~$X$ of this SDP is an $n\times n$ positive semidefinite (psd) matrix.
LPs correspond to the case where all matrices are diagonal.
A famous example is the algorithm of Goemans and Williamson~\cite{GoemansWilliamson95} for approximating the size of a maximum cut in a graph~$G=([n],E)$: the maximum, over all subsets $S$ of vertices, of the number of edges between $S$ and its complement~$\bar{S}$. Computing MAXCUT$(G)$ exactly is NP-hard. It corresponds to the following integer program
\begin{align*}
\max \quad &\frac{1}{2}\sum_{\{i,j\}\in E}(1-v_iv_j)\\
\text{s.t.}\ \ \ & v_j\in\{+1,-1\}\quad \text{ for all } j \in [n],\end{align*}
using the fact that $(1-v_iv_j)/2=1$ if $v_i$ and $v_j$ are different signs, and $(1-v_iv_j)/2=0$ if they are the same.
We can relax this integer program by replacing the signs $v_j$ by unit vectors, and replacing the product $v_iv_j$ in the objective function by the dot product $v_i^T v_j$.
We can implicitly optimize over such vectors (of unspecified dimension) by explicitly optimizing over an $n\times n$ psd matrix $X$ whose diagonal entries are~1.
This $X$ is the Gram matrix of the vectors $v_1,\ldots,v_n$, so $X_{ij}=v_i^T v_j$.
The resulting SDP is
\begin{align*}
\max \quad &\frac{1}{2}\sum_{\{i,j\}\in E}(1-X_{ij}) \\
\text{s.t.}\ \ \ &\mbox{\rm Tr}(E_{jj} X) =1 \quad \text{ for all } j \in [n], \\
&X \succeq 0,
\end{align*}
where $E_{jj}$ is the $n\times n$ matrix that has a~1 at the $(j,j)$-entry, and 0s elsewhere.
This SDP is a relaxation of a maximization problem, so it may overshoot the correct value, but Goemans and Williamson showed that an optimal solution to the SDP can be rounded to a cut in $G$ whose size is within a factor $\approx 0.878$ of MAXCUT$(G)$ (which is optimal under the Unique Games Conjecture~\cite{kkmo:maxcutj}). This SDP can be massaged into the form of~\eqref{eq:SDP} by replacing the equality $\mbox{\rm Tr}(E_{jj} X)=1$ by inequality $\mbox{\rm Tr}(E_{jj} X)\leq 1$ (so $m=n$) and letting $C$ be a properly normalized version of the Laplacian of~$G$.
\subsection{Classical solvers for LPs and SDPs}
Ever since Dantzig's development of the simplex algorithm for solving LPs in the 1940s~\cite{Dantzig1947}, much work has gone into finding faster solvers, first for LPs and then also for SDPs. The simplex algorithm for LPs (with some reasonable pivot rule) is usually fast in practice, but has worst-case exponential runtime. Ellipsoid methods and interior-point methods can solve LPs and SDPs in polynomial time; they will typically \emph{approximate} the optimal value to arbitrary precision.
The best known general SDP-solvers~\cite{lsw:faster} approximate the optimal value \mbox{\rm OPT}\ of such an SDP up to additive error $\varepsilon$, with complexity
$$
{\mathcal O}(m(m^2+n^\omega + mns)\, \text{polylog}(m,n,R,1/\varepsilon)),
$$
where $\omega\in[2,2.373)$ is the (still unknown) optimal exponent for matrix multiplication; $s$ is the \emph{sparsity}: the maximal number of non-zero entries per row of the input matrices; and $R$ is an upper bound on the trace of an optimal~$X$.\footnote{See Lee, Sidford, and Wong~\cite[Section~10.2 of arXiv version~2]{lsw:faster}, and note that our $m,n$ are their $n,m$, their $S$ is our $mns$, and their $M$ is our~$R$. The bounds for other SDP-solvers that we state later also include another parameter~$r$; it follows from the assumptions of~\cite[Theorem~45 of arXiv version~2]{lsw:faster} that in their setting $r\leq mR$, and hence $r$ is absorbed in the $\log^{O(1)}(mnR/\varepsilon)$ factor.}
The assumption here is that the rows and columns of the matrices of SDP~\eqref{eq:SDP} can be accessed as adjacency lists: we can query, say, the $\ell$th non-zero entry of the $k$th row of matrix $A_j$ in constant time.
Arora and Kale~\cite{arora&kale:sdp} gave an alternative way to approximate \mbox{\rm OPT}, using a matrix version of the ``multiplicative weights update'' method.\footnote{See also~\cite{ahk:multweights} for a subsequent survey; the same algorithm was independently discovered around the same time in the context of learning theory~\cite{TRW:matrixexp,WK:onlinevarmin}.}
In Section~\ref{sec:classicalAK} we will describe their framework in more detail, but in order to describe our result we will start with an overly simplified sketch here.
The algorithm goes back and forth between candidate solutions to the primal SDP and to the corresponding \emph{dual} SDP, whose variables are non-negative reals $y_1,\ldots,y_m$:
\begin{align} \label{eq:SDP2}
\min \quad & b^T y \\ \notag
\text{s.t.}\ \ \ &\sum_{j=1}^m y_j A_j - C \succeq 0,\\ \notag
&y \geq 0.
\end{align}
Under assumptions that will be satisfied everywhere in this paper, strong duality applies:
the primal SDP~\eqref{eq:SDP} and dual SDP~\eqref{eq:SDP2} will have the same optimal value \mbox{\rm OPT}.
The algorithm does a binary search for \mbox{\rm OPT}\ by trying different guesses $\alpha$ for it. Suppose we have fixed some $\alpha$, and want to find out whether $\alpha$ is bigger or smaller than $\mbox{\rm OPT}$. Start with some candidate solution $X^{(1)}$ for the primal, for example a multiple of the identity matrix ($X^{(1)}$ has to be psd but need not be a \emph{feasible} solution to the primal). This $X^{(1)}$ induces the following polytope:
\begin{align*}
\mathcal{P}_{\varepsilon}(X^{(1)}) := \{ y \in \mathbb R^m:\ &b^T y \leq \alpha, \\
&\tr{\Big(\sum_{j=1}^m y_j A_j - C\Big) X^{(1)}} \geq -\varepsilon,\\
& y \geq 0 \}.
\end{align*}
This polytope can be thought of as a relaxation of the feasible region of the dual SDP with the extra constraint that $\mbox{\rm OPT}\leq\alpha$: instead of requiring that $\sum_j y_j A_j - C$ is psd, we merely require that its inner product with the particular psd matrix $X^{(1)}$ is not too negative.
The algorithm then calls an ``oracle'' that provides a $y^{(1)}\in \mathcal{P}_{\varepsilon}(X^{(1)})$, or outputs ``fail'' if $\mathcal{P}_{0}(X^{(1)})$ is empty (how to efficiently implement such an oracle depends on the application).
In the ``fail'' case we know there is no dual-feasible $y$ with objective value $\leq\alpha$,
so we can increase our guess $\alpha$ for \mbox{\rm OPT}, and restart.
In case the oracle produced a $y^{(1)}$, this is used to define a Hermitian matrix $H^{(1)}$ and a new candidate solution $X^{(2)}$ for the primal, which is proportional to $e^{-H^{(1)}}$.
Then the oracle for the polytope $\mathcal{P}_{\varepsilon}(X^{(2)})$ induced by this $X^{(2)}$ is called to produce a candidate $y^{(2)}\in\mathcal{P}_{\varepsilon}(X^{(2)})$ for the dual (or ``fail''), this is used to define $H^{(2)}$ and $X^{(3)}$ proportional to $e^{-H^{(2)}}$, and so on.
Surprisingly, the average of the dual candidates $y^{(1)},y^{(2)},\ldots$ converges to a nearly-dual-feasible solution.
Let $R$ be an upper bound on the trace of an optimal $X$ of the primal, $r$ be an upper bound on the sum of entries of an optimal $y$ for the dual, and $w^*$ be the ``width'' of the oracle for a certain SDP: the maximum of $\nrm{\sum_{j=1}^m y_j A_j - C}$ over all psd matrices $X$ and all vectors $y$ that the oracle may output for the corresponding polytope~$\mathcal{P}_{\varepsilon}(X)$. In general we will not know the width of an oracle exactly, but only an upper bound $w \geq w^*$, that may depend on the SDP; this is, however, enough for the Arora-Kale framework. In Section~\ref{sec:classicalAK} we will show that without loss of generality we can assume the oracle returns a $y$ such that $\nrm{y}_1 \leq r$. Because we assumed $\nrm{A_j},\nrm{C} \leq 1$, we have $w^* \leq r+1$ as an easy width-bound.
General properties of the multiplicative weights update method guarantee that after $T=\widetilde{\mathcal O}(w^2R^2/\varepsilon^2)$ iterations\footnote{The $\widetilde{\mathcal O}(\cdot)$ notation hides polylogarithmic factors in all parameters.}, if no oracle call yielded ``fail'', then the vector $\frac{1}{T}\sum_{t=1}^T y^{(t)}$ is close to dual-feasible and satisfies $b^T y\leq\alpha$. This vector can then be turned into a dual-feasible solution by tweaking its first coordinate, certifying that $\mbox{\rm OPT}\leq\alpha+\varepsilon$, and we can decrease our guess $\alpha$ for $\mbox{\rm OPT}$ accordingly.
The framework of Arora and Kale is really a meta-algorithm, because it does not specify how to implement the oracle. They themselves provide oracles that are optimized for special cases, which allows them to give a very low width-bound for these specific SDPs. For example for the MAXCUT SDP, they obtain a solver with near-linear runtime in the number of edges of the graph. They also observed that the algorithm can be made more efficient by not explicitly calculating the matrix $X^{(t)}$ in each iteration: the algorithm can still be made to work if instead of providing the oracle with $X^{(t)}$, we feed it good estimates of $\mbox{\rm Tr}(A_j X^{(t)})$ and $\mbox{\rm Tr}(C X^{(t)})$. Arora and Kale do not describe oracles for general SDPs, but as we show at the end of Section~\ref{sec:runtime} (using Appendix~\ref{app:trace} to estimate $\mbox{\rm Tr}(A_j X^{(t)})$ and $\mbox{\rm Tr}(C X^{(t)})$), one can get a general classical SDP-solver in their framework with complexity
\begin{equation}\label{eq:AKgeneralupperbound}
\bOt{nms\left(\frac{Rr}{\varepsilon}\right)^{4}+ns\left(\frac{Rr}{\varepsilon}\right)^{7}}.
\end{equation}
Compared to the complexity of the SDP-solver of~\cite{lsw:faster}, this has much worse dependence on $R$ and~$\varepsilon$, but better dependence on $m$ and $n$. Using the Arora-Kale framework is thus preferable over standard SDP-solvers for the case where $Rr$ is small compared to $mn$, and a rough approximation to \mbox{\rm OPT}\ (say, small constant $\varepsilon$) is good enough.
It should be noted that for many specific cases, Arora and Kale get significantly better upper bounds than~\eqref{eq:AKgeneralupperbound} by designing oracles that are specifically optimized for those cases.
\subsection{Quantum SDP-solvers: the Brand\~ao-Svore algorithm}
Given the speed-ups that \emph{quantum} computers give over classical computers for various problems~\cite{shor:factoring,grover:search,dhhm:graphproblemsj,ambainis:edj,hhl:lineq}, it is natural to ask whether quantum computers can solve LPs and SDPs more efficiently as well. Very little was known about this, until recently Brand\~ao and Svore~\cite{brandao&svore:qsdp} discovered quantum algorithms that significantly outperform classical SDP-solvers in certain regimes. Because of the general importance of quickly solving LPs and SDPs, and the limited number of quantum algorithms that have been found so far, this is a very interesting development.
The key idea of the Brand\~ao-Svore algorithm is to take the Arora-Kale approach and to replace two of its steps by more efficient quantum subroutines. First, given a vector $y^{(t-1)}$, it turns out one can use ``Gibbs sampling'' to prepare the new primal candidate $X^{(t)}\propto e^{-H^{(t-1)}}$ \emph{as a $\log(n)$-qubit quantum state $\rho^{(t)}:=X^{(t)}/\mbox{\rm Tr}(X^{(t)})$} in much less time than needed to compute $X^{(t)}$ as an $n\times n$ matrix.
Second, one can efficiently implement the oracle for $\mathcal{P}_{\varepsilon}(X^{(t)})$ based on a number of copies of $\rho^{(t)}$, using those copies to estimate $\mbox{\rm Tr}(A_j \rho^{(t)})$ and $\mbox{\rm Tr}(A_j X^{(t)})$ when needed (note that $\mbox{\rm Tr}(A\rho)$ is the expectation value of operator~$A$ for the quantum state~$\rho$). This is based on something called ``Jaynes's principle.'' The resulting oracle is weaker than what is used classically, in the sense that it outputs a sample $j\sim y_j/\nrm{y}_1$ rather than the whole vector~$y$. However, such sampling still suffices to make the algorithm work (it also means we can assume the vector $y^{(t)}$ to be quite sparse).
Using these ideas, Brand\~ao and Svore obtain a quantum SDP-solver of complexity
$$
\widetilde{\mathcal O}(\sqrt{mn}s^2 R^{32} /\delta^{18}),
$$
with \emph{multiplicative} error $1\pm\delta$ for the special case where $b_j\geq 1$ for all $j\in[m]$, and $\mbox{\rm OPT}\geq 1$ (the latter assumption allows them to convert additive error~$\varepsilon$ to multiplicative error~$\delta$)~\cite[Corollary~5 in arXiv version 4]{brandao&svore:qsdp}. They describe a reduction to transform a general SDP of the form~\eqref{eq:SDP} to this special case, but that reduction significantly worsens the dependence of the complexity on the parameters $R$, $r$, and $\delta$.
Note that compared to the runtime~\eqref{eq:AKgeneralupperbound} of our general instantiation of the original Arora-Kale framework, there are quadratic improvements in both $m$ and $n$, corresponding to the two quantum modifications made to Arora-Kale. However, the dependence on $R,r,s$ and $1/\varepsilon$ is much worse now than in~\eqref{eq:AKgeneralupperbound}. This quantum algorithm thus provides a speed-up only in regimes where $R,r,s,1/\varepsilon$ are fairly small compared to $mn$ (finding good examples of SDPs in such regimes is an open problem).
\subsection{Our results}
In this paper we present two sets of results: improvements to the Brand\~ao-Svore algorithm, and better lower bounds for the complexity of quantum LP-solvers (and hence for quantum SDP-solvers as well).
\subsubsection{Improved quantum SDP-solver}
Our quantum SDP-solver, like the Brand\~ao-Svore algorithm, works by quantizing some aspects of the Arora-Kale algorithm. However, the way we quantize is different and faster than theirs.
First, we give a more efficient procedure to estimate the quantities $\mbox{\rm Tr}(A_j\rho^{(t)})$ required by the oracle. Instead of first preparing some copies of Gibbs state $\rho^{(t)}\propto e^{-H^{(t-1)}}$ as a mixed state, we coherently prepare a purification of $\rho^{(t)}$, which can then be used to estimate $\mbox{\rm Tr}(A_j\rho^{(t)})$ more efficiently using amplitude-estimation techniques.
Also, our purified Gibbs sampler has logarithmic dependence on the error, which is exponentially better than the Gibbs sampler of Poulin and Wocjan~\cite{PoulinWocjan:GibbsEval} that Brand\~ao and Svore invoke. Chowdhury and Somma~\cite{ChowdhurySomma:GibbsHit} also gave a Gibbs sampler with logarithmic error-dependence, but assuming query access to the entries of $\sqrt{H}$ rather than $H$ itself.
Second, we have a different implementation of the oracle, without using Gibbs sampling or Jaynes's principle (though, as mentioned above, we still use purified Gibbs sampling for approximating the $\mbox{\rm Tr}(A\rho)$ quantities). We observe that the vector $y^{(t)}$ can be made very sparse: \emph{two} non-zero entries suffice.\footnote{Independently of us, Ben David, Eldar, Garg, Kothari, Natarajan, and Wright (at MIT), and separately Ambainis observed that in the special case where all $b_j$ are at least~1, the oracle can even be made 1-sparse, and the one entry can be found using one Grover search over $m$ points (in both cases personal communication 2017). The same happens implicitly in our Section~\ref{sec:oracle} in this case. However, in general 2 non-zero entries are necessary in~$y$.}
We then show how we can efficiently find such a 2-sparse vector (rather than merely sampling from it) using two applications of a new generalization of the well-known quantum minimum-finding algorithm of D{\"u}rr and H{\o}yer~\cite{durr&hoyer:minimum}, which is based on Grover search~\cite{grover:search}.
These modifications both simplify and speed up the quantum SDP-solver, resulting in complexity
$$
\widetilde{\mathcal O}(\sqrt{mn}s^2 (Rr/\varepsilon)^{8}).
$$
The dependence on $m$, $n$, and $s$ is the same as in Brand\~ao-Svore, but our dependence on $R$, $r$, and $1/\varepsilon$ is substantially better.
Note that each of the three parameters $R$, $r$, and $1/\varepsilon$ now occurs with the same 8th power in the complexity. This is no coincidence: as we show in Appendix~\ref{app:reductions}, these three parameters can all be traded for one another, in the sense that we can massage the SDP to make each one of them small at the expense of making the others proportionally bigger.
These trade-offs suggest we should actually think of $Rr/\varepsilon$ as \emph{one} parameter of the primal-dual pair of SDPs, not three separate parameters. For the special case of LPs, we can improve the runtime to
$$
\widetilde{\mathcal O}(\sqrt{mn} (Rr/\varepsilon)^{5}).
$$
Like in Brand\~ao-Svore, our quantum oracle produces very sparse vectors $y$, in our case even of sparsity~2. This means that after $T$ iterations, the final $\varepsilon$-optimal dual-feasible vector (which is a slightly tweaked version of the average of the $T$ $y$-vectors produced in the $T$ iterations) has only $\mathcal O(T)$ non-zero entries. Such sparse vectors have some advantages, for example they take much less space to store than arbitrary $y\in\mathbb{R}^m$.
In fact, to get a sublinear running time in terms of~$m$, this is necessary.
However, this sparsity of the algorithm's output also points to a weakness of these methods: if \emph{every} $\varepsilon$-optimal dual-feasible vector $y$ has many non-zero entries, then the number of iterations needs to be large. For example, if every $\varepsilon$-optimal dual-feasible vector $y$ has $\Omega(m)$ non-zero entries, then these methods require $T=\Omega(m)$ iterations before they can reach an $\varepsilon$-optimal dual-feasible vector. Since $T = {\mathcal O}\left(\frac{R^2r^2}{\varepsilon^2} \ln(n)\right)$ this would imply that $\frac{Rr}{\varepsilon} = \Omega(\sqrt{m/\ln(n)})$, and hence many classical SDP-solvers would have a better complexity than our quantum SDP-solver.
As we show in Section~\ref{sec:downside}, this will actually be the case for families of SDPs that have a lot of symmetry.
\subsubsection{Tools that may be of more general interest}
Along the way to our improved SDP-solver, we developed some new techniques that may be of independent interest. These are mostly tucked away in appendices, but here we will highlight two.
\paragraph{Implementing smooth functions of a given Hamiltonian.}
In Appendix~\ref{apx:LowWeight} we describe a general technique to apply a function $f(H)$ of a sparse Hamiltonian~$H$ to a given state $\ket{\phi}$. Roughly speaking, what this means is that we want a unitary circuit that maps $\ket{0}\ket{\phi}$ to $\ket{0}f(H)\ket{\phi}+\ket{1}\ket{*}$.
If need be, we can then combine this with amplitude amplification to boost the $\ket{0}f(H)\ket{\phi}$ part of the state.
If the function $f:\mathbb{R}\to\mathbb{C}$ can be approximated well by a low-degree Fourier series, then our preparation will be efficient
in the sense of using few queries to $H$ and few other gates.
The novelty of our approach is that we construct a good Fourier series from the polynomial that approximates~$f$ (for example a truncated Taylor series for~$f$). Our Theorem~\ref{thm:Taylor} can be easily applied to various smooth functions without using involved integral approximations, unlike previous works building on similar techniques. Our most general result Corollary~\ref{cor:patched} only requires that the function $f$ can be nicely approximated locally around each possible eigenvalues of $H$, improving on Theorem~\ref{thm:Taylor}.
In this paper we mostly care about the function $f(x)=e^{-x}$, which is what we want for generating a purification of the Gibbs state corresponding to~$H$; and the function $f(x)=\sqrt{x}$, which is what we use for estimating quantities like $\mbox{\rm Tr}(A\rho)$.
However, our techniques apply much more generally than these two functions. For example, they also simplify the analysis of the improved linear-systems solver of Childs et al.~\cite{ChildsKothariSomma:lse}, where the relevant function is $f(x)=1/x$. As in their work, the Linear Combination of Unitaries technique of Childs et al.~\cite{ChildsWiebeLCU,BerryChilds:hamsim,BerryChilds:hamsimFOCS} is a crucial tool for us.
\paragraph{A generalized minimum-finding algorithm.}
D{\"u}rr and H{\o}yer~\cite{durr&hoyer:minimum} showed how to find the minimal value of a function $f:[N]\to\mathbb{R}$ using ${\mathcal O}(\sqrt{N})$ queries to~$f$,
by repeatedly using Grover search to find smaller and smaller elements of the range of~$f$. In Appendix~\ref{app:genMinFind} we describe a more general
minimum-finding procedure. Suppose we have a unitary $U$ which prepares a quantum state $U\ket{0}=\sum_{k=1}^{N}\ket{\psi_k}\ket{x_k}$, where the $\ket{\psi_k}$ are unnormalized states. Our procedure can find the minimum value $x_{k^*}$ among the $x_k$'s that have support in the second register, using roughly ${\mathcal O}(1/\nrm{\psi_{k^*}})$ applications of $U$ and~$U^{-1}$. Also, upon finding the minimal value $k^*$ the procedure actually outputs the normalized state proportional to $\ket{\psi_{k^*}}\ket{x_{k^*}}$. This immediately gives the D{\"u}rr-H{\o}yer result as a special case, if we take $U$ to produce $U\ket{0}=\frac{1}{\sqrt{N}}\sum_{k=1}^{N}\ket{k}\ket{f(k)}$ using one query to~$f$. Unlike D{\"u}rr-H{\o}yer, we need not assume direct query access to the individual values $f(k)$.
More interestingly for us, for a given $n$-dimensional Hamiltonian~$H$, if we combine our minimum-finder with phase estimation using unitary $U=e^{iH}$ on one half of a maximally entangled state, then we obtain an algorithm for estimating the smallest eigenvalue of~$H$ (and preparing its ground state) using roughly ${\mathcal O}(\sqrt{n})$ applications of phase estimation with~$U$. A similar result on approximating the smallest eigenvalue of a Hamiltonian was already shown by Poulin and Wocjan~\cite{PoulinWocjan:GroundMany}, but we improve on their analysis to be able to apply it as a subroutine in our procedure to estimate~$\mbox{\rm Tr}(A_j\rho)$.
\subsubsection{Lower bounds}
What about lower bounds for quantum SDP-solvers? Brand\~ao and Svore already proved that a quantum SDP-solver has to make $\Omega(\sqrt{n}+\sqrt{m})$ queries to the input matrices, for some SDPs. Their lower bound is for a family of SDPs where $s,R,r,1/\varepsilon$ are all constant, and is by reduction from a search problem.
In this paper we prove lower bounds that are quantitatively stronger in $m$ and~$n$, but for SDPs with non-constant $R$ and $r$.
The key idea is to consider a Boolean function $F$ on $N=abc$ input bits that is the composition of an $a$-bit majority function with a $b$-bit OR function with a $c$-bit majority function. The known quantum query complexities of majority and OR, combined with composition properties of the adversary lower bound, imply that every quantum algorithm that computes this function requires $\Omega(a\sqrt{b}c)$ queries. We define a family of LPs, with constant $1/\varepsilon$ but non-constant $r$ and $R$ (we could massage this to make $R$ or $r$ constant using the results of Appendix~\ref{app:reductions}, but not $Rr/\varepsilon$), such that constant-error approximation of \mbox{\rm OPT}\ computes~$F$. Choosing $a$, $b$, and $c$ appropriately, this implies a lower bound of
$$
\Omega\left(\sqrt{\max\{n,m\}} \left( \min\{n,m\} \right)^{3/2} \right)
$$
queries to the entries of the input matrices for quantum LP-solvers. Since LPs are SDPs with sparsity $s=1$, we get the same lower bound for quantum SDP-solvers. If $m$ and $n$ are of the same order, this lower bound is $\Omega(mn)$, the same scaling with $mn$ as the classical general instantiation of Arora-Kale~\eqref{eq:AKgeneralupperbound}. In particular, this shows that we cannot have an $O(\sqrt{mn})$ upper bound without simultaneously having polynomial dependence on~$Rr/\varepsilon$.
The construction of our lower bound implies that for the case $m\approx n$, this polynomial dependence has to be at least $(Rr/\varepsilon)^{1/4}$.
\paragraph{Subsequent work.}
Following the first version of our paper, improvements in the running time were obtained in~\cite{brandao2017QSDPSpeedupsLearning,apeldoorn2018ImprovedQSDPSolving}, the latter providing a runtime of $\bOt{(\sqrt{m}+\sqrt{n} \frac{Rr}{\epsilon}) s \left(\frac{Rr}{\epsilon}\right)^4}$. More recently, a quantum interior point method for solving SDPs and LPs was obtained by Kerenidis and Prakash~\cite{kerenidis2018QIntPoint}. It is hard to compare the latter algorithm to the other SDP-solvers for two reasons. First, the output of their algorithm consists only of almost-feasible solutions to the primal and dual (their algorithm has a polynomial dependence on the distance to feasibility). It is therefore not clear what their output means for the optimal value of the SDPs. Secondly, the runtime of their algorithm depends polynomially on the condition number of the matrices that the interior point method encounters, and no explicit bounds for these condition numbers are given.
Our results on implementing smooth functions of a given Hamiltonian have been extended to more general input models (block-encodings) in~\cite{gilyen2018QSingValTransf}. This recent paper builds on some of our techniques, but achieves slightly improved complexities by directly implementing the transformations without using Hamiltonian simulation as a subroutine.
Very recently van Apeldoorn et al.~\cite{apeldoorn2018ConvexOptUsingQuantumOracles} and Chakrabarti et al.~\cite{chakrabarti2018QuantumConvexOpt} developed quantum algorithms for general black-box convex optimization, where one optimizes over a general convex set~$K$, and the access to $K$ is via membership and/or separation oracles. Since we work in a model where we are given access directly to the constraints defining the problem, our results are incomparable to theirs.
\paragraph{Organization.}
The paper is organized as follows. In Section~\ref{sec:upperbounds} we start with a description of the Arora-Kale framework for SDP-solvers, and then we describe how to quantize different aspects of it to obtain a quantum SDP-solver with better dependence on $R$, $r$, and $1/\varepsilon$ (or rather, on $Rr/\varepsilon$) than Brand\~ao and Svore got. In Section~\ref{sec:downside} we describe the limitations of primal-dual SDP-solvers using general oracles (not optimized for specific SDPs) that produce sparse dual solutions~$y$: if good solutions are dense, this puts a lower bound on the number of iterations needed. In Section~\ref{sec:lowerbounds} we give our lower bounds. A number of the proofs are relegated to the appendices:
how to classically approximate $\mbox{\rm Tr}(A_j\rho)$ (Appendix~\ref{app:trace}),
how to efficiently implement smooth functions of Hamiltonians on a quantum computer (Appendix~\ref{apx:LowWeight}),
our generalized method for minimum-finding (Appendix~\ref{app:genMinFind}),
upper and lower bounds on how efficiently we can query entries of sums of sparse matrices (Appendix~\ref{app:sparsematrixsum}),
how to trade off $R$, $r$, and $1/\varepsilon$ against each other (Appendix~\ref{app:reductions}),
and the composition property of the adversary method that we need for our lower bounds (Appendix~\ref{app:adversarycomposition}).
\section{An improved quantum SDP-solver}
\label{sec:upperbounds}
Here we describe our quantum SDP-solver. In Section~\ref{sec:classicalAK} we describe the framework designed by Arora and Kale for solving semidefinite programs. As in the recent work by Brand\~ao and Svore, we use this framework to design an efficient quantum algorithm for solving SDPs. In particular, we show that the key subroutine needed in the Arora-Kale framework can be implemented efficiently on a quantum computer. Our implementation uses different techniques than the quantum algorithm of Brand\~ao and Svore, allowing us to obtain a faster algorithm. The techniques required for this subroutine are developed in Sections~\ref{sec:trCalc} and~\ref{sec:oracle}. In Section~\ref{sec:runtime} we put everything together to prove the main theorem of this section (the notation is explained below):
\begin{restatable}{theorem}{upperbound}
\label{thm:upperbound}
Instantiating Meta-Algorithm~\ref{alg:AKSDP} using the trace calculation algorithm from Section~\ref{sec:trCalc} and the oracle from Section~\ref{sec:oracle} (with width-bound $w:=r+1$), and using this to do a binary search for $\mbox{\rm OPT}\in[-R,R]$ (using different guesses $\alpha$ for $\mbox{\rm OPT}$), gives a quantum algorithm for solving SDPs of the form~\eqref{eq:SDP}, which (with high probability) produces a feasible solution $y$ to the dual program which is optimal up to an additive error $\varepsilon$, and uses
\[
\bOt{\sqrt{nm} s^2\left( \frac{Rr}{\varepsilon}\right)^{\!\!8} }
\]
queries to the input matrices and the same order of other gates.
\end{restatable}
\paragraph{Notation/Assumptions.}
We use $\log$ to denote the logarithm in base~$2$.
We denote the all-zero matrix and vector by~$0$.
Throughout we assume each element of the input matrices can be represented by a bitstring of size $\text{poly}(\log n,\log m)$. We use $s$ as the sparsity of the input matrices, that is, the maximum number of non-zero entries in a row (or column) of any of the matrices $C, A_1,\ldots, A_m$ is $s$. Recall that for normalization purposes we assume $\nrm{A_1}, \ldots, \nrm{A_m}, \nrm{C} \leq 1$. We furthermore assume that $A_1 = I$ and $b_1 = R$, that is, the trace of primal-feasible solutions is bounded by $R$ (and hence also the trace of primal-optimal solutions is bounded by $R$).
The analogous quantity for the dual SDP~\eqref{eq:SDP2}, an upper bound on $\sum_{j=1}^m y_j$ for an optimal dual solution~$y$, will be denoted by~$r$. However, we do not add the constraint $\sum_{j=1}^m y_j \leq r$ to the dual.
We will assume $r\geq 1$. For $r$ to be well-defined we have to make the explicit assumption that the optimal solution in the dual is attained.
In Section~\ref{sec:downside} it will be necessary to work with the best possible upper bounds: we let $R^*$ be the smallest trace of an optimal solution to SDP~\eqref{eq:SDP}, and we let $r^*$ be the smallest $\ell_1$-norm of an optimal solution to the dual. These quantities are well-defined; indeed, both the primal and dual optimum are attained: the dual optimum is attained by assumption, and due to the assumption $A_1 = I$, the dual SDP is strictly feasible, which means that the optimum in~\eqref{eq:SDP} is attained.
Unless specified otherwise, we always consider \emph{additive} error. In particular, an $\varepsilon$-optimal solution to an SDP will be a feasible solution whose objective value is within additive error~$\varepsilon$ of the optimum.
\paragraph{Input oracles:} We assume sparse black-box access to the elements of the matrices $C, A_1,\ldots, A_m$ defined in the following way: for input $(j, k,\ell) \in (\{0\}\cup[m]) \times [n] \times [s]$ we can query the location and value of the $\ell$th non-zero entry in the $k$th row of the matrix $A_j$ (where $j=0$ would indicate the $C$ matrix).
Specifically in the quantum case, as described in~\cite{BerryChilds:hamsimFOCS}, we assume access to an oracle $O_I$ which serves the purpose of sparse access. $O_I$ calculates the $\text{index}_{A_j}: [n] \times [s] \to [n]$ function, which for input $(k,\ell)$ gives the column index of the $\ell$th non-zero element in the $k$th row of $A_j$. We assume this oracle computes the index ``in place":
\begin{equation}
O_I\ket{j,k,\ell} = \ket{j,k,\text{index}_{A_j}(k,\ell)}.
\label{eq:oracleind}
\end{equation}
(In the degenerate case where the $k$th row has fewer than $\ell$ non-zero entries, $\text{index}_{A_j}(k,\ell)$ is defined to be $\ell$ together with some special symbol.) We also assume we can apply the inverse of~$O_I$.
We also need another oracle $O_M$, returning a bitstring representation of $(A_j)_{ki}$ for any $j \in \{0\}\cup[m]$ and $ k,i \in [n]$:
\begin{equation}
O_M\ket{j,k,i,z} = \ket{j,k,i,z \oplus{(A_j)_{ki}}}.
\label{eq:oraclemat}
\end{equation}
The slightly unusual ``in place'' definition of oracle $O_I$ is not too demanding. In particular, if instead we had an oracle that computed the non-zero entries of a row in order, then we could implement both $O_I$ and its inverse using $\log(s)$ queries (we can compute $\ell$ from $k$ and $\text{index}_{A_j}(k,\ell)$ using binary search)~\cite{BerryChilds:hamsimFOCS}.
\paragraph{Computational model:}
As our computational model, we assume a slight relaxation of the usual quantum circuit model: a classical control system that can run quantum subroutines.
We limit the classical control system so that its number of operations is at most a polylogarithmic factor bigger than the gate complexity of the quantum subroutines, i.e., if the quantum subroutines use $C$ gates, then the classical control system may not use more than ${\mathcal O}(C\,\text{polylog}(C))$ operations.
When we talk about gate complexity, we count the number of two-qubit quantum gates needed for implementation of the quantum subroutines.
Additionally, we assume for simplicity that there exists a unit-cost QRAM gate that allows us to store and retrieve qubits in a memory, by means of a swap of two registers indexed by another register:
\[
QRAM : \ket{i,x,r_1,\dots,r_K} \mapsto \ket{i,r_i,r_1,\dots,r_{i-1},x,r_{i+1},\ldots,r_K},
\]
where the registers $r_1,\dots,r_K$ are only accessible through this gate. The QRAM gate can be seen as a quantum analogue of pointers in classical computing.
The only place where we need QRAM is in Appendix~\ref{app:sparsematrixsum}, for a data structure that allows efficient access to the non-zero entries of a sum of sparse matrices; for the special case of LP-solving it is not needed.
\subsection{The Arora-Kale framework for solving SDPs} \label{sec:classicalAK}
In this section we give a short introduction to the Arora-Kale framework for solving semidefinite programs. We refer to~\cite{arora&kale:sdp,ahk:multweights} for a more detailed description and omitted proofs.
The key building block is the Matrix Multiplicative Weights (MMW) algorithm introduced by Arora and Kale in~\cite{arora&kale:sdp}. The MMW algorithm can be seen as a strategy for you in a game between you and an adversary. We first introduce the game.
There is a number of rounds $T$. In each round you present a density matrix $\rho$ to an adversary, the adversary replies with a loss matrix $M$ satisfying $-I \preceq M \preceq I$. After each round you have to pay $\tr{M \rho}$. Your objective is to pay as little as possible. The MMW algorithm is a strategy for you that allows you to lose not too much, in a sense that is made precise below.
In Algorithm~\ref{alg:MMW} we state the MMW algorithm, the following theorem shows the key property of the output of the algorithm.
\begin{algorithm}[ht]
\begin{description}
\item[Input] Parameter $\eta \leq 1$, number of rounds $T$.
\item[Rules] In each round player $1$ (you) presents a density matrix $\rho$, player $2$ (the adversary) replies with a matrix $M$ satisfying $-I \preceq M \preceq I$.
\item[Output] A sequence of symmetric $n \times n$ matrices $M^{(1)},\ldots, M^{(T)}$ satisfying $-I \preceq M^{(t)} \preceq I$, for $t \in [T]$ and a sequence of $n \times n$ psd matrices $\rho^{(1)},\ldots, \rho^{(T)}$ satisfying $\tr{\rho^{(t)}}=1$ for $t \in [T]$.
\item[Strategy of player $1$:]
\end{description}
\begin{algorithmic}
\State Take $\rho^{(1)} := I/n$
\State In round $t$:
\begin{enumerate}
\item Show the density matrix $\rho^{(t)}$ to the adversary.
\item Obtain the loss matrix $M^{(t)}$ from the adversary.
\item Update the density matrix as follows:
\[
\rho^{(t+1)}:= \left.\exp\left(- \eta \sum_{\tau=1}^t M^{(\tau)}\right)\right/\tr{\exp\left(- \eta \sum_{\tau=1}^t M^{(\tau)}\right)}
\]
\end{enumerate}
\end{algorithmic}
\caption{Matrix Multiplicative Weights (MMW) Algorithm}
\label{alg:MMW}
\end{algorithm}
\begin{theorem}[{\cite[Theorem~3.1]{arora&kale:sdp}}]
For every adversary, the sequence $\rho^{(1)}, \ldots, \rho^{(T)}$ of density matrices constructed using the Matrix Multiplicative Weights Algorithm~\eqref{alg:MMW} satisfies
\[
\sum_{t=1}^T \tr{M^{(t)} \rho^{(t)}} \leq \lambda_{\min}\left(\sum_{t=1}^T M^{(t)}\right) + \eta \sum_{t=1}^T \tr{ (M^{(t)})^2 \rho^{(t)}} + \frac{\ln(n)}{\eta}.
\]
\end{theorem}
Arora and Kale use the MMW algorithm to construct an SDP-solver.
For that, they construct an adversary who promises to satisfy an additional condition: in each round~$t$, the adversary returns a matrix $M^{(t)}$ whose trace inner product with the density matrix $\rho^{(t)}$ is non-negative.
The above theorem shows that then, after $T$ rounds, the average of the adversary's responses satisfies the stronger condition that its smallest eigenvalue is not too negative: $\lambda_{\min}\left(\frac{1}{T} \sum_{t=1}^T M^{(t)}\right) \geq - \eta - \frac{\ln(n)}{\eta T}$. More explicitly, the MMW algorithm is used to build a vector $y \geq 0$ such that
\[
\frac{1}{T} \sum_{t=1}^T M^{(t)} \propto \sum_{j=1}^m y_j A_j -C
\]
and $b^T y \leq \alpha$.
That is, the smallest eigenvalue of the matrix $\sum_{j=1}^m y_j A_j -C$ is only slightly below zero and $y$'s objective value is at most $\alpha$. Since $A_1=I$, increasing the first coordinate of $y$ makes the smallest eigenvalue of $\sum_j y_j A_j -C$ bigger, so that this matrix becomes psd and hence dual-feasible. By the above we know how much the minimum eigenvalue has to be shifted, and with the right choice of parameters it can be shown that this gives a dual-feasible vector $\overline{y}$ that satisfies $b^T \overline{y} \leq \alpha + \varepsilon$. In order to present the algorithm formally, we require some definitions.
Given a candidate solution $X \succeq 0$ for the primal problem~\eqref{eq:SDP} and a parameter $\varepsilon \geq 0$, define the polytope
\begin{align*}
\mathcal{P}_{\varepsilon}(X) := \{ y \in \mathbb R^m:\ &b^T y \leq \alpha, \\
&\tr{\Big(\sum_{j=1}^m y_j A_j - C\Big) X} \geq -\varepsilon,\\
& y \geq 0 \}.
\end{align*}
One can verify the following:
\begin{lemma}[{\cite[Lemma~4.2]{arora&kale:sdp}}] \label{lem:xfeas}
If for a given candidate solution $X \succeq 0$ the polytope $\mathcal P_0(X)$ is empty, then a scaled version of $X$ is primal-feasible and of objective value at least $\alpha$.
\end{lemma}
The Arora-Kale framework for solving SDPs uses the MMW algorithm where the role of the adversary is taken by an $\varepsilon$-approximate oracle:
\begin{algorithm}[h!]
\begin{description}
\item[Input] An $n \times n$ psd matrix $X$, a parameter $\varepsilon$, and the input matrices and reals of~\eqref{eq:SDP2}.
\item[Output] Either the \textsf{Oracle}$_{\varepsilon}$ returns a vector $y$ from the polytope $\mathcal P_\varepsilon(X)$ or it outputs ``fail''. It may only output fail if $\mathcal P_0(X) = \emptyset$.
\end{description}
\caption{$\varepsilon$-approximate \textsf{Oracle}$_{\varepsilon}$ for maximization SDPs}
\label{alg:Oracle}
\end{algorithm}
\noindent As we will see later, the runtime of the Arora-Kale framework depends on a property of the oracle called the \emph{width}:
\begin{definition}[\emph{Width} of \textsf{Oracle}$_{\varepsilon}$]
The \emph{width} of \textsf{Oracle}$_{\varepsilon}$ for an SDP is the smallest $w^* \geq 0$ such that for every primal candidate $X \succeq 0$, the vector $y$ returned by \textsf{Oracle}$_{\varepsilon}$ satisfies $\nrm{\sum_{j=1}^m y_j A_j- C} \leq w^*$.
\end{definition}
In practice, the width of an oracle is not always known. However, it suffices to work with an upper bound $w \geq w^*$: as we can see in Meta-Algorithm~\ref{alg:AKSDP}, the purpose of the width is to rescale the matrix $M^{(t)}$ in such a way that it forms a valid response for the adversary in the MMW algorithm.
\begin{metaalgorithm}[ht]
\begin{description}
\item[Input] The input matrices and reals of SDP~\eqref{eq:SDP} and trace bound~$R$. The current guess $\alpha$ of the optimal value of the dual~\eqref{eq:SDP2}. An additive error tolerance $\varepsilon>0$. An $\frac{\varepsilon}{3}$-approximate oracle \textsf{Oracle}$_{\varepsilon/3}$ as in Algorithm~\ref{alg:Oracle} with width-bound~$w$.
\item[Output] Either ``Lower'' and a vector $\overline{y} \in \mathbb R^{m}_{+}$ feasible for~\eqref{eq:SDP2} with $b^T \overline{y} \leq \alpha+\varepsilon$ \\
or ``Higher'' and a symmetric $n \times n$ matrix $X$ that, when scaled suitably, is primal-feasible with objective value at least $\alpha$.
\end{description}
\begin{algorithmic}
\State $T := \left\lceil \frac{9 w^2 R^2 \ln(n)}{\varepsilon^2}\right\rceil$.
\State $\eta := \sqrt{\frac{\ln (n)}{T}}$.
\State $\rho^{(1)} := I/n$
\For{ $t = 1,\dots,T$}
\State Run \textsf{Oracle}$_{\varepsilon/3}$ with $X^{(t)} = R\rho^{(t)}$.
\If{\textsf{Oracle}$_{\varepsilon/3}$ outputs ``fail''}
\State \Return ``Higher'' and a description of $X^{(t)}$.
\EndIf
\State Let $y^{(t)}$ be the vector generated by \textsf{Oracle}$_{\varepsilon/3}$.
\State Set $M^{(t)} = \frac{1}{w} \left( \sum_{j=1}^m y_j^{(t)} A_j - C\right)$.
\State Define $H^{(t)} = \sum_{\tau=1}^t M^{(\tau)}$.
\State Update the state matrix as follows: $\rho^{(t+1)}:= \exp\left(- \eta H^{(t)}\right)/\tr{\exp\left(- \eta H^{(t)}\right)}$.
\EndFor
\State If \textsf{Oracle}$_{\varepsilon/3}$ does not output ``fail'' in any of the $T$ rounds, then output the dual solution
$\overline{y} = \frac{\varepsilon}{R} e_1 + \frac{1}{T} \sum_{t=1}^T y^{(t)}$ where $e_1 = (1,0,\ldots, 0) \in \mathbb R^m$.
\end{algorithmic}
\caption{Primal-Dual Algorithm for solving SDPs}
\label{alg:AKSDP}
\end{metaalgorithm}
The following theorem shows the correctness of the Arora-Kale primal-dual meta-algorithm for solving SDPs, stated in Meta-Algorithm~\ref{alg:AKSDP}:
\begin{theorem}[{\cite[Theorem~4.7]{arora&kale:sdp}}]
Given an SDP of the form~\eqref{eq:SDP} with input matrices $A_1 = I, A_2, \ldots, A_m$ and $C$ having operator norm at most $1$, and input reals $b_1=R, b_2, \ldots, b_m$. Assume Meta-Algorithm~\ref{alg:AKSDP} does not output ``fail'' in any of the rounds,
then the returned vector $\overline{y}$ is feasible for the dual~\eqref{eq:SDP2} with objective value at most $\alpha+\varepsilon$.
If \textsf{Oracle}$_{\varepsilon/3}$ outputs ``fail'' in the $t$-th round then a suitably scaled version of $X^{(t)}$ is primal-feasible with objective value at least~$\alpha$.
\end{theorem}
The SDP-solver uses $T = \left\lceil \frac{9 w^2 R^2 \ln(n)}{\varepsilon^2}\right\rceil$ iterations. In each iteration several steps have to be taken. The most expensive two steps are computing the matrix exponential of the matrix $- \eta H^{(t)}$ and the application of the oracle.
Note that the only purpose of computing the matrix exponential is to allow the oracle to compute the values $\tr{A_j X}$ for all $j$ and $\tr{CX}$, since the polytope depends on $X$ only through those values.
To obtain faster algorithms it is important to note, as was done already by Arora and Kale, that the primal-dual algorithm also works if we provide a (more accurate) oracle with approximations of $\tr{A_j X}$.
Let $a_j:=\tr{A_j\rho} = \tr{A_jX}/\tr{X}$ and $c:=\tr{C\rho} = \tr{CX}/\tr{X}$. Then, given a list of reals $\tilde{a}_1, \ldots, \tilde{a}_m, \tilde{c}$ and a parameter $\theta \geq 0$, such that $|\tilde{a}_j - a_j| \leq \theta$ for all $j$, and $|\tilde{c}- c| \leq \theta$, we define the polytope
\begin{align*}
\tilde{\mathcal{P}}(\tilde{a}_1, \ldots, \tilde{a}_m,\tilde{c}-(r+1)\theta) := \{ y \in \mathbb R^m:\ & b^T y \leq \alpha,\\&\sum_{j=1}^m y_j \leq r, \\
&\sum_{j=1}^m \tilde{a}_j y_j \geq \tilde{c} - (r+1)\theta\\&y \geq 0\}.
\end{align*}
For convenience we will denote $\tilde{a} = (\tilde{a}_1, \ldots, \tilde{a}_m)$ and $c^{\prime} := \tilde{c}- (r+1) \theta$. Notice that $\tilde{\mathcal{P}}$ also contains a new type of constraint: $\sum_j y_j \leq r$. Recall that $r$ is defined as a positive real such that there exists an optimal solution $y$ to SDP~\eqref{eq:SDP2} with $\nrm{y}_1 \leq r$. Hence, using that $\mathcal P_0(X)$ is a \emph{relaxation} of the feasible region of the dual (with bound $\alpha$ on the objective value), we may restrict our oracle to return only such~$y$:
\[
\mathcal P_0(X) \neq \emptyset \Rightarrow \mathcal P_0(X) \cap \{y \in \mathbb R^m: \sum_{j=1}^m y_j \leq r\} \neq \emptyset.
\]
The benefit of this restriction is that an oracle that always returns a vector with bounded $\ell_1$-norm automatically has a width $w^* \leq r+1$, due to the assumptions on the norms of the input matrices. The downside of this restriction is that the analogue of Lemma~\ref{lem:xfeas} does not hold for $\mathcal P_0(X) \cap \{y \in \mathbb R^m: \sum_{j} y_j \leq r\}$.\footnote{Using several transformations of the SDP, from Appendix~\ref{app:reductions} and Lemma 2 of~\cite{brandao&svore:qsdp}, one can show that there is a way to remove the need for this restriction. Hence, after these modifications, if for a given candidate solution $X \succeq 0$ the oracle outputs that the set $\mathcal P_0(X)$ is empty, then a scaled version of $X$ is primal feasible for this new SDP, with objective value at least $\alpha$. This scaled version of $X$ can be modified to a near-feasible solution to the original SDP (it will be psd, but it might violate the linear constraints a little bit) with nearly the same objective value.}
The following shows that an oracle that always returns a vector $y \in \tilde{\mathcal{P}}(\tilde{a},c')$ if one exists, is a $4Rr\theta$-approximate oracle as defined in Algorithm~\ref{alg:Oracle}.
\begin{lemma} \label{lem:approxP}
Let $\tilde{a}_1, \ldots, \tilde{a}_m$ and $\tilde{c}$ be $\theta$-approximations of $\tr{A_1 \rho}, \ldots, \tr{A_m \rho}$ and $\tr{C\rho}$, respectively, where $X = R \rho$. Then the following holds:
\[
\mathcal P_0(X) \cap \{y \in \mathbb R^m: \sum_{j=1}^m y_j \leq r\} \subseteq \tilde{\mathcal{P}}(\tilde{a},c') \subseteq \mathcal P_{4Rr\theta}(X).
\]
\end{lemma}
\begin{proof}
First, suppose $y \in \mathcal P_{0}(X) \cap \{y \in \mathbb R^m: \sum_{j} y_j \leq r\}$.
We now have
\[
\sum_{j=1}^m \tilde{a}_j y_j - c \geq - \sum_{j=1}^m |\tilde{a}_j - \tr{A_j \rho}| y_j - |\tilde{c} -\tr{C \rho}| \geq - \theta \nrm{y}_1 - \theta \geq - (r+1) \theta
\]
which shows that $y \in \tilde{\mathcal{P}}(\tilde{a},c')$.
Next, suppose $y \in \tilde{\mathcal{P}}(\tilde{a},c^{\prime})$. We show that $y \in \mathcal P_{4Rr \theta}(X)$. Indeed, since $|\tr{A_j\rho} - \tilde{a}_j| \leq \theta$ we have
\[
\tr{\!\!\left(\sum_{j=1}^m y_j A_j -C\right)\rho\!} \geq \left(\sum_{j=1}^m \tilde{a}_jy_j + \tilde{c} \right) -(r+1)\theta \geq -(2+r + \nrm{y}_1) \theta \geq\! -4r \theta \]
where the last inequality used our assumptions $r\geq 1$ and $\nrm{y}_1\leq r$.
Hence
\[
\tr{\left(\sum_{j=1}^m y_j A_j -C\right)X} \geq -4r\tr{X} \theta = -4Rr\theta.
\]
For the latter inequality we use $\tr{X} = R$.
\end{proof}
We have now seen the Arora-Kale framework for solving SDPs. To obtain a quantum SDP-solver it remains to provide a quantum oracle subroutine. By the above discussion it suffices to set $\theta = \varepsilon/(12Rr)$ and to use an oracle that is based on $\theta$-approximations of $\tr{A\rho}$ (for $A \in \{A_1, \ldots, A_m, C\}$), since with that choice of $\theta$ we have $\mathcal P_{4Rr \theta}(X)=\mathcal P_{\varepsilon/3}(X)$.
In the section below we first give a quantum algorithm for approximating $\tr{A\rho}$ efficiently (see also Appendix~\ref{app:trace} for a classical algorithm). Then, in Section~\ref{sec:oracle}, we provide an oracle using those estimates. The oracle will be based on a simple geometric idea and can be implemented both on a quantum computer and on a classical computer (of course, resulting in different runtimes). In Section~\ref{sec:runtime} we conclude with an overview of the runtime of our quantum SDP-solver.
We want to stress that our solver is meant to work for any SDP. In particular, our oracle does not use the structure of a specific SDP. As we will show in Section~\ref{sec:downside}, any oracle that works for all SDPs necessarily has a large width-bound. To obtain quantum speedups for a \emph{specific} class of SDPs it will be necessary to develop oracles tuned to that problem, we view this as an important direction for future work. Recall from the introduction that Arora and Kale also obtain fast classical algorithms for problems such as MAXCUT by developing specialized oracles.
\subsection{Approximating the expectation value \texorpdfstring{$\tr{A\rho}$}{traces} using a quantum algorithm}\label{sec:trCalc}
In this section we give an efficient quantum algorithm to approximate quantities of the form $\tr{A\rho}$. We are going to work with Hermitian matrices $A,H\in\mathbb{C}^{n\times n}$, such that $\rho$ is the Gibbs state $e^{-H}/\tr{e^{-H}}$. Note the analogy with quantum physics: in physics terminology $\tr{A\rho}$ is simply called the ``expectation value" of $A$ for a quantum system in a thermal state corresponding to $H$.
The general approach is to separately estimate $\tr{A e^{-H}}$ and $\tr{e^{-H}}$, and then to use the ratio of these estimates as an approximation of $\tr{A\rho}=\tr{A e^{-H}}/\tr{e^{-H}}$. Both estimations are done using state preparation to prepare a pure state with a flag, such that the probability that the flag is~0 is proportional to the quantity we want to estimate, and then to use amplitude estimation to estimate that probability.
Below in Section~\ref{ssec:generalapproach} we first describe the general approach.
In Section~\ref{sec:lptrace} we then instantiate this for the special case where all matrices are diagonal, which is the relevant case for LP-solving. In Section~\ref{sec:estTrArhogeneral} we handle the general case of arbitrary matrices (needed for SDP-solving); the state-preparation part will be substantially more involved there, because in the general case we need not know the diagonalizing bases for $A$ and~$H$, and $A$ and $H$ may not be simultaneously diagonalizable.
\subsubsection{General approach}\label{ssec:generalapproach}
To start, consider the following lemma about the multiplicative approximation error of a ratio of two real numbers that are given by multiplicative approximations:
\begin{lemma}\label{lemma:trTogether}
Let $0\leq \theta \leq 1$ and let $\alpha, \tilde{\alpha}, Z, \tilde{Z}$ be positive real numbers
such that $|\alpha - \tilde{\alpha}| \leq \alpha\theta / 3$ and $|Z - \tilde{Z}| \leq Z\theta / 3$. Then
\[
\left|\frac{\alpha}{Z}-\frac{\tilde{\alpha}}{\tilde{Z}}\right| \leq \theta\frac{\alpha}{Z}
\]
\end{lemma}
\begin{proof} Observe $|\tilde{Z}|\geq |Z|2/3$, thus
\begin{align*}
\left|\frac{\alpha}{Z}-\frac{\tilde{\alpha}}{\tilde{Z}}\right|
&= \left|\frac{\alpha\tilde{Z}-\tilde{\alpha}Z}{Z\tilde{Z}}\right|
= \left|\frac{\alpha\tilde{Z}-\alpha Z+\alpha Z-\tilde{\alpha}Z}{Z\tilde{Z}}\right|\\
&\leq \left|\frac{\alpha\tilde{Z}-\alpha Z}{Z\tilde{Z}}\right|+\left|\frac{\alpha Z-\tilde{\alpha}Z}{Z\tilde{Z}}\right|
\leq\frac{\alpha}{Z} \left|\frac{\tilde{Z}-Z}{\tilde{Z}}\right|+\left|\frac{\theta\alpha }{3\tilde{Z}}\right|\\
&\leq \frac{3}{2}\left(\frac{\alpha}{Z}\left|\frac{\tilde{Z}-Z}{Z}\right|+\frac{\theta\alpha }{3Z}\right)
\leq \frac{3}{2}\left(\frac{2\theta}{3}\frac{\alpha}{Z}\right)=\theta\frac{\alpha}{Z}.
\end{align*}
\vskip-5mm
\end{proof}
\begin{corollary}\label{col:split}
Let $A$ be such that $\nrm{A}\leq 1$. A multiplicative $\theta/9$-approximation of both $\tr{\frac{I+A/2}{4}e^{-H}}$ and $\tr{\frac{I}{4}e^{-H}}$ suffices to get an additive $\theta$-approximation of $\frac{\tr{Ae^{-H}}}{\tr{e^{-H}}}$.
\end{corollary}
\begin{proof}
According to Lemma~\ref{lemma:trTogether} by dividing the two multiplicative approximations we get \[
\frac{\theta}{3}\frac{\tr{\frac{I+A/2}{4}e^{-H}}}{\tr{\frac{I}{4}e^{-H}}}
= \frac{\theta}{3}\left(1+\frac{\tr{\frac{A}{2}e^{-H}}}{\tr{e^{-H}}}\right)
\leq \frac{\theta}{3}\left(1+\frac{\nrm{A}}{2}\right)
\leq \theta/2,
\]
i.e., an additive $\theta/2$-approximation of
$$
1+\frac{\tr{\frac{A}{2}e^{-H}}}{\tr{e^{-H}}},
$$
which yields an additive $\theta$-approximation to $\tr{A\rho}$.
\end{proof}
It thus suffices to approximate both quantities from the corollary separately. Notice that both are of the form $\tr{\frac{I+A/2}{4}e^{-H}}$, the first with the actual $A$, the second with $A=0$.
Furthermore, a multiplicative $\theta/9$-approximation to both can be achieved by approximating both up to an additive error $\theta \tr{e^{-H}} / 72$, since $\tr{\frac{I}{8} e^{-H}}\leq \tr{\frac{I+A/2}{4}e^{-H}}$.
For now, let us assume we can construct a unitary $U_{A,H}$ such that if we apply it to the state~$\ket{0\ldots 0}$ then we get a probability $\frac{\tr{(I+A/2)e^{-H}}}{4n}$ of outcome~0 when measuring the first qubit. That is:
\[
\nrm{(\bra{0}\otimes I)U_{A,H}\ket{0\ldots 0}}^2 = \frac{\tr{(I+A/2)e^{-H}}}{4n}.
\]
In practice we will not be able to construct such a $U_{A,H}$ exactly, instead we will construct a $\tilde{U}_{A,H}$ that yields a sufficiently close approximation of the correct probability.
Since we have access to such a unitary, the following lemma allows us to use amplitude estimation to estimate the probability and hence $\tr{\frac{I+A/2}{4}e^{-H}}$ up to the desired error.
\begin{lemma}\label{lem:ampest}
Suppose we have a unitary $U$ acting on $q$ qubits such that $U\ket{0\ldots 0}=\ket{0}\ket{\psi}+\ket{\Phi}$ with $(\bra{0}\otimes I)\ket{\Phi}=0$ and $\nrm{\psi}^2 =p\geq p_{\min}$ for some known bound $p_{\min}$. Let $\mu \in(0,1]$ be the allowed multiplicative error in our estimation of $p$. Then with ${\mathcal O}\left(\frac{1}{\mu\sqrt{p_{\min}}}\right)$ uses of $U$ and $U^{-1}$ and using ${\mathcal O}\left(\frac{q}{\mu\sqrt{p_{\min}}}\right)$ gates on the $q$ qubits we obtain a $\tilde{p}$ such that $|p-\tilde{p}|\leq \mu p$ with probability at least $4/5$.
\end{lemma}
\begin{proof}
We use the amplitude-estimation algorithm of~\cite[Theorem~12]{bhmt:countingj} with $M$ applications of $U$ and $U^{-1}$. This provides an estimate $\tilde{p}$ of $p$, that with probability at least $8/\pi^2>4/5$ satisfies
\begin{equation*}
|p-\tilde{p}|
\leq2\pi\frac{\sqrt{p(1-p)}}{M}+\frac{\pi^2}{M^2}
\leq\frac{\pi}{M}\left(2\sqrt{p}+\frac{\pi}{M}\right).
\end{equation*}
Choosing $M$ the smallest power of $2$ such that $M \geq 3\pi/(\mu\sqrt{p_{\min}})$, with probability at least $4/5$ we get
\begin{equation*}
|p-\tilde{p}|
\leq\mu\frac{\sqrt{p_{\min}}}{3}\left(2\sqrt{p}
+\mu\frac{ \sqrt{p_{\min}}}{3}\right)
\leq\mu\frac{\sqrt{p}}{3}\left(3\sqrt{p}\right)
\leq \mu p.
\end{equation*}
The $q$ factor in the gate complexity comes from the implementation of the amplitude amplification steps needed in amplitude-estimation. The gate complexity of the whole amplitude-estimation procedure is dominated by this contribution, proving the final gate complexity.
\end{proof}
\begin{corollary}\label{col:main22}
Suppose we are given the positive numbers $z\leq \tr{e^{-H}}$, $\theta\in(0,1]$, and unitary circuits $\tilde{U}_{A',H}$ for $A'=0$ and $A'=A$ with $\nrm{A}\leq 1$, each acting on at most $q$ qubits such that
$$
\left|\nrm{(\bra{0}\otimes I)\tilde{U}_{A',H}\ket{0\ldots 0}}^2-\frac{\tr{(I+A'/2)e^{-H}}}{4n}\right| \leq \frac{\theta z}{144n}.
$$
(To clarify the notation: if $\ket{\psi}$ is a 2-register state, then $(\bra{0}\otimes I)\ket{\psi}$ is the (unnormalized) state in the 2nd register that results from projecting on $\ket{0}$ in the 1st register.)
Applying the procedure of Lemma~\ref{lem:ampest} to $\tilde{U}_{A',H}$ (both for $A'=0$ and for $A'=A$) with $p_{\min}=\frac{z}{9n}$ and $\mu=\theta/19$, and combining the results using Corollary~\ref{col:split} yields an additive $\theta$-approximation of $\tr{A\rho}$ with high probability.
The procedure uses
\[
{\mathcal O}\left(\frac{1}{\theta}\sqrt{\frac{n}{z}}\right)
\]
applications of $\tilde{U}_{A,H}$, $\tilde{U}_{0,H}$ and their inverses, and ${\mathcal O}\left(\frac{q}{\theta}\sqrt{\frac{n}{z}}\right)$ additional gates.
\end{corollary}
\begin{proof}
First note that since $I+A'/2 \succeq I/2$ we have
\begin{equation*}
p:=\frac{\tr{(I+A'/2)e^{-H}}}{4n} \geq \frac{\tr{e^{-H}}}{8n}
\end{equation*}
and thus
\begin{equation}\label{eq:pDiff}
\left| \nrm{(\bra{0}\otimes I)\tilde{U}_{A',H}\ket{0\ldots 0}}^2 - p\right|
\leq \frac{\theta z }{144n}
\leq \frac{\theta}{18}\cdot\frac{\tr{e^{-H}}}{8 n}
\leq \frac{\theta}{18} p
\leq \frac{p}{18}.
\end{equation}
Therefore
\[
\nrm{(\bra{0}\otimes I)\tilde{U}_{A',H}\ket{0\ldots 0}}^2
\geq \left(1-\frac{1}{18}\right)p
\geq \left(1-\frac{1}{18}\right)\frac{\tr{e^{-H}}}{8n}
> \frac{\tr{e^{-H}}}{9 n}
\geq \frac{z}{9 n}=p_{\min}.
\]
Also by \eqref{eq:pDiff} we have
\begin{equation*}
\nrm{(\bra{0}\otimes I)\tilde{U}_{A',H}\ket{0\ldots 0}}^2
\leq \left(1+\frac{\theta}{18}\right) p
\leq \frac{19}{18} p.
\end{equation*}
Therefore using Lemma~\ref{lem:ampest}, with $\mu=\theta/19$, with high probability we get a $\tilde{p}$ satisfying
\begin{equation}\label{eq:ptDiff}
\left| \tilde{p} - \nrm{(\bra{0}\otimes I)\tilde{U}_{A',H}\ket{0\ldots 0}}^2 \right|
\leq \frac{\theta}{19}\cdot\nrm{(\bra{0}\otimes I)\tilde{U}_{A',H}\ket{0\ldots 0}}^2
\leq \frac{\theta}{18} p.
\end{equation}
By combining \eqref{eq:pDiff}-\eqref{eq:ptDiff} using the triangle inequality we get
\begin{equation*}
\left| p - \tilde{p}\right|
\leq \frac{\theta}{9}p,
\end{equation*}
so that Corollary~\ref{col:split} can indeed be applied.
The complexity statement follows from Lemma~\ref{lem:ampest} and our choices of $p_{\min}$ and $\mu$.
\end{proof}
Notice the $1/\sqrt{z}\geq 1/\sqrt{\tr{e^{-H}}}$ factor in the complexity statement of the last corollary. To make sure this factor is not too large, we would like to ensure $\tr{e^{-H}}=\Omega(1)$. This can be achieved by substituting $H_+ = H-\lambda_{\min}I$, where $\lambda_{\min}$ is the smallest eigenvalue of $H$. It is easy to verify that this will not change the value $\tr{Ae^{-H}/\tr{e^{-H}}}$.
It remains to show how to compute $\lambda_{\min}$ and how to apply $\tilde{U}_{A,H}$. Both of these steps are considerably easier in the case where all matrices are diagonal, so we will consider this case first.
\subsubsection{The special case of diagonal matrices -- for LP-solving}\label{sec:lptrace}
In this section we consider diagonal matrices, assuming oracle access to $H$ of the following form:
\[
O_H\ket{i}\ket{z} = \ket{i}\ket{z\oplus H_{ii}}
\]
and similarly for $A$. Notice that this kind of oracle can easily be constructed from the general sparse matrix oracle~\eqref{eq:oraclemat} that we assume access to.
\begin{lemma}\label{lem:diagU}
Let $A,H \in \mathbb{R}^{n\times n}$ be diagonal matrices such that $\nrm{A}\leq 1$ and $H\succeq 0$, and let $\mu > 0$ be an error parameter. Then there exists a unitary $\tilde{U}_{A,H}$ such that
\[
\left| \nrm{ (\bra{0}\otimes I)\tilde{U}_{A,H}\ket{0\ldots0} }^2 - \tr{\frac{I+A/2}{4n}e^{-H}}\right| \leq \mu,
\]
which uses $1$ quantum query to $A$ and $H$ and ${\mathcal O}(\log^{{\mathcal O}(1)}(1/\mu) + \log(n))$ other gates.
\end{lemma}
\begin{proof}
For simplicity assume $n$ is a power of two. This restriction is not necessary, but makes the proof a bit simpler to state.
In the first step we prepare the state $\sum_{i=1}^{n}\ket{i}/\sqrt{n}$ using $\log(n)$ Hadamard gates on $\ket{0}^{\otimes\log(n)}$.
Then we query the diagonal values of $H$ and $A$ to get the state $\sum_{i=1}^{n}\ket{i}\ket{H_{ii}}\ket{A_{ii}}/\sqrt{n}$. Using these binary values we apply a finite-precision arithmetic circuit to prepare
$$
\frac{1}{\sqrt{n}} \sum_{i=1}^{n} \ket{i}\ket{H_{ii}}\ket{A_{ii}}\ket{\beta_i} \text{, where } \beta_i:=\arcsin\left(\sqrt{\frac{1+A_{ii}/2}{4}e^{-H_{ii}}+\delta_i}\right)/\pi \text{, and } |\delta_i|\leq \mu.
$$
Note that the error $\delta_i$ comes from writing down only a finite number of bits $b_1.b_2b_3\dots b_{\log(8/\mu)}$. Due to our choice of $A$ and $H$, we know that $\beta_i$ lies in $[0,1]$.
We proceed by first adding an ancilla qubit initialized to $\ket{1}$ in front of the state, then we apply $\log(8/\mu)$ controlled rotations to this qubit: for each $b_j=1$ we apply a rotation by angle $\pi2^{-j}$. In other words, if $b_1 = 1$, then we rotate $\ket{1}$ fully to $\ket{0}$. If $b_2 = 1$, then we rotate halfway, and we proceed further by halving the angle for each subsequent bit.
We will end up with the state:
$$
\frac{1}{\sqrt{n}}\sum_{i=1}^{n} \left(\sqrt{\frac{1+A_{ii}/2}{4}e^{-H_{ii}}+\delta_i}\ket{0}+\sqrt{1-\frac{1+A_{ii}/2}{4}e^{-H_{ii}}-\delta_i}\ket{1}\right) \ket{i}\ket{A_{ii}}\ket{H_{ii}}\ket{\beta_i}.
$$
It is now easy to see that the squared norm of the $\ket{0}$-part of this state is as required:
$$
\nrm{\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\sqrt{\frac{1+A_{ii}/2}{4}e^{-H_{ii}}+\delta_i}\ket{i}}^2
=\frac{1}{n}\sum_{i=1}^{n}\left(\frac{1+A_{ii}/2}{4}e^{-H_{ii}}+\delta_i\right)
=\frac{\tr{(I+A/2)e^{-H}}}{4n}+\sum_{i=1}^{n}\frac{\delta_i}{n},
$$
which is an additive $\mu$-approximation since $\left| \sum_{i=1}^{n}\frac{\delta_i}{n} \right| \leq \mu$.
\end{proof}
\begin{corollary}\label{col:traceDiagCalc}
Let $A,H \in \mathbb{R}^{n\times n}$ be diagonal matrices, with $\nrm{A}\leq 1$. An additive $\theta$-approximation of
\[
\tr{A\rho} = \frac{\tr{Ae^{-H}}}{\tr{e^{-H}}}
\]
can be computed using ${\mathcal O}\left(\frac{\sqrt{n}}{\theta}\right)$ queries to $A$ and $H$ and $\bOt{\frac{\sqrt{n}}{\theta}}$ other gates.
\end{corollary}
\begin{proof}
Since $H$ is a diagonal matrix, its eigenvalues are exactly its diagonal entries. Using the quantum minimum-finding algorithm of D{\"u}rr and H{\o}yer~\cite{durr&hoyer:minimum} one can find (with high success probability) the minimum $\lambda_{\min}$ of the diagonal entries using ${\mathcal O}(\sqrt{n})$ queries to the matrix elements.
Applying Lemma~\ref{lem:diagU} and Corollary~\ref{col:main22} to $H_+ = H-\lambda_{\min}I$, with $z=1$, gives the stated bound.
\end{proof}
\subsubsection{General case -- for SDP-solving}\label{sec:estTrArhogeneral}
In this section we will extend the ideas from the last section to non-diagonal matrices. There are a few complications that arise in this more general case. These mostly follow from the fact that we now do not know the eigenvectors of $H$ and $A$, which were the basis states before, and that these eigenvectors might not be the same for both matrices. For example, to find the minimal eigenvalue of $H$, we can no longer simply minimize over its diagonal entries. To solve this, in Appendix~\ref{app:genMinFind} we develop new techniques that generalize minimum-finding.
Furthermore, the unitary $\tilde{U}_{A,H}$ in the LP case could be seen as applying the operator
$$
\sqrt{\frac{I+A/2}{4}e^{-H}}
$$
to a superposition of its eigenvectors. This is also more complicated in the general setting, due to the fact that the eigenvectors are no longer the basis states. In Appendix~\ref{apx:LowWeight} we develop general techniques to apply smooth functions of Hamiltonians to a state. Among other things, this will be used to create an efficient purified Gibbs sampler.
Our Gibbs sampler uses similar methods to the work of Chowdhury and Somma~\cite{ChowdhurySomma:GibbsHit} for achieving logarithmic dependence on the precision. However, the result of~\cite{ChowdhurySomma:GibbsHit} cannot be applied to our setting, because it implicitly assumes access to an oracle for $\sqrt{H}$ instead of $H$. Although their paper describes a way to construct such an oracle, it comes with a large overhead: they construct an oracle for $\sqrt{H'}=\sqrt{H + \nu I}$, where $\nu\in\mathbb{R}_+$ is some possibly large positive number. This shifting can have a huge effect on $Z'=\tr{e^{-H'}}=e^{-\nu}\tr{e^{-H}}$, which can be prohibitive due to the $\sqrt{1/Z'}$ factor in the runtime, which blows up exponentially in $\nu$.
In the following lemma we show how to implement $\tilde{U}_{A,H}$ using the techniques we developed in Appendix~\ref{apx:LowWeight}.
\begin{lemma}\label{lemma:trPreEst}
Let $A,H\in \mathbb{C}^{n\times n}$ be Hermitian matrices such that $\nrm{A}\leq 1$ and $I\preceq H\preceq K I$ for a known $K\in \mathbb{R}_+$. Assume $A$ is $s$-sparse and $H$ is $d$-sparse with $s\leq d$. Let $\mu > 0$ be an error parameter. Then there exists a unitary $\tilde{U}_{A,H}$ such that
\[
\left| \nrm{ (\bra{0}\otimes I)\tilde{U}_{A,H}\ket{0\ldots0} }^2 - \tr{\frac{I+A/2}{4n}e^{-H}}\right| \leq \mu
\]
that uses $\bOt{Kd}$ queries to $A$ and $H$, and the same order of other gates.
\end{lemma}
\begin{proof}
The basic idea is that we first prepare a maximally entangled state $\sum_{i=1}^{n}\ket{i}\ket{i}/\sqrt{n}$, and then apply the (norm-decreasing) maps $e^{-H/2}$ and $\sqrt{\frac{I+A/2}{4}}$ to the first register. Note that we can assume without loss of generality that $\mu\leq 1$, otherwise the statement is trivial.
Let $\tilde{W}_0=\left(\bra{0}\otimes I\right)\tilde{W}\left(\ket{0}\otimes I\right)$ be a $\mu/5$-approximation of the map $e^{-H/2}$ (in operator norm) implemented by using Theorem~\ref{thm:emH}, and let $\tilde{V}_0=\left(\bra{0}\otimes I\right)\tilde{V}\left(\ket{0}\otimes I\right)$ be a $\mu/5$-approximation of the map $\sqrt{\frac{I+A/2}{4}}$ implemented by using Theorem~\ref{thm:Taylor}. We define $\tilde{U}_{A,H}:=\tilde{V}\tilde{W}$, noting that there is a hidden $\otimes I$ factor in both $\tilde{V}$ and $\tilde{W}$ corresponding to each other's ancilla qubit. As in the linear programming case, we are interested in $p$, the probability of measuring a $00$ in the first register (i.e., the two ``flag'' qubits) after applying $\tilde{U}_{A,H}$. We will analyze this in terms of these operators below. We will make the final approximation step precise in the next paragraph.
\begin{align}
p'&:=\nrm{\left(\bra{00}\otimes I\right) \tilde{U}_{A,H}\left(\ket{00}\otimes I\right)\sum_{i=1}^{n}\frac{\ket{i}\ket{i}}{\sqrt{n}}}^2 \nonumber\\
&=\nrm{\tilde{V}_0\tilde{W}_0\sum_{i=1}^{n}\frac{\ket{i}\ket{i}}{\sqrt{n}}}^2\nonumber\\
&=\frac{1}{n}\sum_{i=1}^{n}\bra{i}\tilde{W}_0^\dagger\tilde{V}_0^\dagger\tilde{V}_0\tilde{W}_0\ket{i}\nonumber\\
&=\frac{1}{n}\tr{\tilde{W}_0^\dagger\tilde{V}_0^\dagger\tilde{V}_0\tilde{W}_0}\nonumber\\
&=\frac{1}{n}\tr{\tilde{V}_0^\dagger\tilde{V}_0\tilde{W}_0\tilde{W}_0^\dagger}\label{eq:exactTrace}\\
&\approx\frac{1}{n}\tr{\frac{I+A/2}{4}e^{-H}}. \label{eq:approxTrace}
\end{align}
Note that for all matrices $B,\tilde{B}$ with $\nrm{B}\leq 1$, we have
\begin{align*}
\nrm{B^\dagger B - \tilde{B}^\dagger \tilde{B}}& = \nrm{(B^\dagger-\tilde{B}^\dagger)B + B^\dagger(B-\tilde{B}) - (B^\dagger-\tilde{B}^\dagger)(B-\tilde{B})}\\
& \leq \nrm{(B^\dagger-\tilde{B}^\dagger)B} + \nrm{B^\dagger(B-\tilde{B})} +\nrm{(B^\dagger-\tilde{B}^\dagger)(B-\tilde{B})}\\
& \leq \nrm{B^\dagger-\tilde{B}^\dagger}\nrm{B} + \nrm{B^\dagger}\nrm{B-\tilde{B}} +\nrm{B^\dagger-\tilde{B}^\dagger}\nrm{B-\tilde{B}}\\
& \leq 2\nrm{B-\tilde{B}}+\nrm{B-\tilde{B}}^2.
\end{align*}
Since $\mu\leq 1$, and hence $2\mu/5 + (\mu /5)^2\leq \mu/2$, this implies (with $B=e^{-H/2}$ and $\tilde{B}=\tilde{W}_0^\dagger$) that $\nrm{e^{-H} - \tilde{W}_0\tilde{W}_0^\dagger}\leq \mu / 2$, and also (with $B=\sqrt{(I+A/2)/4}$ and $\tilde{B}=\tilde{V}_0$) $\nrm{(I+A/2)/4 - \tilde{V}_0^\dagger\tilde{V}_0}\leq \mu/2$. Let $\nrm{\cdot}_1$ denote the trace norm (a.k.a.\ Schatten 1-norm). Note that for all $C,D,\tilde{C},\tilde{D}$:
\begin{align*}
\left|\tr{CD}-\tr{\tilde{C}\tilde{D}}\right|
&\leq \nrm{CD-\tilde{C}\tilde{D}}_1\\
&= \nrm{ (C-\tilde{C})D + C(D-\tilde{D})-(C-\tilde{C})(D-\tilde{D}) }_1\\
&\leq \nrm{(C-\tilde{C})D}_1+\nrm{C(D-\tilde{D})}_1+\nrm{(C-\tilde{C})(D-\tilde{D})}_1\\
& \leq \nrm{C-\tilde{C}}\nrm{D}_1+\nrm{D-\tilde{D}}\left(\nrm{C}_1+\nrm{C-\tilde{C}}_1\right).
\end{align*}
Which, in our case (setting $C=(I+A/2)/4$, $D=e^{-H}$, $\tilde{C}=\tilde{V}_0^\dagger\tilde{V}_0$, and $\tilde{D}=\tilde{W}_0\tilde{W}_0^\dagger$) implies that
\[
\left|\tr{\left(I+A/2\right) e^{-H}/4}-\tr{\tilde{V}_0^\dagger\tilde{V}_0\tilde{W}_0\tilde{W}_0^\dagger}\right| \leq (\mu/2) \tr{e^{-H}} +(\mu/2)(1/2+\mu / 2)n.
\]
Dividing both sides by $n$ and using equation~\eqref{eq:exactTrace} then implies
\begin{align*}
\left|\tr{\left(I+A/2\right) e^{-H}}/(4n)-p'\right| & \leq \frac{\mu}{2} \frac{\tr{e^{-H}}}{n}+\frac{\mu}{2}\left(\frac{1}{2}+\frac{\mu}{2}\right) \\
&\leq \frac{\mu}{2}+ \frac{\mu}{2}\\
&= \mu.
\end{align*}
This proves the correctness of $\tilde{U}_{A,H}$. It remains to show that the complexity statement holds.
To show this we only need to specify how to implement the map $\sqrt{\frac{I+A/2}{4}}$ using Theorem~\ref{thm:Taylor}, since the map $e^{H/2}$ is already dealt with in Theorem~\ref{thm:emH}. To use Theorem~\ref{thm:Taylor}, we choose $x_0:=0$, $K:=1$ and $r:=1$, since $\nrm{A}\leq 1$. Observe that $\sqrt{\frac{1+x/2}{4}}=\frac{1}{2}\sum_{k=0}^{\infty}\binom{1/2}{k}\left(\frac{x}{2}\right)^k$ whenever $|x|\leq 1$. Also let $\delta=1/2$, so $r+\delta=\frac{3}{2}$ and $\frac{1}{2}\sum_{k=0}^{\infty}\left|\binom{1/2}{k}\right|\left(\frac{3}{4}\right)^k\leq 1=:B$.
Recall that $\tilde{V}$ denotes the unitary that Theorem~\ref{thm:Taylor} constructs.
Since we choose the precision parameter to be $\mu/5=\Theta(\mu)$, Theorem~\ref{thm:Taylor} shows $\tilde{V}$ can be implemented using ${\mathcal O}\left(d\log^2\left(1/\mu\right)\right)$ queries and ${\mathcal O}\left(d\log^2\left(1/\mu\right)\left[\log(n)+\log^{2.5}\left(1/\mu\right)\right]\right)$ gates. This cost is negligible compared to our implementation cost of $e^{-H/2}$ with $\mu/5$ precision: Theorem~\ref{thm:emH} uses ${\mathcal O}\left(Kd\log^2\left(K/\mu\right)\right)$ queries and ${\mathcal O}\left(Kd\log^2\left(Kd/\mu\right)\left[\log(n)+\log^{2.5}\left(Kd/\mu\right)\right]\right)$ gates to implement~$\tilde{W}$.
\end{proof}
\begin{corollary}\label{col:traceCalc}
Let $A,H\in \mathbb{C}^{n\times n}$ be Hermitian matrices such that $\nrm{A}\leq 1$ and $\nrm{H}\leq K$ for a known bound $K\in \mathbb{R}_+$. Assume $A$ is $s$-sparse and $H$ is $d$-sparse with $s\leq d$. An additive $\theta$-approximation of
\[
\tr{A\rho} = \frac{Ae^{-H}}{\tr{e^{-H}}}
\]
can be computed using $\bOt{\frac{\sqrt{n}dK}{\theta}}$ queries to $A$ and $H$, while using the same order of other gates.
\end{corollary}
\begin{proof}
Start by computing an estimate $\tilde{\lambda}_{\min}$ of $\lambda_{\min}(H)$, the minimum eigenvalue of $H$, up to additive error $\varepsilon = 1/2$ using Lemma~\ref{lemma:normEst}.
We define $H_+ := H-(\tilde{\lambda}_{\min}-3/2)I$, so that $I\preceq H_+$ but $2I\nprec H_+$.
Applying Lemma~\ref{lemma:trPreEst} and Corollary~\ref{col:main22} to $H_+$ with $z=e^{-2}$ gives the stated bound.
\end{proof}
\subsection{An efficient 2-sparse oracle} \label{sec:oracle}
Recall from the end of Section~\ref{sec:classicalAK} that $\tilde{a}_j$ is an additive $\theta$-approximation to $\tr{A_j\rho}$, $\tilde{c}$ is a $\theta$-approximation to $\tr{C\rho}$ and $c^{\prime} = \tilde{c} - r \theta - \theta$. We first describe our quantum 2-sparse oracle assuming access to a unitary which acts as $\ket{j}\ket{0}\ket{0} \mapsto \ket{j} \ket{\tilde{a}_j} \ket{\psi_j}$, where $\ket{\psi_j}$ is some workspace state depending on~$j$. We then briefly discuss how to modify the analysis when we are given an oracle which acts as $\ket{j}\ket{0}\ket{0} \mapsto \ket{j} \sum_i \beta_j^{i}\ket{\tilde{a}^{i}_j} \ket{\psi^{i}_j}$ (where each $\tilde{a}^{i}_j$ is an additive $\theta$-approximation to $\tr{A_j \rho}$), since this is the output of the trace-estimation procedure of the previous section.
Our goal is to find a $y \in \tilde{\mathcal{P}}(\tilde{a},c^{\prime})$, i.e., a $y$ such that
\begin{align*}
\nrm{y}_1&\leq r\\
b^T y &\leq \alpha\\
\tilde{a}^T y &\geq c^{\prime}\\
y &\geq 0
\end{align*}
If $\alpha \geq 0$ and $c^{\prime} \leq 0$, then $y=0$ is a solution and our oracle can return it.
If not, then we may write $y = Nq$ with $N=\nrm{y}_1 > 0$ and hence $\nrm{q}_1 = 1$.
So we are looking for an $N$ and a $q$ such that
\begin{align}
b^T q &\leq \alpha / N\label{eq:Nbound}\\
\tilde{a}^T q &\geq c^{\prime} / N\notag\\\notag
\nrm{q}_1 &= 1\\\notag
q &\geq 0\\\notag
0 &< N \leq r
\end{align}
We can now view $q \in \mathbb{R}_+^m$ as the coefficients of a convex combination of the points $p_i = (b_i,\tilde{a}_i)$ in the plane. We want such a combination that lies to the upper left of $g_N = (\alpha/N,c^{\prime} / N)$ for some $0 < N\leq r$. Let $\mathcal{G}_N$ denote the upper-left quadrant of the plane starting at $g_N$.
\begin{lemma}\label{lem:2DArg}
If there is a $y \in \tilde{\mathcal{P}} (\tilde{a},c^{\prime})$, then there is a $2$-sparse $y^{\prime} \in \tilde{\mathcal{P}} (\tilde a,c^{\prime})$ such that $\nrm{y}_1 = \nrm{y^{\prime}}_1$.
\end{lemma}
\begin{proof}
Consider $p_i = (b_i,\tilde{a}_i)$ and $g = (\alpha / N,c^{\prime} / N)$ as before, and write $y = Nq$ where $\sum_{j=1}^m q_j =1$, $q \geq 0$. The vector $q$ certifies that a convex combination of the points $p_i$ lies in $\mathcal{G}_N$. But then there exist $j,k \in \lbrack m \rbrack$ such that the line segment $\overline{p_jp_k}$ intersects $\mathcal{G}_N$. All points on this line segment are convex combinations of $p_j$ and $p_k$, hence there is a convex combination of $p_j$ and $p_k$ that lies in~$\mathcal{G}_N$. This gives a $2$-sparse $q^{\prime}$, and $y' = N q' \in \tilde{\mathcal{P}} (\tilde{a},c^{\prime})$.
\end{proof}
We can now restrict our search to $2$-sparse $y$.
Let $\mathcal{G} = \bigcup_{N \in (0, r\rbrack } \mathcal{G}_N$. Then we want to find two points $p_j,p_k$ that have a convex combination in $\mathcal{G}$, since this implies that a scaled version of their convex combination gives a $y \in \tilde{\mathcal{P}} (\tilde{a},c^{\prime})$ and $\nrm{y}_1 \leq r$.
\begin{lemma}\label{lem:oracle}
There is an oracle that returns a 2-sparse vector $y\in \tilde{\mathcal{P}}(\tilde{a},c^{\prime})$, if one exists, using one search and two minimizations over the $m$ points $p_j=(b_j,\tilde{a}_j)$.
This gives a classical algorithm that uses ${\mathcal O}(m)$ calls to the subroutine that gives the entries of~$\tilde{a}$ and ${\mathcal O}(m)$ other operations, and a quantum algorithm that (in order to solve the problems with high probability) uses ${\mathcal O}(\sqrt{m})$ calls to the subroutine that gives the entries of~$\tilde{a}$ and $\bOt{\sqrt{m}}$ other gates.
\end{lemma}
\begin{proof}
The algorithm can be summarized as follows:
\begin{enumerate}
\item Check if $\alpha \geq 0$ and $c^{\prime}\leq 0$. If so, output $y=0$. \label{it:step1}
\item Check if there is a $p_i\in \mathcal G$. If so, let $q = e_i$ and $N = \frac{c^{\prime}}{\tilde{a}_i}$. \label{it:step2}
\item Find $p_j,p_k$ so that the line segment $\overline{p_jp_k}$ goes through $\mathcal G$. This gives coefficients $q$ of a convex combination that can be scaled by $N = \frac{c^{\prime}}{\tilde{a}^T q}$ to give $y$. The main realization is that we can search separately for $p_j$ and $p_k$. \label{it:step3}
\end{enumerate}
First we will need a better understanding of the shape of $\mathcal G$ (see Figure~\ref{fig:graph} for illustration). This depends on the sign of $\alpha$ and $c^{\prime}$. If we define $\mbox{\rm sign}(0) = 1$:
\begin{itemize}
\item[(a)] If $\mbox{\rm sign}(\alpha) = -1, \mbox{\rm sign}(c^{\prime}) = -1$. The corner point of $\mathcal{G}$ is $(\alpha / r,c^{\prime} / r)$. One edge goes up vertically and an other follows the line segment $\lambda \cdot (\alpha,c^{\prime})$ for $\lambda \in [ 1/r,\infty)$ starting at the corner.
\item[(b)] If $\mbox{\rm sign}(\alpha) = -1, \mbox{\rm sign}(c^{\prime}) = 1$. Here $\mathcal G_N \subseteq \mathcal G_r$ for $N\leq r$. So $\mathcal G = \mathcal G_r$. The corner point is again $(\alpha / r,c^{\prime} / r)$, but now one edge goes up vertically and one goes to the left horizontally.
\item[(c)] If $\mbox{\rm sign}(\alpha) = 1, \mbox{\rm sign}(c^{\prime}) = -1$. This is the case where $y=0$ is a solution, $\mathcal G$ is the whole plane and has no corner.
\item[(d)] If $\mbox{\rm sign}(\alpha) = 1, \mbox{\rm sign}(c^{\prime}) = 1$. The corner point of $\mathcal G$ is again $(\alpha / r,c^{\prime} / r)$. From there one edge goes to the left horizontally and one edge follows the line segment $\lambda \cdot (\alpha,c^{\prime})$ for $\lambda \in [ 1/r,\infty)$.
\end{itemize}
\begin{figure}[hbt]
\renewcommand{\scale}{1}
\renewcommand{\sfw}{.49}
\centering
\begin{subfigure}[b]{\sfw \linewidth}
\centering
\renewcommand{\cx}{-1}
\renewcommand{\cy}{-.5}
\input{Ggraph.txt}
\caption{$\mbox{\rm sign}(\alpha) = -1,\mbox{\rm sign}(c^{\prime}) = -1$}
\end{subfigure}
\begin{subfigure}[b]{\sfw \linewidth}
\centering
\renewcommand{\cx}{-1}
\renewcommand{\cy}{.5}
\input{Ggraph.txt}
\caption{$\mbox{\rm sign}(\alpha) = -1,\mbox{\rm sign}(c^{\prime}) = 1$}
\end{subfigure}
\vskip .5cm
\begin{subfigure}[b]{\sfw \linewidth}
\centering
\renewcommand{\cx}{1}
\renewcommand{\cy}{-.5}
\input{Ggraph.txt}
\caption{$\mbox{\rm sign}(\alpha) = 1,\mbox{\rm sign}(c^{\prime}) = -1$}
\end{subfigure}
\begin{subfigure}[b]{\sfw \linewidth}
\centering
\renewcommand{\cx}{1}
\renewcommand{\cy}{.5}
\input{Ggraph.txt}
\caption{$\mbox{\rm sign}(\alpha) = 1,\mbox{\rm sign}(c^{\prime}) = 1$}
\end{subfigure}
\caption{The region $\mathcal G$ in light blue. The borders of two quadrants $\mathcal G_N$ have been drawn by thick dashed blue lines. The red dot at the beginning of the arrow is the point $(\alpha/r,c^{\prime}/r)$.}
\label{fig:graph}
\end{figure}
Since $\mathcal G$ is always an intersection of at most $2$ halfspaces, steps~\ref{it:step1}-\ref{it:step2} of the algorithm are easy to perform. In step~\ref{it:step1} we handle case (c) by simply returning $y=0$. For the other cases $(\alpha /r ,c^{\prime} /r)$ is the corner point of $\mathcal G$ and the two edges are simple lines. Hence in step~\ref{it:step2} we can easily search through all the points to find out if there is one lying in $\mathcal G$; since $\mathcal G$ is a very simple region, this only amounts to checking on which side of the two lines a point lies.
\begin{figure}[hbt]
\renewcommand{\scale}{.75}
\renewcommand{\sfw}{.49}
\centering
\centering
\renewcommand{\cx}{1}
\renewcommand{\cy}{-1}
\input{angles.txt}
\caption{Illustration of $\mathcal G$ with the points $p_j,p_k$ and the angles $\angle \ell_j L_1,\angle L_1 L_2,\angle L_2\ell_k$ drawn in. Clearly the line $\overline{p_jp_k}$ only crosses $\mathcal{G}$ when the total angle is less than $\pi$.}
\label{fig:angles}
\end{figure}
Now, if we cannot find a single point in $\mathcal{G}$, we need a combination of two points in step~\ref{it:step3}. Let $L_1, L_2$ be the edges of $\mathcal G$ and let $\ell_j$ and $\ell_k$ be the line segments from $(\alpha/r,c^{\prime}/r)$ to $p_j$ and $p_k$, respectively.
Then, as can be seen in Figure~\ref{fig:angles}, the line segment $\overline{p_jp_k}$ goes through $\mathcal G$ if and only if (up to relabeling $p_j$ and $p_k$) $\angle \ell_j L_1 + \angle L_1 L_2 + \angle L_2 \ell_k \leq \pi$. Since $\angle L_1 L_2$ is fixed, we can simply look for a $j$ such that $\angle \ell_j L_1$ is minimized and a $k$ such that $\angle L_2\ell_k$ is minimized. If $\overline{p_jp_k}$ does not pass through $\mathcal G$ for this pair of points, then it does not for any of the pairs of points.
Notice that these minimizations can be done separately and hence can be done in the stated complexity. Given the minimizing points $p_j$ and $p_k$, it is easy to check if they give a solution by calculating the angle between $\ell_j$ and $\ell_k$. The coefficients of the convex combination $q$ are then easy to compute.
It remains to compute the scaling $N$. This is done by rewriting the constraints of~\eqref{eq:Nbound}:
\[
\frac{c^{\prime}}{q^T \tilde{a}}\leq N \leq \frac{\alpha}{q^T b}
\]
So we can pick any value in this range for~$N$.
\vskip -0.5cm
\end{proof}
The analysis above applies if we are given an oracle which acts as $\ket{j}\ket{0}\ket{0} \mapsto \ket{j} \ket{\tilde{a}_j} \ket{\psi_j}$. However, the trace estimation procedure of Corollary~\ref{col:traceCalc} acts as $\ket{j}\ket{0}\ket{0} \mapsto \ket{j} \sum_i \beta_j^{i}\ket{\tilde{a}^{i}_j} \ket{\psi^{i}_j}$ where each $\ket{\tilde{a}^{i}_j}$ is an approximation of $a_j$ and the amplitudes $\beta_j^{i}$ are such that measuring the second register with high probability returns an $\tilde{a}^{i}_j$ which is $\theta$-close to $a_j$. Since we can exponentially reduce the probability that we obtain an $\tilde{a}^{i}_j$ which is further than $\theta$ away from $a_j$, we will for simplicity assume that for all $i,j$ we have $|\tilde{a}^{i}_j - a_j| \leq \theta$; the neglected exponentially small probabilities will only affect the analysis in negligible ways.
Note that while we do not allow our oracle enough time to obtain classical descriptions of all $\tilde a_j$s (we aim for a runtime of $\bOt{\sqrt{m}}$), we do have enough time to compute $\tilde{c}$ once initially. Knowing $\tilde c$, we can compute the angles defined by the points $(b_j,\tilde{a}_j^{i})$ with respect to the corner point of $(\alpha/r,(\tilde{c}-\theta)/ r-\theta)$ and the lines $L_1, L_2$ (see Figure~\ref{fig:angles}). We now apply our generalized minimum-finding algorithm with runtime $\bOt{\sqrt{m}}$ (see Theorem~\ref{thm:genMin}) starting with a uniform superposition over the $j$s to find $\ell,k \in [m]$ with points $(\tilde a_\ell,b_\ell)$ and $(\tilde a_k, b_k)$ approximately minimizing the respective angles to lines $L_1,L_2$. It follows from Lemma~\ref{lem:approxP} and Lemma~\ref{lem:2DArg} that if $\mathcal P_{0}(X) \cap \{y \in \mathbb R^m: \sum_{j} y_j \leq r\}$ is non-empty, then some convex combination of $(\tilde a_\ell, b_\ell)$ and $(\tilde a_k, b_k)$ lies in $\mathcal G$. On the other hand, if $\mathcal P_{4Rr\theta}(X) \cap \{y \in \mathbb R^m: \sum_{j} y_j \leq r\}$ is empty, then by the same lemmas the respective angles will be such that we correctly conclude that $\mathcal P_{0}(X) \cap \{y \in \mathbb R^m: \sum_{j} y_j \leq r\}$ is empty.
\subsection{Total runtime} \label{sec:runtime}
We are now ready to add our quantum implementations of the trace calculations and the oracle to the classical Arora-Kale framework.
\upperbound*
\begin{proof} Using our implementations of the different building blocks, it remains to calculate what the total complexity will be when they are used together.
\begin{description}
\item[Cost of the oracle for $H^{(t)}$.]
The first problem in each iteration is to obtain access to an oracle for $H^{(t)}$. In each iteration the oracle will produce a $y^{(t)}$ that is at most $2$-sparse, and hence in the $(t+1)$th iteration, $H^{(t)}$ is a linear combination of $2t$ of the $A_j$ matrices and the $C$ matrix.
We can write down a sparse representation of the coefficients of the linear combination that gives $H^{(t)}$ in each iteration by adding the new terms coming from $y^{(t)}$. This will clearly not take longer than $\bOt{T}$, since there are only a constant number of terms to add for our oracle. As we will see, this term will not dominate the complexity of the full algorithm.
Using such a sparse representation of the coefficients, one query to a sparse representation of $H^{(t)}$ will cost $\bOt{st}$ queries to the input matrices and $\bOt{st}$ other gates. For a detailed explanation and a matching lower bound, see Appendix~\ref{app:sparsematrixsum}.
\item[Cost of the oracle for $\tr{A_j\rho}$.]
In each iteration $M^{(t)}$ is made to have operator norm at most~$1$.
This means that
\[
\nrm{-\eta H^{(t)}} \leq \eta \sum_{\tau = 1}^t \nrm{M^{(\tau)}} \leq \eta t.
\]
Furthermore we know that $H^{(t)}$ is at most $d:=s(2t+1)$-sparse.
Calculating $\tr{A_j\rho}$ for one index $j$ up to an additive error of $\theta := \varepsilon/(12Rr)$ can be done using the algorithm from Corollary~\ref{col:traceCalc}. This will take
\[
\bOt{\sqrt{n}\frac{\nrm{H}d}{\theta}} = \bOt{\sqrt{n} s\frac{\eta t^2 Rr}{\varepsilon}}
\]
queries to the oracle for $H^{(t)}$ and the same order of other gates.
Since each query to $H^{(t)}$ takes $\bOt{st}$ queries to the input matrices, this means that
\[
\bOt{\sqrt{n}s^2\frac{\eta t^3 Rr}{\varepsilon}}
\]
queries to the input matrices will be made, and the same order of other gates, for each approximation of a $\tr{A_j\rho}$ (and similarly for approximating $\tr{C\rho}$).
\item[Total cost of one iteration.]
Lemma~\ref{lem:oracle} tells us that we will use $\bOt{\sqrt{m}}$ calculations of $\tr{A_j\rho}$, and the same order of other gates, to calculate a classical description of a $2$-sparse $y^{(t)}$.
This brings the total cost of one iteration to
\[
\bOt{\sqrt{nm} s^2\frac{\eta t^3 Rr}{\varepsilon}}
\]
queries to the input matrices, and the same order of other gates.
\item[Total quantum runtime for SDPs.]
Since $w\leq r+1$ we can set $T = \bOt{\frac{R^2r^2}{\varepsilon^2}}$. With $\eta = \sqrt{\frac{\ln(n)}{T}}$,
summing over all iterations in one run of the algorithm gives a total cost of
\begin{align*}
\bOt{\sum_{t=1}^T \sqrt{nm} s^2\frac{\eta t^3 Rr}{\varepsilon}} &= \bOt{\sqrt{nm} s^2\frac{\eta T^4 Rr}{\varepsilon}} \\
&= \bOt{\sqrt{nm} s^2\left( \frac{Rr}{\varepsilon}\right)^{\!\!8} }
\end{align*}
queries to the input matrices and the same order of other gates.
\end{description}
\vskip -0.5cm
\end{proof}
\paragraph{Total quantum runtime for LPs.}
The final complexity of our algorithm contains a factor $\bOt{sT}$ that comes from the sparsity of the $H^{(t)}$ matrix. This assumes that when we add the input matrices together, the rows become less sparse. This need not happen for certain SDPs. For example, in the SDP relaxation of MAXCUT, the $H^{(t)}$ will always be $d$-sparse, where $d$ is the degree of the graph. A more important class of examples is that of linear programs: since LPs have diagonal $A_j$ and $C$, their sparsity is $s=1$, and even the sparsity of the $H^{(t)}$ is always~$1$. This, plus the fact that the traces can be computed without a factor $\nrm{H}$ in the complexity (as shown in Corollary~\ref{col:traceDiagCalc} in Section~\ref{sec:lptrace}), means that our algorithm solves LPs with
\[
\bOt{\sqrt{nm} \left( \frac{Rr}{\varepsilon}\right)^{\!\!5} }
\]
queries to the input matrices and the same order of other gates.
\paragraph{Total classical runtime.}
Using the classical techniques for trace estimation from Appendix~\ref{app:trace}, and the classical version of our oracle (Lemma~\ref{lem:oracle}), we are also able to give a general classical instantiation of the Arora-Kale framework. The final complexity will then be
\[
\bOt{nms\left(\frac{Rr}{\varepsilon}\right)^{\!\!4}+ns\left(\frac{Rr}{\varepsilon}\right)^{\!\!7}}.
\]
The better dependence on $Rr/\varepsilon$ and $s$, compared to our quantum algorithm, comes from the fact that we now have the time to write down intermediate results explicitly. For example, we do not need to recalculate parts of $H^{(t)}$ for every new query to it, instead we can just calculate it once at the start of the iteration by adding $M^{(t)}$ to $H^{(t-1)}$ and writing down the result.
\paragraph{Further remarks.}
We want to stress again that our solver is meant to work for \emph{all} SDPs. In particular, it does not use the structure of a specific SDP. As we show in the next section, every oracle that works for all SDPs must have large width. To obtain quantum speedups for a \emph{specific} class of SDPs, it will be necessary to develop oracles tuned to that problem. We view this as an important direction for future work. Recall from the introduction that Arora and Kale also obtain fast classical algorithms for problems such as MAXCUT by doing exactly that: they develop specialized oracles for those problems.
\section{Downside of this method: general oracles are restrictive}\label{sec:downside}
In this section we show some of the limitations of a method that uses sparse or general oracles, i.e., ones that are not optimized for the properties of specific SDPs. We will start by discussing sparse oracles in the next section. We will use a counting argument to show that sparse solutions cannot hold too much information about a problem's solution.
In Section~\ref{sec: general width bounds} we will show that width-bounds that do not depend on the specific structure of an SDP are for many problems not efficient.
As in the rest of the paper, we will assume the notation of Section~\ref{sec:upperbounds}, in particular of Meta-Algorithm~\ref{alg:AKSDP}.
\subsection{Sparse oracles are restrictive}
\begin{lemma}
If, for some specific SDP, every $\varepsilon$-optimal dual-feasible vector has at least $\ell$ non-zero elements, then the width $w$ of any $k$-sparse oracle for this SDP is such that $\frac{Rw}{\varepsilon} = \Omega\left(\sqrt{\frac{\ell}{k\ln(n)}}\right)$.
\end{lemma}
\begin{proof}
The vector $\bar{y}$ returned by Meta-Algorithm~\ref{alg:AKSDP} is, by construction, the average of $T$ vectors $y^{(t)}$ that are all $k$-sparse, plus one extra $1$-sparse term of $\frac{\varepsilon}{R}e_1$, and hence $\ell \leq kT+1$. The stated bound on $\frac{Rw}{\varepsilon}$ then follows directly by combining this inequality with $T = {\mathcal O}(\frac{R^2w^2}{\varepsilon^2}\ln(n))$.
\end{proof}
The oracle presented in Section~\ref{sec:oracle} always provides a $2$-sparse vector $y$. This implies that if an SDP requires an $\ell$-sparse dual solution, we must have $\frac{Rw}{\varepsilon} = \Omega(\sqrt{\ell / \ln(n)})$. This in turn means that the upper bound on the runtime of our algorithm will be of order $\ell^4 \sqrt{nm} s^2$. This is clearly bad if $\ell$ is of the order $n$ or $m$.
Of course it could be the case that almost every (useful) SDP has a sparse approximate dual solution (or can easily be rewritten so that it does), and hence sparseness might be not a restriction at all. However, this does not seem to be the case. We will prove that for certain kinds of SDPs, no useful dual solution can be very sparse. Let us first define what we mean by \emph{useful}.
\begin{definition}\label{def:problem}
A problem is defined by a function $f$ that, for every element $p$ of the problem domain $\mathcal D$, gives a subset of the solution space $\mathcal S$, consisting of the solutions that are considered correct.
We say a family of SDPs, $\{SDP^{(p)}\}_{p\in \mathcal D}$, solves the problem via the dual if there is an $\varepsilon \geq 0$ and a function $g$ such that for every $p \in \mathcal D$ and every $\varepsilon$-optimal dual-feasible vector $y^{(p)}$ to $SDP^{(p)}$:
\[
g(y^{(p)}) \in f(p).
\]
In other words, an $\varepsilon$-optimal dual solution can be converted into a correct solution of the original problem without more knowledge of $p$.
\end{definition}
For these kinds of SDP families we will prove a lower bound on the sparsity of the dual solutions.
The idea for this bound is as follows. If you have a lot of different instances that require different solutions, but the SDPs are equivalent up to permuting the constraints and the coordinates of $\mathbb{R}^n$, then a dual solution should have a lot of unique permutations and hence cannot be too sparse.
\begin{theorem}
Consider a problem and a family of SDPs as in Definition~\ref{def:problem}. Let $\mathcal T \subseteq \mathcal D$ be such that for all $p,q \in \mathcal T$:
\begin{itemize}
\item $f(p) \cap f(q) = \emptyset$. That is, a solution to $p$ is not a solution to $q$ and vice-versa.
\item Let $A_j^{(p)}$ be the constraints of $SDP^{(p)}$ and $A_j^{(q)}$ those from $SDP^{(q)}$ (and define $C^{(p)}$, $C^{(q)}$, $b_j^{(p)}$, and $b_j^{(q)}$ in the same manner). Then there exist $\sigma \in S_n$, $\pi \in S_m$ s.t.\ $\sigma^{-1} A^{(p)}_{\pi(j)} \sigma = A^{(q)}_j$ (and $\sigma^{-1} C^{(p)} \sigma = C^{(q)}$). That is, the SDPs are the same up to permutations of the labels of the constraints and permutations of the coordinates of $\mathbb{R}^n$.
\end{itemize}
If $y^{(p)}$ is an $\varepsilon$-optimal dual-feasible vector to $SDP^{(p)}$ for some $p\in \mathcal T$, then $y^{(p)}$ is at least $\frac{\log(|\mathcal T|)}{\log m}$-dense (i.e., has at least that many non-zero entries).
\end{theorem}
\begin{proof}
We first observe that, with $SDP^{(p)}$ and $SDP^{(q)}$ as in the lemma, if $y^{(p)}$ is an $\varepsilon$-optimal dual-feasible vector of $SDP^{(p)}$, then $y^{(q)}$ defined by
\[
y^{(q)}_j := y^{(p)}_{\pi(j)} = \pi(y^{(p)})_j
\]
is an $\varepsilon$-optimal dual vector for $SDP^{(q)}$. Here we use the fact that a permutation of the $n$~coordinates in the primal does not affect the dual solutions. Since $f(p)\cap f(q) = \emptyset$ we know that $g(y^{(p)}) \neq g(y^{(q)})$ and so $y^{(p)} \neq y^{(q)}$. Since this is true for every $q$ in $\mathcal T$, there should be at least $|\mathcal T|$ different vectors $y^{(q)} = \pi (y^{(p)})$.
A $k$-sparse vector can have $k$ different non-zero entries and hence the number of possible unique permutations of that vector is at most
\[
\binom{m}{k} k! = \frac{m!}{(m-k)!} = \prod_{t = m-k+1}^m t \leq m^k
\]
so
\[
\frac{\log |\mathcal{T}|}{\log m} \leq k.
\]
\vskip -5mm
\end{proof}
\paragraph{Example.} Consider the $(s,t)$-mincut problem, i.e., the dual of the $(s,t)$-maxflow. Specifically, consider a simple instance of this problem: the union of two complete graphs of size $z+1$, where $s$ is in one subgraph and $t$ in the other. Let the other vertices be labeled by $\{1,2,\dots,2z\}$.
Every assignment of the labels over the two halves gives a unique mincut, in terms of which labels fall on which side of the cut. There is exactly one partition of the vertices in two sets that cuts no edges (namely the partition consists of the two complete graphs), and every other partition cuts at least $z$ edges. Hence a $z/2$-approximate cut is a mincut. This means that there are $\binom{2z}{z}$ problems that require a different output. So for every family of SDPs that is symmetric under permutation of the vertices and for which a $z/2$-approximate dual solution gives an $(s,t)$-mincut, the sparsity of a $z/2$-approximate dual solution is at least\footnote{Here $m$ is the number of constraints, not the number of edges in the graph.}
\[
\frac{\log {\binom{2z}{z}}}{\log m} \geq \frac{z}{ \log{m}},
\]
where we used that $\binom{2z}{z} \geq \frac{2^{2z}}{2\sqrt{z}}$.
\subsection{General width-bounds are restrictive for certain SDPs} \label{sec: general width bounds}
In this section we will show that width-bounds can be restrictive when they do not consider the specific structure of an SDP.
\begin{definition} A function $w(n,m,s,r,R,\varepsilon)$ is called a \emph{general width-bound} for an oracle if $w(n,m,s,r,R,\varepsilon)$ is a correct width-bound for that oracle, for every SDP with parameters $n,m,s,r,R,\varepsilon$. In particular, the function $w$ may not depend on the structure of the $A_j$, $C$, and $b$.
\end{definition}
We will show that general width-bounds need to scale with $r^{*}$ (recall that $r^{*}$ denotes the smallest $\ell_1$-norm of an optimal solution to the dual). We then go on to show that if two SDPs in a class can be combined to get another element of that class in a natural manner, then, under some mild conditions, $r^{*}$ will be of the order $n$ and $m$ for some instances of the class.
We start by showing, for specifically constructed LPs, a lower bound on the width of any oracle. Although these LPs will not solve any useful problem, every general width-bound should also apply to these LPs. This gives a lower bound on general width-bounds.
\begin{lemma}\label{lem:genisrstar}
For every $n \geq 4$, $m\geq 4$, $s\geq 1$, $R^{*}>0$, $r^{*}>0$, and $\varepsilon \leq 1/2$ there is an SDP with these parameters for which every oracle has width at least $\frac{1}{2} r^{*}$.
\end{lemma}
\begin{proof}
We will construct an LP for $n=m=3$. This is enough to prove the lemma since LPs are a subclass of SDPs and we can increase $n$, $m$, and $s$ by adding more dimensions and $s$-dense SDP constraints that do not influence the analysis below.
For some $k> 0$, consider the following LP
\begin{align*}
\max \ \ \ & (1,0,0) x\\
\text{s.t.} \ \ \ & \begin{bmatrix}
1 & 1 & 1\\
1/k & 1 & 0\\
-1 & 0 & -1
\end{bmatrix} x \leq \begin{bmatrix}
R\\
0\\
-R
\end{bmatrix}\\
& x \geq 0
\end{align*}
where the first row is the primal trace constraint.
Notice that $x_1 = x_2 = 0$ due to the second constraint. This implies that $\mbox{\rm OPT} = 0$ and, due to the last constraint, that $x_3\geq R$. Notice that $(0,0,R)$ is actually an optimal solution, so $R^{*} = R$.
To calculate $r^{*}$, look at the dual of the LP:
\begin{align*}
\min \ \ \ & (R,0,-R) y\\
\text{s.t.} \ \ \ & \begin{bmatrix}
1 & 1/k & -1\\
1 & 1 & 0\\
1 & 0 & -1
\end{bmatrix} y \geq \begin{bmatrix}
1\\
0\\
0
\end{bmatrix}\\
& y \geq 0,
\end{align*}
due to strong duality its optimal value is $0$ as well. This implies $y_1 = y_3$, so the first constraint becomes $y_2 \geq k$. This in turn implies $r^{*}\geq k$, which is actually attained (by $y = (0,k,0)$) so $r^{*} = k$.
Since the oracle and width-bound should work for every $x\in \mathbb{R}^3_+$ and every $\alpha$, they should in particular work for $x = (R,0,0)$ and $\alpha = 0$. In this case the polytope for the oracle becomes
\begin{align*}
\mathcal{P}_{\varepsilon}(x) := \{ y \in \mathbb R^m:\ & y_1 - y_3 \leq 0, \\
& y_1-y_3+y_2/k\geq 1-\varepsilon,\\
& y \geq 0 \}.
\end{align*}
since $b^T y = y_1-y_3$, $c^T x = 1$, $a_1^T x = 1$, $a_2^T x = 1/k$ and $a_3^T x = -1$. This implies that for every $y\in \mathcal{P}_{\varepsilon}(x)$, we have $y_2 \geq k/(1-\varepsilon) \geq k/2 \geq r^{*}/2$.
Notice that the term
\[
\nrm{\sum^m_{j=1} y_j A_j - C}
\]
in the definition of width for an SDP becomes
\[
\nrm{ A^T y - c}_{\infty}
\]
in the case of an LP. In our case, due to the second constraint in the dual, we know that
\[
\nrm{ A^T y - c}_{\infty} \geq y_1 + y_2 \geq \frac{r^{*}}{2}
\]
for every vector $y$ from $\mathcal{P}_{\varepsilon}(x)$. This shows that any oracle has width at least $r^{*}/2$ for this LP.
\end{proof}
\begin{corollary}
For every general width-bound $w(n,m,s,r,R,\varepsilon)$, if $n,m\geq 3$, $s\geq 1$, $r>0$, $R>0$, and $\varepsilon\leq 1/2$, then
\[
w(n,m,s,r,R,\varepsilon) \geq \frac{r}{2}.
\]
\end{corollary}
Note that this bound applies to both our algorithm and the one given by Brand\~ao and Svore.
It turns out that for many natural classes of SDPs, $r^{*}, R^{*}$, $\varepsilon$, $n$ and $m$ can grow linearly for some instances. In particular, this is the case if SDPs in a class combine in a natural manner.
Take for example two SDP relaxations for the MAXCUT problem on two graphs $G^{(1)}$ and $G^{(2)}$ (on $n^{(1)}$ and $n^{(2)}$ vertices, respectively):
\noindent\begin{minipage}{\textwidth}
\begin{minipage}{.49\textwidth}
\begin{align*}
\max \quad & \tr{L(G^{(1)})X^{(1)}} \\
\text{s.t.}\ \ \ & \tr{X^{(1)}} \leq n^{(1)}\\
& \tr{E_{jj} X^{(1)}} \leq 1 \text{ for }j=1,\dots,n^{(1)}\\
&X^{(1)} \succeq 0
\end{align*}
\end{minipage}
\begin{minipage}{.49\textwidth}
\begin{align*}
\max \quad & \tr{L(G^{(2)})X^{(2)}} \\
\text{s.t.}\ \ \ & \tr{X^{(2)}} \leq n^{(2)}\\
& \tr{E_{jj} X^{(2)}} \leq 1 \text{ for }j=1,\dots,n^{(2)}\\
&X^{(2)} \succeq 0
\end{align*}
\end{minipage}
\end{minipage}
\noindent Where $L(G)$ is the Laplacian of a graph. Note that this is not normalized to operator norm $\leq 1$, but for simplicity we ignore this here. If we denote the direct sum of two matrices by $\oplus$, that is
\[
A\oplus B = \begin{bmatrix}A &0\\0&B\end{bmatrix},
\]
then, for the disjoint union of the two graphs, we have
\[
L(G^{(1)} \cup G^{(2)}) = L(G^{(1)}) \oplus L(G^{(2)}).
\]
This, plus the fact that the trace distributes over direct sums of matrices, means that the SDP relaxation for MAXCUT on $G^{(1)} \cup G^{(2)}$ is the same as a natural combination of the two separate maximizations:
\begin{align*}
\max \quad & \tr{L(G^{(1)})X^{(1)}} + \tr{L(G^{(2)})X^{(2)}} \\
\text{s.t.}\ \ \ & \tr{X^{(1)}}+ \tr{X^{(2)}} \leq n^{(1)}+n^{(2)}\\
& \tr{E_{jj} X^{(1)}} \leq 1 \text{ for }j=1,\dots,n^{(1)}\\
& \tr{E_{jj} X^{(2)}} \leq 1 \text{ for }j=1,\dots,n^{(2)}\\
&X^{(1)},X^{(2)} \succeq 0.
\end{align*}
It is easy to see that the new value of $n$ is $n^{(1)}+n^{(2)}$, the new value of $m$ is $m^{(1)}+m^{(2)}-1$ and the new value of $R^{*}$ is $n^{(1)}+n^{(2)} = R^{* (1)}+R^{* (2)}$. Since it is natural for the MAXCUT relaxation that the additive errors also add, it remains to see what happens to $r^{*}$, and so, for general width-bounds, what happens to~$w$. As we will see later in this section, under some mild conditions, these kind of combinations imply that there are MAXCUT-relaxation SDPs for which $r^{*}$ also increases linearly, but this requires a bit more work.
\begin{definition} \label{def:combining}
We say a class of SDPs (with associated allowed approximation errors) is \emph{combinable} if there is a $k\geq 0$ so that for every two elements in this class, $(SDP^{(a)},\varepsilon^{(a)})$ and $(SDP^{(b)},\varepsilon^{(b)})$, there is an instance in the class, $(SDP^{(c)},\varepsilon^{(c)})$, that is a combination of the two in the following sense:
\begin{itemize}
\item $C^{(c)} = C^{(a)} \oplus C^{(b)}$.
\item $A^{(c)}_j = A^{(a)}_j \oplus A^{(b)}_j$ and $b_j^{(c)} = b_j^{(a)}+b_j^{(b)}$ for $j \in \lbrack k \rbrack$.
\item $A^{(c)}_{j} = A^{(a)}_{j}\oplus \mathbf{0}$ and $b^{(c)}_{j} = b^{(a)}_{j}$ for $j = k+1,\dots,m^{(a})$.
\item $A^{(c)}_{m^{(a)}+j-k} = \mathbf{0} \oplus A^{(b)}_{j}$ and $b^{(c)}_{m^{(a)}+j-k} = b^{(b)}_{j}$ for $j = k+1,\dots,m^{(b)}$.
\item $\varepsilon^{(c)} \leq \varepsilon^{(a)}+\varepsilon^{(b)}$.
\end{itemize}
In other words, some fixed set of constraints are summed pairwise, and the remaining constraints get added separately.
\end{definition}
Note that this is a natural generalization of the combining property of the MAXCUT relaxations (in that case $k=1$ to account for the trace bound).
\begin{theorem}
If a class of SDPs is combinable and there is an element $SDP^{(1)}$ for which every optimal dual solution has the property that
\[
\sum_{j=k+1}^m y_m \geq \delta
\]
for some $\delta >0$, then there is a sequence $(SDP^{(t)})_{t\in \mathbb{N}}$ in the class such that $\frac{R^{* (t)}r^{* (t)}}{\varepsilon^{(t)}}$ increases linearly in $n^{(t)}$, $m^{(t)}$ and $t$.
\end{theorem}
\begin{proof}
The sequence we will consider is the $t$-fold combination of $SDP^{(1)}$ with itself.
If $SDP^{(1)}$ is
\noindent\begin{minipage}{\textwidth}
\begin{minipage}{.49\textwidth}
\begin{align*}
\max \quad &\mbox{\rm Tr}(CX) \\
\text{s.t.}\ \ \ &\mbox{\rm Tr}(A_j X) \leq b_j \quad \text{ for } j \in [m^{(1)}], \\
&X \succeq 0
\end{align*}\end{minipage}
\begin{minipage}{.49\textwidth}
\begin{align*}
\min \quad &\sum_{j=1}^{m^{(1)}} b_j y_j \\
\text{s.t.}\ \ \ &\sum_{j=1}^{m^{(1)}} y_j A_j - C \succeq 0,\\
&y \geq 0
\end{align*}\end{minipage}
\end{minipage}
then $SDP^{(t)}$ is
\begin{align*}
\max \quad &\ \sum_{i=1}^t \tr{CX_i} & &\\
\text{s.t.}\ \ \ & \sum_{i=1}^t \mbox{\rm Tr}(A_j X_i) \leq t b_j &&\text{ for } j \in [k], \\
& \mbox{\rm Tr}(A_j X_i) \leq b_j &&\text{ for } j = k+1 , \dots , m^{(1)} \text{ and } i = 1 , \dots , t \\
&X_i \succeq 0 &&\text{ for all } i = 1, \dots,t
\end{align*}
with dual
\begin{align*}
\min \quad &\sum_{j=1}^k t b_j y_j + \sum_{i=1}^t \sum_{j=k+1}^{m^{(1)}} b_j y_j^i\\
\text{s.t.}\ \ \ &\sum_{j=1}^k y_j A_j + \sum_{j=k+1}^{m^{(1)}} y_j^i A_j \succeq C \text{ for } i = 1,\dots,t\\
&y,y^i \geq 0.
\end{align*}
First, let us consider the value of $\mbox{\rm OPT}^{(t)}$. Let $X^{(1)}$ be an optimal solution to $SDP^{(1)}$ and for all $i \in \lbrack t \rbrack$ let $X_i = X^{(1)}$. Since these $X_i$ form a feasible solution to $SDP^{(t)}$, this shows that $\mbox{\rm OPT}^{(t)} \geq t\cdot \mbox{\rm OPT}^{(1)}$.
Furthermore, let $y^{(1)}$ be an optimal dual solution of $SDP^{(1)}$, then $(y^{(1)}_1,\dots,y^{(1)}_k)\oplus \left( y^{(1)}_{k+1},\cdots,y^{(1)}_{m^{(1)}}\right)^{\oplus t}$ is a feasible dual solution for $SDP^{(t)}$ with objective value $t\cdot \mbox{\rm OPT}^{(1)}$, so $\mbox{\rm OPT}^{(t)} = t \cdot \mbox{\rm OPT}^{(1)}$.
Next, let us consider the value of $r^{* (t)}$. Let $\tilde{y} \oplus y^1 \oplus \dots \oplus y^t$ be an optimal dual solution for $SDP^{(t)}$, split into the parts of $y$ that correspond to different parts of the combination. Then $\tilde{y} \oplus y^i$ is a feasible dual solution for $SDP^{(1)}$ and hence $b^T ( \tilde{y}\oplus y^i) \geq \mbox{\rm OPT}^{(1)}$. On the other hand we have
\[
t \cdot \mbox{\rm OPT}^{(1)} = \mbox{\rm OPT}^{(t)} = \sum_{i=1}^t b^T (\tilde{y} \oplus y^i),
\]
this implies that each term in the sum is actually equal to $\mbox{\rm OPT}^{(1)}$. But if $(\tilde{y}\oplus y^i)$ is an optimal dual solution of $SDP^{(1)}$ then $\nrm{(\tilde{y}\oplus y^i)}_1 \geq r^{* (1)}$ by definition and $\nrm{y^i}_1 \geq \delta$. We conclude that $r^{* (t)} \geq r^{* (1)} - \delta + t\delta$.
Now we know the behavior of $r^{*}$ under combinations, let us look at the primal to find a similar statement for $R^{*(t)}$. Define a new SDP, $\widehat{SDP}^{(t)}$, in which all the constraints are summed when combining, that is, in Definition~\ref{def:combining} we take $k = n^{(1)}$, however, contrary to that definition, we even sum the psd constraints:
\begin{align*}
\max \quad &\ \sum_{i=1}^t \tr{CX_i} \\
\text{s.t.}\ \ \ & \sum_{i=1}^t \mbox{\rm Tr}(A_j X_i) \leq t b_j \quad \text{ for } j \in [m^{(1)}], \\
&\sum_{i=1}^t X_i \succeq 0.
\end{align*}
This SDP has the same objective function as $SDP^{(t)}$ but a larger feasible region: every feasible $X_1,\dots,X_t$ for $SDP^{(t)}$ is also feasible for $\widehat{SDP}^{(t)}$. However, by a change of variables, $X := \sum_{i=1}^t X_i$, it is easy to see that $\widehat{SDP}^{(t)}$ is simply a scaled version of $SDP^{(1)}$. So, $\widehat{SDP}^{(t)}$ has optimal value $t\cdot \mbox{\rm OPT}^{(1)}$. Since optimal solutions to $\widehat{SDP}^{(t)}$ are scaled optimal solutions to $SDP^{(1)}$, we have $\hat{R}^{*(t)} = t\cdot R^{* (1)}$. Combining the above, it follows that every optimal solution to $SDP^{(t)}$ is optimal to $\widehat{SDP}^{(t)}$ as well, and hence has trace at least $t\cdot R^{*(1)}$, so $R^{* (t)}\geq t\cdot R^{* (1)}$.
We conclude that
\[
\frac{
R^{*(t)} r^{*(t)}
}{
\varepsilon^{(t)}
}
\geq
\frac{
tR^{* (1)}
(r^{* (1)}+(t-1)\delta)
}{
t\varepsilon^{(1)}}
= \Omega\left( t \right)
\]
and $n^{(t)} = t n^{(1)}$, $m^{(t)} = t ( m^{(1)}-k)+k$.
\end{proof}
This shows that for many natural SDP formulations for combinatorial problems, such as the MAXCUT relaxation or LPs that have to do with resource management, $R^{*}r^{*}/\varepsilon$ increases linearly in $n$ and $m$ for some instances. Hence, using $R^{*}\leq R$ and Lemma~\ref{lem:genisrstar}, $Rw/\varepsilon$ grows at least linearly when a general width-bound is used.
\section{Lower bounds on the quantum query complexity}\label{sec:lowerbounds}
In this section we will show that every LP-solver (and hence every SDP-solver) that can distinguish two optimal values with high probability needs $\Omega\left(\sqrt{\max\{n,m\}} \left( \min\{n,m\} \right)^{3/2} \right)$ quantum queries in the worst case.
For the lower bound on LP-solving we will give a reduction from a composition of Majority and OR functions.
\begin{definition}
Given input bits $Z_{ij\ell} \in \{0,1\}^{a\times b\times c}$ the problem of calculating
\begin{align*}
MAJ_a(&\\
&OR_b(MAJ_c(Z_{111},\dots,Z_{11c}),\dots,MAJ_c(Z_{1b1},\dots,Z_{1bc})),\\
&\dots,\\
&OR_b(MAJ_c(Z_{a11},\dots,Z_{a1c}),\dots,MAJ_c(Z_{ab1},\dots,Z_{abc}))\\
)&
\end{align*}
with the promise that
\begin{itemize}
\item Each inner $\mbox{\rm MAJ}_c$ is a boundary case, in other words $\sum_{\ell=1}^c Z_{ij\ell} \in \{c/2,c/2+1\}$ for all $i,j$.
\item The outer $\mbox{\rm MAJ}_a$ is a boundary case, in other words, if $\tilde{Z} \in \{0,1\}^a$ is the bitstring that results from all the OR calculations, then $|\tilde{Z}| \in \{a/2,a/2+1\}$.
\end{itemize}
is called the promise $\MAJ_a\mhyphen\OR_b\mhyphen\MAJ_c$ problem.
\end{definition}
\begin{lemma}\label{lem:knownlow}
It takes at least $\Omega(a\sqrt{b} c)$ queries to the input to solve the promise $\MAJ_a\mhyphen\OR_b\mhyphen\MAJ_c$ problem.
\end{lemma}
\begin{proof}
The promise version of $\mbox{\rm MAJ}_k$ is known to require $\Omega(k)$ quantum queries. Likewise, it is known that the OR$_k$ function requires $\Omega(\sqrt{k})$ queries. Furthermore, the adversary bound tells us that query complexity is multiplicative under composition of functions (even promise functions). For completeness we include a proof of this composition property as Theorem~\ref{thm:adv} in Appendix~\ref{app:adversarycomposition}. The proof is a straightforward modification of a proof of Lee et al.~\cite[Section~5]{lmrss:stateconv} for the case where the outer function is a total Boolean function;
Kimmel~\cite{Kimmel:adv} already gave a similar modification of the proof of Lee et al.\ for the case where a promise function is composed with itself.
\end{proof}
\begin{lemma} \label{lem:calculation}
Determining the value
\[
\sum_{i=1}^a \max_{j \in \lbrack b\rbrack} \sum_{\ell=1}^c Z_{ij\ell}
\]
for a $Z$ from the promise $\MAJ_a\mhyphen\OR_b\mhyphen\MAJ_c$ problem up to additive error $\varepsilon = 1/3$, solves the promise $\MAJ_a\mhyphen\OR_b\mhyphen\MAJ_c$ problem.
\end{lemma}
\begin{proof}
Notice that due to the first promise, $\sum_{\ell=1}^c Z_{ij\ell} \in \{c/2,c/2+1\}$ for all $i\in [a],j\in[b]$. This implies that
\begin{itemize}
\item If the $i$th OR is $0$, then all of its inner MAJ functions are $0$ and hence
\[
\max_{j \in \lbrack b\rbrack} \sum_{\ell=1}^c Z_{ij\ell} = \frac{c}{2}
\]
\item If the $i$th OR is $1$, then at least one of its inner MAJ functions is $1$ and hence
\[
\max_{j \in \lbrack b\rbrack} \sum_{\ell=1}^c Z_{ij\ell} = \frac{c}{2} + 1
\]
\end{itemize}
Now, if we denote the string of outcomes of the OR functions by $\tilde{Z}\in \{0,1\}^a$, then
\[
\sum_{i=1}^a \max_{j \in \lbrack b\rbrack} \sum_{\ell=1}^c Z_{ij\ell} = a \frac{c}{2} + |\tilde{Z}|
\]
Hence determining the left-hand side will determine $|\tilde{Z}|$; this Hamming weight is either $\frac{a}{2}$ if the full function evaluates to $0$, or $\frac{a}{2}+1$ if it evaluates to $1$.
\end{proof}
\begin{lemma} \label{lem:generallp}
For an input $Z\in \{0,1\}^{a\times b\times c}$ there is an LP with $m = c+a$ and $n = c+ab$ for which the optimal value is
\[
\sum_{i=1}^a \max_{j \in \lbrack b\rbrack} \sum_{\ell=1}^c Z_{ij\ell}
\]
Furthermore, a query to an entry of the input matrix or vector costs at most $1$ query to $Z$.
\end{lemma}
\begin{proof}
Let $Z^{(i)}$ be the matrix one gets by fixing the first index of $Z$ and putting the entries in a $c\times b$ matrix, so $Z^{(i)}_{\ell j} = Z_{ij\ell}$.
We define the following LP:
\begin{align*}
\mbox{\rm OPT} = \text{max} \ \ \ & \sum_{k=1}^c w_k\\
\text{s.t.} \ \ \ &
\begin{bmatrix}
I & -Z^1 & \cdots & -Z^a \\
0 & \mathbf{1}^T & &\\
0 & & \ddots &\\
0 & & & \mathbf{1}^T
\end{bmatrix}
\begin{bmatrix}
w\\
v^{(1)}\\
\vdots\\
v^{(a)}\\
\end{bmatrix}
\leq
\begin{bmatrix}
0\\
\mathbf{1}\\
\vdots\\
\mathbf{1}
\end{bmatrix}\\
& v^1,\dots,v^a \in \mathbb{R}_{+}^b ,w \in \mathbb{R}_{+}^c
\end{align*}
Notice every $Z^{(i)}$ is of size $c\times b$, so that indeed $m = c+a$ and $n = c + ab$.
For every $i \in [a]$ there is a constraint that says
\[
\sum_{j=1}^b v^{(i)}_j \leq 1.
\]
The constraints involving $w$ say that for every $k\in[c]$
\[
w_k \leq \sum_{i=1}^a \sum_{j=1}^b v^{(i)}_j Z^{(i)}_{k j} = \sum_{i=1}^a ( Z^{(i)} v^{(i)} )_k
\]
where $( Z^{(i)} v^{(i)} )_k$ is the $k$th entry of the matrix-vector product $Z^{(i)} v^{(i)}$.
Clearly, for an optimal solution these constraints will be satisfied with equality, since in the objective function $w_k$ has a positive weight.
Summing over $k$ on both sides, we get the equality
\begin{align*}
\mbox{\rm OPT} &= \sum_{k = 1}^c w_{k}\\
&= \sum_{k = 1}^c \sum_{i=1}^a ( Z^{(i)} v^{(i)} )_{k}\\
&= \sum_{i=1}^a \sum_{k = 1}^c ( Z^{(i)} v^{(i)} )_{k}\\
&= \sum_{i=1}^a \nrm{ Z^{(i)} v^{(i)}}_1
\end{align*}
so in the optimum $\nrm{ Z^{(i)} v^{(i)}}_1$ will be maximized.
Note that we can use the $\ell_1$-norm as a shorthand for the sum over vector elements since all elements are positive.
In particular, the value of $\nrm{ Z^{(i)} v^{(i)}}_1$ is given by
\begin{align*}
\text{max} \ \ \ & \nrm{ Z^{(i)} v^{(i)}}_1\\
\text{s.t.} \ \ \ & \nrm{v^{(i)}}_1\leq 1\\
& v^{(i)}\geq 0
\end{align*}
Now $\nrm{\smash{ Z^{(i)} v^{(i)}}}_1$ will be maximized by putting all weight in $v^{(i)}$ on the index that corresponds to the column of $Z^{(i)}$ that has the highest Hamming weight.
In particular in the optimum $\nrm{ \smash{Z^{(i)} v^{(i)}}}_1 = \max_{j\in \lbrack b \rbrack} \sum_{\ell = 1}^c Z^{(i)}_{\ell j}$.
Putting everything together gives:
\[
\mbox{\rm OPT} = \sum_{i=1}^a \nrm{ Z^{(i)} v^{(i)}}_1 = \sum_{i=1}^a \max_{j\in \lbrack b \rbrack} \sum_{\ell = 1}^c Z^{(i)}_{\ell j} = \sum_{i=1}^a \max_{j\in \lbrack b \rbrack} \sum_{\ell = 1}^c Z_{i j \ell}
\]
\vskip-5mm
\end{proof}
\begin{theorem}\label{thm:lowbound}
There is a family of LPs, with $m \leq n$ and two possible integer optimal values, that require at least $\Omega(\sqrt{n}m^{3/2})$ quantum queries to the input to distinguish those two values.
\end{theorem}
\begin{proof}
Let $a = c = m/2$ and $b = \frac{n-c}{a} = \frac{2n}{m} - 1$, so that $n =c+ab$ and $m=c+a$.
By Lemma~\ref{lem:generallp} there exists an LP with $n =c+ab$ and $m=c+a$ that calculates
\[
\sum_{i=1}^a \max_{j \in \lbrack b\rbrack} \sum_{\ell=1}^c Z_{ij\ell}
\]
for an input $Z$ to the promise $\MAJ_a\mhyphen\OR_b\mhyphen\MAJ_c$ problem. By Lemma~\ref{lem:calculation}, calculating this value will solve the promise $\MAJ_a\mhyphen\OR_b\mhyphen\MAJ_c$ problem.
By Lemma~\ref{lem:knownlow} the promise $\MAJ_a\mhyphen\OR_b\mhyphen\MAJ_c$ problem takes $\Omega(a\sqrt{b}c)$ quantum queries in the worst case.
This implies a lower bound of
\[
\Omega\left(m^2\sqrt{\frac{n}{m}}\right) = \Omega(m^{3/2}\sqrt{n})
\]
quantum queries on solving these LPs.
\end{proof}
\begin{corollary}
Distinguishing two optimal values of an LP (and hence also of an SDP) with additive error $\varepsilon<1/2$ requires \[\Omega\left(\sqrt{\max\{n,m\}} \left( \min\{n,m\} \right)^{3/2} \right)\] quantum queries to the input matrices in the worst case.
\end{corollary}
\begin{proof}
This follows from Theorem~\ref{thm:lowbound} and LP duality.
\end{proof}
It is important to note that the parameters $R$ and $r$ from the Arora-Kale algorithm are not constant in this family of LPs ($R,r = \Theta(\min\{n,m\}^2)$ here), and hence this lower bound does not contradict the scaling with $\sqrt{mn}$ of the complexity of our SDP-solver or Brand\~{a}o and Svore's. Since we show in the appendix that one can always rewrite the LP (or SDP) so that $2$ of the parameters $R,r,\varepsilon$ are constant, the lower bound implies that any algorithm with a sub-linear dependence on $m$ or $n$ has to depend at least polynomially on $Rr/\varepsilon$. For example, the above family of LPs shows that an algorithm with a $\sqrt{mn}$ dependence has to have an $(Rr/\varepsilon)^\kappa$ factor in its complexity with $\kappa\geq 1/4$. It remains an open question whether a lower bound of $\Omega(\sqrt{mn})$ can be proven for a family of LPs/SDPs where $\varepsilon$, $R$ and $r$ all constant.
\section{Conclusion}
In this paper we gave better algorithms and lower bounds for quantum SDP-solvers, improving upon recent work of Brand\~ao and Svore~\cite{brandao&svore:qsdp}. Here are a few directions for future work:
\begin{itemize}
\item {\bf Better upper bounds.}
The runtime of our algorithm, like the earlier algorithm of Brand\~ao and Svore, has better dependence on $m$ and $n$ than the best classical SDP-solvers, but worse dependence on $s$ and on $Rr/\varepsilon$. It may be possible to improve the dependence on $s$ to linear and/or the dependence on $Rr/\varepsilon$ to less than our current 8th power.
\item {\bf Applications of our algorithm.}
As mentioned, both our and Brand\~ao-Svore's quantum SDP-solvers only improve upon the best classical algorithms for a specific regime of parameters, namely where $mn\gg Rr/\varepsilon$. Unfortunately, we don't know particularly interesting problems in combinatorial optimization in this regime. As shown in Section~\ref{sec:downside}, many natural SDP formulations will not fall into this regime. However, it would be interesting to find useful SDPs for which our algorithm gives a significant speed-up.
\item {\bf New algorithms.} As in the work by Arora and Kale, it might be more promising to look at oracles (now quantum) that are designed for specific SDPs. Such oracles could build on the techniques developed here, or develop totally new techniques. It might also be possible to speed up other classical SDP solvers, for example those based on interior-point methods.
\item {\bf Better lower bounds.}
Our lower bounds are probably not optimal, particularly for the case where $m$ and $n$ are not of the same order. The most interesting case would be to get lower bounds that are simultaneously tight in the parameters $m$, $n$, $s$, and $Rr/\varepsilon$.
\end{itemize}
\paragraph{Acknowledgments.}
We thank Fernando Brand\~ao for sending us several drafts of~\cite{brandao&svore:qsdp} and for answering our many questions about their algorithms, Stacey Jeffery for pointing us to~\cite{Kimmel:adv}, and Andris Ambainis and Robin Kothari for useful discussions and comments. We also thank the anonymous FOCS'17 referees for helpful comments that improved the presentation.
\bibliographystyle{alpha}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,618 |
Lomme est une ancienne commune française située dans le département du Nord et la région Hauts-de-France, associée à Lille depuis .
Géographie
Lomme se trouve à la périphérie de Lille, au nord-ouest. Elle est depuis 2000 une commune associée à cette dernière et est membre de Métropole européenne de Lille.
Située en Flandre romane, dans la vallée de la Deûle, elle jouxte les communes de Lille, Lambersart, Lompret, Pérenchies, Capinghem, Ennetières-en-Weppes, Englos, Sequedin, Loos et Haubourdin.
Urbanisme
Voies de communications et transports
L'axe structurant le plus important est la D933 (avenue de Dunkerque) disposant d'un faisceau métropolitain en infrastructure (ligne ).
La commune est desservie par plusieurs types de transports en commun :
la ligne 2 du métro de Lille Métropole dessert Lomme sur six stations, dont le terminus : Saint-Philibert, Bourg, Maison des Enfants, Mitterie, Pont Supérieur, Lomme - Lambersart.
le réseau de bus Ilévia (lignes 10, 18, , CO3, L99, 61, 62, 64, 75, 76, 80) ;
La gare de Lomme, arrêt ferroviaire de la ligne d'Haubourdin à Saint-André, située rue de la gare de Lomme, n'est plus desservie par les trains de voyageurs et est fermée, après avoir longtemps perduré comme boutique commerciale de la SNCF.
Quartiers
Lomme comprend cinq quartiers : le Bourg, la Délivrance, la Mitterie, le Mont-à-Camp et le Marais.
Le Bourg
C'est le quartier le plus ancien et il était autrefois le cœur de la ville. L'église, Notre-Dame-De-La-Visitation, dont la construction remonte au a la particularité de ne plus avoir de flèche. Celle-ci fut détruite lors de la première guerre mondiale.
La Délivrance
Ce quartier édifié près de la gare de triage construite vers 1920 était destiné à loger les cheminots.
La Mitterie
Presque aussi ancien que le quartier du bourg, il correspond à la villa Ulma, terrain légué aux chanoines de la Collégiale Saint-Pierre de Lille qui donna son nom à la ville. Dans une période plus récente, on y trouvait notamment des ateliers de constructions mécaniques.
Mont-à-Camp
Le nom proviendrait d'une dénivellation du terrain (motte) sur lequel fut installée au une garnison chargée de protéger Lille, d'où le nom de motte-à-camp, transformé en Mont-à-camp. L'église de mont-à-camp construite au début du siècle, qui était surmontée d'une statue du christ, fut démolie en 2005 pour être remplacée par une église plus fonctionnelle.
On y trouve la chapelle de la maladrerie, vestige d'une léproserie, édifiée entre 1461 et 1466. Ce monument est incontestablement aujourd'hui le plus ancien de tout Lomme. À cette époque, ce sont donc seulement les "ladres", les lépreux, qui vivent dans le petit hameau de Canteleu. En 1791, on recense seulement 24 habitants sur l'ensemble du hameau.
Le bourg industriel de Canteleu, qui a laissé son nom à une station de métro proche, était situé en partie sur ce quartier. Le nom "canteleu" signifiant "chante loup" (du picard "cante leu") était à l'origine celui d'un hameau situé à Esquermes (commune annexée à Lille en 1858).
Le bourg apparaît en 1857, date à laquelle Eugène Verstraete rachète les terres de la Maladrerie pour y édifier une usine de lin. Il construit ensuite durant les années 1860 les premières maisons de rangée individuelles du quartier pour y loger ses employés au début de ce qui est aujourd'hui la rue du Marais (à l'origine un très ancien chemin rural de communication, sans habitation, qui partait de la Maladrerie pour rejoindre l'abbaye de Loos). Ces maisons, qui existent toujours du côté impair au début de la rue sont aujourd'hui les plus anciens logements de particuliers de la ville. D'autres industries s'implantent dans le bourg et celui-ci se développe à cheval sur les communes de Lomme, Lambersart et Lille. En 1873, plusieurs notables, dont Eugène Verstraete, portent sans succès un projet visant à constituer le bourg en commune autonome.
Le Marais
Comme son nom l'indique, il était composé de marais qui furent asséchés au cours des siècles par les moines de l'abbaye de Loos.
Une des premières raffineries de France qui fournissait du pétrole pour les lampes d'éclairage s'y installa et ainsi que des filatures aujourd'hui disparues. L'église Notre-Dame-de-Lourdes date de 1895.
Le Marais est également marqué dans sa partie est par le développement de l'industrie du bourg de Canteleu, qui s'étale vers le Marais. En particulier, l'usine Le Blan, repère du quartier, est construite à Lille à la limite de ce quartier. On met en place également des rigoles pour achever d'assécher les derniers marécages. Ces rigoles ont disparu aujourd'hui mais leur cours a défini des axes qui demeurent, comme l'impasse de la tortue. Le quartier s'assainit et prospère. À partir des années 1970-1980, néanmoins, les usines font faillite les unes après les autres et laissent derrière elles des zones désaffectées. Après des décennies d'immobilisme, le quartier bénéficie du rayonnement d'EuraTechnologies, gigantesque pôle d'innovation inauguré en 2009 dans l'usine Le Blan réhabilitée, et voit désormais affluer les investisseurs. Les anciens logements sont enfin remis aux normes et c'est désormais un quartier en pleine mutation qui attire un public de plus en plus jeune, séduit par un quartier d'avenir.
Toponymie
L'origine du nom semble remonter au avec un lieu appelé villa Ulma, retrouvé dans le titre de la fondation de la Collégiale Saint-Pierre de Lille. Il viendrait donc du latin ulmus, orme (puis olm en flamand), indiquant un lieu où poussaient des ormes.
Histoire
Lomme, qui faisait partie de la Flandre wallonne, dépendait de la châtellenie de Lille et du diocèse de Tournai puis de celui de Cambrai. Saint Bernard fonda une "abbaye-fille" de Cîteaux en 1146 dont Jean le Bel fut le premier abbé. À la suite des conquêtes de Louis XIV, Lomme devint française en 1667.
C'est au cours du que la ville va connaître son essor avec l'arrivée de nombreuses industries, notamment textiles, mécaniques, ou chimiques.
Avant la Révolution française, Lomme était le siège de plusieurs seigneuries. L'une d'entre elles, le Grand-Bas (ou le Grand-Bus), était détenue en 1612 par Josse de Parmentier. Le , la gouvernance de Lille a rendu une sentence à propos d'un différend entre Josse de Parmentier, licencié es-droits, seigneur du Grand-Bas à Lomme et les officiers fiscaux de la gouvernance. Ces derniers avançaient que Josse de Parmentier n'était pas noble et ne pouvait jouir des immunités (dont l'exemption d'impôts) liées à cet état. La sentence reconnait la noblesse du plaignant ce que celui-ci a pu prouver. Il est à noter que le père de Josse, Jean était greffier de la dite gouvernance, (ce qui a joué en sa faveur?). Sa mère était Philippine Picavet, dame du (les femmes ne sont pas « seigneur de » mais « dame de ») Grand-Bas.
Lors de la Seconde Guerre mondiale, dans la nuit du 9 au , un bombardement anglais qui visait à quelques semaines du débarquement de Normandie l'important complexe ferroviaire et centre névralgique de la gare de triage de Lille-Délivrance fit plus de 400 victimes parmi la population, touchant plusieurs quartiers, la gare étant enserrée par la ville et cernée par plusieurs quartiers. Les villes voisines, jusqu'à Sequedin, Wambrechies, Marquette-lez-Lille ne sont pas épargnées.
Dans la seconde moitié du , avec la fermeture de ses usines, Lomme s'adapta en accueillant le MIN de Lille, en créant le centre commercial d'Englos, en recevant le nouvel hôpital Saint-Philibert, et en remplaçant ses tramways par la ligne 2 du métro de Lille Métropole (6 stations sur le territoire lommois).
Après des délibérations des conseils municipaux de Lille et de Lomme intervenues en 1999, les deux communes fusionnent le 22 février 2000 et Lomme devient une commune associée de Lille.
Politique et administration
Rattachements administratifs et électoraux
Avant sa fusion dans Lille, Lomme se trouvait dans l'arrondissement de Lille du département du Nord.
Elle faisait partie de 1793 à 1982 du canton d'Haubourdin, année où elle devient le chef-lieu du canton de Lomme. Dans le cadre du redécoupage cantonal de 2014 en France, cette circonscription administrative territoriale a disparu, et le canton n'est plus qu'une circonscription électorale.
Pour les élections départementales, le territoire de Lomme fait partie depuis 2014 du canton de Lille-6
Pour l'élection des députés, son territoire fait partie de la onzième circonscription du Nord.
Intercommunalité
Lomme faisait partie depuis 1967 de la communauté urbaine de Lille (CUDL), un établissement public de coopération intercommunale (EPCI) à fiscalité propre et auquel la commune avait transféré un certain nombre de ses compétences, dans les conditions déterminées par le code des communes et qui est depuis 2015 la Métropole européenne de Lille.
Lomme n'étant plus une commun de plein exercice, elle a cessé d'en être membre en 2000.
Tendances politiques et résultats
Après le premier tour des élections municipales de 2020, le , le confinement lié à la pandémie de Covid-19 a retardé de trois mois la tenue du second tour, le . Celui-ci se solde par une quadrangulaire de laquelle sort gagnant le maire sortant Roger Vicot, poursuivant une série de mandats avec un maire PS.
Liste des maires
.
Distinctions et labels
En 2011, Lomme a été récompensée par le label « Ville Internet @@@ ».
Équipements et services publics
Enseignement
La ville de Lomme compte huit écoles maternelles, onze écoles élémentaires, trois collèges, trois lycées et un centre de formation d'apprentis.
Population et société
Démographie
Vie associative
.
On y trouve aussi :
Le Centre régional des arts du cirque, qui propose des cours pour les amateurs à partir d'un an (avec leurs parents) jusqu'aux adultes, une classe de préparation aux concours, une formation artistique en trois ans et une formation pédagogique au BIAC et au BPJEPS "activités du cirque". Des résidences d'artistes, un studio de création et une programmation de cirque complètent ses activités.
L'odyssée, médiathèque construire en 2002, associée à la Bibliothèque municipale de Lille en 2016.
Sport et loisirs
Équipements sportifs de la commune :
le stade des Ormes
une piscine municipale
onze salles de sport
dix terrains de football
Parmi les différents clubs sportifs de la ville, le Lomme Lille Métropole handball évolue à haut niveau.
Entre 1985 et 1987 fut ouvert un parc de loisirs dénommé "Le parc de Lomme" (ou Lillom) qui fut rapidement fermé à cause de la concurrence avec le parc Bellewaerde.
Le Kinepolis, le plus grand cinéma multiplexe de France, qui compte 23 salles totalisant plus de 6800 places.
Économie
Catégories socio-professionnelles de la population active lommoise en 1999 :
agriculteurs : 0,11 %
artisans, commerçants, chefs d'entreprises : 5,09 %
cadres, professions intellectuelles : 6,08 %
professions intermédiaires : 17,17 %
employés : 33,85 %
ouvriers : 37,71 %
Lomme abrite plus de 600 entreprises sur son réseau "entreprendre", une zone commerciale, une plate-forme multimodale, une clinique et un établissement hospitalier, un Marché d'intérêt national : « Le Marché de Gros – Lille ».
Culture locale et patrimoine
Lieux et monuments
Personnalités liées à la commune
Maximilien Vilain de Gand (1530-1583), seigneur de Lomme.
Louis de Gand-Vilain (1678-1767), prince d'Isenghien, maréchal, seigneur de Lomme.
. Le Géant de Lomme la représente.
Étienne Poulet (1890-1960), aviateur français, né à Lomme au Château d'Isenghien.
Robert Clément (1907-1991), architecte de l'hôtel de ville et d'écoles, professeur d'architecture.
Amandine Henry (1989 - ), internationale française de football, ayant débuté à Lomme.
Marceau Stricanne (1920-2012), international français de football, né à Lomme.
Eugène Descamps (1922-1990), syndicaliste français, né à Lomme.
Marcel Lefebvre (1905-1991), homme d'église, archevêque de Dakar (1947-1962), vicaire à la paroisse Notre-Dame de Lourdes au Marais de Lomme de 1930 à 1931.
Roland Clauws (1933-2004), footballeur professionnel, né à Lomme.
Le Géant de Lomme
Comme beaucoup de villes des Flandres, Lomme possède une géant. Celui-ci représente Anne Delavaux, une Lommoise qui combattit comme porte-étendard pendant la Fronde sous des habits d'hommes dans l'armée espagnole, au temps où Lomme était sous domination espagnole. Elle se signala par de nombreux exploits sous le nom d'Antoine de Bonne-Espérance. C'est au moment où elle fut blessée sur le champ de bataille, qu'on découvrit sa supercherie. Elle fut alors confiée à l'abbaye du repos de Notre-Dame de Marquette, puis partit en retraite à l'abbaye-hôpital de La Byloke à Gand où étaient soignés les soldats malades ou blessés.
On peut le voir dans la salle d'accueil de la mairie de Lomme. Il sort une fois par an au moment du carnaval au mois de juin.
Héraldique
Voir aussi
Articles connexes
Anciennes communes du Nord
Liens externes
.
Histoire de Lomme
Notes et références
Quartier de plus de 10 000 habitants en France
Quartiers de Lille
Commune associée dans le département du Nord
Ville Internet
Flandre romane | {
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} | 3,078 |
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30 Harrowing Pictures From The 9/11 Terrorist Attacks
On the 17th anniversary of the 9/11 attacks, these are the pictures that are forever seared into our memories and hearts. Warning: Some images may be disturbing.
By Gabriel H. Sanchez
Gabriel H. Sanchez BuzzFeed News Photo Essay Editor
Posted on September 11, 2017, at 2:10 p.m. ET
Ezra Shaw / Getty Images
An ambulance rushes toward the World Trade Center area after reports that the twin towers had been struck by a commercial airliner in a suspected terrorist attack on Sept. 11, 2001.
Spencer Platt / Getty Images
Spectators look up at the damaged World Trade Center after two airplanes slammed into the twin towers.
Doug Mills / AP
Chief of Staff Andy Card whispers into the ear of President George W. Bush during a visit to the Emma E. Booker Elementary School in Sarasota, Florida, to inform him of the plane crashes.
A fiery blasts rocks the World Trade Center after it was hit with a second commercial airliner.
This image is graphic
Tap to reveal
Jose Jimenez / Primera Hora / Getty Images
A man falls to his death from the World Trade Center after two planes hit the twin towers.
New York Daily News Archive / Getty Images
New York Daily News staff photographer David Handschuh is carried from danger after his leg was shattered by falling debris while he was photographing the scene.
Angela Jimenez / Getty Images
A survivor kneels in horror and disbelief outside the World Trade Center.
Doug Kanter / AFP / Getty Images
An American flag flies in the foreground as one of the World Trade Center towers burns in the background.
People walk over the Brooklyn Bridge as the World Trade Center towers burn in the distance.
Andrew Lichtenstein / Getty Images
Crowds swarm over the Manhattan Bridge to leave Manhattan on the morning of the Sept. 11 attack.
Chung Sung-jun / Getty Images
A man in Seoul watches a multiscreen broadcast of the attack on the World Trade Center.
Thomas Nilsson / Getty Images
The south tower of the World Trade Center collapses in New York City.
A firefighter runs into the street as the World Trade Center crumbles behind him.
Civilians take cover as a dust cloud from the collapse of the World Trade Center envelops lower Manhattan.
New York Daily News staff photographer David Handschuh takes refuge in a storefront after being injured by falling debris.
Millrock Productions, Inc. / Getty Images
A group of firefighters carry one of their colleagues, Al Fuentes, who was injured in the collapse of the World Trade Center.
Ron Agam / Getty Images
A vehicle sits on its side amid rubble at the World Trade Center.
A firefighter covered with ash is helped toward shelter after the World Trade Center collapse.
Firefighters search for survivors following the collapse of the twin towers.
Capt. Michael Dugan hangs an American flag from a light pole in front of what is left of the World Trade Center.
A firefighter breaks down during a brief moment of rest.
Handout / Getty Images
President George W. Bush, Vice President Dick Cheney, and National Security Advisor Condoleezza Rice meet in the Emergency Operations Center after the attacks.
Alex Wong / Getty Images
Smoke billows from the Pentagon in Arlington, Virginia, after a commercial plane crashed into the building and set off a huge explosion.
A US flag is set outside the Pentagon after a highjacked commercial jetliner crashed into it.
David Maxwell / AFP / Getty Images
An unidentified Pennsylvania state trooper stands guard at the crash site of United Airlines Flight 93 in Shanksville, Pennsylvania. The mound of dirt in the center of the field is the crater made by the plane's impact.
Tim Matsui / Getty Images
Seattle residents wait to donate blood at the Puget Sound Blood Center on Sept. 11, 2001, in the wake of the terrorist attacks.
Joe Raedle / Getty Images
A man looks at photos posted of people missing in the World Trade Center disaster on Sept. 14, 2001, in New York City.
Lucy Nicholson / AFP / Getty Images
American Muslim women are filmed as they pray for the victims of the terrorist attacks at the Islamic Center of Southern California in Los Angeles.
Ullstein Bild / Getty Images
A vigil is held in Berlin a few hours after the terrorist attacks in the US.
Mark Wilson / Getty Images
People cover themselves with a US flag on the US Capitol grounds during a candlelight vigil on Sept. 12, 2001.
Gabriel H. Sanchez is the photo essay editor for BuzzFeed News and is based in New York City.
Contact Gabriel H. Sanchez at gabriel.sanchez@buzzfeed.com. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,423 |
\section{Introduction}
Human computation refers to the practice of tapping into human intelligence and cognition as computational elements within an information processing system design, often done on a large global scale \cite{quinn2011human}. The sheer scale of human computation that the Web enables has made possible things that were previously unimaginable, e.g., \textit{Captchas} digitizing the entire NYTimes historical publications, global participatory platforms for human rights and crises response,\footnote{\url{https://www.ushahidi.com/}} and large-scale data and distributed analyses enabled by citizen science projects.\footnote{\url{https://www.citizenscience.gov/}} In particular, human computation has played a critical role in the research, development, and deployment of modern-day artificial intelligence systems, through the creation of training datasets \cite{imagenet_cvpr09}, and human-in-the-loop systems \cite{Filippova2019, monarch2021human}. By enabling efficient and scalable distribution of data labelling microtasks, crowdsourcing platforms are a natural choice for dataset developers aiming to cheaply and efficiently generate dataset annotations.
In this paper we explore the challenges and decision points inherent to crowdsourced annotation of machine learning datasets and propose a framework, CrowdWorkSheets, for reflecting on data-set annotation decisions, and documenting them in a standardized manner. At a high level, CrowdWorkSheets prompts dataset developers to ask: who is annotating the data, and why is that important? We consider how the ethical concerns of data annotation intersect with the identities of the annotators, the social structures surrounding their work, and how their individual perspectives may become encoded within the dataset labels. In doing so, we push back against the prevalent notion that crowdworkers are interchangeable and instead seek to illuminate why they are not. Data generated in crowdwork tasks is shaped by a range of social factors and the datasets that workers help to build continue to shape systems long after worker engagement ends. Processes of annotation thus impact future models built from this data; therefore, understanding the perspectives captured through data labeling is crucial to fully understanding these models and the potential social impact they can have.
Our work is motivated by, and extends, prior scholarship examining ethical considerations relating to crowdsourcing. For instance, \citet{vakharia2015beyond} outline various kinds of challenges encountered in this space by analyzing and comparing seven different crowdsourcing platforms. In addition, \citet{schlagwein2019ethical} conducted extensive fieldwork, engaging crowdworkers, platform organizers, and requesters over the course of three years to uncover a range of ethical dilemmas relating to gig economy crowdsourcing. \citet{shmueli2021beyond} identified risks of harm to crowdworkers engaged in NLP tasks. Attending to ethical issues more broadly, \citet{kocsis2016towards} used value-sensitive design and transparency literature to develop a taxonomic framework of ethical considerations in crowdsourcing.
Our primary contribution is the introduction of CrowdWorkSheets, a novel framework designed to facilitate critical reflection and transparent documentation of dataset annotation decisions, processes, and outcomes. CrowdWorkSheets complements and extends dataset development and documentation frameworks that have previously been developed in service of transparency, accountability, and reproducibility \citep{Gebru2018, bender2018data, Holland2018, chmielinski2020DNP, hutchinson2021accountability,ramirez2021state,datacards}, but focuses specifically on unique considerations relating to crowdsourced data-set annotation. Similar to recent dataset documentation frameworks that have tailored to specific domains (e.g. \citep{artsheets, healthsheets}), our work starts from a recognition of the limitations of ``one-size-fits-all'' solutions to ethical issues in dataset development. More specifically, we offer CrowdWorkSheets as a targeted intervention to address unresolved ethical problems in crowdsourcing that relate specifically to worker subjectivity and worker experiences.
The remainder of this paper is structured as follows. First, re review literature relating to (1) how annotators' individual and collective social experiences can impact their annotations, and (2) the relationship between the annotators and the crowdsourcing platforms, and what that relationship means for their ability to engage in fair work. Next, we introduce the CrowdWorkSheets considerations and documentation questions.
Finally, we step through a hypothetical case study to illustrate how a dataset developer might use CrowdWorkSheets to document their decisions.
\section{Who is annotating ML datasets and why does it matter?}
\label{sec_challenges}
The historical lineage of crowdsourced labor can be traced back to manufacturing innovation of piecework \citep{Alkhatib2017}---a form of labor that produced the ``unskilled worker'' as the paradigmatic \textit{interchangeable} component and which has been credited with giving rise to the productivity and ingenuity of American manufacturing \citep{freeman2018}. In an analogous manner, crowdwork platforms are often designed to position crowdworkers as \textit{interchangeable} \citep{irani2013turkopticon}.
While some forms of digital work can be decomposed and distributed, the presumption that crowdsourced dataset annotators exercise near-identical capacities of perception and judgement ignores
the fact that social position, identity, and experience shape how annotators apply knowledge.
Yet, recent empirical work has revealed that dataset annotators are often treated as interchangeable in practice. For example, relatively little attention is given or documented about annotator positionality---how annotator social identity shapes their understanding of the world \citep{geiger2020, scheuerman2021}. Crowd workers are often selected by task requesters based on quality metrics, rather than on any socially defining features of their knowledge or experience. This is concerning; when crowd-sourced annotations are used to build datasets capturing subjective phenomena, such as sentiment or hate speech, annotators' values and subjective judgments shape the perspectives that machine learning models learn from in a manner that is wholly unaccounted for.
\subsection{Accounting for the socio-cultural backgrounds of annotators}
Understanding socio-cultural factors of an annotator pool---or even selecting annotators based on these factors---is important because annotator's identity and lived experience can impact how annotation questions are interpreted and responded to. More generally, subjective interpretations of a task can produce divergent annotations across different communities \citep{sen2015turkers}. As \citet{aroyo2015truth} argue, the notion of ``one truth'' in crowdsourcing responses is a myth; disagreement between annotators, which is often viewed as problematic noise, can actually provide a valuable signal.
A variety of social, cultural, economic, and infrastructural factors contribute to the sociodemographic distribution of workers on any given platform.
For example, as \citep{gray2019ghost} points out, the remote nature of crowdwork differentially attracts workers along gender lines, such as mothers who do crowdwork because it allows for an easier balance of childcare in comparison to other work.
Other work similarly notes significant gender differences among workers who report engaging in crowdwork because they are only able to conduct work from their homes \citep{berg2015income}. This leads to a different gender balance among crowdworkers in the United States than in many other parts of the world; crowdworkers in most of the world are disproportionately male, while in contrast over 60\% of U.S. annotators are female \cite{posch2018characterizing}. \citet{ipeirotis2010demographics} hypothesizes this to be due to the remote nature of the work, which attracts stay-at-home parents and unemployed or underemployed adults, who are more likely to be women. Additionally, health problems and disability are also a factor that cause many workers to only be able to work from home and motivates them to pursue crowd work \citep{berg2015income}.
Since many crowdsourced annotator pools are sociodemographically skewed, there are implications for which populations and cultural values are represented in datasets and models \cite{ghosh-etal-2021-detecting} as well as which populations face the challenges of crowdwork \citep{irani2013turkopticon, gray2019ghost}.
Accounting for skews in annotator demographics is critical for contextualizing datasets and ensuring responsible downstream use. In short, there is value in acknowledging, and accounting for, worker's socio-cultural background---both from the perspective of data quality and societal impact.
\subsection{Lived experiences of annotators as expertise}
Just as substantive work experience lends valuable domain expertise for a given problem (e.g., annotation of medical imagery by a medical professional), lived experience with, and proximity to, a problem domain can provide a valuable source of expertise for dataset annotation. For example, women experience higher rates of sexual harassment online compared to men, and among those who have experienced online abuse, women are more likely to identify it as such \citep{vogels2021state}. This underscores the importance of considering raters' experience with gender-based harassment, when using crowdwork to annotate/moderate online harassment. Recent work has highlighted how the ``average'' rater, in terms of gender and other social characteristics, varies dramatically depending on which geographies raters are selected from \cite{posch2018characterizing}. Additionally, as a result of the previously mentioned sociodemographic differences among who is likely to conduct crowdwork, ratings on sexual harassment data, for example, may differ according to the geographic distribution of raters.
At the same time, relevant lived experience among annotators does not always fall along demographic lines. \citet{waseem2016racist} demonstrated that incorporating feminist and antiracist activists' perspectives into hate speech annotations yielded better aligned models. Similarly, \citet{patton2019annotating} demonstrated the importance of situated domain expertise---including contextualized knowledge of local language, concepts, and gang activity---when annotating Twitter images to detect pathways to violence among gang-involved youth in Chicago. They found that expert annotations (i.e., those from individuals situated in Chicago and with community ties) significantly diverged from those of graduate students who were scholars of social work and who were trained to perform the annotation task but who lacked this lived experience.
In summary, a core question to answer in data collection is how much annotator's identity, lived experience, and prior knowledge of a problem space matters for the task at hand, and how it impacts what the resulting dataset is intended to capture. While the aforementioned examples constitute relatively subjective tasks, even seemingly objective tasks such as annotating medical text vary surprisingly with annotator backgrounds and experience. \citet{aroyo2015truth} show that medical experts are more likely to erroneously identify medical relations as being expressed in text compared with non experts because the experts already know the relation is true based on knowledge external to the task. Their work underscores a need to examine annotator experience even in tasks that appear to be unambiguous or objective.
\section{Worker Experiences of Dataset Annotation}\label{experiences}
Another series of considerations are rooted in annotators' experiences with annotation work itself and how those experiences impact how they do their work. These include issues related to worker compensation, power imbalances in between worker and requester, and the structure of annotation work itself---all of which can pose barriers to crowdworker well-being and their ability to produce quality work.
\subsection{Compensation and working conditions}
Compensation policies of crowdwork platforms should be a core aspect to consider when thinking about responsible data collection. For instance, in the U.S., there are currently no regulations around worker pay for crowdwork \citep{berg2015income}, and the Fair Labor Standards Act that established the minimum wage,\footnote{\url{https://www.dol.gov/agencies/whd/flsa}} is not applicable for crowdworkers as they are independent contractors \citep{semuels2018internet}. Reports on how much crowdworkers actually earn vary, but generally show an average lower than minimum wage \cite {irani2013turkopticon}; surveys of workers from Amazon Mechanical Turk and Crowdflower place it on average between \$1 and \$5.5 per hour \cite{berg2015income} with a median wage of roughly \$2 / hour \cite{hara2018data,semuels2018internet}; only a small fraction of workers (4\%) earn more than \$7.25 / hour \cite{hara2018data}.
Recent research has also identified how crowdworking platforms often necessitate various kinds of unpaid labor from crowdworkers, which reduces overall wages.
For example, one report found that for every hour of paid work, workers spend another 18 minutes on unpaid work, including searching for tasks \citep{berg2015income}.
Another recent study found that once daily invisible labor was accounted for, the median hourly wage for crowdworkers on Amazon Mechanical Turk dropped from \$3.76 to \$2.83 \citep{Toxtli2021}.
Workers often invest significant labor outside the platform itself to find tasks, relying on web browsers extensions and participating in crowd work forums \cite{hanrahan2021expertise, kaplan2018striving}. Time spent working is compounded by competition from other crowdworkers \citep{semuels2018internet}, which can pressure workers to be constantly available to look for work \citep{berg2015income}.
The working conditions of crowdworkers are characterized by long working hours, partially as a result of this competition. As \citet{berg2015income} notes, this conflicts with the work flexibility motivates many workers to choose crowd work.
Worker psychological safety is a particular area of concern. Crowd workers who work on content moderation of user generated content often need to look at content that includes violent imagery or sexual and pornographic content \cite{roberts2016digital}, or to transcribe conversations about trafficking children into sexual slavery \cite{emerson2019suffering}.
In many cases, it is impossible to ascertain that a job may contain such content \cite{emerson2019suffering}.
If crowdworkers find themselves upset or disturbed by this content, they have little recourse; often, workers need to sign non-disclosure agreements preventing them from talking to anyone about the awful things they must look at, even for support \cite{roberts2016digital}. Additionally, raising concerns to their employers is quite difficult; both bureaucracy and physical distance (many of these workers are in the Global South) prohibit any direct lines of feedback or complaints. There is research available on the long-term impacts of viewing harmful user-generated content, but it is difficult to assess the full harm this causes to workers' well-being \cite{roberts2016digital}.
\subsection{Power dynamics}
Power dynamics between the requesters and annotators is another major challenge. Annotators are often heavily distanced from those leading the development of datasets are requesting tasks, which can obfuscate working conditions.
Top-down organizational structures often results in the workers viewing requesters as more informed as they are the ones who provided the data and the label schema \citep{miceli2020between}. Hence, instead of resolving ambiguities, workers are more likely to try to judge from the standpoint of the requester, often with limited exposure to the goals of the annotation. This contributes to the \textit{portability trap} \citep{selbst2018fairness}: a ``failure to understand how repurposing algorithmic solutions designed for one social context may be misleading, inaccurate, or otherwise do harm when applied to a different context.''
Power dynamics are also at play in the rejection of work: a large majority of crowdworkers (94\% as per \citep{berg2015income}) have had work that was rejected or for which they were not paid. Yet, some platforms give requesters full rights over the data they receive, regardless of whether they accept or reject it, and workers have no way of taking legal action if requesters use rejected work anyway \cite{irani2013turkopticon}; \citet{roberts2016digital} describes this system as one that ``enables wage theft''. Moreover, rejecting work and withholding pay is painful because rejections are often caused by unclear instructions and the lack of meaningful feedback channels. Many crowdworkers report that poor communication negatively affects their work \citep{berg2015income}. Moreover, requesters get to choose whether the work is up to their standards before choosing whether to pay for it, even though rejections are often caused by unclear instructions and the very lack of feedback channels they refuse to provide \cite{berg2015income}. Workers also feel powerless to speak up about perceived injustices from requesters or the platform; Amazon Mechanical Turk (AMT) users have reportedly had their accounts suspended for speaking negatively about Amazon \cite{semuels2018internet}. Additionally, requesters can block users who offer them feedback without consequence \cite{berg2015income}.
Power asymmetries also reflect global power dynamics. For instance, since technology development happens primarily in the West, human computation from the Global South is often relegated to the margins \citep{sambasivan2021re}. In particular, \cite{sambasivan2021re} points out that the technical, social, ethical, and physical distance between the builders of a technology and the communities it is meant to serve is large, in such settings.
\cite{bott2012role} has pointed out the potential of crowdsourcing to revolutionize civic participation in many developing countries to address complex challenges in governance around global issues such as climate change, poverty, armed conflict, and other crises. They also point out the challenges when it comes to employing crowdsourced interventions on the ground in the Global South. They note that systemic disparities endemic to local contexts are often reflected in who is represented in \textit{crowd}; for instance, the digital crowd in the Global South tends to over-represent the elite, educated, young males who belong to the upper tiers of local social hierarchies.
\citet{roberts2016digital} compares the commercial content moderation work to the practice of developed nations offloading their hazardous e-waste refuse on countries in the Global South. Her interviewees characterized this work as ``akin to being immersed in `a cesspool' -- feeling that they are within a pit of toxic matter and waste day in and day out''. The metaphor goes further in highlighting the fact that digital content moderation when outsourced to countries in the Global South, serves to keep the digital refuse away from the field of vision of those in the Global North who are responsible for its existence, and for whom it was intended, in much the same way the rotting garbage and e-waste produced in the Global North is kept away.
On the other hand, some platforms have geographical blocking, which many non-Americans find problematic \cite{berg2015income} since it can be used to exclude them. This reinforces the dynamic where requesters in the United States get to decide which global perspectives they want to consider for their task, and which they want to disregard.
The anonymous and geographically distributed nature of crowdsourced annotation work imposes significant barriers to collective action on the part of dataset annotators. While some platforms offer communication spaces, such as discussion forums, for workers to communicate with one another, these platform-moderated spaces have been shown to be ineffective at supporting labor organizing or worker power \cite{Gerber21}. In response, several tools have been developed independently from crowdwork platforms to support crowd workers.
For example, TurkerNation, Turk Alert, MTurkGrind, and Reddit's /r/HITsWorthTurkingFor offer online forums for AMT workers to share information about well-paying work and share experiences with different requesters and Turkopticon \cite{irani2013turkopticon, turkopticon} is a browser add-on that enables AMT workers to review and report requesters and view reviews from other workers. These tools can help workers overcome the information asymmetries built into the AMT platform \cite{Martin2014}.
Dynamo is another community platform designed specifically to support and enable collective action for AMT workers, creating ``unities without unions'' \cite{Salehi2015}. A 2015 study of the platform found that twenty-two ideas for action had been generated and two active campaigns had been initiated.
In summary, responsible data annotation requires careful consideration of the power dynamics that structure the working relationship between requesters, annotators, and the platforms.
\cite{semuels2018internet} As a result, workers feel pressured to be constantly available online to look for work and work longer hours \cite{berg2015income}. \cite{berg2015income} notes that this is in conflict with one of the reasons many workers choose crowdwork, which is flexibility in working hours.
\section{CrowdWorkSheets: A Documentation Framework for Crowdsourced Dataset Annotation}
\label{framework}
We now introduce our framework, CrowdWorkSheets, which outlines a series of \textit{considerations} designed to guide the collection, use, and dissemination of crowd-sourced annotations and \textit{questions} designed to elicit information about various decisions and outcomes. We have decomposed the framework into sections based on different parts of a typical dataset construction pipeline, from the formulation of tasks to dissemination of datasets.
\subsection{Task formulation}
First, we must ask: \textit{what are we asking annotators to do?}
Our considerations and documentation questions focus on many aspects of task formulation including which assumptions we make about annotators, how we handle ambiguity and subjectivity within our task, and how our task is ultimately framed and communicated.
While some tasks tend to pose objective questions with a correct answer (\textit{is there a human face in an image?}), oftentimes datasets aim to capture judgement on relatively subjective tasks with no universally correct answer (\textit{is this piece of text offensive?}). Moreover, even seemingly objective tasks can still be rife with ambiguity or corner-cases and ultimately require subjective judgements to be made on the part of annotators. As such, it is important to consider how questions afford varied interpretations or may require subjective judgements on the part of the annotators.
Clarifying such aspects of of an annotation task as critical to ensuring a resulting dataset captures the aspects of human intelligence they are meant to capture. Moreover, as discussed in Section~\ref{experiences}, a survey of crowdworkers on AMT found that many instances of work rejection were due to unclear instructions \citep{berg2015income}.
While we discuss the nuances of annotator selection in greater depth in Section~\ref{selecting_annotators}, tasks should be formulated based on considerations regarding \textit{who} will be annotating data and what perspectives should (or should not) be included. Determinations should be tied to the purpose of dataset creation and the downstream use cases it is meant to serve, rather than what is convenient, efficient, or scalable.
Some tasks may benefit from being informed by the annotators' lived experiences and thus may be designed to explicitly seek out such expertise. On the other hand, a dataset developer may want to frame task instructions so as to restrict the annotators from relying on their lived experiences, e.g. for a dataset meant to capture a set of policies defined by a platform.
Finally, when formulating a task, it is important to consider how much information to disclose to annotators about the task in advance. Some information may be essential to disclose in order to enable annotators to make informed decision regarding whether or not to accept the task. For example, disclosure of how data will be stored, packaged, and potentially published may be particularly important when sociodemographic, or other sensitive information, about annotators is being requested. Similarly, disclosure of risks relating to psychological harm should be included where appropriate.
\textit{Considerations}
\begin{itemize}[itemsep=1pt,topsep=0pt]
\item Consider the role subjectivity plays in your annotation task. Remember that individuals with different social and cultural backgrounds might differ in their judgements.
\item Consider the forms of expertise that should be incorporated through data annotation, including both formal disciplinary training and lived experience with the problem domain. Remember that insufficiently capturing this expertise in the annotator pool may carry risks for downstream model usage.
\item
Make sure task instructions are clear and unambiguous in order to prevent annotators from wasting time on a task where their work will be rejected due to misunderstandings. Consider assessing the task instructions in a small-scale setting prior to launching your full annotation task.
\item Consider the personal information you are collecting from annotators and the potential ethical or privacy risks that may accompany such collection.
\item Consider the amount of information you disclose to annotators prior to engagement with the task and ensure annotators have an opportunity to make informed decisions based on any potential risks the task carries.
\end{itemize}
\textit{Documentation questions}
\begin{enumerate}[itemsep=1pt,topsep=0pt]
\item At a high level, what are the subjective aspects of your task?
\item What assumptions do you make about annotators?
\item How did you choose the specific wording of your task instructions? What steps, if any, were taken to verify the clarity of task instructions and wording for annotators?
\item What, if any, risks did your task pose for annotators and were they informed of the risks prior to engagement with the task?
\item What are the precise instructions that were provided to annotators?
\end{enumerate}
\subsection{Selecting annotators}
\label{selecting_annotators}
Next, we ask: \textit{who is annotating the data?} While there is no single ``correct'' way to assemble an annotator pool, the selection of an annotator pool is a highly consequential decision. Since annotators from different communities can produce significantly different annotations given the same task \cite{sen2015turkers}, it is important to recognize that annotator selection may have a significant impact on the labels of your dataset. With this in mind, it is important to consider the intended use of the datasets---which communities will be most impacted by models built from the data, and which communities could be harmed the most by resulting biases present if they are not represented in the annotator pool?
In some cases, social identities of annotators indicate a form of expertise relevant to our task so it may be prudent to select annotators based on self-identified sociodemographic factors. In other cases, it may be important to select annotators based on other forms of expertise or experience with a problem domain. Understanding one's desired annotator pool may subsequently impact decisions regarding platform selection, as different platforms offer differing degrees of flexibility to assemble custom annotator pools.
While selecting annotators based on sociodemographic factors may help ensure a dataset reflects perspectives of certain groups, targeted data collection efforts---particularly those oriented towards the inclusion of marginalized groups---are not without risk. For example, \citep{Denton2020BringingTP} discuss how the mere inclusion of marginalized groups within a dataset, without sufficient attention to broader considerations of data capture and use, can operate as a form of ``predatory inclusion''\footnote{The term "predatory inclusion" has been used to describes modes of inclusion that are extractive and predatory in nature in other domains (e.g. \cite{seamster2017})}. Discourses of inclusion can serve to ``further rather than subvert vulnerability to what might more broadly be called `data violence'" \citep{termsofinclusion}.
From a privacy perspective, if sociodemograph-ic information is collected and published with a dataset, developers should take extra care to mitigate risks of unintentionally making annotators identifiable.
\textit{Considerations}
\begin{itemize}[itemsep=1pt,topsep=0pt]
\item While there are multiple valid ways to assemble an annotator pool, remember that annotators are not interchangeable, and that
the decisions in this stage can heavily impact the final dataset.
\item Consider the ways in which social identities of annotators may relate to the forms of expertise important for the task.
\item Consider the intended usage contexts of the dataset, and the marginalized communities therein, when choosing which annotators to be prioritized to be included.
\item Consider how labor practices intersect with the choice of who the annotators are. For example: if female annotators make up the majority, as they do in the U.S.\ \citep{posch2018characterizing}, consider how fair payment, or a lack thereof, could impact this group.
\end{itemize}
\textit{Documentation questions}
\begin{enumerate}[itemsep=1pt,topsep=0pt]
\item Are there certain perspectives that should be privileged? If so, how did you seek these perspectives out?
\item Are there certain perspectives that would be harmful to include? If so, how did you screen these perspectives out?
\item Were sociodemographic characteristics used to select annotators for your task? If so, please detail the process.
\item If you have any aggregated sociodemographic statistics about your annotator pool, please describe.
\item Do you have reason to believe that sociodemographic characteristics of annotators may have impacted how they annotated the data? Why or why not?
\item Consider the intended context of use of the dataset and the individuals and communities that may be impacted by a model trained on this dataset. Are these communities represented in your annotator pool?
\end{enumerate}
\subsection{Platform and infrastructure choices}
Next, we ask, \textit{under what conditions are data annotated?} As described in Section~\ref{experiences}, platform policies around compensation and power asymmetries play a huge role in shaping worker experiences and the quality of work that annotators produce. Different platforms offer different affordances for communication between task requesters and annotators, which might impact the extent to which task requesters can incorporate annotator feedback into the task framing or annotator guidelines.
Different platforms also impose different minimum-pay constraints; requesters may want to support platforms that uphold fair pay standards.
Additionally, requesters should be mindful of potential differences between legal minimum wages and a living wage \cite{living-wage}.
Separately from the platform, task creators should be aware of worker pay per hour; some platforms may only offer requesters the option to select pay per item for an annotation task, and the defaults may be set low. Task creators should take care when estimating work time per item to ensure they are paying workers fairly.
Another thing to consider when choosing a platform for data annotation is how well that platform supports rater psychological safety. Some platforms provide more affordances than others for crowdworkers to seek out support if they are experiencing distress, or if they otherwise have questions or feedback for requesters.
\textit{Considerations}
\begin{itemize}[itemsep=1pt,topsep=0pt]
\item Consider platform's underlying annotator pool and the options they provide to source specialized rater pools, and whe-ther they enable you to curate an appropriate pool of annotators (e.g.\ considering sociodemographic factors or domain expertise).
\item Consider comparing and contrasting the minimum pay requirements established across different platforms. You may choose to support a platform that upholds fair pay standards.
\item Consider the extent to which you would like to establish a channel of communication and feedback between your team and the annotators. Platform mediated channels of communication can give annotators an opportunity to provide feedback on confusing instructions, or otherwise seek out support.
\end{itemize}
\textit{Documentation questions}
\begin{enumerate}[itemsep=1pt,topsep=0pt]
\item What annotation platform did you utilize?
\item At a high level, what considerations informed your decision to choose this platform?
\item Did the chosen platform sufficiently meet the requirements you outlined for annotator pools? Are any aspects not covered?
\item What, if any, communication channels did your chosen platform offer to facilitate communication with annotators? How did this channel of communication influence the annotation process and/or resulting annotations?
\item How much were annotators compensated? Did you consider any particular pay standards, when determining their compensation? If so, please describe.
\end{enumerate}
\subsection{Dataset analysis and evaluation}
Once data instances are annotated, \textit{what do we do with the results?} This section focuses on considerations related to the process of converting the ``raw'' annotations into the labels that are ultimately packaged in a dataset. A common practice in building crowdsourced annotations for discrete labeling tasks is to obtain multiple annotator judgements that are then aggregated (e.g.,\ through majority voting) to obtain a single ``ground truth'' that is released in the dataset \citep{sabou2014corpus}. However, the disagreements between annotators may embed valuable nuances about the task \citep{alm2011subjective,aroyo2013crowd}. Aggregation, in such cases may obscure such nuances, and potentially exclude perspectives from minority annotators \citep{prabhakaran2021releasing}.
It is thus critical to consider uncertainty and disagreement between annotators, and potentially leverage this as a signal, to avoid losing nuanced and diverse opinions in the aggregation process. It might be important to analyze how annotators disagree along sociodemographic lines in order to be able to share this information with potential users of the dataset, so they can best understand how to represent these diverse perspectives in their use of the data.
\textit{Considerations}
\begin{itemize}[itemsep=1pt,topsep=0pt]
\item Consider including uncertainty or disagreement between annotations on each instance as a signal in the dataset.
\item Consider analyzing systematic disagreements between annotators of different sociodemographic groups in order to better understand how diverse perspectives are represented.
\item Consider how the final dataset annotations will relate to individual annotator responses. For instance, one option is to release only the aggregated labels, e.g.\ through a majority vote.
Consider what valuable information might be lost through such aggregation.
\end{itemize}
\textit{Documentation questions}
\begin{enumerate}[itemsep=1pt,topsep=0pt]
\item How do you define the annotation quality in your context, and how did you assess quality in your dataset?
\item Have you conducted any analysis on disagreement patterns? If so, what analyses did you use and what were the major findings?
\item Did you analyze potential sources of disagreement?
\item How do the individual annotator responses relate to the final labels released in the dataset?
\end{enumerate}
\subsection{Dataset release and maintenance}
Finally, it is critical to consider \textit{what is the future of the dataset?} Data exists within an ever-changing world, and should be viewed and used in that context. Users of the dataset now and in the future should understand the limitations of the data based on when and how it was collected. For example, a dataset may require periodic updates to remain robust to new slang or changes in language use over time. In addition, annotation tasks may be predicated upon legal definitions or medical standards that may change according to decisions by institutions or governing bodies.
\textit{Considerations}
\begin{itemize}[itemsep=1pt,topsep=0pt]
\item Consider designing and sharing a dataset maintenance plan \citep{Hutchinson2021}.
\item Consider potential conditions under which annotations may become outdated or less useful.
\end{itemize}
\textit{Documentation Questions}:
\begin{enumerate}[itemsep=1pt,topsep=0pt]
\item Do you have reason to believe the annotations in this dataset may change over time? Do you plan to update your dataset?
\item Are there any conditions or definitions that, if changed, could impact the utility of your dataset?
\item Will you attempt to track, impose limitations on, or otherwise influence how your dataset is used? If so, how?
\item Were annotators informed about how the data is externalized? If changes to the dataset are made, will they be informed?
\item Is there a process by which annotators can later choose to withdraw their data from the dataset? Please detail.
\end{enumerate}
\section{Case Study}
We now present a \textit{hypothetical} case study to demonstrate how our considerations outlined in Section ~\ref{framework} might be incorporated in practice and how dataset annotation decisions might be documented using CrowdWorkSheets. Responses to documentation questions are not intended to be prescriptive, nor are they completely comprehensive. Instead, they should be considered as one of many valid responses to this line of inquiry, and as a way to provoke further thought and discussion.
In this hypothetical case study, we take our goal to be the development of a benchmark dataset for public release to support academic research in social media content moderation. A Twitter corpus of 20,000 English-language tweets has been collected and we seek to label each tweet independently on a four-point ``toxicity'' scale defined in \cite{dixon2018annotation}.
\vspace{4mm}
{
\centering
\begin{tabular}{m{.97\columnwidth}c}
\hline
\multicolumn{1}{|c|}{\textbf{Task Formulation}} \\
\hline
\\
\end{tabular}
}
{
\small
\textbf{At a high level, what are the subjective aspects of your task?} \\
Judgements of toxicity of online comments is highly subjective. What makes a tweet harmful or hurtful varies greatly not only by the literal content of the tweet, but by the context surrounding it. In our task setup, tweets are presented to annotators in isolation, so they do not have access to the overall context of the online conversation. As such, we anticipate that annotators may infer surrounding context and make subjective judgements based on this inference. \\
\textbf{What assumptions do you make about annotators?} \\
Some of the key assumptions we make of our annotators:
\begin{itemize}[after=\vspace{\baselineskip}]
\item Annotators that claim proficiency in English and familiarity with social media have enough context to reasonably interpret the task.
\item By giving a clear understanding of the goals of this work and explicitly indicating that this is a subjective task where disagreement is expected, we will increase the likelihood that annotators will allow their lived experiences to inform how they label toxicity.
\item By paying well, we will increase the likelihood that annotators will take time to think through particularly challenging examples.
\end{itemize}
\textbf{How did you choose the specific wording of your task instructions? What steps, if any, were taken to verify the clarity of task instructions and wording for annotators?} \\
To align with existing research in the area, we've chosen to give annotators an existing definition of toxicity, as ``rude, disrespectful or otherwise likely to make someone leave a discussion'' \cite{aroyo2019crowdsourcing}.
To settle on a final task wording, our research team first completed 50 annotation tasks each to identify any obvious challenges applying this definition. We then ran several small pilot studies with slightly varying task instructions, and allowed annotators the option to give feedback on aspects that were unclear. Looking over these results, we settled on the question phrasing that yielded the least reported confusion.
We intentionally chose to leave our definition of toxicity somewhat open to interpretation, operating under the understanding that being overly specific in task instructions for subjective work does not improve response quality \cite{aroyo2015truth}. We also explicitly informed annotators that we expect a variety of interpretations of each comment, and that we were looking for their personal best judgements in the given situation.
To motivate thoughtful responses, we chose to pay well above minimum page and gave annotators a clear idea of the ultimate purpose of their work.
However, we know that it is inevitable that some annotators will simply give answers they think we want as quickly as possible. While we screen out responses below a minimum duration, it's impossible to ensure every answer is honest and thoughtful. We assume that these responses are randomly distributed; we leave it up to dataset users to do further analysis.\\
\textbf{What, if any, risks did your task pose for annotators and were they informed of the risks prior to engagement with the task?}\\
Our task required annotators to read text that potentially contained hate speech, slurs, and other harmful content. As such, the task posed a risk of psychological harm to annotators. Moreover, given that we selected annotators who had previously experienced online harassment, there is a potential for the task to trigger an emotional response related to past trauma. We informed annotators about this risk prior to the start of the task. We also informed annotators that we would be requesting sociodemographic information in order to assess disagreement across different groups. We outlined our data storage policy and steps we took to prevent responses from being linked to sociodemographic information. \\
\textbf{What are the precise instructions that were provided to annotators?}\\
The final task instructions used for data collection reflected in the released data is available at HypotheticalTaskInstructions.com.
}
\vspace{4mm}
{
\centering
\begin{tabular}{m{.97\columnwidth}c}
\hline
\multicolumn{1}{|c|}{\textbf{Selecting Annotations}} \\
\hline
\\
\end{tabular}
}
{
\small
\textbf{Are there certain perspectives that should be privileged? If so, how did you seek these perspectives out?}\\
We want to privilege the perspectives of annotators who have personally experienced online harassment or hold marginalized identities that are often targeted online. To this end, we included screening questions such that our annotator pool consisted of raters who have direct experience with online harassment. We intentionally defined ``direct experience'' very broadly to capture a wide range of experiences, intending to include annotators who've been personally harassed by others via online channels, who've encountered online content that threatened or disparaged identities they share, who have experience moderating online forums, or who have felt otherwise personally affected by harmful online content.\\
\textbf{Are there certain perspectives that would be harmful to include? If so, how did you screen these perspectives out?}\\
We believe that there are many harmful worldviews annotators might hold that we do not want captured by our annotations; we do not want to employ annotators who participate in hateful online communities, for example. To attempt to account for this, we identified several tweets that we agreed were unambiguously toxic, and screened out any annotators that did not label these as toxic.\\
\textbf{Were sociodemographic characteristics used to select annotators for your task? If so, please detail the process.}\\
In addition to screening for annotators who have previously experienced online harassment, we selected annotators based on self-identified gender and age. We aimed for an approximately gender balanced pool and we selected for at least 10\% of the annotators to be older than 65 years old. Because annotators were sourced from multiple geographic regions, we could not easily specify thresholds for racial or ethnic diversity; however, because we are screening for annotators who have experienced harassment online, we achieved decent representation among marginalized groups.\\
\textbf{If you have any aggregated sociodemographic statistics about your annotator pool, please describe.}\\
We first selected annotators who indicated that they had previously experienced online harassment. This resulted in a pool that is disproportionately composed of women and people of color compared with platform demographics. More specific demographic breakdowns are available with the released dataset.\\
\textbf{Do you have reason to believe that sociodemographic characteristics of annotators may have impacted how they annotated the data? Why or why not?}\\
Yes, we believe that annotators who have themselves experienced online harassment may be more likely to identify tweets as toxic. Based on rates of reported experience with hate speech attacks, we also expect that these annotators will disproportionately be members of marginalized social groups in their respective geographic region. \\
\textbf{Consider the intended context of use of the dataset and the individuals and communities that may be impacted by a model trained on this dataset. Are these communities represented in your annotator pool?}\\
Our intended audience is researchers studying English-language online content moderation, although we can anticipate that our work may have impact within industry. Content moderation has far-reaching and pervasive influence on online discourse, which impacts a wide range of individuals and communities. Not everyone is equally vulnerable to the worst impacts of toxic language online, so we specifically selected for an annotator pool where this more vulnerable population is represented.\\
}
\vspace{4mm}
{
\centering
\begin{tabular}{m{.97\columnwidth}c}
\hline
\multicolumn{1}{|c|}{\textbf{Platform and Infrastructure Choices}} \\
\hline
\\
\end{tabular}
}
{
\small
\textbf{What annotation platform did you utilize?}\\
We're using HypotheticalPlatform.\\
\textbf{At a high level, what considerations informed your decision to choose this platform?}\\
We have selected HypotheticalPlatform for several reasons: First, they are a generally reliable platform with a history of high data quality. Second, they are able to guarantee that annotators are paid at or above a living wage. Third, their platform's interface allows annotators to easily communicate feedback and concerns. And finally, their platform allows us to make ample use of screening questions to select the annotator pool for our main body of work.\\
\textbf{Did the chosen platform sufficiently meet the requirements you outlined for annotator pools? Are any aspects not covered?}\\
We were able to meet all of our requirements for annotator pools through the use of many screening and demographic questions. The main trade-off we made to accomplish this is in cost; to pay annotators well, including for their time answering screening questions, we set a limit on the number of tweets we could label.\\
\textbf{What, if any, communication channels did your chosen platform offer to facilitate communication with annotators? How did this channel of communication influence the annotation process and/or resulting annotations?} \\
We included a free response section at the end of our survey to allow feedback from annotators. In our pilot studies, we used this to clarify our task instructions. In the full study, most annotators left this blank, so we chose to leave them out of the final dataset.\\
\textbf{How much were annotators compensated? Did you consider any particular pay standards, when determining their compensation? If so, please describe.} \\
Informed by the 2020 results of the MIT Living Wage Calculator \cite{living-wage}, we aimed for annotators to take home at least \$25/hr on our work, with the goal of comfortably reaching a living wage for a single adult with no dependents, and decrease the pressure to complete tasks as quickly as possible. Annotators were paid \$6.25 for labeling a batch of 40 tweets, designed to take no more than 15 minutes, and verified over the course of the annotation job.
}
\vspace{4mm}
{
\centering
\begin{tabular}{m{.97\columnwidth}c}
\hline
\multicolumn{1}{|c|}{\textbf{Dataset Analysis and Evaluation}} \\
\hline
\\
\end{tabular}
}
{
\small
\textbf{How do you define the quality of annotations in your context, and how did you assess the quality in the dataset you constructed?}\\
We assessed quality along several dimensions, each of which had an associated question in each 40-question batch:
\begin{itemize}
\item Attention: We included 1 attention check question was introduced that instructs the annotator to give a particular response so ensure annotators are reading each question;
\item Self-consistency: We included 2 duplicated questions within each batch, to ensure annotators were actually reading each tweet and being self-consistent in their responses.
\item Alignment with pre-defined ratings: We included several 2 tweets that the research team had pre-labeled as unoffensive and highly offensive. We chose tweets for which we would expect no disagreement from annotators.
\end{itemize}
We removed from the final dataset all batches where 2 or more of these 5 data quality questions were incorrectly answered. This ultimately accounted for 12\% of our data.\\
\textbf{Have you conducted any analysis on disagreement patterns? If so, what analyses did you use and what were the major findings?}\\
While the main purpose of this work is data collection and not analysis, we did conduct very preliminary analyses as a starting point for dataset users. We ran standard inter-annotator agreement metrics and found a relatively low interannotator agreement across all raters (Fleiss' $\kappa = 0.25$ \cite{fleiss1971measuring}). However, we do not believe this to be an issue of data quality---when we looked at the data aggregated along different demographic axes, we found many demographic groups with high interannotator agreement whose annotations differ significantly from the majority opinion.\\
\textbf{Did you analyze potential sources of disagreement?} \\
In our preliminary analysis, we looked at a few annotator demographics as a source of disagreement. There are a myriad of other factors one could analyze with respect to disagreement---tweet topic, presence or absence of particular words, or how quickly annotators responded, for example---but as this is intended to be released as a research dataset, we have not conducted all of these analyses.\\
\textbf{How do the individual annotator responses relate to the final labels released in the dataset?}\\
After bucketing annotator demographics such that no annotator was uniquely identifiable, we released all responses, attached to the demographics of the annotator that gave each response. We chose not to aggregate responses into final tweet toxicity labels, and instead leave this to dataset users to aggregate in a way that's appropriate for their use case.
}
\vspace{4mm}
{
\centering
\begin{tabular}{m{.97\columnwidth}c}
\hline
\multicolumn{1}{|c|}{\textbf{Dataset Release and Maintenance}} \\
\hline
\\
\end{tabular}
}
{
\small
\textbf{Do you have reason to believe the annotations in this dataset may change over time? Do you plan to update your dataset?}\\
The relevancy of and perceptions about tweets will certainly change over time. In an effort to remind dataset users that this data should be taken in its temporal context, we include the month and year that each tweet was (a) written and (b) annotated as meta-data. However, as a longer-term strategy, we are also open-sourcing and making public all parts of our annotation pipeline, including rater instructions, data formatting schemes, and information on how to coordinate with our data labeling partners. We will publicly extend an open invitation to future collaborators who want to reuse our pipeline to annotate more data. If this pipeline is used and our guidelines followed satisfactorily, we will append future annotations to our existing dataset.\\
\textbf{Are there any conditions or definitions that, if changed, could impact the utility of your dataset?}\\
Over time we expect societal views to deviate somewhat from the annotations collected. For example, it will not capture any shifts in attitude regarding language targeting social groups that may be considered marginalized in the future but that are not considered marginalized today. \\
\textbf{Will you attempt to track, impose limitations on, or otherwise influence how your dataset is used? If so, how?}\\
To access the data, we require dataset users indicate their affiliation, contact information, and use case. The research team will be assessing uses on a case-by-case basis, with particular attention given to risks associated with use cases that explicitly include sociodemographic data in their modeling. We also ask that any publications cite our dataset release paper so we can track academic uses of the dataset. Our full data license if available at HypotheticalDataLicense.com.\\
\textbf{Were annotators informed about how the data is externalized? If changes to the dataset are made, will they be informed?}\\
Annotators were informed that this data will be released as a research dataset prior to engaging in the task. We allowed raters to opt in to an email list that with share updates about data release an availability. This site will contain an automatically-updated list of papers that cite our dataset release paper. \\
\textbf{Is there a process by which annotators can later choose to withdraw their data from the dataset? If so, please detail.}\\
By design, we have no mechanisms of linking individual annotators to specific responses, and so have no option for annotators to withdraw their annotations form our dataset. We make this explicit to the annotators, and allow them to stop answering questions at any point if they decide they no longer want to continue.
}
\section{Conclusion}
In this work, we challenge the common portrayal of dataset annotators as interchangeable. Rather, we argue, their individual histories and experiences bring unique perspectives to the table that can become encoded in the overall dataset in a significant ways. Therefore, it becomes imperative to consider how the process of selecting annotators, and their experience working on annotation, is documented alongside other aspects of dataset development. Towards this end, we introduced CrowdWorkSheets, a framework for reflecting on and documenting key decision points of crowdsourced dataset development, and a set of recommendations for dataset developers. While this framework is oriented towards individual dataset developers, we also recognize the role large institutions can play in shifting incentives to engage with these recommendations, e.g. incentivizing transparent dataset documentation through conference submission and reviewer guidelines.
\newline \newline
\textbf{Funding:} This research was supported by Google.
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\bibliographystyle{ACM-Reference-Format}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,482 |
One word is sufficient AMAZING..
bought it for morning walk. Thanks team Amazon..
Good for the price lets how it turns out after using them for few weeks .
Fitting is ok. As mentioned in the description.
As expected in this price range.
Shoes are quite worth at this price. Sole seems quite hard and not made up of rubber. looks very sturdy.
Light Weight. Comfortable For Running.
Really loved the product. Bought this one for my father. Fits really well. He loved it too. Exactly like the picture. Comfortable,soft and light weighted.
Good and best in this range.
I got a different colour other than what i ordered. | {
"redpajama_set_name": "RedPajamaC4"
} | 764 |
La Coppa di Svizzera 1966 è stata la 9ª edizione della principale coppa nazionale svizzera di hockey su pista. La competizione ha avuto inizio il 2 luglio ed è terminata l'8 ottobre 1966. Il trofeo è stato vinto dall' per la seconda volta nella sua storia.
Risultati
Primo turno
|colspan="4" style="background-color:#D0D0D0" align=center|2 luglio 1966
Quarti di finale
|colspan="4" style="background-color:#D0D0D0" align=center|17 settembre 1966
Semifinali
|colspan="4" style="background-color:#D0D0D0" align=center|24 settembre 1966
Finale
Collegamenti esterni
Edizioni della Coppa svizzera di hockey su pista
Tornei per club di hockey su pista nel 1966 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,614 |
\section{Introduction}
It is well known that higher-order theories are of interest in theoretical physics. In 1850 Ostrogradski developed works concerning the hamiltonian formalism for systems with higher derivatives \cite{1}. Since then, the research of systems such as generalizations of electrodynamics \cite{2, 3, 4, 5}, string theory \cite{6}, and dark energy physics \cite{7, 8} has led to interesting results about gravity; where higher-order Lagrangians, which include quadratic products of the curvature tensor, have ensured the renormalization of these theories \cite{9, 10}. Moreover, when it comes to a theory with gauge symmetries, the standard study of higher-order theories is done by using the so-called Ostrogradski-Dirac framework. \\
The Ostrogradski-Dirac framework is based on the extension of the phase space and on the choice of the fields and their temporal derivatives as canonical variables; the identification of the constraints is then performed as usual \cite{11}. Nonetheless, the classification of the constraints into first or second class is a difficult task. Alternative approaches can be employed, like the one developed by G\"uler \cite{F17} based on the identification of the constraints, called Hamiltonians, and the construction of a fundamental differential. These Hamiltonians can be either involutive or non-involutive and are used to obtain the characteristic equations, the gauge symmetries, and the generalized $HJ$ brackets of the theory. Using this approach, the construction of the fundamental differential is straightforward and the identification of symmetries is, in general, more economical than other approaches \cite{F17, F18, F19, F20, F21a, 12, 13}. \\
There is also a generalization of Ostrogradski's framework, the so-called $GLT$ formalism \cite{14, 15}. Based on the introduction of the canonical momenta as Lagrange multipliers, this framework allows one to reformulate the problem to one with only first-order time derivatives. Then, through a proper gauge fixing and making use of the Dirac brackets, the unphysical degrees of freedom can be removed. At the end of the calculations the identification of the constraints is less complicated than in Ostrogadski's method. \\
In this paper we will analyze the higher order Maxwell-Chern-Simons gauge theory \cite{16, 17}. We start with the G\"uler-$HJ$ approach, we will introduce additional fields to reduce the problem to a first-order time derivative one. Due to this, non-physical degrees of freedom will appear, which are associated with non-involutive Hamiltonians; however, these will be removed with the introduction of the generalized $HJ$ brackets. In this manner, the identification of the Hamiltonians will be straightforward, we also extend the results reported in \cite{ 16, 17}. Incidentally, we report an alternative study beyond the Ostrogradski-Dirac framework in search of the best alternative for analyzing higher-order singular systems. Then, we will finish our work by performing a $GLT$ analysis. In fact, we will analyze the higher order Maxwell-Chern-Simons from two different perspectives and we will compare our results. \\
The paper is organized as follows. In the Section II we develop the $HJ$ analysis for the higher order Maxwell-Chern-Simons gauge theory. We construct a fundamental differential, where the characteristics equations and all symmetries of the theory are found. Then, we reproduce and extend the results reported in \cite{16, 17}. In Section III the $GLT$ formalism is implemented; we reduce the higher-order theory to a first-order one, then we identify all constraints of the theory and present a complete description of the Dirac algebra.
\section{The Hamilton-Jacobi analysis}
The action under consideriation is given by \cite{16}
\begin{eqnarray}
\mathcal{L}=-\frac{1}{4} F^{\mu \nu} F_{\mu \nu}+\frac{\theta}{4} \epsilon^{\mu \nu \lambda} A_{\mu} F_{\nu \lambda}+\frac{1}{4 m} \epsilon^{\mu \nu \lambda}\left(\Box A_{\mu}\right) F_{\nu\lambda},
\label{Lag1}
\end{eqnarray}
where $A_\mu$ is the gauge potential, $F_{\mu \nu}$ is the curvature tensor, and $\epsilon^{\mu \nu \lambda}$ is the Levi-Cevita antisymmetric tensor. Throughout this paper we will use the following metric convection $\eta_{\mu \nu}= (-1, 1, 1)$, spacetime indices will be represented by greek alphabet $\alpha, \beta = 0, 1, 2$, and space indices by the Latin one $i, j, k =1, 2$. \\
By performing the $2+1$ decomposition we can write the action as
\begin{eqnarray}
\mathcal{L}&=&\frac{1}{2}\dot{A}^{i}\dot{A}_{i} - \dot{A}_{i}\partial^{i}A_{0} - \frac{1}{2}\partial^{i}A^{0}\partial_{i}A_{0} - \frac{1}{4}F^{i j}F_{i j} + \frac{\theta}{2}\epsilon^{i j}A_{0}\partial_{i}A_{j} - \frac{\theta}{2}\epsilon^{i j}A_{i}\dot{A}_{j} + \frac{\theta}{2}\epsilon^{i j}A_{i}\partial_{j}A_{0} \nonumber \\[5pt]
&-& \frac{1}{2m}\epsilon^{i j}\ddot{A}_{0}\partial_{i}A_{j} + \frac{1}{2m}\epsilon^{i j}\nabla^{2}A_{0}\partial_{i} A_{j} + \frac{1}{2m}\epsilon^{i j}\ddot{A}_{i}\dot{A}_{j} - \frac{1}{2m}\epsilon^{i j}\nabla^{2}A_{i}\dot{A}_{j} - \frac{1}{2m}\epsilon^{i j}\ddot{A}_{i}\partial_{j}A_{0} \nonumber \\[5pt]
&+& \frac{1}{2m}\epsilon^{i j}\nabla^{2}A_{i}\partial_{j} A_{0},
\label{Lag2}
\end{eqnarray}
For the purpose of analysis, we will write the Lagrangian (\ref{Lag2}) in a new fashion by introducing the following variables $A_{\mu} \rightarrow \xi_{\mu}$, $\dot{A}_{\mu} \rightarrow v_{\mu}$. By doing this, the following constraints, given by $\dot{\xi}_{\mu}- v_{\mu}=0$, will be added to the Lagrangian by means of new unphysical variables $\psi^{\mu}$. Thus, the Lagrangian takes the form
\begin{eqnarray}
\mathcal{L}&=& \frac{1}{2}v^{i}v_{i} - v_{i}\partial^{i}\xi_{0} - \frac{1}{2}\partial^{i}\xi^{0}\partial_{i}\xi_{0} - \frac{1}{4}F^{i j}F_{i j} + \frac{\theta}{2}\epsilon^{i j}\xi_{0}\partial_{i}\xi_{j} - \frac{\theta}{2}\epsilon^{i j}\xi_{i}v_{j} + \frac{\theta}{2}\epsilon^{i j}\xi_{i}\partial_{j}\xi_{0} \nonumber \\
&+& \frac{1}{2m}\epsilon^{i j}\left(-\dot{v}_{0} + \nabla^{2}\xi_{0}\right)\partial_{i}\xi_{j} - \frac{1}{2m}\epsilon^{i j}\left(-\dot{v}_{i} + \nabla^{2}\xi_{i}\right)v_{j} + \frac{1}{2m}\epsilon^{i j}\left(-\dot{v}_{i}+\nabla^{2}\xi_{i}\right)\partial_{j}\xi_{0} \nonumber \\[5pt]
&+& \psi^{0}\left(v_{0} - \dot{\xi}_{0}\right) + \psi^{i}\left(v_{i} - \dot{\xi}_{i}\right). \label{Eq:eq1}
\end{eqnarray}
We can observe that the theory is now linear in the temporal derivatives and we can apply the $HJ$ analysis. From the definition of the momenta
\begin{eqnarray*}
P^{\mu}=\frac{\partial \mathcal{L}}{\partial \dot{Q}_{\mu}},
\end{eqnarray*}
where $Q_{\mu} = (\xi_{0}, \xi_{i}, v_{0}, v_{i}, \psi_{0}, \psi_{i})$ are the canonical variables and $P^{\mu} = (\pi^{0}, \pi^{i}, \tilde{\pi}^{0}, \tilde{\pi}^{i}, p^{0}, p^{i})$ their corresponding momenta, we find the following Hamiltonians \cite{F17, F18, F19, F20, F21a, 12, 13}
\begin{eqnarray}
\Omega_{1}^{0} &=& \pi^{0} + \psi^{0} = 0, \nonumber \\
\Omega_{1}^{i} &=& \pi^{i} + \psi^{i} = 0, \nonumber \\
\Omega_{2}^{0} &=& \tilde{\pi}^{0} + \frac{1}{2m}\epsilon^{i j}\partial_{i}\xi_{j} = 0, \nonumber \\
\Omega_{2}^{i} &=& \tilde{\pi}^{i} - \frac{1}{2m}\epsilon^{i j}\left(v_{j} - \partial_{j}\xi_{0}\right) = 0, \nonumber \\
\Omega_{3}^{0} &=& p^{0} = 0, \nonumber \\[5pt]
\Omega_{3}^{i} &=& p^{i} = 0;
\label{const}
\end{eqnarray}
and the canonical Hamiltonian, given by
\begin{eqnarray}
\mathcal{H} &=& \dot{\xi}_{\mu}\pi^{\mu} + \dot{v}_{\mu}\tilde{\pi}^{\mu} + \dot{\psi}_{\mu}p^{\mu} - \mathcal{L} \nonumber \\[5pt]
&=& -\frac{1}{2}v^{i}v_{i} + v_{i}\partial^{i}\xi_{0} + \frac{1}{2}\partial^{i}\xi^{0}\partial_{i}\xi_{0} + \frac{1}{4}F^{i j}F_{i j} - \frac{\theta}{2}\epsilon^{i j}\xi_{0}\partial_{i}\xi_{j} + \frac{\theta}{2}\epsilon^{i j}\xi_{i}v_{j} - \frac{\theta}{2}\epsilon^{i j}\xi_{i}\partial_{j}\xi_{0} \nonumber \\[5pt]
&+& \tilde{\pi}^{0}\nabla^{2}\xi_{0} + \tilde{\pi}^{i}\nabla^{2}\xi_{i} + \pi^{0}v_{0} + \pi^{i}v_{i}.
\end{eqnarray}
Thus, with the Hamiltonians identified, we construct the fundamental differential, which describes the evolution
of any function, say $F$, on the phase space \cite{F17, F18, F19, F20, F21a, 12, 13}
\begin{eqnarray}
dF &=& \int \Big[ \{F \;,\; \mathcal{H}\} dt^{0} + \{ F\;,\;\Omega_{1}^{0}\} d \omega^{1}_0 + \{F \;,\; \Omega_{1}^{i}\} d \omega^{1}_i + \{F \;,\; \Omega_{2}^{0} \} d \omega^{2}_0 + \{F \;,\; \Omega_{2}^{i} \} d \omega^{2}_ i \nonumber \\
&+& \{F \;,\; \Omega_{3}^{0} \} d \omega^{3}_ 0 + \{F \;,\; \Omega_{3}^{i} \} d \omega^{3}_i \Big] \;\; d^{2}y,
\end{eqnarray}
where $ \omega^{1}_0, \omega^{1}_i, \omega^{2}_0, \omega^{2}_ i, \omega^{3}_ 0, \omega^{3}_i $ are parameters associated with the Hamiltonians. To this end we separate the Hamiltonians into involutive and non-involutive. Involutive Hamiltonians are those whose Poisson brackets with all Hamiltonians, including themselves, vanish; otherwise, they are called non-involutive. These will be labeled by $\Gamma$ and $\Lambda$, respectively. The Poisson algebra between the Hamiltonians in (\ref{const}) is given by
\begin{eqnarray}
\begin{matrix}
\begin{array}{ll}
\{\Omega_{1}^{0}(x) \;,\; \Omega_{2}^{i}(y)\} = -\frac{1}{2m}\epsilon^{i j}{\partial_{j}}\delta^{2}(x-y), & \{\Omega_{1}^{0}(x) \;,\; \Omega_{3}^{0}(y)\} = -\delta^{2}(x-y) \\[5pt]
\{\Omega_{1}^{i}(x) \;,\; \Omega_{2}^{0}(y)\} = \frac{1}{2m}\epsilon^{i j}{\partial_{j}}\delta^{2}(x-y), & \{\Omega_{1}^{i}(x) \;,\; \Omega_{3}^{j}(y)\} = \eta^{i j}\delta^{2}(x-y) \\[5pt]
\{\Omega_{2}^{i}(x) \;,\; \Omega_{2}^{j}(y)\} = -\frac{1}{m}\epsilon^{i j}\delta^{2}(x-y)
\end{array}
\end{matrix}
,\end{eqnarray}
hence, we observe that all the Hamiltonians are non-involutive; particularly those related to the unphysical fields $\psi^{\mu}$ and their momenta $p_\mu$. The matrix composed of these Poisson brackets, namely
\begin{eqnarray}
\Delta_{a b}=
\begin{pmatrix}
0 & 0 & 0 & -\frac{1}{2m}\epsilon^{j k}{\partial_{k}}& -1& 0 \\[5pt]
0 & 0 & \frac{1}{2m}\epsilon^{i j}{\partial_{j}} & 0 & 0 & \eta^{i j} \\[5pt]
0 & -\frac{1}{2m}\epsilon^{j k}{\partial_{k}}& 0 & 0 & 0 & 0 \\[5pt]
\frac{1}{2m}\epsilon^{i j}{\partial_{j}}& 0 & 0 & -\frac{1}{m}\epsilon^{i j} & 0 & 0 \\[5pt]
1 & 0 & 0 & 0 & 0 & 0 \\[5pt]
0 & -\eta^{i j} & 0 & 0 & 0 & 0 \\[5pt]
\end{pmatrix} \delta^2(x-y) \nonumber
,\end{eqnarray}
is not invertible, which means that the Hamiltonians are not independent. We will use the null vectors $\zeta$ of the matrix $\Delta_{a b}$ to identify the independent ones, such as it is done in a pure Dirac framework \cite{18}
\begin{eqnarray}
\int d^{2} y \Delta^{\mu \nu} \zeta(y)_{\mu}=0.
\end{eqnarray}
Then a null vector is found, $\zeta_{\mu} = (0, 0, w, 0, 0, -\frac{1}{2m}\epsilon_{l j}\partial^jw)$, where $w$ is an arbitrary function. Contracting $\zeta_{\mu}$ with a vector composed of the non-involutive Hamiltonians $\Omega^{\mu} = (\Omega_{1}^{0}, \Omega_{1}^{i}, \Omega_{2}^{0}, \Omega_{2}^{i}, \Omega_{3}^{0}, \Omega_{3}^{i})$ yields a new Hamiltonian, given by
\begin{eqnarray}
\zeta_{\mu}\Omega^{\mu}&=&0, \nonumber \\
\rightarrow \Gamma_1 &:& \tilde{\pi}^{0} + \frac{1}{2m}\epsilon^{i j}\partial_{i}\xi_{j} -\frac{1}{2m}\epsilon^{l j}\partial_{j}p_{l} = 0.
\label{invo1}
\end{eqnarray}
Since it's Poisson brackets with all other Hamiltonians (\ref{const}) vanishes, this new Hamiltonian is an involutive one. In this manner, the complete set of non-involutives Hamiltonians is given by
\begin{eqnarray}
\nonumber \Lambda_{1} &=& \pi^{0} + \psi^{0} = 0, \label{Eq:eq24} \\ \nonumber
\Lambda_2^ {i} &=& \pi^{i} + \psi^{i} = 0, \label{Eq:eq25} \\ \nonumber
\Lambda_3^{i} &=& \tilde{\pi}^{i} - \frac{1}{2m}\epsilon^{i j}\left(v_{j} - \partial_{j}\xi_{0}\right) = 0, \label{Eq:eq26} \\ \nonumber
\Lambda_4 &=& p^{0} = 0, \label{Eq:eq27} \\[5pt]
\Lambda_5^{ i} &=& p^{i} = 0,. \label{Eq:eq28}
\end{eqnarray}
Thus, the new $\Delta_{a b}$ matrix, whose entries will be the Poisson brackets between the new non-involutive Hamiltonians (\ref{Eq:eq28}), takes the form
\begin{eqnarray}
\Delta_{a b}=
\begin{pmatrix}
0 & 0 & -\frac{1}{2m}\epsilon^{j k}\partial_{k} & -1 & 0 \\[5pt]
0 & 0 & 0 & 0 & \eta^{i j} \\[5pt]
\frac{1}{2m}\epsilon^{i j}\partial_{j} & 0 & -\frac{1}{m}\epsilon^{i j} & 0 & 0 \\[5pt]
1 & 0 & 0 & 0 & 0 \\[5pt]
0 & -\eta^{i j} & 0 & 0 & 0 \\[5pt]
\end{pmatrix} \delta^{2}(x-y).
\end{eqnarray}
Which is found to not be singular; therefore it has an inverse, given by
\begin{eqnarray}
\Delta_{ab}^{-1}(x, y) =
\left(\begin{array}{ccccc}
0 & 0 & 0 & 1 & 0 \\[5pt]
0 & 0 & 0 & 0 & -\eta_{j l} \\[5pt]
0 & 0 & m\epsilon_{j l} & \frac{1}{2}{\partial_{j}} & 0 \\[5pt]
-1 & 0 & -\frac{1}{2}{\partial_{l}} & 0 & 0 \\[5pt]
0 & \eta_{j l} & 0 & 0 & 0 \\[5pt]
\end{array}\right) \delta^{2}(x-y). \nonumber
\end{eqnarray}
With this inverse matrix at hand we introduce the generalized brackets, defined as
\begin{eqnarray}
\{A(x), B(x^{\prime})\}^* = \{A(x), B(x^{\prime})\} - \int\int \{A(x), \xi^{a}(y)\} \Delta^{{a b}^{-1}}(y, z) \{\xi^{b}(z), B(x^{\prime})\} \; d^{2}y \; d^{2}z
\label{gbra}
,\end{eqnarray}
where $\xi^{\mu}$ represent the non-involutive Hamiltonians and $\Delta^{{a b}^{-1}}$ is the inverse of the matrix $\Delta_{a b}$, whose entries are the Poisson brackets between the non involutive Hamiltonians. Hence, by using the generalized brackets (\ref{gbra}) we can calculate the nontrivial ones between the phase space variables, these are
\begin{eqnarray}
\begin{matrix}
\begin{array}{ll}
\{\xi_{\mu} \;,\; \pi^{\nu}\}^{*} = \delta_{\mu}^{\nu}\delta^{2}(x-y), & \{\xi_{\mu} \;,\; \psi_{\nu}\}^{*} = -\eta_{\mu \nu}\delta^{2}(x-y), \\[5pt]
\{\pi^{0} \;,\; v_{k}\}^{*} = \frac{1}{2}{\partial_{k}}\delta^{2}(x-y), & \{\pi^{0} \;,\; \tilde{\pi}^{k}\}^{*} = -\frac{1}{4m}\epsilon^{k l}{\partial_{l}}\delta^{2}(x-y), \\[5pt]
\{v_{0} \;,\; \tilde{\pi}^{0}\}^{*} = \delta^{2}(x-y), & \{v_{i} \;,\; v_{k}\}^{*}= m\epsilon_{i k} \delta^{2}(x-y), \\[5pt]
\{v_{i} \;,\; \tilde{\pi}^{k}\}^{*}= \frac{1}{2}\delta_{i}^{k}\delta^{2}(x-y), & \{v_{i} \;,\; \psi_{0}\}^{*} = \frac{1}{2}{\partial_{i}}\delta^{2}(x-y), \\[5pt]
\{\tilde{\pi}^{i} \;,\; \tilde{\pi}^{k}\}^{*} = \frac{1}{4m}\epsilon^{i k}\delta^{2}(x-y), & \{\tilde{\pi}^{i} \;,\; \psi_{0}\}^{*} = -\frac{1}{4m}\epsilon^{i j}{\partial_{j}}\delta^{2}(x-y),
\end{array}
\end{matrix}
\end{eqnarray}
These generalized brackets will coincide with those of Dirac, which are calculated in the next section. In particular, we can observe that the generalized $HJ$ bracket between the Hamiltonian (\ref{invo1}) with itself gives
\begin{eqnarray}
\{\Gamma_1(x) \;,\; \Gamma_1(y)\}^* =0
,\end{eqnarray}
confirming that $\Gamma_1$ is indeed involutive. In this manner, the introduction of the $HJ$ brackets removes the non-involutive Hamiltonians and leaves us with a new fundamental differential, given by
\begin{eqnarray}
dF&=& \int \Big[ \{F \;,\; \mathcal{H} (y)\}^* dt^{0} + \{ F\;,\;\Gamma_{1}(y)\}^* d \sigma^{1} \Big] \;\; d^{2}y.
\label{dfg}
\end{eqnarray}
By using the fundamental differential we have removed the unphysical degrees of freedom $\psi^{0}$ and $\psi^{i}$, making the results in this section match those of the next section. In this regard, once the generalized brackets are introduced, we could perform the substitution of the fields $\psi 's$ by the momenta $\pi's$ in the action (\ref{Eq:eq1}), the result would be that the $HJ$ and $GLT$ actions are equivalent. In other approaches, see the ref. \cite{17}, the unphysical degrees of freedom are removed until the end of the calculations, because the separation of the constraints into first and second class allows the introduction of the Dirac brackets; in contrast, in the $HJ$ framework, the elimination of unphysical degrees of freedom is more convenient. We also have to take into account the Frobenius integrability conditions, which ensure that system is integrable. Applying this conditions to the Hamiltonian $\Gamma_1$ the following Hamiltonian arises
\begin{eqnarray}
d\Gamma_1(x)&=& \int \Big[ \{\Gamma_1(x) \;,\; \mathcal{H} (y)\}^* dt^{0} + \{ \Gamma_1(x) \;,\;\Gamma_{1}(y)\}^* d \sigma^{1} \Big] \;\; d^{2}y=0 \nonumber \\
&\rightarrow& \Gamma_2 \equiv \pi^{0} - \partial_{i}\tilde{\pi}^{i}=0, \nonumber
\label{invo2}
\end{eqnarray}
we observe that, since $\{\Gamma_2(x), \Gamma_2(y) \}^*=\{\Gamma_2(x), \Gamma_1(y) \}^*=0$, $\Gamma_2$ is an involutive Hamiltonian. We add this new involutive Hamiltonian to the fundamental differential and then calculate it's integrability, obtaining a new involutive Hamiltonian
\begin{eqnarray}
d\Gamma_2(x)&=& \int \Big[ \{\Gamma_2(x) \;,\; \mathcal{H} (y)\}^* dt^{0} + \{ \Gamma_2 (x)\;,\;\Gamma_{1}(y)\}^* d \sigma^{1} + \{ \Gamma_2(x)\;,\;\Gamma_{2}(y)\}^* d \sigma^{2} \Big] \;\; d^{2}y=0 \nonumber \\
&\rightarrow& \Gamma_3 \equiv \partial_{i}\pi^{i}+ \frac{\theta}{2}\epsilon^{i j}\partial_{i}\xi_{j} +\frac{1}{2m}\epsilon^{i j}\nabla^{2}\partial_{i}\xi_{j} =0. \nonumber
\label{invo3}
\end{eqnarray}
No further Hamiltonians emerge from the integrability conditions of $\Gamma_{3}$. As a result, the complete set of involutive Hamiltonians is given by
\begin{eqnarray}
\nonumber \Gamma_{1} &=& \tilde{\pi}^{0} + \frac{1}{2m}\epsilon^{i j}\partial_{i}\xi_{j}=0, \\ \nonumber
\Gamma_2 &=& \pi^{0} - \partial_{i}\tilde{\pi}^{i}=0, \\
\Gamma_{3} &=& \partial_{i}\pi^{i}+ \frac{\theta}{2}\epsilon^{i j}\partial_{i}\xi_{j} +\frac{1}{2m}\epsilon^{i j}\nabla^{2}\partial_{i}\xi_{j}=0.
\label{f1}
\end{eqnarray}
And the complete fundamental differential becomes
\begin{eqnarray}
dF&=& \!\! \int \Big[ \{F \;,\; \mathcal{H} (y)\}^* dt^{0} + \{ F\;,\;\Gamma_{1}(y)\}^* d \sigma^{1} + \{ F\;,\;\Gamma_{2}(y)\}^* d \sigma^{2} + \{ F\;,\;\Gamma_{3}(y)\}^* d \sigma^{3} \Big]\; d^{2}y
\label{dfg}
,\end{eqnarray}
where $\sigma^1, \sigma^2, \sigma^3$ are parameters associated to the Hamiltonians.
Therefore, we have presented an alternative for studying higher-order theories in the context of $HJ$ theory, which is more economical than those previously reported in the literature. With the fundamental differential we can obtain the characteristic equations and then identify the symmetries, which is done in appendix A.
\section{The Gitman-Lyakhovich-Tyutin framework }
In order to complete our analysis, the $GLT$ formalism will be performed. Starting with the Lagrangian (\ref{Lag2}) and following the formalism, we introduce the following variables \cite{14, 15}
\begin{eqnarray}
v_{\mu} = \dot A_{\mu},
\quad \quad
\beta_{\mu} = \dot v_{\mu},
\label{eq:var}
\end{eqnarray}
and their conjugated canonical momenta, satisfying
\begin{eqnarray}
\nonumber
\left\{A_{\mu}, \pi^{\nu}\right\} &=& \delta_{\mu}^{\nu}\delta^{2}(x-y), \\
\left\{v_{\mu}, \tilde{\pi}^{\nu}\right\} &=& \delta_{\mu}^{\nu}\delta^{2}(x-y).
\label{bra}
\end{eqnarray}
Thus, the Lagrangian (\ref{Lag2}) can be written as
\begin{eqnarray}
\mathcal{\tilde{L}} = \mathcal{L}+ \pi^{\mu}\left(\dot{A}_{\mu} - v_{\mu}\right) + \tilde{\pi}^{\mu}\left(\dot{v}_{\mu} - \beta_{\mu}\right), \nonumber
\end{eqnarray}
this is
\begin{eqnarray}
\nonumber
\mathcal{\tilde{L}}&=& \frac{1}{2}v^{i}v_{i} - v_{i}\partial^{i}A_{0} - \frac{1}{2}\partial^{i}A^{0}\partial_{i}A_{0} - \frac{1}{4}F^{i j}F_{i j} + \frac{\theta}{2}\epsilon^{i j}A_{0}\partial_{i}A_{j} - \frac{\theta}{2}\epsilon^{i j}A_{i}v_{j} + \frac{\theta}{2}\epsilon^{i j}A_{i}\partial_{j}A_{0} \\ \nonumber
&-& \frac{1}{2m}\epsilon^{i j}\beta_{0}\partial_{i}A_{j} + \frac{1}{2m}\epsilon^{i j}\nabla^{2}A_{0}\partial_{i} A_{j} + \frac{1}{2m}\epsilon^{i j}\beta_{i}v_{j} - \frac{1}{2m}\epsilon^{i j}\nabla^{2}A_{i}v_{j}
- \frac{1}{2m}\epsilon^{i j}\beta_{i}\partial_{j}A_{0} \\
&+& \frac{1}{2m}\epsilon^{i j}\nabla^{2}A_{i}\partial_{j} A_{0} + \pi^{\mu}\left(\dot{A}_{\mu} - v_{\mu}\right) + \tilde{\pi}^{\mu}\left(\dot{v}_{\mu} - \beta_{\mu}\right).
\label{Lag3}
\end{eqnarray}
We can observe that the theory is now first-order in the time derivatives, as well as that the canonical momenta have been introduced from the beginning. It is worth mentioning that the introduction of the momenta allows us to more easily identify the constraints compared to Ostrogradski's formalism. In fact, in the $GLT$ framework it is not necessary to introduce a generalized canonical momenta for the higher-order time derivatives of the fields, as is done in Ostrogradski's framework \cite{14, 15, 16, 17}. Subsequently, the canonical Hamiltonian is given as usual
\begin{eqnarray}
\mathcal{H} &=& \dot{A}_{\mu }\pi^{\mu } + \dot{v}_{\mu }\tilde{\pi}^{\mu } - \mathcal{\tilde{L}}, \nonumber \\
&=& v_{0}\pi^{0} + v_{i}\pi^{i} + \beta_{0}\tilde{\pi}^{0} + \beta_{i}\tilde{\pi}^{i} - \frac{1}{2}v^{i}v_{i} + v_{i}\partial^{i}A_{0} + \frac{1}{2}\partial^{i}A^{0}\partial_{i}A_{0} + \frac{1}{4}F^{i j}F_{i j} - \frac{\theta}{2}\epsilon^{i j}A_{0}\partial_{i}A_{j} \nonumber \\
&+& \frac{\theta}{2}\epsilon^{i j}A_{i}v_{j} - \frac{\theta}{2}\epsilon^{i j}A_{i}\partial_{j}A_{0} + \frac{1}{2m}\epsilon^{i j}\beta_{0}\partial_{i}A_{j} - \frac{1}{2m}\epsilon^{i j}\nabla^{2}A_{0}\partial_{i} A_{j} - \frac{1}{2m}\epsilon^{i j}\beta_{i}v_{j} + \frac{1}{2m}\epsilon^{i j}\nabla^{2}A_{i}v_{j} \nonumber \\
&+& \frac{1}{2m}\epsilon^{i j}\beta_{i}\partial_{j}A_{0} - \frac{1}{2m}\epsilon^{i j}\nabla^{2}A_{i}\partial_{j} A_{0}.
\label{Ham1}
\end{eqnarray}
Thus, the primary constraints (called $\phi$) are given by \cite{14, 15}
\begin{eqnarray}
\phi^{3} &=& \frac{\partial\mathcal{L}}{\partial \beta_{0}} - \tilde{\pi}^{0} = -\frac{1}{2m}\epsilon^{i j}\partial_{i}A_{j} - \tilde{\pi}^{0}\approx 0, \label{Eq:RestriccionGLT3} \\[5pt]
\phi^{i} &=& \frac{\partial\mathcal{L}}{\partial \beta_{i}} - \tilde{\pi}^{i} = \frac{1}{2m}\epsilon^{i j}v_{j} - \frac{1}{2m}\epsilon^{i j}\partial_{j}A_{0} - \tilde{\pi}^{i} \approx0. \label{Eq:RestriccionGLTi}
\end{eqnarray}
They coincide only with $\Omega_{2}^{0}$, and $\Omega_{2}^{i}$ from equation \ref{const}. Their algebra is given by
\begin{eqnarray}
\nonumber \{\phi^{3} , \phi^{i}\} &=& 0, \\
\{\phi^{i} , \phi^{j}\} &=& - \frac{1}{m}\epsilon^{i j}\delta^{2}(x-y).
\label{Eq:Poisson00}
\end{eqnarray}
At this point, it is important to comment the differences between the $HJ$ formalism and the GLT formulation. On one hand, in $GLT$'s formulation we must identify future constraints through consistency, then perform the classification of the constraints into first and second class, then Dirac's brackets are introduced and second class constraints can be taken strongly as zero. Only at the end of the calculations we can compare both formalisms. On the other hand, in the $HJ$ scheme the generalized brackets, which has an equivalent construction just as the Dirac ones, are introduced from the beginning. At the end of the calculations one ends up only with involutive Hamiltonians; which will agree with the set of first-class constraints of the $GLT$ formalism. \\
We continue with the classification of the constraints. By using the primary constraints we introduce the primary Hamiltonian
\begin{eqnarray}
\mathcal{H}'= \mathcal{H}+\lambda_{3}\phi^{3}+\lambda_{i}\phi^{i},
\end{eqnarray}
where $\lambda_{3}$ and $\lambda_{i}$ are Lagrange multipliers, thus, by using (\ref{Eq:Poisson00}) and by requiring consistency of the primary constraints we obtain a secondary constraint
\begin{eqnarray}
\nonumber \chi^{0}: \dot{\phi}^{3} &=& \{\phi^{3} \;,\;\mathcal{H}'\} \\
&=& \pi^{0}-\frac{1}{2m}\epsilon^{i j}\partial_{i}v_{j} \approx 0
,\end{eqnarray}
Consistency of $\phi^i$ provides a relation between the Lagrange multipliers, this is
\begin{eqnarray}
\nonumber \dot{\phi}^{i} &=& \{\phi^{i} \;,\;\mathcal{H}'\} \\
&=&-\epsilon^{i j} \lambda_{j} + \epsilon^{i j}\beta_{j} -\frac{1}{2}\epsilon^{i j}\partial_{j}v_{0} + m\pi^{i} - mv^{i} + m\partial^{i}A_{0} - \frac{\theta m}{2}\epsilon^{i j}A_{j} - \frac{1}{2}\epsilon^{i j}\nabla^{2}A_{j} \approx0.
\label{12a}
\end{eqnarray}
Now, by demanding consistency of the secondary constraint $\chi^0$ we find
\begin{eqnarray}
\nonumber \dot{\chi^0}&=& \{\chi^{0} \;,\;\mathcal{H}'\} \\
&=&-\epsilon^{i j}\partial_{i}\lambda_{j} + \epsilon^{i j}\partial_{i}\beta_{j} - m\partial^{i}v_{i} + m\nabla^{2}A^{0} - \theta m\epsilon^{i j}\partial_{i}A_{j} - \epsilon^{i j}\nabla^{2}\partial_{i}A_{j}\approx0,
\label{Eq:consist-chi0-GLT}
\end{eqnarray}
which also contains relations between the Lagrange multipliers. Furthermore, from (\ref{12a}) and (\ref{Eq:consist-chi0-GLT}) we can eliminate the Lagrange multipliers to obtain yet another secondary constraint
\begin{eqnarray}
\chi^{1} = \partial_{i}\pi^{i} + \frac{\theta }{2}\epsilon^{i j}\partial_{i}A_{j} + \frac{1}{2m}\epsilon^{i j}\nabla^{2}\partial_{i}A_{j}\approx0
\label{Eq:Chi5-GLT}
,\end{eqnarray}
From consistency of $\chi^1$ no more constraints are found. In this manner, the complete set of $GLT$ constrains is given by
\begin{eqnarray}
\nonumber \phi^{3} &=& \tilde{\pi}^{0}+\frac{1}{2m}\epsilon^{i j}\partial_{i}A_{j} \approx 0, \nonumber \\[5pt]
\phi^{i} &=& \tilde{\pi}^{i} - \frac{1}{2m}\epsilon^{i j}v_{j} + \frac{1}{2m}\epsilon^{i j}\partial_{j}A_{0} \approx0, \nonumber \\[5pt]
\chi^{0} &=& \pi^{0} -\frac{1}{2m}\epsilon^{i j}\partial_{i}v_{j} \approx 0, \nonumber \\[5pt]
\chi^{1} &=& \partial_{i}\pi^{i} + \frac{\theta }{2}\epsilon^{i j}\partial_{i}A_{j} + \frac{1}{2m}\epsilon^{i j}\nabla^{2}\partial_{i}A_{j}\approx0.
\label{fullcons}
\end{eqnarray}
Notice that $\phi^{i}$ is actually two constraints, so there are five in total. To separate them into first and second class we calculate the $5 \times 5$ matrix whose entries are the Poisson brackets between all constraints. This, in compact form, is
\begin{eqnarray}
\boldsymbol{A} =
\begin{pmatrix}
-\frac{1}{m}\epsilon^{i j} & 0 & \frac{1}{m}\epsilon^{i j}\partial_{j} & 0 \\[8pt]
0 & 0 & 0 & 0 \\[8pt]
-\frac{1}{m}\epsilon^{j i}\partial_{i} & 0 & 0 & 0 \\[8pt]
0 & 0 & 0 & 0
\end{pmatrix} \delta^{2}(x-y)
\label{Eq:matrixPoissonGLT}
,\end{eqnarray}
we observe that this matrix has a rank=2 and 3 null vectors, this means that there will be two second class constraints and three first class ones \cite{18}. The contraction of the null vectors with the constraints (\ref{fullcons}) allows us identify the following first class constraints
\begin{eqnarray}
\nonumber \gamma^{1} &=& \tilde{\pi}^{0}+ \frac{1}{2m}\epsilon^{i j}\partial_{i}A_{j}, \nonumber \\[5pt]
\gamma^{2} &=& \pi^{0} -\partial_{i}\tilde{\pi}^{i}, \nonumber \\[5pt]
\gamma^{3} &=& \partial_{i}\pi^{i} + \frac{\theta }{2}\epsilon^{i j}\partial_{i}A_{j} + \frac{1}{2m}\epsilon^{i j}\nabla^{2}\partial_{i}A_{j}
\label{first}
,\end{eqnarray}
e.g. one null vector is given by $\tilde{v}= (0, \partial_i w, w, 0)$, and from the contraction with (\ref{fullcons}) we obtain $\gamma^2$. We observe that the constraints (\ref{first}) coincide with the Hamiltonians (\ref{f1}) obtained with $HJ$ framework in the previous section. The two second class constraints are
\begin{eqnarray}
\xi^{i} = \tilde{\pi}^{i}- \frac{1}{2m}\epsilon^{i j}v_{j} + \frac{1}{2m}\epsilon^{i j}\partial_{j}A_{0},
\label{second}
\end{eqnarray}
these constraints are removed from the beginning in $HJ$ approach, in this sense the $HJ$ is more economical. It is worth commenting that the constraints have been obtained in consistent form by using the ideas presented in \cite{18} and it is not necessary to fix them by hand such as has been done previously in the literature \cite{17}. With the identification of the correct constraints, we can carry out the counting of physical degrees of freedom as follows: there are $12$ canonical variables $\left\{A_{\mu}, \pi^{\nu}\right\}, \left\{v_{\mu}, \tilde{\pi}^{\nu}\right\}$, three first class constraints $( \gamma^1, \gamma^2, \gamma^3)$ and two second class constraints $(\xi^i)$, therefore, there are two physical degrees of freedom, as expected \cite{16}.\\
The second class constraints can be removed by means of the Dirac bracket
\begin{eqnarray}
\{A(x), B(x^{\prime})\}_{D} = \{A(x), B(x^{\prime})\} - \int\int \{A(x), \xi^{a}(y)\} \Delta_{a b}(y, z) \{\xi^{b}(z), B(x^{\prime})\} \; d^{2}y \; d^{2}z
,\end{eqnarray}
where $\Delta_{a b}$ is the inverse of $\Delta^{a b}$, which consists of Poisson brackets among the second class constraints: $\Delta^{a b}=\{\xi^{a}, \xi^{b}\}$. This $2\times2$ matrix being as follows
\begin{eqnarray}
\Delta_{i j}(x, y) = m\epsilon_{i j}\delta^{2}(x-y)
\end{eqnarray}
This results in the following non-trivial Dirac's brackets
\begin{eqnarray}
\begin{matrix}
\begin{array}{ll}
\{A_{0} \;,\; \pi^{0}\}_{D} = \delta^{2}(x-y), & \{A_{i} \;,\; \pi^{j}\}_{D} = \delta_{i}^{j}\delta^{2}(x-y), \\[5pt]
\{\pi^{0} \;,\; v_{i}\}_{D} = \frac{1}{2}\partial_{i}\delta^{2}(x-y), & \{\pi^{0} \;,\; \tilde{\pi}^{i}\}_{D}= -\frac{1}{4m}\epsilon^{i j}\partial_{j}\delta^{2}(x-y), \\[5pt]
\{v_{0} \;,\; \tilde{\pi}^{0}\}_{D} = \delta^{2}(x-y), & \{v_{i} \;,\; \tilde{\pi}^{j}\}_{D} = \frac{1}{2}\delta_{i}^{j} \delta^{2}(x-y), \\[5pt]
\{v_{i} \;,\; v_{j}\}_{D} = m \epsilon_{i j} \delta^{2}(x-y), & \{\tilde{\pi}^{i} \;,\; \tilde{\pi}^{j}\}_{D} = \frac{1}{4m} \epsilon^{i j} \delta^{2}(x-y). \\[5pt]
\end{array}
\end{matrix}
\label{DiracGLT}
\end{eqnarray}
Using these we see that the constraints $\gamma^{1}$, $\gamma^{2}$, and $\gamma^{3}$ are still first class.
We will now fix the gauge in order to remove all first class constraints, turning them into second class. It is important to comment that the gauge-fxing condition removes the redundant degrees of freedom \cite{11}. Demanding consistency of the Coulomb gauge $\gamma^{4} = \partial_{i}A^{i}$ results in
\begin{eqnarray}
\gamma^5: =\{\partial_{i}A^{i} \;,\; H\}_{D} = \partial_{i}v^{i}.
\end{eqnarray}
Demanding consistency of $\gamma^{5}$ yields $\gamma^{6}$
\begin{eqnarray}
\gamma^{6}: = \{\partial_{i}v^{i} \;,\; H\}_{D} = \frac{1}{2}\nabla^{2}v_{0} + m\epsilon^{i j}\partial_{i}\pi_{j} - m\epsilon^{i j}\partial_{i}v_{j},
\end{eqnarray}
preservation in time of $\gamma^6$ gives no new constraints. Below we present the nontrivial brackets among all constraints
\begin{eqnarray}
\{\gamma^{4} \;,\; \gamma^{3}\}_{D} &=& -\nabla^{2}\delta^{2}(x-y), \nonumber \\[5pt]
\{\gamma^{5} \;,\; \gamma^{2}\}_{D} &=& \nabla^{2}\delta^{2}(x-y), \nonumber \\[5pt]
\{\gamma^{6} \;,\; \gamma^{1}\}_{D} &=& \nabla^{2}\delta^{2}(x-y), \nonumber \\[5pt]
\{\gamma^{6} \;,\; \gamma^{3}\}_{D} &=& \frac{\theta m}{2}\nabla^{2}\delta^{2}(x-y) + \frac{1}{2}\nabla^{4}\delta^{2}(x-y), \nonumber \\[5pt]
\{\gamma^{5} \;,\; \gamma^{6}\}_{D} &=& m^{2}\nabla^{2}\delta^{2}(x - y).
\label{alge}
\end{eqnarray}
Since, as can be easily seen, $\gamma_1,.., \gamma_6$ are all second class constraints, a new Dirac bracket can be introduced. In fact, by using (\ref{alge}) the new Dirac's brackets, say $\{, \}_{D_2}$, are given by
\begin{eqnarray}
\begin{matrix}
\begin{array}{ll}
\{A_{0} \;,\; v_{0}\}_{D_{2}} = m^{2}\frac{1}{\nabla^{2}}\delta^{2}(x-y), & \{A_{0} \;,\; v_{i}\}_{D_{2}} = m\epsilon_{i j}\frac{\partial^{j}}{\nabla^{2}}\delta^{2}(x-y), \\[5pt]
\{A_{0} \;,\; \pi^{0}\}_{D_{2}} = \delta^{2}(x-y), & \{A_{0} \;,\; \pi^{i}\}_{D_{2}} = -\frac{m}{2}\epsilon^{i j}\frac{\partial_{j}}{\nabla^{2}}\delta^{2}(x- y), \\[5pt]
\{A_{0} \;,\; \tilde{\pi}^{i}\}_{D_{2}} = -\frac{1}{2}\frac{\partial^{i}}{\nabla^{2}}\delta^{2}(x-y), & \{A_{i} \;,\; v_{0}\}_{D_{2}} = -m\epsilon_{i j}\frac{\partial^{j}}{\nabla^{2}}\delta^{2}(x-y), \\[5pt]
\{A_{i} \;,\; \pi^{j}\}_{D_{2}} = \frac{1}{2}\left(\delta_{i}^{j} - \frac{\partial^{j}\partial_{i}}{\nabla^{2}}\right)\delta^{2}(x-y), & \{v_{0} \;,\; \pi^{i}\}_{D_{2}} = -\left(\theta m\frac{1}{2 \nabla^{2}} + \frac{1}{2}\right)\partial^{i}\delta^{2}(x-y), \\[5pt]
\{v_{0} \;,\; \tilde{\pi}^{0}\}_{D_{2}} = \frac{1}{2}\delta^{2}(x-y), & \{v_{0} \;,\; \tilde{\pi}^{i}\}_{D_{2}} = -\frac{m}{2}\epsilon^{i j}\frac{\partial_{j}}{\nabla^{2}}\delta^{2}(x-y), \\[5pt]
\{v_{i} \;,\; \tilde{\pi}^{j}\}_{D_{2}} = \frac{1}{2}\left(\delta_{i}^{j} - \frac{\partial_{i}\partial^{j}}{\nabla^{2}}\right)\delta^{2}(x- y), & \{v_{i} \;,\; \pi^{0}\}_{D_{2}} = -\frac{1}{2}\partial_{i}\delta^{2}(x-y), \\[5pt]
\{\pi^{0} \;,\; \tilde{\pi}^{i}\}_{D_{2}} = -\frac{1}{4 m} \epsilon^{i j} {\partial_{j}} \delta^{2}\left(x-y \right), & \{\pi^{i} \;,\; \tilde{\pi}^{0}\}_{D_{2}} = \frac{1}{4m}\epsilon^{i j}\partial_{j}\delta^{2}(x-y), \\[5pt]
\{\pi^{i} \;,\; \tilde{\pi}^{j}\}_{D_{2}} = \frac{1}{4}\left(\delta^{i j} - \frac{\partial^{i}\partial^{j}}{\nabla^{2}}\right)\delta^{2}(x-y). &
\end{array}
\end{matrix}
\end{eqnarray}
These brackets were not reported in \cite{16, 17}. They can be used for quantization of the theory by using the methods reported in \cite{24}, where a procedure of gauge fixing is developed in the path integral approach. In this manner, our results extend those reported in the literature.
\section{Conclussions}
A detailed $HJ$ and $GLT$ analysis for higher-order Maxwell-Chern-Simons theory was developed. Regarding the $HJ$ study, with the introduction of auxiliary fields the theory was written as a first-order time derivative Lagrangian, and by means of the null vectors all Hamiltonians were identified. Then with the introduction of the generalized $HJ$ brackets, all unphysical fields were removed. We then constructed a fundamental differential given in terms of the generalized brackets and involutive Hamiltonians. This allowed us to identify the characteristic equations of the theory, where the equations of motion and the gauge transformations were reported. In this manner, we showed that the $HJ$ is an excellent framework for analyzing higher-order systems. \\
On the other hand, from the $GLT$ we report the complete structure of the constraints. We observed that the constraints were obtained in a consistent way, and there was no need to fix their structure by hand, as developed previously in the literature. Additionally, by fixing the gauge the complete structure of the Dirac brackets was presented. Therefore, our analysis extend those results presented in \cite{16, 17}, where different approaches were used. Finally, the study developed in this paper can be extended to theories with a more extensive structure, such as gravity and string theory. However, all those results are in progress and will be the subject of forthcoming works \cite{19}. \\
\section{Appendix: Gauge transformations}
\subsection{$HJ$ formalism}
We start by calculating the characteristic equations from the fundamental differential, which will reveal the symmetries of the theory. Using (\ref{dfg}), we find them to be
\begin{eqnarray}
\nonumber d\xi_{0} &=& v_{0}dt - d\sigma^{2}, \\ \nonumber
d\xi_{i} &=& v_{i}dt + \partial_{i}d\sigma^{3}, \\ \nonumber
d\pi^{0} &=& \left[ \frac{1}{2}{\partial_{i}}v^{i} - \frac{1}{2}\nabla^{2}\xi_{0} + \frac{3\theta}{4}\epsilon^{i j}{\partial_{i}}\xi_{j} - \nabla^{2}\tilde{\pi}^{0} + \frac{1}{4m}\epsilon^{i j}\nabla^{2}{\partial_{i}}\xi_{j} + \frac{1}{2}{\partial_{i}}\pi^{i} \right]dt, \\ \nonumber
d\pi^{i} &=& \left[ -\partial_{j}F^{i j}- \frac{\theta}{2}\epsilon^{i j}v_{j} - \nabla^{2}\tilde{\pi}^{i} \right] dt - \frac{1}{2m}\epsilon^{i j}\partial_{j}d\sigma^{1} + \left[\frac{\theta}{2}\epsilon^{i j}\partial_{j} + \frac{1}{2m}\epsilon^{i j}\nabla^{2}\partial_{j}\right]d\sigma^{3}, \\ \nonumber
dv_{0} &=& \nabla^{2}\xi_{0}dt + d\sigma^{1}, \\ \nonumber
dv_{i} &=& \left[\frac{1}{2} \nabla^{2}\xi_{i}+ \frac{1}{2}\partial_{i}v_{0} -m\epsilon_{i j}v^{j} +m \epsilon_{i j}\partial ^{j}\xi_{0} + \frac{\theta m}{2}\xi_{i} + m\epsilon_{i j}\pi^{j}\right] dt - \partial_{i}d\sigma^{2}, \\ \nonumber
d\tilde{\pi}^{0} &=& -\pi^{0}dt, \\
d\tilde{\pi}^{i} &=& \left[ \frac{1}{2}v^{i} - \frac{1}{2}\partial^{i}\xi_{0} + \frac{\theta}{4}\epsilon^{i j}\xi_{j} - \frac{1}{4m}\epsilon^{i j}\partial_{j}v_{0} + \frac{1}{4m}\epsilon^{i j}\nabla^{2}\xi_{j} - \frac{1}{2}\pi^{i} \right] dt.
\end{eqnarray}
The evolution of the dynamical variables with respect to our parameters $\sigma^{i}$ is understood as canonical transformations, with the corresponding hamiltonians $\Gamma^{i}$ as generators \cite{20, 21}. Due to Frobenius' theorem \cite{21}, the transformation with respect to one of these parameters is independent of the evolution along the others. To relate these canonical transformations to the gauge ones we set $dt=0$ \cite{F20}, obtaining
\begin{eqnarray}
\delta \xi_{0} &=& -\delta \sigma^{2}, \nonumber \\
\delta \xi_{i} &=& \partial_{i} \delta \sigma^{3}, \nonumber \\
\delta \pi^{0} &=& 0, \nonumber \\
\delta \pi^{i} &=& -\frac{1}{2m} \epsilon^{i j} \partial_{j} \delta \sigma^{1} + \left[\frac{\theta}{2} \epsilon^{i j} \partial_{j}+\frac{1}{2m} \epsilon^{i j} \nabla^{2} \partial_{j}\right] \delta \sigma^{3}, \nonumber \\
\delta v_{0} &=& \delta \sigma^{1}, \nonumber \\
\delta v_{i} &=& -\partial_{i} \delta \sigma^{2}, \nonumber \\
\delta \tilde{\pi}^{0} &=& 0, \nonumber \\
\delta \tilde{\pi}^{i} &=& 0.
\label{eq42}
\end{eqnarray}
In $HJ$, to find the gauge transformations it is necessary to see the specific conditions in which (\ref{eq42}) acts into the Lagrangian. Thus, the Lagrangian (\ref{Lag1}) becomes invariant under these transformations if $\delta L =0$. This will result in relations between the parameters $\sigma^{2}, \sigma^3$. The variation of the Lagrangian is
\begin{eqnarray*}
\delta L = \int dt\;d^{2}x\; \left[\frac{\partial \mathcal{L}}{\partial A_{\mu}}\delta A_{\mu} + \frac{\partial \mathcal{L}}{\partial (\partial_{\nu}A_{\mu})}\delta (\partial_{\nu}A_{\mu}) + \frac{\partial \mathcal{L}}{\partial (\partial_{\nu}\partial^{\mu}A_{\mu})}\delta (\partial_{\nu}\partial^{\mu}A_{\mu})\right],
\end{eqnarray*}
here we use $A_{\mu}$ instead of $\xi_{\mu}$ to more easily compare both formalisms. This, up to a total time derivative, is found to be
\begin{eqnarray}
\delta L = \int dt\;d^{2}x\; \left[\theta\epsilon^{\sigma \nu \lambda}\partial_{\nu} A_{\lambda} + \partial_{\rho}F^{\rho \sigma} - \frac{1}{2m}\epsilon^{\sigma \rho \mu}\left(\partial_{0}\partial^{0}\partial_{\rho}A_{\mu}\right) + \frac{1}{m}\epsilon^{\sigma \nu \lambda}\nabla^{2}\partial_{\nu} A_{\lambda}\right] \delta A_{\sigma} = 0.
\label{varia}
\end{eqnarray}
We can combine the first and second equations in (\ref{eq42}) to write the variation of $A_{\sigma}$ as
\begin{eqnarray}
\delta A_{\sigma} = - \delta_{\sigma}^{0}\delta \sigma^{2} + \delta_{\sigma}^{i} \partial_{i} \delta \sigma^{3},
\label{gauge1}
\end{eqnarray}
thus, by using (\ref{gauge1}) into (\ref{varia}) the variation of the action takes the form
\begin{eqnarray}
\delta L = -\int dt\;d^{2}x\;\left(\theta\epsilon^{i j}\partial_{i} A_{j} + \partial_{i}F^{i 0} - \frac{1}{2m}\epsilon^{i j}\partial_{i}\ddot{A}_{j} + \frac{1}{m}\epsilon^{i j}\nabla^{2}\partial_{i} A_{j}\right) \left(\delta \sigma^{2} + \partial_{0} \delta \sigma^{3}\right) = 0.
\end{eqnarray}
The theory will be invariant under (\ref{eq42}) if the parameters $\sigma^{i}$ obey
\begin{eqnarray}
\delta \sigma^{2} = -\partial_{0} \delta \sigma^{3},
\end{eqnarray}
hence, from (\ref{gauge1}) the gauge transformations are given by
\begin{eqnarray}
\delta A_{\mu} = \partial_{\mu} \delta \sigma^{3}.
\label{gauge}
\end{eqnarray}
Additionally, since $v_{\mu} = \dot{A}_{\mu}$, it can be seen that $\delta \sigma^{1} = \partial_{0}\partial_{0} \delta \sigma^{3}$.
\subsection{GLT formalism}
In this section we use Castellani's procedure \cite{17, 22, 23} to obtain the gauge transformations. We start this calculation with the hamiltonian (\ref{Ham1}), the constraints given in (\ref{first}), and the Dirac brackets (\ref{DiracGLT}). First, we define the gauge generator as
\begin{eqnarray}
G=\int \epsilon_{a} \gamma^{a} d^{2}x,
\end{eqnarray}
where $\epsilon_{a}$ are the gauge parameters and $a=1,2,3$. This generates infinitesimal gauge transformations on pase space variables, say $F$, through
\begin{eqnarray}
\delta F=\int \delta\epsilon_{a}(y)\left\{F(x), \gamma^{a}(y)\right\}_{D} d^{2} y.
\label{GLTgenerator}
\end{eqnarray}
In particular, the generator obeys the following equation, called the master equation,
\begin{eqnarray}
\frac{\partial}{\partial t} G+\left\{G, \mathcal{H}_{T}\right\}_{D}=0.
\end{eqnarray}
Where $\mathcal{H}_{T} = \mathcal{H} + u_{a}\gamma^{a}$ is the total hamiltonian. From the algebra of the constraints and the canonical hamiltonian $\mathcal{H}$ we can obtain the structure functions $V_{b}^{a}$, $C_{c}^{a b}$, given by
\begin{eqnarray}
\left\{\mathcal{H}, \gamma^{a}(\mathrm{x})\right\}_{D} &=& \int d^{2}y\; V_{b}^{a}(x, y) \gamma^{b}(y), \\
\left\{\gamma^{a}(x), \gamma^{b}(y)\right\}_{D} &=& \int d^{2}z\; C_{c}^{a b}(x, y, z) \gamma^{c}(z).
\end{eqnarray}
Using these, the master equation becomes
\begin{eqnarray}
\frac{d \epsilon_{a}(x)}{d t}-\int d^{2}y\; \epsilon_{b}(y) V_{a}^{b}(x, y) - \int d^{2}y\; d^{2}z\; \epsilon_{b}(y) \gamma_{c}(z) C_{a}^{c b}(x, y, z) = 0.
\end{eqnarray}
Since the only non-zero structure functions are
\begin{eqnarray}
V_{2}^{1}&=& -\delta^{2}(x-y) \quad, \quad V_{3}^{2} = -\delta^{2}(x-y), \nonumber
\end{eqnarray}
with all the $C_{c}^{a b}=0$. We obtain the following relations between the generators.
\begin{eqnarray}
\epsilon_{1} &=& \ddot{\epsilon}_{3}, \nonumber \\
\epsilon_{2} &=& -\dot{\epsilon}_{3}.
\label{relationsGLT}
\end{eqnarray}
Therefore, the generator has only one parameter and can be written as
\begin{eqnarray}
G=\int d^{2}x\; \left(\delta \ddot{\epsilon}_{3} \gamma^{1} - \delta \dot{\epsilon}_{3} \gamma^{2} + \delta \epsilon_{3} \gamma^{3}\right),
\end{eqnarray}
using (\ref{GLTgenerator}) the gauge transformations of the variables are
\begin{eqnarray}
\delta A_{0} &=& \int \delta \epsilon_{2}(y)\left[\delta^{2}(x-y)\right] d^{2}y, \nonumber \\
\delta A_{i} &=& \int \delta \epsilon_{3}(y)\left[\frac{\partial}{\partial y^{i}} \delta^{2}(x-y)\right] d^{2}y, \nonumber \\
\delta \pi^{0} &=& \int 0 d^{2}y, \nonumber \\
\delta \pi^{i} &=& \int \delta \epsilon_{1}(y)\left[\frac{1}{2m}\epsilon^{i j}\frac{\partial}{\partial x^{j}}\delta^{2}(x-y)\right] + \delta \epsilon_{3}(y)\left[-\frac{\theta}{2}\epsilon^{i j}\frac{\partial}{\partial x^{j}}\delta^{2}(x-y) - \frac{1}{2m}\epsilon^{i j}\nabla_{y}^{2}\frac{\partial}{\partial x^{j}} \delta^{2}(x-y)\right] d^{2}y, \nonumber \\
\delta v_{0} &=& \int \delta \epsilon_{1}(y)\left[-\delta^{2}(x-y)\right] d^{2}y, \nonumber \\
\delta v_{i} &=& \int \delta \epsilon_{2}(y)\left[\frac{\partial}{\partial x^{i}} \delta^{2}(x-y)\right] d^{2}y, \nonumber \\
\delta \tilde{\pi}^{0} &=& \int 0 d^{2}y, \nonumber \\
\delta \tilde{\pi}^{i} &=& \int 0 d^{2}y,
\end{eqnarray}
and by using (\ref{relationsGLT}) the following gauge transformations are found
\begin{eqnarray}
\delta A_{\mu} &=& -\partial_{\mu} \delta \epsilon_{3}, \nonumber \\
\delta \pi^{\mu} &=& \epsilon^{0 \mu j}\left(- \frac{\theta}{2} + \frac{1}{2m} - \frac{1}{2m}\nabla^{2}\right)\partial_{j} \delta \epsilon_{3}, \nonumber \\
\delta v_{\mu} &=& -\partial_{\mu} \delta \dot{\epsilon}_{3}, \nonumber \\
\delta \tilde{\pi}^{\mu} &=& 0.
\end{eqnarray}
By identifying $\sigma^{3} = - \epsilon_{3}$ both formalisms agree (see equations (\ref{gauge}) and (\ref{eq42})).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,478 |
/* -*- mode:CSharp; coding:utf-8-with-signature -*-
*/
using UnityEngine;
using System.Collections;
namespace UTJ {
public struct MyCollider
{
private static MyCollider player_;
const int POOL_BULLET_MAX = 128;
private static MyCollider[] pool_bullet_;
private static int pool_bullet_index_;
const int POOL_ENEMY_MAX = 1024;
private static MyCollider[] pool_enemy_;
private static int pool_enemy_index_;
const int POOL_ENEMY_HOMING_MAX = 1024;
private static MyCollider[] pool_enemy_homing_;
private static int pool_enemy_homing_index_;
const int POOL_ENEMY_BULLET_MAX = 2048;
private static MyCollider[] pool_enemy_bullet_;
private static int pool_enemy_bullet_index_;
private static int nearest_enemy_index_;
public static void createPool()
{
player_.alive_ = false;
player_.id_ = 0;
player_.type_ = Type.Player;
pool_bullet_ = new MyCollider[POOL_BULLET_MAX];
for (var i = 0; i < POOL_BULLET_MAX; ++i) {
pool_bullet_[i].alive_ = false;
pool_bullet_[i].id_ = i;
pool_bullet_[i].type_ = Type.Bullet;
}
pool_bullet_index_ = 0;
pool_enemy_ = new MyCollider[POOL_ENEMY_MAX];
for (var i = 0; i < POOL_ENEMY_MAX; ++i) {
pool_enemy_[i].alive_ = false;
pool_enemy_[i].id_ = i;
pool_enemy_[i].type_ = Type.Enemy;
}
pool_enemy_index_ = 0;
pool_enemy_homing_ = new MyCollider[POOL_ENEMY_HOMING_MAX];
for (var i = 0; i < POOL_ENEMY_HOMING_MAX; ++i) {
pool_enemy_homing_[i].alive_ = false;
pool_enemy_homing_[i].id_ = i;
pool_enemy_homing_[i].type_ = Type.EnemyHoming;
}
pool_enemy_homing_index_ = 0;
pool_enemy_bullet_ = new MyCollider[POOL_ENEMY_BULLET_MAX];
for (var i = 0; i < POOL_ENEMY_BULLET_MAX; ++i) {
pool_enemy_bullet_[i].alive_ = false;
pool_enemy_bullet_[i].id_ = i;
pool_enemy_bullet_[i].type_ = Type.EnemyBullet;
}
pool_enemy_bullet_index_ = 0;
}
private static void clear(ref MyCollider[] pool)
{
for (var i = 0; i < pool.Length; ++i) {
pool[i].alive_ = false;
pool[i].disabled_ = false;
pool[i].id_ = 0;
pool[i].opponent_info_.clear();
}
}
public static void restart()
{
clear(ref pool_bullet_);
clear(ref pool_enemy_);
clear(ref pool_enemy_homing_);
clear(ref pool_enemy_bullet_);
}
private static void create(ref MyCollider[] pool, ref int pool_index)
{
int cnt = 0;
while (pool[pool_index].alive_) {
++pool_index;
if (pool_index >= pool.Length)
pool_index = 0;
++cnt;
if (cnt >= pool.Length) {
Debug.LogError("EXCEED Collider POOL!");
Debug.Assert(false);
break;
}
}
pool[pool_index].alive_ = true;
pool[pool_index].disabled_ = false;
pool[pool_index].opponent_info_.clear();
pool[pool_index].phase_ = 0;
pool[pool_index].power_ = 0f; // set later
}
public static int createPlayer()
{
player_.alive_ = true;
player_.disabled_ = false;
player_.opponent_info_.clear();
player_.phase_ = 0;
player_.power_ = 0f;
return 0;
}
public static void setPowerForBullet(int id, float power)
{
Debug.Assert(pool_bullet_[id].alive_);
pool_bullet_[id].power_ = power;
}
public static int createBullet()
{
create(ref pool_bullet_, ref pool_bullet_index_);
return pool_bullet_index_;
}
public static int createEnemy()
{
create(ref pool_enemy_, ref pool_enemy_index_);
return pool_enemy_index_;
}
public static int createEnemyHoming()
{
create(ref pool_enemy_homing_, ref pool_enemy_homing_index_);
return pool_enemy_homing_index_;
}
public static int createEnemyBullet()
{
create(ref pool_enemy_bullet_, ref pool_enemy_bullet_index_);
return pool_enemy_bullet_index_;
}
public static void initSpherePlayer(int id, ref Vector3 pos, float radius)
{
player_.initSphere(ref pos, radius);
}
public static void initSphereBullet(int id, ref Vector3 pos, float radius)
{
Debug.Assert(pool_bullet_[id].alive_);
pool_bullet_[id].initSphere(ref pos, radius);
}
public static void initSphereEnemy(int id, ref Vector3 pos, float radius)
{
Debug.Assert(pool_enemy_[id].alive_);
pool_enemy_[id].initSphere(ref pos, radius);
}
public static void initSphereEnemyHoming(int id, ref Vector3 pos, float radius)
{
Debug.Assert(pool_enemy_homing_[id].alive_);
pool_enemy_homing_[id].initSphere(ref pos, radius);
}
public static void initSphereEnemyBullet(int id, ref Vector3 pos, float radius)
{
Debug.Assert(pool_enemy_bullet_[id].alive_);
pool_enemy_bullet_[id].initSphere(ref pos, radius);
}
public static void updatePlayer(int id, ref Vector3 pos)
{
player_.update(ref pos);
}
public static void updateBullet(int id, ref Vector3 pos)
{
Debug.Assert(pool_bullet_[id].alive_);
pool_bullet_[id].update(ref pos);
}
public static void updateEnemy(int id, ref Vector3 pos)
{
Debug.Assert(pool_enemy_[id].alive_);
pool_enemy_[id].update(ref pos);
}
public static void updateEnemyHoming(int id, ref Vector3 pos)
{
Debug.Assert(pool_enemy_homing_[id].alive_);
pool_enemy_homing_[id].update(ref pos);
}
public static void updateEnemyBullet(int id, ref Vector3 pos)
{
Debug.Assert(pool_enemy_bullet_[id].alive_);
pool_enemy_bullet_[id].update(ref pos);
}
public static void destroyPlayer(int id)
{
player_.alive_ = false;
player_.opponent_info_.clear();
}
public static void destroyBullet(int id)
{
Debug.Assert(pool_bullet_[id].alive_);
pool_bullet_[id].alive_ = false;
pool_bullet_[id].opponent_info_.clear();
}
public static void destroyEnemy(int id)
{
Debug.Assert(pool_enemy_[id].alive_);
pool_enemy_[id].alive_ = false;
pool_enemy_[id].opponent_info_.clear();
}
public static void destroyEnemyHoming(int id)
{
Debug.Assert(pool_enemy_homing_[id].alive_);
pool_enemy_homing_[id].alive_ = false;
pool_enemy_homing_[id].opponent_info_.clear();
}
public static void destroyEnemyBullet(int id)
{
Debug.Assert(pool_enemy_bullet_[id].alive_);
pool_enemy_bullet_[id].alive_ = false;
pool_enemy_bullet_[id].opponent_info_.clear();
}
public static void calculate()
{
// clear
player_.opponent_info_.clear();
for (var i = 0; i < pool_bullet_.Length; ++i) {
pool_bullet_[i].opponent_info_.clear();
}
for (var i = 0; i < pool_enemy_.Length; ++i) {
pool_enemy_[i].opponent_info_.clear();
}
for (var i = 0; i < pool_enemy_homing_.Length; ++i) {
pool_enemy_homing_[i].opponent_info_.clear();
}
for (var i = 0; i < pool_enemy_bullet_.Length; ++i) {
pool_enemy_bullet_[i].opponent_info_.clear();
}
// player - enemy
nearest_enemy_index_ = -1;
float nearest_dist2 = System.Single.MaxValue;
for (var i = 0; i < pool_enemy_.Length; ++i) {
if (pool_enemy_[i].alive_ && !pool_enemy_[i].disabled_) {
var dist2 = check_intersection(ref player_, ref pool_enemy_[i]);
if (dist2 < nearest_dist2) {
nearest_dist2 = dist2;
nearest_enemy_index_ = i;
}
}
}
// player - enemy_bullet
for (var i = 0; i < pool_enemy_bullet_.Length; ++i) {
if (pool_enemy_bullet_[i].alive_) {
check_intersection(ref player_, ref pool_enemy_bullet_[i]);
}
}
// enemy - bullet
for (var i = 0; i < pool_enemy_.Length; ++i) {
if (pool_enemy_[i].alive_ && !pool_enemy_[i].disabled_) {
for (var j = 0; j < pool_bullet_.Length; ++j) {
if (pool_bullet_[j].alive_) {
check_intersection(ref pool_enemy_[i], ref pool_bullet_[j]);
}
}
}
}
// enemy_homing - bullet
for (var i = 0; i < pool_enemy_homing_.Length; ++i) {
if (pool_enemy_homing_[i].alive_ && !pool_enemy_homing_[i].disabled_) {
for (var j = 0; j < pool_bullet_.Length; ++j) {
if (pool_bullet_[j].alive_ && !pool_bullet_[j].disabled_ && pool_bullet_[j].phase_ == 0) {
if (pool_enemy_homing_[i].center_.y > 0f) {
check_homing(ref pool_enemy_homing_[i], ref pool_bullet_[j]);
}
}
}
}
}
}
private static float check_intersection(ref MyCollider col0, ref MyCollider col1)
{
var diff = col1.center_ - col0.center_;
var len2 = (diff.x * diff.x +
diff.y * diff.y +
diff.z * diff.z);
var rad2 = col0.radius_+col1.radius_;
rad2 = rad2 * rad2;
if (len2 < rad2) {
float len = Mathf.Sqrt(len2);
var intersect_point = col0.center_ + (diff * (col0.radius_/len));
col0.opponent_info_.set(col1.type_, ref intersect_point, col1.power_);
col1.opponent_info_.set(col0.type_, ref intersect_point, col0.power_);
}
return rad2;
}
private static void check_homing(ref MyCollider col0, ref MyCollider col1)
{
var diff = col1.center_ - col0.center_;
var len2 = (diff.x * diff.x +
diff.y * diff.y +
diff.z * diff.z);
var rad2 = col0.radius_+col1.radius_;
rad2 = rad2 * rad2;
if (len2 < rad2) {
col0.opponent_info_.set(col1.type_, ref col1.center_, col1.power_);
++col0.phase_;
col1.opponent_info_.set(col0.type_, ref col0.center_, col0.power_);
++col1.phase_;
}
}
public static Type takeEnemyDamageForPlayer(int id, ref Vector3 pos)
{
Debug.Assert(player_.alive_);
pos = player_.opponent_info_.intersect_point_;
return player_.opponent_info_.type_;
}
public static Type getHitOpponentForBullet(int id)
{
Debug.Assert(pool_bullet_[id].alive_);
return pool_bullet_[id].opponent_info_.type_;
}
public static void getHitOpponentInfoPositionForBullet(int id, out Vector3 pos)
{
Debug.Assert(pool_bullet_[id].alive_);
pos = pool_bullet_[id].opponent_info_.intersect_point_;
}
public static Type getHitOpponentForEnemy(int id)
{
Debug.Assert(pool_enemy_[id].alive_);
return pool_enemy_[id].opponent_info_.type_;
}
public static float getHitPowerForEnemy(int id)
{
Debug.Assert(pool_enemy_[id].alive_);
return pool_enemy_[id].opponent_info_.power_;
}
public static void getIntersectPointForEnemy(int id, out Vector3 pos)
{
Debug.Assert(pool_enemy_[id].alive_);
pos = pool_enemy_[id].opponent_info_.intersect_point_;
}
public static void getIntersectPointForEnemyBullet(int id, out Vector3 pos)
{
Debug.Assert(pool_enemy_bullet_[id].alive_);
pos = pool_enemy_bullet_[id].opponent_info_.intersect_point_;
}
public static Type getHitOpponentForEnemyHoming(int id)
{
Debug.Assert(pool_enemy_homing_[id].alive_);
return pool_enemy_homing_[id].opponent_info_.type_;
}
public static Type getHitOpponentForEnemyBullet(int id)
{
Debug.Assert(pool_enemy_bullet_[id].alive_);
return pool_enemy_bullet_[id].opponent_info_.type_;
}
public static bool isDisabledBullet(int id)
{
return pool_bullet_[id].disabled_;
}
public static bool getNearestEnemyPosition(out Vector3 pos)
{
if (nearest_enemy_index_ >= 0) {
pos = pool_enemy_[nearest_enemy_index_].center_;
return true;
} else {
pos = CV.Vector3Zero;
return false;
}
}
public static void disableForBullet(int id, bool flg)
{
pool_bullet_[id].disabled_ = flg;
pool_bullet_[id].opponent_info_.clear();
}
public static void disableForEnemy(int id)
{
pool_enemy_[id].disabled_ = true;
pool_enemy_[id].opponent_info_.clear();
}
public static void disableForEnemyHoming(int id)
{
pool_enemy_homing_[id].disabled_ = true;
pool_enemy_homing_[id].opponent_info_.clear();
}
public enum Type {
None,
Player,
Bullet,
Enemy,
EnemyHoming,
EnemyBullet,
}
public enum Shape {
Sphere,
}
public struct OpponentInfo {
public Type type_;
public Vector3 intersect_point_;
public float power_;
public void clear() {
type_ = Type.None;
}
public void set(Type type, ref Vector3 pos, float power) {
type_ = type;
intersect_point_ = pos;
power_ = power;
}
}
public bool alive_;
public bool disabled_;
public int id_;
public Type type_;
public OpponentInfo opponent_info_;
public Vector3 center_;
public float radius_;
public Shape shape_;
public int phase_;
public float power_;
public void initSphere(ref Vector3 pos, float radius)
{
center_ = pos;
radius_ = radius;
shape_ = Shape.Sphere;
}
public void update(ref Vector3 pos)
{
center_ = pos;
}
}
} // namespace UTJ {
/*
* End of MyCollider.cs
*/
| {
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{"url":"https:\/\/www.mathway.com\/examples\/algebra\/functions\/determining-odd-and-even-functions?id=1063","text":"# Algebra Examples\n\nDetermine if Odd, Even, or Neither\nDetermine if the function is even.\nTo check if a function is even, substitute in for and see if the resulting function is the same as the original. In other words, .\nRemove parentheses around .\nThe function is not even because the resulting function (after substituting in ) is not the same as the original.\nis not even\nis not even\nDetermine if the function is odd.\nTo check if the function is odd, substitute in for and check if the resulting function is the opposite of original function. In other words, determine if .\nRemove parentheses around .\nThe function is not odd because does not product the opposite function of . In other words, .\nis not odd\nis not odd\nSince and , the function is not even and not odd.\nThe function is neither even nor odd\n\nWe're sorry, we were unable to process your request at this time\n\nStep-by-step work + explanations\n\u2022 \u00a0\u00a0\u00a0Step-by-step work\n\u2022 \u00a0\u00a0\u00a0Detailed explanations\n\u2022 \u00a0\u00a0\u00a0Access anywhere\nAccess the steps on both the Mathway website and mobile apps\n$--.--\/month$--.--\/year (--%)","date":"2018-02-21 15:14:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.9238423109054565, \"perplexity\": 1282.493733661455}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-09\/segments\/1518891813626.7\/warc\/CC-MAIN-20180221143216-20180221163216-00093.warc.gz\"}"} | null | null |
Paryphoconus fittkaui är en tvåvingeart som beskrevs av Gustavo R. Spinelli och Wirth 1984. Paryphoconus fittkaui ingår i släktet Paryphoconus och familjen svidknott. Inga underarter finns listade i Catalogue of Life.
Källor
Svidknott
fittkaui | {
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Q: Need a help in design approach in android app i'm developing app for school management where their are three different menus principal,teacher,student .
here i had done it by keeping all screens and code in one app and displaying respective screens according to login details.by this size of app increased and when teacher is login he don't want the screens of principal and student.
is this a correct way to represent respective screens by using login details or is their another approach like downloading java project form server based on login details?
A: You should be developing 3 different apps.
Mobiles are personal devices so a student will not be using teachers module at any given point of time. and same goes for other roles as well.
A: You can provide an option for selecting the user at starting screen and proceed with the screens for the selected user.Eg.If user is a teacher,you can show login screen for teacher and likewise
A: Honestly I'm not sure this question belongs to SO.
If anything this is purely a matter of opinion;
If the authentication process itself is secure enough and the methods it used are also safe, it should be no problem to have them all in one app.
| {
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{"url":"https:\/\/istopdeath.com\/identify-the-sequence-1-2-3\/","text":"# Identify the Sequence 1 , 2 , 3\n\n, ,\nThis is an arithmetic sequence since there is a common difference between each term. In this case, adding to the previous term in the sequence gives the next term. In other words, .\nArithmetic Sequence:\nThis is the formula of an arithmetic sequence.\nSubstitute in the values of and .\nMultiply by .\nCombine the opposite terms in .\nSubtract from .","date":"2022-12-01 00:08:16","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\": 0.8158842921257019, \"perplexity\": 514.0852652325161}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446710777.20\/warc\/CC-MAIN-20221130225142-20221201015142-00437.warc.gz\"}"} | null | null |
<?xml version="1.0"?>
<Textures>{% load hdscales %}
<Texture filename="{{texture.trimmedName}}" dynamic_load="1">{% for sprite in allSprites %}
<Image name="_{{sprite.trimmedName}}" x="{{sprite.frameRect.x|hdscales}}" y="{{sprite.frameRect.y|hdscales}}" w="{{sprite.frameRect.width|hdscales}}" h="{{sprite.frameRect.height|hdscales}}"{% if sprite.rotated %} vertical="1"{% endif %}/>{% endfor %}
</Texture>
{% for sprite in allSprites %}
<CompositeImage name="{{texture.trimmedName}}/{{sprite.trimmedName}}" w="{{sprite.untrimmedSize.width|hdscales}}" h="{{sprite.untrimmedSize.height|hdscales}}"><ImageRef name="{{texture.trimmedName}}/_{{sprite.trimmedName}}" x="{{sprite.sourceRect.x|hdscales}}" y="{{sprite.sourceRect.y|hdscales}}" w="{{sprite.sourceRect.width|hdscales}}" h="{{sprite.sourceRect.height|hdscales}}"/></CompositeImage>{% endfor %}
</Textures>
| {
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È apparso in molti ruoli, sia come protagonista che come non protagonista, in film come Serpico (1973), Tutti gli uomini del presidente (1976), Scarface (1983), Il nome della rosa (1986), Last Action Hero - L'ultimo grande eroe (1993), Star Trek - L'insurrezione (1998), Scoprendo Forrester (2000), A proposito di Davis (2013) e Grand Budapest Hotel (2014). Inoltre è stato un membro del cast regolare, tra la terza e la sesta stagione, della serie televisiva Homeland - Caccia alla spia.
Biografia
Abraham è nato a Pittsburgh, in Pennsylvania, il 24 ottobre 1939, figlio di Frederick Abraham, un meccanico siriano di etnia assira e di religione siro-ortodossa, emigrato negli Stati Uniti nel corso degli anni venti, e di Josephine Stello, una casalinga statunitense, figlia a sua volta di immigrati italiani originari della provincia di Reggio Calabria, Bruno Stilo di Condofuri e Caterina "Katherine" Cosmano di Staiti. È cresciuto ad El Paso (in Texas). L'attore si chiama semplicemente "Murray Abraham", ma è sempre stato accreditato come "F. Murray Abraham". Per anni si è speculato sul significato della "F", tanto che sul famoso sito IMDb, è stato rivelato che la "F", sta per "Fahrid", ma l'attore interpellato sull'argomento ha negato il tutto, dichiarando che la "F", non è parte del suo nome, ma si fa accreditare così, in onore del padre che si chiamava "Frederick".
Abraham ha raggiunto grande notorietà in Italia per aver interpretato i ruoli di Jacopo nel Marco Polo (1982) di Giuliano Montaldo e dell'Innominato ne I Promessi Sposi (1989) di Salvatore Nocita. Ha esordito al cinema con una piccola parte in They Might Be Giants (1971) di Anthony Harvey, con protagonisti Joanne Woodward e George C. Scott. In seguito ha preso parte a film di vario genere, come I ragazzi irresistibili (1975) di Herbert Ross, Tutti gli uomini del presidente (1976) di Alan J. Pakula e Il vizietto americano (1976) di Richard Lester, ma raramente nel ruolo di protagonista e spesso in quello del villain, come Omar Suarez, il trafficante di droga in Scarface (1983) di Brian De Palma.
Grazie alla sua interpretazione di Antonio Salieri nel film Amadeus (1984) di Miloš Forman si è aggiudicato l'Oscar al miglior attore protagonista. Ha in seguito interpretato il terribile inquisitore Bernardo Gui ne Il nome della rosa (1986) di Jean-Jacques Annaud. Nel 1989 ha recitato nel film Un uomo innocente di Peter Yates, al fianco di Tom Selleck, dove interpreta un ergastolano detenuto in un carcere di massima sicurezza. Tuttavia, nonostante il successo e il premio Oscar, in campo hollywoodiano ha proseguito sporadicamente la propria carriera, privilegiando invece il teatro e preferendo produzioni più modeste, soprattutto negli anni della maturità.
In Star Trek - L'insurrezione (1998) di Jonathan Frakes ha interpretato Ahdar Ru'afo, l'antagonista principale del film. A partire dai primi anni duemila, ha stretto un sodalizio artistico con il regista italiano Renzo Martinelli, con cui ha girato cinque film: Piazza delle Cinque Lune (2003), Il mercante di pietre (2006), Carnera - The Walking Mountain (2008), Barbarossa (2009) e 11 settembre 1683 (2013).
Vita privata
È sposato dal 1962 con Kate Hannan e ha due figli: Jamili e Mick.
Filmografia
Attore
Cinema
They Might Be Giants, regia di Anthony Harvey (1971)
Serpico, regia di Sidney Lumet (1973) - non accreditato
Prigioniero della seconda strada (The Prisoner of Second Avenue), regia di Melvin Frank (1975)
I ragazzi irresistibili (The Sunshine Boys), regia di Herbert Ross (1975)
Il vizietto americano (The Ritz), regia di Richard Lester (1976)
Tutti gli uomini del presidente (All the President's Men), regia di Alan J. Pakula (1976)
Moses Wine detective (The Big Fix), regia di Jeremy Kagan (1978)
Madman, regia di Dan Cohen (1978)
Scarface, regia di Brian De Palma (1983)
Amadeus, regia di Miloš Forman (1984)
Il nome della rosa (Der Name Der Rose), regia di Jean-Jacques Annaud (1986)
Russicum - I giorni del diavolo, regia di Pasquale Squitieri (1988)
Slipstream, regia di Steven Lisberger (1989)
Tramonto di un eroe (Beyond the Stars), regia di David Saperstein (1989)
La favorita (The Favorite), regia di Jack Smight (1989)
Un uomo innocente (An Innocent Man), regia di Peter Yates (1989)
Performance Pieces, regia di Tom Abrams - cortometraggio (1989)
Uomini al passo (Cadence), regia di Martin Sheen (1991) - cameo non accreditato
La battaglia dei tre tamburi di fuoco (La batalla de los Tres Reyes), regia di Souheil Ben-Barka e Uchkun Nazarov (1990)
Il falò delle vanità (The Bonfire of the Vanities), regia di Brian De Palma (1990) - cameo non accreditato
Money - Intrigo in nove mosse (Money), regia di Steven Hilliard Stern (1991)
L'impero del crimine (Mobsters), regia di Michael Karbelnikoff (1991)
Eye of the Widow, regia di Andrew V. McLaglen (1991)
Sfida d'onore (By the Sword), regia di Jeremy Paul Kagan (1991)
Palle in canna (Loaded Weapon 1), regia di Gene Quintano (1993)
Last Action Hero - L'ultimo grande eroe (Last Action Hero), regia di John McTiernan (1993)
Sweet Killing, regia di Eddy Matalon (1993)
Fresh, regia di Boaz Yakin (1994)
L'affaire, regia di Sergio Gobbi (1994)
Sopravvivere al gioco (Surviving the Game), regia di Ernest R. Dickerson (1994)
Nostradamus regia di Roger Christian (1994)
Jamila regia di Monica Teuber (1994)
Dillinger and Capone regia di Jon Purdy (1995)
La dea dell'amore (Mighty Aphrodite), regia di Woody Allen (1995)
Figli della rivoluzione (Children of the Revolution), regia di Peter Duncan (1996)
Baby Face Nelson, regia di Scott P. Levy (1996)
Riccardo III - Un uomo, un re (Looking For Richard), regia di Al Pacino (1996)
Mimic, regia di Guillermo del Toro (1997)
Una vacanza all'inferno, regia di Tonino Valerii (1997)
Eruption, regia di Gwyneth Gibby (1997)
Star Trek - L'insurrezione (Star Trek: Insurrection), regia di Jonathan Frakes (1998)
I Muppets venuti dallo spazio (Muppets from Space), regia di Tim Hill (1999)
The All New Adventures of Laurel & Hardy in For Love or Mummy, regia di John R. Cherry III e Larry Harmon (1999)
Scoprendo Forrester (Finding Forrester), regia di Gus Van Sant (2000)
I cavalieri che fecero l'impresa, regia di Pupi Avati (2001)
I tredici spettri (Thir13en Ghosts), regia di Steve Beck (2001)
Joshua, regia di Jon Purdy (2002)
Piazza delle Cinque Lune, regia di Renzo Martinelli (2003)
My Father, regia di Egidio Eronico (2003)
Another Way of Seeing Things, regia di Cory Taylor - cortometraggio (2004)
Peperoni ripieni e pesci in faccia, regia di Lina Wertmüller (2004)
Il ponte di San Luis Rey (The Bridge of San Luis Rey), regia di Mary McGuckian (2004)
Il mercante di pietre, regia di Renzo Martinelli (2006)
Come le formiche, regia di Ilaria Borrelli (2007)
Bloodmonkey - Le scimmie assassine (Bloodmonkey), regia di Robert Young (2007)
Carnera - The Walking Mountain, regia di Renzo Martinelli (2008)
A House Divided, regia di Mitch Davis (2008)
Perestroika, regia di Slava Tsukerman (2009)
Barbarossa, regia di Renzo Martinelli (2009)
The Unseen World, regia di Liana Marabini (2010)
11 settembre 1683, regia di Renzo Martinelli (2012)
Goltzius and the Pelican Company, regia di Peter Greenaway (2012)
Dead Man Down - Il sapore della vendetta (Dead Man Down), regia di Niels Arden Oplev (2013)
A proposito di Davis (Inside Llewyn Davis), regia di Joel ed Ethan Coen (2013)
Ti ho cercata in tutti i necrologi (I Looked in Obituaries), regia di Giancarlo Giannini (2013)
Grand Budapest Hotel (The Grand Budapest Hotel), regia di Wes Anderson (2014)
Il mistero di Dante, regia di Louis Nero (2014)
Max nella città degli scacchi (A Little Game), regia di Evan Oppenheimer (2014)
Robin Hood - L'origine della leggenda (Robin Hood), regia di Otto Bathurst (2018)
Lilli e il vagabondo (Lady and the Tramp), regia di Charlie Bean (2019)
L'apparenza delle cose (Things Heard & Seen), regia di Shari Springer Berman e Robert Pulcini (2021)
The Magic Flute, regia di Florian Sigl (2022)
Televisione
Nightside, regia di Richard Donner - film TV (1973)
How to Survive a Marriage - serie TV (1974)
Kojak - serie TV, 2 episodi (1975-1977)
Arcibaldo (All in the Family) - serie TV, 1 episodio (1976)
Mike Andros (The Andros Targets) - serie TV, 1 episodio (1977)
A.E.S. Hudson Street - serie TV, 1 episodio (1977)
Sex and the Married Woman, regia di Jack Arnold - film TV (1977)
Marco Polo - sceneggiato TV, 6 puntate (1982)
Dream West - miniserie TV (1986)
I promessi sposi, regia di Salvatore Nocita - sceneggiato TV (1989)
Eppur si muove!, regia di Ivo Barnabò Micheli - film TV (1989)
La primavera di Michelangelo (A Season of Giants), regia di Jerry London - film TV (1989)
Largo Desolato, regia di Jiri Zizka - film TV (1990)
Il flauto magico (Die Zauberflöte), regia di Brian Large - film TV (1991)
The First Circle, regia di Sheldon Larry - film TV (1992)
Viaggio al centro della Terra, regia di William Dear - film TV (1993)
Il caso Dozier, regia di Carlo Lizzani - film TV (1994)
Dead Man's Walk - miniserie TV (1996)
Color of Justice, regia di Jeremy Kagan - film TV (1997)
I giudici - Excellent Cadavers, regia di Ricky Tognazzi - film TV (1999)
L'arca di Noè (Noah's Ark), regia di John Irvin - film TV (1999)
Ester (Esther), regia di Raffaele Mertes - film TV (1999)
The Darkling - Il lato oscuro dell'anima (The Darkling), regia di Po-Chi Leong - film TV (2000)
Un dono semplice, regia di Maurizio Zaccaro – film TV (2000)
Dead Lawyers, regia di Paris Barclay - film TV (2004)
Il placido Don (And Quiet Flows the Don) - miniserie TV (2006)
L'inchiesta, regia di Giulio Base - miniserie TV (2006)
Shark Swarm - Squali all'attacco (Shark Swarm), regia di James A. Contner - film TV (2008)
Saving Grace – serie TV, episodio 3x04 (2009)
Law & Order: Criminal Intent – serie TV, episodio 9x16 (2010)
Bored to Death - Investigatore per noia (Bored to Death) - serie TV, 1 episodio (2010)
The Good Wife – serie TV, 4 episodi (2011-2014)
Louie – serie TV, episodi 2x13-3x08-4x12 (2011-2014)
Beauty and the Beast, regia di Yves Simoneau - film TV (2012)
Blue Bloods - serie TV, 1 episodio (2012)
Homeland - Caccia alla spia (Homeland) - serie TV, 43 episodi (2012-2018)
Elementary - serie TV, 1 episodio (2013)
Do No Harm - serie TV, 1 episodio (2013)
Inside Amy Schumer - serie TV, 1 episodio (2016)
Taxi 22 - serie TV (2016)
Curb Your Enthusiasm - serie TV, 1 episodio (2017)
The Good Fight - serie TV, 1 episodio (2018)
The Orville - serie TV, 1 episodio (2019)
Chimerica - serie TV, 4 episodi (2019)
Mythic Quest - serie TV (2020-in corso)
Cabinet of Curiosities - serie TV (2022)
The White Lotus – serie TV, 7 episodi (2022)
Doppiatore
The Little Match Girl, regia di Michael Sporn (1990) - narratore
Through an Open Window, regia di Eric Mendelsohn - cortometraggio (1992)
David Proshker, regia di Larry Eisenberg - cortometraggio (2000)
L'ultimo giorno di Pompei (Pompeii: The Last Day), regia di Peter Nicholson (2003) - narratore nella versione in inglese
411, regia di Oliver Power - cortometraggio (2015)
L'isola dei cani (Isle of Dogs), regia di Wes Anderson (2018)
Dragon Trainer - Il mondo nascosto (How to Train Your Dragon: The Hidden World), regia di Dean DeBlois (2019)
Moon Knight - miniserie TV (2022)
Teatro
The Fantasticks, Off Broadway (1967)
The Ritz, Broadway (1975–76)
Legend (1976)
Landscape of the Body Off Broadway (1977)
Zio Vanja (Zio Vanya) Off Broadway (1983–84)
La dodicesima notte (Twelfth Night) Off Broadway (1986)
Teibele and Her Demon (1979–80)
Macbeth (1986–87)
Aspettando Godot (Waiting for Godot) Off Broadway (1988)
A Life in the Theatre, Off Broadway (1992)
Angels in America - Fantasia gay su temi nazionali (Angels in America) Broadway (1994)
Un mese in campagna (A Month in the Country) Broadway (1995)
Tolstoj, Bath e Plymouth (1995-1996)
Re Lear (King Lear) Off Broadway (1996)
Triumph of Love (1997–98)
Mauritius (2007)
Il mercante di Venezia (The Merchant of Venice) Stratford (2007)
L'ebreo di Malta (The Jew of Malta) Off Broadway (2007)
Almost an Evening, Off Broadway (2008)
Il mercante di Venezia (The Merchant of Venice) Off Broadway (2011)
Golden Age, Off Broadway (2012)
It's Only a Play (2014–15)
Nathan the Wise, Off Broadway (2016)
The Mentor, Londra (2017)
Riconoscimenti
Premio Oscar
1985 – Miglior attore protagonista per Amadeus
Golden Globe
1985 – Miglior attore in un film drammatico per Amadeus
2023 – Candidatura alla miglior attore in una miniserie o film televisivo per The White Lotus
British Academy Film Awards
1986 – Candidatura per il miglior attore protagonista per Amadeus
Los Angeles Film Critics Association Awards
1984 – Miglior attore per Amadeus (ex aequo con Albert Finney per Sotto il vulcano)
Kansas City Film Critics Circle Awards
1985 – Miglior attore per Amadeus
Detroit Film Critics Society Awards
2014 – Miglior cast per Grand Budapest Hotel (condiviso con il resto del cast)
Florida Film Critics Circle Awards
2014 – Miglior cast per Grand Budapest Hotel (condiviso con il resto del cast)
Phoenix Film Critics Society Awards
2014 – Candidatura per il miglior cast per Grand Budapest Hotel (condivisa con il resto del cast)
San Diego Film Critics Society Awards
2014 – Candidatura per il miglior cast per Grand Budapest Hotel (condivisa con il resto del cast)
Screen Actors Guild Award
2014 – Candidatura per il miglior cast cinematografico per Grand Budapest Hotel (condivisa con il resto del cast)
Southeastern Film Critics Association Awards
2014 – Miglior cast per Grand Budapest Hotel (condiviso con il resto del cast)
Washington DC Area Film Critics Association Awards
2014 – Candidatura per il miglior cast per Grand Budapest Hotel (condivisa con il resto del cast)
Central Ohio Film Critics Association Awards
2015 – Miglior cast per Grand Budapest Hotel (condiviso con il resto del cast)
Georgia Film Critics Association
2015 – Miglior cast per Grand Budapest Hotel (condiviso con il resto del cast)
Gold Derby Awards
2015 – Candidatura per il miglior cast per Grand Budapest Hotel (condivisa con il resto del cast)
2020 – Candidatura per il cast del decennio per Grand Budapest Hotel (condivisa con il resto del cast)
Screen Actors Guild Award
2014 – Candidatura per il miglior cast in una serie drammatica per Homeland – Caccia alla spia (condivisa con il resto del cast)
2016 – Candidatura per il miglior cast in una serie drammatica per Homeland – Caccia alla spia (condivisa con il resto del cast)
Online Film & Television Association
2015 – Candidatura per il miglior attore ospite in una serie drammatica per Homeland – Caccia alla spia
Primetime Emmy Awards
2015 – Candidatura per il miglior attore ospite in una serie drammatica per Homeland – Caccia alla spia
2018 – Candidatura per il miglior attore ospite in una serie drammatica per Homeland – Caccia alla spia (episodio All In)
Drama Desk Award
1980 – Candidatura per il miglior attore in un'opera teatrale per Teibele and Her Demon
1992 – Candidatura per il miglior attore in un'opera teatrale per A Life in the Theater
2015 – Candidatura per il miglior attore in un'opera teatrale per It's Only a Play
Doppiatori italiani
Nelle versioni in italiano delle opere in cui ha recitato, F. Murray Abraham è stato doppiato da:
Dario Penne in Scoprendo Forrester, Piazza delle Cinque Lune, Il mercante di pietre, Saving Grace, Carnera - The Walking Mountain, Barbarossa, The Good Wife, Blue Bloods, 11 Settembre 1683, Dead Man Down - Il sapore della vendetta, Ti ho cercata in tutti i necrologi, Do No Harm, Il mistero di Dante, The Good Fight
Michele Kalamera in Amadeus, Dead Man's Walk, Color of Justice, Il ponte di San Luis Rey, Bored to Death - Investigatore per noia, Louie, Homeland - Caccia alla spia, Elementary, Grand Budapest Hotel, Max nella città degli scacchi
Ennio Coltorti in Figli della rivoluzione, Riccardo III - Un uomo, un re, Mimic, I giudici - Excellent Cadavers, Peperoni ripieni e pesci in faccia, L'inchiesta, A proposito di Davis, The Orville
Sandro Iovino in Un uomo innocente, Uomini al passo, Last Action Hero - L'ultimo grande eroe, Sopravvivere al gioco, Viaggio al centro della Terra, Ester, Bloodmonkey - Le scimmie assassine
Michele Gammino in Star Trek - L'insurrezione, L'apparenza delle cose, Cabinet of Curiosities, The White Lotus
Pietro Biondi in L'impero del crimine, Homeland - Caccia alla spia (ep. 3x11-3x12)
Luciano De Ambrosis in Prigioniero della seconda strada, L'arca di Noè
Luca Biagini in Amadeus (ridoppiaggio), My Father
Pino Colizzi ne I promessi sposi
Angelo Nicotra in Scarface
Rodolfo Bianchi in Kojak (ep. 3x01)
Dario Mazzoli in Kojak (ep. 4x15)
Sandro Sardone ne Il falò delle vanità
Oreste Rizzini in La dea dell'amore
Ferruccio Amendola in Marco Polo
Renato Cortesi in Palle in canna
Diego Reggente in Magma
Pino Ammendola ne Il vizietto americano
Sergio Rossi ne Il nome della rosa
Sergio Graziani in Russicum - I giorni del diavolo
Leslie La Penna ne Il caso Dozier
Vittorio Di Prima in Un dono semplice
Omero Antonutti ne I cavalieri che fecero l'impresa
Massimo Pizzirani ne I Muppets venuti dallo spazio
Paolo Buglioni in La primavera di Michelangelo
Giorgio Lopez ne I tredici spettri
Stefano De Sando in Come le formiche
Franco Zucca in Shark Swarm - Squali all'attacco
Domenico Brioschi in Law & Order: Criminal Intent
Angelo Maggi in Robin Hood - L'origine della leggenda
Elio Zamuto in Lilli e il vagabondo
Paolo Marchese in Mythic Quest
Da doppiatore è sostituito da:
Michele Kalamera in L'isola dei cani
Massimo Lodolo in Dragon Trainer - Il mondo nascosto
Ennio Coltorti in Moon Knight
Note
Altri progetti
Collegamenti esterni
Attori teatrali statunitensi
Attori televisivi statunitensi
Attori italoamericani
Golden Globe per il miglior attore in un film drammatico
Membri della Royal Shakespeare Company | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,790 |
{"url":"https:\/\/www.techwhiff.com\/issue\/an-athelete-runs-some-distance-before-taking-a-long--196048","text":"An athelete runs some distance before taking a long jump because : give answer\n\nQuestion:\n\nAn athelete runs some distance before taking a long jump because : give answer\n\nSenath Company's annual report reveals net credit sales of $266,000 and average accounts receivable of$46,000. The report also shows an average inventory balance of $16,500 and cost of goods of$265,000. Based on this information (treat any partial day as a whole day) :_________ a. the average number of days to collect receivables is 6. b. the average number of days to collect receivables is 64. c. the accounts receivable turnover is 16. d. the accounts receivable turnover is 15.\n\nSenath Company's annual report reveals net credit sales of $266,000 and average accounts receivable of$46,000. The report also shows an average inventory balance of $16,500 and cost of goods of$265,000. Based on this information (treat any partial day as a whole day) :_________ a. the average numb...\n\n2. In paragraphs 49-56, what causes the conflict between the narrator and her mother? [RL.3] A. The mother disapproves of the narrator using post-it notes to cheat on the test. B. The mother disapproves of her relationship with Cat because of its negative influence. C. The mother disapproves of her relationship with Cat because Cat doesn't take the SAT seriously. D. The mother disapproves of the narrator spending all of her time on art projects instead of studying.\n\n2. In paragraphs 49-56, what causes the conflict between the narrator and her mother? [RL.3] A. The mother disapproves of the narrator using post-it notes to cheat on the test. B. The mother disapproves of her relationship with Cat because of its negative influence. C. The mother disapproves of her ...\n\nWhat was some bad things about Giuseppe Garibaldi? I have a debate to prepare for and I have to prove he wasn\u2019t a good leader.\n\nWhat was some bad things about Giuseppe Garibaldi? I have a debate to prepare for and I have to prove he wasn\u2019t a good leader....\n\nA(n) ___________________ refers to an explanation of a poem that takes into consideration the words, punctuation, and visual representation of the poem. a. alliteration c. explication b. metaphor d. assonance\n\nA(n) ___________________ refers to an explanation of a poem that takes into consideration the words, punctuation, and visual representation of the poem. a. alliteration c. explication b. metaphor d. assonance...\n\nWhich 2 countries opposed each other during the cold war\n\nwhich 2 countries opposed each other during the cold war...\n\nWrite any two factors that affect the efficiency of machine\u200b\n\nWrite any two factors that affect the efficiency of machine\u200b...\n\nDo you think it was difficult for Ponyboy to change his hairstyle? Why or why not?\n\nDo you think it was difficult for Ponyboy to change his hairstyle? Why or why not?...\n\nWhich of the following is a solution to the equation x - 5\/7 = -3\/7 A.2\/7 B.-2\/7 C.2 D.1 1\/7\n\nWhich of the following is a solution to the equation x - 5\/7 = -3\/7 A.2\/7 B.-2\/7 C.2 D.1 1\/7...\n\nWhat can you use deductive rules to\n\nWhat can you use deductive rules to...\n\nWhat should you do when you\u2019re asked to contrast two stories? Summarize each of the story. Find differences in the two stories. Review key events in the stories. Talk about how the two stories are alike.\n\nWhat should you do when you\u2019re asked to contrast two stories? Summarize each of the story. Find differences in the two stories. Review key events in the stories. Talk about how the two stories are alike....\n\n1. When coming to a stop behind a vehicle in city traffic, you should be how far back? (1 point) 5 to 10 feet 10 to 20 feet 20 to 30 feet 40 to 45 feet 2. Entrances to expressways (1 point) allow you to enter traffic at speeds less than 25 mph. allow you to enter traffic at speeds between 25 and 45 mph. are usually in the left lane. are usually used for carpools. 3. A strategy for managing time in urban areas is to (1 point) always wear a watch. use jackrabbit starts when traffic signals first\n\n1. When coming to a stop behind a vehicle in city traffic, you should be how far back? (1 point) 5 to 10 feet 10 to 20 feet 20 to 30 feet 40 to 45 feet 2. Entrances to expressways (1 point) allow you to enter traffic at speeds less than 25 mph. allow you to enter traffic at speeds between 25 and 45 ...\n\nA mass of 13.9 kg bounces up and down from a spring with constant 9.3 N\/m. Toward the bottom of its motion the mass dips into a pool of water and comes back out. The wave created by this process travels away at 5 m\/s. What is the associated wavelength of this water wave measured in meters?\n\nA mass of 13.9 kg bounces up and down from a spring with constant 9.3 N\/m. Toward the bottom of its motion the mass dips into a pool of water and comes back out. The wave created by this process travels away at 5 m\/s. What is the associated wavelength of this water wave measured in meters?...\n\nHow can i find x here 2-2x[5+4]\n\nhow can i find x here 2-2x[5+4]...\n\nHow did the Black Death change the European economy? by depleting the continent\u2019s wealth and turning most people back to subsistence farming by creating a labor shortage that pushed the continent toward a market economy by damaging crops and making the continent dependent on imported food and textiles by shutting down trade routes and isolating the continent\n\nHow did the Black Death change the European economy? by depleting the continent\u2019s wealth and turning most people back to subsistence farming by creating a labor shortage that pushed the continent toward a market economy by damaging crops and making the continent dependent on imported food and tex...\n\nFor the snowy tree cricket,if n is the number of chirps per minute what is the number of chirps in 14 seconds\n\nFor the snowy tree cricket,if n is the number of chirps per minute what is the number of chirps in 14 seconds...","date":"2022-11-29 07:46:02","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\": 0.23675599694252014, \"perplexity\": 2382.1509885051664}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446710690.85\/warc\/CC-MAIN-20221129064123-20221129094123-00147.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.clutchprep.com\/physics\/practice-problems\/138309\/a-parallel-plate-capacitor-is-formed-from-two-9-1-cm-diameter-electrodes-spaced-","text":"Parallel Plate Capacitors Video Lessons\n\nConcept\n\n# Problem: A parallel-plate capacitor is formed from two 9.1 cm-diameter electrodes spaced 1.1 mm apart. The electric field strength inside the capacitor is 6.0\u00d7106 N\/C. What is the charge (in nC) on each electrode?\n\n###### FREE Expert Solution\n\nCharge:\n\n$\\overline{){\\mathbf{Q}}{\\mathbf{=}}{\\mathbit{E}}{\\mathbit{A}}{{\\mathbit{\\epsilon }}}_{{\\mathbf{0}}}}$\n\nA = \u03c0r2 = (\u03c0)(d\/2)2 = (\u03c0)[(9.1 \u00d7 10-2)\/2]2 = 6.5 \u00d7 10-3\u00a0 m2\n\n90% (371 ratings)\n###### Problem Details\n\nA parallel-plate capacitor is formed from two 9.1 cm-diameter electrodes spaced 1.1 mm apart. The electric field strength inside the capacitor is 6.0\u00d7106 N\/C. What is the charge (in nC) on each electrode?","date":"2020-11-25 10:51:31","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 1, \"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\": 0.5741965174674988, \"perplexity\": 6845.940520863146}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-50\/segments\/1606141182776.11\/warc\/CC-MAIN-20201125100409-20201125130409-00483.warc.gz\"}"} | null | null |
\section{Introduction}
Graph neural networks (GNNs), and its numerous variants, have shown to be successful in graph representation learning by extracting high-level features for nodes from their topological neighborhoods. GNNs have boosted the state-of-the-art performance in a variety of graph analytic tasks, such as semi-supervised node classification and link prediction \citep{kipf2016semi, kipf2016variational,hasanzadeh2019sigvae, hajiramezanali2019vgrnn}.
Despite their successes, GNNs have two major limitations: 1) they cannot go very deep due to \textit{over-smoothing} and \textit{over-fitting} phenomena \citep{li2018deeper, kipf2016semi}; 2) the current implementations of GNNs do not provide uncertainty quantification~(UQ) of output predictions.
There exist a variety of methods to address these problems.
For example, DropOut \citep{srivastava2014dropout} is %
a popular regularisation technique with deep neural networks~(DNNs) to avoid over-fitting, where network units are randomly masked during training. In GNNs, %
DropOut is realized by randomly removing the node features during training \citep{rong2019dropedge}. Often, the procedure is independent of the graph topology. However, empirical results have shown that, due to the nature of Laplacian smoothing in GNNs, graph convolutions have the over-smoothing tendency of mixing representations of adjacent nodes so that, when increasing the number of GNN layers, all nodes' representations will converge to a stationary point, making them unrelated to node features
\citep{li2018deeper}. While it has been shown in~\citet{kipf2016semi} that DropOut alone is ineffectual in preventing over-fitting, partially due to over-smoothing, the combination of DropEdge, in which a set of edges are randomly removed from the graph, with DropOut has recently shown potential to alleviate these problems \citep{rong2019dropedge}.
On the other hand, with the development of efficient posterior computation algorithms, there have been successes in learning with uncertainty by Bayesian extensions of traditional deep network architectures, including convolutional neural networks~(CNNs). However, for GNNs, deriving their Bayesian extensions is more challenging due to their irregular neighborhood connection structures.
In order to account for uncertainty in GNNs, \citet{zhang2019bayesian} present a Bayesian framework where the observed graph is viewed as a realization from a parametric family of random graphs. This allows joint inference of the graph and the GNN weights, leading to resilience to noise or adversarial attacks. Besides its prohibitive computational cost, the choice of the random graph model is important and can be inconsistent for different problems and datasets. Furthermore, the posterior inference in the current implementation only depends on the graph topology, but cannot consider node features.
In this paper, we introduce a general stochastic regularization technique for GNNs by adaptive connection sampling---Graph DropConnect (GDC). We show that existing GNN regularization techniques such as DropOut \citep{srivastava2014dropout}, DropEdge \citep{rong2019dropedge}, and node sampling \citep{chen2018fastgcn} are special cases of GDC. GDC regularizes neighborhood aggregation in GNNs at each channel, separately. This prevents connected nodes in graph from having the same learned representations in GNN layers; hence better improvement without serious over-smoothing can be achieved. Furthermore, adaptively learning the connection sampling or drop rate in GDC enables better stochastic regularization given graph data for target graph analytic tasks. In fact, our ablation studies show that only learning the DropEdge rate, without any DropOut, already substantially improves the performance in semi-supervised node classification with GNNs. By probabilistic modeling of the \emph{connection} drop rate, we propose a hierarchical beta-Bernoulli construction for Bayesian learnable GDC, and derive the solution with both continuous relaxation and direct optimization with Augment-REINFORCE-Merge~(ARM) gradient estimates. With the naturally enabled UQ and regularization capability, our learnable GDC can help address both over-smoothing and UQ challenges to further push the frontier of GNN research.
We further prove that adaptive connection sampling of GDC at each channel can be considered as random aggregation and diffusion in GNNs, with a similar Bayesian approximation interpretation as in Bayesian DropOut for CNNs \citep{gal2015bayesian}.
Specifically, Monte Carlo estimation of GNN outputs can be used to evaluate the predictive posterior uncertainty. An important corollary of this formulation is that any GNN with neighborhood sampling, such as GraphSAGE \citep{hamilton2017inductive}, could be considered as its corresponding Bayesian approximation.%
\section{Preliminaries}
\subsection{Bayesian Neural Networks}
Bayesian neural networks (BNNs) %
aim to capture model uncertainty of DNNs by placing prior distributions over the model parameters %
to enable posterior updates during DNN training. It has been shown that these Bayesian extensions of traditional DNNs can be robust to over-fitting and provide appropriate prediction uncertainty estimation \citep{gal2016dropout,boluki2020learnable}.
Often, the standard Gaussian prior distribution is placed over the weights.
With random weights $\{\mathbf{W}^{(l)}\}_{l=1}^L$, the output prediction given an input $\mathbf{x}$ can be denoted by $\widehat{\mathbf{f}} \big( \mathbf{x}, \{\mathbf{W}^{(l)}\}_{l=1}^L \big)$, which is now a random variable in BNNs, enabling uncertainty quantification (UQ).
The key difficulty in using BNNs is that Bayesian inference is computationally intractable. There exist various methods that approximate BNN inference, such as Laplace approximation \citep{mackay1992bayesian}, sampling-based and stochastic variational inference \citep{paisley2012variational,rezende2014stochastic,hajiramezanali2020semi,dadaneh2020pairwise}, Markov chain Monte Carlo (MCMC) \citep{neal2012bayesian}, and stochastic gradient MCMC \citep{ma2015complete}.
However, their computational cost is still much higher than the non-Bayesian methods, due to the increased model complexity and slow convergence \citep{gal2016dropout}.%
\subsection{DropOut as Bayesian Approximation}
Dropout is commonly used in training many deep learning models as a way to avoid over-fitting. Using dropout at test time enables UQ with Bayesian interpretation of the network outputs as Monte Carlo samples of its predictive distribution \citep{gal2016dropout}. %
Various dropout methods have been proposed to multiply the output of each neuron by a random mask drawn from a desired distribution, such as Bernoulli \citep{hinton2012improving, srivastava2014dropout} and Gaussian \citep{kingma2015variational,srivastava2014dropout}. Bernoulli dropout and its extensions are the most commonly used in practice due to their ease of implementation and computational efficiency in existing deep architectures.
\subsection{Over-smoothing \& Over-fitting in GNNs}
It has been shown that graph convolution in graph convolutional neural networks (GCNs) \citep{kipf2016semi} is simply a special form of Laplacian smoothing, which mixes the features of a node and its nearby neighbors. Such diffusion operations lead to similar learned representations when the corresponding nodes are close topologically with similar features,
thus greatly improving node classification performance.
However, it also brings potential concerns of \textit{over-smoothing} \cite{li2018deeper}. If a GCN is deep with many convolutional layers, the learned representations may be over-smoothed and nodes with different topological and feature characteristics may become indistinguishable. More specifically, by repeatedly applying Laplacian smoothing many times, the node representations within each connected component of the graph will converge to the same values. %
Moreover, GCNs, like other deep models, may suffer from \textit{over-fitting} when we utilize an over-parameterized model to fit a distribution with limited training data, where the model we learn fits the training data very well but generalizes poorly to the testing data.
\vspace{-.02 in}
\subsection{Stochastic Regularization \& Reduction for GNNs}
Quickly increasing model complexity and possible over-fitting and over-smoothing when modeling large graphs, as empirically observed in the
GNN literature, have been conjectured for the main reason of limited %
performance
from deep GNNs \citep{kipf2016semi,rong2019dropedge}. Several stochastic regularization and reduction methods in GNNs have been proposed to improve the deep GNN performance. For example, \textit{stochastic regularization techniques}, such as DropOut \citep{srivastava2014dropout} and DropEdge \citep{rong2019dropedge}, have been used to prevent over-fitting and over-smoothing in GNNs. %
Sampling-based \textit{stochastic reduction by} random walk neighborhood sampling \citep{hamilton2017inductive} and node sampling \citep{chen2018fastgcn} %
has been deployed in GNNs to reduce the size of input data and thereafter model complexity. %
Next, we review each of these methods and show that they can be formulated in our proposed adaptive connection sampling framework.
Denote the output of the $l$th hidden layer in GNNs by $\mathbf{H}^{(l)} = [\mathbf{h}^{(l)}_0, \dots, \mathbf{h}^{(l)}_n]^T\in \mathbb{R}^{n \times f_l}$ with $n$ being the number of nodes and $f_l$ being the number of output features at the $l$th layer. Assume $\mathbf{H}^{(0)} = \mathbf{X} \in \mathbb{R}^{n \times f_0}$ is the input matrix of node attributes, where $f_0$ is the number of nodes features. Also, assume that $\mathbf{W}^{(l)} \in \mathbb{R}^{f_l \times f_{l+1}}$ and $\sigma(\,\cdot\,)$ are the GNN parameters at the $l$th layer and the corresponding activation function, respectively. Moreover, $\mathcal{N}(v)$ denotes the neighborhood of node $v$; $\hat{\mathcal{N}}(v) = \mathcal{N}(v) \cup \{v\}$; and $\mathfrak{N}(.)$ is the normalizing operator, i.e., $\mathfrak{N}(\mathbf{A}) = \mathbf{I}_N + \mathbf{D}^{-1/2}\, \mathbf{A}\, \mathbf{D}^{-1/2}$. Finally, $\odot$ represents the Hadamard product.
\subsubsection{DropOut \citep{srivastava2014dropout}}
In a GNN layer, DropOut randomly removes output elements of its previous hidden layer $\mathbf{H}^{(l)}$ based on independent Bernoulli random draws with a constant success rate at each training iteration. This can be formulated as follows:
\begin{equation}
\mathbf{H}^{(l+1)} = \sigma \left(\mathfrak{N}(\mathbf{A})(\mathbf{Z}^{(l)} \odot \mathbf{H}^{(l)})\, \mathbf{W}^{(l)} \right),
\label{eq:dropout}
\end{equation}
where $\mathbf{Z}^{(l)}$ is a random binary matrix, with the same dimensions as $\mathbf{H}^{(l)}$, whose elements are samples of $\mathrm{Bernoulli}(\pi)$. Despite its success in fully connected and convolutional neural networks, DropOut has shown to be ineffectual in GNNs for preventing over-fitting and over-smoothing.
\subsubsection{DropEdge \citep{rong2019dropedge}}
DropEdge randomly removes edges from the graph by drawing independent Bernoulli random variables (with a constant rate) at each iteration. More specifically, a GNN layer with DropEdge can be written as follows:
\begin{equation}
\mathbf{H}^{(l+1)} = \sigma \left(\mathfrak{N}(\mathbf{A} \odot \mathbf{Z}^{(l)})\, \mathbf{H}^{(l)} \, \mathbf{W}^{(l)} \right),
\label{eq:dropedge}
\end{equation}
Note that here, the random binary mask, i.e. $\mathbf{Z}^{(l)}$, has the same dimensions as $\mathbf{A}$. Its elements are random samples of $\mathrm{Bernoulli}(\pi)$ where their corresponding elements in $\mathbf{A}$ are non-zero and zero everywhere else. It has been shown that the combination of DropOut and DropEdge reaches the best performance in terms of mitigating overfitting in GNNs.
\subsubsection{Node Sampling \citep{chen2018fastgcn}}
To reduce expensive computation in batch training of GNNs, due to the recursive expansion of neighborhoods across layers, \citet{chen2018fastgcn} propose to relax the requirement of simultaneous availability of test data. Considering graph convolutions as integral transforms of embedding functions under probability measures allows for the use of Monte Carlo approaches to consistently estimate the integrals. This leads to an optimal node sampling strategy, FastGCN, which can be formulated as
\vspace{-.03 in}
\begin{equation}
\label{eq:nodesample}
\mathbf{H}^{(l+1)} = \sigma \left(\mathfrak{N}(\mathbf{A})\, \mathrm{diag}(\mathbf{z}^{(l)}) \mathbf{H}^{(l)} \, \mathbf{W}^{(l)} \right),
\end{equation}
where $\mathbf{z}^{(l)}$ is a random vector whose elements are drawn from $\mathrm{Bernoulli}(\pi)$. This, indeed, is a special case of DropOut, as all of the output features for a node are either completely kept or dropped while DropOut randomly removes some of these related output elements associated with the node.
\section{Graph DropConnect}%
We propose a general stochastic regularization technique for GNNs---Graph DropConnect (GDC)---by adaptive connection sampling, which can be interpreted as an approximation of Bayesian GNNs.
In GDC, we allow GNNs to draw different random masks for each channel and edge independently. More specifically, the operation of a GNN layer with GDC is defined as follows:
\vspace{-.05 in}
\begin{gather}
\mathbf{H}^{(l+1)}[:,j] = \sigma \left(\sum_{i=1}^{f_l}\mathfrak{N}(\mathbf{A} \odot \mathbf{Z}_{i,j}^{(l)})\, \mathbf{H}^{(l)}[:,i] \, \mathbf{W}^{(l)}[i,j] \right),\nonumber\\
\text{for}\quad j = 1,\, \dots \,, f_{l+1}
\label{equ: gdc}
\end{gather}
where $f_l$ and $f_{l+1}$ are the number of features at layers $l$ and $l+1$, respectively, and $\mathbf{Z}_{i,j}^{(l)}$ is a sparse random matrix (with the same sparsity as $\mathbf{A}$) whose non-zero elements are randomly drawn by $\mathrm{Bernoulli}(\pi_l)$. Note that $\pi_l$ can be different for each layer for GDC instead of assuming the same constant drop rate for all layers in previous methods.
As shown in~(\ref{eq:dropout}), (\ref{eq:dropedge}), and (\ref{eq:nodesample}), DropOut \cite{srivastava2014dropout}, DropEdge \citep{rong2019dropedge}, and Node Sampling \cite{chen2018fastgcn} have different sampling assumptions on channels, edges, or nodes, yet there is no clear evidence to favor one over the other in terms of consequent graph analytic performance. In the proposed GDC approach, there is a free parameter $\{\mathbf{Z}_{i,j}^{(l)} \in \{0, 1\}^{n \times n}\}_{i=1}^{f_l}$ to adjust the binary mask for the edges, nodes and channels. Thus the proposed GDC model has one extra degree of freedom to incorporate flexible connection sampling.
The previous stochastic regularization techniques can be considered as special cases of GDC. %
To illustrate that, we assume $\mathbf{Z}_{i,j}^{(l)}$ are the same for all $j \in \{1, 2, \dots, f_{l+1}\}$, thus we can omit the indices of the output elements at layer $l+1$ and rewrite~(\ref{equ: gdc}) as
\vspace{-.1 in}
\begin{gather}
\mathbf{H}^{(l+1)} = \sigma \left(\sum_{i=1}^{f_l}\mathfrak{N}(\mathbf{A} \odot \mathbf{Z}_i^{(l)})\, \mathbf{H}^{(l)}[:,i] \, \mathbf{W}^{(l)}[i,:] \right)
\raisetag{22pt}
\label{equ: gdc_red}
\end{gather}
Define $\mathbf{J}_n$ as a $n \times n$ all-one matrix.
Let $\mathbf{Z}^{(l)}_{DO} \in \{0, 1\}^{n \times f_l}$, $\mathbf{Z}^{(l)}_{DE} \in \{0, 1\}^{n \times n}$, and $\mathrm{diag}(\mathbf{z}_{NS}^{(l)}) \in \{0, 1\}^{n \times n}$ be the random binary matrices corresponding to the ones adopted in DropOut \cite{srivastava2014dropout}, DropEdge \citep{rong2019dropedge}, and Node Sampling \cite{chen2018fastgcn}, respectively. The random mask $\{\mathbf{Z}_i^{(l)} \in \{0, 1\}^{n \times n}\}_{i=1}^{f_l}$ in GDC become the same as those of the DropOut when $\mathbf{Z}_i^{(l)} = \mathbf{J}_n \, \mathrm{diag}(\mathbf{Z}^{(l)}_{DO}[:,\,i])$, the same as those of DropEdge when $\{\mathbf{Z}_i^{(l)}\}_{i=1}^{f_l} = \mathbf{Z}^{(l)}_{DE}$, and the same as those of node sampling when $\{\mathbf{Z}_i^{(l)}\}_{i=1}^{f_l} = \mathbf{J}_n \mathrm{diag}(\mathbf{z}_{NS}^{(l)})$.
\subsection{GDC as Bayesian Approximation}
In GDC, random masking is applied to the adjacency matrix of the graph to regularize the aggregation steps at each layer of GNNs. In %
existing Bayesian neural networks, the model parameters, i.e. $\mathbf{W}^{(l)}$, are considered random to enable Bayesian inference based on predictive posterior given training data \cite{gal2017concrete,boluki2020learnable}. %
Here, we show that connection sampling in GDC can be transformed from the output feature space to the parameter space so that it can be considered as appropriate Bayesian extensions of GNNs.
First, we rewrite \eqref{equ: gdc_red} to have a node-wise view of a GNN layer with GDC. More specifically,
\vspace{-.07 in}
\begin{equation}
\mathbf{h}^{(l+1)}_v = \sigma \left(\frac{1}{c_v}\big(\sum_{u \in \hat{\mathcal{N}}(v)} \mathbf{z}_{vu}^{(l)} \odot \mathbf{h}^{(l)}_u\big)\, \mathbf{W}^{(l)} \right),
\label{equ: gdc_nw}
\end{equation}
where $c_v$ is a constant derived from the degree of node $v$, and $\mathbf{z}_{vu}^{(l)} \in \{0, 1\}^{1 \times f_l}$ is the mask row vector corresponding to connection between nodes $v$ and $u$ in three dimensional tensor $\mathcal{Z}^{(l)} = [\mathbf{Z}_1^{(l)},\dots,\mathbf{Z}_{f_l}^{(l)}]$. For brevity and without loss of generality, we ignore the constant $c_v$ in the rest of this section. We can rewrite and reorganize \eqref{equ: gdc_nw} to transform the randomness from sampling to the parameter space as
\vspace{-.1 in}
\begin{equation}
\begin{split}
\mathbf{h}^{(l+1)}_v &=\sigma \left(\big(\sum_{u \in \hat{\mathcal{N}}(v)} \mathbf{h}^{(l)}_u\, \mathrm{diag}(\mathbf{z}_{vu}^{(l)})\big)\, \mathbf{W}^{(l)} \right)\\
&=\sigma \left(\sum_{u \in \hat{\mathcal{N}}(v)} \mathbf{h}^{(l)}_u\, \big(\mathrm{diag}(\mathbf{z}_{vu}^{(l)})\, \mathbf{W}^{(l)}\big) \right).
\end{split}
\label{equ: noise}
\end{equation}
Define $\mathbf{W}_{vu}^{(l)} := \mathbf{z}_{vu}^{(l)}\, \mathbf{W}^{(l)}$. We have:
\begin{equation}
\mathbf{h}^{(l+1)}_v = \sigma \left(\sum_{u \in \hat{\mathcal{N}}(v)} \mathbf{h}^{(l)}_u\, \mathbf{W}_{vu}^{(l)} \right).
\label{equ: noise_last}
\end{equation}
$\mathbf{W}_{vu}^{(l)}$, which pairs the corresponding weight parameter with the edge in the given graph. The operation with GDC in \eqref{equ: noise_last} can be interpreted as learning different weights for each of the message passing along edges $e = (u, v) \in \mathcal{E}$ where $\mathcal{E}$ is the union of edge set of the input graph and self-loops for all nodes.
Following the variational interpretation in~\citet{gal2017concrete}, GDC can be seen as an approximating distribution $q_\theta(\boldsymbol{\omega})$ for the posterior $p(\boldsymbol{\omega} \, |\, \mathbf{A}, \mathbf{X})$ when considering a set of random weight matrices $\boldsymbol{\omega} = \{\boldsymbol{\omega}_e\}_{e=1}^{|\mathcal{E}|}$ in the Bayesian framework, where $\boldsymbol{\omega}_e = \{\mathbf{W}_{e}^{(l)}\}_{l=1}^L$ is the set of random weights for the $e$th edge, $|\mathcal{E}|$ is the number of edges in the input graph, and $\theta$ is the set of variational parameters.
The Kullback--Leibler (KL) divergence $\mathrm{KL}(q_\theta(\boldsymbol{\omega}) || p(\boldsymbol{\omega}))$ is considered in training as a regularisation term, which ensures that the approximating $q_\theta(\boldsymbol{\omega})$ does not deviate too far from the prior distribution. To be able to evaluate the KL term analytically, the discrete quantised Gaussian can be adopted as the prior distribution as in \citet{gal2017concrete}. Further with the factorization $q_\theta(\boldsymbol{\omega})$ over $L$ layers and $|\mathcal{E}|$ edges such that $q_\theta(\boldsymbol{\omega}) = \prod_l \prod_e q_{\theta_l}(\mathbf{W}_{e}^{(l)})$ and letting $q_{\theta_l}(\mathbf{W}_{e}^{(l)}) = \pi_l \delta(\mathbf{W}_{e}^{(l)}- 0) + (1 - \pi_l) \delta(\mathbf{W}_{e}^{(l)}- \mathbf{M}^{(l)})$, where $\theta_l = \{\mathbf{M}^{(l)}, \pi_l\}$, the KL term can be written as $\sum_{l=1}^L \sum_{e=1}^{|\mathcal{E}|} \mathrm{KL}(q_{\theta_l}(\mathbf{W}_{e}^{(l)}) \, ||\, p(\mathbf{W}_{e}^{(l)}))$ and approximately
\begin{equation*}
\mathrm{KL}(q_{\theta_l}(\mathbf{W}_{e}^{(l)}) \,||\, p(\mathbf{W}_{e}^{(l)})) \propto \frac{(1-\pi_l)}{2} \,||\, \mathbf{M}^{(l)} ||^2 - \mathcal{H}(\pi_l),
\end{equation*}
where $\mathcal{H}(\pi_l)$ is the entropy of a Bernoulli random variable with the success rate $\pi_l$.
Since the entropy term does not depend on network weight parameters $\mathbf{M}^{(l)}$, it can be omitted when $\pi_l$ is not optimized. But we learn $\pi_l$ in GDC, thus the entropy term is important. %
Minimizing the KL divergence with respect to the drop rate $\pi_l$ is equivalent to maximizing the entropy of a Bernoulli random variable with probability $1 - \pi_l$. This pushes the drop rate towards 0.5, which may not be desired in some cases where higher/lower drop rate probabilities are more appreciated. %
\subsection{Variational Inference for GDC}
We consider $\mathbf{z}_e^{(l)}$ and $\mathbf{W}^{(l)}$ as local and global random variables, respectively, and denote $\mathbf{Z}^{(l)} = \{\mathbf{z}_e^{(l)}\}_{e=1}^{|\mathcal{E}|}$ and $\boldsymbol{\omega}^{(l)} = \{\mathbf{W}^{(l)}_e\}_{e=1}^{|\mathcal{E}|}$. For inference of this approximating model with GDC, we assume a factorized variational distribution $q(\boldsymbol{\omega}^{(l)}, \mathbf{Z}^{(l)}) = q(\boldsymbol{\omega}^{(l)}) \, q(\mathbf{Z}^{(l)})$.
Let the prior distribution $p(\mathbf{W}_e^{(l)})$ be a discrete quantised Gaussian and $p(\boldsymbol{\omega}^{(l)}) = \prod_{e=1}^{\mathcal{E}} p(\mathbf{W}_e^{(l)})$.%
Therefore, the KL term can be written as $\sum_{l=1}^L \mathrm{KL}(q(\boldsymbol{\omega}^{(l)}, \mathbf{Z}^{(l)}) \, || \, p(\boldsymbol{\omega}^{(l)}, \mathbf{Z}^{(l)}))$, %
with
\begin{equation}
\begin{split}
&\mathrm{KL}\left(q(\boldsymbol{\omega}^{(l)} , \mathbf{Z}^{(l)}) \, || \, p(\boldsymbol{\omega}^{(l)}, \mathbf{Z}^{(l)}) \right) \propto \nonumber \\
& \quad \frac{|\mathcal{E}| (1-\pi_l) }{2} ||\mathbf{M}^{(l)}||^2 + \, \sum_{e=1}^{|\mathcal{E}|} \mathrm{KL}\left(q(\mathbf{z}_{e}^{(l)}) \, || \, p(\mathbf{z}_{e}^{(l)})\right). \nonumber
\end{split}
\end{equation}
The KL term consists of the common weight decay in the non-Bayesian GNNs with the additional KL term $\sum_{e=1}^{|\mathcal{E}|} \, \mathrm{KL}(q(\mathbf{z}_{e}^{(l)}) \, || \, p(\mathbf{z}_{e}^{(l)}))$ that acts as a regularization term for $\mathbf{z}_{e}^{(l)}$. %
In this GDC framework, the variational inference loss, for node classification for example, can be written~as
\begin{equation}
\begin{split}
&\mathcal{L}(\{\mathbf{M}^{(l)},\pi_l\}_{l=1}^{L}) =\\
&\quad\mathbb{E}_{q(\{\boldsymbol{\omega}^{(l)} , \mathbf{Z}^{(l)}\}_{l=1}^{L} )}[\mathrm{log}P(Y_{o}|X,\{\boldsymbol{\omega}^{(l)}, \mathbf{Z}^{(l)}\}_{l=1}^{L})] \\
&\quad - \sum_{l=1}^L \mathrm{KL}(q(\boldsymbol{\omega}^{(l)}, \mathbf{Z}^{(l)}) \, || \, p(\boldsymbol{\omega}^{(l)}, \mathbf{Z}^{(l)})),
\end{split}
\raisetag{21pt}
\label{eq:varloss}
\end{equation}
where $Y_{o}$ denotes the collection of the available labels for the observed nodes.
The optimization of \eqref{eq:varloss} with respect to the weight matrices can be done by a Monte Carlo sample, i.e. sampling a random GDC mask and calculating the gradients with respect to $\{\mathbf{M}^{(l)}\}_{l=1}^{L}$ with stochastic gradient descent. It is easy to see that if $\{\pi_l\}_{l=1}^{L}$ are fixed, implementing our GDC is as simple as using common regularization terms on the neural network weights.
We aim to optimize the drop rates $\{\pi_l\}_{l=1}^{L}$ jointly with the weight matrices. This clearly provides more flexibility as all the parameters of the approximating posterior will be learned from the data instead of being fixed \emph{a priori} or treated as hyper-parameters, often difficult to tune. However, the optimization of \eqref{eq:varloss} with respect to the drop rates is challenging. Although the KL term is not a function of the random masks, the commonly adopted reparameterization techniques \citep{rezende2014stochastic,kingma2013auto} are not directly applicable here for computing the expectation in the first term since the drop masks are binary. Moreover, score-function
gradient estimators, such as REINFORCE \cite{williams1992simple, fu2006gradient}, possess high variance.
One potential solution is continuous relaxation of the drop masks. This approach has lower variance at the expense of introducing bias. Another solution is the direct optimization with respect to the discrete variables by the recently developed Augment-REINFORCE-Merge~(ARM) method \cite{yin2018arm}, which has been used in BNNs \cite{boluki2020learnable} and information retrieval \cite{icassp_arsm,dadaneh2020pairwise}. In the next section, we will discuss in detail about our GDC formulation with more flexible beta-Bernoulli prior construction for adaptive connection sampling and how we solve the joint optimization problem for training GNNs with adaptive connection sampling.
\section{Variational Beta-Bernoulli GDC}
The sampling or drop rate in GDC can be set as a constant hyperparameter as commonly done in other stochastic regularization techniques. In this work, we further enrich GDC with an adaptive sampling mechanism, where the drop rate is directly learned together with GNN parameters given graph data. In fact, in the Bayesian framework, such a hierarchical construct may increase the model expressiveness to further improve prediction and uncertainty estimation performance, as we will %
show empirically in Section~\ref{sec:exp}.
Note that in this section, for brevity and simplicity we do the derivations for one feature dimension only, i.e. $f_l=1$. Extending to multi-dimensional features is straightforward as we assume the drop masks are independent across features. Therefore, we drop the feature index in our notations.
Inspired by the beta-Bernoulli process \citep{thibaux2007hierarchical}, whose marginal representation is also known as the Indian Buffet Process~(IBP) \citep{ghahramani2006infinite}, we impose a beta-Bernoulli prior to the binary random masks as%
\begin{eqnarray}
a_{e}^{(l)} &=& z_e^{(l)} a_e , \quad
z_{e}^{(l)} \sim \mathrm{Bernoulli}(\pi_l), \nonumber \\
\pi_l &\sim& \mathrm{Beta}(c/L, \, c (L - 1)/L),
\label{eq:beta-bernoulli}
\end{eqnarray}
where $a_e$ denotes an element of the adjacency matrix $\mathbf{A}$ corresponding to an edge $e$, and $\hat{a}_e^{(l)}$ an element of the matrix $\hat{\mathbf{A}}^{(l)} = \mathbf{A} \odot \mathbf{Z}^{(l)}$. Such a formulation is known to be capable of enforcing sparsity in random masks \citep{zhou2009non,hajiramezanali2018bayesian}, which has been shown to be necessary for regularizing deep GNNs as discussed in DropEdge~\cite{rong2019dropedge}.
With this hierarchical beta-Bernoulli GDC formulation, inference based on Gibbs sampling can be computationally demanding for large datasets, including graph data \citep{hasanzadeh2019sigvae}. In the following, we derive efficient variational inference algorithm(s) for learnable GDC.
To perform variational inference for GDC random masks and the corresponding drop rate at each GNN layer together with weight parameters, we define the variational distribution as $q(\mathbf{Z}^{(l)}, \pi_l) = q(\mathbf{Z}^{(l)} \,|\, \pi_l) q(\pi_l)$.
We define $q(\pi_l)$ to be Kumaraswamy distribution \cite{kumaraswamy1980generalized}; as an alternative to the beta prior factorized over $l$th layer
\begin{equation}
q(\pi_l; a_l, b_l) = a_l b_l \pi_l^{a_l -1}(1 - \pi_l^{a_l})^{b_l - 1},
\end{equation}
where $a_l$ and $b_l$ are greater than zero. Knowing $\pi_l$ the edges are independent, thus we can rewrite $q(\mathbf{Z}^{(l)} \,|\, \pi_l) = \prod_{e=1}^{|\mathcal{E}|}q(z_e^{(l)} \,|\, \pi_l)$. We further put a Bernoulli distribution with parameter $\pi_l$ over $q(\mathbf{z}^{(l)}_{e}|\pi_l)$. The KL divergence term can be written as
\begin{equation}
\begin{split}
&\mathrm{KL}\left(q(\mathbf{Z}^{(l)}, \pi_l) \,||\, p(\mathbf{Z}^{(l)}, \pi_l)\right) = \nonumber \\
&\,\,\,\sum_{e=1}^{|\mathcal{E}|} \mathrm{KL}\left(q(z_e^{(l)} \,|\, \pi_l) \,||\, p(z_e^{(l)} \,|\, \pi_l)\right) + \mathrm{KL}\left(q(\pi_l) \,||\, p(\pi_l)\right). \nonumber
\end{split}
\end{equation}
While the first term is zero due to the identical distributions, the second term can be computed in closed-form as
\begin{equation}
\begin{split}
&\mathrm{KL}\left(q(\pi_l) \,||\, p(\pi_l)\right) = \nonumber \\
&\quad\frac{a_l - c/L}{a_l}\left(- \gamma - \Psi(b_l) - \frac{1}{b_l} \right) + \mathrm{log} \frac{a_l b_l}{c/L} - \frac{b_l - 1}{b_l},
\end{split}
\end{equation}
where $\gamma$ is the Euler-Mascheroni constant and $\Psi(\cdot)$ is the digamma function.
The gradient of the KL term in \eqref{eq:varloss} can easily be calculated with respect to the drop parameters. However, as mentioned in the previous section, due to the discrete nature of the random masks, we cannot directly apply reparameterization technique to calculate the gradient of the first term in \eqref{eq:varloss} with respect to the drop rates (parameters). One way to address this issue is to replace the discrete variables with a continuous approximation. We impose a concrete distribution relaxation \citep{jang2016categorical,gal2017concrete} for the Bernoulli random variable $z^{(l)}_{uv}$, leading to an efficient optimization by
sampling from simple sigmoid distribution which has a
convenient parametrization
\begin{equation}
\tilde{z}^{(l)}_{e} = \mathrm{sigmoid}\left(\frac{1}{t} \mathrm{log}\big(\frac{\pi_l}{1 - \pi_l}\big) + \mathrm{log}\big(\frac{u}{1-u}\big)\right),
\end{equation}
where $u \sim \mathrm{Unif}[0, 1]$ and $t$ is temperature parameter of relaxation. We can then use stochastic gradient variational Bayes to optimize the variational parameters $a_l$ and $b_l$.
Although this approach is simple, the relaxation introduces bias. Our other approach is to directly optimize the variational parameters using the original Bernoulli distribution in the formulation as in \citet{boluki2020learnable}. We can calculate the gradient of the variational loss with respect to $\boldsymbol{\alpha}=\{\text{logit}(1-\pi_l)\}_{l=1}^{L}$ using ARM estimator , which is unbiased and has low variance, by performing two forward passes as %
\vspace{-0.05cm}
\begin{equation}
\begin{split}
\nabla_{\boldsymbol{u}}\mathcal{L}&(\mathbf{\alpha}) = \mathbb{E}_{u \sim \prod_{l=1}^{L}\prod_{e=1}^{|\mathcal{E}|}\mathrm{Unif}[0, 1](u_e^{(l)})}\Big[\big( \mathcal{L}(\{\mathbf{M}^{(l)}\}_{l=1}^L,\\ &1_{[\boldsymbol{u}>\sigma(-\boldsymbol{\alpha})]})-\mathcal{L}(\{\mathbf{M}^{(l)}\}_{l=1}^L,1_{[\boldsymbol{u}<\sigma(\boldsymbol{\alpha})]})\big) \big(\boldsymbol{u} - \frac{1}{2} \big) \Big],\nonumber
\end{split}
\end{equation}
where $\mathcal{L}(\{\mathbf{M}^{(l)}\}_{l=1}^L,1_{[\boldsymbol{u}<\sigma(\boldsymbol{\alpha})]})$ denotes the loss obtained by setting
$\mathbf{Z}^{(l)}=1_{[\boldsymbol{u}^{(l)}<\sigma(\boldsymbol{\alpha}_l)]}:=\big(1_{[u^{(l)}_1<\sigma(\alpha_l)]},\ldots, 1_{[u^{(l)}_{|\mathcal{E}|}<\sigma(\alpha_l)]}\big)$ for $l=1,\ldots, L$. The gradient with respect to $\{a_l,b_l\}_{l=1}^{L}$ can then be calculated by using the chain rule and the reparameterization for $\pi_l=(1-u^{\frac{1}{b_l}})^{\frac{1}{a_l}}, u\sim \text{Unif}[0,1]$.
It is worth noting that although the beta-Bernoulli DropConnect with ARM is expected to provide better performance due to the more accurate gradient estimates, it has slightly higher computational complexity as it requires two forward passes.
\begin{table*}[!t]
\caption{Semi-supervised node classification accuracy of GCNs with our adaptive connection sampling and baseline methods.}
\label{tab:results}
\centering
\resizebox{2.0\columnwidth}{!}{
\begin{tabular}{@{}l | c c c c c c}
\toprule
\multirow{1}{*}{\textbf{Method}} & \multicolumn{2}{c}{\textbf{Cora}} & \multicolumn{2}{c}{\textbf{Citeseer}} &
\multicolumn{2}{c}{\textbf{Cora-ML}} \\
& \multicolumn{1}{c}{2 layers} & \multicolumn{1}{c}{4 layers} & \multicolumn{1}{c}{2 layers} & \multicolumn{1}{c}{4 layers} & \multicolumn{1}{c}{2 layers} & \multicolumn{1}{c}{4 layers} \\
\midrule
GCN-DO & $80.98\pm0.48$ & $78.24\pm2.4$ & $70.44\pm0.39$ & $64.38\pm0.90$ & $83.45\pm0.73$ & $81.51\pm1.01$ \\
GCN-DE & $78.36\pm0.92$ & $73.40\pm2.07$ & $70.52\pm0.75$ & $57.14\pm0.90$ & $83.30\pm1.37$ & $68.89\pm3.37$ \\
GCN-DO-DE & $80.58\pm1.19$ & $79.20\pm1.07$ & $70.74\pm1.23$ & $64.84\pm0.98$ & $83.61\pm0.83$ & $81.21\pm1.53$ \\
\midrule
\textbf{GCN-BBDE} & $\mathbf{81.58}\pm0.49$ & $\mathbf{80.42}\pm0.25$ & $\mathbf{71.46}\pm0.55$ & $\mathbf{68.58}\pm0.88$ & $\mathbf{84.62}\pm1.70$ & $\mathbf{84.73}\pm0.52$ \\
\textbf{GCN-BBGDC} & $\mathbf{81.80}\pm0.99$ & $\mathbf{82.20}\pm0.92$ & $\mathbf{71.72}\pm0.48$ & $\mathbf{70.00}\pm0.36$ & $\mathbf{85.43}\pm0.70$ & $\mathbf{85.52}\pm0.83$ \\
\bottomrule
\end{tabular}
}
\end{table*}
\section{Connection to Random Walk Sampling}
Various types of random walk have been used in graph representation learning literature to reduce the size of input graphs. In GNNs, specifically in GraphSAGE \citep{hamilton2017inductive}, random walk sampling has been deployed to reduce the model complexity for very large graphs. One can formulate a GNN layer with random walk sampling as follows:
\vspace{-.02in}
\begin{equation}
\mathbf{h}^{(l+1)}_v = \sigma \left(( \sum_{u \in \hat{\mathcal{N}}(v)} (z_{vu}^{(l)}\, |\, \mathbf{Z}^{(l-1)})\, \mathbf{h}^{(l)}_u )\, \mathbf{W}^{(l)} \right).
\end{equation}
Here, $\mathbf{Z}^{(l)}$ is the same as the one in DropEdge except that it is dependent on the masks from the previous layer. This is due to the fact that random walk samples for each node are connected subgraphs. %
In this setup, we can decompose the variational distribution of the GDC formulation in an autoregressive way. Specifically, here we have $q(z_{uv}^{(l)} | \mathbf{Z}^{(l-1)}) = \mathrm{Bernoulli}(\pi_l) 1_{\sum_{u \in \hat{\mathcal{N}}(v)} z_{vu}^{(l-1)}>0}$. With fixed Bernoulli parameters, we can calculate the gradients for the weight matrices with Monte Carlo integration. Learning Bernoulli parameters is challenging and does not allow direct application of ARM due to the autoregressive structure of the variational posterior. We leave sequential ARM for future study.
\vspace{-.1 in}
\begin{corollary}
Any graph neural network with random walk sampling, such as GraphSAGE, is an approximation of a Bayesian graph neural network as long as outputs are calculated using Monte Carlo sampling.
\end{corollary}
\section{Sampling Complexity}\label{sec:samp_comp}
The number of random samples needed for variational inference in GDC, \eqref{equ: gdc}, at each layer of a GNN is $|\mathcal{E}|\times f_{l} \times f_{l+1}$. This number would reduce to $|\mathcal{E}|\times f_{l}$ in the constrained version of GDC as shown in \eqref{equ: gdc_red}. These numbers, potentially, could be very high specially if the size of the graph or the number of filters are large, which could increase the space complexity and computation time. To circumvent this issue, we propose to draw a single sample for a block of features as oppose to drawing a new sample for every single feature. This would reduce the number of required samples to $|\mathcal{E}|\times \mathrm{nb}$ with $\mathrm{nb}$ being the number of blocks. In our experiments, we have one block in the first layer and two blocks in layers after that.
In our experiments, we keep the order of features the same as the original input files, and divide them into $\mathrm{nb}$ groups with the equal number of features.
While in our GDC formulation, as shown in \eqref{equ: gdc} and \eqref{equ: gdc_red}, the normalization $\mathfrak{N}(\cdot)$ is applied after masking, one can multiply the randomly drawn mask with the pre-computed normalized adjacency matrix. This relaxation reduces the computation time and has negligible effect on the performance based on our experiments. An extension to the GDC sampling strategy is asymmetric sampling where the mask matrix $\mathbf{Z}$ could be asymmetric. This would increase the number of samples by a factor of two; however it increases the model flexibility. In our experiments, we have used asymmetric masks and multiplied the mask with the normalized adjacency matrix.
\section{Numerical Results}
\label{sec:exp}
We test the performance of our
adaptive connection sampling framework, learnable GDC, on semi-supervised node classification using real-world citation graphs. In addition, we compare the uncertainty estimates of predictions by Monte Carlo beta-Bernoulli GDC and Monte Carlo Dropout. We also show the performance of GDC compared to existing methods in alleviating the issue of over-smoothing in GNNs. Furthermore, we investigate the effect of the number of blocks on the performance of GDC. We have also investigated learning separate drop rates for every edge in the network, i.e. \emph{local} GDC, which is included in the supplementary materials.
\subsection{Semi-supervised Node Classification}
\subsubsection{Datasets and Implementation Details}
We conducted extensive experiments for semi-supervised node classification with real-world citation datasets.
We consider \emph{Cora}, \emph{Citeseer}
and \emph{Cora-ML}
datasets, and preprocess and split them same as \citet{kipf2016semi} and \citet{bojchevski2018deep}. We train beta-Bernoulli GDC (BBGDC) models for 2000 epochs with a learning rate of 0.005 and a validation set used for early stopping. All of the hidden layers in our implemented GCNs have 128 dimensional output features. We use $5\times10^{-3}$, $10^{-2}$, and $10^{-3}$ as L2 regularization factor for Cora, Citeseer, and Cora-ML, respectively. For the GCNs with more than 2 layers, we
use warm-up during the first 50 training epochs to gradually impose the beta-Bernoulli KL term in the objective function.
The temperature in the concrete distribution is set to 0.67. For a fair comparison, the number of hidden units are the same in the baselines and their hyper-parameters are hand-tuned to achieve their best performance. Performance is reported by the average accuracy with standard deviation based on 5 runs on the test set. The dataset statistics as well as more implementation details are included in the supplementary materials.
\subsubsection{Discussion}
Table \ref{tab:results} shows that BBGDC outperforms the state-of-the-art stochastic regularization techniques in terms of accuracy in all benchmark datasets. DO and DE in the table stand for DropOut and DropEdge, respectively.
Comparing GCN-DO and GCN-DE, we can see that DropEdge alone is less effective than DropOut in overcoming over-smoothing and over-fitting in GCNs. The difference between accuracy of GCN-DO and GCN-DE is more substantial in deeper networks (5\% in Cora, 7\% in Citeseer, and 13\% in Cora-ML), which further proves the limitations of DE. Among the baselines, combination of DO and DE shows the best performance allowing to have deeper models. However, this is not always true. For example in Citeseer, 4-layer GCN shows significant decrease in performance compared to 2-layer GCN.
To show the advantages of learning the drop rates as well as the effect of hierarchical beta-Bernoulli construction, we have also evaluated beta-Bernoulli DropEdge (BBDE) with the concrete approximation, in which the edge drop rate at \emph{each layer} is learned using the same beta-Bernoulli hierarchical construction as GDC. We see that GCN with BBDE, without any DropOut, performs better than both GCNs with DE and DO-DE. By comparing GCN with BBDE and GCN with BBGDC, it is clear that the improvement is not only due to learnable sampling rate but also the increased flexibility of GDC compared to DropEdge. We note that GCN-BBGDC is the only method for which the accuracy improves by increasing the number of layers except in Citeseer.
\subsubsection{Concrete relaxation versus ARM}
To investigate the effect of direct optimization of the variational loss with respect to the drop parameters with ARM vs relaxation of the discrete random variables with concrete, we construct three ARM optimization-based variants of our method: Learnable Bernoulli DropEdge with ARM gradient estimator (BDE-ARM) where the edge drop rate of the Bernoulli mask at each layer is directly optimized; beta-Bernoulli DropEdge with ARM (BBDE-ARM); and beta-Bernoulli GDC with ARM (BBGDC-ARM). We evaluate these methods on the 4-layer GCN setups where significant performance improvement compared with the baselines has been achieved by BBDE and GDC with concrete relaxation. Comparing the performance of BBDE-ARM and BBGDC-ARM in Table \ref{tab: acc_ssnc} with the corresponding models with concrete relaxation, suggests further improvement when the drop parameters are directly optimized. Moreover, BDE-ARM, which optimizes the parameters of the Bernoulli drop rates, performs better than DO, DE, and DO-DE.
\begin{table}[!t]
\caption{Accuracy of ARM optimization-based variants of our proposed method in semi-supervised node classification.}
\vspace{0.25em}
\centering
\resizebox{1.0\columnwidth}{!}{
\begin{tabular}{@{}l | c c c }
\toprule
{\textbf{Method}} & {\textbf{Cora} (4 layers)} & {\textbf{Citeseer} (4 layers)} \\
\midrule
\textbf{GCN-BDE-ARM} & $79.95\pm0.79$ & $67.90\pm0.15$\\
\textbf{GCN-BBDE-ARM} & $81.78\pm0.81$ & $69.43\pm0.45$\\
\textbf{GCN-BBGDC-ARM} & $82.40\pm0.60$ & $70.25\pm0.07$\\
\bottomrule
\end{tabular}
}
\label{tab: acc_ssnc}
\vspace{-0.2in}
\end{table}
\subsection{Uncertainty Quantification}
To evaluate the quality of uncertainty estimates obtained by our model, we use the Patch Accuracy vs Patch Uncertainty (PAvPU) metric introduced in \cite{mukhoti2018evaluating}. PAvPU combines $p(\mathrm{accurate|certain})$, i.e. the probability that the model is accurate on its output given that it is confident on the same, $p(\mathrm{certain|inaccurate})$, i.e. the probability that the model is uncertain about its output given that it has made a mistake in its prediction, into a single metric. More specifically, it is defined as $\mathrm{PAvPU} = (n_{ac}+n_{iu})/(n_{ac}+n_{au}+n_{ic}+n_{iu}),$
where $n_{ac}$ is the number of accurate and certain predictions, $n_{au}$ is the number of accurate and uncertain predictions, $n_{ic}$ is the number of inaccurate and certain predictions, and $n_{iu}$ is the number of inaccurate and uncertain predictions. Higher PAvPU means that certain predictions are accurate and inaccurate predictions are uncertain.
We here demonstrate the results for uncertainty estimates for a 4-layer GCN-DO and a 4-layer GCN-BBGDC with random initialization for semi-supervised node classification on Cora. We have evaluated PAvPU using 20 Monte Carlo samples for the test set where we use predictive entropy as the uncertainty metric. The results are shown in Figure \ref{fig: uncertain}. It can be seen that our proposed model consistently outperforms GCN-DO on every uncertainty threshold ranging from 0.5 to 1 of the maximum predictive uncertainty. While Figure \ref{fig: uncertain} depicts the results based on one random initialization, other initializations show the same trend.
\begin{figure}[!t]
\centering
\includegraphics[width=1.0\columnwidth,keepaspectratio,trim={0 0 1cm 1cm},clip]{pavpu_v2.pdf}
\vspace{-0.2in}
\caption{Comparison of uncertainty estimates in PAvPU by a 4-layer GCN-BBGDC with 128-dimensional hidden layers and a 4-layer GCN-DO 128-dimensional hidden layers on Cora.}%
\label{fig: uncertain}
\end{figure}
\begin{figure*}[!t]
\centering
\begin{subfigure}{}
\includegraphics[width=.47\textwidth,trim={0 0 1cm 1cm},clip]{do_gdc_tv.pdf}
\end{subfigure}
\begin{subfigure}{}
\includegraphics[width=.47\textwidth,trim={0 0 1cm 1cm},clip]{cora_mullays_v3.pdf}
\end{subfigure}
\vspace{-0.2in}
\caption{From left to right: a) Total variation of the hidden layer outputs during training in a 4-layer GCN-BBGDC with 128-dimensional hidden layers and a 4-layer GCN-DO 128-dimensional hidden layers on Cora; b) Comparison of node classification accuracy for GCNs with a different number of hidden layers using different stochastic regularization methods. All of the hidden layers are 128 dimensional.}%
\label{fig: totvar}
\end{figure*}
\subsection{Over-smoothing and Over-fitting}
To check how GDC helps alleviate over-smoothing in GCNs, we have tracked the total variation (TV) of the outputs of hidden layers during training. TV is a metric used in the graph signal processing literature to measure the smoothness of a signal defined over nodes of a graph \cite{chen2015signal}. More specifically, given a graph with the adjacency matrix $\mathbf{A}$ and a signal $\mathbf{x}$ defined over its nodes, TV is defined as $\mathrm{TV}(\mathbf{x}) = \Vert \mathbf{x} - (1/|\lambda_{max}|)\mathbf{A}\,\mathbf{x} \Vert_{2}^{2}$,
where $\lambda_{max}$ denotes the eigenvalue of $A$ with largest magnitude. Lower TV shows that the signal on adjacent nodes are closer to each other, indicating possible over-smoothing.
We have compared the TV trajectories of the hidden layer outputs in a 4-layer GCN-BBGDC and a 4-layer GCN-DO normalized by their Frobenius norm, depicted in Figure \ref{fig: totvar}(a). It can be seen that, in GCN-DO, while the TV of the first layer is slightly increasing at each training epoch, the TV of the second hidden layer decreases during training. This, indeed, contributed to the poor performance of GCN-DO. On the contrary, the TVs of both first and second layers in GCN-BBGDC is increasing during training. Not only this robustness is due to the dropping connections in GDC framework, but also is related to its learnable drop rates.
With such promising results showing less over-smoothing with BBGDC, we further investigate how our proposed method works in deeper networks. We have checked the accuracy of GCN-BBGDC with a various number of 128-dimensional hidden layers ranging from 2 to 16. The results are shown in Figure \ref{fig: totvar}(b). The performance improves up to the GCN with 4 hidden layers and decreases after that. It is important to note that even though the performance drops by adding the 5-th layer, the degree to which it decreases is far less than competing methods. For example, the node classification accuracy with GCN-DO quickly drops to 69.50\% and 64.5\% with 8 and 16 layers. In addition, we should mention that the performance of GCN-DO only improves from two to three layers. This, indeed, proves GDC is a better stochastic regularization framework for GNNs in alleviating over-fitting and over-smoothing, enabling possible directions to develop deeper GNNs.
\begin{table}[!b]
\caption{Accuracy of 128-dimensional 4-layer GCN-BBGDC with different number of blocks on Cora in semi-supervised node classification.}
\centering
\resizebox{0.9\columnwidth}{!}{
\begin{tabular}{@{}l | c c c }
\toprule
{\textbf{Method}} & {2 blocks} & {16 blocks} & {32 blocks}\\
\midrule
\textbf{GCN-BBGDC} & $82.2$ & $83.0$ & $83.3$\\
\bottomrule
\end{tabular}
}
\label{tab: acc_nblock}
\end{table}
\subsection{Effect of Number of Blocks}
In GDC for every pair of input and output features, a separate mask for the adjacency matrix should be drawn. However, as we discussed in Section \ref{sec:samp_comp}, this demands large memory space. We circumvented this problem by drawing a single mask for a block of features. While we used only two blocks in our experiments presented so far, we here investigate the effect of the number of blocks on the node classification accuracy. The performance of 128-dimensional 4-layer GCN-BBGDC with 2, 16, and 32 blocks are shown in Table~\ref{tab: acc_nblock}. As can be seen, the accuracy improves as the number of blocks increases. This is due to the fact that increasing the number of blocks increases the flexibility of GDC. The choice of the number of blocks is a factor to consider for the trade off between the performance and memory usage as well as computational complexity.
\section{Conclusion}
In this paper, we proposed a unified framework for adaptive connection sampling in GNNs that generalizes existing stochastic regularization techniques for training GNNs. Our proposed method, Graph DropConnect (GDC), not only alleviates over-smoothing and over-fitting tendencies of deep GNNs, but also enables learning with uncertainty in graph analytic tasks with GNNs. %
Instead of using fixed sampling rates,
our GDC technique parameters can be trained jointly with
GNN model parameters.
We further
show that training a GNN with GDC is equivalent to an approximation of training Bayesian GNNs. %
Our experimental results shows that GDC boosts the performance of GNNs in semi-supervised classification task by alleviating over-smoothing and over-fitting. We further show that the quality of uncertainty derived by GDC is better than DropOut in GNNs.
\section*{Acknowledgements}
The presented materials are based upon the work supported by the National Science Foundation under Grants ENG-1839816, IIS-1848596, CCF-1553281, IIS-1812641, IIS-1812699, and CCF-1934904.
\end{document}
\section{Ablation Study: Global versus Local}
We further investigate our learnable GDC, in which for each edge at each layer a different connection sampling distribution is learned. We refer to this scenario as the \textit{local} learnable GDC. This, indeed, is a more general case than learning a single distribution for all edges in a layer.
Expanding the variational beta-Bernoulli GDC to local learnable GDC is straightforward. Note that the KL term in the loss function can be derived in the same manner as in the global learnable GDC -- as described in Section 4 of the paper -- except that it will include the sum of $\mathrm{num\_layers} \times \mathrm{num\_edges}$ terms as opposed to the $\mathrm{num\_layers}$ terms in the global GDC.
By training the aforementioned model on the citation datasets, we find that the accuracy degrades and the KL divergence reduces to zero for every choice of prior. This phenomenon, which is known as \textit{posterior collapse} or \textit{KL vanishing}, is a common problem in variational auto-encoders for language modeling \citep{bowman2015generating, goyal2017zforcing,liu2019cyclical}. It is often due to over-parametrization in the model, which is indeed the case in the local learnable GDC. A solution to this issue could be making the parameters of the distribution dependent on the graph topology and/or node attributes. We leave this for future studies.
\section{Datasets and Implementation Details}
All of the models are implemented in PyTorch \citep{paszke2017automatic}. All of the simulations are conducted on a single NVIDIA GeForce RTX 2080 GPU node. We evaluate our proposed methods, GCN-BBDE and GCN-BBGDC, and baselines on three standard citation network benchmark datasets. We preprocess and split the dataset as done in \cite{kipf2016semi} and \cite{bojchevski2018deep}. For Cora and Cora-ML, we use 140 nodes for training, 500 nodes for validation and 1000 nodes for testing. For Citeseer, we use 120 nodes for training and the same number of nodes as Cora for validation and testing. Table~\ref{tab:stats} provides the detailed statistics of the graph datasets used in our experiments. The warm-up factor used in GCN-BBGDC with more than 2 layers for Cora and Cora-ML is $\mathrm{min}(\{1,\, \mathrm{epoch}/20\})$, and for Citeseer is $\mathrm{min}(\{1,\, \mathrm{epoch}/40\})$. We have deployed Adam optimizer \citep{kingma2014adam} in all of our experiments.
\begin{table}[!t]
\centering
\caption{Graph dataset statistics.}
\vspace{0.25em}
\resizebox{1.0\columnwidth}{!}{
\begin{tabular}{@{}l | c c c c}
\toprule
Dataset & \# Classes & \# Nodes & \# Edges & \# Features\\ \midrule
\textbf{Cora} & 7 & 2,708 & 5,429 & 1,433\\
\textbf{Citeseer} & 3 & 3,327 & 4,732 & 3,703\\
\textbf{Cora-ML} & 7 & 2,995 & 8,416 & 2,879\\
\bottomrule
\end{tabular}
}
\label{tab:stats}
\end{table}
\section{GDC versus Other Stochastic Regularization Techniques}
To further clarify the differences of our proposed GDC from existing stochastic regularization techniques, we draw the schematics of a GCN layer to which DropOut, DropEdge, Node Sampling, and our GDC are applied; shown in figures below. The input graph topology for the GCN layer is depicted in \ref{fig:gcn}. The number of input and output features are both two in this toy example.
\begin{figure*}[!h]
\centering
\vspace{0.25cm}
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\path (vec8.east) edge [nand,color=gray!20] (mec8.west);
\end{tikzpicture}
\caption{\textbf{Top}: Schematic of a GCN layer on a graph with 4 nodes. Number of both input and output features are two. The connections are localized as explicitly depicted for node 2. \textbf{Bottom}: The same GCN layer shown in a more conventional way, i.e. each layer is a vector of neurons or features. Each circle is a feature and each square represents a node. The connections are sparse and the sparsity is based on the input graph topology. The connections for node 2 in layer $l+1$ are highlighted.}
\label{fig:gcn}
\end{figure*}
\begin{figure*}[!h]
\centering
\vspace{0.25cm}
\begin{tikzpicture}[start chain=1 going below,start chain=2 going below]
\node [obs, fill=bleudefrance] (vec1) {\tiny{$h_{11}^{l}$}};
\node [obs,yshift=-.75cm, fill=bleudefrance!40] (vec2) {\tiny{$h_{12}^{l}$}};
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\node [obs,yshift=-3.85cm, fill=bleudefrance!60] (vec6) {\tiny{$h_{32}^{l}$}};
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\node [obs,yshift=-5.4cm, fill=bleudefrance!40] (vec8) {\tiny{$h_{42}^{l}$}};
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\path (vec6.east) edge [nand,color=purple!60,dashed] (mec6.west);
\path (vec7.east) edge [nand,color=gray!60] (mec1.west);
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\path (vec7.east) edge [nand,color=gray!60] (mec7.west);
\path (vec7.east) edge [nand,color=gray!60] (mec8.west);
\path (vec8.east) edge [nand,color=gray!60] (mec1.west);
\path (vec8.east) edge [nand,color=purple!60,dashed] (mec2.west);
\path (vec8.east) edge [nand,color=purple!60,dashed] (mec7.west);
\path (vec8.east) edge [nand,color=gray!60] (mec8.west);
\end{tikzpicture}
\caption{Schematic of our proposed GDC. Each circle is a feature and each square represents a node. GDC drops connections independently across layers. The dashed lines show dropped connections and the gray ones show the kept connections.}
\label{fig:gdc}
\end{figure*}
\clearpage
\begin{figure*}[!h]
\centering
\vspace{0.25cm}
\begin{tikzpicture}[start chain=1 going below,start chain=2 going below]
\node [obs, fill=bleudefrance,opacity=.2] (vec1) {\tiny{$h_{11}^{l}$}};
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\node[minimum size=.2cm, yshift=-1.9cm,xshift=8.5cm] (v2p) {\small{$n_2$}};
\node[minimum size=.2cm, yshift=-3.45cm,xshift=8.5cm] (v3p) {\small{$n_3$}};
\node[minimum size=.2cm, yshift=-4.95cm,xshift=8.5cm] (v4p) {\small{$n_4$}};
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\node [draw,below=of mm3, yshift=1.02cm,minimum height=1.53cm, minimum width=.8cm,very thick,color=orange] (mm4) {};
\node[yshift=1cm] (l1) {Layer $l$};
\node[yshift=1cm,xshift=7.5cm] (l2) {Layer $l+1$};
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\path (vec7.east) edge [nand,color=gray!60] (mec8.west);
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\end{tikzpicture}
\caption{Schematic of DropOut \citep{srivastava2014dropout}. Each circle is a feature and each square represents a node. DropOut drops features at each layer. The faded circles represent dropped features while the other ones are kept. The dashed lines show dropped connections and the gray ones show the kept ones.}
\label{fig:dropout}
\end{figure*}
\begin{figure*}[!h]
\centering
\vspace{0.25cm}
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\path (vec4.east) edge [nand,color=purple!60, dashed] (mec2.west);
\path (vec4.east) edge [nand,color=gray!60] (mec3.west);
\path (vec4.east) edge [nand,color=gray!60] (mec4.west);
\path (vec5.east) edge [nand,color=gray!60] (mec1.west);
\path (vec5.east) edge [nand,color=gray!60] (mec2.west);
\path (vec5.east) edge [nand,color=gray!60] (mec5.west);
\path (vec5.east) edge [nand,color=gray!60] (mec6.west);
\path (vec6.east) edge [nand,color=gray!60] (mec1.west);
\path (vec6.east) edge [nand,color=gray!60] (mec2.west);
\path (vec6.east) edge [nand,color=gray!60] (mec5.west);
\path (vec6.east) edge [nand,color=gray!60] (mec6.west);
\path (vec7.east) edge [nand,color=purple!60, dashed] (mec1.west);
\path (vec7.east) edge [nand,color=purple!60, dashed] (mec2.west);
\path (vec7.east) edge [nand,color=gray!60] (mec7.west);
\path (vec7.east) edge [nand,color=gray!60] (mec8.west);
\path (vec8.east) edge [nand,color=purple!60,dashed] (mec1.west);
\path (vec8.east) edge [nand,color=purple!60,dashed] (mec2.west);
\path (vec8.east) edge [nand,color=gray!60] (mec7.west);
\path (vec8.east) edge [nand,color=gray!60] (mec8.west);
\end{tikzpicture}
\caption{Schematic of DropEdge \citep{rong2019dropedge}. Each circle is a feature and each square represents a node. DropEdge drops edges between nodes hence all of the connections between their corresponding channels are dropped. Note that the mask in DropEdge is symmetric. In this example, the edge between nodes 1 and 2 as well as the edge between nodes 1 and 4 are dropped. The dashed lines show dropped connections and the gray ones show the kept ones.}
\label{fig:dropedge}
\end{figure*}
\clearpage
\begin{figure*}[!t]
\centering
\vspace{0.25cm}
\begin{tikzpicture}[start chain=1 going below,start chain=2 going below]
\node [obs, fill=bleudefrance] (vec1) {\tiny{$h_{11}^{l}$}};
\node [obs,yshift=-.75cm, fill=bleudefrance!40] (vec2) {\tiny{$h_{12}^{l}$}};
\node [obs,yshift=-1.55cm, fill=bleudefrance!30, opacity=.2] (vec3) {\tiny{$h_{21}^{l}$}};
\node [obs,yshift=-2.3cm, fill=bleudefrance!60, opacity=.2] (vec4) {\tiny{$h_{22}^{l}$}};
\node [obs,yshift=-3.1cm, fill=bleudefrance!10] (vec5) {\tiny{$h_{31}^{l}$}};
\node [obs,yshift=-3.85cm, fill=bleudefrance!60] (vec6) {\tiny{$h_{32}^{l}$}};
\node [obs,yshift=-4.65cm, fill=bleudefrance!20, opacity=.2] (vec7) {\tiny{$h_{41}^{l}$}};
\node [obs,yshift=-5.4cm, fill=bleudefrance!40, opacity=.2] (vec8) {\tiny{$h_{42}^{l}$}};
\node [draw, yshift=-.36cm,minimum width=.8cm,minimum height=1.55cm, very thick,color=orange] (vv1) {};
\node [draw,minimum width=.8cm,minimum height=1.53cm, very thick,color=orange,below=of vv1, yshift=1.02cm,opacity=.2] (vv2) {};
\node [draw,minimum width=.8cm,minimum height=1.53cm, very thick,color=orange,below=of vv2,yshift=1.02cm] (vv3) {};
\node [draw,minimum width=.8cm,minimum height=1.55cm, very thick,color=orange,below=of vv3, yshift=1.02cm,,opacity=.2] (vv4) {};
\node [obs, xshift=7.5cm, minimum width=.6cm,minimum height=.2cm, fill=bleudefrance!10] (mec1) {\tiny{$h_{11}^{l+1}$}};
\node [obs, yshift=-.75cm,xshift=7.5cm,minimum width=.6cm,minimum height=.2cm, fill=bleudefrance!60] (mec2) {\tiny{$h_{12}^{l+1}$}};
\node [obs, yshift=-1.55cm,xshift=7.5cm,minimum width=.6cm,minimum height=.2cm, fill=bleudefrance!80] (mec3) {\tiny{$h_{21}^{l+1}$}};
\node [obs, yshift=-2.3cm,xshift=7.5cm,minimum width=.6cm,minimum height=.2cm, fill=bleudefrance!30] (mec4) {\tiny{$h_{22}^{l+1}$}};
\node [obs, yshift=-3.1cm,xshift=7.5cm,minimum width=.6cm,minimum height=.2cm, fill=bleudefrance!50] (mec5) {\tiny{$h_{31}^{l+1}$}};
\node [obs, yshift=-3.85cm,xshift=7.5cm,minimum width=.6cm,minimum height=.2cm, fill=bleudefrance!10] (mec6) {\tiny{$h_{32}^{l+1}$}};
\node [obs, yshift=-4.65cm,xshift=7.5cm,minimum width=.6cm,minimum height=.2cm, fill=bleudefrance] (mec7) {\tiny{$h_{41}^{l+1}$}};
\node [obs, yshift=-5.4cm,xshift=7.5cm,minimum width=.6cm,minimum height=.2cm, fill=bleudefrance!70] (mec8) {\tiny{$h_{42}^{l+1}$}};
\node [draw, yshift=-.36cm,xshift=7.5cm ,minimum width=.8cm,minimum height=1.55cm, very thick,color=orange] (mm1) {};
\node [draw, below=of mm1, yshift=1.02cm, minimum width=.8cm,minimum height=1.53cm, very thick,color=orange] (mm2) {};
\node [draw,below=of mm2, yshift=1.02cm, minimum width=.8cm,minimum height=1.53cm, very thick,color=orange] (mm3) {};
\node [draw,below=of mm3, yshift=1.02cm,minimum height=1.53cm, minimum width=.8cm,very thick,color=orange] (mm4) {};
\node[yshift=1cm] (l1) {Layer $l$};
\node[yshift=1cm,xshift=7.5cm] (l2) {Layer $l+1$};
\node[minimum size=.2cm, yshift=-.45cm,xshift=-.8cm] (v1p) {\small{$n_1$}};
\node[minimum size=.2cm, yshift=-1.9cm,xshift=-.8cm] (v2p) {\small{$n_2$}};
\node[minimum size=.2cm, yshift=-3.45cm,xshift=-.8cm] (v3p) {\small{$n_3$}};
\node[minimum size=.2cm, yshift=-4.95cm,xshift=-.8cm] (v4p) {\small{$n_4$}};
\node[minimum size=.2cm, yshift=-.45cm,xshift=8.5cm] (v1p) {\small{$n_1$}};
\node[minimum size=.2cm, yshift=-1.9cm,xshift=8.5cm] (v2p) {\small{$n_2$}};
\node[minimum size=.2cm, yshift=-3.45cm,xshift=8.5cm] (v3p) {\small{$n_3$}};
\node[minimum size=.2cm, yshift=-4.95cm,xshift=8.5cm] (v4p) {\small{$n_4$}};
\path (vec1.east) edge [nand,color=gray!60] (mec1.west);
\path (vec1.east) edge [nand,color=gray!60] (mec2.west);
\path (vec1.east) edge [nand,color=gray!60] (mec3.west);
\path (vec1.east) edge [nand,color=gray!60] (mec4.west);
\path (vec1.east) edge [nand,color=gray!60] (mec5.west);
\path (vec1.east) edge [nand,color=gray!60] (mec6.west);
\path (vec1.east) edge [nand,color=gray!60] (mec7.west);
\path (vec1.east) edge [nand,color=gray!60] (mec8.west);
\path (vec2.east) edge [nand,color=gray!60] (mec1.west);
\path (vec2.east) edge [nand,color=gray!60] (mec2.west);
\path (vec2.east) edge [nand,color=gray!60] (mec3.west);
\path (vec2.east) edge [nand,color=gray!60] (mec4.west);
\path (vec2.east) edge [nand,color=gray!60] (mec5.west);
\path (vec2.east) edge [nand,color=gray!60] (mec6.west);
\path (vec2.east) edge [nand,color=gray!60] (mec7.west);
\path (vec2.east) edge [nand,color=gray!60] (mec8.west);
\path (vec3.east) edge [nand,color=purple!60, dashed] (mec1.west);
\path (vec3.east) edge [nand,color=purple!60, dashed] (mec2.west);
\path (vec3.east) edge [nand,color=purple!60, dashed] (mec3.west);
\path (vec3.east) edge [nand,color=purple!60, dashed] (mec4.west);
\path (vec4.east) edge [nand,color=purple!60, dashed] (mec1.west);
\path (vec4.east) edge [nand,color=purple!60, dashed] (mec2.west);
\path (vec4.east) edge [nand,color=purple!60, dashed] (mec3.west);
\path (vec4.east) edge [nand,color=purple!60, dashed] (mec4.west);
\path (vec5.east) edge [nand,color=gray!60] (mec1.west);
\path (vec5.east) edge [nand,color=gray!60] (mec2.west);
\path (vec5.east) edge [nand,color=gray!60] (mec5.west);
\path (vec5.east) edge [nand,color=gray!60] (mec6.west);
\path (vec6.east) edge [nand,color=gray!60] (mec1.west);
\path (vec6.east) edge [nand,color=gray!60] (mec2.west);
\path (vec6.east) edge [nand,color=gray!60] (mec5.west);
\path (vec6.east) edge [nand,color=gray!60] (mec6.west);
\path (vec7.east) edge [nand,color=purple!60, dashed] (mec1.west);
\path (vec7.east) edge [nand,color=purple!60, dashed] (mec2.west);
\path (vec7.east) edge [nand,color=purple!60, dashed] (mec7.west);
\path (vec7.east) edge [nand,color=purple!60, dashed] (mec8.west);
\path (vec8.east) edge [nand,color=purple!60,dashed] (mec1.west);
\path (vec8.east) edge [nand,color=purple!60,dashed] (mec2.west);
\path (vec8.east) edge [nand,color=purple!60, dashed] (mec7.west);
\path (vec8.east) edge [nand,color=purple!60, dashed] (mec8.west);
\end{tikzpicture}
\caption{Schematic of the node sampling strategy in FastGCN \citep{chen2018fastgcn}. Each circle is a feature and each square represents a node. FastGCN drops nodes hence all of the connections to that node are dropped. The faded nodes represents the dropped nodes. The dashed lines show dropped connections and the gray ones show the kept ones.}
\label{fig:ndsamp}
\end{figure*}
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,259 |
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